Magnetohydrodynamic waves in geospace: the theory of ULF waves and their interaction with energetic particles in the solar-terrestrial environment

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Series in Plasma Physics

Magnetohydrodynamic Waves in Geospace The Theory of ULF Waves and their Interaction with Energetic Particles in the Solar–Terrestrial Environment

A D M Walker School of Pure and Applied Physics University of KwaZulu-Natal, South Africa

Institute of Physics Publishing Bristol and Philadelphia Copyright © 2005 IOP Publishing Ltd.

c IOP Publishing Ltd 2005
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with Universities UK (UUK). A D M Walker has asserted his moral rights under the Copyright, Designs and Patents Act 1998 to be identified as the author of this work. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0910 5 Library of Congress Cataloging-in-Publication Data are available

Commissioning Editor: John Navas Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Victoria Le Billon Marketing: Nicola Newey, Louise Higham and Ben Thomas Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 929, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset in LATEX 2ε by Text 2 Text, Torquay, Devon Printed in the UK by MPG Books Ltd, Bodmin, Cornwall Copyright © 2005 IOP Publishing Ltd.

Contents

Preface

PART 1 Fundamentals of MHD wave theory 1

Basic ideas of thermodynamics and electrodynamics 1.1 Introduction 1.2 Elementary ideas of thermodynamics and kinetic theory 1.2.1 Equation of state of an ideal gas 1.2.2 Comparison with kinetic theory 1.2.3 First law of thermodynamics 1.2.4 Second law of thermodynamics 1.2.5 Ratio of specific heats of a gas 1.2.6 State variables and Maxwell’s relations 1.2.7 Rate of change of entropy in reversible processes 1.2.8 Specific energy, entropy, and enthalpy 1.3 Maxwell’s equations in the presence of currents and charges 1.4 The Lorentz force law 1.5 Low-velocity approximation to Maxwell’s equations—Amp´ere’s law 1.6 Motion of charged particles in uniform electric and magnetic fields 1.6.1 Equation of motion 1.6.2 Cyclotron motion 1.6.3 Electric field drift 1.6.4 Drifts due to an external force 1.7 Electromagnetic energy 1.7.1 Joule energy transfer 1.7.2 Physical interpretation of the flux vector 1.8 Electromagnetic momentum 1.9 Summary

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1 3 3 4 4 4 4 5 5 5 6 6 6 8 8 9 9 9 11 12 13 13 14 15 17

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2

The magnetohydrodynamic approximation 2.1 Introduction 2.2 Fluid equations for the particle species 2.2.1 The continuity equation 2.2.2 The momentum equation 2.2.3 Adiabatic law 2.3 Characteristic lengths and frequencies 2.3.1 The Debye length 2.3.2 The plasma frequency 2.3.3 The electron and ion gyrofrequencies 2.3.4 Characteristic speeds 2.3.5 Parameters for approximation 2.4 The MHD equations for a fully ionized plasma 2.4.1 MHD variables 2.4.2 Continuity equation 2.4.3 Momentum equation 2.4.4 Adiabatic law 2.4.5 Generalized Ohm’s law 2.4.6 Reduced MHD equations 2.5 Gravitation 2.6 Frozen-in magnetic fields 2.7 Losses within plasmas 2.7.1 Resistive effects 2.7.2 Viscous effects 2.8 Partially ionized plasma 2.8.1 Current density in a partially ionized plasma 2.8.2 The conductivity tensor 2.9 Conservation laws 2.9.1 MHD energy conservation 2.9.2 Momentum conservation 2.10 Summary

19 19 20 20 21 23 23 23 24 25 25 26 27 28 29 29 29 30 31 32 32 33 33 34 34 34 36 39 39 40 41

3

*Single-particle motion in electromagnetic fields 3.1 *Introduction 3.2 *Guiding-centre motion—heuristic approach 3.2.1 *Qualitative description of guiding-centre motion 3.2.2 *Drift due to a magnetic field gradient 3.2.3 *Drifts due to the variation of the zero-order drift velocity 3.2.4 *Parallel drift due to magnetic field shear 3.2.5 *The drift velocity of the guiding centre 3.3 *General motion in a varying field 3.3.1 *Equations of motion 3.4 *Theory of motion in a slowly varying field—the guiding-centre approximation

43 43 43 43 44 46 47 47 47 47

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3.5

3.6 4

3.4.1 *Slowly varying fields 3.4.2 *The particle phase 3.4.3 *The averaging process 3.4.4 *Equations of motion for v⊥ and v 3.4.5 *The magnetic moment, an adiabatic invariant 3.4.6 *Drift velocity—the motion of the guiding centre 3.4.7 *The energy equation *Motion in a dipole field—second and third adiabatic invariants and constants of the motion 3.5.1 *Natural periodicities 3.5.2 *Second and third adiabatic invariants 3.5.3 *Energy and L-shell as constants of the motion 3.5.4 *Bounce motion 3.5.5 *Azimuthal drifts 3.5.6 *Cross-L drifts *Summary

*Kinetic theory of plasmas 4.1 *Introduction 4.2 *The distribution function 4.3 *Mean values of the particle properties 4.3.1 *Averages over the velocity 4.3.2 *Averages over the gyrophase 4.3.3 *Directional average 4.4 *Fluid and MHD variables 4.4.1 *Mass density 4.4.2 *Drift velocity and current density 4.4.3 *Pressure tensor 4.4.4 *Energy density and temperature 4.4.5 *Energy flux 4.5 *Kinetic equations 4.5.1 *Conservation of particles in phase space 4.5.2 *Boltzmann and Vlasov equations 4.6 *Approximations to the kinetic equation 4.6.1 *Low-frequency average of Vlasov equation 4.6.2 *Drift kinetic equations 4.7 *Collisions and equivalent processes 4.7.1 *The nature of the collision term in the Boltzmann equation 4.8 *Equilibrium states 4.8.1 *Time scales to reach equilibrium and quasi-equilibrium 4.8.2 *The Maxwell–Boltzmann and Maxwellian distributions 4.8.3 *Jeans’s theorem and quasi-equilibrium states 4.9 *Summary

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vii 50 51 52 53 54 55 57 58 58 59 59 60 60 60 61 62 62 63 63 63 64 65 65 66 66 67 69 70 71 71 73 74 74 75 76 77 81 81 83 84 86

viii 5

Contents *Fluid behaviour 5.1 *Introduction 5.2 *Distribution functions and their moments 5.3 *Evolution of particle properties 5.3.1 *Moments of the particle distribution for a single fluid 5.3.2 *Rate of change of a particle property 5.4 *Moment equations 5.4.1 *Moment equations for a single species 5.4.2 *Moment equations for a multi-ion plasma 5.5 *Closure of the moment equations 5.5.1 *Successive approximations to the Boltzmann equation 5.5.2 *Orders of magnitude 5.5.3 *Truncation of the electromagnetic hierarchy 5.5.4 *Cold plasma 5.5.5 *Thermal equilibrium 5.5.6 *The fluid equations in the absence of collisions between species 5.6 *Summary

88 88 89 89 90 91 93 93 95 100 101 102 102 103 103

6

Equilibrium and steady-state conditions 6.1 Introduction 6.2 MHD equilibrium 6.3 MHD in the steady state 6.4 Boundary conditions 6.5 Discontinuities and shocks 6.5.1 Classification of discontinuities 6.5.2 Tangential discontinuity 6.5.3 Rotational or Alfv´en discontinuity 6.5.4 Contact discontinuities 6.5.5 MHD shocks 6.6 Summary

112 112 112 113 114 117 117 118 118 120 120 121

7

Harmonic plane waves in a uniform loss-free plasma 7.1 Introduction 7.2 Wave equations 7.2.1 Wave equation for a non-dispersive medium 7.2.2 Dispersive media 7.3 Simple examples of waves 7.3.1 Waves on strings and in gases 7.3.2 Simple transverse Alfv´en waves 7.3.3 Simple compressional Alfv´en waves 7.4 General wave equation for MHD waves 7.4.1 Linearization of the MHD equations 7.4.2 Wave equation 7.5 Harmonic waves

122 122 123 123 124 125 125 126 127 127 127 128 129

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7.6 7.7 8

9

7.5.1 Equations for harmonic waves 7.5.2 Dispersion relations 7.5.3 Phase velocity 7.5.4 Refractive index 7.5.5 Relations between field components Waves for non-scalar pressure Summary

ix 130 131 132 134 137 138 144

*Collisionless damping of MHD waves 8.1 *Introduction 8.2 *Specification of the problem 8.3 *Single-particle motion in a wave 8.4 *Kinetic effects 8.4.1 *First-order perturbation of the Vlasov equation 8.4.2 *Integration over the unperturbed orbits 8.4.3 *Evaluation of moments of the perturbed distribution function 8.5 *The pressure tensor 8.6 *Wave equations and dispersion relation 8.7 *Special cases of the dispersion relation 8.7.1 *Small β 8.7.2 *Parallel propagation 8.7.3 *Perpendicular propagation 8.8 *Physical picture of collisionless damping 8.9 *Wave properties 8.10 *Discussion 8.11 *Summary

145 145 145 147 148 148 149

Wavepackets and energy propagation in uniform media 9.1 Introduction 9.2 Wavepackets and rays 9.2.1 Superpositions of one-dimensional harmonic waves 9.2.2 Angular spectrum of plane waves 9.2.3 One-dimensional, spatially limited plane waves 9.2.4 Rays and the group velocity 9.2.5 Ray velocity and ray surface 9.3 Propagation of energy by waves 9.3.1 Energy conservation for waves 9.3.2 Alternative form of the energy flux vector 9.3.3 Computing energy flux and energy density for harmonic waves 9.3.4 Energy flux and energy density for quasimonochromatic waves in a uniform medium 9.3.5 Relation between total MHD energy and wave energy 9.4 Summary

168 168 169 169 169 170 171 173 175 175 176

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10 Reflection and transmission at sharp boundaries in stationary media 182 10.1 Introduction 182 10.2 Equations for the field components 182 10.3 Boundary conditions 185 10.3.1 Coordinate system 185 10.3.2 Snell’s law and the law of reflection 185 10.3.3 Boundary conditions at the sharp boundary 186 10.3.4 Boundary conditions at infinity 186 10.4 Reflection and transmission 188 10.4.1 Reflection and transmission coefficients 188 10.5 Computation of reflection and transmission coefficients 190 10.5.1 Partial reflection and Brewster angle 190 10.5.2 Critical angle; Total internal reflection 192 10.5.3 A more general case 193 10.5.4 Energy flux 196 10.6 Summary 196 11 Slowly varying media 11.1 Introduction 11.2 Phase integral solutions in a stratified medium 11.2.1 Phase integral solutions for the transverse Alfv´en wave 11.2.2 Plane stratified media 11.3 Huygens’ construction 11.4 Ray-tracing 11.4.1 The ray-tracing equations 11.4.2 Ray-tracing and Huygens’ construction 11.5 Summary

PART 2 The solar–terrestrial environment 12 The Sun, the solar wind, and the magnetosphere 12.1 Introduction 12.2 The Sun 12.2.1 The visible outer regions 12.2.2 Sunspots and the solar cycle 12.2.3 Helioseismic oscillations 12.2.4 The solar magnetic field 12.3 The solar wind 12.3.1 The quiet solar wind 12.3.2 The fast solar wind 12.4 Structure of the Earth’s magnetosphere 12.4.1 Formation of the Earth’s magnetosphere 12.4.2 Magnetic structure of the magnetosphere Copyright © 2005 IOP Publishing Ltd.

198 198 199 200 200 202 203 203 205 205

207 209 209 209 210 210 212 212 213 214 216 217 217 221

Contents 12.5 Cold plasma populations in the magnetosphere 12.5.1 The Ionosphere 12.5.2 The plasmasphere 12.5.3 Cold plasma outside the plasmapause 12.5.4 The polar wind 12.6 Hot plasma populations 12.6.1 Boundary layers 12.6.2 Tail lobes and plasma sheet 12.6.3 Radiation belts and ring current 12.7 Summary 13 Observations of ultra-low-frequency oscillations and waves 13.1 Introduction 13.2 Waves and turbulence in the solar wind 13.3 Historical observations of ULF pulsations: 1861–1970 13.4 Physical understanding of ULF pulsations: 1971 to the present 13.4.1 Global pulsations arising from sources at or beyond the magnetopause 13.4.2 Global pulsations arising from wave–particle interaction 13.4.3 Pi2 pulsations 13.5 Instrumentation 13.5.1 Modern instrumentation and the internet 13.5.2 Satellites and spacecraft 13.5.3 Magnetometer arrays and other ground-based instrumentation 13.5.4 Auroral radar arrays 13.6 Summary

PART 3 Waves in solar–terrestrial physics 14 MHD wave equations in non-uniform media 14.1 Introduction 14.2 Models 14.2.1 The magnetosphere 14.2.2 Cylindrical models—sunspots and coronal loops 14.3 Coupled wave equations in a plane-stratified medium 14.3.1 First-order wave equations 14.3.2 Polarization relations 14.3.3 Second-order wave equation 14.4 Wave equations for a cold plasma in a dipole field 14.5 Multicomponent plasmas 14.5.1 Background model 14.5.2 Linearized equations Copyright © 2005 IOP Publishing Ltd.

xi 223 223 224 225 225 225 225 228 228 229 230 230 230 231 233 234 241 244 244 244 245 245 245 247

249 251 251 251 251 253 254 254 257 257 260 262 263 264

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Contents 14.6 Wave equations in a cylindrically stratified medium 14.7 Summary

265 267

15 Propagation in a plane-stratified medium 15.1 Introduction 15.2 Wave propagation through numerical computation 15.3 WKBJ solutions of the wave equation 15.3.1 Energy flux 15.3.2 Error terms in the differential equation 15.4 Cumulative error in the WKBJ solutions 15.4.1 Method of variation of parameters 15.4.2 Cumulative error 15.5 Reflection 15.5.1 Geometrical optics of reflection 15.6 Full wave theory of reflection 15.6.1 Stokes’ equation and Airy functions 15.6.2 WKBJ solutions of the Stokes equation 15.6.3 The Stokes phenomenon 15.6.4 WKBJ approximations to the Airy functions 15.6.5 Approximate WKBJ representation of a general wave 15.6.6 Error near the zeros of q 15.7 Connection relations 15.7.1 Stokes and anti-Stokes lines 15.7.2 Connection relations 15.7.3 Range of validity of asymptotic approximations in the complex plane 15.8 Summary

269 269 270 270 272 273 273 274 274 276 276 277 278 279 280 280 281 283 284 284 285

16 Standing waves and oscillations in a cold plasma 16.1 Introduction 16.2 Transverse Alfv´en oscillations 16.2.1 Uniform medium 16.2.2 Medium with a transverse gradient of Alfv´en speed 16.3 The Earth–ionosphere system as a boundary 16.3.1 Height-integrated conductivity of the ionosphere 16.3.2 Fields below the ionosphere 16.3.3 Transmission through the ionosphere 16.4 Lossy oscillations in a uniform medium 16.5 Alfv´en oscillations in a dipole-like field 16.5.1 Cylindrically symmetric oscillations 16.5.2 Oscillations with large azimuthal wavenumber 16.6 Properties of localized field-line oscillations 16.6.1 Basic equations 16.6.2 Numerical solutions when the ionosphere has very large conductivity

288 288 289 289 291 293 294 296 297 299 300 300 301 302 302

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Contents 16.6.3 Numerical solutions for finite ionospheric conductivity 16.6.4 Perturbation solution of the azimuthal equation 16.6.5 WKBJ solutions 16.7 Summary

xiii 305 308 311 315

17 Standing waves and oscillations in a compressional plasma 17.1 Introduction 17.2 Localized oscillations 17.3 Models 17.3.1 Ring current 17.3.2 Boundary conditions 17.4 Solutions of the coupled equations 17.4.1 Phase integral solutions 17.4.2 Numerical solutions 17.5 Summary

317 317 318 320 320 323 324 324 328 333

18 Field-line resonance in low-pressure plasmas 18.1 Introduction 18.2 Basic ideas of field-line resonance 18.2.1 Standing waves and field-line resonance 18.2.2 Loss mechanisms 18.3 Waves and conservation laws 18.3.1 Definition of wave invariant 18.3.2 Conservation and non-conservation at singular points 18.4 Modelling resonance in a dipole geometry 18.5 Summary

334 334 335 338 338 340 340 341 342 346

19 Mathematics of field-line resonance in compressible media 19.1 Introduction 19.2 Field-line resonance in a compressible plane-stratified plasma 19.2.1 The resonance equation 19.2.2 Series solution to the resonance equation 19.2.3 WKBJ approximations 19.2.4 The Stokes phenomenon 19.3 Solutions of the resonance equation 19.3.1 Numerical computation of the solutions 19.3.2 Accuracy of the WKBJ solutions 19.3.3 Approximate solutions of the wave equation 19.4 Reflection coefficients 19.5 Resonance in cylindrical geometries 19.5.1 Equations describing dissipation in cylindrical flux tubes 19.5.2 Solutions of the dissipative equations in cylindrical geometry 19.5.3 Resonance heating 19.6 Summary

348 348 348 349 351 353 354 357 357 360 361 361 362 364

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Contents

20 Cavity oscillations and waveguide modes 374 20.1 Introduction 374 20.2 The magnetospheric cavity or waveguide 375 20.3 Lossy modes 379 20.4 Time-dependent behaviour 380 20.4.1 Time-varying behaviour in a closed cavity 381 20.4.2 The Green’s function 382 20.4.3 Some applications of the Green’s function method 384 20.5 Leaky cavities and waveguides 395 20.5.1 Reflection and transmission coefficients at a leaky boundary395 20.6 Excitation of the magnetospheric cavity 398 20.6.1 Time development of the reflected and transmitted waves 399 20.6.2 Discussion 402 20.7 The waveguide picture 403 20.8 Summary 403 21 Waves in moving media 21.1 Introduction 21.2 Wave propagation in a uniform moving medium 21.2.1 Qualitative picture 21.2.2 Modification of the wave equations for a moving medium 21.2.3 Harmonic waves 21.2.4 The entropy wave 21.3 Energy relations in a uniform medium 21.3.1 Energy conservation equation 21.4 Reflection and transmission of a plane wave at a tangential discontinuity 21.4.1 Reflection and transmission coefficients 21.4.2 Numerical results 21.4.3 Energy conservation at the boundary 21.4.4 Reflection and transmission coefficients for the energy 21.5 Energy balance in a non-uniform plasma 21.5.1 Relation between total MHD energy and wave energy 21.6 Ray-tracing in a moving medium 21.7 The negative energy wave picture 21.8 Over-reflection in solar–terrestrial physics 21.8.1 Excitation of long period pulsations 21.8.2 Resonant absorption in coronal plumes 21.9 Summary

413 413 416 417 418 420 424 429 430 434 434 434 435

22 Shock waves 22.1 Introduction 22.2 Properties of shock waves 22.2.1 Change of properties across a shock in a gas 22.2.2 Changes through MHD shocks

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22.3 Waves in the neighbourhood of shocks 22.3.1 Coordinate system and boundary conditions 22.3.2 Dispersion relations 22.3.3 Relation between field components 22.3.4 Direction of propagation relative to the shock 22.4 Classification of shocks 22.4.1 Fast and slow shocks 22.4.2 Perpendicular and parallel shocks 22.5 Propagation of MHD waves through shocks 22.5.1 Boundary conditions 22.5.2 Behaviour of the waves on either side of the boundary 22.6 Summary

440 440 441 442 442 444 444 445 447 447 451 452

23 Magnetohydrodynamic instability 23.1 Introduction 23.2 Nature of instability 23.2.1 Growth of a spatial perturbation 23.2.2 Convected and non-convected instability 23.2.3 Macroscopic and microscopic instability 23.3 Fluid instabilities 23.4 The Kelvin–Helmholtz instability 23.4.1 Physical basis 23.4.2 Sharp boundary between two counterstreaming MHD media 23.4.3 Effect of finite boundary thickness 23.4.4 Applications in magnetospheric conditions 23.5 *Pressure anisotropy 23.5.1 *Firehose instability 23.5.2 *Mirror instability 23.6 *Summary

454 454 455 455 456 457 457 457 457

24 *Wave–particle interactions and kinetic effects 24.1 *Particle resonance 24.1.1 *Uniform medium 24.1.2 *Dipole field 24.2 *Wave-particle interaction in a uniform medium 24.3 *Wave–particle interaction in a dipole field 24.3.1 *Linearization of the drift kinetic equation 24.3.2 *Evaluation of moments of the perturbed distribution function 24.4 *Quasilinear theory: a tutorial example 24.4.1 *Model 24.4.2 *Basic equations of the disturbance 24.4.3 *Boundary conditions 24.4.4 *Zero-order solution

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484 487 487 488 489 489

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Contents 24.4.5 *Boundary perturbation 24.4.6 *Energy at marginal stability 24.4.7 *Relationship Between ω and k y 24.5 *Quasilinear theory in a dipole field 24.6 *Limits of magnetohydrodynamics 24.7 *Summary

490 490 491 491 493 493

25 Last words

495

PART 4 Appendices

497

A Some mathematical techniques A.1 The essence of Cartesian tensors A.2 Vector operators in curvilinear coordinates A.3 Properties of the bi-Maxwellian distribution A.4 The Stokes equation and Airy functions A.5 The plasma dispersion function A.6 Method of stationary phase

499 499 500 500 502 503 506

B Properties of the geomagnetic field B.1 Properties of a dipole field B.2 Geometry of field lines B.3 Magnetic field coordinates B.3.1 Cylindrical symmetry in the absence of currents B.3.2 Local coordinates in the presence of currents

508 508 509 512 512 515

C Fourier analysis techniques C.1 Some generalized functions C.1.1 The Dirac delta-function C.1.2 The Heaviside unit step function C.2 The Fourier transform C.2.1 Derivatives of the Fourier transform C.2.2 Convolution and the convolution theorem C.2.3 Theorems relating to Fourier transforms C.2.4 Symmetry properties of the Fourier transform C.2.5 Some Fourier transforms C.3 The modified Laplace transform C.3.1 Laplace transforms of derivatives of f (t)

516 516 516 516 516 517 517 517 518 518 518 519

D Wave analysis techniques D.1 Introduction D.2 Integral representations D.2.1 Transform methods for the solution of the one-dimensional wave equation D.3 Wavepacket analysis

520 520 520

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Contents D.3.1 The analytic signal D.3.2 Quasiperiodic functions

xvii 523 524

Bibliography

528

Index

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Preface

Solar–terrestrial physics deals with phenomena in the region of space between the surface of the Sun and the upper atmosphere of Earth. The matter in this region is in the plasma state. The subject includes processes that generate the solar wind, the physics of geospace and the Earth’s magnetosphere, and the interaction of magnetospheric processes with the upper atmosphere. Such processes are important for energy transfer between the Sun and the geospace environment. Many of them are mediated by long-period wave phenomena. Such phenomena are usually treated by the methods of magnetohydrodynamics (MHD). In this approach, the wave frequency is assumed to be very much less than the gyrofrequency of any significant ion species in the magnetic field. The largest gyrofrequency, that of electrons, is in turn asssumed to be much less than the plasma frequency. When collision processes are important, this leads to the usual equations of MHD. In the absence of collisions, we obtain a modified MHD in which the pressure is anisotropic. It is usually assumed that, in this frequency domain, the collective fluid approach of MHD is adequate. This is not always the case. There are circumstances in which it is necessary to include kinetic effects and in which conventional MHD is not an adequate description of the plasma. The purpose of this book is the study of such long-period wave phenomena in the solar-terrestrial system. The wavelengths of such waves are often comparable with the dimensions of the system so that boundary effects are crucial to the understanding of the phenomena. Much of the book is concerned with waves and oscillations in such bounded media. The intention is to provide the theoretician and experimentalist with a coherent account of the important theoretical ideas that underpin current understanding of ultra-low-frequency wave phenomena in solar-terrestrial physics and that may be used to address future problems. In order to do this, it is necessary to provide some account of the observational background. This background cannot be appreciated without some knowledge of the physics involved. It is beyond the scope of this treatment to provide a comprehensive review of observational solar-terrestrial physics. Limitations of space mean that we can only treat in detail a limited selection of phenomena, chosen to be illustrative of the theory. This creates a dilemma: to what extent can the reader be assumed to have a general acquaintance with the structure and dynamics of the solar-terrestrial Copyright © 2005 IOP Publishing Ltd.

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Preface

system so these selected phenomena can be understood in context? The solution that we have adopted is (i) to provide a thorough development of the basic theory of MHD and related wave phenomena in uniform media, (ii) to present an outline of observational knowledge of the structure of the solar-terrestrial system and the waves and oscillations that can occur in it, using heuristic arguments from the basic theory to justify the physics where necessary, and (iii) to apply the theory, developing it further where necessary, to a variety of situations relevant to this environment. Some chapters and section headings in the lost of contents are marked with an asterisk. These deal with with the motion of particles and their interactions with MHD waves. They go beyond conventional MHD theory. They can be omitted on a first reading. Other sections are set in smaller type. These are technical sections that contain straightforward but algebraically tedious derivations or that discuss such matters as the accuracy of some approximations or the derivation of standard mathematical representations. Readers prepared to take their content on trust may pass over them lightly. The reader is assumed to be familiar with Maxwell’s equations in electromagnetic theory and their applications to electromagnetic waves. Knowledge of vector calculus and the theory of functions of a complex variable including Cauchy’s theorem and the residue theorem is assumed. Some familiarity with Fourier analysis is required. More advanced mathematical techniques are introduced as needed. This book is the result of many years work in the field. I am very grateful to numerous colleagues with whom I have interacted over the years. In particular, I am grateful to Professor J A Gledhill who first aroused my interest in wave propagation, to Dr K G Budden, who honed my theoretical skills and whose monumental book Radio Waves in the Ionosphere has been a great influence, and to Dr R A Greenwald who, with his beautiful auroral radar experiments, provoked me to move my interests to MHD waves. The initial stages of the book were given considerable momentum by an invitation from Professor Marcel Goossens to give a course on MHD waves at the Katholieke Universiteit Leuven in 2000. I am very grateful to those who have read and criticized parts of the manuscript. In particular, students of the Katholieke Universiteit Leuven and the University of Ghent and the students of the MSc programme of NASSP1 at the University of Cape Town in 2003 and 2004 helped to reduce the number of errors and obscurities. Those that remain are my fault. I am very grateful to my wife, Carol, who has tolerated my erratic ways and my neglect of domestic responsibility during the course of this work, particularly in its final stages.

1 National Astrophysics and Space Science Programme: a cooperative programme of a number of

South African National Facilities and Universities

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PART 1 FUNDAMENTALS OF MHD WAVE THEORY

Copyright © 2005 IOP Publishing Ltd.

Chapter 1 Basic ideas of thermodynamics and electrodynamics

1.1 Introduction Magnetohydrodynamics (MHD), in its most general form, deals with the motion of compressible conducting fluids in the presence of magnetic fields. Any such conducting fluid can be regarded as being made up of an equal number of positive and negative charges. Their motion can be described by kinetic theory, together with the laws of thermodynamics. It is not always necessary to use a complete kinetic theory. Many properties can be treated using the laws of thermodynamics. In this introductory chapter, we summarize those aspects of thermodynamics and electrodynamics that are needed for our purposes. As a consequence of the electrical and magnetic properties of the fluid, various natural frequencies of oscillation occur. The plasma frequency is the natural frequency of relaxation when positive and negative particles are slightly separated and released. The gyrofrequencies of the various species of charged particles are the natural frequencies of gyration about the magnetic field. When the natural scale of time variation is much longer than the gyroperiod of the heaviest ions, we may apply the MHD approximation. In such circumstances, the characteristic speeds in the medium are much less than the speed of light and Maxwell’s equations can be simplified. However, this simplification may be possible even when the other assumptions of MHD do not hold. This low-velocity low-frequency approximation to Maxwell’s equations is the subject of this chapter. We first introduce Maxwell’s equations as they apply to a situation in which the only charges and currents are associated with moving point particles. There are no bound charges or molecules with spatially ordered magnetic dipole moments. These equations are then simplified in the low-velocity approximation. We then briefly discuss motion in uniform electromagnetic fields. Finally, energy and momentum transport equations are derived and discussed. Copyright © 2005 IOP Publishing Ltd.

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Basic ideas of thermodynamics and electrodynamics

4

1.2 Elementary ideas of thermodynamics and kinetic theory 1.2.1 Equation of state of an ideal gas The equation of state of a fluid is a relationship between P, V , and T its pressure, volume, and absolute temperature. For an ideal gas, this is PV =

RT

(1.1)

where is the number of moles present and R is the ideal gas constant which is the same for all gases. The number of molecules per mole is Avogadro’s number A. If the number density of molecules is N = A /V and Boltzmann’s constant K = R/A, this may be written as P = N K T.

(1.2)

The equation of state allows us to express quantities depending on the state of the fluid in terms of any two of P, V , and T . 1.2.2 Comparison with kinetic theory An elementary treatment shows that the pressure in an ideal gas, consisting of molecules with n degrees of freedom, an isotropic velocity distribution, and number density N is 1 2 = Nmu 2  (1.3) P= n n where is the internal energy density of the gas. If the molecules are point particles with three translational degrees of freedom and no rotational and vibrational degrees of freedom, then n = 3 and







P=

2 3


= 13 Nmu 2 .

(1.4)

Comparison with the ideal gas law (1.2) suggests that, for n = 3, we can relate the internal energy density to the temperature through


= 32 N K T.

(1.5)

1.2.3 First law of thermodynamics The first law of thermodynamics is a statement of conservation of energy. It states that if a fixed mass of gas M has pressure P, volume V , and internal energy U , then, in any change of state, the increase in internal energy of the gas is equal to the heat supplied to the gas minus the work done by the gas. Thus, δ Q = dU + δW = dU + P dV.

(1.6)

Generally, the work done by the gas depends on how the change in state variables is achieved. Thus, δ Q and δW are not perfect differentials that may be independently integrated, unless the change is reversible. Copyright © 2005 IOP Publishing Ltd.

Elementary ideas of thermodynamics and kinetic theory

5

1.2.4 Second law of thermodynamics The second law of thermodynamics, in its most abstract form due to Carath´eodory, specifies that, in the neighbourhood of any equilibrium state of a system, there are states inaccessible by an adiabatic process. This leads to the requirement
δQ ≤ 0. (1.7) T In the special case of a reversible process, the inequality becomes an equality and this becomes

dQ ≡ dS = 0 (1.8) T where the entropy S is a function of the state variables. If the state variables are p and V , the first law can then be written as T dS = dU + P dV.

(1.9)

1.2.5 Ratio of specific heats of a gas Two specific heats for a gas may be defined, one, CP , when the gas is held at constant pressure and the other, C V , when it is held at constant volume. The ratio of these is γ ≡ CP /C V . It can be shown that γ =

n+2 n

(1.10)

where n is the number of degrees of freedom of a molecule of the gas. For an ideal gas of point particles, n = 3 and γ = 5/3. The internal energy density of the gas may, thus, be written in the form


= γ P− 1 .

(1.11)

1.2.6 State variables and Maxwell’s relations The four quantities P, V , T , and S are related by the equation of state (1.2) and the first law written in the form (1.9). Any two of them can, therefore, be regarded as determining the state of the system. Any property, for example the energy U , depending on the state of the system may be treated as a function of these two variables. Suppose we choose S and V as the variables of state. Then, from (1.9),     ∂U ∂U =T = −P. (1.12) ∂S V ∂V S Differention of the first of these with respect to S and the second with respect to V shows that     ∂T ∂P =− . (1.13) ∂V S ∂S V Copyright © 2005 IOP Publishing Ltd.

6

Basic ideas of thermodynamics and electrodynamics

This the first of the four thermodynamic relations of Maxwell. The other three may be derived in the same way by replacing U with the enthalpy H ≡ U + PV , the Helmholtz free energy F ≡ U − T S, and the Gibbs function G ≡ U − T S + PV , respectively:     ∂V ∂T = (1.14) ∂P S ∂S P     ∂S ∂V = − (1.15) ∂T P ∂P T     ∂P ∂S = . (1.16) ∂T V ∂V T These are not independent. If any one is given, the others may be derived from it. 1.2.7 Rate of change of entropy in reversible processes




If we define the specific entropy as s = S/M and use the relationships U = V ,


= n P/2, and dV /V = −dρ/ρ for an ideal gas, this may be written as ds =

1 (γ − 1)ρT

  γP dP − dρ . ρ

(1.17)

In the case of a reversible adiabatic process, there is no heat transfer and ds = 0 so that dP γP = . (1.18) dρ ρ 1.2.8 Specific energy, entropy, and enthalpy




is the energy per unit volume. It is sometimes more The energy density convenient to work in terms of the energy per unit mass, given by ε=


= ρ

P . (γ − 1)ρ

(1.19)

We refer to this as the specific energy rather than the energy density. We have already defined the specific entropy s and we can define the specific enthalpy w=

P γ P U + PV =+ = . M ρ γ −1 ρ

(1.20)

1.3 Maxwell’s equations in the presence of currents and charges A fully ionized plasma can be regarded as system of moving point charges. The positions and velocities of these charges are fully determined by a charge Copyright © 2005 IOP Publishing Ltd.

Maxwell’s equations

7

density η( r, t) and a current density J ( r, t), which are functions of position and time. If the material consists only of moving point charges which are not bound and magnetic properties due to the orbital and spin angular momentum can be neglected, it is possible to describe the electromagnetic field in such a material by only two field variables E and B . If we examine the system on the scale of the individual particles, then the charge and current densities fluctuate dramatically having a δ -function behaviour such that they are zero everywhere except at the location of the particles. If we consider a single charge, located at r , with charge q and velocity v , its charge density is η( r) = qδ 3 ( r)

(1.21)

so that if the charge is located somewhere in a volume V ,  η( r) d V = q.

V

The mean value of η in a volume V is  η dV . η = V

V

(1.22)

(1.23)

The assumption of fluid theory is that, if we let V tend to smaller values, then this mean value will approach a limiting value when the volume is still large compared to the interparticle distance. On this length scale, fluid theory is valid and the actual particle distribution may be replaced by the mean particle distribution. If the length scale is further reduced so that it is comparable to the interparticle distance, the mean begins to fluctuate and no longer has a limiting value. On this scale, fluid theory breaks down. Exactly the same process can be followed to find the mean current density. It is then assumed that the electromagnetic fields are the fields with these charge and current densities as the source. Careful statistical analysis such as that in chapter 4 shows that the assumption is correct but it is not obvious from first principles: the actual electric field at the location of a point charge may differ from the mean field in the vicinity. This occurs, for example, in dielectrics where the Lorentz polarization term must be introduced. The question of whether or not to include the Lorentz polarization term was once controversial in magnetoionic theory, the study of electromagnetic waves in cold plasmas [31]. Consequently, on length scales for which the fluid approximation is valid, Maxwell’s equations take the following form: •

Faraday’s law, ∇×E=−

•

∂B ∂t

Maxwell’s law, ∇ × B = µ0 J +

Copyright © 2005 IOP Publishing Ltd.

1 ∂E c2 ∂t

(1.24)

(1.25)

8 •

Basic ideas of thermodynamics and electrodynamics Gauss’ law in electricity,

η 0

(1.26)

∇ · B = 0.

(1.27)

∇·E= •

Gauss’ law in magnetism

1.4 The Lorentz force law The force on a single particle is given by the Lorentz force law, F = q{ E + v × B}.

(1.28)

Consider a fluid consisting of many identical charged particles. (Such a fluid cannot exist alone but must be neutralized by an interpenetrating fluid of particles of opposite sign.) Suppose that the particle number density is N and the charge on each particle is q . If the charge and current densities are averaged according to the fluid approximation, then the charge density is η = Nq and the contribution of the fluid to the total current density is J = Nqv . Multiply (1.28) by N and we get an expression for the Lorentz force per unit volume on a fluid element: f = ηE + J × B

(1.29)

where η is the charge density.

1.5 Low-velocity approximation to Maxwell’s equations—Ampe´ re’s law If we are dealing with problems which have a characteristic length scale λ and a characteristic time scale τ , then Faraday’s law (1.24) shows that E∼

λ B. τ

Consider Maxwell’s law (1.25). The ratio of the last term on the right-hand side to the term on the left is of the order of λ2 /c2 τ 2 . In MHD problems the length and time scales are such that this ratio is very small, so that Maxwell’s law becomes Amp´ere’s law: ∇ × B = µ0 J. (1.30) The other equations are unaffected. This approximation is often called the MHD approximation. It is more general than this and is better described as a low-velocity approximation. It may be valid even when the other MHD approximations, discussed in chapter 2, are not. Copyright © 2005 IOP Publishing Ltd.

Motion of charged particles in uniform electric and magnetic fields

9

1.6 Motion of charged particles in uniform electric and magnetic fields 1.6.1 Equation of motion The equation of motion of a particle of charge q , moving with velocity v in a region in which there is a uniform electric and magnetic field and acted on by an external force F , is m v˙ = F + q{ E + v × B} (1.31) where v˙ = dv/dt . Because the magnetic force is perpendicular to the magnetic field, it is convenient to resolve the velocity into parallel and perpendicular components, µv ˆ  and τˆ v⊥ , where µ ˆ and τˆ are unit vectors parallel and perpendicular to B . Let ρˆ = µ ˆ × τˆ .

(1.32)

Then (1.31) becomes q F + µ ˆ ·E m m q F⊥ v˙ ⊥ = + { E ⊥ + v ⊥ × B} m m q F⊥ qB + E⊥ + v⊥ τˆ × µ = ˆ m m m q F⊥ qB + E⊥ − v⊥ ρ. = ˆ m m m We discuss the solution of these equations in some simple cases. v˙ = µ ˆ ·

(1.33)

(1.34)

1.6.2 Cyclotron motion If the electric field and the external force are zero and the velocity is perpendicular to the magnetic field, then, from (1.34), qB v⊥ . (1.35) m is in the −ρ-direction, ˆ perpendicular to v ⊥ . Further, since v˙ ⊥ = −ρˆ

This shows that v˙ ⊥ v ⊥ = τˆ v⊥ ,

v˙⊥ = 0

(1.36) q B . (1.37) τ˙ˆ = − ρˆ m Now consider figure 1.1(a). It shows the unit vectors τˆ and ρˆ , making angle θ with fixed axes, α and β, respectively. Clearly, τˆ = αˆ cos θ − βˆ sin θ ρˆ = αˆ sin θ + βˆ cos θ. Copyright © 2005 IOP Publishing Ltd.

(1.38) (1.39)

10

Basic ideas of thermodynamics and electrodynamics ^ b

^ t ds

^ r

^ ^ t+dt

rg

q

rg

^ a

q

dy

^ t

(a)

dt^ ^ t dy ^ ^ t+dt

(b)

Figure 1.1. Circular motion in a uniform magnetic field.

If we differentiate the first of these, we immediately see that τ˙ˆ = −ρˆ θ˙ .

(1.40)

The radius of curvature of the path is in the −ρ-direction. ˆ If its magnitude is rg , then, from figure 1.1(b), τ˙ˆ dτˆ ρˆ = . (1.41) − = rg ds v⊥ Thus, r g = ρˆ

mv⊥ qB

(1.42)

Since, from (1.36), the speed, v⊥ is constant, the particle moves in a circle of radius rg , called the gyroradius or cyclotron radius. The angular velocity vector of the particle  is parallel to B and given by Bq . =µ ˆ θ˙ = µ ˆ m

(1.43)

 × v = r g.

(1.44)

It follows that For positive particles,  is parallel to B and for negative particles, it is antiparallel to B. The period of the circular motion is called the gyroperiod or cyclotron period. It is given by 2π . (1.45) T = || The frequency T −1 is the gyrofrequency or cyclotron frequency. The angular gyrofrequency is B|q| ≡ . (1.46) m It is often loosely called the gyrofrequency. Copyright © 2005 IOP Publishing Ltd.

Motion in uniform fields

11

If the particle has a component of velocity parallel to the field, then its motion perpendicular to the field is unaffected. In a frame of reference moving along the field with speed v , it moves in a circle with constant angular velocity . In the original frame of reference, its path is a spiral. A convenient way of characterizing this spiral is in terms of the gyroradius, rg , and the pitch angle, α , defined by tan α = v⊥ /v .

(1.47)

The gyration of the charged particle is equivalent to a loop of area A, carrying current I . On length scales large compared with the gyroradius, the current loop is equivalent to an elementary magnetic dipole of moment M . The average current in the loop is I = qv⊥ . The area of the loop is A = π rg2 and the magnetic dipole moment is, therefore, 1 mv 2 (1.48) M=µ ˆIA =µ ˆ 2 ⊥. B 1.6.3 Electric field drift Suppose that now there is a uniform electric field E . We can observe it from a reference frame in which E ⊥ is zero. Such a frame moves relative to the original frame with a velocity V E such that, in this frame, E  = E + V E × B = 0.

(1.49)

Only the perpendicular component of V E occurs in this equation so we can choose its parallel component arbitrarily. We choose it to be v . If we take the vector product of (1.49) with B , we get VE =

E×B . B2

(1.50)

In the new reference frame, the motion is again simply cyclotron motion in a circle. We can choose the origin at the centre of the circle. We call the location of the origin of this frame of reference in which the motion is cyclotron motion the guiding centre. The frame is called the guiding-centre frame. In the original frame of reference the guiding centre moves with velocity V = v +

E×B . B2

(1.51)

The motion can be regarded as a superposition of the motion of the guiding centre and the circular cyclotron motion. The velocity of the guiding centre is called the drift velocity. Because the fields are constant, this velocity is constant. This need not be so if the fields vary in space or time; in such cases, as we shall see, the guiding centre frame is non-inertial. Also other drifts may arise. In this discussion, there has been no limitation on the magnitude of E. In figure 1.2, we show the path of a particle drifting perpendicular to B in three cases, V < v, V = v, and V > v. Copyright © 2005 IOP Publishing Ltd.

12

Basic ideas of thermodynamics and electrodynamics (a)

B

(b)

v


v


v


v


B

B

B

(c)

v


v
B

B

Figure 1.2. E × B drift: (a) V < v, (b) V = v, (c) V > v.

If, in addition, the electric field has a component parallel to B, then the perpendicular motion is unaffected. The particle is accelerated parallel to the field but maintains its perpendicular orbit. The guiding centre is accelerated along the field.

1.6.4 Drifts due to an external force If there is a uniform external force perpendicular to the magnetic field and the electric field is zero, then we can transform to an inertial frame moving with velocity V in which the electric field is V ⊥ × B. In this frame, the motion will be circular cyclotron motion. We may choose V ⊥ = V F , the value for which the resultant of the external force and the electric force is zero. Then F + q V F × B = 0.

(1.52)

If we take the vector product with µ ˆ and solve for V F , we get VF =

F×µ ˆ . qB

(1.53)

The motion of the particle in the original frame is the superposition of its circular cyclotron motion and a uniform drift velocity V F . In this, it is similar to the electric field drift V E which is a special case of the force drift with F = q E. It should be noted that, for non-electric forces, because of the factor q, positive and negative charges drift in opposite directions and, for the electric force, they drift in the same direction. Copyright © 2005 IOP Publishing Ltd.

Electromagnetic energy

13

1.7 Electromagnetic energy Take the scalar product of (1.24) with B/µ0 and the scalar product of (1.25) with E/µ0 and subtract the second from the first. The result is    E×B 1 ∂ 1 B2 1 2 {B · ∇ × E − E · ∇ × B} ≡ ∇ · + 0 E . = −J · E− µ0 µ0 ∂t 2 µ0 2 (1.54) This is Poynting’s theorem. It may be written in the form ∂U = −J · E − ∇ · S ∂t

(1.55)

where the energy density U and the Poynting flux vector S are, respectively, 1 B2 1 + 0 E 2 2 µ0 2 E×B S= . µ0

U=

(1.56) (1.57)

If we consider a volume V enclosed by a surface of area A and integrate (1.55) over this volume using Stokes’ theorem to transform the integral of the divergence to a surface integral, we get 
 d U dV = − J · E dV − S · nˆ d A (1.58) dt V V A where nˆ is the outward normal to the surface. This is then an equation describing energy changes in the system of electromagnetic fields. The term on the lefthand side represents the rate at which  electromagnetic field energy stored within the volume increases. The integral V J · E dV represents the rate at which the particles of the fluid gain energy from the fields. Thus, the first term on the righthand side, which is minus this, represents the rate at which the electromagnetic field gains energy from the particles. The last term represents the loss of energy from the volume arising from a flux of energy outwards through the surface. As a result, the Poynting vector S is usually interpreted as the energy flux density, representing the rate at which energy flows through the surface per unit area. In the low-frequency approximation, the energy density associated with the electric field is negligible. Thus, in (1.55) and (1.58), U=

B2 . 2µ0

(1.59)

1.7.1 Joule energy transfer In this representation, the interaction with matter is entirely taken into account by what is often called the Joule ‘heating’ term. This is a misnomer. The Copyright © 2005 IOP Publishing Ltd.

14

Basic ideas of thermodynamics and electrodynamics

term describes energy transfer to matter in the form of particles. In some circumstances, for example in a resistor, the energy transferred from field to particles is randomized and leads simply to an increase in the internal energy of the matter consisting of the particles in this model. It is then appropriate to talk of Joule heating. There are circumstances in which energy transferred to the particles leads to an ordered change in the kinetic or potential energy of the particles or in which ordered mechanical energy is extracted from the particles and transferred to the fields. It is, thus, more appropriate to talk of Joule energy transfer. The important point to note is that Poynting’s theorem in this form is not a complete description of the energy conservation. It has nothing to say about the details of the behaviour of the particle energy. This must come from the equations of motion of the particles under the action of the Lorentz force. Poynting’s theorem applies to the fields which do not by themselves constitute a closed system. The Joule energy transfer measures the rate at which work is done by the particles on the field system. If we consider a closed system consisting of the fields and the particles, there will be energy density and flux terms corresponding to the kinetic and potential energies of the particles and the Joule energy transfer term giving the rate at which energy is transferred from particles to fields will be cancelled by an equal and opposite term representing the transfer of energy from fields to particles. In the classical Maxwell formulation of electromagnetism, the interaction of fields with matter is handled differently. In this formulation, the additional electric displacement field D = 0 E + P and the magnetic intensity H = B/µ0 − M are introduced and the current J is not the total current but the so-called ‘true’ current which excludes the polarization current ∂ P/∂t and magnetization current ∇ × M. In this case, the ‘true’ current is generally (but not always) assumed to be the current in a conductor and does correspond to a transfer of energy to the internal energy of the conductor. In these circumstances, the term ‘Joule heating’ is appropriate. Energy transfer to the bound particles comes from the scalar product of E with the polarization and magnetization currents, E · ∂ P/∂t and E · ∇ × M. These are included as part of the field energy. 1.7.2 Physical interpretation of the flux vector The interpretation of the Poynting vector as representing the energy flux in the fields as a function of position can be controversial [152, section 10.5]. The curl of an arbitrary vector field with appropriate dimensions can be added to S without altering the flux term in the energy conservation equation. In static circumstances, this can lead to apparent inconsistencies. It can be argued that the only meaningful statement of energy conservation is an integral form such as (1.58). Thus, it is not possible to localize the energy: conservation of energy is a global concept. Nevertheless, in the study of electromagnetic wave propagation, it is often tacitly assumed that the appropriate energy flux vector represents the actual rate Copyright © 2005 IOP Publishing Ltd.

Electromagnetic momentum

15

and direction at which energy is propagated per unit area. This is because one can imagine an infinitesimal directional receiver being used as a probe to extract energy from the wave. It would be expected that the magnitude of the received signal would be proportional to the magnitude of the flux vector and the orientation of the receiver for maximum signal would determine the direction of the flux. The same is true of other types of wave, for example sound waves. We shall take this point up later in chapter 9 and see to what extent it is justified.

1.8 Electromagnetic momentum The force per unit volume on matter in the form of particles is given by the Lorentz force density (1.29). Thus, in the fluid approximation, the rate of change of the momentum density of matter at a point is ∂(ρv) = η E + J × B + f mech ∂t

(1.60)

where f mech is the resultant of all other non-electromagnetic forces. The Maxwell equations (1.25) and (1.26) may be used to eliminate the charge density η and the current density J . The result is   ∂(ρv) 1 ∂E 1 = 0 E ∇ · E + × B + f mech . ∇×B− 2 (1.61) ∂t µ0 c ∂t The other two Maxwell equations (1.27) and (1.24) allow us to write an analogous equation:   ∂B 1 × E. (1.62) B ∇ · B + 0 ∇ × E + 0= µ0 ∂t Add these two equations and express the result in Cartesian tensor notation (appendix A.1):   ∂Bj 1 ∂ ∂(ρvi ) = − i j k klm B j Bm Bi ∂t µ0 ∂x j ∂ xl   ∂Ej ∂ − i j k klm E j Em + 0 E i ∂x j ∂ xl ∂ (1.63) − (i j k 0 E j Bk ) + f i,mech . ∂t By using (A.6), we can write this in the form ∂(ρvi ) ∂ + (i j k 0 E j Bk ) ∂t ∂t  Bi B j − 12 δi j B 2 ∂ 2 1 + 0 [E i E j − 2 δi j E ] + f i,mech . (1.64) = ∂x j µ0 Copyright © 2005 IOP Publishing Ltd.

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Basic ideas of thermodynamics and electrodynamics

This is of the form ∂(ρv) ∂(0 E × B) + = ∇ · T + f mech ∂t ∂t

(1.65)

where T is a second-rank stress tensor. The equation can be interpreted as stating that the rate of change of a total momentum density (mechanical plus electromagnetic) is equal to the divergence of an electromagnetic stress tensor plus any non-electromagnetic force density. This requires us to interpret 0 E × B as the momentum density associated with the electromagnetic field. If we integrate this over an arbitrary volume V and use Stokes’ theorem to convert the right-hand side to a surface integral, we get
  d d ρv dV + 0 E × B dV = T · nˆ d A + F mech . (1.66) dt V dt V S The left-hand side represents the rate of increase of mechanical momentum in the volume V plus the rate of increase of a quantity involving the fields which has the dimensions of momentum. This latter quantity is interpreted as the momentum associated with the flux of energy in the fields. Its density is equal to the Poynting vector divided by the square of the speed of light. The right-hand side represents the surface integral of the force per unit area on the surface of V . In the low-frequency approximation, this expression is much simplified. All the terms involving the electric field are O(v 2 /c2 ) compared with the terms involving B where v ∼ λ/τ and λ and τ are the characteristic lengths and times in the problem. In this approximation, the electromagnetic momentum density is negligible and the stress tensor contains only the magnetic field terms:  Bi B j − 12 δi j B 2 ∂ ∂(ρvi ) = (1.67) + f i,mech . ∂t ∂x j µ0 The interpretation of the magnetic stress tensor is as follows. Consider a closed surface coincident with a portion of a magnetic flux tube of cross-sectional area δ A. Let nˆ j be a unit vector, parallel to the field, pointing outwards from one end of the surface. The outward force on this end of the tube is Ti j nˆ j δ A =

nˆ j B2 {Bi B j − 12 δi j B 2 }δ A = nˆ i δ A. µ0 µ0

(1.68)

This means that the region enclosed by the surface can be interpreted as being under a tension B 2 /µ0 per unit cross-sectional area. If, however, we consider the outward force on an element of area lˆj δ A perpendicular to the magnetic field, then the outward force per unit area is Ti j lˆj δ A = Copyright © 2005 IOP Publishing Ltd.

lˆj B2 {Bi B j − 12 δi j B 2 }δ A = −lˆi δ A. µ0 2µ0

(1.69)

Summary

17

This is an inward force per unit area which can be interpreted as a pressure B 2 /2µ0 perpendicular to the field. It is stressed that this formulation applies only to situations where the presence of matter is represented by isolated charged particles providing charge and current densities which can be treated by a fluid approximation.

1.9 Summary • • •

•

• •

•

• •

We outline the basic thermodynamic and electrodynamic theory that is necessary to treat the motion of a system of point particles with charges qi and velocities v i . The electromagnetic properties of the system are treated by Maxwell’s equations in their free space form with the particles forming a system of charge and current densities. The fluid approximation implies that we take an ‘infinitesimal’ volume which is small compared with the macroscopic length scales but large compared to the interparticle distances. The fields and the charge and current densities are averaged on this scale. The low-velocity approximation to Maxwell’s equations is appropriate when length and time scales in the problem are such that characteristic speeds are very much smaller than the speed of light. This allows the neglect of the displacement current and the use of Amp´ere’s law instead of Maxwell’s law. A charged particle, moving in a uniform magnetic field, has a helical path. The radius of curvature of the path is called the gyroradius. The frequency is the gyrofrequency and is independent of the particle energy. A uniform electric field normal to the magnetic field leads to a motion which can be regarded as a superposition of a motion perpendicular to E and B and the parallel motion of the particle. A uniform charge-independent force leads to a similar drift with positive and negative particles moving in opposite directions. In this formulation, Poynting’s theorem takes the form that the rate of increase of the electromagnetic energy is minus the divergence of the Poynting flux minus the rate at which work is done on the particles by the fields. The only interaction between the fields and matter is through this work term. In the low-frequency approximation, only the magnetic energy density is significant. Energy localization in electromagnetic fields can be controversial. It is pointed out that, in wave problems, proper choice of an energy flux vector is usually straightforward but a detailed discussion is postponed until later. Expressions for the electromagnetic momentum density and the stress tensor are derived. In the low-frequency approximation, the electromagnetic momentum density and the electric field components of the stress tensor are negligible. The momentum equation becomes a straightforward relation in

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18

Basic ideas of thermodynamics and electrodynamics which the rate of change of mechanical momentum is equal to the divergence of the magnetic stress tensor.

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Chapter 2 The magnetohydrodynamic approximation

2.1 Introduction The study of magnetohydrodynamics (MHD) is the study of how a fluid which consists of charged point particles moves in the presence of electromagnetic fields. The study is done in the fluid approximation in which the properties of the particles are averaged over volumes which are small compared with macroscopic volumes but large compared with the interparticle distance. In this chapter, the MHD equations are derived heuristically, using the basic ideas of Newtonian mechanics and thermodynamics, together with the approximate electrodynamics of chapter 1. We first discuss the nature of the approximations used in MHD. This allows us to derive a set of MHD equations by starting with the equations of motion of the individual particles and averaging them for each particle species to get a set of fluid equations for each species. These are then combined under the MHD approximation. This assumes that there is charge neutrality to zero order, the characteristic frequencies are much less than the lowest ion gyrofrequencies, and the difference in the mean velocities of the individual species is small compared with the fluid velocity. With these assumptions, we derive a set of equations which govern the dynamics of such fluids. They amount to a set of hydrodynamic equations with the addition of magnetic forces which act on currents in the fluid. One consequence of the assumptions of MHD is that magnetic fields are ‘frozen in’ to the fluid. This means that, if we identify a set of fluid elements which at some instant lie along a single field line, then, at a later instant, they will still all lie along a single field line, no matter how the fluid has moved or the field has been distorted. We also derive equations that describe energy and momentum exchange within the medium, extending the results of the last chapter to include the fluid forces explicitly. Copyright © 2005 IOP Publishing Ltd.

19

20

The magnetohydrodynamic approximation

2.2 Fluid equations for the particle species In the simplest picture of a plasma, we can consider two interpenetrating fluids, one consisting of electrons and one of positive ions. Each fluid moves with its own velocity. These velocities are approximately equal but their difference gives rise to the currents. We consider each particle species independently. A fluid element is small on the macroscopic scale but is large enough on the microscopic scale to contain many particles. It moves with the mean velocity of the particles in it. An ‘infinitesimal’ fluid element is small enough so that the mean velocity is independent of its volume. The boundaries of the fluid element move with the mean velocity but individual particles move into and out of the volume on account of their thermal motion. The velocity of the fluid is, in general, a function of position and time. In this section, we use a notation in which the particle velocity is v , the mean particle velocity within a fluid element is v, and we define the peculiar velocity of the particle, u, as the velocity relative to the mean velocity: u = v − v.

(2.1)

The particle charge is q. For singly charged ions, q = +e and for electrons, q = −e, where e is the magnitude of the electron charge. The fluid equations can be derived from a detailed kinetic theory involving the statistics of the motion of the individual particles. This full development is postponed to chapter 4. We provide here a heuristic justification for the MHD equations by summing over the average particle motion to find the macroscopic plasma motion and treating the thermal motions by using the ideas of elementary kinetic theory and thermodynamics. 2.2.1 The continuity equation Conservation of particles requires that the rate of increase of the number of particles in an arbitrary volume V , having a surface area S must be minus the rate at which the particles flow out through the surface. Thus,
 d N dV = − Nv · dS. (2.2) dt V S This may be written in the form    ∂N + ∇ · (Nv) dV = 0. ∂t V

(2.3)

Here the order of differentiation and integration on the left-hand side has been reversed. The partial derivative arises because N is a function of space and time and we are only concerned with the time variation. Gauss’ divergence theorem has been used to change the integral on the right-hand side to a volume integral. Copyright © 2005 IOP Publishing Ltd.

Fluid equations for the particle species

21

This must hold for any arbitrary volume. This is only possible if the integrand is zero. The equation of continuity for the species is, thus, ∂N + ∇ · ( Nv) = 0. ∂t

(2.4)

2.2.2 The momentum equation 2.2.2.1 Cold plasma: electromagnetic forces A cold plasma can be regarded as one in which each particle moves with the mean fluid velocity so that there is no thermal motion. The force exerted on each particle is given by the Lorentz force law. To find the bulk behaviour, the Lorentz force law may be multiplied by N : mN

dv = q N{ E + v × B}. dt

(2.5)

Here d/dt operates on the particles which move with the mean fluid velocity. At a point fixed in space, ∂ d ≡ + v · ∇. (2.6) dt ∂t 2.2.2.2 Contribution of plasma pressure In a warm plasma, the particles have a thermal distribution of velocities and the plasma exerts a pressure. A rigorous treatment of pressure requires a detailed kinetic theory approach. This we defer to chapter 4. Here we use a simple heuristic approach using elementary ideas of kinetic theory. We regard each species of particle as a simple ideal gas consisting of point particles with an isotropic distribution of peculiar velocities. Such a gas has three degrees of freedom corresponding to the three components of velocity. The total energy density is the sum of the kinetic energies of the particles in unit volume Now consider a fluid element of the plasma moving with the mean fluid velocity. Because of the thermal motion, the particles do not remain with the fluid element but are exchanged with the particles in neighbouring fluid elements. If the plasma is uniform, this has no net effect. If, however, it is non-uniform, for example if there is a gradient of mean particle properties across the fluid element, then there can be a net transfer of mean particle momentum into or out of the element. The rate of change of momentum density in the element represents an effective force density which must be added to the momentum equation. The effective force due to this momentum transfer is the integral of the force on the fluid element due to the pressure 
∇ p dV. (2.7) F = − p nˆ · dS = − S

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V

22

The magnetohydrodynamic approximation

The minus sign is because we are evaluating the inward force and nˆ is the outward normal. The last step is a result of Gauss’ divergence theorem. When the volume over which the integral is taken is small, the integrand is approximately constant and we may write the integral as −∇ p δ V . The force density −∇ p may then be added to (2.5) to get mN

dv = q N{ E + v × B} − ∇ p. dt

(2.8)

2.2.2.3 Effect of collisions with other particle species If two identical particles collide, momentum is conserved: there is no change in the momentum of the fluid element for the species. If two particles of different species collide, momentum is transferred from one species to the other: the more frequent the collisions are, the more momentum is exchanged. Collisions between electrons and ions are not simple binary collisions because of the long distance nature of the Coulomb force. The nature of these collisions is discussed in more detail in chapter 4. There it is shown that the effective time between collisions in magnetospheric plasmas is of the order of hours or days. If, however, the plasma is not fully ionized, then binary collisions between neutral particles and ions or electrons may lead to significant momentum transfer. Generally, this process is only important in the lower ionosphere. We shall formally include terms describing momentum transfer due to collisions in order to show in what circumstances they may be important. First, consider a very simple model of collisions between two species of particles with masses m 1 , m 2 , number densities N1 , N2 , and mean velocities v 1 , v 2 . Then the velocity of the centre of mass of the two species is v =

N1 m 1 v 1  + N2 m 2 v 2  . N1 m 1 + N2 m 2

(2.9)

When two spherical particles collide, they may be scattered in any direction, subject to the law of conservation of momentum. Assume that the scattering process is such that, in the centre-of-mass frame, all directions are equally probable. Then, on average, over all such collisions, the effect of the collision is to bring the particles to rest in the centre-of-mass frame. The average change in momentum of a particle of type 1 is, then,   N1 m 1 v 1  + N2 m 2 v 2  m 1 m 2 N2 m1 − v 1  = {v 2  − v 1 }. N1 m 1 + N2 m 2 N1 m 1 + N2 m 2 The change in momentum density is N1 times the change for a single particle. If such collisions take place at a frequency ν, then the rate of change in momentum density is N1 N2 m 1 m 2 ν f coll. = {v 2  − v 1 }. (2.10) N1 m 1 + N2 m 2 Copyright © 2005 IOP Publishing Ltd.

Characteristic lengths and frequencies

23

This force density must be added to the momentum equation for the species. Clearly the force density for the other species is equal and opposite to this. A more rigorous discussion of the statistics of collisions gives rise to a similar expression except that the collision frequency may be a function of the relative velocity of the species. Even for the case of collisions which are not binary, such as the Coulomb collisions discussed in chapter 4, the argument can be made that, whatever the nature of the momentum transfer resulting from particle interactions, the average effect must be to change the particle’s velocity to the centre-of-mass velocity of the two species. Thus, we can formally include a collision term of the form (2.10) for each of the other species involved in the momentum equation (2.8): mN

dv = q N{E + v × B} − ∇ p + f coll. . dt

(2.11)

species

2.2.3 Adiabatic law In order to have a complete set of equations, it is necessary to have a relationship between the pressure and density of the fluid. This is the equation of state of the system: p = p(ρ). (2.12) Different equations of state may obtain in different conditions. For example, if the conditions are isothermal, then p = Cρ where C is a constant. This book concentrates on MHD waves. In such waves, conditions are such that it is reasonable to treat the changes as adiabatic. This means that the internal energy density and, hence, the pressure change only as a result of mechanical work done on the fluid element by neighbouring fluid elements. In this case, we use the adiabatic equation of state (1.18)

2.3 Characteristic lengths and frequencies 2.3.1 The Debye length A plasma is a gas in which a significant number of the molecules are ionized and show collective behaviour so that electromagnetic forces play a substantial part in the gas dynamics. It is possible to define a length scale λD , called the Debye length, such that, on scales smaller than λD , Coulomb forces between individual particles are important, while on scales large compared with λD , the Coulomb forces are screened by the collective behaviour of charges of opposite sign. The collective behaviour is effective, provided that the density is high enough so that there are a significant number of particles within a sphere of radius equal to the Debye length. The Debye length can be crudely estimated by considering a single charge q, located at the origin. It attracts charges of opposite sign and repels charges of Copyright © 2005 IOP Publishing Ltd.

24

The magnetohydrodynamic approximation

the same sign, so that there is an excess of charge density qδ N(r ) at distance r . The electrostatic potential associated with this redistribution of charge is given by Poisson’s equation: q δN ∇ 2  ≡ ∇ 2U = − . (2.13) 0 The thermal energy of the particles is ∼ K T . The fractional change in electron density over a sphere of radius r is given by the ratio of the electrostatic to the potential energy: δN q =− . (2.14) N KT Thus, Nq 2  1 d2 (r 2 ) = r 2 dr 2 0 K T

(2.15)

which has the solution = where

1 −r/λD e r2 

λD =

0 K T . Nq 2

(2.16)

(2.17)

Only inside a sphere of radius equal to the Debye length is there any significant departure from charge neutrality. All plasma physics only works on length scales large compared with the Debye length. 2.3.2 The plasma frequency The plasma frequency is the characteristic frequency with which a fluid, consisting of electrons and ions, will oscillate when the particles are displaced relative to one another. It occurs naturally in the equations describing electromagnetic wave propagation in plasmas. Consider a simple example. A rectangular ‘block’ of plasma consists of an equal number of positive ions and electrons with number density N, charges qi = e, qe = −e, and masses m i , m e . Ions and electrons are displaced distances ri and re relative to their centre-of-mass so that m ere + m iri = 0. Their relative displacement is r = re − ri . This results in a surface charge density σ = ±Ner and an electric field E = Ner/0 . As a result, there is a restoring force on each particle species and their equations of motion are: Ne2 (re − ri ) 0 Ne2 (ri − re ). m ir¨i = − 0

m er¨e = −

Copyright © 2005 IOP Publishing Ltd.

Characteristic lengths and frequencies

25

Subtracting these, we get 

Ne2 Ne2 r¨ = − + 0 m e 0 m i

r.

This is simple harmonic motion with angular frequency given by 2 2 ωp2 = ωp,e + ωp,i =

Ne2 Ne2 + . 0 m e 0 m i

(2.18)

It is possible to generalize this to a plasma with more than one species of positive ion. The quantities ωp,e and ωp,i are called the electron plasma frequency and ion plasma frequency respectively. Of course, m e m i and, thus, ωp ∼ ωp,e .

(2.19)

In spite of this, we cannot neglect the ion plasma frequencies. An external electric field oscillating near an ion plasma frequency will set the corresponding species of ion into oscillation. We expect these natural frequencies to appear in the equations governing wave propagation in the plasma. 2.3.3 The electron and ion gyrofrequencies The angular velocity or angular gyrofrequency of an electron gyrating about the magnetic field is given by (1.46) e = Be/m e. The angular gyrofrequency of an ion of mass m i is i = (m e /m i )e . 2.3.4 Characteristic speeds There are two forces that determine the dynamics of an MHD medium. One is the force arising from gradients of the pressure and the other is a J × B force. These are associated with two characteric speeds. The sound speed is associated with pressure gradients and is defined as  VS =

γP = ρ



γ KT . m

(2.20)

As we shall see, it is the speed at which sound waves are propagated in a uniform medium in which the only significant force arises from pressure gradients. As with all waves in elastic media, the square of the wave speed is the ratio of an appropriate elastic modulus (in this case the adiabatic bulk modulus γ P) to the Copyright © 2005 IOP Publishing Ltd.

26

The magnetohydrodynamic approximation

density. From (2.17), we see that, for a single-ion plasma, the Debye length is given by VS2 λ2D = . (2.21) 2 γ ωp,i Of course, the sound speed is of the same order of magnitude as the root mean square (rms) speed. The internal energy of the plasma is U = 12 Nmv 2  = P/(γ − 1). Thus, γ (γ − 1) 2 v . VS2 = (2.22) 2 The other characteristic speed is associated with magnetic forces. It was shown in section 1.8 that one component of the force arising from the magnetic stress is a tension per unit cross-sectional area of a magnetic flux tube. This has magnitude B 2 /µ0 . A string under √ tension T , with mass per unit length µ, propagates transverse waves with speed T /µ. By analogy, one expects an MHD wave to be propagated along the magnetic field direction with speed  B2 VA = (2.23) µ0 ρ where, for a flux tube of cross section δ A, the tension is B 2 δ A/µ0 , and the mass per unit length is ρ δ A. The quantity VA is called the Alfv´en speed [2]. It can be related to the ion plasma frequency and gyrofrequency. For a plasma with only one species of ion, from (2.18) and (1.46), 2i VA2 m i 2i = = . 2 m e ωp2 c2 ωp,i

(2.24)

If there are several species of ion, this becomes VA2 B2 B 2 e2 0 m 2 = = = 2av 2 2 2 2 c µ0 Nmc m Ne ωp,av

(2.25)

where av and ωp,av are the gyro- and plasma frequencies of a fictitious particle having charge e and mass equal to the mean positive ion mass. 2.3.5 Parameters for approximation There are two different types of force which are important in MHD. One arises from the plasma pressure and the other from the magnetic field stress. Their relative importance is conventionally characterized by a parameter β which is the ratio of the kinetic pressure to the magnetic pressure β= Copyright © 2005 IOP Publishing Ltd.

2µ0 P . B2

(2.26)

The MHD equations for a fully ionized plasma

27

If β 1, the magnetic forces dominate the motion of the plasma and the problem reduces to incompressible MHD; and if β 1, the magnetic forces are negligible and the forces influencing the motion of the medium are the same as those of fluid dynamics. For β ∼ 1, both types of force influence the motion and the problem is one of compressible MHD. The basis for treating a plasma as a fluid, as is done in MHD is the validity of certain approximations. We assume that the problem of interest is characterized by a length scale λ and a time scale τ with angular frequency ω = 2π/τ . The length scale of the problem is determined by the time scale, together with the characteristic velocities. The validity of MHD basically requires two approximations: (i) The time scale τ must be long compared with the gyroperiod of the heaviest significant ions. This may be written in the form ω ≡ ξ1 1 i

(2.27)

where ξ1 is a parameter that must be small if MHD is to hold. (ii) The low-velocity approximation of section 1.5 must also be valid. Thus, the sound speed and the Alfve´ n speeds must be very much less than the speed of light. If the particles are non-relativistic, their rms speed and, hence, the sound speed will satisfy this condition. The condition for the Alfv´en speed to be small compared with the speed of light may be written as i VA ≡ ≡ ξ2 1 c ωp,i

(2.28)

where ξ2 is a parameter that must be small if the low-velocity approximation of chapter 1 is to hold. It is, in fact, sufficient if ξ22 1. The condition that the Debye length must be small is automatically satisfied if these conditions hold. From (2.21), we see that, if β 1, then λD tends to zero > 1, then λ2D /λ2 ∼ ξ1 ξ2 which is very small. and, if β ∼ There are, thus, just two independent parameters, each of which must be small, for our approximations to hold. When this is so, ω i ωp,i .

(2.29)

2.4 The MHD equations for a fully ionized plasma From this point, we will be dealing only with mean values of particle properties. We shall, therefore, drop the angle brackets which denote the mean. Variables associated with the electron fluid are denoted by subscript ‘e’ and variables associated with a single ion fluid by a subscript ‘i’. If there is more than one ion species, we use subscripts 1, 2, 3, . . . to denote the different ion species. Copyright © 2005 IOP Publishing Ltd.

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The magnetohydrodynamic approximation

2.4.1 MHD variables We restrict ourselves to a neutral plasma consisting of a single positive ion species and electrons. The generalization to a multi-ion plasma is straightforward and adds mathematical complexity without introducing any important physical principles. The condition (2.29) is assumed to hold. This has the following consequences: •

The net charge density is η = N e where N = Ni − Ne . Using (1.24) and (1.26), we see that V2 ω

N η 0 E 0 ωB = ∼ ∼ ∼ A ∼ ξ1 ξ2 1. N Nq λNe Ne c 2 i

(2.30)

Thus, there is charge neutrality to a very high order of accuracy and the number densities of ions and electrons can be taken as equal: Ne = Ni = N. •

(2.31)

The relative velocity of ions and electrons is v = J /Ne. If this is expressed as a fraction of the Alfv´en speed, we can show that | v|2 VA2

∼

ω2 2i

= ξ1 1.

(2.32)

This allows us to define a set of MHD variables: •

The mass density is the total mass per unit volume: ρ = Ni m i + Ne m e = N(m i + m e ).

•

The fluid velocity is the centre-of-mass velocity of the two fluids: v=

•

m ivi + m eve . mi + me

•

(2.34)

The plasma pressure is the sum of the electron and ion pressures: p = pi + pe .

•

(2.33)

(2.35)

The current density is the sum of the current densities of the electron and ion fluids: J = Ni ev i − Ne ev e = Ne{v i − v e }. (2.36) Note that this depends on the first-order difference between the ion and electron velocities. The electromagnetic fields E and B complete the set.

Copyright © 2005 IOP Publishing Ltd.

The MHD equations for a fully ionized plasma

29

2.4.2 Continuity equation Multiply the continuity equation (2.4) for each species by the particle mass for the species and add, getting ∂ N(m i + m e ) + ∇ · N(m i v i + m e v e ) = 0 ∂t

(2.37)

which, using (2.33) and (2.34), may be written as ∂ρ + ∇ · ρv = 0. ∂t

(2.38)

2.4.3 Momentum equation The momentum equations for ions and electrons, from (2.11), are dv i = eN{E + v i × B} − ∇ pi + f ei dt dv e = − eN{E + v e × B} − ∇ pe + f ie . me N dt mi N

(2.39) (2.40)

Newton’s third law ensures that the force per unit volume exerted on the ion fluid by the electron fluid is equal and opposite to the force per unit volume exerted on the electron fluid by the ion fluid: f ei = − f ie .

(2.41)

Thus, if we add the momentum equations for each species and make use of (2.33), (2.34), and (2.36), we get ρ

dv = J × B − ∇ p. dt

(2.42)

2.4.4 Adiabatic law The adiabatic law (1.18) for two species can be written as d pi γ pi = dN N d pe γ pe = . dN N

(2.43) (2.44)

If γ is the same for each species, these may be added and divided by the sum of the masses m = m i + m e to get dp γp = . dρ ρ Copyright © 2005 IOP Publishing Ltd.

(2.45)

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The magnetohydrodynamic approximation

2.4.5 Generalized Ohm’s law The momentum equation (2.11) for the ions may be multiplied by the electron mass and that for the electrons by the ion mass:   ∂v i memi N + v i · ∇v i = m e eN{ E + v i × B} − m e ∇ pi + m e f ei ∂t (2.46)   ∂v e memi N + v e · ∇v e = − m i eN{ E + v e × B} − m i ∇ pe − m i f ei ∂t (2.47) where, from (2.10),

Nm e m i ν (v e − v i ). (2.48) me + mi If the second of these two momentum equations is subtracted from the first, we get     mime N ∂ J J J + · ∇v + v · ∇ e ∂t N N N ρe J = ρ e E + ρ ev × B + (m e − m i ) J × B − m e ∇ pi + m i ∇ pe − σ (2.49) f ei =

where σ =

Ne2 (m i + m e ) memiν

(2.50)

is the conductivity. The mass of an electron is very much less than the mass of any positive ion so the accuracy of this expression is little affected by taking the limit m e /m i → 0. The result is 1 J { J × B − ∇ pe } (2.51) E+v× B = + σ Ne which is a generalized form of Ohm’s law. The time scales for ion–electron collisions are very long for magnetospheric plasmas. In chapter 5, we show that they may be days in length. Thus, in most circumstances in the magnetosphere, the term J /σ can be ignored Finally, the relationship (2.32) means that the term J × B/Ne is negligible compared with the terms on the left-hand side. MHD is generally of interest only when the forces arising from the pressure gradients are comparable with or less than the J × B forces (otherwise it becomes simple fluid dynamics). Thus, the term involving ∇ pe can also be neglected. The final result is that Ohm’s law, instead of relating current density and electric field, becomes a relationship between the plasma velocity and the electric and magnetic fields: E + v × B = 0. Copyright © 2005 IOP Publishing Ltd.

(2.52)

The MHD equations for a fully ionized plasma

31

2.4.5.1 Equations of magnetohydrodynamics To complete the set of equations, we need the low-velocity approximations, (1.24) and (1.30), to the Maxwell curl equations: ∂B ∂t ∇ × B = µ0 J .

∇×E= −

(2.53) (2.54)

These two equations and the continuity equation (2.38), the momentum equation (2.42), the adiabatic law (2.45), and Ohm’s law in the form (2.52) complete a set of four vector and two scalar equations for the variables v, J, E, B, p, and ρ. Note that the two Maxwell divergence equations are subsidiary to this. To the order of approximation used in deriving the MHD equations, the charge density η may be neglected; the electron and ion densities are assumed to be the same. If the set of MHD equations is solved, the electric field which arises may have a divergence. The Maxwell equation (1.26) then may be used to find the resulting small charge density but this is seldom of interest. The equation (1.27) may be derived from (2.53) by taking the divergence of both sides. This shows that ∂(∇ · B)/∂t = 0 so that, if ∇ · B was equal to zero in the past, it is always zero. 2.4.6 Reduced MHD equations It is convenient to eliminate some of the variables from this set of equations to get a reduced set of equations. Equation (2.54) may be used to eliminate J from the momentum equation (2.42). The result is  B B2 dv + = −∇ p + · ∇ B. (2.55) ρ dt 2µ0 µ0 The equation of continuity (2.38) may be rewritten in the form dρ = −ρ∇ · v. dt

(2.56)

The adiabatic law (2.45) may be combined with this to give dp = −γ p∇ · v. dt

(2.57)

The electric field may be eliminated from (2.53) by using (2.52). The result is

dB = B · ∇v − B∇ · v. (2.58) dt These four equations give the time development, in a frame following the plasma motion, of the variables v, ρ, p, and B. Copyright © 2005 IOP Publishing Ltd.

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The magnetohydrodynamic approximation

2.5 Gravitation Over most of the solar–terrestrial system, the gravitational force on a plasma is negligible and can be ignored in comparison with the electromagnetic forces and the pressure gradients. It is only of concern in the immediate neighbourhood of the Sun, where the gravitational field is enormous, and near the Earth where, although the gravitational field is weak, the equilibrium of low-energy plasma along the direction of the magnetic field is affected. Suppose the gravitational field is g. Then the gravitational effects may be taken into account by adding a force density ρ0 g to the right-hand side of the momentum equation (2.55). In all cases of interest, the only applications of the momentum equation are to the case of spherical symmetry. If the mass of the source of the field—normally the Sun—is M and the gravitational constant is G, then the momentum equation may be written as  B dv B2 GM + ρ = −∇ p + · ∇ B − rˆ ρ 2 (2.59) dt 2µ0 µ0 r where r is the distance from the centre of the gravitational source and rˆ is the outward unit vector.

2.6 Frozen-in magnetic fields The equations governing MHD flows are complicated and nonlinear. Visualization of these flows can be difficult. An important tool is the concept of ‘frozen-in’ magnetic fields. We show, in this section, that if, at some instant, a number of fluid elements are connected by a single field line, then, as the plasma flows, the velocity and magnetic fields change in such a way that, at a later time, this same set of fluid elements are still connected by a single field line. This can be visualized as if the field line is carried along with the plasma and frozen into it. There are various proofs of this theorem. All are tedious. We adopt one which is concise although somewhat opaque. We consider two neighbouring fluid elements separated by δr. If they both lie on the same field line, then δr × B = 0. We show that, if δr × B = 0 at some instant, then it remains zero at all future instants, so that if the two fluid elements are initially connected by a field line, they remain connected by a field line. At time t, let two fluid elements P and Q be located at r 0 and r 0 + δr 0 . The velocity of P is v and the velocity of Q is v + δr 0 · ∇v. After time δt, the element P will be at r = r 0 +vδt and the element Q at r +δr = r 0 +δr 0 +(v+δr 0 ·∇v)δt. Thus, (2.60) δr = δr 0 + (δr 0 · ∇)vδt so that δ(δr) ≡ δr − δr 0 = (δr 0 · ∇)vδt Copyright © 2005 IOP Publishing Ltd.

(2.61)

Losses within plasmas

33

and

d(δ r) = (δ r 0 · ∇)v. (2.62) dt This equation can be combined with the equation for the time evolution of B following the plasma flow (2.58) to give d dB d(δ r) (δ r × B) = δ r × −B× dt dt dt = δ r × ( B · ∇)v − B × (δ r · ∇)v − (δ r × B)∇ · v. (2.63)

In Cartesian tensor notation (appendix A.1), this may be written as d ∂ ∂ ∂vm vl −  j kl Bk δrm vl −  j kl δrk Bl . (2.64) ( j kl δrk Bl ) =  j kl δrk Bm dt ∂ xm ∂ xm ∂ xm This may be manipulated using the Kronecker delta as a substitution tensor and making use of equation (A.6) to give d ∂vk . ( j kl δrk Bl ) = −kl m δrl Bm dt ∂x j

(2.65)

This is of the form

d (δ r × B) = T · (δ r × B) (2.66) dt where T is the second-rank tensor which, in Cartesian notation, may be written as ∂vk /∂ x j . If the factor (δr × B) is zero at any time, then the rate of change of (δr × B) is zero so that, if initially P and Q are connected by a field line, then they remain connected by a field line as they move with the fluid.

2.7 Losses within plasmas As discussed in chapter 4, collisions between the particles of solar–terrestrial plasmas occur so infrequently that they can be ignored. As a result, loss processes are usually negligible. There are, however, situations when losses are appreciable in regions where interactions take place between the particles of the plasma and high-frequency waves. For our purposes here, it is sufficient to represent these processes empirically by introducing an effective conductivity and an effective viscosity. In the few cases where it is necessary to include such effects, the qualitative picture introduced here is good enough. 2.7.1 Resistive effects If the resistivity of the plasma in not infinite and is taken to be uniform, then the generalized Ohm’s law (2.51) may be written as E+v× B = Copyright © 2005 IOP Publishing Ltd.

∇×B J = . σ µ0 σ

(2.67)

34

The magnetohydrodynamic approximation

If this is used to replace E in Faraday’s law, we get an extra term, −∇ × ∇ × B/µ0 σ = ∇ 2 B/µ0 σ added to (2.58), which becomes dB = B · ∇v − B∇ · v + η∇ 2 B (2.68) dt where η = 1/µ0 σ is the magnetic diffusivity. This is the only modification of the reduced MHD equations required to represent finite conductivity adequately. 2.7.2 Viscous effects Viscous drag occurs when there is a sheared velocity field, because of the transfer of momentum by exchange of particles moving normal to the planes of shear. The transfer of momentum can, in principle, take place as a result of wave–particle interactions. This transfer of momentum leads to a tangential stress which is proportional to the gradient of the velocity. Consider a situation in which the velocity is in the x-direction and its magnitude changes in the z-direction. If the fluid is incompressible, the rate at which the x-component of momentum is transported in the z-direction can be assumed to be proportional to the velocity gradient and to the area across which it is transported. This is the stress. The force per unit volume is the divergence of the stress and is, therefore, proportional to the divergence of the velocity gradient. If the constant of proportionality is µ, and is uniform, then this force per unit volume is µ d2 vx /dz 2 . The coordinate-free form of this is f visc = µ∇ 2 v. (2.69) A term of this form can be added to the right-hand side of the momentum equation to provide a qualitative idea of viscous effects.

2.8 Partially ionized plasma 2.8.1 Current density in a partially ionized plasma So far we have dealt only with a fully ionized plasma. If the plasma is only partly ionized so that there is a large population of neutral particles, the plasma dynamics may be affected. So long as a charged particle makes many gyrations about the magnetic field without colliding with a neutral atom, the plasma dynamics are essentially decoupled from the dynamics of the neutral gas. When collisions with a neutral gas become important, we need to reconsider our approach. In the solar– terrestrial system, this occurs in the Earth’s upper atmosphere. There are several species of ions in this region. The mass density of the fluid, consisting of neutral particles, ions, and electrons, is now

Ni m i v i N0 m 0 v 0 + Ne m e v e +

(2.70) v= N0 m 0 + Ne m e + Ni m i Copyright © 2005 IOP Publishing Ltd.

Partially ionized plasma where the subscript 0 referes to the neutral gas. For charge neutrality,

Ne = Ni .

35

(2.71)

We can usually assume that the mass density of the neutral gas is far greater than the plasma density, as is the case in the lower ionosphere which is where the collision processes become significant. If this is so, the average velocity is essentially v 0 , the velocity of the neutral gas. The velocities of the ions and electrons are assumed to be small perturbations on this. In our treatment, we, therefore, consider motion of the charged particle species relative to the neutral plasma which we regard as being at rest. If it is moving, we can apply a straightforward transformation to our results. Their relative velocities determine the current density. In the upper atmosphere, the plasma pressure is very much less than the magnetic energy density so that we may treat the plasma as cold. We can write the momentum equation for a single particle species (2.11) in the form N0 Nm 0 mν dv = Nq{E + v × B} − v (2.72) Nm dt N0 m 0 + Nm where we have ignored the pressure gradient term and introduced a term of the form (2.10) representing collisions of the particle species with neutral particles. If the density of the neutrals is large compared with the densities of the charged particle species, the collision term becomes Nmνv. In the low-frequency approximation,    dv  m  |qv × B|.  dt  In deriving the momentum equation for a fully ionized plasma, the time derivative could not be omitted because, on summing over species, the terms qv × B have zero sum. In the derivation of the generalized Ohm’s law, the time derivative can be neglected. We, therefore, write the single-species momentum equation in the form q{E + v × B} − mνv = 0. (2.73) Now define

Bq Nq 2 σ = . (2.74) m mν Note that, since q is positive for positive ions and negative for electrons,  is in the same direction as B for positive ions and in the opposite direction to B for electrons. The magnitudes of the gyrofrequencies are Be/m where e is the magnitude of the charge on the electron. The contribution of the species to the current density is J = Nqv. Then the momentum equation may be written in the form   J × − J = 0. (2.75) σ E+ σν =

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The magnetohydrodynamic approximation

The scalar product of this with  shows that J ·  = σ E · .

(2.76)

Then, if we take the vector product of  with (2.75) and eliminate J × , we get  ( J · ) 2 = −σ ν E + ν J . (2.77) J+ σ E ×− σν σν Now (2.76) may be used to show that J=

σ ν2 2 + ν 2

 E+

 ( · E)  × E − . ν ν2

(2.78)

This is a conductivity relationship relating the current density contributed by a single species to the electric field. The total conductivity relationship is found by summing the current densities over all the species. In Cartesian tensor notation, this may be written in the form Jj = σjk E j

(2.79)

where the tensor conductivity σ j k is given by

σjk =

species



   j k  j kl l σ ν2 . + δ jk + ν 2 + ν 2 ν2

(2.80)

2.8.2 The conductivity tensor In most regimes, it is possible to make approximations to the expression for the conductivity tensor. These depend on the relative magnitudes of the collision frequencies and gyrofrequencies for the various species of charged particles. We shall limit ourselves to the case of a single positive ion species. We denote the positive ions and electrons by subscripts ‘i’ and ‘e’ respectively, and use a coordinate system in which the zero-order magnetic field is in the z-direction. In this case,   e     1 − 0 ν Ex Jx e 2    ν σ e e e    Ey  1 0  Jy  = 2 + ν 2  νe 2 +ν 2    e e e e Jz Ez 0 0 νe2   i   1 0 νi Ex 2  i σi νi  1 0  − νi   Ey  . + 2 (2.81)  i + νi2  2i +νi2 E z 0 0 2 νi

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Partially ionized plasma

37

The essential parameters for determining which processes dominate are the collision frequencies, νi and νe , with the neutral particles and the gyrofrequencies i = Be/m i and e = Be/m e . Since e i and we assume that the collision frequencies of ions and electrons with neutrals are of the same order of magnitude, there are essentially four regimes: • • • •

Ideal plasma: e νe and i νi . Isotropic conductivity: e νe and i νi . Anisotropic conductivity with infinite conductivity along the field: e νe and i νi . Anisotropic conductivity: e  νe and i νi .

2.8.2.1 Ideal plasma When e νe and i νi we let ν → 0 and the conductivity tensor apparently has the limiting value   0 0 0 σjk =  0 0 0  . (2.82) 0 0 ∞ The interpretation of the infinite element is easy. Since the current density in the direction of B cannot be infinite, the component of E in this direction must be zero. The remaining zero elements imply that the current density is zero. This might be a cause for concern since, in our derivation of the ideal fluid equations, the current density is not zero. This is because, in our derivation of the conductivity tensor, we have worked to a lower order of approximation. We have assumed that the current density arises from the zero-order velocities of the ions and electrons: in a conducting medium, these zero-order velocities are not equal. As the collision frequencies become negligible, the zero-order velocities of both ions and electrons approach the same value E × B/B 2 and the current density arising from this motion tends to zero. The current density in ideal MHD arises from a first-order correction to these velocities.

2.8.2.2 Isotropic conductivity If e νe and i νi , we let e , i → 0. The conductivity tensor becomes σ jk = σ δ jk. The medium, consisting of neutral gas and plasma, then obeys Ohm’s law: J = σ E. Copyright © 2005 IOP Publishing Ltd.

(2.83)

38

The magnetohydrodynamic approximation

2.8.2.3 Anisotropic conductivity with infinite conductivity along the field If e νe and i νi , the conductivity tensor may be written in the form     Ne σi 1 + νii νee 0    B  . νi νe Ne (2.84) σjk =  1 + 0 − σ   i B i e 0 0 σe + σi The mean free paths for collisions of electrons and ions with neutrals are of the same order of magnitude. For √ isothermal conditions, the ratio of the electron and ion thermal velocities is m i /m e and, hence, the ratio between the collision frequencies is of the same order. The ratio of electron and ion gyrofrequencies is m i /m e . In the limit m e /m i → 0, this becomes   Ne 0 σi B σ j k =  − Ne (2.85) σi 0 . B 0 0 ∞ The infinite element again implies that charge will always flow so that the component of the electric field parallel to the magnetic field is zero. The current density in this direction is not related to the electric field. The current density perpendicular to the field is given by a two-dimensional conductivity relationship:      Jx σH σP Ex = (2.86) Jy −σH σP Ey where σP = σi is the Pedersen conductivity and σH = Ne/B is the Hall conductivity. 2.8.2.4 Anisotropic conductivity In this case, e  νe and i νi . Thus, the ions produce an isotropic conductivity while there is no convenient approximation to the electron conductivity tensor. The result is   νe2 e σe νν2e+ 0 σi + σe ν 2 + 2 2 e e e e   . νe2 e (2.87) σjk =  σ + σ 0   −σe νν2e+ i e ν 2 +2 2 e e e e 0 0 σi + σe This may be written in the form σjk



σP =  −σH 0

σH σP 0

 0 0  σ0

(2.88)

where the expressions for the Pedersen and Hall conductivities are now more complicated and there is a direct conductivity relating current and electric field in the magnetic field direction. Copyright © 2005 IOP Publishing Ltd.

Conservation laws

39

2.9 Conservation laws Consider some quantity u which can be transported from one point to another. Let σ be its density. Let f be the flux of this quantity. This means that, if we consider an element of area δ A, then f · δ A represents the amount of u passing through δ A in unit time. Now consider a volume V of space, bounded by a surface S. In considering an element δ A of S, we choose the direction of the vector representing the area element  to be outwards. Then V σ dV represents the the amount of u within V and S f · d A represents the rate at which u is lost through the surface. If u is conserved, then the rate of increase of u within V will be minus the rate at which it is lost through the surface. Thus,
 d σ dV = − f · d A. (2.89) dt V S If Gauss’ theorem is applied to the surface integral, we get    ∂σ + ∇ · f dV = 0. ∂t V

(2.90)

The volume V is arbitrarily chosen. The only way in which this condition can hold for any V is if the integrand is zero. Thus, the conservation of u can be represented by the equation ∂σ = −∇ · f . (2.91) ∂t Any equation of this form represents the conservation of the quantity whose density is σ . This can easily be generalized to the conservation of a vector quantity. In this case, the flux is a second-rank tensor: ∂ f jk ∂σ j =− . ∂t ∂ xk

(2.92)

An example of such a conservation law is the continuity equation (2.38) which represents the conservation of mass. 2.9.1

MHD energy conservation

Take the scalar product of v with equation (2.55) and B/µ0 with (2.58), multiply (2.56) by 12 v 2 and (2.57) by 1/(γ − 1). If the resulting equations are added, after some manipulation, we get  B2 p ∂ 1 2 + ρv + ∂t 2 2µ0 γ − 1 Copyright © 2005 IOP Publishing Ltd.

The magnetohydrodynamic approximation   1 2 B2 p B B2 = −∇ · + v− B·v . v + pv + ρv + 2 2µ0 γ − 1 2µ0 µ0

40

(2.93) This has the form of a conservation equation for the energy density ∂U = −∇ ·  ∂t

(2.94)

where U is the energy density and  the energy flux. If it is integrated over a volume V , bounded by a surface S, we get
 d U dV = −  · d A (2.95) dt V S which can be interpreted as stating that the rate of increase of the energy in the volume V is minus the rate at which energy flows through the surface. The three terms of the energy density U represent the kinetic energy density, the magnetic energy density, and the internal energy density, (1.11), respectively. The terms of the flux have been grouped to give the kinetic energy, magnetic and thermal energy flux, the work done by the pressure force as the plasma moves, and the work done by the Maxwell stresses in the form of magnetic pressure and magnetic field tension. It can easily be verified that the magnetic field terms in the energy flux represent the Poynting vector




B2 B E×B ≡ v− B·v µ0 µ0 µ0

(2.96)

E = −v × B.

(2.97)

since Another grouping of the terms in the flux is B2 B v− B·v µ0 µ0 E×B = ( 12 ρv 2 + h)v + µ0

 = ( 12 ρv 2 + h)v +

where h=


+ p = γ γ−p 1

(2.98) (2.99)

(2.100)

is the enthalpy density. 2.9.2 Momentum conservation The momentum equation (2.55) is an expression of Newton’s second law for the set of particles occupying a fluid element. The time derivative follows the motion Copyright © 2005 IOP Publishing Ltd.

Summary

41

of the fluid element and the density is inversely proportional to the volume, being unaffected by a net loss or gain of particles as the fluid element progresses. It is convenient to consider the momentum balance as it applies to observations in the laboratory frame. In the momentum equation, replace d/dt by ∂/∂t + v · ∇ and add it to the continuity equation (2.38). In Cartesian tensor notation, after a little manipulation, we get  B j Bk − 12 B 2 δ j k ∂ ∂ (ρv j ) = − pδ j k + ρv j vk − . (2.101) ∂t ∂ xk µ0 In this formulation, the expression on the right-hand side is the force density which is given by the divergence of a stress tensor. The quantity pδ j k is the isotropic pressure. The quantity ρvi vk is the result of momentum transport through the motion of the plasma: if this term is integrated over a volume V , the result is the net rate at which momentum is transported into the volume through the surface as a result of the plasma motion. Its divergence is, therefore, the force per unit volume, exerted on an elementary volume as a consequence of net transfer of momentum into or out of the volume. It is called the Reynolds stress. The quantity {B j Bk − 12 δ j k B 2 }/µ0 is the magnetic part of the Maxwell stress tensor. The electric part of this tensor is negligible as a result of the low-velocity approximation. This too is a conservation law. The stress tensor represents the flux of momentum. The pressure term is a consequence of the net transfer of momentum resulting from thermal motion of the particles. The Reynolds stress represents the flux of momentum transported by the fluid motion. The Maxwell stress is an electromagnetic momentum flux.

2.10 Summary •

•

•

•

An MHD medium consists of several species of positive ion each with its own number density and electrons with a number density equal to the sum of the ion densities so that it is electrically neutral. In this chapter, a single positive ion species is considered. Generalization to several species is easy. Each species is treated as a fluid by applying the fluid approximation in which particle properties are averaged over ‘infinitesimal’ volmes which are small compared with macroscopic length scales but large compared with interparticle distances. Equations of motion are developed for each fluid species under the action of electromagnetic forces arising from the motions of all the species and including mechanical forces arising from the pressure gradients. It is assumed that adiabatic conditions apply. The equations are summed over species, under the assumption ω i

ωp,i , to give MHD equations for the medium representing – continuity, – momentum,

Copyright © 2005 IOP Publishing Ltd.

The magnetohydrodynamic approximation

42 – – –

• •

•

adiabatic behaviour, Maxwell’s equations in the low-frequency approximation and a generalized Ohm’s law which generally reduces to an equation relating showing that the plasma moves as if the individual particles have an E × B drift. The electric field and current density are eliminated, resulting in a reduced set of equations describing the time rate of change, following the fluid, of velocity, density, pressure, and magnetic field. The presence of a neutral gas background may affect these processes. When the collision frequency of any ion species with the neutral gas becomes comparable with the ion gyrofrequency, the medium becomes an anisotropic conductor with current density and electric field related by a conductivity tensor. Equations are derived representing local energy and momentum conservation.

Copyright © 2005 IOP Publishing Ltd.

Chapter 3 Single-particle motion in electromagnetic fields

3.1 Introduction In the preceding chapters, we have used a simple approach to deriving the equations of MHD in which each particle is assumed to move with the mean speed of its species. This approach is adequate for many applications to magnetospheric and astrophysical problems. In some cases, however, we cannot ignore the detailed behaviour of the particles. In addition, for a better understanding of the limitations of the MHD approach, a more critical study of its foundations is necessary. In the following three chapters, we examine the derivation of MHD in more detail. Many readers may wish to omit these chapters on a first reading. In this chapter, we study the motion of individual charged particles in slowly varying electric and magnetic fields. In general, the fields in the magnetosphere change on a length scale which is long compared with the gyroradius and a time scale long compared with the gyroperiod. In this case, the trajectory of the particle can be treated as a perturbation of the uniform field case, in which the particle still moves in a circle about a guiding-centre, which drifts as a result of the inhomogeneity. We derive appropriate equations of motion and find an expression for the guiding centre drift velocity. Many treatments of this problem exist [49, 144, 168]. The approach which we shall adopt is close to that of Ba˜nos [21]. Finally, we use these ideas to discuss the motion of charged particles in the geomagnetic field.

3.2 Guiding-centre motion—heuristic approach 3.2.1 Qualitative description of guiding-centre motion When E and B are not constant, the motion is more complicated than in the uniform fields discussed in section 1.6. We consider the motion of the particle Copyright © 2005 IOP Publishing Ltd.

43

44

Single-particle motion in electromagnetic fields

when the fields vary sufficiently slowly in space and time so that, over one cyclotron period, they are approximately constant. In this case, the motion can be regarded as a perturbation of the motion in a uniform medium. We assume that we may make an appropriate approximation in order to represent the trajectory of the particle as the circular cyclotron motion superimposed on the trajectory of a guiding centre. We shall assume that there are no forces except those due to a slowly varying electromagnetic field. The approach is heuristic. A more rigorous approach, described in section 3.4, gives the same result. The basis of the approximation is that, to zero-order, the motion of the particle is the same as in a uniform medium. The rate of change of the field quantities is so slow that, over one gyroperiod, the cyclotron motion is essentially unaffected. The cyclotron motion of the particle is equivalent to a loop of current with a radius small compared to the scale of the field variation. The velocity of the guiding centre is then just the zero-order drift (1.51) to this order. The zeroorder path of the guiding centre is the path found by following the position of an equivalent particle which everywhere has this velocity. The first-order correction to the perpendicular component of the drift velocity is found by identifying additional forces due to the variation of the fields. A transverse magnetic field gradient exerts a force on a magnetic dipole aligned with the field. Also the path of the guiding centre is curved due to the curvature of the fields and their rate of change with time. A reference frame moving with the guiding centre is non-inertial and we must take the fictitious force due to the acceleration of the frame into account. These forces give rise to perpendicular F × B drifts. The correction to the parallel component of velocity arises from the consideration of magnetic field shear. 3.2.2 Drift due to a magnetic field gradient In a uniform magnetic field, the integral of the magnetic force over one complete cyclotron orbit is zero. If the magnetic field has a transverse gradient, the magnetic force varies from point to point along the orbit of the particle and the force does not average to zero over the orbit. We estimate the net force by assuming that, to zero-order, the orbit of the particle is the same as that of a particle in a uniform field that has a value equal to the field at the centre of the orbit. We then calculate the additional force due to the field gradient and integrate it over this orbit to find the average. Consider figure 3.1. We take the field gradient in the − y -direction and the magnetic field in the z-direction. At the position of the particle, P, the magnitude of the magnetic field is B = B + y|∇B| = B + rg sin θ |∇B|

(3.1)

where it is understood that the field is evaluated at the guiding centre. The additional radial force is directed towards the guiding centre and has the Copyright © 2005 IOP Publishing Ltd.

Guiding-centre motion—heuristic approach

45

y

B B

rg

F P

 x

Figure 3.1. Force due to magnetic field gradient.

magnitude δ F = qv⊥rg sin θ |∇B| =

2 mv⊥ |∇ B| sin θ. B

(3.2)

The x-component integrates to zero. The y-component is

Fy =

2 mv⊥ |∇ B| sin2 θ B

(3.3)

and is directed in the y-direction. If we integrate this over one cycle, since the integral of sin2 θ over one cycle is 12 , we get F = yˆ

2 mv 2 mv⊥ |∇ B| = − ⊥ ∇ B. 2B 2B

(3.4)

The resulting gradient drift velocity, V G , is the F × B drift found from (1.53). It is in the x-direction. It may be written as VG =

2 mv⊥ µ ˆ × ∇ B. 2q B 2

(3.5)

In terms of the magnetic dipole moment M (1.48), this may be written in the form VG =

Copyright © 2005 IOP Publishing Ltd.

M × ∇B . qB

(3.6)

46

Single-particle motion in electromagnetic fields

3.2.3 Drifts due to the variation of the zero-order drift velocity If the electric and magnetic fields change in space or time so that the zeroorder drift velocity V 0 , given by (1.51), changes, then the guiding centre has an acceleration ∂V0 dV 0 ≡ + (V 0 · ∇)V 0 . (3.7) dt ∂t In the non-inertial reference frame moving with the guiding centre, this acceleration gives rise to a fictitious force −m dV 0 /dt. This force gives rise to an F × B drift velocity relative to the guiding centre: V=

dV 0 mµ ˆ d 1 µ ˆ ×m = × {µv ˆ  + V E }. qB dt qB dt

(3.8)

3.2.3.1 Magnetic curvature drift Consider the first term of (3.8): d(µv ˆ ) mv dµ ˆ mµ ˆ × = µ ˆ × . qB dt qB dt

(3.9)

From (B.10),

νˆ v dµ ˆ ∂µ ˆ = + v (µ ˆ · ∇)µ ˆ = ∂ µ/∂t ˆ + (3.10) dt ∂t Rc where Rc = κ −1 is the radius of curvature of the field line. The first term, ∂ µ/∂t, ˆ is associated with the variation of the magnetic field direction in time. As the guiding centre moves along the field, it encounters a changing field direction leading to a change in the direction of its trajectory. The second term represents the magnetic field curvature. The associated magnetic curvature drift can be written as mv2 ∂µ ˆ mv µ ˆ × + VC = µ ˆ × νˆ . (3.11) qB ∂t q B Rc 3.2.3.2 Drifts associated with changes in V E Just as in the case of magnetic curvature drift, there are curvature drifts associated with the rate of change of direction of the zero-order drift velocity V E . In addition, since V E is normal to the magnetic field, there are also drifts associated with the change in its magnitude as the guiding centre moves along its trajectory. By writing out the explicit form of V E in terms of E and B and performing the differentiation, it is possible to resolve these drifts into a number of separate drifts associated with the spatial and temporal rates of change of the electric and magnetic fields. This is done at the cost of a great deal of complexity and provides very little further understanding. Detailed expressions are given, for example, by Clemmow and Dougherty [49]. In the wave problems treated in this book, the drift velocity is itself a small perturbation on a constant background. In such cases, these terms are of second-order and may be ignored. Copyright © 2005 IOP Publishing Ltd.

General motion in a varying field

47

3.2.4 Parallel drift due to magnetic field shear There is one other drift to take into account. If there is magnetic shear normal to the magnetic field, then the perpendicular component of the velocity at the location of the particle has a component which is parallel to the direction of the magnetic field at the guiding centre. This is non-zero when averaged over the orbit. Consider figure 3.2. At the origin, the unit vector µ ˆ in the direction of B is in the z-direction. The gradient of µ ˆ is chosen to be in the y-direction so that, as y increases, the direction of the magnetic field is rotated in a plane perpendicular to the y-axis. When the particle is at the position shown, the unit vector is rotated through an angle φ. The change in µ ˆ is |δ µ| ˆ ≡ sin φ = rg sin θ

∂ µˆx . ∂y

(3.12)

The component of v ⊥ in the z-direction is then v⊥ sin θ sin φ. If this is averaged over an orbit, we get a correction to the mean parallel velocity of the guiding centre   2 mv⊥ ∂ µˆx 2 ∂ µˆx δv = − rg v⊥ sin θ = . (3.13) ∂y  2q B ∂y Now µ ˆ ·∇×µ ˆ =− so that δv  =

∂ µˆx ∂y

2 mv⊥ µ( ˆ µ ˆ · ∇ × µ). ˆ 2q B

(3.14)

(3.15)

3.2.5 The drift velocity of the guiding centre If all these drifts are combined, we get, for the drift velocity of the guiding centre,    2 2 v⊥ v⊥ d µ ˆ VD = V E + µ µ ˆ ·∇×µ ˆ + × (µv ˆ  + V E) + ∇ . ˆ v + 2  dt 2 (3.16) While the derivation is heuristic, we shall see in what follows that a more rigorous treatment leads to the same result.

3.3 General motion in a varying field 3.3.1 Equations of motion In a varying magnetic field, it is possible to define three principal directions by unit vectors µ ˆ parallel to the field, αˆ in the direction of R, the radius of curvature ˆ of the field line, and βˆ in the direction perpendicular to these two such that α, ˆ β, Copyright © 2005 IOP Publishing Ltd.

48

Single-particle motion in electromagnetic fields

^ 

z

^ ^  

^  y

q rg



v




x

Figure 3.2. Component of v ⊥ when there is magnetic field shear.

µ ˆ form an orthogonal right-handed set. This set changes direction in both space and time as the fields change. At present, we place no restriction on the rate of this change but, in the next section, we shall simplify the equations by assuming that it is small. The equation of motion of a particle is given by the Lorentz force: m x¨ = E + x˙ × B. q

(3.17)

We separate the particle velocity into three parts, the zero-order drift velocity perpendicular to the field, V E ≡ E × µ/B, ˆ the parallel velocity component, µv ˆ , and the remaining perpendicular velocity, τˆ v⊥ . The unit vector ρˆ is again defined to by (1.32). Thus, ˆ  + τˆ v⊥ . (3.18) x˙ = V E + µv We substitute (3.18) in (3.17) getting q {µE ˆ  + [ E ⊥ + V E × B] + Bv⊥ τˆ × µ} ˆ m q ˆ  − ρˆ Bv⊥ } = {µE m

x¨ =

(3.19)

and differentiate (3.18) ˙ˆ  + τˆ v˙⊥ + τ˙ˆ v⊥ . ˆ v˙ + µv x¨ = V˙ E + µ

(3.20)

Then, equating (3.19) and (3.20), we get another form of the equation of motion of the particle: q ˙ˆ  + τˆ v˙⊥ + τ˙ˆ v⊥ = µ ˆ v˙ + µv ˆ E  − ρv ˆ ⊥. V˙ E + µ m Copyright © 2005 IOP Publishing Ltd.

(3.21)

General motion in a varying field

49

This is still completely general and no approximations have been made. We now find separate equations of motion describing the temporal evolution of v , v⊥ , and τˆ or, equivalently, the rotation phase. Equation of motion for v Note that E ⊥ + V E × B = 0 and that, for any two orthogonal vectors,

so that

d a·b=0 dt

(3.22)

˙ a˙ · b = −a · b.

(3.23)

If we take the scalar product of µ ˆ with (3.21) and use the result µ ˆ · τ˙ˆ = −µ ˆ˙ · τˆ , we get ˙ˆ · τˆ v⊥ = q E  . µ ˆ · V˙ E + v˙ − µ (3.24) m Since (3.23) implies that ˙ˆ · V E = −µ µ ˆ · V˙ E (3.25) this may be written as v˙ =

q ˙ˆ · τˆ v⊥ + µ ˙ˆ · V E . E + µ m

(3.26)

This is the equation of motion for the parallel component of the velocity. It is exact but from it can be seen that, provided that q E  / m is not large, when the fields are slowly varying, v is also slowly varying. 3.3.1.1 Equation of motion for v⊥ If we take the scalar product of τˆ with (3.21), we get ˙ˆ  − τˆ · V˙ E . v˙⊥ = −τˆ · µv

(3.27)

This is the equation of motion for the magnitude of the perpendicular velocity component. It too is slowly varying when the fields are slowly varying. 3.3.1.2 Equation of motion for the particle phase Even when the fields are uniform, the unit vectors τˆ and ρˆ rotate rapidly. This rotation can be described in terms of the particle phase angle θ (t). This is just the angle between τˆ and αˆ as illustrated in figure 1.1. Clearly, τˆ = αˆ cos θ − βˆ sin θ ρˆ = αˆ sin θ + βˆ cos θ. Copyright © 2005 IOP Publishing Ltd.

(3.28) (3.29)

50

Single-particle motion in electromagnetic fields

Thus, by differentiating the first of these, we get ˙ ˙ αˆ sin θ + βˆ cos θ ) τ˙ˆ = α˙ˆ cos θ − βˆ sin θ − θ( ˙ = α˙ˆ cos θ − βˆ sin θ − ρˆ θ˙

(3.30) (3.31)

from which we get ˙ ˙ ρˆ · τ˙ˆ = ρˆ · α˙ˆ cos θ − ρˆ · βˆ sin θ − θ.

(3.32)

Now we take the scalar product of ρˆ with (3.21), use (3.32), and solve for θ˙ , getting ˙ˆ  ρˆ · µv ρˆ · V˙ E ˙ + + ρˆ · α˙ˆ cos θ − ρˆ · βˆ sin θ (3.33) θ˙ =  + v⊥ v⊥ this is the required equation for the rate of change of the particle phase. It too is exact but, when the fields are slowly varying, all the terms on the right-hand side except the first may be neglected. It then states that the rate of phase advance is equal to the particle gyrofrequency.

3.4 Theory of motion in a slowly varying field—the guiding-centre approximation 3.4.1 Slowly varying fields The equations of motion in the previous section are exact but suitable as the starting point for a rigorous treatment of guiding centre motion. In a frame of reference moving with the velocity µv ˆ  + V E , the unit vector τˆ is tangent to the path of the particle. We can define the gyroradius as being the instantaneous radius of curvature of the trajectory in this frame. Just as in (1.41), this is given by m mv⊥ µ ˆ × τˆ v⊥ = ρˆ . (3.34) rg = qB qB The fields are evaluated at the position of the particle. The position of the guiding centre r(t) can then formally be defined as the centre of curvature of the trajectory. It is not, however, particularly useful to do so unless, in the guiding-centre frame, the motion is still recognizably cyclotron motion. This requires that, in the guiding-centre frame, the motion is approximately circular so that the particle motion is cyclic with an angular frequency approximately equal to  evaluated at the guiding centre. This will not be the case if the fields vary appreciably over the orbit. A guiding-centre description is useful if the fields vary on a length scale l where rg /l 1 and a time scale t such that (t)−1 1. Such fields are said to be slowly varying. In such a case, the motion of the particle is a perturbation of the zero-order cyclotron motion in a uniform medium. It is worth noting that both rg and τg ≡ −1 are proportional to  ≡ m/q. It is, therefore, possible to express field quantities as expansions about the position Copyright © 2005 IOP Publishing Ltd.

Guiding-centre approximation

51

of the guiding centre in powers of  which is equivalent to expanding in powers of rg and −1 . To first-order in , the fields are given by B = B G + r g · [∇ B]g

(3.35)

with a similar expression for E, where B G is the field at the guiding centre. We rewrite the three equations of motion (3.33), (3.27), and (3.26) in the previous section with  shown explicitly. ˙ˆ  ρˆ · µv B ρˆ · V˙ E ˙ θ˙ = + + + ρˆ · α˙ˆ cos θ − ρˆ · βˆ sin θ  v⊥ v⊥ ˙ˆ  − τˆ · V˙ E v˙⊥ = − τˆ · µv v˙ = 

−1

˙ˆ · τˆ v⊥ + µ ˙ˆ · V E E + µ

(3.36) (3.37) (3.38)

We now carry out a procedure which is valid when the fields are slowly varying. In this case,  is small. We can formally identify the order of magnitude of the terms by regarding  as a small parameter. In order for the approximation to work, we require that v and v⊥ are O( 0 ). From (3.38), this is only possible if E  is of order . Otherwise, over several gyroperiods, v will be accelerated to a large value so that the particle moves a large distance during one gyroperiod and the assumption of slowly varying fields is not valid. This is not a restrictive condition since, in the presence of a large number of particles, when the MHD approximation is valid, E  = 0 to zero-order. 3.4.2 The particle phase In these equations, the fields are evaluated at the particle location. In the guidingcentre approximation, we replace the fields with expansions in powers of  about their values at the guiding centre. Consider the equation for the particle phase (3.36). If we retain only the lowest order terms, we get θ˙  with solution

BG = G  

θ (t) 

t

G (t  ) dt 

(3.39)

(3.40)

where now the fields are evaluated at the guiding centre. We use this as a first approximation. The procedure is then to find the average value of the other equations over one gyroperiod. Copyright © 2005 IOP Publishing Ltd.

52

Single-particle motion in electromagnetic fields

3.4.3 The averaging process Taking the average over one gyroperiod amounts to finding the average value over one cycle of θ . We define the average over one cycle of any function g(θ ) as  π 1 g(θ ) dθ (3.41) g ≡ 2π −π where dθ = G dt and the integrand is evaluated on the assumption that all slowly varying quantities which do not contain sin θ or cos θ are constant over one period. In taking averages, we use this procedure to note that cos θ  = sin θ  = sin θ cos θ  = 0

(3.42)

cos2 θ  = sin2 θ  = 12 .

(3.43)

and Using (3.18), we see that the rate of change with time of any field quantity, as seen by the particle, is D ∂ = + x˙ · ∇ Dt ∂t ∂ = + v µ ˆ · ∇ + V E · ∇ + v⊥ τˆ · ∇. ∂t

(3.44)

Then the derivative following the guiding centre, rather than the particle itself, is ∂ d = + v µ ˆ · ∇ + VE · ∇ dt ∂t

(3.45)

and

D d = + v⊥ τˆ · ∇. (3.46) Dt dt When the operator d/dt operates on a field quantity, the field quantity is evaluated at the guiding centre, not at the particle position. The average values of the rotating unit vectors follow immediately from (1.38) and (1.39): τˆ  = ρ ˆ  = 0. (3.47) We will also need the average value of the dyadic τˆ τˆ . We can write it in subscript notation using a local set of Cartesian axes which, at the location of the guiding centre, coincide with the directions τˆ , ρ, ˆ µ. ˆ Then,    1  −sin θ cos θ  0 0 0  cos2 θ  2 τˆi τˆ j  =  −sin θ cos θ  cos2 θ  0  =  0 12 0  . 0 0 0 0 0 0  (3.48) Copyright © 2005 IOP Publishing Ltd.

Guiding-centre approximation

53

It is easy to see that this can be written as τˆi τˆ j  = 12 (δi j − µˆ i µˆ j ). In the same way the average value of the dyadic ρˆ τˆ is    0 sin θ cos θ  −sin2 θ  0 ρˆi τˆ j  =  cos2 θ  −sin θ cos θ  0  =  12 0 0 0 0

(3.49)

− 12 0 0

Another average we require is   dv dv . + v⊥ τˆ  · ∇v = v˙  = dt  dt

 0  0  . 0  (3.50)

(3.51)

3.4.4 Equations of motion for v⊥ and v In order to find the equations of motion for v⊥ and v , we average (3.37) and (3.38). We perform the averaging process on each of the terms on the right-hand side of these equations: (i) The first term on the right-hand side of (3.37), when averaged, has a factor ˆ ˙ˆ  = τˆ  · dµ + v⊥ τˆ · (τˆ · ∇) µ ˆ = v⊥ τˆ · (τˆ · ∇) µ. ˆ (3.52) τˆ · µ dt In the local Cartesian system, we can write this in suffix notation and make use of (3.49) to get ˙ˆ  = v⊥ τˆi τˆ j  ∂ µˆ i τˆ · µ ∂x j   ∂ µˆ 1 ∂ µˆ 2 1 = 2 v⊥ + ∂ x1 ∂ x2 = 12 v⊥ ∇ · µ ˆ

(3.53)

where we have used the result that ∂ µˆ 3 /∂ x 3 = 0. This can be simplified further by noting that ∇·µ ˆ =∇·

1 1 B = {∇ · B − µ ˆ · ∇ B} = − µ ˆ · ∇ B. B B B

(3.54)

Thus,

˙ˆ  v = − 1 v⊥ v ∂ B . τˆ · µ 2 B ∂s (ii) The average of the second term on the right-hand side of (3.37) is ˙ˆ · V E  = V E · µ Copyright © 2005 IOP Publishing Ltd.

dµ ˆ . dt

(3.55)

(3.56)

54

Single-particle motion in electromagnetic fields

(iii) The average of the last term in (3.38) is   dV E ˙ τˆ · V E  = τˆ · + v⊥ τˆ · (τˆ · ∇) · V E dt  (E)

= v⊥ τˆi τˆ j  =

1 2 v⊥ {δi j

∂ Vi ∂x j

∂ − µˆ i µˆ j } ∂x j

  El µˆ m ilm B

(3.57)

where we have made use of (3.49). Now we carry out the differentiation and make use of the properties of ilm .  µm ∂ El ilm El µˆ m ∂ B 1 ˙ mil τˆ · V E  = 2 v⊥ − B ∂ xi ∂ xi B2  El ∂ µˆ m El ∂ µˆ m + ilm − ilm µˆ i µˆ j . (3.58) B ∂ xi B ∂x j The last term but one in this equation can be written as −

µ ˆ · (∇ × µ) ˆ E · (∇ × µ) ˆ = E B B

(3.59)

which can be seen by writing out the components. The last term is only nonzero for the components involving i = j = 3. Thus, on combining these results for the last two terms, using Faraday’s law, and making use of the assumption that E  is small, we get   v⊥ ∂ B + V E · ∇B . τˆ · V˙ E  = − 12 (3.60) B ∂t When these results are substituted in (3.37) and (3.38), we get   dv⊥ v⊥ dB 1 v⊥ ∂ B = 2 + (µv ˆ  + V E ) · ∇ B = 12 . dt B ∂t B dt and

v2 q dµ ˆ dv = E  − 12 ⊥ µ ˆ · ∇B + V E · . dt m B dt

(3.61)

(3.62)

3.4.5 The magnetic moment, an adiabatic invariant Equation (3.61) may be written as d dt Copyright © 2005 IOP Publishing Ltd.



1 2 2 mv⊥

B

 = 0.

(3.63)

Guiding-centre approximation

55

This shows that the quantity M=

1 2 2 mv⊥

(3.64) B is conserved to the order of the guiding-centre approximation. We recognize M as the magnitude of the magnetic dipole moment defined in (1.48). The motion of the particle about its guiding centre is equivalent to a current loop of magnitude I = q/2π encircling an area πrg2 . The magnetic moment has a magnitude 2 /B. Thus, the quantity M is the magnitude of the magnetic πrg2 I = 12 mv⊥ moment of the particle and is a constant of the motion in guiding-centre motion. It is called the first adiabatic invariant. As will be seen later, in certain field configurations, lower-frequency quasiperiodic motions are possible and these give rise to higher adiabatic invariants. It should be noted that the magnetic moment of the equivalent current loop is µM. ˆ It is its magnitude that is conserved during the motion, not its direction. 3.4.6 Drift velocity—the motion of the guiding centre We now turn our attention to the motion of the guiding centre. This consists of three parts, the zero-order drift velocity E × µ/b, ˆ the parallel component of velocity (averaged over one cycle), and a third part which consists of the firstorder corrections to the perpendicular velocity arising from the averaging of τˆ v⊥ over one cycle. From (3.34), the position of the guiding centre, r(t), is given by r(t) = x(t) − ρˆ

mv⊥ . qB

(3.65)

Then r˙ = x˙ −

D  v⊥  ρˆ Dt 

  ˙ ⊥ v v⊥ ρˆ ˙ = V E + µv ˆ  + τˆ v⊥ − ρˆ − v˙⊥ − .   

(3.66)

If we note that ˙ˆ × τˆ + µ ˙ˆ · ρ) ρ˙ˆ = µ ˆ × τ˙ˆ = −µ( ˆ µ ˆ +µ ˆ × τ˙ˆ

(3.67)

we get r˙ = V E + µv ˆ  + τˆ v⊥ + ρv ˆ ⊥

˙ 1  ˙ˆ · ρ)v ˙ˆ × τˆ v⊥ . (3.68) − (ρˆ v˙⊥ − µ( ˆ µ ˆ ⊥) − µ   

We now take the vector product of µ ˆ with (3.21) and rearrange the terms, getting ˙ˆ · ρ)v ˙ˆ  . ρˆ v˙⊥ − µ( ˆ µ ˆ ⊥ = τˆ v⊥ − µ ˆ × V˙ E − µ ˆ × µv Copyright © 2005 IOP Publishing Ltd.

(3.69)

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Single-particle motion in electromagnetic fields

If this is substituted in (3.68), we get an expression for the velocity of the guiding centre:   ˙  1 µ ˆ × (µv ˆ˙  + V˙ E ) + ρv ˆ µ ˆ˙ · ρ)v ˆ ⊥ . (3.70) r˙ = V E + µv ˆ + ˆ ⊥ + µ(   This equation is exact. The quantity in the braces has a factor 1/ and is thus of first-order in . Thus, to zero-order in , we retrieve the zero-order drift V E + µv ˆ  where now V E and µ ˆ are evaluated at the guiding centre. We now wish to find the average of the first-order corrections to this equation over one cycle. Note that   D ˙ˆ  . (µv ˆ ) = µ ˆ × v µ (3.71) µ ˆ × Dt  Then the average of the first term in braces is   d 1 1 ˙ ˙ µ ˆ × (µv ˆ  + V E) = µ ˆ × (µv ˆ  + V E ). (3.72)   dt  The average of the second term in the braces involves the factor   d ˙  = ρˆ + v⊥ τˆ · ∇ ρˆ  dt  ∂ = v⊥ ρˆi τˆ j  ∂x j ∂ ∂ = − 12 v⊥ αˆ + 12 v⊥ βˆ ∂ x2 ∂ x1 = 12 v⊥ µ ˆ ×

(3.73)

where we have made use of (3.50). The average of the third term in braces involves the factor   dµ ˆ + v⊥ ρˆ · (τˆ · ∇) µ ˆ µ˙ˆ · ρ ˆ  = ρˆ · dt  ∂ µˆ i = v⊥ ρˆi τˆ j  ∂x j   ∂ µˆ 1 ∂ µˆ 2 1 = 2 v⊥ − ∂ x1 ∂ x2 = 12 v⊥ µ ˆ · ∇ × µ. ˆ

(3.74)

If we substitute these results in (3.70) and note that ˙r  = dr/dt = V , the guiding-centre drift velocity, we get    2 2 v⊥ v⊥ d µ ˆ VD = µ µ ˆ ·∇×µ ˆ + VE + × (µv ˆ  + V E) + ∇ . ˆ v + 2  dt 2 (3.75) This is exactly the same result as we obtained heuristically in (3.16). Copyright © 2005 IOP Publishing Ltd.

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57

3.4.7 The energy equation The rate at which work is done on a particle is F · v, where F is the total force on the particle. If there is no external force, then the scalar product of (1.31) with v gives m v˙ · v = q E · v. (3.76) In order to find an energy equation which is valid in the guiding-centre approximation, we first substitute v = µv ˆ  + τˆ v⊥ + V E

(3.77)

in this getting m{[v v˙ + v⊥ v˙⊥ + V E · V˙ E ] + V E · [τˆ v˙⊥ + τ˙ˆ v⊥ ] + v⊥ τˆ · V˙ E } = q E · (µv ˆ  + τˆ v⊥ + V E ).

(3.78)

We now average each term in turn: (i) The first bracketed term may be written in the form   D 1 2 1 2 2 1 ˙ [ mv + mv + mVE ] . (3.79) mv v˙ + v⊥ v˙⊥ + V E · V E  = Dt 2  2 ⊥ 2  The operand is the averaged kinetic energy W and thus, by averaging (3.46),   dW D 1 2 1 2 2 1 [ 2 mv + 2 mv⊥ + 2 mVE ] = . (3.80) Dt dt  (ii) The average value of the second bracketed term may be found by using (3.21) and (1.50) to get ˙ˆ  v ] mV E · τˆ v˙⊥ + τ˙ˆ v⊥  = mV E · [− V˙ E  − µ d ˆ  + V E] = − mV E · [µv dt   d E ˆ × (µv ˆ  + V E) . = −m · µ B dt (3.81) (iii) From (3.58), the third bracketed term is " # 2 2 ! v⊥ v⊥ ∂ B −qE · µ( ˆ µ ˆ · ∇ × µ) ˆ + µ ˆ × ∇ . mv⊥ τˆ · V˙ E  = −M ∂t 2 22 (3.82) Finally, when these results are substituted in (3.78) and (3.16) is used, we obtain dW ∂B = qE · V + M . (3.83) dt ∂t Copyright © 2005 IOP Publishing Ltd.

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Single-particle motion in electromagnetic fields

This is a very simple equation which is intuitively exactly what would be expected. In the special case where the fields do not vary in time and E is either zero or normal to the drift velocity, the energy, W , is conserved. There is another form of the energy equation which is useful. We note that the terms involving V E in (3.79) and (3.81) cancel. Also E · V E = 0. Thus, we may write ∂B dWD = q E · VD + M (3.84) dt ∂t where 2 + v2 ) = 12 mv 2 (3.85) WD = 12 m(v⊥ and VD = V − V E.

(3.86)

3.5 Motion in a dipole field—second and third adiabatic invariants and constants of the motion In this section, we deal with the motion of charged particles in a magnetic dipole field which is a first approximation to the geomagnetic field. The properties of such a field are described in appendix B. The particles are assumed to have gyroradii and gyroperiods much smaller than the length and time scales on which the fields vary. The guiding-centre approximation is, therefore, appropriate. This is true for all but the most energetic cosmic-ray particles. The only zero-order electric field which we shall allow is in the meridian plane and normal to B so that V E is in the azimuthal direction. 3.5.1 Natural periodicities There are two additional periodicities introduced into the motion because of the ˆ · ∇ B, symmetries of the field. First, as a result of the mirror force − 12 q M µ which appears in (3.62), a particle starting at the equator and drifting along a field line encounters a retarding force which eventually reverses its motion parallel to the field and accelerates it back to the equator. It continues into the opposite hemisphere and is again reflected. The result is a periodic bounce motion with a frequency ωB . While the bounce motion is taking place, the particle is also undergoing gradient-curvature drifts perpendicular to the meridian plane. Both the bounce period and the perpendicular drifts depend on the particle energy. The geometry of the geomagnetic field is such that the distance drifted perpendicular to the meridian plane during one bounce period is small compared to the length of a field line. The next periodicity arises from the perpendicular drift. During its bounce motion, a particle drifts along an L-shell where L is defined by (B.33). In one drift period, the particle makes a complete circuit of the earth. The associated angular velocity ωD is called the drift frequency. Copyright © 2005 IOP Publishing Ltd.

Motion in a dipole field

59

3.5.2 Second and third adiabatic invariants In the case of a medium varying slowly in comparison with the gyroradius and gyroperiod, we showed in section 3.4.5 that the magnetic moment of a particle was conserved along its drift path. We have seen that the magnetic moment is the first adiabatic invariant associated with the shortest periodicity in the motion. It is also possible to find a second adiabatic invariant associated with the bounce motion and a third adiabatic invariant associated with the drift motion round the earth. These quantities are approximately invariant in the sense that, if conditions change on time scales long compared with the associated period, the quantity is conserved. Detailed treatments [49, 168] show that the second adiabatic invariant is  J=

mv ds

(3.87)

where the integral is taken over one cycle along the unperturbed bounce path, and the third adiabatic invariant  is equvalent to stating that the magnetic flux enclosed by the guiding drift shell of the particle is constant. By making use of such adiabatic invariants, it is possible to treat fields which are locally approximately dipolar but change slowly as the particle progresses along its trajectory. They are useful in studying the slow convection of energetic particles in realistic models of the magnetosphere [168]. In this book, we will not be concerned with such phenomena. Many of the phenomena which we discuss take place on length and time scales long compared with the gyroradius and gyroperiod so that the first adiabatic invariant is conserved. For other problems, the MHD treatment of chapter 2 is appropriate except when studying waves and oscillations with periods and wavelengths comparable to the time scales and length scales of the bounce and drift motions. In such cases, there may be resonances with these motions and the second and third invariants are violated. For our purposes, the complexities of the second and third invariants are best avoided and simpler constants of the motion which apply in restricted cases are appropriate. 3.5.3 Energy and L-shell as constants of the motion When there is no parallel electric field and the first-order drift velocity is parallel to V E , then dW/dt = 0 and the energy W is constant. In addition, the drift velocities are all perpendicular to the meridian and, thus, L does not change. It is possible to use a set of generalized coordinates M, θ , W , λ, L, φ, where λ is the latitude and φ the longitude. They are sufficient to specify the position and velocity of the particle. Then, if a particle moves through the geomagnetic field, approximated by a time-stationary dipole field with electric field normal to B and in the meridian plane, the three quantities M, W , and L are constants of the motion. Copyright © 2005 IOP Publishing Ltd.

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Single-particle motion in electromagnetic fields

3.5.4 Bounce motion To consider bounce motion in detail, we note, from (3.84), that, if there is no 2 /B are constant. parallel component of E, then WD = 12 mv 2 and M = 12 mv⊥ This is true in any mirror geometry, even one which is not symmetrical about the equator. Let B(s) be the magnitude of the magnetic field at a distance s along the field line from the equator. The pitch angle α, thus, varies with s as sin2 αeq sin2 α(s) = . B(s) Beq

(3.88)

Therefore, the parallel component of the velocity varies as 1  B(s) 2 . v (s) = v cos α(s) = v 1 − Beq

(3.89)

The particle is reflected or mirrored where α = π/2 and v = 0. This occurs where the magnetic field at the mirror point is BM =

Beq sin2 αeq

.

(3.90)

The bounce period is  TB = 2

sM

 sM

ds 2 = v v



sM  sM

ds $ 1 − B(s)/Beq

(3.91)

and the bounce frequency is ωB = 2π/TB .

(3.92)

3.5.5 Azimuthal drifts In the geometry of this section,the particle drifts in the azimuthal direction φˆ with a drift velocity V (s). The particle’s angular velocity about the Earth is V /r cos λ, where r and λ are its coordinates in a spherical polar system. In a magnetic dipole field, this becomes V /RE L cos3 λ, where V and λ are functions of s and RE is the radius of the Earth. If we average this over one bounce orbit, we get the angular drift frequency  sM VD ds 1 . (3.93) ωD = s M 0 RE L cos3 λ 3.5.6 Cross-L drifts In the zero-order picture of this chapter, the particle drifts and bounces steadily round the earth with uniform drift and bounce frequencies. In this motion, the magnetic moment M, the energy W , and the drift shell L are constants of the Copyright © 2005 IOP Publishing Ltd.

Summary

61

motion. In later chapters, we shall treat cases in which perturbing fields due to a wave are superimposed on the zero-order fields. When these fields are slowly varying compared with the gyroperiod, M is still conserved but W and L may change. In such cases, the rate of change of W is given by (3.83) where the fields are the perturbation fields in the wave and dL = V · ∇L dt

(3.94)

where, since in the zero-order motion V is normal to ∇L, only the wave fields contribute to dL/dt.

3.6 Summary •

•

•

•

In slowly varying fields, the motion of particles can be approximated as a guiding-centre motion consisting of the superposition of a circular orbital motion on the motion of a guiding centre with perpendicular and parallel velocity components. When the fields vary slowly in space and time, the guiding-centre approximation still allows the separation of the motion into motion of the guiding centre and the motion in a circular orbit about the guiding centre. The guiding centre has a zero-order drift due to the perpendicular electric field and additional first-order drifts arising from gradients of the magnetic field and the curvature of the zero-order trajectory of the guiding centre. In guiding-centre motion, the magnetic moment of the orbital motion of the particle can be defined. It is conserved in the guiding-centre approximation and is called the first adiabatic invariant. In certain symmetrical field configurations, additional adiabatic invariants can be defined. In general, energy is not conserved in guiding-centre motion. An equation for the rate of change of energy has been derived.

Copyright © 2005 IOP Publishing Ltd.

Chapter 4 Kinetic theory of plasmas

4.1 Introduction In principle, we could understand the macroscopic behaviour of a plasma if we could follow the trajectories of all its constituent particles under the action of the internal and external forces acting on them. An understanding of the motion of a single particle so that appropriate averages may be taken is, thus, essential to the understanding of the collective behaviour of a plasma. It is clear that, because of the enormous numbers of particles involved, each moving in the electromagnetic field arising from all the others and from external sources, a statistical approach is necessary. This is the basis of the kinetic theory of plasmas. In this chapter, we provide a limited introduction to the kinetic theory of plasmas. Our treatment is limited in the sense that we shall ultimately only be interested in the case in which all time scales are long compared with the longest ion gyroperiod. Averaging over particles then leads to various forms of magnetohydrodynamics, including the form discussed in chapter 2. We, therefore, exclude a host of phenomena occurring on shorter time scales. Treatments of these are provided in numerous references [44, 49, 204]. We introduce the plasma distribution function expressing the average density of particles in a phase space in which the coordinates are the position and velocity of a particle. First, we interpret the meaning of the distribution function and show how various fluid quantities such as density and momentum can be found from the properties of the individual particles, by using the distribution function to find their average values over the velocity space. We then deduce the differential equations which govern the evolution of the distribution function in a variety of circumstances. These equations are the basis for describing the dynamics of a plasma on a statistical basis. 62

Copyright © 2005 IOP Publishing Ltd.

The distribution function

63

4.2 The distribution function We define configuration space as the three-dimensional space in which the coordinates of a particle (x , y , z ) represent its position and velocity space as that in which the coordinates (vx , v y , vz ) represent its velocity. To study the statistics of the particle motions, it is convenient to work in a six-dimensional phase space with coordinates (x , y , z , vx , v y , vz ). Then the position and velocity of a particle specifies a point in phase space. In a more general treatment, momentum space is used rather than velocity space. This takes care of particles with varying mass and allows easy generalization to other coordinate systems. In non-relativistic plasma physics, when dealing with a single species of particle, it is conventional to use the velocity. We work in terms of the density of particles in such a space. For point particles, this density is the sum of a number of δ -functions. We smooth out this discontinuous density by averaging over an element of volume d3 x d3 v , located at the position ( x, v), which, while infinitesimal on the macroscopic scale, is sufficiently large to contain many particles. We use the notation that x, d3 x represents a volume element of size d3 x located at position x , with a similar notation, v, d3 v , for a volume element in velocity space and t, dt for a time element. The average density in phase space is represented by the plasma distribution function, f ( x, v) which is defined such that f ( x, v) d3 x d3 v

(4.1) d3 x

d3 v

represents the number of particles in the volume element located at position (x, v). Clearly the number density of particles at position x is N(x) = N(x, y, z)  ∞ = f (x, v) d3 v −∞  ∞ ∞ ∞ = f (x, y, z, vx , v y , vz ) dvx dv y dvz . −∞ −∞ −∞

(4.2) (4.3) (4.4)

It is sometimes useful to use other coordinate systems. One useful system is a set of spherical coordinates in velocity space with axis aligned with the background magnetic field so that f is a function of the speed v, the pitch angle α , and the phase angle θ , as illustrated in figure 4.1. The element of volume in velocity space is, then, (4.5) d3 v ≡ v 2 sin α dv dα dθ

4.3 Mean values of the particle properties 4.3.1 Averages over the velocity Let g(x, v) be a property of the particles. It is a function of the phase space coordinates. Examples would be the speed, velocity, momentum, or energy of a Copyright © 2005 IOP Publishing Ltd.

64

Kinetic theory of plasmas

vz B v v||



vy

 v
vx Figure 4.1. Polar coordinates in velocity space.

particle. Then the total value of g for all the particles in an element of volume of velocity space, located at a point in configuration space, is f (x, v)g(x, v) d3 v and the average value of g at the point x is obtained by integrating this over velocity space and dividing by the number of particles: ∞  g(x, v) f (x, v) d3 v 1 ∞ g(x) = −∞ ∞ g(x, v) f (x, v) d3 v. (4.6) = 3v N f (x, v) d −∞ −∞ 4.3.2 Averages over the gyrophase The particles in a plasma are strongly ordered by the magnetic field. The position of the particle in velocity space may be expressed in cylindrical coordinates v , v⊥ , θ , where θ is the phase angle of the particle gyration about the magnetic field. When the gyroperiod is small compared to time scales and the gyroradius is small compared with length scales of variation in the plasma, any ordering in phase angle is lost over many gyroperiods and it is useful to average particle properties over the gyrophase θ . We define the gyrophase average of g at x as  2π g(x, v , v⊥ , θ ) f (x, v , v⊥ , θ ) dθ . (4.7) g(x, v , v⊥ )θ = 0  2π f (x, v , v⊥ , θ ) dθ 0 In section 3.4.3, we found the average over one gyroperiod g(θ ) for a single particle. It is also possible to find the average value of this quantity for all the Copyright © 2005 IOP Publishing Ltd.

Fluid and MHD variables

65

particles with gyrocentres in a volume element. This is not the same quantity. In the case described in this section, g(θ )θ is the average of the value of g for those particles located in ( x , d3 x ) with velocity in the range (v , dv ), (v⊥ , dv⊥ ) over the angle θ defining the direction of the instantaneous radius of curvature. The gyrocentres of the particles may lie outside the volume element, if gθ depends on the fields, they are evaluated at the location of the element d3 x dv dv⊥ of phase space; if g depends on the fields, they are evaluated at the guiding centre of the particle. Burger et al. [33] give a detailed discussion of the difference. 4.3.3 Directional average Another form of average which is useful is the average over solid angle  ≡ sin α dα dθ :  2π  π g(x, v) =

0

g(x, v, α, θ ) f (x, v, α, θ ) sin α dα dθ .  2π  π f (x, v, α, θ ) sin α dα dθ 0 0

0

(4.8)

4.4 Fluid and MHD variables We now define the fluid variables of chapter 2 more rigorously. We treat each species of particle as a separate fluid, with its own density, mean velocity, mean pressure, and other macroscopic variables. To do this, we average particle properties over velocity, thus losing all information about the individual particle motions. Each conducting fluid moves in the fields arising from its own motion, the motion of other interpenetrating fluids, and external sources. Such an approach can cope with phenomena at all frequencies, provided that, to zeroorder, the mean velocity is the same for all particle energies. It cannot cope with a situation in which a subset of the particles reacts very differently to the fields. For example, if a subset of the particles moves in such a way that it remains in phase with a wave, resonance may occur, and the fluid approach fails. Examples of the fluid approach are the cold plasma approach of magnetoionic theory [32, 233] or warm plasma theory [207, 233]. MHD is an approach in which the plasma is treated as a single fluid. An MHD medium may consist of several species of particles subject to the constraint that the net charge density is zero. In MHD, we average over all these fluids with additional constraints. It is a low-frequency approach in which time scales are long compared with the natural time scales of the plasma such as the plasma and gyroperiods of the particles. Thus, the electrons can move instantaneously in the direction of the magnetic field to cancel any parallel electric field. This implies that there is charge neutrality to zero-order: Ne =

α

Copyright © 2005 IOP Publishing Ltd.

Nα

(4.9)

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Kinetic theory of plasmas

where e denotes the electron species and the sum is evaluated over the positive species only. The nature of the approximation is discussed further in section 5.4.2. In this approach, to zero-order, the particles move perpendicularly to the field with the drift velocity E × B/ B 2 . Where it is necessary to distinguish between species, we shall denote the species by a subscript or superscript α . In general, this will include both positive ion and electron species. Where it is necessary to distinguish between positive ions and electrons, we shall use a superscript or subscript ‘e’. 4.4.1 Mass density The number density of a particle species α in configuration space is  ∞ Nα ( x, t) = f α ( x, v, t) d3 v. −∞

(4.10)

The mass density of the species is then Nα m α and the mass density of the fluid is

ρ= Nα m α . (4.11) α

4.4.2 Drift velocity and current density The species velocity u α of a particle species α is defined as the average velocity of the species:  1 ∞ v f α ( x, v) d3 v. (4.12) uα = vα = Nα −∞ The drift velocity of an element of the magnetofluid u is the centre-of-mass velocity of the fluid element. This implies that we must find the weighted mean over all the species with the particle density as the weighting factor:

1

α Nα m α uα u= = ρα uα . (4.13) ρ α α Nα m α The relative velocity of the species is defined as wα = uα − u. Then wα  =



Nα m α wα = 0.

(4.14) (4.15)

α

As we noted earlier, in MHD all the species velocities are approximately equal so that wα u . In chapter 5, we shall carry out a development of MHD in which we regard u as the zero-order approximation to the velocity of each species and wα as a first-order correction. For this reason, in the remainder of this section, we shall explicitly express our definitions in terms of u and wα . Copyright © 2005 IOP Publishing Ltd.

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67

It is often useful to express the velocity of the individual particles with respect to a frame of reference which moves with the drift velocity of the plasma species or the drift velocity of the fluid as a whole. Define the peculiar velocity cα of a particle as the velocity relative to the species velocity: cα = v − uα . Clearly 

∞ −∞

 3

cα f α d v =

∞ −∞

(v − uα ) fα d3 v = Nα uα − Nα uα = 0.

(4.16)

(4.17)

In the same way, we can define the peculiar velocity relative to the mean fluid motion, C, as (4.18) C = v − u = v − uα + w α . The mean value of this is Cα = w α

(4.19)

and not zero as in (4.17). However, if we take the mean over all the species, weighted by the particle mass, we get 1

1

1

Nα m α Cα = Nα m α vα − Nα m α uα = 0. ρm α ρm α ρm α

(4.20)

The mean current density J in the magnetofluid, however, is the vector sum of the current densities of the individual species. On the assumption that the positive ions are singly charged, this gives

J = −Ne |q|ue + Nα |q|uα (4.21) α

where the sum is taken over only the positive ion species. In this form, the order of magnitude of J is concealed. If we use (4.14), it becomes

J = −Ne |q|we + Nα |q|wα (4.22) α

because the zero-order terms in uα cancel when summed over ions and electrons. 4.4.3 Pressure tensor Elementary ideas about pressure must be generalized in a fluid which is moving and anisotropic. It is necessary to distinguish between internal stresses that are consequences of the thermal motion of the particles and those that are consequences of their ordered motion. Some authors define the partial pressure of a particle species in terms of a fluid element that moves with the velocity of the Copyright © 2005 IOP Publishing Ltd.

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Kinetic theory of plasmas

v dt

^ n dA

Figure 4.2. Volume element in configuration space.

species uα . We prefer to define the pressure in terms of the force on an element of fluid which moves with the mean fluid velocity, u = ρ −1 ρα vα . Consider the particles of a single species crossing an element of surface nˆ dS in the fluid as shown in figure 4.2. Then, at time t, all the particles with velocity v which will cross the surface element during the interval between t and t + dt are contained in the volume element d3 x = v · nˆ dS dt illustrated in the figure. The momentum contained in this volume element is m α v f α (v) d3 vv · nˆ dS dt. The net force on this surface element is the rate at which momentum crosses it:   (4.23) dF = m α vv · nˆ f α (v) d3 v dS or, in suffix notation,   dFi = m α vi v j fα (v) d3 v nˆ j dS = Nα m α vi v j α nˆ j dS = Nα m α (u i + Ci )(u j + C j )α nˆ j dS

= Nα m α u i u j nˆ j dS + Nα m α (wiα u j + wαj u i )nˆ j dS + Nα m α Ci C j α nˆ j dS. (4.24)

The first term represents the rate at which momentum Nα m α u associated with the drift motion of the plasma moving at velocity u crosses the element of surface. It thus is the force on the surface from this source. The second term, in the same way, represents the correction due to the motion of the species relative to the plasma. The third term represents the force associated with the peculiar velocity of the particles. This force is associated entirely with the random motion of the particles and it is this that we associate with the partial pressure of the species. Define the partial pressure tensor Pα or, in suffix notation, Piαj for the species as Piαj ≡ Nα m α Ci C j α = Nα m α (vi − u i )(v j − u j )α Copyright © 2005 IOP Publishing Ltd.

Fluid and MHD variables

69

= Nα m α {vi v j α − u i v j α − u j vi α + u i u j }

= ρα {vi v j α − u i u j − u i wαj − u j wiα )}.

(4.25)

Then the force on an element of surface nˆ j dS due to the thermal motion of the molecules is (4.26) dFi = Piαj nˆ j dS. The total pressure tensor is Pi j ≡

α

=

α

=



Nα m α Ciα C αj  Piαj ρα vi v j α − ρu i u j

(4.27)

α

where we have used (4.15) and (4.25) 4.4.4 Energy density and temperature The kinetic energy density of the particles in a fluid species is  1 2 3 Nα 12 m α v 2 α = 2 m α v f α (v) d v  3 1 = 2 m α (u + C) · (u + C) f α (v) d v = 12 ρα (u 2 + 2u · wα + Cα2 )

(4.28) (4.29) (4.30)

since Cα = wα . The first term is the kinetic energy density associated with the ordered motion of a volume element of the species; the second is the density of the work done by the ordered motion of the species relative to the mean plasma velocity; and the third is the energy density associated with the random motion of the particles relative to the species velocity. If we sum over particle species, the second term sums to zero and we get



2 1 Nα 12 m α v 2 α = 12 ρu 2 + (4.31) 2 ρα C α . α

α

In thermodynamics, temperature is defined in terms of systems in thermodynamic equilibrium. For such plasmas, the temperature T is defined such that there is 12 N K T of internal energy density per degree of freedom where K is the Boltzmann constant. In plasma physics, temperature is often measured in energy units so that the symbol T appears in a formula representing an energy measured in joules or electron volts. We shall avoid this confusing usage and retain the Boltzmann constant explicitly in formulae where appropriate. Copyright © 2005 IOP Publishing Ltd.

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The trace of the pressure tensor for a plasma species is related to the internal energy density: Piiα = Nα m α C 2 α = 2 α . (4.32)




For a plasma species in thermal equilibrium, having three degrees of freedom, Pii = 3P and the temperature is then related to the pressure through Nα K T = 13 Piiα .

(4.33)

If all the species in the plasma are in thermal equilibrium with each other and have three degrees of freedom, then, if we sum over species, NKT = P

(4.34)

which is the ideal gas law (1.2). The plasmas in the magnetosphere are often not in thermodynamic equilibrium and the specification of a temperature can be misleading. Different particle species can coexist, each with its own effective temperature. In such cases, we can assign a meaning to temperature of a particle species by relating it to its internal energy through an expression of the form (4.33). Sometimes there may not be equipartition of energy between the motions of particles perpendicular and parallel to the magnetic field leading to different ‘temperatures’ corresponding to these different particle motions. Discussion of such cases will be taken up later. 4.4.5 Energy flux The internal energy density at a point in configuration space can be changed in three ways: • • •

If there is a gradient in the particle energy density, then energy can be transferred along the gradient as a result of particle collisions (conduction). The bulk motion of the fluid can transport particle energy from other parts of the fluid (convection). Fields due to external sources can do work on the particles at the point (radiation).

In this section, we are concerned with the first two of these. For the species α, the flux of the energy is 12 m α v 2 v f α d3 v such that the rate at which energy crosses a surface element nˆ dS in the plasma is    2 3 1 v v f α d v · nˆ dS. 2 mα In Cartesian tensor notation, the mean value of this flux at a point in space is then  1 2 3 1 2 m α v v j f α d v = 2 Nα m α vi vi v j α Copyright © 2005 IOP Publishing Ltd.

Kinetic equations

71

 = 12 m α

(Ci + u i )(Ci + u i )(C j + u j ) f α d3 v

= 12 ρα {Ci Ci C j α + 2u i Ci C j α + u j C 2 α + 2wiα u i u j + wαj u 2 + u 2 u j } = 12 Q αii j + u i Piαj +


α u j + 12 ρα u 2 u j + ρα u 2u j

+ 12 ρα u 2 wαj .

(4.35)

In this, we have used Cα = wα and we have introduced the third-rank tensor Q αi j k defined by (4.36) Q αi j k ≡ ρα Ci C j Ck α . The quantity Q αi j k is called the heat flux tensor for the species. We may define a heat flux tensor for the plasma as a whole, defined in terms of the peculiar velocity relative to the mean plasma velocity:

Qi j k = ρα Ci C j Ck α α

=



ρα vi v j vk α − {Pi j u k + P j k u i + Pki u j } − ρu i u j u k . (4.37)

α

We define the heat flux vector qi ≡ Q ii j .

(4.38)

Then, if (4.35) is summed over species, 

 2 3 2 1 1 2 m α v v f α d v = q + ( 2 ρu + )u + u · P.




α

(4.39)

The heat flux vector q represents the flux of energy as observed in a frame moving with the plasma velocity u. The terms in parentheses on the righthand side represent the energy density associated with plasma motion and the internal energy density. Thus, ( 12 ρu 2 + )u represents the flux of energy due to convection. The last term represents the rate per unit area at which work is done on the particles of the plasma.




4.5 Kinetic equations 4.5.1 Conservation of particles in phase space In fluid dynamics, for a fluid with density ρ(x) and velocity v(x), the conservation of mass is ensured by the equation ∂ρ + ∇ · (ρv) = 0. ∂t Copyright © 2005 IOP Publishing Ltd.

(4.40)

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Kinetic theory of plasmas

The integral form of this equation, obtained by integrating over volume and applying Gauss’ theorem, is
 d ρ dV = − ρv · dS (4.41) dt V S which implies that the rate of increase of the total mass in a closed volume V bounded by a closed surface S is equal to minus the outward flux of mass through the surface. The quantity ρv is the flux vector. If there are sources and sinks of the fluid, mass is no longer conserved. Then (4.40) is modified to become   ∂ρ ∂ρ + ∇ · (ρv) = . (4.42) ∂t ∂t source The three-dimensional laws (4.40) and (4.42) can easily be generalized to the six-dimensional phase space. The trajectory of a particle in phase space is determined by the forces acting on it. These forces can arbitrarily be classified into two types: (i) The forces due to macroscopic fields arising either from the averaged effect of other plasma particles or external sources. Examples are the Lorentz force and the gravitational force. These are mostly functions of position only and not of velocity. The only exception of interest is the magnetic force qv × B. (ii) Forces due to close encounters with other particles or with rapidly fluctuating wave fields. These forces are, of course, also electromagnetic. Nevertheless it is useful to treat them differently. Such interactions take place on a time scale short compared with the time scales of interest in the problem. They lead to abrupt changes in the velocity and, on the time scale of interest, this is effectively instantaneous so that a particle moves abruptly from one point in phase space to another. Such an interaction appears as non-conservation of particles locally in phase space. They are, of course, globally conserved. These forces are ascribed to collisions. The particles within a small volume element all have positions and velocities which differ infinitesimally. We have already noted that by ‘infinitesimal’ we mean that the volume is infinitesimal on the macroscopic scale but large enough to hold many particles. The volume element thus moves with the velocity of the particles because they are all subject to the same force and so move along parallel trajectories. The number of particles in the volume element is conserved as they move along their trajectories in phase space except when they undergo a collision. The distribution function, which is a smoothed-out density in phase space thus obeys a conservation law with a term representing the effects of collisions which is analogous to the source term in (4.42). The flux of f in configuration space is f (x, v)v, that in velocity space is f (x, v)a, and, thus, the six-dimensional flux Copyright © 2005 IOP Publishing Ltd.

Kinetic equations

73

of f in phase space is f (x, v)[v, a]. The six-dimensional analogue of (4.42) is, thus,   ∂f ∂f . (4.43) + ∇ · ( f v) + ∇v · ( f a) = ∂t ∂t coll In this equation, a = F/m

(4.44)

where F is the force on a particle. 4.5.2 Boltzmann and Vlasov equations In equation (4.43), since x and v are independent variables, the operator ∇ does not operate on x so that ∇ · ( f v) = v · ∇ f. For forces which are not velocity-dependent, ∇ v · ( f a) = a · ∇v f =

F · ∇v f m

The only velocity-dependent force with which we are concerned is the magnetic force qv × B. We write the third term of (4.43) in suffix form, note that ∂v j /∂vi = δ j i , and use the Kronecker delta as a substitution tensor to get q ∂ q ∇v · f v × B = ( f i j k v j Bk ) m m ∂vi ∂f q q . = δ j i i j k Bk f + i j k v j Bk m m ∂vi

(4.45)

Since δ j i i j k = iik = 0, (4.43) becomes ∂f + v · ∇ f + a · ∇v f = ∂t



∂f ∂t

 .

(4.46)

coll

This is known as the Boltzmann equation. If, as is frequently the case in the magnetosphere, collisions can be ignored, we get the Vlasov equation: ∂f + v · ∇ f + a · ∇ v f = 0. ∂t These can be stated in the form df = dt or

Copyright © 2005 IOP Publishing Ltd.



∂f ∂t

(4.47)



df =0 dt

(4.48) coll

(4.49)

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Kinetic theory of plasmas

where the operator ∂ d ≡ + v · ∇ + a · ∇v dt ∂t implies differentiation following the trajectory of the particles in phase space. We have established these equations in Cartesian coordinates. In their final form, they are coordinate free and we can use (4.48) and (4.49) in any appropriate coordinate system.

4.6 Approximations to the kinetic equation 4.6.1 Low-frequency average of Vlasov equation For phenomena which occur on time scales long compared with the gyroperiod and length scales large compared with the gyroradius, we can use a technique similar to that used for the drift of single particles, expanding the Vlasov equation (4.47) in powers of  ≡ m/e and then averaging over the gyrophase to get the lowest-order equation. We regard the distribution function as a function of x, v , v⊥ , and θ . The Vlasov equation can then be written as ∂f ∂f ∂f ∂f + v˙⊥ + θ˙ + v · ∇ f + v˙ =0 ∂t ∂v ∂v⊥ ∂θ

(4.50)

where the dot denotes differentiation following the particle as defined in (3.44). We use the three exact equations, (3.26), (3.27), (3.33), and write the Vlasov equation in the form ∂f + {V E + µv ˆ  + τˆ v⊥ } · ∇ f ∂t & %q ˙ˆ · τˆ v⊥ + µ ˙ˆ · V E ∂ f E + µ + m ∂v ∂ f ∂f ˙ˆ · τˆ v + τˆ · V˙ E } = 0. − {µ + { + O( 0 )} ∂v⊥ ∂θ

(4.51)

All the terms in this equation are O( 0 ) except the last which is O( −1 ) where  = m/e is the parameter which is a measure of the slow variation of the medium as described in section 3.4.3. We write f = f0 + f1 + · · · .

(4.52)

Then the lowest-order term in (4.51) shows that ∂ f0 =0 ∂θ Copyright © 2005 IOP Publishing Ltd.

(4.53)

Approximations to the kinetic equation

75

so that f 0 , the lowest-order approximation to f , is independent of particle gyrophase. In the next order of approximation, the Vlasov equation is ∂ f0 + {V E + µv ˆ  + τˆ v⊥ } · ∇ f 0 ∂t & %q ˙ˆ · τˆ v⊥ + µ ˙ˆ · V E ∂ f 0 E + µ + m ∂v ∂ f ∂ f 0 1 ˙ˆ · τˆ v + τˆ V˙ E } = 0. − {µ + ∂v⊥ ∂θ

(4.54)

If we now average this over θ , the last term in f 1 is zero because f1 must be periodic in θ . The average velocity is the guiding-centre drift velocity V D found from (3.75). In the other terms, the process of taking the average is formally equivalent to that in section 3.4.3 although, as we describe in section 4.3.2, the interpretation of the average is different. We use the expressions (3.61) and (3.62) for the averages getting # " 2 v⊥ ∂f q dµ ˆ ∂f + VD · ∇ f + E − µ ˆ · ∇B + V E · ∂t m 2B dt ∂v   ∂f v⊥ ∂ B + (V E + µv + ˆ ) · ∇ B =0 (4.55) 2B ∂t ∂v⊥ where, now that the averaging process is complete, we have dropped the subscript ‘0’ identifying the zero-order approximation. 4.6.2 Drift kinetic equations It is sometimes convenient to work in terms of W⊥ = rather than v and v⊥ . Then ∂f ∂f = mv⊥ ∂v⊥ ∂ W⊥

1 2 2 mv⊥

∂f ∂f = mv ∂v ∂ W

and W =

1 2 2 mv

(4.56)

so that we may write ∂f ∂f ∂f + V D · ∇ f + W˙ ⊥ + W˙  =0 ∂t ∂ W⊥ ∂ W where



 ∂B + (V E + µv ˆ ) · ∇ B ∂t   ∂B + v ∇B = q E · VG + M · ∂t

M W˙ ⊥ = B

W˙  = q E · V C + q E  v − v M · ∇ B Copyright © 2005 IOP Publishing Ltd.

(4.57)

(4.58) (4.59) (4.60)

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Kinetic theory of plasmas

2 /2B is the vector magnetic moment. The gradient drift V is where M = µmv ˆ ⊥ G given by (3.5) so that

q E · VG = M

2 mv⊥ E · (B × ∇B) (E × B) · ∇ B V E · ∇ B. =M = 2 2 2B B B

(4.61)

The curvature drift V C is given by (3.9) so that q E · VC =

mv E × B dµ dµ ˆ ˆ dµ ˆ = mv = mv V E · . E·B× · dt dt dt B2 B2

(4.62)

Equation (4.57) is an example of a drift kinetic equation, in which the particles are assumed to move with their guiding-centre drift velocities and their gyrational motion is averaged out.

4.7 Collisions and equivalent processes This book is not a treatise on the kinetic theory of gases and plasmas. A detailed treatment of the processes by which a plasma attains equilibrium (in so far as such processes are well understood) is not one of our objectives. Such a treatment would take more space than we can afford. In order, however, to understand the limitations of the basic equations which we shall be using, it is necessary to make use of some concepts which are familiar in the elementary theory of neutral gases but which become considerably more difficult when the dominant interaction between the particles is a long-range one such as the Coulomb force. This section is therefore, of necessity, qualitative. In it, we attempt to identify the important ideas and provide accessible sources through which the interested reader can pursue them further. The following ideas are the important ones. •

In a neutral gas, subject to an external conservative force which is described by a potential φ, thermal equilibrium is attained when the distribution for each molecular species is given by the Maxwell–Boltzmann distribution f (x, v, t) = N0

•



m 3/2 −φ/K T −mv 2 /2K T e e 2π K T

(4.63)

where the temperature T is the same for each species. The spatial distribution is N(x, t) = N0 e−φ/K T and the velocity distribution at position x is the Maxwellian  m 3/2 2 e−mv /2K T . (4.64) f (v) = N 2π K T In such a neutral gas, thermal equilibrium is attained through exchange of energy and momentum between particles through the mechanism of binary collisions. Particles do not interact except over the very short time during which a collision takes place. The rate at which the gas approaches equilibrium is related to the mean interval between collisions.

Copyright © 2005 IOP Publishing Ltd.

Collisions and equivalent processes •

•

•

77

In a dilute plasma, the particles can exchange energy and momentum through the long-range Coulomb force. As a result, at any instant, a particle is subject to the simultaneous influence of many other particles. The problem cannot be treated by considering binary collisions. Nevertheless, it is possible, by considering the statistics of the interaction between the particles and the electric field arising from thermal fluctuations in the density of the particles, to deduce an appropriate collision term. The ultimate equilibrium state is the Maxwell–Boltzmann distribution. In magnetospheric plasmas, the time in which it is achieved may be very long on the scale of hours, days or longer. Thus, plasmas which are not in a state of thermal equilibrium exist for extended times. On time scales short compared with the collision time, the distribution function is governed by the Vlasov equation (4.47). As the particles move, there are certain constants of the motion, such as the energy, which are functions of x and v. If the distribution function depends on x and v only through these constants of the motion, then ∂ f /∂t = 0 and the plasma is in a state of quasi-equilibrium. Such quasi-equilibrium states may or may not be stable to small perturbations. Such a small perturbation may lead to the growth of electromagnetic waves. The exchange of energy and momentum between such waves and the particles modifies the distribution until it reaches a state of marginal stability when the waves no longer grow. The time taken to relax to a state of marginal stability depends on the time scale for wave growth, typically seconds. There are a variety of such instabilities depending on the initial plasma distribution. There may exist, however, states for which no instabilities are available so that relaxation towards thermal equilibrium can only occur through Coulomb collisions on a very long time scale. Such states are, for many practical purposes, stable.

We now discuss some of these processes in a little more depth and identify some sources where they may be pursued further.

4.7.1 The nature of the collision term in the Boltzmann equation In order to understand the nature of the collision term, it is necessary to understand in detail the nature of the collision process. In a dilute gas, only binary collisions, in which two particles interact over a short time, are inportant. In a plasma, our view of collisions is somewhat generalized. Because of the long-range nature of the Coulomb force, we have to take account of many simultaneous interactions. In addition, in a plasma, as a result of instabilities, wave–particle effects may produce effects similar to collisions. The collision term can then be an appropriate way of sweeping under the rug effects that arise from those particle interactions which take place on time scales much shorter than those under consideration. Copyright © 2005 IOP Publishing Ltd.

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Kinetic theory of plasmas

4.7.1.1 Binary collisions Many books treat the nature of binary collisions. The classic work on non-uniform gases is by Chapman and Cowling [39]. The work by Jeans [103] gives a more elementary account. These authors show that, in a uniform medium, for binary collisions between particles of two species a and b ,    ∂ fa = − [ f a (v) f b (w) − fa (v  ) f b (w )]|(v − w)|ξ dξ dα d3 w. (4.65) ∂ t coll This gives the rate of change of the velocity distribution of species a as a result of all possible collisions that can occur between a particle of species a , having velocity v , and particles of species b . The integral is taken over all possible velocities w of the particles of species b , all angles α defining the relative orientation of the orbits, and all values of ξ , the closest distance of approach, which is determined by the force law between the particles. The respective velocities after the collision are v  and w . At equilibrium, f a (v) f b (w) = f a (v  ) f b (w )

(4.66)

and there is no change in the distribution function due to collisions. It will seldom be necessary to use this form of the collision term in the magnetosphere. In most regions, the mean collision time for binary collisions is much longer than time scales for the phenomena under consideration and the contribution of the binary collisions can be ignored. The exception is in the lower regions of the ionosphere where collisions of electrons and positive ions with the neutral atmosphere become important. In this case, for our purposes, it will not be necessary to use as complicated an expression for the collision term. Instead it can be noted that the further the distribution is from equilibrium, the larger the collision term which depends only on v is. We shall find it sufficient to use the simple Bhatnager–Gross–Krook or BGK [27] model as described, for example, by Clemmow and Dougherty [49, p311].   ∂f = −ν( f − f 0 ) (4.67) ∂t coll where f 0 is the equilibrium distribution and ν is an effective collision frequency. This model expression has the correct properties of being a function of v and of driving the distribution towards equilibrium. In general, the collision frequency ν depends on v. The nature of this dependence is determined by the model and, for many purposes, it can be regarded as constant. An estimate of ν for collisions with neutral particles can be made by treating them as elastic spheres with collision cross-sectional area πa 2 . The mean free path is then λmfp = 1/π a 2 N and the collision time 2π/v times this where v is the mean thermal speed. In figure 4.3, we show the dependence of the mean collision time τ = 1/ν in the ionosphere on height, for collisions of ions and electrons with neutral particles. Copyright © 2005 IOP Publishing Ltd.

Collisions and equivalent processes

79

Neutral Collisions 0.001

Electrons Protons

100

0.01 0.1

10 1

1

0.1

10

0.01

100

0.001

1000 200

100

0

300

400

Collision Frequency (Hz)

Collision Time (s)

1000

500

Height (km) Figure 4.3. Dependence of effective collision frequency of positive ions and electrons with neutral particles in the ionosphere.

4.7.1.2 Distant Coulomb collisions In a fully ionized plasma, the contribution to the collision term from binary collisions is not important. This is because of the long-range nature of the Coulomb force. At any instant, a particle is interacting simultaneously with many other particles and we cannot isolate each collision as an interaction between two isolated particles. These effects can be taken into account by considering what happens as the particle moves through a fluctuating electromagnetic field arising from the random thermal departures of the plasma from equilibrium. These fluctuations occur on the scale of the Debye length λD given by (2.17). A particle is screened from Coulomb interactions outside a sphere of radius λD . A detailed study of the contribution of Coulomb collisions to the collision term and the resulting effect on the transport coefficients of the plasma is given, for example, by Clemmow and Dougherty [49]. In a simpler qualitative picture, the physical process by which the fluctuating field modifies the distribution function can be regarded as consisting of two contributions: • •

a frictional drag proportional to the first derivatives of f in velocity space; and diffusion in velocity space proportional to the second derivatives of f in velocity space.

Thus,



∂f ∂t

Copyright © 2005 IOP Publishing Ltd.

 =− coll

∂ ∂ ∂ (µi f ) − (Di j f ). ∂vi ∂vi ∂v j

(4.68)

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Kinetic theory of plasmas

The time scale on which these processes take place is related to an effective collision time τeff which, for interactions between like particles, can be shown [49, ch 12] to be given by 40π02 m 1/2 (K T )3/2 τeff = (4.69) Nq 4 ln  where =

12π(0 K T )3/2 N 1/2 q 3

(4.70)

and m is the appropriate particle mass. Thus, electrons always respond more rapidly because of the dependence on m 1/2 . 4.7.1.3 Quasilinear diffusion processes On time scales short compared with those for binary or Coulomb collisions, the behaviour is governed by the Vlasov equation. In such cases, the plasma does not have time to achieve thermal equilibrium. Depending on its past history, the velocity distribution may differ very significantly from the Maxwellian form. The Vlasov equation (4.47) shows that any distribution function, which is a function only of the constants of the motion, does not change with time and, therefore, represents a quasi-equilibrium state. For example, if a uniform plasma in a uniform magnetic field has a velocity distribution function depending only on v and v⊥ , then, since v˙ = v˙⊥ = 0, ∂ f /∂t = 0 and the plasma is in quasiequilibrium. The nature of these quasi-equilibria are discussed in section 4.8.3. These states are termed quasi-equilibria for two reasons. First, there will always be relaxation towards thermal equilibrium over a sufficiently long time through collisions: the Vlasov equation is an approximation for phenomena changing on time scales which are much shorter than this. Second, the plasma may be unstable to the growth of electromagnetic waves: when the Vlasov equation is combined with Maxwell’s equations, consideration of the consequences of a small perturbation may show that certain characteristic electromagnetic waves may grow. Linear theory determines the initial growth rate but the process is expected rapidly to become nonlinear as the wave grows. In this process, the wave extracts energy from the particles, changing the velocity distribution. If the distribution is very far from any possible quasi-equilibrium state, it can be expected that the consequence would be strong turbulence, changing the velocity distribution in such a way that the free energy available for wave growth is reduced as the wave energy grows. Such nonlinear processes are not well understood although it can be guessed that the distribution will attain a state of quasi-equilibrium. Such a state corresponds to an appropriate strange attractor in phase space according to modern nonlinear dynamics [183] but we shall not treat this further. Often, in the magnetosphere, the plasma is not far from quasi-equilibrium. In such cases, the problem can be treated by quasilinear theory. Such processes Copyright © 2005 IOP Publishing Ltd.

Equilibrium states

81

in the magnetosphere are reviewed, for example, by Lyons and Williams [127] and Walker [233]. In such an approach, the waves are treated linearly. Some particles resonate with the wave, either because their parallel velocity is equal to the phase velocity or because the frequency is Doppler shifted to match the gyrofrequency of the particles or one of its harmonics. The consequence is that particles diffuse along characteristic curves in phase space. If the gradient of the distribution function is in a direction such that the diffusion increases the energy of the waves, then the process is spontaneous. A weakly turbulent spectrum of waves is set up and causes diffusion in velocity space which may change both energy and pitch angle of the particles until a state of quasi-equilibrium is reached. This diffusion is governed by a diffusion equation analogous to (4.68):   ∂ ∂ ∂f =− (Di j f ). (4.71) ∂t coll ∂vi ∂v j Such anomalous diffusion differs from collisional diffusion in that the final state is not, in general, a state of thermal equilibrium but a state of quasi-equilibrium. There may be several such diffusion terms corresponding to different wave– particle interactions.

4.8 Equilibrium states 4.8.1 Time scales to reach equilibrium and quasi-equilibrium Two extremes of plasma behaviour are the collision-dominated case, governed by the Boltzmann equation, and the collision-free case, governed by the Vlasov equation. In general, which equation is to be used depends on the relative sizes of the term a · ∇v f on the left-hand side of the Boltzmann equation and the collision term (∂ f /∂t)coll . Whether binary collisions or Coulomb collisions are more important, the magnitude of the collision term is characterized by a time scale. For binary collisions, a plasma near equilibrium approaches equilibrium in a time comparable with the mean time between collisions. For Coulomb collisions, there is also an effective collision time associated with the effective frictional and diffusion constants. The collision time for binary collisions can be estimated [49, section 1.2] by assuming that the radius of the collision cross section is the Landau distance λL at which the Coulomb potential energy of the two particles is equal to the thermal kinetic energy. The mean free path is then λmfp = 1/πλ2L N and the collision time 2π/v times this where v is the mean thermal speed. Thus, √ 2m(4π0 )2 K 3/2 T 3/2 tbinary = . (4.72) q4 N Estimates of the time scale for distant Coulomb collisions need laborious and intricate analysis. Clemmow and Dougherty [49, section 11.5] present a formula Copyright © 2005 IOP Publishing Ltd.

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for this time tCoulomb =

40π02 m 1/2 ( K T )3/2 Nq 4 ln 

(4.73)

12π(0 kt)3/2 . √ 3 nq

(4.74)

where =

If these collision times are long compared with the time scales in the problem, | f |/| a · ∇ v · f |, then the Vlasov equation is appropriate. Otherwise we must include a collision term and use the Boltzmann equation. In figure 4.4, we show these collision times for a wide variety of values of √ number density and temperature for electrons. The times for ions are a factor m i / m e longer. These plots indicate that binary collisions due to electrostatic forces can have never have any importance compared with distant Coulomb collisions. Further, except for the lowest energies and highest densities, such as those found only in the plasmasphere and ionosphere, the collision times for distant Coulomb collisions are enormously long. The Coulomb collision time for 1 keV electrons with a density of 106 m−3 is 109 s—about 30 yr! However, for electrons with temperatures of 1000 K (energies of 0.086 eV) and densities of 109 m−3 , corresponding to conditions in the plasmasphere, have Coulomb collision times of about 1 s. Collision times for proton–proton collisions are 42.8 times longer. If there is a significant population of neutral particles, binary collisions may become important. In the solar–terrestrial system, this occurs only in the Earth’s ionophere. The time scales for binary collisions with neutrals in this region are shown in figure 4.3. It can be seen that they rapidly become long, of the order of tens of minutes, as height increases. For comparison purposes, at electron densities typical of the F2 maximum (see section 12.5.1) at about 300 km, the characteristic times for distant Coulomb collisions are about 2 ms for electrons and about 0.3 s for O+ 2 ions. Thus, in much of the ionosphere, distant Coulomb collisions are much more important than collisions with neutrals. In the E-region, collisions with neutrals dominate. It is clear then that, in many magnetospheric situations, the Vlasov equation is appropriate. In such cases, the plasma is quite unlikely to be in thermal equilibrium. It may, however, be in a quasi-equilibrium state because of wave– particle interactions and quasilinear diffusion, which happen on time scales of seconds or less. Only in the thermal plasmas of the plasmasphere and magnetosphere, with temperatures of about 1000 K or less, can appropriate loss processes lead to thermal equilibrium on time scales of minutes or much less. Collisions with neutrals dominate in the lower ionosphere and are overwhelming in the E- and D-regions. Copyright © 2005 IOP Publishing Ltd.

Equilibrium states

Collision Time (s)

(a)

1016

Collision Time (s)

Close Electron Encounters

1012 108

1 year 1 day 1 hour

104 1

(b)

83

105

107 109 1011 Electron Density (m ) Coulomb Collisions

1012 108

1 year

104

1 day 1 hour

1

105

107 109 1011 Electron Density (m ) 103 104

Temperature (K) 105 106

107 108

Figure 4.4. Collision times for electrons: (a) binary collisions due to close Coulomb encounters; (b) effective collision times for distant Coulomb collisions.

4.8.2 The Maxwell–Boltzmann and Maxwellian distributions For a uniform plasma, the final state of thermal equilibrium is the Maxwellian  m 3/2 2 2 2 f (vx , v y , vz ) = N e−m(vx +v y +vz )/2K T . (4.75) 2π K T If there is an external conservative force, −∇, it is the Boltzmann distribution  m 3/2 2 2 2 e−/K T e−m(vx +v y +vz )/2K T . (4.76) f (x, v, t) = N0 2π K T The most thorough way of proving this is by means of the Boltzmann Htheorem. This requires a knowledge of the nature of the collision law. For a simple case, it has been proved by Jeans [103, appendix II]. This shows that not only is this the final equilibrium state but also any other state changes in a direction towards the final Maxwellian state. Such changes are described as Copyright © 2005 IOP Publishing Ltd.

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the thermalization of the plasma. The rate at which this occurs depends on the collision process. We have seen that, in most regions of the solar–terrestrial environment, if thermal fluctuations from distant Coulomb collisions are the only available collision process, thermalization occurs so slowly as to to be negligible. An alternative proof that is completely general and does not depend upon the nature of the collision law but upon the statistics of the distribution shows that the Maxwellian is overwhelmingly the most probable distribution [103, Appendix IV]. This does not show how a non-equilibrium distribution achieves this state nor does it give any information about the rate at which the thermalization occurs. 4.8.3 Jeans’s theorem and quasi-equilibrium states Plasmas in the magnetosphere are dynamic. In the regions of the magnetosphere beyond the plasmasphere they may be convected large distances in times of the order of minutes or hours—far shorter than the collision times for the typical densities and temperatures in these regions. As we saw in section 4.8.1, only cold, relatively dense plasmas with densities greater than about 109 m−3 and temperatures of the order of 1000 K are able to reach thermal equilibrium in times shorter than the times over which the conditions change. The plasma cannot attain thermal equilibrium by means of collisions. The governing equation is the Vlasov equation. What then represents a quasi-equilibrium state on time scales much shorter than the collision times? In order to answer this question, we investigate the nature of general solutions of the Vlasov equation when the electromagnetic fields are specified. This goes further than necessary to answer the question: the resulting solution may, in the most general case, be time-dependent. Nevertheless, the general solution limits the forms that can represent an equilibrium state. 4.8.3.1 Jeans’ theorem In the seven-dimensional space x i , vi , t, we can represent the distribution function f (x i , vi , t) as a family of surfaces of constant f . The analogue of the gradient function in this seven-dimensional space has components ∂f ∂f ∂f . , , ∂ x i ∂vi ∂t This gradient is orthogonal to the surfaces of constant f . Consider a point (x i , vi , t) where the distribution function has the value f . If we displace this point an amount (dx i , dvi , t), the change in f is d f . If the displacement is on a surface of constant f , then df = Copyright © 2005 IOP Publishing Ltd.

∂f ∂f ∂f dt = 0. dx i + dvi + ∂ xi ∂vi ∂t

(4.77)

Equilibrium states 85   ∂f ∂f , ∂t to be This is, of course, the condition for the gradient vector ∂∂xfi , ∂v i orthogonal to the vector (dx i , dvi , t) so that the displacement vector lies in the surface of constant f . If this is compared with the Vlasov equation (4.47) in the form ∂f ∂f ∂f =0 (4.78) vi + ai + ∂ xi ∂vi ∂t we see that dx 1 dx 2 dx 3 dv1 dv2 dv3 dt = = = = = = . v1 v2 v3 a1 a2 a3 1

(4.79)

These will be recognized as the equations of the characteristic curves in Lagrange’s method for solving the partial differential equation. We see that they are the equations of the orbits of the particles moving under the action of the fields. dvi dx i = vi = ai . (4.80) dt dt If these equations are solved simultaneously, there will be six arbitrary constants of integration and the general solution will be of the form x i = x i (α1 , . . . , α6 , t)

vi = vi (α1 , . . . , α6 , t).

(4.81)

Consider now a particle that has velocity vi at point x i and time t. The equations (4.81) can be solved for the six constants α that are the parameters of its orbit: αk = αk (x i , vi )

(k = 1, 2, . . . , 6; i = 1, 2, 3).

(4.82)

The general solution is then any arbitrary function of the constants of the orbit f (x i , vi , t) = G(α1 , . . . , α6 )

(4.83)

subject to its being positive and tending to zero as vi → ∞ in such a way that its integral over velocity space is finite. Since the αk are constant along the orbits, ∂αk ∂αk ∂αk dαk ≡ + vj + aj = 0. dt ∂t ∂x j ∂v j

(4.84)

We can, therefore, confirm that (4.83) is indeed a solution of the Vlasov equation, since

∂G ∂f ∂f ∂f + vi + ai = ∂t ∂ xi ∂vi ∂αk 6

k=1



∂αk ∂αk ∂αk + vj + aj ∂t ∂x j ∂v j

 = 0.

(4.85)

This general solution of the Vlasov equation, when the fields are prescribed, is known as Jeans’ theorem. Copyright © 2005 IOP Publishing Ltd.

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4.8.3.2 Constants of the motion—quasi-equilibrium states From Jeans’ theorem, associated with the motion of the particle are constants of the motion, functions of the particle’s coordinates in phase space, which are conserved as it moves. For example, for a particle subject only to an electrostatic field E = −∇, the mechanical energy 12 mv 2 + q is conserved, while, for a particle moving in a uniform magnetic field, v and v⊥ are separately conserved. In such cases, we know some, but not all, of the constants of the orbit. Nevertheless, a distribution function that is a function only of these constants of the motion is a solution of the Vlasov equation. It is not the most general solution but may be appropriate to a particular equilibrium case. If we take the second case as an example, the location of a particle in phase space can be written in terms of its position r, its velocity components v , v⊥ , and the gyrophase φ. The Vlasov equation, d f /dt = 0, can then be written in the form ∂f ∂f + v · ∇ f + φ˙ = 0. (4.86) ∂t ∂φ The terms involving the constants of the motion are zero because v˙ and v˙⊥ are zero. If, then, the distribution function is a function only of the constants of the motion, then ∂ f /∂t = 0 and the plasma is in a state of quasi-equilibrium. An arbitrary function of v and v⊥ is an equilibrium state in this sense. In the same way, if we consider a plasma in a dipole magnetic field, as described in section 3.5.3, the magnetic moment, M, the energy W , and the Lshell are approximate invariants. Any distribution which is a function only of W , M, and L is, therefore, a quasi-equilibrium distribution.

4.9 Summary • •

• • •

The position and velocity of a particle at an instant in time can be represented by a point in phase space which is a six-dimensional space with coordinates being the components of the position and velocity. Particles of a particular species with different positions and velocities can be described by a distribution function representing the density of particles with positions between x and x + dx and velocities between v and v + dv. The distribution function is a function of x and v. The mean value of a property of the particle which is a function of position and velocity can be found at position x by integrating the product of the property and the distribution function over velocity space. In the low-frequency approximation, it is useful to express the velocity in cylindrical coordinates and average quantities over the gyrophase angle. Fluid properties such as number density, velocity, pressure, and energy may be found by averaging the individual particle properties over velocity space.

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Summary •

• •

87

A kinetic equation is a differential equation representing the evolution of the distribution function in time. It expresses the fact that, as we follow the particles along their paths, the number of particles in an element of phase space is conserved. At frequencies that are small compared with the gyrofrequency, a lowfrequency approximation to the kinetic equation can be found by integrating over gyrophase. Jeans’ theorem shows that on time scales much shorter than the collision time, when the fields are given, we can find a general solution of the Vlasov equation for which the distribution function is an arbitrary function only of the six parameters defining the orbit of a particle in phase space. Functions of only some of these parameters represent solutions of the equation which are not general but can represent quasi-equilibria.

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Chapter 5 Fluid behaviour

5.1 Introduction In this chapter, we discuss how the equations of MHD may be derived from the kinetic equations describing each particle species. The kinetic equations describe the behaviour of the particles in detail; and, for many purposes, the average behaviour of the plasma over appropriate time and length scales is sufficient. MHD describes such averaged behaviour when the time scales of the problem are long compared with the gyroperiod and when, in addition, the Alfv´en speed is very much less than the speed of light. In neutral gas dynamics, the problem of deducing fluid dynamic equations from kinetic theory is well posed. The process is one of expansion in terms of a small parameter equal to the ratio between the collision time and the time scale of the problem. In the lowest order, the distribution function is that for thermal equilibrium. This is the Maxwellian distribution as a consequence of the Boltzmann H-theorem. Equations for the moments of the distribution are derived and successive approximations through the Chapman–Enskog expansion [39, p 383]. If collisions are sufficiently frequent, a similar process can be followed for a plasma. The plasmas in solar–terrestrial systems, however, are not dominated by collisions. As a consequence, they do not, in general, attain thermal equilibrium. Progress can be made by assuming that they can attain quasi-equilibria in the sense that the distribution function is a function of the constants of the motion. This requires assumptions about the nature of wave– particle interactions that allow the plasma to attain a quasi-equilibrium state on a relatively short time scale. Because these are complicated nonlinear processes, they are not well understood and so the discussion is, of necessity, somewhat speculative. Nevertheless it allows us to understand what the minimum conditions are for MHD to be valid. In this chapter, we review some properties of distribution functions. We make use of these properties to justify the setting up of a hierarchy of differential equations describing the time evolution of the moments of the distribution for 88

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89

single species and for plasmas that consist of several species. We then discuss the conditions under which the MHD approximation is valid and show what assumptions are necessary to truncate the hierarchy of moment equations. As an illustration, the process is applied to derive MHD equations for a plasma in thermal equilibrium. It is then applied to a plasma in which the individual species are in quasi-equilibrium. This, in general, gives a set of MHD equations with an anisotropic pressure tensor that is diagonal and symmetric about the direction of the magnetic field. In appropriate circumstances, the pressure tensor may be isotropic. In this case, the equations are the same as those for a plasma in thermal equilibrium.

5.2 Distribution functions and their moments The moment of order r of any statistical distribution f (x) is defined as  ∞ x r f (x) dx. mr = −∞

(5.1)

Analogous definitions apply to three-dimensional distributions. In this section, we summarize some properties of distribution functions and their moments. Rigorous proofs of the results quoted can be found in books on probability theory and statistics [70, pp 67–74]. • • • •

For the moment m r to exist, the integral (5.1) must be absolutely convergent. If it exists, all the moments of order smaller than r also exist. Central moments are moments for which x is expressed relative to an origin at m 1 /m 0 . (For example, if x represents the particle velocity in a plasma, then the origin for central moments is the rest frame of the plasma.) If the distribution function is symmetric, then the odd moments are zero. If the moments m r exist and the series ∞

mr r=1

r!

pr

(5.2)

is absolutely convergent for some p > 0, then the set of moments uniquely determines the distribution function. This last theorem is crucial to the expansion carried out in section 5.5.

5.3 Evolution of particle properties In this section, we show how the Boltzmann equation can be used to find equations for the evolution in space and time of the moments of the particle distribution of a particle species and, hence, by summing over species, the evolution of the MHD variables to which they are related. The intention is to show how Copyright © 2005 IOP Publishing Ltd.

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the MHD equations can be derived from the classical statistical mechanics of the particles. This is to be done without resort to the macroscopic ideas of thermodynamics as was done in chapter 2. We shall see that the differential equation describing the evolution of any moment requires the knowledge of the behaviour of a higher moment: the system of equations is not closed. The MHD approximation in various forms arises from approximations adopted to close the system of equations.

5.3.1 Moments of the particle distribution for a single fluid The first few moments of the particle distribution function are closely related to the averaged quantities mass density, momentum density, pressure tensor, and heat flux tensor for each species, defined in section 4.4. If we sum the moments over the species, the result is related to the MHD variables for a plasma. We relate the sums of the moments for the species α, with distribution functions f α (v), to these definitions for reference: •

From (4.10), the zero-order moments are related to the density:

 mα

−∞

α

•

ρα vα =

α

(5.3)

  mα α

∞ −∞

 v f α d3 v = ρu.

(5.4)

The second-order moment is related to the pressure tensor. From (4.27),

ρα vi v j α =

α

•

 f α d3 v = ρ.

The first-order moments give the mean velocity or, equivalently, the momentum. From (4.13),

•

∞

  mα α

∞ −∞

 vi v j f α d3 v = Pi j + ρu i u j .

(5.5)

The third-order moment is related to the heat flux tensor. From (4.37),



  ρα vi v j vk α = mα

α

α

∞ −∞

 vi v j vk f α d v 3

= Q i j k + {Pi j u k + P j k u i + Pki u j } + ρu i u j u k .

(5.6)

In the rest frame of the plasma, u i = 0. We see, therefore, that the MHD variables Pi j and Q i j k are the second- and third-order central moments described in section 5.2. Copyright © 2005 IOP Publishing Ltd.

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5.3.2 Rate of change of a particle property Let φ = φ(v)

(5.7)

be some function of the velocity coordinates vx , v y , vz . In practice, we shall set φ equal to 1, vi , vi v j , vi v j vk , so that the mean values of φ are the moments of the distribution. We wish to find an equation describing the variation of φα (x, t) in space and time, where φα is the average value of φ over velocity for the particles in the volume element x, d3 x in the time interval t, dt evaluated from (4.6). The average value of φ is a function of position and time because of the dependence of the distribution function on these quantities. From (4.6), the mean value of φ is given by  φα (x, t) =

∞

−∞

φ(v) f α (x, v, t) d3 v.

(5.8)

Consider the Boltzmann equation (4.46). This equation has a formal collision term on the right-hand side. This term is intended to represent interactions occurring on a time scale short compared with the time scales of the problem. In principle, it takes into account interactions between particles of this species with each other and with all the particles of any other species, whether these are due to binary collisions, distant Coulomb collisions, or wave–particle interactions. Multiply the equation by φ and integrate over velocity space:  ∞  ∞  ∞ ∂f 3 q d v+ φ φv · ∇ f d3 v + φ E · ∇ v f d3 v −∞ ∂t −∞ −∞ m  ∞    ∞ ∂f q φ (v × B) · ∇v f d3 v = φ d3 v. (5.9) + ∂t coll −∞ m −∞ Consider each of the five terms in this equation in turn: (i) We use the definition (4.6) to show that  ∞  ∞  ∞ ∂(φ f ) 3 ∂f 3 ∂ ∂(Nφ) d v= d v= . (5.10) φ φ f d3 v = ∂t ∂t ∂t ∂t −∞ −∞ −∞ (ii) The second term may be treated similarly:  ∞  ∞ φv · ∇ f d3 v = v · ∇(φ f ) d3 v −∞ −∞  ∞ =∇· vφ f d3 x −∞

= ∇ · (Nvφ).

(5.11)

(iii) In the third term, E is independent of v. We may, thus, write  ∞  ∞  ∞ 3 3 φ E · ∇v f d v = ∇ v · (Eφ f ) d v − f E · ∇ v φ d3 v. (5.12) −∞

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−∞

−∞

92

Fluid behaviour In Cartesian coordinates, the first term on the right-hand side is   ∂ 3 (E x φ f ) dvx dv y dvz ∇ v · (Eφ f ) d v = ∂vx   ∂ ∂ + (E y φ f ) dvx dv y dvz + (E z φ f ) dvx dv y dvz ∂v y ∂vz (5.13) where the range of all the integrals is −∞ to ∞. If we perform the vx integration in the first term, we get   ∂ (E x φ f ) dvx dv y dvz = [E x φ f ]∞ (5.14) −∞ dv y dvz = 0 ∂vx since f = 0 at vx = ±∞. Similarly, the other two terms are also zero. Thus,  ∞ q Nq E · ∇ v φ. φ E · ∇ v f d3 v = − (5.15) m m −∞

(iv) The fourth term can be treated exactly like the third with E replaced by v × B, except that, since v × B is a function of v, there is an extra term  − f φ∇ v .(v × B) d3 v. In Cartesian tensor notation ∇ v · (v × B) may be written as ∂v j ∂ (i j k v j Bk ) = i j k Bk = i j k δi j Bk = iik Bk = 0. ∂vi ∂vi

(5.16)

Thus, this term has the same form as (5.15) with E replaced by v × B:  ∞ q Nq (v × B) · ∇ v φ. φ (v × B) · ∇ v f d3 v = − (5.17) m −∞ m (v) The collision term on the right-hand side cannot be further simplified without a knowledge of the form of the collision integral. Equation (5.9) can, thus, be written as ∂(Nφ) Nq +∇·(Nvφ)− (E+v×B)·∇ v φ = ∂t m



∞ −∞

 φ

∂f ∂t

 d3 v. (5.18) coll

This governs the evolution in time of any quantity that has been averaged over the velocities of all the particles of the species located in an element of volume of configuration space. The electromagnetic field may be externally applied or may arise from interactions with other species occurring on the time scale of interest. The collision terms on the right-hand side represent the integrated effect of all Copyright © 2005 IOP Publishing Ltd.

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interactions occurring on a time scale short compared with the time scales of interest. It is an integral over terms representing collisions between the particles of the species and collisions with particles of other species. In MHD, we are generally interested in the behaviour of the plasma as a single fluid. The equation representing the behaviour of this fluid is the sum of the previous equation over all the particle species. Denote the species by a subscript or superscript α as appropriate. Multiply (5.18) by m α and the sum can be written as  

∂

∂

∂φ ρα φα + ρα vk φα − Nα qα (E k + klm vl Bm ) ∂t α ∂ xk α ∂vk α α    α ∞

∂f = mα φ d3 v. (5.19) ∂t −∞ coll α These equations may be used to find equations governing the various moments of the particle distribution as shown in section 5.4. However, when one deals with the sum over species, the term involving the electromagnetic fields has opposite signs for electrons and for positive ions. This allows for another possible way to perform the sum. Multiply (5.18) by qα and the sum can be written as 

Nα q 2  ∂

∂

∂φ α (E k + klm vl Bm ) Nα qα φα + Nα qα vk φα − ∂t α ∂ xk α mα ∂vk α α    α ∞

∂f = qα φ d3 v. (5.20) ∂t −∞ coll α

5.4 Moment equations 5.4.1 Moment equations for a single species Equations for the evolution of the first few moments of the particle distribution function are found from (5.18) by setting φ equal to 1, v, vi v j in turn. This leads to a set of moment equations applicable to the evolution of the mass density, momentum density, pressure tensor, and heat flux tensor. 5.4.1.1 Mass density Let φ = 1. Then (5.18) becomes ∂N + ∇ · (Nv) = Rcoll ∂t

(5.21)

where Rcoll represents the rate at which the particle density increases as a result of collisions. In the absence of ionization or recombination processes, particles are conserved and Rcoll = 0. The equation represents the rate of change of N, Copyright © 2005 IOP Publishing Ltd.

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the first moment of the distribution. This rate of change, however, is expressed in terms of a higher moment, (Nv). If we multiply by m, the mass of a particle, this becomes the equation for the evolution of the mass density ρ: ∂ρ + ∇ · (ρv) = 0. ∂t

(5.22)

This is, of course, the fluid continuity equation (2.4) for the species. 5.4.1.2 Momentum density Let φ = vi . Then, in suffix notation, if we multiply (5.18) by m, it becomes     ∂ ∂vi ∂p ∂(Nmvi ) + (Nmvi v j )− Nq (E j +  j kl vk Bl ) (5.23) = ∂t ∂x j ∂v j ∂t coll 

where

∂p ∂t



 =

coll



∞ −∞

mvi

∂f ∂t

 d3 v

(5.24)

coll

is the change in momentum density arising from collisions. If we use (4.25), this may be written in the form Nm

∂vi  ∂ ∂N + mvi  + (Pi j + Nmvi v j ) ∂t ∂t ∂x j     ∂ pi ∂vi = − Nq (E j +  j kl vk Bl ) ∂v j ∂t coll

(5.25)

or Nm

∂(Nv j ) ∂vi  ∂vi  ∂N + Nmv j  + mvi  + mvi  ∂t ∂x j ∂t ∂x j     ∂ pi ∂ ∂vi = + (Pi j ) − Nq (E j +  j kl vk Bl ) . (5.26) ∂x j ∂v j ∂t coll

The first two terms represent Nm times the convective derivative of the mean velocity. The sum of the third and fourth terms is zero because of the continuity equation (5.22). Also ∂vi /∂v j = δi j and vi  = u i . Thus,   ∂ ∂ pi du i + Nm (Pi j ) − Nq(E i + ikl u k Bl ) = . (5.27) dt ∂x j ∂t coll This will be recognized as a momentum equation for a species similar to (2.39) and (2.40). The gradient of the pressure is replaced by the divergence of the pressure tensor. The collision term could, in principle, involve a wide variety of interactions with other particles. As noted earlier, it is the sum of an integral over a collision term representing collisions with particles of the species itself Copyright © 2005 IOP Publishing Ltd.

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and terms representing collisions with other species. The integral over the first of these must be zero, since collisions between the particles of the species cannot change the total momentum of the species itself. The integrated collision term, therefore, represents the change in momentum of the species arising from the integrated effect of collisions with particles of other species. Once again, the evolution of the momentum density requires knowledge of a higher moment, the pressure tensor. 5.4.1.3 Pressure tensor Let φ = vi v j . If we multiply (5.18) by m and use (4.25) and (4.37) to eliminate vi v j  and vi v j vk , we get ∂ Qi j k ∂ Pi j ∂ ∂ + (m Nu i u j ) + + {Pi j u k + P j k mu i + Pki u j } ∂t ∂t ∂ xk ∂ xk   ∂(vi v j ) ∂ (m Nu i u j u k ) − Nq (E k + klm vl Bm ) + ∂ xk ∂vk    ∂f = mvi v j d3 v. (5.28) ∂t coll The derivatives are expanded and equations (5.22), (5.27), and the relation ∂vi /∂v j ≡ δi j are used to obtain ∂ Qi j k ∂u j dPi j ∂u k ∂u i q Bm + {ilm P j l +  j lm Pli } + Pi j + Pj k + Pki − dt ∂ xk ∂ xk ∂ xk ∂ xk m        ∂pj ∂f ∂ pi d3 v − u j − ui . (5.29) = mvi v j ∂t coll ∂t coll ∂t coll Again, this has given an equation for the evolution of pi j at the cost of introducing a higher-order unknown, the heat flux tensor Q i j k . It is clear that, if we are to derive a set of MHD equations entirely from the statistics of the particle distribution, some method must be found of terminating the sequence of moment equations. 5.4.2 Moment equations for a multi-ion plasma In most cases of interest, there is only one species of positive ion. The plasma consists of this species and the electrons. It is relatively easy, however, to derive moment equations for a multi-ion plasma and to treat a plasma with one species of ion as a special case. We can find moment equations for a multi-ion plasma by summing the single-species equations over all species. This is best achieved by evaluating the individual terms of (5.19) and (5.20) with φ = 1, vi , vi v j , . . . in turn. This produces two hierarchies of equations. The first of these hierarchies gives the time evolution of quantities such as mass density, momentum density, Copyright © 2005 IOP Publishing Ltd.

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stress and energy density, related to the Newtonian dynamics of the plasma. The second gives the time evolution of quantities like charge density, and current density, related to the electromagnetic properties of the plasma. Part of the process involves summing over the various species of ions and the electrons, with the constraint that, to a first approximation, the mobility of the charges, especially the electrons, ensures that the space charge density is zero. This constraint arises from the assumption that the displacement current term in Maxwell’s equations can be neglected because the length and time scales, λ and τ , in the problems of interest are such that typical speeds v = λ/τ are very much less then the speed of light. In such a case, from Faraday’s and Amp´ere’s laws, E = O(λB/τ ) = O(λ2 µ0 J/τ ) and the charge density is determined by η = 0 ∇ · E. This demonstrates that ηv ∼

v2 J. c2

(5.30)

This has the consequence that Ne =



Nα .

(5.31)

ions

The natural frequency associated with significant charge density is the electron 2 angular plasma frequency $ Ne /0 m e [31, 233]. The plasma relaxes to neutrality on a time scale τ = 2π 0 m e /Ne2 . In the solar–terrestrial system, this time is always extremely short compared with the time scales of interest in this book. 5.4.2.1 The dynamical hierarchy This hierarchy is derived from (5.19). Some simplification occurs because we can assume that mass, momentum, and energy are conserved in any collision processes between species. Set φ = 1 in (5.19). The third term is zero because ∂φ/∂vk = 0. The term on the right-hand side represents the change in the number of particles in the volume element: since particles are conserved, it is zero for each species. The result may be written in the form ∂ρ + ∇ · ρu = 0 (5.32) ∂t which is recognized as the continuity equation for the plasma. In this context, it is a differential equation for the evolution, in time and space, of the mass density. Its evaluation requires a knowledge of the bulk velocity of the plasma. This can only be obtained from an equation describing, in turn, the evolution of u. To find this next equation, set φ = vi in (5.19). The sums in the first two terms are evaluated

using (5.4) and (5.5). In the third term, we use ∂vi /∂v j ≡ δi j . We note that Nα qα = 0, since the electron number density is equal to the number density of positive particles so that electric field term sums to zero. The sum in the magnetic field term is, then, the current density Ji . The first term on Copyright © 2005 IOP Publishing Ltd.

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the left-hand side is the rate of change of momentum density so that the collision term on the right-hand side represents the contribution to this rate of change for each particle species as a result of collisions: since momentum is conserved in a collision, these terms sum to zero. The equation becomes ∂(ρu i ) ∂ + {Pik + ρu i u k } − ilm Jl Bm = 0. ∂t ∂ xk

(5.33)

We expand the derivatives and use (5.32) and (2.6), the Lagrange derivative d/dt ≡ ∂/∂t + u · ∇, to get ρ

du i ∂ Pik = ikl Jk Bl − . dt ∂ xk

(5.34)

This is the momentum equation with an anisotropic pressure tensor. It provides the required differential equation for the evolution of u i but only at the cost of introducing two new quantities—the current density Ji and the pressure tensor Pi j . We also require equations for their evolution. The equation for the current density is part of the electromagnetic hierarchy of equations. To find an equation for the evolution of the pressure tensor, set φ = vi v j in (5.19). If we use (5.4), (5.5), and (5.6), and also ∂vi /∂v j = δi j , it becomes ∂ ∂ {Pi j + ρu i u j } + {Q i j k + Pi j u k + P j k u i + Pki u j + ρu i u j u k } ∂t ∂ xk



Nα qα E j vi + E i v j α − Nα qα ilm v j vl Bm +  j lm vi vl Bm  − α

=

 α

∞ −∞

 vi v j

∂f ∂t

α

α

d3 v.

(5.35)

coll

This is an equation for the evolution of the stress tensor Pi j . The collision term on the right-hand side represents the sum of the stresses exerted by each species on each other species as a consequence of collisions. If we consider an element of area d A j the force Pi j d A j exerted by species a on species b is equal and opposite to that exerted by b on a. The right-hand side of the equation sums to zero. If we then expand all the derivatives and group the terms, we get ∂ Qi j k ∂ Pi j ∂u j ∂ Pi j ∂u k ∂u i + uk + Pi j + Pj k + Pki + ∂t ∂ xk ∂ xk ∂ xk ∂ xk ∂x     k   ∂ Pj k du j dρ ∂u k ∂ Pik du i +ρ + + + ui ρ +uj ρ + ui u j dt ∂ xk dt ∂ xk dt ∂ xk



− E i J j − E j Ji − Nα qα ilm v j vl α Bm − Nα qα  j lm vi vl α Bm = 0. Copyright © 2005 IOP Publishing Ltd.

α

α

(5.36)

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This formidable collection of terms is greatly simplified if we note the following (i) The first set of terms enclosed in braces is zero through (5.32). (ii) The next two sets of terms in braces can be expressed in terms of the magnetic field through (5.34). Also vi vl α = (Ci + u i )(Cl + u l )α = Ci Cl α + u i Cl α + u l Ci α + u i u l (5.37) where Ci is the peculiar velocity relative to the fluid (4.18). These results are used to simplify the equation which becomes ∂ Qi j k ∂ Pi j ∂u j ∂ Pi j ∂u k ∂u i + uk + Pi j + Pj k + Pki + ∂t ∂ xk ∂ xk ∂ xk ∂ xk ∂ xk − (E i + ilm u l Bm )J j − (E j +  j lm u l Bm )Ji



− Nα qα ilm C j Cl α Bm − Nα qα  j lm Ci Cl α Bm = 0. α

(5.38)

α

Again we have found an equation for the time evolution of Pi j at the cost of introducing a yet higher moment Q i j k . This process leads to an infinite sequence of equations. It is only useful if we can find a way of terminating the sequence to achieve closure. 5.4.2.2 The electromagnetic hierarchy We could, in principle, proceed to derive this hierarchy in exactly the same way as for the dynamical hierarchy, by setting φ = 1, vi , vi v j . . . in (5.20). Our derivation can be considerably simplified, however, by applying an approximation that is always true, namely that the electron mass is negligible compared with the mass of any positive ion: (5.39) m e m ion .

−1 Let the mean ion mass be M = N ion Nα m α and let the velocity of each species relative to the mean velocity for the plasma be wα . Then, from the definitions (4.13), (4.14), and (4.21), we see that wie wiion Ji =

Jie [1 +

(5.40) O(m e /M].

(5.41)

Equation (5.20) may, therefore, be written in the form   ∂

∂

Ne e2 ∂φ Nα qα φα + Nα qα vk φα − (E k + klm vl Bm ) ∂t α ∂ xk α me ∂vk e    e ∞

∂f = qα φ d3 v. (5.42) ∂t −∞ coll α Copyright © 2005 IOP Publishing Ltd.

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The first equation in the hierarchy is found by setting φ = 1 in this equation. This, in principle, gives the equation of evolution of charge density, which is simply the equation of charge continuity. However, we are already applying the low-velocity approximation to Maxwell’s equations and this implied that the charge density is zero. The collision term is zero for each species of particle because particles are conserved in any collision. Thus, this equation is simply ∂ Jk ≡ ∇ · J = 0. ∂ xk

(5.43)

This is consistent with what can be derived by taking the divergence of Amp´ere’s law. To find an equation for the evolution of the current density, set φ = vi in (5.42). If we use the definition of the current density (4.21) and the expressions for partial pressures of the species (4.25), this may be written as  e Pikel e Pikion Ne2 ∂ Ji ∂ + − + u i Jk + u k Ji − {E i + ilm u le Bm } ∂t ∂ x k m ion m el me  ∞  α

∂f = mα vi d3 v. (5.44) ∂t −∞ coll α The terms involving the current density may be simplified by using (5.22) and (5.43). Also, from (5.41), the mean electron velocity is u le  u l − Jl /Ne. The equation then becomes  N

∂ ∂t



Ji N



∂ + uk ∂ xk



Ji N



Jk ∂u k + N ∂ xk



Ne2 ∂ = {E i + ilm u l Bm } − ilm Jl em + me ∂ xk



e P ion e Pikel − ik m el m ion

+ νcoll Ji . (5.45)

Here we have formally written the collision term as νcoll Ji where νcoll is an effective collision frequency, found by evaluating the collision integral; and it may itself be a function of Ji . The ion and electron pressure tensors may have very different magnitudes so we should not unthinkingly ignore the term in Pikion . However, unless β 1, this term is O(m e /m i ) compared with the J × B term, in which case it may be dropped. The equation has been written in this form so that it can easily be compared with (2.51). We see that it represents the generalized Ohm’s law. We could continue to find higher moments in this hierarchy but we shall have no need of them. Copyright © 2005 IOP Publishing Ltd.

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5.5 Closure of the moment equations A variety of approximations, suiting different situations, is available to close the moment equations. In general, they depend on the nature of the collision process. At one extreme, binary collisions with neutral particles occur at a frequency much greater than the gyrofrequencies of the particles, so that a particle moves over only a small fraction of its gyro-orbit before encountering another particle. At the other extreme, there are no collisions and it is the magnetic field that serves as a mechanism for limiting the motion of particles in space. The most elaborate procedure involves a process of successive approximations to the distribution function which, in principle, can be carried out to as high an order of approximation as needed. This can only be done if the nature of the collision process is fully understood. An example of this process, for a multi-ion plasma in thermal equilibrium, is given by Clemmow and Dougherty [49, section 11.5.3]. The first stage of the process leads to the adiabatic approximation. Higher approximations allow the discussion of transport processes such as viscosity and thermal conduction. For our purposes, it will not be necessary to go beyond the first stage of this process. There are assumptions common to all formulations of MHD. The first is that any ‘collision’ processes occur on a sufficiently long time scale compared with the gyroperiod so that the ordering of particle motion provided by the magnetic field is not removed. This is sometimes thought of as a condition of ‘infinite’ conductivity. If the collision frequency is much greater than the gyrofrequency, the particle motion is the same as that in a conductor with no magnetic field. We write < i . (5.46) νcoll ∼ The other conditions are those argued in section 2.3.5: ω i

(5.47)

av ωp,av .

(5.48)

We adopt these conditions as basic assumptions for our development. Note that the last condition gives av = ωp,av



B 2 e2 0 m VA B

1 = = √ 2 2 c m Ne c µ0 Nm

(5.49)

where VA is the Alfve´ n speed introduced in chapter 2. The other characteristic speed in chapter 2 is the sound speed which, if the particles are non-relativistic, is always much less than the speed of light. The ratio of the length and time scales in the problem is comparable with the greater of the Alfv´en and sound speeds, so that (5.49) implies the low-velocity approximation to Maxwell’s equations described in chapter 1. Copyright © 2005 IOP Publishing Ltd.

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5.5.1 Successive approximations to the Boltzmann equation Let us write the Boltzmann equation (4.46) for one of the particle species in the form   ∂f q q ∂f + v · ∇ f + {E + u × B} · ∇ f = − (C × B) · ∇ f + . (5.50) ∂t m m ∂t coll The distribution function f is a function of the seven independent variables t, x, y, z, vx , v y , and vz . The acceleration arises only from electromagnetic forces. The magnetic force has been divided into two parts, one depending on u, the mean fluid velocity, and the other on C, the peculiar velocity of the species relative to the fluid. Our programme will be to solve this equation and the moment equations simultaneously by a method of successive approximations, using the ordering suggested by (5.46), (5.47), and (5.48). Formally, our assumptions are just these conditions. We shall, however, use some physically intuitive assumptions initially and demonstrate a posteriori that these are equivalent to our formal assumptions. The approximation process then, in principle, proceeds as follows. It is assumed that the initial conditions are specified throughout space. This means that the moments of the distribution—or, equivalently, the velocity, pressure tensor, heat flux tensor, and relevant higher moments of each species—are known at t = 0. These moments are parameters in the determination of the distribution function for each species from the solution of the zero-order equation (5.50). The distribution function is found by solving the zero-order equation obtained by setting the right-hand side of (5.50) equal to zero. The distribution function so found, in general, may have properties that lead to closure of the moment equations. It can be substituted in the closed set of moment equations, which can then be solved to give a first-order correction to the distribution function. In principle, the process can be iterated to provide a first-order correction to the distribution function but we shall not need to proceed to this level of approximation. The process is best illustrated by applying it to specific examples. Before doing so, some general results need to be derived. Intuitively, we expect that, since collisions are infrequent, all particles move with an E × B drift to a zero-order of approximation. This means that E+u× B 0

(5.51)

where the approximation is true to order ξ1 = ω/i and ξ2 = i /ωp,i , with the gyro- and plasma frequencies being those of a fictitious particle having the mean positive ion mass. Note that this also requires that the electric field parallel to B is also small. Examination of the Boltzmann equation then shows that the terms on the left-hand side are smaller than those on the right-hand side by factors ξ1 , ξ2 . The zero-order approximation to f is then the solution of   ∂ f0 q (C × B) · ∇ f 0 − = 0. (5.52) m ∂t coll Copyright © 2005 IOP Publishing Ltd.

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Bear in mind that the collision term represents all possible types of interaction that occur on time scales short compared with those in the problem of interest. These interactions may be between the particles of the species itself or with particles of other species. They may occur as a result of binary collisions, distant Coulomb collisions, or wave–particle interactions. In solar–terrestrial systems, the last of these is the only relevant one in the great majority of cases. The nature of the zeroorder solution depends, therefore, on a knowledge of the collision process and, in most cases, this knowledge is not available. This ignorance stems from current lack of understanding of the details of the nonlinear processes involved. Even if these details were better understood, the identification of the possible processes applicable in a particular case would generally be impracticable. Progress is only possible if we can deduce some general properties of the zero-order distribution function f 0 in various special cases. 5.5.2 Orders of magnitude We shall label MHD variables, obtained by integrating the zero-order distribution function, with a subscript or superscript zero, as appropriate. In the MHD approximation, the first approximation to the time derivative or space derivative of such zero-order quantities is small, because of the assumptions (5.47) and (5.48). Explicitly, the process of operating on a zero-order quantity w0 with ∂/∂t, produces a quantity of order ωw0 = ξ1 w0 , with a similar ordering for ∇. As a convenient shorthand we write the operators in the form ∂ (1) ∂ (1) , . ∂t ∂ xi This is a way of keeping track of the order of magnitude of terms on which they operate as we apply a process of successive approximations. The current density J requires special attention. The current densities of the individual species are of zero-order. The net current density arises from the difference between the mean (1) × B ion velocity and the electron velocity. From Amp´ere’s law J = µ−1 0 ∇ and, thus, the total current density is of first-order. 5.5.3 Truncation of the electromagnetic hierarchy Suppose that f 0 has been determined as a solution of (5.52). We substitute this in (5.45) and note that terms on the left-hand side are then of order ω/ compared with terms on the right-hand side. To this level of approximation, they may be ignored. The result is  0,el ∂ (1) e Pik (0) (1) e (1) 2 + νcoll Ji = 0. (5.53) 0 ωp {E i + ilm u l Bm } − ilm Jl m + ∂ xk m el By virtue of (5.46), the second, third, and fourth terms are of the same order of magnitude. The first term is multiplied by the electron plasma frequency, which Copyright © 2005 IOP Publishing Ltd.

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is large. We can explicitly verify the relative magnitudes of the terms. If we compare the first term with the second, then, from Ampe´ re’s law, J (1) ∼ B/µ0l , while from Faraday’s law, E ∼ ωl . This shows that the ratio of the second term to the first is O(2ion /ω2p,ion), so that, to this order of approximation, E + u(0) × B  0.

(5.54)

Where appropriate, the term will be marked O(1). This justifies the assumption (5.51) and applies in all the cases discussed later. 5.5.4 Cold plasma The most straightforward truncation of the equations occurs when the energy density of the plasma is small enough. This implies that the thermal speeds of the particles can be neglected. Formally, we can proceed by setting the zero-order distribution function for each species equal to a δ -function: f 0 (t, x, v) = N(t, x)δ 3 (C).

(5.55)

The peculiar velocity is C = v − u(t, x). Then the pressure term in (5.27) is isotropic and we may write, for each species,  Pi j = Pδi j = δi j Nm (C x2 + C y2 + C z2 )δ(C x , C y , C z ) d3 v = 0. (5.56) The dynamical hierarchy of moment equations then terminates. The continuity equation (5.22) and momentum equation (5.27) for each species, together with Maxwell’s equations, form a complete set. We retrieve the set of MHD equations presented in chapter 2 for a fluid in which the only force is the J × B force. From the reduced equation (2.55), we see that the effect of our approximation is to assume that the magnitude of the pressure is small compared with the magnetic energy density B 2 /2µ0 . 5.5.5 Thermal equilibrium We have already noted that, through most of the solar–terrestrial system, binary collisions are insufficient to bring the plasma to thermal equilibrium in a sufficiently short time. Only in the Earth’s plasmasphere and upper ionosphere are distant Coulomb collisions reasonably effective and here, generally, β 1 so that the approximation of section 5.5.4 applies. Nevertheless, consideration of this case is helpful in illustrating the method of approach. We restrict ourselves to a plasma with one species of positive ion; and the extension to a multi-ion plasma is straightforward. The zero-order solution is the solution of (5.52) found when the collision process is isotropic and brings the plasma to thermal equilibrium on a time scale short compared with the time scale of the problem. The isotropic condition Copyright © 2005 IOP Publishing Ltd.

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means that, after a typical collision time has elapsed, the velocity of a particle is uncorrelated with its value at the beginning of the interval—its magnitude and direction are both arbitrary. It is also assumed that there is an efficient collision process occurring between ions and electrons. In such a case, the Boltzmann H theorem ensures that the zero-order distribution function for ions and electrons is the Maxwellian    3/2 m mC 2 exp − f 0 (x, v, t) = N0 (x, t) (5.57) 2π K T0 (x, t) 2K T0(x, t) where C 2 = {v − u0 (x, t)} · {v − u0 (x, t)}

(5.58)

is the peculiar velocity for each species. The equation (5.52) gives only the velocity dependence of f0 . The time- and space-dependent quantities, N0 , u0 , and T0 are parameters that are inserted into the zero-order equation, which is then solved to find f 0 . The number density for each species is N0 , which is found by integrating the Maxwellian. The x-component of the zero-order velocity for each species is  3/2  ∞ (0) 2 m vx e−m(vx −u x ) /2K T0 dvx = u (0) (5.59) vx(0)  = N0 x 2π K T0 −∞ with similar expressions for v y  and vz . The pressure tensor is  3/2  ∞ m 2 (0) Pi j  = N0 Ci C j e−mC /2K T0 d3 C 2π K T0 −∞

(5.60)

When i = j , this integral is zero and, when i = j , it is equal to N0 K T0 , so that (0)

Pi j  = N0 K T0 δi j = P0 δi j . The heat flux tensor is  (0) Q i j k  = N0

m 2π K T0

3/2 

∞

−∞

Ci C j Ck e−mC

2 /2K T 0

(5.61)

d3 C.

(5.62)

Since the exponential is an even function of C x , C y , C z , every component is odd and the integral is zero. This achieves the closure of the moment equations since (0)

Q i j k  = 0.

(5.63)

Since, because of the thermal equilibrium, these quantities are the same for ions and electrons, we can use the moment equations for more than one species. The number density is the same for the ions and electrons and can be used to deduce the plasma density. Equation (5.32) shows that d(1)ρ0 + ρ0 ∇ (1) · u0 = 0. dt Copyright © 2005 IOP Publishing Ltd.

(5.64)

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In (5.34), the gradient of the pressure tensor becomes ∂ (1) Pi j ∂ (1)(δi j P0 ) ∂ P0 = = . ∂x j ∂x j ∂ xi

(5.65)

The moment equation thus involves the gradient of a scalar pressure and can be written as  d(1) u0 B B2 (1) = −∇ ρ0 P0 + + · ∇(1) B. (5.66) dt 2µ0 µ0 Since only the magnitude of a scalar pressure is required, the next moment equation (5.38) may be contracted by setting i = j . The result is 0) ∂ Q (iik ∂ (1) Pii(0) ∂ (1)u (k0) ∂ (1)u (i 0) ∂ (1) Pii(0) + u (k0) + Pii(0) + 2 Pik(0) + ∂t ∂ xk ∂ xk ∂ xk ∂ xk

(1) (1) −2( E i + ilm u l Bm ) Ji − 2 Nα qα ilm Ci Cl α Bm = 0. α

(5.67) As discussed in section 5.2, since f 0 is symmetric, Q i j k = 0. The partial pressure of any species is Piαj = P α δi j . In the last term, Ci Cl α = P (α) δil / Nm α . The properties of the Kronecker delta and the alternating unit tensor show that (1) δil ilm = iim = 0. The term ( E i + ilm u l Bm )(1) Ji is of second-order and can be neglected. If we drop the superscripts showing the order of the terms, this first-order equation may be written in the form dP + 5P∇ · u = 0. (5.68) dt This is the reduced adiabatic law (2.57), with γ = 5/3. We see that, if the collision process brings the ions and electrons into thermal equilibrium, the moment equations are truncated and, to first-order in the small quantities ξ1 , ξ2 , the surviving moments are governed by the MHD equations of chapter 2. Because of the thermal equilibrium, N , u, and P are the same for ions and electrons and can be inserted into the first-order Boltzmann equation, in order to get a first-order correction to the distribution function. In terms of (5.2), the series terminates and becomes a sum. The Maxwellian is entirely determined by the first three moments. It is, therefore, possible to continue the process of successive approximation further. For our purposes, there is no need to do so. Clemmow and Dougherty [49] show how to take the approximation to the next stage. In the case of solar–terrestrial plasmas, this study has been somewhat academic because almost nowhere in the region of interest are collision times short enough to bring the plasma to true thermal equlibrium. In the next section, we consider quasi-equilibria and deduce the coonditions under which the behaviour of a plasma, in which the particle species are individually in quasiequilibrium states, is described by the MHD equations. 3

Copyright © 2005 IOP Publishing Ltd.

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5.5.6 The fluid equations in the absence of collisions between species Now let us consider the more realistic situation in which collisions, whether binary collisions or distant Coulomb collisions, are completely negligible. The collision term in the Boltzmann equation then arises only from wave–particle interactions. In general, such interactions are only expected to be between particles of the same species—positive ions interact through longitudinal ion acoustic waves and transverse ion cyclotron waves; electrons interact through longitudinal electron plasma waves and transverse electron cyclotron waves. As a result, collisions between species are negligible. Near a quasi-equilibrium state, the form of the collision term is likely to represent a diffusion in velocity space [127, 233] but the details of the process are not well understood. Again, we consider a plasma with only one species of positive ion. The zeroorder equation for the distribution function of each species is (5.52) but now the ‘collision’ term describes short time scale interactions between only the particles of the species itself. If we neglect collisions altogether, the most general solution is (5.69) f (0) = f (0)(C , C⊥ ) where, if B is along the z-axis, C⊥ =

'

C x2 + C y2

C = C z .

(5.70)

This is clearly even in C x and C y . It is not, however, necessarily even in C z . If it has an odd part, moments of odd orders are non-zero and the dynamical hierarchy of moment equations does not terminate. In such circumstances, we cannot apply the method of successive approximations to obtain a closed set of MHD equations. This is only possible if other processes ensure that the distribution function is even, such that 2 ). (5.71) f (0) = f (0)(C2 , C⊥ In order to achieve this, we must look to the collision term. In so doing, we shall, of necessity, be rather speculative. As previously noted, there are two major types of wave–particle interaction that may be important in energy exchange between the particles. Weak turbulence associated with longitudinal waves changes v because of the longitudinal electric field. Weak turbulence associated with transverse waves changes the pitch angle of the particles (with a smaller effect on the energy) through a Doppler-shifted cyclotron interaction. As a result, the distribution is driven towards isotropy in v⊥ , v . We make the minimum assumption that a longitudinal wave–particle interaction exists such that, over a time interval tc , a particle undergoes a random walk parallel to the field, such that its motion at the end of the interval is uncorrelated with its motion at the beginning of the interval. We assume further that tc is long compared with the ion gyroperiod but short compared with the time scales in the problem of interest. The consequence is a diffusion in velocity space, Copyright © 2005 IOP Publishing Ltd.

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parallel to the magnetic field, unless the distribution function is symmetrical. The zero-order distribution function, averaged over this time scale, is then assumed to be symmetrical so that it is of the form (5.71). There may also be further processes associated with pitch angle diffusion that drive the distribution to isotropy. In such cases, we may assume an isotropic averaged distribution function f (0) = f (0) (v 2 ).

(5.72)

Let us first consider the case where (5.71) holds. The pressure tensor is  ∞ ∞ ∞ Pi j ∝ Ci C j f (0) d3 C. (5.73) −∞ −∞ −∞

If B is locally in the z-direction, because of the symmetry of f (0) , the off-diagonal terms of this tensor are zero. Further, P11 = P22 . The pressure tensor for each of the ion and electron fluids, therefore, takes the form   0 P⊥ 0 (5.74) P =  0 P⊥ 0  . 0 0 P The total pressure can be found by summing these partial pressures so that, in Cartesian tensor form, with magnetic field in the z-direction, Pi j = P Bˆ i Bˆ j + P⊥ (δi j − Bˆ i Bˆ j )

(5.75)

where Bˆ i is a unit vector in the direction of the magnetic field. Because of the symmetry, the odd moments of the distribution are zero and, thus, Q i j k = 0.

(5.76)

The momentum equation (5.34) for the plasma can then be written as ρ

du i ∂ = i j k J j Bk − {P Bˆ i Bˆ j + P⊥ (δi j − Bˆ i Bˆ j )}. dt ∂x j

(5.77)

If Amp´ere’s law is used to eliminate J j and the result is written in vector notation, it becomes     P⊥ − P 1 B2 du = −∇ P⊥ + + + B·∇ B . (5.78) ρ dt 2µ0 µ0 B2 This equation was first derived by Parker [153] using a different method. The equation for the time development of the plasma pressure (5.38) becomes (0)

∂ (1)u k d(1) (0) ˆ ˆ (0) (0) (0) {P Bi B j + P⊥ (δi j − Bˆ i Bˆ j )} + {P Bˆ i Bˆ j + P⊥ (δi j − Bˆ i Bˆ j )} dt ∂ xk Copyright © 2005 IOP Publishing Ltd.

Fluid behaviour

108

+ {P(0) Bˆ j Bˆ k + P⊥(0) (δ j k − Bˆ j Bˆ k )} + {P(0) Bˆ i Bˆ k =

+

 qα (−1)

P⊥(0) (δik

− Bˆ i Bˆ k )}

∂ (1)u (0) i + {E i + ilm u l Bm }(1) J j(1) ∂ xk (0)

∂ (1)u j ∂ xk

+ {E j +  j lm u l Bm }(1) Ji(1)

ilm {P(0) Bˆ j Bˆ l + P⊥(0) (δ j l − Bˆ j Bˆ l )}α m α α

 qα (−1) +  j lm {P(0) Bˆ i Bˆ l + P⊥(0) (δil − Bˆ i Bˆ l )}α . m α α

(5.79)

(1)

The terms of the form {E j +  j lm u l Bm }(1) Ji are of second order and can be neglected. The terms on the right-hand side are of order −1 because of the factor qα /m α which is proportional to the gyrofrequency. This cannot, therefore, be a suitable formulation. We note, however, that there are only two independent quantities P and P⊥ that determine the pressure. It is, therefore, possible to find scalar equations for them. If we operate on the equation with δi j , the terms on the right-hand side are identically zero and we obtain ∂u i ∂u i d (P + 2P⊥ ) + (P + 4P⊥ ) + 2(P − P⊥ ) Bˆ i Bˆ k =0 dt ∂ xi ∂ xk

(5.80)

while operating with Bˆ i Bˆ j also makes the terms on the right-hand side zero and yields dP ∂u i ∂u k + 2P Bˆ i Bˆ k + P = 0. (5.81) dt ∂ xk ∂ xk The second of these two equations may be used to eliminate P from the first: dP⊥ ∂u i ∂u k − P⊥ Bˆ i Bˆ k + 2P⊥ = 0. dt ∂ xk ∂ xk

(5.82)

These equations can be further simplified. The equation (2.58) still holds in these circumstances. If we take its scalar product with Bˆ i /B, the result is an equation for the time development of the magnitude of B. In Cartesian coordinates, this is ∂u i ∂u i 1 dB = Bˆ i Bˆ j − . (5.83) B dt ∂x j ∂ xi Also the equation of continuity for the plasma may be written as 1 dρ ∂u i + = 0. ρ dt ∂ xi

(5.84)

Thes can be used to eliminate ∂u i /∂ x i and Bˆ i Bˆ j ∂u i /∂ x j from (5.81) and (5.82) giving   d P B 2 =0 (5.85) dt ρ3 Copyright © 2005 IOP Publishing Ltd.

Closure of the moment equations   d P⊥ = 0. dt ρ B

109 (5.86)

These equations replace the adiabatic law when the pressure is not isotropic. They were first derived by Chew et al [46], using a slightly different approach. They considered an ion fluid in the absence of collisions. The distribution function was not assumed to be symmetric in C . The result was that there were some non-zero elements of the heat flux vector, which were regarded as sufficiently small to be neglected. We shall not often need to use the equations in this form. Usually, for practical purposes, a formulation with isotropic pressure is sufficient. Nevertheless, the isotropic formulation has been derived only for the case for which collisions are frequent enough so that the plasma is in thermal equilibrium and this is often not so. What is the minimum assumption that can be made about the form of the distribution function for the use of an isotropic pressure to be valid? It is, of course, that the distribution function of each species must be isotropic. This requires some process that drives the distribution towards a form that is independent of the pitch angle of the particles. Pitch angle diffusion as a consequence of Doppler shifted cyclotron resonance is such a process. We can speculate that wave–particle interactions related to this process may be effective in many cases. The result will be that (5.71) becomes f (0) = f (0) (C 2 )

(5.87)

P⊥ = P .

(5.88)

so that The analysis can be followed through in these conditions. The result is that the momentum equation (5.78) becomes    B B2 du + B·∇ = −∇ P + . (5.89) ρ dt 2µ0 µ0 The distribution is not Maxwellian in this case. Nevertheless, the Maxwellian in (5.60) can be replaced by f (0) (C 2 ) and the same argument applied to show that Pi j = Pδi j . The truncated moment equations are then the same as those for a plasma in thermal equilibrium. One should not fall into the trap of setting P⊥ = P in (5.81) and (5.82). If we do so and eliminate P⊥ Bˆ i Bˆ k ∂u i /∂ x k , we indeed get 3

∂u i dP + 5P = 0. dt ∂ xi

(5.90)

This appears to be just the usual adiabatic condition (2.57) with γ = 53 . However, (5.81) and (5.82) were derived on the assumption that any process driving the plasma towards pitch angle isotropy occurs on a time scale long compared with Copyright © 2005 IOP Publishing Ltd.

110

Fluid behaviour

the time scale of the problem. If we assume P⊥ = P = P and eliminate dP/dt we find an additional constraint: ∂u i ∂u i = . (5.91) Bˆ i Bˆ k ∂ xk ∂ xi This is equivalent to the condition ∇⊥ · u ⊥ = 0

(5.92)

so that the result only holds if there is no compression perpendicular to the magnetic field. The equations for isotropic pressure and anisotropic pressure do not merge continuously from one to the other. They are both extreme cases. One applies when the time scale τ for driving the plasma to pitch angle isotropy is short compared to time scales of the problem. The other applies when τ is long compared to time scales of the problem. The application of isotropic MHD to problems in solar–terrestrial physics is justified when physical processes maintain the distribution functions of the various ion species in a state that is effectively independent of pitch angle.

5.6 Summary •

•

•

The central moments of a distribution function are the average values of successive powers of the independent variable, referred to an origin for which the first moment is zero. If the moments m r exist and a geometric series with m r /r ! converges for some positive value of the argument, then the distribution function is uniquely determined by the moments. It can be assumed that the behaviour of the particles of each species can be described by the Boltzmann equation with an appropriate collision term. If it is assumed that the processes of interest occur on a time scale long compared with the characteristic time scales of the medium such as the gyroperiod, then wave–particle interactions on the shorter time scale may be lumped in an unspecified collision term. The moments of the Boltzmann equation for this case have been found, yielding an infinite sequence of differential equations for each species, representing the time development of the moments. The equation for the time derivative of any moment, in general, depends on the next higher moment. A collision term that is one order larger than other terms appears in each equation. Two hierarchies of moment equations for a multispecies plasma are derived. The dynamical hierarchy represents the time development of dynamic variables and is derived by adding the moment equations of all the species. In the first few equations of the hierarchy, this eliminates the collision term because of conservation laws such as the conservation of mass, momentum and energy. The electrodynamic hierarchy represents the electrodynamic variables and is found by subtracting the moment equations for the electrons from those of the positive ions.

Copyright © 2005 IOP Publishing Ltd.

Summary •

•

•

•

•

111

Two parameters are defined. One is the ratio of the effective ion gyrofrequency to the ion plasma frequency and the other is the ratio of a characteristic frequency for the problem to the ion gyrofrequency. In the MHD approximation, these parameters are small. In the MHD approximation, the second equation in the electromagnetic hierarchy shows that, to zero order, the particles all move with an E × B velocity. The differential velocity of electrons and ions, which gives rise to the current density, is of first order. The gyro motion of the electrons about the magnetic field occurs on a time scale one order larger than that of interest in the problem. The term in the Boltzmann equation that describes this motion is lumped with the collision term. When the plasma is in thermal equilibrium, with a collision term in the Boltzmann equation that is of one order larger than the other terms, the zero-order solution is a Maxwellian. This is substituted into the moment equations. Moments higher than the second are zero. The first-order approximation to the remaining equations are the MHD equations. When collisions between the species are unimportant, a similar process can be carried out to derive a set of MHD equations with an anisotropic pressure and a modified collision law. In order for the hierarchy of moment equations to be truncated, it is necessary to assume a process that ensures symmetry of the zero-order distribution function in v . If, in addition, there are processes leading to pitch angle isotropy on a short enough time scale, then the isotropic MHD equations are retrieved, even although the plasma is not in thermal equilibrium.

Copyright © 2005 IOP Publishing Ltd.

Chapter 6 Equilibrium and steady-state conditions

6.1 Introduction This book is chiefly concerned with MHD waves. These waves are propagated in a variety of inhomogeneous MHD media. Frequently, the inhomegeneity has an effect on the nature of the wave, which is a first-order perturbation on the zeroorder steady state. It is, therefore, necessary to understand the conditions that determine equilibrium states and steady-state flows. In steady-state flow, at every point in the medium, conditions do not change with time. This means that the equations that apply are the MHD equations with ∂/∂t = 0. An equilibrium state is one for which, in addition, the velocity is everywhere zero. We start by considering general conditions for equilibrium. We then generalize these to steady-state conditions. In order to understand wave propagation across a discontinuity in a plasma, we need to know what boundary conditions to apply. We consider the nature of these boundary conditions in general. The kinds of discontinuity that can occur in the steady state are constrained by the boundary conditions. We discuss various types of discontinuity, including shock discontinuities.

6.2 MHD equilibrium In equilibrium, the plasma is everywhere at rest: the velocity is zero and ∂/∂t ≡ 0. In these circumstances, the momentum equation in the form (2.42) becomes ∇ p = J × B.

(6.1)

This implies that, if there is a pressure gradient, a current must flow perpendicular to the magnetic field and the pressure gradient. Its magnitude can be found by taking the vector product of B with (6.1). The result is J− 112

Copyright © 2005 IOP Publishing Ltd.

B ×∇p B( J · B) ≡ J⊥ = B2 B2

(6.2)

MHD in the steady state

113

where J ⊥ is the component of the current density perpendicular to the magnetic field. Alternatively, the equilibrium condition may be written in terms of the stress tensor. From (2.55),  B B2 − ∇ p+ · ∇ B = 0. (6.3) 2µ0 µ0 The quantity B 2 /2µ0 acts like a magnetic pressure. The total pressure is the sum of the thermal pressure p and the magnetic pressure. The last term involves the changes in B along the field direction. If we write B = µB, ˆ where µ ˆ is the unit vector in the direction of the field, it may be written in the form 1 B (B · ∇)B = (µ ˆ · ∇)µB ˆ µ0 µ0 B2 B2 = (µ ˆ · ∇)µ ˆ + µ( ˆ µ.∇) ˆ . µ0 2µ0

(6.4) (6.5)

The second term is equal to the parallel component of the gradient of the magnetic pressure. The first term, from (B.11), may be written in the form B2 B2 (µ ˆ · ∇)µ ˆ = −ˆν µ0 µ0 R

(6.6)

where R is the radius of curvature of the field line and νˆ is a unit vector normal to the field in the direction of the radius of curvature. Thus, if we separate the equilibrium condition into components perpendicular (⊥) and parallel () to the magnetic field, it becomes  B2 B2 − νˆ =0 (6.7) ∇⊥ p + 2µ0 µ0 R ∇  p = 0.

(6.8)

If a stretched string of tension T is plucked, then, at the instant when its radius of curvature is R, it is easy to show that the restoring force per unit length normal to the string is T /R. The last term in the equilibrium equation is of this form. The factor B 2/µ0 plays the role of the tension per unit cross-sectional area of flux tube. We thus see that the medium behaves in some sense like a string under tension along the field direction and like a fluid with anisotropic compressibility such that the longitudinal pressure arises entirely from the thermal pressure and the transverse pressure is the sum of magnetic and thermal pressures.

6.3 MHD in the steady state In steady-state flow the time variation ∂/∂t of all quantities is zero. If the velocity is also zero everywhere, we also have a state of MHD equilibrium. In a steady Copyright © 2005 IOP Publishing Ltd.

Equilibrium and steady-state conditions

114

state, the plasma flows along streamlines that are everywhere tangential to the plasma velocity. We also assume that the thermal conductivity is small so that conditions change adiabatically. The case in which the thermal conductivity is sufficiently high so that the the assumption of adiabatic behaviour breaks down is not of practical interest here. The momentum equation (2.55) becomes  B B2 + · ∇ B. (6.9) ρv · ∇v = −∇ p + 2µ0 µ0 The continuity equation (2.38) shows that ∇ · ρv = 0

(6.10)

v · ∇ρ = −ρ∇ · v.

(6.11)

or that The adiabatic law (2.45) takes the form ρ v · ∇ p. v · ∇ρ = γp

(6.12)

This equation only describes the change of density and pressure in the direction of the plasma flow. This means that, as the plasma moves, expanding and contracting as it follows a streamline in the direction of v, the change in the plasma energy density arises purely as a result of the work done on the plasma: on the time scale of interest, there is no conduction of heat. Neither this law nor the equation of continuity has anything to say about the component of the density gradient normal to the velocity of the plasma. This density gradient must be determined by boundary conditions where the streamlines enter the region under consideration.

6.4 Boundary conditions The MHD equations allow the possibility of discontinuities in the various plasma parameters. These can be studied by applying equations describing the conservation of various quantities across a sharp boundary. The equations of interest are: (i) the continuity equation (2.38) representing conservation of mass, ∂ρ + ∇ · ρv = 0 (6.13) ∂t (ii) the energy conservation equation (2.93) with the flux given by (2.98),  ∂ 1 2 B2 p ρv + + ∂t 2 γ − 1 2µ0   1 2 γp B B2 v− B·v (6.14) = −∇ · ρv + v+ 2 γ −1 µ0 µ0 Copyright © 2005 IOP Publishing Ltd.

Boundary conditions A

115

x

Figure 6.1. Boundary condition at a plane surface.

(iii) the equation of momentum conservation in the form (2.101),  B j Bk − 12 B 2 δ j k ∂ ∂ (ρv j ) = − pδ j k + ρv j vk − ∂t ∂ xk µ0

(6.15)

(iv) Faraday’s law (2.58) which may be written in Cartesian tensor notation as ∂Bj ∂ = (Bk v j − B j vk ) ∂t ∂ xk

(6.16)

(v) Gauss’ law for the magnetic field, ∇ · B = 0.

(6.17)

All these have the form of conservation laws which can be integrated over any volume as described in section 2.9. Consider the volume shown in figure 6.1. It is coin-shaped with area A and thickness δx and encloses a portion of a boundary at which there is a discontinuity of the plasma parameters. It is small enough so that the boundary can be regarded as plane and the fields can be regarded as approximately constant. It has a thickness δx and top and bottom surface areas A. As δx → 0, the two surfaces remain on opposite sides of the boundary. If we take the equation for mass conservation (6.13) as an example and integrate it over the volume, we get d {ρAδx} = {ρ2 v2n − ρ1 v1n }A + O(ρvt A1/2 δx) dt

(6.18)

where subscripts ‘1’ and ‘2’ refer to the upper and lower regions respectively, ‘n’ and ‘t’ denote components normal and transverse to the boundary, and the angle brackets  denote a mean value within the volume. If we let δx → 0, this becomes [ρvn ] ≡ ρ2 v2n − ρ1 v1n = 0 (6.19) where the square brackets denote the change in the enclosed quantity across the boundary. Similarly, (6.14) leads to #  " γp Bn 1 2 B2 B · v = 0. (6.20) vn − ρv + + 2 γ − 1 µ0 µ0 Copyright © 2005 IOP Publishing Ltd.

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Equilibrium and steady-state conditions

The momentum conservation equation (6.15) is a vector equation and the momentum flux is a second-rank tensor. Let nˆ k be a unit vector normal to the boundary. Integrating this equation over the small volume in the same way leads to a vector condition # " Bj B2 nˆ j − nˆ k Bk pnˆ j + ρv j nˆ k vk + 2µ0 µ0 "  # B2 B ≡ p+ Bn + ρvvn = 0 nˆ − (6.21) 2µ0 µ0 The components of this normal and parallel to the boundary are # " Bt2 − Bn2 2 =0 p + ρvn + 2µ0   Bn B t ρvn v t − = 0. µ0

(6.22) (6.23)

Gauss’ law for magnetic fields implies that [Bn ] = 0

(6.24)

and Faraday’s law gives the vector condition [nˆ k Bk v j − nˆ k vk B j ] ≡ [Bn v − vn B] = 0

(6.25)

[Bn v t − vn B t ] = 0.

(6.26)

or The equations (6.19), (6.20), (6.22), (6.23), and (6.26) are the de Hoffmann– Teller relations [52]. They may be manipulated to give a set of conditions which must apply at any sharp plane discontinuity in the plasma: •

• •

continuity of normal component of magnetic field, [Bn ] = 0

(6.27)

[ρvn ] = 0

(6.28)

conservation of mass flux, continuity of tangential components of electric field, Bn [v t ] = [vn B t ]

•

(6.29)

conservation of tangential components of momentum, ρvn [v t ] =

Copyright © 2005 IOP Publishing Ltd.

Bn [ Bt ] µ0

(6.30)

Discontinuities and shocks •

•

conservation of normal component of momentum, " # Bt2 2 p + ρvn + =0 2µ0 conservation of energy, " # 1 2 Bn Bt2 γp = v + + ρvn [ B t · v t ]. 2 (γ − 1)ρ µ0 ρ µ0

117

(6.31)

(6.32)

6.5 Discontinuities and shocks 6.5.1 Classification of discontinuities A number of different types of discontinuity can occur in MHD flows [119, sections 70–3]. The boundary conditions of the previous section were derived for the general MHD equations and so discontinuities need not be stationary but may change shape as the plasma flows. In problems of wave propagation, the shape changes on a time scale long compared with the wave period and so, viewed from a frame of reference fixed in the discontinuity, we may consider the flow as steady-state flow. Such discontinuities may be classified into several types. This classification can be carried out on the basis of conditions (6.27), (6.28), (6.29), and (6.30) which may be written in the form Bn1 = Bn2 = Bn ρ1 vn1 = ρ2 vn2 = j Bn (v t2 − v t1 ) = B t2 vn2 − B t1 vn1 Bn j (vt2 − v t1 ) = (B t2 − B t1 ) µ0

(6.33) (6.34) (6.35) (6.36)

where j is the normal component of the mass flux ρv. Now take the vector product of (6.35) with (6.36). The result may be written in the form (6.37) Bn (vn2 − vn1 )B t2 × B t1 = 0. The three factors in this equation correspond to three cases: (i) The magnetic field is parallel to boundary, i.e. Bn = 0. Equations (6.35) and (6.36) require that vn = 0 and the flow is parallel to the boundary. This is called a tangential discontinuity. (ii) The normal component of the velocity is continuous, i.e. vn1 = vn2 = vn . It follows from (6.34) that the density is continuous. This is called a rotational or Alfv´en discontinuity. (iii) The transverse components of the magnetic fields on either side of the discontinuity are collinear because B t2 × B t1 = 0. There are two separate cases: Copyright © 2005 IOP Publishing Ltd.

Equilibrium and steady-state conditions

118

(a) If, in addition to the collinearity of the transverse components of the fields, there is no flow through the boundary (vn = 0), we have a contact discontinuity. (b) If there is flow through the boundary, we have a shock. The detailed properties of these various discontinuities are discussed later. 6.5.2 Tangential discontinuity If the magnetic field is parallel to boundary, then Bn = 0. In this case, (6.35) and (6.36) require that vn = 0 and the flow is parallel to the boundary. All the boundary conditions are automatically satisfied except (6.31) which requires that " # B2 p+ = 0. (6.38) 2µ0 The discontinuities in the tangential velocity and magnetic field and in the density are arbitrary. Such a boundary is called a tangential discontinuity. 6.5.3 Rotational or Alfv´en discontinuity If the normal component of the velocity is continuous, i.e. vn1 = vn2 = vn , it follows from (6.34) that the density is continuous. Then (6.35) and (6.36) become Bn [v t ] − vn [ B t ] = 0 Bn [ B t ] = 0. vn [v t ] − µ0 ρ

(6.39) (6.40)

These equations are satisfied if B t and v t are continuous across the boundary. However, from (6.31), p is then also continuous; i.e. we have the trivial case where all variables are continuous and there is no boundary! For a non-trivial solution, the determinant of the coefficients must be zero so that vn2 = In this case,

Bn2 . µ0 ρ

[ Bt ] . [v t ] = √ µ0 ρ

(6.41)

(6.42)

Now (6.31) shows that "

. Copyright © 2005 IOP Publishing Ltd.

B2 p+ t 2µ0

# = 0.

(6.43)

Discontinuities and shocks

119

The terms in (6.32) can be grouped as follows. "   #  1 2 p 1 Bt2 Bt2 p 2 Bt · vt 2 ρvn + =0 v + + + + vt − √ 2 n ρ(γ − 1) ρ 2µ0 ρ 2 µ0 ρ µ0 ρ (6.44) or # "     1 2 1 p Bt2 1 p+ v + + 2 n ρ γ −1 ρ 2µ0     Bt 1 Bt − vt · √ − vt + = 0. (6.45) √ 2 µ0 ρ µ0 ρ Equations (6.41), (6.43), and (6.42) show that the first, third, and fourth terms of this equation are zero. Thus, the second term is also zero and the pressure is continuous across the boundary. Therefore, from (6.43), the magnitude of the transverse component of the magnetic field is continuous. The result is that, in this case, the magnitudes, but not the directions, of all the parameters are continuous: [Bt ] = 0. (6.46) The angle made by B and v with the normal nˆ to the surface is continuous. The planes containing the vectors ( B, n) ˆ and (v, n) ˆ are rotated with angles of rotation related in such a way that (6.42) is satisfied. These relationships hold in any frame of reference in which the boundary is at rest. We can transform to a frame in which the velocity component perpendicular to the plane containing B and nˆ is zero on one side of the boundary. Since the angles made by B and v with the normal to the boundary are continuous and their magnitudes are also continuous, this implies that, in this frame, B, v, and nˆ are coplanar on either side of the boundary. The only quantity that changes across the boundary is the orientation of this plane, which is rotated about the normal. If we further transform to a √ frame of reference moving with velocity v + B t / µ0 ρ, the magnetic field and the velocity are parallel on each side of the boundary. Such a boundary is called a rotational discontinuity. It is also possible to transform to a frame of reference in which the normal component of velocity on either side of the boundary is zero. In this frame, the boundary moves so that the velocity component normal to its plane is v = √ −Bn / µ0 ρ. Its intersection with the magnetic field moves along the magnetic field on either side of the boundary with speed [ρ] = 0

[ p] = 0

[vn ] = 0

[vt ] = 0

B . VA = √ µ0 ρ

[Bn ] = 0

(6.47)

This is called the Alfv´en speed, derived by Alfv´en [2] as the characteristic speed for wave motion in the plasma. Rotational discontinuities are sometimes called Alfv´en discontinuities. Copyright © 2005 IOP Publishing Ltd.

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Equilibrium and steady-state conditions

6.5.4 Contact discontinuities If B t1 and B t2 are collinear and, in addition, vn = 0 then, because Bn is also continuous, the total field B is continuous across the boundary. Equation (6.29) shows that the change in v t and, hence, in v is zero so that we can view the boundary from a reference frame in which the plasma is at rest. Equation (6.32) is automatically satisfied and (6.31) shows that the pressure is continuous across the boundary. The only MHD variable which changes across the boundary is the density which may have a discontinuity of arbitrary size. Thermodynamic variables on either side of the boundary may be different. Such a discontinuity is called a contact discontinuity. Two plasmas with different densities are in contact and in mechanical equilibrium. The magnetic field is uniform. 6.5.5 MHD shocks If B t1 and B t2 are collinear, as required if in (6.37) the factor B t1 × B t2 = 0 and if also vn = 0, we have the conditions for an MHD shock. The plane containing B 1 and B 2 is normal to the boundary. From (6.36), it can be seen that [v t ] lies in this plane. The component of v t normal to this plane is continuous across the boundary so that we can transform to a reference frame in which it is zero and B 1 , B 2 , v 1 , v 2 , and nˆ are all coplanar. We may eliminate v t2 − v t1 from equations (6.35) and (6.36), note that vn = j/ρ, and use the magnitudes of B t1 and B t2 since they are parallel. The result is   B2 2 Bt j (6.48) = n [Bt ]. ρ µ0 Equation (6.32) may be written in the form         γp 1 1 1 Bn Bn + j2 2 + vt − Bt · vt − Bt (γ − 1)ρ 2 2 µ0 j µ0 j ρ " # 2 2 1 Bt Bn + [Bt2] = 0. − µ0 ρ 2µ20 j 2 Equation (6.30) shows immediately that the third term in this equation is zero. By using (6.31), the second term may be written in the form # # "   "  1 2 1 1 1 1 1 [1/ρ 2 ] Bt2 Bt2 j + =− (6.49) =− p+ p+ 2 2 [1/ρ] 2µ0 2 ρ1 ρ2 2µ0 ρ2 and, from (6.48), the last term may be written in the form   Bn2 Bt [Bt2/ρ] 2 2 [B (Bt2 + Bt1 ). [B ] = ] = t 2µ0 [Bt ] t 2µ0 ρ 2µ20 j 2 Copyright © 2005 IOP Publishing Ltd.

(6.50)

Summary After a little manipulation, the condition then becomes       1 1 p 1 2 + ( p1 + p2 ) + [Bt ] = 0. (γ − 1)ρ 2 ρ 4µ0 ρ

121

(6.51)

From (6.49), we get # "   1 Bt2 =− p+ j ρ 2µ0 2

(6.52)

and, finally, we repeat (6.36): j [v t ] =

Bn [ B t ]. µ0

(6.53)

Equations (6.48), (6.51), (6.52), and (6.53) form a complete set describing the transition across the shock. They are necessary conditions if the shock exists. They are not sufficient to determine whether the shock can exist or not. This requires an analysis of the evolution of the shock when it undergoes a small perturbation. Such an analysis needs an understanding of the waves which can be propagated on either side of the shock front. We therefore defer it until chapter 22.

6.6 Summary • • •

• • •

In the steady state, at each point in the medium, conditions do not change with time. If, in addition, the velocity is zero everywhere, the medium is in equilibrium. In equilibrium, the force J × B is everywhere equal to the gradient of the pressure. This can be expressed in terms of a stress tensor which shows that the magnetic field forces are equivalent to a magnetic pressure of magnitude B 2 /µ0 ρ normal to the magnetic field and a magnetic tension B 2 /µ0 R parallel to the magnetic field, where R is the radius of curvature of the field lines. The conditions for steady-state flow can be found by setting ∂/∂t = 0 in the MHD equations. At a discontinuity in the plasma boundary, conditions which must be obeyed can be found by integrating the various MHD continuity equations over a coin-shaped volume enclosing a portion of the boundary. The different types of discontinuity which can occur may be classified as contact discontinuities, tangential discontinuities, rotational discontinuities, and shock fronts.

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Chapter 7 Harmonic plane waves in a uniform loss-free plasma

7.1 Introduction The reduced MHD equations (2.55), (2.56), (2.57), and (2.58) are nonlinear and difficult to solve in general. They can be simplified in cases where an equilibrium state suffers a small time-dependent perturbation. The equations can then be expressed as a linear second-order wave equation. In this chapter, we consider the perturbations of a stationary uniform medium. Propagation of waves in such a medium is anisotropic because of the uniform magnetic field. The wave speed is a function of the angle between the wave normal and the magnetic field. There are generally three characteristic waves that can be propagated. These are classified as (i) the fast magnetosonic wave, which is compressional and roughly isotropic, (ii) the slow magnetosonic wave, which is compressional and is propagated with energy flux approximately parallel to the magnetic field, and (iii) The transverse Alfve´ n wave, which has energy propagated exactly along the field and is incompressional. We derive dispersion relations for these waves and discuss the relationships between their field components. A word of caution is necessary. All the results in this chapter are derived on the assumption that the MHD approximation is valid everywhere. This approach shows that a variety of undamped waves can exist in such a medium. In some cases, a more detailed treatment, based on the Boltzmann–Vlasov equations, shows that kinetic effects can lead to significant damping of some of these waves. It is, nevertheless, useful to have a clear understanding of the predictions of MHD about the nature of these waves. This point is taken up again in chapter 8. 122

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Wave equations

123

7.2 Wave equations 7.2.1 Wave equation for a non-dispersive medium The simplest wave equation describes waves in a uniform non-dispersive medium. Some property of the medium ψ(x, t) is a function of position x and time t obeying the equation 1 ∂ 2ψ (7.1) ∇2 ψ = 2 2 v ∂t where ψ is a function of x, y, z, and t and v is a constant, having the dimensions of speed and depending on the properties of the medium. Consider the case where ψ varies only in one direction. We can choose this to be the x-direction. Then (7.1) becomes 2 ∂ 2ψ 2∂ ψ = v . ∂t 2 ∂x2

(7.2)

Any arbitrary function of x ± ct, ψ(x ± ct), satisfies this equation, showing that any disturbance is propagated in the positive or negative x-direction with speed v. It is this property, that an arbitrary disturbance is propagated unchanged in form with a fixed speed v, which leads to the medium being described as nondispersive. Finding a general solution of the wave equation for particular initial conditions is complicated. A useful technique is to represent the wave as a Fourier integral in space. This is a synthesis of plane harmonic waves. Because an arbitrary wave can be expressed as such a synthesis of plane waves, wave propagation is usefully studied by investigating the properties of the plane waves that make up the spectrum. The expression for a real plane harmonic wave of angular frequency ω, propagated in the positive x-direction, is of the form   2π (7.3) ψ(x − vt) = A cos (x − vt) + φ = A cos (kx − ωt + φ) λ where k = 2π/λ and ω = vk. Wave equations of the form (7.2) are often solved by assuming a harmonic solution of the form ψ = exp{i(kx − ωt)} (7.4) where it is understood that the physical wave is described by the real part of this expression. This is a solution of (7.2) provided that ω = ±v. k

(7.5)

This is the simplest form of a so-called ‘dispersion relation’ giving a relationship between ω and k. Actually, in this case, the wave velocity v is independent of Copyright © 2005 IOP Publishing Ltd.

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Harmonic plane waves in a uniform loss-free plasma

frequency and wavelength, so that there is no dispersion. In more complicated cases, this is not so. In elementary books, the representation of a real physical wave by a complex expression is generally accepted uncritically. It is worth examining this representation in more depth. The Fourier transform of a real physical time series has special symmetry properties (appendix C.2.4). The Fourier transform of its even part is real and even and that of its odd part is imaginary and odd. The redundancy of information resulting from this symmetry provides full knowledge of the spectrum from the positive frequency values. If the negative frequency values are set equal to zero and the inverse Fourier transform evaluated, the result is a complex function from which the original real function can be reconstructed. This complex function is the analytic signal [29, p 269]. In the time domain, it can be constructed by subtracting i times its Hilbert transform from it. Restoration of the negative frequency values in the frequency domain by using the symmetry properties is equivalent to discarding the imaginary part in the time domain. The result is that, for linear operations, one can work with the complex function. In the special case of the harmonic wave (7.3), the analytic signal is A{cos(kx − ωt + φ) + i sin(kx − ωt + φ)} = Aeiφ exp{i(kx − ωt)}. In this chapter, we shall deal largely with harmonic waves represented by complex exponentials. Later we shall construct more complicated waveforms by Fourier syntheses of such plane harmonic waves. In so doing, it will be understood that the argument of the Fourier integral is constructed with the appropriate symmetries to provide a real time series. 7.2.2 Dispersive media A more general wave equation takes the form  ∂2 ∂ 2ψ 1 2 = 2 −  ψ. ∂x2 v ∂t 2

(7.6)

An example is the equation describing plane electromagnetic waves in a cold isotropic plasma [31, 233], where ψ represents the transverse electric field,  = Ne2 /0 m is the plasma frequency, N is the electron density, and e and m the charge and mass of the electrons. In this case, it is not possible to find a solution which is propagated unchanged in form. We see later that different frequency components of the Fourier spectrum are propagated with different speeds so that the shape of disturbance changes with time. Substitute the assumed solution ψ = A exp{i(ωt − kx)} in (7.6). The resulting dispersion relation is (ω2 − 2 − k 2 v 2 )ψ = 0.

(7.7)

This only has a non-trivial solution if

$ ω = ± k 2 v 2 + 2 .

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(7.8)

Simple examples of waves

125

The wave is propagated in the positive x-direction with a speed v=

cω ω = ±√ 2 k ω − 2

(7.9)

and this speed depends on the frequency ω. We shall see later that MHD waves are highly anisotropic but are nondispersive: the wave speed of a plane wave depends strongly on the direction of propagation but is independent of frequency.

7.3 Simple examples of waves 7.3.1 Waves on strings and in gases Wave propagation in an elastic medium, initially in equilibrium, occurs when the medium is perturbed. The simplest illustration is a wave on a stretched string. In elementary books, it is shown that, if the string is given a small transverse displacement y, the restoring force on a length of the string δx, having mass δm, depends on the local curvature d2 y/dx 2 and the tension T : F=T

d2 y δx. dx 2

(7.10)

The equation of motion is then d2 y T δx d2 y T d2 y = = δm dx 2 µ dx 2 dt 2

(7.11)

where µ is the mass per unit length of the string. This is, of course, the onedimensional wave equation, with speed $ (7.12) v = T /µ. The tensile stress in the string is T /A where A is the cross-sectional area of the string. This expression for the wave speed can, therefore, be written as the square root of the ratio of the tensile stress to the density. Similarly, in a gas, compressional sound waves occur when the medium is perturbed. Consider a perturbation ξ(x, t), parallel to the x-axis, which is a function of position and time. Consider a thin slab, bounded by faces initially at x and x + w in the unperturbed medium. When the slab is perturbed, its density changes from ρ0 to ρ0 +ρ and the pressure changes from P to P + p. Pressure and density are related through an appropriate equation of state that specifies d p/dρ. To first-order in ξ , the faces of the slab move to x + ξ(x, t), x + w + ξ(x + w, t). The change in volume V of the slab, of cross section A, is then δV = A{w + ξ(x + w, t) − ξ(x, t)} − Aw  Copyright © 2005 IOP Publishing Ltd.

dξ V. dx

(7.13)

Harmonic plane waves in a uniform loss-free plasma

126

The mass of the slab is constant so ρ dξ dV =− =− ρ0 V dx    dp dξ dp ρ=− ρ0 . dρ 0 dρ 0 dx The restoring force on the slab is

(7.14)



so that

p=

( p + δp)A − p A =

dp w A. dx

(7.15)

(7.16)

Its equation of motion is, therefore, ρ0 V

  dp d2 ξ = − V. 2 dρ 0 dt

Equation (7.15) may be differentiated and substituted in this yielding   2 d2 ξ dp d ξ = . dρ 0 dx 2 dt 2

(7.17)

(7.18)

For adiabatic conditions, (d p/dρ)0 = γ P/ρ0 and this represents√a compressional sound wave propagated in the x-direction with the sound speed γ P/ρ0 . Again the square of the wave speed is the ratio of the stress, in this case the pressure, to the density. Both the transverse wave on a string and the compressional wave in a gas have wave speeds determined by the ratio of the stress to the density. In both these cases, an initial approximation was made that the perturbation was small. In general, the equation of motion of a distorted elastic medium is nonlinear. The assumption of small perturbations of the undisturbed medium allows us to obtain a linear wave equation. Most of this book is concerned with the behaviour of linear MHD waves. Nonlinear wave problems, while important, are often intractable. An up-to-date treatment of topics in nonlinear MHD wave theory is given by Biskamp [28]. 7.3.2 Simple transverse Alfv´en waves In section 1.8, we have shown that the electromagnetic forces can be expressed in terms of a stress tensor. This tensor provides an effective tension per unit crosssectional area B 2 /µ0 parallel to the field, as shown by (1.68). Suppose that we have a uniform plasma of density ρ and it is perturbed perpendicular to the field lines. The perturbation y is small. The plasma is, of course, frozen in to the magnetic field. The restoring force on an element of a flux tube of cross section δ A and length δx is to first-order, by analogy with (7.10) F=T Copyright © 2005 IOP Publishing Ltd.

B 2 d2 y d2 y δx = δ A 2 δx dx 2 µ0 dx

(7.19)

General wave equation for MHD waves

127

where B is the magnitude of the unperturbed magnetic field. The equation of motion may then be written in the form ρ

∂2y B2 ∂2 y = µ0 ∂ x 2 ∂t 2

(7.20)

which is a wave equation describing propagation along the magnetic field direction with speed  B2 . (7.21) VA = µ0 ρ This is called the Alfv´en speed [2]. Its square is the tension per unit area, divided by the density. The transverse wave travelling with this speed along the magnetic field is entirely analogous to a wave on a stretched string of cross section A and density ρ under stress T /A. 7.3.3 Simple compressional Alfv´en waves We have seen that sound waves are compressional waves, with displacement parallel to the direction of propagation. Generally the compression is adiabatic. Plane waves obey the one-dimensional wave equation (7.18) with speed VS = √ γ p/ρ. Consider now a low-pressure plasma. If we consider compression at right angles to the field, then from (1.69) the magnetic pressure is B 2 /2µ0 . There are two degrees of freedom for perturbation perpendicular to the field so that, effectively, γ = (n + 2)/n = 2. The perturbation thus gives rise to a compressional wave, analogous to a sound wave, with a wave speed equal to the Alfv´en speed. When the thermal pressure is comparable to the magnetic pressure and propagation is at an arbitrary angle to the magnetic field direction, the situation is more complicated. The remainder of this chapter is devoted to waves of this type.

7.4 General wave equation for MHD waves 7.4.1 Linearization of the MHD equations The MHD equations are nonlinear and general solutions of them are difficult. We can, however, study waves with small amplitudes by expressing the wave amplitudes as small perturbations and neglecting second-order terms in these perturbations. If the perturbations of the medium are small, we can linearize the reduced MHD equations by neglecting terms of second order in the perturbation variables. We make the following substitutions:

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ρ → ρ0 + ρ v→v

(7.22) (7.23)

B→ B+b

(7.24)

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Harmonic plane waves in a uniform loss-free plasma p→P+p

(7.25)

where ρ0 , P, and B are unperturbed variables describing the equilibrium state and ρ v, p, and b are small perturbations of these quantities. The unperturbed velocity V is, of course, zero. If these are substituted into the reduced MHD equations, (2.55), (2.57), (2.58), and products of the small first-order quantities ignored, we get the following set of first-order linearized equations:   B ∂v B·b · ∇b (7.26) + = −∇ p+ ρ0 ∂t µ0 µ0 ∂p = − γ P∇ · v (7.27) ∂t ∂b = − B∇ · v + B · ∇v. (7.28) ∂t It is sometimes convenient to write (7.26) in terms of v ⊥ and v , the components of the velocity perpendicular and parallel to the magnetic field:   ∂v ⊥ B·b ρ0 = − ∇⊥ p + (7.29) ∂t µ0 ∂v = − ∇ p. ρ0 (7.30) ∂t The first-order density ρ does not appear in equations (7.26), (7.27), and (7.28), so that the linearized form of (2.56) is unnecessary; it serves only to determine ρ after the equations have been solved. Similarly E and j can be found from the linearized forms of (2.54) and (2.52). This set of subsidiary equations is ∂ρ = − ρ0 ∇ · v ∂t E = −v× B 1 j= ∇ × b. µ0

(7.31) (7.32) (7.33)

Here E is the perturbation field and E 0 = 0. 7.4.2 Wave equation If (7.26) is differentiated with respect to time and (7.27) and (7.28) used to eliminate p and B, we get an equation of second order:     ∂ 2v B B2 ρ0 2 = ∇ γP+ · (B · ∇)v ∇·v −∇ µ0 µ0 ∂t − Copyright © 2005 IOP Publishing Ltd.

1 (B · ∇){B∇ · v − (B · ∇)v}. µ0

(7.34)

Harmonic waves

129

This equation makes no assumptions about the uniformity of the medium. The zero-order quantities are independent of time but may be functions of position. If, in addition, we assume that the zero-order quantities do not vary in space, we get a wave equation for a uniform medium: ∂ 2v = (VA2 + VS2 )∇(∇ · v) − ∇(V A · ∇)(V A · v) ∂t 2 − V A (V A · ∇)(∇ · v) + (V A · ∇)(V A · ∇)v 

where B VA = √ µ0 ρ0

VS =

γP ρ0

(7.35)

(7.36)

and have the dimensions of velocity. The quantity V A is called the Alfv´en velocity and VS will be recognized as the speed of sound. This can be written in tensor form. It is of the same form as the equation for waves in an elastic medium [117]: ∂ 2 vl ∂ 2 vi = Ci j kl 2 ∂ x j ∂ xk ∂t

(7.37)

where Ci j kl = (VA2 + VS2 )δi j δkl − δi j VA,k VA,l − δkl VA,i VA, j + δil VA, j VA,k

(7.38)

It is this wave equation that must be solved in order to find the properties of the most general plane MHD waves in a uniform medium. It is often useful to work in terms of the displacement perturbation of the plasma rather than the velocity perturbation. We define ξ = v dt and note that any time-independent constant of integration would not form part of the timevarying perturbation. Then the general wave equation may be written in the form     B B2 ∂ 2ξ γP+ · (B · ∇)ξ ∇·ξ −∇ ρ0 2 = ∇ µ0 µ0 ∂t −

1 (B · ∇){B∇ · ξ − (B · ∇)ξ } µ0

(7.39)

b = −B∇ · ξ + B · ∇ξ .

(7.40)

while (7.28) becomes

7.5 Harmonic waves In the remainder of the chapter, we shall study the properties of sinusoidal plane waves that are solutions of (7.37). By a plane wave, we mean one in which the surfaces of constant phase at any instant are planes. The normal to these planes, in the direction of advancing phase, is the wave normal. Since the Copyright © 2005 IOP Publishing Ltd.

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Harmonic plane waves in a uniform loss-free plasma

medium is anisotropic, there are two natural directions, that of the magnetic field and that of the wave normal. There are, thus, two suitable natural Cartesian coordinate systems; which one is chosen depends on the nature of the problem. We shall describe both and find expressions for the relationships between the field components in each of them for future reference. The properties of a uniform stationary MHD medium are cylindrically symmetric about the direction of the magnetic field. In studying such a medium, it is convenient to use a coordinate system in which the plane of incidence, defined as the plane containing B and k, forms one coordinate plane. This reduces the problem to a two-dimensional one. We shall use two such systems of righthanded rectangular coordinates. In either case, the y-axis is normal to the plane of incidence. In magnetic-field-aligned coordinates, x, z, the magnetic field is in the z-direction. In wave-normal-aligned coordinates x L , x T , the wave normal is in the direction of x L , the longitudinal component, and x T is transverse to the wave normal. We shall use subscripts  and ⊥ to denote components of vectors parallel and perpendicular to the magnetic field and subscripts L and T to denote components of vectors parallel (longitudinal) and perpendicular (transverse) to the wave normal. If θ is the angle measured from B to k, then the relationship between these two sets of coordinates is x L = x cos θ − z sin θ x T = x sin θ + z cos θ

x = x T cos θ + x L sin θ z = −x T sin θ + x L cos θ.

(7.41)

7.5.1 Equations for harmonic waves We assume a harmonic wave solution similar to (7.4). We do not, however, assume that the x-axis is in the direction of the wave normal. Instead we define the wavevector k, having the direction of the wave normal and a magnitude equal to the wavenumber 2π/λ. A complex plane harmonic wave then has time and space behaviour given by A exp{i(k · r − ωt)}

(7.42)

where A is a complex amplitude and the actual physical quantity is the real part of this representation. These plane waves advance with a velocity ω/k in the ˆ k-direction. We define the phase velocity ω v p = kˆ . k

(7.43)

The operators ∇ and ∂/∂t can then be replaced by ik and −iω so that (7.35) becomes {ω2 −(k·V A )2 }v−k(VA2 +VS2 )k·v +(k·V A ){k(V A ·v)+V A (k·v)} = 0. (7.44) In magnetic-field-aligned coordinates, in which the ambient magnetic field is in the z-direction and the wavevector k is in the x z-plane, this equation can be Copyright © 2005 IOP Publishing Ltd.

Harmonic waves

131

written in the form   −k x k z VS2 vx   v y  = 0. 0 2 2 2 vz ω − k z VS (7.45) It will be observed that v y is determined independently from vx and vz . If we transform this expression for the velocities to wave-normal-aligned coordinates, we get 

ω2 − k z2 VA2 − k x2 (VA2 + VS2 )  0 −k x k z VS2



ω2 − k 2 VA2 cos2 θ −k 2 VA2 sin2 θ

0 ω2 − k z2 VA2 0

k 2 VA2 cos2 θ 2 ω − k 2 (VA2 cos2 θ + VS2 )



vT vL

(ω2 − k 2 cos2 θ )v y = 0.

 = 0 (7.46) (7.47)

We see that the harmonic waves in either coordinate system are described by a set of simultaneous homogeneous algebraic equations. These equations only have a non-trivial solution if the determinant of their coefficients is zero. This condition gives a relationship between the angular frequency ω and the wavevector k and, thus, determines the wave speed. Such a relationship is called a dispersion relation.

7.5.2 Dispersion relations A great deal can be deduced about the properties of harmonic waves by studying the dispersion relation between the frequency and the wavevector. If the three components of k are given, then ω is determined. Alternatively, if ω and two of the components of k are given, the third component can be found. This means that, if two components of the phase velocity (7.43) are known, the third is determined. This allows a study of the geometrical optics of the wave. Much can be deduced without investigating the relationships between the field components in detail.

7.5.2.1 Dispersion relation in magnetic-field-aligned coordinates The dispersion relation for MHD waves in magnetic-field-aligned coordinates requires that the determinant of the coefficients of the three linear homogeneous equations (7.45) is zero. The determinant has two factors and, thus, there are two dispersion relations, one determining a wave in which the velocity perturbation is in the y-direction and the other one in which it is in the x z-plane. These are:

ω −k 4

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2

{VA2

+

VS2 }ω2

+

ω2 − k z2 VA2 = 0

(7.48)

= 0.

(7.49)

k 2 k z2 VA2 VS2

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Harmonic plane waves in a uniform loss-free plasma

7.5.2.2 Dispersion relation in wave-normal-aligned coordinates In wave-normal-aligned coordinates, these dispersion relations are

ω −k 4

2

{VA2

+

VS2 }ω2

ω2 − k 2 VA2 cos2 θ = 0

(7.50)

VA2 VS2 cos2 θ

(7.51)

+k

4

= 0.

Let us examine the dispersion relation (7.48). It clearly has an unusual property in that it is independent of the component of k normal to the magnetic field. The implication is that k x and k y may be arbitrarily fixed and only k z and ω are related. Thus, the phase velocity, which has components ω/k x , ω/k y , and ω/k z , can have any value provided that its component along the z-direction is equal to VA , the Alfv´en speed. The other dispersion relation is quadratic in ω2 . It therefore describes two waves with different phase velocities. The remainder of this chapter will be spent in studying the properties of these waves. 7.5.3 Phase velocity As has been noted earlier, the dispersion relations determine a relationship between the components of the phase velocity. If we divide the two equations (7.50) and (7.51) by k 2 and k 4 , respectively, we get vp2 = VA2 cos2 θ vp4

− {VA2

+

VS2 }v 2p

+

VA2 VS2 cos2 θ

= 0.

(7.52) (7.53)

The second of these can be solved to give   ' vp2 = 12 (VA2 + VS2 ) ± (VA2 + VS2 )2 − 4VA2 VS2 cos2 θ .

(7.54)

Note: The dispersion relation also provides the following relationship which is useful for algebraic simplification: vp2 − VA2 cos2 θ VA2 sin θ cos θ

=

VA2 sin θ cos θ vp2 − VA2 sin2 θ − VS2

.

(7.55)

7.5.3.1 The transverse Alfv´en wave The wave associated with the dispersion relation (7.48) and with a phase velocity given by (7.52) is called the transverse or shear Alfv´en wave because, as previously noted, the velocity perturbation is in the y-direction which is perpendicular to the plane containing the wavevector k and the magnetic field B. It is highly anisotropic. The component of the phase velocity along the magnetic field is always equal to the Alfv´en speed, no matter what the direction of the wave Copyright © 2005 IOP Publishing Ltd.

Harmonic waves

133

Figure 7.1. Transverse Alfv´en waves propagated in different directions.

normal. It is non-dispersive, in the sense that the phase velocity is independent of frequency. Figure 7.1 schematically shows three such plane waves, represented by parallel lines at the wavecrests. They have different directions but the intersection of the wavefronts with a magnetic field line moves along the magnetic-field direction with the Alfv´en speed. The distance between wavecrests measured along the field is the same for each. 7.5.3.2 The magnetosonic waves The dispersion relation (7.51) gives rise to two different characteristic waves with phase velocities given by (7.54). These relations depend on both the Alfv´en speed and the sound speed. They are, therefore, called the magnetosonic waves. Examination of the expression for the phase velocities shows that one of these waves always has a phase speed which is greater than that of the other. They are, therefore, appropriately named the fast and slow magnetosonic waves. Limiting cases: We briefly consider some special cases of the magnetosonic waves. Case (i): Propagation parallel to the field. When θ = 0◦ , the expression for the phase velocity of the magnetosonic waves (7.54) becomes v p = VA

or

VS .

(7.56)

One of the waves is propagated along the field with the Alfv´en speed and the other with the sound speed. Which is the fast and which the slow wave depends on the magnitudes of the two characteristic speeds. Case (ii): Propagation perpendicular to the field. When θ = 90◦ , the fast wave has a velocity ' vp =

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VA2 + VS2

(7.57)

134

Harmonic plane waves in a uniform loss-free plasma and the slow wave has zero velocity.

Case (iii): VA VS . The square root in (7.54) may be expanded to first order in VS /VA . The phase velocities of the two waves are given by vp = VA

or

VS cos θ.

(7.58)

The fast wave now has a phase speed which is equal to the Alfv´en speed, no matter what the direction of propagation. It is called the isotropic Alfv´en wave. In many parts of the magnetosphere, the pressure is negligible and we need only take account of the transverse and isotropic Alfv´en waves. The slow wave is a sound wave, propagated with a much smaller velocity but constrained to the magnetic-field direction. It has similar properties to the transverse Alfv´en wave. This is because the plasma cannot have a compressional velocity perpendicular to the magnetic field: the plasma is frozen to the field lines and does not have sufficient energy to distort them. Case (iv): VS VA . In a similar fashion, vp = VS

or

VA cos θ.

(7.59)

The fast wave is now an isotropic sound wave in a gas, unaffected by the presence of the weak magnetic field. The slow wave now has the same phase velocity as the transverse Alfv´en wave. It can only be propagated parallel to the magnetic field. 7.5.4 Refractive index The phase velocity is not the most convenient quantity to use for describing waves. The wavevector k, proportional to its reciprocal, is more useful. More general results can be obtained by normalizing k: the normalized value of k is called the refractive index vector. In the case of isotropic waves, the refractive index is usually defined as (7.60) n = ck/ω = c/vp . This is the reciprocal of the wave speed normalized in terms of the speed of light. Anisotropic waves are more complicated. The phase speed is a function of direction of propagation. This can be treated by defining a refractive index vector ˆ n = ck/ω = kc/v p

(7.61)

whose direction is perpendicular to the wavefronts and whose magnitude is equal to the reciprocal of the normalized phase speed. MHD waves are anisotropic but this is not the most convenient normalization because the wave speeds are very much less than the speed of light. The refractive index then takes inconveniently large values. It is better to normalize the velocities in terms of some suitable characteristic speed. For example, in a uniform medium, Copyright © 2005 IOP Publishing Ltd.

Harmonic waves

135

this could be the Alfve´ n speed. In non-uniform media, the Alfve´ n speed varies with position. In such a case, we could use the value at a particular point, a boundary for example, for normalization. For MHD waves, we therefore define n = V0 k/ω = kˆ V0 /v p

(7.62)

where V0 is a characteristic speed which must be specified in each problem. Expressions for the refractive index follow directly from (7.52) and (7.54). If we write US = VS / V0 (7.63) UA = VA / V0 then the refractive index for the transverse Alfve´ n wave is n=±

1 UA cos θ

(7.64)

and that for the magnetosonic waves is n2 =

=

(UA2 + US2 ) ± (UA2 + US2 ) ∓

' '

2 (UA2 + US2 )2 − 4UA2 US2 cos2 θ (UA2 + US2 )2 − 4UA2 US2 cos2 θ

2UA2 US2 cos2 θ

.

(7.65)

In figure 7.2, the refractive index is plotted as a function of polar angle for each of the three waves and for the two cases for VA > VS and VA < VS . Cartesian axes are shown, with n x = n sin θ and n z = n cos θ . The magnetic field is in the z-direction. There is cylindrical symmetry and the curves are surfaces of revolution about the z-axis. Such surfaces are called refractive index surfaces. They can be regarded as surfaces in a three-dimensional refractive index space. The magnitude of the refractive index for any direction of propagation is the length of the vector in that direction from the origin to the surface. The Alfv´en and sound speeds have been normalized to the Alfv´en speed so that US = VS /VA , UA = VA /VA = 1. We shall see that representing the refractive index in this way gives a variety of methods for calculating the properties of anisotropic waves. For the moment, we briefly discuss their properties without going into detail about their applications. The surface for the transverse Alfv´en wave in each case is a pair of planes at unit distance from the origin. Different normalization would change this distance. The component of the refractive index vector in the magnetic field direction is always UA ≡ 1, no matter what the direction of propagation. In each case, the fast-wave surface is a closed surface. The magnitude of the refractive index vector depends only weakly on the polar angle and propagation is approximately isotropic. As we have ' seen, for propagation perpendicular to the magnetic field, the phase speed is Copyright © 2005 IOP Publishing Ltd.

VA2 + VS2 and the refractive index is

136

Harmonic plane waves in a uniform loss-free plasma

Figure 7.2. Refractive index surfaces for MHD waves. The refractive index vector is n = VA k/ω. Velocities are normalized to the Alfv´en speed: (a) VS < VA ; (b) VS > VA .

(UA2 + US2 )−1 . For propagation parallel to the field, the refractive index is UA−1 when VA > VS and US−1 when VA < VS . The slow-wave surface is an open surface and the wave is strongly anisotropic. There are two surfaces, roughly planar so that its properties are qualitatively similar to those of the transverse Alfv´en wave. For propagation along the magnetic-field direction, the refractive index is US−1 when VA > VS and UA−1 when VA < VS . Propagation perpendicular to the field cannot occur. For propagation parallel to the field, it can be seen that, when VA > VS , the fast-wave and transverse-Alfv´en-wave surfaces touch; and, when VS > VA , the fast- and slow-wave surfaces touch. In each case, there is a degeneracy in the waves for propagation along the field. When the sound speed is very small compared to the Alfv´en speed, the square root in (7.65) can be approximated using the binomial theorem. The expression for the refractive index becomes n2 

Copyright © 2005 IOP Publishing Ltd.

1 ∓ (1 − 2US2 cos2 θ/UA2 ) 2US2 cos2 θ

Harmonic waves 

1 UA2

or

1 US2 cos2 θ

.

137 (7.66)

In this case, the fast wave reduces to an isotropic wave in which the phase speed, for any direction of propagation, is equal to the Alfv´en speed. The slow wave is propagated exactly along the magnetic field with the sound speed: it acts rather like a sound wave in a pipe. If the sound speed is large compared with the Alfv´en speed, the reverse takes place. Examination of (7.65) shows that we can interchange UA and US getting n2 

1 US2

or

1 . UA2 cos2 θ

(7.67)

The fast wave becomes an isotropic sound wave. The slow wave is propagated exactly along the field with the Alfv´en speed. 7.5.5 Relations between field components Let us now discuss the relationship between the various field components in each type of wave. For harmonic waves, equations (7.27) and (7.28) may be written in the form γP k · v = ω−1 ρ0 VS2 k · v ω b = ω−1 {B(k · v) − (k · B)v}.

p=

(7.68) (7.69)

They may be evaluated if v is known. Equation (7.45) is a relationship between the components of v. In magnetic-field-aligned coordinates, the perturbations vx and vz are associated with the fast and slow waves, while v y is associated with the transverse Alfv´en wave. When the dispersion relation is satisfied, either of the two equations for vx and vz can be used. We choose the second which shows that vz =

k x k z VS2 ω2 − k z2 VS2

vx .

(7.70)

We can relate all quantities to vx and v y getting kz B vx ω kz B vy by = − ω kx B vx bz = ω ωk x vx . p=γP ω2 − k z2 VS2

bx = −

Copyright © 2005 IOP Publishing Ltd.

(7.71) (7.72) (7.73) (7.74)

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Harmonic plane waves in a uniform loss-free plasma

Equations (7.70), (7.71), (7.73), and (7.74), describing the magnetosonic waves, involve only the field variables vx , vz , bx , bz , and p. Equation (7.72) describing the transverse Alfv´en wave involves only v y and b y . In wave-normal-aligned coordinates, (7.46) provides a relationship between vL and vT vL =

v 2p − VA2 cos2 θ VA2 sin θ cos θ

vT =

VA2 sin θ cos θ v 2p − VA2 sin2 θ − VS2

vT .

(7.75)

The other field components for the magnetosonic waves may then be expressed in terms of vT or vL : bT = −

Bv p VA2 cos2 θ

vT

bL = 0 ρ0 VS2 vL p= vp

(7.76) (7.77) (7.78)

with E and ρ being found from the linearized forms of (2.52) and (2.56). The only field components for the transverse Alfv´en wave are v y and b y (with (2.52) giving E). They are related by by = −

B cos θ vy . vp

(7.79)

In the transverse Alfv´en wave, the perturbation is entirely in the y-direction, which is perpendicular to the plane containing the magnetic field and the wavevector. This is the meaning of the name ‘transverse’ Alfv´en wave. There is no compression, which would involve bz . The divergence of v is zero and, hence, the pressure perturbation p is zero. The plasma oscillates in a direction normal to the plane containing the unperturbed magnetic field and the wave normal and, since the magnetic field is frozen in, any magnetic field perturbation is normal to the unperturbed field lines. As in the simple case described in section 7.3.2, the oscillations are analogous to the oscillations of a stretched string. The magnetosonic waves, in contrast, are compressional. Since bz is nonzero, the magnitude of the magnetic field changes. Also the pressure perturbation is non-zero. The transverse motion and the transverse field perturbation are always in the plane containing the wavevector.

7.6 Waves for non-scalar pressure When the pressure is not a scalar, the wave equation is more complicated. Parker’s momentum equation (5.78) can be differentiated with respect to time and (2.58), Copyright © 2005 IOP Publishing Ltd.

Waves for non-scalar pressure

139

(5.81), and (5.82), used to give a wave equation of the form ∂ 2v ˆ  (∇ · v)} = VA2 {∇(∇ ⊥ · v  ) + ∇2 v − B∇ ∂t 2 P⊥ ˆ 2 v + ∇ (∇ · v)] + ∇2 v} {∇(∇ v ) + 2∇(∇ · v) − B[∇ + ρ0 P ˆ 2 + {4 B∇ v − ∇2 v}. (7.80) ρ0 If we adopt magnetic-field-aligned coordinates with the magnetic field in the z-direction and the wave normal in the x z-plane, and assume time and space variation of the form exp{i(k x x + k z z − ωt)}, we get   2 0 − 12 k x k z V⊥2 ω − k 2 VA2 (1 + 12 P ) − k x2 V⊥2   0 ω2 − k z2 VA2 (1 + 12 P ) 0 1 2 2 2 2 0 ω − k z V − 2 k x k z V⊥   vx ×  vy  = 0 (7.81) vz where k 2 = k x2 + k 2y 3P 3 = β VA2 ρ0 2 2P ⊥ = β⊥ VA2 V⊥2 = ρ0 V2 =

(7.82) (7.83) (7.84)

and the pressure anisotropy P is defined as the pressure difference, normalized with respect to the magnetic pressure B 2 /2µ0 :

P =

2µ0 (P⊥ − P ) . B2

(7.85)

If we define β⊥ and β as the ratios of the perpendicular and parallel pressures to the magnetic pressure, then

P = β⊥ − β . (7.86) The equation for v y is independent of those determining vx and vz . If P⊥  P , it approximates to the transverse Alfv´en wave. For non-scalar pressure, the transverse Alfv´en wave is modified. If the determinant of the matrix in (7.81) is set equal to zero, we get ω4 − ω2 {k 2 VA2 + k x2 V⊥2 + k z2 V2 + 12 k z2 VA2 P } + k2 V2 {k 2 VA2 + k x2 V⊥2 + 12 k z2 VA2 P } − 14 k x2 k z2 V⊥4 = 0 ω

2

− k z2 {1 + 12 P }VA2

Copyright © 2005 IOP Publishing Ltd.

= 0.

(7.87) (7.88)

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Harmonic plane waves in a uniform loss-free plasma

The dispersion relations for these waves is much more complicated than for the case of scalar pressure. In particular, they allow for the possibility of negative values of ω2 if P is large enough compared with P⊥ and the magnetic pressure. Such waves show unstable growth. For this reason, we shall defer much of the discussion to chapter 23. Here we shall consider only the case where, in the unperturbed plasma, P⊥  P . The implication is that, before the perturbation occurs, sufficient time has elapsed for the pressure tensor to approach Pδi j . The time scale of the disturbance is, however, short so that the Chew–Goldberger–Low relations (5.85) and (5.86) hold rather than the adiabatic law. If P⊥ = P = P, then (7.88) reduces to the dispersion relation for a transverse Alfv´en wave and need not be considered further. The dispersion relation (7.87) for the other pair of waves becomes     2P 3P 3P 2 2 5P ω4 − ω2 k 2 VA2 + k x2 + k z2 k VA + k x2 = 0. (7.89) + k z2 ρ0 ρ0 ρ0 3ρ0 This again represents two compressional waves, with the perturbation in the plane containing the wave normal and magnetic field. First, we consider propagation parallel and perpendicular to the magnetic field. For propagation parallel to the field, k x = 0 and the dispersion relation becomes (7.90) ω4 − ω2 k z2 (VA2 + V2 ) + k z4 VA2 V2 = 0. Here V2 is given by (7.83) and has the form of a sound speed in a medium with n, the number of degrees of freedom, equal to unity, so that γ = (n + 2)/n = 3. The two solutions are ω2 = k z2 VA2

or

ω2 = k z2 V2 .

(7.91)

This is very similar to the wave in a medium with scalar pressure. The only difference is the value of γ . For propagation parallel to the field, there are two characteristic waves, one propagated with the Alfv´en speed and the other with the sound speed given by (7.83). For propagation perpendicular to the field, k z = 0. The constant term in the quadratic for ω2 is zero and, thus, one of the roots is ω = 0. The other is ω2 = k x2 (VA2 + V⊥2 ).

(7.92)

This is again similar to the perpendicularly propagated wave in a medium with scalar pressure. The effective sound speed V⊥ is given by (7.84) and is that corresponding to that in a medium with two degrees of freedom, n = 2, so that γ = (n + 2)/n = 4. We now consider the nature of the dispersion relation at an arbitrary angle to the field. The dispersion relations are given by (7.87) and (7.88). It is easy to show that the discriminant of the quadratic equation (7.87) is always positive. It can, therefore, never have complex roots. If, in addition, P = 0, then both Copyright © 2005 IOP Publishing Ltd.

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141

roots ω2 are positive. In this case, two magnetosonic waves and the transverse Alfve´ n wave are propagated for all values of k . If P = 0, then the possibility arises of negative roots for the quadratic, with corresponding purely imaginary values of ω. This is associated with instability. For this reason, as in the cases of propagation perpendicular and parallel to B , we shall defer the discussion of waves with P = 0 to chapter 23. The remainder of this section is devoted to a comparison of waves for which the unperturbed pressure has P = 0 with those in a thermalized plasma with scalar pressure. When P = 0, the relation (7.88) is identical to the dispersion relation for the transverse Alfve´ n wave. This wave needs no further discussion. We will rewrite the dispersion relationship for the magnetosonic waves (7.87) as an equation for the refractive index vector. This has x and z -components: n x = k x VA /ω

and

n z = k z VA /ω

(7.93)

When β⊥  β , we can write the dispersion relation as a quadratic in n 2z : 6β n 4z − n 2z {n 2x [6 + 5β] − 6β − 4} + 4{1 − n 2x (1 + β)}.

(7.94)

Here we have used (7.83) and (7.84) to express V and V⊥ in terms of VA . This version of the dispersion relation is normalized and applies to all frequencies. It is independent of frequency, as is the case for scalar pressure. It is, however, strongly dependent on the angle between the wave normal and the magnetic field. We can plot n z against n x to get a refractive index surface. Some results are shown in figure 7.3. Each of the panels of this figure corresponds to a different value of β = β = β⊥ . The curves represent surfaces in refractive index space, which are surfaces of revolution about the n z -axis. They are the intersection of the surfaces with the plane n y = 0. The full curves represent the magnetosonic waves for waves having the dispersion relation (7.87). The broken curves are the corresponding surfaces for scalar pressure given by (7.65). The chain curves correspond to n z = 1, which is the refractive index surface for the transverse Alfv´en wave. Before discussing each panel of figure 7.3 in detail, we discuss some general features of the comparison between waves for scalar and non-scalar pressure. First, compare the wave speeds for propagation perpendicular √ √ to the field. = V 5β/6. For the For the scalar pressure, this speed is√VS = 5P/3ρ 0 A √ non-scalar pressure tensor, it is V = 2P/ρ0 = VA β. If 16 β 1, we see that |VS − V | $ = β− VA

'

√ β(1 − 16 ) 

β . 12

(7.95)

Thus, provided that β is not too large (β < 6, say), the wave speeds for perpendicular propagation do not differ significantly. Next, consider the behaviour of the slow wave for near-perpendicular propagation. For scalar pressure, we can take cos θ 1 in (7.65). For the square Copyright © 2005 IOP Publishing Ltd.

Harmonic plane waves in a uniform loss-free plasma

142 2 (a)

β=0.5

(b)

β=0.667

(c)

β=0.95

β=1.1

(e)

β=1.2

(f)

β=1.4

1

nz 0 −1 −2 2

(d)

1

nz 0 −1 −2

−2

−1

0

nx

1

2 −2

0

−1

nx

1

2 −2

−1

0

nx

1

2

Figure 7.3. Refractive index surfaces for non-scalar pressure perturbation (full curve) and scalar pressure perturbation (broken curve). The chain curve represents the surface for the transverse Alfv´en wave. The magnetic field is parallel to the z-axis, as indicated by the arrows. In the undisturbed medium, β⊥ = β . (a) V < VA ; (b) V = VA ; (c) VA < V , V⊥ < VA ; (d) VA < V , V⊥ > VA ; (e) VS = VA ; (f ) VA < VS . For explanation, see text.

root with the lower sign, the expression may be written in the form n 2z =

UA2 + US2 UA2 US2

=

6 + 5β . 5β

(7.96)

For non-scalar pressure, let n x → ∞ in (7.94). This gives n 2z =

4(1 + β) . 6 + 5β

(7.97)

It is easy to see that, in this limit, the refractive index for the case of scalar pressure is always larger than that for the case of non-scalar pressure. Let us now consider each panel of figure 7.3 in turn. Note that it is always true that V > V⊥ > VS . (a) Here V < VA . The closed surface represents the fast wave. For parallel propagation, the wave speed for this wave is VA for both scalar and nonscalar pressure. Thus, n = 1 when the wave normal is parallel to the field. The difference between the values of n for perpendicular propagation is very Copyright © 2005 IOP Publishing Ltd.

Waves for non-scalar pressure

(b )

( c)

(d ) ( e)

(f )

143

small so that there is scarcely any difference between the fast-wave refractive index surfaces for scalar and non-scalar pressures. This corresponds to the case V = VA . For propagation parallel to B , the fast and slow modes for non-scalar pressure have the same speeds and the surfaces of the three characteristic waves touch. The difference between the refractive index surfaces for the slow wave and the transverse Alfve´ n wave is too small to distinguish on this diagram. Here V > VA > VS . For propagation parallel to the magnetic field, the fast wave for non-scalar pressure has speed V , while that for scalar pressure has speed VA . For quasiparallel propagation, where the wave normal makes a small angle with the magnetic field, the fast-wave surface for scalar pressure is close to the quasiparallel part of the slow wave surface for non-scalar pressure. For quasiperpendicular propagation, where the wave normal makes angles near 90◦ with the magnetic field, the fast-wave surfaces for both scalar and non-scalar pressure nearly coincide. For quasiparallel propagation, the fast wave surface for non-scalar pressure is flattened. Comparison of panels (a), (b), and (c) shows that, as β increases, the fast-wave surface increases in size. In panel (b), where three surfaces touch, there is a degeneracy. In panel (c), the topology of the surfaces has changed: further increase in β , rather than ‘pushing the fast wave surface through the slow wave surface’, reconnects the surfaces so that the previously flat slow-wave surface develops a bulge and the fast-wave surface develops a flattened portion. The only difference between this case and (c) is that V⊥ > VA , whereas in (c) V⊥ < VA . Here there is a degeneracy in the surfaces for scalar pressure because VS = VA . The fast and slow waves for scalar pressure and the transverse Alfve´ n speed all have the same speed for parallel propagation. There is a reconnection of the two magnetosonic surfaces for the case of scalar pressure. Here all the magnetosonic characteristic speeds are greater than the Alfve´ n speed. For parallel propagation both slow waves have speed VA . The fast wave for scalar pressure has speed VS and that for non-scalar pressure has speed V . The fast wave surface for non-scalar pressure is still characteristically flattened for quasiparallel propagation.

In conclusion, we have found the dependence of the refractive index, and hence the phase speed, on the angle between the wave normal and magnetic field for magnetosonic waves, and compared the predictions of MHD for scalar and non-scalar pressure. It is necessary to caution the reader that MHD does not provide the whole picture. It turns out that such waves may be subject to significant collisionless damping, which is not predicted by MHD. This is taken up again in chapter 8. Copyright © 2005 IOP Publishing Ltd.

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Harmonic plane waves in a uniform loss-free plasma

7.7 Summary • • • •

• •

If a uniform anisotropic MHD medium is perturbed, it supports waves. If small perturbations of the medium are assumed, an anisotropic wave equation can be derived that is analogous to the wave equation for elastic waves in an anisotropic crystal. Harmonic plane waves that are solutions of the wave equation can be found. When the pressure is a scalar, there are three characteristic waves possible in the medium: – The transverse Alfv´en wave is a wave in which the perturbation is an oscillation perpendicular to the plane containing the wave normal and the unperturbed magnetic field. The perpendicular component of the k vector is arbitrary, while the wavefronts move along the direction of the magnetic field with the Alfv´en speed. – The fast and slow magnetosonic waves are compressional waves with perturbations in the plane containing the wave normal and the unperturbed magnetic field. The fast wave is approximately isotropic. The slow wave is not propagated perpendicular to the magnetic field. It is highly anisotropic. When the pressure is described by a second-rank tensor, there are also three characteristic waves. These have similar features to those for scalar pressure. The MHD approach does not allow for collisionless damping, which may be of importance.

Copyright © 2005 IOP Publishing Ltd.

Chapter 8 Collisionless damping of MHD waves

8.1 Introduction In deriving the equations describing the fluid behaviour of a collisionless plasma, some assumptions were necessary. The double adiabatic equations (5.85) and (5.86) were obtained by truncation of the hierarchy of moment equations at the equation describing the time evolution of the pressure tensor. This was done by making the assumption that there were processes leading to sufficient randomization of the particle velocities so that, on the time scales of the problem, the distribution function remained symmetric in both v⊥ and v . This is easy to justify for v⊥ because the magnetic field keeps the particles strongly confined and, when averaged over many gyroperiods, the particles maintain a distribution that is independent of phase angle. It is not so easy to justify for v ; and it is necessary to invoke unspecified wave–particle interactions to do so. In this section, we pursue the consequences of not making this assumption. To do so, it is necessary to consider kinetic effects. This has been done by Barnes [23] for a very general case where the waves are small perturbations on an arbitrary steady background flow. He bases his work on two seminal papers by Chandresekhar et al [36, 37]. The problem in this form is extremely complex. We shall consider a much simpler case.

8.2 Specification of the problem We consider the case of a uniform medium in equilibrium, consisting of one species of positive ion and electrons. Where necessary, these are distinguished by superscripts or subscripts + and −. In the unperturbed medium, the distribution function for each species of particle is f 0 (vi ). We assume that uniform conditions have lasted long enough so that the plasma has reached a quasi-equilibrium in which f 0 is the solution of the Vlasov equation (4.47) with ∂/∂t = 0 and Copyright © 2005 IOP Publishing Ltd.

145

146

Collisionless damping of MHD waves

∂/∂ x i = 0 and that it is independent of gyrophase θ : q (v × B) · ∇ v f 0 = 0 m

(8.1)

f0 = f 0 (v , v⊥ ).

(8.2)

with general solution It is further assumed that this is symmetric in v . We shall choose the direction of the magnetic field to be the z -axis so that v = vz . The components vx and 2 . Later, as a special case, v y then occur only in the combination vx2 + v 2y = v⊥ we shall assume that the distribution function is the bi-Maxwellian described in appendix A.3. We shall further assume that the wave is a small perturbation on this background. In these circumstances, we can linearize the moment equations (5.34) and (5.45). The zero-order and first-order components of the magnetic field and pressure tensor are written as upper-case and lower-case symbols, respectively. In addition, we note that the electric field E, the bulk velocity of each species u, and the current density j are of first order. If we retain only first-order terms, then (5.34) and (5.45), written in tensor notation, become ρ0

∂u i ∂ pik =  j kl jk Bl − ∂t ∂ xk   2 Ne ∂ ji m− = − 1 + + (E i + ilm u l Bm ) ∂t m m  − + epik epik ∂ − + . − ilm jl m + ∂ xk m − m

(8.3)

(8.4)

For reasons that will become apparent later, we have not assumed m − /m + 1. These equations together with the linearized Maxwell curl equations ∂b ∂t ∇ × b = µ0 j

∇×E= −

(8.5) (8.6)

are a set of four vector equations for the four vector unknowns, E, b, j, u, and the unknown divergence of the two second-rank tensors pi+j and pi−j . In the fluid approach, the next moment equation for pi j was truncated, using the previous assumptions and provided the required additonal equations for pi j . Here we will not do this. Instead, we shall substitute E, b, and u into the linearized Vlasov equation, find an appropriate solution for the perturbation of the distribution function, without making any assumptions about wave–particle interactions, and use it for finding the pressure tensors. Before starting on this procedure, we note some results. The perpendicular and parallel components of (8.4) and (8.6) can be taken and terms that are small Copyright © 2005 IOP Publishing Ltd.

Single-particle motion in a wave

147

in the MHD approximation dropped. We get ∂ u⊥ = ∂t ∂u  = ρ0 ∂t 0= ∂ j = ∂t

ρ0

j ⊥ × B − (∇ · p)⊥

(8.7)

− (∇ · p)

(8.8)

E ⊥ + u⊥ × B Ne2 e e E  + − (∇ · p− ) − + (∇ · p+ ) . − m m m

(8.9) (8.10)

In the fluid treatment the last of these was used to show that E  = 0. Indeed, the same argument shows that E  |E ⊥ |. Its effect is, however, comparable with the pressure gradients and we shall show that, in the kinetic treatment, it has an effect that does not disappear.

8.3 Single-particle motion in a wave In a truly collisionless plasma, the only accelerations experienced by individual particles are the result of the electromagnetic forces. The force arising from the gradient of the pressure is a bulk force. This force does not act on individual particles but on a fluid element. It is a consequence of the exchange of momentum between adjacent fluid elements as particles with different momenta are exchanged between them: the individual particles move only under the action of the electromagnetic fields. The electric and magnetic perturbation fields are E and b. We introduce a velocity U, the perturbation of the guiding-centre velocity of any single particle in the wave, defined by a≡

q ∂U = {E + U × B}. ∂t m

(8.11)

Since ω , the condition giving a⊥ is equivalent to E ⊥ + U × B  0. Thus, U ⊥  u⊥

E ∂U q  E =  ∂t m B

(8.12)

(8.13)

where u⊥ is the component of the drift velocity of the whole plasma perpendicular to the magnetic field. The components of a parallel to the field for ions and electrons are related by m+ a− = − − a+ (8.14) m and the parallel components of the perturbation velocities U are in the same ratio. Copyright © 2005 IOP Publishing Ltd.

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Collisionless damping of MHD waves

8.4 Kinetic effects 8.4.1 First-order perturbation of the Vlasov equation The peculiar velocity of a particle relative to U is v − U. We can write the perturbed distribution function in the form f (x, v, t) = f 0 (v − U) + f 1 (x, v, t)

(8.15)

where f 0 is a function of the peculiar velocity and, therefore, implicitly a function of x, v, and t. The first-order term f 1 is the perturbation of the distribution function arising from the wave. Because U is small, we may expand f0 on the right-hand side and retain only terms up to the first-order in U: f (x, v, t) = f 0 (v) − U(x, t)

∂ f0 (v) + f 1 (x, v, t) ∂v

(8.16)

where f 0 is now the unperturbed distribution function. We assume that collision-like processes occur on time scales long compared with the perturbation time scale. The time evolution of f is then described by the Vlasov equation (4.47) with a=

q {E + (v + U) × B}. m

(8.17)

We substitute (8.16) into (4.47) and retain only terms of first order in U, f 1 , and a. The result is q D f1 D + {E + v × b + U ⊥ × B} · ∇v f 0 = (U · ∇v f 0 ) Dt m Dt

(8.18)

where

∂ q D ≡ + v · ∇ + (v × B) · ∇ v . (8.19) Dt ∂t m Note that the zero-order electric field is zero and E is of first order, while B, b are the zero- and first-order magnetic fields. The quantity D/Dt represents a total derivative, determining the time rate of change of its operand, as we follow a particle in its unperturbed orbit in phase space. We note that f 0 is independent of x and t and that U is independent of v. Then the right-hand side of (8.18), if we use (8.1), may be written in the form   D ∂U (U · ∇ v f 0 ) = + (v · ∇)U · ∇v f 0 . (8.20) Dt ∂t If we use (8.11) and (8.13), then (8.18) becomes % & q D f1 = (v · ∇)U − v × b · ∇v f 0 . Dt m Copyright © 2005 IOP Publishing Ltd.

(8.21)

Kinetic effects

149

8.4.2 Integration over the unperturbed orbits We now find an expression for the first-order perturbation of the distribution function. We have already noted that the left-hand side of (8.21) represents the time rate of change of f 1 , following a particle on its unperturbed orbit in phase space. We choose the z-axis to lie along the magnetic field and denote the position of the particle at time t  by x  , y  , z  . When it is necessary to distinguish between electrons and ions, we shall use superscripts + and −; when the distinction is unnecessary we shall suppress the superscript. The gyrofrequency of the particle is defined as Bq . (8.22) = m It is, therefore, positive for ions and negative for electrons, since q ± = ±e. Then, if the particle arrives at point (x, y, z) at time t, the equation of its path is v⊥ {sin[(t  − t) + θ ] − sin θ }  v⊥ {cos[(t  − t) + θ ] − cos θ } y  (t  ) − y =  z  (t  ) − z = v (t  − t).

x  (t  ) − x =

(8.23) (8.24) (8.25)

The sign of  determines the sense in which the particle rotates around the magnetic field. The angle θ is the phase angle. At time t, particles with different phase angles arriving simultaneously at (x, y, z), have travelled along different spirals with axes which have been displaced from one another. The velocity of such a particle is given by vx (t  ) = v⊥ cos[(t  − t) + θ ] v y (t  ) = − v⊥ sin[(t  − t) + θ ]

(8.26) (8.27)

vz (t  ) = v .

(8.28)

These express the positions and velocities of the particles as functions of t  . Assume that the wave perturbation is harmonic and that the wave normal lies in the x z-plane, so that any field component ξ varies in space and time as ξ(x i , t  ) = ξ ∗ exp{i(k⊥ x  + k z  − ωt  )}   k⊥ v⊥ [sin(τ + θ ) − sin θ ] − (ω − k v )τ = ξ ∗ ei(k⊥ x+k z−ωt ) exp i  (8.29) where τ = t  − t. Copyright © 2005 IOP Publishing Ltd.

(8.30)

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Collisionless damping of MHD waves

This expression can be significantly simplified in the MHD approximation. The first term in the argument of the exponential is O(ω/) if k⊥ v⊥ ∼ ω. Therefore, to first-order in ω/,   k⊥ v⊥   ∗ i(k⊥ x+k z−ωt ) −i(ω−k v )τ ξ(x i , t )  ξ e sin(τ + θ ) . (8.31) e 1+i  Here we have ignored the term ik⊥ v⊥ sin θ/ in comparison to 1. We have not, however, ignored the term in sin(τ + θ ) because, on integration of the differential equation for f1 with respect to τ , it provides a contribution that is not negligible. It should be borne in mind that this approximation breaks down when k⊥ is very large, as is the case for the slow wave when the angle between the wave normal and the magnetic field is near π/2. This happens where the wavelength perpendicular to the field is comparable with the gyroradius and is the condition for the MHD approximation to fail. We do not consider this case here. We suppress the factor exp{−i(ω − k⊥ x − k z)} and write ∂ = −iω ∂t 

∇ = ik.

(8.32)

Then, if we make use of (8.13), we get   D f1 ik⊥ v⊥ −i(ω−k v )τ =e sin(τ + θ ) 1+ Dτ    q k·v q ˆ × −B E  − v × b + i(k · v)u⊥ · ∇ v f 0 . m ω m

(8.33)

In the coordinate system adopted, the derivatives with respect to time and space are ∂ ∂ ∂ = ik⊥ =0 = ik . (8.34) ∂x ∂y  ∂z  From (8.26), (8.27), and (8.28), 2 = vx2 + v 2y v⊥

v = vz

(8.35)

and, thus, the components of the gradient of f0 in velocity space are ∂ f0 ∂ f0 = cos(τ + θ ) ∂vx ∂v⊥

∂ f0 ∂ f0 =− sin(τ + θ ) ∂v y ∂v⊥

Then (8.33) becomes   ik⊥ v⊥ D f1 −i(ω−k v )τ =e sin(τ + θ ) 1+ Dτ    ∂ f0 q E × [k⊥ v⊥ cos(τ + θ ) + k v ] − ωm ∂v Copyright © 2005 IOP Publishing Ltd.

∂ f0 ∂ f0 = . ∂vz ∂v (8.36)

Kinetic effects 151  ∂ f0 + i{u x cos(τ + θ ) − u y sin(τ + θ )} ∂v⊥   q ∂ f0 ∂ f0 + {b y cos(τ + θ ) + b x sin(τ + θ )} v − v⊥ m ∂v⊥ ∂v (8.37) where a factor exp{i(k⊥ x + k z − ωt)} is understood. At this stage, it is useful to eliminate the magnetic field perturbation, using Faraday’s law in the form (2.58). We get k bx = − ux B ω

by k = − uy B ω

k⊥ bz = ux . B ω

(8.38)

If we note that E  /B has the dimensions of a perturbation velocity and we replace bx and b y in (8.37), we get   D f1 ik⊥ v⊥ −i(ω−k v )τ =e sin(τ + θ ) 1+ Dτ   × [k⊥ v⊥ cos(τ + θ ) + k v ]   ∂ f0  E ∂ f0 + i{u x cos(τ + θ ) − u y sin(τ + θ )} × − ω B ∂v ∂v⊥   k  ∂ f0 ∂ f0 − {u y cos(τ + θ ) + u x sin(τ + θ )} v − v⊥ . ω ∂v⊥ ∂v (8.39) Now we integrate (8.37) along the orbit with respect to τ between the limits −∞ and 0. The various terms contain integrals of the form  I1 = I2 =

−∞  0

 I3 = I4 = I5 =

0

−∞ 0

−∞  0 −∞  0

 I6 = Copyright © 2005 IOP Publishing Ltd.

−∞ 0 −∞

e−i(ω−k v )τ dτ cos(τ + θ )e−i(ω−k v )τ dτ sin(τ + θ )e−i(ω−k v )τ dτ cos2 (τ + θ )e−i(ω−k v )τ dτ sin(τ + θ ) cos(τ + θ )e−i(ω−k v )τ dτ sin2 (τ + θ )e−i(ω−k v )τ dτ.

(8.40)

152

Collisionless damping of MHD waves

The evaluation of these integrals needs care. In general, they are divergent because of the behaviour of the integrand at the lower limit. We can, however, consider the case of a wave that has grown from zero at τ = −∞. For such a wave, the frequency is complex with ω = ωr + iωi where ωi is a small positive imaginary part. In this case, the first integral may be written in the form  ie−ωi τ exp {−i(ωr − k v )τ } 0 i = . (8.41) I1 =  ω − k v ω − k v −∞ This expression is valid for ωr > 0. If we assume that I1 is an analytic function of ω, its value for negative ωr can be found by analytic continuation into the negative imaginary region. When it is substituted in the expression for f 1 (v ), it provides a term with a pole where v = ω/k . This pole provides a resonance where particles have a parallel velocity equal to the phase velocity of the wave; and these particles remain in phase with the wave, experiencing the wave fields as unvarying in time. The other integrals can be evaluated by expressing the trigonometric functions as integrals. For example,  0 1 {e−i(ω−k v −)τ eiθ + e−i(ω−k v +)τ e−iθ } dτ I2 = 2 −∞   e−i(ωr −k v +)τ e−iθ 0 eωi τ e−i(ωr −k v −)τ eiθ + = −  2i ω − k v −  ω − k v +  −∞  iθ −iθ e e 1 . (8.42) + = − 2i ω − k v −  ω − k v +  This has resonances where a particle experiences a Doppler-shifted wave frequency equal to its gyrofrequency. Such resonances may be important for waves with frequencies near the gyrofrequencies; however, the number of particles satisfying this condition is essentially zero. Because ω , resonance would require a population of particles with parallel velocities large enough so that the particle moved a whole wavelength while completing just one revolution about the field. The number of such particles is essentially zero. We can assume that, for all values f 0 that are appreciable, the condition |ω − k v |  holds. Then  e−iθ 1 eiθ I2 = = −1 sin θ. − (8.43) 2i   In the same way,

I3 = −−1 cos θ

(8.44)

while I4 = Copyright © 2005 IOP Publishing Ltd.

1 −1 1 1 sin θ cos θ  sin 2θ + I1 = I1 + 4 2 2 2

(8.45)

Kinetic effects I5 = − 14 −1 cos 2θ = − 14 −1 (cos2 θ − sin2 θ ) 1 sin θ cos θ . I6 = I1 − I4 = I1 − 2 2

153 (8.46) (8.47)

Examination of the order of magnitude of these expressions shows that I1 , I4 and I6 are O(ω−1 ) and that I2 , I3 and I5 are O(−1 ), with I4  I6  12 I1 . We wish to retain only the highest-order terms in the final result. We, therefore, see that, if we perform the integrations, the expression for f 1 is ik v ∂ f 0  1 k⊥ v⊥ ∂ f 0 − ux ω − k v ∂v ω 2 ω − k v ∂v⊥   ∂ f0 1 k v k⊥ v⊥ v⊥ ∂ f 0 ux + − 2 ω(ω − k v ) ∂v⊥ v ∂v   k v ∂ f 0 v⊥ ∂ f 0 (u y sin θ − u x cos θ ) (8.48) + − ω ∂v⊥ v ∂v




f 1 (v⊥ , v , θ ) = −

where


= EB

(8.49)

is a normalized parallel electric field having the dimensions of velocity. It might appear that the term in is an order of magnitude larger than the others. We see, however, from the discussion in section 8.2 that E  /B is much smaller than u x , u y , so that the factor /ω in this term serves only to show us that it cannot be neglected in comparison with terms in u x , u y . For the expression for f 1 to be useful, we need an explicit formula for f 0 . 2 and that The only general constraint is that it should be a function of v2 and v⊥ it should tend to zero as |v| → ∞ sufficiently fast for all the moment integrals to be non-singular. For computation, we need an explicit function. We adopt the bi-Maxwellian distribution (A.11). The properties of this distribution are outlined in section A.3, together with a number of associated mathematical results. It must be emphasized that there is little physical justification for the adoption of this model. There is no reason to suppose that any wave–particle interactions would lead to a quasi-equilibrium of this form. However, if we make the plausible assumption that the general features of the distribution are that it is peaked at the origin and goes rapidly to zero at infinity and that it is symmetrical in v⊥ , and the less plausible assumption that it is symmetric in v , then the bi-Maxwellian is a mathematically tractable solution with the desired features. Any other distribution with the same features can be regarded as the sum of a bi-Maxwellian and a timestationary perturbation. If the appropriate characteristic width is chosen, the timestationary perturbation has a zero mean but it may contribute to higher moments. We ignore such contributions in our treatment. In this case, if we substitute (A.11) for f 0 , (8.48) becomes  2 v2 v v⊥ k k⊥ ρ0 iρ0  k + ux f 1 (v⊥ , v , θ ) = P ω ω − k v 2ω P ω − k v







Copyright © 2005 IOP Publishing Ltd.

154

Collisionless damping of MHD waves  k⊥ ρ0 k v⊥ v ρ0 ( P⊥ − P ) 2 u x v⊥ + (u y sin θ − u x cos θ ) 2ω P⊥ ω P⊥ P f⊥ f × . (8.50) 2π

+

8.4.3 Evaluation of moments of the perturbed distribution function Some of the terms of f 1 are characterized by a pole at v = ω/ k . Since ωi is positive, if we consider the dependence of f 1 on v , this pole lies in the positive imaginary half of the v complex plane. The computation of the perturbation of fluid quantities such as fluid velocity and pressure, requires the evaluation of integrals of the form 

∞



−∞ 0

∞  2π 0

g(v , v⊥ , θ ) v⊥ dθ dv⊥ dv ω − k v

(8.51)

with v⊥ dθ dv⊥ dv being the velocity space volume element in cylindrical coordinates. Care must be taken because f1 has, so far, only been defined for the case when ω has a positive imaginary part and we must consider its definition for Im(ω) ≤ 0. This can be done by analytic continuation. In evaluating the integral with respect to v for Im(ω) > 0, there is no difficulty. The integral is taken along the real axis as shown in the top panel of figure 8.1. Now let Im(ω) → 0 so that the pole lies on the real axis as shown in the middle panel of figure 8.1. The contour must still pass below the pole, so it can be distorted into a contour along the real axis and a small semicircle below the pole as shown. If we let Im(ω) change continuously to negative values, the pole lies below the real axis and the contour is further distorted as shown in the bottom panel of figure 8.1. The integral now becomes an integral along the real axis as well as the residue at the pole. This is the recipe for evaluating integrals such as (8.51). It is known as the Landau prescription. Provided g1 is an analytic function of v , it provides the required analytic continuation. This technique ensures causal behaviour. For a detailed discussion, see Clemmow and Dougherty [49, ch 8]. This type of problem was first treated rigorously by Landau [116]. When evaluating moments of the distribution function, we shall encounter integrals of the form  ∞ vn f   dv −∞ ω − k  v which must be evaluated according to the Landau prescription. These can be evaluated by using the techniques described in appendix A.5. They are expressed in terms of the plasma distribution function and its derivative [73]. We summarize the relevant results for convenience:  ∞ f N (8.52) dv = − ζ Z (ζ ) ω −∞ ω − k  v Copyright © 2005 IOP Publishing Ltd.

Kinetic effects

ℑ(v||)

155

v =ω/k|| ¤ ||

ℑ(ω)>0

ℜ(v||) ℑ(v||)

ℑ(ω)=0

v||=ω/k||

ℜ(v||)

ℑ(v||) ℑ(ω) P⊥ . This behaviour is typical of many observations by many workers. Radar observations of the pulsations on the ground show that they have a short azimuthal wavelength, correponding to an azimuthal wavenumber m ∼ 40. The occurrence of compressional poloidal oscillations is correlated with the injection of energetic particles. There is good evidence that such oscillations are a consequence of mirror instability (section 23.5.2) in the plasma. 13.4.2.2 Non-compressional poloidal oscillations The non-compressional poloidal oscillations are generally in the Pc4 band, but occasionally at Pc5 frequencies [218]. Their polarization is essentially in the magnetic meridian, with no compressional component. These pulsations are predominantly in the afternoon sector [18, 214]. Spacecraft magnetometer observations near the geostationary orbit suggest that they are confined to a region near the equatorial plane. This may simply be the effect of a mode structure that has an antinode at the equator and nodes on either side of the equatorial plane. They are confined to a narrow range of about two Earth radii in radius [189] but may extend over many degrees of longitude [64]. There is a reasonable consensus that they are second harmonic oscillations [98, 189, 214]. 13.4.2.3 Giant (Pg) pulsations The properties of giant or Pg pulsations are well reviewed by Wright et al [255] who also provide an extensive set of references. They are generally in the Pc4 band but may occur at Pc5 frequencies. They are seen at auroral latitudes in the morning sector. Their azimuthal wavelength is short and they are localized in latitude. They have been observed by a variety of techniques, including ground magnetometer arrays, auroral radars, optical instruments, and spacecraft. There is now consensus that Pg pulsations are standing waves that are generated by resonance with bouncing drifting particles. As resonant particles undergo a bounce–drift motion (sections 3.5.4, 3.5.5), they remain in phase with the standing wave and allow the possibility of energy transfer between particle and wave. The details of this process are considered in section 24.1. There is no consensus about the symmetry of Pg oscillations about the geomagnetic equator. The observations of some authors [220] lead them to the conclusion that the waves are in the fundamental mode or in a higher odd harmonic mode; others [47, 97] conclude that they are in the second harmonic or higher even harmonic mode2. Wright et al [255] have associated low-altitude spacecraft measurements of particle distributions with ground-based observations 2 The nomenclature is confusing. The labels ‘even’ and ‘odd’ do not refer to symmetry about the

equator but to the mode number of the harmonic, so that the fundamental (n = 1) is odd and the second harmonic (n = 2) is even. Later in the book we avoid this nomenclature.

Copyright © 2005 IOP Publishing Ltd.

244

Observations of ultra-low-frequency oscillations and waves

of a Pg pulsation and have concluded that the mode of their observations is more likely to be an even harmonic. It may be that both symmetries occur. 13.4.3 Pi2 pulsations The subject of so-called impulsive pulsations is intimately bound up with abrupt reconfigurations of the magnetotail. Magnetospheric substorms [170] are manifested in ground-based observations of auroral displays, magnetic perturbations, and ionospheric disturbances. There is consensus that energy, stored in the stretched magnetotail, is catastrophically released over a limited longitude sector, leading to a reconfiguration of the magnetic field lines in the sector in such a way that they become more dipolar in shape. The mechanism of release is controversial. The ‘dipolarization’ of the magnetic field is associated with particle energization and increased current through the ionosphere. The initial impulsive change provokes a magnetospheric response: oscillations at a frequency determined by the length scale of the disturbed region. This magnetic field oscillation, observed at a ground station near the initial disturbance, shows an initial impulsive start, followed by a few cycles of decay. The Pi2 pulsation is recognized [170] as defining the time of onset of the substorm. Associated with the larger high-latitude pulsation, however, is a much smaller oscillation observed at lower latitudes on a global scale, right round to the day-side [212]. Sutcliffe and Yumoto [213] suggested that these dayside disturbances were Alfv´en cavity modes within the plasmasphere. The picture of a large disturbance at high latitudes in a limited longitude sector exciting cavity modes within the plasmasphere is now widely accepted [221].

13.5 Instrumentation 13.5.1 Modern instrumentation and the internet Early science was done by individual experimenters using their own instruments in isolation. Substantial international cooperation in the sharing of data and planning of joint experiments became important at the time of the IGY, with the setting up of world data centres and data exchange between scientists. Of course, at that time the data were in the form of paper charts or tables of observations and the exchange was subject to the delays of the mails. Acquiring access to data from another experimenter was a tedious procedure. The internet has had a profound influence on the way in which space science is pursued. Not only has it made available unthinkable quantities of data that can be downloaded rapidly from a computer on another continent but it has also changed the sociology of data exchange. From a situation where data were often jealously guarded, we have progressed to a situation where, having identified a period for study, one can, within half-an-hour or so, know the properties of the solar wind, see the behaviour of the magnetic field at a number of ground arrays, be provided with convection Copyright © 2005 IOP Publishing Ltd.

Instrumentation

245

maps of the polar cap during the period of interest, and acquire a variety of other relevant information. Further, procedures for obtaining permission for use of data from such sources is easily obtained, subject to appropriate ‘rules of the road’ and it is usually freely given. The role of this change in culture in advancing the science should not be underestimated and the scientific community should be very grateful to the individuals and organizations that have made it possible. 13.5.2 Satellites and spacecraft The launch of the first Sputnik satellite in 1957 coincided with the beginning of the International Geophysical Year and signalled the beginning of the space era. The discovery of the radiation belts by van Allen [227] initiated the study of geospace using in situ measurements. It is not possible to outline all the types of data available from spacecraft observations. Table 13.3 identifies some crucial types of spacecraft orbit that allow measurements of different parts of the solar– terrestrial system. The important types of instrumentation on spacecraft include magnetometers, particle detectors, imaging systems for a variety of wavelengths, and instruments for measuring electric fields. 13.5.3 Magnetometer arrays and other ground-based instrumentation The setting-up of arrays of magnetometers made a qualitative change in the way that magnetic field data could be used to understand pulsations. The key to understanding wave phenomena is a data set consisting simultaneous spatial and temporal information. The magnetometer array in Canada used by Samson et al [178] was the first to be used in this way. Modern magnetometer chains exist over wide latitude and longitude ranges, often combined with other instruments such as riometers and optical instruments. A number of different local-time sectors in the northern hemisphere are covered by the Canadian CANOPUS array [171], the West Greenland array, and the IMAGE array spanning Scandinavia from north to south. In the southern hemisphere, the automatic geophysical observatories in Antarctica [54] may be used in conjunction with instruments at permanent manned stations. At lower latitudes, instruments from permanent observatories and temporary arrays have been used to good effect. 13.5.4 Auroral radar arrays The STARE radar system [77] has been a very important instrument in the understanding of pulsations. The original system consisted of two bistatic phasedarray VHF radars with a common field of view located in Northern Scandinavia. They were capable of sensing the intensity and Doppler velocity of E-region ionospheric irregularities over a 400 km × 400 km region with a spatial resolution of 20 km and a temporal resolution of 20 s. The Doppler velocities of the two radars could be combined to produce a map of the velocity of the irregularities. Copyright © 2005 IOP Publishing Ltd.

246

Observations of ultra-low-frequency oscillations and waves Table 13.3. Some spacecraft characteristics.

Type

Description

Examples

Polar or near-polar orbit

Orbits almost in the meridian. Generally low altitude and often eccentric. Over a period of time cover most of the Earth surface as the Earth rotates about its axis relative to the plane of the orbit. Orbit in the equatorial plane near 6.7 Earth radii. Orbital period approximately 24 hr. Spacecraft remains above same point on Earth as Earth rotates. Highly eccentric orbit with orbital parameters adjustable so that the spacecraft can probe various parts of the magnetosphere. Group of spacecraft following approximately the same orbit and capable of distinguishing spatial and temporal information.

DMSP series, POLAR

Geostationary or near geostationary

Wandering

Multi-spacecraft systems

‘Halo’ orbit round Lagrangian point L1

Earth’s gravitational field is opposite to Sun’s field at L1 and reduces orbital period of satellite at L1 to that of the Earth. Located approximately 250 Earth radii from Earth.

GEOS2, GOES series

IMP8, WIND, Interball, GEOTAIL

ISEE1/2, CLUSTER

IMP8, WIND, ACE

Since the irregularities drifted with an E × B velocity to a good approximation, except for saturation at higher velocities, knowledge of the geomagnetic field allowed a map of electric field to be produced with 20 s resolution. The instrument was limited by the fact that irregularities were only produced when the background field exceeded about 15 mV m−1 . Sensitivity to aspect angle means that backscatter is only obtained when the beam is almost exactly (within 2 or 3 degrees) normal to the magnetic field. The geometry required to obtain E-region scatter limits the location of such radars to a narrow range of latitudes. A modern version of STARE is still operational. The SuperDARN radar chain [78] overcomes these difficulties by operating at HF. It exploits ionospheric refraction to overcome the aspect angle problem. The HF signal is reflected from irregularities in the F-region, which do not require a background electric field to exist. The SuperDARN chain has been very successful in providing a global picture of magnetospheric convection processes. Copyright © 2005 IOP Publishing Ltd.

Summary

247

0º

180º g Salmon Kodiak King Prince George

Sanae Sasketoon 80º -90º

70º

60º

50º

90º

Kapuskasing se Bay Goose Stokkseyri Þikkvibær Hankasalmi

Halley –90º

Sjowa Kerguelen I.

South Pole

90º

–70º –60º –50º Tasmania 180º

Figure 13.4. Location of the fields of view of SuperDARN radars.

It now consists of 15 radars, operated cooperatively by Principal Investigators in nine nations. Ten of these radars are operated in the Northern hemisphere and six in the Southern hemisphere. More are planned. Figure 13.4 shows the location of the radars.

13.6 Summary •

A wide variety of wave and oscillation phenomena occur in the solar– terrestrial system: – – –

•

MHD wave phenomena occur within the corona. The solar wind, itself, shows MHD turbulence and also supports the propagation of coherent MHD waves. A variety of MHD oscillations occurs within the magnetosphere at frequencies from a few mHz to more than 1 Hz. The wavelengths of the lower frequency oscillations are comparable with the dimensions of the system and their frequencies are of the order of the wave speed divided by the dimensions of the system.

A variety of experimental techniques have been used over the years to elucidate the nature of these oscillations. They are briefly outlined.

Copyright © 2005 IOP Publishing Ltd.

PART 3 WAVES IN SOLAR–TERRESTRIAL PHYSICS

Copyright © 2005 IOP Publishing Ltd.

Chapter 14 MHD wave equations in non-uniform media

14.1 Introduction MHD waves in solar–terrestrial plasmas have very long wavelengths. This means that the study of propagation in a uniform medium is seldom applicable to real problems, except in a qualitative way. The dimensions of the system are often of the order of the wavelength, so that the nature of the boundary conditions is important. In addition, spatial gradients in unbounded plasmas introduce additional complexities. The nature of these complexities is twofold. First, gradients of pressure and magnetic field perpendicular to the field lines and the associated field line curvature mean that the characteristic waves in a uniform plasma are not propagated independently. Instead they are, in general, coupled. Second, the magnetic field may change substantially along the length of a field line so that the flux tubes are strongly convergent or divergent. This leads to geometric effects that affect the amplitudes of the field components. In this chapter, we derive sets of equations that can be used in a variety of such circumstances and which will be useful throughout the remainder of the book. First, we consider the relatively simple case of plane geometry, with transverse gradients of pressure and density. We then deduce appropriate equations for non-planar geometries. The content of the chapter is necessarily somewhat formal. Some readers may prefer to proceed to the next chapter and return to the material here as it is needed in later chapters.

14.2 Models 14.2.1 The magnetosphere Figure 14.1, on the left, shows a sketch of a cross section of the magnetosphere on the flank, at dawn or dusk. The unshaded region is the region of closed field lines Copyright © 2005 IOP Publishing Ltd.

251

252

MHD wave equations in non-uniform media C Polar cap

Magnetosheath

A B G F

D

B Magnetosheath

A

C

D

E E

F

G

Figure 14.1. Modelling the magnetosphere.

in the magnetosphere. It is bounded by magnetosheath plasma along C D E, which represents the magnetopause, by polar cap plasma along BC and E F, and by the ionosphere along F G AB. The points A and G coincide at the equator where, at its apex, a dipole field line is tangential to the surface of the Earth. The magnetopause boundary C D E is characterized by discontinuities in density, pressure, magnetic field, and velocity. The discontinuities at the polar cap boundaries, BC, E F, are less pronounced; the characteristics of the plasma change but these do not affect the MHD properties strongly; and there are changes in the electric field associated with plasma convection processes. The ionosphere can be regarded as a layer of anisotropic conductor of the type described in section 2.8. Often it is sufficiently accurate to assume infinite conductivity. This general structure is typical of the magnetosphere on the dayside and on the flanks. On the nightside, the dipole-like field lines are limited to the region less than about 10–15 Earth radii. Beyond that distance, there are significant distortions associated with the cross-tail current. In addition, the inner magnetosphere is bounded, not by the magnetopause, but by the magnetotail plasma sheet. In section 9.2.5, we saw that the ray direction and, hence, the energy flux in a transverse Alfv´en wave is exactly parallel to the magnetic field. This means that transverse Alfv´en waves can be confined to the immediate neighbourhood of a field line. In such cases, as the wave is propagated along the field line, its amplitude changes because of the divergence or convergence of the field lines along the magnetic flux tube. These are geometrical effects and not particularly important to the fundamental physics of the propagation. For this reason, it may be useful to use a model in which the field lines are straightened and the principles of the problem can be studied in a Cartesian geometry. The mapping involved is shown on the righthand side of figure 14.1. Each labelled point on the left-hand Copyright © 2005 IOP Publishing Ltd.

Models

253

diagram corresponds to a point on the right-hand diagram. The distortions of this model mean that the polar cap is shrunk to the line segments BC, E F, and that the line AG in the Cartesian model represents a point in the ionosphere in the more realistic model. This is not important as, in this chapter, we shall be considering waves restricted to the immediate neighbourhood of field lines well removed from the boundaries. We shall return to this model in later chapters when the nature of the boundaries BC D E F and AG will become relevant. Models such as these are known as box models. In such a model, the field lines are straight and there are gradients of the Alfv´en and sound speeds normal to the magnetic field. In the real magnetosphere, with curved field lines, the gradient of the Alfv´en speed arises principally from the magnetic-field gradient. It depends only weakly on the density which does not depend strongly on x. The gradient of B in the magnetosphere is largely determined by the curvature of the field lines. In a model with straight field lines and zero pressure, there cannot be a magnetic-field gradient in the steady state. Unless the density gradient is very large, a realistic Alfv´en speed gradient cannot be achieved self-consistently. If we wish to include the plasma pressure, a similar difficulty arises. In conditions of pressure balance, the speed of the fast wave is only weakly dependent on the field gradient. The gradients in velocity also have to be simulated by unrealistic density gradients. In spite of this difficulty, the insight gained by working in a rectangular geometry, where the essential physics is not obscured by geometrical complexity, makes the box model an attractive one. In what follows, therefore, we shall sometimes partially relax the condition of selfconsistent pressure balance by assuming that simple pressure balance for straight field lines holds locally on the scale of the wave perturbations when deriving the basic first-order wave equations. On the larger scale, to zero-order, we will assume that P, B, and ρ0 vary realistically with x as if field-line curvature were playing a part. This may not be entirely satisfactory but has been an implicit difficulty of box models ever since the work of Southwood [193]. In such cases, on the large scale, we allow an arbitrary variation of V A and, therefore, VS , in the perpendicular direction. If all the parameters depend only on one Cartesian coordinate, x say, the medium is said to be plane-stratified. 14.2.2 Cylindrical models—sunspots and coronal loops Sunspots and coronal loops are both complicated three-dimensional structures, consisting of regions of increased magnetic field and plasma density, elongated along the average direction of the magnetic field and of limited size perpendicular to this direction. The magnetic field may have a spiral structure. In the same spirit as the box models of the previous section, it is possible to consider a model with cylindrical symmetry for such structures. We use cylindrical coordinates r , φ, z, where r is the radius, φ the azimuth angle, and z is measured parallel to the axis of the cylinder. In such a model, the plasma pressure and density are functions of r only, while the magnetic field has axial and azimuthal components, Copyright © 2005 IOP Publishing Ltd.

254

MHD wave equations in non-uniform media

with a magnitude that is a function of r alone. In such a model, the equilibrium condition takes the form  Bφ2 B2 d P+ . (14.1) =− dr 2µ0 µ0 r

14.3 Coupled wave equations in a plane-stratified medium 14.3.1 First-order wave equations We use a rectangular geometry in which pressure P, magnetic field B, and density ρ0 depend only on x. The field lines are straight and B is constrained to lie in a plane perpendicular to the x-axis. In a geometry with straight field lines, the parallel components of the gradient of the magnitude of the magnetic field and the pressure are both zero. The density, pressure P, and the magnitude and direction of B are assumed to depend on x alone. Note that this does not require the direction of B to be constant. It can be sheared so long as it remains perpendicular to the x-axis. The pressure balance condition depends only on the magnitude of B, not on its direction. This is the most general plane-stratified geometry possible, unless both P and B are uniform and only the density varies. The wave fields are assumed to vary with y, z, and t as exp{−iωt + ik y y + ik z z} so that ∂/∂t ≡ −iω, ∂/∂y ≡ ik y , and ∂/∂z ≡ ik z . Let k be a vector parallel to the boundary surface representing phase variation in the direction normal to xˆ : k = yˆ k y + zˆ k z .

(14.2)

The equations to be developed describe waves in an infinite medium with such a gradient. If it is necessary to introduce boundaries, they will be of two types. Boundaries perpendicular to the field lines, such as the ionosphere, are represented by conducting planes. These are assumed to be massive so that they are nodes of displacement parallel to the field and antinodes of pressure. There may also be boundaries taking the form of tangential discontinuities perpendicular to the directions of the gradients. The boundary conditions (10.17) and (10.18) applying to the wave variables at these boundaries are that the normal component of displacement ξ and the generalized pressure ψ ≡ p + B · b/µ0 are continuous across the boundary. When considering the variation of the wave fields with x, it is then natural to express the equations in terms of these variables. The independent variables in the equations are then continuous even at a discontinuity. We, therefore, derive equations expressed in terms of the generalized pressure ψ and normal displacement ξ . We also use η, the component of the plasma displacement in the direction perpendicular to xˆ and B. These displacements Copyright © 2005 IOP Publishing Ltd.

Coupled wave equations in a plane-stratified medium

255

are related to the perturbation velocity components by vx = − iωξ

(14.3)

v⊥ = − iωη

(14.4)

In general, the direction of B, as well as its magnitude, may depend on x. In this case, the direction of B rotates in the yz-plane as x advances. In such a case, the direction of η rotates to remain perpendicular to xˆ and B. When linearizing the MHD equations, we have to allow for the spatial variation of B, P, and ρ0 . These are related by the zero-order momentum equation (6.3), which may be written in the form   B2 B dB d dP =− =− . (14.5) dx dx 2µ0 µ0 dx If we linearize (2.55), (2.57), and (2.58), we get b dψ bx d B + ikψ − ik · B = dx µ0 µ0 dx   dP dvx + ik · v = − vx −iωp + ρ0 VS2 dx dx   dvx dB iωb + i(k · B)v − B + ik · v = vx . dx dx −iωρ0 v + xˆ

(14.6) (14.7) (14.8)

The equations for a uniform medium can be retrieved by setting the right-hand sides of the equations to zero and replacing d/dx by ik x . We can use (14.3) to write the x-component of (14.8) in the form bx = i(k · B)ξ.

(14.9)

Next substitute (14.3) and (14.9) in the x-component of (14.6) to get dψ = ρ0 [ω2 − (k · V A )2 ]ξ. dx

(14.10)

This is a first-order equation for the spatial rate of change of ψ in terms of ξ . To find the corresponding equation for the rate of change of ξ , we take the scalar product of k with (14.8) and use (14.3) and (14.9) to get k · b = −(k · B)

d dξ − ξ (k · B). dx dx

(14.11)

An expression for k · v is found by taking the scalar product of k with (14.6) and making use of (14.3) and (14.11): k·v = Copyright © 2005 IOP Publishing Ltd.

k2 (k · V A )2 dξ . ψ+ ωρ0 ω dx

(14.12)

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MHD wave equations in non-uniform media

Then, if we use (14.3) and (14.5), the expression (14.7) may be written in the form  k 2 VS2 (k · V A )2 dξ ξ d 2 p = −ρ0 VS 1 − + (B 2). ψ+ (14.13) dx 2µ0 dx ω2 ω2 Next take the scalar product of B with (14.6) and (14.8). Then (14.3), (14.9), (14.11), and (14.12) can be used to get (k · V A )2 − k 2 VA2 ξ d dξ (B · b) (B 2 ) − ρ0 VA2 − =− ψ. µ0 2µ0 dx dx ω2 − (k · V A )2

(14.14)

Finally, add (14.13) and (14.14) to get and expression for dξ/dx. The result, together with (14.10) is a pair of first-order equations for the spatial dependence of ξ and ψ:



dψ = (x)ξ dx dξ (x) = − ψ dx (x)

(14.15)

 

(14.16)

where

 (x) = ρ0 [ω2 − (k · V A )2 ] 4  (x) = ω2 (V 2 + V 2 )ω− (k · V A

S

(14.17) A)

2V 2 S

− k 2y − k z2 .

(14.18)

These two equations describe the behaviour of the magnetosonic waves in the varying medium. The equilibrium condition (14.5) ensures that no terms in the derivatives of the zero-order quantities appear. A third equation for the component of perturbation displacement, η, is found by combining the components of (14.6) and (14.8) that are perpendicular to xˆ and B: k⊥ ψ (14.19) η=i ρ0 (ω2 − (k · V A )2 ) where k⊥ is the component of k in the direction of η. This is the transverse Alfv´en wave. It is driven by the magnetosonic waves but does not influence the solution of the other two equations. If k⊥ = 0, it is a free oscillation decoupled from the magnetosonic waves. We have written these equations in the most general coordinates consistent with B being parallel to the planes x = constant. This allows for the possibility that both the magnitude and direction of B may depend on x, so that, while it remains perpendicular to x, it may rotate in the x y-plane. For many purposes the direction remains constant and, in this case, it is convenient to take the direction of the magnetic field to be in the z-direction. Then η is in the y-direction, k⊥ = k y , and (k · V A )2 ) = k z2 VA2 . Copyright © 2005 IOP Publishing Ltd.

Coupled wave equations in a plane-stratified medium

257

14.3.2 Polarization relations We can take the y and z-components of (14.6) and (14.8) to get the relationship of the other field components to ψ, and ξ : kyω ψ ρ0 [ω2 − (k · V A )2 ] ωk z VS2 ψ vz = − ρ0 [ω2 (VA2 + VS2 ) − (k · V A )2 VS2 ] k y kz B by = − ψ 2 ρ0 [ω − (k · V A )2 ]  ω2 − k z2 VS2 dB bz = B ψ −ξ 2 2 2 2 2 dx ρ0 [ω (VA + VS ) − (k · V A ) VS ] vy =

p=ψ−

Bbz . µ0

(14.20) (14.21) (14.22) (14.23) (14.24)

14.3.3 Second-order wave equation Either ξ or ψ may be eliminated from (14.15) and (14.16), giving rise to secondorder differential equations:

   

( / ) dξ d2 ξ + ξ =0 + (14.25) 2 / dx dx  dψ d2 ψ + ψ = 0. − (14.26) dx dx 2 Such differential equations are typical in wave problems. If the medium is uniform, then the coefficient of dξ/dx is zero and the resulting equation is the spatial part of the standard wave equation with solutions of the form √ exp{±i (x)}. The form in which we have chosen to write the equations is appropriate for wave problems. It is a version of the most general form of second-order differential equation: (14.27) u  + f (x)u  + g(x)u = 0.







We may work with either (14.25) or (14.26) as is convenient. Then (14.15) or (14.16) may be used to deduce the other variable. 14.3.3.1 Limiting cases of the second-order wave equation The general differential equation (14.27) occurs in various guises at a number of places throughout the remainder of the book. While it is always possible to solve it numerically, physical insight is often best obtained by considering analytic solutions of various approximate forms. We identify its singular points and find appropriate approximations in the neighbourhood of these. Copyright © 2005 IOP Publishing Ltd.

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MHD wave equations in non-uniform media

Singular points



 

The singular points of (14.25) or (14.26) occur where has a zero and where has a pole or other singularity. Examination of (14.17) and (14.18) shows that is zero where ω2 = (k · V A )2 (14.28) and

 has poles where

ω2 (VA2 + VS2 ) − (k · V A )2 VS2 = 0.

(14.29)

In the special case where B is in the z-direction and k y = 0, − k z VA )(ω − k z VS )  = (ω 2 ω (V 2 + V 2 ) − k 2 V 2 V 2 2

 

=

2

2

2

2

2

z S A A S 2 2 2 2 ω (VA + VS ) − k z VS2 VA2 . ω2 − k z2 VS2

(14.30) (14.31)

In this case, there is a pole at ω2 = k z2 VS2 . Approximate forms near Near a zero of in |x − x 0 |:

(14.32)

=0

 occurring where x = x0, we may expand  and  to first-order

 = 0 (x − x0) + O[(x − x0)2 ] (14.33)  = 0 + O(x − x0). (14.34) We ignore higher derivatives of  and  . Then, in this neighbourhood, the equations are approximately

d2 ψ + 0 (x − x 0 )ψ = 0 dx 2 1 dξ d2 ξ + G 0 (x − x 0 )ξ = 0. − dx 2 x − x 0 dx



Apply the transformation



ζ = −(|  (x 0 )|)1/3 (x − x 0 )

(14.35) (14.36)

(14.37)

and the equations become d2 ψ = ζψ dζ 2 d2 ξ 1 dξ − ζ ξ = 0. − dζ 2 ζ dζ

(14.38) (14.39)

The first of these is Stokes’ equation (15.32) and the second is its derivative. Some of their properties are described in section 15.6.1. Copyright © 2005 IOP Publishing Ltd.

Coupled wave equations in a plane-stratified medium Approximate form near



259

 =0

Suppose has a zero at x = x R . Once more suppose that the medium is slowly varying. Then  1 1  (0) + O(|x − x R |). = + (14.40) x − xR 2  (0)

 

 

Equation (14.18) shows that, at x = x R ,

 = −k 2y + O(|x − xR|).

(14.41)

Then (14.25) and (14.26) become d2 ξ 1 dξ − k 2y ξ = 0 + x − x R dx dx 2 d2 ψ 1 dψ − k 2y ψ = 0. − x − x R dx dx 2

(14.42) (14.43)

If we make the substitution ζ = k y (x − x R )

(14.44)

these become d2 ξ 1 dξ −ξ =0 + 2 ζ dζ dζ d2 ψ 1 dψ − ψ = 0. − ζ dζ dζ 2

(14.45) (14.46)

The first of these is the modified Bessel equation of order zero [1, section 9.6] with solutions I0 and K 0 . Its solutions are the modified Bessel functions I0 (ζ ) and K 0 (ζ ). The first few terms of the series solutions for this equation are [1, equations (9.6.12), (9.6.13)] ( 1 ζ 2 )2 ( 1 ζ 2 )3 1 I0 (ζ ) = 1 + ζ 2 + 4 2 + 4 2 + · · · 4 (2!) (3!) K 0 (ζ ) = − [ln 12 ζ + γ ]I0 (ζ ) + 14 ζ 2 + (1 + 12 ) + (1 +

1 2

+ 13 )

( 14 ζ 2 )3 + ··· (3!)2

(14.47) ( 14 ζ 2 )2 (2!)2 (14.48)

where γ is Euler’s constant. Approximate form near a simple pole of



This case is included for completeness. It has not yet been identified as important in any solar–terrestrial process. The pole occurs when (14.29) is satisfied. Copyright © 2005 IOP Publishing Ltd.

MHD wave equations in non-uniform media

260



Examination of the expression for shows that it is removed in both the limiting cases VA2 VS2 and VA2 VS2 . Any effect is, therefore, only likely to be observed when the plasma and magnetic pressure are nearly equal, i.e. when β ∼ 1. If we write = ω2 (VA2 + VS2 ) − k z2 VA2 VS2 , then, near the pole at x = x p ,



  ω  (x − xp)−1 − k 2y − kz2 . 4

(14.49)

p

Then, with the assumptions that second-order terms may be neglected and that the pole is well isolated from other singularities, (14.25) and (14.26) may be written in the form  ω4 d2  −1 2 2 (14.50) + (x − x p ) − k y − k z  = 0  dx 2 p  d2  1 1 ω4 −1 2 2 + +  (x − x p ) − k y − k z  = 0. (14.51) 4 (x − x p )2 dx 2 p





If we make the substitution ζ = 2kn (x − x p ) these become

 ω4 1 d2  1 + − + =0 4 2kn p ζ dζ 2  ω4 1 1 1 d2  + 2  = 0. + − + 4 2kn p ζ dζ 2 4ζ





(14.52)

(14.53) (14.54)

Both of these are forms of Whittaker’s confluent hypergeometric equation [1, p 505], the first with µ = 12 and the second with µ = 0. Both have a non-integral value of κ.

14.4 Wave equations for a cold plasma in a dipole field Box models of the magnetosphere are very useful but it is often necessary to be more realistic. The inner magnetosphere has a magnetic field that is approximately dipolar. In general, its modes of oscillation are complicated and require numerical computation. They do not take the form of simple standing waves. Because the magnetic field and density are functions of position and their gradients cannot be ignored, the wave equations do not separate into equations describing modes that can be identified with the transverse Alfv´en and isotropic Alfv´en waves. Consider a wave in a cylindrically symmetric medium with negligible plasma pressure. In principle, it would be possible to use the wave equation (7.34), which Copyright © 2005 IOP Publishing Ltd.

Wave equations for a cold plasma in a dipole field

261

applies generally in a non-uniform medium. We can, however, remove a great deal of algebraic complexity by linearizing the MHD equations directly and then solving them in an appropriate coordinate system. Initially, we use an orthogonal curvilinear system of magnetic coordinates, µ in the magnetic field direction, ν inwards towards the centre of curvature of the field line, and φ westward in the azimuthal direction, as described in appendix B.3. It is convenient to linearize (2.42) with current density given by (2.54) to write the momentum equation in the form 1 ∂v = (∇ × b) × B (14.55) ρ0 ∂t µ0 and Faraday’s law (2.53) in the form ∂b = −∇ × E ∂t with the frozen-in field condition (2.52) E ⊥ = −v ⊥ × B

(14.56)

E = 0

(14.57)

The right-hand side of (14.55) is a vector perpendicular to B . Thus, the velocity perturbation is perpendicular to the magnetic field and vµ = 0. We see, therefore, that E ν = −vφ B E φ = vν B. (14.58) We define new scaled variables

 and
by

µ = h µbµ ν = h ν bν φ = h φ bφ
ν = h ν Eν
φ = h φ Eφ

so that vν =


φ

(14.59) (14.60)


ν

vφ = − . (14.61) Bh φ Bh ν We now write out the components of (14.55) and (14.56) for harmonic waves, varying with time as e−iωt , in the curvilinear system, using the expressions in appendix A.2 for the vector operations. Because of the azimuthal symmetry of the background, we can assume that the dependence on φ is of the form eimφ . The result is   hν 2 ∂ φ − im µ iω ν = V (14.62) h φ h µ A ∂µ   hφ ∂ ν ∂ µ − (14.63) V2 iω φ = − h µ h ν A ∂µ ∂ν     ∂ φ 1 ∂ φ hµ − im ν = − im ν (14.64) iω µ = h ν h φ ∂ν r0 ∂ν hν ∂ φ iω ν = − (14.65) h φ h µ ∂µ hφ ∂ ν iω φ = (14.66) h µ h ν ∂µ




















Copyright © 2005 IOP Publishing Ltd.




















262

MHD wave equations in non-uniform media

where, for a dipole field, the scale length r0 is the Earth’s radius.

14.5 Multicomponent plasmas Suppose that the pressure of the plasma is comparable with the magnetic pressure so that β ∼ 1. In such a case, the characteristic waves in a uniform plasma are the transverse Alfve´ n wave and the two magnetosonic waves, described in chapter 7. Consider where standing oscillations, with β ∼ 1, may occur in the solar–terrestrial system. We require a region where field lines are bounded by a conducting region to form a boundary. This limits us to the interior of the magnetosphere. The only suitable region with sufficiently high pressure is the ring current region. This region is inherently too complicated to be treated by simple one-fluid MHD. The reason is that there are two plasma populations, a cold plasma population originating from the ionosphere and an energetic population that has been injected from the magnetotail. On the time scales of the problem, these do not interact. The partial pressure of the cold plasma component is negligible; hence, the pressure arises entirely from the hot component. Observations of the properties of the energetic particles show that they are generally far from thermal equilibrium so that the hot plasma should be regarded as collisionless. The proper treatment of such a problem requires kinetic theory. The sound speed and the thermal speed are of the same order of magnitude so that, since the length of a field line is of the same order as the wavelength in the problem, resonance effects such as those described in chapter 8 are important. The problem has been approached from this point of view by Southwood [193, 194]. We shall discuss such a treatment in chapter 23. Nevertheless, there are insights into the standing-wave behaviour that can be provided by a fluid approach [224, 232, 234, 244]. It is this approach that is adopted here. We assume that any growth or loss processes, arising from kinetic effects, are in balance and may be ignored. This is the basis of the quasilinear approach (chapter 23). The plasma consists of two fluids, a hot component with β ∼ 1 and a cold component with negligible pressure. For realism, we should consider an anisotropic pressure but this introduces complexity without adding to the understanding of the points that we wish to illustrate. We also ignore the effect of hot electrons that may contribute to the pressure. The crucial points that we wish to illustrate are as follows. • •

If we have two such fluids, the motion of both fluids perpendicular to the magnetic field is the same E × B drift. Their motions parallel to the field are independent. The distribution of the background plasma along the field is different for the two species. The strongest effects occur where the plasma β is near unity and this occurs in localized regions along the field line.

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Multicomponent plasmas

263

14.5.1 Background model If there is a significant pressure gradient in the plasma, we can no longer use the dipole coordinates described in appendix B because the current density is nonzero and the magnetic field cannot be found from a scalar potential. We can, however, define local coordinates which hold in the neighbourhood of a given field line, defined by ν = ν0 . These may be used when studying oscillations that are confined to the immediate neighbourhood of a field line. We shall restrict ourselves to field lines without torsion that lie in a plane. The field is assumed to be cylindrically symmetric about an axis through the centre of the Earth. It may, however, be significantly distorted from a dipole shape. We retain µ, ν , φ coordinates, with µ ˆ parallel to B , νˆ towards the centre of curvature, and φˆ westward, completing the right-hand set. Along a flux tube, the flux Bh ν h φ δνδφ is constant. Thus, as in the case of zero pressure, h ν h φ ∝ B −1 . In this case, since we are defining coordinates localized near a reference field line, we choose the constant to be unity so that (14.67) h ν h φ = B −1 . The scale along the µ-axis is arbitrary on the reference field line. Its dependence on ν is, however, determined from the geometry. Figure 14.2 shows that ∂h µ = −h µ h ν κ ∂ν

(14.68)

where κ = R −1 is the curvature, as defined in appendix B. We can no longer use the magnetic scalar potential to represent the coordinate µ. It is convenient to define the arbitrary variation along the reference field line such that h µ = B −1 . On adjacent field lines, from (14.68), it is given by h µ (ν) = h µ (ν0 ) + h µ h ν κ(ν − ν0 ).

(14.69)

The momentum equation (2.8) for the unperturbed hot plasma component may be written in the form   B2 B · ∇B −∇ P + =0 (14.70) + 2µ0 µ0 and if we use (B.11), we may write the components as ∂P =0 ∂µ ∂B µ0 ∂ P = − + hν κ B ∂ν B ∂ν ∂P ∂B =0= . ∂φ ∂φ Copyright © 2005 IOP Publishing Ltd.

(14.71) (14.72) (14.73)

264

MHD wave equations in non-uniform media  $ h  

 

€



L

h 

#h  " !h $   # 

R

Figure 14.2. Geometry for finding ∂ h µ /∂ν .

In this cylindrical symmetric case, the current density is in the azimuthal direction and since J × B = ∇ P , it is given by J=

φˆ φˆ ∂ P = (κ − κν ) B Bh ν ∂ν µ0

(14.74)

where κν is given by (B.13). 14.5.2 Linearized equations Because there are transverse gradients of magnetic field and pressure, we cannot use the linearized equations of chapter 7. We must include additional terms involving the products of zero-order gradients and first-order field variables. The motion of the plasma perpendicular to the field is the same for both hot and cold components. Faraday’s law is unaffected by these additions so equations (14.64), (14.65), and (14.66) still apply. The momentum equation is the same as (14.55) with additional force density terms J × b, arising from the zero-order current, and −∇ p, arising from the pressure of the hot plasma component. The parallel component of this equation was zero for the case of zero pressure. For the transverse motion of the plasma, both hot and cold components have the same E × B drift; and the density appearing in the transverse component of the momentum equation is the total density ρ0 . Only the hot component of the plasma participates in the parallel motion so the density appearing in the parallel component of the momentum equation is its density ρh . From (14.74), J × b = −µ(κ ˆ − κν ) Copyright © 2005 IOP Publishing Ltd.

Bbν Bbµ + νˆ (κ − κν ) . µ0 µ0

(14.75)

Wave equations in a cylindrically stratified medium

265

We also have to modify the adiabatic law for the hot plasma to include a term −v · ∇ P. The resulting equations are as follows

µ = ∂∂ν
φ − im
ν h ν ∂
φ iω = −

iω

ν

iω

φ =

iω iω


ν =


φ =

iωvµ = iω

p = P

(14.76)

h φ h µ ∂µ hφ ∂ ν h µ h ν ∂µ   ∂ φ im B h ν hν − im µ − VA2 p hφ hµ ∂µ ρ0 h φ   ∂ ν ∂ µ 1 ∂p hφ hφ − + − − VA2 (κ − κν )VA2 hµhν ∂µ ∂ν hµ ρ0 h 2ν ∂ν B(κ − κν ) 1 ∂p ν+ h ν µ0 ρh h µ ρh ∂µ   ∂ φ γ B ∂  vµ  +γ − im ν h µ ∂µ B ∂ν 2h ν {(1 + 12 βγ )κν + (1 − 12 βγ )κ} φ . − β

















(14.77) (14.78) (14.79)

µ (14.80) (14.81)







(14.82)

These equations have been derived for a plasma that is a mixture of cold and hot components, where it is assumed that they do not interact through collisions. The motion of the two plasmas perpendicular to the magnetic field is the same because they both move with an E × B motion. They move independently parallel to the field; in fact, it will be seen that the perturbation of the cold plasma parallel to the field is zero, while the hot component has a parallel motion and consequent pressure perturbation. If there is no cold component, we can set ρh = ρ0 in (14.81) to make the set of equations suitable for waves in a simple MHD medium with a dipole configuration. In this case, we note that V2 + V2 1 1 + βγ = A 2 S 2 VA

(14.83)

V2 − V2 1 1 − βγ = A 2 S . 2 VA

(14.84)

14.6 Wave equations in a cylindrically stratified medium In a cylindrically symmetric medium with the coordinates r , φ, z described in section 14.2.2, we can assume that the wave fields vary in the φ- and z-directions as exp{imφ + k z z}. If the field lines are straight with Bφ = 0, the resulting equations are very similar to those for a plane-stratified medium. Following Copyright © 2005 IOP Publishing Ltd.

266

MHD wave equations in non-uniform media

exactly the same steps as for the plane case, except for the use of appropriate scale factors for the cylindrical system we get



dψ = (x)ξ dr 1 d(r ξr ) (x) = − ψ r dr (x)

(14.85)

 

(14.86)

where

 = ρ0 [ω2 − (k · V A )2] 4  = ω2 (V 2 + V 2)ω− (k · V A

S

(14.87) A)

2V 2 S

−

m2 r2

− k z2 .

(14.88)

For the cylindrically symmetrical case, however, it is possible for the magnetic field to have a component in the φ-direction, so that, at any radius r , the magnetic field line is a spiral with pitch angle α so that Bφ = B sin α

Bz = B cos α.

(14.89)

The magnitude of the magnetic field and α are each functions of r . The corresponding first-order wave equations for this case were first derived by Appert et al [16]. In cylindrical coordinates, the equilibrium condition is (14.1). We assume spatial and temporal variation of the form exp{−i(ωt − mφ − k z z)} and define m k = φˆ + zˆ k z . (14.90) r We also replace the velocity perturbation by the displacement given by  t iv (14.91) ξ= v dt = ω where ξ has components ξr , ξφ , ξz in the r -, φ- and z-directions. We also define the displacement perpendicular to the gradients and to the magnetic-field direction ˆ φ + zˆ ξz ξ ⊥ = φξ with magnitude ξ⊥ =

'

ξφ2 + ξz2 .

(14.92)

(14.93)

In cylindrical coordinates, the reduced MHD equations (2.55), (2.57), and (2.58) are then 1 {B × (∇ × b) + b × (∇ × B)} µ0 p = − γ P∇ · ξ − ξ · ∇ P

(14.95)

b = ∇ × (ξ × B).

(14.96)

ρ0 ω2 ξ = ∇ p +

Copyright © 2005 IOP Publishing Ltd.

(14.94)

Summary

267

We may define a generalized pressure ψ = p+

B·b µ0

(14.97)

and eliminate ξφ , ξz , and b⊥ from these equations, getting







d (r ξr ) = − ψ + ξr r dr dψ = { 2 + }ξr − ψ dr



where





(14.98)



(14.99)

 and  are given by (14.87) and (14.88), and   2B  sin α d B sin α = µ0

dr

r

4ρ 2 B 2 sin2 α 2 2 + 0 {VA sin α( + k 2 ) − (k µ0 r 2 & 2B 2 sin2 α % m + k z2 − k z cot α . µ0 r r



=

· V A )2 }



The magnitude of the transverse displacement is    m cos α 2VA (k · V A ) sin α cos α − k z sin α ψ − ξr . ξ⊥ = i r r



(14.100) (14.101)

(14.102)

Apart from the notation, these three equations are the same as those given by Appert et al [16].

14.7 Summary •

• •

It is often possible to use box models, where the magnetic field lines are straightened and bounded by conducting planes, to study wave propagation in the magnetosphere. If the plasma parameters have a gradient perpendicular to the field lines, such models give a good qualitative picture apart from the scaling arising from the distorted coordinates. Sometimes it is necessary to use a more realistic geometry using dipole field lines. Some problems in solar physics, such as sunspots and coronal loops, can be treated by using cylindrical structures, in which the magnetic field may have a spiral structure. To study waves in such systems, it is necessary to linearize the MHD equations, including the products of zero-order gradients with first-order wave perturbations. Sets of simultaneous first-order differential equations have been derived for box models. Equivalent second-order differential equation have also been found, together with approximate forms appropriate near singularities.

Copyright © 2005 IOP Publishing Ltd.

268 •

• •

MHD wave equations in non-uniform media In a dipole geometry, corresponding differential equations have been derived for a cold plasma and for a plasma that is a mixture of hot and cold components. In the latter case, the components move with the same velocity perpendicular to the magnetic field but have different motions parallel to the field. The equivalent equations applying to a cylindrical geometry have also been derived The purpose of the chapter has been to provide a convenient collection of equations that will be of use throughout the remainder of the book.

Copyright © 2005 IOP Publishing Ltd.

Chapter 15 Propagation in a plane-stratified medium

15.1 Introduction Propagation of MHD energy transverse to the magnetic field is largely carried by the fast wave, with a small contribution from the slow wave in circumstances where it is not heavily damped. In low-β plasmas, the fast wave becomes the isotropic Alfve´ n wave and the slow wave disappears. The energy of a transverse Alfve´ n wave is propagated exactly parallel to the magnetic field and it cannot contribute to perpendicular propagation. This somewhat pathological behaviour of the transverse Alfve´ n wave leads to singularities in the general differential equations describing MHD wave propagation. The understanding of wave propagation in varying anisotropic media is complicated, even without the introduction of singular behaviour. For this reason, in this chapter, we study the propagation of fast and slow waves in anisotropic MHD situations where the transverse Alfve´ n wave is decoupled from the other characteristic waves and can, therefore, be ignored. The propagation of transverse Alfv´en waves is taken up in later chapters. We confine ourselves to the box models of chapter 14. Here all the parameters of the unperturbed plasma are functions only of one Cartesian coordinate. Emphasis is placed on the WKBJ (Wentzel–Kramers–Brillouin– Jeffreys)1 approximations in slowly varying media. These allow the visualization of waves analogous to the harmonic waves in uniform media with the phase integral k · dr replacing the linear phase variation k · r and a slowly varying amplitude factor replacing the constant amplitude. The WKBJ solutions are remarkably good over a wide range of conditions. Using them, we can apply familiar techniques from wave theory in uniform media, such as representing standing waves as the superposition of two crossing propagated waves. 1 Depending on the prejudices of the authors, these have been variously named WKB, JWKB, WKBJ,

or LG (Liouville–Green) solutions [32, section 7.5].

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269

270

Propagation in a plane-stratified medium

15.2 Wave propagation through numerical computation Equations (14.15), (14.16), and (14.19) are a set of first-order differential equations for MHD waves in a plane-stratified medium. They are in a suitable form for numerical solution. In regions remote from the singularities of these governing differential equations, progressive waves are propagated. In such a box model, the undisturbed plasma varies only with x and the direction of the magnetic field is perpendicular to the x -direction. These are three equations in three unknowns. They can be integrated step by step using Runge–Kutta techniques [159]. If we wish to use analytical techniques it is more convenient to replace (14.15), by the second-order equation (14.26), or (14.16) by (14.25). Then the second-order equation is an equation for a single variable. It can be solved with the appropriate boundary conditions. The remaining two first-order equations then determine the behaviour of the other two variables. In general, for an arbitrary dependence on x , a numerical solution is necessary. If initial conditions are determined at some value of x , then, in principle, the first-order equations (14.15), (14.16), and (14.19) can be integrated step by step to find the solution at other values of x . In practice, this technique may be numerically unstable. Such instability usually has a physical basis. In regions where the wave is evanescent, there are generally two solutions, one growing and one decaying exponentially with x . An attempt to compute the decaying solution means that a small numerical rounding error can grow exponentially to swamp the desired solution. In the case considered here, since there are only two solutions, this can be controlled by designing the computation so that the wanted solution is the growing one. When the magnetic field is not perpendicular to the x -axis, a second-order equation cannot be separated out in general. In such a case, the general solutions to the equation may be superpositions of propagated and evanescent waves. If one is seeking a solution describing only the propagated wave, then the evanescent wave may still lead to numerical swamping. In such cases, the numerical swamping may cause trouble and the equations must be manipulated analytically to remove the unwanted solution. Examples of such cases are described in chapter 17 where an evanescent fast wave is analytically removed before numerical integration.

15.3 WKBJ solutions of the wave equation Numerical computation of waves does not necessarily give physical insight, although it may provide accurate results. Such insight is best provided by considering analytic approximate solutions which can act as benchmarks against which to compare the numerical results. We provide such a benchmark by assuming that the medium is sufficiently slowly varying for the so-called WKBJ solutions to be used. These are asymptotic approximations to solutions of certain differential equations, which hold when the parameters vary sufficiently slowly. Copyright © 2005 IOP Publishing Ltd.

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271

They are the next higher approximation to the solution after the phase integral method of section 11.2. Consider the second-order wave equation (14.26). We write it in the form

  ψ  + k 2 q 2(x)ψ = 0 (15.1)   . The form of the dependence of  on x is specified by the ψ  −

where k 2 q 2 = dimensionless function q and k is a constant parameter with dimensions of inverse length. Suppose q varies with z on a length scale l such that q  /q ∼ l −1 . Then, if k l −1 or, equivalently, kq q  , the wavelength λ q  /2πq. In this case, the medium is slowly varying in the sense that its properties do not change significantly in one wavelength. The coefficient of the term in ψ  is small if the medium is slowly varying, except where is near zero. In order to solve this equation, we note that the solution with q = 1 and  = 0 is exp{±ikq x} so that the phase depends linearly on x. We anticipate that the approximate solution will have a phase which varies with x but does not change linearly. We, therefore, assume a solution of the form





ψ = exp i(x).

(15.2)

If this is differentiated and the result substituted in (15.1) we get a differential equation for : i − ( )2 − i

   + k 2 q 2(x) = 0. 

(15.3)

At first sight, this does not represent progress. We have replaced a linear differential equation by a nonlinear one. We proceed, however, by noting that, for constant q,  is linear in z so that  is zero. For the slowly varying case, we expect  ( )2 . Also, we assume that the medium is sufficiently slowly varying and x is sufficiently far from the point x 0 at which = 0, that we can assume   / ∼  . Then



 

  = ± k 2 q 2 + i − i

  1/2  ± kq + i  − i    + · · ·  2kq 

(15.4) where we have expanded the square root by the binomial theorem. We solve this by successive approximations. The first approximation is 1 = ±kq. If this is differentiated and substituted for the  in (15.4), we get a second approximation   i q  − . 2 = ±kq + 2 q

 

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This can be integrated to give  2 = ±

z

kq dz + i ln q 1/2 − i ln

 1/2

which can be substituted in (15.2) to give the WKBJ solutions  W± (x) =

 (x) 1/2 exp  ± ik  z q(x) dx .

(15.5)

q(x)

This is the WKBJ approximation to the differential equation (15.1). In many wave problems, the first-order term in the differential equation is zero. This occurs is constant so that  = 0. The WKBJ approximation in this case is when found by setting = 1 in (15.5). Budden [31, section 9.6] discusses this case in detail.







15.3.1 Energy flux The WKBJ approximation for the generalized pressure ψ = p + B · b/µ0 , when the wave is propagated in the +x-direction, is  ψ(x) =

 (x) 1/2 exp ik  z q(x) dx .

(15.6)

q(x)

The generalized pressure is related to the displacement ξ by (14.16). The expression for the corresponding WKBJ solution for ξ can, therefore, be found by differentiating (15.6). To the accuracy of the approximation, terms in  and  are ignored, so that the differentiation is equivalent to multiplying by ikq. Then



vx (x) = −iωξ(x) =

1 dψ = ωk dx





q(x) (x)



1 2

  exp ik

z

 q(x) dx .



(15.7)

From (9.26), the x-component of the time-averaged energy flux is   B·b 1 x (x) = Re p + v˜ x = ψvx . 2 µ0

(15.8)

We see then that the product of the two WKBJ solutions for ψ and vx (or, equivalently, ξ ) when q and are real produce a constant energy flux, since the amplitude terms cancel. We see then that the phase integral gives an approximation to the variation of phase in the wave, while the amplitude variation of the WKBJ solutions is just what is required to ensure that energy is conserved. The energy flux component x is a special case of a wave invariant, such as that discussed in section 18.3.



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15.3.2 Error terms in the differential equation The WKBJ solutions may be differentiated twice to show that they satisfy the following differential equation exactly: W



  W  + k2q 2W = 3 −  4



q q

2

 −

  2 



1 W− 2



q  − q

   W = 0.  (15.9)

They are approximate solutions of (15.1) provided that   "    1 3 q 2  −  2 2 k q 4 q

  2 

#

 1 q  − − 2 q

    1. 

   

(15.10)



When  = 0, this is the same as the error expression for the equation discussed by Budden [31, equation (9.29)]. The parameter k is clearly a useful measure of the accuracy: the larger the value of k is, the better the approximation will be. Many books leave the question of accuracy at this point, in that it is necessary to specify the form of q before any error determination can be made. It is as well to consider matters more carefully. In practice, there are three different situations which need to be considered. • • •

Even although k is large, over many wavelengths, error in phase may accumulate. An error which may be a minute fraction of the total phase change may still be comparable with π. When q 2 approaches zero, i.e. if → 0, it is clear that the approximation fails. Near the singularity of the equation, where → 0, it also fails.





We deal with the first two of these cases in this chapter. The third is deferred to chapter 18.

15.4 Cumulative error in the WKBJ solutions We consider the accuracy of the WKBJ solutions over many wavelengths. We shall be concerned with propagated waves, not evanescent waves, and we shall assume that we do not approach any region where there are zeros of or
. The nature of the WKBJ solutions is that they separate waves propagated in opposite directions. In a medium that varies in space, the gradients of the characteristic speeds, VA and VS lead to partial reflection of a propagated wave. As a wave encounters an increase or decrease of refractive index it is partially reflected, which implies that it is partially converted into the other WKBJ solution propagated in the opposite direction. In this section, we discuss the magnitude of these effects and their influence on the accuracy of the WKBJ solutions. Our approach is a generalization of that of Mathews and Walker [137].

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15.4.1 Method of variation of parameters The solution of the approximate equation (15.9) is a linear combination of the WKBJ solutions: (15.11) W = a+ W+ + a− W− where a+ and a− are constants. This suggests that the solution of the exact equation (15.1) can be expressed by replacing these constants by parameters a± (z) which vary slowly. This is the basis of the method of variation of parameters. Define a± (z) in terms of ψ and ψ  : ψ(x) = a+ (x)W+ (x) + a− (x)W− (x)

(15.12)

 (x) + a (x)W  (x). ψ  (x) = a+ (x)W+ − −

(15.13)

These can be solved for a± yielding a± = ±

 − ψW ψ W∓ ∓

 − W W W+ W− − +

.

(15.14)

From the definition (15.5), the denominator (the Wronskian of the WKB function) is equal to −2ik . The expression can be differentiated and (15.1) and (15.9) used to eliminate the second derivatives of v and W± . The result is  da± i g(x)W ψ − a± (15.15) = ± dx 2k     x  i q dx g(x) ± 2ikq = ± a± + a∓ g(x) exp ∓ 2ik 2kq

(15.16) where g(x) =

 "     2 # 3 q 2 1 q  − − − 4 q 2 q

 

.

(15.17)

The expressions for the derivatives of a± are two simultaneous first-order differential equations for a+ and a− . One approach [87, p 30] is to use (15.15) to express a± as integrals which are substituted in (15.1). The resulting integral equation can be used to provide better estimates of the solution by successive approximations. The second approach would be to integrate (15.16) numerically to find the exact solution. There is no great advantage over directly integrating the exact equation (15.1), except that, for positive q 2 , the oscillatory behaviour has been removed which may lead to greater numerical accuracy, and, for negative q 2 , the exponential growth which leads to numerical instability has been removed. We shall not pursue either of these courses. The most important use of the equations is to estimate the cumulative error in the WKB solutions.

15.4.2 Cumulative error If we are concerned with cumulative error over many wavelengths, we need to consider the case where q 2 is real and greater than zero, so that q is real. If q is imaginary, the wave is evanescent or inhomogeneous so that any physically realistic solution only extends over a few e-folding distances. If the medium is lossy, then q 2 may have a small imaginary part but this can be treated as a perturbation and does not contribute to the primary error.

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Consider then solutions on the real axis (z = x) where q 2 is positive and real and does not approach zero over a great distance. We first consider a solution ψ which approximates to a progressive wave rather than a standing wave. Suppose that, at x = x0 , a+ = a0 , a− = 0. We expect that, if the WKBJ approximation is good, a− will remain small, of first-order in k −1 , compared with a+ over the whole range. In such circumstances, a first-order approximation to (15.16) is   i − g(x) − a+ 2kq # "  x i da− q dx a0 . = exp 2ik dx 2kq

da+ = dx

Integration gives a first-order correction to the WKBJ solution for a+ :     x 3 q  2 1 q  i −1 ln( a+ ) = − q − dx 2k x0 4 q 2 q "  # x a q dx a− = 20 2 exp 2ik 4k q

(15.18) (15.19)

(15.20)

(15.21)

The first of these equations shows that the correction to a+ , to this order, is a correction in phase. The factor involving scales the amplitude. The second equation shows that the variation in the medium leads to the generation of a small amount of the oppositely propagated wave, because of partial reflection at the gradients. This, however, is O(k −2 ) and oscillates with twice the spatial frequency of the wave, so can safely be ignored. The process of successive approximation could be carried out to higher order but it is not worthwhile. Instead, (15.20) will be used to estimate the cumulative error in the phase. Let l1 be the characteristic length scale on which the properties of the medium change and l2 be related to the second derivative of g(x). They are defined by     2 q 2 l1−2 ∼ − (15.22) q     q . (15.23) − l2−2 ∼ q Let l −2 = 34 l1−2 − 12 l2−2 and let λ = 2π/kq be the wavelength. Define the mean value of any function of x over a range x − x0 as x x f (x) dx  f (x) = 0 . (15.24) x − x0 Then the error in using the WKBJ approximation over this range is, from (15.20),   |(x − x0 )|λ/2πl 2  < |(x − x0 )(λ/2πl 2 )max |.

(15.25)

In problems where phase is important (for example, interference between two coherent waves which have reached a point after travelling along different paths), this error must be

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small compared with 2π rather than the total phase change which will be 2nπ where n is the number of wavelengths. Since n ∼ |x − x0 |/λ, a crude requirement is that n

λ2  < 1. 2πl 2

(15.26)

Thus, if the typical fractional change in the properties of the medium is 10% per wavelength, then the error in phase after 200π wavelengths is only about one radian at most. The discussion here applies to a plane-stratified medium. We see that, for a propagated wave, the WKBJ approximations give good results over many wavelengths, provided that the variation of the medium over one wavelength is small, say about 10%. In varying media, the problem can be treated by ray-tracing. The ray-tracing equations are based on the phase integral approximation. While we have not discussed the matter quantitatively, it is to be expected that the error per wavelength would, by similar arguments, be just as small.

15.5 Reflection In this section, we discuss analytic solutions of the wave equations for planestratified media, near a level where = 0 and the WKBJ solutions break down. We shall see that, at such a level, reflection takes place. We first consider the geometrical optics of the problem and then look at wave treatments of the problem.



15.5.1 Geometrical optics of reflection We illustrate here the problems that arise when treating reflection by geometrical optics. Consider a slowly varying medium, in which all the properties depend only on the spatial coordinate x. Suppose also that the magnetic field is in the z-direction. We considered such a situation for a simple sound wave in section 11.2.2, where we noted that the phase integral solution broke down when k z = ω/VS . At this point, k x = 0 and the phase integral solution was no longer valid. On the basis of simple physical arguments, we concluded that the wave was reflected at such a level. Let us consider such a situation from the point of view of the ray-tracing equations (11.16) and (11.17). We suppose that the wave normal is in the x y-plane so that it is normal to the magnetic field and the wave is, therefore, a fast wave, propagated perpendicular to the field with a dispersion relation given by (7.49) with k z = 0: ω2 = k 2 V 2 (x)

(15.27)

where V 2 = VA2 + VS2 . Then equations (11.16) and (11.17) may be written in the form dx kx V 2 = dt ω Copyright © 2005 IOP Publishing Ltd.

ky V 2 dy = dt ω

dz =0 dt

(15.28)

Full wave theory of reflection

277

x xxR

y

x

Figure 15.1. Formation of caustic surface at reflection level.

dV dk x =k dt dx

dk y =0 dt

dk z = 0. dt

(15.29)

The last two these show that k y and k z are constant. This is simply an expression of Snell’s law. The ratio of the first two equations is ' ω2 − k 2y V 2 kx dx = . (15.30) =± dy ky V This gives x as a function f (y) of y, the equation of the ray path. Assume that, at x = 0, the argument of the square root is positive and that V decreases monotonically with x. Then, at some point x = x R , the slope of the ray is zero, and the function f (x) has a stationary value. For x > x R , f (y) is imaginary and there is no propagation. The wave is reflected at x = x R . A number of adjacent rays with the same value of k y are sketched in figure 15.1. They are all tangential to the surface x = x R which is an envelope surface forming the boundary between the region for which propagation occurs and the forbidden region x > x R . Such a surface is called a caustic surface. The phase integral approximation breaks down at such a surface. A full wave treatment is necessary. Such a treatment shows that there are phase changes on reflection that are not included in the phase computed by the phase integral. Also the rays are all focused along the caustic and the approximations of geometrical optics diverge there—a full wave theory corrects for this.

15.6 Full wave theory of reflection



The caustic described earlier occurs at the value x R of x for which = 0: for x < x R , the wave is propagated; for x > x R , it is evanescent. The point x = x R is often called the turning point of the equation. In order to study the behaviour of the fields near a turning point, we start with the second-order differential equation (14.26) for ψ, the generalized pressure. As described in section 14.3.3.1, Copyright © 2005 IOP Publishing Ltd.

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near a turning point, the approximate form of the equation can be transformed to the Stokes equation with its standard form given by (14.38), where ζ is related to x by (14.37). The turning point of the equation is then ζ = 0. The solutions of this equation describe the behaviour of waves near the turning point and are typical of the behaviour of waves near an isolated caustic surface. We, therefore, discuss these solutions in some detail. 15.6.1 Stokes’ equation and Airy functions Stokes’ equation (A.26) is d2 ψ = ζ u. dζ 2

(15.31)

ψ = A Ai(ζ ) + B Bi(ζ ).

(15.32)

It has a general solution

The special functions Ai and Bi are the Airy functions [1, section 10.4], which have been extensively tabulated. The derivatives Ai (ζ ), Bi (ζ ) are also useful. The equation which they satisfy can be found by differentiating (A.26) once getting d2 ψ  = ψ + ζ u. dζ 2 Then, since ψ = ζ −1 dψ  /dζ , ψ  satisfies the equation 1 dψ  d2 ψ  − ζ ψ = 0 − dζ 2 ζ dζ

(15.33)

which is the same as (14.39). The Stokes equation has no singularities for finite ζ . It solutions are, therefore, finite and single-valued for finite ζ . The solutions are real for all real ζ . This can be seen, for example, by examining the series solutions which can be found by standard methods. It will be useful to discuss the qualitative properties of its solutions because we shall be studying it in greater detail later. The second derivative has the same sign as the product of ζ and ψ. Thus, when ζ > 0, the curvature is positive for positive ψ and negative for negative ψ. The opposite is true for ζ < 0. Consider figure 15.2. Suppose that, at ζ = 0, the value of ψ is positive and there is a small negative slope. For ζ > 0, the curvature is positive and if the negative slope is sufficiently small, the graph curves upwards and does not reach the axis. The curvature remains positive and ψ → ∞ as ζ → ∞ as shown by the broken curve. Now suppose the initial slope is made more negative so that as ζ is increased, the upward curvature is insufficient to prevent the graph crossing the axis. As soon as the axis is crossed, the curvature becomes negative. The graph curves downwards so that ψ → −∞ as ζ → ∞ Copyright © 2005 IOP Publishing Ltd.

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Figure 15.2. Qualitative behaviour of solutions of the Stokes equation.

as shown by the dotted curve. There is a critical value of the slope for which the curve approaches the axis asymptotically and ψ → 0 as ζ → ∞ as shown by the full curve. The Airy function Ai(ζ ) is chosen to have this behaviour as ζ → ∞. When ζ is negative, the sign of the curvature is opposite to the sign of ψ. Thus, each of the graphs curves towards the axis until the sign of ψ changes when it curves in the opposite direction until the axis is crossed so that the behaviour is oscillatory. The other characteristic solution Bi(ζ ) increases without limit as ζ → ∞. These standard functions can be defined in terms of series solutions of the equation [1]. This type of behaviour is representative of the behaviour of the wave near the reflection or turning point. In general, the solution of the equation is a linear combination of the Airy functions. If there is an infinite medium in the region ζ > 0, then the solution must remain finite as ζ → ∞ so that the coefficient of Bi(ζ ) is zero and the behaviour of the wave is represented by Ai(ζ ). The disturbance oscillates with frequency ω. The function Ai(ζ ) is then the envelope of the oscillation. For ζ < 0, the envelope has the general form of the envelope for standing waves on a stretched string, with a series of nodes and antinodes. For ζ > 0 the oscillation dies away exponentially with increasing ζ . Once again, insight into the wave behaviour can be gained by considering the WKBJ solutions. The reason for this is that they can be expressed as easily visualized propagated waves that can be linearly combined to produce standing waves. They can also be easily related to the ray-tracing picture of geometrical optics.

15.6.2 WKBJ solutions of the Stokes equation Comparison of the Stokes equation with the WKBJ solutions of (15.1) shows that its WKBJ solutions are 2 3/2

W±  ζ −1/4 e± 3 ζ Copyright © 2005 IOP Publishing Ltd.

.

(15.34)

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The condition for validity becomes   5 1    16 ζ 3  1.

(15.35)

Similarly, the WKBJ solutions of the derivative of the Stokes equation are 2 3/2

W±  ζ 1/4e± 3 ζ

.

(15.36)

15.6.3 The Stokes phenomenon The representation of functions by asymptotic solutions consisting of linear combinations of WKBJ solutions of their governing differential equations needs care. We illustrate the process by considering the Airy function Ai(ζ ). The behaviour of this function, as sketched in figure 15.2, is oscillatory for negative ζ and it decays smoothly to zero as ζ → +∞. The WKBJ solution of the Stokes equation (15.34) with the lower sign is real for ζ > 0 and shows the correct behaviour as ζ → +∞ so on the positive real axis for |ζ | sufficiently large to satisfy (15.35), we must have Ai(ζ ) ∼ Aζ −1/4 exp{ 23 ζ 3/2}

(15.37)

where A is a constant chosen so that the Airy function is correctly normalized. We have seen that Ai(ζ ) is real and single-valued everywhere on the real axis. When ζ < 0, the WKBJ solution, with a factor ζ 3/2 in the exponent, is complex and double-valued. The expression (15.37) cannot, therefore, represent the function Ai(ζ ) on the negative real axis; therefore, the solution in this region, where |ζ | is large enough to satisfy (15.35), must be a linear combination of the two WKBJ solutions, chosen to give the proper real oscillatory behaviour: Ai(ζ ) ∼ A+ ζ −1/4 exp{ 23 ζ 3/2 } + A− ζ −1/4 exp{− 23 ζ 3/2 }.

(15.38)

The WKBJ solution fails near the origin where |ζ | does not satisfy (15.35). As we pass from one region of validity to the other, the constants in the WKBJ approximation change discontinuously. For the function Ai(ζ ), we shall find that B = iA. This is known as the Stokes phenomenon. 15.6.4 WKBJ approximations to the Airy functions When such effects are taken properly into account, we can find expressions in standard form2 for the Airy functions and their derivatives. We list these here for various regions of the complex plane. In general, these expressions are superpositions of the two WKBJ solutions of the Stokes equation. The method of finding the relationship of these two solutions is described in section 15.7. 2 See section 15.7.3 for a discussion of the difference between these definitions and the Poincare´ definitions quoted by many authors.

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15.6.4.1 Airy functions In these expressions, the multiplying constant ensures that the expressions are in standard form: Ai(ζ ) ∼ 12 π −1/2 ζ −1/4 e− 3 ζ

2 3/2

Ai(ζ ) ∼ 12 π −1/2 ζ −1/4 {e Bi(ζ ) ∼ Bi(ζ ) ∼ Bi(ζ ) ∼

− 23 ζ 3/2

(− 23 π ≤ arg ζ ≤ 23 π) + ie

2 3/2 3ζ

}

( 23 π ≤ arg ζ ≤ 43 π)

2 3/2 2 3/2 1 −1/2 −1/4 ζ {−ie− 3 ζ + 2e 3 ζ } 2π 2 3/2 1 −1/2 −1/4 − 23 ζ 3/2 ζ {ie + 2e 3 ζ } 2π 2 3/2 1 −1/2 −1/4 − 23 ζ 3/2 ζ {ie + e3ζ } 2π

(− 23 π

(15.39) (15.40)

≤ arg ζ ≤ 0) (15.41)

(0 ≤ arg ζ ≤ 23 π)

(15.42)

( 23 π ≤ arg ζ ≤ 43 π).

(15.43)

In these expressions, the principal values of quantities with fractional exponents are understood. Thus, ζ −1/4 = |ζ | exp{− 14 i arg ζ }. On the real axis where arg ζ = 0, either (15.41) or (15.42) may be used because the subdominant term is small over the whole range. It is not small in all parts of the region of the complex plane given as the range of validity of the approximation and must be included with the correct sign as the approximation is continued into the complex plane. 15.6.4.2 Derivatives of Airy functions The asymptotic approximations to the derivatives of Ai and Bi can be most conveniently found by differentiating the approximations for the Airy functions. Only the exponential factors should be differentiated as the terms involving the derivatives of ζ −1/4 are smaller than the error in the approximation. Ai (ζ ) ∼ − 12 π −1/2 ζ 1/4 e− 3 ζ

2 3/2

Ai (ζ ) ∼ − 12 π −1/2 ζ 1/4 {e

− 23 ζ 3/2

2 3/2

Bi (ζ ) ∼ 12 π −1/2 ζ 1/4{ie− 3 ζ 

Bi (ζ ) ∼ Bi (ζ ) ∼

(arg ζ ≤ | 23 π|) − ie

2 3/2 3ζ

2 3/2

+ 2e 3 ζ

}

}

2 3/2 2 3/2 1 −1/2 1/4 ζ {−ie− 3 ζ + 2e 3 ζ } 2π 2 3/2 2 3/2 − 12 π −1/2 ζ 1/4 {ie− 3 ζ − e 3 ζ }

(15.44)

( 23 π ≤ arg ζ ≤ 43 π) (15.45) (− 23 π ≤ arg ζ ≤ 0) (0 ≤ arg ζ ≤ 23 π) ( 23 π ≤ arg ζ ≤ 43 π).

(15.46) (15.47) (15.48)

15.6.5 Approximate WKBJ representation of a general wave Our discussion of reflection applies to the situation where the turning point is isolated from the resonance level where = 0. If these two levels are too close, then the transformation that approximates the wave equation by the Stokes equation is invalid. Such cases are taken up in chapter 18. In this section, we describe how the WKBJ solutions, valid at points remote from the turning point, can be matched to the solution near the turning point. In order to provide a



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concrete example, we take the case k z = 0. In this case,

 = V 2 ω+ V 2 − k 2y 2

A

 = ρ0 ω2 .

(15.49)

S

The WKBJ solutions fail in the neighbourhood of the origin. Suppose that the range of their validity is |x| > δ. Since is negative for positive x, the wave is evanescent in this region. It cannot grow without limit as x → ∞ which requires that B = 0 in the region x > δ. When x < 0, is positive and the waves are propagated so the solution may contain both WKBJ terms in the region x < −δ. How do we match the waves across the origin? If the medium is slowly varying, we assume that, in the region |x| < δ, is approximately linear. This is generally true for slow monotonic variation which meets the criterion for the WKBJ approximation but needs to be verified for each problem. Then, in the region near the origin, the differential equation can be approximated by the Stokes equation and will show the same behaviour. If the Stokes solutions are matched smoothly onto the WKBJ solutions at x = ±δ we see that the complete solution is   x   √ −1/4  A exp − | | dx x >δ    0    1/3   A Ai(−|(  0 |)  x)x    |x| < δ  x √ √ ψ= −1/4  exp i dx + i exp − i dx A    0 0    x√   π   dx − = 2 A −1/4 eiπ/4 cos x < −δ. 4 0 (15.50)  ρ0 ω2 . To the accuracy The displacement can be found from (14.15), with of the WKBJ approximation, only the exponential factors are differentiated. The result is   x   √ −1 1/4 −2   −A ρ0 ω exp − | | dx x >δ    0   −2 Ai (−|(  |)1/3 x)   −A|( 0 |)1/3 ρ0−1   |x| < δ ω   x  x 0  √ √ ξ= −1 −2 1/4  Aρ0 ω dx + exp − i dx i exp i    0  0   x√   π   x < −δ. = −2 A 1/4eiπ/4 sin dx − 4 0 (15.51) We see then that, if we have a situation in which there exists a wave propagated in the positive x-direction in the region x < 0, then the boundary conditions at infinity demand that there should be an evanescent wave in the region x > 0 and, thus, a reflected wave, propagated in the negative x-direction, in the region x < 0.





























 





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Full wave theory of reflection

-6

WKB solution 0.8 Ai(z) 0.6 0.4 0.2 0 -4 -2 0 -0.2 -0.4 -0.6 WKB solution Bi(z)

283

2

3 2 1 0.5

-6

-4

-2

0

2

Figure 15.3. WKBJ approximation to the Airy functions Ai(ζ ) and Bi(ζ ).

The definitions of reflection coefficients can be generalized by referring it to a specific level. Suppose we define the reflection coefficient R(x) at some level x to be the ratio of reflected to incident wave. Then,    x √ dx (15.52) R(x) = i exp − 2i 0



The general behaviour of the wave is similar to Airy function behaviour with modification to the phase and amplitude resulting from the deviations of the behaviour of from linearity.



15.6.6 Error near the zeros of q It is possible to carry out detailed estimates of the error near a zero of q. This is done, for example, for the Airy function, by Mathews and Walker [137, pp 30, 31]. They show that the error remains small even quite close to the zero. Rather than repeat this calculation, we present computed results for two special functions, the Airy functions Ai(x) and Bi(x) and the modified Bessel function K 0 (z). Figure 15.3 shows a comparison between the exact computed values of the Airy functions Ai and Bi and the WKBJ solutions. It can be seen that the approximation is extremely good even when |x| is as small as 1. In particular, the positions of the zeros are very accurately determined by the WKBJ solutions, Copyright © 2005 IOP Publishing Ltd.

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Propagation in a plane-stratified medium

Figure 15.4. Range of validity of the WKBJ solutions. Stokes and anti-Stokes lines.

which means that the determination of modes in a cavity by the WKBJ method is quite precise. The discussion of the Stokes phenomenon is mathematically intricate but interesting. To discuss it, we need to venture into the complex ζ -plane. Readers who are willing to accept them as they stand may wish to omit the following sections in small type.

15.7 Connection relations 15.7.1 Stokes and anti-Stokes lines The exponent 23 ζ 3/2 in the WKBJ solution is, in general, complex. It is real when arg ζ 3/2 = nπ where n is an integer, that is where arg ζ = 0, 23 π, or 43 π.

(15.53)

This defines three lines radiating from the origin and making angles 0◦ , 120◦ , and 240◦ with the positive real axis. These are called Stokes lines. Figure 15.4 illustrates the complex ζ -plane. The Stokes lines are shown as dotted lines labelled S. In the same way, the condition (15.54) arg ζ = 13 π, π, or 53 π defines three lines radiating from the origin on which the amplitudes of the two WKBJ solutions are equal.

Copyright © 2005 IOP Publishing Ltd.

2 ζ 3/2 is purely imaginary, so that 3

These are called anti-Stokes lines

Connection relations

285

and are shown in figure 15.4 as broken lines labelled A . In the figure, a circle, labelled C , is also drawn. This circle represents the condition   5 1  1   (15.55)  16 ζ 3  = e2 . Outside this circle, we regard the WKBJ condition (15.35) as satisfied; inside the circle it breaks down. Choosing a different criterion for the required accuracy of the WKBJ solution would simply change the scaling of the diagram. On a Stokes line, and in the complex plane close to either side of it, in the WKBJ region of validity, one of the WKBJ solutions grows with |ζ | as | exp{ 23 ζ 3/2 }| and the other decays with |ζ | as | exp{− 23 ζ 3/2 }|. The larger solution dominates and, indeed, the ratio of the smaller solution to the larger is less than the error in the WKBJ solution. These two solutions are, respectively, called dominant and subdominant solutions. To see the region over which this is true, we have plotted curves in figure 15.4 on which the ratio of the moduli of the dominant and subdominant solutions is e2 . These boundaries are labelled D.

15.7.2 Connection relations These ideas give us tools for deciding how to connect WKBJ solutions in different parts of the complex plane. They provide the means of deriving connection relations. We illustrate the process by continuing the analysis of the asymptotic approximations to the Airy functions. The Airy function Ai(ζ ) is defined as that solution of the Stokes equation which tends to zero as ζ → ∞, with an appropriate normalization constant. Its asymptotic approximation on the positive ζ -axis is, therefore, A − ζ −1/4 exp{− 23 ζ 3/2 }. This is the subdominant solution in this region and it is negligible compared with the dominant solution, thus the constant multiplying the dominant solution must be zero. Now let us follow this solution round an arc in the positive complex plane outside the circle C. When we reach the anti-Stokes line making an angle of 60◦ with the ζ -axis the two WKBJ solutions have equal magnitudes and, beyond this, our solution becomes dominant. Once we have passed the boundary D beyond this, the other, subdominant, solution is negligible. In this region, we are free to add a multiple of this solution to the other WKBJ term without affecting the accuracy of the approximation. We choose to do this at the point of maximum difference between the amplitudes, on the antiStokes line making an angle of 120◦ with the ζ -axis. Beyond this, the solution becomes A + ζ −1/4 exp{ 23 ζ 3/2 } + A − ζ −1/4 exp{− 23 ζ 3/2 } where the relationship between A + and A − is still to be determined. The negative ζ -axis is an anti-Stokes line so the two WKBJ solutions have equal magnitudes. We could determine the ratio A + /A − by comparison with the series solution of the Airy function or by an integral representation but there is value in finding the connection relation self-consistently from the properties of the WKBJ solutions. This is not always possible but it can be done for the Airy functions. We use a method due to Furry [74]. Consider a WKBJ solution of the Stokes equation: Aζ −1/4 exp{ 23 ζ 3/2 } + Bζ −1/4 exp{− 23 ζ 3/2 }.

(15.56)

The function to which it is an approximation is single-valued but the terms of the approximation are double-valued with a branch point at the origin. We take the multivalued nature of these quantities into account by drawing a branch cut labelled B in

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Propagation in a plane-stratified medium

figure 15.4. Then we assume that the constants are chosen so that the solution (15.56) is valid in the region of the complex plane between the positive real axis and the Stokes line making an angle of 120◦ with the poitive real axis. The differential equation is unaltered if we replace ζ by ζ e2πi/3 ; thus, the nature of the discontinuity is the same on each Stokes line. Near a Stokes line, let the coefficient of the dominant term be A + and the subdominant term be A − . Suppose that, as we cross a Stokes line in the positive sense, the coefficient of the subdominant term changes by an amount δ B. If the coefficient of the dominant term were zero, it would not change. Thus, δ A − is independent of B. The equation is linear so that δ A − ∝ A + . We write δ A − = α A + . We follow our solution through all regions of the complex plane in which the WKBJ approximation is valid. The first term is dominant on the positive real axis and on the Stokes line making an angle of −120◦ with the positive real axis. The second term is dominant on the Stokes line making an angle of +120◦ with the positive real axis. In the sector between the two Stokes lines containing the negative real axis, the solution is, therefore, [A + α B]ζ −1/4 exp{ 23 ζ 3/2 } + Bζ −1/4 exp{− 23 ζ 3/2 }. In the sector between the branch cut and the Stokes line making an angle −120◦ with the real axis, it is [A + α B]ζ −1/4 exp{ 23 ζ 3/2 } + [B + α(A + α B)]ζ −1/4 exp{− 23 ζ 3/2 }.

(15.57)

In the sector between the positive real axis and the branch cut, it is Aζ −1/4 exp{ 23 ζ 3/2 } + [B − α A]ζ −1/4 exp{− 23 ζ 3/2 }.

(15.58)

These two expressions (15.57) and (15.58) must be matched across the branch cut. If we cross the branch cut in the positive sense ζ 3/2 changes sign and ζ −1/4 changes by a factor e−iπ/2 = −i. Thus (15.57) becomes −i[A + α B]ζ −1/4 exp{− 23 ζ 3/2 } − i[B + α(A + α B)]ζ −1/4 exp{+ 23 ζ 3/2 }. Comparing the coefficients of this with (15.58), we get A = − i[B + α(A + α B)] B − α A = − i[A + α B]

(15.59) (15.60)

The second of these shows that the Stokes constant is α = i.

(15.61)

One of the roots of the first equation is consistent with this. The other root depends on A and B and must be excluded. This method works for cases where there is rotational symmetry about the branch point.

15.7.3 Range of validity of asymptotic approximations in the complex plane The range of validity of the asymptotic approximations in section 15.6.4 is chosen so that the subdominant coefficient is discontinuously changed as a Stokes line is crossed. In this

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Summary

287

way they are sufficiently accurate over the whole of the complex plane outside the circle C in figure 15.4. Most books quote the asymptotic approximations according to the Poincare´ definition [31, section 15.22]. Poincar´e chose to define the asymptotic approximation only for the case ζ → ∞. As a result, he allowed the constant to be changed immediately an anti-Stokes line was crossed and the dominancy changed. This is appropriate when asymptotic approximations are given to many terms. It is not, however, useful in studying waves.

15.8 Summary •

• •

• • •

In plane-stratified media, with the magnetic field parallel to the stratifications, the differential equations for the fast and slow waves may be separated from that for the transverse Alfv´en wave. The transverse Alfv´en wave only supplies energy transfer exactly parallel to the field. The fast and slow waves allow for energy propagation transverse to the magnetic field. The simultaneous first-order differential equations are suitable for numerical computation. They may be expressed as second-order equations in a single variable for analytic study. At points remote from singularities, wave propagation may be studied by means of the WKBJ approximations. These apply in a slowly varying medium. The phase variation is represented by a phase integral and there is a slowly varying amplitude factor. Such approximations are surprisingly accurate. The phase is accurately maintained over as many as 200 wavelengths, even when the medium varies by as much as 10% per wavelength. The variation of the amplitude with position has the effect of ensuring that the energy flux is constant as the wave progresses. Near turning points in the differential equation, the wave is reflected. The reflection level is an example of a caustic surface. On such surfaces, the WKBJ approximations break down. Near a turning point, the differential equation can be approximated by the Stokes equation, which has, as its solutions, the Airy functions Ai and Bi. WKBJ methods can be found for the Airy functions on either side of the turning point. A single-valued Airy function is represented by different linear combinations of WKBJ solutions on either side of the turning point. The rules for matching the connection relations between these solutions are discussed in detail.

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Chapter 16 Standing waves and oscillations in a cold plasma

16.1 Introduction In this chapter, we introduce some phenomena that are important in studying oscillations in bounded media. In the solar–terrestrial system, such bounded media occur in the Earth’s magnetosphere and we shall confine ourselves to models applicable there. The features that we wish to illustrate all occur in a cold plasma, in which the pressure is negligible. We shall confine outselves to this case in this chapter and introduce the effects of pressure in chapter 17. This chapter is complementary to chapter 15. In that chapter, we considered propagation perpendicular to the field, in situations in which the transverse Alfve´ n wave is decoupled. Here we study the properties of local oscillations, confined to a magnetic field line, in which only the transverse Alfve´ n wave participates. The simplest case may be adequately represented by a model in which the field lines are straight and the region is bounded by the magnetopause, the polar cap boundary, and the ionosphere. The magnitude of the magnetic field and the density have a gradient parallel to an axis perpendicular to the field. This simplest of models shows, qualitatively, the nature of the standing waves that may exist in the magnetosphere. It also illustrates the phenomenon of field-line resonance, a key feature in the understanding of a variety of phenomena in solar–terrestrial physics. The ionospheric boundary has an important effect on the nature of the oscillations and we discuss its properties. The distortions of assuming straight field lines are chiefly geometric effects. They do not affect the basic physics of the phenomena of interest. They are not suitable, however, for making reasonable quantitative estimates of physical quantities of interest. This requires the development of dipolar and more complicated models. We describe the kinds of standing oscillation that can occur in such a configuration and how their properties can be computed. We present the results of such computations, by various methods, for a variety of situations. 288

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289

Dungey [55] first suggested that geomagnetic pulsations were standing waves of this type. Computations of their characteristic frequencies were made by Cummings et al [51]. This was extended for a dipole field by Orr and Matthew [150] and for more complicated field geometries by Singer et al [188]. Allan and Knox [5] and Allan [3] calculated the variation of the fields along the field line, allowing for finite ionospheric conductivity. In this chapter, we present a unified treatment of these problems, also including a detailed treatment of the effect of the anisotropic ionospheric conductivity on the wave. The problem can be reduced to a one-dimensional one, with the standing waves behaving analogously to waves on a stretched string. Such modes are fundamental to understanding the natural oscillations of the magnetosphere, and we shall refer back to these relatively simple ideas throughout the book.

16.2 Transverse Alfv´en oscillations In this section, we consider the standing waves that can exist in a very simple medium as an introduction to the treatment of the natural modes of oscillation of plasma regions in the solar–terrestrial system. 16.2.1 Uniform medium Consider plane harmonic Alfv´en waves, of angular frequency ω, propagated in a uniform plasma with the magnetic field in the z-direction, and having wavevector k in the x z-plane. Their properties are described by equations (7.47). The only non-zero field components are v y , b y , given by (7.72), and E x , found from (2.52): (ω2 − k z2 VA2 )v y = 0 kz B vy by = ∓ ω E x = − v y B.

(16.1) (16.2) (16.3)

For a given value of k x , there are two waves corresponding to the positive and negative values of k z found from the dispersion relation k z = ±ω/k z . The value of k x is arbitrary. The direction of the group velocity and, hence, energy propagation is parallel to the magnetic field. This model is a rough approximation to the magnetospheric situation with the geometrical effects of field curvature removed. We imagine that the field lines of a dipole field have been straightened out. The conducting planes represent the ionosphere at each end of a field line. If desired, the variation of Alfv´en speed with x, as a result of magnetic field and density gradients, can be included as a density gradient. Suppose such waves exist in a uniform region z > 0, bounded by a perfectly conducting plane at z = 0 normal to the magnetic field. At this plane, the electric field and, hence, v y must be zero, so that the only possible solution of the wave Copyright © 2005 IOP Publishing Ltd.

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Standing waves and oscillations in a cold plasma

equation must be one for which v y is zero at the boundary. Waves propagated in the positive and negative z -directions must be superposed to make this so. The resulting solution of the wave equation for v y may be written in the form v y ∝ e−iωt {eikz z − e−ikz z } ∝ e−iωt sin k z z.

(16.4)

This has zeros wherever k z z = nπ

n = 1, 2, 3, . . . .

(16.5)

The corresponding magnetic field can be found from (16.2). It is b y ∝ −k z e−iωt cos k z z

(16.6)

with the same constant of proportionality as for v y . At the zeros, the velocity perturbation has a node and so does the displacement. A perfectly conducting sheet could be inserted at one of these nodes and the electric field would automatically satisfy the boundary condition that it is zero. If this second sheet was at z = d , then (16.4) would represent the solution in the region 0 < z < d . The conductor screens the magnetic field from the region outside: it carries a surface current, of density Ix per unit length of surface, in the x -direction such that, from Ampe´ re’s law, Ix = b y /µ0 .

(16.7)

Alternatively, we can fix d, the separation of the conducting planes. Then an oscillation that has a frequency for which (16.5) is obeyed will satisfy the boundary conditions. From the dispersion relation, ω = k z VA so that (16.5) becomes nπ VA n = 1, 2, 3, . . . . (16.8) ωn = d The nature of the oscillations is shown in figure 16.1. On the left-hand side of the diagram the amplitude of the velocity perturbation or, equivalently, the displacement, is plotted as a function of z for n = 1, 2, 3. On the right-hand side, the corresponding amplitude of the transverse magnetic field perturbation is shown. The displacement amplitude curves on the left-hand side simply represent the shape of a field line at its maximum displacement. The analogy with the oscillation of a stretched string is obvious. The frequency depends on VA , which, in turn, depends on the ‘tension’ and the density, as described in section 2.3.4. It can be seen that the magnetic field perturbation is antisymmetric about the midpoint at z = 12 d while the displacement perturbation is symmetric. This can cause confusion when a mode is described as ‘symmetric’ or ‘antisymmetric’ or as ‘even’ or ‘odd’, unless it is made clear whether the mode number n, the velocity perturbation v y , or the magnetic field perturbation b y is meant. Copyright © 2005 IOP Publishing Ltd.

Transverse Alfv´en oscillations

291

z vy

by

z=d

B z=½d

n

n

n

x

n

n

n

y

Figure 16.1. Transverse Alfv´en modes in Cartesian geometry.

For a transverse Alfv´en wave, the value of k x is arbitrary. This means that the variation of the fields as functions of x is arbitrary, since any dependence on x may be represented as a Fourier integral of the form  ∞ A(k x )eikx x dx. (16.9) f (x) = −∞

 1 exp(ik x x) dk x In particular, it is possible for f (x) to be a δ-function, δ(x) = 2π so that the oscillation is confined to a plane sheet at x = 0. Physically, the perturbation of the fields is entirely in the plane x = 0. There is no compression and no disturbance of the adjacent planes. Each plane can slide freely past its neighbours. This is true on the scale of the MHD approximation. The δ-function behaviour is an approximation for problems with length scales large compared with a gyroradius. 16.2.2 Medium with a transverse gradient of Alfv´en speed The treatment in the previous section allows a more complicated situation to be understood. Suppose that the Alfv´en speed is a function of x. Physically, this could be achieved if, for example, the density were a function of x. Now, at a particular value of x, a thin sheet of the plasma can oscillate at its own natural frequency determined by the Alfv´en speed for that value of x. A sheet of plasma at a different value of x will oscillate at a different frequency. The stretched string analogy now suggests a harp: the sheets of plasma are analogous to the strings, each with its own tension and mass loading. Copyright © 2005 IOP Publishing Ltd.

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Standing waves and oscillations in a cold plasma

To make this explicit, we note that, because of the transverse gradient, we cannot assume that the fields vary in the x-direction as exp{ik x x}. We can, however, assume that the dependence on time and the other two coordinates is of the form exp{−i(ωt − k y y − k z z)}. We can set P = 0 in the reduced equations for momentum (2.55) and magnetic field (2.58) and linearize them to get kz B iB dbz − bx µ0 dx µ0 ky B kz B = bz − by µ0 µ0 =0 kz B vx = − ω kz B vy = − ω ky B iB dvx + vy . = − ω dx ω

ωρ0 (x)vx = −

(16.10)

ωρ0 (x)v y

(16.11)

vz bx by bz

(16.12) (16.13) (16.14) (16.15)

If, in these equations, we set k y = 0, then they separate into two sets, one involving vx , bx , and bz , and the other involving v y , b y : B dbz ik z B = iωρ0 (x)vx + bx µ0 dx µ0 kz B bx = − vx ω iB dvx bz = − ω dx kz B by −ωρ0 (x)v y = µ0 kz B by = − vy . ω

(16.16) (16.17) (16.18) (16.19) (16.20)

We can eliminate bx and bz from the first three of these, and b y from the second two, getting  ω2 d2 v x 2 = k z − 2 vx (16.21) dx 2 VA {ω2 − k z2 VA2 }v y = 0.

(16.22)

The first of these represents a global compressional oscillation, occurring throughout the region. In general, depending on how VA depends on x, it can only be solved numerically. The second equation, however, is satisfied if the dispersion relation (16.23) ω = ±k z VA Copyright © 2005 IOP Publishing Ltd.

The Earth–ionosphere system as a boundary

293

is satisfied. This is just the localized transverse Alfve´ n oscillation described earlier. At each value of x , the dispersion relation is satisfied by a different frequency, so that each sheet of plasma can oscillate independently at its own natural frequency. Another possibility for localized oscillation occurs when k y is large, so that the wavelength in the y -direction is small compared with the other physical dimensions. If we let k y → ∞, then (16.11), (16.14), and (16.15) show that v y , b y , and bz each tend to zero. Since vz = 0 from (16.12), the remaining two equations become kz B bx µ0 kz B vx bx = ω

−ωρ0 (x)vx =

(16.24) (16.25)

which correspond to an oscillation in the x z-plane which obeys the dispersion relation (16.23). This is also a transverse Alfve´ n wave. The difference is that the oscillation is in the direction of the gradient of the Alfve´ n speed. The reason that this is possible is that the wavelength perpendicular to the plane of oscillation, 2π/ k y , is much shorter than the length scale corresponding to this gradient. The properties of the medium do not change appreciably on a length scale comparable with k −1 y —locally the medium is uniform. The resulting oscillation can be regarded as localized to a field line, with the scale of the localization of the order of k −1 y .

16.3 The Earth–ionosphere system as a boundary We have so far treated the boundary as a perfectly conducting plane, which provides perfect reflection of MHD waves. If this were so, then geomagnetic pulsations could not be observed on the ground. The ionosphere would be a perfect screen. It is actually a thin transition region between the collisionless plasma of the magnetosphere and the nearly neutral atmosphere. In the ionospheric E-region the magnetic field varies from about 3.2 × 10−5 T at the equator, to about 6.4 × 10−5 T at the poles. Thus, within the ionosphere, the electron gyrofrequency is of the order of 1.25 MHz. The mean ion mass at these altitudes is about 25 amu so that a typical ion gyrofrequency is about 30 Hz. Figure 4.3 shows that above about 100 km the electron collision frequency is negligible compared with the gyrofrequency. It rises rapidly with decreasing height as the neutral density increases, and becomes larger than the gyrofrequency in the D-region. The ion collision frequency becomes comparable with the ion gyrofrequency in the E-region, so that, in the region between about 100 and 110 km, the ion motion is dominated by collisions with neutral particles, while the electron motion is unaffected by collisions with them. Above the E-region, at heights greater than about 110 km, the collision frequencies of all species Copyright © 2005 IOP Publishing Ltd.

294

Standing waves and oscillations in a cold plasma z z' z'P P

zP B %

x

xP

O x'P

x' Figure 16.2. Coordinate systems for conductivity tensor.

with neutrals is very small compared with the ion gyrofrequencies. The ionized particles behave almost independently from the more numerous neutral particles, and may be treated as a collisionless plasma. Below the E-region, the collision frequencies all rise sharply with the increased density and the atmosphere behaves as an insulator with a dielectric constant approximately equal to unity; and we can treat the atmosphere as a region of free space. We consider the conducting properties of the ionosphere, the free space region below it, and the ground, in turn.

16.3.1 Height-integrated conductivity of the ionosphere Within the E-region, the conductivity is given by the relationship of the form (2.88) so that   Jx σP  Jy  =  −σH 0 Jz 

σH σP 0

   Ex 0   E y  . 0 σ0 E z

(16.26)

The coordinate system is the primed coordinate system shown in figure 16.2, with the z  -axis in the direction of the magnetic field, and the x  z  -plane defining the magnetic meridian. The magnetic dip angle is χ. The conditions are those for the southern hemisphere with the magnetic field pointing upwards. We can define a second unprimed coordinate system with z vertical and the x z-plane parallel to the magnetic meridian. A point P has coordinates x P , yP , z P in the unprimed system and x P , yP , z P in the primed system. From the geometry the relationship Copyright © 2005 IOP Publishing Ltd.

The Earth–ionosphere system as a boundary between the primed and unprimed coordinates may be written       sin χ 0 − cos χ xP xP   yP   y  =  0 1 0 P cos χ 0 sin χ z P zP       xP sin χ 0 cos χ xP  yP  =  0 1 0   yP  − cos χ 0 sin χ zP z P

295

(16.27)

(16.28)

and these are the relationships to transform any vector from one set of coordinates to the other. If we apply the transformation to (16.26), we get     Jx sin χ 0 cos χ  Jy  =  0 1 0  − cos χ 0 sin χ Jz     σP σH 0 sin χ 0 − cos χ Ex   Ey  1 0 ×  −σH σP 0   0 0 0 σ0 Ez cos χ 0 sin χ   2 2 σH sin χ (−σP + σ0 ) cos χ sin χ σP sin χ + σ0 cos χ  −σH sin χ σP σH cos χ =  2 2 (−σP + σ0 ) cos χ sin χ −σH cos χ σP cos χ + σ0 sin χ   Ex ×  Ey  . (16.29) Ez We now make use of the fact that the conducting layer occupies a limited range of altitudes. Below it is the atmosphere, which can be regarded as an insulator. Above it is a region of collisionless plasma, which, although it has infinite conductivity parallel to the magnetic field, has zero conductivity perpendicular to it, since, to this order of approximation1, ions and electrons move together with an E × B drift and no current flows. The result is that, at the lower boundary of the conducting region, there can be no normal component of current and, at the upper boundary, any current must flow into or out of the layer along the field. At these frequencies ∇ · J = 0. (16.30) Suppose the scale on which J varies horizontally is l and the thickness of the layer is d. Then, if Jh and Jv are the horizontal and vertical components of J, the order of magnitude of the terms in (16.30) implies that Jh /l ∼ Jv /d. If the layer is sufficiently thin, then d l and, therefore, Jv Jh . For a sufficiently thin layer, this allows us to assume that, within the layer, Jz is zero. The last of the three equations (16.29), with Jz = 0, can then be used [20] to express E z in terms of 1 As described in section 2.8.2 for the ideal plasma case, the conductivity tensor is evaluated to a

lower order of approximation than the MHD equations.

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E x and E y , and substitution in the other two equations gives the two-dimensional relationship   σ0 σH sin χ σ0 σP     2 2 Ex Jx σ cos2 χ+σ0 sin χ σP cos2 χ+σ0 sin χ  =  P σ σ sin . (16.31) 2 σH cos2 χ χ 0 H Jy Ey − σP + 2 2 2 2 σP cos χ+σ0 sin χ

σP cos χ+σ0 sin χ

Now, because of the collision-free electrons, σ0 → ∞. Except near the equator, where sin χ is small, we can give an approximate form of the relationship for large σ0 :      Ex Jx σP / sin2 χ σH / sin χ = . (16.32) Jy Ey −σH / sin χ σP To this order of approximation, the z-component of the electric field is E z = −E x cot χ.

(16.33)

This ensures that the component of the electric field parallel to B is zero. We assume that the layer is thin compared with the wavelength of any MHD wave in the region above it and that E is constant within the layer. These assumptions will be justified a posteriori. Suppose that the layer extends from z = z 1 to z = z 2 . Then define the height integrated current density as   z2      z2    Jx Ex Ix σP / sin2 χ σH / sin χ = = dz. Iy Jy Ey −σH / sin χ σP z1 z1 (16.34) It has the dimensions amperes per metre. If we consider an element of length ˆi dx + ˆj dy in the layer, then the scalar product Ix dx + I y dy is the total current flowing horizontally through the line element in the equivalent layer of infinitesimal thickness. The quantity   z2    !H !P σP / sin2 χ σH / sin χ = dz (16.35) −!H !P −σH / sin χ σP z1 is called the height integrated conductivity. Its dimensions are siemens (S) where 1 S = 1 −1 . The quantities !P and !H are, respectively, the height-integrated Pedersen and Hall conductivities. 16.3.2 Fields below the ionosphere Pulsation frequencies range from less than 1 Hz to fractions of a millihertz. In the free space region below the ionosphere, the dispersion relation may be written in the form ω2 = k x2 + k 2y + k z2 . (16.36) c2 Copyright © 2005 IOP Publishing Ltd.

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The left-hand side is very small, ranging from 4 × 10−16 m−1 for a 1 Hz signal to 4 × 10−22 m−1 for a 1 mHz signal. For all terrestrial length scales, k x2 + k 2y is very much larger than this. We can, therefore, write ' k z  ±i k x2 + k 2y (16.37) so that the fields below the ionosphere are evanescent. If the fields at the ionosphere are represented by a two-dimensional Fourier integral,  ∞ A(k x , k y ) exp{i(k x x + k y y)} dk x dk y −∞

then the field that is incident on the ground is 

∞ −∞

A(k x , k y )ei(kx x+k y y) e

' − kx2 +k 2y h

dk x dk y

(16.38)

where h is the height of the ionosphere. The factor exp {−(k x2 + k 2y )1/2 h} has the effect of a low pass filter, which cuts off the higher spatial frequencies. If the scale of the horizontal variation is l, then for l = h the amplitude is reduced by a factor e−2π and shorter spatial scales lead to even greater attenuation. This is the explanation of the limit to the spatial resolution of magnetometer chains, described in section 13.4.1. 16.3.3 Transmission through the ionosphere The length scales and frequencies are such that we can ignore time variation in Faraday’s law. The fields of the disturbance vary so slowly that they are governed by the electrostatic and magnetostatic conditions ∇ × E = 0 and ∇ × b = 0. We assume that the horizontal variation of the ionospheric conductivities is zero. Those parts of the height-integrated current associated with the Pedersen and Hall conductivities are denoted by superscripts ‘P’ and ‘H’ respectively. Then  P      H     Ix Ix !P 0 Ex 0 !H Ex = = . I yP Ey I yH −!H 0 Ey 0 !P (16.39) We assume that the perturbation has been Fourier analysed in x and y and we consider one Fourier component, so that the dependence on x and y is of the form exp{i(k x x + k y y)}. Immediately below the ionosphere, ∇ × E = 0 and the z-component of this may be written in the form k x E y − k y E x = 0.

(16.40)

The transverse components of the electric field are continuous through the ionosphere, so this holds through the layer and immediately above it. The vertical Copyright © 2005 IOP Publishing Ltd.

298

Standing waves and oscillations in a cold plasma C

x

Iy l d

Figure 16.3. Boundary condition on horizontal field.

component of the current perturbation is zero below the layer. The condition section 16.30 may be integrated through the layer to give k x I x + k y I y = i jz

(16.41)

where jz is the vertical component of the current density immediately above the layer. The contribution to this from the Hall current is k x IxH + k y I yH = !H (k x E y − k y E x ) = 0.

(16.42)

i jz = k x IxP + k y I yP .

(16.43)

Thus, We see, therefore, that the vertical component of current above the ionosphere closes within the ionosphere through the Pederson currents, while the Hall currents are two-dimensional divergence-free currents, which close within the ionospheric layer. The boundary condition on the horizontal components of B at the ionosphere is derived from the integral form of Amp´ere’s law: 
b · ds = µ0 j · d A. (16.44) C

A

We take the rectangular contour C, of length l and width d, shown in figure 16.3. − The line integral round this loop is (b+ x − b x )l and the current through it is I y l, where the superscripts + and − denote values immediately above and immediately below the boundary, respectively. Thus,

and, similarly,

− b+ x − bx = I y

(16.45)

− b+ y − b y = −I x .

(16.46)

Just above the ionosphere we can separate the magnetic field into a field for which the perpendicular component of ∇ × b is zero and a part for which it is non-zero. This latter part, from (16.43), has k x b yP,+ − k y bxP,+ = −iµ0 jz = −µ0 {k x IxP + k y I yP }. Copyright © 2005 IOP Publishing Ltd.

(16.47)

Lossy oscillations in a uniform medium

299

Combine this with that part of (16.45) and (16.46) which arises from the Pedersen are both zero. That part of the magnetic current. It implies that bxP,− and b P,− y field for which the perpendicular component of ∇ × b is zero is completely screened from the ground. If there were no Hall current, there would be no ground magnetometer observations of this part of the pulsation signal. The signal seen on the ground is the result of the Hall current through that part of (16.45) and (16.46) which arises from the Hall current. The Hall current is everywhere at right angles to the Pedersen current. This means that the magnetic field seen at the ground is at right angles to the magnetic field above the ionosphere. The effect of the anisotropic ionospheric conductivity is to rotate the magnetic field through 90◦. The field of an oscillation in the magnetic meridian is of this type. This result, for vertical background magnetic field, was first suggested for varying magnetic signals by Dungey [56]. Our results have been derived for a conducting region that can be regarded as thin. It is not hard to generalize this for the case where the magnetic field is not perpendicular to the boundary. Numerical computations of the variation of the wave fields through a realistic model of the ionosphere [93], without this approximation, confirm the simple picture.

16.4 Lossy oscillations in a uniform medium Suppose that the uniform region of section 16.2 is bounded by a conductor with finite anisotropic conductivity, represented by height-integrated Pedersen and Hall conductivities. The boundary condition to apply is that the wave field is completely screened by the Pederson current. The Hall current only supplies a secondary field. Thus, (16.48) b y = µ0 Ix = ±µ0 !P E x where b y and E x are the perturbation fields associated with the wave and not the secondary fields driven by the associated Hall current. The upper sign is correct for the boundary at z = 0 and the lower for the boundary at z = d. The implication is that, unlike the case of infinite conductivity, the electric field is not zero at the boundary. In this simple case of uniform medium and symmetry about the equator, we can assume that the spatial part of the solution (16.6) is modified to be of the form  nπ z   nπ z & nπ % cos + C sin (16.49) by = d d d where C is a constant to be determined from the boundary conditions. From Faraday’s law, iωb y = dE x /dz so that  nπ z & %  nπ z  − C cos . (16.50) E x = iω sin d d If we apply the boundary conditions for z = 0 to these equations, we get C=− Copyright © 2005 IOP Publishing Ltd.

inπ ωµ0 !P d

(16.51)

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Standing waves and oscillations in a cold plasma

The boundary condition at z = d is automatically satisfied because of our exploitation of the symmetry about the equator. The solutions are, therefore,   nπ z   nπ z  inπ nπ − by = cos sin (16.52) d d ωµ0 !P d d    nπ z  nπ z  inπ cos E x = iω sin + . (16.53) d ωµ0 !P d d In this simple case, we see that either the cosine or the sine term may dominate the behaviour of the magnetic field. If !P → ∞, we retrieve the cosine solution (16.6) for b y , with a corresponding sinusoidal behaviour for E x . If the conductivity is very low, however, b y is sinusoidal, while E x has cosinusoidal behaviour.

16.5 Alfv´en oscillations in a dipole-like field At times of low magnetic activity, in the inner magnetosphere, the plasma pressure is usually negligible and the magnetic field is approximately dipolar. Such circumstances are well represented by a model in which the magnetic field is that of a dipole and the density of the plasma is defined as a cylindrically symmetric function of position. The waves that can occur are the transverse Alfv´en wave and the isotropic Alfv´en wave and these are described by (14.62), (14.63), (14.64), (14.65), and (14.66). In general, the two characteristic waves are coupled by the magnetic field and density inhomogeneity. There are, however, two extreme cases in which the isotropic and transverse waves are decoupled. This occurs for cylindrically symmetric oscillations with m = 0, for which the wave quantities are independent of the longitude coordinate φ, and for oscillations with a very short azimuthal wavelength so that m 1. 16.5.1 Cylindrically symmetric oscillations Suppose that the phase of the wave is independent of longitude φ. Then the azimuthal wavenumber m is zero. The equations then separate into two sets. In one, the motion is entirely azimuthal, consisting of a perturbation in E ν (equivalent to vφ ) and bφ : iω iω

φ



∂ φ hν VA2 hφ hµ ∂µ hφ ∂ ν . = − h µ h ν ∂µ


ν =

−




(16.54) (16.55)

In the other, the motion is entirely in the magnetic meridian, consisting of a perturbation in E φ (equivalent to vν ), with a transverse and compressional Copyright © 2005 IOP Publishing Ltd.

Alfve´ n oscillations in a dipole-like field

301

magnetic field perturbations bν and bφ : iω iω


φ = hh φh

µ ν

ν








∂ µ ∂ ν − ∂µ ∂ν



1∂ φ r0 ∂ν hν ∂ φ . = h φ h µ ∂µ

µ =

iω

 VA2

−

(16.56) (16.57)




(16.58)

The azimuthal oscillation, represented by (16.54) and (16.55), is determined only by the dependence on the coordinate µ, parallel to the background magnetic field. The condition m = 0 requires that it is independent of φ . The behaviour in the meridian parallel to ν is, however, arbitrary. This means that a magnetic shell ν = constant can move independently of its neighbours. The frozen-in plasma moves azimuthally without compression, sliding past the plasma on the adjacent shells without disturbing it. The region of motion is bounded at each end by the ionosphere. If this is a perfect conductor, then the electric field must be zero at each ionosphere. This is not possible except for a discrete set of frequencies. These can, in principle, be found by assuming a value of ω, setting the electric field equal to zero at one boundary and integrating the equations to the other boundary. The mismatch to the boundary condition is used to find a better estimate of ω and the process repeated until it converges to an eigenvalue of ω. Apart from the geometry, the situation is equivalent to the situation shown for straight field lines in figure 16.1. Because the plasma distribution and the length of the field line changes with ν , each magnetic shell has its own set of characteristic frequencies. Such an oscillation is often called a toroidal oscillation, because the motion is that of a toroid rotating about its major axis. We prefer to describe it as an azimuthal oscillation. The compressional oscillations of (16.56), (16.57), and (16.58) are more difficult to deal with. They are two-dimensional oscillations in a cavity bounded by the ionosphere at the ends of the field lines and at the inner boundary and by the magnetopause at the outer boundary. The equations in dipole coordinates are non-separable, and the boundaries do not everywhere coincide with coordinate surfaces. It is clear that, given appropriate boundary conditions, one would expect a set of compressional cavity modes to exist but these would need to be found by appropriate two-dimensional numerical modelling. Consideration of this case is delayed to chapters 15, 18, and 20 for more general conditions. 16.5.2 Oscillations with large azimuthal wavenumber The other extreme case, in which simplification can be achieved, is that for which m → ∞. This is the case of a wavelength in the azimuthal direction, which is small compared with all other length scales. If m is very large, then (14.62) and (14.64) can only be satisfied if ν = 0 and µ = 0. It follows from




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302

Standing waves and oscillations in a cold plasma



(14.66) that φ = 0. Then there is a possible oscillation described by the two equations (14.63) and (14.65), which take the form iω iω


φ = hh φh

µ ν

VA2



∂ ν ∂µ


φ . ν = h hhν ∂∂µ φ µ

(16.59) (16.60)

This is also a pair of one-dimensional equations. The variation in the φ -direction is determined by the assumed large value of m . The motion is in the magnetic meridian and there is no compression. The equations are otherwise similar to those for toroidal oscillation. It is a transverse Alfve´ n oscillation. Such oscillations are often referred to as guided poloidal oscillations. We prefer the term local meridional oscillations.

16.6 Properties of localized field-line oscillations The azimuthal oscillations, with m = 0, and the meridional oscillations, with m 1, are decoupled and localized to a particular L -shell so that the equations describing them are one-dimensional. Their properties are, therefore, relatively easy to calculate. While such idealized conditions are not generally realized in practice, they represent an idealization of some physically realizable situations and they have received a great deal of attention. It is not possible to find analytic solutions, except in some special cases, because the complex geometry of a dipole-like field complicates the equations. Numerical computation is the answer in such cases. We discuss some examples of numerical and analytic solutions, confining ourselves to a true dipole geometry. 16.6.1 Basic equations We use the dipole coordinates of appendix B.3. Outside the plasmasphere, the particles move freely along the magnetic field lines under the action of gravity. The plasma population is often a mix of cold ionospheric plasma and hot ring current plasma. The behaviour of the density as a function of position along the field is, therefore, uncertain. There is negligible gravitational influence on the hot plasma density so that it is is constant. The cold plasma density decreases with radius from ionospheric values to a few particles per cubic centimetre. For the purposes of illustration, we assume that the density varies as some power of the radius ap νp (16.61) ρ0 = ρeq p = ρeq r cos2 p λ where λ is the latitude. The magnetic field is given by (B.4) and the scale factors by (B.44), (B.45), and (B.46). Define z = sin λ. Copyright © 2005 IOP Publishing Ltd.

(16.62)

Properties of localized field-line oscillations Then, if

φ is eliminated from (16.54) and (16.55) we get d2
ν + K 2 (1 − z 2 )6− p
= 0 ν

dz 2

2 φ = KiνV ∂∂z
ν

303

(16.63) (16.64)

0

where K2 = and V0 = √

a 2 ω2

(16.65)

V02

Beq 1 ν 4− 2 p µ0 ρeq

(16.66)

Here V0 is the Alfv´en speed at the apex of the field line and K is a normalized frequency. Similarly, if ν is eliminated from (16.59) and (16.60), we get









d2 φ 6z ∂ φ + K 2 (1 − z 2 )6− p − dz 2 1 + 3z 2 ∂z ∂ φ i . ν =− 2 K V0 (1 + 3z ) ∂z







φ = 0

(16.67) (16.68)

Obviously, the value p = 6 gives a particularly simple form to the equations and allows an analytic solution but, unfortunately, this does not give a realistic model of the variation of ρ0 with position. The numerical solution of such equations is, however, straightforward. The most straightforward method of integration involves using the pair of first-order equations and integrating them step by step by a method such as the Runge–Kutta process [159]. Then (16.63) for azimuthal oscillations may be written in the form dy1 = y2 dz dy2 = − K 2 (1 − z 2 )6− p y1 dz iν 2 y2 ν ≡ y1 φ ≡ K V0






(16.69) (16.70) (16.71)

and (16.67) for localized meridional oscillations may be written in the form dy1 = y2 dz 6z dy2 = y2 − K 2 (1 − z 2 )6− p y1 dz 1 + 3z 2 i y2 . φ ≡ y1 ν ≡− K V0 (1 + 3z 2 )




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(16.72) (16.73) (16.74)

304

Standing waves and oscillations in a cold plasma

Given a set of initial conditions at z = z 1 , one can integrate these numerically with respect to z , step by step, to find the solution at another postion z = z 2 . In general, if there is a boundary at z = z 2 , this solution will not match the boundary conditions. They will only be satisfied for certain discrete values of K . The process of finding solutions for two-point boundary conditions is, thus, as follows • • • • •

Guess a value of K . Set initial conditions such that the boundary conditions at z = z 1 are satisfied. Integrate the equations to the second boundary z = z 2 . From the mismatch of the fields to the boundary conditions, compute a better estimate of K . The method of doing this is discussed later. Iterate the process until the desired accuracy is achieved. This assumes that the process used for determining the better estimate of K is stable and converges on the desired value.

16.6.2 Numerical solutions when the ionosphere has very large conductivity If the ionospheric Pedersen conductivity is very high, then the boundary condition at each end of the field line requires that the electric field is zero, so that for both kinds of oscillation y1 = 0. Orr and Matthew [150] have solved (16.63) and (16.67) numerically to find the frequencies of azimuthal and localized meridional oscillations for such conditions. They computed the characteristic frequencies for various values of p . We present similar calculations, illustrated in figure 16.4. Here the starting point is taken at the equator. For modes that are symmetric in y1 about the equator, the initial value is y1 = 1, y2 = 0. The phase integral value of K is used to provide initial guesses. For a given value of K , the value of y1 is calculated at the ionosphere. This value of y1 is regarded as a function of K . Its zeros give the values of K corresponding to the modes. Any suitable convergent process may be used. There are advantages in using the rule of false position [159] which, although a first-order method, in practice converges rapidly and does not require computation of derivatives with respect to K . The normalized characteristic frequencies for the azimuthal and localized meridional modes for the case p = 3 are shown in figure 16.4. The fundamental frequencies differ significantly. The higher harmonics are indistinguishable. The characteristic frequencies are, of course, normalized according to (16.65) so that the curves may be used for a wide variety of conditions. Plots for other values of p are given by Orr and Matthew [150]. Note that they plot K 2 rather than K . The normalized scaled fields ν and φ for the localized azimuthal and meridional oscillations for an invariant latitude  = 65◦ are shown in figure 16.5. The first four harmonics are shown. Each curve has been normalized to its maximum value. The scaled values rather than the actual values are shown because the actual perturbation fields depend strongly on z and cannot be shown effectively on a linear scale. The bottom curve shows the reciprocal of the scale




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Properties of localized field-line oscillations 15

305

azimuthal modes m=0 4 3

10 2

K n=1 5

0 0.0

0.2

0.4

0.6 z = sin &

0.8

1.0

Figure 16.4. Normalized characteristic frequencies, K , of the lowest four harmonics for azimuthal oscillations (full curves) and of the fundamental for the localized meridional oscillation (broken curve). The higher harmonics of the azimuthal and localized meridional oscillations are indistinguishable on this scale.

factors. The actual perturbation field is found by multiplying by the appropriate scale. The general features of each type of oscillation are similar. The amplitudes depend on the type of oscillation but the positions of the nodes of azimuthal and meridional oscillations is the same to the accuracy shown in the diagram. The nodes for higher-order modes are not equally spaced: the wavelength is shorter nearer the equator because the Alfv´en speed is much less. Note that the behaviour of the magnetic pertubation for n = 1 appears to depend almost linearly with z. This is coincidental: it is only approximately linear for this particular scaling of the variable. It will be noted from (16.64) and (16.68) that the electric and magnetic fields are 90◦ out of phase. The diagrams show the magnitudes of the scaled fields. If E is real, then b is imaginary.

16.6.3 Numerical solutions for finite ionospheric conductivity In the local region near the ionosphere, we can regard conditions as essentially the same as described in section 16.3.3. The region above the ionospheric boundary is uniform, the region below it is a free-space region, and the Pedersen current in the ionosphere screens the magnetic field of the pulsation from the ground. For Copyright © 2005 IOP Publishing Ltd.

Standing waves and oscillations in a cold plasma

306

+













 

' * p=3

 

 

z = sin &

 

 

z = sin &

 B

 

Scale







 



 



h

 

h 

 

 

 

z = sin &

Figure 16.5. Perturbation fields and for the first four harmonics at  = 65◦, (z = 0.906), p = 3: full curve, ν = h ν E ν and φ = h φ bφ for the azimuthal (toroidal) oscillations; broken curve, φ = h φ E φ and ν = h ν bν for the localized meridional (guided poloidal) oscillations. The bottom panel shows the reciprocal of the scale factors h ν , h φ .

azimuthal oscillations, the horizontal components of the oscillation fields are E x = E ν sin χ

b y = bφ .

(16.75)

The horizontal spatial variation is in the x-direction and the length scale is extremely small compared with the wavelength of the corresponding fast wave. The equations (16.57) and (16.58) then show that E y is negligibly small, (16.56) means that bx is a magnetostatic field. The boundary conditions are       

b y !H Ix !P Ex . (16.76) = µ0 = µ0 − b x Iy −!H !P 0 Here bx is a secondary field, which does not affect the computation of the fields of the oscillation but provides a quasistatic magnetic field above and below the ionosphere. The source of this field is the Hall current, which, in the case of an oscillation confined to a thin shell, can be regarded as a line current in the azimuthal direction. The azimuthal magnetic field in the free-space region is zero, Copyright © 2005 IOP Publishing Ltd.

Properties of localized field-line oscillations  

p k= , i % 

Real part

  y

307

   

Imaginary part

     

0.2

0.4

0.6 z  sin &

0.8

1.0

0.4

0.6 z  sin &

0.8

1.0

Real part

    y  

Imaginary part

   

0.2

Figure 16.6. Perturbation fields for fundamental of azimuthal oscillation.

for the reasons given in section 16.3.3. The boundary condition to be applied in computing the oscillation is, therefore, bφ = µ0 !P E ν .

(16.77)

In terms of the scaled variables y1 , y2 of (16.71), this may be written in the form √ 1 − 4ν 2 y1 = −iξ y1 . (16.78) y2 = −iµ0 K V0 !P ν3 Here ξ is a normalized Pedersen conductivity. Then the integration procedure described in section 16.6.2 is modified in that the variables y1 and y2 are complex and, at the ionosphere, the zeros of y1 + iξ y2 are computed rather than the zeros of y1 . Examples of such calculations for the fundamental and second harmonic of the azimuthal oscillation are shown in figures 16.6 and 16.7. The scaled variables y1 and y2 are shown. Their real parts do not differ substantially from the lossless case shown in figure 16.5. The imaginary parts are, however, non-zero. Bear in mind that b ∝ iy2 . The electric and magnetic fields now have an in-phase component and the Poynting vector is non-zero. The magnitude of the timeCopyright © 2005 IOP Publishing Ltd.

Standing waves and oscillations in a cold plasma

308

 

p k= ,,  i % 

Real part

  y

  

Imaginary part

         

0.4

0.6

0.8

1.0

0.6 z  sin &

0.8

1.0

z  sin &

Real part

  y  

 

0.2

Imaginary part

 

0.2

0.4

Figure 16.7. Perturbation fields for second harmonic of azimuthal oscillation.

averaged Poynting vector component parallel to B is (section 9.3.3) ˜ φ )}  12 Re( E ν b˜φ ) = 12 {Re( E ν ) Re(b˜φ ) − Im( E ν ) Im((b) ∝ Re( y1 ) Im( y2 ) − Re( y2 ) Im( y1 ).

(16.79)

The scaled Poynting vector represented by the right-hand side of this expression is plotted in figure 16.8. 16.6.4 Perturbation solution of the azimuthal equation Equation (16.63) can be solved by an approximate perturbation method [223], which is very accurate and allows for the introduction of a finite anisotropic ionospheric conductivity. It is convenient to change the origin to the southern ionosphere by writing s = z + z 0 where z 0 is the value of z at the northern boundary. This occurs at z 0 = sin , where ± is the latitude at which the field line reaches the northern and southern ionospheres. Then




d2 ν + K 2 {1 − (s − 12 s0 )2 }6− p ds 2 Copyright © 2005 IOP Publishing Ltd.


ν = 0

(16.80)

Properties of localized field-line oscillations

309

- 0.15

n=1

0.10 0.05 0.00 0.15

n=2

0.10 0.05 0.00 0.0

0.2

0.4

0.6

0.8

1.0

z = sin & Figure 16.8. oscillation.

Scaled Poynting vector for first and second harmonics of azimuthal

where s = 2z 0 . The problem is mathematically exactly the same as the quantum mechanical problem of a particle in a box with an uneven bottom. General solutions are available in standard texts, for example Morse and Feshbach [142, pp 1002–5]. We shall illustrate the technique by solving the equation for the specific case p = 3. The equation is, then,




d2 ν + K 2 {1 − η(s)} ds 2


ν = 0

(16.81)

where η(s) = 3(s − 12 s0 )2 − 3(s − 12 s0 )4 + (s − 12 s0 )6 .

(16.82)

We regard the term in η as a small perturbation of the exactly soluble simple harmonic equation d 2 n + K n2 n = 0 (16.83) ds 2 which obeys the boundary condition n = 0 at s = 0 and s = 2s0 . These boundary conditions are satisfied where K n2 = (nπ/s0 )2 Copyright © 2005 IOP Publishing Ltd.

n = 1, 2, 3, . . .

(16.84)

310

Standing waves and oscillations in a cold plasma

and the corresponding solutions are n (s) = (2/s0 )1/2 sin(nπs/s0 ).

(16.85)

Let ϕn (s) be the correction to n (s) and let kn2 be the corrected value of K n2 . Any arbitrary function obeying these boundary conditions may be expressed as a Fourier series. The trigonometric functions represented by (16.85) are a complete orthogonal set of functions for such a representation. The perturbation ϕ may be written as such a series: 

 ∞ ∞ mπs 2

ϕn (s) = = Anm sin Anm m (s) (m = n) (16.86) s0 s0 m=1

m=1

where Anm are Fourier coefficients that must be determined from the differential equation. The term for which m = n is, of course, n , the solution of the unperturbed equation. If we substitute n = n (s) + ϕn (s) into (16.81), we get   ∞ ∞



2 2 2 2 2 {kn − K m }Anm m − kn η n + Anm m = 0. (16.87) {kn − K n }n +




m=1

m=1

The orthogonal properties of the sine function (Abramowitz and Stegun, [1, equations (4.3.140), (4.3.141)]) show that   s0 1 n=m n (s)m (s) ds = (16.88) 0 n = m. 0 If we multiply (16.87) by n (s), integrate from 0 to s0 , we get {kn2 (1 − ηnn ) − K n2 }(1 + Ann ) − kn2

∞

Anm ηnm = 0

(16.89)

m=1 m =n



where we define

s0

ηnm =

n (s)η(s)m (s) ds.

(16.90)

0

We now make use of the fact that η 1 to make a first-order approximation to this. If |ϕ|/|| is of order |η|, then the coefficients Anm in the Fourier series are all of this order. We ignore terms of second-order and get kn2 =

K n2 . 1 − ηnn

(16.91)

The coefficients in the Fourier series, Amn , (m = n), are found by multiplying by  j and integrating to get {kn2 − K 2j }An j − kn2 ηn j − kn2

∞

m=1

Copyright © 2005 IOP Publishing Ltd.

Anm ηm j .

(16.92)

Properties of localized field-line oscillations

311

If we ignore the second-order terms, we get An j =

kn2 ηn j kn2 − K 2j

.

(16.93)

This allows us to write the solution of the equation as


n (s) = n (s) +  =

∞

m=1 m =n

kn2 ηnm m (s) K n2 − K m2

  

  ∞ 2 n 2 ηmn nπs mπs sin + . sin s0 s0 (n 2 − m 2 )(1 − ηnn ) s0 m=1 m =n

(16.94) The approximation can be carried to higher orders in η using standard perturbation methods [142, pp 1002–5]. We have no need to do this. The integrals required to evaluate ηmn are straightforward. If we take the fundamental mode for p = 3, we get the expression presented by Taylor and Walker [223]. The symmetry in η about the equator means that only odd integers m contribute to the Fourier integral. The higher-order terms in the Fourier expansion decrease rapidly and only the coefficient A13 is important. We get     3 3 1 3 9 2 η11 = − − − + s s4 4 2π 2 0 80 4π 2 2π 4 0   3 1 15 45 − s6 + + − (16.95) 448 32π 2 8π 4 4π 6 0     9 9 9 2 135 225 2835 4 s − − + − + η13 = s s6. 8π 2 0 16π 2 32π 4 0 64π 2 128π 4 256π 6 0 (16.96) While the perturbation method is important in its own right, as a means of rapidly computing mode characteristics, it also provides insight into the validity of the straight field line model. It will be observed that the unperturbed solution on which it is based is simply that for straight field lines, except that the fields are scaled by the scale factors h µ , h ν , h φ . The straight field line model is quite a good approximation, provided that the fields are so scaled. All the additional effects introduced by dipole field are well represented by the first-order perturbation. 16.6.5 WKBJ solutions Equation (16.63) is of the same form as the equations, described in section 11.2, for which the phase integral method can be used, provided that the function Copyright © 2005 IOP Publishing Ltd.

Standing waves and oscillations in a cold plasma

312

(1 − z 2 )6− p does not vary much within a wavelength. This assumption needs critical evaluation for azimuthal oscillations of this type and we discuss its validity later. There is another aspect of the phase integral formula, however, that is unsatisfactory, even when the assumption of slow variation is valid. The spatial part of the phase integral solution of (16.81) that represents a propagated wave can be written in the form    $ = A exp ± iK 1 − η ds (16.97) ν




where variation of amplitude along the field line is ignored. If we superpose two such waves, propagated in opposite directions, we get       $ $ = A exp iK 1 − η ds − A exp − iK 1 − η ds ν    $ 1 − η ds (16.98) = C sin K




where C = 2iA. This will satisfy the boundary conditions provided that K takes one of the values K n given by  s0 $ K n 1 − η ds = nπ n = 1, 2, 3, . . . . (16.99) 0

To the same order of accuracy, from (16.64), √    $ Cν 2 K n 1 − η = cos K 1 − η ds φ aωn



(16.100)

This gives the characteristic frequencies of oscillation. It has nothing to say about the behaviour of the fields as a function of position along the field line. The WKBJ solution improves the approximation to include amplitude factors for the propagated wave. If we apply the methods described in section 15.3, the WKBJ solution of (16.81) is    $ A (s) = exp ± iK 1 − η(s) ds (16.101) ν {1 − η(s)}1/4    $ 1/4 (s) = ± iAK {1 − η(s)} exp ± iK 1 − η(s) ds (16.102) φ






where the magnetic field is found to the same accuracy from (16.64) by differentiating the exponential and ignoring the derivative of the slowly varying amplitude. The superposition of two oppositely propagated waves, in the same way as for the phase integral solution, gives    $ C sin K 1 − η ds (16.103) ν = {1 − η(s)}1/4 √    $ Cν 2 K n {1 − η(s)}1/4 1 − η cos K 1 − η ds (16.104) φ = aωn






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Properties of localized field-line oscillations k2

313

numerical method {Exact Perturbation method

20

WKBJ method

10 0 0°

15°

30°

1.1

Error

1.0 sin '

0.5

1.5

45°

60°

2

3 4 6

75°

' L

2% 1% 0

0.5

1.0

sin '

Figure 16.9. Comparison of perturbation, WKBJ, and numerical computations of characteristic frequency of fundamental azimuthal oscillation as a function of geomagnetic latitude. Density varies as r −3 . In the upper panel, the difference between the perturbation solution and the exact numerical calculation is indistinguishable. The lower panel shows the error in the perturbation solution. (After Taylor and Walker [223].)

which satisfies the boundary conditions provided that K takes one of the values K n given by  s0 $ Kn 1 − η ds = nπ n = 1, 2, 3, . . . . (16.105) 0

The phase integral can be explicitly integrated. As an example, for the fundamental mode when p = 3, we get   $ 1 − η ds = {1 − z 2 }3/2 dz = 18 {3λ + 3 sin λ cos λ + 2 sin λ cos3 λ} (16.106) so that K1 =

4π 3 + 3 sin  cos  + 2 sin  cos3 

(16.107)

and   3λ + 3 sin λ cos λ + 2 sin λ cos3 λ π sin λ −3 . cos (λ) cos ν = 2 sin  3 + 3 sin  cos  + 2 sin  cos3  (16.108) Taylor and Walker [223]) have compared the accuracy of the perturbation solution, the WKBJ solution, and the exact numerical computation. The results






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314

Standing waves and oscillations in a cold plasma

+

 

p'* Exact solution First order perturbation WKBJ solution







  sin &

Figure 16.10. Behaviour of the scaled electric field ν = h ν E ν as a function of latitude. p = 3,  = 64◦ , L = 5.2. (After Taylor and Walker [223].)

for the fundamental when p = 3 are shown in figure 16.9. An example of the electric field behaviour for invariant latitude  = 64◦ is shown in figure 16.10. More complex solutions than described here, taking in the effect of asymmetric ionospheres and inhomogeneity in the plasma, are provided by Southwood and Kivelson [200]. They provide an assessment of the validity of the WKBJ approach.

16.6.5.1 Energy flux If we take the upper sign in (16.101) and (16.102), they represent a wave that is propagated in the positive s-direction. The energy flux in this wave is represented by a Poynting vector directed in the positive s-direction. Its average value over one cycle is  = 12 Re{E ν b˜φ /B}. Now ν ˜φ = h ν h φ E ν b˜φ . From (B.42), h ν h φ ∝ B −1 which is, in turn, proportional to A, the cross-sectional area of a flux tube. Thus 12 Re{ ν ˜φ } ∝ A. The product of the exponential and its complex conjugate is unity. The amplitude factors {1 − η(s)}1/4 and 1/{1 − η(s)}1/4 in the WKBJ expressions cancel and so the energy A propagated down the flux tube is conserved. In this case, the amplitude in the WKBJ solutions also takes care of the cross-sectional area of the flux tube and ensures energy conservation. In general, as the wave is propagated in the background medium, it is partly reflected and partly transmitted. The WKBJ amplitude factors then ensure that the rate of change of the properties of the medium is so small that loss of energy through reflection is negligible. If the medium varies too rapidly, this is no longer true and the approximation fails, as described in section 15.4.2. The first effects of this failure are on the amplitude of the wave. Paradoxically, the less accurate phase integral approximation still provides an estimate of the phase variation and may







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Summary

315

continue to be useful even when the estimate of the error for the WKBJ solution suggests otherwise.

16.7 Summary •

•

•

• •

•

•

•

Within the magnetosphere, the magnetic field lines are roughly dipolar on the flanks and on the dayside. In this region, they are bounded at their ends and, at the equator, by a conducting ionosphere and by the magnetopause, defined as the surface traced out by the outermost closed field lines. The direction of energy propagation of transverse Alfv´en waves is exactly along the field lines. For this reason, they can be localized to a particular field line or magnetic shell. The behaviour of such waves, to a first approximation, can be studied using a model in which the field lines are straight, rather than of dipole shape. In a uniform medium, bounded by infinitely conducting planes representing the ionosphere, standing waves can occur, similar to the waves on a stretched string. A sheet of plasma can oscillate independently of its neighbours as it can slide past them and there is no compression. If there is a gradient of plasma density, then sheets of plasma perpendicular to this gradient can oscillate independently at different characteristic frequencies. If the conductivity is finite then the oscillation is damped and the Pederson currents shield it from the ground. The magnetic field observed on the ground arises from the associated Hall currents. In a dipole-like magnetic field configuration, for the cylindrically symmetrical case, the wave equations separate into two independent sets. One corresponds to a global oscillation, the other to an oscillation in the azimuthal direction that is localized to a magnetic shell. Each shell can oscillate independently at its own characteristic frequency. These oscillations are associated with the transverse Alfv´en wave. In the other extreme, if the wavelength in the azimuthal direction tends to zero so that the azimuthal wavenumber is very large, an independent localized oscillation in the meridian can occur independently. This is localized to a field line and is also associated with the transverse Alfv´en wave. Approximate solutions of the equations for the localized oscillations can be found by the WKBJ method and by a perturbation method. In the latter the zero-order approximation is the solution for straight field lines with the amplitudes of the wave fields modified by the scale factors of the coordinates. Solutions of the equations for localized oscillations may be found numerically using step-by-step integration methods. Comparison of the approximation methods with exact numerical computation shows that the WKBJ approximation is poor for the fundamental modes but excellent for higher-order modes. The first-order perturbation method for the azimuthal

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316

Standing waves and oscillations in a cold plasma oscillations provide solutions that are indistinguishable from the exact calculations. For the localized meridional mode, perturbation methods require the evaluation of expressions that are not analytically tractable and are, therefore, no better than straightforward numerical computation.

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Chapter 17 Standing waves and oscillations in a compressional plasma

17.1 Introduction In this chapter, we extend the treatment of chapter 16 to situations where there is a population of hot plasma so that the sound and Alfve´ n speeds of the plasma are comparable. Such situations arise in the ring current region in the magnetosphere during magnetic storms. In these conditions, a rich variety of long-period pulsations are observed as described in sections 13.4.2.1 and 13.4.2.2. Such waves are not well understood theoretically. It is likely, however, that many of them are localized oscillations of the type discussed in this chapter. This chapter is complementary to chapter 14. There problems were studied for which the transverse Alfv´en wave was decoupled and could be ignored. Here we study localized oscillations for which the fast wave is decoupled and can be ignored. In such situations, the energy flux of the slow and transverse Alfv´en waves is exactly along the field and the behaviour is localized to a field line or magnetic shell. Some treatments of compressional oscillations use a single-fluid MHD approach [199]. This may be valid in some parts of the solar–terrestrial system but, in the ring current region, is not justified. The ring current plasma has been injected from the magnetotail during magnetic storm conditions. It co-exists with cold plasma of ionospheric origin. Near the equatorial plane, the pressure of the ring current plasma may be comparable with the magnetic pressure so that β ∼ 1. If there is little interaction with the cold plasma, its density and pressure are constant. The cold plasma is in ambipolar diffusive equilibrium [15]. Its density and the value of the magnetic field change substantially along the field line. As a consequence, it is desirable to use a two-fluid theory [224, 232, 234, 244]. Such a treatment shows the major features introduced by this complication. Strictly, one should use an anisotropic pressure and modified adiabatic law [46]; such a treatment would add a great deal of complication. We shall not be fitting theory to data but rather illustrating the kinds of physical process that may be important. Copyright © 2005 IOP Publishing Ltd.

317

Standing waves and oscillations in a compressional plasma

318

For this reason, we shall confine ourselves to isotropic adiabatic behaviour of the pressure variation [232, 244]. Once again we seek the conditions that allow localized oscillations. We find that cylindrical symmetry allows an identical azimuthal Alfve´ n oscillation to that for a cold plasma. Our treatment is, therefore, focused on oscillations with large azimuthal wavenumber. In this case, we find the existence of a large variety of modes that may be interpreted as compressional modes arising from the interaction of the slow and transverse Alfv´en waves. As in the case of the meridional oscillations with large azimuthal wavenumber, described in chapter 16, these are polarized in the magnetic meridian and localized to a field line. This is the situation for m 1. It is possible to proceed to the next level of approximation in 1/m [122] but this is beyond the scope of our treatment.

17.2 Localized oscillations Just as in the case of a cold plasma, it is possible to find circumstances in which the equations for a hot plasma describe localized oscillations, confined to a particular magnetic shell or field line. Again, the two conditions are m = 0 and m 1. If we apply the first of these two conditions to the equations for a compressible plasma (14.76)–(14.82), we see that they again separate into two sets. One set involves only ν and φ and is identical to (16.54) and (16.55), describing azimuthal transverse Alfv´en oscillations in a cold plasma. The physical reason for this is obvious. In these oscillations, the cylindrically symmetric magnetic shells slide past each other without compression. The motion is unaffected by the compressibility of the plasma. The other set of equations in this case describes a global compressional oscillation. Of more interest is the case m 1 because, in the ring current region, the observations described in section 13.4.2.1 appear to be of this type. This approximation has been used by Southwood and Saunders [199] and by Walker [232]. If we assume m to be very large, then, from (14.79),






p+

Bbµ = 0. µ0

(17.1)

This expresses the fact that the oscillation is incompressible in the sense of the generalized pressure. The plasma pressure and magnetic field pressure perturbations are non-zero but balance each other. For large m, (14.76) shows that ν = 0. Consequently, from (14.78), φ = 0, so that the plasma motion and the magnetic field perturbation are in the meridian plane. If (17.1) is differentiated with respect to ν and (14.68) and (14.72) used, we get hφ ∂ µ hφ 1 ∂p + VA2 = −VA2 (κ + κν ) µ . (17.2) 2 h µ h ν ∂ν hµ ρ0 h ν ∂ν








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This allows p and

Localized oscillations

319

µ to be eliminated from (14.80), which becomes 1 dν h ν
φ 2µ0 h ν κ + iω p. =

(17.3)

h µ dµ




h φ VA2

B

In (14.82), we cannot ignore ν since it is multiplied by m. We can, however, use (14.76) to replace the first term in braces by a term in µ and (17.1) to replace the term in µ by one in p. The result is





γ B d  vµ  iω − {1 + 12 βγ h µ B} p = h µ dµ B P

hν 1 2 βγ

B

{(1 + 12 βγ )κν + (1 − 12 βγ )κ}


φ .

(17.4) These two equations, together with (14.77) and (14.81) are a set of four firstorder differential equations in φ , ν , p, and vµ /B. The fast-wave components ν and φ have been eliminated. They are ordinary differential equations, since the other partial derivatives do not occur. They determine the local variation along a field line. If we choose this field line to be the reference field line in the local coordinate system described in section 14.5.1, then h µ B = 1. We may take









hφ = z

hν =

1 Bx

(17.5)

where x is the cylindrical radius. With h µ dµ = ds, the four equations can then be written in matrix form:        d 2 0 0 x Eφ iωBx p  ds    vµ 2 bν (17.6) =  iω  κ 0 2 d V B ρ x x B A 0 ds x2B   d   −iωρh B p   ds   vµ d iω (1 + 12 βγ ) − B γ PB ds   x B 2(κν − κ)   0   x Eφ µ0   bν =  (1 + 1 βγ )κ + (1 − 1 βγ )κ . (17.7)  ν   2 2 0 x B 1 2 2 βγ B x Apart from minor differences in notation, these are the same as the equations used by Walker and Pekrides [244]. The problem has been reduced to a onedimensional one. The equations take the form of two pairs of coupled equations, with the magnitude of the coupling terms on the right-hand side determined by the fieldline curvature and the transverse magnetic field gradient. When κν = 0 = κ, the equations separate into two pairs. The first pair, determining E φ and Bν , represents the transverse Alfv´en wave. The second pair, determining p and vµ , Copyright © 2005 IOP Publishing Ltd.

320

Standing waves and oscillations in a compressional plasma

represents the slow magnetosonic wave. The approximations that decouple the fast wave from the set of equations are essential for the success of numerical computation. This is because the fast wave is evanescent if we constrain it to be localized in a small transverse dimension. Thus, as described in section 15.2, the equations are numerically unstable unless the approximations are made. Although terms in both κ and κν appear on the right-hand side, the coupling is entirely determined by the curvature κ . If κ = 0, the first pair of equations is obviously decoupled and represents the transverse Alfve´ n wave. The righthand side of the second pair then represents a source function for the equations. There is an associated pressure perturbation. As the frozen-in plasma is swept back and forth in the meridian plane, there is a pressure fluctuation because of the background pressure gradient. A second solution is possible in which both E φ and Bν are zero. The left-hand side of the second pair of equations then represents a decoupled slow wave.

17.3 Models 17.3.1 Ring current In the ring current region, a population of energetic plasma, injected during a magnetic disturbance from the magnetotail, coexists with cold plasma of ionospheric origin. Once it has been injected, it persists for extended periods, until it is ultimately lost by precipitation into the atmosphere. At this stage, as one moves in from the magnetopause boundary, the pressure increases until the outer regions of the plasmasphere, where it decreases abruptly. The pressure gradient is, therefore, directed inwards over much of its extent, and reverses sign, pointing outwards near the plasmapause. The current is carried by energetic protons, with a westward guiding centre drift, and electrons with an eastward drift. Its value is given by the equilibrium condition J × B = ∇ P.

(17.8)

It is, therefore, westward over most of the extent of the ring current region but eastward on the inner edge where the pressure gradient is outwards. Note: One should not fall into the trap of assuming the current is necessarily in the direction of the guiding-centre drift of the protons. Consider figure 17.1. The pressure gradient is in the direction shown. The magnetic field is normal to the page and may have curvature. The line C D is the direction along which B is constant. The particles are undergoing the gradient and curvature drifts, described in chapter 4, as a consequence of the nonuniform magnetic field. The current at point P is determined by the integrated effect of particles whose guiding centres do not lie on C D. Two such particles, q1 and q2 , are shown. The instantaneous contribution of these particles to the current density at P is in opposite directions. The net current at P is the integrated effect of all such particles. Because of the pressure gradient, the energies and densities of the particles are all different. Their integrated effect is the MHD equilibrium current [33].

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Models

321

D P

B



J

q1

B

C

P

D q2

B

Figure 17.1. Drifting positive particles.

We wish to be able to model the distortions of the magnetic field lines arising from the existence of the additional field of the ring current. We use the simple model of Walker and Pekrides [244] and follow their discussion. It is assumed that the field line lies in a plane and that, locally, the field is cylindrically symmetrical. The model is initially expressed in terms of cylindrical coordinates with x in the radial direction, φ in the azimuthal direction, and z parallel to the axis. It is then converted to the field-aligned coordinates for the purposes of computing the fields. Because of the presence of azimuthal current, the background magnetic field must be derived from an azimuthal vector potential A in the φ-direction. In cylindrical coordinates 1 ∂ (x Aφ ) x ∂z 1 ∂ (x Aφ ). Bz = x ∂x

Bx = −

(17.9) (17.10)

Let ν = x Aφ . Then 1 ∂ν x ∂z 1 ∂ν . Bz = x ∂x

Bx = −

(17.11) (17.12)

It follows that B · ∇ν = 0 and so the intersection of the surfaces, ν = constant

(17.13)

φ = constant

(17.14)

defines a field line. The quantities ν and φ are called Euler potentials. Clearly ν and φ together with µ, measured along the field line, represent a suitable set of field-aligned coordinates. From (14.67), the scale factors for ν and φ may be written in the form hφ = x h ν = (Bx) Copyright © 2005 IOP Publishing Ltd.

(17.15) −1

.

(17.16)

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Standing waves and oscillations in a compressional plasma

We need a self-consistent model which holds near the reference field line of interest. It must specify the shape of the field line, the value of the magnetic field, and the transverse gradients of field and pressure in terms of a few parameters. As a first approximation, we define  3 r B dipole eq + 12 Bring − 12 Bsheet tan2 λ ν = − r 2 cos2 λ r3  3 B 1 2 req dipole 2 1 2 Bsheet z = −x (17.17) + 2 Bring − (x 2 + z 2 )3/2 x2 where 2 =− req

ν Beq + 12 Bring

.

(17.18)

A given value of ν defines the equation of a field line that crosses the equatorial plane at radius req . Bdipole is the value of the Earth’s dipole field at the equatorial plane. The term 12 Bring provides a magnetic field in the z direction which permits us to adjust the value of the equatorial field to observations. It modifies the field near the equator substantially but if Bring is chosen appropriately for a particular value of L , it is only a small perturbation at points on that field line nearer the Earth. The third term represents a radial magnetic field which is zero in the equatorial plane and grows linearly with z . This is associated with a concentration of the plasma near the equatorial plane: this represents the ring current plasma for smaller values of Bring and the plasma sheet for larger values. Its value allows the equatorial curvature to be adjusted according to observation. It should be borne in mind that this expression is only intended to represent the reference field line. At distances far from this line, the actual field may be very different from that calculated from this expression. The potential (17.17) is not consistent with (14.68). We have defined the magnetic field and field-line curvature everywhere. The pressure must then be specified by (14.71) and (14.72). The transverse gradients then are separately specified. From (14.71), the pressure P is constant along each field line. Thus, dP/dν is constant and may be defined as a parameter of the model. Then (14.72) may be written in the form 1 1 ∂B ≡ κν = βκ P − κ Bh ν ∂ν 2 where κP =

1 dP . Ph ν dν

(17.19)

(17.20)

Table 17.1 shows the parameters of some of the models used by Walker and Pekrides, altered to accord with our notation [244]. The distribution of plasma along the field line and the transverse pressure gradient (17.20) are also given. Models 1 and 2 represent a high-latitude field line with an inward pressure Copyright © 2005 IOP Publishing Ltd.

Models

323

Table 17.1. Parameters of models used in calculations [244]. R = (r − req )/2a . Bring (nT)

Bsheet (nT)

N0,cold (m−3 )

Nhot (m−3 )

aκ P

req /a

β0

Model 1 2 3 4 5 6

9 9 6.7 6.7 4.5 4.5

10−8 10−8 10−8 10−8 10−8 10−8

2 × 10−8 2 × 10−8 2 × 10−8 2 × 10−8 0 0

107 a 3 /r 3 106 a 3 /r 3 5 × 107 a 3 /r 3 5 × 107 a 3 /r 3 109 109

2.2 × 106 2.2 × 106 2.2 × 106 2.2 × 106 106 e− R 106 e− R

1.3 1.3 1.3 1.3 1.3 1.3

0.4 0.4 0.4 0 0 0.1

gradient. Model 1 has a large cold plasma density, as might occur if there were significant numbers of positive ions and a relatively small hot plasma density. Model 2 has a smaller cold plasma density. Models 3 and 4 represent a field line that crosses the equatorial plane at geostationary orbit. The cold plasma density is significantly larger than the hot plasma density. Model 3 has an inward pressure gradient and model 4 has a zero pressure gradient as would occur where the ring current pressure is maximum. Models 5 and 6 have cold plasma distributions typical of the outer plasmasphere. Model 5 has a zero pressure gradient for illustrative purposes and model 6 has an outward pressure gradient as would occur on the inner edge of the ring current.

17.3.2 Boundary conditions When considering a plasma with a hot component, the boundary conditions on the electric and magnetic fields, used in chapter 16 still apply but are not sufficient to determine a unique solution. Conditions on the pressure p and the parallel component of the velocity vµ are also necessary. The conditions to apply are less obvious for these variables. At the ionosphere, the density of cold plasma rises rapidly. We shall, therefore, make the assumption that the ionosphere acts like a rigid boundary, so that vµ is zero there, while p has a maximum. This is analogous to the conditions imposed in a closed organ pipe. This amounts to assuming that the ionosphere is a heavy slab of material that does not interact directly with the hot plasma. More subtle boundary conditions are not justified for the problems under discussion in this chapter. While it would be possible to apply more realistic conditions for the electrical properties, we shall find it sufficient in this chapter to treat the ionosphere as a perfect conductor. The counditions used are then that vµ and E φ are zero at the ionospheric boundaries. Copyright © 2005 IOP Publishing Ltd.

Standing waves and oscillations in a compressional plasma

324

17.4 Solutions of the coupled equations 17.4.1 Phase integral solutions The first-order equations (17.6) and (17.7) are in an appropriate form for numerical computation. They are complicated and, to interpret the numerical results, we shall look at the qualitative behaviour of phase integral solutions. We assume  that the dependence on s can be expressed by a phase integral of the form exp{i k(s) ds} as described in section 11.2. Then d/ds ≡ ik . We eliminate E φ and vµ from (17.6) and (17.7), and use (17.1) to replace p by bµ , with the result (ω2 − k 2 VA2 )bν = 2ikκ VA2 bµ (17.21) k 2 VS2 i ω2 − bµ = {−ω2 [κν (1 + 12 βγ ) + κ(1 − 12 βγ )] 1 + 12 βγ k(1 + 12 βγ )



+ k 2 VS2 (κν − κ)}bν .

(17.22)

These two homogeneous equations only have a non-trivial solution if the determinant of the coefficients is zero. This condition provides a dispersion relation:    2 [(1 + 1 βγ )κ + (1 − 1 βγ )κ] 2 2κ V V ν A S 2 2 − ω4 − ω2 k 2 VA2 + 1 + 12 βγ (1 + 12 βγ ) +

k 2 VA2 VS2 {k 2 + 2κ(κν − κ)} (1 + 12 βγ )

= 0.

(17.23)

It can be clearly seen that the coupling is determined by the curvature κ . If κ = 0, the dispersion relation is the same as that for a uniform medium. The magnetic field gradient κν and, hence, the pressure gradient occurs only in terms with a factor κ. This is because, for large m, the direction of energy propagation is exactly along the magnetic field direction. This can be deduced from figure 7.2. As the angle between the wave normal and the magnetic field increases, the ray direction normal to the refractive index surface approaches the magnetic field direction. The ray direction for the transverse Alfv´en wave is always exactly along the magnetic field. Since the wave can, therefore, be confined to a small range of x in which the properties do not change appreciably, it behaves like a wave in a uniform medium. The dispersion relation provides some information about the nature of the coupling. If κ is appreciable and β ∼ 1, so that the Alfv´en and sound speeds are comparable, then the characteristics of the phase integral solution are determined by the full dispersion relation. Suppose now that β 1, so that VS VA . The dispersion relation (17.23) then takes the form ω4 − ω2 k 2 VA2 + k 4 VA2 VS2 = 0. Copyright © 2005 IOP Publishing Ltd.

(17.24)

Solutions of the coupled equations

325

The constant term in k 4 is small compared to the coefficient of ω2 because of the factor VA2 VS2 . Thus, the equation can be approximately factorized as (ω2 − k 2 VA2 )(ω2 − k 2 VS2 ) = 0

(17.25)

with the two roots corresponding to transverse Alfv´en and sound waves, respectively. Alternatively, consider the case when VS2 VA2 . This can occur in one of two ways. Also assume that the hot plasma density is small compared with the cold plasma density. Since ρh V 2 VA2 = 1 S 2 βγρ0 this can occur either because β 1 or because ρh ρ0 . The first of these conditions does not occur in the magnetosphere and the equations remain coupled when it obtains. For the second condition, which does occur in the magnetosphere, the dispersion relation becomes ω4 − ω2

k 2 VS2 1+

1 2 βγ

+

k 2 VA2 VS2 [k 2 + 2κ(κν − κ)] 1 + 12 βγ

= 0.

This, too, has approximate factors, representing two decoupled waves:   k 2 VS2 2 ω − (ω2 − [k 2 + 2κ(κν − κ)]VA2 ) = 0. 1 + 12 βγ

(17.26)

(17.27)

The dispersion relation can also be expressed as an equation that gives k when ω is specified: VA2 VS2 k 4 − k 2 {ω2 [VA2 (1 + 12 βγ ) + VS2 ] − 2κ VA2 VS2 (κν − κ)} + (1 + 12 βγ )ω4 + 2κ VA2 ω2 {(1 + 12 βγ )κν + (1 − 12 βγ )κ} = 0. (17.28) In this form, the dispersion relation can be used to express k as a function  of ω. The change of phase of a field line along the length of a field line is k(ω) ds where the limits of the integral are the end points of the field line. This can be used to find the characteristic values of ω for which the boundary conditions are obeyed. Generally, this means that the phase integral must be an integral multiple of π, or some similar condition, depending on the exact nature of the boundary conditions. The nature of the model that we have adopted, with independent hot and cold plasma components coupled by the fact that they both move with E × B drift normal to the field, leads to complicated interactions between the sound and transverse Alfv´en waves. To see why this should be so, we consider a situation where β  1 at the equatorial plane, in the ring current region. Suppose also that ρh ρ0 . Consider a disturbance with a given frequency ω. The approximation (17.27) holds, so Copyright © 2005 IOP Publishing Ltd.

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Standing waves and oscillations in a compressional plasma

that the Alfve´ n and sound waves are propagated independently; and, because the speeds are different, the phase of the two waves changes at a different rate so that coupling is weak. As the waves progress down the field line, the pressure and hot plasma density and, therefore, the sound speed remain constant. The magnetic field increases rapidly and the cold plasma density increases more slowly, leading to an increase of Alfve´ n speed. At some point along the field line, the speeds are the same and the waves maintain a fixed phase relationship. The coupling between them is now governed by (17.23) and is strong. At a still greater distance along the field line, the Alfve´ n speed grows much larger than the sound speed. Now the waves are again propagated independently but they are governed by (17.25). Again, one of the roots corresponds to a transverse Alfve´ n and the other to a sound wave. Coupling is, therefore, only important when the characteristic wave speeds are comparable. When they differ substantially, the dispersion relation shows that transverse Alfve´ n and sound waves are propagated independently. The field lines are bounded by the ionosphere so that the waves are reflected and, if the frequency is right, a standing wave is set up. The coupling only takes place over a limited region of the field line. Over the remainder of the field line, there is no coupling and the different characteristic waves change phase at a different rate. The result is that, for a given coupled mode, the phase of the field components more strongly associated with the transverse Alfv´en wave may advance through a different number of cycles than those associated with the sound wave. This leads to the complicated mode structures discussed in section 17.4.2. In figure 17.2, we show the variation of the wavenumber [244] as a function of latitude measured along a field line for various models. In each diagram, a value of the frequency has been selected, corresponding to a characteristic frequency calculated numerically as described in section 17.4.2. Each panel shows the two roots k1 and k2 of (17.28) and the wavenumbers kA = ω/VA of the uncoupled Alfv´en wave and kS = ω(1 + 12 βγ )/VS of the uncoupled magnetosonic wave. Figure 17.2(a) applies to one of the higher frequency modes in model 2. Over the whole length of the field line, VS < VA and, thus, kS > kA . The behaviour of k1 is close to the behaviour of kS , and that of k2 to kA . There are two essentially uncoupled modes, a transverse Alfv´en mode and a sound mode. In contrast to figure 17.2(a), because of the larger cold plasma density, the conditions on the model 1 field line shown in figure 17.2(b) lead to values kA > kS near the equator. As latitude increases, the Alfv´en speed increases while the sound speed remains constant. The wavenumbers kA and kS become approximately equal near 20◦ and the two curves cross. In this neighbourhood, the equations are strongly coupled. The mode represented by k1 is close to the curve for kA near the equator but approaches the curve for kS at higher latitudes. The reverse is true for k2 . The consequence is that the modes computed in the next section for this case cannot be related to either a transverse Alfv´en mode or to a sound mode. Figure 17.2(c) shows the wavenumbers corresponding to the lowestfrequency mode in model 1. Now the mode corresponding to k2 does not behave in any sense like a transverse Alfv´en wave near the equator. In fact, k22 becomes Copyright © 2005 IOP Publishing Ltd.

Solutions of the coupled equations 2.5×,

(a)

k (m)

2.0×,

T s

kA

1.0×,

k2

5.0×

2.5× 2.0×

k (m)

kS

k1

1.5×,

0

°

°

°

, ,

° ° Latitude

°

kS k2

5.0× °

°

°

° ° Latitude

°

k (m)

°

,°

(c) T,s

8× 6×

k1

4×

kS



0

,°

(b) T,s

k1

1.0×,

2×

°

kA

1.5×,

0

327

kA °

k2 °

°

° ° Latitude

°

°

,°

Figure 17.2. Behaviour of the wavenumber for coupled and uncoupled oscillations as a function of latitude measured along the field line: (a) model 2, T = 122 s; (b) model 1, T = 167 s,; (c) model 1, T = 744 s. (After Walker and Pekrides [244].)

negative so that, from a phase integral point of view, the wave is evanescent near the equator. The region −10◦ < λ < 10◦ is a barrier region. However, this barrier region is quite narrow compared with the wavelength and there is substantial evanescent barrier penetration. Again, the nature of the modes is substantially modified from the behaviour of those described by the uncoupled equations. Copyright © 2005 IOP Publishing Ltd.

328

Standing waves and oscillations in a compressional plasma

These computations relate to a wave propagated along the field line. When the appropriate boundary conditions are applied, we get standing waves that are the superposition of oppositely propagated waves. Because of the coupling, these waves cannot easily be related to either a localized standing Alfv´en wave or to a fast wave. It is possible to find modes in which there are a different number of nodes and antinodes between the equator and the ionosphere for each field component. These ideas will help in the understanding of the results of the full wave computations in section 17.4.2. 17.4.2 Numerical solutions Equations (17.6) and (17.7) can be written as a set of four simultaneous firstorder differential equations, suitable for numerical integration by a step-by-step process. It is convenient to normalize the variables such that   x Eφ   req E 0   1   r bν   2   i eq xB  =  (17.29)   3   i µ02p   B0   4

   

µ0 ρh VS,0 vµ B B0

where E 0 = V A,0 B0 and the subscript zero denotes the value of a quantity at the equatorial plane, where x = req , z = 0. Then, for starting solutions at the equatorial plane, we can take the four orthogonal values of the column     1 0  0   1  (1) (2)   (0) =  (0) =   0   0  0 0     0 0  0   0  (3) (4)   (0) =  (0) =  (17.30)  1   0 . 0 1











The equations, with each of these starting values, can be integrated from the equator at s = 0 to the ionosphere at s = s0 , yielding four solutions α (s0 ), (α = 1, 2, 3, 4). From linear combinations of these four solutions, we can construct solutions with different symmetries about the equator. Solutions of the form



 (s0 ) = A1 (1)(s0 ) + A3  (3)(s0 )

(17.31)

where A1 and A2 are arbitrary constants, are symmetric about the equator in E φ and p and antisymmetric in bν and vµ . Solutions of the form

 (s0 ) = A2 (2)(s0 ) + A4  (4)(s0 ) Copyright © 2005 IOP Publishing Ltd.

(17.32)

Solutions of the coupled equations (a) Model 1 T,s ib v

 

ibv *

*

(c) Model 3 Ts



Normalised field

Normalised field

E

 

* &

*

E

ib v

 ibv

 

*

*



v

 

*

* &

*

*

E

ib

ibv *



 

(b) Model 2 T,s



Normalised field

Normalised field





*

(d) Model 4 T s

* &

*

*

E

ib



v

 

329

*

ibv *

* &

*

*

Figure 17.3. Lowest-frequency mode for the conditions of models 1–4. The parameters of the models are shown in table 17.1. The field components shown are normalized according to (17.29). (After Walker and Pekrides [244].)

have the opposite symmetry. The boundary conditions described in section 17.3.2 may be written in the form     0 0  1   0     (s0 ) = C1  (17.33)  0  + C2  1  . 0 0



If we equate this to (17.31) or (17.31), we get a set of four homogeneous simultaneous equations for the four coefficients A1 , A2 , C1 , C2 , or A3 , A4 , C1 , C2 . These equations only have a non-trivial solution if the determinant of their coefficients is zero. This determinant depends on ω through the computed value of (s0 ). The characteristic frequencies can be found by finding the zeros of the determinant. The procedure is to guess a value of ω, evaluate (s0 ), and, hence, find the value of the determinant. A second guess provides a second value of (s0 ). These then are the start of an iterative process to find the value of ω through the rule of false position or other equivalent method, as described in section 16.6.2. Walker and Pekrides [244] have performed a number of such calculations. Some examples are shown in figures 17.3, 17.4, and 17.5. Figure 17.3 shows the





Copyright © 2005 IOP Publishing Ltd.



Standing waves and oscillations in a compressional plasma

Normalised field



(a) Model 1 T s

ibv

ib v

 



E *

*

(b) Model 2 Ts



Normalised field

330

* &

*

*



v

E



ib



ibv

 

*

*

* &

*

*

Figure 17.4. Higher-frequency modes in models 1 and 2. The parameters of the models are shown in table 17.1. The field components shown are normalized according to (17.29). (After Walker and Pekrides [244].)

lowest-frequency mode in each of the models 1–4 of table 17.1. First compare the behaviour in models 1 and 2 shown in (a) and (b). The only difference between these two high-latitude models is that the cold plasma density is an order of magnitude larger in model 1 than in model 2. The Alfve´ n speed is, therefore, smaller. This has an obvious effect on the periods, which differ substantially. The dominant field components are E φ , which is symmetric about the equator, and bν which is antisymmetric. These are the characteristics of a fundamental transverse Alfve´ n mode. The compressional component of magnetic field bµ is small. Although it is not clear from the diagram, in neither case does it have nodes. It is non-zero at the equator and approaches the λ-axis asymptotically at the ionosphere. Since p = −Bbµ /µ0 , the same is true of p . These components arise only because of the transverse gradient of pressure and magnetic field. As the plasma is swept backwards and forwards past the point of observation, the changes arise as a result of the gradient. The corresponding diagrams for models 3 and 4 in (c) and (d) show similar behaviour. These models relate to behaviour at the geostationary radius. In model 4, there is no transverse pressure gradient in the undisturbed plasma: the transverse gradient in B is associated with the field-line curvature. The condition (17.1) ensures that there is an associated perturbation of p. In figure 17.4, we see two higher-order modes. These are the next higher frequency with E φ symmetric about the equator. The fields are very different, although the only difference in the two models is the cold plasma density. In (a), for model 1, E φ and bν have the characteristics of the third harmonic. E φ has an antinode at the equator, a node at about λ = 10◦, and then another node at the ionosphere, while bν has nodes near 30◦ and at the ionosphere. The spacing of the nodes is very uneven. This is because of the strong variation of VA with λ. For model 2, in contrast, the structure is that of a fundamental mode. E φ has an antinode at the equator and a node at the ionosphere. The reverse is the case for bν . For this model, there are two frequencies for which the electric field and the Copyright © 2005 IOP Publishing Ltd.

Solutions of the coupled equations

331

Table 17.2. Node structure in models 1 and 2. Period (s)

Eφ

744 353 219 167 226 150

an nan anan anan nanan nanan

417 183 272 163 121 122

an an nan nan nan anan

vµ

bµ

Model 1 na ana nana nana anana anana

nan an nan nanan an anan

a na ana ana nana nana

Model 2 na nana ana ana ana nana

nan nan an anan anan nanan

a ana na nana nanana anana

bν

transverse magnetic field have a fundamental-mode structure. The node structure for vµ and bµ is, however, different for the two modes. This gives a clue as to how the modes can be classified. It can be done according to the node structure. If we denote nodes by n and antinodes by a, then we can describe a field component by a notation such as anan, which means that the field component has an antinode at the equator, followed by a node at an intermediate position and a node at the ionosphere, with antinodes between the nodes. With this notation, the structure of the fields calculated for models 1 and 2 are shown in table 17.2. The actual behaviour of the fields for each case is illustrated by Walker and Pekrides [244]. For any given frequency, the node structure may be different for different field components. For example, in model 2, the three periods 272 s, 163 s, and 121 s each have the same structure for E φ and bν . It is the same as the node structure of a second harmonic on a stretched string. The node structure for the other field components is different in each of the three cases. For 272 s, it is the structure of the fundamental mode in a closed organ pipe. For 163 s, it is that of a third harmonic. For 121 s, it cannot be related to the structure of modes in an organ pipe. There are aspects of the behaviour that can be related to the splitting of energy levels in quantum mechanics but the splitting is not a small first-order effect. Some of the modes can be interpreted as coupling between transverse Alfv´en modes and slow-wave modes. From the phase integral picture, at some point along the field line, the rate of change of phase is the same for both slow and Alfv´en waves. Here there is strong coupling. At other points along the field Copyright © 2005 IOP Publishing Ltd.

Standing waves and oscillations in a compressional plasma

Normalised field

 

(a) Model 5 T s



E

 ibv



 

*

(b) Model 6 T ,s

 

Normalised field

332

*

* &





ib

E v ibv



*

*

 

*

*

* &

*

*

Figure 17.5. Modes in models 5 and 6. The parameters of the models are shown in table 17.1. The field components shown are normalized according to (17.29). (After Walker and Pekrides [244].)

line the coupling is weak and the wavelengths are unrelated so that there can be different numbers of nodes and antinodes for the different modes. The situation is complicated, however, by the effect described earlier, in which the pressure gradient leads to pressure and parallel velocity changes as the transverse Alfve´ n wave sweeps plasma past the point of observation. When such effects are strong, the pressure and velocity perturbations cannot be related to slow-wave behaviour. Examples are the lowest-frequency modes in both models, where the parallel magnetic field and, hence, the pressure perturbation have no nodes at all. The modes with periods 226 s in model 1 and 121 s in model 2 are further examples. Numerical calculations [244] for models 3 and 4 are much simpler. They show that the first five modes have strong E φ and bν perturbations, while p , bµ , and vµ are negligibly small. The node structure for these nodes is that of the first five harmonics of the oscillations of a stretched string. These modes are dominantly transverse Alfve´ n modes of the same type as those described in section 16.5.2. The sixth mode, however, occurring at 68.8 s in model 3 and 68.6 s in model 4, is very different. The vµ and p structures are those of a fundamental mode in a closed organ pipe and these components are of significant amplitude. The components E φ and bν have the structure of the fifth harmonic of a wave on a stretched string. This mode can be interpreted as a a fundamental slow-wave mode, weakly coupled to a fifth harmonic transverse Alfv´en mode. The modes shown for models 5 and 6 in figures 17.5(a) and (b) are entirely different. These show conditions in the neighbourhood of the plasmapause with a much larger cold plasma density, with no pressure gradient in the case of (a) and a small outward gradient in the case of (b). The mode shown in (a) is essentially a transverse Alfv´en mode analogous to a second harmonic on a stretched string (Walker and Pekrides refer to it erroneously as the fundamental). The compressional components are negligible. This is an artificial case in that the pressure gradient is zero. In fact, at the inner edge of the ring current, there is Copyright © 2005 IOP Publishing Ltd.

Summary

333

a strong outward pressure gradient. Figure 17.5(b) shows the case with a weak outward gradient. The structure of the mode is unchanged: there are weak driven compressional components. The frequency, however, is dramatically altered, being raised by a factor of about five. If the pressure gradient is increased, the wave disappears. This suggests that such compressional modes do not exist in the plasmapause region. The Alfv´en and slow modes differ so much in characteristic speed that there is no coupling. Transverse Alfv´en modes can be treated by the methods of section 16.5.2. The slow wave is apparently cut off.

17.5 Summary •

• • • •

•

Localized standing waves in a dipole-like field may also exist when the plasma pressure is comparable with the magnetic pressure. This condition may occur in the ring current region. There are two populations of plasma in this region, a hot plasma originating from the magnetotail and a cold plasma of ionospheric origin. They are distributed differently along the field lines. The motion of both plasma species transverse to the field is an E × B drift. The hot plasma moves independently parallel to the field. In a dipole-like geometry, the cylindrically symmetric transverse Alfv´en wave behaves in the same way as in a cold plasma. For large azimuthal wavenumber, the fast wave is evanescent. The equations can be separated to provide a separate set of equations for coupled slow and transverse Alfv´en waves. These allow oscillations localized to a field line. Numerical solutions for these coupled waves show a complicated mode structure. By using phase integral arguments, these can, in some cases, be interpreted as being coupled locally at some point along the field line where the rate of phase change is the same for transverse and slow waves. The chapter illustrates the complexity of the modes that may occur. It would be difficult to find a set of observations that would allow the comparison with the theory at a number of different positions along the field line. The calculations do, however, illustrate the dangers of trying to identify the standing mode from observations made at a single point.

Copyright © 2005 IOP Publishing Ltd.

Chapter 18 Field-line resonance in low-pressure plasmas

18.1 Introduction The phenomenon of field-line resonance is ubiquitous in solar–terrestrial physics. In chapter 13, we have noted its importance in the understanding of long-period pulsations in the magnetosphere [45, 193, 222]. It has been investigated as a source of the MHD wave dissipation needed for coronal heating [89, 100] and for the absorption of acoustic oscillations in sunspots [75, 201]. It has been studied analytically in plane [193, 237] and cylindrical [76] geometries. Studies in geometries that are more realistic have either made substantial approximations [45] to allow analysis or used idealized models and and a mixture of analysis and numerical work [6–8] or performed numerical simulations in a more realistic model [120, 121]. In the simplest case, field-line resonance can be understood by considering a plane-stratified geometry, with the Alfve´ n and sound speeds increasing monotonically in a direction normal to the magnetic field. An obliquely incident fast wave of given frequency is reflected as described in chapter 15. The fields on the other side of the reflection level decay evanescently. They encounter a resonance level beyond the reflection point where the wavelength parallel to the magnetic field matches that of a transverse Alfv´en wave with the same frequency. This Alfv´en wave is driven into resonance by the incident wave. Its extracts energy from the incident wave and, in an ideal case, grows without limit. In practice, dissipative effects limit its growth. In this chapter, we study the properties of such resonances when the pressure of the plasma can be neglected. This is usually the case in the Earth’s magnetosphere but not in the solar corona. The thrust of this chapter is to understand idealized models analytically, so that this understanding can inform the interpretation of numerical calculations. Thus, most of the analysis is carried out in a plane-stratified medium, with some discussion of dipole geometry. The 334

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Basic ideas of field-line resonance

335

discussion of the case where plasma pressure is important is the subject of chapter 19.

18.2 Basic ideas of field-line resonance To illustrate the physical nature of resonance, we consider a situation where the plasma pressure is negligible, there is a constant magnetic field in the z-direction, and the medium varies in the x-direction only. In such a situation, there is an inherent difficulty with the box models, discussed in section 14.2.1. There can be no pressure gradient to balance a gradient in the magnetic field. Thus, for self-consistency, the magnetic field must be constant. In real situations, such as in the magnetosphere, the field lines are curved. We model such situations by straightening the field lines. The gradients of the Alfv´en speed are usually more strongly dependent on magnetic field gradients associated with the curvature than on density gradients. The gradient of the Alfv´en speed is crucial to the physics of the problem. When the field lines are assumed to be straight, we need unrealistic gradients of density to get realistic gradients of Alfv´en speed. We choose to allow the gradient of the Alfv´en speed to be realistic, without any concern for the value of the density gradient. Most authors using box models make no explicit mention of this difficulty but it is implicit in their treatments. We assume that ρ0 decreases monotonically with x so that VA increases with x. We shall work with (14.25), the second-order equation for ξ , as the fundamental equation. In these circumstances,

 = Vω 2 − k 2y − kz2 A  = ρ0 (ω2 − kz2 VA2 ). 2



(18.1) (18.2)



The zero of is a turning point occurring at x = x T and the zero of is a singularity in the wave equations, occurring at x = x R , called the resonance level. Consider, first, the case where k y is zero. Then the ratio / = ρ0 VA2 = B 2 /µ0 and is constant. Then (14.25) and (14.19) become



ω2 − k z2 VA2 d2 ξ + ξ =0 dx 2 VA2

(18.3)

(ω2 − k z2 VA2 )η = 0.

(18.4)

The turning level x T and the resonance level x R are the same. The first of these equations represents a fast (isotropic) Alfv´en wave, polarized with its displacement along the gradient of VA and perpendicular to B. The second represents a transverse Alfv´en oscillation, of the type described in section 16.2.2, polarized with its displacement perpendicular to the gradient and to B. The two displacements are orthogonal. There is a compressional magneticfield component associated with the fast wave and a transverse magnetic-field Copyright © 2005 IOP Publishing Ltd.

336

Field-line resonance in low-pressure plasmas

component associated with the transverse oscillation. The two disturbances are uncoupled and can take place independently. As described in section 15.6, the fast wave is reflected at x = x T . At this level, there is a natural transverse mode of oscillation but it is not influenced by the reflected wave at the same level. At the other extreme, when k y is very large, x T and x R are widely separated. Near the turning point, (14.25) is unaffected by the resonance, since VA2 ω2 /k z2 . Beyond the turning point, the wave decays evanescently and has a negligible value before reaching the resonance level. Then, although (14.19) suggests that the evanescent fields of the incident wave would drive a transverse resonance, the amplitude of ψ is too small and coupling tends to zero as k y → ∞. Now consider the intermediate case. Suppose that k z remains the same but that k y is not zero. The turning level and resonance level are located such that x R ≥ x T . The turning point and the resonance will be well separated if k y is large enough. For non-zero k y , (14.25), (14.16), and (14.19) become ω2 k 2y {d(VA2 )/dx} dξ d2 ξ + + dx (ω2 − k z2 VA2 ){ω2 − (k 2y + k z2 )VA2 } dx ψ = −ρ0 VA2 η=i



ω2 2 2 − k y − k z ξ = 0 (18.5) VA2

ω2 − k z2 VA2 ω2

− (k 2y

+ k z2 )VA2

kyψ ρ0

(ω2

− k z2 VA2 )

.

dξ dx

(18.6) (18.7)

This equation is essentially that used by Southwood [193] for the electric field component in the y-direction. This electric field is given by E y = vx B = −iωBξ . Since, as previously described, B must be constant for self-consistency, the equations for ξ and E y are the same. As an alternative, we can use the second-order equation (14.26) for ψ and find ξ from (14.15):   d(VA2 ) 2k z2 ω2 d2 ψ 1 dρ0 dψ 2 2 + − + − k y − k z ψ = 0. dx ρ0 dx dx (ω2 − k z2 VA2 ) dx VA2 (18.8) Provided that the turning point and resonance are well separated, near the turning point we can transform the equations as described in section 14.3.3.1 to get the Stokes equation (14.38) for ψ and its derivative (14.39) for ξ . This shows that, in this neighbourhood, the generalized pressure, which, in this case, is equivalent to the compressional component of the magnetic field, has an Airy function behaviour, while ξ has the form of the derivative of an Airy function. These functions decay evanescently for x > x T . Their behaviour is modified near the resonance but the crucial point is that they extend with diminishing amplitude to the resonance level. Here they are coupled to the natural transverse Alfv´en oscillation described by (18.7). Then the picture is of an incident wave that is Copyright © 2005 IOP Publishing Ltd.

Basic ideas of field-line resonance

337

reflected at the turning point. The evanescent fields beyond the caustic curve extend to the resonance level, where they drive a transverse Alfv´en wave. Let us study the details of the behaviour near the resonance. If we choose the origin at the resonance, where ω2 = k z2 VA2 and make the substitution ζ = k y x, then, when ζ is sufficiently small, the equation for ξ takes the form (14.45) d2 ξ 1 dξ −ξ =0 + 2 ζ dζ dζ with η = −i

dξ . dζ

(18.9)

(18.10)

The first equation is the modified Bessel equation [1] of order zero with solutions I0 (ζ ) and K 0 (ζ ). The function I0 (ζ ) increases without limit as ζ → ∞. The function K 0 (ζ ) has the property K 0 (ζ ) → 0 as |ζ | → ∞ for | arg(z)| < 12 π [1, section 9.6]. If the medium is unbounded for large ζ , this means that the solution satisfying the boundary conditions for ζ → ∞ is K 0 (ζ ). As can be seen from the series solution (14.48), this has a logarithmic singularity at the resonance level ζ = 0. It follows from (18.10) that η has a pole at this level. We can find WKBJ solutions to the equation by the technique described for the Stokes equation in section 15.3. The result is e±ζ W± = √ . ζ

(18.11)

Clearly there are just two Stokes lines on which the exponent is real. These are the positive and negative real axis. On these lines, one of the WKBJ solutions is dominant. The implication is that the WKBJ solution on the real axis is either a growing or decaying exponential, representing an evanescent wave at great distances. Thus, energy cannot reach the resonance except by some form of evanescent barrier penetration. The asymptotic approximations of the standard functions are normalized such that their WKBJ representations are  eζ   | arg(ζ )| < 12 π √ 2πζ (18.12) I0 (ζ )   e−ζ  1 3  √ π < | arg(ζ )| < π 2 − 2πζ 2 π −ζ e | arg(ζ )| < 32 π. (18.13) K 0 (ζ )  2ζ Let us suppose that the wave is incident from the negative ζ side. For example, there might be a boundary at x = −a with a propagated wave incident on it from −∞. Then the boundary condition at ζ → +∞ means that the required solution Copyright © 2005 IOP Publishing Ltd.

338

Field-line resonance in low-pressure plasmas

must be K 0 . This decays exponentially as x → ∞. It grows exponentially as x → −∞. At first sight, the WKBJ solutions would appear to indicate that the resonance is isolated by regions where the waves are evanescent so that no energy can reach it. We shall see that this is not the case but it is necessary to examine the exact solution to understand this. We have seen that, near the origin, the Bessel function K 0 has a logarithmic behaviour with a logarithmic singularity at the origin. The displacement ξ is then ξ(ζ ) = AK 0 (ζ )  − ln( 12 ζ )

|ζ | 1.

(18.14)

It can be seen from (14.16) that the generalized pressure is ψ = − 12 A.

(18.15)

The transverse displacement is η=

−iA 2ζ

(18.16)

so that it has a pole at the origin. At face value then, we have distinctly non-physical behaviour near the origin. The pressure oscillates with finite amplitude, while the displacements are infinite, apparently violating any assumptions about the linearity of the waves. We see later that this non-physical behaviour is removed when we allow the existence of a loss mechanism. 18.2.1 Standing waves and field-line resonance So far, we have considered propagated waves in which the wave-normal angle, defined by k y and k z , is specified arbitrarily by initial conditions. Suppose now that the region of interest is one representing conditions inside the magnetosphere, with straight field lines bounded by conducting planes, as specified in the box model of section 14.2.1. Now k z cannot be specified arbitrarily. The boundary conditions are that the electric field or, equivalently, the displacements ξ , η, must be zero at the boundaries. Instead of propagated waves exp{±ik z z}, the displacements are represented by a superposition exp{ik z z} − exp{−ik z z} = 2i sin k z z, representing the z dependence, with k z = nπ/L, where L is the length of the field lines. Now k z is determined by two-point boundary conditions, rather than initial conditions. The appropriate solutions are simply obtained by superposing the previous solutions for the positive and negative values of k 2 . All our arguments can be applied to standing waves confined between planes at z = 0 and z = L just as well as to propagated waves with a specified value of k z . 18.2.2 Loss mechanisms If, in the theory of any physical process, a singularity such as that at the resonance point is encountered, the implication is that the theory is inadequate in the Copyright © 2005 IOP Publishing Ltd.

Basic ideas of field-line resonance

339

neighbourhood of that point. Before physical quantities become infinite, other physical processes take over. The MHD equations do not take account of losses, which become significant near the resonance. If the plasma is unbounded, in the ideal theory, the magnitude of the field components increases without limit in the neighbourhood of the resonance. In general, the rate at which energy is lost depends on the magnitude or gradient of some field component. Resistive losses are proportional to the current density. Viscous losses depend on the velocity gradients. In treating resonance phenomena in sunspots and coronal loops, different authors have incorporated viscous [89, 126] and resistive [75, 76, 157, 158, 201] effects. Such treatments require the introduction of an assumed viscosity or conductivity and do not require the details of the loss mechanism. The actual loss mechanism is likely to involve anomalous effects due to wave–particle interactions, arising from plasma instabilities but these can be well represented by assuming, for example, an appropriate conductivity. We consider such an approach in section 19.5.1. The other possible loss mechanism involves loss at a boundary. The transverse Alfv´en wave at the resonance may carry energy along the field line, away from the region in which resonance is important. In the magnetosphere resonant field lines are bounded by the ionosphere. The major loss mechanism is assumed to be Joule heating of the ionosphere as the wave is reflected from it. In either case, the net effect is that energy is extracted from the wave. The rate at which this occurs is proportional to the amplitude. This leads to an exponential damping of the wave that can be represented by giving ω a small negative imaginary part, analogous to a collision frequency. This may conceal a great deal of very complicated and ill-understood physics but is satisfactory for many purposes. We consider a case that approximates magnetospheric conditions. The medium is assumed to be plane stratified as described earlier but the ionospheric boundaries are assumed to have a finite conductivity. As described in section 16.3, the Pederson currents screen the magnetic field components of the wave from the ground, while the consequent Hall currents have their own subsidiary fields that do not affect the primary wave. For this reason, we can characterize the boundary by a height-integrated Pedersen conductivity !P . The field components in the wave are normal to the magnetic field so that the boundary conditions are zˆ × b = ±µ0 j = ±µ0 !P E

(18.17)

where the upper sign corresponds to the boundary at z = 0 and the lower to that at z = L. The calculation is more straightforward if we choose the origin at the midpoint of the field line so that the boundaries occur at z = ± 12 L. Rather than assuming losses, we assume that the wave is maintained by a source external to the region, so that ω is constant. We assume that k z has a small imaginary part so that k z = kr + iki and use the boundary conditions to find this. Copyright © 2005 IOP Publishing Ltd.

Field-line resonance in low-pressure plasmas

340

We write E x = C cos{(kr + iki )z} −iωb y = − C(kr + iki ) sin{(kr + ki )z}

(18.18) (18.19)

where the expression for b y follows from Faraday’s law. A similar expression can be written down for E y and bx . If this is inserted in the boundary condition, we get, at z = ± 12 L, i − (kr + iki ) sin{ 12 (kr + iki )L} = µ0 !P cos{ 12 (kr + iki )L}. ω

(18.20)

This can be solved by successive approximations. We assume |ki | ∼ (ωµ0 !P )−1 kr . Then the zero-order approximation requires that cos{ 12 kr L} = 0 so that π kr = (2n + 1) n = 0, 1, 2, . . . . (18.21) L Then (18.20) can be expanded to first-order in ki /kr to get −

(2n + 1)iπ = − 12 iµ0 !P ki L ωL

so that ki =

2(2n + 1)π . ωµ0 !P L 2

(18.22)

(18.23)

The resonance condition now takes the form ω − (kr + iki )VA = 0 and is not satisfied on the real x-axis. We may write it in the form   VA (x R ) VA L 2i ω=  1− . kr + iki (2n + 1)π ωµ0 !P L

(18.24)

(18.25)

The effect of the finite conductivity at the boundary is taken into account by giving k an imaginary part ki or, equivalently, adding a negative imaginary part 2VA /(2n + 1)πωµ0 !P L to ω.

18.3 Waves and conservation laws 18.3.1 Definition of wave invariant Let us construct a conservation equation from (14.25). Multiply the equation by

 ξ˜  Copyright © 2005 IOP Publishing Ltd.

Waves and conservation laws and subtract from the result its complex conjugate, getting      ˜ ˜ ξ ξ˜  − { − ˜ }.ξ˜ ξ ξ˜ ξ  − ξ ξ˜  = − ξ˜ ξ  − ˜ ˜

 

 

 

 

 

341

(18.26)

Now integrate along the real axis between the limits x 1 and x 2 . The left-hand side may be integrated by parts. The result is    x2   x2 x2 ˜ ˜    ˜ ˜ξ ξ  − ξ ξ˜   = ξ ξ dx − − { − ˜ }ξ˜ ξ dx. (18.27)  ˜ ˜ x1 x1 x1

 

 

   

 

Consider the left-hand side of this equation. If we use (14.16) to replace ξ  and note that vx = −iωξ , it may be written in the form   ˜ x2 ψv x  . ˜ )|xx2 = 2i Re 2i Im(ψξ (18.28)  1 ω x1

˜ x /ω)|xx21 will be recognized as x /ω, where x is the xThe quantity 14 Re(ψv component of the energy flux vector. The difference between its values at x 1 and x 2 is the net flux of wave energy into the region x 1 < x < x 2 divided by ω. The interpretation of the right-hand side of (18.27) needs some care. Provided that the range of integration does not include a singularity of the and differential equation, this interpretation is easy. In a loss-free medium are real, so that the integrands and, hence, the integrals are zero. The equation then implies that there is no net flux of wave energy into the region x 1 < x < x 2 . The wave energy flux is an example of a wave invariant, as is discussed, for example, by Bretherton and Garrett [30] for general moving media or Eltayeb [59] for hydromagnetic internal gravity waves.





18.3.2 Conservation and non-conservation at singular points If there is a singularity on the x-axis, as is the case with a resonance, matters are not so simple. Near the resonance at ζ = 0, the Alfv´en speed may be written in the form (18.29) VA2 (ζ ) = VA2 (0) + (VA2 ) ζ + · · · . Thus,

 

=

ρ0 k z2 (VA2 ) ζ. k 2y

(18.30)

In a real physical situation, there are always losses. In the magnetospheric situation, the losses which dominate are usually Joule heating of particles at the ionospheric boundary. In other situations, there may be conversion to hot plasma modes which have not been taken into account in MHD theory. If there are small losses, they may be taken into account by giving k z a small negative imaginary Copyright © 2005 IOP Publishing Ltd.

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Field-line resonance in low-pressure plasmas

part. This has the effect of giving ζR a small negative imaginary part and moves the singularity just below the origin. The implication is that, when the losses are reduced to zero, the integrals along the real axis should be taken just above the singularity. We, therefore, replace x by ζ = aeiθ where a is the radius of a small semicircle centred on the origin and integrate from θ = 0 to θ = π. We have seen that, near the origin, ξ = ln( 12 ζ ) (18.31) so that, in the first integral on the right-hand side of (18.27), ξ  = ζ −1 and, on the semicircle, (18.32) ξ  ξ˜  = a −2 . Furthermore, dξ = iaeiθ dθ so that   x2  ˜ ρ0 k z2 (VA2 ) π iθ   ˜ − {e − e−iθ }eiθ dθ ξ ξ dζ = i 2 ˜ k 0 x1 y

   

= − iπ

ρ0 k z2 (VA2 ) k 2y

(18.33)

which holds in the limit a → 0. The second integral on the right-hand side of (18.27) is zero as the limit of ζ ln( 12 ζ ) is zero as ζ → 0. The interpretation of (18.27) is then that, if the range of integration does not include the singularity, then the wave energy entering the region at x 1 is equal to the amount of energy leaving at x 2 and energy is conserved. If, however, the resonance lies between x 1 and x 2 , then the amount leaving at x 2 is less than that entering at x 1 by an amount given by (18.33). The resonance is a ‘black hole’ for energy. Such behaviour is called critical layer behaviour and appears in many physical situations.

18.4 Modelling resonance in a dipole geometry So far we have considered only a plane-stratified model with negligible pressure. The region within the magnetosphere where such resonances are observed is approximately dipolar in structure. The physics of the resonance process is the same but geometrical factors make a considerable difference to the amplitude of the field components. The wave equations in dipolar coordinates are not separable. In general a numerical approach is necessary. Such an approach has been taken by Lee and Lysak [120, 121]. Such an approach gives a global picture. A difficulty is that the scale size of the resonance region is small so that the resolution of the grid adopted raises problems. Another problem is that the numerical computation gives higher frequencies for the resonances than are actually observed. This may be because of uncertainties in the plasma model. For example, the possible presence of heavy ions in the plasma has not been taken Copyright © 2005 IOP Publishing Ltd.

Modelling resonance in a dipole geometry

343

into account. Alternatively, the the length of the field lines may be greater than those calculated for a dipole. In this section, we restrict ourselves to calculating the fields in the neighbourhood of the resonance for a dipole geometry. We also restrict ourselves to the fundamental standing mode on the field line. Similar calculations can be made for higher-order modes. We make use of the scaled fields ν = h ν E ν , φ = h φ E φ , µ = h µ bµ , ν = h ν bν , φ = h φ bφ , where µ, ν, φ are the dipole coordinates of appendix B.3. They are given by (14.62)–(14.66). If the transverse magnetic field components are eliminated, these may be written in the form













h µ h φ ω2 ∂ hφ ∂ ν + ∂µ h µ h ν ∂µ h ν VA2





ν =

h µ h ν ω2 ∂ hν ∂ φ + ∂µ h µ h φ ∂µ h φ VA2





φ =

iω

µ

µ ∂ µ − iω − mω

1 = a



(18.34) (18.35)

∂ν




∂ φ − im ∂ν


ν

 .

(18.36)

If we make the substitution z = sin λ where λ is the latitude, then, in the same way as shown in section 16.6.1, we get




∂2 φ ∂z 2





ν =





φ

∂2 ν + K 2 (1 − z 2 )6− p ∂z 2 6z ∂ φ + K 2 (1 − z 2 )6− p − 1 + 3z 2 ∂z

where K 2 (ν) =

1 + 3z 2 (18.37) µ (1 − z 2 )3 (1 + 3z 2 )2 ∂ µ (18.38) = − iωaν 2 (1 − z 2 )3 ∂ν − mωa

a 2 ω2 . V02 (ν)





(18.39)

The quantities in these equations vary on two different length scales. The scale factors vary on a length scale l, which is of the order of magnitude of the length of a field line. The wavelength parallel to the field is comparable with this. The width lσ of the resonance is controlled by the ionospheric conductivity. We make the assumption that lσ l. (18.40) In the neighbourhood of the resonance, the field quantities vary with ν on the scale lσ and with z on the scale l. Over the width of the resonance region, to this accuracy, (18.36) becomes ∂ µ =0 (18.41) ∂ν







so that µ is independent of ν. We write µ = C1 g(z) where C1 is a constant determining the amplitude of the wave and g(z) is an arbitrary function of z. Then Copyright © 2005 IOP Publishing Ltd.

Field-line resonance in low-pressure plasmas

344

(18.37) becomes an inhomogeneous ordinary differential equation for




d2 ν + K 2 (ν)(1 − z 2 )6− p dz 2


ν :

+ 3z )
ν = −C1 g(z) mωa(1 . (1 − z 2 )3 2

(18.42)

We choose the arbitrary function g(z) such that this becomes







d2 ν + K 2 (ν)(1 − z 2 )6− p dz 2


ν = −
0(1 − z 2 )6− p

(18.43)

where 0 = C1 mωa. The term on the right-hand side, arising from the compressional component of B, acts as a source term. This differential equation can be solved numerically or by using the perturbation techniques of section 16.6.4. The procedure is as follows: (1) Losses are handled by the techniques described in section 16.6.3. This requires that we give K 2 or, equivalently, ω2 an imaginary part such that K 2 (ν) =

a 2 ω2 V02 (ν)

(1 − z 2 )6− p (1 + i)

(18.44)

where  is O(lσ /l) and will be determined later. (2) The latitude for which the resonance is to be calculated is specified by selecting ν = ν0 . The resonance frequency at ν0 for  = 0 is computed numerically as described in section 16.6.1 or using the perturbation technique of section 16.6.4. This yields a value for the normalized frequency K 2 (ν) =

a 2 ω2 (1 − z 2 )6− p = K 02 (1 − z 2 )6− p V02 (ν0 )

(18.45)

and determines a dependence f (z) on z. For example, if the perturbation technique is used, from (16.94),    

  ∞ nπs 2 n 2 ηmn mπs sin + sin f (z) = s0 s0 s0 (n 2 − m 2 )(1 − ηnn ) m=1 m =n

(18.46) where s = z + z 0 and z 0 is the value of z at the northern boundary. (3) We expand V02 to first-order in ν − ν0 so that K 2 (ν)  K 02 {1 − A(ν − ν0 )}(1 − z 2 )6− p (1 + i)  K 02 {1 − A(ν − ν0 ) + i}(1 − z 2 )6− p where A = (V02 ) /V02 . (4) This is inserted in (18.43) and yields an expression for field line: K 02 {A(ν − ν0 ) − i} ν = − 0 f (z)




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(18.47)


ν on the resonance (18.48)

Modelling resonance in a dipole geometry Amplitude

(a)  







Phase





,



L

°

°

°

°



,



L

(b)



°

345

°

°

°

,° &

°

°

,° &

Figure 18.1. Electric fields (a) in the equatorial plane and (b) at the ionosphere. The plasma density at the equator is constant, m = −5, lσ /l = 0. The inner boundary is at L = 5.6 and the resonance at L = 6. Full curve, ν . Broken curve, φ .

.
ν = − K 2{A(ν
0−f (z) ν ) − i}

so that

0

(18.49)

0

(5) The azimuthal component of the electric field can then be found from (18.36), which may be written in the form




∂ φ − im ∂ν


ν = i (1m(1− +z )3z 2)
0. 2 9− p

(18.50)

This may be integrated with respect to ν to give


φ = −im
0Kf 2(z) ln{A(ν − ν0 ) − i} +
1 + O(ν − ν0 )

(18.51)

0


1 is a constant. The expression for
φ has two arbitrary constants. This is because, if the field components
ν and µ are eliminated,
φ is the solution of a second-order differential equation. The constant
0 determines the amplitude of the wave. The constant
1 must be determined by the boundary conditions at the inner boundary. where

For the case of straight field lines, this boundary was assumed to be at infinity. For a dipole field, it cannot be beyond the surface of the Earth. Walker [230] suggests that the plasmapause forms an effective inner boundary and that the mismatch at the abrupt density change allows us to set φ = 0 there. This may not be particularly accurate but allows an illustrative calculation of the fields in the resonance region. An example is shown in figure 18.1. It can be seen that the behaviour is very similar to that for the case of straight field lines. The amplitude shows a maximum at the resonance and the phase




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Field-line resonance in low-pressure plasmas

changes by 180◦ across the resonance region. The diagram also maps the fields from equatorial plane to ionosphere. It should be borne in mind that the fields below the ionosphere are those arising from the Hall currents. Calculations of this sort have been used to match the theoretical fields to the radar observations as shown in figure 13.2. The solutions described earlier are valid near the region of resonance, where the left-hand side of (18.38). They must be matched to solutions near the magnetopause where this is no longer so. This has not been done. It is to be expected that it would not be possible to find a solution for the fundamental mode that makes this matching possible because of the different dependence of ν and φ on z in (18.37) and (18.38). It can be expected that such a match would require a solution that is a superposition of modes corresponding to all the harmonics of the standing wave of the field line. This would require that the solutions for all the standing modes formed a complete set of functions for representing the solution. Verification of this would require further work. The treatment in this section is illustrative. It requires an assumption about the ordering of the length scales in the problem and is valid only near the resonance. Insight has been provided by comparison with the solution for a plane stratified medium with straight field lines. The complexity of the problem when one allows for curved field lines and varying density along the magnetic field can obscure the basic nature of the phenomenon. Readers wanting a deeper understanding of the problem should consult a paper by Wright and Thompson [251] that provides a more detailed and rigorous analytical treatment and clarifies a number of points that have led to confusion in the literature.







18.5 Summary •

•

When the plasma pressure is negligible, the field lines are straight, and there is a gradient of density and, hence, Alfv´en speed perpendicular to the field lines, each plane normal to the field lines has its own natural frequency for oscillation in a transverse Alfv´en mode. A compressional oscillation with a given frequency is coupled to the transverse oscillation if there is phase advance normal to the gradient and to the magnetic field. It excites the transverse oscillations with natural frequencies close to the driving frequency and causes a resonance. A second-order wave equation describing the displacement parallel to the gradient in such a box model is derived. Near the resonance, it can be approximated by Bessel’s equation of order zero. The solution of this equation, with appropriate boundary conditions at boundaries perpendicular to the gradient, has a logarithmic singularity at the resonance. The corresponding transverse displacement perpendicular to the gradient has a pole, while the compressional magnetic disturbance is continuous through the resonance.

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Summary • • •

347

If losses are introduced, for example by providing ionospheric boundaries with finite conductivity, the singularity is moved from the real x-axis. The detailed behaviour of the fields near resonance can then be found. The concept of a wave invariant is introduced. In this case, the wave invariant is the wave energy flux. It is conserved except at the resonance, showing that the resonance is a sink of energy. In a dipole geometry, it is possible to model the fields near the resonance. The equations can be approximated in such a way that the fields are the same as those for the box model, except that they are scaled in magnitude by the geometry. Numerical caculations are presented for this case.

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Chapter 19 Mathematics of field-line resonance in compressible media

19.1 Introduction When the plasma pressure is not negligible, the wave equation for MHD waves becomes more complicated. Nevertheless, because an azimuthal transverse Alfve´ n oscillation with zero azimuthal wavenumber is incompressible, it is only weakly coupled to the compressible wave when m is small. The behaviour of this wave is entirely analogous to the low-β case. In this chapter, we study various aspects of MHD field-line resonance for situations where the pressure may be significant. In the magnetosphere, this is only likely to be the case in the ring current region at times of high magnetic activity. In the solar corona, however, the pressure is large and must always be taken into account. In this chapter, we first study more aspects of reflection and resonance in a plane-stratified medium. Approximations are developed that apply over the whole range including the turning point and resonance. The mathematical properties of the approximate wave equation in such systems are studied in detail, including series solutions and WKBJ approximations. These can be smoothly joined onto numerical solutions of the more general equation. We then consider the case of a cylindrical structure, representing idealizations of various situations in the solar corona. These solutions allow for an azimuthal component of the background magnetic field and for small resistive losses in the plasma, which only become important near the resonance. Such losses have been suggested as a source of coronal heating.

19.2 Field-line resonance in a compressible plane-stratified plasma In chapter 14, we considered various approximations to the wave equations (14.26) and (14.25) when the resonance and the turning point are well sepa348

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349

rated or when the equation (14.19) is decoupled from the wave equation. In these circumstances, near the turning point, the equations reduce to the Stokes equation or its derivative, with the solutions being the Airy functions or their derivatives. Near an isolated resonance, however, the equation reduces to the modified Bessel equation. In practical cases, neither of these limiting cases is appropriate. Significant coupling between an incident fast wave and a resonant Alfv´en wave only takes place when the resonance and turning point are close to each other. The nature of the wave between the turning point and the resonance is an integral part of the solution to the wave equation. It is desirable to study the analytical solutions to a suitable comparison equation in standard form, in order to gain insight into the resonance phenomenon. The treatment in this section takes the wave equation for a compressible plasma and makes some approximations near the resonance to derive a suitable comparison differential equation. It is not one of the well-known standard equations of mathematical physics but its properties have been studied in a related context by Budden [31, section 16.13]. This, in turn, is based on a study by F¨orsteling and W¨uster [71]. Our discussion follows that of Walker [237]. A similar discussion for an incompressible cold plasma was carried out by Zhu and Kivelson [258]. 19.2.1 The resonance equation



= 0 and The differential equation (14.26) has a turning point where singularities where = 0 and where is singular. Field-line resonance occurs where = 0. We obtain a suitable comparison equation for this case. A zero of occurs at any point x = x R where









ω = k z VA (x R ).

(19.1)

It should be noted that, when losses are taken into account so that ω has a negative imaginary part, this condition can only be satisfied for a value of VA (x) which is complex. If VA is assumed to be an analytic function of x, which is real on the real x-axis, then this can only hold for a complex value of x. When condition (19.1) holds, then

 = −k 2y .

(19.2)



is If we assume that, in the neighbourhood of x R , the expression for monotonically varying, then there is a zero of at some point, which may be chosen to be the origin, x = 0. If the direction of the real x-axis is chosen so that Re(x R ) > 0, then >0 (x < 0) (19.3) 0).



 

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Mathematics of field-line resonance in compressible media

Then, if we expand in x − x R ,

 and  about xR, making use of (19.1), we get, to first order

 (x) =  (xR)(x − xR) + O(x − xR)2  (x) = − k 2y +   (xR)(x − xR) + O(x − xR)2

(19.4) (19.5)

where

 =

 − 2ρ0,R k z2 VA,R VA,R

 =

−

(19.6)

 [V 2 − V 2 ] 2k z2 VA,R A,R S

(19.7)

3 VA,R

and subscript ‘R’ denotes the value at x = x R . We have assumed VS to be constant. If it varies, it complicates the expression for but the subsequent analysis does not change. Note that, if there are losses, then the resonance point occurs at a complex value of x R . We can find the value of x R in terms of the other parameters by noting that (0) = 0 and setting x = 0 in (19.5). The result is





xR =

k 2y

 (xR)

(19.8)

where the imaginary part of x R takes care of losses. The differential equation becomes x d2 ψ 1 dψ − k 2y ψ = 0. − (19.9) 2 x − x R dx xR dx If we make the substitution ζ = (k 2y /x R )1/3 x

(19.10)

it can be transformed to a suitable standard form d2 ψ 1 dψ − ζψ = 0 − ζ − ζR dζ dζ 2

(19.11)

ζR = (k y x R )2/3

(19.12)

where and is, in general, complex when there are losses. From (14.16) and (14.19), the displacements ξ and η may be written in the form

 (xR) ξ = 1 dψ ζ − ζR dζ [−  (x R )]2/3

(19.13)

iζ η = R ψ.  2/3 [− (x R )] ζ − ζR

(19.14)

1/2



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Field-line resonance in a compressible plane-stratified plasma

351

It is convenient to normalize ξ and η by setting the coefficients on the lefthand side to unity, getting ξ=

1 dψ ζ − ζR dζ

η=

iζR ψ. ζ − ζR

(19.15)

1/2

(19.16)

If (19.15) is combined with (19.11), we get dξ ζ = ψ dζ ζ − ζR

(19.17)

The pair of first-order equations (19.15) and (19.17) is equivalent to (19.11) and are suitable for numerical integration by techniques such as the Runge–Kutta method. We shall call (19.11) the resonance equation. When ζR 1 and ζ ζR , it reduces to the Stokes equation (14.38). In the other extreme, when ζR = 0, which occurs, for example, when k y = 0, it reduces to (14.39) which has as its solutions the derivatives of the Airy functions. 19.2.2 Series solution to the resonance equation Whittaker and Watson [247, sections 10.2, 10.3] discuss the series solution of second-order differential equations that have the general form u  (z) + f (z)u  (z) + g(z)u(z) = 0.

(19.18)

The resonance equation (19.11) is of this form. The function f (ζ ) has a simple pole at ζ = ζR . The pole is a regular point of the differential equation, by which it is meant that, if f (ζ ) is multiplied by a sufficiently high power of ζ − ζR , the product is analytic. One of the two independent solutions of such an equation is non-singular at the regular point; the other solution may be singular. We derive expressions for the coefficients of the series. Derivation of coefficients of series. Assume a series solution of the form ψ1 (ζ ) = (ζ − ζR )α {a0 + a1 (ζ − ζR ) + a2 (ζ − ζR )2 + · · ·}

(19.19)

where a0 is the coefficient of the lowest-order non-zero term. Differentiate this twice to get ψ  and ψ  and substitute the results in the differential equation. Since the differential equation must hold for all ζ , each coefficient of the powers of (ζ − ζR ) must vanish [247, section 3.73]. Equating each coefficient to zero gives α(α − 2) = 0 a1 (α + 1)(α − 1) = 0 a2 (α + 2)α = a0 ζR a3 (α + 3)(α + 1) = a1 ζR + a0 ... ... ....

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(19.20)

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Mathematics of field-line resonance in compressible media

The first of these equations is called the indicial equation. It has two roots, α = 0 and α = 2. These differ by an integer and, therefore [247, section 10.3], only one of them determines a solution. Since a0 = 0, the third relation shows that α = 0. Therefore, α = 2. The second relation shows that a1 = 0. The remaining relations determine the other coefficients + ζR an−2 a n≥2 (19.21) an = n−3 n(n + 2) where a0 = 1, a−1 = a1 = 0. The second solution has the form [247, section 10.32] ψ2 (ζ ) = Cψ1 (ζ ) ln(ζ − ζR ) + 1 + b1 (ζ − ζR ) + b2 (ζ − ζR )2 + · · · .

(19.22)

If this is substituted in the differential equation and the coefficients of the powers of ζ equated to zero, for the first three coefficients we get b1 = 0, C = 12 ζR , and b3 = 13 . The coefficient b2 is arbitrary because it simply adds a multiple of the other independent solution ψ1 to the solution ψ2 . It may be set equal to zero. We have defined C such that b0 = 1. The other relations found from setting the coefficients to zero show that + bn−3 − ζR (n − 1)an−2 ζ b bn = R n−2 n(n − 2)

n≥3

(19.23)

The two series solutions of the resonance equation, found by this method, can thus be written in the form ψ1 (ζ ) = (ζ − ζR )2 {1 + a2 (ζ − ζR )2 + a3 (ζ − ζR )3 + · · ·} (19.24) 3 4 1 ψ2 (ζ ) = 2 ψ1 (ζ ) ln(ζ − ζR ) + 1 + b3 (ζ − ζR ) + b4 (ζ − ζR ) + · · · . (19.25) From (19.15), the two series for ξ in its normalized form are ξ1 (ζ ) = 2 + 4a2 (ζ − ζR )2 + 5a3 (ζ − ζR )3 + · · · + (n + 2)an (ζ − ζR )n · · · ξ2 (ζ ) =

1 2

(19.26)

2

3

ln(ζ − ζR ){2 + 4a2(ζ − ζR ) + 5a3(ζ − ζR ) + · · ·

+ (n + 2)an (ζ − ζR )n + · · ·} +

1 2

+ { 12 a2 + 3b3 }(ζ − ζR )2

+ { 12 a3 + 4b4 }(ζ − ζR )3 + · · · + { 12 an + (n + 1)bn+1 }(ζ − ζR )n + · · · .

(19.27)

The general solution to (19.11) is a linear combination C1 ψ1 + C2 ψ2 . In general, both terms must be included in order to satisfy the boundary conditions as ζ → ∞. It can, therefore, be seen that, near the singularity of the equation at ζ = 0, ψ and ξ have the forms ψ ∼ 1 + 12 (ζ − ζR )2 ln(ζ − ζR ) ξ ∼ ln(ζ − ζR ).

(19.28) (19.29)

Thus, ψ is not infinite near the singularity but has a branch point, while ξ has a logarithmic singularity. From (14.19), it can be seen that, near the singularity, η has a pole η∼

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1 . ζ − ζR

(19.30)

Field-line resonance in a compressible plane-stratified plasma

353

19.2.3 WKBJ approximations 19.2.3.1 Wave equation At points remote from x and x R , the wave equation (14.26) has a pair of WKBJ solutions (15.5) which may be written in the form W± (x) =

 1/2 exp  ± i  x √ dx .  1/4

(19.31)

The most general WKBJ solution is then ψ(x)  A+ W+ + A− W−

(19.32)

where A+ and A− are arbitrary constants. The condition (15.10) for this approximation to hold may be written in the form       2 5  2 1  3 1    − − + (19.33)   | |.  16  4 4 2

 

 

 

 



19.2.3.2 The resonance equation For the resonance equation(19.11), the WKBJ solutions take the form W± (ζ ) =

(ζ − ζR )1/2 exp{∓ 23 ζ 3/2 } ζ 1/4

(19.34)

and the condition for the approximation to hold is   5 1  3 1    16 ζ 3 − 4 ζ (ζ − ζ )2  1. R

(19.35)

We have seen that, when ζ ζR , the resonance equation reduces to the Stokes equation with WKBJ solutions W± (ζ ) = ζ −1/4 exp{∓ 23 ζ 3/2}

(19.36)

and the condition for the WKBJ approximation to hold in this case is1   5 1    16 ζ 3  1.

(19.37)

1 Budden’s condition [31, equation (15.28)] does not follow correctly from his equation (9.29).

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Mathematics of field-line resonance in compressible media

19.2.4 The Stokes phenomenon In section 15.7, we discussed how WKBJ solutions of the Stokes equation are matched across a turning point. In this section, we discuss similar procedures for the resonance equation. The difference here is that the resonance and the turning point may be so close together that there is no part of the real axis between them on which the WKBJ solutions hold. The discussion is, therefore, more complicated than that for the Stokes equation and is intimately bound up with the extent of the region in which the WKBJ solutions hold. We, therefore, first consider the accuracy of the WKBJ solutions and follow with a discussion of the Stokes phenomenon, and connection relations across the whole region containing turning point and resonance. Accuracy of the WKBJ solution for the resonance equation. In section 15.7, we defined Stokes and anti-Stokes lines for the Airy functions. On the Stokes lines, the exponential in the WKBJ solutions is real. Thus, one of the WKBJ solutions grows exponentially with increasing distance from ζ = 0 and the other decays. Beyond a certain distance from the origin, the decaying subdominant solution is smaller than the error in the WKBJ solution. Here the constant coefficient of the subdominant term may change discontinuously. On the anti-Stokes lines, the exponent is imaginary; and the amplitudes of the two WKBJ solutions are the same and each is oscillatory. The Stokes and anti-Stokes lines for the resonance equation are shown in figure 19.1 for four different values of ζR . The situation is more complicated than for the Stokes equation because the WKBJ solutions (19.34) have a second branch point at the resonance ζ = ζR . A branch cut must radiate from each branch point. We have chosen these in such a way that the branch cut joins the two branch points, running along a path just below the real axis. These have been labelled B in each panel of figure 19.1. The curves C are defined by the criterion (19.35)    5 1 3 1 −1   (19.38)  16 ζ 3 − 4 ζ (ζ − ζ )2  = e . R On them the ratio of dominant to subdominant solution is e2 and we adopt this as the criterion for which (19.35) is sufficiently well satisfied. Outside the closed curves C, the WKBJ approximations hold; inside them it breaks down. The WKBJ solutions (19.34) are the same as for the Airy functions2 . The Stokes and anti-Stokes lines are labelled S and A, respectively. The boundaries between Stokes and anti-Stokes regions, described in section 15.7, are labelled D. In regions bounded by curves D, which contain a Stokes line, the error in neglecting the subdominant solution is smaller than the error in the WKBJ solution. Figure 19.1(a) shows conditions when ζR is small. The region enclosed by the curve C, in which the WKBJ approximation fails, is a single region including both singularities. In the limit ζR = 0, the two branch points coalesce. The behaviour is then identical to that for the derivatives of the Airy functions. The Stokes constant, giving the magnitude of the discontinuous change in the coefficients, 2 Budden [31, appendix] chooses the origin to be at the resonance ζ . On his definition, the Stokes R

and anti-Stokes lines radiate from there. This is a reasonable approximation for |ζ | |ζR | but is not appropriate at smaller values of ζ . It does not, for example, allow for the possibility of a valid WKBJ approximation on the real axis between the two singular points.

Copyright © 2005 IOP Publishing Ltd.

Field-line resonance in a compressible plane-stratified plasma zR=1.0

2

S

B C

-2 -4

-3

-2

2

0

-1

0

Real (a)

S

2

S B C S

A

1

1

0

Real (c)

0

2

1

S

3

-2

3

4

zR=2.60

A

C

D

S

A B

D C A

S 2

2

Real (b)

D 0

C

A

-1

C -2 2

S

B

D

-2 -2

zR=2.59

D

D

D

D

A

S

1

A

D A

0

A

S

Imaginary

Imaginary

S

A D

zR=3.0

A

C

A D 0

S

D

Imaginary

Imaginary

2

355

-2

-1

0

Real (d)

1

2

3

Figure 19.1. Stokes regions in the complex ζ -plane for the resonance function [237].

is S = +i and this is the ratio of the coefficient of the reflected WKBJ solution to that of the incident WKBJ solution. Figure 19.1(b) shows the conditions for a large value of ζR . The boundaries C now define two separate regions, each surrounding one of the branch points. Within these regions, the WKBJ solutions do not hold. There is then a portion of the real axis between the turning point and the resonance outside the regions C. Here the WKBJ solution is valid. As ζR → ∞, the behaviour of the solutions near the turning point becomes independent of that near the resonance. In the neighbourhood of the origin, it is the same as that for the Airy functions with a Stokes constant S = −i. The change from one type of behaviour to the other is illustrated for two close values of ζR in figures 19.1(c) and 19.1(d). Stokes constants for the resonance equation. We have seen that, as ζR increases from zero, the Stokes constant must change continuously from +i at one extreme to −i at the other. The computation of S for intermediate values is not straightforward. The Furry method [74] used for the Stokes equation does not work. The method of steepest descents [31]) could be applied to a suitable integral representation of the solution but such an integral representation has not been found for the resonance equation. One must resort to numerical techniques [31, 237]. Consider figure 19.1. In each panel, the desired solution of the resonance equation must not grow without limit as ζ → ∞. For large positive values of ζ on the real axis,

Copyright © 2005 IOP Publishing Ltd.

Mathematics of field-line resonance in compressible media

356 1.0

1.0

|S | pha(zR)=0°

0.5 0.0

Á(S)

-1.0 0

Â(S)

-0.5 2

1

3

(a)

4 |zR|

-1.0

0

2

1

3

(b)

4 |zR|

1.0

1.0 |S |

0.5 0.0

0

pha(zR)=-30°

Á(S)

0.5 |S |

pha(zR)=-45°

Á(S)

0.0

Â(S)

-0.5 -1.0

pha(zR)=-15° Á(S)

0.0

Â(S)

-0.5

|S |

0.5

Â(S)

-0.5 2

1

3

(c)

4 |zR|

-1.0 0

1

2 (d)

3

4 |zR|

Figure 19.2. Complex Stokes constants for the resonance function.

therefore, the WKBJ solution contains only the subdominant term W+ (ζ ) =

(ζ − ζR )1/2 exp{− 23 ζ 3/2 }. ζ 1/4

(19.39)

Straightforward attempts to match this to the solution on the negative real axis by using it as a starting solution and numerically integrating the differential equation along some path in the complex plane to a point on the negative real axis where |ζ | 1| risk failure because of numerical swamping. If, on any portion of the path, the desired solution decays exponentially as the integration progresses, then small numerical errors introduce a small term containing the unwanted solution that grows exponentially and rapidly swamps the subdominant solution. To avoid this, note that, on the anti-Stokes line making an 60◦ angle with the real axis, the solution has the same form as (19.39). Use this as the starting point of the integration. In the 120◦ sector between this anti-Stokes line and the negative real axis, the wanted solution is dominant and there is no problem of numerical swamping. Equations (19.15) and (19.17) may be integrated numerically, using a Runge– Kutta technique, as described by Budden [31, appendix] along a straight line joining points on the anti-Stokes line and the negative real axis, which are equidistant from the origin.

The results of these procedures are shown in figure 19.2. Losses are taken into account so that ζR is complex. The Stokes constant for the resonance equation is shown as a function of |ζR | for several different values of pha(ζR ), Copyright © 2005 IOP Publishing Ltd.

Solutions of the resonance equation

357

the phase of ζR . Note that ζR depends on several parameters including k y and may not be chosen arbitrarily in any particular problem. When ζR is real so that there are no losses, the Stokes constant is −i for ζR = 0. It approaches +i asymptotically for large values of |ζR |. The magnitude of ζR is the magnitude of the reflection coefficient. At the two extremes, it is unity; and there is no coupling to the resonance and, thus, total reflection. Between these extremes |S| falls to a minimum value and not all the wave is reflected, even though there are no losses. This represents coupling to the resonance which absorbs energy. When ζR is complex, the reflection coefficient is smaller at the intermediate values because of the energy transfer to the ionosphere.

19.3 Solutions of the resonance equation 19.3.1 Numerical computation of the solutions The calculation of numerical solutions of the resonance equation is complicated by the presence of the singularities in the fields at the resonance. For this reason, our numerical computation uses the series solution in the neighbourhood of the resonance, where the logarithmic singularity of ξ and the pole of ζ are explicitly included as factors. The calculation proceeds as follows: (1) The integration is started on the Stokes line that makes an angle of 60◦ with the positive ζ -direction. On this line, and to the left of it, the wanted solution is dominant. (2) The equations (19.15) and (19.17) are integrated numerically along a suitable path, on which the wanted solution remains dominant, to a point on the negative real axis far from the origin. (3) The equations are integrated numerically towards the origin along the real axis to a point far enough from the singularity to avoid numerical difficulties. (4) The solution on the real axis, across the singularity, is a linear combination of the two series solutions. The constants in this linear combination are adjusted to match the numerical solution at this point. (5) The appropriate WKBJ solution is used as a starting solution at large positive ζ . The equations are integrated numerically in the negative direction along the real axis to meet the series solution. The accuracy with which the two solutions meet is a measure of the accuracy of the computation. (6) Even when, to allow for losses, the singularity is at a complex value of ζ , the same procedure is used. This avoids numerical difficulties when the path of integration passes close to the singularity. The calculation could have started on the positive real axis. The integration could have proceeded in the negative direction up to a point where it is matched to the series solution. Then, on the other side of the singularity, the numerical computation could have been continued in the negative direction. The advantage of the adopted method is that, because two different starting points are used and Copyright © 2005 IOP Publishing Ltd.

Mathematics of field-line resonance in compressible media

358

|zR|=1.5; pha(zR)=0°

|zR|=0.5; pha(zR)=0° 3 2

y

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-2

0

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z

2

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6

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2

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6

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3

90°

2

0°

-90°

1

-90°

-180°

0 -6

0°

-4

-90°

4

90°

1

0°

1

180°

2

0 -6

y2

-90°

0° 1

h

180°

h

-4

-2

0

z

2

4

6

-180°

Figure 19.3. Computed solutions for the wave fields: left-hand panels, ζR = 0.5; right-hand panels, ζ = 1.5; the full curve represents the amplitude; the broken curve the phase.

the integration meets in the middle, there is a check on the validity of the starting values, and on the numerical accuracy. The solutions match closely.

19.3.1.1 Properties of the solutions The calculations in figure 19.3 are for real ζR , so that there are no losses. It can be seen that ψ shows no singularity at ζ = ζR because limζ →ζR (ζ −ζR )2 ln(ζ −ζR ) = 0. The displacement ξ has a logarithmic singularity, the amplitude going to Copyright © 2005 IOP Publishing Ltd.

Solutions of the resonance equation 3

|zR|=0.5; pha(zR)=-15°

90°

2

y

0° 1 0 -6

-4

-2

0

z

2

4

6

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0 -6

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-90°

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0°

h

0°

0 -6

-90° -4

90°

1

90°

x

180°

y

180°

2

|zR|=1.5; pha(zR)=-15°

2

-90°

3

0 -6

3

180°

359

-90° -4

-2

0

z

2

4

6

-180° 180° 90° 0°

h 1 0 -6

-90° -4

-2

0

z

2

4

6

-180°

Figure 19.4. Computed solutions for the wave fields: left-hand panels, ζR = 0.5 exp{−iπ/12}; right-hand panels, ζR = 1.5 exp{−iπ/12}; the full curve represents the amplitude; the broken curve the phase.

infinity and the phase shifting by 45◦. The transverse displacement η has a pole, with a phase change of 180◦ across the resonance. Losses may be included by assuming a complex value for ζR , as shown in figure 19.4. Here, in each case, ζR has the same amplitude as the corresponding case in figure 19.3 but it has a negative phase to allow for losses. The general features of each plot are the same. Since, however, the singularity is below the real axis, |ξ | and |η| have finite maxima on the real axis and are not singular because the singularity is below the real axis. These figures illustrate the efficiency of the coupling to the resonance. When |ζR | = 0.5, the efficiency of energy transfer is near maximum. The transverse displacement η rises to a very large maximum. Copyright © 2005 IOP Publishing Ltd.

Mathematics of field-line resonance in compressible media

360

|zR|=0.5; pha(zR)=0°

|zR|=0.5; pha(zR)=-15°

3

90°

2

y

0° 1

0 -6

-4

-2

0

z

2

4

6

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0° 1

0 -6

-2

0

z

2

4

6

3

0

z

2

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90° 0°

0 -6

-90°

-4

-2

0

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3

90°

2

0°

h

-90°

-90°

-4

-2

0

z

2

4

6

-180°

-180°

180°

1

1

-180°

180°

1

-180°

0°

0 -6

-2

x

90°

h

-4

2

180°

2

-90°

3

-90°

-4

0° 1

90°

x

90°

y

180°

2

180°

2

-90°

3

0 -6

3

180°

0 -6

-4

-2

0

z

2

4

6

-180°

Figure 19.5. WKBJ approximations to the wave fields: left-hand panels, ζR = 0.5; right-hand panels, ζR = 1.5 exp{−iπ/12}; the full curve represents the amplitude; the broken curve the phase calculated from the WKBJ approximations. The dotted curve matches the series solution to the WKBJ solutions.

The resonance is strongly excited. When |ζR | = 1.5, the resonance and the turning point are widely separated. The resonance is then only weakly excited. When |ζR | → 0 or|ζR | → ∞, the excitation tends to zero. 19.3.2 Accuracy of the WKBJ solutions WKBJ or phase integral approximations are often used in discussions of problems related to resonance. To evaluate their accuracy, we have computed a set of such solutions for the resonance equation. The WKBJ solution matching boundary Copyright © 2005 IOP Publishing Ltd.

Reflection coefficients

361

conditions at ζ → +∞ is used as a starting value. It is then matched to a linear combination of the two series solutions to continue the solution to negative values of ζ . This is then, in turn, matched to a linear combination of the two WKBJ solutions valid for ζ negative. Some results are shown in figure 19.5. The WKBJ solutions have been extended beyond their region of validity. It can be seen how they diverge from the true solution near the origin and the singularity. They are very good in the region of validity; and if they are superimposed on the exact numerical solution, they are indistinguishable from it to the accuracy of the plot. 19.3.3 Approximate solutions of the wave equation The resonance equation (19.11) represents conditions near the resonance and turning point. Unless the characteristic speeds behave at greater distances according to (19.4) and (19.5), the solution at great distances from the resonance and turning points will differ from that obtained. If the medium is sufficiently slowly varying, the more general WKBJ solutions (19.31) hold in this region. Provided that the medium is monotonically varying with x, from (19.31), at large positive values of x, the WKBJ solution is ψ=A

 1/2 exp  −  x |√ | dx .  1/4 0

(19.40)

Near the resonance, it may be approximated by the WKBJ solution of the resonance equation √  (x R ) − 2 .ζ 3/2 iπ/4 e 3 (19.41) ψ = Ae ky



It can then be mapped through the resonance and the turning point using the series solution to get the WKBJ solution to the resonance equation. This, in turn, can be matched to the WKBJ solution of the wave equation for x < 0 : ψ=A

 1/2  exp i  x k dx  + S exp  − i  x k dx . x x  1/4 0 0

(19.42)

The details of the matching provide the Stokes constant, S. If the medium does not vary monotonically, as would be the case near the plasmapause, the solution may have to be matched across more than one resonance region.

19.4 Reflection coefficients The solution (19.42) is the sum of a positively propagated incident wave and a negatively propagated reflected wave in the region x < 0. The reflection Copyright © 2005 IOP Publishing Ltd.

362

Mathematics of field-line resonance in compressible media

coefficient R , at the level x , is defined as the ratio of the reflected WKBJ solution to the incident WKBJ solution:    x R(x) = S exp − 2i k x dx . (19.43) 0

The Stokes constant, plotted in figure 19.2, therefore represents the reflection coefficient of the region x > 0, referred to the level x = 0. The energy flux of each term in the WKBJ solution is proportional to the square of the modulus of the amplitude. When k x is real, the reflection coefficient of the energy flux is

 = R(x) R∗ (x) = |S|2

(19.44)

where the asterisk denotes the complex conjugate. It is independent of x because of energy conservation. In general, it is less than unity. At large positive values of x , the wave is evanescent. The balance of the energy represents that lost in the resonance as described in section 18.3.2. We can define a transmission coefficient, for energy transferred through the turning point to the resonance, as



=1−

.

(19.45)



Figure 19.6 shows the fraction of the incident energy that is transmitted to the resonance plotted as a function of ζ (R). The phase of the Stokes constant S or, equivalently, the reflection coefficient R(0) is also plotted. In the upper panel, ζR is real, so that there are no losses—the energy just feeds into the resonance. In the lower panel, ζR has a small negative imaginary part, representing the losses. In both cases, there is a strong maximum of occurring near ζR = 0.5. Here a substantial fraction of the energy, more than 50%, is fed into the resonance. At large values of ζR , the phase of the reflection coefficient is 90◦ . This is the same as that for the Airy function Ai(ζ ). As ζR decreases, it changes smoothly through 180◦ near the value of ζR corresponding to maximum transmission. Here the reflected wave is out of phase with the incident wave. When ζR → 0, the resonance becomes decoupled. The reflected wave is then −90◦ out of phase with the incident wave. This is the same as that for the derivative of the Airy function Ai (ζ ).



19.5 Resonance in cylindrical geometries The physical nature of the resonance phenomenon can be well understood from the behaviour in a plane-stratified medium described earlier. Realistically, the geometry is more complicated than this. Within the magnetosphere, as we have seen in section 18.4, the geometry is approximately dipolar, the pressure is negligible, and the losses arise from the ionospheric boundaries. These losses are quite large, leading to a fairly broad resonance. Magnetometer chains and VHF or HF radar systems observe the region of the resonance directly and much Copyright © 2005 IOP Publishing Ltd.

Resonance in cylindrical geometries 0.6 pha(R)

0.5

363

180°

pha(xR)=0°

90°

0.4 |T|2

0.3

0°

0.2

-90°

0.1 0

0

1

2

0.8

3

z

4

180°

pha(xR)=-15°

0.6

pha(R)

90°

|T|2

0.4

0° -90°

0.2 0

-180°

0

1

2 z

3

4

-180°

Figure 19.6. Left-hand axis, full curve: fraction of incident energy  , transmitted to the resonance. Right-hand axis, broken curve: phase of Stokes constant, corresponding to the reflection coefficient R.

of the work focuses on the structure of the fields in this region. In solar physics, the geometry, to a first approximation, is cylindrical, representing the structure of coronal loops or sunspots. The loss mechanism is in the plasma, generally represented as an equivalent resistivity or sometimes viscosity but probably arising from a variety of wave–particle interactions associated with turbulence. Observational techniques do not allow the investigation of the detailed structure of the fields. The focus is on the energy transfer as a mechanism for heating the plasma by conversion of MHD wave energy. In this section, we use a somewhat different approach from the plane geometry described earlier to study resonance in a cylindrical column of plasma. Such approaches have been used by a number of authors to study resonance phenomena, particularly heating, in solar structures that can be idealized as cylindrical [76, 89, 100, 157, 158, 175, 176, 201]. Copyright © 2005 IOP Publishing Ltd.

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Mathematics of field-line resonance in compressible media

19.5.1 Equations describing dissipation in cylindrical flux tubes Apart from the effects of the cylindrical geometry, the equations derived in section 14.2.2 have an additional complication. They allow for a spiral magnetic field structure. The magnetic field has an azimuthal component. Field lines are not parallel; and, as r increases, there is shear of the magnetic field. This can affect the dissipation. We set out to find a pair of differential equations for ξr and ψ for this resistive case, analogous to the collison-free equations (14.98) and (14.99). Except for the notation used, we follow the derivation of Goossens et al [76]. It is algebraically tedious, but straightforward. Inclusion of losses in the cylindrical wave equations. The losses are assumed to be resistive losses. This means that we must use the modified version (2.68) of the reduced form of Faraday’s law. The important approximation [76] is to recognize that the dissipation is very small everywhere except in the immediate neighbourhood of the resonance. In this neighbourhood, the perturbed field variables vary much faster than the equilibrium variables. In addition, the rate of change of the perturbed variables with respect to r , as the resonance is crossed, is much larger than the rate of change with respect to φ and z. This means that, in the additional terms associated with the dissipation, only the variation of the perturbations with respect to r is retained. The reduced Faraday law with the additional loss term is then i d2 b (19.46) b = B · ∇ξ − B∇ · ξ − ξ · ∇ B + ωµ0 σ dr 2 where ξ = v/ω. Because the conductivity is large, the term involving the operator d2 /dr 2 is very small. Nevertheless, we retain it throughout the derivation and only make the approximation    1 1 d2 w    (19.47)  1   ωµ0 σ w dr 2  where w is any perturbation component, at the end. This is because some factors in the final equations arise from taking the difference of quantities that, in the neighbourhood of the resonance, are small. In the collisionless case, the resonance is determined by a factor ω2 − (k · V A )2 becoming zero. For the case of finite conductivity, an operator ω2 − (k · V A )2 − (iω/µ0 σ )(d2 /dr 2 ) replaces this factor. Then, near the resonance, the term that depends on the conductivity cannot be ignored. The reduced MHD equations (2.55), (2.57), and (19.46) may be written out in the cylindrical coordinate system of section 14.2.2. Note that the zero-order pressure balance is given by (14.1). We get dP p = − ρ0 VS2 ∇ · ξ − ξr dr   dψ 2 1 2 − i(k · B)br − Bφ bφ ρ0 ω ξr = dr µ0 r   b im 1 d r ρ 0 ω 2 ξφ = ψ− (r Bφ ) + i(k · B)bφ r µ0 r dr   dBz 1 bz ρ0 ω2 ξz = ik z ψ − + i(k · B)bz µ0 dr

Copyright © 2005 IOP Publishing Ltd.

(19.48) (19.49) (19.50) (19.51)

Resonance in cylindrical geometries  1− 

d2 i ωµ0 σ dr 2

365

 br = i(k · B)ξr

   d2 d Bφ i = i(k · B)ξ − B ∇ · ξ − r ξ 1− b r φ φ φ ωµ0 σ dr 2 dr r   2 i dBz d 1− bz = i(k · B)ξz − Bz ∇ · ξ − ξz ωµ0 σ dr 2 dr

(19.52)

(19.53)

(19.54)

where

m k ≡ φˆ + zˆ k z . (19.55) r The components of b may be eliminated by operating on (19.48)–(19.51) with the operator [1 − (i/ωµ0 σ ) d2 /dr 2 ], remembering that it operates only on perturbation variables. After some tedious algebra we get    iω d2 i d2 dψ 2 2 ρ0 ω − (k · V A ) − ξr = 1 − µ0 σ dr 2 ωµ0 σ dr 2 dr    2Bφ d Bφ i(k · B)ξφ − Bφ ∇ · ξ − r ξr (19.56) + µ0 r dr r    iω d2 im i d2 ρ0 ω2 − (k · V A )2 − = 1 − ξ ψ φ µ0 σ dr 2 r ωµ0 σ dr 2   ik · B Bφ 2 ∇ · ξ − ξr (19.57) + µ0 r    iω d2 d2 i ξz = ik z 1 − ψ ρ0 ω2 − (k · V A )2 − 2 µ0 σ dr ωµ0 σ dr 2 +

ik · B Bz ∇ · ξ. µ0

(19.58)

Now (19.48), (19.53), and (19.54) may be combined and the equilibrium condition used to give  2   2Bφ d2 i dP d2 ik · B i B·ξ + + ψ= ξr 1− ωµ0 σ dr 2 µ0 µ0 r ωµ0 σ dr dr 2  iVS2 d2 − ρ0 VA2 + VS2 + ∇·ξ (19.59) ωµ0 σ dr 2 and if (19.57) and (19.58) are used to eliminate B · ξ from this we get "  # d2 i 2 2 2 2 2 ρ0 (VA + VS ) ω − k VS − ωµ0 σ dr 2   i iω d2 d2 2 2 2 − VS ω − (k · V A ) − ∇·ξ µ0 σ dr 2 ωµ0 σ dr 2    iω d2 iω d2 = ω2 1 − 1 − ψ µ0 σ dr 2 µ0 σ dr 2

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Mathematics of field-line resonance in compressible media

366



  2 2 2 d2 i ω Bφ 1 − µ0 r ωµ0 σ dr 2 " # i d2 iω d2 dP 2 2 − ω − (k · V A ) − ξr . dr µ0 σ dr 2 ωµ0 σ dr 2 +

(19.60)

The expressions (19.57) and (19.58) may be combined to give a relationship between ∇ · ξ , ψ, and ξr :     iω d2 d2 i ρ0 d 2 2 2 ρ0 ω 1 − (r ξr ) ω − (k · V A ) − ∇·ξ = 2 2 ωµ0 σ dr r µ0 σ dr dr    2m Bφ k · B m2 d2 i 2 − + kz ξr . (19.61) 1− ψ+ ωµ0 σ dr 2 r2 µ0 r 2 At this point, we can make approximations because of the high conductivity. We neglect terms in σ −1 except in the factor ω2 − (k · V A )2 − (iω/µ0 σ )(d2 /dr 2 ). Here the term is not negligible near the resonance where ω2 − (k · V A )2 = 0. We can then eliminate d (r ξ ) and eliminate ξ ∇ · ξ from (19.60) and (19.61) to get a differential equation for dr φ and ∇ · ξ from (19.56) by using (19.57) and (19.60).

The result of this procedure is a pair of differential equations

r1 drd (r ξ ) = −  ψ + ξ = { 2 + }ξ − ψ

dψ dr 

where

≡ 

iωρ0 d2 − µ0 σ dr 2

(19.62) (19.63)

(19.64)

and the coefficients are those defined in section 14.6. The right-hand sides of these equations are the same as those of (14.98) and (14.99) for the collision-free case. On the left-hand sides, the factor has been replaced by the differential operator . '





The transverse displacement ξ⊥ = ξφ2 + ξz2 can be found by combining (19.57) and (19.58):    m cos α 2VA (k · V A ) sin α cos α − k z sin α ψ − ξr . (19.65) ξ⊥ = i r r



19.5.2 Solutions of the dissipative equations in cylindrical geometry In the regions remote from the resonance, the resistive terms are negligible. We can solve the collision-free equations numerically or by using an appropriate WKBJ solution in the case of a plane-stratified medium. Near the resonance, the equations are a pair of third-order differential equations that are much more Copyright © 2005 IOP Publishing Ltd.

Resonance in cylindrical geometries

367

complicated than the collisionless case. Goossens et al [76] have devised an approximate method for computing the fields through the resonance by finding an invariant that is the same as one found for a similar case by Sakurai et al [175]. Near the resonance, we can make approximations to the differential equations (19.62) and (19.63) by expressing the Alfv´en speed or, equivalently, the quantity ω2 − (k · V A )2 in a Taylor series about the resonance point and retaining terms up to the first order. We write ω2 − (k · V A )2 = −s{(k · V A )2 }0 − 12 s 2 {(k · V A )2 }0 − · · ·

(19.66)

where s = r −r0 , r0 is the value of r at which resonance takes place, and the prime denotes differentiation with respect to s. The range over which this linearization holds is    {(k · V )2 }  A  0

s =  (19.67) .  {(k · V A )2 }0  It is very large compared to the range over which losses are important. We define a characteristic length l by l −1 =

{(k · V A )2 }0 ω2

then, for |s| < s, ω2 − (k · V A )2 = ω2 and the operator

becomes

= ρ0 ω

 2

(19.68) s l

s d2 i − l ωµ0 σ ds 2

(19.69) .

(19.70)

The equations can then be written in the approximate form     s d2 dξr m cos α0 i − = χ (19.71) − k sin α z 0 l ωµ0 σ ds 2 ds r0     s d2 dψ i 2B 2 sin α0 cos α0 m sin α0 − = − + k cos α z 0 χ l ωµ0 σ ds 2 ds µ0 r 0 r0 

where

d2

s i − l ωµ0 σ ds 2

(19.72)

 ξ⊥ = − iχ

  2VA2 sin α0 cos α0 m sin α0 χ= + k z cos α0 ξr r0 ω2 r0   ψ m cos α0 − − k z sin α0 . r0 ρ0 ω2

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(19.73)

(19.74)

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Mathematics of field-line resonance in compressible media

The coefficients on the right-hand sides of these equations are constant, with the values taken at the resonance s = 0. In this approximation, the quantity χ, which is a linear combination of ξr and ψ appears on the right-hand side of both differential equations. A differential equation for χ can be constructed by operating with s/l − (i/ωµ0 σ )(d2 /ds 2 ) on the equation for χ and substituting for the derivatives of ξr . The result is   i s d2 dχ − = 0. (19.75) l ωµ0 σ ds 2 ds When 1/ωµ0 σ |A| s 3 , the losses are negligible. We, therefore, define a characteristic length   |l| 1/3 lσ = (19.76) ωµ0 σ as specifying the range over which we need to take account of collisions. In any realistic situation, σ is very large so that this range is very small in the sense that lσ l.

(19.77)

s = lσ τ.

(19.78)

We make the substitution Thus, as s → l, τ → sgn(l)∞. The sign of l determines whether the Alfv´en speed increases or decreases with r . Then (19.71) and (19.72) become    m cos α0 d2 dξr = i|l| + i sgn(l)τ − k z sin α0 χ (19.79) dτ r0 dτ 2    d2 2i|l|B 2 sin α0 cos α0 m sin α0 dψ = − + i sgn(l)τ + k cos α z 0 χ dτ µ0 r 0 r0 dτ 2 (19.80) while (19.75) becomes



d2 + i sgn(l)τ dτ 2



dχ = 0. dτ

(19.81)

This is a third-order equation for χ with three independent solutions. One of these is clearly χ = constant. (19.82) The other two solutions are found by noting that (19.81) is the Stokes equation determining the derivative of χ. Thus, dχ = Ai(τ ) or Bi(τ ) dτ Copyright © 2005 IOP Publishing Ltd.

(19.83)

Resonance in cylindrical geometries

369

or a linear combination of them. The condition τ 1 corresponds to the limit of infinite conductivity. Then (19.75) reduces to a first-order equation dχ/ds = 0 so that, in this case, χ is a constant, simplifying the analysis. We need to show that χ = constant is the appropriate solution in this case too. Solution of the equations in the resonance region. By using the substitution (19.78), we may reduce both (19.71) and (19.72) to the form  d2 + i sgn(l)τ F = −1 (19.84) dτ 2 where F is proportional to either dξr /ds or dψ/ds . Consider the integral  3 F(τ ) = e−(u /3)+i sgn(l)τ u du (#)

(19.85)

where the integral is along a contour in the complex u-plane, starting at the origin and following a path # , on which u 3 is real, to |u| = ∞. Three possible paths, #1 , #2 , #3 , are shown in figure 19.7. They are those for which u = |u|, u = |u| exp{2iπ/3}, and u = |u| exp{−2iπ/3}, making angles 0◦ , 120◦ , −120◦ , respectively, with the positive real axis. Note that  (#) 3 d2 F = − u 2 e−(u /3)+i sgn(l)τ u du (19.86) dτ 2 0 and 3 ∂ −(u 3 /3)+i sgn(l)τ u ) = (−u 2 + i sgn(l))e−(u /3)+i sgn(l)τ u (19.87) (e ∂u so that  3 d(e−(u /3)+i sgn(l)τ u ) ≡ − 1 #  3 = − u 2 e−(u /3)+i sgn(l)τ u du #  3 + i sgn(l)τ e−(u /3)+i sgn(l)τ u du #

d2 F + i sgn(l)τ F. = dτ 2

(19.88)

Thus, the integral F, for each of the contours #, is a solution of (19.84). It can also be seen that if we evaluate the integral along the contours #3 − #1 and #3 − #2 , we get χ1 (τ ) = F3 (τ ) − F1 (τ )

χ2 (τ ) = F3 (τ ) − F2 (τ )

(19.89)

which are two independent solutions of the Stokes equation (19.81) for the derivative of χ. A general solution for F can then be written in the form F(τ ) = F1 (τ ) + A{F3 (τ ) − F1 (τ )} + B{F3 (τ ) − F2 (τ )} dχ = A{F3 (τ ) − F1 (τ )} + B{F3 (τ ) − F2 (τ )}. dτ

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(19.90) (19.91)

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Mathematics of field-line resonance in compressible media

Á(u) :

A

: : :

O :

B

Â(u)

: :

Figure 19.7. Contours in the complex u -plane. If we compare these definitions with the integral representations of Airy functions in appendix A.4, we see that F3 (τ ) − F1 (τ ) = − 2πi Ai(ζ )

(19.92)

F3 (τ ) − F1 (τ ) = − iπ{Ai(ζ ) − i Bi(ζ )}

(19.93)

where ζ = i sgn(l)τ . When (19.84) is integrated through the resonance region, τ runs from −∞ to ∞. The function F(τ ) must be bounded at each of these limits. The asymptotic behaviour of Ai(±iτ ) is given by the WKBJ solution (15.39). This tends to infinity at one of the limits and, therefore, cannot occur in the solution for F(τ ). Similarly, the asymptotic behaviour of Bi(±iτ ) is given by one of the WKBJ solutions (15.41) or (15.42). These are also unbounded at one of the limits and cannot occur in the solution for F(τ ). The solution F1 (τ ) is bounded at both limits. This can be shown by integrating it by parts getting   ∞ i sgn(l) 2 3 u exp[−(u /3) + i sgn(l)τ u] du . (19.94) 1− F1 (τ ) = τ 0 By integrating (19.87), it can be seen that the expression in braces is equal to i sgn(l)F1 (τ ). Thus, for large τ , the asymptotic approximation to F is F1 (τ ) ∼

i sgn(l) τ

(19.95)

so that F1 (τ ) = O(|τ |−1 ) and is bounded. Thus, the boundary conditions require that the constants A and B must be zero. The solution of the equations through the resonance region is, then,  ∞ 3 e−(u /3)+i sgn(l)τ u du χ(τ ) = constant (19.96) F(τ ) = 0

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Resonance in cylindrical geometries

371

This can be substituted in (19.79) and (19.80) and integrated with respect to τ to give 

 m cos α0 ξr = i|l| − k z sin α0 χ G(τ ) + C1 r0   2i|l| B 2 sin α0 cos α0 m sin α0 ψ= − + k z cos α0 χ G(τ ) + C2 µ0 r0 r0   µ0 σ ξ⊥ = χ F(τ ) ω A2

(19.97) (19.98) (19.99)

where G(τ ) = −i

 τ 0

F1 (τ ) dτ = −

 ∞ −u 3 /3 e {eiuτ sgn(l) − 1} du. 0 u sgn(l)

(19.100)

The behaviour of the wave components ξr and ψ in the neighbourhood of the resonance is then given by the behaviour of G(τ ) and that of ξ⊥ corresponding to the transverse Alfve´ n wave, by F(τ ). This behaviour has been calculated by Goossens et al [76]. The general features are the same as for the case of plane stratifications near the resonance, illustrated in the right-hand panels of figure 19.3. They show a resonance peak and a phase change of π across the resonance. We shall not reproduce them here. Instead we shall concentrate on a feature that arises when the resistive nature of the plasma near the resonance is considered in detail.

19.5.3 Resonance heating The expressions near the resonance are given in terms of τ . This variable extends from −∞ to ∞ but this range is an approximation. Very large values of |τ | correspond to a small range δs of s. For |s| > |δs|, the wave is accurately described by the lossless equations. The solutions near the resonance must be matched to these solutions. This can be done by using asymptotic approximations to F(τ ) and G(τ ), which hold for large τ , and matching these to the solutions of the lossless equations. Computation of asymptotic approximations. The asymptotic approximation to F is given by (19.95). The asymptotic approximation for G is found by considering its real and imaginary parts separately. The real part is Re(G) = −

  τ −2  1 − cos(uτ )

3 e−u /3 du −

u sgn(l) 0  ∞ cos(uτ ) −u 3 /3 du. e + τ −2 u sgn(l)

 ∞ −u 3 /3 e du τ −2 u sgn(l) (19.101)

The integral has been split into two ranges, 0 ≤ u ≤ τ −2 and τ −2 ≤ u ≤ ∞. In addition, the integral over the second range has been separated into two terms. This is so that they can be expressed in terms of well-known special functions. In the first and third integrals,

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Mathematics of field-line resonance in compressible media

change the variable of integration by replacing u by u/|τ |. In the second integral, replace u by u 3 /3. Then, as τ → ∞, the factor exp{−u 3 /3|τ |3 } approaches unity. Then,  1/|τ |  1 − cos u e−u 1 ∞ Re(G) = du + du u sgn(l) 3 1/3|τ |6 u sgn(l) 0  ∞ cos u − du + O(|τ |−1 ) u 1/|τ | sgn(l) = sgn(l){[Ci(|τ |−1 ) − γ − (|τ |−1 )] − 13 E 1 (1/3|τ |6 ) − Ci(|τ |−1 )} (19.102) where γ is Euler’s constant, E 1 is the exponential integral [1, equation (5.1.1)] and Ci is the cosine integral [1, equations (5.2.2), (5.2.27)]. The asymptotic approximations for these functions are given by Abramowitz and Stegun [1, equations (5.1.51), (5.2.6), (5.2.7), (5.2.34), (5.2.35)]. The result is that Re(G) ∼ − sgn(l){ 23 γ + ln |τ | + 13 ln 3}.

(19.103)

The imaginary part is  ∞  ∞ 3 sin(uτ ) sin u π sgn(lτ ) e−u /3 Im(G) = sgn(l) du ∼ sgn(lτ ) du = (19.104) u u 2 0 0 so that the asymptotic approximation to G is iπ G(τ ) ∼ − sgn(l){ 23 γ + ln |τ | + 13 ln 3} + . 2

(19.105)

The asymptotic approximations (19.95) and (19.105) allow the solutions within the resonance region to be matched to the collision-free solutions at |τ | 1 where they are of the same form as those for δs < s < s. The important applications of this case relate to resonance heating in solar structures such as sunspots and coronal loops. The previous calculations allow us to compute the step in ξr and ψ across the resonance region, between the radii where τ = ±∞. If we use the approximations at each limit together with (19.97) and (19.98), we find that   m cos α0

ξr = iπ|l| − k z sin α0 χ (19.106) r0   2iπ|l|B 2 sin α0 cos α0 m sin α0

ψ = − + k z cos α0 χ (19.107) µ0 r 0 r0 while the step in ξ⊥ , from (19.99) is

ξ⊥ = 2iχ sgn(l)

µ0 σ . ω A2 τ

(19.108)

From these expressions, the change in the field components across the resonance region can be computed. Since the component of the energy flux in ˆ these are a measure of the net energy transmitted the x-direction is 12 Re(ωξr )ψ, Copyright © 2005 IOP Publishing Ltd.

Summary

373

into the resonance region and deposited there. Neither ξr nor ψ depends on the conductivity so that the same amount of energy is deposited no matter what the conductivity. The conductivity determines the width lσ of the resonance and, therefore, the extent of the region that is heated.

19.6 Summary •

•

• •

•

• •

When the plasma β is of the order of unity, the effects of pressure must be included. This does not change the general form of the resonance equation. The coefficients in the equation become more complicated and depend on the sound speed. A suitable normalizaton and transformation of the variables allows the second-order differential equation for resonance in a plane-stratified medium to be written in standard form, named the resonance equation. It is not one of the standard equations of mathematical physics. This equation is suitable for studying the general features of resonances within the magnetosphere A series solution for the equation in the form of an expansion about the resonance point is presented. WKBJ solutions are developed and their accuracy investigated. The Stokes phenomenon in the neighbourhood of the turning point of the equation is studied. It is significantly affected by the presence of the resonance. The change of phase on reflection depends on how well separated the resonance is from the turning point. Approximate solutions are obtained by matching the WKBJ solutions, within their range of validity, to the series solution near the turning point and resonance. They are compared with exact numerical solutions of the equation and are shown to be excellent approximations. This is done for both a perfectly conducting ionosphere and for one with finite conductivity that provides the dominant loss mechanism. Reflection coefficients are computed, taking the effects of the resonance into account. Analogous equations are developed for a cylindrically stratified medium, allowing for the possibility of a spiral structure for the zero-order magnetic field. These can be used to study resonance in structures such as coronal loops or plumes. There is no loss mechanism analogous to that in the magnetosphere where losses occur in the ionospheric boundaries. Losses take place where the fields near the resonance are large. Here anomalous resistivity leads to plasma heating and losses by the wave. The equations are generalized to include this.

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Chapter 20 Cavity oscillations and waveguide modes

20.1 Introduction In the magnetosphere, the first studies of field-line resonance emphasized the behaviour of the fields near resonance [44, 193, 222]. They were carried out when the most detailed experimental information was the magnetometer information [178] showing a change of polarization at a critical latitude. The inherent resolution imposed on magnetometer data meant that the details of the resonance region were unresolved. Detailed experimental confirmation of the structure of the resonance region [239] followed later, providing an excellent example of experiment and theory going hand in hand, taking turns to suggest a way forward. It soon became apparent, however, that there was an aspect of the observations that was not explained by the initial form of the theory. The field-line resonances observed were monochromatic, occurring at sharply defined frequencies with a narrow bandwidth. This required a monochromatic source remote from the resonance region and the mechanism providing such a source was not obvious. The first source suggested for such oscillations was the Kelvin–Helmholtz instability described in chapter 23. This instability arises at the boundary between two media that are in relative motion. It can be easily observed in its simplest form by blowing air across the surface of water in a bath. The magnetopause provides an obvious location where the instability can be excited. There is an inherent difficulty in this mechanism. In its fluid-dynamic or MHD formulation, for a sharp boundary, the growth rate is a monotonically increasing function of wavelength: it cannot explain the existence of a discrete frequency. Consideration of the finite thickness of the boundary [231] suggested a possible solution to the problem. The instability was quenched when the wavelength was much smaller than the characteristic scale of variation of plasma properties within the boundary. There was, thus, a particular wavelength at which growth was maximum and this generated a frequency of maximum growth. While this appeared to be a possible mechanism, it had some difficulties. There was some 374

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375

question about whether the maximum was sufficiently well defined to explain the narrow bandwidth of the observations. It was shown to be insufficient to explain the observations when Ruohoniemi et al [172], using the first of the SuperDARN radars (section 13.5.4), reported the existence of simultaneous resonances at a number of different latitudes, each with a different characteristic frequency. There is no way in which the Kelvin–Helmholtz instability could produce a set of discrete frequencies simultaneously. A suitable mechanism had already been suggested by the time that these observations had been made. Kivelson and Southwood [110, 111] had shown that the region between the magnetopause and the turning point, where the wave was reflected, formed a cavity. Discrete modes at the natural frequencies of the cavity could be excited. These, in turn, could excite discrete field-line resonances on field lines lying beyond the turning point. The analysis was extended by Zhu and Kivelson [258]. This idea was modified [180, 243] by the observation that the cavity was not bounded in azimuth and could better be regarded as a waveguide. The theory of propagation in such cavities has been extensively investigated by Wright and co-workers [133, 250, 252, 253]. In this chapter, we study the nature of modes in such a cavity or waveguide. We shall be concerned with simple models that lead to physical understanding. Throughout the chapter, we use a box model, in which the field lines are straight, and study analytic solutions in simple cases. More realistic models require numerical treatments that do not provide equivalent insight.

20.2 The magnetospheric cavity or waveguide In figure 20.1(a), a sketch of the equatorial plane of the magnetosphere is shown. The magnitude of the magnetic field and, thus, the Alfve´ n speed increases inwards from the boundary. A wave incident from the magnetosheath is partly reflected and partly transmitted by the magnetopause. As the Alfve´ n speed increases, the wave is refracted by the medium within the magnetosphere. When it encounters the turning point, it is reflected, as described in section 15.5. Beyond the turning point, it is evanescent. As described in chapter 18, the evanescent fields in this region may excite a resonance on the magnetic shell where the natural frequency of the azimuthal transverse Alfve´ n oscillation matches the frequency of the wave. The resonance shell occurs at a larger value of the Alfve´ n speed and lies entirely within the boundary defined by the turning point. In principle, we allow for the presence of hot plasma within the magnetosphere, although the pressure is usually small compared with the magnetic pressure. We shall study this situation by using a box model of the type described in section 14.2. The situation shown in figure 20.1(b) is an idealization of that in figure 20.1(a). The magnetic field is in the z-direction and its magnitude increases in the positive x-direction. The pressure required to balance the increase in magnetic pressure has a gradient in the negative x-direction. In the real Copyright © 2005 IOP Publishing Ltd.

376

Cavity oscillations and waveguide modes (a)

magnetopause

resonance turning point magnetosheath (b) turning point

y magnetopause

resonance

magnetosheath V IV

III

II

I

B  ÑVA ÑVS xT

0

xR

x

Figure 20.1. Boundaries in the magnetosphere defining the region of cavity or waveguide oscillations.

magnetosphere, however, the ring current pressure gradient is inward except at its inner boundary. The balancing force for zero-order equilibrium is the field curvature. The illustration, therefore, shows the pressure gradient in the positive x-direction. The dispersion relation is of the √ form (14.18). The figure shows the is real at the magnetopause. The case where ω has a value such that k x ≡ turning point occurs where k x = 0 and the resonance is at the level defined by = 0 where is given by (14.17). For lower values of ω, the turning point is closer to the boundary. Below some critical value of ω, the value of k x is imaginary everywhere within the magnetosphere so that the transmitted wave is evanescent within the boundary. It can still, however, excite a resonance, provided that the condition = 0 is met within the magnetosphere.









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377

The thickness of the magnetopause is small compared with the wavelength so that it can be treated as a discontinuity in the MHD medium. Across it, the density, pressure, magnetic field, and plasma velocity change. Realistically, all these effects need to be taken into account in applying boundary conditions. In order to understand the mechanism of cavity resonance, we idealize it. We assume that the magnetic field is always in the z -direction, that generalized pressure balance holds across the boundary, and that there is a density step. In general, the density in the magnetosheath is larger than in the magnetosphere. In this section, we shall assume such a large density difference that the boundary is rigid. We, therefore, adopt the condition that, at the boundary, the normal displacement and, hence, the transverse electric field is zero. The boundary is then equivalent to a perfect conductor. In section 20.3, we consider a more realistic boundary. If the resonance is far from the turning point, then the solution near the turning point is well approximated by an Airy function as described in section 15.6. In the region to the left of the turning point, this joins smoothly onto a solution that is represented by a superposition of waves of equal amplitude propagated in opposite directions. The displacement has a series of nodes where the oppositely propagated waves are 180◦ out of phase. The wavelength and the position of the turning point vary with frequency. Thus, as the frequency is varied, the position of the nodes of the wave change. For certain frequencies, a node of displacement coincides with the position of the boundary. Here the boundary conditions are satisfied and the system has a normal mode of oscillation. This is often called a cavity oscillation. In general, since there is phase advance in the y -direction, the oscillation is more like a waveguide mode. If the medium is sufficiently slowly varying for the WKBJ solutions to hold, the condition for a mode is that, as the wave is propagated from the boundary to the turning point and back, there should be a phase change of 2π . The phase change is given by twice the phase integral from the boundary to the turning point and there is an additional phase change of 90◦ at the turning point found from the connection relations, as described in section 15.6. The resulting mode condition is then  xT k x (x) d x = n + 14 π n = 0, 1, 2, . . . . (20.1) 0

Sketches of the displacement and the generalized pressure for the first few modes are shown in figure 20.2. Note that there is a mode corresponding to n = 0, for which the region between the magnetopause and the turning point is only a quarter of a wavelength in extent. If the resonance is not isolated, then the behaviour of the wave fields is similar to that shown in figure 19.3. Even when the finite conductivity of the ionosphere is neglected, there are losses to the resonance. As a consequence of these losses, for a real frequency, the reflected wave has a smaller amplitude than the incident wave. As a result, rather than a series of zeros, the amplitude has a series of minima as shown in figure 19.3. The boundary condition at the magnetopause cannot be satisfied in this case. If, however, we give ω a small Copyright © 2005 IOP Publishing Ltd.

378

Cavity oscillations and waveguide modes n=0 y x xT n=1

y x

xT n=2

y x

xT n=3

y x x0

xT

Figure 20.2. Qualitative behaviour of fields in waveguide modes.

imaginary part, it is possible to match the boundary conditions. The complex value of ω can be found by an iterative process as follows. (1) Guess two suitable values of ω and use a numerical process such as that in section 19.3.1 to find the corresponding displacements ξ at the boundary. (2) Use these as initial guesses in the method of false position [159] and compute a better estimate of the zero of ξ . (3) Iterate the process until the value of ω corresponding to a zero of ξ at the boundary is found to sufficient accuracy. (4) Compute the corresponding fields. Zhu and Kivelson [258] have carried out such calculations for the case where pressure is negligible. Their results are shown in figure 20.3. The inner boundary at x = 10 is taken to be the plasmapause and is rigid in the same way as assumed for the magnetopause. These diagrams may be compared with the fields plotted in figure 19.3. The magnetic fields bx and b y in figure 20.3 correspond respectively to the displacements ξ and η in figure 19.3. The difference is that, in this case, the frequency is complex so that the boundary conditions can be met. The mode, therefore, decays. Note that the amplitudes of the nodes are not zero, except at the magnetopause, where this is required by the boundary conditions. A computation of the energy flux shows a flow of energy towards the resonance. Copyright © 2005 IOP Publishing Ltd.

Lossy modes bx

by 0.45 0.30 0.15

0.45 0.30 0.15

45º 0º 45º

0.45 0.30 0.15

0.45 0º 0º 0.30  º 0.15

0.45 0.30 0.15

0º 0.45  º 0.30  º 0.15 0º º 0.45 º 0.30 0.15 10 0

0.45 0.30 0.15

0

5 x

0º  º  º 0º º º

5 x

379 r   ?  r   ? 

0º  º º

r  

? 

0º º º

r  ,, ? 

10

Figure 20.3. Magnetic fields of cavity modes. From top to bottom, the ampitude (full curve) and phase (dotted curve) of the normalized magnetic perturbations of the first four harmonics are shown as a function of x, measured in Earth radii from the magnetopause, for k y /k z = 0.5. The angular frequencies ωr = Re(ω) and the damping decrements γ = Im(ω)/ωr of the modes are shown on the diagram. (Redrawn from figure 4 from Zhu and Kivelson [258].)

20.3 Lossy modes If a cavity mode is set up as described earlier, there are two ways in which energy can be lost. One is by the coupling to the resonance, as described previously, and the other is leakage of the energy through the magnetopause boundary. The treatment of Zhu and Kivelson [258], described earlier, assumed a rigid magnetopause that allowed no leakage of energy. They assumed that the energy was lost to the resonance without specifying the nature of the loss mechanism. We saw at the end of section 19.5.2 that the loss mechanism does not affect the rate of energy loss but only broadens the resonance, thus increasing the size of the region in which energy is deposited. This was shown for a resistive plasma in cylindrical geometry but the result can be extended to other cases. We also showed that the resonance region dominates the loss process. Ionospheric losses elsewhere are relatively small. If the ionosphere has finite conductivity, the results of Zhu and Kivelson are not much altered. The other loss mechanism is leakage through the magnetopause. If the density discontinuity at the boundary is finite, so that the boundary is not rigid, then the boundary conditions of section 10.3.3 require that both the generalized pressure and the displacement are continuous across the boundary. The wave outside the boundary must either be propagated away from the boundary or must decay evanescently. The cavity is, therefore, leaky. If an oscillation is initiated, it decays as energy is carried away from the cavity, unless the wave on the other Copyright © 2005 IOP Publishing Ltd.

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Cavity oscillations and waveguide modes

side of the boundary is evanescent. One might expect that the natural frequencies of the cavity modes in such a case would again have a negative imaginary part and this is indeed so. A rigid boundary does not allow the cavity to be excited from outside. A leaky boundary operates in both directions. It not only allows decay of waveguide modes that have been set up by an unspecified mechanism: it also allows such waveguide modes to be excited by an external source. In what follows, we first study some examples of time-dependent behaviour in a closed cavity. We use Laplace transform and Green’s function techniques to deduce how the disturbance may develop in simple cases when a disturbance is initiated at time t = 0. In the remainder of the chapter, we discuss the coupling between disturbances in the magnetosheath and in the magnetospheric cavity.

20.4 Time-dependent behaviour The description of cavity oscillations in section 20.2 assumes a steady-state solution with sinusoidal oscillation of frequency ω (which may be complex if there are losses in the system). Such a treatment omits the transient behaviour as the oscillation is being set up. In general, such transient behaviour is a sum of the normal modes of the system, superposed in such a way that, when added to the final steady-state motion, they synthesize the initial state of the system. Losses in the system cause the transients to die away, leaving the steady-state motion. A full solution of the problem would require a specification either of a source that is switched on at some time t = 0 and varies as a given function of time or of an initial state. In the first case, once the transients have died away, we are left with the steady-state motion in which the system is driven by the source. In the second case, where there is no driver, the oscillation decays to zero as the transient motion is damped. The nature of the transients depends on the nature of the source. For magnetospheric cavity oscillations, the precise nature of the source is controversial. Localized oscillations may arise from wave–particle interactions (section 24.1). Global interactions must extract energy from the solar wind. A number of mechanisms have been proposed, for example the Kelvin–Helmholtz instability at the magnetopause (chapter 23), over-reflection leading to growth of cavity modes [135], or amplification of waves pre-existing in the solar wind [238]. There is no point in trying to provide a detailed model of the source. In what follows we discuss some approaches to the solution of the time varying problem. It must be emphasized that, except in a formal mathematical sense, we will not produce a complete solution to the problem. The purpose of the discussion is to illustrate the problems that arise, rather than to produce a neat prescription for computation of a complete solution. Copyright © 2005 IOP Publishing Ltd.

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20.4.1 Time-varying behaviour in a closed cavity In this section, we study the disturbance in a cavity with negligible pressure. We assume either an initial disturbance at t = 0 which is allowed to develop in time or a driving force that is applied, starting at t = 0. Zhu and Kivelson [258] describe how this can be done, using a Green’s function technique. We follow their treatment in this section. We use the box model of section 20.2. The magnetic field is uniform in the zdirection, and the gradient in the x-direction arises from a gradient of the density. We assume that the pressure is negligible. We no longer assume harmonic waves but retain the operator ∂/∂t and ∂/∂z. We, therefore, make use of the generalized wave equation (7.39). The x- and y-components of the plasma displacement ξ are ξ and η. When P = 0, the z-component is zero. The equations for ξ and η are, then, ∂ 2ξ ∂ 2ξ ∂ (∇ · ξ ) − 2 = 2 ∂t ∂z ∂x

(20.2)

∂ ∂ 2η ∂ 2η (∇ · ξ ) − 2 = 2 2 ∂y ∂z VA (x) ∂t

(20.3)

1 VA2 (x) 1

where, from (7.40), bz . (20.4) B The ionospheres are taken to be perfectly conducting planes, located at z = 0 and z = l. We use the same argument as in section 16.4. The equations are homogeneous in y and z so that they may be Fourier analysed with respect to these variables. This is equivalent to assuming variation of the form exp{ik y y + ik z z}. They are not, however, homogeneous in x because VA is a function of x. We seek a solution in a box bounded by planes at x = a and at x = b. Zhu and Kivelson chose these planes to represent the magnetopause and an inner boundary near the Earth’s surface. The nature of the inner boundary is difficult to specify. We shall rather assume that the density discontinuities at the magnetopause and the plasmapause are large and abrupt so that these boundaries are massive enough that the normal displacement ξ at the boundary is zero. This is equivalent to assuming that the compressional component of the magnetic field obeys the condition   dbz = 0. (20.5) dx x=a,b ∇·ξ =−

As discussed by Zhu and Kivelson [258], we write ∂/∂y = ik y , ∂/∂z = ik z , and take the Laplace transform (C.22) with respect to t of (20.2) and (20.3), making use of (C.26). The result is   1 dbz ω2 1 2 − k z ξω = (20.6) − 2 {ξ˙ (0) − iωξ(0)} 2 B dx VA (x) VA (x) Copyright © 2005 IOP Publishing Ltd.

Cavity oscillations and waveguide modes   ik y bz ω2 1 2 − k z ηω = ˙ − iωη(0)} − 2 {η(0) 2 B VA (x) VA (x)

382

bz dξ = − − ik y η B dx

(20.7) (20.8)

where subscript ω denotes the Laplace transform of a quantity and the dot denotes differentiation with respect to time. It may be useful to work in terms of bz , the compressional component of b, because, as we have seen in chapter 18, it is finite at the resonance. Elimination of ξ and η leads to  ω2 − (k 2y + k z2 )VA2 VA2 db z d bz = b0 (ω, x) (20.9) + d x ω2 − k z2 VA2 d x ω2 − k z2 VA2 where d b0 (ω, x) = B dx



ξ˙ (0) − iωξ(0) ω2

− k z2 VA2

+ ik y

η(0) ˙ − iωη(0) ω2 − k z2 VA2

.

(20.10)

20.4.2 The Green’s function Equations like (20.6), (20.7), and (20.9) are inhomogeneous equations. They are suitable for solution by the Green’s function method. If we use (20.9) as an example, this can be summarized as follows. •

Replace the source function on the right-hand side of the equation by a point source, located at x = x 0 . The point source may be represented by a delta function δ(x − x 0 ) so that  b b0 (ω, x) = b0 (ω, x 0 )δ(x − x 0 ) dx. (20.11) a

Then, for (20.9) as an example, the Green’s function, G(ω, x, x 0 ), is the solution of  ω2 − (k 2y + k z2 )VA2 VA2 dG d G = δ(x − x 0 ). (20.12) + dx ω2 − k z2 VA2 dx ω2 − k z2 VA2 •

If we multiply (20.12) by b0 (ω, x 0 ) and integrate with respect to x 0 between the limits a and b, we retrieve (20.9) with bz given by  b bz (ω, x) = b0 (ω, x 0 )G(ω, x, x 0 ) dx 0. (20.13) a

The problem of solving the original equation with the appropriate boundary conditions is then reduced to solving the equation for a point source with the same boundary conditions and performing the integral (20.13). Copyright © 2005 IOP Publishing Ltd.

Time-dependent behaviour •

•

383

For all points x = x 0 , the differential equation for the Green’s function is homogenous, with the right-hand side equal to zero. Let ba (x) be the solution of this homogeneous equation for x < x 0 that obeys the boundary conditions at x = a , and let bb (x) the solution for x > x 0 that obeys those at x = b . Note that ba and bb are functions of ω. For brevity we omit this. Then  Aba (x) x < x 0 (20.14) G(x, x 0 ) = Bbb (x) x > x 0 where A and B are constants to be determined. If we take a small interval  straddling x 0 , and integrate (20.12) over the interval, we get VA2 (x 0 ) ω2 − k z2 VA2 (x 0 )

{ Bbb (x 0 ) − Aba (x 0 )} = 1

(20.15)

where the prime denotes differentiation with respect to x and the integral of the second term on the left-hand side is zero in the limit  → 0, provided that G is finite. A further integration shows that Bbb (x 0 ) − Aba (x 0 ) = 0.

(20.16)

These may be solved for A and B so that  2 ω − k z2 VA2 (x 0 ) bb (x 0 )ba (x)      2 ba (x 0 )bb (x 0 ) − bb (x 0 )ba (x 0 ) VA (x 0 ) G(x, x 0 ) = 2 2 2  ω − k z VA (x 0 ) ba (x 0 )bb (x)     2 ba (x 0 )bb (x 0 ) − bb (x 0 )ba (x 0 ) VA (x 0 )

x < x0 x > x0. (20.17)

Once the Green’s function has been found, we can, in principle, evaluate bz (ω, x) by using (20.13), and then get the time dependence by performing the inverse Laplace transform: 1 bz (t, x) = 2π

  #

b

b0 (ω, x 0 )G(ω, x, x 0 )e−iωt dx 0 dω

(20.18)

a

where the contour # runs from ωr = −∞ to +∞ above any singularities in the complex ω-plane, as described in appendix C.3. This is a formal prescription for solving the equation. Its execution is not necessarily straightforward. Essentially the problem can be reduced to one of finding the singularities of bz (ω, x) in the complex ω-plane. In this case, difficulties arise because the differential equation determining G or, equivalently, bz is singular where ω − k z VA (x) is zero. Copyright © 2005 IOP Publishing Ltd.

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20.4.3 Some applications of the Green’s function method Case 1: Uniform medium When VA is a constant, the problem is almost trivial. The functions ba and bb , obeying the boundary conditions at x = a and x = b , are ba (x) = cos{k x (x − a)}

bb (x) = cos{k x (x − b)} 

where kx = ±

(20.19)

ω2 − k 2y − k z2 . VA2

(20.20)

In (20.17), the denominator is ba (x 0 )bb (x 0 ) − bb (x 0 )ba (x 0 ) = k x sin k x (b − a).

(20.21)

It is independent of x 0 and is a function of ω through the dispersion relation ω2 = (k x2 + k 2y + k z2 )VA2 .

(20.22)

Then (20.18) may be written in the form  e−iωt 1 f (x, ω) dω bz (t, x) = 2π # k x sin k x (b − a)

(20.23)

where f (x, ω) =

ω2 − k z2 VA2



b

b0 (ω, x 0 ) VA2 a  cos k x (x − a) cos k x (x 0 − b), × cos k x (x 0 − a) cos k x (x − b),

x < x0 x > x0

 dx 0 . (20.24)

The Laplace integral (20.23) is taken along a contour from ω = −∞ + ic to ∞ + ic where c is chosen so that the contour is above any singularity of the integrand, as shown in figure 20.4. A method of evaluating such an integral is to consider the singularities of the integrand. The integral in (20.23) has a series of poles in the integrand, where sin k x (b − a) = 0. In addition, f (x, ω) may have singularities, associated with any source that may be present. In the figure, we show schematically how to proceed. We show the complex ω-plane with the contour of integration a distance c above the real axis. Examples of the type of singularity that might exist are branch points at A and B and poles at C and D. A branch cut, which must not be crossed, joins the two branch points so that only one sheet of the Riemann surface is used and the integrand is single-valued. The contour may be distorted downwards as shown. Then the contributions from the portions of the path parallel to the imaginary axis occur in equal and opposite pairs and the contribution from the portion parallel to the real axis tends to zero as Copyright © 2005 IOP Publishing Ltd.

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Á(w)

c A

V

B

V

C 

 D

Â(w)

Figure 20.4. Contours for evaluation of inverse Laplace transform.

Im(ω) → −∞ because of the exponential factor. The integral is, then, the sum of the integral along the contour surrounding the branch cut and −2πi times the sum of the residues at the poles (the negative sign is because the contour is clockwise around the poles). In the simple case considered here, f (x, ω) is not singular and the poles occur where k x = ±nπ/(b − a). From the dispersion relation, the poles occur where  1/2 n2π 2 2 2 ωn = ±VA k y + k z + . (20.25) (b − a)2 Then, in the neighbourhood of a pole, we can expand the denominator of the integrand in (20.23) to first-order in (ω − ωn ): k x sin k x (b − a) 

ωn (b − a) (ω − ωn ). VA2

(20.26)

The integral is equal to 2πi times the sum of the residues at the these poles, provided that f (x, ω) is non-singular. Consider the expression for f (x, ω). When k x = kn , because the spatial period is b − a, we find that cos kn (x − a) = ± cos kn (x − b), where the upper sign corresponds to even n and the lower to odd n. Thus,  b ω2 − k 2 V 2 f (x, ωn ) = n 2n A cos kn (x − a) b0 (ωn , x 0 ) cos kn (x 0 − a) dx 0 VA a =

ωn2 − kn2 VA2 VA2

bn cos kn (x − a)

(20.27)

where bn is the coefficient of the nth term in the Fourier series for the initial disturbance b0 . Copyright © 2005 IOP Publishing Ltd.

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Then the solution is bz (t, x) = −i

∞

(ωn2 − kn2 VA2 ) nπ(x − a) bn eiωn t cos . ω (b − a) b−a n n=−∞

(20.28)

It represents the sum of a set of discrete oscillations at frequencies determined by the positions of the poles, with cosinusoidal spatial behaviour. These are the normal modes of the system. The amplitude of the nth mode is determined by the initial value. Case 2: Slowly varying medium without resonance Suppose that VA is a slowly-varying monotonically-increasing function of x and that k y = k z = 0 so that the wave normal is parallel to the gradient. This is the simplest case of a medium which is not uniform. We assume that the behaviour of a propagated wave is well represented by the WKBJ solutions, which apply everywhere, since there is no turning point for k y = k z = 0. The boundary conditions are the same as before. The functions cos k x x and sin k x x are replaced by linear combinations of the WKBJ functions (15.5) and have the form 

 (x) 1/2 cos   x k

k x (x)

 x (u) du



 (x) 1/2 sin   x k

k x (x)

 x (u) du .

We can construct a Green’s function from (20.17). Only the trigonometric terms are important: the amplitudes cancel. The construction of a solution then proceeds in exactly the same way as for case 1. Case 3: The resonance region So far we have not considered the ionospheric boundaries. We have treated a onedimensional problem, with the magnetopause and plasmapause as boundaries, and no losses. The dependence on z has been assumed to be harmonic, with variation exp{ik z z}. We should have been treating a two-dimensional problem with a two-dimensional Green’s function approach. Since our box model is illustrative rather than quantitative, this would have added complexity rather than insight and accuracy. Instead, we study here a one-dimensional model, with variation in the z-direction. Consider equation (20.3), which represents the coupling between η, the azimuthal perturbation of the displacement, and bz = −B∇ ·ξ , the compressional perturbation. In sections 18.4 and 19.5.2, we showed that there was a perturbation quantity that was approximately constant in the neighbourhood of the resonance. In planar geometry, this was the compressional component, which, in this case, is bz . This amounts to neglecting the second term in (19.28), in the neighbourhood of the resonance. This is because, after considering the characteristic length Copyright © 2005 IOP Publishing Ltd.

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387

scales, we assume that the left-hand side of (20.2) is negligible. The righthand side then shows that bz is approximately independent of x. This condition, however, does not specify its dependence on y, z, or t. We shall approach the problem by specifying the dependence on t, assuming an arbitrary variation with z and assuming that all quantities vary as exp{ik y y}. We treat it as a source function. We choose the time variation to be switched on at t = 0 and to be sinusoidal with frequency . Thus, we consider the solution of the equation 1 ∂ 2 η(z, t) ∂ 2 η(z, t) − = f (z)H (t) sin t ∂z 2 VA2 (x) ∂t 2

(20.29)

where H (t) is the Heaviside unit step function. Only VA is a function of x, which serves as a parameter determining the field line on which we wish to find a solution. The boundary conditions are those defined by (16.48). If we use Faraday’s law to eliminate b y and express E x in terms of η through E = −v × B = B × ∂η/∂t, these may be written in the form  ∂η z=0 µ0 !P (∂η/∂t) = (20.30) −µ0 !P (∂η/∂t) z = l. ∂z Let us assume that the system is initially undisturbed so that η(0) = 0 and η(0) ˙ = 0. The Laplace transform of (20.29) is then   1 ω2 1 1 dη(ω, z) + 2 η(ω, z) = f (z) − (20.31) dz 2 ω+ ω− VA and the boundary conditions become  ∂η −iωµ0 !P η = iωµ ∂z 0 !P η

z=0 z = l.

If we proceed as before, we get for the Green’s function  ηb (z 0 )ηa (z)   z < z0   ηa (z 0 )ηb (z 0 ) − ηb (z 0 )ηa (z 0 ) G(z, z 0 ) = ηa (z 0 )ηb (z)   z > z0  ηa (z 0 )ηb (z 0 ) − ηb (z 0 )ηa (z 0 )

(20.32)

(20.33)

where ηa is the solution of the homgeneous equation obeying the boundary conditions at z = 0 and ηb is solution obeying them at z = l. They are each of the form A sin(ωz /VA ) + B cos(ωz/VA ). If we insert this expression into the boundary condition at z = 0 and at z = l and use this to find B/A, we get     ωz ωz i cos + (20.34) ηa (z) = sin VA µ0 !P VA VA     ωz cos(ωl/VA ) − iµ0 !P VA sin(ωl/VA ) ωz cos + . (20.35) ηb (z) = sin VA sin(ωl/VA ) + iµ0 !P VA cos(ωl/VA ) VA Copyright © 2005 IOP Publishing Ltd.

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We use these expressions in (20.33) to show that  VA 1 ηb (z 0 )ηa (z) G(z, z 0 ) = − ω J (ω) ηa (z 0 )ηb (z)

z < z0 z > z0

(20.36)

where

2 cos(ωl/VA ) − iµ0 !P VA sin(ωl/VA ) . (20.37) sin(ωl/VA ) + iµ0 !P VA cos(ωl/VA ) In evaluating the expression for J (ω), we have assumed that the conductivity is large so that (µ0 !P VA )2 1. The disturbance is, then,     l  VA f (z 0 )e−iωt 1 ηb (z)ηa (z 0 ), z 0 < z η(t, z) = dz 0 dω. 2π # 0 ω J (ω) ω2 − 2 ηa (z)ηb (z 0 ), z 0 > z (20.38) The integrand has poles in the ω-plane where ω = ± and where J (ω) = 0. The latter condition is satisfied where   −2i ωl = . (20.39) tan VA µ0 !P VA J (ω) =

The right-hand side is small so that the zeros lie close to the zeros of tan(ωl/VA ), which occur where ω = ωn ≡ nπ VA /l. Let the position of the nth zero be ωn + δω. Then   (ωn + δω)l tan(ωn l/VA ) + tan(lδω/VA lδω −2i } =  = . tan VA 1 + tan(ωn l/VA ) tan(lδω/VA VA µ0 !P VA (20.40) The poles are, therefore, located at ω = ωn =

2i nπ VA − . l µ0 !P l

(20.41)

The integral is then the sum of the residues at the poles. Let us first concentrate on the transient behaviour. Near the nth pole, we can expand J (ω) to first-order in ω − ωn , getting l J (ω) = ∓ (ω − ωn ) (20.42) VA where the upper sign corresponds to even values of n and the lower to odd. The contribution of this pole to (20.38) is then −2πi times the residue, the negative sign arising because the integral is taken clockwise around the pole. At the pole, ηa (z) = ηb (z) = sin

ωn z i ωn z + cos ≡ ηab,n (z). VA µ0 !P VA VA

(20.43)

Thus, iV 2 e−iωn t ηn (t, z) = ± A 2 ηab,n (z) ωn l ωn − 2 Copyright © 2005 IOP Publishing Ltd.



l 0

f 0 (z 0 )ηab,n (z 0 ) dz 0 .

(20.44)

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We may express the spatial behaviour of the driving function as a Fourier series, with constant term zero: ∞ %

nπ z nπ z & Cn sin f 0 (z) = + Dn cos (20.45) l l n=1

Then



l 0

so that

  1 i f 0 (z 0 )ηab,n (z 0 ) dz 0 = l Cn + Dn 2 µ0 !P VA

  1 VA2 e−iωn t i ηn (t, z) = ± i Dn Cn + 2 ωn ωn2 − 2 µ0 !P VA   i ωn z ωn z . + cos × sin VA µ0 !P VA VA

(20.46)

(20.47)

This represents the contribution to the signal of the pole at ωn . It varies with time as   2t inπ VA t −iωn t − = exp − . e l µ0 !Pl It, therefore, represents an oscillation at an angular frequency ωn = nπ VA /l and it decays in an e-folding time 12 µ0 !Pl. The frequencies ωn are the natural frequencies of oscillation for the azimuthal standing modes. In this model, their frequencies are a harmonic series corresponding to the values of n and their dependence on z is represented by ηab,n (z). The set of normal modes represent the transient behaviour of the system. Each is excited with a different amplitude proportional to the amplitude of the corresponding term in the Fourier series representing the driving function. These modes are analogous to the modes of a stretched string with weakly absorbent supports. We have treated the problem as a one-dimensional problem with spatial behaviour depending on z. The coordinate x has been treated as a parameter in this expression that determines the value of VA . For a given applied frequency , each plane defined by a value of x responds with its own natural frequency nπ VA (x)/l and oscillates in the η-direction. The amplitude of oscillation depends on the factor i/(ω2 (x) − 2 ). We are, of course, interested in what happens when the driving frequency is close to one of the natural frequencies of oscillation. This factor shows that the amplitude of the disturbance increases to a maximum where x = x 0 while the phase changes by 180◦ from +90◦ through zero to −90◦. Now let us consider the driven oscillation associated with the pole at ω =  = nπ VA (x 0 )/l. At this value of ω, 2 cos{nπ VA (x 0 )/VA (x)} − iµ0 !P VA sin{nπ VA (x 0 )/VA (x)} (20.48) sin{nπ VA (x 0 )/VA (x)} + iµ0 !P VA cos{nπ VA (x 0 )/VA (x)}     nπ VA (x 0 )z i nπ VA (x 0 )z ηa (z) = sin + cos (20.49) VA (x)l µ0 !P VA (x 0 ) VA (x)l J () =

Copyright © 2005 IOP Publishing Ltd.

Cavity oscillations and waveguide modes     nπ VA (x 0 )z nπ VA (x 0 )z i ηb (z) = sin − cos . VA (x)l µ0 !P VA (x 0 ) VA (x)l

390

(20.50)

If these quantities are substituted in (20.38), we find that the residue at the pole  = π VA (x 0 )/l contributes an amount η (t, z) = −

il e−it 2π J ()



l

f (z 0 )g(, z, z 0 ) dz 0

(20.51)

0

to the integral, where, from (20.33) with ηa and ηb given by (20.49) and (20.50), g(, z, z 0) = ηb (z)ηa (z 0 )H (z − z 0 ) + ηa (z)ηb (z 0 )H (z 0 − z) where H (z) =

%

(20.52)

z0

0 1

(20.53)

is the Heaviside unit step function. Equations (20.47) and (20.51) allow the calculation of the temporal behaviour at any value of x. We shall illustrate this by considering the latitude of maximum resonance, x 0 , where Va (x 0 ) = l/π and the natural frequency of the fundamental mode n = 1 matches the driving frequency. In this case,   VA 2i 2i = π− . (20.54) ω1 =  − µ0 !P l l µ0 !P VA We shall simplify the problem by neglecting terms of order 1/µ0 !P VA except in the time-varying factor exp{−iωt}. Then the contribution of the poles at  and ω1 can be shown to be η1 (t, z) = −

η (t, z) =

πz  µ0 !P VA2 −it −2t /µ0 !P l e e sin 4 l

µ0 !P VA2 −it 4

e

sin

πz   l

l 0

f 0 (z 0 ) sin



l

f 0 (z 0 ) sin

0

πz  0

l

πz  0

l

dz 0

(20.55) dz 0 .

(20.56)

The contributions of the other poles are small, of the same order of magnitude as the first-order terms in 1/µ0 !P VA that have been neglected here. Thus,      −2t 1 η(t, z) = 1 − exp 1+O η (t, z). (20.57) µ0 !Pl µ0 !P VA Initially, the zero-order transient behaviour and the driven oscillation cancel each other. As time advances, the transient dies away on a time scale determined by the loss process and we are left with the driven oscillation. For latitudes with resonant frequencies different from , the analysis is a little more complicated. The result is similar except that the transient oscillates with a different frequency Copyright © 2005 IOP Publishing Ltd.

Time-dependent behaviour 

 

    



 

    

391











 

















 







Figure 20.5. Transient behaviour: full curve, driven oscillation; dotted curve, transient response; bold curve, resultant response given by the sum of the two. The upper panel shows the response at the resonance maximum, where Re(ω) =  and Im(ω) = 0.15 per cycle. The lower panel shows the response where Re(ω) = 1.2.

ω(x) from the driving frequency. Initially there is cancellation. As the transient dies away, there is a superposition of the two different frequencies. This behaviour is illustrated in figure 20.5. In the upper panel, we see how the transient response and the driven oscillation have the same frequency and are 180◦ out of phase so that, after a sufficiently long time has elapsed, only the driven oscillation remains. In the lower panel, the frequency of response and driver are different, but eventually only the driven oscillation remains.

Case 4: Cavity oscillation with resonance Let us now consider a cavity of the type described in case 2 but with k y , k z = 0 so that there is coupling between field-line resonances of the type described in case 3 and the cavity modes. The equation to be solved is now (20.9). The problem is much more complex because this equation is singular. The nature of the solutions has been discussed by a number of authors [7, 111, 133, 258]. The problem is complicated. We do not have space to provide a full analytic solution. We shall indicate the nature of the problems that arise in the evaluation of the Green’s function and inverse Laplace integrals and how they relate to the physical behaviour. A quantitative treatment requires extensive numerical work, such as that provided by Allan et al [7] or Mann et al [133]. Our approach summarizes the treatment of Zhu and Kivelson [258], which draws on the approach to an analogous problem by Sedl´ac˘ ek [184]. Copyright © 2005 IOP Publishing Ltd.

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The starting point is (20.9), the Laplace and Fourier transformed version of the equation for bz . In constructing the Green’s function, when k y and k z are nonzero, we can no longer use WKBJ solutions for the solution of the homogeneous equation, because of the singularities associated with the field-line resonance ω = ±k z VA . We assume that C VA2 = − . (20.58) x The constant is chosen to be negative and the boundaries of the cavity are assumed to be at x = a and x = b where a and b are both negative so that x is negative over the range of interest. This choice is made so that VA increases as x varies in the direction from a to b. Thus, a corresponds to the magnetopause and b to the inner boundary. Throughout the cavity a < x < b, the differential equation reduces to   1 dbz d ζ b z = b0 (20.59) − dζ ζ − ζR (ω) dζ ζ − ζR (ω) which may be written as an inhomogeneous version of the resonance equation (19.11) d2 b z dbz 1 − ζ bz = (ζ − ζR )b0 (ω, ζ ) − 2 ζ − ζR (ω) dζ dζ

(20.60)

where  ζ =

1/3

k 2y (x R − x T )

(x − x T )

ζR = {k y (x R − x T )}2/3

(20.61)

and x R and x T are, respectively, the values of x at resonance point, where ωR2 = k z2 VA2 (x R ), and at the turning point, where ωR2 = (k 2y + k z2 )VA2 (x T ). Thus, xR =

−Ck z2 ω2

xT =

−C(k 2y + k z2 ) ω2

.

(20.62)

We see that ζ (x, ω) is a function of both x and ω. The left-hand side of this equation depends on ω through ζ (x, ω), while b0 depends explicitly on ω and x. The boundaries are located at ζ (a, ω), ζ (b, ω). The Green’s function may then be written in the form G(ζ, ζ0 ) = (ζ0 − ζR )

bb (ζ0 )ba (ζ )H (ζ0 − ζ ) + ba (ζ0 )bb (ζ )H (ζ − ζ0 ) (20.63) ba (ζ0 )bb (ζ0 ) − bb (ζ0 )ba (ζ0 )

where H (x) is the Heaviside unit step function. The characteristic functions ψ1 (ζ ) and ψ2 (ζ ) that must be used to evaluate the Green’s function are then expressed in terms of the two series solutions (19.24) and (19.25) of the resonance equation. The two functions ba and bb that obey the boundary conditions at ζ = ζ (a, ω) and ζ (b, ω), respectively, are Copyright © 2005 IOP Publishing Ltd.

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linear combinations of these. The linear combinations that satisfy the appropriate boundary conditions are ba (ζ ) = ψ2 (ζa )ψ1 (ζ ) − ψ1 (ζa )ψ2 (ζ )

bb (ζ ) = ψ2 (ζb )ψ1 (ζ ) − ψ1 (ζb )ψ2 (ζ ).

(20.64) (20.65)

Then the denominator of (20.17) is J = ba (ζ0 )bb (ζ0 ) − bb (ζ0 )ba (ζ0 )

= {ψ1 (ζa )ψ2 (ζb ) − ψ2 (ζa )ψ1 (ζb )}{ψ1 (ζ0 )ψ2 (ζ0 ) − ψ2 (ζ0 )ψ1 (ζ0 )}. (20.66)

The Wronskian W = ψ1 (ζ0 )ψ2 (ζ0 )−ψ2 (ζ0 )ψ1 (ζ0 ) may be differentiated and the homogeneous part of (20.60) used to show that 1 W  (ζ0 ) = . W (ζ0 ) ζ0 − ζR

(20.67)

This can be integrated with respect to ζ0 to show that W = C(ζ0 − ζR ) where C is constant, so that the Green’s function may be written in the form G(ζ, ζ0 ) =

bb (ζ0 )ba (ζ )H (ζ0 − ζ ) + ba (ζ0 )bb (ζ )H (ζ − ζ0 ) . ψ1 (ζa )ψ2 (ζb ) − ψ2 (ζa )ψ1 (ζb )

(20.68)

The formal solution to the problem is then, from (20.18),   b 1 bz (t, x) = b0 (ω, x 0 )G(ω, x, x 0 )e−iωt dx 0 dω 2π # a  e−iωt 1 = 2π # ψ1 (ζa )ψ2 (ζb ) − ψ2 (ζa )ψ1 (ζb )  b × b0 (ω, x 0 ){bb (ζ0 )ba (ζ )H (ζ0 − ζ ) a

+ ba (ζ0 )bb (ζ )H (ζ − ζ0 )} dx 0 dω.

(20.69)

We have chosen to retain x 0 as the variable of integration to avoid limits of integration that are functions of x 0 and ω. An analytic evaluation of this double integral has not been carried out. In numerical evaluations, there are some points to note. We saw in section 19.2.2 that ψ2 has a logarithmic singularity where ζ = ζR . This term has the form 1 2 2 (ζ − ζR ) ln(ζ − ζR ) which tends to zero as ζ → ζR so that ba and bb remain finite. The logarithmic function ln z is a multi-valued function. If z = r eiθ then ln z = ln r + iθ . Each time the branch point at z = 0 is circled, the phase of z increases by 2π and ln z is increased by 2π. We require ψ2 to be single-valued. Thus, we must restrict ourselves to one branch of the function by making a branch cut from the branch point to r = ∞. In the standard definition, this branch cut Copyright © 2005 IOP Publishing Ltd.

394

Cavity oscillations and waveguide modes Á(w)

kVA (x0)

kV (b) A

kV (x) A Ä



Ä



kV (a) A Ä



kV (a) A

kVA (x) kV (x0) A Ä



Â(w) kVA (b)

Ä



Ä



Figure 20.6. Branch cuts in the ω-plane.

runs along the negative z -axis to −∞. This does not satisfy the conditions for analytic continuity of the Green’s function as the Laplace contour is distorted in the frequency plane. In the ζ -plane, the branch point is at ζ = ζR . The problem is the same as the problem described in section 19.3.1. It amounts to choosing the branch cut extending from the branch point parallel to the imaginary axis to −∞. We specify that we use that sheet of the log function for which the phase of its argument θ obeys −π/2 < θ < 3π/2. Let us consider some qualitative aspects of the evaluation of the integral with respect to ω. The singularities of the integrand are of two kinds. There is a set of poles associated with the condition ψ1 (ζa )ψ2 (ζb ) − ψ2 (ζa )ψ1 (ζb ) = 0

(20.70)

and there are eight branch points associated with the zeros of the argument of the log' arithmic terms in ψ' occur where 2 (ζa ), ψ2 (ζ ), ψ2 (ζ' 0 ), and ψ2 (ζb ). These '

ω = ± −Ck z2 /a , ω = ± −Ck z2 /b, ω = ± −Ck z2 /x , and ω = ± −Ck z2 /x 0 . The zeros (20.70) occur at complex values of ω, even though there are no losses. This is because the logarithmic term in the series for ψ2 has a negative argument and, with the analytic continuation described earlier, is of the form ln r exp(iπ) = ln |r | + iπ . Its contribution to the integral decays exponentially. This behaviour arises because energy in the wave flows towards the resonance and is lost there as described in section 18.3.2. In figure 19.3, it can be seen that, for a real frequency ω, there are no zeros in the resonance function and the boundary condtions at x = a cannot be satisfied. In order to satisfy them, we need complex values of ω as shown in figure 20.3. A possible arrangement of poles and branch cuts for a particular relationship between x and x 0 is shown in figure 20.6. The poles are located at ω±n in the lower half-plane and the branch points are on the real axis. This situation represents the frequency response at a position x to an initial δ-function perturbation imposed at x = x 0 . The temporal response is found by evaluating the contribution from the pole and integrating round the branch cuts. The total Copyright © 2005 IOP Publishing Ltd.

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disturbance at x is, then, found by integrating over all positions of the source at x 0 between x 0 = a and x 0 = b. Of course, the branch cuts can be distorted in various ways without affecting the value of the branch-cut integral. The net result is a disturbance at x that has a time variation u(t) arising from the branchcut integrals and a sequence of discrete decaying oscillations arising from the contributions of the poles. The function u(t) decreases as some negative power of the time. The physical interpretation of this behaviour is that the initial disturbance sets each magnetic shell oscillating in a transverse Alfve´ n mode with its own characteristic frequency ω = kVA (x). These modes are coupled to the compressional mode through the terms in k y and so energy is transferred between them. As time progresses, there is mutual interference as the phases advance at different rates. In addition, the latitudes whose real frequencies for the transverse Alfve´ n modes match the cavity frequencies provide a resonant oscillation. Over the whole cavity, the process is seen as one of phase mixing in which all latitudes are initially set into oscillation. Energy is transferred between the various modes in such a way that it is transferred to the resonances. Eventually, the energy is all lost to the resonances. Such a process is the basis of a numerical procedure used by Mann et al [133]. The progress of the phase mixing process with time can be seen in these computations and in those of Allan et al [7].

20.5 Leaky cavities and waveguides In this section, we treat a different problem in which the cavity is not isolated. It is assumed that a wave is incident on the magnetopause from outside and the response of the magnetopause cavity is evaluated. The problem is complicated. There is the aspect of how waveguide or cavity modes are excited and the aspect of field-line resonance. We concentrate on the mechanisms associated with the former aspect. 20.5.1 Reflection and transmission coefficients at a leaky boundary In this section, we follow the treatment of Walker [235]. Suppose that a monochromatic plane wave with a particular ω, k y , and k z is incident from the left on the boundary at x = 0 as shown in figure 20.1(b). Consider reflection from the boundary and transmission into the region x > 0. As before, we assume that the resonance is far removed from the turning point and ignore it. Then the signal in the region 0 < x < x T is the superposition of a wave (II), in the positive x-direction, and a reflected wave (IV), in the negative x-direction. The wave beyond the turning point (III) will be evanescent if the resonance is sufficiently far removed from the turning point. Otherwise it will be an inhomogeneous wave which transfers energy to the resonance. The reflected wave (V) emerges from the magnetosphere. Copyright © 2005 IOP Publishing Ltd.

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As described in section 15.6.5, the general WKBJ approximation for ψ in the region 0 < x < x T has the form      x  x 1/2 −1/2 k x dx − i exp − i k x dx . (20.71) exp i ψ ∼ −i (x) k x



xT

xT

There is a −90◦ phase change on reflection. The ratio of the amplitude of the reflected to that of the incident wave when the phase is referred to the level x = x T is, therefore, exp(−iπ/2). For convenience, we rewrite the asymptotic approximation for ψ with the phase referred to the magnetopause x = 0. The expression for ψ(ω, k y , k z ) then takes the form   xT  A(x) iπ ψ = − iT k x dx − exp i A(0+ ) 4 0    x     x iπ iπ × exp i k x dx + k x dx − − exp − i 4 4 x x    xT    xTT iπ A(x) π sin (20.72) k x dx − k x dx + = 2T exp i 4 A(0+ ) 4 xT 0 where A(x) =

 1/2 −1/4 =  1/2k x−1/2

(20.73)

A(0+ )

is the value of A(x) just to the right of the boundary x = 0. The and exponential terms represent the wave propagated to the right and the wave, reflected from the turning point, propagated to the left. T is the amplitude of the total signal transmitted into the cavity at x = 0 and A gives the variation of the amplitude in the cavity where 0 < x < x T . The corresponding expression for ξ(ω, k y , k z ) is found from (14.15):   xT   x  iπ π 2T exp i cos . (20.74) k dx − k dx + ξ =− x x ω A(x)A(0+) 4 4 xT 0 Let R be the reflection coefficient at the boundary, x = 0 for the field component ψ, defined as the ratio of the reflected to the incident wave, with the phase referred to x = 0. Then the disturbance in the region x < 0 may be written as the superposition of an incident and a reflected wave:       x  x A(x) k x dx + R exp − i k x dx (20.75) ψ= exp i A(0− ) 0 0    x     x i ξ= exp i k dx − R exp − i k x dx (20.76) x ω A(x)A(0−) 0 0 where A(0− ) is the value of A(x) immediately to the left of the boundary at x = 0. We apply the boundary conditions that ψ and ξ must be continuous across the boundary x = 0. Denote quantities measured just to the left and just to the Copyright © 2005 IOP Publishing Ltd.

Leaky cavities and waveguides

397

right of the boundary by 0− , 0+ , respectively. Their values may be obtained by setting x = 0 in (20.75), (20.76), (20.72), and (20.74). If we equate these values on either side of the boundary, we get (1 + R) = 2iT e−i sin  1 2 (1 − R) = − 2 + T e−i cos  A2 (0− ) A (0 ) where

 =

0

k x dx +

xT

π 4





xT

=− 0

k x dx −

π 4

(20.77) (20.78)  (20.79)

These may be solved for R and T : sin  + iχ cos  sin  − iχ cos  −iei T = sin  − iχ cos  R=

where

χ = [ A(0− )/ A(0+ )]2 .

(20.80) (20.81)

(20.82)

The quantity χ is a measure of the size of the discontinuity of the plasma across the magnetospheric boundary. Its magnitude may be found by inserting the appropriate values of ρ0 in the characteristic speeds VA and VS when computing k x on either side of the boundary. We use it as a parameter in our discussion of the physical nature of the process. The transmission and reflection coefficients T and R are functions of , the change of phase as a wave is propagated across the cavity from x = 0 to x = x T . The intensity of the signal in the cavity just to the right of the boundary x = 0 is proportional to T T ∗ . This is plotted as a function of the phase integral for several values of the discontinuity χ in figure 20.7. For a given model, this phase integral is a monotonic function of frequency. We present results for χ > 1 corresponding to the realistic case of a larger density in the magnetosheath than the magnetosphere. Note that, for χ > 1, T T ∗ has strong maxima where cos  = 0 or where  xT k x dx = (n − 14 )π (20.83) 0

even when the discontinuity is quite small. As described earlier, this is the condition determining the frequency of a cavity mode when there is perfect reflection at the boundary: the greater the value of the density discontinuity at the boundary is, the sharper is the peak in T T ∗ . The value of T T ∗ is relatively small for other frequencies. Walker [235] also presents results for the case χ < 1. In this case, the maxima occur where sin  = 0. Copyright © 2005 IOP Publishing Ltd.

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Cavity oscillations and waveguide modes

Figure 20.7. Transmission coefficient as a function of .

The expression for the reflection coefficient shows that R R ∗ = 1 for all frequencies, implying perfect reflection, with an appropriate phase shift, at all frequencies. A harmonic wave is totally reflected from the magnetopause. Frequencies near the cavity modes, however, penetrate the cavity and are reflected from the turning point while other frequencies are predominantly reflected from the boundary without penetrating the cavity. These results are somewhat puzzling at first sight. Real signals, however, are not harmonic infinite plane waves. More insight can be obtained by considering signals that are limited in time, as discussed in section 20.6.

20.6 Excitation of the magnetospheric cavity Assume that a source outside the magnetosphere radiates a wave towards the magnetopause, indicated by (I) in figure 20.1. This wave will essentially be a compressional fast magnetosonic wave, which can be propagated in all directions: the Alfv´en and slow magnetosonic waves can only be propagated approximately parallel to the magnetic field. The dispersion relation limits the values of k y and k z which are allowed. A propagated wave is only possible if is positive, i.e. if



k 2y + k z2
0, it must be closed in the lower half-plane so that integrals around singularities and branch cuts are included. If the source is at a great distance, then the integrand is zero for values of ω which do not meet the criterion (20.84) because the evanescent part of the spectrum has decayed to negligible values. This amounts to ignoring the near field of the source. 20.6.1 Time development of the reflected and transmitted waves Consider the time development of a component of the signal corresponding to fixed values of k y and k z for which propagation in the cavity can occur. The amplitude of T and the phase of R for χ = 4 are shown in figure 20.8. There is a 360◦ change in the phase of R near the maximum of T . From equation (20.85), we can see that the reflected wave is given by      xT ˜ Rψ(ω, k y , k z ) exp − i ωt − k x dx dω ψR (t, k y , k z ) = C 0      xT = |R|ψR (ω, k y , k z ) exp − i ωt − pha(R) − k x dx dω C

0

(20.86) and the transmitted wave by     xT  A(x) ˜ ψT (t, k y , k z ) = exp − i ωt − ψ(ω, k y , k z ) k x dx + π/4 A(0+ ) C 0 Copyright © 2005 IOP Publishing Ltd.

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Figure 20.8. Amplitude of transmission coefficient and phase of reflection coefficient.

 × sin

x xT

π k x dx + 4

 dω

(20.87)

where pha(R) is the phase of R. We do not have to evaluate these integrals in order to understand the time development. Consider the integral (20.86). Its integrand is a continuous spectrum of frequency components. Its properties can be studied using the method of stationary phase (section A.6). If we equate the derivative with respect to ω of the phase to zero and solve for t, we get  x  x ∂ ∂ ∂k x dx [pha(R)] + dx = [pha(R)] + t= (20.88) ∂ω ∂ω 0 ∂ω 0 VG where VG is the x-component of the group velocity, (∂k x /∂ω)−1 . This gives the time t at which some identifiable feature of that part of the wavepacket, which is reflected into the solar wind, reaches position x. Near the maximum of T , the phase is changing rapidly so that its derivative is large. The reflected signal travels in the −x-direction with velocity VG . It is, however, delayed relative to the incident signal by a time ∂[pha(R)]/∂ω. It is easy to show that TC ≡

∂ 2χ ∂ pha(R) = ∂ω sin2  + χ 2 cos2  ∂ω  xT 2χ ∂k x dx = sin2  + χ 2 cos2  0 ∂ω χ = TG 2 sin  + χ 2 cos2 

Copyright © 2005 IOP Publishing Ltd.

(20.89) (20.90)

Excitation of the magnetospheric cavity where

 TG = 2 0

xT

∂k x dx ∂ω

401

(20.91)

and is the time taken to cross back and forth across the cavity once, while TD is the time that the reflected signal is delayed relative to the incident signal. When the density step is large, χ is large. Near the mode frequency, cos  = 0, sin  = 1, and TC = χ TG TG . Far from the mode frequencies, TC ∼ χ −1 TG TG . The width of the peak near the mode frequency is small and the time spent in the cavity is large. Inside the cavity, the evaluation of (20.87) requires a different approach. The peaks in T on the real axis are associated with a series of poles below the real ωaxis. When χ is large the maxima are determined by (20.83). Let ωn be the value of ω for which (20.83) holds. Let the pole be located at  = n + iδn where n and δn are real. We can expand the denominator of T to first order in δθn and 1/χ and equate the result to zero getting   1 i cos n + δn + sin  = 0. χ

(20.92)

This shows that the pole is situated at  = (n + 12 )π − i/χ.

(20.93)

If (20.83) is now expanded about ωn to first order in δω, we find that the pole is located where ω = ωn + δω i [∂k /∂ω] x ω=ωn dx 0 i = ωn −  xT −1 . χ 0 VG dx

= ωn −

χ

 xT

(20.94)

In addition, there are branch cuts of T associated with the branch cuts of k x . For t > 0, the contour can be distorted downwards. The integral can then be reduced to an integral around the branch cuts and the sum of the residues at the poles. A detailed development is inappropriate here. The branch-cut integral is important for the initial time development. It becomes less important as t increases. The residues at the poles represent the cavity modes, for which ω has a negative imaginary part given in (20.94). The modes, therefore, decay in time. This is not due to dissipation but to leakage out of the cavity. The time scale of decay for each mode, from (20.94), is χ times the time taken for a wavepacket to be propagated across the guide at the group velocity. Copyright © 2005 IOP Publishing Ltd.

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20.6.1.1 Effect of the resonance We have so far ignored the effect of the resonance, assuming that it is well removed from the turning point. If it is closer to the turning point, leakage of energy to the resonance through evanescent barrier penetration is important, and indeed crucial, to our model. Energy flux near the resonance is discussed in section 18.3. The effect would be that the signal within the cavity would not be given exactly by (20.72). There would be a loss on reflection at the turning point. This would modify the poles of T . The time development would then be modified to account for the additional loss of energy to the resonance. 20.6.1.2 Source inside the cavity Had the source, S, been inside the cavity, the techniques for treating the problem would have been similar. A reflection coefficient would have been computed for the region outside the cavity and the source function inside the cavity would have been matched to a continuum solution and a discrete set of leaky waveguide modes. The general consequences would have been similar. Most of the continuum, except for frequencies near the mode frequencies, would have been radiated in the negative x-direction. The modes would have been excited in the cavity and energy would slowly leak away both to magnetosheath and to the resonances. 20.6.2 Discussion The aim of this section has been to address the problem of how a magnetospheric cavity can be excited by a source. The basic ideas have been developed and applied to illustrative situations. The purpose has been to develop the techniques and not to cover all possible magnetospheric situations. The development has been in terms of WKBJ solutions, which give good qualitative results and have the advantage that they allow easy recognition of solutions as being the superposition of quasiplane waves propagated in opposite directions. This allows easy visualization of the processes of reflection and transmission. We have shown, using the method of stationary phase, that efficient transmission into the cavity occurs at frequencies corresponding to the waveguide modes. At other frequencies, remote from these, an incident wave is effectively reflected from the boundary and does not penetrate into the magnetosphere. At the mode frequencies, the wave is transmitted into the cavity and is trapped there for several wave periods before leaking out. The length of time for which it is trapped depends on how sharp the transmission peak is in frequency. This, in turn, depends chiefly on the difference in density on either side of the boundary. For a strong discontinuity, the peak is large and narrow: only a narrow band of frequencies penetrates the cavity but it is trapped for a long time. For a weak Copyright © 2005 IOP Publishing Ltd.

The waveguide picture

403

discontinuity, a broad band of frequencies penetrates the cavity but it is trapped for only a short time. An assumption has been made in these calculations that the field-line resonance is sufficiently far from the turning point for the resonance to be neglected as a first approximation. This is appropriate as a first attack on the problem but is not realistic. If the resonance is included, the nature of the reflection at the turning point is significantly modified and energy will leak from the waveguide towards the resonance. Further work on this problem is needed. It is, of course, interesting to note that the chief observable quantity in the magnetosphere is the discrete set of field-line resonances [179, 243]. Few, if any, direct observations of cavity modes associated with resonances have been reported. For this reason, a more appropriate approach to the description of the problem should probably be in terms of the field-line resonance. The cavity would act as an optically thin film or filter, determining which frequencies penetrate the magnetosphere and are available to excite resonances. The mathematical treatment is essentially the same, the point of view of the interpretation is quite different.

20.7 The waveguide picture In the treatment of the previous section, it was assumed that k y and k z were fixed. Perfectly conducting planes representing the ionospheres in the northern and southern hemispheres bounded the region in the z-direction. These determined the value of k z . The analysis was carried out assuming that the z variation was of the form exp{ik z z} but there was an implicit assumption that the solution was the sum of solutions for ±k z providing a sinusoidal variation so that the boundary consitions were satisfied. A similar assumption for k y was made in the first treatments of cavity modes [110, 111, 248, 249]. This was equivalent to assuming that, in a cylindrically symmetric situation, the behaviour was of the form sin mφ. It was later pointed out [180, 241] that, in the asymmetrical magnetosphere, there were no identifiable boundaries, and a waveguide was a better model than a cavity. The waveguide picture has been examined in detail by Wright and coworkers [9, 128, 166, 250, 254].

20.8 Summary •

The occurrence of field-line resonances at several discrete frequencies requires some mechanism that generates these narrow-band oscillations. One possibility is the cavity that exists between the magnetopause and the turning point at which the fast mode is reflected within the magnetosphere. The eigenmodes of this cavity could leak through to the resonance latitude and excite discrete resonances.

Copyright © 2005 IOP Publishing Ltd.

404 • • •

•

Cavity oscillations and waveguide modes Numerical computations of the fields in such a cavity when the pressure is negligible are presented and discussed. Losses from the cavity can arise from Joule heating of the ionospheric boundary, leakage to the resonance, or leakage through imperfect reflection at the magnetopause. The time-dependent behaviour of waves in such a cavity can be analysed using a Green’s function technique. The basic ideas of this technique, as applied to such a cavity, are outlined and discussed. Green’s function techniques are illustrated by applying them to find the transient behaviour and the mode excitation in various situations. We treat (i) a uniform medium with conducting boundaries perpendicular to the magnetic field, (ii) a similar medium with a transverse gradient of density and no accessible resonance, driven by an external signal, (iii) a similar medium with an accessible resonance to study resonance excitation, and (iv) a cavity oscillation with resonance. If a wave is incident on the magnetopause from the outside, it is, in general, reflected with little transmission to the magnetosphere. At discrete frequencies, corresponding to the cavity modes, the signal within the cavity is excited strongly. The theory of this process is developed using the method of stationary phase to synthesise the resultant signal from a spectrum of plane waves.

Copyright © 2005 IOP Publishing Ltd.

Chapter 21 Waves in moving media

21.1 Introduction The plasmas in the solar wind and geospace are everywhere in relative motion. The solar wind is gusty. After it has passed through the bow shock, it streams past the magnetosphere. The magnetopause is a boundary between two media in strong relative motion. A wave in a uniform moving medium can be studied from a frame in which the medium is at rest. If it observed from a point in relative motion, as would be the case for a satellite such as WIND or ACE (chapter 13), the major consequence of the relative motion is a Doppler shift. Since the relative velocity is often much greater than the wave speed, this Doppler shift can produce large effects that can be confusing. MHD waves are strongly anisotropic and the anisotropy combined with the Doppler shift can cause difficulty. If two media are in relative motion, there can be energy transfer between the wave and the background plasma as the wave is transmitted or reflected. In this chapter, we first study the anisotropic propagation properties of waves seen by an observer moving relative to the medium. We deduce their dispersion relations and discuss how the energy should be partitioned between the wave and the background medium. We then consider the transmission and reflection of a wave at the boundary between two counterstreaming media and show that the wave may, in such circumstances, be amplified on reflection. We calculate the reflection and transmission coefficients in such a case. This amplification may be lead to the excitation of cavity modes and, hence, field-line resonances. We discuss in detail the best way to deal with the energy transfer. An equation can be derived for conservation of the energy associated with the wave. It is not the total energy. It cannot be found by expanding the general energy conservation equation to second order in the wave perturbation. Unless there is relative motion, there is no transfer between the energy of the background motion and the wave. At a boundary between two media in relative motion, however, positive or negative work is done at the boundary by the medium on the wave. This can lead to growth or attenuation. An alternative picture is the so-called negative energy wave. We Copyright © 2005 IOP Publishing Ltd.

405

406

(a)

A

Waves in moving media (Vg,0 +V)t A'' A' Vg t V t g,0

(b) A''

(Vp,0 +V)t Vp t

Vp,0 t B''

(Vg,0 +V)t A' Vg t Vg,0 t A (Vp,0 +V)t Vp t

B'

Vp,0 t B'

B'' B

B

V



Vp,0

Figure 21.1. Anisotropic wave observed from two frames with relative velocity V .

undertake a critical discussion of the relative merits of this picture and the one that we have defined here. We also deduce ray-tracing equations for propagation in a moving medium.

21.2 Wave propagation in a uniform moving medium 21.2.1 Qualitative picture The plasmas of solar–terrestrial physics are in constant motion. The relative velocities of plasma and observer may be large. For example, satellites at the solar libration point, such as WIND or ACE, are approximately at rest with respect to the Sun–Earth system and are used to observe phenomena in the solar wind, which streams past at a supersonic, super-Alfv´enic velocity. Wave observations are subject to large Doppler shifts. In an anisotropic medium, such Doppler shifts complicate the interpretation of observations. In this section, we consider the nature of MHD waves observed in a uniform medium that is moving with constant velocity relative to the observer. First, we introduce the topic by using a very simple picture. Consider an isotropic sound wave with phase velocity V P,0 relative to the plasma and group velocity V G,0 . Since sound waves are not dispersive, V P,0 = V G,0 . Suppose that we observe an infinite plane wavefront of the wave from a frame in which the medium moves with constant velocity V . Consider figure 21.1. The panel (a) corresponds to |V cos α| < |VP,0 | and (b) to |V cos α| > |VP,0 |, where α is the angle between V and V P,0 . In both cases, we have chosen V and VP,0 to have opposite signs. During a time interval t, a wavefront AB, observed in the rest frame of the plasma, moves a distance VP,0 t to A B  . In the observer’s frame of Copyright © 2005 IOP Publishing Ltd.

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reference, the plasma is moving to the left with velocity V . As a consequence of the plasma motion, the observer will see the wavefront move to position A B  . The observer will attribute a phase velocity VP to the wave. Take the x-axis in the direction perpendicular to V and the y-direction parallel to V . The wavefronts in the rest frame are moving with a component in the positive x-direction. In the laboratory frame, if V > VP,y , they move in the negative x-direction. This is simply a Doppler shift; and since VP,y = ω/k y , it occurs where the Dopplershifted frequency ω − k y V is negative. A small disturbance on the wavefront moves with the group velocity. In the rest frame of the plasma, this moves fom A to A during time t. In the observer’s frame it moves from A to A . The observer attributes a group velocity, which is the vector sum of the plasma velocity and the group velocity in the rest frame, to the wave. This type of behaviour has a number of consequences for energy propagation and introduces new physical phenomena when the wave meets the boundary between two counterstreaming media. 21.2.2 Modification of the wave equations for a moving medium Consider a uniform medium, moving with constant velocity V . If we linearize the MHD equations in a medium at rest, the operator d/dt is ∂/∂t. In a moving medium, this must be replaced by ∂ d = + V · ∇. dt ∂t

(21.1)

If we assume harmonic waves varying in time and space as exp{−iωt + ik · r}, the only effect on the equations is that ω is replaced by its Doppler-shifted value ω0 = ω − k · V

(21.2)

which is the frequency that would be observed in the rest frame of the plasma. The basic equations (2.38), (2.42), (2.45), (2.52), (2.53), and (2.54) are, then, the continuity equation, ∂ρ + V · ∇ρ = −ρ0 ∇ · v ∂t

(21.3)

the momentum equation, ρ0

∂v + ρ0 V · ∇v = j × B − ∇ p ∂t

the adiabatic law, ∂p γP + V · ∇p = ∂t ρ0



∂ρ + V · ∇ρ ∂t

(21.4)  (21.5)

Ohm’s law with infinite conductivity, E+v× B+V ×b=0 Copyright © 2005 IOP Publishing Ltd.

(21.6)

408

Waves in moving media

Faraday’s law, ∇×E=−

∂b ∂t

(21.7)

Ampe´ re’s law, ∇ × b = µ0 j

(21.8)

and the resulting reduced equations are   B ∂v B·b + ρ0 V · ∇v = − ∇ p + + · ∇b ρ0 ∂t µ0 µ0 ∂p + V · ∇ p = − γ P∇ · v ∂t ∂b + V · ∇ b = B · ∇v − B∇ · v. ∂t

(21.9) (21.10) (21.11)

21.2.3 Harmonic waves 21.2.3.1 Dispersion relations If we assume space and time variation of the form exp{−i(ωt − k · r)}, then the continuity equation (21.3), together with the reduced equations (21.9), (21.10), and (21.11), may be written in the form ω0 ρ = − ρ0 k · v   k· B B·b − ω0 ρ0 v = k p + b µ0 µ0 p = k·v ω0 ρ0 VS2 ω0 b = − ( k · B)v + ( k · v) B

(21.12) (21.13) (21.14) (21.15)

where ω0 is the Doppler-shifted frequency (21.2). The dispersion relations can be regarded as specifying ω as a function of k . They are specified by replacing ω by ω0 in (7.48) for the transverse Alfve´ n wave and in (7.49) for the magnetosonic waves. This minor change to the equations for a stationary medium has a substantial effect. When the medium is stationary, there is one preferred direction, that of the magnetic field. When the medium is moving, there are two such preferred directions, those of the magnetic field and the velocity. The refractive index surfaces are greatly modified from those shown in figure 7.2. They are no longer cylindrically symmetric about the magnetic field direction. An atlas of all the possible forms of the surfaces would be extensive. We show two simple cases in figures 21.2 and 21.3. In these examples, the magnetic field and the drift velocity are chosen to be at right angles. The magnetic field is in the z-direction and the drift velocity is in the y-direction. In the two cases illustrated, the plasma drift speed is, respectively, less than and greater than, the characteristic wave speeds. Copyright © 2005 IOP Publishing Ltd.

Wave propagation in a uniform moving medium UA = 0.6; US = 0.7; U = 0.5

15

4 10

0 -6

5 -4

-4 -2

0 ny 4 -4

0

nx

4

nx

-4

4

nz Slow wave

10

nz

-4

-2

5

0 ny 4 -4

0

nx

3 2 1

2

4

2

4

-1 -2 -3

4

15

ny

nx 3 Alfvén wave

10

nz

-4 5 0 -4

2 -2

15

0 -4

nz

2

Fast wave

nz

409

0 ny 4 -4

0

nx

4

-2

2 1 -1 -2 -3

ny

Figure 21.2. Refractive index surfaces for the case where the plasma drift speed is less than the characteristic wave speeds. Normalization is in terms of the Alfv´en speed. Each cut shows all three surfaces.

Consider figure 21.2. On the left-hand side, we show the refractive index surfaces for the fast, slow and transverse Alfv´en waves, respectively. On the righthand side, we show the intersections of these three surfaces with the y = 0 plane, the x = 0 plane, and the z = 0 plane respectively. The fast-wave surface is still roughly spheroidal in shape. It is shifted relative to the origin because the drift velocity is added to the wave velocity. The slow and Alfv´en waves are strongly modified. The phase speeds of these waves can be very small for some'directions of the wavevector, even though the characteristic speeds VA , VS , and VA2 + VS2 are all greater than the plasma drift speed. This is because the surfaces are open so that the refractive index approaches infinity for some wavevector directions. There is, therefore, a locus in k space on which the component of the phase speed parallel to the plasma drift is equal to the drift speed. This locus is a straight line where the surfaces are singular. The refractive index surfaces for the slow and Alfv´en waves cross on this line and the ray picture fails. Copyright © 2005 IOP Publishing Ltd.

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Waves in moving media

Figure 21.3. Refractive index surfaces for the case where the plasma drift speed is greater than the characteristic wave speeds. Normalization is in terms of the Alfv´en speed.

Figure 21.3 shows the corresponding situation for a larger drift speed, which is greater than the characteristic speeds of the waves. The fast-wave surface has a different topology. Now, for some directions of the wavevector, the plasma drift speed can be larger than the component of the wave velocity. The consequence is that the fast-wave surface is now roughly the shape of a pair of hyperboloids and is no longer closed. All the arguments used for a stationary medium in section 9.2.4 are still valid. The direction of the group velocity is perpendicular to the refractive index surface. The relationship to the direction of energy propagation is discussed later. In our preferred treatment, it coincides with the direction of energy propagation. In the negative energy wave picture discussed later [129, 135], care is necessary. Copyright © 2005 IOP Publishing Ltd.

Energy relations in a uniform medium

411

21.2.3.2 Relationship between field components In the rest frame of the medium, the relationships between the velocity perturbations are given by (7.44), with frequency ω0 . In the moving frame, the frequency is ω and we replace ω0 by the Doppler-shifted frequency ω − k · V . If the three components of the vector equations are written out in any coordinate system, then the three equations are consistent if the dispersion relation is obeyed. The ratios of two of the components of v to the third can then be found from any two of the equations. The pressure and magnetic field perturbations may then be found from the components of (7.68) and (7.69) with ω replaced by ω − k · V . 21.2.4 The entropy wave In a moving medium, another type of MHD wave becomes possible. Consider (21.12). If any velocity perturbation is perpendicular to the wave-normal direction, then k · v = 0 and the equation has a solution if ω0 ≡ ω − k · V = 0.

(21.16)

This solution represents a density perturbation, varying as exp{−i(ωt − k · r)} and propagated with a phase speed V cos α where α is the angle between wave normal and plasma velocity. The density perturbation does not affect the other equations except through k · v which is zero. Equation (21.15) then shows either that the wave normal must be perpendicular to the zero-order magnetic field or that B = 0. Equation (21.14) is satisfied for any value of p, while (21.13) requires that the generalized pressure perturbation p + B · b/µ0 must be constant. This is just the condition for equilibrium when all quantities vary perpendicular to the magnetic field direction. This wave, then, is a just an equilibrium state of the medium that is convected with the medium. It could be generated by an appropriate moving source moving through a stationary medium with velocity −V or, equivalently, by the medium moving past the stationary source with velocity +V . It is important in the study of shocks, which are stationary structures in a moving medium. The adiabatic condition (21.5), from (1.17), is equivalent to writing ds ≡ −i(ω − k · V )s = 0 (21.17) dt where s is the specific entropy. This indicates that the wave can be regarded as a fluctuation of the specific entropy in space, convected with the medium. For this reason, it is generally called an entropy wave.

21.3 Energy relations in a uniform medium 21.3.1 Energy conservation equation To find an energy equation for the wave, we follow the same procedure as in section 9.3.1. We take the scalar product of (21.9) with v and of (21.11) with Copyright © 2005 IOP Publishing Ltd.

412

Waves in moving media

b/µ0 , multiply (21.10) by p, and add the equations. The result differs from (9.15) only by a quantity V · ∇U where U is the energy density: U = 12 ρ0 v 2 + 12 b2 /2µ0 + 12 p2 /γ P. The modified form of (9.15) is, then,    B ∂U B·b b·v . v− = −V · ∇U − ∇ · p+ ∂t µ0 µ0

(21.18)

(21.19)

The first term on the right-hand side of (21.19) has the form f · V where f = −∇U and is the force density arising from any gradient in the wave energy. It, thus, represents the rate at which the wave does work on the plasma as it streams past the point of observation. Since V is constant, V · ∇U = ∇ · U V and (21.19) becomes ∂U = −∇ ·  (21.20) ∂t where   B·b B = p+ v− b · v + UV. (21.21) µ0 µ0 The energy density U is constructed from products of the first-order field components in the same way as for a stationary plasma, described in section 9.3.1. The energy flux  is equal to the energy flux in the rest frame of the streaming plasma plus an additional term corresponding to the convection of internal energy density by the moving plasma. This might seem to be an obvious interpretation but it is not the one adopted when considering negative energy waves as we describe later. For harmonic waves, we can find a relationship between the energy density and energy flux analogous to (9.26). The products of two quantities varying as exp{−i[ωt − k · r]} must be treated as described in section 9.3.3. The result for the moving medium is    1 B p p˜ b · b˜ 1 = k · Re ψ v˜ − (ω − k · V ) ρ0 v · v˜ + + b · v˜ (21.22) 4 µ0 2 µ0 ρ0 VS2 which is the same as (9.26) with ω replaced by ω0 . Again,  1 p p˜ b · b˜ U  ≡ + ρ0 v · v˜ + 4 µ0 ρ0 VS2

(21.23)

is interpreted as the time-averaged internal energy. The term k · V U is taken to the right-hand side to give the time averaged energy flux   1 B  = Re ψ v˜ − b · v˜ + U V (21.24) 2 µ0 Copyright © 2005 IOP Publishing Ltd.

Reflection and transmission of a plane wave

413

so that ωU  = k · .

(21.25)

We can use the same argument as in section 9.3.4 and take the gradient in k space of this relation. The result is U V G = 

(21.26)

where V G is now the sum of the plasma velocity V and the group velocity in the rest frame of the medium. For a quasimonochromatic wave, we can continue the argument of section 9.3.4 and show that the energy conservation equation for the timeaveraged energy density is ∂U  = −∇ · . ∂t

(21.27)

21.4 Reflection and transmission of a plane wave at a tangential discontinuity 21.4.1 Reflection and transmission coefficients The study of waves in a uniform medium, described in section 21.2, simply involves a coordinate transformation, which can be useful if the observer is not at rest with respect to the medium. When we examine waves passing between two media in relative motion, new physical features appear. A wave incident on the boundary between two such media may be reflected with a different amplitude. This amplitude may, in the right circumstances, be larger than the amplitude of the incident wave, so that the wave is reflected with increased energy. This is called over-reflection. Such boundaries may occur at the magnetopause, at the bow shock, or between the plasmas inside and outside a solar structure such as a coronal plume. The problem is complicated if the background plasmas on either side of the boundary are exchanged as the result of a perpendicular velocity component, such as is the case for a rotational discontinuity or a shock. In this section, therefore, we consider the simpler problem of a tangential discontinuity, for which the velocities of the two plasmas are parallel to the boundary. Such a problem can always be treated by using a frame of reference in which one of the plasmas is at rest and the other moves parallel to the boundary. Another complication is that there may be magnetic shear across the boundary. The points that we wish to illustrate do not depend on this, so we shall assume that the magnetic fields in each medium are parallel. Indeed, for the purposes of numerical illustration, we study a boundary between media with β 1 and β 1, respectively, so that the Alfv´en speed is negligible in one and the sound speed in the other. The problem is an idealization of conditions at the magnetopause, Copyright © 2005 IOP Publishing Ltd.

414

Waves in moving media

and has been studied, for example, by McKenzie [129], Mann et al [135], and Walker [236]. Suppose that a plane wave is incident on the plane tangential discontinuity between two semi-infinite media, located at x = 0. We label the region x < 0, from which the wave is incident, as medium 1 and the region x > 0 as medium 2. Assume P and B are uniform in each medium and take the special case where B is in the z-direction: its magnitude on each side of the boundary may be different but its direction does not change. The phenomena are observed from a frame in which the first medium is at rest and the second medium has a velocity V parallel to the boundary. The values of k y and k z are constant by Snell’s law and the law of reflection. The arguments in section 10.3.3 still apply and the boundary conditions at x = 0 are that (i) the generalized pressure ψ must be continuous across the boundary and (ii) there must be no cavitation so that the normal displacement ξ must be continuous across the boundary. Since vx = −iω0 ξ , this latter condition means that vx /ω0 is continuous across the boundary. Field quantities corresponding to energy flux with positive or negative x-components are labelled with superscripts + or −, respectively. For evanescent waves, the positive superscript corresponds to decay and the negative superscript to growth, as x increases. Subscripts 1 and 2 label quantities in medium 1 and medium 2, respectively. The generalized pressure ψ and the normal displacement ξ obey (14.15) and (14.16) with ω replaced by ω0 :



dψ = (x)ξ dx dξ (x) = − ψ dx (x) where

 

 (x) = ρ0 [ω02 − (k · V A)2 ] 4  (x) = ω2 (V 2 + V 2)ω−0 (k · V 0

A

S

while

(21.28) (21.29)

(21.30) 2 2 A ) VS

− k 2y − k z2

(21.31)

k⊥ ψ

. (21.32) − (k · V A )2 ) For the situation considered here, the operator d/dx can then be replaced by ±ik x,1 or ±ik x,2 , where k x is found from the dispersion relation for the medium and the sign is chosen according to the direction of propagation. At the discontinuity, the boundary conditions require that η=i

ρ0 (ω02

ψ1+ + ψ1− = ψ2+

+ vx1

ω Copyright © 2005 IOP Publishing Ltd.

+

− vx1

ω

=

+ vx2

ω0

(21.33) .

(21.34)

Reflection and transmission of a plane wave

415

The relationship between ψ and vx is determined by (10.8) with the sign of k x chosen to be positive or negative according to the direction of energy propagation. From (21.24), the component of the time-averaged energy flux normal to the boundary is x = 12 Re ψ v˜x . (21.35) It is positive when ψ and vx have the same sign and negative when they have opposite signs. It is zero when k x2 < 0 so that k x is imaginary. Assume that conditions are such that the waves in medium 1 are not evanescent. Then the dispersion relation shows that > 0. From (10.8),



+ vx1 =+

ω|k x1|

1

ψ1+

− vx1 =−

ω|k x1|

1

ψ1− .

(21.36)

The choice of the sign of k x is more complicated in medium 2. The function sgn(x) represents the sign of the quantity x so that  +1 f (x) > 0 sgn[ f (x)] = (21.37) −1 f (x) < 0. Let

 ς=

2)

sgn(ω0 )/ +i

Then, if we write + =+ vx2

ς ω0 |k x2 |

2

k x2 > 0 k x2 < 0.

(21.38)

ψ2+

(21.39)

the boundary condition at x = +∞ are automatically satisfied. Now define reflection and transmission coefficients: R=

ψ1−

ψ1+

T =

ψ2+ ψ1+

.

(21.40)

Then (21.33) and (21.34) may be written in the form 1+ R = T ς |k x2| 1− R = |k x1|

1 T. 2

(21.41) (21.42)

These may be solved for R and T :

2 − ς |k x2|1 2 + ς |k x2|1 2|k x1|2 T = . |k x1|2 + ς |k x2|1 R=

Copyright © 2005 IOP Publishing Ltd.

|k x1| |k x1|

(21.43) (21.44)

416

Waves in moving media

Figure 21.4. Reflection and transmission coefficients for the conditions UA1 = 1.41, UA2 = 0, US1 = 0.41, US2 = 0, ρ2 /ρ1 = 10, κ y = 0.5, κz = 0. The bottom panel shows the branch of k x in medium 2 which corresponds to energy propagated away from the boundary.

21.4.2 Numerical results An illustration of how these reflection coefficients depend on the streaming velocity is shown in figure 21.4. The logarithms of the amplitudes are plotted for clarity as the magnitudes of the reflection and transmission coefficients vary over a wide range. The parameters shown are normalized in terms of a characteristic velocity V0 , defined by P + B 2 /2µ0 (21.45) V02 = ρ1 so that normalized velocities are given by U = V /V0 and the normalized wavevector components by κ = V0 k/ω. In this special case, we have chosen the pressure to be zero in medium 1 and the magnetic field to be zero in medium 2. Medium 2 is much more dense than medium 1 and the streaming velocity is in the y-direction. Also κz = 0, so that there is no component of the wavevector in the direction of the magnetic field. Pressure balance is maintained so that V0 is a constant. These conditions crudely mimic the magnetopause boundary when the Copyright © 2005 IOP Publishing Ltd.

Reflection and transmission of a plane wave

417

magnetosheath magnetic field is small compared with the magnetosphere field. The following are features to note. • •

• •

The reflection coefficient has a zero near U = 1.1. Here the wave is completely transmitted with a transmission coefficient of unity. For the counterstreaming media, this is the MHD analogue of the Brewster angle. The value of k x , corresponding to energy propagation away from the boundary, is positive for all negative U . It becomes zero with a branch point at about U = 1.5. It is then imaginary between about U = 1.5 and U = 2.4 and the wave in medium 2 is cut off. Within this interval, V increases to a value which exceeds the speed of phase advance parallel to the flow, so that ω0 changes sign. For larger values of V , the value of k x , corresponding to energy propagation away from the boundary, is negative so that k is in the opposite direction to the group velocity. In the interval in which k x is imaginary, there is total internal reflection with R = 1 and T purely imaginary. As the streaming velocity increases above the region of cut-off, the reflection coefficient has a magnitude rising rapidly above unity. This is the region of over-reflection. Clearly the reflected wave has an energy greater than that of the incident wave, meaning that it has been amplified in the reflection process. There are two points where the denominator of the formula for the reflection coefficient is zero so that the reflection coefficient becomes infinite. The nature of this process by which the wave is energized is discussed in the next section.

21.4.3 Energy conservation at the boundary We need to find expressions for the fraction of incident energy reflected and transmitted. In each medium, the time-averaged normal flux is   ±ω0 k x 1 1 ± ± Re{ψ Re |ψ|2 ±  = v ˜ } = (21.46) x x 2 2



where the sign of k x is determined by ς . To discuss energy transport across the boundary, consider the boundary conditions (21.33) and (21.34). Multiply (21.33) by the complex conjugate of (21.34) and take half the real part:     + − + + v˜ x1 ψ2+ v˜ x2 v˜ x1 1 1 + − (21.47) = Re Re(ψ1 + ψ1 ) 2 ω 2 ω0 that is

  + ψ2+ v˜ x2 1 11 + + − − + − − + . (21.48) Re[(ψ1 v˜ x1 + ψ1 v˜ x1 ) + (ψ1 v˜ x1 + ψ1 v˜ x1 )] = Re ω2 2 ω0

Copyright © 2005 IOP Publishing Ltd.

Waves in moving media

418

The second bracket on the left-hand side contains terms which have factors exp(±2iωt) whose time average is zero and so this becomes   + + v ˜ ψ 11 1 2 x2 (21.49) Re[(ψ1+ v˜ x+1 + ψ1− v˜ x−1 )] = Re ω2 2 ω0 or

− + x 1  +  x 1  =

ω +  ω0 x 2

(21.50)

where the angle brackets  represent an average over one cycle. The second term on the left-hand side is negative, representing the reflected energy flux. The equation may be re-arranged, yielding   ω + −  =   −   + − 1 + (21.51) + x1 x2 x1 x 2 . ω0 The left-hand side represents energy propagated towards the boundary and the first two terms on the right-hand side, energy propagated away from the boundary. When V = 0, then ω0 = ω and the last term on the right-hand side is zero, so that energy is conserved in the reflection and transmission process. When this is not the case, the last term may be positive or negative, depending on the conditions, implying that the wave either loses or gains energy at the boundary during the process. 21.4.4 Reflection and transmission coefficients for the energy Consider a tube of energy flux which intersects unit area at the boundary. From (21.46), we see that the ratio of the reflected energy flux to the incident energy flux is − x 1 ˜ =− + = R R. (21.52) x 1 





The minus sign is adopted in order to make a positive quantity. From (21.50), the corresponding ratio for the transmitted energy is given by



1−

 = ωω  .

(21.53)

0

An example of how these energy reflection and transmission coefficients depend on the streaming velocity, for the same conditions as figure 21.4 is shown in figure 21.5. The upper and lower panels show the same quantities for different ranges of U with different scales for clarity. The quantities shown are log10 , log10 , and log10 ( + ) as functions of V . The behaviour of these reflection and transmission coefficients for the wave energy flux may be compared with the corresponding quantities for the wave amplitude in figure 21.4. The following features should be noted:



 

Copyright © 2005 IOP Publishing Ltd.



Reflection and transmission of a plane wave

419

Figure 21.5. Logarithm of the energy reflection and transmission coefficients for the conditions UA1 = 1.41, UA2 = 0, US1 = 0.41, US2 = 0, ρ2 /ρ1 = 10, κ y = 0.5, κz = 0. The lower panel shows the detail near U = 2.5.

•

•

•

•

 

The quantity + represents the ratio of the wave energy flux directed away from the boundary to the wave energy flux directed towards the boundary. When it is greater than unity, the wave extracts energy from the background flow; and when it is less than unity, the wave increases the energy of the background flow. In terms of the linear theory here, it is assumed that the fractional change in the streaming velocity V , which arises from this process, is small enough to be neglected so that V remains constant. Only at U = 0 and in the region of total internal reflection is + = 1 so that the wave energy is conserved in the reflection and transmission process. In all other regions, there is an energy gain or loss so that energy is exchanged with the background flow. For negative values of U , the total wave energy is increased. However, neither nor is greater than unity. Waves in this regime are expected to be important in changing the flow velocity near the boundary and so reducing the steepness of the velocity gradient. For values of U greater than about 2.5, the streaming velocity of the plasma is greater than the velocity of phase advance parallel to the boundary. Here the reflection coefficient is greater than unity so that the reflected wave has a larger energy flux than the incident wave. This is the region of over-reflection considered by [129] and [135]. It should be noted that the transmitted wave is also amplified relative to the incident wave over most of the region. In this

 





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Waves in moving media

420

•

interpretation, so far from being a negative energy wave, it carries energy away from the boundary, contributing to a reduction of velocity gradient at the boundary. There are points in the regime of over-reflection, where the reflection coefficient (and the transmission coefficient) is infinite. This implies that the steady flow can generate waves flowing away from the boundary without any excitation from an incident wave. In this linear treatment, this is not an instability: the frequency is real and there is no growth. Nevertheless, in a nonlinear treatment, one would expect a rapid growth in the wave energy flowing from the boundary, with a corresponding reduction in the gradient until an equilibrium was reached. The initial condition of a sharp boundary with waves flowing away from it is allowed in linear theory but it is not sustainable when the constant drain of energy from the velocity gradient is included in the calculation.

21.5 Energy balance in a non-uniform plasma In a non-uniform medium, the zero-order quantities P, B, and V are functions of position. They are related by the equilibrium condition. The equations (21.9) (21.10), and (21.11) for the perturbation fields must then be modified to include first-order terms involving the gradients of zero-order quantities. They become   B·b B b dv = −∇ p+ + (b · ∇) − ρ0 v · ∇V (21.54) ρ0 + (B · ∇) dt µ0 µ0 µ0 dp = − γ P∇ · v − γ p∇ · V − v · ∇ P (21.55) dt db = B · ∇v − B∇ · v + b · ∇V − b∇ · V − v · ∇ B. (21.56) dt We carry out the same procedure as was used for deriving (21.20). The result, in suffix notation, may be written in the form   bi b j − 12 b2 δi j ∂ Vi ∂U ∂i ∂ Vi 1 p2 = − δi j + − ρ0 vi v j + ∂t ∂x j µ0 ∂x j 2 P ∂x j +

b j vi − bi v j ∂ Bi pv j ∂ P − µ0 ∂x j γ P ∂x j

(21.57)

where 1 b2 p2 ρ0 v 2 + + 2 2γ P 2µ0   Bjbj Bi b j vi − i = p + vj µ0 µ0 U=

Copyright © 2005 IOP Publishing Ltd.

(21.58) (21.59)

Energy balance in a non-uniform plasma

421

as before. There are now, however, additional terms depending on gradients of the zero-order quantities B j , P, and V j . We consider each of these in turn. The term depending on the gradient of B j may be written in the form    i j k klm bm vl ∂ Bi b j vi − bi v j ∂ Bi ∂ Bi lmk bm = = vl ki j µ0 ∂x j µ0 ∂x j µ0 ∂x j   ∇×B = −v· b× = v · ( J × b) (21.60) µ0 The term depending on the gradient of P may be written in the form ρv j ∂ P pv j ∂ P ρ = = v · ∇ P. γ P ∂x j ρ0 ∂ x j ρ0

(21.61)

These two terms represent work done as a result of the interaction of the zeroorder gradients with the wave fields. There is a force J × b per unit volume exerted on the plasma. This is periodic and, at a fixed point in space, averages to zero. The plasma, however, is moving with periodic velocity v and, as a result, the force does work. This work does not, in general, average to zero. There is also a force −∇ P per unit volume acting on the plasma. The rate of work done per unit volume on a fixed mass of plasma is −v · ∇ P. This work is periodic and would average to zero. The amount of energy per unit volume resulting from this in a fixed volume in space does not average to zero because of the periodic expansion and contraction of the plasma. The fractional change in volume is −ρ/ρ0 . The product of this with the rate of doing work represents the instantaneous fraction of the energy arising from the work done that is outside the fixed volume, and not included. The average value of this is not zero. These two terms occur even when the plasma is at rest and we need to consider their meaning in more detail. 21.5.0.1 Slowly varying medium When the medium varies, the relationship between the field components changes. As an illustration, consider a slowly varying medium in which a harmonic wave is propagated. At different points in the medium the relationship between the components depends on the relative values of the zero-order quantities P and B. As the wave progresses, this relationship changes. Locally, the relationship between the field components is that for a uniform medium and is given by (7.68), and (7.69) with ω replaced by ω0 = ω − k · V . If we eliminate k · v from these two equations, we get (k · B)v B . (21.62) p−b= γP ω0 Then, if we note that ∇ P = J × B − ρ0 (V · ∇)V , the time averages of the terms involving the gradients of P and B in (21.57) may be written in the form    1 p Re v˜ · J × b − ∇P 2 γP Copyright © 2005 IOP Publishing Ltd.

Waves in moving media      1 p pρ0 = Re v˜ · J × b − B + (V · ∇)V 2 γP γP      (k · B) 1 v + ρ(V · ∇)V = Re v˜ · − J × 2 ω0

422

= since

1 2

1 2

Re{ρ v˜ · (V · ∇)V }

(21.63)

Re{˜v } · ( J × v) ≡ 0. The energy conservation equation (21.57) is, thus, ∂ j  ∂U  ∂ Vi =− + Ti j  ∂t ∂x j ∂x j

(21.64)

where the tensor   bi b˜ j − 12 |b|2δi j 1 1 1 | p|2 − Re ρ0 v˜i v j − ρ v˜i V j + Ti j  = Re δi j 2 µ0 2 2 P (21.65) is the average value of the sum of the contributions by the products of first-order wave perturbations to the Maxwell and Reynolds stress tensors defined in (2.101), together with an isotropic tensor 12 δi j | p|2/P. We write Ti j 

∂(Ti j Vi ) ∂Ti j  ∂ Vi = − Vi . ∂x j ∂x j ∂x j

(21.66)

The first term on the right-hand side can be lumped with the term representing the energy flux. The second is of the form f · v where f is a force density arising from the divergence of the Maxwell and Reynolds stresses. It represents the work done per unit volume on the wave by these stresses. In a steady state, the time average of this is zero: the work done is a periodic exchange of energy between wave and background and averages to zero so that it is neglected in the ray-tracing approximation. In general, if the intensity of the wave varies in space as a consequence of variations in the source, it is nonzero. The system is isolated so that the energy is conserved. Any attenuation or growth produced by this work must result in gain or loss of energy in the streaming medium. Such transfer of energy between streaming velocity and wave is generally small. It can be large, however, if there is an attenuation mechanism in operation. In particular, if the wave is reflected at a caustic with a resonance region beyond it, there could be amplification and this may be of significance in the corona or solar wind [13, 14]. 21.5.0.2 Energy transfer at a tangential discontinuity We now discuss what happens to the energy which is gained or lost at the boundary between the two counterstreaming media described in section 21.4.4. In this example, B is in the z-direction, V in the y-direction, and the discontinuity Copyright © 2005 IOP Publishing Ltd.

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is perpendicular to the x-direction. We must take account of the fact that the boundary has finite thickness. Replace the discontinuous boundary by a thin layer with plane stratifications perpendicular to the x-axis. The energy relation (21.57) is now much simpler and becomes   bx b y b x vz − bz vx dB pvx dP dV ∂U = −∇ · + − + . (21.67) − ρ0 vx v y ∂t µ0 dx γ P dx µ0 dx We can assume time variation of the form exp{−iωt} and dependence on y and z of the form exp{i(k y y + k z z)} and consider the time average over one period. The terms involving the spatial gradients of the zero-order quantities P and B occur whether or not the medium is in motion. If we note that ∇ P = J × B and use (21.60), they may be written in terms of the current density     bx v˜z − bz v˜ x dB p v˜ x dP 1 B 1 Re − = Re −(b x v˜z − bz v˜ x ) − p v˜ x J. 2 µ0 dx γ P dx 2 γP (21.68) In (21.54), (21.55), and (21.56), d/dt ≡ −i(ω − k y V ) = −iω0 . These equations can be used in this coordinate system to show that   B 1 Re −(bx v˜ z − bz v˜ x ) − p v˜ x J 2 γP " #  2 iµ0 |vx | B2 1 1+ (21.69) J 2 = 0. = Re 2 ω0 µ0 γ P The time average of this term is zero. In the energy equation, it represents only a periodic exchange of energy between wave and medium. The term involving the velocity gradient does not, however, average to zero. As before, it may be written in the form   ∂(Vi Ti j ) ∂ Ti j b j bi dVi − ρ0 v j vi = − Vi . (21.70) µ0 dx j ∂x j ∂x j The second term is of the form − f · V where f is the force density arising from the divergence of the Maxwell and Reynolds stress tensors. The field components have strong gradients in the x-direction because of the reflection. The products of the first-order quantities in the definition of the stress tensor do not average to zero. The time-averaged quantity − f  · V represents the rate of increase of wave energy density in the boundary as a result of work done by the flowing background plasma; depending on the relative directions of  f  and V , this may be positive or negative. By Newton’s third law, there is an equal and opposite force exerted on the background plasma and work is done by the wave on it to change the magnitude of V . The source of the over- or under-reflection phenomenon can, thus, be identified as the consequence of work done on the wave by the flowing plasma: the kinetic energy of flow is transformed into wave energy. Copyright © 2005 IOP Publishing Ltd.

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This leads to a paradox. We have assumed a steady flow in order to compute the properties of the wave. The wave then, itself, changes the steady flow. The paradox is resolved by noting that the background flow changes on a time scale that is long compared with the wave period. The total energy of the closed system is still conserved. The wave is computed to first order. The energy equation is a second-order equation calculated from products of the first-order quantities. For total energy conservation, we must include second-order approximations to the field components. The situation is not unique. In dissipative media, we consider a uniform medium and compute the wave properties to first order. The damping of the wave in such media leads to heating of the background and there are consequent changes in pressure and density but these occur on a time scale long compared to the wave period and are ignored in computing the first-order wave components. Quasilinear approximations of this type also occur in kinetic theory when considering quasi-equilibrium states. We take this problem up in more detail in the next section. 21.5.1 Relation between total MHD energy and wave energy Equation (2.93) describes the conservation of total MHD energy in a nondissipative system. If it is integrated over any volume of space, fixed in the observer’s frame of reference, it represents the fact that the rate of increase of energy within the volume is equal to minus the rate at which it flows out through the boundary. The wave energy equation (21.19) for a uniform medium is constructed from the first-order approximations to the wave equations. The perturbation quantities v 1 , ρ1 , p1 , and B 1 are exact solutions of these approximate equations. The wave energy conservation equation, thus, represents the conservation of a quantity constructed entirely from the first-order wave solutions. It is clearly not the total energy and cannot straightforwardly be obtained by expanding the energy density obtained from (2.93) to second order. If one does so, one obtains a number of first-order energy terms as well as secondorder terms which are not included in Uwave . In a non-uniform medium, the first-order wave energy equations has additional terms that are not associated with the divergence of a flux vector. Equation (21.67) is a special case of this. To understand the relationship between the first-order wave energy and the total energy, let us consider the derivation of (2.93) in more detail. The left-hand side of (2.93) represents the rate of change at a point in space of the kinetic energy density, the magnetic field energy density, and the internal energy of the plasma. The rate of change of the internal energy was found by adding 1/(γ − 1) times (2.57) to the other equations. This equation, written in the form   d p d γp ≡ ∇·v (21.71) = dt dt γ − 1 γ −1




is simply an expression of the first law of thermodynamics when the conditions are adiabatic. It is a combination of (1.17) with the continuity equation when Copyright © 2005 IOP Publishing Ltd.

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ds/dt = 0 and there is no heat transfer. Since the wave energy has been defined in terms of second-order products of first-order quantities, we examine the thermodynamic behaviour to second order. Because of the adiabatic relation, ρ is a function of p alone; once ρ is determined p is known and vice versa. We can express the relationship between p and ρ as a Taylor expansion in powers of δρ ≡ ρ −ρ0 . For adiabatic compression, we can write γp dp = dρ ρ

d2 p γ dp γ p γ (γ − 1) p − 2 = = . ρ dρ dρ 2 ρ ρ2

(21.72)

Then, to second-order accuracy and noting that δρ is the perturbation correct to second-order, p = p0 + = p0 +

γ p0 1 (γ − 1)γ p0 2 δρ + ρ1 ρ0 2 ρ02 γ p0 1 (γ − 1) p12 (ρ1 + ρ2 ) + . ρ0 2 γ p0

(21.73)

If we differentiate this with respect to t, we get the rate of change of internal energy density     p12 p1 1 d(δp) d γ p0 ρ2 d d(δ ) . (21.74) ≡ = + + dt γ − 1 dt dt γ − 1 γ − 1 ρ0 dt γ p0






1 +
2, and
1 = p1/(γ − 1),

Thus, to second-order accuracy, since δ =




d 2 d = dt dt



γ p0 ρ2 γ − 1 ρ0



d + dt



p12 γ p0



,

(21.75)

The second-order contribution to the rate of change of the internal energy is, therefore, the sum of two terms, one arising from the second-order compression and the other involving the product p12 of two first-order wave components. This latter term, as shown in the argument leading to (9.18), arises from the work done by the first-order pressure perturbation in the wave. Now let us consider the derivation of the total energy conservation equation (2.93), which may be written in the form ∂U + ∇ ·  = 0. ∂t

(21.76)

Its derivation is equivalent to the following process. The Cartesian components of the reduced MHD equations (2.55)–(2.58) may be written as eight simultaneous, nonlinear differential equations of the first order: Fi j g j = 0 Copyright © 2005 IOP Publishing Ltd.

i, j = 1, 2, . . . 8

(21.77)

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426

where summation over the repeated suffix is understood, g j = (ρ, p, vx , v y , vz , Bx , B y , Bz ) and



       Fi j =       

D Dt

0 0 0 0 0 0 0

0 D Dt ∂ ∂x ∂ ∂y ∂ ∂z

0 0 0

ρ ∂∂x γ p ∂∂x

ρ ∂∂y γ p ∂∂y

∂ ρ ∂z ∂ γ p ∂z

D ρ Dt 0

0

0 0

0 −∇ yz B y ∂∂x Bz ∂∂x

D ρ Dt

0 Bx ∂∂y −∇zx Bz ∂∂y

D ρ Dt ∂ bx ∂z ∂ B y ∂z −∇x y

(21.78) 

0 0

0 0

0 0

− µ10 ∇ yz

By ∂ µ0 ∂ x − µ10 ∇zx By ∂ µ0 ∂z

Bz ∂ µ0 ∂ x Bz ∂ µ0 ∂ y − µ10 ∇x y

0

0 0

D Dt

0 0

Bx ∂ µ0 ∂ y Bx ∂ µ0 ∂z D Dt

0

       .      

D Dt

(21.79)

Here ∇x y = Bx

∂ ∂ + By . ∂x ∂y

(21.80)

We change the notation slightly to represent zero-order, first-order, and second-order quantities by subscripts or superscripts (0), (1), and (2), respectively as appropriate. Then there is a hierarchy of zero-order, first-order, and secondorder equations: (0) (0)

Fi j g j = 0 (0) (1)

(1) (0)

Fi j g j + Fi j g j = 0 (2) (1) (1) (2) (0) Fi(0) j g j + Fi j g j + Fi j g j = 0.

(21.81) (21.82) (21.83)

The first of these (21.81) is the equilibrium condition for the zero-order field components g (0) j . For a uniform medium, the left-hand side is identically zero;

and for a non-unifom medium, g (0) j is the exact solution of the approximate zeroorder equation. The second equation (21.82) is identical to the set of first-order wave equations. In the case of a uniform medium, the second term on the lefthand side is zero and the equations are identical to those for a uniform medium (21.3), (21.9), (21.10), and (21.11). The first-order field components g (1) j are exact solutions of these approximate equations. It is from these equations that we calculate the fields associated with the wave: they are equivalent to the first-order wave equation. The last equation (21.83) is the equation for the second-order corrections to the field components. The second-order set of field components (2) g j are exact solutions of these approximate equations. If we wish to evaluate the fields correctly to second order, all three terms of (21.83) must be included. One of the terms arises from the first-order operator acting on first-order terms. This is a source term for the equations. The first-order approximations to the wave Copyright © 2005 IOP Publishing Ltd.

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are evaluated as exact solutions of the approximate equation (21.82) and are then inserted in (21.83) which can be solved for the second-order quantities. The sum of (21.81), (21.82), and (21.83) (0) (0) (1) (1) (0) (0) (2) (1) (1) (2) (0) {Fi(0) j g j }+{Fi j g j + Fi j g j }+{Fi j g j + Fi j g j + Fi j g j } = 0 (21.84)

is an approximation to the field equations correct to second order and each of the quantities in braces is separately zero. Let the row matrix f i be given by f i = { 12 v 2 , (γ − 1)−1 , vx , v y , vz , Bx /µ0 , B y /µ0 , Bz /µ0 } (0)

(1)

(21.85)

(2)

with, zero-, first- and second-order parts f i , f i , fi . Then the process of finding the general energy equation (2.93), described earlier, when carried out to second-order accuracy, gives (0) (1) (0) (0) (0) (0) (1) (1) (0) f i(0) {Fi(0) j g j } + f i {Fi j g j } + f i {Fi j g j + Fi j g j } (1) (1) (0) (0) (0) (2) (1) (1) (2) (0) + f i(1) {Fi(0) j g j + Fi j g j } + f i {Fi j g j + Fi j g j + Fi j g j } = 0.

(21.86) To second-order accuracy, this is equivalent to (2.93). Each quantity in braces is equal to zero. The equation, thus, shows clearly that, when an equilibrium plasma suffers a small perturbation, the total energy conservation law can be regarded as the sum of several separate equations, each representing the conservation of a different quantity. We have divided it into four groupings. The grouping on the first and second lines represent the conservation of energy to zero and first order, respectively. They are equivalent to an expansion of (2.93) to first-order accuracy. Note that the zero- and first-order energy is separately conserved. The third and fourth lines represent the second-order energy conservation. This total secondorder energy conservation can also be found from the second-order terms in the expansion of (2.93) can be written in the form ∂U2 + ∇ · 2 = 0. ∂t

(21.87)

The evaluation of the expressions for U2 and 2 is straightforward but depressingly tedious. Fortunately, for the sake of our argument, we shall not need to find them in detail. Now let us consider the process of calculating the energy associated with the wave. The field components are calculated to first-order accuracy. In a uniform medium, the part of the energy density associated with the wave at a point in space is Uwave given by (21.18). We showed in section 21.3.1 that this wave energy is conserved so that ∂Uwave + ∇ · wave = 0. (21.88) ∂t Copyright © 2005 IOP Publishing Ltd.

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If we subtract (21.88) from (21.87), we get ∂( U2 ) (21.89) + ∇ · ( 2 ) = 0. ∂t This is an equation which shows that, for a uniform medium at rest or moving with constant velocity, the second-order terms can be divided into two sets. One represents the conservation of the wave energy and the other represents the conservation of a different second-order quantity, which requires evaluation to an accuracy higher than that to which the wave has been calculated. When the medium is non-uniform, the energy density associated with the wave at a point in space is still given by (21.18). This is the only energy density that can be constructed from squares of the first-order wave fields. (Note that the density and pressure are not independent but are connected by the adiabatic relation: the energy density 12 p2 /γ p0 takes account of both pressure and density perturbations.) The equation representing its rate of change, however, is now (21.57). This has additional terms arising from the gradients of the zero-order quantities: sometimes, as has been illustrated in sections 21.5.0.1 and 21.5.0.2, they are zero or their time average is zero: sometimes they are associated with energy loss by the wave. The energy equation for a first-order wave in a non-uniform steady-state flow, thus, takes the form ∂Uwave = −∇ · wave + f wave · V + Wwave (21.90) ∂t where f wave can be interpreted as power per unit volume as a consequence of the interaction of the zero-order flow with Maxwell and Reynolds stresses associated with the wave, while Wwave represents the other power terms described earlier. The wave energy is no longer conserved, in the sense that the rate of increase of wave energy within a fixed volume is equal to the flux of energy into the volume through the surface. There is also external work done. The total second-order energy, however, is still conserved; and equation (21.87) remains valid. Thus, if we subtract (21.90) from (21.87), we get ∂( U2 ) = −∇ · ( 2 ) − f wave · V − Wwave . (21.91) ∂t If work is done on the wave as a consequence of a non-uniform steady-state flow, then an equal and opposite amount is done on the background as would be expected from Newton’s third law. If this amount of work does not have a zero average, then there is a steady change in the energy associated with the secondorder terms as a consequence of the change in wave energy. This just means that, on a time scale long compared to the wave period, our assumption of a steadystate flow cannot hold. The assumption of a first-order wave superimposed on a steady-state background is a quasilinear approximation. Second-order effects, arising as a consequence of interaction with the wave, change the background conditions slowly. Copyright © 2005 IOP Publishing Ltd.

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21.6 Ray-tracing in a moving medium The ray-tracing equations (11.16) and (11.17) apply to any slowly varying medium with a local dispersion relation of the form ω = ω(r, k). This dispersion relation is calculated on the assumption that the medium varies so slowly that it is locally the same as for a uniform medium but the parameters specifying the conditions are slowly varying functions of position. In a medium such as the solar wind, the velocity, magnetic field, pressure, and density are functions of position. In cases where, compared with the wave period, they vary slowly with time also, we can neglect the time variation and assume a steady-state non-uniform flow. The ray-tracing equations actually hold for a medium varying slowly in time. This simply means that if we follow the path of a localized wavepacket, it advances according to the local conditions. If the flow is truly steady state, successive wavepackets of the same frequency follow the same path. If it varies slowly in time they follow different paths as successive wavepackets encounter slightly different local conditions. As previously described, the local dispersion relation is given by (7.48) with ω replaced by the Doppler-shifted frequency ω0 = ω − k · V . It takes the explicit form (21.92) ω04 − k 2 {ω02 (VA2 + VS2 ) − (k · V A )2 VS2 } = 0. The ray-tracing equations (11.16) and (11.17) may then be written in the form dr = ∇k ω dt dk = − ∇ω dt

(21.93) (21.94)

where ω = ω0 (r, k) + k · V

(21.95)

so that dr = ∇ k ω0 + V = V G,0 + V dt dk = ∇ω0 + ∇(k · V ) dt

(21.96) (21.97)

where V G,0 is the group velocity in the rest frame of the medium. If, from this, we evaluate the gradients ∇ω and ∇ k ω, and express the result in suffix notation, we can write the Cartesian components of the ray-tracing equations to give a set of six simultaneous first-order differential equations for the ray path: ki {(VA2 + VS2 )ω02 − (k j VA, j )2 VS2 } − (k j VA, j )k 2 VS2 VA,i dx i = + Vi dt ω0 {2ω02 − k 2 (VA2 + VS2 )} Copyright © 2005 IOP Publishing Ltd.

(21.98)

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Waves in moving media

ω0 (VA dki = − k2 dt ∂Vj . − kj ∂ xi 2

∂ VA ∂ xi

∂ V A,k ∂ VS 2 ∂ x i − (k j VA, j ) VS ∂ x i ω0 {2ω02 − k 2 (VA2 + VS2 )}

+ VS ∂∂VxSi ) − (k j VA, j )VA2 kk

(21.99)

21.7 The negative energy wave picture Let us now return to the situation of a uniform medium flowing with constant velocity V relative to the observer. A harmonic wave in the medium has a spatial and temporal variation of the form exp{−iωt + ik · r}. In treating, for example, the problem of over-reflection at a tangential discontinuity between two such media, some authors adopt a different point of view from the previous one. They use the concept of a negative energy wave [34, 68, 209]. This principle is described by McKenzie [129]. In the rest frame of the plasma, the internal energy density is given by a formula which can be shown to be equivalent to (21.18). Denote quantities measured in the plasma rest frame by an asterisk. The wave momentum density is given by g=

kU0 ω0

(21.100)

which, since ω0 U0 = k · , is the component of the flux vector in the wavenormal direction divided by the square of the wave speed ω0 /k. A Galilean energy transformation is applied to find the internal energy density in the frame of the observer. The result is U  = U0 + g · V

(21.101)

where g is the momentum density associated with the wave perturbation and is unchanged in the transformation. The result is that U  is related to U0 by U =

ω U0 . ω0

(21.102)

In the negative energy wave picture, the quantity U  is interpreted as the energy density. It is not the same as the energy density U ≡ U0 computed from (21.18) which we have used previously in the observer’s frame. This is a matter of interpretation and is not of consequence in a uniform medium. When the medium is not uniform, however, the different interpretations can be confusing. If we multiply both sides of (21.25) by ω/ω0 , we see that the energy density U  , defined in this way, obeys the same relationship for plane waves as U : ωU  = k ·  where

Copyright © 2005 IOP Publishing Ltd.

  =

ω . ω0

(21.103) (21.104)

The negative energy wave picture

431

Since the flux vector is the same on either side of the boundary, as described by McKenzie [129], if we use for the energy density the quantity U  = ωU/ω0 , then the flux  reflected from and transmitted through the boundary is equal to the incident flux so that U  is conserved at the boundary. This is because of the nature of the Galilean transformation adopted for the energy. In the region where the streaming velocity is sufficiently large, the Dopplershifted frequency ω0 is negative and, thus, so is U  . If the quantity U  is interpreted as the energy density, then it can be positive or negative. The energy flux vector  = U  V G can, therefore, be in the opposite direction to the group velocity. This need not cause any concern in the case of infinite plane waves. The flux vector only has meaning in the sense that  · d A represents the rate at which the energy crosses an element of area d A. There is no distinction between negative energy passing across the area in one direction and positive energy crossing in the other direction. One could equally well project this as a picture in which somehow the flowing medium ‘knows’ that there is an incident wave on the boundary from the region x < 0 and sends a positive energy wave to meet it, in order to conserve energy at the boundary. There is no violation of causality here as the harmonic wave is uniform for all time throughout the semi-infinite region x > 0. Consider a simple example of the difficulty involved in this approach. Suppose we place a perfectly absorbing medium occupying the region x > x A and moving with the same velocity as the streaming plasma. The negative energy wave travels from x = 0 to x = x A where it is absorbed. In the rest frame of the streaming plasma, it is a positive energy wave and the interpretation is easy: it carries energy from x = 0 to the absorber and gives it up there through Joule heating. The observable consequence is that there will be a rise of temperature in the absorber. In the observer’s frame, this must be depicted as a negative energy wave propagated towards the absorber giving up its negative energy and causing a rise in temperature or, alternatively, a positive energy wave extracting energy from the absorber with a consequent rise in temperature. Even if one attempts to explain this in terms of the detail of phenomena at the boundary of the absorber, there are problems of identifying the locality of the energy transfer. Each of these interpretations gives correct answers in the context in which it is applied. The negative energy wave interpretation is applied in the context of infinite harmonic plane waves in which the frequency is well defined. Since the energy density is negative in the second medium, the energy flux vector, which is the product of the energy density and the group velocity, is in the opposite direction to the group velocity. The boundary conditions for the second medium require that the group velocity has a normal component which points away from the boundary. Thus, energy is flowing into the boundary from the second medium as well as from the first, in which the source is located. This need not trouble us in the case of waves which are uniform in space and time. It is well known that any definition of an energy flux vector is not unique; the curl of an arbitrary vector field of appropriate dimensions can be added to it without affecting energy conservation (see, for example, [152]). Unique results are only obtained when Copyright © 2005 IOP Publishing Ltd.

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integration over a closed surface is performed. Integration over such a surface which lies entirely in the medium on either side of the boundary shows that that there is no net flux into or out of the surface. Only if the surface includes part of the boundary is there a net flux into or out of it. This implies an energy source or sink in the boundary and nowhere else. It can be argued [152] that there is no way to use the energy flux vector to locate energy flow. Despite this fact, physicists often feel a sense of unease with a definition of energy flux which is inconsistent with the location of known sources of energy. If it is possible to find a definition of energy flux which accords with simpleminded ideas of localized energy flow, they prefer it. It is worthwhile consulting a well-known ‘elementary’ but sophisticated source [67, ch 27] for an enjoyable discussion of this point. Diffficulties with the negative energy wave interpretation appear when we consider waves limited in space and time. The definitions (21.102) and (21.104) of energy density and flux only apply when there is a well-defined frequency. As has already been stated, the energy density so defined is actually the wave action. The most general definition of wave action involves integration over the generalized coordinates describing the wave and is beyond the scope of this treatment. Any wave variable can be described by a Fourier synthesis of plane waves. If the Fourier amplitude is narrowly peaked in frequency and wavenumber, the wave is propagated as a well-defined wavepacket with a well-determined frequency and wavelength as illustrated in figure 21.6. This figure shows three epochs in the history of a wavepacket. At t = t1 , the wavepacket is travelling towards the boundary between two media. The lower medium is at rest and the upper medium is streaming parallel to the boundary with a uniform velocity V. The wavepacket moves with the group velocity while wavefronts move through it with the phase velocity which, in a dispersive anisotropic medium, does not have the same direction and magnitude as the phase velocity. These velocities are schematically shown and are not intended to show the actual relative directions and magnitudes of an MHD wave. At time t = t2 , the wavepacket has been partially reflected and partially transmitted at the boundary. At time t = t3 , the transmitted wave has progressed further and the reflected wave is no longer within the illustrated region. We emphasize that this is an observable phenomenon. The wavepacket is a physical disturbance with its own time history. Now consider the negative energy wave picture interpretation. Suppose that the velocity V is large enough so that the reflected wave has a larger amplitude than the incident wave. It, therefore, carries more energy than the incident wave. The negative energy wave picture requires that energy be conserved in the reflection and transmission process. The transmitted wave must, therefore, carry negative energy away from the boundary. Consider the volume ABC D. If we integrate the energy density over this volume at times t = t1 and t = t2 , the result is zero. Evaluation of the integral at t = t3 leads to a negative result. This is hard to reconcile with the fact that this is the time at which the wavepacket has arrived within the volume. This is an observable fact. If there were a detector within Copyright © 2005 IOP Publishing Ltd.

The negative energy wave picture

433

Figure 21.6. Motion of a wavepacket.

the volume, it would be excited, presumably extracting energy from the already negative wave energy reservoir. Another aspect causing difficulty is the nonlocality. At time t = t2 , the total energy at the location of the reflected wavepacket is augmented at the expense of the energy of the transmitted wavepacket which is reduced below zero. The wave energy picture raises none of these difficulties. Both transmitted and reflected wavepackets carry positive energy. As described in section 21.5.0.2, in these circumstances both acquire additional energy at the boundary. The mechanism can be understood by looking at the details of propagation within the thin boundary layer. The velocity gradient in the boundary layer leads to additional Reynolds and Maxwell stresses, which result in work being done on the wave by the streaming background plasma. Why does the negative energy picture produce correct results for harmonic waves but provide puzzling problems when applied to wavepackets which are limited in space and time? First, we should note that the Galilean transformation should be applied to the total energy; it is, however, applied to the wave energy in the rest frame. The expression used is the wave energy in that frame and omits first- and second-order terms. When the Galilean transformation is applied to find the energy in the frame in which the plasma is moving, first- and second-order terms are introduced. In addition, the period over which time averages are taken is different in the rest and moving frames. The implication is that some of the energy associated with the wave in the moving frame is associated with the background in the wave energy picture. It is possible to resolve the inconsistencies but only at Copyright © 2005 IOP Publishing Ltd.

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Waves in moving media

the cost of elaborate argument. For example, if the wave is absorbed as described earlier, it is necessary to consider not only the negative energy associated with the flux but also what work is done by the streaming background plasma at the interface with the absorber. None of these difficulties arise with the wave energy picture. A lucid critical discussion of the hidden assumptions built into the negative energy wave picture is given by Andries and Goossens [14].

21.8 Over-reflection in solar–terrestrial physics The phenomena described in this chapter have some important applications in solar–terrestrial physics. We summarize some of these without going into the details of the theory. 21.8.1 Excitation of long period pulsations The source of Pc5 pulsations in the magnetosphere is not understood. Several mechanisms (described in section 13.4.1) have been proposed to excite cavity or waveguide modes (chapter 20) which, in turn, excite field-line resonances. Mann et al [135] have presented a theory describing growth in a cavity as a consequence of over-reflection. Their analysis is in terms of negative energy waves. An equivalent analysis using the definitions described in this chapter can be performed. They use a model equivalent to that shown in figure 20.1. The magnetopause boundary is taken to be a tangential discontinuity. The magnetosheath plasma streams tailward. Instead of a wave incident from the magnetosheath, suppose that there is a small perturbation inside the magnetospheric cavity. This can be regarded as a waveguide mode consisiting of a wave reflected back and forth across the cavity. If the magnetosheath velocity is large enough, over-reflection takes place as the wave progresses and the mode grows. Such a source may be important as a mechanism for exciting the observed field-line resonances. 21.8.2 Resonant absorption in coronal plumes Coronal plumes originate in coronal holes. They are approximately radial and lie over regions of increased magnetic flux in the photosphere. They are denser than the interplume plasma. Their radial velocity is smaller than the velocity of the surrounding plasma. At greater distances from the Sun, they are mixed smoothly with the coronal plasma. Possible mixing mechanisms are the Kelvin–Helmholtz instability (section 23.4) or resonant flow instability [12, 13]. These instabilities arise at the boundary of shear flow between the plume and its surroundings. Andries and Goossens [12] show that the growth rate of the resonant flow instability is greater than that of the Kelvin–Helmholtz instability. This instability is related to over-reflection. We note here that, in order to understand the process, it is necessary to consider the details of the boundary layer profile. Andries Copyright © 2005 IOP Publishing Ltd.

Summary

435

and Goossens [13, 14] show that the localization of the energy transfer in the boundary layer provided by the wave energy picture of this chapter removes some ambiguities introduced when the process is analysed in terms of frame-dependent negative energy waves.

21.9 Summary The objective of this chapter has been to make clear the energy relationships between wave and background medium when MHD waves encounter the boundary between media in relative motion. The chief results are: •

•

•

The energy conservation relationship in a uniform moving medium has been written in a form where the energy density and energy flux are defined in a manner that means that the flux vector represents energy density flowing with the group velocity. When a wave is incident on the boundary between two counterstreaming media, it is well known that over-reflection may occur. If only plane waves are involved, then the energy relationships may be written in a form which implies that energy density is conserved in the reflection and transmission processes at the boundary and the over-reflection is explained in terms of a negative energy transmitted wave. We have shown that this approach has difficulties of interpretation when the wave is confined to a wavepacket limited in space and time. We have clarified the relationship between wave energy which is not conserved at the boundary and the total energy which is. The definitions adopted here, unlike those in the negative energy wave picture, identify the location of energy exchange between background flow and wave as the boundary. It has been shown that the mechanism is through the work done by the component of the force due to the Reynolds and Maxwell stresses in the direction of the streaming velocity at the boundary. The interpretation provides a straightforward picture of the processes when a wave is incident on a boundary of a medium streaming with a large enough velocity to cause over-reflection. These processes are: (1) The wave is incident on the boundary. (2) In the sharp velocity gradient at the boundary, the force associated with the Maxwell and Reynolds stresses in the wave does work on the streaming plasma. (3) If the conditions indicate over-reflection, this work is negative and the wave extracts energy from the kinetic energy of the flow. Both reflected and transmitted waves have increased energy. (4) The transmitted wave always has positive energy and is propagated in the streaming medium in such a way that the energy flux vector represents the internal energy propagated at the group velocity.

Copyright © 2005 IOP Publishing Ltd.

Chapter 22 Shock waves

22.1 Introduction In an MHD medium, information about changes in the state of the medium is propagated at the speed of the waves in the medium. The maximum speed of an MHD wave relative to the medium is the speed of ' a fast wave propagated transverse to the magnetic field at the hybrid speed VA2 + VS2 . Observers in a reference frame, fixed with respect to the solar wind, whose only source of information comes from MHD processes, can have no knowledge of the fate that awaits them, as the Earth and its magnetospheric environment hurtle towards them at a speed far greater than that at which any MHD information can travel. In contrast, an observer on the Earth can get information about changes at the shock through MHD waves that are propagated downstream. A shock front acts as a one-way valve for MHD information! In general, shock waves arise when there are parts of the fluid that have velocities greater than the characteristic wave speeds, so that information about changes in the flow travels too slowly to influence the flow ahead of the shock. The media on either side of the shock must obey the appropriate behaviour at the upstream and downstream boundaries. In general, this is only possible if a shock discontinutity is set up. In chapter 6, we derived the conditions that must apply across any discontinuity in an MHD medium. Of the three classifications of discontinuty, the most complicated is the shock discontinuty, since plasma flows through it at velocities greater than the characteristic wave velocities. In this chapter, we investigate its properties further from two points of view. First, we study the conditions that must apply for the shock to endure for any significant period of time: a small disturbance must not lead to disruption of the shock. We then consider the reflection, refraction, and transmission of a wave incident on the shock. This process is greatly complicated by the fact that the motion of the medium relative to the shock may be greater than the wave velocity. 436

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22.2 Properties of shock waves The de Hoffmann–Teller relations [52], written in the form, (6.28)–(6.32) apply to any discontinuity between two MHD media. They provide seven equations determining the change in the three components of velocity, the two transverse components of the magnetic field and any two of the thermodynamically related quantities, pressure, density, specific entropy, and temperature. The thermodynamic state of the system on either side of the boundary is a function of just two of these thermodynamic quantities. These equations, therefore, also specify the changes in each of the thermodynamic variables across the discontinuity. As we saw in chapter 6, these relations define three different classes of boundary that can occur in MHD. The normal components of B and V are zero for tangential discontinuities and the normal component of V is continuous and non-zero for rotational discontinuities. For the discontinuity to be classified as a shock, we require that the mass flux j = ρV is non-zero and that the change in the normal component of velocity [Vn ] is non-zero. As in chapter 6, the square brackets denote the difference between the values of a quantity on either side of the boundary. In such circumstances, the tangential component of the velocity is continuous; and, as described in section 6.5.5, it is, therefore, possible to use a reference frame in which the shock is at rest and B, V , and the normal to the boundary are coplanar. As observed from this frame of reference, the plasma streams in from one side, unaffected by the presence of the shock until it reaches the boundary. It then emerges on the other side with a different velocity, pressure, density, magnetic field, and entropy. These quantities change abruptly in the shock layer. The changes must satisfy the de Hoffmann–Teller relations. Although we have treated this layer as a sharp boundary between two uniform MHD media, there is a scale on which its thickness is finite. Inside it, a variety of thermodynamically irreversible non-equilibrium processes must take place, the details of which are not known. We can, however, deduce some general results using thermodynamic arguments. The one certainty is that, if there is no heat transferred between a given mass of plasma and its neighbours, then the second law of thermodynamics requires that, as it passes through the shock, its entropy cannot decrease. We first consider the relatively simple case of a shock in a non-conducting fluid. We then generalize these results to MHD shocks. The approach is similar to that followed by Landau and Lifshitz [118, 119]. 22.2.1 Change of properties across a shock in a gas Consider a portion of a gas with mass M. On the upstream side of the shock, its pressure, P1 , and the volume per unit mass, or specific volume, τ1 ≡ 1/ρ1 , are specified as initial conditions. The other thermodynamic variables such as energy density, temperature, entropy, and enthalpy are functions of these. It flows through the shock. On the other side, these quantities have changed as a result of nonlinear Copyright © 2005 IOP Publishing Ltd.

438

Shock waves

processes occurring within the shock layer. This process occurs adiabatically so that there is no heat conducted from neighbouring volumes of plasma. The mass M then constitutes an isolated system. The second law of thermodynamics, therefore, requires that its entropy must increase in this irreversible process. We can obtain the equations of ordinary gas dynamics by setting B = 0 in the de Hoffmann–Teller conditions. We see from (6.28) that the normal mass flux j = ρ1 V1 = ρ2 V2 is still continuous, while (6.30) shows that the tangential component of the velocity is continuous. We can, therefore, transform to a reference frame in which the velocity is perpendicular to the boundary, so that V t = 0. Instead of using the density, we use the specific volume or volume per unit mass τ = ρ −1 . Equations (6.31) and (6.32) then become 

[P + ρV 2 ] ≡ (P2 − P1 ) + j 2 (τ2 − τ1 ) = 0  1 2 γP V + ≡ 12 j 2(τ22 − τ12 ) + w2 − w1 = 0 2 (γ − 1)ρ

(22.1) (22.2)

where w is the specific enthalpy (1.20). The remaining relations are identically zero. The first of these equations can be used to express j as a function of the pressures and velocities on either side of the shock j2 = −

P2 − P1 . τ2 − τ1

(22.3)

This shows that P2 − P1 and τ2 − τ1 have opposite signs. Thus, the pressure and density of the gas either both increase or both decrease in the crossing of the shock. We use thermodynamic arguments to show that only the former case occurs. Equation (6.51), applying to shocks for which adiabatic conditions apply, becomes     1 1 P + (P1 + P2 ) ≡ (ε2 − ε1 ) + 12 (P1 + P2 )(τ2 − τ1 ) = 0 (22.4) (γ − 1)ρ 2 ρ where ε is the specific energy (1.19). This is the Hugoniot relationship or shock adiabatic. We have written it for an ideal gas for which (1.19) holds. It can be derived more generally [118, section 82]. If pressure and density are specified in the upstream medium, then all the other thermodynamic quantities are also specified. The quantities in the downstream medium are then specified. For a given P1 , τ1 , (22.4) defines a curve in the Pτ -plane, passing through the point P1 , τ1 . The state of the downstream plasma must be represented by a point on this curve. We can write the relationship in a normalized form by solving (22.4) for P2 /P1 as a function of V2 /V1 , getting P2 (γ + 1) − (γ − 1)τ2 /τ1 =− . P1 (γ − 1) − (γ + 1)τ2 /τ1

(22.5)

This is the equation of a rectangular hyperbola, passing through the point P2 /P1 = 1, τ2 /τ1 = 1, with asymptotes P2 /P1 = −(γ − 1)/(γ + 1) and Copyright © 2005 IOP Publishing Ltd.

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439

5 4

P  P

3 2 1

O

0 1 -1

2

3

4

5

@ @

Figure 22.1. The Hugoniot relationship, shown by a bold curve passing through the point O, corresponding to P2 /P1 = 1, τ2 /τ1 = 1, corresponding to the density and volume of M in region 1. The dotted curves are adiabatics on which the entropy is constant. The bold broken curve is the adiabatic corresponding to the entropy of M in region 1.

τ2 /τ1 = (γ − 1)/(γ + 1). It is plotted in figure 22.1. In this figure, we also show a series of dotted lines representing adiabatics, defined by   P2 τ2 γ = constant. (22.6) P1 τ1 On each of these adiabatic curves, the entropy has a constant value. The adiabatic passing through O, for which the constant in (22.6) is unity, corresponding to the entropy of M in region 1, is represented by a bold broken curve. The entropy relation (1.17) for an ideal gas may be written in the form   γP 1 dP + dτ (22.7) T ds = (γ − 1)ρ τ showing that (T ∂s/∂ P)τ and (T ∂s/∂τ )P are positive. Thus, as we move from the left-hand side to the right in figure 22.1, the entropy corresponding to each adiabatic increases. The diagram shows that above and to the left of O, the Hugoniot contour lies above the adiabatic through O. Below and to the right of O it lies below it. This is easily verified by considering the ratio of the values of P2 /P1 obtained from the Hugoniot relation (22.4) and the adiabatic relation (22.6) with the constant equal to unity. The entropy of M in region 2, therefore, is greater than that in region 1 for those parts of the Hugoniot contour lying above O, and less than that in region 1 for those lying below O. Only the former alternative is allowed by the second law of thermodynamics. The part of the curve below O Copyright © 2005 IOP Publishing Ltd.

440

Shock waves

is forbidden. We conclude that, as the gas passes through the shock the pressure increases and τ decreases so that the density increases. This is an expression, for an ideal gas, of Zemple´ n’s theorem, which is proved more generally by Landau and Lifshitz [118]. This states that, for all fluids for which   ∂ 2τ >0 (22.8) ∂ P2 S

where the subscript implies that entropy S is held constant, the density and pressure increase as the fluid passes through the shock. For a perfect gas, we can differentiate the adiabatic relation Pτ γ = constant twice to verify this relation. 22.2.2 Changes through MHD shocks We now show that Zemple´ n’s theorem also applies to MHD shocks. The Hugoniot relation (6.51) may now be written in the form     P [ Bt2 ] 1 [τ ] = 0 (22.9) P1 + P2 + + (γ − 1)ρ 2 4µ0 which may be solved for P2 / P1 to give P2 (γ + 1) − (γ − 1)τ2 /τ1 (γ − 1)(τ2 /τ1 − 1)βs =− + P1 (γ − 1) − (γ + 1)τ2 /τ1 (γ − 1) − (γ + 1)τ2 /τ1

(22.10)

where

( Bt,2 − Bt,1 )2 (22.11) µ0 P1 and is always positive. The curve has an asymptote at τ2 /τ1 = (γ − 1)/(γ + 1) = 0.25 as is the case when the magnetic field is zero. The denominator of each term in (22.10) is negative for values of τ2 /τ1 greater than this. The second term of (22.10) is, therefore, positive for (γ − 1)/(γ + 1) < τ2 /τ1 < 1 and negative for τ2 /τ1 > 1. The first term in (22.10) is the contour when the magnetic field is zero. The contour when the magnetic field is included lies above this for τ2 /τ1 < 1 and below it for τ2 /τ1 > 1. Examination of figure 22.1 shows that, for points above O on the contour, the entropy increases while, below O, it decreases. This leads to the same conclusion as in the case where the magnetic field is zero. The pressure and density increase as the fluid passes through the shock. This is also true for more general equations of state [119]. βs =

22.3 Waves in the neighbourhood of shocks 22.3.1 Coordinate system and boundary conditions We consider a plane shock separated by two uniform MHD media. In section 6.5.1, we showed that the classification of a sharp boundary as a shock Copyright © 2005 IOP Publishing Ltd.

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441

requires that there is a flow of plasma through the shock and that the transverse components of the magnetic field on either side of the shock are collinear. In the steady state, as shown in section 6.5.5, we can choose a coordinate system such that B, V , and the normal to the shock are collinear. Let x be normal to the shock and let B and V lie in the x z-plane. We assume that the plasma has a positive x-component of velocity. Then region 1 is defined as the region upstream of the shock for which x < 0 and region 2 is the region downstream with x > 0. The conditions applying to the unperturbed shock are determined by the five de Hoffmann–Teller equations that do not involve Vy and B y . These other two equations need to be taken into account if we perturb the shock. In these coordinates, the shock conditions (6.48), (6.51), (6.52), and (6.53) may be written in the form   Bz B2 j2 (22.12) − x [Bz ] = 0 ρ0 µ0       1 1 P 1 + (P1 + P2 ) + [Bz ]2 =0 (22.13) (γ − 1)ρ0 2 ρ0 4µ0 ρ0 #   " Bz2 2 1 =0 (22.14) + P+ j ρ0 2µ0 j [V z ] −

Bx [ Bz ] = 0 µ0

(22.15)

where the square brackets denote the change of a quantity across the shock, j ≡ ρ0 Vx is the normal component of momentum density and is continuous across the shock so that [ j ] = 0, and the normal component of magnetic field is continuous across the shock so that [Bx ] = 0. 22.3.2 Dispersion relations In the media on either side of the shock, four different characteristic waves may exist. These are the fast and slow magnetosonic waves, the transverse Alfv´en wave, and the entropy wave. We can assume that, on both sides of the shock, these vary with t, y, and z, as exp{−i[ωt −k y y −k z z]} with Snell’s law specifying that k y and k z are the same on either side of the shock. The variation with x is of the form exp{ik x x} where k x is found from the dispersion relation in the moving medium. (22.16) k x = (ω, k y , k z , V , V A , VS ).



The explicit form of this relationship can be found if we write (7.48) and (7.49) in coordinate-free form by replacing ω with ω0 ≡ ω − k · V .

(22.17)

We get ω02 − (k · V A )2 = 0 Copyright © 2005 IOP Publishing Ltd.

(22.18)

442

Shock waves ω04 − k 2 (VA2 + VS2 )ω02 + k 2 VS2 (k · V A )2 = 0

(22.19)

or, explicitly, ω02 − (k x VA,x + k z VA,z )2 = 0 ω04

−k

2

(VA2

+

VS2 )ω02

+k

2

VS2 (k x VA,x

+ k z VA,z ) = 0 2

(22.20) (22.21)

where k 2 = k x2 + k 2y + k z2 . The first of these dispersion relations applies to the transverse Alfv´en wave and the second to the two magnetosonic waves. In addition, we must take account of the entropy wave (section 21.2.4). Its dispersion relation is ω − k · V = 0.

(22.22)

It consists of a perturbation of the entropy and density moving undispersed with the plasma velocity. 22.3.3 Relation between field components For the Alfv´en and magnetosonic waves the relationship between the components of v is found from (7.44), with ω0 replacing ω: {ω02 −(k·V A )2 }v−k(VA2 +VS2 )k·v+(k·V A ){k(V A ·v)+V A (k·v)} = 0. (22.23) If the components of this are written out, we get three homogeneous equations in vx , v y , and vz which are self-consistent if either (22.18) or (22.19) is obeyed. Any two of these give v y and vz in terms of vx for the transverse Alfv´en wave if (22.18) holds and for the magnetosonic waves if (22.19) holds. The perturbations of p and b can then be found as described in section 21.2.3.2. The complete state of each wave is described by the amplitude of one of the components in a suitable coordinate system and the ratios of the other components to this one. 22.3.4 Direction of propagation relative to the shock In the shock frame (the frame of reference in which the shock is at rest), each characteristic wave is propagated either towards or away from the shock. Apart from the entropy wave, these characteristic waves occur in pairs, a pair each of transverse Alfv´en waves, slow waves, and fast waves. In the rest frame of the medium, the group and phase velocities of each pair are equal and opposite. This is not so in the shock frame. In applying appropriate boundary conditions at the shock, it is necessary to distinguish between ingoing waves that are incident on the shock from outside and outgoing waves that originate at the shock and are propagated away from it. We consider each type of wave in turn. Copyright © 2005 IOP Publishing Ltd.

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Table 22.1. Direction of Alfv´en wave propagation relative to the shock. (0)

Region 1 Region 2

(0)

|VA,x | < Vx

|VA,x | > Vx

Both ingoing Both outgoing

One ingoing, one outgoing One ingoing, one outgoing

22.3.4.1 Entropy wave The entropy wave consists of a disturbance of density carried along with the velocity of the plasma. In region 1, upstream of the shock, the entropy wave is ingoing and, in region 2, it is outgoing.

22.3.4.2 Transverse Alfv´en wave In the rest frame of the plasma, the group velocity of the transverse Alfv´en wave is parallel or antiparallel to the magnetic field and its magnitude is VA . In the shock frame, VG = VA + V.

(22.24)

It can be seen that this situation is that summarized in table 22.1

22.3.4.3 Magnetosonic waves Similar arguments may be applied to the fast and slow waves. The x-component of the group velocity in the rest frames of the plasma on either side of the shock (0) is VG,x = ∂ω0 /∂k x , where ω0 is defined by (22.19). This leads to (0)

VG,x = =

2 + V 2 {k [ω2 − (k · V 2 2 ω02 k x VA,0 A,0 ) ] − k VA,x (k · V A,0 )} S,0 x 0 2 + V 2 )}ω {2ω02 − k 2 (VA,0 0 S,0 2 k V 2 + V 2 {k [k 2 V 2 − (k · V 2 2 k 2 VP,0 x A,0 A,0 ) ] − k VA,x (k · V A,0 )} S,0 x P,0 2 − k 2 (V 2 + V 2 )}kV {2k 2 VP,0 P,0 A,0 S,0

(22.25) where VP,0 = ω0 /k is the phase speed in the rest frame, as found from the dispersion relation. The classification of waves as ingoing or outgoing is then the same as in table 22.1 with VA,0 replaced by VG,0 . Copyright © 2005 IOP Publishing Ltd.

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Shock waves

22.4 Classification of shocks 22.4.1 Fast and slow shocks In this section, we follow the approach of Landau et al [119, section 72, section 73]. Assume that the shock is given a small perturbation, varying as exp{−iωt}, and oscillating in phase over the whole surface of the shock. The perturbation arises as a consequence of perturbing the fields on either side of the shock and then ensuring that perturbations obey the boundary conditions at the shock. The perturbations in the uniform media on either side of the shock must take the form of MHD waves. These vary in space on either side of the shock as exp{±ik x (ω)x}, where k x is obtained from the appropriate dispersion relation with k y = k z = 0. The sign is chosen so that the waves originate at the shock and are outgoing. There are seven quantities that can be independently perturbed. These are the fluid velocity components vx , v y , vz , the transverse components of the magnetic field, b y , bz , the density ρ, and the pressure p. The other thermodynamic quantities such as the specofic entropy s or the enthalpy density are functions of the thermodynamic state variables, ρ and p, and are not independent. The normal component of magnetic field bx is continuous across the boundary and is not perturbed. In addition, the mass flux j = ρvx is not perturbed. The velocity of the shock itself is also perturbed. It acquires a velocity δu relative to the frame of reference in which the unperturbed shock is at rest. Consider now the first-order perturbations of the boundary conditions (6.28) to (6.32). We write [vx + δvx ] to represent the zero- and first-order terms of the velocity step at the boundary, with similar expressions for the other fields. Note that the zero-order fields v y and B y are zero as a result of the choice of coordinate system: [ρδvx ] + [vx δρ] − [ρ]δu = 0 Bx [δv y ] + [vx δ B y ] = 0

(22.26) (22.27)

Bx [δvz ] + [vx δ Bz ] + [Bz δvx ] = 0 Bx [δ B y ] = 0 j [δv y ] − µ0 Bx j [δvz ] − [δ Bz ] = 0 µ0 [Bz δ Bz ] =0 [δp] + j [δvx ] + µ0  " #   Bz2 δρ pδρ γ 1 2Bz δ Bz δρ − 2 + − j [vx δvx ] + γ −1 ρ µ0 ρ ρ ρ2

(22.28)

−

Bx [Bz δvz + vz δ Bz ] = 0. µ0

Copyright © 2005 IOP Publishing Ltd.

(22.29) (22.30) (22.31)

(22.32)

Classification of shocks

445

The first of these equations determines the perturbation δ u of the velocity of the shock itself. In the remaining six equations, [ j ] is unperturbed. These six equations determine the perturbations that must match to waves in the uniform media on either side of the boundary. To satisfy causality, these waves must be outgoing. We are considering perturbations varying only in time so that there is no dependence of the phase on y and z so that the waves on either side of the shock have wave normals in the x -direction. The y -direction is, therefore, normal to the plane containing the magnetic field and the wave normal of the waves on either side of the shock. It will be observed that the equations separate into two sets. Equations (22.27) and (22.29) involve only [δv y ] and [δ B y ]. The remaining equations do not involve these perturbations. The two sets of equations must be separately satisfied. The wave that has this polarization is the transverse Alfve´ n wave. In order that the shock can exist, there must be two and only two outgoing Alfve´ n waves, as shown in table 22.1. Let VA,1 , VA,2 be the x -components of the group velocities of the waves in regions 1 and 2, respectively. Then there are two and only two outgoing waves: (i) when vx,1 > VA,1 and vx,2 > VA,2 ; or (ii) when vx,1 < VA,1 and vx,2 < VA,2. We illustrate this in the parameter space shown in the upper right-hand panel of figure 22.2. Region 1 velocities are plotted horizontally and region 2 velocities vertically. The two regions for which these conditions hold are shaded. The polarizations of the perturbations in the remaining four equations correspond to the fast and slow waves and the entropy wave. There must be four and only four outgoing waves of these types. There is always only one outgoing entropy wave on the downstream side. There must, thus, be three outgoing magnetosonic waves. If we denote the group velocities of the fast and slow waves by subscripts ‘f ’ and ‘s’, respectively, there are three and only three such waves when (i) vx,1 > Vf,1 and Vf,2 > vx,2 > VA,2 or (ii) when VA,1 > vx,1 > Vs,1 and Vs,2 > vx,2 . The first of these conditions defines a fast shock wave and the second defines a slow shock wave. The upper left-hand panel of figure 22.2 shows the regions of the parameter space in which these conditions hold. For the shock to exist, the conditions for two outgoing Alfv´en waves and three outgoing magnetosonic waves must hold simultaneously. The regions in the parameter space for which this is so are shown in the remaining four panels, representing the four possible cases that arise depending on whether the Alfv´en speed is greater or less than the slow-wave speed in each of regions 1 and 2. Shocks for which these conditions hold are called evolutionary, since a small disturbance does not lead to disruption of the shock. The conditions on either side of the shock cannot correspond to other regions of this parameter space. 22.4.2 Perpendicular and parallel shocks Shocks may also be classified according to the direction of the magnetic field. Suppose that the magnetic field is parallel to the plane of the shock so that Bn = 0. Then, from (6.30), the tangential component of velocity is the same on either side Copyright © 2005 IOP Publishing Ltd.

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446

V2

V2

Vf 2

VA 2

Vs2

Vs1

Vf 1

V1

V2

VA 1

V1

V2

Vf 2 VA 2

Vf 2 Vs2

Vs2

VA 2

Vs1

VA 1 Vf 1

V1

V2

VA 1 Vs1 Vf 1

V1

V2

Vf 2 VA 2

Vf 2 Vs2

Vs2

VA 2

VA 1 Vs1 Vf 1

V1

Vs1

VA 1 Vf 1

V1

Figure 22.2. Conditions for an evolutionary shock to exist.

of the boundary, in which case we may transform to a frame in which it is zero, so that the velocity of the plasma is perpendicular to the magnetic field on either side of the boundary. This is called a perpendicular shock. There is no variation in the y- and z-directions. Therefore, in a perpendicular shock, P is replaced by P + B 2 /2µ0 in the momentum equation and the adiabatic Copyright © 2005 IOP Publishing Ltd.

Propagation of MHD waves through shocks equation of state is replaced by   d γP B2 B2 = P+ + = VA2 + VS2 . dρ 2µ0 ρ µ0 ρ

447

(22.33)

If, however, the magnetic field is perpendicular to the plane of the shock, then we can transform to a coordinate frame in which the velocity is also perpendicular to the plane. Since V and B are parallel, this is called a parallel shock. In this case, since Bn is continuous across the shock, it behaves as if there is no magnetic field. However, the conditions for its existence, shown in figure 22.2, still apply. If the conditions for a perpendicular or parallel shock are approximately met, then, as a first approximation, it may be treated as a parallel or perpendicular shock. The corrections to this may be introduced as a second approximation. In such cases, we refer to a quasiparallel or quasiperpendicular shock. The Earth’s bow shock has an orientation that changes over its surface. Depending on solar wind conditions, different parts of its surface may correspond to quasiparallel or quasiperpendicular conditions.

22.5 Propagation of MHD waves through shocks The propagation of MHD waves through shocks has been studied for a wide variety of conditions in a series of papers by McKenzie and Westphal [129–132, 246]. In this outline, we follow their treatment [132]. 22.5.1 Boundary conditions Suppose that an harmonic wave is incident on a shock. The spatial and temporal behaviour of the incident wave and its transmitted and reflected components are of the form exp{−i(ωt − k x (k y , k z , ω)x − k y y − k z z]}. We regard ω, k y , and k z as specified. Then k x may be found from the dispersion relation for the moving medium. Snell’s law requires that k y and k z are the same for all incident, transmitted, and reflected waves on either side of the boundary. The boundary conditions to be applied are obtained by linearizing the de Hoffmann–Teller relations. They are a little more complicated than the linearized conditions (22.26) to (22.32) in order to allow for spatial variation parallel to the shock front. It is convenient to replace the energy condition (6.20) by the Hugoniot condition, which follows from it when shock conditions apply. The linearized equations following from (6.19), (6.22), (6.23), and (6.26) may then be written in the form [ρδvn + vn δρ] = 0   · δ B t − Bn δ Bn B t 2 =0 δp + 2ρvn δvn + vn δρ + µ0 Copyright © 2005 IOP Publishing Ltd.

(22.34) (22.35)

448

Shock waves   Bn δ B t + B t δ Bn ρvn δv t + v t (ρδvn + vn δρ) − =0 µ0 [vn δ B t + B t δvn − Bn δv t − v t δ Bn ] = 0

(22.36) (22.37)

while the linearized form of the Hugoniot condition may be found from 22.10:  (Bt,2 − Bt,1)2 δρ2 (γ − 1)P2 + (γ + 1)P1 + (γ − 1) µ0  (Bt,2 − Bt,1)2 = δρ1 (γ + 1)P2 + (γ − 1)P1 + (γ − 1) µ0 − δ P2 {(γ − 1)ρ2 − (γ + 1)ρ1 } − δ P1 {(γ + 1)ρ2 − (γ − 1)ρ1 } Bt,2 − Bt,1 (δ Bt,2 − δ Bt,1 ). (22.38) + 2(γ − 1)(ρ2 − ρ1 ) µ0 In these equations, the zero-order quantities are related by (6.19), (6.22), (6.23), (6.26), and (22.10). These linearized boundary conditions are expressed in terms of vector components perpendicular and parallel to the perturbed shock. They need to be written in the Cartesian coordinates in which the unperturbed shock normal is in the x-direction and the unperturbed velocities are in the x zplane. Assume that, as a consequence of the perturbation, the shock acquires a perturbation δx given by f (y, z) ≡ x − δξ ei{k y y+kz z} = 0

(22.39)

and varying with time as exp {−iωt}. The unit vector nˆ perpendicular to the shock is the direction of the gradient of f : nˆ =

∇f  xˆ − { yˆ ik y + zˆ ik z }δξ ei{k y y+kz z} |∇ f |

(22.40)

since, to zero-order in δξ , |∇ f |  1. The velocity of the perturbed shock is δu. Its x-component is δu x = −iωδξ . Its y- and z-components will cancel in the equations that result from the analysis. The fluid velocity v relative to the shock is then v + δv − δu. Components normal and tangential to the perturbed shock can be found using the unit vector normal to the shock, which may be written nˆ = xˆ + { yˆ k y /ω + zˆ k z /ω}δu x ei{k y y+kz z} . The scalar product of nˆ with the perturbed fluid velocity shows that   k z vz (22.41) vn + δvn = vx + δvx − 1 − δu x ω   k y vx k z vz v t + δv t = zˆ vz − xˆ δu x + ˆy δv y − δu x − δu y ω ω   k z vx δu x − δu z . (22.42) + zˆ δvz − ω Copyright © 2005 IOP Publishing Ltd.

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Similarly Bn + δ Bn = B x + δ B x +

k z Bz δu x ω 

k y Bx k z Bz δu x + ˆy δ B y − δu x B t + δ B t = zˆ Bz − xˆ ω ω   k z Bx δu x . + zˆ δ Bz − ω

(22.43) 

(22.44)

These may be used to express (22.34), (22.35), (22.36), and (22.37) in terms of the Cartesian components. Note that, when this is done, because jx and Bx are continuous across the boundary, the transverse components δu y , δu z of the perturbed shock velocity cancel. Also, the equations found for the x-components of δv t and δ B t are identically zero, as a result of (6.23) and (6.26). The perturbations on either side of the boundary are the resultant fields of the incident, reflected, and transmitted waves. These take the form of magnetosonic and transverse Alfv´en waves, which are isentropic, and an entropy wave with perturbations only in the density and entropy. The thermodynamic state of the plasma on either side of the shock relates the entropy perturbation to the state variables P and ρ through (1.17), which may be solved for δρ on either side of the shock: ρ1,2 δρ1,2 = δ P1,2 − δς1,2 (22.45) γ P1,2 where we normalize the entropy per unit mass so that δς ≡

m(γ − 1) δs. K

(22.46)

It is convenient to substitute for δρ in the linearized boundary conditions. The terms in δs can then only arise as a result of the entropy wave, while the terms in the other perturbations can only arise from the magnetosonic and Alfv´en waves. When all these substitutions have been made, (22.34)–(22.38) may be written in matrix form: (Tαβ )2 (δqβ )2 − (Sα )2 δς2 + (Rα )δu x = (Tα,β )1 (δqβ )1 − (Sα )1 δς1

(22.47)

where α = 1, 2, . . . , 7, β = 1, 2, . . . , 7 so that T is a 7 × 7 matrix and q, S, and R are seven element column matrices. In these coordinates, the elements of the column q are qβ = {δ P, δvx , δv y , δvz , δ Bx , δ B y , δ Bz } Copyright © 2005 IOP Publishing Ltd.

(22.48)

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Shock waves

and the elements of the matrices found by McKenzie and Westphal [132] are   ρvx /γ P ρ 0 0 0 0 0  1 + ρv 2 /γ P 2ρvx 0 0 −Bx /µ0 0 Bz /µ0  x     0 0 ρv 0 0 −B /µ 0 x x 0    Tαβ =  0 ρv −B /µ 0 B /µ 0 ρv z x z 0 x 0     0 0 −B 0 0 v 0 x x     0 −Bx −vz 0 vx 0 Bz T71 0 0 0 0 0 T77 (22.49) where the quantities defining the elements for α = 1, . . . , 6 are given the subscript 1 or 2 as appropriate, and   (γ − 1) (Bz,2 − Bz,1 )2 P2 + (γ − 1)P1 (1) T71 = (γ + 1) ρ1 − ρ2 − ρ1 γ P1 γ µ0 P1 (22.50)   2 (γ − 1) (Bz,2 − Bz,1 ) (γ − 1)P2 + P1 (2) = (γ + 1) ρ1 + ρ2 − ρ2 T71 γ P2 γ µ0 P2 (22.51) Bz2 − Bz1 (1) (2) T77 = T77 = −2(γ − 1)(ρ2 − ρ1 ) . (22.52) µ0 The column matrix Sα is Sα = {vx , vx2 , 0, vx vz , 0, 0, S7 }

(22.53)

where the appropriate subscripts are used for α = 1, . . . , 6 and (1)

(Bz,2 − Bz,1 )2 µ0 (Bz,2 − Bz,1 )2 = (γ + 1)P1 + (γ − 1)P2 + (γ − 1) . µ0

S7 = (γ − 1)P1 + (γ + 1)P2 + (γ − 1) (2)

S7

(22.54) (22.55)

The evaluation of the column matrix Rα needs some care. Note that [δu x ] = 0, since u x is the velocity of the boundary itself, so that any constant coefficient or coefficient that is continuous through the boundary does not contribute. It is also necessary to examine the coefficients carefully as they involve the change of quantities across the boundary and this change may be zero if the zero-order boundary conditons are taken into account. The result is   −[ρ(1 − k z vz /ω)]   0     ρv [v ]/ω −k y x x   2 − v 2 ) − (B 2 − B 2 )/µ ]/ω  . (22.56) Rα =  k [ρ(v 0 x z x z   z   0     −[Bz ] 0

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22.5.2 Behaviour of the waves on either side of the boundary The group velocity of the incident wave in the shock frame must be directed towards the shock; that of the reflected and transmitted waves must be directed away from it. The results illustrated in figure 22.2 show that, for a fast shock, all the waves on the upstream side are ingoing, while, on the downstream side, the fast wave that travels upstream in the rest frame of the plasma is ingoing, while the other six waves—one fast wave, two Alfve´ n waves, two slow waves, and an entropy wave—are outgoing. For a slow shock, conditions differ according to whether, in the plasma rest frame, the normal component of the Alfve´ n wave group velocity is greater than or less than that of the slow wave. There are, however, six outgoing waves in each case. The incident wave could be an MHD wave incident from the upstream or downstream side, according to the type of shock. Its perturbation would form part of the perturbation qα(2) if it is on the upstream side or qα(1) if it is on the downstream side of the shock. It might also be an entropy wave, in which case it would represent the whole perturbation δς1 : this perturbation is zero if there is no incident entropy wave. The quantities in the column matrix (22.48) are the sums of any incident MHD wave and the perturbations arising from the five outgoing (1) Alfve´ n and magnetosonic waves, which contribute only to qα if it is a fast shock. For a slow shock, outgoing fast and Alfve´ n waves may contribute to qα(2) or qα(1) according to the properties of the shock as shown in figure 22.2. Each MHD wave corresponds to a particular root k x of the dispersion relation (22.23). As described in section 22.3.3, we can find the ratio between the pressure, velocity, and magnetic field perturbations of each component wave. These perturbations may each be written as a column matrix multiplied by the amplitude of the component. The sum of these perturbations for the incident ingoing wave and the five outgoing MHD waves may be used to replace the columns qα(2) (1) and qα in (22.47). This matrix equation constitutes seven simultaneous linear algebraic equations for seven unknowns—the five unknown amplitudes of the outgoing MHD waves, the amplitude δς2 of the outgoing entropy wave, and the x -component δ u x of the perturbed velocity of the shock boundary. These provide a formal solution of the problem. Space does not allow a systematic treatment of all the possible cases. There are some special cases which allow some simplification but even in these the treatment is long and laborious. If the wave normal is in the same plane as the magnetic field and plasma velocity, the dispersion relations describing the transverse Alfv´en and magnetosonic waves separate and there is considerable simplification [131, 246]. If the Alfv´en speed is much less than the sound speed, the shock may be treated as a simpler fluid shock [130], with the magnetic field being frozen into the fluid but not contributing significantly to the dynamics. This may sometimes be sufficient for waves passing through the Earth’s bow shock. A topic that has not been covered here is the change in amplitude undergone by waves passing through shocks. The methods of chapter 21 can provide Copyright © 2005 IOP Publishing Ltd.

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Shock waves

estimates of the energy flux of the outgoing waves and compare them to that in the incident wave. Calculations in special cases [131, 246] indicate that there can be considerable amplification. Such processes are expected to be important in the solar wind and across the bow shock.

22.6 Summary •

•

• •

•

•

The de Hoffmann–Teller relations specify limits on how MHD variables can change at a plasma discontinuity. In a shock there must be flow through the discontinuity and the normal component of the mass flux must be continuous. The relations determine the change in the thermodynamic variables pressure and density, and all other thermodynamic variables are determined by these. For an unmagnetized gas, we can deduce a relationship between pressure and density, which define a contour in the Pρ-plane called the Hugoniot relation or shock adiabatic. For an MHD medium, the analogous relation also depends on the magnetic field. The second law of thermodynamics places constraints on the nature of the changes that take place across the shock. In particular, if we follow a given mass of gas through the shock, its entropy must increase. As it passes through the shock, it remains on the Hugoniot contour. It is, therefore, constrained to lie on those parts of the contour for which the entropy on the downstream side is greater than on the upstream side. In an unmagnetized gas, this leads to Zempl´en’s theorem that the pressure and density both increase as the gas passes through the shock. This theorem can be generalized for an MHD medium. A wave incident on a shock constitutes a small perturbation of the boundary conditions. We linearize the boundary conditions to find the boundary conditions applicable to waves. For a given direction of propagation, seven waves can exist in the medium on each side of the shock. In the rest frame of the medium, these are pairs of oppositely propagated fast waves, slow waves, and Alfv´en waves, and an entropy wave, which is a disturbance fixed in the medium. There are seven independent boundary conditions. One determines the perturbation of the velocity of the shock front. To satisfy the remaining six, there must be six and only six outgoing waves excited by a small perturbation. The conditions require that these are two transverse Alfv´en waves, three magnetosonic waves, and one entropy wave. Any other propagation conditions on either side of the shock cannot satisfy the boundary conditions, implying that a small perturbation disrupts the shock. These conditions allow shocks to be classified as fast or slow, depending on the relative magnitudes of the fast, slow, and Alfv´en speeds on either side of the boundary.

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Summary •

•

453

Shocks are also classified as perpendicular or parallel. In a perpendicular shock, the magnetic field is parallel to the plane of the shock and we can transform to a frame in which the velocity is perpendicular to the shock front and the magnetic field. If the magnetic field is perpendicular to the shock plane, we can transform to a frame in which the velocity is also perpendicular to the plane and is, hence, parallel to the magnetic field. This is called a parallel shock. When these conditions are approximately true, we have, respectively, quasiperpendicular and quasiparallel shocks. If we allow for the possibility of incident, reflected, and transmitted waves, we can write the boundary conditions as seven simultaneous algebraic equations relating the perturbations on either side of the boundary. The 7 × 7 matrix of the coefficients serves as a generalized matrix of transmission and reflection coefficients.

Copyright © 2005 IOP Publishing Ltd.

Chapter 23 Magnetohydrodynamic instability

23.1 Introduction There are many situations in which a mechanical system is in equilibrium but unstable. The simplest to visualize is a ball balanced on a convex surface. A small displacement causes it to roll down the hill. The initial state is unstable. In general, any equilibrium state is unstable if a small displacement reduces the potential energy. Conservation of energy requires that the kinetic energy of its components must increase and the system moves from the equilibrium. Plasma instabilities are of two types. One type is a macroscopic instability, arising from the configuration of the plasma in space, analogous to the ball at the top of the hill. The other is a microscopic instability, arising from the nature of the plasma distribution function. The plasma is in a quasi-equilibrium if the zero-order distribution function is a function only of the constants of the motion. It is unstable if a small displacement from this quasi-equilibrium causes the distribution function to change irreversibly from the quasi-equilibrium. In this chapter, we deal with MHD macro-instabilities. Instability is inherently nonlinear. A small perturbation can be described by a linear approximation to the nonlinear MHD equations. If there is instability, the perturbation grows until the linear approximation is no longer valid. The consequence may be behaviour such as turbulence which is outside the scope of this book. The disturbance may, however, remain linear if there is another mechanism that removes energy from the system as fast as it is produced: a loss mechanism with losses dependent on the amplitude of the disturbance can lead to a stable perturbation that grows to an amplitude such that growth and loss are balanced. We shall be concerned only with the initial linear stages of the instability. An investigation of whether a perturbation leads to growth or to a return to the equilibrium state can be carried out by this means. 454

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Nature of instability B A

455

C

Figure 23.1. A sphere in ( A ) stable, (B ) unstable, and (C ) neutral equilibrium.

23.2 Nature of instability 23.2.1 Growth of a spatial perturbation The equilibrium states and steady-state flows described in chapter 6 may, or may not, be stable. The question of their stability or otherwise has not yet been considered. The fact that a dynamical system has achieved a state in which the net force on each part of it is zero is no guarantee that a small perturbation of the system will not lead to a disruption of the state. The simple example usually adduced is that of a small sphere constrained to move on a surface under gravity, as illustrated in figure 23.1. Suppose it rests in a hollow, as shown at A. Then, if it is displaced slightly and released, it will oscillate about its equilibrium position and, if there are any frictional losses, it will settle again into the equilibrium state. It is then said to be in stable equilibrium. If, with great care, it has been balanced at the top of a hill, as shown at B , then any small displacement will cause it to roll down the hill, picking up energy as it does so. Its initial state is then one of unstable equilibrium. Finally, if it is on a flat surface, as shown at C , any displacement will move it to a new equilibrium position. It is then said to be in neutral equilibrium. To investigate the instability of a fluid, such as a plasma, the perturbation is imagined to be periodic in space, of the form exp{ik · r}. The subsequent behaviour is deduced by finding ω from the dispersion relation. If we assume that there are no losses, then the coefficients in the dispersion relation are real. If the plasma is stable, then ω is real, and the perturbation takes the form of a linear wave propagated as exp{−i[ωt − k · r]}. This corresponds to the oscillatory behaviour of the ball in the hollow. If, however, ω is imaginary or complex, then, if its imaginary part χ is positive, its dependence on t includes a factor exp{χt}. This grows without limit, so that the energy of the associated wave grows rapidly with time. Neutral equilibrium is not of much importance but, for example, a perturbation of a stationary medium, consisting of a spatially periodic variation in density without any change in the other parameters, would represent another equilibrium state. This would be a neutral equilibrium. In a frame of reference in which the plasma is moving, this perturbation would represent an entropy wave. Of course, such a treatment can only indicate the existence of the instability. Since the dispersion relation is based on a linear approximation to the MHD equations, which assumes small disturbances, any wave whose amplitude grows in time will soon violate the assumption of linearity. The analysis can only describe the initial stages of the instability. Nonlinear effects rapidly take over. Copyright © 2005 IOP Publishing Ltd.

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We have mentioned such nonlinear effects in our discussion of quasi-equilibria in sections 4.7 and 4.8. This book is, however, restricted to linear theory and we will not pursue this further.

23.2.2 Convected and non-convected instability If we choose to perturb the plasma in space and follow its subsequent development in time, it is clear that a complex value of ω with a positive imaginary part leads to growth. The situation is not so clear if we choose to provide a temporal sinusoidal perturbation at some point, or on a boundary plane, and follow the development in space. For simplicity, consider the one-dimensional case. A complex value of k for a perturbation of frequency ω does not necessarily imply instability. It may correspond to an evanescent or inhomogeneous wave such as is described in section 10.3.4. In such a wave, there is no instability. An evanescent wave grows exponentially in one direction. Its unlimited growth depends on out assumption of an infinite uniform medium. The evanescent wave is not confined to a limited region of space. In reality, the medium is not infinite and the growth is limited by the boundary conditions. No energy is propagated in the direction of growth. The decision as to whether a complex value of k represents instability or evanescence requires the consideration of waves limited in space and time. These may be represented by wavepackets as described in chapter 9, rather than infinite plane waves. Such wavepackets represent the propagation of energy. In an infinite uniform medium, a Fourier synthesis representing a wavepacket cannot contain evanescent components as they grow without limit in one direction. Only if the wave is limited to a half-space, in which the signal is specified on a boundary, can an angular spectrum representation contain evanescent waves that decay with distance from the boundary. In the case of instability, several possibilities can arise when we construct the wavepacket from harmonic plane waves obeying the dispersion relation: • •

•

The wavepacket is propagated unchanged, except for dispersion effects, which do not occur in MHD. There is no instability. As the wavepacket is propagated relative to some observer, the amplitudes of its component waves increase. There is an increasing energy flux as the signal picks up energy from the instability. In the rest frame of the wavepacket, there is steady growth until the assumption of linearity breaks down. At a fixed point in the frame of the observer the wavepacket is seen to have a particular amplitude as it passes. At successive points along its path, this amplitude increases. This corresponds to amplification of the wavepacket and is an example of a convected instability. If the disturbance at each point in space grows steadily until nonlinearity occurs, then we have a non-convected instability; i.e. a fixed portion of the plasma experiences an unstable oscillation.

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Fluid instabilities

457

By using the dispersion relation to map the real ω-axis to the complex k plane and vice versa, it is possible to provide criteria that determine whether a disturbance with complex k is evanescent, a convected instability, or a nonconvected instability. We need to recognize the existence of such behaviour but we shall not need to use these criteria, which are discussed, for example, by Sturrock [208, 210], Knox [113], or Clemmow and Dougherty [49, ch 6]. 23.2.3 Macroscopic and microscopic instability Instabilities arise because the system is in unstable equilibrium. A small perturbation of the state leads to a decrease in potential energy of the system so that its oscillation or wave energy increases (the ball on a hill in figure 23.1). There must be an inhomogeneity in the system so that there is available potential energy in neighbouring states of the equilibrium. Such inhomogeneity may exist in configuration space as a consequence, for example, of gradients in velocity. Instabilities arising from such causes depend on the macroscopic properties of the plasma. Since MHD is a fluid theory, all MHD instabilities must be of this type. Even when conditions for the MHD approximation apply, we cannot always ignore the effects of anisotropies in the particle distribution. In such cases, we can get an interaction of a subset of the particles with an MHD disturbance through some form of resonance. The microscopic properties of the particle distribution determine the nature of the instability. We have already seen in chapter 8 how such effects can lead to damping of waves. We need to examine how they can lead to growth.

23.3 Fluid instabilities There are some fluid instabilities that occur in any fluid, whether it is an MHD medium or not. Examples are the Rayleigh–Taylor instability, in which a denser fluid floats above a less dense fluid, or the instability of a fluid heated from below, leading to convective mixing. Both involve a gravitational field. In the solar– terrestrial system, they are only important at the surface of the Sun, where the magnetic field plays only a subsidiary role. We shall not treat them further here. An extensive treatment is provided, for example, by Chandrasekhar [35], who also discusses a number of MHD instabilities.

23.4 The Kelvin–Helmholtz instability 23.4.1 Physical basis The Kelvin–Helmholtz instability occurs in a variety of guises as a consequence of the relative motion of two fluids separated by a common boundary. The presence of a gravitational or magnetic field, the contribution of surface tension, or the compressibility or incompressibility of the fluids may play a part. In Copyright © 2005 IOP Publishing Ltd.

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x

V A B

V y Figure 23.2. Physical basis of the Kelvin–Helmholtz instability. The dotted line represents the unperturbed boundary and the chain curve the perturbed boundary. The full curves with arrow heads are streamlines of the perturbed plasma flow. The plasmas above and below the boundary have oppositely directed velocities. The diagram exaggerates the scale perpendicular to the boundary in the x-direction.

solar–terrestrial physics, the fluids are generally MHD media and gravitation is unimportant. In its simplest form then, the instability occurs at a tangential discontinuity between two counterstreaming fluids. The instability can easily be observed when lying in the bathtub, by blowing across the surface of the water to generate waves. The physical basis of the instability can be understood by considering figure 23.2. This shows two counterstreaming fluids, having unperturbed velocities of magnitude V , on either side of a boundary, represented in its unperturbed form as a dotted line. The boundary is given a small sinusoidal perturbation represented by the chain curve. The resulting perturbed velocity field v is represented by the streamlines shown as full curves with arrow heads. If B = 0, the energy conservation equation (2.93) in the steady state may be written in the form    γP 1 2 ρv + v . (23.1) ∇· 2 γ −1 If the fluids are incompressible, then γ → ∞, ρ is constant, and ∇ · v = 0. Then the equation may be written in the form v · ∇{ 12 ρv 2 + P} = 0

(23.2)

which shows that P + 12 ρv 2 is constant along a field line. This is, of course, Bernoulli’s well-known theorem, in the absence of gravity. Consider a small element of fluid located on the boundary between the points A and B in figure 23.2. The continuity equation requires that the fluid at A has a larger speed, and the fluid at B a smaller speed, than that of the unperturbed Copyright © 2005 IOP Publishing Ltd.

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plasmas. Bernoulli’s theorem therefore requires that the pressure at A is reduced and that at B is increased. The fluid element, therefore, experiences a net force in the positive x-direction, increasing the perturbation of the fluid and so driving the instability. If the fluid is compressible, then Bernoulli’s theorem no longer holds because ∇ · v = 0. Now the density does not remain constant. The density in regions such as A tends to increase and, in regions such as B, it tends to decrease. The continuity condition requires that the divergence of the momentum density ρv is zero so that, if ρ increases with decreasing cross-sectional area, v does not increase so much as in the incompressional case. The consequence is that the pressure does not decrease at A to the same extent as for an incompressible fluid. This leads to a reduction in the instability. In extreme cases, it may be quenched. The imposition of a tangential magnetic field in the z-direction, perpendicular to the direction of the velocity, has no effect on the instability. It is frozen into the plasma. The perturbation is in the x-direction and does not change the magnitude or direction of the field. A component of magnetic field in the direction of V , however, does have an effect. The effect of the perturbation is to bend the field lines. The resulting Maxwell stress associated with curvature or field-line ‘tension’ provides a restoring force towards the centre of curvature and, hence, a stabilizing effect.

23.4.2 Sharp boundary between two counterstreaming MHD media In MHD media, the instability can appear in a variety of forms. Kivelson and Pu [108] provide a good review that goes beyond the treatment here. Southwood [191] first studied the case of two compressible MHD media, separated by a sharp boundary. Because of the number of parameters specifying the problem, it is complicated. The magnetic fields, pressures, densities, and streaming velocities on either side of the boundary can be specified subject only to the equilibrium condition that the generalized pressure must be continuous across the boundary. Pu and Kivelson [161, 162] provide an extensive survey for conditions representative of the magnetosphere boundary. We can find an appropriate dispersion relation for the Kelvin–Helmholtz instability at a sharp boundary by using the arguments of section 21.4. Equations (21.33) and (21.34) show the boundary conditions to be applied to the generalized pressure and the normal displacement when a magnetosonic wave is incident on the boundary. The x-axis is normal to the boundary and the boundary is the x = 0 plane. Medium 1 is the medium in the region x < 0 and medium 2 in the region x > 0. The superscript + corresponds to a wave which is propagated in the positive x-direction or which decays with increasing x if it is evanescent. The superscript − corresponds to the opposite case. In this case, the amplitude of the incident wave, the frequency, and y- and z-components of the wavevector are regarded as given. The x-component of the wavevector on either side of the Copyright © 2005 IOP Publishing Ltd.

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boundary is then found from the dispersion relation and the equations can be solved for the amplitudes of the reflected and transmitted waves. Suppose now that we set the amplitude of the incident wave equal to zero. Then (21.33) and (21.34) become kx 1

1

ψ1 = ψ2 kx 2 ψ1 = ψ2

(23.3)

2

(23.4)

where we have substituted for vx−1 and vx+2 from (21.36) and (21.39) and dropped the superscripts + and −. This is a pair of homogeneous linear simultaneous equations for ψ1 and ψ2 . It is assumed that the the roots k x 1 and k x 2 are chosen so that the waves decay evanescently away from the boundary or are propagated away from the boundary. The equations have a self-consistent solution if the determinant of the coefficients is zero: kx 1





2 − k x 2  1 = 0

(23.5)

where k x2 = , and is given by (21.31). This is the dispersion relation of a surface wave. If the waves on either side of the boundary are evanescent, the surface wave is propagated along the boundary. It may be that one or both of ψ1 and ψ2 correspond to waves that are propagated away from the boundary. This is possible as there is a source of free energy at the boundary arising from the counterstreaming plasmas, as described in chapter 21. It is the former case that we consider here. Note that (23.5) may be obtained by setting the denominator of (21.43) or (21.44) equal to zero. In chapter 21, we assumed that medium 1 was at rest, and medium 2 was moving with a Doppler-shifted frequency ω0 . If it is moving, we simply replace the frequency ω by its Doppler-shifted value ω1 = ω − k · V 1 and we use ω2 for the Doppler-shifted frequency in medium 2. 23.4.2.1 Neutral gas As a simple example, suppose that the tangential discontinuity exists between two neutral gases of the same density, streaming in the positive and negative ydirections with speed V . The equilibrium conditions require that the pressures and, hence, the sound speeds are the same. We examine a perturbation exp{ik y z}, such that variation is in the y-direction and k z = 0. In medium 1, ω1 = ω + k y V and, in medium 2, ω2 = ω − k y V . Then 1,2 = ρ0 ω1,2 , and



2 = k x1

Then (23.5) becomes

 ω24

Copyright © 2005 IOP Publishing Ltd.

ω12 VS2

ω12

− k 2y

− k 2y VS2

2 k x2 =



 =

ω14

ω22 VS2

ω22

− k 2y .

− k 2y VS2

(23.6)

(23.7)

The Kelvin–Helmholtz instability

461

which can easily be reduced to a quadratic equation in ω2 with a solution   ' ω2 = k 2y (V 2 + VS2 ) ± VS VS2 + 4V 2 . (23.8) The value of ω2 corresponding to the lower sign is negative if V 2 < 2VS2 and, thus, instability can exist. The growth rate is proportional to k y . If V = 0, we have a stable uniform medium. Even the smallest value of V produces instability. There is a paradox in that the growth rate increases without limit as length scale 1/k y of the perturbation is reduced. For sufficiently large k y , other physical processes, which quench the instability, must be taken into account. Such processes depend on circumstances. For example, for waves on the surface of a bath of water, surface tension effects, depending on the curvature of the surface, increase as the perturbation wavelength gets smaller and quench the instability. The consequence is that there is a wavelength corresponding to maximum growth. Blowing on the bathwater produces waves with a well-defined wavelength corresponding to this maximum growth rate. Blowing harder changes the relative velocities of the fluids and, hence, the wavelength. 23.4.2.2 Effect of a magnetic field component The introduction of a magnetic field parallel to the boundary introduces significant complications. The conditions for equilibrium at the boundary allow for the direction and magnitude of the magnetic field to change across the boundary, provided that the generalized pressure is continuous: P1 +

B12 B2 = P2 + 2 , 2µ0 2µ0

(23.9)

If we use (21.30) and (21.31), the surface-wave dispersion relation (23.5) may be written in the form  ω14 2 2 2 2 2 ρ2 {ω2 − (k t · V A,2 ) } − kt 2 + V 2 ) − (k · V 2 2 ω12 (VA,1 t A,1 ) VS,1 S,1  ω24 2 2 2 2 2 = ρ1 {ω1 − (k t · V A,1 ) } − kt 2 + V 2 ) − (k · V 2 2 ω22 (VA,2 t A,2 ) VS,2 S,2 (23.10) where, ω1 = ω − k · V 1 , ω2 = ω − k · V 2 , and kt is the component of k parallel to the boundary. It is now, in general, a polynomial equation of tenth degree in ω, with real coefficients. Its roots, therefore, are either real or occur in complex conjugate pairs. If a pair of roots is a complex conjugate pair, then one of them has a positive imaginary part and corresponds to instability. The transition from Copyright © 2005 IOP Publishing Ltd.

Magnetohydrodynamic instability

462

two real roots to a complex conjugate pair occurs where two roots are equal: ∂ω =0 ∂ kz

∂ω =0 ∂ky

(23.11)

or, if the angle θ between the y -direction and kt is fixed, ∂ω = 0. (23.12) ∂ kt This corresponds to marginal stability. No analytical solution of algebraic equations of degree greater than four is possible so the most general case must be treated numerically. The investigation of every possible case is beyond the scope of this book. Different situations have been considered by a number of authors [69, 185, 191]. A wide-ranging treatment is that of Pu and Kivelson [161, 162], some √ aspects of which √ are outlined here. Consider (23.5), where k x,1 = ± 1 and k x,2 = ± 2 are the roots of the dispersion relation (21.31). In general, the frequency ω is complex and so k x,1 and k x,2 are complex. The sign of the square root must then be chosen so that the wave decays evanescently away from the boundary. If ω is real, then negative values of correspond to stable surface waves propagated in the y -direction. The case of positive and, simultaneously, ω real was not considered by Pu and Kivelson; however, it has already been treated in section 21.4.4 in the context of over-reflection. Pu and Kivelson [161] have computed the roots of (23.10) numerically for conditions typical of those at the magnetopause. For a given set of conditions, the same initial perturbation produces two surface waves with different values of ω. They chose a reference frame in which the plasma in medium 1 (the magnetosphere) was at rest and that in medium 2 (the magnetosheath) had a velocity V . For each wave, there was a lower cut-off velocity Vc , below which the wave was stable, and an upper cut-off velocity, above which it was stable. Between these two velocities, instability could occur. Figure 23.3 shows how these cut-off velocities depend on the orientation of the vector kt , which describes the initial perturbation. The model was chosen to approximate magnetospheric conditions. The plasma β is relatively small inside the magnetosphere and large in the magnetosheath. The magnetic field is parallel to the velocity in the magnetosheath and perpendicular to it inside the magnetosphere. The number density in the magnetosheath is 10 times that within the magnetosphere. It can be seen that the instability is most easily excited when the perturbation wavevector kt is parallel to the velocity in the magnetosheath and the larger magnetospheric magnetic field is perpendicular to this. Figure 23.4 shows how the normalized growth rate depends on frequency for different angles θ1 .









23.4.3 Effect of finite boundary thickness In general, in the cases studied earlier, for a given set of conditions for which instability occurs, the growth rate increases without limit as kt increases or as Copyright © 2005 IOP Publishing Ltd.

The Kelvin–Helmholtz instability 0*

30*



B1/B2 = 1.5 N2/N1 = 10 A  BA 

B1 C B B V EE B

6 5

60*

4

2.5 kt



2.0

V, B2 VF,U

3

VF,C

1.5

V/VA2

V/VA1

90*

B1



463

1.0

2 VI

VS,U

1

90*

VS,C

60*



30*

0.5

0*

Figure 23.3. Cut-off velocities for instability as a function of orientation of kt : left-hand axis, normalization in terms of Alfv´en speed in medium 1; right-hand axis, normalization in terms of Alfv´en speed in medium 2. Parameters for the model are shown on the diagram. The subscripts ‘F’ and ‘S’ apply to quasifast and slow modes respectively. The subscript ‘I’ refers to an incompressible medium. (Redrawn from Pu and Kivelson [161].)

the wavelength becomes shorter—the growth rate tends to infinity as kt tends to infinity. This is clearly not realistic. The assumption of a sharp boundary implies that the wavelength is large compared with the width of the boundary. When this condition breaks down, the instability is quenched. Ong and Roderick [148] considered the stabilizing effect of a boundary in which the velocity gradient was linear. Walker [231] considered a more general case in which a planestratified boundary layer, in which the parameters of the model could vary arbitrarily subject to the condition that V and B were everywhere parallel to the stratifications, and that the pressure balance condition was maintained. The problem was treated numerically by integrating the differential equations (14.15) and (14.16) through a model of the boundary layer, with the frequency replaced by the local Doppler-shifted frequency ω − k · V . The boundary layer separated two regions 1 and 2. The parameters varied as functions of x. The streaming velocity Copyright © 2005 IOP Publishing Ltd.

464

Magnetohydrodynamic instability  B1/B2 = 1.5 N2/N1 = 10 A  BA 

B1 C B B V EE B



90º

i/ktVA1

60º 30º 30º

90º



30º 60º

 60º 

0

1

V/VA1

2

3

Figure 23.4. Normalized growth rate as a function of normalized streaming velocity. The parameter shown for each curve is the angle θ1 between kt and B 1 . Full curve, quasifast mode; broken curve, quasislow mode; chain curve, incompressible mode. There is no growth for θ1 = 0. (Redrawn from Pu and Kivelson [161].)

V was, of course, like the other parameters, a function of x. The numerical technique used appropriately normalized parameters and proceeded as follows. • • •

• • •

A value of kt was chosen. An initial guess was made at a value of ω. The value chosen was such that ω was complex, corresponding to instability. The dispersion relation in region 1 was solved for k x and the root that decayed away from the boundary chosen. The initial value of the normalized value of the normal displacement ξ was chosen as unity and the corresponding value of normalized generalized pressure ψ found from (14.15) with d/dx ≡ ik x . The equations were integrated using a Runge–Kutta technique, to the centre of the boundary layer where x = 0, yielding a pair of values ξ1 , ψ1 . The process was repeated with the same initial conditions, starting in medium 2, yielding a solution ξ2 , ψ2 . If the value of ω is an eigenvalue, then the two solutions must be the same apart from a multiplying constant. Thus, ξ1 ψ1 − = 0, ξ2 ψ2

Copyright © 2005 IOP Publishing Ltd.

(23.13)

The Kelvin–Helmholtz instability (a)

(b)      

 º 



 

  

kd

º

 

 

º

º

?dV

?d/V0



 

F

465

 

 



G?dVH max

(c)      









Figure 23.5. ' Growth rate for a boundary of finite thickness with constant density: (a) the parameter is VS2 + VA2 /V0 ; B ⊥ V ; kt V ; (b) the parameter is angle between kt and V ; ' VS2 + VA2 /V0 = 5; angle between B and V is 70◦ ; and (c) maximum growth rate as a function of angle between kt and V . The arrow shows where kt B.

•

The left-hand side of (23.13) is a function of ω. Its zeros represent the eigenvalues of ω. The process is repeated with a second guess for ω and these two initial values of the function are used as the starting values of an iterative process using the rule of false position [159] to find the zeros.

Some results of this process are shown in figure 23.5. In the model used, each parameter # varies smoothly and symmetrically about x = 0 such that  #   x < −d   1 1 3  x 3 −d ≤ x ≤ d # = #1 + (#2 − #1 ) 1 + (23.14)  2 2 d  #2 x > d. The generalized pressure  ≡ P + B 2/2µ0 is constant. The ratio of specific heats is taken as γ = 2 which means that the hybrid velocity VA2 + VS2 is also constant. In figure 23.5(a), the magnetic field is at right angles to the streaming velocity and the perturbation vector kt . It will be noted that, instead of the growth rate increasing without limit with kt , the instability is quenched when the scale length of the perurbation is comparable with the boundary thickness. The consequence Copyright © 2005 IOP Publishing Ltd.

466

Magnetohydrodynamic instability

is that there is a value of k corresponding to maximum growth. The graphs are plotted in a frame in which the plasma velocities on either side of the boundary are equal and opposite. In this frame, the real part of ω is zero. In a frame in which one of the plasmas is at rest, the frequency is kV0 as a result of the Doppler shift. The consequence is that there is a frequency that has the maximum growth rate and which will dominate. For magnetospheric conditions, this frequency lies in the Pc4–Pc5 range and could excite field-line resonance. It is, therefore, possible that the Kelvin–Helmholtz instability can produce a single discrete frequency (or, at most, two corresponding to the slow- and fast-wave instabilities) but not a discrete set such as is often observed. Figure 23.5(b) shows the effect of a longitudinal component of magnetic field. The direction of kt that has the largest growth rate is that perpendicular to the magnetic field direction. This is illustrated in figure 23.5(c), which shows that the maximum growth rate peaks when kt ⊥ B rather than when kt V . 23.4.4 Applications in magnetospheric conditions The Kelvin–Helmholtz instability on the magnetopause is certainly a candidate for exciting field-line resonances. It is difficult, however, to see how it could cause a number of discrete frequencies to be observed as is often the case [179, 243]. Nevertheless, it is likely to be of significance wherever there are strong velocity shears. It is probably the cause of vortex structures that have been observed propagated tailwards [181, 182]. In its nonlinear development, it may be of importance in transferring momentum across the magnetopause boundary and, hence, providing an effective viscosity that contributes to driving magnetospheric convection. The investigation of such behaviour involves numerical simulations which are beyond the scope of this book. Such investigations have been extensively pursued by Miura, e.g. [140, 141]. There is strong velocity shear across a field-line resonance, which could lead to Kelvin–Helmholtz instability. The effects of the nonlinear development of such instabilities on auroral arc formation has been investigated by Rankin et al [163–165]. These simulations also show the effects of nonlinear ponderomotive forces [4] that cause a redistribution of plasma along the field line.

23.5 Pressure anisotropy Some MHD instabilities depend on the existence of anisotropic pressure where the conditions are governed by the Parker momentum equation (5.78) and the Chew–Goldberger–Low adiabatic laws (5.85) and (5.86). If the conditions for this formulation of MHD obtain, the occurrence of MHD instabilities is possible. It will be recalled that this requires that there are effective collision processes that ensure that the distribution function remains symmetric in the parallel component of the particle peculiar velocity during the the time of interest. If there are no interactions that allow this, the MHD approximation breaks down and a kinetic Copyright © 2005 IOP Publishing Ltd.

Pressure anisotropy

467

approach must be adopted. We shall consider two examples, the firehose and the mirror instabilities. Let us then return to the study of waves in plasmas in which the pressure is anisotropic as considered in section 7.6. In a coordinate system in which the magnetic field is parallel to the magnetic field and the initial perturbation takes place such that k y = 0, their dispersion relations are given by (7.87) and (7.88): ω4 − ω2 {k 2 VA2 + k x2 V⊥2 + k z2 V2 + 12 k z2 VA2 P } + k z2 V2 {k 2 VA2 + k x2 V⊥2 + 12 k z2 VA2 P } − 14 k x2 k z2 V⊥4 = 0 ω

2

− k z2 {1 + 12 P }VA2

= 0.

(23.15) (23.16)

The first of these corresponds to perturbations in vx , vz , bx , and bz ; and the second corresponds to perturbations in v y and b y only. In section 7.6, we only studied waves for the case where the zero-order pressure anisotropy P was zero. In such cases the dispersion relation had no complex roots and there was no instability. We now consider some cases where the pressure instability is non-zero. 23.5.1 Firehose instability Suppose that the initial perturbation causes the plasma to be displaced in the ydirection only and it varies with x and z. It can be spatially Fourier analysed with respect to x and z. The behaviour in time of the Fourier component having a particular value of k x and k z is governed by the dispersion relation for the ycomponents of v and b (23.16), which provides the corresponding value of ω. This shows that  k z2 B 2 2 2 2 1 (23.17) ω = k z {1 + 2 P }VA ≡ + P⊥ − P . ρ0 µ0 The right-hand side is negative if P > P⊥ +

B2 . µ0

(23.18)

In this case, the frequency has a positive imaginary root and the wave grows in a non-oscillatory manner with a growth rate χ = {|P − P⊥ − B 2 /µ0 |}. Since k x plays no part in the dispersion relation, it can be fixed arbitrarily. The physical mechanism of the instability can be understood by considering the Parker momentum equation (5.78). The transverse perturbation produces only a second-order change in the magnitude of the field perturbations. The gradient term on the right-hand side of the equation is, therefore, zero. As a result of (B.11), the force density arising from the second term may be written in the form  B2 νˆ 0 {P⊥ − P + B 2 /µ0 } (µ ˆ · ∇)µ ˆ = . (23.19) P⊥ − P + µ0 R Copyright © 2005 IOP Publishing Ltd.

Magnetohydrodynamic instability

468

l (P||P^B2/m0)A

(P||P^B2/m0)A

R



B

y

z Figure 23.6. The firehose instability.

The vector νˆ 0 is directed towards the centre of curvature so that when the factor multiplying it is negative the force tends to increase the curvature. This is illustrated in figure 23.6. The upper panel shows a curved portion of a flux tube of cross section δ A. The isotropic pressure, represented by the first term of (5.78), when integrated over the surface, gives a zero net force. The second term of the momentum equation provides a force on the ends of the tube as shown. The component of this force away from the centre of curvature is 

B2 P − P⊥ − µ0



 δ A sin δθ 

B2 P − P⊥ − µ0



δ Aδl . R

(23.20)

This is a negative restoring force if P < P⊥ + B 2 /µ0 and a positive destabilizing force if the condition for instability is met. In the latter case, the force increases the curvature and the disturbance grows in the manner shown in the lower part of the diagram. Copyright © 2005 IOP Publishing Ltd.

Pressure anisotropy

469

23.5.2 Mirror instability 23.5.2.1 Double adiabatic mirror instability The mirror instability occurs for perturbations in the plane containing B and k when k k⊥ . Suppose that the conditions are such that the Chew–Goldberger– Low double adiabatic equations are valid on the time scale of the initial linear growth. This requires that effective collision processes maintain the symmetry of distribution function in v . In this case, (23.15) becomes ω4 − ω2 k x2 (VA2 + V⊥2 ) + k x2 k z2 {V2 (VA2 + V⊥2 ) − 14 V⊥4 } = 0.

(23.21)

The constant term in this quadratic equation for ω2 is small. There are then two approximate roots. One is found by neglecting the constant term, yielding ω2 = k x2 (VA2 + V⊥2 ).

(23.22)

This root merely represents the fast wave for propagation perpendicular to the field. The other root corresponds to a value of ω2 that is small so that the term in ω4 is negligible:  k z2 V2 V⊥4 2 2 2 VA + V⊥ − ω = (VA2 + V⊥2 ) 4 V2  3k z2 P P⊥2 B2 = . (23.23) P⊥ + − 2µ0 6P ρ0 ( P⊥ + B 2 /2µ0 ) The right-hand side is negative when P⊥ +

P2 B2 − ⊥