Turbulence and Diffusion: Scaling Versus Equations (Springer Series in Synergetics)

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Springer Complexity Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems – cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science. Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse “real-life” situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications. Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence. The two major book publication platforms of the Springer Complexity program are the monograph series “Understanding Complex Systems” focusing on the various applications of complexity, and the “Springer Series in Synergetics”, which is devoted to the quantitative theoretical and methodological foundations. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works.

Editorial and Programme Advisory Board ´ P´eter Erdi Center for Complex Systems Studies, Kalamazoo College, USA and Hungarian Academy of Sciences, Budapest, Hungary

Karl Friston Institute of Cognitive Neuroscience, University College London, London, UK

Hermann Haken Center of Synergetics, University of Stuttgart, Stuttgart, Germany

Janusz Kacprzyk System Research, Polish Academy of Sciences, Warsaw, Poland

Scott Kelso Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, USA

J¨urgen Kurths Nonlinear Dynamics Group, University of Potsdam, Potsdam, Germany

Linda Reichl Center for Complex Quantum Systems, University of Texas, Austin, USA

Peter Schuster Theoretical Chemistry and Structural Biology, University of Vienna, Vienna, Austria

Frank Schweitzer System Design, ETH Zurich, Zurich, Switzerland

Didier Sornette Entrepreneurial Risk, ETH Zurich, Zurich, Switzerland

Springer Series in Synergetics Founding Editor: H. Haken

The Springer Series in Synergetics was founded by Herman Haken in 1977. Since then, the series has evolved into a substantial reference library for the quantitative, theoretical and methodological foundations of the science of complex systems. Through many enduring classic texts, such as Haken’s Synergetics and Information and Self-Organization, Gardiner’s Handbook of Stochastic Methods, Risken’s The Fokker Planck-Equation or Haake’s Quantum Signatures of Chaos, the series has made, and continues to make, important contributions to shaping the foundations of the field. The series publishes monographs and graduate-level textbooks of broad and general interest, with a pronounced emphasis on the physico-mathematical approach.

Oleg G. Bakunin

Turbulence and Diffusion Scaling Versus Equations

123

Oleg G. Bakunin Kurchatov Institute Nuclear Fusion Institute 123182 Moskva Russia oleg [email protected]

ISBN: 978-3-540-68221-9

e-ISBN: 978-3-540-68222-6

Springer Series in Synergetics ISSN: 0172-7389 Library of Congress Control Number: 2008929627 c 2008 Springer-Verlag Berlin Heidelberg  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

This book is dedicated with love to my wife and our children Irina, Constantine, and Mikhail

Preface

This book is intended to serve as an introduction to the multidisciplinary field of anomalous diffusion in complex systems such as turbulent plasma, convective rolls, zonal flow systems, stochastic magnetic fields, etc. In spite of its great importance, turbulent transport has received comparatively little treatment in published monographs. This book attempts a comprehensive description of the scaling approach to turbulent diffusion. From the methodological point of view, the book focuses on the general use of correlation estimates, quasilinear equations, and continuous time random walk approach. I provide a detailed structure of some derivations when they may be useful for more general purposes. Correlation methods are flexible tools to obtain transport scalings that give priority to the richness of ingredients in a physical problem. The mathematical description developed here is not meant to provide a set of “recipes” for hydrodynamical turbulence or plasma turbulence; rather, it serves to develop the reader’s physical intuition and understanding of the correlation mechanisms involved. The text, although rich in quantitative analysis, reduces the mathematical discussion to its essentials. The level of presentation is not excessively technical. There is no intention to give a full account of all aspects (many of which are of fundamental importance) of interest in this broad research area; the reader is referred to the many authoritative books already available. Instead, this book tries to capture a lively synthesis to arouse curiosity in readers who are not already professionally involved in this area of turbulence. Turbulent transport theory is a vast and rapidly developing field, and the present volume is by no means complete. I chose for presentation topics that contribute most significantly to an understanding of the basic physics of transport processes and also illustrate a wide variety of mathematical methods that prove useful in turbulent diffusion theory. The text is basically theoretical. However, a number of references to pertinent computer simulation experiments and laboratory experiments are included. The core of the book is a synthesis of several of my review papers and lectures (given at the Moscow State University and at the Nuclear Fusion Institute in Moscow) addressed to students and young researchers interested in undertaking

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research in currently advanced and “hot” areas of turbulence and plasma physics. In these courses, it was assumed that students already had a reasonable background in classical physics, turbulence, and plasma physics but otherwise there were no special prerequisites. The text is divided into four parts. Part 1 provides a brief introduction to the diffusion equation formalism and turbulence phenomenology. The general terminology, methods, and basic equations are summarized in Chaps. 1 and 2. Chapter 3 points out the importance of the integral representation for the description of nonlocal effects. Part 2 provides a farly informative treatment of seed diffusion effects in the framework of the correlation description, quasilinear equations, and scaling. The relationship between Lagrangian and Eulerian correlation functions is discussed in Chap. 4. The quasilinear equations are derived in Chap. 5. Chapter 6 considers anomalous transport in the system of random shear flows. The quasilinear approximation to describe stochastic magnetic field is presented in Chapter 7. Chapter 8 analyzes the problems of relationships between stochastic instability and transport effects in the stochastic magnetic field. The focus of Chapter 9 is the derivation of the effective diffusion coefficient for a system of convective cells. Part 3 is devoted to the percolation description of turbulent transport. Necessary definitions are introduced in Chap. 10. Chapter 11 observes the percolation methods to describe transport in random two-dimensional flows on the ground of the monoscale representation. Chapter 12 deals with the multiscale approach to turbulent transport. The relationships between the transport and correlation exponents are derived. Part 4 analyzes trapping effects in terms of the continuous time random walk concept. Chapter 13 treats the problem of subdiffusive regimes. Chapter 14 discusses nonlocal and memory effects in the framework of the continuous time random walk model. Chapter 15 is devoted to the kinetic (phase-space) approach describing ballistic modes of anomalous transport. Since 1905, an enormous amount of literature on the subject has evolved. The many references provided in this book should not be interpreted as an attempt at a thorough investigation of all the relevant papers related to the various topics covered; thus the history of the results shown is not discussed. Many important articles have probably been missed and others may not be properly emphasized. The references here are meant to provide the reader with a rather rich framework of research papers within which the issues that are only briefly discussed in this book can be found discussed in much greater detail than is possible here. They generally reflect personal experience. In this sense, it should be clear at the outset that the bibliography may be rather incomplete. The author thanks R. Balescu, N. Erochin, G. Golitsin, E. Kusnetsov, V. Lisitsa, D. Morozov, F. Parchelly, T. Schep, V. Shafranov, D. Stauffer, A. Timofeev, E. Yurchenko, Yu. Yushmanov, G. Zaslavsky, and Hugo de Blank for useful discussions and support. Nieuwegein, The Netherlands

O.G. Bakunin

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General References Diffusion Concept Balescu, R. (1997). Statistical Dynamics. Imperial College Press, London. Ben-Avraham, D. and Havlin, S. (1996). Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge, U.K. Gardiner, C.W. (1985). Handbook of Stochastic Methods. Springer-Verlag, Berlin. Mazo, R.M. (2002). Brownian Motion, Fluctuations, Dynamics and Applications. Clarendon Press, Oxford. Montroll, E.W. and Shlesinger, M. F. (1984). On the wonderful world of random walks, in Studies in Statistical Mechanics, 11, 1. Elsevier, Amsterdam. Montroll, E.W. and West, B. J. (1979). On an enriched collection of stochastic processes, in Fluctuation Phenomena. Elsevier, Amsterdam. Pecseli, H.L. (2003). Fluctuations in Physical Systems. Cambridge University Press, Cambridge, U.K. Shiesinger, M.F. and Zaslavsky, G.M. (1995). Levy Flights and Related Topics in Physics. SpringerVerlag, Berlin.

Correlations in Complex Systems Erdi, P. (2008). Complexity Explained. Springer-Verlag, Berlin. Haken, H. (1978). Synergetics. Springer-Verlag, Berlin. Nicolis, J.S. (1989). Dynamics of Hierarchical Systems. An Evolutionary Approach. Springer-Verlag, Berlin. Pekalski, A. and Sznajd-Weron, K., eds. (1999). Anomalous Diffusion. From Basics to Applications. Springer-Verlag, Berlin. Reichl, L.E. (1998). A Modern Course in Statistical Physics. Wiley-Interscience, New York. Schweitzer, F. (2003). Brownian Agents and Active Particles. Springer-Verlag, Berlin. Sornette, D. (2006). Critical Phenomena in Natural Sciences. Springer-Verlag, Berlin. Zeldovich, Ya.B., Ruzmaikin, A.A., and Sokoloff, D.D. (1990). The Almighty Chance. World Scientific, Singapore.

Hydrodynamics and Turbulence Barenblatt, G.I. (1994). Scaling Phenomena in Fluid Mechanics. Cambridge University Press, Cambridge, U.K. Batchelor, G.K., Moffat, H.K., and Worster, M.G. (2000). Perspectives in Fluid Dynamics. Cambridge University Press, Cambridge, U.K. Bohr, T., Jensen, M.H., Giovanni, P., and Vulpiani, A. (2003). Dynamical Systems Approach to Turbulence. Cambridge University Press, Cambridge, U.K. Davidson, P.A. (2004). Turbulence: An Introduction for Scientists and Engineers. Oxford University Press, Oxford. Frisch, U. (1995). Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge, U.K. Frost, W. and Moulden, T.H., eds. (1977). Handbook of Turbulence. Plenum Press, New York. Lesieur, M. (1997). Turbulence in Fluids. Kluwer Academic, Dordrecht. McComb, W.D. (1994). The Physics of Fluid Turbulence. Clarendon Press, Oxford. Monin, A.S. and Yaglom, A.M. (1975). Statistical Fluid Mechanics. MIT Press, Cambridge, MA. Oberlack, M. and Busse, F.H., eds. (2002). Theories of Turbulence. Springer-Verlag, Vienna.

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Peinke, J., Kittel, A., Barth, S., and Oberlack, M., eds. (2005). Progress in Turbulence. SpringerVerlag, Berlin. Pope, S.B. (2000). Turbulent Flows. Cambridge University Press, Cambridge, U.K. Tabeling, P. and Cardoso, O. (1994). Turbulence: A Tentative Dictionary. Plenum Press, New York. Ting, L. Klein, R., and Knio, O.M. (2007). Vortex Dominated Flows. Springer-Verlag, Berlin. Tsinober, A. (2004). An Informal Introduction to Turbulence. Kluwer Academic, Dordrecht. pt

Correlation Functions and Geophysical Turbulence Csanady, G.T. (1972). Turbulent Diffusion in the Environment. D. Reidel, Dordrecht. Cushman-Roisin, B. (1994). Introduction to Geophysical Fluid Dynamics. Prentice-Hall, Englewood Cliffs, NJ. Frenkiel, N.F., ed. (1959). Atmospheric Diffusion and Air Pollution. Academic Press, New York. Nieuwstadt, F.T.M. and Van Dop, H., eds. (1981). Atmospheric Turbulence and Air Pollution Modeling. D. Reidel, Dordrecht. Panofsky, H.A. and Dutton, I.A. (1970). Atmospheric Turbulence, Models and Methods for Engineering Applications. Wiley-Interscience, New York. Pasquill, F. and Smith, F.B. (1983). Atmospheric Diffusion. Ellis Horwood Limited, Halsted Press, New York. Squires, T. and Quake, S. (2005). Reviews of Modern Physics, 77, 986. Zeldovich, Ya.B., Ruzmaikin, A.A., and Sokoloff, D.D. (1990). The Almighty Chance. World Scientific, Singapore.

Plasma Physics and Magnetohydrodynamic Turbulence Balescu, R. (2005). Aspects of Anomalous Transport in Plasmas. IOP, Bristol and Philadelphia. Biskamp, D. (2004). Magnetohydrodynamic Turbulence. Cambridge University Press, Cambridge, U.K. Childress, S. and Gilbert, A.D. (1995). Stretch, Twist, Fold: The Fast Dynamo. Springer-Verlag, Berlin. Dandy, R. (2001). Physics of Plasma. Cambridge University Press, Cambridge, U.K. Horton, W. and Ichikawa, Y.-H. (1994). Chaos and Structures in Nonlinear Plasmas. World Scientific, Singapore. Kadomtsev, B.B. (1976). Collective Phenomena in Plasma. Nauka, Moscow. Kadomtsev, B.B. (1991). Tokamak Plasma: A Complex System. IOP, Bristol. Kingsep, A.S. (1996). Introduction to the Nonlinear Plasma Physics. Moskovskiy FizikoTekhnichesky Institute, Moscow. Mikhailovskii, A. (1974). Theory of Plasma Instabilities. Consultant Bureau, New York. Rosenbluth, M.N. and Sagdeev, R.Z., eds. (1984). Handbook of Plasma Physics. North-Holland, Amsterdam. Tsytovich, V.N. (1974). Theory of Turbulent Plasma. Plenum Press, New York. Wesson, J.A. (1987). Tokamaks. Oxford University Press, Oxford.

Chaos and Mixing Aref, H. and El Naschie, M.S. (1994). Chaos Applied to Fluid Mixing. Pergamon Press, Oxford. Beck, C. and Schlogl, F. (1993). Thermodynamics of Chaotic Systems. Cambridge University Press, Cambridge, U.K. Berdichevski, V. (1998). Thermodynamics of Chaos and Order. Longman, White Plains, NY.

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Dorfman, J.R. (1999). An Introduction to Chaos in Nonequilibrium Statistical Mechanics. Cambridge University Press, Cambridge, U.K. Guyon, E., Nadal, J.-P., and Pomeau, Y., eds. (1988). Disorder and Mixing. Kluwer Academic, Dordrecht. Kuramoto, Y. (1984). Chemical Oscillations, Waves and Turbulence. Springer-Verlag, Berlin. Manneville, P. (2004). Instabilities, Chaos and Turbulence. An Introduction to Nonlinear Dynamics and Complex Systems. Imperial College Press, London. Mikhailov, A. (1995). Introduction to Synergetics, Part 2. Springer-Verlag, Berlin. Moffatt, H.K., Zaslavsky, G.M., Comte, P., and Tabor, M. (1992). Topological Aspects of the Dynamics of Fluids and Plasmas. Kluwer Academic, Dordrecht. Ott, E. (1993). Chaos in Dynamical Systems. Cambridge University Press, Cambridge, U.K. Ottino, J. (1989). The Kinematics of Mixing. Cambridge University Press, Cambridge, U.K. Pismen, L.M. (2006). Patterns and Interfaces in Dissipative Dynamics. Springer-Verlag, Berlin.

Fractals and Percolation Bunde, A. and Havlin, S., eds. (1995). Fractals and Disordered Systems. Springer-Verlag, Berlin. Bunde, A. and Havlin, S., eds. (1996). Fractals in Science. Springer-Verlag, Berlin. Chorin, A.J. (1994). Vorticity and Turbulence. Springer-Verlag, Berlin. Feder, J. (1988). Fractals. Plenum Press, New York. Gouyet, J.-F. (1996). Physics and Fractal Structure. Springer-Verlag, Berlin. Hunt, A. (2005). Percolation Theory for Flow in Porous Media. Springer-Verlag, Berlin (LNP674). Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman, San Francisco. Pietronero, L. (1988). Fractals’ Physical Origin and Properties. Plenum Press, New York. Sahimi, M. (1993). Application of Percolation Theory. Taylor & Francis, London. Schroeder, M. (2001). Fractals, Chaos, Power Laws. Minutes from an Infinite Paradise.W.H. Freeman, New York. Stanley, H.E. (1971). Introduction to Phase Transitions and Critical Phenomena. Clarendon Press, Oxford. Stauffer, D. (1985). Introduction to Percolation Theory. Taylor and Francis, London. West, B.J., Bologna, M., and Grigolini, P. (2003). Physics of Fractal Operators. Springer-Verlag, New York. Ziman, J.M. (1979). Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems. Cambridge University Press, Cambridge, U.K.

The Fokker-Planck Equation and Kinetic Theory Coffey, W.T., Kalmykov, Yu.P., and Waldron, J.T. (2005). The Langevin Equation. World Scientific, Singapore. Haken, H. (1983). Advanced Synergetics. Springer-Verlag, Berlin. Hanggi, P. and Talkner, P., eds. (1995). New Trends in Kramer’s Reaction Rate Theory. Kluwer Academic, Boston. Malchow, H. and Schimansky-Geier, L. (1985). Noise and Diffusion in Bistable Nonequilibrium Systems. Teuber, Leipzig. Risken, H. (1989). The Fokker-Planck Equation. Springer-Verlag, Berlin. Van Kampen, N.G. (1984). Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam.

Contents

Part I Turbulent Diffusion Concepts 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Brownian Motion, Random Walks, and Correlation Scales . . . . . . . . 3 1.2 The Fick Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Diffusion and the Characteristic Velocity Scale . . . . . . . . . . . . . . . . . . 10 1.4 Lagrangian Description of Turbulent Diffusion . . . . . . . . . . . . . . . . . . 13

2

Turbulent Diffusion and Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Correlation Functions and Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Richardson Law and Anomalous Transport . . . . . . . . . . . . . . . . . 2.3 The Kolmogorov Description of Turbulence . . . . . . . . . . . . . . . . . . . . 2.4 Relative Diffusion and Scalings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Cascade Phenomenology and Scalar Spectrum . . . . . . . . . . . . . . . . . .

21 21 23 26 32 34

3

Nonlocal Effects and Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Einstein Functional Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Nonlocality and Levy-Stable Distributions . . . . . . . . . . . . . . . . . . . . . . 3.3 Fractional Derivatives and Anomalous Diffusion . . . . . . . . . . . . . . . . 3.4 The Monin Nonlocal Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 44 47 51

Part II Correlation Effects and Scalings 4

Diffusive Renormalization and Correlations . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Corrsin Independence Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Correlation Function and Anomalous Diffusion . . . . . . . . . . . . . 4.3 Seed Diffusivity and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Effective Diffusivity and the Peclet Number . . . . . . . . . . . . . . . . . . . . 4.5 Diffusive Renormalization and the Correlation Function . . . . . . . . . .

57 57 60 62 64 66

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5

Diffusion Equations and the Quasilinear Approximation . . . . . . . . . . . 5.1 The Taylor Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Advection and Scalar Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Zeldovich Flow and the Kubo Number . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Short-Range and Long-Range Correlations . . . . . . . . . . . . . . . . . . . . . 5.6 The Telegraph Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 75 77 81 83 84

6

Return Effects and Random Shear Flows . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 “Returns” and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Superdiffusion and Return Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Random Shear Flows and Stochastic Equations . . . . . . . . . . . . . . . . . 6.4 The “Manhattan-Grid” Flow and Turbulent Transport . . . . . . . . . . . .

87 87 90 93 95

7

Turbulence of Magnetic Field Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.1 Basic Equations of Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.2 Magnetic Field Evolution and Magnetic Reynolds Number . . . . . . . . 104 7.3 Magnetic Diffusivity and the Quasilinear Approach . . . . . . . . . . . . . . 106 7.4 Stochastic Magnetic Field and Transport Scalings . . . . . . . . . . . . . . . 110 7.5 Diffusive Renormalization and a Braded Magnetic Field . . . . . . . . . . 112

8

Stochastic Instability and Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.1 Stochastic Instability and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.2 Quasilinear Scaling for the Stochastic Instability Increment . . . . . . . 119 8.3 The Rechester-Rosenbluth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.4 Collisional Effects and Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8.5 The Quasi-Isotropic Stochastic Magnetic Field . . . . . . . . . . . . . . . . . . 127

9

Anomalous Transport and Convective Cells . . . . . . . . . . . . . . . . . . . . . . . 131 9.1 Convective Cells and Turbulent Diffusion . . . . . . . . . . . . . . . . . . . . . . 131 9.2 Complex Structures and the Statistical Topography . . . . . . . . . . . . . . 135 9.3 Fluctuation–Dissipative Relation and Turbulent Mixing . . . . . . . . . . . 136 9.4 Bohm Scaling and Electric Field Fluctuations . . . . . . . . . . . . . . . . . . . 138 9.5 Diffusive Renormalization and Correlations . . . . . . . . . . . . . . . . . . . . 141

Part III Fractals and Percolation Transport 10

Fractal and Percolation Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 10.1 Self-Similarity and the Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . 147 10.2 Fractality and Anomalous Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 10.3 Turbulence Scalings and Fractality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 10.4 Percolation Transition and Correlations . . . . . . . . . . . . . . . . . . . . . . . . 157 10.5 Continuum Percolation and Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 160 10.6 Finite Size Renormalization and Scaling . . . . . . . . . . . . . . . . . . . . . . . 163

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11

Percolation and Turbulent Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 11.1 Random Steady Flows and Seed Diffusivity . . . . . . . . . . . . . . . . . . . . 169 11.2 Reorganization of Flow Topology and Percolation Scalings . . . . . . . 174 11.3 Spatial and Temporal Hierarchy of Scales . . . . . . . . . . . . . . . . . . . . . . 178 11.4 Percolation in Drift Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 11.5 Drift and Low-Frequency Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 11.6 Renormalization and the Stochastic Instability Increment . . . . . . . . . 187 11.7 Stochastic Magnetic Field and Percolation . . . . . . . . . . . . . . . . . . . . . . 189

12

Multiscale Approach and Scalings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 12.1 The Nested Hierarchy of Scales and Drift Effects . . . . . . . . . . . . . . . . 193 12.2 The Brownian Landscape and Percolation . . . . . . . . . . . . . . . . . . . . . . 196 12.3 Correlations and Transport Scalings . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 12.4 Diffusive Approximation and the Multiscale Model . . . . . . . . . . . . . . 201 12.5 Stochastic Instability and the Temporal Hierarchy of Scales . . . . . . . 203 12.6 Isotropic and Anisotropic Magnetohydrodynamic Turbulence . . . . . . 204

Part IV Trapping and the Escape Probability Formalism 13

Subdiffusion and Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 13.1 Diffusion in the Presence of Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 13.2 Trapping and Strong Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 13.3 Comb Structures and Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 13.4 Double Diffusion and Return Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 220

14

Continuous Time Random Walks and Transport Scalings . . . . . . . . . . . 223 14.1 The Montroll and Weiss Approach and Memory Effects . . . . . . . . . . 223 14.2 Fractional Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 14.3 Correlation Function and Waiting Time Distribution . . . . . . . . . . . . . 228 14.4 The Klafter Blumen and Shlesinger Approximation . . . . . . . . . . . . . . 230 14.5 Stochastic Magnetic Field and Balescu Approach . . . . . . . . . . . . . . . . 233 14.6 Longitudinal Correlations and the Diffusive Approximation . . . . . . . 235 14.7 Vortex Structures and Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

15

Correlation and Phase-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 15.1 Kinetics and the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 15.2 Phase Space and Transport Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 15.3 The One-Flight Model and Anomalous Diffusion . . . . . . . . . . . . . . . . 247 15.4 Correlations and Nonlocal Velocity Distribution . . . . . . . . . . . . . . . . . 249 15.5 The Corrsin Conjecture and Phase-Space . . . . . . . . . . . . . . . . . . . . . . . 252

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Part I

Turbulent Diffusion Concepts

Chapter 1

Introduction

1.1 Brownian Motion, Random Walks, and Correlation Scales The aim of this chapter is to provide some basic knowledge of the diffusive motion of particles for further use and reference within this book. Diffusion is the random migration of small particles arising from motion due to thermal energy. A particle at absolute temperature T has a kinetic energy related to movement along each axis of kB T /2, where kB is the Boltzmann constant. A particle of mass m and velocity vx on the x-axis has a kinetic energy of mv2x /2. This energy fluctuates; however, on average it is:  2 kB T mvx = . (1.1.1) 2 2 Here, the symbol . . . denotes an average over time or aggregate of similar particles. Such a random migration was first described by the Dutch physician Jan Ingenhousz (1785), who observed that finely powered charcoal floating on an alcohol surface exhibited a highly erratic random motion (see Fig. 1.1). This process was named after the observations of the English botanist Robert Brown (1829), who noted the erratic motion of pollen grains suspended in fluids. Brownian motion in water was experimentally investigated by Perrin [1]. To characterize diffusive spreading, it is convenient to reduce the problem to its barest essentials and treat particle motion along the x-axis (see Fig. 1.2). The particles begin their motion at time t = 0 at position x = 0 and execute random walks. Here, each particles steps to the right or to the left once every τ seconds, moving at velocity ±vx a distance δ = ±vx τ . We consider τ and δ as constants [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The probability of moving to the right at each step is 1/2, and the probability of moving to the left at each step is 1/2. Successive steps are statistically independent. The particles do not interact with one another and move independently of all the other particles. By these suppositions, the particles go nowhere on the average, and their root-mean-square displacement is proportional not to time, but to the square root of the time. Let us establish these propositions by applying an iterative procedure. Consider an aggregate of Np particles. After N steps, the i-th particle will remain at position

O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008 

3

4

1 Introduction

Fig. 1.1 Path of a two-dimensional Brownian motion (Brownian flight)

xi (N), but the position of a particle after the N-th step differs from its position after the (N − 1)th step by ±δ : xi (N) = xi (N − 1) ± δ .

(1.1.2)

A + sign will apply to roughly half of the particles and a – sign to the other half. The mean displacement of the particles after the N-th step is found by summing over the particle index i and dividing by N p : N

x (N) =

1 p ∑ xi (N). Np i=1

(1.1.3)

t

4τ 3τ 2τ

Fig. 1.2 Construction of the one-dimensional random walk in the space–time representation

τ

x −2δ

−δ

δ

2δ

1.1 Brownian Motion, Random Walks, and Correlation Scales

5

Expressing xi (N) in terms of xi (N − 1), we can write the expression: N

x (N) =

N

1 p 1 p [xi (N − 1) ± δ ] = ∑ ∑ xi (N − 1) = x (N − 1). Np i=1 Np i=1

(1.1.4)

The second term in the brackets averages to zero, because its sign is positive for roughly half of the particles and negative for the other half. From this equation, we see that the mean position of the particle does not change from step to step. Since all the particles start at the origin, where the mean position is zero, the mean position remains zero. The spreading of the particles is symmetrical about the origin. A more informative measure of spreading is the root-mean-square displacement 1/2  2 . Because the square of a negative x (N)  number is positive, the result must be finite and cannot be zero. To obtain x2 (N) , we rewrite xi (N) in terms of xi (N − 1) as in Eq. (1.1.2), and take the square: xi2 (N) = xi2 (N − 1) ± 2δ xi (N − 1) + δ 2 .

(1.1.5)

Calculation of the mean yields the relation: 

N  1 p 2 x2 (N) = x (N), ∑ Np i=1 i

(1.1.6)

which is given by: 

N     1 p 2 x (N − 1) ± 2δ xi (N − 1) + δ 2 = x2 (N − 1) + δ 2 . (1.1.7) x2 (N) = ∑ Np i=1 i

The second term in the brackets averages to zero, because its sign is positive for roughly half of the particles and negative for the other half. Because xi (0) = 0 for  all particles i, x2 (0) = 0. Thus, one obtains:      x2 (1) = δ 2 , x2 (2) = 2δ 2 , . . . , and x2 (N) = N δ 2 .



(1.1.8)

Note that the mean-square displacement increases with step number N and the rootmean-square displacement with the square root of N. From the above suppositions we see that the particles execute N steps in a time t = N τ . Hence, N is proportional to time t, the mean-square displacement to t, and the root-mean-square displacement to the square root of t. The spreading increases with square root of the time. Analyzing this more carefully, note that N = t/τ , so that:

2  2  t 2 δ x (t) = t, (1.1.9) δ = τ τ where we write x(t) rather than x(n) to denote the fact that x is now considered a function of t. It is convenient to define a diffusion coefficient in the form

6

1 Introduction

D=

δ2 2τ

in cm2 / sec. This gives us an important relation:  2 x = 2Dt,

(1.1.10)

(1.1.11)

or in terms of the transport scaling:  1/2 = (2Dt)1/2 ∝ t 1/2 . R = x2

(1.1.12)

The diffusion coefficient D characterizes the migration of particles of a given kind in a given medium at a given temperature. It depends on the size of the particle, the structure of the medium, and the absolute temperature (for a small molecule in water at room temperature D ≈ 10−5 cm2 / sec). Displacement is not proportional to time but rather to the square root of the time; therefore, there is no such notion as a diffusion velocity. This is an important result. Trying to define a diffusion velocity by dividing the root-mean-square displacement by time, we obtain the explicit function of the time. Dividing both sides of Eq. (1.1.11) by t, we find:  2 1/2 1/2 x 2D R = = . t t t

(1.1.13)

Thus, the shorter period of observation t corresponds to the larger apparent velocity. For values of t smaller than τ , the apparent velocity is larger than δ /τ = vx , the instantaneous velocity of the particle. This is an unreasonable estimate and we discuss the problem in the next section. The suppositions apply for each dimension. Furthermore,  assert that motions    2we = 2Dt, then y2 = in the x, y, and z directions are statistically independent. If x  2 2Dt and z = 2Dt. In two dimensions, the square of the distance from the origin to the point (x, y) is r2 = x2 + y2 , and therefore:  2 r = 4Dt. (1.1.14) Analogously, for a three-dimensional space, r2 = x2 + y2 + z2 , and  2 r = 6Dt.

(1.1.15)

The definition of the diffusion coefficient (1.1.10) is based on using the notions of the correlation length ΔCOR = δ and the correlation time τCOR = τ . If the values of time and length are smaller than the correlation values, then the motion of particles has a ballistic character, whereas if these values are larger than the correlation scales, we are dealing with the diffusion mechanism (1.1.12). The key problem in investigating the turbulent diffusion is the choice of correlation scales responsible for the effective transport. This is not surprising, because turbulent diffusion models

1.2 The Fick Transport Equation

7

differ significantly from one-dimensional transport models. Often, several different types of transports are present simultaneously in turbulent diffusion. Therefore, accounting for initial diffusivity (seed diffusion), anisotropy, and stochastic instability reconnection of streamlines is important. Moreover, turbulent transport could have a nondiffusive character where the scaling R2 ∝ t is not correct. To describe the anomalous diffusion, it is convenient to use scaling with an arbitrary exponent H: R2 ∝ t 2H ,

(1.1.16)

where H is the Hurst exponent. The case H = 1/2 corresponds to classical diffusion (1.1.12). The values 1 > H > 1/2 describe superdiffusion, whereas the values 1/2 > H > 0 correspond to the subdiffusive transport. The case H = 1 corresponds to the ballistic motion of particles. Calculating the Hurst exponent H and determining the relationship between transport and correlation characteristics underlie the anomalous diffusion theory.

1.2 The Fick Transport Equation The famous Fick equations [8, 9, 10, 11] describing the spatial and temporal variation of nonuniform particle distributions is derived from the model of random walks. If we know the number of particles at each point along the x-axis at time t, then we can find how many particles will move across unit area in unit time from point x to point (x + δ ), and the net flux in the x direction qx . At time (t + τ ), after the next step, half the particles at x will have stepped across the dashed line from left to right, and half the particles at (x + δ ) will have stepped across the dashed line from right to left. The net number crossing to the right will be: 1 − [Np (x + δ ) − Np (x)] . 2

(1.2.1)

Dividing by the area normal to the x-axis, Sa , and by the time interval τ , yields the net flux qx : [N p (x + δ ) − Np (x)] . (1.2.2) qx = − 2Sa τ Simple calculations yield: qx = −



δ 2 1 Np (x + δ ) Np (x) . − 2τ δ Sa δ Sa δ

(1.2.3)

2 In this expression, the quantity δ 2τ is the diffusion coefficient D. The term  Np (x + δ ) Sa τ is the number of particles per unitvolume at the point (x + δ ), i.e., the concentration n (x + δ ), and the term Np (x) Sa τ is the concentration n(x). Then, we can rewrite the relation for the net flux (see Fig. 1.3):

8 Fig. 1.3 Schematic illustration of particle flux due to a concentration gradient

1 Introduction n2

qx = − D

n2 − n1 lx

n1 lx

1 qx = −D [n (x + δ ) − n (x)] . δ

(1.2.4)

It was assumed that the quantity δ is very small. In the limit δ → 0, by the definition of a partial derivative, we obtain the well-known Fick law: qx = −D

∂n . ∂x

(1.2.5)

If the concentration n is expressed as particles/cm3 , then qx is expressed in particles/ cm2 sec. The diffusion equation follows from Fick’s law. This equation shows that the total number of particles is conserved (particles are neither created nor destroyed). Consider the volume of the box Sa δ (see Fig. 1.4). In a period of time τ , qx (x) Sa τ particles will enter from left and qx (x + δ ) Sa τ particles will leave from the right. If particles are neither created nor destroyed, the number of particles per unit volume in the box must increase at the rate: [qx (x + δ ) − qx (x)] Sa τ 1 1 [n (t + τ ) − n (t)] = − = − [qx (x + δ ) − qx (x)] . τ τ Sa δ δ (1.2.6) In the limit τ → 0 and δ → 0 one obtains:

∂n ∂ qx =− . ∂t ∂x

(1.2.7)

After substitution of (1.2.5) we arrive to the classical diffusion equation:

∂n ∂ 2n =D 2. ∂t ∂x

(1.2.8)

This equation states that the time rate of change in concentration is proportional to the curvature of the concentration function with the diffusion coefficient D. We now find the simplest solution of the diffusion equation for a point source. Suppose that at time t = 0, particles of the dye are injected into the water at the rate Qs per sec for an infinitesimal period of time dt. The total number of particles

1.2 The Fick Transport Equation

9

Fig. 1.4 Schematic illustration of particle fluxes through the faces of a thin box

x +δ

x

qx ( x + δ , t )

qx ( x, t )

x

injected is N = Qs dt. With these boundary conditions, the one-dimensional diffusion equation has the Gaussian solution: n (x,t) =

Np

e−x

(4π Dt)

1/2

2 /4Dt

.

For the three-dimensional case, the diffusion equation takes the form:

2 ∂n ∂ n ∂ 2n ∂ 2n 2 = D∇ n(x, y, z,t) = D , + + ∂t ∂ x2 ∂ y2 ∂ z2

(1.2.9)

(1.2.10)

where the particle flux in the xi direction is given by: qi = D

∂n . ∂ xi

(1.2.11)

Here ∇2 is the Laplace operator. In the case of spherical symmetry one obtains:

∂n ∂n 1 ∂ =D 2 r2 . ∂t r ∂r ∂r

(1.2.12)

Then, we find the point source solution in the well-known form: n (r,t) =

Np (4π Dt)

3/2

e−r

2 /4Dt

.

(1.2.13)

This is a three-dimensional Gaussian distribution. The concentration remains highest at the source, but it decreases there as the three-halves power of the time. As observer at radius r sees a wave that peaks at t = r2 /6D. The further extension to flowing fluids is easily accomplished [11, 12, 13] if we merely replace the partial derivative with respect to time ∂ /∂ t by the total derivative ∂ /∂ t +Vi ∂ /∂ xi , which takes into account the effects of convection on the time dependence. It follows that Eq. (1.2.10) becomes:

10

1 Introduction

∂ n ∂ (nVi ) + = D∇2 n. ∂t ∂ xi

(1.2.14)

In the next parts of the book, we concentrate mainly on the underlying phenomenon of the diffusive action of turbulence. Indeed, we shall be concerned with the subject of passive scalar transport, where by “scalar” we mean something like small particle or chemical species concentration and by “passive” we mean that the added substance does not change the nature of fluid to the point where the turbulence is appreciably affected. It is important to note that in spite of the oversimplified char→ acter of the convection–diffusion equation, the use of the model functions for V ( r ,t) allows one to describe nontrivial correlation mechanisms responsible for the scalar transport in the presence of complex structures such as system of zonal flows, convective cells, braded magnetic fields, etc. [14, 15, 16, 17, 18, 19, 20].

1.3 Diffusion and the Characteristic Velocity Scale The classical diffusion equation is an approximation only. Indeed, the dimensional analysis of Eq. (1.2.8) leads to the diffusive scaling R2 ∝ D t. Such a character of the dependence for the mean-square displacement of particles leads to difficulties (see (1.1.13)) when considering problems that require accounting for finite velocities of tracer [21, 22, 23, 24, 25, 26]. The example of interest is the diffusion of the chimney plume. Let us consider a chimney of the height hA . It is known that if Ux is the horizontal velocity of wind and Uy is the characteristic vertical velocity of the smoke, then the smoke will reach the ground at a distance not closer than the value Lm (see Fig. 1.5): Ux Lm = hA . (1.3.1) Uy

Uy Ux

h

Fig. 1.5 Smoke dispersion from an elevated source (chimney) in a given wind

Lm

1.3 Diffusion and the Characteristic Velocity Scale

11

This contradicts the diffusion equation, from which it follows that the smoke could be found as much closer to the chimney. Indeed, it follows from the conventional diffusion equation that the scalar is distributed instantly and its concentration is nonzero everywhere. This makes it impossible to investigate transport near the cloud boundary. Actually, we must take into account the limited velocity of particle propagation, which is related to the limitation on wind velocity fluctuations creating turbulent mixing. One of the first models to describe finite velocity effects in turbulent diffusion was the Davydov model [27], which is based on the telegraph equation:

∂ 2n 1 ∂ n ∂ 2n + 2 = V2 2 . τ ∂t ∂t ∂x

(1.3.2)

Here, V is the velocity scale and τ is the correlation time. To derive this modification of the conventional diffusion equation, we introduce the following values: n(+) (x,t) is the density of particles moving to the right with velocity +V and n(−) (x,t) denote the density of left-moving particles with velocity −V . The coupled conservation laws are:  ∂ n(+) ∂ n(+) 1  +V = n − n(+) , ∂t ∂x 2τ (−)  ∂ n(−) ∂ n(−) 1  −V = n(+) − n(−) . ∂t ∂x 2τ

(1.3.3) (1.3.4)

These equations are self-evident. Let us rewrite these equations in an alternative form by defining the total concentration n(x,t), and the flux q(x,t) as: n ≡ n(+) + n(−) ,   q ≡ V n(+) − n(−) .

(1.3.5) (1.3.6)

In terms of these new variables we have the particle conservation equation:

∂n ∂q + = 0, ∂t ∂x

(1.3.7)

and the phenomenological flux-gradient relation: q ∂q ∂n = − −V 2 . ∂t τ ∂x

(1.3.8)

Eliminating the particle flux q from this relation, we find:

∂ 2n ∂ 2n ∂n + τ 2 = V 2τ 2 . ∂t ∂t ∂x

(1.3.9)

This equation can be regarded as an interpolation between the wave and diffusion equations, since when τ → ∞, with the parameter V remaining finite, it reduces to

12

1 Introduction

the wave equation, and when τ → 0 and V 2 τ → D = const it reduces to the diffusion equation. The telegraph equation is known by this name because it was first derived by Kelvin in his analysis of signal propagation in the first transatlantic cable and then was often applied to describe turbulent diffusion [28, 29, 30, 31]. The physical meaning of the new particle flux representation can be easily clarified by writing the formal solution of the linear equation:

∂ q q0 − q = . ∂t τ where q0 (x,t) = −V 2 τ

(1.3.10)

∂ n(x,t) . ∂x

(1.3.11)

The solution has the form: t

q(x,t) = 0

dt  q0 (x,t ) exp(−(t − t )) =− τ 



t

V 2τ

0

∂n dt  exp(−(t − t  )) . (1.3.12) ∂x τ

Obviously, such an expression for the particle flux contains memory effects. The telegraph equation is linear and its solution in free space can be found by means of the Fourier-Laplace transform. This solution can be obtained in terms of the Bessel functions [10, 28, 32]. However, here we discuss only a qualitative pattern related to the modification of the conventional diffusion equation. The finite cutoff is apparent in the particle distributions, as well as in the bell shape that approaches the Gaussian form at very large values of t. Solutions to the telegraph equation shape the property inherent in the wave equation, that signals are propagated with finite velocity, which contrasts with solutions of the diffusion equation which are said to propagate signals at infinite speed. Because of the first time derivative in the telegraph equation, the shape of the profile is modified as it travels. In the limit t → ∞ the profile tends toward a Gaussian shape (see Fig. 1.6). n

T1

T2

T3

Fig. 1.6 A typical plot of the solution of the telegraph equation

y=

x V0τ

1.4 Lagrangian Description of Turbulent Diffusion

13

Note that the particle flux relation was generalized in many studies in such a way as to replace the exponential function by an arbitrary memory function Mm (t − t  ): t

q(x,t) =

q0 (x,t  )Mm (t − t  )

0

dt  . τ

(1.3.13)

In this general case one obtains the diffusion equation in the form:

∂ n(x,t) = ∂t

t

D 0

 ∂ 2 n(x,t  )  dt M (t − t ) . m ∂ x2 τ

(1.3.14)

Maxwell [18] was the first to suggest the hyperbolic model of heat-conductivity for the description of the finite velocity of perturbation spreading. This corresponds fairly well to his investigations of electromagnetic theory. From the modern point of view, such an approach to the turbulent transport looks fairly naive. However, in essence, the idea of using the additional derivative in the equations describing the anomalous character of turbulent diffusion was clearly formulated as early as 1934 [32–33]. At present, not only are conventional partial derivatives used, but even fractional derivatives, ∂ξn ∂ζn , , (1.3.15) ∂ xξ ∂ t ς are used, better mirroring the essence of the nonlocality and memory effects because they have the integral character of the operator [34, 35, 36]. Here, ξ and ζ are the fractional parameters of the problem. Moreover, this approximation method is also applied to the description of strong nonequilibrium processes in the framework of the kinetic equation [37, 38, 39]. From the dimensional standpoint, the use of the telegraph equation permits one to obtain scaling laws for the smoke front propagation in the ballistic form, RF (t) ∝ t,

(1.3.16)

and the new scaling for the diffusion coefficient, DT = V 2 τ ,

(1.3.17)

differs significantly from the random walks estimate D = δ 2 /2τ . Now we can account for the scale of velocity fluctuations, which is more convenient for describing the statistical properties of turbulence than the correlation length.

1.4 Lagrangian Description of Turbulent Diffusion In the previous consideration, the diffusion was discussed in terms of Eulerian (laboratory) coordinate frame. Let us now consider the Lagrangian description of particle movements [40, 41]. In Fig. 1.7, we denote the position of the marked fluid particle

14

1 Introduction

Fig. 1.7 Schematic illustration of the Lagrangian coordinate system and a particle path

z

δx x (t + δt) x (t )

x

y at any time by x(t), and this is the Lagrangian position coordinate of the particle. Let introduce the Lagrangian velocity V (t) of the particle by the usual rules: V (t) = limΔt→0

Δx dx = . Δt dt

(1.4.1)

That is, the Lagrangian velocity is the instantaneous rate of change of position with respect to time. In order to introduce a statistical treatment, we must average over many such particle paths. This would imply that the Lagrangian coordinates should be tagged in order to distinguish which coordinate applies to which particle. Following the Taylor statistical approach, we consider the displacement x(t) of a marked fluid particle in one dimension. From Eq. (1.4.1), the displacement (in one dimension) can be expressed in terms of the velocity field as: t

x(t) =

V (x0 ,t  ) dt  ,

(1.4.2)

0

the ideas of Langevin’s and Einstein’s [42]. The displacement will be positive as often as it is negative, therefore its mean value will be zero. In these conditions, the lowest-order statistical moment, which does not vanish, is the variance (or mean square) of particle position. Squaring x and averaging, we find: ⎫⎧ ⎫   ⎧t t t t ⎨     ⎬ ⎨    ⎬        2    V t dt dt V t dt dt V t V t x (t) = = ⎩ ⎭⎩ ⎭ 0

t

= 0

dt 

0

t 0

     dt  V t  V t  .

0

0

(1.4.3)

1.4 Lagrangian Description of Turbulent Diffusion

15

The averaging procedure is based on the supposition that one considers simultaneously released a large number of particles at t = 0, at different points in the fluids, and averaged over all the particle tracks. However, this requires the turbulence field to be spatially homogeneous. Expression (1.4.3) is symmetric under the interchange of t  and t  . The integration over the rectangular field of integration (see Fig. 1.8) specified by the limits 0 ≤ t  , t  ≤ t can be replaced by twice the integration over the triangular field of integration specified by the limits 0 ≤ t  ≤ t, 0 ≤ t  ≤ t  . Then, we rewrite Eq. (1.4.3) in the form [43]: 



t t t t         x (t) = 2 dt  dt  V (t  )V t  = 2 dt  dz V t  V t  − z . (1.4.4)



2

0

0

0

0

Here, we change the variable t  = t  − z. Let us introduce the Lagrangian correlation function C(t):   C(t) = V (x0 , z)V (x0 , z + t) = V02 RL (t),

(1.4.5)

where V0 is the characteristic scale of velocity fluctuations:  2   2   2 V (t) = V (0) = V = V02 .

(1.4.6)

Integrating with respect to t  by parts, we find ⎡  ⎤t  t t t t  2     ⎣ ⎦ x = 2 dt dzC (z) = 2 t C (z) dz − 2 t C t  dt  0

0

0

t

= 2t 0

C (z) dz − 2

0

0

t

zC (z) dz

(1.4.7)

0

This yields the Kampe de Feriet mean square distance traveled by a diffusing marked particle [44]:

t ''

Fig. 1.8 Field of integration for the Taylor diffusion coefficient

t '' = t '

t'

16

1 Introduction

        t − t  C t  dt  . R2 = x2 (t) = 2 V 2 t

(1.4.8)

0

Then, one obtains an important relationship, which will be used in the subsequent discussions: d2  2  x (t) = 2C(t). (1.4.9) dt 2 t=τ Estimates of the turbulent diffusion coefficient in the Taylor approach lead to the expression: 1 d d  x2 (t)  = DT = 2 dt dt

t







(t − t )C(t ) dt =

o

t

C(t  ) dt  ≈ V02 τ .

(1.4.10)

0

Here, τ is the Lagrangian correlation time, which is given by (see Fig. 1.9):

τ=

1 V02

∞

C (t) dt.

(1.4.11)

0

Such a definition is especially relevant for the description of turbulent transport where velocity fluctuates in a fairly unpredictable way, whereas in a steady laminar flow the velocity does not change with time. Figure 1.10 summarizes this difference between laminar conditions and turbulence. The specific form of the expression for the turbulent diffusion coefficient DT (t) depends on the behavior of the correlation function C(t). From the following properties of the correlation coefficient, we can derive two important asymptotic results: C (0) = 1,

and C (t) → 0,

as t → ∞.

(1.4.12)

The first of these was obtained from the definition of C (t), and the second follows from the obvious physical fact that events widely separated in time or space become uncorrelated. Usually, experimental results and theoretical arguments suggest that C(t)

V02

Fig. 1.9 A typical plot of the Lagrangian correlation function

τ

t

1.4 Lagrangian Description of Turbulent Diffusion Fig. 1.10 A typical plot of the dependence of the velocity in turbulent and in laminar flow

17

V(t)

Turbulent

Laminar t

the latter condition could be expressed more strongly. That means C (t) tends to zero faster than a power of t as t tends to infinity. We analyze the limiting cases of t → 0 and t → ∞. First we consider the case of short diffusion times where C (t) ≈ V02 . Integrating the general relation for R2 (t) leads to the ballistic scaling:  2   2 2 x (t) = V t . (1.4.13) Now consider the case of long diffusion times. We assume times long enough for the correlation coefficient to have fallen to zero. This is of the order of the Lagrangian correlation time τ . Note that for t > τ the correlation coefficient falls rapidly to zero and cuts off the integral relation for R2 :  2    x (t) = 2 V 2 (τ t − const) for t >> τ . (1.4.14) When t increases, the second term can be neglected in comparison to the first term, and the mean-square particle displacement becomes (see Fig. 1.11):     (1.4.15) x2 (t) = 2 V 2 τ t ∝ t. In terms of the diffusion coefficient one obtains:  2 1/2 x (t) = (2DT t)1/2 ,

(1.4.16)

where the turbulent diffusion coefficient is given by: DT = V02 τ .

(1.4.17)

18 Fig. 1.11 Comparison of the mean square displacement of the particle with a constant drift

1 Introduction R2(t) 2Dt V02t2

t >> τ

t

This differs significantly from the random walks result in which the diffusion coefficient scales as the correlation length squared and does not depend on the characteristic velocity scale. Even from the general considerations, it is clear that the integral relationship between the diffusion coefficient and the Lagrangian correlation function of velocity is a more relevant tool of investigation than the constant diffusion coefficient. In the next sections, we will show that the development of correlation ideas had an essential influence on the form of diffusion equations as well as on the choice of the effective correlation length and correlation time. The simple estimates show that diffusion related to turbulence is much greater than the molecular diffusion. Thus, for molecular diffusion in the atmospheric boundary layer we have V0 ≈ 104 cm/s, l ≈ 10−5 cm, D0 ≈ 0.1 cm2 /s, whereas for turbulent transport the following estimates are given by V0 ≈ 10 cm/s, l ≈ 10−2 –10−3 cm, D0 ≈ 103 –104 cm2 /s [23, 28]. Indeed, from the probabilistic point of view one can consider turbulent diffusion as random walk on a random walk, which could considerably change the effective transport in comparison with the seed (molecular) diffusivity. On the other hand, the relationship between the flows and gradients of parameters is of fundamental importance. The linear Fick approximation is useful for the first step of an analysis of effective transport. The density gradient and the particle flux are chosen as typical examples. The relationship between the flows and gradients of parameters has been intensively studied theoretically and experimentally [45, 46, 47, 48, 49, 50]. Thus, for strong turbulence it was shown that the flux is not a linear function of the gradient. Its nonlinearity increases as the gradient becomes stronger, which is of physical interest, because turbulent flows are really the target medium of nonlinear–nonequlibrium physics.

Further Reading

19

Further Reading Diffusion Concept Ben-Avraham, D. and Havlin, S. (1996). Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge, U.K. Berg, H.C. (1969). Random Walks in Biology. Princeton University Press, Princeton, NJ. Gardiner, C.W. (1985). Handbook of Stochastic Methods. Springer-Verlag, Berlin. Mazo, R.M. (2002). Brownian Motion, Fluctuations, Dynamics and Applications. Clarendon Press, Oxford. Montroll, E.W. and Shlesinger, M.F. (1984). On the wonderful world of random walks, in Studies in Statistical mechanics 11, 1. Elsevier, Amsterdam. Montroll, E.W. and West, B.J. (1979). On an enriches collection of stochastic processes, in Fluctuation Phenomena. Elsevier, Amsterdam. Pecseli, H.L. (2003). Fluctuations in Physical Systems. Cambridge University Press, Cambridge, U.K.

Correlations in Complex Systems Haken, H. (1978). Synergetics. Springer-Verlag, Berlin. Hanggi, P. and Thomas, H. (1982). Physics Reports, 88, 207. Mikhailov, A. (2006). Physics Reports, 425, 79. Moffatt, H.K. (1981). Journal of Fluid Mechanics, 106, 27. Schweitzer, F. (2003). Brownian Agents and Active Particles. Springer-Verlag, Berlin. Sornette, D. (2006). Critical Phenomena in Natural Sciences. Springer-Verlag, Berlin. Zeldovich, Ya. B., Ruzmaikin, A.A., and Sokoloff, D.D. (1990). The Almighty Chance. World Scientific, Singapore.

Chapter 2

Turbulent Diffusion and Scaling

2.1 Correlation Functions and Scaling The idea of using the Lagrangian correlation functions for the analysis of turbulent transport has appeared very fruitful. Although the foregoing treatment of diffusion from a continuous point source in homogeneous turbulence requires, for its full exploitation, an explicit formulation of the Lagrangian spectrum, a good deal of clarification may be achieved merely by examining the consequences of assuming various possible functional forms. The exponential form, 

t , (2.1.1) C (t) = V02 exp − TL was used by Taylor [43] in his original discussion of the turbulent diffusion coefficient (see Fig. 2.1). Here, TL is the Lagrangian characteristic time, which is given by: 1 TL = 2 V0

∞





∞

C(t ) dt = 0

  RL t  dt  ,

(2.1.2)

0

where RL is the normalized correlation function. On the other hand, considerable effort has been devoted to the development of correlation scalings. Thus, at an early stage of the turbulent diffusion investigation a simple approximation was suggested [52]:

 t −αC 2 , (2.1.3) C (t) = V0 1 + T0 where T0 is the characteristic time. This formula differs significantly from the exponential dependence, which is also applied in the framework of the Langevin approach [53, 54, 55, 56, 57]. Justification of this scaling form was provided by reference to aerodynamic experiments. Actually, the power dependence contains an additional parameter αC that enables one to interpret experimental results. Here, the value of the characteristic time T0 is interpreted in terms of the viscosity νF :

O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008 

21

22

2 Turbulent Diffusion and Scaling

Fig. 2.1 Lagrangian velocity autocorrelation function in isotropic turbulence. (After S.B. Pope [61], with permission.)

T0 =

νF , V02

(2.1.4)

which allows one to fit the experimental data on atmosphere diffusion [22, 23, 24, 25, 26]. Simple calculations lead to the expression for the mean square displacement R2 . Based on the Taylor formula, we obtain: t 2

R =

dt 0



t



t

C (z) dz

= V02

0

dt



0

Then we can rewrite: V02 T02 R2 = (1 − αC ) (2 − αC )

!

t 1− T0

2−αC

 t

1+ 0

z T0

−αC dz.

" t − 1 + (αC − 2) . T0

(2.1.5)

(2.1.6)

For t >> T0 and αC < 2, we find: R2 (t) ≈

V02 T02 (1 − αC ) (2 − αC )



t T0

2−αC ,

(2.1.7)

where αC  1 and αC  2. This scaling actually relates the Hurst exponent H to the correlation exponent αC , R2 (t) ∝ t 2H ∝ t 2−αC ,

(2.1.8)

2.2 The Richardson Law and Anomalous Transport

H=

23

2 − αC , 2

(2.1.9)

where from the condition 0 < H < 1 it follows that 0 < αC < 2. More detailed calculations yield the exact formulas for αC = 1, 2, 3: !

" 

 t t t ln 1 + − (2.1.10) for αC = 1, R2 (t) = V02 T02 1 + T0 T0 T0 ! R2 (t) = V02 T02

"

t t for αC = 2. − ln 1 + T0 T0

(2.1.11)

 2 R2 (t) = V02 T02

t T0

1 + Tt0

for αC = 3.

(2.1.12)

It is pertinent to note that in the case αC = 3, when the Lagrangian correlation time is chosen as the characteristic time T0 = τ , we can derive simple asymptotic formulas to describe long-range correlation effects. For t >> τ = T0 , we find the diffusive scaling, R2 ≈ V02 τ t, whereas for t → 0, we obtain the ballistic motion R2 ≈ V02 t 2 . Naturally, such an approach is just a simple approximation. In reality, turbulent transport is considerably different from conventional diffusive mechanisms and one has to look for both new theoretical conceptions and phenomenological models. This is not surprising since turbulent motions, which mixes passive scalar, appear to have a multiscale nature. This is a result of the presence of numerous nonlinear interactions and instabilities.

2.2 The Richardson Law and Anomalous Transport We now consider the important differences between the diffusion from a continuous source, in which particles are released in sequence at a fixed position, and that of a single puff of particles. The particle nature of the latter type of diffusion was recognized by Richardson [58] at a very early stage (1926). He considered the dispersion of pairs of particles passively advected by an homogeneous, isotropic, fully developed turbulent type. Due to the incompressibility of the velocity field the particles will, on average, separate from each other (see Fig. 2.2). There are several stages in the process of relative diffusion. At the first stage, the particles are initially close together and only the smallest eddied can increase their separation. At the next stage particles move further apart, a greater range of eddy sizes becomes important, with, at all times, the eddies comparable in size to the interparticle separation having the dominant effect. The last stage is when the distance between particles becomes greater than the largest turbulent eddy, and the motion of each particle becomes independent of the other. The separation between them is

24

2 Turbulent Diffusion and Scaling

t2

t3

t0

t1

Fig. 2.2 Schematic illustration of the tracer relative dispersion due to turbulent mixing

then determined by their own individual random walks. This stage is characterized by the largest energy-containing eddies. To find out how the coefficient of eddy diffusivity DR varies with scale lR Richardson plotted DR versus lR ranging from 0.05 to 108 cm. The logarithms of DR and lR were found to lie on a line of slight curvature in the sense that d (log DR ) /d (log lR ) increased with lR , but all except the extreme points could be represented with good approximation by the relation (see Fig. 2.3): 4

DR = CR lR 3 ,

(2.2.1)

where CR ≈ 0.2 is the Richardson constant and l ranges from 102 to 106 cm. This law was formulated on the basis of the scaling representation of the diffusion coefficient and by analogy with the conventional diffusion. Indeed, the results of various experiments on diffusion in the atmosphere lead to the empirical formula:    2 1 d lR 2 (t) = const lR 2 (t) 3 . (2.2.2) 2 dt Alternatively, by integrating once, we can write the mean square separation of the particle as:  2  3 lR (t) ∝ t . (2.2.3) Result (2.2.3) is not trivial because it differs significantly from the ballistic scaling. Indeed, from the conventional point of view we can consider that particle 1 and particle 2 are released simultaneously at time t = 0 and at positions x1 and x2 respectively. Let the distance between the two particles be l(t). Then we shall set:

2.2 The Richardson Law and Anomalous Transport

25

Fig. 2.3 Richardson scaling. (After L.F. Richardson [58], with permission.)

Y (t) = x2 (t) − x1 (t) and the mean square separation is given by:         Y 2 (t) = x12 (t) − 2 x1 (t)x2 (t) + x22 (t) .

(2.2.4)

(2.2.5)

Destroying correlations in time, 

 x1 (t)x2 (t) = 0,

(2.2.6)

leads to a result that is in accord with the following estimate:  2  Y (t) ≈ 2(2DT )t. (2.2.7)   The mechanism behind lR2 (t) ∝ t 3 pair separation in turbulent flows has been a puzzle since it was reported, and understanding the particle pair dispersion in turbulent velocity fields is of great interest for both theoretical and practical implications. Richardson introduced the fundamental notion that the rate of separation of a pair of particles at any instant is dependent on the separation itself (acceleration process): 2 dlR (t) ∝ lR (t) 3 , (2.2.8) dt

26

2 Turbulent Diffusion and Scaling

and that as separation increases so also does the rate of separation. This meant that the spread of a large cloud of particles could not be built up by superimposing the growths of component elements of the cloud treated separately. Richardson was concerned with finding a diffusion equation to describe the concentration field relative to the center of mass of a moving cloud. He suggested using the diffusion equation for the description of the probability density evolution F to find two initially close particles at distance l from one another at the moment t:

∂ F(lR ,t) ∂ ∂ F(lR ,t) = DR . ∂t ∂ lR ∂ lR

(2.2.9)

In the framework of the scaling law (2.2.3), the expression for DR (lR ) takes the scaling form: (2.2.10) DR (lR ) ≈ CR lR 4/3 . In principle, the possibility to describe the dispersion process by means of a conventional diffusion equation is based on two important physical assumptions, which can be verified a posteriori. The first is that the dispersion process is selfsimilar in time, which is probably true in a nonintermittent velocity field. The second is that the velocity field is short correlated in time. We will discuss these aspects of the problem below. The really important feature, which was introduced in [58] was the idea of a virtually continuous range of eddy sizes, with turbulent energy being handed down from larger to smaller eddies and ultimately dissipated in viscous action. Specific expression of the concept came considerably later in the parallel developments by Kolmogorov, on the basis of his similarity theory, and Obukhov, on the basis of the equation of energy balance in the spectrum. Concluding this section, recall Taylor’s words [23]: “Since the curve shown when here seems to contain all the observational data that Richardson had when he announced the remarkable Richardson law, it reveals a well-developed physical intuition that he chose as his index 4/3 instead of, say, 1.3 or 1.4 but he had the idea that the index was determined by something connected with the way energy was handed down from larger to smaller and smaller eddies. He perceived that this is a process which, because of its universality, must be subject to some simple universal rule.”

2.3 The Kolmogorov Description of Turbulence Richardson’s ideas of the interactions between different scales of eddy motion influenced the development of the Kolmogorov-Obukhov similarity theory. To introduce these scaling ideas of well-developed turbulence, we will examine the Navier-Stokes equation of motion for a Newtonian fluid: 1 ∂P ∂ ui ∂ ui ∂ 2 ui +uj =− + νF , ∂t ∂xj ρ ∂ xi ∂ x j∂ x j

(2.3.1)

2.3 The Kolmogorov Description of Turbulence

27

where ui is the velocity in the xi direction, ρ is the density, and νF is the kinematic viscosity. We perform the conventional scale analysis and represent a characteristic velocity by V0 and length scale by L0 . The terms in the Navier-Stokes equation can be estimated as follows:

 V2 V2 V0 V0 , 0 , 0 , νF 2 . (2.3.2) V0 L0 L0 L0 L0 The relative effect of friction is given by dividing the advection term by the viscous friction term: V02 L0 νF LV02 0

=

V0 L0 = Re. νF

(2.3.3)

This is the Reynolds number. A universal law of fluid dynamics is the Reynolds similarity law, which states that the dynamical behaviors of two fluids with identical Reynolds numbers are similar, independent of their constituent molecules. To consider a dissipation rate of kinetic energy εDis , we introduce a different length scale δν into the viscous term. The viscous term is balanced by the other terms in the equation of motion: V02 V0 ∝ νF 2 . L0 δν

(2.3.4)

Then, we write the expression for the viscous length scale: L0 δν ∝ √ . Re

(2.3.5)

From this expression we see that viscosity plays an important role in only a small fraction of the field. Such an approach was developed into the classical Blasius boundary-layer theory [59, 60, 61]. Consider now a laminar boundary layer over a flat plate (see Fig. 2.4). The dissipation rate of kinetic energy εDis can be estimated as: y

V (y)

Fig. 2.4 Schematic illustration of the laminar boundary layer

x

28

2 Turbulent Diffusion and Scaling



εDis = νF

∂u ∂z

2

∝ νF

V0 δν

2 .

Using the expression for the viscous length scale, we obtain:

 V03 Re 2 = εDis ∝ νF V0 . δν L0 2

(2.3.6)

(2.3.7)

This result demonstrates that the dissipation rate is determined by the large-scale parameters of the flow, and not by the viscosity. The properties of the flow on all scales depend on Re. Flows with Re < 100 are laminar; chaotic structures develop gradually as Re increases, and those with Re ∼ 103 are appreciably less chaotic than those with Re ∼ 107 (see Fig. 2.5). Indeed, when the Reynolds number is small, viscosity stabilizes the flow. On the other hand, when it is greater than 104 , the flow is unstable and becomes turbulent. For water at room temperature, νF is about 102 cm2 /s, hence flow becomes turbulent for relatively small L and V0 , for example, L0 ≈ 10 cm and V0 ≈ 10 cm/s. Nearly the same estimate can be made for air. Most of the water and air around us is in turbulent states. The complexity of the shape of cigarette smoke is also due to turbulence. On the other hand, observed features such as star-forming clouds and accretion discs are very chaotic with Re ≥ 108 . The Kolmogorov phenomenological turbulence model [62] regards a turbulent velocity field as the superimposition of structures (eddies) characterized by a spatial →

scale l and the associated velocity field increment: #→  →  $ → → → → V r + l −V r l . δ Vl = l

(2.3.8)

Because of additionally assumed statistical isotropy, the field increments depend solely on l, which allows one to define the characteristic eddy velocity: 1/2  Vl = δ Vl2

(2.3.9)

or, in terms of spectral terminology, 1/2  , Vk = δ Vk2

(2.3.10)

1 1 = . l (k) lk

(2.3.11)

where k is the wave number, k≈

In such an approach there are three scale ranges (see Fig. 2.6): • The energy-containing scales, driving the flow • The inertial range where nonlinear interactions govern the dynamics and the influence of driving and dissipation are negative

2.3 The Kolmogorov Description of Turbulence

29

Fig. 2.5 Example of a turbulent flow at about Re = 5500. (After D.P. Papailiou and P.S. Lyykoudis [74], with permission.)

• The dissipation range at smallest scales, where dissipative effects dominate, removing energy from the system. Here, the rate of dissipation of energy per unit mass by viscosity εK is the most important parameter. Suppose that the fluid motion is excited at scales LE and greater. Then, the energy cascades through nonlinear interactions to progressively smaller and smaller scales at the eddy turnover rate, τk ≈ V1k , with insignificant energy k losses along the cascade. The energy reaches the molecular dissipation scale lν ,

30

2 Turbulent Diffusion and Scaling

Fig. 2.6 A typical plot of the Kolmogorov-Obukhov energy spectrum

E (k )

E (k ) ∝ k −5/ 3

Energy cascade

kE

kv

k

i.e., the scale where the local Re 1, and is dissipated there. The scales between LE and lν are called the internal range and it typically covers many decades. Indeed, the magnitude lν is determined in order of magnitude by εK and νF only, and hence on dimensional grounds:

3 1/4 νF . (2.3.12) lν ∝ εK Moreover, εK satisfies the important semiempirical relationship:

εK ∝

V03 , LE

(2.3.13)

 1/2 . Then we rewrite the expression for lν as: where V0 = V 2 lν ∝

LE Re3/4

> 1.

(4.5.16)

Note that Ref. [126] does not contain a reference to Howells’ paper [125]. Apparently, this result has become widely known more recently due to the Moffat analysis of turbulent transport problems [12, 13]. In the framework of this diffusion approximation, the value D is the effective diffusivity, which differs essentially from the concepts based on the use of “seed” diffusivity as the “decorrelation mechanism.” Such methods have been applied to a large variety of problems. Thus, Wang and co-workers [130] suggested a modification of the seed diffusion representation of correlation function (4.3.5) by a substitution of the Taylor correlation expression for the mean square displacement: RD (t) ≈ 2

t

2

(t − t  )C(t  )dt.

(4.5.17)

0

Formal calculations lead to a complex nonlinear integral equation for C(t): ⎡ ⎤− d 2 t V02 ⎣ V02   ⎦ ≈ (t − t )C(t )dt . C(t) ≈ n nRD (t)d

(4.5.18)

0

To illustrate this idea, it is sufficient to consider the simplified model of diffusive evolution of the “correlation cloud” where the simplest case corresponds to the twodimensional space d = 2. The approximative equation to define the Lagrangian correlation function takes the form: t

C(t) o

(t − t  )C(t  )dt  =

V02 . n

(4.5.19)

Unfortunately, even this version of the nonlinear equation cannot be analytically solved. However, numerical simulations carried out in [130] permit one to consider complex correlation effects in a stochastic magnetic field. Corrsin introduced a fruitful method to connect Eulerian and Lagrangian correlation functions. Thus, in the Eulerian frame, velocity correlations decay in space and in time as well. In a turbulent flow, the velocities at a single location will become decorrelated after a period of time, which is usually called the Eulerian integral time. Conversely, two observers separated by the integral (characteristic) scale will see uncorrelated velocities. From the point of view of a Lagrangian approach one, by

Further Reading

69

drifting, observes both the spatial and temporal decorrelation simultaneously. This means that the characteristic time measured by the Lagrangian observer will be less than that measured by a fixed observer. Corrsin’s famous conjecture states that the Lagrangian autocorrelation function could be expressed via the Eulerian spatial– temporal autocorrelation function in terms of the probability density function of particle displacements. Indeed, the integral over the displacement probability density function mirrors how far the particles wander from their initial positions. From the point of view of the correlation approach discussed above, this shows how much the spatial decorrelation affects the Lagrangian ones. If we suppose that the spatial scale is the correlation scale and the diffusive displacement at the same time, then we find fairly a universal approximation to treat turbulent transport.

Further Reading Correlations and Scaling Falkovich, G., Gawedzki, K., and Vergassola, M. (2001). Reviews of Modern Physics, 73, 913. Gurbatov, S., Malahov, A., and Saichev, A. (1990). Nonlinear Random Waves. Nauka, Moscow. Jovanovic, J. (2004). The Statistical Dynamics of Turbulence. Springer-Verlag, Berlin. Klyatskin, V.I. (2001). Stochastic Equations by the Physicist’s Eyes. Fizmatlit, Moscow. Moffatt, H.K. (1981). Journal of Fluid Mechanics, 106, 27. Monin, A.S. and Yaglom, A.M. (1975). Statistical Fluid Mechanics. MIT Press, Cambridge, MA. Pomeau, Y. and Resibois, P. (1975). Physics Reports, 19, 63.

Correlation Functions and Geophysical Turbulence Csanady, G.T. (1972). Turbulent Diffusion in the Environment. D. Reidel, Dordrech-Holland, Boston. Cushman-Roisin, B. (1994). Introduction to Geophysical Fluid Dynamics. Prentice-Hall, Englewood Cliffs, NJ. Frenkiel, N.F., ed. (1959). Atmospheric Diffusion and Air Pollution. Academic Press, New York. Panofsky, H.A. and Dutton, I.A. (1970). Atmospheric Turbulence, Models and Methods for Engineering Applications. Wiley Interscience, New York. Pasquill, F. and Smith, F.B. (1983). Atmospheric Diffusion. Ellis Horwood Limited, Halsted Press, New York.

Chapter 5

Diffusion Equations and the Quasilinear Approximation

5.1 The Taylor Dispersion In spite of the effectiveness of scaling to treat correlation effects and transport, the equations describing the evolution of tracer distribution are also promising tool for investigating passive scalar diffusion. Here, we discuss the effective transport of a tracer in a laminar shear flow in the presence of seed diffusivity (see Fig. 5.1). Such a scalar dispersion provides a classic example of the role of convection in dispersing inhomogeneous flows. Taylor suggested [131, 132] a fruitful method of obtaining the effective diffusion coefficient, which is based on averaging the transport equation: ∂n ∂n +Vx (y, z) = D0 ∇2 n. (5.1.1) ∂t ∂x Here, n is the scalar density, Vx is the longitudinal (along the x-axis) velocity, and D0 is the seed diffusivity. In this approach, the influence of molecular diffusion on longitudinal convective transport is analyzed. Let us consider the Poiseuille flow in a cylindrical tube, but in order to simplify calculations we will analyze a flat model. Suppose that the profile of the longitudinal flow has the form: Vx (z) =

V0  2 2  L −z , L2

(5.1.2)

where L is the characteristic spatial scale. Now we can consider the scalar transport problem in the framework of the decomposition method, where the density field n can be represented as a sum of the mean density n and the fluctuation component n1 : n = n + n1 (x, z,t) = n0 + n1 (x, z,t), Vx = Vx  +V1 ≡ V0 +V1 .

(5.1.3) (5.1.4)

Here, use is made of the expression for mean values: n ≡

1 2L

L

n(x, z,t)dz ≡ n0 (x,t),

(5.1.5)

−L

O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008 

71

72

5 Diffusion Equations and the Quasilinear Approximation

Fig. 5.1 Schematic illustration of the Poiseuille two-dimensional flow

z

Vx(z)

L

x

0

1 Vx  = 2L

L −L

2 V (z)dz = V0 , 3

(5.1.6)

where V0 is the characteristic velocity. Hence, one finds: V1 (z) = V0

1  z 2 − . 3 L

(5.1.7)

The substitution of the expression for n and Vx into the convection–diffusion equation (5.1.1) yields:

∂ ∂ ∂ n0 + n1 + (V0 +V1 ) (n0 + n1 ) = D0 ∇2 [n0 + n1 ] . ∂t ∂t ∂x

(5.1.8)

Taking the average of this equation, we arrive at an expression for the mean density evolution:   ∂ ∂ ∂ ∂ 2 n0 n0 +V0 n0 + V1 n1 = D0 2 . (5.1.9) ∂t ∂x ∂x ∂x To derive a closed-equation for the scalar mean density, it is necessary to find an expression for n1 (x, z,t). Subtracting (5.1.9) from (5.1.8) leads to the equation for the evolution of density perturbation n1 :  

2  ∂ n1 ∂ n0 ∂ n1 ∂ n1 ∂ n1 ∂ n1 ∂ 2 n1 +V1 +V0 +V1 − V1 =D . (5.1.10) + ∂t ∂x ∂x ∂x ∂x ∂ x2 ∂ z2 The expression obtained is too complex; therefore, we use, as suggested by Taylor, the heuristic method to find the estimate of effective diffusion, which is based on

5.1 The Taylor Dispersion

73

several hypotheses: quasi-steadiness of n1 , i.e., in comparison with

∂ n0 ∂x

and

∂ n1 ∂t

≈ 0; smallness of

2 ∂ 2 n1 . We keep the term ∂∂ zn21 ∂ z2

∂ n1 ∂x

and

∂ 2 n1 ∂ x2

in order to take into account

the density gradient in the direction of the walls, which has to be grater to satisfy the no-flux condition:

∂ n1 = 0 at z = L and z = −L. ∂z

(5.1.11)

Next, solving the equation obtained from (5.1.10): D0 where the term

∂ n0 ∂x

∂ 2 n1 ∂ n0 (x,t) , = V1 (z) ∂ z2 ∂x

(5.1.12)

is considered as a parameter, we easy find the expression for n1 :

z4 ∂ n0 V0 z2 − + const. n1 = ∂ x 3D0 2 4L2

(5.1.13)

Note that the order of n1 is given by the scaling n1 ∝ n0

V0 L ≈ n0 Pe, D0

(5.1.14)

where Pe is the Peclet number. Applying the condition n1  = 0 we express const through the problem parameter ∂∂nx0 :

∂ n0 V0 const = ∂ x 3D0



7 − L2 60



(5.1.15)

and then after substitution one arrives at

∂ n1 ∂ 2 n0 V0 z2 z4 7 2 = − L . (5.1.16) − ∂x ∂ x2 3D0 2 4L2 60   Now, the term V1 ∂∂nx1 , which defines an additional contribution in longitudinal diffusive transport, can be rewritten in the form:   8 (V0 L)2 ∂ 2 n0 ∂ n1 ∂ 2 n0 =− . V1 = −D ∗ ∂x 945 D0 ∂ x2 ∂x

(5.1.17)

The equation for n0 takes the following form:

∂ ∂ ∂ 2 n0 n0 +V0 n0 = (D0 + D∗ ) 2 . ∂t ∂x ∂x

(5.1.18)

This method is a good example of a general mathematical technique: the simplification of a complicated system by the elimination of “fast modes.”

74

5 Diffusion Equations and the Quasilinear Approximation

The result obtained is not trivial:

D∗ =

8 945



V02 L2 D0

(5.1.19)

because the additional diffusive contribution D∗ depends inversely on seed diffusivity D0 . The physical interpretation of this result is the limitation of the influence of nonuniformity of the longitudinal velocity profile Vx (z) by transverse diffusion. Hence, nonuniform longitudinal convection in combination with transverse diffusion leads to longitudinal diffusion. Naturally, the new diffusive mechanism manifests itself at a large distance downstream only, since the equation obtained is correct only for: L2 . (5.1.20) t >> τD ≈ D0 On the other hand, the condition of smallness of the transverse spatial scale in comparison with the longitudinal one l was used: L