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
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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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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
<< LE .
(2.3.14)
Here, Re is the Reynolds number of the turbulence, assumed large. In spectral terminology, one obtains: kE =
1 , LE
kν =
1 ∝ Re3/4 kE lν
(2.3.15)
and defines the internal range: kE << k << kv .
(2.3.16)
This phenomenological theory also provides a simple scaling for hydrodynamic motions. If the velocity at a scale k from the inertial range is Vk , then the kinetic energy is transferred to the next scale within one eddy turnover time τk ≈ V1k . k Within the internal range the statistical properties of the turbulence are determined by the local wave number k and εK , the rate of cascade energy, which is scaleindependent:
2.3 The Kolmogorov Description of Turbulence
εK =
31
Vk3 Vk2 Vk2 = = . lk τk (kVk )−1
(2.3.17)
From this relation we obtain: 1
Vk ≈ (εK lk ) 3 ≈
ε 13 K
k
.
(2.3.18)
The one-dimensional energy spectrum E(k) is the amount of energy between the wave number k and (k + dk) divided by dk:
Vk2 =
E (k) dk ≈ kE (k).
(2.3.19)
Δk
Since Vk2 ≈ kE (k) ≈ (εK /k)2/3 , we find E(k) as: E (k) = CK εK 2/3 k−5/3 ,
(kE << k << kv ),
(2.3.20)
where CK ≈ 1.6 is the Kolmogorov constant. This scaling is the energy spectrum for isotropic turbulence, for which there is much experimental support (see Fig. 2.7), particularly from atmospheric and oceanographic turbulence [63, 64, 65, 66, 67, 68, 69, 70]. Based on the theory of phenomenological turbulence, Obukhov suggested [71] a theoretical interpretation of the Richardson law for relative diffusion. Indeed, it is possible to compose the scaling for $ diffusion coefficient based on the dimen# the 2 sional character of the value εK = TL 3 and the variable k that characterizes the 1 = L1 . Then, simple calculations yield the dimensional estispatial scale k ≈ l(k) mate for the Richardson coefficient:
Fig. 2.7 A logarithmic plot of a one-dimensional spectrum. The straight line has a slope of −5/2. (After A.E. Gargett et al. [63], with permission.)
32
2 Turbulent Diffusion and Scaling
2 2 1/3 k εK L 1 ≈ DR (l) = ≈ εK 1/3 4/3 ≈ εK 1/3 l 4/3 ∝ l 4/3 . 2 T k k
(2.3.21)
This coincides exactly with the Richardson relative diffusion coefficient. Thus, the idea of describing turbulence in terms of the hierarchy of eddies of different scales has obtained its first experimental confirmation in the framework of the scalar transport.
2.4 Relative Diffusion and Scalings Approximation (2.2.1) suggested by Richardson corresponds to his notion of the hierarchical character of turbulent transport. Thus, he related nonlocality to increases in the scale of eddies taking part in transport processes. Therefore, in his approach the diffusion coefficient DR is the function of the interparticle distance l. However, there exist alternative possibilities. Batchelor [72] considered the problem from a different point of view. In his model, the diffusion coefficient DR is the result of statistical averaging over the group of different scales. Hence, he proposed using the temporal dependence for the definition of DR . In the framework of this approach, the dimensional consideration yields the expression: 2 2/3 Y (t) ≈ lR 2 (t) ∝ t 2. (2.4.1) DR (t) ≈ t Then, the equation for the probability density takes the following form, which is similar to the Richardson equation but with the time-dependent coefficient of diffusion:
∂ F(lR ,t) ∂ ∂F = DR (t) . ∂t ∂ lR ∂ lR
(2.4.2)
After the simplest analysis, it becomes clear that these models lead to different results in spite of the underlying law (2.2.3). Thus, in the conventional diffusive equation the law of temporal relaxation of the function F in the Fourier form corresponds to the relation: F%k (t) ∝ exp(−t),
(2.4.3)
whereas in the case of the time-dependent diffusion coefficient D (t) ∝ t 2 we deal with a stronger decay: (2.4.4) F%k (t) ∝ exp(−t 3 ). Here, F%k (t) is the Fourier transformation of the function F(x,t) over the variable x. It is obvious that the characters of those solutions describing the probability density evolution are also different. Considering the model with a point-source of particles, one can obtain for the Richardson model [58]:
2.4 Relative Diffusion and Scalings
F(lR ,t) =
33
8 315π 8/2
9 4t
&
9/2
exp −
9lR 2/3 4t
' .
(2.4.5)
Under analogous conditions (the model with a point source) for the Batchelor model one finds [72]: & F(lR ,t) =
'3/2
1
2π lR 2 (t)
&
lR 2 exp − 2 2 lR (t)
' .
(2.4.6)
Note that the arguments in favor of one type or another of the diffusion coefficient have a qualitative character in both of these cases. Moreover, the “combination” of both of these approaches is possible if one supposes that DR can depend on both the interparticle distance lR and time t: DR (t, lR ) ≈
lR 2 ≈ t φ lR ϕ . t
(2.4.7)
To conserve the relative diffusion law lR 2 ∝ t 3 we must take into account the relationship between exponents φ and ϕ : 2 2−ϕ = , 1+φ 3
or
2φ + 3ϕ = 4.
(2.4.8)
Then, the case φ = 0, ϕ = 4/3 corresponds to the Richardson form and the case φ = 2, ϕ = 0 corresponds to the Batchelor supposition. Thus, a mixed algebraic representation for the diffusion coefficient was suggested in [73]: DR (t, lR ) ≈ t lR 2/3 .
(2.4.9)
The three-dimensional solution of the diffusion equation for the point source at t = 0 and interrelated exponents ϕ and φ is given by:
3(1+φ ) lR 2−ϕ − 2−ϕ (2.4.10) exp −const 1+φ . F(lR ,t) = const t t Unfortunately, it is impossible to decide what is a correct equation, if one looks at this problem from the conventional diffusion point of view, because the physical arguments from Kolmogorov and Obukhov lead to an explanation in terms of the hierarchy of scales, whereas Richardson and Batchelor deal with the local diffusive equation with partial differentials. Moreover, the Reynolds numbers of both direct numerical simulations and laboratory experiments is at present too low to exhibit an inertial subrange even for Eulerian statistics, and it seems likely that even larger Reynolds numbers are needed to produce an inertial subrange in Lagrangian statistics. On the other hand, field experiments are subject to great uncertainty in both the reliability of the measurements and in the extent to which the turbulence can be regarded as locally isotropic [28, 65]. Nevertheless, these classical papers
34
2 Turbulent Diffusion and Scaling
[58, 62, 72] formulated problems that allow us to develop theoretical methods of anomalous transport description that are based on the analysis of correlation effects and scaling laws. Modern theoretical models related to the ideas of intermittency and fractality permit more exact approximations for DR to be obtained. It was found in [75] that data are best fitted by a relation:
4 2 + μi (2.4.11) DR (lR ) ∝ lR 3 3 with the intermittency exponent μi ≈ 0.2. Similar results were obtained in [76–79] by using the Levy walks approach. One can see that the correlation term 2μi /3 is approximately 10 times less than the Richardson exponent, but the physical meaning of the fractal representation of intermittency phenomena is of great importance. The concepts underlying these models are discussed in Chaps. 3 and 14.
2.5 Cascade Phenomenology and Scalar Spectrum The scalar power spectrum can also be obtained by simple phenomenological argu ments in terms of the dissipation rate of the scalar “energy” θ 2 . Indeed, consider the evolution of the tracer that is described by the advection–diffusion equation:
∂θ → + U · ∇θ = D0 ∇2 θ , ∂t
(2.5.1)
→
where D0 is the molecular diffusivity and U is the advection velocity, which is nondivergent. This equation is truly linear in θ . Depending on the particular experimental situation, the tracer θ could be density or temperature (see Fig. 2.8).
Fig. 2.8 Scalar (temperature) fluctuations in time recorded at a fixed point in a turbulent flow. (After P. Mestayer [81], with permission.)
2.5 Cascade Phenomenology and Scalar Spectrum
35
Following the line of argument of Obukhov and Batchelor, we suppose that the seed diffusivity D0 is so small as to make the effect of diffusion appreciable only at the large wave number end of the spectrum [12, 13, 28]. The part of the equilibrium → range of wave numbers for which the Fourier components of U are independent of viscosity is usually termed the “inertial subrange” and an appropriate term for the part of the equilibrium range for which the Fourier components of θ are independent of molecular diffusion is the “advection subrange.” No actual destruction of tracer variance takes place at high wave numbers as a result of the action of molecular diffusion. The total rate of destruction of variance per-unit volume is calculated by integrating the advection–diffusion equation over the whole domain: ∂ θ2 = −2D0 |∇θ |2 = −εθ . (2.5.2) ∂t This expression is similar to the energy conservation law: ∂ V2 = −εK ∂t
(2.5.3)
∂ θ 2 and shows that the advection term in (2.5.1) makes no contribution to ∂ t . Thus, when one Fourier component of the spectrum of θ is changed by the interaction → between θ and U, other Fourier components are changed simultaneously in such a way that the sum of the contributions to θ 2 from all Fourier components remains the same. This shows that θ variance is simply transferred from small to large wave numbers in the advection subrange and εθ is agiven constant quantity. The dissipation rate of the scalar “energy” θ 2 is also the spectral transfer rate. The linearity of this equation requires that the scalar spectrum,
Eθ (k) =
dΩk |θk |2 ,
(2.5.4)
is proportional to εθ . Then in the internal-convective subrange, where neither viscosity nor diffusion is important, dimensional analysis gives the spectrum (see Fig. 2.9):
where:
Eθ (k) = Cθ εθ εK −1/3 k−5/3 ,
(2.5.5)
) ( k < min kν = lν−1 , kθ = lθ−1 .
(2.5.6)
In the framework of cascade phenomenology, the increase of gradients of θ accompanying the stirring action of the velocity field, which is a consequence of the quadratic term of the scalar advection–diffusion equation, can also be thought of as a transfer between different Fourier components of the spectrum of θ . If both → → U and θ are written in the form of Fourier integrals, the term U · ∇θ leads to the generation of new harmonics of θ and the growth of ever-increasing wave numbers. The transfer of tracer variance from low wave numbers to high wave numbers is
36
2 Turbulent Diffusion and Scaling
Fig. 2.9 A typical plot of the scalar spectrum for Pr > 1
Eθ (k )
Pr > 1 Eθ (k ) ∝ k −5/ 3 Eθ (k ) ∝ k −1
kν
kθ
k
mathematically similar to that suggested by Kolmogorov for the velocity variance in a turbulent field. The k−5/3 temperature spectrum has been observed experimentally in turbulence of sufficiently high Reynolds number [25, 26, 28]. The parameter Cθ called the Obukhov-Corrsin constant, is found in the range Cθ ≈ 0.45–0.55 [65]. The diffusion scale lθ is determined by the balance of advection and diffusion in (2.5.1):
δ Vl δ θl δ θl ∝ D0 2 . l l If the Prandtl number Pr =
(2.5.7)
v < 1, D0
(2.5.8)
then we have lθ > lν , where
lν ≈
v3 εK
14
≈
1 kν
(2.5.9)
is the molecular dissipation scale (the Kolmogorov microscale), such that lθ lies in the inertial range of the velocity field, where: 1/3
Vl ∝ εK l 1/3 .
(2.5.10)
The diffusion scale lθ is given by the scaling:
lθ =
D3 εK
14
= lK Pr−3/4 ,
Pr < 1.
(2.5.11)
For lθ−1 < k < lν−1 , the inertial–diffusive subrange, the scalar energy spectrum falls off steeply. In this subrange, where the scalar modes are no longer governed by
2.5 Cascade Phenomenology and Scalar Spectrum
37
spectral transfer, Eθ (k) is driven locally by the velocity. Here, the scalar spectrum 2/3 Eθ (k) should be proportional to the velocity spectrum E(k), in particular to εK , and independent of the viscosity ν . One can represent the scalar spectrum in the following form: 5 −1 Eθ (k) ∝ εθ εk 3 k− 3 f klθ , Pr .
(2.5.12)
f = (klθ )αθ Prβθ ,
(2.5.13)
Then using factorization,
we obtain the exponents αθ = −4 and βθ = 0. This gives the internal-diffusive-range spectrum (see Fig. 2.10): Eθ (k) ∝ εθ εK D0 −3 k−17/3 ,
(2.5.14)
lθ−1 = kθ < k < kν = lν−1 ,
(2.5.15)
2/3
where
In the opposite case, Pr > 1, we have the viscous-convective subrange lν−1 < k < lθ−1 , where the scalar energy spectrum is flatter than k−5/3 . Here, the smallest velocity structures have the scale lν , producing the vorticity: V (l = lν ) Vlν εK 2 = ≈ . lν lν v 1
(2.5.16)
A fluid element of smaller scale is therefore strained by a uniform shear flow ε 12 K , such that the scalar transfer rate is proportional to this strain, v
Eθ (k )
Pr < 1 Eθ (k ) ∝ k
−5/ 3
Eθ (k ) ∝ k −17 / 3
Fig. 2.10 A typical plot of the scalar spectrum for Pr < 1
kθ
kν
k
38
2 Turbulent Diffusion and Scaling
εθ ∝ kEθ (k)
ε 12 K
, (2.5.17) v and one obtains the Batchelor viscous-convective subrange spectrum [12, 13, 28],
v εK
1 2
k−1 ,
(2.5.18)
lν−1 = kν < k < kθ = lθ−1 .
(2.5.19)
Eθ (k) ∝ εθ where
In this case, the diffusion length lθ is larger and it is determined by the consistency condition: kθ kθ 12 v εθ ∝ k k2 Eθ (k)dk ∝ k εθ k dk, (2.5.20) εK 0
where
kθ = lθ−1 .
0
The scaling for the diffusion length lθ is given by:
lθ =
vk2 εK
14
= lν Pr− 2 , Pr > 1. 1
(2.5.21)
In turbulence, the scalar field is highly intermittent and the scalar spectrum does not provide important information on the spatial distributions [82, 83, 84, 85, 86, 87, 88]. On the other hand, even the modification of the conventional diffusion equation to describe turbulent transport is a nontrivial problem because there is no unique recipe to combine memory effects, nonlocality, and cascade phenomenology and the best procedure depends on the application. We conclude this chapter with some notes in favor of cascade phenomenology. Indeed, the scaling behavior is one of the most intriguing aspects of fully developed turbulence. In this chapter, we derived the Kolmogorov spectrum for the inertial range on the basis of purely dimensional considerations. The basic hypothesis was that the kinetic energy was injected at a statistically steady rate. It was also conjectured that this same rate εK is the rate at which the energy is transferred through wave number space until it reaches the place where it can be dissipated. In other words, the energy is transferred and dissipated at the same rate at which it is injected. This is an important property of turbulent Navier-Stokes fluids that is universally agreed on; that the dissipation rate is not determined by any microscopic or molecular occurrence. There is no parameter that governs the dissipation rate; rather the fluid dissipates whatever you subject it to. If you stir the fluid harder, the spectrum just moves slightly farther out in k space until it finds a place where the energy can be dissipated at the same rate it is being injected. This is a fundamental and widely accepted assumption about the behavior of a Navier-Stokes fluid, although it has never been proved from first principles.
Further Reading
39
Further Reading Hydrodynamics and Scaling Barenblatt, G.I. (1994). Scaling Phenomena in Fluid Mechanics. Cambridge University Press, Cambridge, U.K. Doering, C.R. and Gibbon, J.D. (1995). Applied Analysis of the Navier-Stokes Equations. Cambridge University Press, Cambridge, U.K. Golitsyn, G.S. (1995). Convection in Rotating Fluids. Springer-Verlag, Berlin. Majda, A.J. and Bertozzi, A.L. (2002). Vorticity and Incompressible Flow. Cambridge University Press, Cambridge, U.K.
Turbulence Berge, P., Pomeau, Y., and Vidal, C. (1988). L’ordre dans le chaos. Hermann, Editeurs des sciences et des arts. 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. Gibson, C.H. (1996). Applied Mechanics Reviews, 49, 299. Herring, J.R. and McWilliams, J.C. (1987). Lecture Notes on Turbulence. World Scientific, Singapore. Lesieur, M. (1997). Turbulence in Fluids. Kluwer Academic, Dordrecht. McComb, W.D. (1994). The Physics of Fluid Turbulence. Clarendon Press, Oxford. Pope, S.B. (2000). Turbulent Flows. Cambridge University Press, Cambridge, U.K. Sreenivasan, K.R. (1999). Reviews of Modern Physics, 71, S 383. Tabeling, P. and Cardoso, O. (1994). Turbulence A Tentative Dictionary. Plenum Press, New York.
Turbulence and Correlation Effects Falkovich, G., Gawedzki, K., and Vergassola, M. (2001). Reviews of Modern Physics, 73, 913. Hunt, J.C.R. (1985). Annual Reviews of Fluid Mechanics, 17, 447. Moffatt, H.K. (1983). Reports on Progress in Physics, 621, 3. Tsinober A., (2004). An Informal Introduction to Turbulence. Kluwer Academic, Dordrecht. Vassilicos, J.C., ed. (2001). Intermittency in Turbulent Flows. Cambridge University Press, Cambridge, U.K. Warhaft, Z. (2000). Annual Reviews of Fluid Mechanics, 32, 203.
Chapter 3
Nonlocal Effects and Diffusion Equations
3.1 The Einstein Functional Equation The nonlocal nature of relative diffusion has stimulated the search for transport equations that differ significantly from conventional diffusive description. We return to the relative diffusion problem defined in the previous section where the dimensional estimates, which give qualitative explanations of the nonlocal transport effects in terms of interaction of different scales, were obtained. Besides the different phenomenological methods of modification of the diffusion equation, the integral equation can be used to describe the random walk processes. The use of such an equation in combination with the scaling ideas has led to the necessity of considering a distribution function that differs essentially from the Gauss function (1.2.9). A new type of distributions called the Levy-Khintchine distribution is now one of the basic tools for researching anomalous transport. As early as 1905, Albert Einstein obtained a functional equation for the particle density solely on the basis of the general ideas about the process of random walk [89]: n(x,t + τ ) =
+∞
WE (y)n(x − y,t) dy,
(3.1.1)
−∞
where WE (y) is the probability density of undergoing a jump y. This fundamentally nonlocal equation can be made local by reducing it to a diffusion equation. Assuming that the time scale τ is short and the jump y is small, we arrive at the classical diffusion equation [90, 91, 92, 93, 94, 95, 96, 97, 98]. In this way, we use the expansions:
∂n τ +... . ∂t
(3.1.2)
y2 ∂ 2 n ∂n y+ +... . ∂x 2 ∂ x2
(3.1.3)
n(x,t + τ ) = n(x,t) + n(x + y,t) = n(x,t) + Simple calculations yield:
O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008
41
42
3 Nonlocal Effects and Diffusion Equations
∂n n+ τ = n ∂t
∞
−∞
∂n WE (y) dy + ∂x
∞ −∞
∂ 2n WE (y)y dy + 2 ∂x
∞
WE (y) −∞
y2 + . . . . (3.1.4) 2
Assuming that the function W is symmetric, WE (y) = WE (−y),
(3.1.5)
and specifying the normalization condition, +∞
WE (y) dy = 1,
(3.1.6)
−∞
one obtains the conventional diffusive equation:
∂n ∂ 2n =D 2, ∂t ∂x
(3.1.7)
where the diffusion coefficient is given by: D=
1 τ
+∞
WE (y) −∞
y2 dy. 2
(3.1.8)
Note that the number of terms in expansions (3.1.2), (3.1.3) was chosen in a physically meaningful way. Based on the relationship characterizing the average behavior of Brownian particles, x2 ≈ R2 ∝ t. We can estimate the orders of the terms for t → ∞ in the expansions as follows: n(r,t) ∝
Np . R(t)d
(3.1.9)
Here, Np is the number of particles and d is the dimensionality of the space. In the one-dimensional case (d = 1), one can obtain the scaling estimates: n∝
Np 1/2
(3Dt)
∝ t −1/2 ,
(3.1.10)
∂n n ∂ 2n n ∝ ∝ t −3/2 , ∝ ∝ t −5/2 , ∂t t ∂ t2 t2
(3.1.11)
∂n n ∂ 2n ∝ ∝ t −1 , ∝ t −3/2 . ∂x R ∂ x2
(3.1.12)
Retaining only two terms in expansions (3.1.2) and (3.1.3), each results in a telegraph equation. However, this does not indicate that the telegraph equation is invalid. The reason is that, in this case, the effects of the finite propagation velocity of the perturbations come into play, which are absent in the classical diffusion model.
3.1 The Einstein Functional Equation
43
The integral approach was further developed in the papers by Smoluchowski, Chapman, and Kolmogorov [99, 100, 101]. A key element in this approach is Markov’s postulate [2, 5, 6, 7] that the length of the jump y is independent of the prehistory of motion. To describe the nonlocal effects the integral form of the equations is very important. Using expansion (3.1.2) of functional (3.1.1), we can readily obtain the Smoluchowski equation [99]:
∂ n(x,t) = ∂t
+∞#
$ K(x , x)n(x ,t) − K(x, x )n(x,t) dx .
(3.1.13)
−∞
Here, K(x, x )dx dx is the probability for a particle at position x at time t to pass over to the interval x + dx during the time interval dt. We introduce the functional:
G(x , x) = K(x , x) − δ (x − x )
+∞
K(x, x ) dx .
(3.1.14)
−∞
For a uniform isotropic medium, we have: G(x − x) = G( x − x ).
(3.1.15)
In the simplest case under consideration, this functional has the form:
∂n = ∂t
+∞
G(x − x )n(t, x )dx .
(3.1.16)
−∞
This representation reflects the nonlocal character of transport and at the same time it has a close relation to the conventional diffusion equation (see Fig. 3.1). It is t
4τ 3τ 2τ
Fig. 3.1 Construction of one-dimensional random flights in the space–time representation
τ
x
44
3 Nonlocal Effects and Diffusion Equations
possible to consider several analytical functions G(x) to find some solution of this equation [102, 103]. As an example one can form such an approximation on the basis of Poisson’s probabilistic law [2, 3, 4, 5], which describes the probability to make a jump λ , whereas the mean jump is λ :
dλ λ . (3.1.17) PV (λ ) = exp − λ λ The case of such an exponential kernel is characterized by the single characteristic scale λ , which makes it impossible to describe the turbulent cascade. However, another good way is to consider the Fourier transform of the functional kernel, which leads to new and very fruitful trends that are especially relevant for turbulent diffusion problems where the scaling representation plays an essential role.
3.2 Nonlocality and Levy-Stable Distributions Linear equations always provide the best conditions for an analysis. The Einstein functional is linear and it is more convenient here to switch to the Fourier representation for the particle density n(x, t) and the functional kernel G(x) with respect to the variable x. Formal manipulations yield the equation:
∂ n˜ k (t) %k n˜ k (t), =G ∂t
(3.2.1)
which indicates the absence of memory effects for the Fourier harmonics. Here, %k and n˜ k (t) are the Fourier transformations of the functions G(x) and n(x,t) with G respect to the variable x. The expression %k n˜ k = −Dk2 n˜ k G
(3.2.2)
corresponds to the classical diffusion equation, where D is the conventional diffusion coefficient. In the case of the telegraph equation, the memory effects were taken into account (see Eq. (1.3.13)):
∂ n˜ k (t) = −k2 ∂t
t 0
n˜ k (t)Mm (t − t )
dt = −k2 Mm (t)∗ n˜ k (t), τ
(3.2.3)
where the asterisk indicates the convolution operation. Applying the Laplace transformation in time, we obtain the following expression for the telegraph equation with memory: 2 % %M (k, s) = − Dk . (3.2.4) G 1 − is τ
3.2 Nonlocality and Levy-Stable Distributions
45
% % s) will denote both the Fourier and Laplace transformations of the Hereafter G(k, function GM (x,t) with respect to the variables x and t. It is an easy matter to combine the memory and nonlocality effects into a common expression containing a convolution:
∂ n˜ k (t) = −k2 ∂t
t
% k (k,t − t ) n˜ k (t)D
0
dt % k (k,t)∗ n˜ k (t). = −k2 D τ
(3.2.5)
Performing the Laplace transformation in time gives the transition from the conventional result to the new one: % % −Dk2 → −k2 D k,s (k, s).
(3.2.6)
In the framework of the probability theory, such heuristic methods are unsatisfactory. Below, we consider this point in more detail. The approach based on the Fourier representation of nonlocal functional equation was developed by Levy and Khintchine, who used the approximate equation of the scaling form [104]:
∂ n˜ k (t) = −const |k|αL n˜ k (t); ∂t
0 < αL ≤ 2.
(3.2.7)
It is easy to see that, for αL = 2, we are dealing with a Gaussian distribution (corresponding to a conventional diffusion equation). Indeed, the formal approach based on the scale invariant behavior of the probability density leads to the relation [102, 103, 104]: n (x,t) = t
− α1
L
g (x, αL ) .
Then, for the Gaussian distribution with αL = 2 we obtain:
2 1 x . g (x, 2) = √ exp − 4 2 π
(3.2.8)
(3.2.9)
Some other analytic distributions are also known. For the case αL = 1, we obtain the Cauchy distribution [105]: g (x, 1) =
1 . π (1 + x2 )
For the case αL = 3/2, one arrives at the Holtsmark distribution [3]:
2 3 4 3 1 1 exp − x W1/2,1/6 x . g (x, 3/2) = √ 27 27 3π x For the case αL = 1/2, we have the Levy-Smirnov distribution [6]:
(3.2.10)
(3.2.11)
46
3 Nonlocal Effects and Diffusion Equations
1 1 1 = √ . g x, exp − 2 4x 2 π x3
(3.2.12)
For the case αL = 2/3, we obtain the Smirnov distribution [7]:
2 1 4 1 1 1 exp W , g (x, 2/3) = √ −1/2,1/6 27 x2 27 x2 2 3π x
(3.2.13)
where the Whittaker function Wθ ,ϑ (z) is given by: z− 2 − z e 2 Γ (7/6) 1
W−1/2,1/6 (z) =
∞
e−t t 6 1
1+
t z
− 5 6
dt.
(3.2.14)
0
All the probability densities with αL < 2 have power-law tails and corresponding Levy flights differ significantly from Brownian walks (see Fig. 3.2). Another important feature is that the second and higher order of moments of the distributions with 1 ≤ αL < 2 and all moments of the distributions with 0 < αL < 1 diverge. There is also an important result that follows from the Fourier representation of density n(x,t):
2 1/2 ∂ ∂ n˜ k (t) |t=0 . (3.2.15) =− x ∂k ∂k This expression is useful for relating the transport scaling laws to probabilistic approximations. Now it is easy to find a relationship between the Hurst exponent H that describes anomalous transport and the Levy-Khintchine exponent aL that is the parameter of the power approximation (3.2.7): H=
1 . αL
(3.2.16)
These results were represented schematically and the reader can find more detailed information on these topics in numerous publications [2, 5, 6, 7, 34, 35, 36, 102, 103].
Fig. 3.2 Schematic illustration of the Levy flight steps with αL → 0
3.3 Fractional Derivatives and Anomalous Diffusion
47
3.3 Fractional Derivatives and Anomalous Diffusion One of the valuable concepts used to study various transport processes is scaling. Scaling has a surprising power of prediction and simple manipulations, allowing one to connect apparently independent quantities and exponents. Note that the power form of the Fourier representation for the kernel (3.2.7): % (k) = const |k|αL , G
(3.3.1)
where 0 < αL < 2 can be interpreted in terms of fractional derivatives that are widely used for the description of nonlocality effects. Indeed, for the common derivative, we have, by definition (see Fig. 3.3): Δy = μ Δx.
(3.3.2)
For a fractal function, we have (see Fig. 3.4): Δy = μH (Δx)αH ,
(3.3.3)
where μ and μH are the ordinary derivative and Holder derivative, respectively. More exactly, for Δx < 0 and Δx > 0 the left-hand and right-hand derivatives, μH− and μH+ respectively, must be introduced. A relevant example of physical objects with deficient smoothness is related to the Kolmogorov turbulence. Turbulent pulsations of the velocity field in the inertial range are of the order: δ V ∝ (δ r)1/3 , (3.3.4) and the velocity field has only one-third of the spatial derivative and αH = 1/3. Here, we are dealing with an unsmooth vector field. Another example is mathematical idealization of the Brownian motion (the Wiener process). In this case, the particle mass and the time between collisions tend to zero. Here, the displacement is given by:
Fig. 3.3 A smooth curve in the vicinity of some point has a tangent line
48
3 Nonlocal Effects and Diffusion Equations
Fig. 3.4 A fractal curve in the vicinity of some point has a curvilinear cone as the tangent
xt+dt − xt = xdt = Δxt ∝ (Δt)1/2 ,
(3.3.5)
where the Brownian particle trajectory has half a derivative with the Holder exponent αH = H = 1/2. Fractional derivatives provide a rather effective description of long-range correlations and memory effects. Here, we use the symmetric fractional derivative of an arbitrary order α > 0 that can be defined, for a “sufficiently well-behaved” function f (x), where −∞ < x < ∞, as the pseudo-differential operator characterized by its Fourier representation, dα f (x) = − |k|α f˜ (k) , d |x|α
(3.3.6)
where α > 0 and −∞ < k < ∞. On the left-hand side, we adopt the notation introduced in (3.2.7). To treat this kind of fractional derivatives property, we recall the definition of the left and right Liouville derivatives on the infinite axis [14, 34, 35, 36],
∂+α
f
(x) = Dα+ f
1 d (x) = Γ (1 − α ) dx
∂−α f (x) = Dα− f (x) = −
x
f (z) dz , (y − z)α
(3.3.7)
f (z) dz , (y − z)α
(3.3.8)
−∞ ∞
d 1 Γ (1 − α ) dx
x
where 0 < α < 1. For α ≥ 1:
∂±α
f
(x) = Dα± f
(±1)n d n (x) = Γ (n − α ) dxn
∞ 0
zn−α −1 f (x ∓ z) dz,
(3.3.9)
3.3 Fractional Derivatives and Anomalous Diffusion
49
n = [α˙ ] + 1, where the square brackets denote the integer part. Derivatives (3.3.7), (3.3.8), (3.3.9) are characterized by their Fourier representation in the form:
∂±α f (x) = Dα± f (x) = (∓ik)α f˜ (k) ,
α ≥ 1,
iαπ sgnk . (∓ik) = |k| exp ∓ 2
where
α
α
(3.3.10)
(3.3.11)
Thus, the symmetric fractional space derivative can be written as: 1 dα πα Dα+ f (x) + Dα− f (x) , where α 1, 3, . . . α f (x) = − d |x| 2 cos 2
(3.3.12)
Fractional integrals as well as derivatives are convolution operators with a power law kernel. This makes them convenient tools in scaling theory. Let us consider a real function f mapping the interval [a,b] onto the real axis. Its n-th order integral have the following form: yn−1 x y1 n ... f (yn ) dyn . . . dy1 = Ia+ f (x) = a a
a
1 (n − 1)!
x
(x − y)α −1 f (y) dy (x > a)
a
(3.3.13) as is readily proven by induction. Generalizing this expression to noninteger n defines the Riemann-Liouville fractional integral of order α > 0 as follows: n Ia+ f (x) =
1 Γ (α )
x
(x − y)α −1 f (y) dy,
(3.3.14)
a
for x > a and f being L1 integrable in the interval [a,b]. For 0 < a < 1, these definitions are extended to the whole real axis for f being L p integrable with 1 ≤ p < 1/α . A natural approach to define a fractional derivative tries to replace α with −α directly in expression (3.3.14):
∂+α f (x) =
d 1−α I f (x) , dx +
(3.3.15)
Let us consider some simple formulas to help perform calculations with a timedependent function: 1 d d β f (t) = Γ (1 − β ) dt dt β
t −∞
f (τ ) dz (t − τ )β
.
(3.3.16)
For a few time differentiations one obtains: d α +β dα dβ dβ dα = α β = β α. α + β dt dt dt dt dt
(3.3.17)
50
3 Nonlocal Effects and Diffusion Equations
Next, for a power function a simple scaling can be obtained: dα β Γ (1 + β ) β −α t t = ∝ t β −α . α dt Γ (1 + β − α )
(3.3.18)
There is also an important relation: limα →1
dα 1 = δ (t) . dt α
(3.3.19)
As an example of the fractional derivative application to nonlocal transport, let us consider a fractional generalization of the diffusion equation. Suppose X(t) is the position of a randomly selected particle along a line at time t ≥ 0 and n(x,t) is the density of X(t). If a sufficiently large aggregate of independent particles evolves according to this model, then n(x,t) defines the relative concentration of particles at locations x at time t > 0. Let us introduce the modeling equation to describe anomalous diffusion:
∂ n (x,t) ∂ αL n (x,t) ∂ αL n (x,t) = Dd βd , αL + Dd αd ∂t ∂ x αL ∂ (−x)
(3.3.20)
α
∂ L where 0 ≤ αd , βd ≤ 1, αd + βd = 1, Dd > 0, and ∂ (±x) αL are fractal derivatives of order 1 ≤ αL ≤ 2, which are easily defined in terms of Fourier transforms. After Fourier transforms we find:
d n˜ (k,t) = Dd βd (−ik)αL n˜ (k,t) + Dd αd (ik)αL n˜ (k,t) dt
(3.3.21)
so that, for the initial condition n˜ (k, 0) = 1, we obtain: n˜ (k,t) = exp[Dd βd t (−ik)αL + Dd αd t (ik)αL ].
(3.3.22)
This initial condition corresponds to the assumption X(0) = 0 with probability 1. Using & ' # πα $ αL # πα $ αL αL iπ /2 1 + i sign (k) tan (3.3.23) = |k| cos (ik) = e k 2 2 and
iπ iπ −e 2 = e( 2 −iπ ) = e−
we now rewrite: ! αL
n˜ (k,t) = exp Dd t |k|
cos
# πα $ 2
iπ 2
,
1 + i (αd − βd ) sign (k) tan
(3.3.24) # πα $ 2
" , (3.3.25)
and recognize the Fourier transform of a stable density with index αL . Hence, Dd (x,t) is the density of the Levy motion X(t), where 0 ≤ αL ≤ 1.
3.4 The Monin Nonlocal Equation
51
Of course, this approach is fairly formal. The model of greatest interest for which fractional derivatives are a natural tool for investigating turbulent transport is elaborated upon in the following section.
3.4 The Monin Nonlocal Equation The Einstein-Smoluchowski functional given in Eq. (3.1.13) could be applied to describe nonlocal effects of turbulent diffusion. This functional equation allows one to incorporate not only the whole hierarchy of spatial scales but also the hierarchy of characteristic times, which is related to the hierarchy of spatial scales. Monin was guided by ideas about the hierarchical properties of well-developed isotropic turbulence [106]. In the corresponding formulation of the problem, all sta1 tistical parameters are determined exclusively # $ by the scale length lk ≈ k ≈ [L] and 2
the mean energy dissipation rate εK = TL 3 . Here, L and T characterize the physical dimensionality of space and time. In the framework of Fourier’s representation for the generalized diffusion equation (3.2.7) there is only the “uncertain parameter” aL , which defines the power form of the kernel of the nonlocal functional. Based on dimensional considerations, it is possible to compose the approximation % for the power function G(k), which has T1 order that is in agreement with relaxation law (3.2.1):
∂ n˜ k (t) nk (t) ∝ . ∂t τRel (k)
(3.4.1)
Here τRel is the characteristic relaxation time. Then, the following expression for the kernel of the nonlocal functional describing turbulent diffusion is given by: 1 % = G( % εK , k) = ε 1 k 23 . = G(k) K3 τRel (k)
(3.4.2)
The expression has a dimensionality that is inversely proportional to the relaxation time τRel . Here, we are dealing with the relationship between the temporal and spatial hierarchies. This fact reflects the essential difference of such a model from the Batchelor approach [72] discussed above. This new representation is consistent with the approximation derived by Richardson [58] under the assumption that: % = −D(k)k2 , G(k)
(3.4.3)
where the dependence D(k) is given by: D(k) ≈
l2 ∝ l 4/3 ∝ k−4/3 . t
Also, in modern terminology [6, 7, 14, 34, 35, 36], the equation
(3.4.4)
52
3 Nonlocal Effects and Diffusion Equations 1 2 ∂ n˜ k (t) = −εK3 k 3 n˜ k (t) ∂t
(3.4.5)
is the one with the fractional derivative with respect to x, n ∂ αL n ∝ ∝ kαL n, α L ∂x (Δx)αL
(3.4.6)
where αL = 2/3 corresponds to the Smirnov distribution (see formula (3.2.13)). However, Monin was unsatisfied with the Fourier form of the transport equation. In fact, it is an equation with fractional derivatives:
∂n ∂ 2/3 n = εK 1/3 2/3 . ∂t ∂x
(3.4.7)
It is only recently that the idea of using fractional derivatives has come to be recognized [6, 34]. In an effort to derive an equation that would be as clear as the telegraph equation (the equation in partial derivatives), one can differentiate a new nonlocal equation twice with respect to time:
∂ 3n ∂ 2n = ε . K ∂ t3 ∂ x2
(3.4.8)
The solution in terms of the Whittaker functions behaves asymptotically as: n(x → ∞) ∝ x− 13 . 11
(3.4.9)
The problem of relaxation to a self-similar regime was discussed in detail in [28]. Note that Monin suggested a nonlocal equation to describe the diffusing particle probability density evolution n. However, this idea can be used to describe the probability density evolution F(lR ,t), which describes relative diffusion. Such a version was considered in refs. [107, 108]. Moreover, in those papers [107, 108], use was made of the modern terminology and the fractional differential is represented as the nonlocal integral operator: √ → ∂ F( l R ,t) 3 = Γ (2/3) 2 ΔL ∂t 4π
→
F( l ,t) 5/3 → → lR− lR
→
d3 l .
(3.4.10)
Here, Γ is the Gamma function and ΔL is the Laplace operator. It is natural that the use of the nonlocal operator leads to the distribution function, which differs significantly from the diffusive Richardson and Batchelor models. Nevertheless, convincing arguments in favor of choice of the specific type of equation describing the behavior of the distribution function are absent and the search for adequate theoretical models and experimental proofs has continued. One can see that there exist distributions in which neither the first nor the second moments exist and the limiting distribution is not a Gaussian distribution, but
3.4 The Monin Nonlocal Equation
53
a so-called stable or Levy distribution [3, 6, 7, 14, 34, 35, 36, 102, 103, 104]. The Gaussian distribution is just a special case of these stable distributions. The most striking property of a Levy distribution is the presence of long-range power law tails, which often lead to epy divergence of even the lowest order moments. Thus, both first and second moments are infinite if the characteristic exponent αL < 1, as already mentioned. It is essential that in physical terms, the divergence of the first and second moments for certain Levy distributions indicates the absence of underlying physical characteristic scales. This is often interpreted as the scale invariance, which is the manifestation of self-similarity. Thus, Levy’s infinite moment random walk trajectories considered above are now called Levy flights and, at present, they have been expanded into areas such as turbulent diffusion, polymer transport, and Hamiltonian chaos.
Further Reading Anomalous Diffusion Ben-Avraham, D. and Havlin, S. (1996). Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge, UK. Bouchaud, G.P. and Georges A. (1990). Physics Reports, 195, 132–292. Haus J.W. and Kehr, K.W. (1987). Physics Reports, 150, 263.. Metzler, R. and Klafter, J. (2000). Physics Reports, 339, 1. 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. Shiesinger, M.F. and Zaslavsky, G.M. (1995). Levy Flights and Related Topics in Physics. SpringerVerlag, Berlin..
Fractional Differential Equations Sornette, D. (2006). Critical Phenomena in Natural Sciences. Springer-Verlag, Berlin. Uchaikin V. (2003). Physics-Uspekhi, 173, 765. West, B.J., Bologna, M., and Grigolini, P. (2003). Physics of Fractal Operators. Springer-Verlag, New York. Zumofen, G., Klafter, J., and Shlesinger, M.F. (1997). Physics Reports, 290, 157.
Part II
Correlation Effects and Scalings
Chapter 4
Diffusive Renormalization and Correlations
4.1 The Corrsin Independence Hypothesis The classical Taylor definition of the turbulent diffusion coefficient, which is based on the Lagrangian correlation function, does not contain any information on molecular (seed) diffusion. It is obvious that a serious problem arises when we analyze the passive tracer transport. Here, we provide a brief exposition of several important models in which seed diffusion and correlation effects play a significant role. Thus, the definition of the correlation function is based on using Lagrangian velocities V (x0 , y), but their experimental determination is a serious problem [21, 23, 24, 25, 26, 109, 110]. For this reason, use is made of the Eulerian representation for the correlation function, which takes into account the velocity correlation at points separated by a distance λ : (4.1.1) CE (λ ,t) = u(x0 , T )u(x0 + λ , T + t) . Here, u(x0 , T ) is the Eulerian velocity at point x0 and time T . This form of the correlation function is more convenient for experimenters. We can also express the Lagrangian correlation function through the Eulerian velocities: (4.1.2) C (t) = u(x0 ; T )u(x(x0 , T + t); T + t) . However, there is no simple relationship between the Lagrangian correlation function and the Eulerian one. Actually, there is no Lagrangian relationship between the points x0 and x0 + λ in expression (4.1.1), where λ is merely some arbitrary displacement. Corrsin [23] suggested an approximation formula in terms of the randomization of the Lagrange correlation function with the probability density ρ (λ ,t): ∞
C (t) =
ρ (λ ,t)CE (λ ,t)d λ ,
(4.1.3)
−∞
O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008
57
58
4 Diffusive Renormalization and Correlations
where the Lagrangian correlation function is expressed through the Eulerian one. However, a more important point is the idea of the diffusive nature of the displacement λ , because for the probability density ρ (λ ,t) it is natural to use the Gaussian distribution that is the solution of the diffusion equation in three-dimensional space:
1 λ2 . (4.1.4) ρ (λ ,t) = exp − 4D0t (4π D0t)3/2 This formula includes the molecular diffusion coefficient D0 . Finally, the integral expression for the Lagrangian correlation function is given by: ∞
C(t) = −∞
CE (λ ,t) λ2 dλ . exp − 4D0t (4π D0t)3/2
(4.1.5)
From this point of view, one can note that λ is the distance and the diffusive displacement at the same time. Hence, we treat the turbulent transport in terms of the molecular diffusion. In fact, instead of formal averaging in the form ∞ V (0)V (y)δ (y − x(t)) dy, V (x(0))V (x(t)) =
(4.1.6)
−∞
the factorization approach was used (the “independence hypothesis”): V (0)V (y)δ (y − x(t)) = V (0)V (x) δ (y − x(t)) .
(4.1.7)
Moreover, Corrsin applied Gaussian distribution (3.1.4) to describe trajectory correlations: δ (y − x(t)) ≈ ρ (y,t). (4.1.8) A rigorous analysis by Weinstock [111] and Kraichnan [112] showed that the Corrsin conjecture is equivalent to a first-order truncation of the renormalization expansion, which can be considered systematically. The Corrsin conjecture has been tested against kinematical simulations of two- and three-dimensional flows with a peak energy spectrum sharply about one well-determined length scale [113] with the conclusion that it is valid for all times (not only for large times), provided that there is no helicity and that the flow is not frozen in time. To investigate the structure of atmospheric turbulence, Hay and Pasquill [114] suggested a somewhat different approach. They took into account that Eulerian and Lagrangian correlation functions have similar shapes, but at the same time, their characteristic scales are different. Thus, the characteristic temporal scale, which corresponds to the Lagrangian correlation function, is defined by the expression: ∞ 1 CL (t)dt. (4.1.9) TL = 2 V 0
4.1 The Corrsin Independence Hypothesis
59
The characteristic Eulerian temporal and spatial scales are defined analogously: 1 τE = 2 U lE =
1 U 2
∞
CE (Δ,t)dt,
(4.1.10)
CE (Δ,t)dΔ.
(4.1.11)
0 ∞ 0
Suppose that there exists a certain universal constant βC that allows us to relate Lagrangian and Eulerian scales (see Figs. 4.1 and 4.2): TL = βC τE , lL (βC t) = βC lE (t).
(4.1.12) (4.1.13)
The variety of turbulence types leads to the fact that the values of βC defined experimentally lie in a wide interval [21, 25, 26]: 1 ≤ βC < 8.5.
(4.1.14)
In spite of the obvious simplicity of the above approach, recent investigations showed that when considering one-particle vertical diffusion in strongly stratified turbulence, the Eulerian and Lagrangian velocity correlation functions are almost the same: (4.1.15) Vi (0)V j (t) = Ui (x, 0)U j (x,t) . To explain this, Kaneda and Ishida [113, 115] considered the Fourier transformation of the Corrsin conjecture (4.1.7) in the form → → → e−i k (x(t )−x(t)) , Vi (t)V j t = d 3 k R%i j k,t,t (4.1.16)
CE ( ζ ) C(t)
Fig. 4.1 Scale relationship between Lagrangian and Eulerian correlation functions
t’
ζ ' = βt '
t
60
4 Diffusive Renormalization and Correlations
Fig. 4.2 Scale relationship between Lagrangian and Eulerian spectra
EL ( n) E E ( n) nE(n)
n’
where → R%i j k,t,t =
1
βn '
Log n
→→ → → → → Ui x + r ,t U j x,t e−i k r d 3 r .
(2π )3
(4.1.17)
From the physical point of view, the Eulerian velocity correlations must be domi→
→
nated by large eddies. This corresponds to small values of k. Therefore, for k ≈ 0 one can expect that → e−i k (x(t )−x(t)) ≈ 1. (4.1.18) This phenomenological estimate gives the simplified form of the interrelation between the Lagrangian and Eulerian correlation functions: → → (4.1.19) Vi (t)V j t = Ui x,t U j x,t . Note that such a representation: CL (τ ) ≈ CE (λ , τ ) |λ =0
(4.1.20)
was first discussed by the author of [116]. In the framework of the Boussinesq approximation of strongly stratified flows, the validity of this simplified conjecture was checked by direct numerical simulations (see Fig. 4.3), and the simulation results agree well with the hypothesis (4.1.19).
4.2 The Correlation Function and Anomalous Diffusion The Corrsin conjecture appears fairly formal; however, it allows us to visualize correlation effects and to take into account the effects of molecular diffusion. Moreover, such a representation offers an additional possibility of developing the scaling
4.2 The Correlation Function and Anomalous Diffusion
61
Fig. 4.3 Comparison of Lagrangian correlations with Eulerian correlations from kinematic simulations. (After C. Cambon et al. [115], with permission)
approximation of transport by the power approximations of both the Eulerian correlation function and different kinds of probability density [2, 3, 4, 5, 6, 7]. Thus, Koch and Brady made a rigorous analysis of equations for a random noncompressible flow in which the mean velocity is zero and the spatial correlation function decays as the power function: CE (λ ) ∝
1 , λ αE
(4.2.1)
and obtained an expression that connects the Hurst exponent H that describes the transport character with the correlation exponent αE : H=
2 . 2 + αE
(4.2.2)
Here, αE describes the power behavior of the spatial correlation function of velocity:
αE 2 λ0 . (4.2.3) CE (λ ) = V (x)V (x + λ ) ∝ V0 λ The values V0 and λ0 are the dimensional parameters of the model. Note that this relationship for the Hurst exponent can be obtained by simple calculations based on both the dimensional consideration of the correlation function: CE ≈ V 2 ≈
λ2 , t2
(4.2.4)
and the power dependence (4.2.3). Then, comparison of (4.2.4) with (4.2.3) yields:
αE λ2 2 λ0 ≈ V0 . (4.2.5) t2 λ
62
4 Diffusive Renormalization and Correlations
If we suppose that the spatial scale λ is the correlation scale and the diffusive displacement at the same time (as in the Corrsin conjecture), then it is possible to treat such a dimensional estimate as the transport scaling. Let us express the value λ through the time t. Now, the transport estimate is given by: 2 1 λ ∝ V0 2 λ0 αE 2+αE t 2+αE
and hence: H (αE ) =
2 . 2 + αE
(4.2.6) (4.2.7)
Here, 0 < αE < 2 because this result was obtained for the two-dimensional incompressible flow, where the subdiffusive transport is absent [117, 118, 119, 120, 121, 122]. Actually, this simple scaling allows us to analyze the turbulent transport in a random two-dimensional flow basing on the scaling representation for the Eulerian velocity correlation function. We can see that long-range correlations (the small values of αE ) lead to the higher values of the Hurst exponent (large convective contribution). Moreover, in the framework of the multiscale ballistic estimate λ (t) ≈ V (λ )t, the turbulent transport in the presence of random shear flows could be considered by the modified expression λ⊥ (t) ≈ V⊥ λ// t, where longitudinal and transverse correlation scales are separated.
4.3 Seed Diffusivity and Correlations We now go one step further in scaling modeling the correlation effects. It is well known that interactions both create and destroy correlations. There is a useful example that illustrates this in terms of the correlation function. It was assumed [123, 124] that the number of interactions NI is proportional to the number of particles that are located in the correlation region WD (see Fig. 4.4): NI ≈ nWD ≈ nRD d .
(4.3.1)
Here, n is the density of particles in this region, RD is the spatial scale of this region, and d is the space dimensionality. Then from the dimensional aspect, the correlation function can be expressed in the form: C(t) = V (0)V (t) ≈ V0
V02 V0 ≈ , NI (t) nWD (t)
(4.3.2)
where V (t) = V0 /NI (t) is the velocity at the moment t and V0 is the characteristic scale of the velocity. The estimate of WD (t) can be obtained from the conventional Gaussian distribution:
4.3 Seed Diffusivity and Correlations
63
Fig. 4.4 Schematic illustration of the ensemble of interacting particles and correlation cloud
V (t ) ∝
V0 2dD0t n
R ∝ 2 D0t
ρ (x,t) =
x2 . exp − 4D0t (4π D0t)d/2 1
(4.3.3)
Here, D0 is the molecular coefficient of diffusion. To derive the correlation scaling, suppose that the correlation scale RD has the diffusive nature: RD (t) ∝ (D0t)1/2
for
t → ∞.
(4.3.4)
In principle, this is in accord with the Corrsin assumptions. Simple algebra then yields: V02 1 ∝ . (4.3.5) C(t) = V (0)V (t) ≈ n(D0t)d/2 t d/2 In spite of the difference between this result and the exponential approximation for the correlation function, the obtained power dependence is not meaningless. The “long tails” of correlation functions: C(t) = V02
τ 3/2 0
t
∝ t −3/2 ,
(4.3.6)
where V0 is the characteristic velocity and τ0 is the characteristic time, are being investigated in molecular dynamics and related to “the collective” (hydrodynamic) nature of the evolution of a system [123, 124]. The correlation function is related to the diffusion coefficient (1.4.10), which in our case leads to the relation: R2 1 d (4.3.7) C(t) ∝ DT ∝ 2 ∝ d/2 . dt t t This yields the transport scaling: H (d) =
4−d , 4
which differs significantly from the classical diffusive one.
(4.3.8)
64
4 Diffusive Renormalization and Correlations
4.4 Effective Diffusivity and the Peclet Number The “seed diffusivity” concept can be successfully applied to the consideration of complex correlation effects. The natural generalization of this approach is the application of the “self-consistent” diffusion coefficient that describes both the turbulent transport and molecular diffusion effects. On the other hand, the estimate of the turbulent diffusion coefficient in terms of the Lagrangian correlation function, DT ≈ V 2 τ , alludes to the relationship between diffusivity and the turbulent energy spectrum E(k). Here, we consider very briefly an interesting model [125] that will be repeatedly discussed in the context of scaling arguments. For a fuller treatment of this problem we refer the reader to the well known Moffat reviews [12, 13]. The Howells analysis of turbulent transport is based on the spectral energy function E(k) that plays a significant role in the framework of the phenomenological theory of isotropic turbulence [28, 64, 65]: 2 ∞ V V02 = = E(k)dk. 2 2
(4.4.1)
0
Here, V0 is the velocity scale and k is the wave number. To establish the direct relationship between the turbulent diffusion coefficient DT and the energy spectrum E(k), let us consider a “local” diffusion coefficient δ D(k) related to the specific scale length lk ≈ 1k of eddies with the characteristic velocity Vk :
δ D(k) ≈ Vk2 τ0 ,
(4.4.2)
where the velocity scale is expressed in terms of the energy spectrum: Vk2 ≈ E(k)δ k.
(4.4.3)
Here, δ k is a small interval of wave numbers and τ0 is the characteristic correlation time. Suppose that τ0 is related to the molecular diffusion Do :
τ0 ≈
1 k2 D0
.
(4.4.4)
Thus, we obtain the expression that is differential in the form:
δ D(k) E(k) = 2 . δk k D0
(4.4.5)
Note that the value of D(k) should be taken into account along with the molecular diffusion D0 . This is the simplest example of diffusive renormalization. One then finds:
4.4 Effective Diffusivity and the Peclet Number
65
E(k) dD(k) = 2 . dk k (D0 + D(k))
(4.4.6)
On solving this differential equation, we obtain the expression for the turbulent diffusion coefficient. This expression describes the influence of different scales in the framework of the Komlogorov phenomenology: ∞
(D(k) + D0 )2 =
E(k) dk + D20 , k2
(4.4.7)
k
where by assumption, D(∞) = 0. In this formula the integral term plays the main role for scales that are larger than the characteristic turbulent scale lT that enters into the expression for the Reynolds number: Re = V0νlT . Here, V0 is the characteristic velocity and νF is the coefficient of viscosity. Neglecting the molecular diffusion effects, we derive the Howells expression: ∞
DH 2 =
E(k) dk. k2
(4.4.8)
k
From the standpoint of dimensional estimates, the expression obtained, * +∞
+ E (k) V0 , DH = , dk ∝ k2 k
(4.4.9)
k
differs significantly from the quasilinear scaling DT ∝ V02 τ as well as from 2 the random walk representation D0 ≈ Δτ . Here, Δ is the characteristic spatial correlation scale length and τ is the characteristic correlation time. Indeed, the new scaling yields a different type of estimate for the diffusion coefficient: DH (V0 ) ∝ V0 λ ∝ D0 Pe,
(4.4.10)
where λ is the transport spatial scale and Pe is the Peclet number: Pe =
V0 λ . D0
(4.4.11)
This dimensionless quantity characterizes a convective contribution to transport in comparison with the diffusive one. In terms of the Peclet number, we obtain the relationships between these transport scalings: D0 = D0 Pe0 ≡ D0 ,
DH = D0 Pe.
(4.4.12)
The result presented in this form with different exponents θ is now in wide use [3, 12, 13, 18]: (4.4.13) De f f = D0 PeθT .
66
4 Diffusive Renormalization and Correlations
Here, De f f is the effective diffusion coefficient. It permits us to describe transport in terms of the exponent θT , and such a representation reflects the nature of the dependence on the velocity amplitude. Of course, there is no unique recipe to obtain the estimates of turbulent transport and the best way depends on a flow character. Thus, the diffusive renormalization considered above was confirmed by direct calculation of the correlation function, and the linear scaling DH ∝ V0 λ was repeatedly used to describe scalar transport in the system of convective cells, where the characteristic spatial scale λ is equal to the stochastic layer width.
4.5 Diffusive Renormalization and the Correlation Function There is a different way to derive the Howells formula in the framework of the correlation approach. Let us consider a fruitful method based on the Fourier representation of Lagrangian velocities appearing in the correlation function: C(t) = V (x(t);t)V (x(0); 0) = ∑ V%k (t) exp [ikx(t)]V%k (0) exp[ik x(0)] . (4.5.1) k,k
Here, < . . . > is the averaging symbol and V%k (t) is the Fourier transformation of the velocity V (x,t) over the spatial x-coordinate. The next step in correlation decomposition is the “independence conjecture” (compare with (4.1.7)): (4.5.2) C(t) = ∑ V%k (t)V%k (0) exp{ik[x(t) − x(0)]} . k
Then, Taylor and McNamara [126] suggested using “the diffusive behavior of trajectories” (the Corrsin conjecture): [x(t) − x(0)]2 ∝ Dt.
(4.5.3)
Here, D is the diffusivity, which depends on the model. This is in fact a “recipe for calculating the average” of the value exp [ikΔx(t)] in accordance with the conventional formula [5, 11]: ! " A2 exp A = exp . (4.5.4) 2 Calculations yield:
exp [ikΔx(t)] = exp −k2 Dt .
The expression for the correlation function was obtained in the form: 2 C(t) = ∑ V%k exp(−k2 Dt). k
(4.5.5)
(4.5.6)
4.5 Diffusive Renormalization and the Correlation Function
67
Let us consider this formula from the “correlation” standpoint. This expression can be interpreted as the sum of the Gaussian exponential correlation functions with “weight factors,” which is proportional to the turbulence spectrum E(k): C(t) ∝ ∑ E(k) exp(− k
t ). τk
(4.5.7)
Here, the characteristic time corresponding to the scale k is given by:
τk = 1/(k2 D).
(4.5.8)
It is necessary to take into account that this sum of the large number of exponents can turn out to be a nonexponential function. Similar expressions are most extensively employed to obtain correlation functions with power tails [34, 35]. Another important feature of expression (4.5.6) is the following formula for the diffusion coefficient: ∞
D= 0
E(k) 1 C(t)dt = ∑ 2 D k k
∞
exp(−k2 Dt)d(k2 Dt) ≈
1 D
E(k) dk. k2
(4.5.9)
0
We note the similarity of the resulting expression to the Howells result: ∞ 2
D =
E(k) dk. k2
(4.5.10)
k
One can see from the above analysis that the ideas of the interaction of scales and the correlation ideas of “diffusive behavior of trajectory” [23, 127, 128, 129] are closely interrelated. To complete the analysis we introduce a quantity: S(t) = [Δx(t)]2 ∝ R2 .
(4.5.11)
Then the expression for the correlation function takes the form: C(t) ≈
d Δx2 d2 d D≈ ≈ 2 S(t). dt dt t dt
(4.5.12)
On the other hand, from formula (4.5.9) we obtain: C(t) =
-
. E(k){1 − exp [−k2 S(t)]} dk.
(4.5.13)
In fact, one has a “Newtonian”-type differential equation for the mean square displacement: d2 S(t) = F{S(t)}. (4.5.14) dt 2
68
4 Diffusive Renormalization and Correlations
After a simple transformation, we find the final solution:
D2 =
. dk E(k){1 − exp [−k2 S(t)]} . k2
(4.5.15)
To obtain (4.5.10) it is sufficient to consider the case: k2 S(t) >> 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.”
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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 << l. The effective diffusion coefficient obtained can be rewritten in terms of the Peclet number Pe = VD00L as follows:
Deff = D0 +
8 945
V02 L2 = D0 + D0 D0
8 Pe2 . 945
(5.1.21)
Taylor’s development laid the foundation for an extensive literature on convective dispersion. However, it remained to study the approximations made and to determine their range of validity. Despite the fact that Taylor considered a laminar flow, his paper helps to understand the renormalization methods widely used at present for the turbulent transport description. Indeed, the similar scaling: (5.1.22) Deff ≈ D0 1 + const · Pe2 for the effective diffusivity characterizes the upper limit of turbulent transport in steady regimes [133]. On the other hand, the heuristic method considered here allows investigation of more complex flows. From the formal standpoint, the equation describing density perturbation: D0
∂ 2 n1 ∂ n0 = V1 (z) 2 ∂z ∂x
(5.1.23)
can easily be supplemented with the terms omitted before. Thus, including the term describing the time-dependent character of density perturbations ∂∂nt1 or using the Laplace operator to describe diffusion in a cylindrical tube of radius R0 retains the advantages of the algorithm of the solution, because the equation for density perturbation n1 has, as before, a linear form:
∂ n1 ∂ n0 = D0 ∇2 n1 −V1 . ∂t ∂x
(5.1.24)
5.2 Advection and Scalar Transport
75
This equation has a compelling physical interpretation: transverse variations in concentration are created by the shear flow tilting and stretching the averaged concentration and these variations are destroyed by transverse molecular diffusion. Note that there is also some arbitrariness in the choice of conditions for solving the partial differential equation. The solutions founded in the framework of a more precise formulation of the problem enable us to see that the nontrivial dependence obtained by Taylor of the effective diffusion Deff on the flow parameters is correct: Deff =
R20 V02 Pe2 + D0 = D0 1 + . D0 48 48
(5.1.25)
By such a method the scalar dispersion has also been investigated in a more complex flow topology and in the presence of memory effects. Thus, below we consider the model profiles Vx = Vx (z,t), to analyze time-dependent velocity fields and tracer diffusivity in the system of random shear flows [134, 135]. Taylor made a fundamental contribution to turbulence, championing the need for developing a statistical theory, and performing the first measurements of the effective diffusivity and viscosity of the atmosphere.
5.2 Advection and Scalar Transport Based on the Taylor renormalization method, it is appropriate to raise a question about the estimation of effective transport in a turbulent flow. Zeldovich [136] suggested the method for the investigation of scalar transport in a turbulent flow in the presence of seed diffusion based on the advection equation: → → ∂n = ∇ (D0 ∇n) − V r ,t ∇n. ∂t
(5.2.1)
→ → Here, the vector V r ,t describes the turbulent velocity field. Multiplying the equation by n and using the Gauss theorem, one finds the equation that describes the approach to equilibrium under diffusion for a single closed volume W : 1∂ 2 ∂t
n dW = −2D
2
W
(∇n)2 dW.
(5.2.2)
W
This equation is correct even when the liquid within the inner vessel is kept in motion. Here we consider an incompressible fluid. The motion of the liquid does not / affect the value of n2 dW in the absence of diffusion, as well as of any quantity of the type
/
W
f (n)dW . In the course of the motion, the position of a small volume dW
W
in which the concentration is n changes but neither n nor dW varies. On the other hand, we can omit the term describing density evolution in quasi-steady turbulence:
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5 Diffusion Equations and the Quasilinear Approximation
0=
1 ∂ 2 ∂t
n2 dW = W
nD0 (∇n)N dS −
D0 (∇n)2 dW.
(5.2.3)
W
S
The flux D0 (∇n)N characterizes the contribution of external sources inside the volume W , which is bounded by the surface S, whereas the term D0 (∇n)2 is related to the scalar redistribution inside the considered volume W . Therefore, it is convenient to introduce here the effective diffusive coefficient in the form: Deff =
1 n2 L 0
D0 (∇n)2 dW,
(5.2.4)
W
where L0 is the system characteristic size. Then, the minimum condition for Deff (the condition of minimizing functional) becomes a purely diffusive equation: ∇ (D0 ∇n) = 0,
where
min Deff = D0 .
(5.2.5)
Applying a method analogous to the above-considered Taylor approach, it is easy to obtain the upper estimate of the effective diffusion coefficient in a quasi-steady turbulent flow. Consider the steady scalar density equation: →
D0 Δn − V ∇n = 0
(5.2.6)
n = n + n1 = n0 + n1 , V = V + v1 = v1 ,
(5.2.7) (5.2.8)
using the perturbation method:
where V = 0, n1 << n0 , and D0 Δn0 = 0. Simple calculations lead to the equation for density perturbation n1 , which actually coincides with the Taylor “renormalization” formulated now for a turbulent velocity field: D0
∂ 2 n1 ∂ n0 . = v1 ∂ x2 ∂x
(5.2.9)
For the sake of simplicity, this equation is presented in the one-dimensional form. In the framework of the dimensional estimate, we obtain: n1 ≈ ν1
L0 n0 ≈ n0 Pe, D0
(5.2.10)
where the Peclet number is small, Pe << 1, which corresponds to weak turbulence. Deriving this relationship, we use the condition of smallness of the term v1 ∇n1 in comparison with ν1 ∇n0 . The expression for the effective diffusion coefficient is given by Deff ≈
1 n20 L 0
W
D0 (∇n0 )2 1 + A0 · Pe2 dW ≈ D0 1 + A0 · Pe2 ,
(5.2.11)
5.3 Zeldovich Flow and the Kubo Number
77
where A0 is the dimensionless constant. The reader can find a more rigorous analysis in [114]. Note that the term ∇n0 ∇n1 is illuminated because of the extreme properties of the distribution n0 . This upper estimate of transport Deff in the steady turbulent flow coincides with the Lagrangian representation Deff ≈ V02 τ when the correlation time τ has the diffusive nature:
τ ≈ τD ≈
L02 . D0
(5.2.12)
We discuss such a transition later in relation to the turbulent transport in the presence of complex structures.
5.3 Zeldovich Flow and the Kubo Number The time dependence of a flow is an important factor that leads to a reconstruction of the streamline topology and has an influence on transport processes. To describe such complex regimes we need a new dimensionless parameter, which includes the characteristic time T0 ≈ ω1 . In the case of high frequencies ω , the path of a test particle can be estimated ballistically: lω ≈
V0 ≈ V0 T0 . ω
(5.3.1)
Then, it is convenient to introduce the dimensionless Kubo number: Ku =
lω V0 ≈ , λ ωλ
(5.3.2)
where λ is the spatial scale of the flow under consideration. In the conventional case, we obtain the estimate of the diffusion coefficient with the correlation time τCOR ≈ ω1 and the correlation length ΔCOR ≈ lω as follows: DT ≈
2 ΔCOR V0 2 ≈ ≈ λ 2 ω Ku2 ∝ Ku2 . τCOR ω
(5.3.3)
Note that in the low-frequency regimes, where ω → 0 and Ku >> 1, this estimate is invalid because the real correlation scale ΔCOR could be much less than the formally defined frequency path: ΔCOR << lω ≈
V0 ω
ω →0
→ ∞.
(5.3.4)
To treat time dependence effects in the low-frequency regime Zeldovich considered [134] turbulent diffusion in the two-dimensional system of regular but timedependent flows (compare with the Taylor scalar dispersion). The solved equation has the form:
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5 Diffusion Equations and the Quasilinear Approximation
∂n ∂n +VX (z,t) = D0 ∂t ∂x
∂ 2n ∂ 2n . + ∂ x 2 ∂ z2
(5.3.5)
The expression for the velocity of flows is given by VX (z,t) = 2V0 cos(kz) cos(ω t).
(5.3.6)
The solution of this equation is written in the form: n(z,t) = n0 + n1 = n0 + sin(kz)(nS sin ω t + nC cos ω t + . . .).
(5.3.7)
The amplitudes of the harmonics nS , nC can be defined as a result of the solution of Eq. (5.3.5). It was suggested that n0 = < n > = ca + cb x.
(5.3.8)
The values V0 , ca , and cb are the flow characteristics. Substituting (5.3.7) into (5.3.5) with allowance for the assumption n1 << n0 yields the equation for the average density n0 : ∂ n0 ∂ n1 ∂ 2 n0 = − VX + D0 2 . (5.3.9) ∂t ∂x ∂x The equations for the amplitudes of the harmonics nS , nC have the form: nS +
D0 k2 2V0 ∂ n0 =− nC , w ∂x ω
(5.3.10)
D0 k 2 nS . ω
(5.3.11)
nC =
Simple calculations lead to the diffusive equations:
2 V02 k2 ∂ n0 ∂ n0 ∂ 2 n0 = D0 + D0 2 . 2 2 2 4 ∂t ∂x ∂x ω + D0 k Deff (k, ω ) = D0
(5.3.12)
Ku2 +1 , 1 + (ωτD )−1
(5.3.13)
1 . (D0 k2 )
(5.3.14)
where the characteristic time is given by
τD = For high frequencies ω >
1 τD
we arrive at the formula:
D ≈ D0
V0 2 k2 ≈ D0 Ku2 . ω2
(5.3.15)
5.3 Zeldovich Flow and the Kubo Number
79
In the case of linear dispersion,
ω 2 ≈ β V0 2 k2 ,
for
β → 0,
(5.3.16)
one obtains the scaling: V02 . (5.3.17) k2 The effective coefficient of diffusion takes the form of the Howells diffusion coefficient [72] Deff ≈ Vk0 . However, in contrast to Howells’ isotropic case, we analyze here the anisotropic model. The general case with many harmonics is described by the integral form ! " % V (k, ω ) dkd ω , (5.3.18) Deff = D0 1 + ω 2 + D20 k4 DD0 ≈ D2eff ≈
which is relevant to more complex dependences Vx (z,t). Despite this expression formally allows the estimate for ω → 0 to be obtained (see Fig. 5.2), it appears to be only an intermediate asymptotic and does not take into account effects related to the flow topology reconstruction (see Fig. 5.3), which are significant for Ku >> 1. From the general consideration, it is clear that in the low-frequency region the effective diffusion coefficient has to increase with the frequency (see Fig. 5.4): Deff (ω ) ∝ ω ηT ,
where
0 < ηT < 1,
(5.3.19)
since a slow reorganization of the flow topology does not lead to transport increasing. Indeed, simulations [18] confirm this supposition. Such a dependence can be interpret in terms of the Kubo number as follows:
Deff
D0 Pe 2
Deff ∝
Ku>>1
Fig. 5.2 A typical plot of the Zeldovich dependence for the two-dimensional time-dependent flow
λ /V0
1
ω
Ku<<1
ω
80
5 Diffusion Equations and the Quasilinear Approximation
Fig. 5.3 Schematic illustration of vortex reconnection
Deff (Ku) ≈ λ 2 ω
V0 λ ω
σT
∝ KuσT .
(5.3.20)
For the large Kubo numbers (low-frequency regime) this leads to the flat scaling Deff ∝ V0 σT , where the exponent σT is given by the simple relation:
σT + ηT = 1.
(5.3.21)
Below the reader will find a more detailed analysis of turbulent transport processes in low-frequency regimes.
Deff
Deff ∝ 1/ω
V0 λ
Fig. 5.4 A typical plot of the dependence of the effective diffusivity on the frequency
Ku >> 1
V0 /λ
Ku << 1
ω
5.4 Quasilinear Equations
81
5.4 Quasilinear Equations The quasilinear approximation has become widely popular due to its exceptional efficiency to describe correlation effects in the framework of conservation equations. Indeed, the construction of turbulent transport theory based on the continuity equation is more relevant than the consideration of equations incorporating the seed diffusion coefficient. Nevertheless, the Taylor work on scalar dispersion in a laminar flow underlies many papers where use is made of “fast modes” elimination. Note that quasilinear equations were suggested in [137, 138] in connection with the phase space description of the interaction between plasma waves and charged particles. For our purposes here, it is sufficient to consider only some of the ideas associated with the averaging method. We analyze the continuity equation for the density of a passive scalar in an incompressible flow:
∂n ∂n +V = 0, ∂t ∂x
(5.4.1)
where n(x,t) is the spatial density of the passive scalar and V (t) is the random velocity field. We average this continuity equation over the ensemble of realizations, assuming that the density field can be represented as a sum of the mean density n0 and the fluctuation component n1 = n − n: n(x,t) = n0 + n1 .
(5.4.2)
We also set n1 = 0 and the velocity field is represented as a sum of the mean velocity v0 and the fluctuation amplitude v1 , V = v0 + v1 , where v0 = const and v1 = 0. After simple algebra one obtains:
∂ n0 ∂ n1 ∂ n0 ∂ n1 + + (v0 + v1 ) + = 0. (5.4.3) ∂t ∂t ∂x ∂x On averaging this equation, we arrive at the equation for the mean density n0 : ∂ n0 ∂ n0 ∂ n1 + v0 + v1 = 0. (5.4.4) ∂t ∂x ∂x Subtracting this equation from the previous one, we find the equation for the density perturbation n1 : ∂ n1 ∂ n1 ∂ n0 ∂ n1 ∂ n1 + v0 + v1 + v1 − v1 = 0. (5.4.5) ∂t ∂x ∂x ∂x ∂x Here, it was assumed:
∂ n0 v1 ∂x
= 0,
∂ n1 v0 ∂x
= 0.
(5.4.6)
82
5 Diffusion Equations and the Quasilinear Approximation
As a result of these manipulations we arrive at the following set of equations for both the mean density n0 and the density perturbation n1 : ∂ n0 ∂ n0 ∂ n1 + v0 + v1 = 0; (5.4.7) ∂t ∂x ∂x ∂ n1 ∂ n1 ∂ n0 ∂ n1 ∂ n1 + v0 + v1 + v1 − v1 = 0. ∂t ∂x ∂x ∂x ∂x
(5.4.8)
Let us introduce a small parameter ε . We assume that the fluctuations n1 and v1 are as small as ε in comparison with the mean density n0 , n1 ∼ ε n0 and v1 ∼ ε v0 . The quasilinear character of the approximation indicates that, in the equation for n0 , we keep the nonlinear term of the order of ε 2 but, in the equation for n1 , we keep only the terms that are of the first order of ε . As a result, the transformations put the equation for density fluctuations n1 into the form:
∂ n1 ∂ n1 ∂ n0 + v0 = −v1 . ∂t ∂x ∂x
(5.4.9)
This equation could be solved via the Green function method. We consider this equation to be a first-order linear hyperbolic equation with the source term I(x,t) = −v1 ∂∂nx0 , where the derivative ∂∂nx0 is the parameter of the equation. We also supplement the equation with the uniform initial condition n1 (x, 0) = 0. Then, we obtain the expression for the Green function G:
∂G ∂G + v0 = δ (x − x1 )δ (t − t1 ). ∂t ∂x
(5.4.10)
It is easy to solve this equation by applying the Laplace transformation in time t and the Fourier transformation in the spatial coordinate x: exp(−t1 s) % exp(ikx). G%k,s = s + ikv0
(5.4.11)
Here and below, the tildes mark the Fourier- or Laplace-transformed quantities. The solution has a simple physical meaning: it describes a perturbation propagating along the characteristic z = x − v0 (t − t1 ): G(x,t, x1 ,t1 ) = δ (x − x1 − v0 (t − t1 ))Θ(t − t1 ),
(5.4.12)
where we have used Θ(t) to denote the Heaviside function. The solution for n1 (x,t) has the form: t ∂ n0 (z,t) dt1 . n1 (x,t) = − v1 (t1 ) (5.4.13) ∂z 0
Substituting this expression for n1 into the equation for the mean density n0 and performing simple manipulations yields:
5.5 Short-Range and Long-Range Correlations
∂ n0 ∂ n0 + v0 = ∂t ∂x
t
83
v1 (t)v1 (t1 )
0
∂ 2 n0 (z,t1 ) dt1 . ∂ z∂ x
(5.4.14)
The integral nature of this equation reflects the Lagrangian character of the relationships between the derivatives of n0 (x,t). In this respect, the new quasilinear equation is quite different from the initial continuity equation. The characteristic that appeared in our analysis relates the derivatives at different times. The left-hand side of the quasilinear equation contains the partial derivatives with respect to x and 2 t. On the right-hand side, we sum the values of the derivative ∂∂ xn20 calculated along the characteristic with a weighting factor, which is the autocorrelation function of velocity, C(t,t1 ) = v1 (t)v1 (t1 ) . Thus in the case of a steady random process, the function C(t,t1 ) ≈ C(t −t1 ) in the equation under analysis plays the role of the memory function. The particular form of the transport equation is governed by the choice of the correlation function C(τ ).
5.5 Short-Range and Long-Range Correlations Developing the quasilinear approach, we discuss different approximations of correlation effects. In the simplest, physically meaningful case, Eq. (5.4.14) reduces to the classical diffusion equation:
∂ n0 ∂ n0 ∂ 2 n0 (x,t) + v0 =D . ∂t ∂x ∂ x2
(5.5.1)
This is possible only if the main contribution to the integral on the right-hand side of Eq. (5.4.14) comes from a short interval (t −t0 ;t) such that t0 << t. If the second derivative changes insignificantly over the short interval, we obtain: t 0
v(t)v(t1 )
∂ 2 n0 (z,t) ∂ 2 n0 (x,t) dt1 ≈ ∂ z∂ x ∂ x2
t
C(t − t1 ) dt1 .
(5.5.2)
t−t0
In fact, we are assuming that the correlations are short-range. Thus, in this approximation, we arrive at the familiar Kubo-Green formula for the diffusion coefficient [139, 140, 141, 142]: ∞
D=
C(τ ) d τ .
(5.5.3)
0
In terms of the δ -correlations [5, 40],C(t − t1 ) ≈ C0 τδ (t − t1 ), quasilinear equation (5.4.14) takes the conventional form with the Taylor estimate of diffusivity, DT ≈ V02 τ ≈ C0 τ : ∂ 2 n0 (x,t) ∂ n0 ∂ n0 + v0 = C0 τ . (5.5.4) ∂t ∂x ∂ x2
84
5 Diffusion Equations and the Quasilinear Approximation
In the case of long-range correlations, we could assume that C (t) ≈ const for t1 >> 0, and Eq. (5.4.14) reduces to:
∂ n0 ∂ n0 + v0 = C0 ∂t ∂x
t
∂ 2 n0 (z,t1 ) dt1 . ∂ z∂ x
0
(5.5.5)
This equation can be further simplified by using the properties of the characteristic z. Differentiating this equation with respect to x gives:
∂ 2 n0 ∂ 2 n0 + v0 2 = C0 ∂ t∂ x ∂x
t 0
∂ 3 n0 (z,t1 ) dt1 . ∂ x3
(5.5.6)
Differentiating the same equation with respect to t gives:
∂ 2 n0 ∂ 2 n0 ∂ 2 n0 = C + v − v0 0 0 ∂ t2 ∂ x∂ t ∂ x2
t 0
∂ 3 n0 (z,t1 ) dt1 . ∂ x3
(5.5.7)
Eliminating the integral in expression (5.5.6) and (5.5.7) yields:
∂ 2 n0 ∂ 2 n0 ∂ 2 n0 + (v20 −C0 ) 2 = 0. + 2v0 2 ∂t ∂ x∂ t ∂x
(5.5.8)
This equation differs markedly from the classical diffusion equation. For C0 > 0, it is a hyperbolic equation, possessing the corresponding properties. Thus, a complete solution to this equation can be represented as a superposition of two initial distributions n0 (x, 0) moving at different velocities. As is known, the fact that hyperbolic equations have characteristics opens new possibilities for describing nonlocal effects. It should be noted, however, that, from a rigorous point of view, the above passage from a parabolic to a hyperbolic equation is incorrect since the Cauchy problems for these two types of equations are radically different.
5.6 The Telegraph Equation The well-known example of the transport equation, which has a hyperbolic form, is the telegraph equation [27, 28, 29, 30, 31, 143]:
∂ 2n ∂n ∂ 2n +τ 2 = D 2 . ∂t ∂t ∂x
(5.6.1)
This equation was one of the first so-called nondiffusion equations to describe turbulent transport. One can obtain this equation in the framework of the quasilinear approach. Indeed, in the theory of random processes [5, 40], one of the most widely used correlation functions is an exponential one:
5.6 The Telegraph Equation
85
C(t) = C0 exp(− |t| /τ ).
(5.6.2)
Here, τ is the characteristic time. This choice is quite natural because it is in this form that the correlation function is applied in the rigorous description of randomwalk processes. By means of this exponential function, we can transform integral equation (5.4.14) to a partial differential equation:
∂ n0 ∂ n0 + v0 = ∂t ∂x
t
C (t,t1 ) 0
∂ 2 n0 (z,t1 ) dt1 . ∂ z∂ x
(5.6.3)
To do this, we set C(t,t1 ) = C(t − t1 ). Differentiating this equation with respect to x gives:
∂ 2 n0 ∂ 2 n0 + v0 2 = ∂ t∂ x ∂x
t
C(t − t1 )
0
∂ 3 n0 (z,t1 ) dt1 . ∂ x3
(5.6.4)
Differentiating the same equation with respect to t gives:
∂ 2 n0 1 = C0 2 − ∂x τ0
t 0
∂ 2 n0 ∂ 2 n0 + v 0 ∂ t2 ∂ t∂ x ∂ 2 n(z,t1 ) C(t − t1 ) dt1 − v0 ∂ x2
(5.6.5) t
C(t − t1 )
0
∂ 3 n0 (z,t1 ) dt1 . ∂ x3
Eliminating the integral in these two equations yields:
2 ∂ n0 ∂ n0 ∂ n0 ∂ 2 n0 ∂ 2 n0 2 + v0 + τ0 + (v0 −C0 ) 2 = 0. + 2v0 ∂t ∂x ∂ t2 ∂ x∂ t ∂x
(5.6.6)
In accordance with the hyperbolic nature of the problem, we introduce the new set of variables:
ξ = t; η = x − v0t
(5.6.7) (5.6.8)
to obtain the telegraph equation form:
∂ n0 ∂ 2n ∂ 2 n0 + τ0 2 = C0 τ0 , ∂ξ ∂ξ ∂ η2
(5.6.9)
√ where C0 is the propagation velocity. This is actually the telegraph equation (1.3.2) in a frame of reference related to the coordinates ξ , η . One can see that the quasilinear approach offers the possibility to bring into accord both the hydrodynamic equation for particle density and correlation meaning of the problem. The solvable theoretical models considered above are often rather simplified, but at the same time there are many advantages in their treatments. Naturally, in theoretical treatments we are free from many artificial restrictions and we could extract the
86
5 Diffusion Equations and the Quasilinear Approximation
essence of scientific problems without the distraction of different unknown elements such as numerical errors or experimental noise. Another merit is the generality of theoretical treatments of fluid dynamics problems whereas in experiments and simulations the number of realizations and the size of the system are often limited by both finances and time.
Further Reading Seed Diffusion Effects Brenner, M.P. (2000). Classical Physics Through the Work of GI Taylor. MIT Lectures. Crisanti, A., Falcioni, M., and Vulpiani, A. (1991). Rivista Del Nuovo Cimento, 14, 1–80. Mazo, R.M. (2002). Brownian Motion, Fluctuations, Dynamics and Applications. Clarendon Press, Oxford. 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. Nieuwstadt, F.T.M. and Van Dop, H., eds. (1981). Atmospheric Turbulence and Air Pollution Modeling. D. Reidel, Dordrecht. 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.
The Quasilinear Approach Kadomtsev, B.B. (1976). Collective Phenomena in Plasma. Nauka, Moscow. Kingsep, A.S. (1996). Introduction to the Nonlinear Plasma Physics. Mosk. Fiz.-Tekh. Inst., Moscow. Krommes, J.A. (2002). Physics Reports, 360, 1–352. Tsytovich, V.N. (1974). Theory of Turbulent Plasma. Plenum Press, New York.
The Telegraph Equation Joseph, D.D. and Prezioso, L. (1989). Reviews of Modern Physics, 61, 41. Uchaikin, V. (2003). Physics-Uspekhi, 173, 765.
Chapter 6
Return Effects and Random Shear Flows
6.1 “Returns” and Correlations Long-range correlation effects are responsible for anomalous transport in complex systems. In everyday language, “correlation” means a relationship between events. The probability theory employs the rigorous mathematical notion of “return of a walking particle” to the initial point [3, 4] to describe simple correlation effects (see Fig. 6.1). This is best illustrated by considering the problem of one-dimensional random walks at the very beginning of the process. In the problem as formulated, the particle will definitely return to its initial position, thereby providing a clear realistic interpretation of the abstract notion of correlations. Rigorous analysis of returns on complicated spatial grids is necessarily based on the chain functional equation for the return probability P0 (t). Recall that most of the fundamental problems in the theory of random-walk processes can be formulated in terms of chain functional equations [6, 7]. However, we restrict ourselves here to the brief consideration of return effects. Qualitative estimates for these effects can be obtained from the classical solution to the equation for the probability density function ρ (x,t) describing the random walks of a particle. For a space of dimensionality d, one obtains the distribution:
(δ x)d x2 . (6.1.1) PG (x,t) = ρ (x,t) (δ x)d = exp − 4Dt (4π Dt)d/2 Here, (δ x)d is the small area around the point x and D is the diffusion coefficient. The probability for a particle returning to point x = 0 at time t has the form: P0 (t) ∝
(δ x)d . (4π Dt)d/2
(6.1.2)
In the one-dimensional case we arrive at: P0 (t) ∝
δx 1 ∝ 1/2 . 1/2 (4π Dt) t
O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008
(6.1.3)
87
88
6 Return Effects and Random Shear Flows
Fig. 6.1 Schematic illustration of the return times t1 , t2 to the origin of a random walk
t
t2
t1 x
Generally, this simple (although rather efficient) formula serves merely to provide estimates. It has the same drawbacks as the conventional diffusion model. However, for our purposes here, this solution is important because it provides evidence that the dimensionality of the space, d, which was used above as a formal parameter, plays a significant role. It turns out [3, 4, 14, 16] that, for grids of dimensionality d ≤ 2, the particle will inevitably return to its initial position. For grids with d > 2, the particle can execute random walks without returning. We thus see that the case d = 2 is intermediate and, as such, attracts much attention among both mathematicians and physicians. Note that the correct dependence for P0 when d = 2 and d = 3 is given by P0 (N) ∝
1 1 << d/2 . N2 N
(6.1.4)
Along with the return probability P0 , use is made of the number of returns and the number of visited grid points. Usually, the task is to express these numbers as certain scaling laws and to establish their relationships to other scaling laws. Using such probabilistic approximations, we derive an important scaling relation for particles executing random motion with no self-intersections. A self-avoiding random walk is a random walk that never intersects its own trajectory (see Fig. 6.2). Though this condition is very simple, theoretical treatment becomes extremely difficult, since the whole past trajectory affects the present motion. We introduce the probability p(N) of self-intersection after N random walks: p(N) ≈
N , Rd
(6.1.5)
where R2 (N) is the root-mean-square displacement, d is the space dimensionality, t and N = is the number of random walks. Here, t is the time and τ is the correτ lation time. In fact, we are assuming that the probability of the particle trajectory intersecting itself is proportional to the number of visited grid points within the region of random particle motion. Then, the probability for a particle to execute N
6.1 “Returns” and Correlations
89
Fig. 6.2 An example of the two-dimensional self-avoiding random walk
self-avoiding random walks can be estimated as PS (N) ≈ (1 − p)N |N→∞ ≈ exp(−pN) ≈ exp(−
N2 ). Rd
(6.1.6)
Taking into account the fact that the relationship between the quantities R and N is of a diffusive nature, we can estimate the effective probability of self-avoiding random walks by averaging the probability PS (N) with the Gauss distribution: ∞
PS (t) = −∞
1 1 t 2 R2 (dR)d . exp − d exp − 4Dt R τ (4π Dt)d/2
(6.1.7)
We assume that the main contribution to the integral comes from the extremum of the integrand:
2 R2 1 t (6.1.8) + min 4Dt Rd τ and perform simple manipulations to obtain the scaling law: 3
1
R (t) ∝ t 2+d >> t 2 ,
(6.1.9)
for d ≤ 3. Here, we must take into account the fact that, in a space of dimensionality d = 1, non-self-intersecting random walks can occur only for the particles moving in one direction, which indicates that R ∝ t. We see that this scaling satisfies this condition automatically. The following relation for the Hurst exponent: H (d) =
3 , 2+d
(6.1.10)
90
6 Return Effects and Random Shear Flows
which was first obtained in the theory of polymers by Flory [3, 4], is very important in describing the properties of different systems with non conventional correlation properties and complex topology. This scaling for H(d) gives a correct value in the case d = 1, d = 2, and d = 4. For d = 3, the mean field theory does not give an exact value. A renormalization group method gives: 1 1 = = 1.701 ± 0.003, H (3) 0.588 ± 0.001
(6.1.11)
which is in good agreement with computer simulations [4]. Note that in the famous de Genes approach the self-avoiding random walk generating function can be mapped exactly onto the portion function of the vectorial spin Heisenberg model in the limit when the number of component k of the order parameter goes to zero [144].
6.2 Superdiffusion and Return Effects In 1971, Corrsin published an interesting report [145] devoted to the probabilistic description of turbulence, wherein he formulated several problems calling for solution. One was the inclusion of “returns” of particles, which diffuse in the turbulent flow. Virtually simultaneously a paper by Dreizin and Dykhne [135] was published where a physically clear model of a strongly anisotropic transport was investigated. Here, we consider this model in terms of return effects. First, we select the longitudinal direction and assume that a “seed” diffusion with the coefficient D0 acts on the plane. In the transverse direction, the diffusing particle experiences random pulsations, which produce narrow convective flow with a velocity V0 and a width a0 (Fig. 6.3). Here, the velocity field has a “quenched” randomness. In the transverse direction the diffusion can be neglected compared to the velocity drift carrying the molecule with the flow. In the framework of this consideration there is no drift in the direction perpendicular to the layers. Let us consider a simple model for calculating the diffusion coefficient in the transverse direction D⊥ : λ2 (6.2.1) D⊥ ≈ ⊥ , t where the transverse displacement λ⊥ is given by the quasi-ballistic expression:
λ⊥ ≈ V0t P∞ ,
(6.2.2)
and P∞ is the relative number of the small fraction of “noncompensated” fluctuations δ N: δN P∞ = . (6.2.3) N The number of shear flows intersected by the particle during its longitudinal motion is defined as:
6.2 Superdiffusion and Return Effects
91
Fig. 6.3 The system of random shear flows (the Dreizin-Dyhne model)
z
a0
V0
√ 2D0t N (t) ≈ . a0
(6.2.4)
Using the Gauss representation for the number of “noncompensated” fluctuations δ N: √ δ N ≈ N ∝ t 1/4 , (6.2.5) we obtain the following formula for transverse diffusion coefficient: 0 t 2 D⊥ ∝ V0 a0 . D0
(6.2.6)
This leads to transport scaling in the form:
1/2
λ⊥ ∝ V0 a0
t D0
3/4 .
(6.2.7)
In the superdiffusive case under consideration, it was found that the Hurst exponent is H = 3/4 > 1/2. To explain this result in the framework of the correlation approach, we consider the correlation function of shear flows in the following form: C(t1 ,t2 ) =
∞
Vx (0)Vx (z) ρ (z,t2 − t1 ) dz.
−∞
(6.2.8)
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6 Return Effects and Random Shear Flows
Suppose that the probability density has the Gaussian form:
1 z2 . ρ= exp − 4D0 (t2 − t1 ) (4π D0 (t2 − t1 ))1/2
(6.2.9)
Here, Vx (z) is the velocity of the flow at the point z. This representation corresponds exactly to the Corrsin idea about the diffusive nature of decorrelation with allowance for the anisotropy of the model. However, using the conjecture about the significant role of returns has become the main step in the description of anomalous diffusion, since the condition z → 0 for the probability density ρ (z,t) corresponds to the return to the initial point. Thus, we have the expression: V 2 a0 , C(t1 ,t2 ) = C(τ ) ≈ √ 0 4π D0 τ
τ = t2 − t1 .
(6.2.10)
Using the classical Taylor expression from the turbulent diffusion theory (1.4.4), one defines the mean square displacement in the perpendicular direction:
λ⊥2
V 2 a0 ≈ √0 4π D0
t t 0 0
V 2 a0 3/2 dt dt √1 2 ≈ √0 t . t1 − t2 4π D0
(6.2.11)
Here denotes the averaging symbol. From the physical point of view, such a representation is valid only if the perpendicular spatial displacement is no larger than the perpendicular correlation length:
λ⊥ (t) < ΔCOR ≈ a0 .
(6.2.12)
V 2a λ⊥ 2 ≈ √0 τCOR 3/2 ≤ a0 2 , D0
(6.2.13)
This restriction yields:
and hence one obtains:
τCOR ≈
D0 a0 2 V0 4
13 .
(6.2.14)
If time scales t > τCOR , then we are dealing with conventional diffusion: 4/3
D⊥ ∝
V a0 2 ∝ 01/3 . 2τCOR D 0
(6.2.15)
Analyses carried out in [146, 147] in the framework of the renormalization theory confirms that if the shear flow velocity V0 is not strictly parallel to the “layers” and if the sample is finite in the longitudinal directions, then the superdiffusive regime D⊥ (t) ∝ t 3/4 is destroyed.
6.3 Random Shear Flows and Stochastic Equations
93
6.3 Random Shear Flows and Stochastic Equations The Dreizin and Dykhne result became well known after the paper [148] by Matheron and De Marsily in connection with learning transport in a porous medium. Their analysis is based on the consideration of stochastic equations of motion in longitudinal and transverse directions: dx = VX (z (t)) , dt
(6.3.1)
dz = η (t) . dt
(6.3.2)
Here, Vx and the shear velocity distribution are taken to be white noise: VX (z)VX z = σV δ z − z .
(6.3.3)
At the same, time the motion along the z-axis is the conventional Brownian motion is given by η (t) η t = 2D0 δ t − t .
(6.3.4)
Note that in the framework of the Dreisin-Dykhne superdiffusive regime, the Brownian motion in the transverse direction can be neglected. Then, longitudinal and transverse displacements can be rewritten in the form: t
z (t) =
η t dt or Δz2 = 2D0t, t
x (t) =
VX z t dt .
(6.3.5) (6.3.6)
To obtain the scaling describing transport in the transverse direction, let us introduce a distribution function J(z,t) of the number of visits that were made by a test particle into different “jets” by the moment t: t
J(z,t) =
δ z − z t dt.
(6.3.7)
0
The formal expressions for the displacement x(t) can then be represented in the form ∞
x(t) =
Vx (z)J(z,t) dz.
(6.3.8)
−∞
Since in the transport analysis the mean values play the main role, we consider the simplest Gaussian approximation of the mean probability density J:
94
6 Return Effects and Random Shear Flows
t
J(z,t) = 0
|z| 1 z2 , ρ (z,t )dt = √ Γ − ; 2 4D0t 4 π D0
(6.3.9)
where Γ − 12 ; x is the incomplete gamma function. Then the mean displacement is given by the expression: 1 x(t) = √ 4 π D0
∞ −∞
1 z2 Vx (z) dz |z| Γ − ; 2 4D0t .
(6.3.10)
Using the averaging over environments, we obtain the expression for the mean square displacement that takes into account the statistical properties of the field Vx (z): ( ) x(t)2 = dz dz VX (z)VX (z ) J(z,t)J(z ,t) . (6.3.11) Applying the statistical properties of the velocity field V (z)V (z ) = σV δ (z − z ) it is easy to find a final solution:
x (t) 2
σV t 3/2 = σV dzJ(z,t) = √ D0 π √ 4 σV = √ 2 − 1 √ t 3/2 . 3 π D0
∞
2
2
1 s2 Γ − ; s2 ds 2
0
(6.3.12)
The averaging method used here is not traditional; however, the scaling obtained is correct: λ⊥ 2 ∝ t 3/2 . (6.3.13) It is worth pointing out the nontrivial character of the different ensemble averages. Thus, the alternative possibility is the definition of xm (t) by the expression:
t m−1 t t1 x (t) = m! dt1 dt2 . . . dtm VX (z(t1 )) . . .VX (z(tm )) , m
0
0
(6.3.14)
0
where VX (z(t1 )) . . .VX (z(tm )) ∞
=
dz1 . . . dzm−1 dzm VX (z1 ) . . .VX (zm−1 )VX (zm ) ρ (zm ,tm ) ρ
−∞
× (zm−1 − zm ,tm−1 − tm ) . . . ρ (z1 − z2 ,t1 − t2 )
(6.3.15)
To calculate xm it is convenient to use the Laplace transformation in time:
6.4 The “Manhattan-Grid” Flow and Turbulent Transport
xm (s) =
m! s
∞
95
dz1 . . . dzm VX (z1 ) . . .VX (zm ) ρ (zm , s) ρ (zm−1 − zm , s) . . . ρ (z1 − z2 , s).
−∞
(6.3.16)
Using the Gaussian representation for ρ (z,t) allows us to carry out the necessary calculations. Thus, for the mean square displacement one obtains:
∞ ∞ 2σV x (s) = dz1 dz2 δ (z1 − z2 ) ρ (z2 , s) ρ (z1 − z2 , s) s 2
−∞ −∞
=
2σV ρ (0, s) s
∞
−∞
σV dzρ (z, s) = √ D0
1
s5/2
which by the Laplace inversion yields:
x2 (t) =
4σ √ V t 3/2 . 3 π D0
(6.3.17)
Actually, in such an approach we consider the averages, which are first taken over the walks . . .W and then over the shear flow configurations . . .C , i.e., . . . ≡ . . .W C .
6.4 The “Manhattan-Grid” Flow and Turbulent Transport There are several ways to generalize the random shear flow model. The above consideration showed that even a small fraction of noncompensated flows, P∞ (t) =
1 δN ∝ 1 , N N (t)
(6.4.1)
leads to a considerable deviation of transport from the diffusive behavior. Moreover, several other important models of anomalous and percolation transport can be described in the framework of a similar approach. Thus, formula (6.2.10) for the correlation function includes the number of returns NB or the number of intersected flows NI for the particle walking along the z-axis during time t as: √ 2D0t N(t) ∝ . (6.4.2) a0 The number of returns NB in the interval δ z can be estimated in terms of return probability ρ (0,t)δ z ≈ ρ (0,t)a0 . Thus, we obtain the estimate: √ t D0t ρ (0,t)a0 ∝ . (6.4.3) NB (t) ∝ τ// a0
96
6 Return Effects and Random Shear Flows
Here, τ// is the longitudinal correlation time: D0 . a0 2
τ// ∝
(6.4.4)
In the case of the one-dimensional longitudinal diffusion (d = 1), which we are considering, we have: (6.4.5) NB (t) ∝ NI (t). From this point of view, the use of the number of “events” as the growing function NE = NE (t) allows the construction of the correlation function in the form: C (t) = V (0)V (t) ∝ V0
V02 V0 ≈ . NE (t) NE (t)
(6.4.6)
Indeed, we already considered the models where the number of “events” NE was interpreted in terms of the number of interactions NI , the number of returns to the initial point NB , and the number of visited points N. Thus, the Dreizin-Dykhne result could be generalized for the case of a more complex “topology” of flows with the value d > 1 as follows: V2 (6.4.7) C(t) ∝ 0 , NI (t) where NI is the number of interactions discussed above. The correlation function can also be represented in terms of returns NB : C(t) ∝
V02 . NB (t)
(6.4.8)
In the isotropic case, for a system of random flows, it is possible to use the Alexander-Orbach conjecture [3, 16, 91] for the number of visited sites: NI (t) ∝ t 2/3 , for 2 ≤ d ≤ 6.
(6.4.9)
Hence, we obtain the following scaling law for the particle displacement R: R2 (t) ∝ t
C(t)dt.
(6.4.10)
Then, one finds the transport scaling in the form: 2
R (t) ∝ t 3 .
(6.4.11)
Here, the Hurst exponent is denoted by H = 2/3. Redner [149] and Bouchaud et al. [150] obtained such a superdiffusion regime for the “Manhattan grid” flow, which is a generalization of the shear flow model [135]. Thus from the formal standpoint a shear flow system is described by the velocity field:
6.4 The “Manhattan-Grid” Flow and Turbulent Transport
→ V = − ∇z × Ψ(z) = (u(z), 0),
97
(6.4.12)
where u(z) is a random function. A flow with the stream function: Ψ = Ψ(x) + Ψ(z)
(6.4.13)
is then a two-dimensional generalization of the shear flows model (see Fig. 6.4). It was assumed that in the case under consideration transport is described by the Hurst scaling R ≈ t H . On the other hand, scaling arguments yield: R (t) ≈ V t.
(6.4.14)
The mean velocity V for the case of Gaussian statistics has the form: V =
V0 V0 δ N 1 N ∑ Vi = N ≈ N 1/2 . N i=1
(6.4.15)
Here, the value N corresponds to the number of “layers” intersected by the test particle. Let us consider a simplest approximation (analogous to the one-dimensional case): N (t) ∝ R (t) ∝ t H ,
(6.4.16)
R (t) ≈ V0t 1−H/2 .
(6.4.17)
which leads to the scaling: A comparison between the last expressions yields the Hurst exponent H = 2/3. Note that this expression implies that the auto-correlation function of velocity has a power tail: 1 (6.4.18) C(t) ≈ H , t
Fig. 6.4 Schematic illustration of the “Manhattan-grid” flow
98
6 Return Effects and Random Shear Flows
which is in accordance with the “event” approximation of the correlation function C (t) ∝ V02 /NE (t). Redner [149] also supposed a “hyper scaling” for models with the dimensionality d ≤ 3: 2 , (6.4.19) H (d) = d +1 which implies the accuracy of the result obtained for both the one-dimensional (d = 1) and the isotropic three-dimensional (d = 3) cases. The “Manhattan grid” flow could be a simple model for the two-dimensional turbulent transport. Thus, the analysis of turbulent transport experimental data in a tokamak (where turbulence can be considered as two-dimensional due to the presence of a strong magnetic field) yields the following estimate for the Hurst exponent: 0.6 < H < 0.75, which approximately corresponds to H = 2/3 [17, 45, 49, 151]. The simplest three-dimensional generalization of the random shear flow model [3] is represented in Fig. 6.5. In this case, the correct transport scaling is given by: R2 ∝ D (t)t ∝ C (t)t 2 ∝
t2 , NI (d = 2)
(6.4.20)
where NI describes the average number of visits of a site. To treat this threedimensional flow system, we must consider correlations between different “cells” on a two-dimensional cross-section. Then, one can use the formula [3]: NI (d = 2) =
t . lnt
(6.4.21)
This gives the expression for anomalous diffusion: R2 (t) ∝ t lnt.
(6.4.22)
Note that two statistical ensembles are involved in all these random flow models, namely the distribution of velocities and the different walks for a given random velocity distribution. The effective transport depends on both, and we must take into account this fact in discussing such nontrivial correlation effects.
Fig. 6.5 Three-dimensional generalization of the random shear flow model.
Further Reading
99
We can conclude that the investigation of complex random walks provides a foundation for understanding a very wide range of transport phenomena. In particular, they play an important role in turbulent transport, kinetics, polymer physics, biology, etc. Random walks can be generated on simple lattices or in continuous spaces. Thus, the well-known example is the nearest neighbor walk on a square lattice, where the random walk starts on a site that can be at the origin. Return of particles to the initial point is one of the important and nontrivial properties of random walk models. There are many cases in which such correlation effects predominate. For instance, this related to random walks on random substrates, including fractal substrates, as models for transport phenomena in disordered systems. Below, we extensively investigate random walks on fractal and percolation clusters in connection with turbulent transport in random two-dimensional flows, where an extract enumeration approach is valuable.
Further Reading Random Flows and Transport Batchelor, G.K., Moffat, H.K., and Worster, M.G. (2000). Perspectives in Fluid Dynamics. Cambridge University Press, Cambridge, U.K. Bouchaud, G.P. and Georges, A. (1990). Physics Reports, 195, 132–292. Childress, S. and Gilbert, A.D. (1995). Stretch, Twist, Fold: The Fast Dynamo. Springer-Verlag, Berlin. Crisanti, A., Falcioni, M., and Vulpiani, A. (1991). Rivista Del Nuovo Cimento, 14, 1–80. Haus, J.W. and Kehr, K.W. (1987). Physics Reports, 150, 263. Horton, W. and Ichikawa, Y.-H. (1994). Chaos and Structures in Nonlinear Plasmas. World Scientific, Singapore. Isichenko, M.B. (1992). Reviews of Modern Physics, 64, 961. Majda, A. and Kramer, P. (1999). Physics Reports, 314, 237–574. Moffatt, H.K., Zaslavsky, G.M., Comte, P., and Tabor, M. (1992), Topological Aspects of the Dynamics of Fluids and Plasmas. Kluwer Academic, Dordrecht. Sornette, D. (2006). Critical Phenomena in Natural Sciences. Springer-Verlag, Berlin.
Chapter 7
Turbulence of Magnetic Field Lines
7.1 Basic Equations of Plasma It is known that plasma often shows chaotic motion. Such motion, when many degrees of freedom are excited to levels considerably above the thermal level, is called turbulent. This section reviews very briefly the basic transport equations of plasma, which allow us to apply the methods considered above to turbulent transport in plasma and stochastic magnetic fields. Indeed, motion of charged particles in random electromagnetic fields is a classical problem in plasma physics and satisfactory understanding of statistical particle behavior in random fields should help elucidate various anomalous transport processes, such as turbulent diffusion in magnetically confined plasmas, solar wind fluctuations, cosmic rays propagation, etc. In general, plasma dynamics is determined by complicated correlations between particle and fields. The full set of basic equations of a plasma consists of two Maxwell equations: →
1 ∂E , ∇ × B = μ0 j + 2 c ∂t →
→
(7.1.1)
→
→
∇×E = −
∂B , ∂t
(7.1.2)
and Poisson’s equation for the electric charge density, ρE : →
∇ · B = 0, → ρE ∇·E = ε0 →
→
(7.1.3) .
(7.1.4) →
Here, B is the magnetic field, E is the electric field, j is the electric current density in the plasma, and ε0 and μ0 are the vacuum permittivity and susceptibility, respectively. These equations demonstrate that the electric and magnetic fields are not independent, but are coupled by their spatial and temporal variations. Moreover, the electric current density turns out to be the source of the magnetic field and of fast fluctuations of the electric field. Since ε0 μ0 = 1/c2 equal the inverse square of the
O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008
101
102
7 Turbulence of Magnetic Field Lines
light velocity, the latter will be negligible in a plasma as long as we do not consider propagation of electromagnetic waves. The current and charge densities are defined as the sums over the current and charge densities of all species: →
→
j = ∑ qs nsV s
(7.1.5)
s
ρE = ∑ qs ns .
(7.1.6)
s
→
Here, qs is the charge, ns is concentration, and V s is the characteristic velocity of component s. The bulk velocities and densities must be calculated from the basic equations determining the dynamics of the plasma. In a purely collisionless state the most fundamental equation describing the plasma dynamics is the Vlasov equation, taken separately for each species [45, 50]: → → → → q → → → ∂ f s → ∂ fs x, V ,t x, V ,t → → s + V · ∇ fs x, V ,t + , (7.1.7) E +V × B → ∂t ms ∂V which is a scalar equation for the particle distribution function fs . For its justification and derivation we refer the reader to numerous publications [45, 49, 50, 56, 57, 123, 151]. This kinetic approach is related to the consideration of particle motion in a plasma. Such a motion is strongly distorted by the presence of all other particles, the propagation of disturbances across a plasma, and a number of other effects. However, due to the Debye screening [48, 151], the particles move approximately freely in a dilute collisionless and hot plasma for distances larger than one Debye length [48, 151]. The assumption is made that the small distortions of the particles caused by their participation in the Debye screening of the Coulomb fields of other particles. In principle, such a perturbation can be described more precisely in terms of the thermal fluctuations of the particle density and velocity. In the framework of our simplified consideration, one may easily calculate the particle orbits. The particle orbits satisfy the single particle equation of motion in which all the collisional interactions with other particles and fields are neglected → → (see Fig. 7.1). Given external magnetic, B, and electric fields, E, this equation of motion reads [49, 50]: → → → → dV s = qs E + V s × B . (7.1.8) ms dt The motion of the particles along the field lines is independent of the magnetic field and, in the absence of a parallel electric field component, E// = 0, the parallel particle velocity remains constant, V// = const. The transverse particle motion can be split into a number of independent velocities if it is assumed that the gyromotion is sufficiently fast with respect to a bulk speed perpendicular to the magnetic field. Averaging over the circular gyromotion, the particle itself can be replaced by its guiding center, i.e., the center of its
7.1 Basic Equations of Plasma Fig. 7.1 Schematic illustration of the charge particle trajectory in space
103 Sun
Earth
gyrocircle. The velocity of the guiding center [49, 50] may be decomposed into a number of particle drifts. In a stationary perpendicular electric field the Lorentz force term in the above equation of motion describes that a simple transformation of the whole plasma into a coordinate system moving with the convection or → → E × B drift: → → → E ×B VE = (7.1.9) B2 cancels the electric field. In this co-moving system the particle motion is indepen→ dent of E ⊥ , which is the perpendicular component of the electric field. In the framework of the kinetic description, the densities and bulk velocities entering the current and charges are determined as the moments of the distribution function, fs , as solutions of the Vlasov equation: → → (7.1.10) ns = d 3V fs x, V ,t → → → → (7.1.11) nsV s = d 3 wV fs x, V ,t . The Vlasov equation together with the system of field equations and definition of densities and currents turns out to be a highly nonlinear system of equations, in which the fields determine the behavior of the distribution function and the field themselves are determined by the distribution function through the charges and currents. This self-consistent system of equations forms the basis for collisionless plasma physics. The Vlasov equation (7.1.7) may be used to derive fluid equations for the different particle components. It is based on a moment integration technique of the Vlasov equation, which is well known from general kinetic theory [45, 123]. One multiplies the Vlasov equation successively by rising powers of the velocity V and integrates the resulting equation over the entire velocity space. The system of hydrodynamic equations obtained consists of an infinite set for the infinitely many
104
7 Turbulence of Magnetic Field Lines
possible moments of the one-particle distribution function fs . The first two moment equations are the continuity equation for the particle density and the momentum conservation equation: → ∂ ns + ∇ · nsV s = 0, ∂t → → → ∂ nsV s qs → → → 1 + ∇ · nsV sV s = ns E + V s × B − ∇ ps , ∂t ms ms
(7.1.12) (7.1.13)
where, for simplicity, the pressure has been assumed to be isotropic. These equations have to be completed by another equation for the pressure or by an energy law. Note that many of the plasma turbulence problems cannot be treated in the framework of the precise kinetic approximation. In these cases, one is forced to return to the fluid description in order to treat the nonlinear effects at least in an approximative way similar to the Kolmogorov fluid turbulence phenomenology.
7.2 Magnetic Field Evolution and Magnetic Reynolds Number To study the turbulent transport in plasma in the presence of stochastic magnetic fields, one may use the generalized Ohm’s law: → → → → j = σ0 E + V × B (7.2.1) to estimate the electric field in Faraday’s law (7.1.2), obtaining & → → ' → → ∂B j = ∇× V ×B− . ∂t σ0 →
(7.2.2) →
Using Ampere’s law (7.1.1) without the ∂ E/∂ t term and noting that ∇ · B = 0, we get a general induction equation for the magnetic field [19, 48, 152, 153, 154]: → → → ∂B 1 = ∇× V ×B + ∇2 B, ∂t μ0 σ0 →
(7.2.3)
where σ0 is the plasma conductivity due to Coulomb or neutral collisions. The magnetic field at a point in a plasma can be changed by the motion of the plasma, which is described by the first term on the right-hand side. It can also be changed by the diffusion due to the second term on the right-hand side. First, let us consider the field evolution assuming the plasma to be at rest and dropping the first term on the right-hand side. This equation for the magnetic field has the well-known form of the diffusion equation:
7.2 Magnetic Field Evolution and Magnetic Reynolds Number
105
→
→ ∂B = ηm ∇2 B ∂t
(7.2.4)
with the magnetic diffusion coefficient given by:
ηm =
1 . μ0 σ0
(7.2.5)
Under the influence of a finite resistance in the plasma, the magnetic field tends to diffuse across the plasma and to smooth out any local inhomogeneities. Consider the sketch in Fig.7.2, where we start out with magnetic field lines confined to small regions of space at time T = 0. With time, the field lines expand away in a diffusive manner, moving through the plasma. By dimensional estimates, one can treat convective effects basing on the analogy with hydrodynamic turbulence. The field evolution equation (7.2.3) can be rewritten in simple dimensional form as: V0 B0 B0 B = + . τB LB τd
(7.2.6)
In this equation, B0 is the average magnetic field strength and V0 represents the average plasma velocity perpendicular to the field, while τB denotes the characteristic time of magnetic field variations, and LB is again the characteristic length over which the field varies. The second term on the right-hand side describes the diffusion of the magnetic field through the medium, while the first term has the form of a convective derivative, which describes the convective motion of the field with the plasma. The ratio of the first and second term yields the magnetic Reynolds number: Rem = μ0 σ0 LBV.
(7.2.7)
T=0
B
T>0
Fig. 7.2 The magnetic field lines are spread out and go to zero if there are no field generation processes
106
7 Turbulence of Magnetic Field Lines
This Reynolds number is very useful in deciding if a medium is diffusion or flow dominated. Thus, when Rem >> 1 the diffusion term in the induction equation can be entirely neglected. In this case the flow dominates, and the magnetic field simply moves together with the flow: it is frozen-in into the flow. The magnetic Reynolds number Rem is proportional to the conductivity, the length scale, and the velocity. Increase in any of these quantities will therefore lead to the dominance of frozen-in conditions. For instance, the solar wind turbulence magnetic Reynolds number is about Rem ≈ 7 · 1016 and justifying above estimates on the possibility of negligible diffusion of the magnetic field. Note that only the perpendicular velocity enters the frozen-in convective term. Any flow parallel to the magnetic field has no consequences. On the other hand, when Rem ≤ 1, diffusion becomes important and may even dominate. The magnetic field is not any more frozen into the plasma and may slip across the plasma. In particular, in a diffusion dominated regimes, the plasma can freely stream across the magnetic field. Plasma turbulence experiments typically provide an enormous date base relevant to the state of the MHD (magnetohydrodynamic) field variables. These data include → the magnetic field vector B along with the plasma ion density, ρI , and the ion fluid → velocity vector V . Thus, the observation data provides a basis for evaluating the state. For example, one can typically extract covariances such as the magnetic autocorrelation function: → → → → (7.2.8) Ci j r ≡ Bi x,t B j x + r ,t . In this expression, Bi is the fluctuating magnetic field, evaluated through an ensemble average (space/time) operation indicated by . . .. These observed averages, correlation, and spectral quantities are highly relevant to characterizing the plasma turbulence. Of course, the simplest one-fluid MHD models described here represent vast oversimplifications of the dynamics of plasma. On the other hand, there is some basis for optimism in setting out to describe turbulence through these simple models. The very appearance of Kolmogorov-like spectral laws indicates some relevance of incompressible turbulence ideas. In the following sections, we briefly review some of the research on the turbulent transport in a stochastic magnetic field, which has been viewed in a simple way of field lines random walk.
7.3 Magnetic Diffusivity and the Quasilinear Approach The interaction of magnetic fields with turbulence is a basic feature of many complex physical systems. Indeed, magnetic fields permeate the universe. They are found in intergalactic media, galaxies, and stars. While there is often a component of the field that is spatially coherent at the scale of the astrophysical object, the field lines are tangled chaotically and there are magnetic fluctuations at scales that range
7.3 Magnetic Diffusivity and the Quasilinear Approach Fig. 7.3 Parker’s model of a stochastic magnetic field in interplanetary space
107 Star
probe
over orders of magnitude. Thus, in cosmic ray description, anomalous transport is thought to regulate the high-energy particle density in galactic magnetic fields (see Fig. 7.3). On the other hand, the turbulent diffusion in a stochastic magnetic field is a key issue in plasma magnetic confinement for controlled thermonuclear fusion. In many physical cases, conventional collisional diffusion processes cannot explain experiments. Thus, one is led to the consideration of stochastic magnetic fields in order to generate anomalous transport mechanisms. Many papers are devoted to this problem, but its complete solution is still a long way off. One of the basic concepts in the study of random or turbulent magnetic fields is the concept of field-line random walks (see Fig. 7.4). This means that the stochastic meandering of a magnetic line of force brought about by turbulent motions of the medium. Quasilinear ideas were used in the first papers [155, 156] devoted to the description of both the stochastic magnetic field and turbulent transport in plasma. The quasilinear approach allows the consideration of the magnetic diffusion coefficient Dm from the correlation point of view. This analysis is based on the stochastic equation for force lines: →
dr⊥ → → = b(z, r ⊥ ), dz
(7.3.1)
→
→
B b= ≈ b0 . B0
(7.3.2)
Here, a weak random field, →
B = Bx e x + By e y = (Bx , By , 0), →
→
(7.3.3)
is superimposed on a strong constant field, →
→
B = B0 e z = (0, 0, B0 ),
(7.3.4)
108
7 Turbulence of Magnetic Field Lines
Fig. 7.4 Schematic illustration of the force lines random walks
aligned with the z-axis and b0 is the characteristic relative scale of perturbations. For instance, in high-temperature magnetized plasma (tokamak plasma) the order of magnetic fluctuations is b0 ≈ 10−4 − 10−3 [46, 47, 48, 49, 50]. In the case of reversed field pinches, the low-mode activity has a higher amplitude, the magnetic fluctuations b0 being of order 10−2 [46]. This representation is analogous to the continuity equation of a test particle motion in a random flow but the analysis of transport in an anisotropic medium leads to the necessity of considering the interplay of both the longitudinal and transverse correlation mechanisms. Jokipii and Parker [155] suggested a quasilinear expression for the transverse diffusion coefficient of the force lines of a magnetic field: 1 Dm = 4
∞
→
→
dz b(z, 0) b(0, 0) ∝ b20 LZ .
(7.3.5)
−∞
Here, LZ is the longitudinal correlation length: 1 LZ = 2 b0
∞
→ → dz b(z, 0) b(0, 0) .
(7.3.6)
−∞
Such a representation is valid only for b0 LZ << Δ⊥ ,
(7.3.7)
where Δ⊥ is the transverse correlation scale. The determination of the interrelation between the magnetic diffusion coefficient Dm and the diffusion coefficient of particles in a braided magnetic field is a complex problem. Fortunately, the quasilinear representation of the magnetic diffusion coefficient permits us to obtain an estimate of particle diffusion in the stochastic magnetic field. In principle, the transport of the charged particles normal to the average magnetic field direction may be regarded as composed of two parts. The first part consists of the actual transfer of particles across the local magnetic field, as a
7.3 Magnetic Diffusivity and the Quasilinear Approach
109
result of either scattering or drift. The second corresponds to the individual particles having a tendency to stay with the individual field lines, but since the field lines themselves are braided or mixed in the transverse direction, this results in particles moving in the direction normal to the average magnetic field direction. First we assume that charges in their motion strictly follow the field line random walk. Thus, from the standpoint of the dimensional analysis we can consider the expression for the particle transverse diffusion coefficient: D⊥ ≈
Δ2⊥ Δ2 LCOR LCOR LCOR ≈ ⊥ ≈ Dm ≈ b0 2 LZ , τ LCOR τ τ τ
(7.3.8)
where the quasilinear magnetic diffusivity is interpreted in terms of the field line random walk: Δ2⊥ /τ ≈ Dm , LCOR is the particle longitudinal correlation length, and τ is the correlation time that describes the particle transport. However, if the longitudinal correlation length LZ that characterizes the stochastic magnetic field is comparable with the longitudinal correlation length LCOR , then using the expression for the longitudinal diffusion coefficient: 2 LCOR , τ
(7.3.9)
D⊥ ≈ b20 D// .
(7.3.10)
D// ≈ we obtain the relation:
Such a representation for the effective diffusion coefficient of particles in the stochastic magnetic field is called a “fluid limit” and appears to be one of the widely used estimates of transport related to the stochastic magnetic field. It is important that the dependence on the amplitude of the magnetic field fluctuation b0 is the same as in the Taylor formula for the passive scalar transport DT ≈ V02 τ , where the velocity scale V0 characterizes turbulent fluctuations. This simple quasilinear estimate demonstrates the necessity of a careful analysis of longitudinal and transverse correlation effects. Thus, neglecting the transverse displacement Δ⊥ in the quasilinear expression: →
→
b(z, Δ⊥ ) ≈ b(z, 0)
(7.3.11)
is a serious drawback. A similar situation arises also in the theory of turbulent diffusion of a passive scalar. Many authors have tried to improve the quasilinear approximation, since it has a limited region of applicability. Thus, the representation of the diffusion coefficient in the form (7.3.5) will be valid only for the case when the diffusive displacement in the transverse direction is much less than the transverse correlation scale length b0 LZ << Δ⊥ . The case of greatest interest arises when transverse correlation effects play a central role: (7.3.12) b0 LZ ≥ Δ⊥ .
110
7 Turbulence of Magnetic Field Lines
Fig. 7.5 A typical plot of the dependence of the magnetic diffusivity on the magnetic Kubo number
Dm
Rm << 1
Rm << 1
Dm ∝ Rm2
0
Rm = 1
Dm ∝ RmσT
Rm
Kadomtsev and Pogutse [157] suggested the use of a new approach and formulated a criterion of its applicability in terms of the dimensionless parameter (the magnetic Kubo number) that characterizes the ratio of longitudinal and transverse correlation effects: b0 LZ . (7.3.13) Rm = Δ⊥ Now the quasilinear expression for the magnetic diffusivity can be rewritten in terms of this parameter: Δ2 Dm ≈ b20 LZ ≈ ⊥ R2m ∝ R2m , (7.3.14) LZ where Rm << 1. The analogy between the Kubo number Ku = V0 T0 /λ and the magnetic Kubo number Rm will be used repeatedly to obtain scalings. The dependences of the effective diffusivity De f f on the Kubo number Ku or the magnetic Kubo number Rm have a similar character (see Fig. 7.5). If the Kubo numbers are small, we deal with the quasilinear dependence De f f ∝ Ku2 . If the Kubo numbers Ku ≈ 1, one has the linear dependence De f f ∝ Ku. The case with Ku >> 1 corresponds to the smooth scaling De f f ∝ KuσT , where 0 < σT < 1 [17, 18, 91]. Note that the regimes with Rm > 1 are often related to the percolation character of streamlines [91]. In this case, one can use the ideas of long-range correlations and fractality, which are relevant to stochastic magnetic field problems.
7.4 Stochastic Magnetic Field and Transport Scalings In this section, we treat one of the first models of particle transport in a “braided” magnetic field. This problem was formulated in connection with the description of cosmic ray diffusion (see Fig. 7.6) in a galactic magnetic field [155, 158]. However, such a model is also relevant for some mechanisms of anomalous diffusion in
7.4 Stochastic Magnetic Field and Transport Scalings
111
Fig. 7.6 Proton trajectories in stochastic magnetic field. (After J. Giacalone and J. Jokipii [159], with permission)
magnetized plasma [17, 20, 91, 160, 161, 162]. The concept of random walks of magnetic force lines in the transverse direction was the basis of this consideration (see Fig. 7.4). In the diffusive case, it is convenient to introduce a magnetic diffusion coefficient of field lines in terms of transverse correlation length: Dm ∝
Δ⊥ 2 . L//
(7.4.1)
Here, Δ⊥ is the displacement of the perturbed force line in the transverse direction under the condition of displacement along the streamline on the length L// . If we assume that charged particles in their motion strictly follow the displacement of the force line, which was initially chosen by them (particles are skewed on the streamline similar to beads), then it is easy to obtain the expression for the coefficient of the transverse diffusion of particles: D⊥ ∝
L// (t) Δ2⊥ Δ⊥ 2 L// ≈ ≈ Dm . t L// t t
(7.4.2)
In the case of ballistic motion of particles along the streamline, the estimate of the transverse diffusion coefficient has the form: D⊥ ≈ DmV// .
(7.4.3)
Here, V// is the particle velocity under the condition of motion along the force line. A nonstandard situation arises when we consider collisions between particles, which are located in the “braided” field. Here we discuss the limit of particle transport in a turbulent magnetic field in which the charged particles are confined strictly
112
7 Turbulence of Magnetic Field Lines
to lines of force, which might apply to very low rigidity particles. In this limit, the particles are assumed to diffuse back and forth along the magnetic field lines, and to move in the direction normal to the field only as a consequence of the meandering of the field lines. Then it is natural to suppose that the motion in the longitudinal direction has a diffusive character (random walks along the force line): D// ≈
L// 2 LCOR 2 . ≈ 2τ 2t
(7.4.4)
Here, LCOR is the particle longitudinal correlation length and τ is the correlation time. Then the estimate of longitudinal displacement is the value: 2 (7.4.5) L// (t) ≈ 2D//t. The substitution of (7.4.5) in (7.4.2) yields: 1 2 2D//t Dm ≈ 2D// √ . D⊥ ≈ Dm t t
(7.4.6)
This result demonstrates an essential deviation of the transverse transport from the classical diffusion, since: 2 √ (7.4.7) Δ⊥ 2 (t) ≈ Dm 2D// t ∝ t 1/2 << t. This corresponds to the subdiffusive character of transport with the Hurst exponent H = 1/4. In this model particles never leave the force line of magnetic field in which they initially were situated. The result obtained shows the nontrivial character of the relation between longitudinal and transverse correlations in the stochastic magnetic field, which leads to the essentially nondiffusive particle transport. Note that double diffusion is a simplified model of anomalous transport in the stochastic magnetic field. This mechanism is destroyed under the influence of timedependent perturbations and the effect of stochastic instability of nearby streamlines [163, 164, 165, 166, 167, 168, 169]. These effects are discussed in Chap. 8 of this book.
7.5 Diffusive Renormalization and a Braded Magnetic Field In this section, we review the diffusive renormalization of quasilinear equations to describe long-range correlation effects in a braded magnetic field. The obvious drawback of the quasilinear theory is that the nonlinear term in the equation for n0 is retained: ∂n ∂n +V = 0, (7.5.1) ∂t ∂x while the nonlinear terms in the equation for n1 are omitted:
7.5 Diffusive Renormalization and a Braded Magnetic Field
113
∂ n1 ∂ n1 ∂ n0 + v0 = −v1 . ∂t ∂x ∂x
(7.5.2)
Many authors have tried to refine the quasilinear approximation. A detailed analysis of such papers was carried out in [45, 49, 170, 171, 172]. In fact, the equation for n1 (7.5.2) is linear and hyperbolic and it keeps the Lagrangian character of correlations. This opens up the possibility of describing the omitted correlation effects by including the additional diffusive term, which is in agreement with to the Corrsin diffusive renormalization. In this context, it is expedient to present some of the results obtained by Kadomtsev and Pogutse [157] on anomalous electron transport in a magnetic field. They →
considered a three-dimensional problem in which a weak random field B (Bx , By , 0) →
is superimposed on a strong constant field B(0, 0, B0 ) aligned with the z-axis. The formal quasilinear representation is valid only when the diffusion-related displacement in the transverse direction is much less than the transverse correlation length (7.3.5), b0 LZ << Δ⊥ . Kadomtsev and Pogutse considered a more complex case than the quasilinear one. They introduced a continuity equation for the density of the magnetic field lines nb :
∂ nb → → + b∇⊥ nb ( r ⊥ , z) = 0, ∂z →
(7.5.3)
→
B b= ≈ b0 , B0
(7.5.4)
and represented nb as a sum of the mean density of magnetic field lines n0 = nb and the fluctuation component n1 : nb = n0 + n1 .
(7.5.5)
Here, b0 is the relative scale of the fluctuation amplitude and < . . . > is the averaging symbol. The problem as formulated is close to the quasilinear description of passive scalar transport. We rewrite the equation for the mean density of magnetic field lines n0 in the traditional form: → ∂ n0 + ∇⊥ bn1 = 0. ∂z
(7.5.6)
However, in the equation for n1 (5.4.5), we replace the second-order terms: ∂ n1 ∂ n1 − b , (7.5.7) b ∂x ∂x (which were omitted in earlier studies) by a diffusion term Dm ∇2⊥ n1 . In essence, we follow Corrsin’s ideas and relate the discarded correlation effects to the diffusive spreading of trajectories:
114
7 Turbulence of Magnetic Field Lines → ∂ n1 − Dm ∇2⊥ n1 = − b∇⊥ n0 . ∂z
(7.5.8)
Here, the diffusion coefficient is considered as effective diffusivity. The set of renormalized equations (7.5.6) (7.5.8) has retained a convenient form for solution. For the passive scalar equations it corresponds to the set of equations: ∂ n0 ∂ n1 + v1 = 0; (7.5.9) ∂t ∂x
∂ n1 ∂ n0 ∂ 2 n1 + v1 =D 2 , ∂t ∂x ∂x
(7.5.10)
which are similar to (7.5.6) and (7.5.8). Here, D is the turbulent diffusion coefficient. Thus, we kept the equation for the density perturbation n1 linear but passed from a hyperbolic equation of form (5.4.9) to the parabolic equation (7.5.10). Applying the mathematical apparatus of Green’s functions to the equation for the fluctuation density of magnetic field lines n1 (7.5.8), we obtain:
∂ GD → → − Dm ∇2⊥ GD = δ ( r − r ). ∂z
(7.5.11)
Let us derive the following equation for the mean density n0 :
∂ n0 = Dm ∇2⊥ n0 , ∂z
(7.5.12)
where the magnetic diffusion coefficient and the Fourier spectrum of perturbation amplitudes are given by: 1 Dm = 2
→
→ b2 ( k) d k, 2 ikz + k⊥ Dm
(7.5.13)
and the magnetic turbulence spectrum may be written as: →
1 b ( k) = (2π )2 2
→→
→
b(0)b(r) exp(−i k r )d r .
(7.5.14)
In the case of short correlations where 2 Dm , Δkz > k⊥
(7.5.15)
we obtain the quasilinear expression: Dm =
π 2
→
→
d kb2 ( k)δ (kz ) ∝ b20 LZ ,
where LZ is the longitudinal correlation length.
(7.5.16)
Further Reading
115
In the case of strong correlations where 2 Dm , Δkz < k⊥
(7.5.17)
we arrive at the following result: Dm
2
1 = 2
→
b2 ( k) → b20 d k ∝ 2 ≈ b20 Δ2⊥ . 2 k⊥ k⊥
(7.5.18)
→ The expressions obtained include the spectrum of magnetic turbulence b2 k , which is a significant step forward in comparison with dimensional estimates. The scaling for the transverse magnetic diffusion coefficient is given by: Dm (b0 ) ≈ b0 Δ⊥ .
(7.5.19)
For suprathermal (collisionless) charges, this leads to the linear estimate of particle transport. This is similar to the Howells result discussed above [125]: * +∞ + E (k) dk ∝ V0 λ, (7.5.20) DH (V0 ) = , k2 k
but in the case of magnetic diffusivity we are dealing with the isotropic medium. Here, λ is the characteristic spatial scale. It once again shows that, on the one hand, it is important to take into account the long-range correlation effects that are neglected in the quasilinear approach and, on the other hand, these correlations are closely related to interactions between different spatial scales. Note that in contrast to the random shear flow model, here one deals with transverse correlations. In spite of the simple form of the obtained estimate, Dm ≈ b0 Δ⊥ , the linear character of the dependence of the effective diffusion coefficient on the “stochastic layer” width Δ⊥ , De f f ≈ V0 Δ⊥ , is repeatedly used to describe turbulent transport in systems with a complex flow topology such as convective cells, percolation structures, etc.
Further Reading Plasma Physics and Magnetohydrodynamic Turbulence Balescu, R. (2005). Aspects of Anomalous Transport in Plasmas. IOP Publishing, Bristol and Philadelphia. Biskamp, D. (2004). Magnetohydrodynamic Turbulence. Cambridge University Press, Cambridge, U.K. Brandenburg, A. (2005). Physics Reports, 417, 1.
116
7 Turbulence of Magnetic Field Lines
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. Mikhailovskii, A. (1974). Theory of Plasma Instabilities. Consultant Bureau, New York. Proctor, M.R.E. and Gilbert A.D., eds. (1994). Lectures on Solar and Planetary Dynamos. Cambridge University Press, Cambridge, U.K.
Stochastic Magnetic Fields and Transport Balescu, R. (1997). Statistical Dynamics. Imperial College Press, London. Horton, W. and Ichikawa, Y.-H. (1994). Chaos and Structures in Nonlinear Plasmas. World Scientific, Singapore. Isichenko, M.B. (1992). Reviews of Modern Physics, 64, 961. Kadomtsev, B.B. (1991). Tokamak Plasma: A Complex System. IOP Publishing, Bristol. Krommes, J.A. (2002). Physics Reports, 360, 1–352. Rosenbluth, M.N. and Sagdeev, R.Z., eds. (1984). Handbook of Plasma Physics. North-Holland, Amsterdam. Wesson, J.A. (1987). Tokamaks. Oxford University Press, Oxford. Zeldovich, Ya.B., Ruzmaikin, A.A., and Sokoloff, D.D. (1990). The Almighty Chance. World Scientific, Singapore.
Chapter 8
Stochastic Instability and Turbulence
8.1 Stochastic Instability and Correlations The problems of turbulent diffusion and the analysis of transport in a stochastic magnetic field have many traits in common. Thus, in the case of magnetized plasma we deal with the anisotropic medium in the presence of several “seed”-diffusive mechanisms simultaneously. Moreover, the stochastic instability of trajectories also leads to the appearance of new decorrelation mechanisms. For example, it “destroys” the double diffusion regime, which is based on the repeated returns of magnetized particles under conditions of longitudinal diffusive motion along the magnetic field (7.4.7). The stochastic instability of trajectories was first discovered in billiard-like systems (see Fig. 8.1), where the characteristic time of instability τS is expressed in the form [163]:
lC
2r*
Fig. 8.1 A billiard-like system and scattered particle trajectory
O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008
117
118
8 Stochastic Instability and Turbulence
Fig. 8.2 Schematic illustration of the stochastic instability of trajectories
l(t)
l(0)
τ 0 .
τS ≈ ln
lc r∗
(8.1.1)
Here, τ0 ≈ τBal is the characteristic ballistic time in the system of scattering centers, lc is the mean distance between the centers of scattering, and r∗ is the curvature radius of scattering spheres. Investigations showed the universal character of relation (8.1.1) in which the ratio lc (8.1.2) KS = r∗ appears to be the mapping parameter. In the discrete form, it could be represented as:
δ xN ≈ KS N δ x0 ≈ δ x0 exp (N ln KS ) .
(8.1.3)
Here, N is the number of iterations, δ x0 is the initial length of phase element, and δ xN is the length of the element under analysis after iterations. In the description of chaos in complex systems, both characteristic times and spatial scales play an important role. From the formal point of view, we can analyze this phenomena in the framework of the divergence of initially nearby trajectories (see Fig. 8.2):
t . (8.1.4) l (t) ≈ l (0) exp (hK t) ≈ l (0) exp τS Here, l(t) is the distance between trajectories at the moment t and hK is the Kolmogorov entropy expressed in terms of Lyapunov’s exponent [84, 86]: 3 4 | l (t) | 1 ln . (8.1.5) hK = liml(0)→0, t→∞ | l (0)| t The chaotic properties of dynamic systems have been investigated in this context in many papers and books [84, 91, 94, 95, 122, 163]. We look here more closely at the scaling aspect of the problem, since both the Kolmogorov entropy hK and the spatial scale l define decorrelation mechanisms and transport. From the standpoint of the streamline chaos description, the value K can be interpreted through the parameters describing random flow. Thus, considering the
8.2 Quasilinear Scaling for the Stochastic Instability Increment
119
Fig. 8.3 Hydrodynamic evolution of a fluid element V0 L* δL ≈ V0 /ω
evolution of fluid element of size L∗ in a hydrodynamical field with the characteristic velocity scale V0 and characteristic frequency ω , it is easy to estimate stretching fluid element during the characteristic time τ0 : L1 ≈ L∗
δL V0 ≈ L∗ . λ ωλ
(8.1.6)
Here, λ is the characteristic spatial scale. Then, the length
L(t) ≈ L∗
V0 ωλ
t/τ0
= L∗ exp
t t ln Ku ≈ L∗ exp τ0 τS
(8.1.7)
corresponds to the time t (Fig. 8.3). Here, Ku = V0 /ωλ is the Kubo number, which plays the role of the mapping parameter K in analyzing the streamline chaos:
τS ≈
τ0 τ0 ≈ . ln Ku ln V0 λω
(8.1.8)
Naturally, in the hydrodynamic case, the characteristic time τ0 also must be interpreted in terms of flow parameters [91]. This allows us to treat stochastic instability effects in the framework of the scaling approach and to describe correlation and transport properties from a common standpoint.
8.2 Quasilinear Scaling for the Stochastic Instability Increment The problem of the exponential divergences of two neighboring field lines of a stochastic magnetic field has been studied in the context of both astrophysical and plasma physics applications. In the simplest steady case, the equation describing the walks of the separated force line can be expressed in the Lagrangian form: → → B dr⊥ = . (8.2.1) |B| dz
120
8 Stochastic Instability and Turbulence →
Here, z characterizes the distance that was traveled along the force line and B(z) is the value of the magnetic field. In the frameworkof the simplified scenario—the → regular component of the magnetic field is absent, B = 0, and the absolute value of the magnetic field is constant, /B/ = const—we obtain the following expression, which is analogous to the quasilinear formula: z ∞ Bi (z1 )B j (z2 ) ≈ dz dgCi j (g). dz1 dz2 (8.2.2) ri r j = → 2 B −∞ Here, Ci j (g) is the correlation function of the random magnetic field, which characterizes the correlation decay along the chosen force line. Such a consideration is correct only for cases where the force line length is much greater than the correlation length Lz . Then one obtains the relation: d 2 ri ≈ dz
∞
dgCii (g).
(8.2.3)
−∞
The analogy between the equations that describe the particle walk and the force line walk allows us to consider the problem of relative divergences of two force lines in the framework of the quasilinear method. The relative displacement of two nearby force lines during the process of their random walks is given by: →
→(1)
Δr = r
→(2)
−r
z
=
→(1)
dz
→(2)
B (z) − B (z) →
.
(8.2.4)
B It is then easy to obtain the expression describing the divergence of force lines of the stochastic field for small values of r2 − r1 [157]:
∂ ∂b (r2 − r1 ) = b (z, r2 ) − b (z, r1 ) ≈ (r2 − r1 ) . ∂z ∂r Formal calculations yield the exponential dependence: ⎡ z
r2 (z) − r1 (z) ≈ Δr (z = 0) exp ⎣
0
⎤ ∂b ⎦ dz . ∂r
(8.2.5)
(8.2.6)
The increment of stochastic instability can be found by averaging this expression with an assumption about the Gaussian character of a random value b, which makes it possible to calculate an average as follows: 5 6 A2 exp A = exp , (8.2.7) 2
8.2 Quasilinear Scaling for the Stochastic Instability Increment
and hence:
⎡
1 Δ⊥ (z) = r2 (z) − r1 (z) = Δr |z=0 exp ⎣ 2
z z
dz dz
121
0 0
⎤ ∂ b (z , r) ∂ b (z , r) ⎦ . ∂r ∂r
(8.2.8) The integral expression in this formula is analogous to the expression for the quasilinear diffusion coefficient. Simple transformations yield the increment of stochastic instability γz in the form: 1 γz = 2
∞ −∞
∂ b (0, 0) ∂ b (z, 0) ∂r ∂r
dz.
(8.2.9)
In terms of the dimensionless magnetic Kubo number Rm , this result may be written as: b0 2 LZ Dm 1 2 γz ≈ ≈ 2 ≈ R . (8.2.10) 2 LZ m Δ⊥ Δ⊥ Hence, in the framework of streamline chaos one obtains the stochastic instability increment in the quasilinear form γ ∝ Ku2 where the Kubo number is applied. Naturally, the limits of applicability of this estimate coincide with the limits of applicability of the quasilinear approximation: Rm ≈
b0 Lz <1 Δ⊥
or
Ku < 1.
(8.2.11)
Fig. 8.4 Stochastic instability increment as a function of the magnetic turbulence strength. (After A. Barghouty and J. Jokipii [169], with permission)
122
8 Stochastic Instability and Turbulence
The dependence of the stochastic instability increment on the Kubo number (or on the magnetic Kubo number) has a shape similar to the turbulent diffusion coefficient (see Fig. 8.4). These results show that in order to describe turbulent transport in a stochastic magnetic field we must take into account stochastic instability as a decorrelation mechanism.
8.3 The Rechester-Rosenbluth Model Here, we will look more closely at stochastic instability of trajectories as a nontrivial decorrelation mechanism in the framework of the scaling approach. On average, two initially close streamlines diverge from one another according to the law:
z . (8.3.1) Δ(z) = l0 exp LK Here, l0 is the initial separation of the streamlines and z is the distance that was passed along the field line. The magnitude hK = L1K is called the Kolmogorov entropy and defined through
3 4 Δ(z) 1 ln . (8.3.2) hK = liml0→0 ,z→∞ z l0 One can see that correlation scales need not be defined by the seed diffusion process alone. Rechester and Rothenbluth [164] assumed that decorrelation is related to stochastic instability and not to particle collisions. In the problem as formulated the collisional random walks of magnetized particles along and across streamlines of the magnetic field play the role of seed diffusion. The simplest estimate can be obtained by the consideration of the expression for the transverse diffusion coefficient of particles in terms of the magnetic diffusion coefficient Dm (7.4.1): De f f ≈
ΔCOR 2 LCOR ≈ Dm . 2τ τ
(8.3.3)
Here, LCOR is the longitudinal correlation length related to stochastic instability and τ is the correlation time. This approach implies that the magnetic diffusion coefficient Dm , the collisional longitudinal diffusion coefficient D// , and the collisional transverse diffusion coefficient D⊥ are known. The analysis is often carried out in terms of the heat-conduction coefficient in order not to complicate the problem by taking into account ambipolar effects. However, we will keep using the diffusion symbols in order to retain the uniformity of the terminology. The values LCOR and τ are interrelated by the expression for the longitudinal diffusion coefficient:
8.3 The Rechester-Rosenbluth Model
123
LCOR 2 . 2τ Then, we obtain the following estimate for De f f : D// ≈
De f f ≈ Dm
2D// . LCOR
(8.3.4)
(8.3.5)
Here, LCOR is the parameter of the problem. To define the value LCOR we use expression (8.3.1):
Δ(LCOR ) r0 ≈ LK ln . (8.3.6) LCOR ≈ LK ln l0 l0 Here, r0 is the transverse spatial scale. If the values r0 and LK assumed to be known and related to the specific model, then to determine LCOR it is necessary to find l0 only. For this purpose, let us consider a small element of evolving area, which has a spatial scale l0 . Because of stochastic instability there will be two competing processes at the same time: the distance between streamlines will be exponentially increasing and the width of the area will be exponentially decreasing in order to conserve the total area:
z . (8.3.7) δ (z) = l0 exp − LK Here, δ is the width of the area (see Fig. 8.5). It is necessary to take into account the influence of perpendicular diffusion processes that increase δ . The balance between these processes is given by the relation:
δ dδ =− , dz LK
(8.3.8)
(d δ )2 ≈ 2D⊥ dt.
(8.3.9)
Taking into account the expression that describes the longitudinal diffusive behavior: (8.3.10) (dz)2 ≈ 2D// dt, 7
one obtains the formula: l0 ≈ δ ≈ LK
D⊥ . D//
(8.3.11)
This model is based on the assumption of the essential role of the stochastic instability of trajectories. However, there is also an alternative possibility related to the agreement between the longitudinal and transverse diffusive mechanisms in a strongly anisotropic medium. Thus, in the case of strongly magnetized plasma we deal with (8.3.12) D// >> D⊥ .
124
8 Stochastic Instability and Turbulence
δ
Fig. 8.5 Schematic illustration of the stochastic instability of a phase element
Thus, Kadomtsev and Pogutse related longitudinal correlation length LCOR to the longitudinal diffusive process: D// ≈
2 LCOR , τ
(8.3.13)
and the decorrelation time τ is related to the transverse diffusion: D⊥ ≈
r02 . τ
(8.3.14)
Here, r0 is the characteristic scale in the transverse direction. Then calculations lead to the formula: 1 D// D⊥ , (8.3.15) De f f ≈ Dm r0 which combines the different diffusion mechanisms in the anisotropic medium. We have derived a new kind of dependence on the longitudinal diffusivity D// . The particle diffusion in a stochastic magnetic field leads to a large variety of regimes: double diffusion (7.4.7), ballistic regime (7.4.3), fluid limit (7.3.10), the Rechester-Rothenbluth model (8.3.5), and the Kadomtsev-Pogutse regime (8.3.15). Moreover, there are several percolation regimes, which are considered in Chap. 11.
8.4 Collisional Effects and Transport Expression (8.3.5) for the effective diffusion coefficient permits us to carry out an interesting scaling analysis. For example, one can estimate the influence of collisions on the effective diffusion coefficient. Following Stix [167], let us introduce
8.4 Collisional Effects and Transport
125
the NColl factor accounting for the number of collisions during the correlation time (compare with the Kubo number Ku = λVω0 ): NColl ≈
τ . τColl
(8.4.1)
Here, τColl is the collisional timescale and τ is the correlation time. The expression for the coefficient of longitudinal diffusion has the form: D// ≈
λ2 LCOR 2 ≈ Coll . 2τ 2τColl
(8.4.2)
Here, λColl is the collisional longitudinal mean free path. Hence, one can obtain: LCOR ≈ λColl (NColl )1/2 .
(8.4.3)
On the other hand, we can represent the longitudinal diffusion coefficient as: D// ≈ λColl V// .
(8.4.4)
Here, V// is the particle characteristic velocity in the longitudinal direction. Now, we can rewrite expression (8.3.5) in the following form: De f f ≈ DmV// √
1 . NColl
(8.4.5)
We see that collisional effects decrease the effective diffusion coefficient in comparison with the collisionless case: De f f ≈ DmV// .
(8.4.6)
However, in the framework of the approach advocated in [164, 166, 167] it does not lead to the appearance of double diffusion (7.4.7). This expression for De f f (NColl ) is reminiscent of the transport scaling for the system of random shear flows, De f f ≈ V02t
1 , NE (t)
(8.4.7)
where the number of “interactions” between the particle and transverse flows is given by: √ 2D0t NE (t) ≈ , (8.4.8) a0 which have here the “collisional” meaning. Here, D0 is the seed diffusivity and a0 is the characteristic spatial scale. Let us estimate the effective diffusion coefficient for the regime in which stochastic instability plays a key role. Considering NColl as the main parameter of the model,
126
8 Stochastic Instability and Turbulence
we can use a transcendental equation to define NColl . The basis of this equation is the relation:
LCOR . (8.4.9) r0 = l0 exp LK However, now we suppose that the value l0 is related to the correlation time τ : l0 2 ≈ D⊥ τ .
(8.4.10)
Then, using the expression describing the relationship between the longitudinal correlation length LCOR and the correlation time τ yields the estimate for l0 , which differs from the Rechester-Rosenbluth relation: 7 D⊥ (8.4.11) l0 = LCOR << LCOR . D// The equation for the longitudinal correlation length LCOR takes the form: 7
D⊥ LCOR r0 = LCOR . exp D// LK
(8.4.12)
Here, as before, the parameters of the problem are D⊥ , D// , r0 , LK . However, we can re-formulate this equation for LCOR in terms of the parameter NColl . Using the expressions for D⊥ and D// we obtain: D⊥ (Δ⊥ , τ ) ≈
Δ2⊥ , τ
λ2 D// λ// , τColl ≈ // . τColl
(8.4.13) (8.4.14)
Here, Δ⊥ is the transverse correlation scale. Substitution of these relations in (8.4.12) yields:
√ 1 λ// NColl . (8.4.15) r0 = Δ⊥ NColl exp LK This transcendental equation for NColl can be used to calculate De f f in accordance with formula (8.4.5): 1 De f f (NColl ) ≈ DmV// √ . (8.4.16) NColl Concluding this section, note that the Kadomtsev-Pogutse scaling [157] can also be rewritten in a form that uses the collisionless diffusion coefficient with an additional factor. The particle longitudinal velocity enters into the expression for the longitudinal diffusion coefficient: D// ≈ V 2 τColl .
(8.4.17)
8.5 The Quasi-Isotropic Stochastic Magnetic Field
127
On the other hand, the transverse diffusion coefficient has the form D⊥ ≈
Δ2⊥ . τColl
(8.4.18)
Here, Δ⊥ is the collisional transverse correlation scale. As a result of simple calculations we find the expression for the effective diffusion coefficient for the KadomtsevPogutse regime: Δ⊥ . (8.4.19) De f f ≈ DmVe r0 We have considered the Rechester-Rosenbluth and the Kadomtsev-Pogutse models in terms of the correlation scale Δ⊥ and the collisional time τColl . However, it is easy to relate these values to the Larmor radius ρe of electron and the collisional frequency ve , to describe plasma physics problems [157, 164].
8.5 The Quasi-Isotropic Stochastic Magnetic Field In the framework of the Rechester-Rosenbluth approach, the expression for longitudinal correlation length,
r0 , (8.5.1) LCOR ≈ LK ln l0 contains the parameters LK , l0 , and r0 , and their definition depends on a selected physical model. In the case of magnetized plasma, the effective approximation is given by: 7 D⊥ and r0 ≈ ρe , (8.5.2) l0 ≈ LK D// where ρe is the Larmor radius of the electron. However, if the tangled magnetic field is described only by the spatial scale lB , then this estimate becomes incorrect. Quasi-isotropic stochastic magnetic fields play an important role in astrophysical problems. Thus, in describing heat-conduction processes in a stochastic magnetic field in clusters of galaxies, there are serious difficulties because the transport observed considerably exceeds theoretical estimates [173]. Chandran and Cowley suggested an interesting modification of the Rechester-Rosenbluth scaling, in which the characteristic spatial scale of nonuniformity of tangled magnetic field lB is simultaneously the parameter describing electron capture by magnetic traps formed by a significant nonuniformity of a magnetic field in a longitudinal direction. Formally, in the conditions when (8.5.3) lB ≤ λB , where λB is the mean free path, the correlation characteristics of stochastic magnetic fields can be represented in the form:
128
8 Stochastic Instability and Turbulence
Fig. 8.6 Magnetic force lines 1–2, 3–4 and particle trajectory 1–4
Δ
1
3
4
LCOR ≈ lB ln Dm ≈
lB ρe
2
,
lB 2 ≈ lB . lB
(8.5.4) (8.5.5)
Suppose that transverse decorrelation in electron motion arises along the whole distance of order lB (see Fig. 8.6) and, at the same time, the same scale characterizes the sizes of the magnetic traps (see Fig. 8.7), which electrons leave after gaining additional energy in collisions. The situation under analysis is described by the following hierarchy of scales:
ρe << lB ≤ λB << LCOR .
(8.5.6)
The expression for the effective diffusivity in a quasi-isotropic stochastic magnetic field takes the form: Deff ≈ Dm
D// D// LCOR lB . ≈ Dm ≈ D// ≈ τ LCOR LCOR ln (lB /ρe )
lB
Fig. 8.7 Schematic illustration of magnetic traps
(8.5.7)
Further Reading
129
In the framework of plasma physics [45, 46, 47, 48, 49, 50], transport estimates imply the use of the expression for the electron heat-conduction coefficient of SpitserHarm χSp as D// taking into account the one-dimensional character of electron motion along force lines: 2 L// χSp . (8.5.8) ≈ D// ≈ 2τ 3 The estimate of the value ρe in the conditions corresponding to clusters of galaxies yields: (8.5.9) lB /ρe ≈ 103 or LCOR ≈ 30 lB ;
therefore, the suggested approach gives the estimate for the effective heat conduction coefficient:
χe f f ≈ De f f ≈ 10−2 χSp .
(8.5.10)
From the point of view of the explanation of astrophysical observations, this estimate is not fully correct, however (as discussed below in the framework of anisotropic magnetohydrodynamic turbulence), the above method needs only insignificant modification. Thus in Chap. 12, we will consider the multiscale approach to study stochastic instability effects. Indeed, the more recent discovery of coherent and organized structures in plasma is reminiscent of the variety of scale lengths present in ordinary fluid turbulence.
Further Reading Chaos and Mixing Aref, H. and El Naschie, M.S. (1994). Chaos Applied to Fluid Mixing. Pergamon, 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. Dorfman, J.R. (1999). An Introduction to Chaos in Nonequilibrium Statistical Mechanics. Cambridge University Press, Cambridge, U.K. Moffatt, H.K., Zaslavsky, G.M., Comte, P., and Tabor, M. (1992). Topological Aspects of the Dynamics of Fluids and Plasmas. Kluwer Academic, Dordrecht. Nicolis, J.S. (1989). Dynamics of Hierarchical Systems. An Evolution Approach. Springer-Verlag, Berlin. 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. Reichl, L.E. (1998). A Modern Course in Statistical Physics. Wiley-Interscience, New York.
130
8 Stochastic Instability and Turbulence
Dynamo Childress, S. and Gilbert, A.D. (1995). Stretch, Twist, Fold: The Fast Dynamo. Springer-Verlag, Berlin. Proctor, M.R.E. and Gilbert A.D., eds. (1994). Lectures on Solar and Planetary Dynamos. Cambridge University Press, Cambridge, U.K. Zeldovich, Ya.B., Ruzmaikin, A.A., and Sokoloff, D.D. (1990). The Almighty Chance. World Scientific, Singapore.
Chapter 9
Anomalous Transport and Convective Cells
9.1 Convective Cells and Turbulent Diffusion The system of convective cells is nontrivial and, at the same time, solvable model, which permit one to investigate transport related to self-organized structure. Thus, it is well known that low-frequency drift waves can act as convective cells and cause rapid particle transport in plasma if they are excited to a large amplitude [45, 46, 47, 48, 49, 50]. The regular character of the location of structure elements simplifies the analysis considerably. However, this model has properties corresponding to a more complex system. The subsequent progress of research on diffusion processes in systems with convective cells [174, 175] has led us to the understanding of the importance of the stochastic layer width Δ in analyzing the convective fraction of the transport (see Fig. 9.1). In the case of a two-dimensional incompressible flow the streamlines coincide with the isolines of Ψ (x, y). Here we will examine the regular cellular system of vortices: Ψ (x, y) = Ψ0 sin (kx x) sin (ky y) .
(9.1.1)
Here, incompressibility implies that the velocity field is given by:
∂Ψ dx = Vx = − , dt ∂y
(9.1.2)
dy ∂Ψ = Vy = . dt ∂x
(9.1.3)
The simplest estimate of the longitudinal convective transport is as usual the quasilinear expression: Δ De f f ≈ V02 τ , (9.1.4) λ which is corrected (renormalized) by the geometrical factor Δ/λ to account for the fraction of space that is responsible for the convection. Here, λ is the cell size and V0 is the characteristic velocity of the convective flow. It is natural to use the characteristic time of leaving the particle from the boundary layer as the correlation time: O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008
131
132
9 Anomalous Transport and Convective Cells
Fig. 9.1 Schematic illustration of the system of convective cells (vortex grid) D0
D0
D0
D0
λ
Δ
τ ≈ Δ2 /D0 , where D0 is the “seed diffusion” coefficient. Then, the scaling for the effective transport takes the form [18]: De f f ≈ V02
Δ3 . λ D0
(9.1.5)
However, the quasilinear character of the dependence of De f f on V0 does not correspond to the results of numerous simulations in the region Pe ≈
λ V0 ≥ 1. D0
(9.1.6)
Here, Pe is the Peclet number. A qualitatively new estimate could be the linear dependence De f f ≈ V0 Δ. Formally, we must consider the effective diffusivity in the convective form: De f f ≈ λ V0 P∞ ,
(9.1.7)
where P∞ is the fraction of space, which is responsible for the convective transport. In the case of convective cells, the value of P∞ can be estimated if we introduce the cell size λ :
λΔ Δ ≈ . (9.1.8) λ2 λ This formula reflects the simple “topology” of the model of convective cells. The diffusive estimate of the stochastic layer width Δ was considered in [174, 175]: 7 1 1 D0 λ ∝ √ . (9.1.9) Δ ≈ D0 τ ≈ V0 V0 P∞ ≈
9.1 Convective Cells and Turbulent Diffusion
133
Δ2 λ ≈ is the correlation timescale. This formula clearly defines the V0 D0 physical significance of the particle number balance. Indeed, the number of particles escaping from a convective cell per unit time is: Here, τ ≈
n ND ∝ D0 λ . Δ
(9.1.10)
The convective flow along the boundary layer carries away a number of particles: NC ∝ nV0 Δ. Since convective flows exist only in the fraction of space De f f ∝ λ V0
Δ (V0 ) = V0 Δ (V0 ) . λ
(9.1.11) Δ , we obtain: λ (9.1.12)
This expression is analogous to the linear scaling, De f f ≈ b0 Δ⊥ , as it was predicted in the framework of the renormalized quasilinear theory, which was derived to explain long-range correlation effects in a braided magnetic field (see Chap. 7). But in our case we arrive at the following estimate for the turbulent diffusion coefficient (see Fig. 9.2): 1 1/2 (9.1.13) De f f = const D0V0 λ ≈ D0 Pe1/2 ∝ V0 . This representation of the result in terms of the Peclet number differs significantly from both the quasilinear De f f ∝ V02 and the linear De f f ∝ V0 estimates.
Fig. 9.2 The effective diffusion coefficient versus the Peclet number Pe. (After M. Osipenko et al. [174], with permission)
134 Fig. 9.3 A concentration profile with the mean concentration gradient imposed. Isoconcentration regions are separated by thin boundary (stochastic) layer of thickness Δ(V0 )
9 Anomalous Transport and Convective Cells λ λ
Δ
Δ
Δ
Note that the quasilinear estimate De f f ∝ V02 DΔ λ transforms into the scaling 0 √ De f f ∝ D0V0 λ , when characteristic time balance Δ2 /D0 = λ /V0 is valid. In fact here we are dealing with the dependence (see Fig. 9.3): 7 λ 1 (9.1.14) Δ (V0 ) = D0 ∝ 1/2 . V0 V 3
0
For large values of the characteristics velocity (large the Peclet number), this gives correlation times τ ≈ Δ2 /D0 , which are less than the initially suggested estimate (9.1.3), where value Δ was just a parameter of a model [18]. Concluding this section, we refer the reader to the experimental results [176, 177] of scalar transport in the system of Rayleigh-Benard convective rolls (see Fig. 9.4). The streamline function in the following form was applied:
2π 2π (x + λ sin ω t) sin y . (9.1.15) Ψ (x, y,t) = Ψ0 sin L0 L0 Here, L0 is the characteristic spatial scale. First, all of the tracer is initially released in a single cell. The main question is: how many cells, NC (t), have been invaded by tracer at time t? If this dispersive process is described by diffusion then we expect that
Fig. 9.4 Schematic illustration of the flow pattern within the Rayleigh-Benard cells
9.2 Complex Structures and the Statistical Topography
NC (t) ∝
1
D0t.
135
(9.1.16)
With certain restriction, this t 1/2 -law is the experimental result [176, 177, 178]. The important conclusion of this research is that the effective diffusion coefficient in low frequency regimes scales with ω .
9.2 Complex Structures and the Statistical Topography A more complex case arises when cells are not regular. In this case the analysis of transport can be based on the statistical properties of the streamline function landscape Ψ (x, y) (see Fig. 9.5). Convective flows of particles along percolation channels contribute most to the effective diffusive coefficient. The topology of these channels is characterized by the second derivative of streamline function Ψ (r). Therefore, introducing a characteristic scale of streamline function Ψ0 and a characteristic spatial scale rc , we obtain an estimate of the channel width responsible for the convection Δ: (9.2.1) Ψ (r) Δ2 ≈ D0 . Applying the scaling approach for Ψ , we obtain an expression for the definition of layer width Δ = Δ (rc , D0 ,V0 ): Ψ0 Δ2 2 ≈ D0 . (9.2.2) rc Simple calculations allow us to derive the dependence of layer width Δ on the flow parameters Ψ0 , D0 , rc :
Fig. 9.5 A chaotic cell flow pattern
136
9 Anomalous Transport and Convective Cells
Δ = rc
D0 Ψ0
1
2
= rc
D0 rcV0
1 2
,
(9.2.3)
or in terms of the Peclet number, Δ = rc
1 Pe1/2
.
(9.2.4)
Here, V0 is the characteristic velocity scale. Using the expression obtained above for effective diffusion in a system of random convective flows in the form: De f f ≈ V0 Δ(Ψ0 , D0 , rc ),
(9.2.5)
we easily find a scaling describing transport: 1
De f f ≈ D0 Pe 2 ,
(9.2.6)
which is in good agreement with the above. As an example of the complex vorticity field we mention two-dimensional turbulence where cell structures have been observed [46, 47, 50, 180, 181, 182, 183, 184, 185]. This approach is an attractive one, which could possibly provide an alternative starting point for the analysis of transport effects in terms of statistical topography [91, 179]. However, it is slightly naive and does not use all the possible information concerning flow topology; therefore, it cannot be applied for a rigorous analysis of fractal streamlines. These problems are developed below in more detail.
9.3 Fluctuation–Dissipative Relation and Turbulent Mixing In this section we briefly consider a fluctuation–dissipative relationship to describe scalar turbulent mixing that allows us to estimate scalar density fluctuations 2 (∇n) [136] in terms of the Peclet number. Indeed, in the framework of quasisteady turbulence, we can write the Zeldovich expression (5.2.3) in the form: 0 = q (n1 − n2 ) − D0
(∇n)2 dW ,
(9.3.1)
W
where the flux q through a boundary is estimated in terms of the mixing length L0 and the velocity fluctuations V0 : q ≈ V0 · L0 .
(9.3.2)
Expression (9.3.1) is fairly universal and valid even for the high Peclet number Pe >> 1. Substitution (9.3.2) into (9.3.1) and simple calculations yield Zeldovich’s fluctuation–dissipative relation:
9.3 Fluctuation–Dissipative Relation and Turbulent Mixing
(∇n)2 =
V0 L0 D0
137
Δn2 .
(9.3.3)
In terms of the Peclet number, Pe = VD0 L00 , one can rewrite this scaling for strong turbulence regimes in the form: (9.3.4) (∇n)2 = Pe Δn2 . This means that when Pe >> 1, two fluid elements having substantially different scalar densities (or temperatures) can appear side by side, which was confirmed by numerous experiments and numerical simulations [12, 13, 18, 19, 82, 83, 84]. It is easy to draw a scaling for density perturbations nλ corresponding to the local scale λ . Let us introduce the local Peclet number Pe = λDV00 , corresponding to the scale λ . Then we can assume that the density perturbation nλ is given by a relation [12, 13]:
λ V0 αn . (9.3.5) nλ ∝ n0 Peαλ n = n0 D0 Supposing that the Peclet number and the Reynolds number are proportional, we can estimate the scale λ basing on the Kolmogorov phenomenology (2.3.14):
λ∝
L0 3/4 Reλ
∝
L0 3/4
,
where
Peλ
Pe ∝ Re >> 1.
(9.3.6)
Then, the expression for the amplitude of scalar density perturbation is given by the scaling 3/4 (9.3.7) nλ (Pe) ∝ (∇n0 ) λ Peαλ n Peλ , which can be represented in the form of (9.3.4):
n 2 (∇n)2 ∝ λ ≈ (∇n0 )2 Pe2(αn +3/4) . λ
(9.3.8)
Comparing (9.3.4) and (9.3.8), we find an expression for the fluctuation exponent αn :
3 = 1. (9.3.9) 2 αn + 4 Hence, a scaling for nλ , for the case of strong turbulence: nλ (Pe) ∝ n0 Peαn ∝
n0 Pe1/4
,
where
Pe >> 1.
(9.3.10)
differs remarkably from the quasilinear limit (week turbulence where Pe << 1) as was shown in Chap. 5 (5.2.10): nλ (Pe) ∝ n0 Pe,
Pe << 1.
(9.3.11)
138
9 Anomalous Transport and Convective Cells
Now it easy to estimate the convective contribution to transport: q (Pe) ∝ nλ (Pe)V0 ∝ n0
V0 Pe
1/4
= D0 Pe3/4
n0 L0
(9.3.12)
and hence the effective diffusivity is given by: De f f (Pe) ∝
q (Pe) L0 ∝ D0 Pe3/4 . n0
(9.3.13)
Remind that in the quasilinear case (Pe << 1) the transport scaling is given by (5.2.11): (9.3.14) De f f (Pe) ∝ D0 1 + const Pe2 . In Fig. 9.6 different turbulent transport regimes (quasilinear mode, regimes with regular structures, and Kolmogorov’s turbulence) are represented. Deff ∝ Pe3/4
Deff
Kolmogorovís regimes
Deff ∝ Pe1/ 2
Regular structures Deff ∝ Pe2
Fig. 9.6 A typical plot of the dependence of the effective diffusivity on the Peclet number
Week turbulence
Pe = 1 Strong turbulence
Pe
9.4 Bohm Scaling and Electric Field Fluctuations Let us now leave the domain of physics of fluids for that of plasma physics. The research of transport in magnetized plasma leads to the necessity of analyzing the → dependence of turbulent diffusion across a magnetic field B on its magnitude (see Chaps. 7 and 8). One of the important approaches is the consideration of drift mo→ → tions of plasma in crossed electric E and magnetic B fields: #→ → $ B×E → ∇ϕ VE = c ∝ . (9.4.1) 2 B B0
9.4 Bohm Scaling and Electric Field Fluctuations
139
→
Here, V E is the drift velocity and ϕ is the electric field potential. Then, in terms of the Kubo number: Ku ≈
k2 ϕ V0 ≈ , λω ω B0
(9.4.2)
we can analyze the dependence: D⊥ ∝ KuσT ≈
k2 ϕ ω B0
σT
∝ VEσT ≈
kϕ B0
σT .
(9.4.3)
Here, B0 is the characteristic magnetic field amplitude, k is the wave number, and ω is the characteristic frequency. The conventional representation of the collisional character of transverse diffusion in a strong magnetic field leads to the estimate [45, 46, 50]: √ ne2 c2 me 1 1 ∝ 2. (9.4.4) D⊥ ≈ B0 B20 Tp Here, e is the electron charge, me is the electron mass, Tp is the plasma temperature, and n is the plasma concentration. Actually, this corresponds to σT = 2. The prognoses based on this formula provide fairly good confinement of plasma in magnetic traps. However, much experimental data point to the incorrectness of this scaling. The main reason that prevents good confinement is strong plasma turbulence. The simplest Bohm consideration [186] of effects of turbulent fluctuations of electric field leads to the linear dependence of the transverse diffusion on the perturbations amplitude with σT = 1. This estimate of plasma diffusivity across a magnetic field is based on the notion of the existence of eddies or nonuniformity in turbulent plasma, which lead to chaotic fluctuations of electric fields. In plasmas, various kinds of electromagnetic fluctuation can be generated. From the impact on transport processes, fluctuations in the frequency range lower than the ion cyclotron frequency are important. When the temporal variation of the electric field is slower than the ion cyclotron frequency, the particle motion is approximately described by the sum → → of gyromotion and E × B motion of the guiding center (see Chap. 7). If we introduce a characteristic scale LB corresponding to the structures under analysis then the simplest correlation estimate of the diffusion coefficient is the expression: DB ≈
LB2
τCOR
.
(9.4.5)
Here, LB plays the role of the spatial correlation scale and τCOR is the characteristic correlation time that can be estimated from a dimensional analysis; introducing into consideration the velocity VE characterizing drift motion in crossed electric and magnetic fields yields:
τCOR ≈
LB L2 ≈ B B0 . δ VE cδ ϕ
(9.4.6)
140
9 Anomalous Transport and Convective Cells
Here, use is made of the expression for the drift velocity:
δ VE ≈
c c δϕ δ Ep ≈ B0 B0 LB
(9.4.7)
and the dimensional estimate of electric field fluctuations through the potential perturbation across the structure, δ ϕ ≈ δ E p LB and δ E p is the electric field perturbation. On substitution we obtain the Bohm scaling for transverse diffusion in magnetized turbulent plasma: cTp c DB ≈ LB δ VE ≈ δ ϕ ≈ B0 eB0
eδ ϕ Tp
2 1/2 ,
(9.4.8)
where Tp is the plasma temperature. Here, electric field fluctuations are normalized in accordance with the dimensional estimate: eδ ϕ ≈ eδ E p LB ≈ Tp .
(9.4.9)
The absence of the characteristic scale of structures LB in the final expression for the transverse diffusion coefficient DB is an important feature of the Bohm scaling, which attaches a “universal” character to this estimate. Let us consider the interpretation of the Bohm scaling De f f ∝ δ VE ∝ Ku in the framework of the correlation time choice. Indeed, in the case of high frequency oscillations, the following estimate is correct:
τCOR ≈
1 ω
(9.4.10)
and one obtains the quasilinear scaling: DT ≈ δ VE2 τCOR ≈
δ VE2 ≈ ω
cδ E B0
2
1 ∝ Ku2 . ω
(9.4.11)
However, if the frequency ω → 0, then DT → ∞ (see Fig. 5.4). To describe low-frequency regimes in the presence of complex structures, such as convective cells, we will consider different approximations for the correlation time τ . In the low-frequency case, the frequency path is much greater than the correlation scale: lω ≈
δ Ep 1 δ VE ∝c >> LB ω B0 ω
(9.4.12)
because correlations are related to turbulent mixing. In such a regime a correct relationship between the correlation length and time is given by the relation:
τCOR ≈
LB2 1 1 ≈ 2 < . DT ω k⊥ DT
(9.4.13)
9.5 Diffusive Renormalization and Correlations
141
This corresponds to the diffusive renormalization and is in agreement with the concepts of the minimum of correlation time [187]. Simple algebra leads to the expression:
δ E p 2 LB2 2 c . (9.4.14) DT ∝ B0 DT Accounting for the relation δ E p LB ≈ δ ϕ , one obtains the transport scaling: DT ∝ c
δ E p LB δϕ ≈c ∝ Ku. B0 B0
(9.4.15)
The transition from the quasilinear regime, De f f ∝ Ku2 , to the linear one, De f f ∝ Ku, when Ku ≈ 1 is shown in Fig. 9.7. In the next section, we will derive the Bohm scaling by using correlation function technique. DT (Ku)
DT ∝ Ku
DT ∝ Ku 2
Fig. 9.7 A typical plot of the dependence of the effective diffusivity on the Kubo number
Ku=1
Ku
9.5 Diffusive Renormalization and Correlations The scaling suggested by Bohm can be interpreted in terms of both the diffusive renormalization method and the interaction of different scales. In the framework of the guiding center approximation, the velocity correlation function for the case of strongly magnetized plasma is given by:
C(τ ) =
c B0
2
δ E⊥ (τ )δ E⊥ (0) ,
(9.5.1)
where δ E⊥ is the electric field fluctuation in the direction perpendicular to the magnetic field. Using the Fourier representation for electric field fluctuations,
142
9 Anomalous Transport and Convective Cells
#→ $ → δ E⊥ (x,t) = ∑ δ Ek (t) exp i k x(t) ,
(9.5.2)
k
and the independence hypothesis [23], we can rewrite the expression for the correlation function in the form: 3
4
2 →→ → → c δ Ek (t)δ Ek (0) exp i k x(t) + k x(0) (9.5.3) C(t) = B0 ∑ kk c 2 #→ $ → = δ E (t) δ E (0) exp i kΔ x(t) . p p B ∑ k Here, Δx(t) is the diffusive particle displacement due to the presence of turbulent fluctuations. On the basis of the Gaussian statistics one obtains the relation: $ #→ → (9.5.4) exp i kΔ x(t) = exp −k2 DT (t) , which is in agreement with the Corrsin diffusive representations. Then, the formal expression for the turbulent diffusion coefficient is given by: ∞
∞
DT =
C(t) dt = 0
0
c B0
2
∑ δ E⊥ (t)δ E⊥ (0)k exp
2 −k DT t dt.
(9.5.5)
k
Taylor and McNamara [126] assumed that the spectrum of statistical fluctuations of electric field δ E 2 k is known and it decays exponentially: δ E⊥ (t)δ E⊥ (0)k = δ E p 2 exp −k2 DT |t| . k
(9.5.6)
with the characteristic correlation time in the diffusive form τ ≈ k21D . Integrating T the expression for the renormalized correlation function over time yields: 2
2 ∞ 2 δ E p c c 1 k DT = dt δ E p 2 exp −2k2 DT t = . ∑ ∑ B0 B0 D⊥ k 2k2 k k 0
(9.5.7)
Applying the spectrum δ E p 2 in the power form [126]: k
2 δ E k = 4π where λD2 =
n p Tp 4 π e2
Tp , 1 + (kλD )2
(9.5.8)
and integrating over k, it is easy to obtain the formula:
D2T
= 4π Tp
c B0
2
→
dk
#
1
(2π ) 2k2 1 + (kλD )2 2
$.
(9.5.9)
Further Reading
143
Here, Tp is the plasma temperature and n p is the plasma density. This expression cT can be transformed into the form containing the factor DB ≈ eBp0 : * + ∞ cTp + dk + e2 $. # DT = , 2 eB0 T k 1 + (k λ ) D 0
(9.5.10)
Here, we have integrated over the azimuth angle θ : →
d k = k sin θ dk.
(9.5.11)
The divergence of this integral in the region of the small wave numbers k reflects the slow correlation decay; therefore we need to consider in more detail different scales interactions, which is relevant for the Coulomb system. Finally, note that at present, the major obstacle on the way to the realization of controlled thermonuclear fusion in closed magnetic configuration devices is commonly attributed to the existence of anomalous energy losses due to particle and energy transport across the confining magnetic field. The anomalous transport of particles is usually related to the turbulent character of plasma behavior [45, 46, 47, 48, 49, 50]. In spite of considerable efforts, this problem still awaits a complete solution.
Further Reading Vortex Structures and Transport Crisanti, A., Falcioni, M., and Vulpiani, A. (1991). Rivista Del Nuovo Cimento, 14, 1–80. Golitsyn, G.S. (2004). Selected Papers. Nauka, Moscow. Guyon, E., Nadal, J.-P., and Pomeau, Y., eds. (1988). Disorder and Mixing. Kluwer Academic, Dordrecht. Holmes, P.J., Lumley, J.L., Berkooz, G., Mattingly, J.C., and Wittenberg, R.W. (1997). Physics Reports, 337–384. Maurel, A. and Petitjeans, P., eds. (2000). Vortex Structure and Dynamics Workshop. SpringerVerlag, Berlin. Moffatt, H.K. (1983). Reports on Progress in Physics, 621, 3.
Two-Dimensional Turbulence and Complex Structures Biskamp, D. (2004). Magnetohydrodynamic Turbulence. Cambridge University Press, Cambridge, U.K. Kraichnan, R.H. and Montgomery, D. (1980). Report on Progress in Physics, 43, 547. Kuramoto, Y. (1984). Chemical Oscillations, Waves and Turbulence. Springer-Verlag, Berlin.
144
9 Anomalous Transport and Convective Cells
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. Pismen, L.M. (2006). Patterns and Interfaces in Dissipative Dynamics. Springer-Verlag, Berlin. Tabeling, P. (2002). Physics Reports, 362, 1–62.
Part III
Fractals and Percolation Transport
Chapter 10
Fractal and Percolation Concepts
10.1 Self-Similarity and the Fractal Dimension There exists a new language for describing scaling behavior, and this is the Mandelbrot language of fractals. In this chapter, we introduce the fractal geometry concepts that provide a language in terms of which to understand better the meaning of turbulence scalings, anomalous diffusion, percolation mechanism, cluster growth, etc. It is well known that an object can be self-similar if it is formed by parts that are similar to the whole. One of the simplest self-similar objects is the Cantor set, whose iterative construction at successive “generations” is shown in Fig. 10.1. If we enlarge the box of generation 3 by a factor of 3, we obtain a set of intervals identical to the generation 2 object. At generation k we can enlarge a part of the object by a factor of 3 and obtain the object of generation (k − 1). Another famous example is the Cantor dust (see Fig. 10.2). Indeed, isotropic fractals are self-similar: they are invariant under isotropic scale transformations (see Fig. 10.3). For the Cantor set, the enlarged part overlaps exactly the original object. We call such an exactly self-similar object a deterministic fractal. However, many natural objects are random. Despite this randomness, such objects could be self-similar in a statistical sense (see Fig. 10.4) (for example, the Brownian particle path, clusters, the coastline of a continent, and etc.). k=0
k=1
k=2
Fig. 10.1 Construction of the Cantor set
O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008
k=3
147
148
10 Fractal and Percolation Concepts
Fig. 10.2 Iteration of the Cantor dust
Fig. 10.3 Schematic illustration of the self-similar structures of a fractal curve
Let us introduce several definitions to describe qualitatively a self-similar system. By embedding dimension, dE , we understand the smallest Euclidean dimension of the space in which a given object can be embedded. To decide on the fractality of an object, we need to measure its Hausdorff dimension. The volume W (δ ) of an arbitrary object can be measured by covering it with balls of linear size δ and volume δ dE . If N (δ ) balls will cover it, then W (δ ) = N (δ ) δ dE .
(10.1.1)
We can expect that for any object, the number of balls is given by: N (δ ) ∝
1
δ dE
,
(10.1.2)
10.1 Self-Similarity and the Fractal Dimension
149
Fig. 10.4 Schematic illustration of a random fractal (cluster)
since the volume of an object does not change if we change the unit of measurement δ . In general, for fractals we can write: N (δ ) ∝
1
δ dF
,
(10.1.3)
where dF is the fractal dimension. Objects with dF < dE are called fractals [188, 189, 190, 191, 192, 193, 194, 195, 196, 197]. From this relation we obtain: ⎧ ⎫ ⎪ ⎪ ⎪ ⎨ ln N (δ ) ⎪ ⎬
. dF = limδ →0 (10.1.4) ⎪ 1 ⎪ ⎪ ⎪ ⎩ ln ⎭ δ For the Cantor set, the natural unit to measure the length of the set at iteration k is the length of the smallest interval:
δk =
k 1 . 3
(10.1.5)
The number of intervals of length δk at level k is given by: N (δk ) = 2k .
(10.1.6)
From (10.1.4) we have for the fractal dimension: dF =
ln (2) = 0.639 . . . ln (3)
Since dF < dE = 1, the Cantor set is a fractal.
(10.1.7)
150
10 Fractal and Percolation Concepts
The fractal concepts appear to be very fruitful to obtain the relationships between the parameters that characterize transport, correlation, and geometric properties of the model under consideration. One of the reasons for such efficiency is the possibility to describe the geometric properties of different natural objects by using scaling terminology. In the framework of the fractal approach, the expression for the very “tortuous” curve length is both the simplest and a very important result. From the formal standpoint, the length of the very tortuous curve (the fractal curve) L(δ ) can be rewritten in the form: L (δ ) ≈ δ N(δ ), where N(δ ) ∝
1
δ dF
.
(10.1.8)
In this fractal approach, the full length L (δ ) is approximated by small segments of size δ ; N(δ ) is the number of these segments, which are necessary for such an approximation; and dF is the fractal dimensionality of the curve (see Fig. 10.5). In the framework of the conventional representation of the geometry of curves, we have to use the value dF = d = 1. However, in this case the drawbacks of the conventional method of length measurement by a “yardstick” (ruler) remain. Mandelbrot considered the problem of the measurement of a tortuous seacoast length in which the increase of measurement accuracy (decrease of the value δ ) leads to a growth of the value N(δ ) (dF > 1). From the formal standpoint this approach yields: (10.1.9) L(δ ) ≈ δ N(δ ) |δ →0 → ∞.
δ2
δ1
Fig. 10.5 Schematic illustration of measurement of the length of a curve in relation to different units
10.2 Fractality and Anomalous Transport
151
This means that such a fractal line embraces “almost” the full plane. There are advantages and disadvantages of such a representation. One of the advantages is the possibility to describe the longest and more complex lines (fractal lines). A similar situation is for the fractal surface area A: A ≈ δ 2 N(δ ) ≈
δ2 . δ dF
(10.1.10)
Here, δ 2 is the small area of size A for measurement and dF > 2 corresponds to the fractal nature of the surface. In the general case, we can obtain the expression for the fractal region in the form: Wd ≈ δ d N(δ ) ≈ δ d−dF .
(10.1.11)
Here, we are dealing with the fractal cases dF > d. It is possible to consider opposite cases when dF < d. This situation can characterize the presence of a large number of voids in the cluster. More detailed information can be found in many books and review articles on fractal geometry and fractal models [188, 189, 190, 191, 192, 193, 194, 195, 196, 197].
10.2 Fractality and Anomalous Transport The discussion of a new geometry of nature, one that embraces the irregular shapes of objects such as coastlines, lighting bolts, cloud surfaces, and molecular trajectories, began in the 1960s [188]. A common feature of these objects is that their boundaries are so irregular that such fundamental concepts as dimension and length measurement must be generalized. Therefore, we shall consider some of the metric peculiarities of a few usual mathematical objects, which we subsequently adopt to describe turbulent transport and anomalous diffusion. The simplest model that permits us to analyze the fractal properties of transport processes is d-dimensional random walks. For the mean square displacement one obtains: (10.2.1) R2 (t) ≈ 2dDt, where the diffusion coefficient is given by: D≈
ΔCOR 2 . 2d τ
(10.2.2)
Here, ΔCOR is the correlation length and τ is the correlation time. For this case, it is easy to obtain an expression that includes the fractal dimensionality of the Brownian trajectory: t 2dDt ΔCOR . L (ΔCOR ) ≈ N (ΔCOR ) ΔCOR ≈ ΔCOR ≈ τ ΔCOR 2
(10.2.3)
152
10 Fractal and Percolation Concepts
Hence, one can obtain a number of “steps” in the scaling form: N (ΔCOR ) ∝
1 ΔCOR 2
,
where
dF = 2.
(10.2.4)
Note that the value dF is independent of the space dimensionality d. Here, we assume that ΔCOR is the small quantity ΔCOR ≈ δ , which corresponds to the definition of the fractal curve (10.1.8). However, the fractal properties can be described on the basis of using “large” parameters such as R, which corresponds to the “cluster” terminology. For example, the expression N (R) ∝ RdF
(10.2.5)
is the equivalent definition of the number of molecules that form the fractal structure of cluster. Here, R is interpreted as a radius of the region occupied by the cluster. In this case, we can rewrite the Brownian scaling law in the following form: N≈
R2 t ≈ ∝ R2 . τ 2dDτ
(10.2.6)
Hence, dF = 2, which corresponds exactly to (10.2.4). From the “fractal” point of view, the scaling laws describing anomalous transport in terms of the Hurst exponent R (t) ∝ t H can be treated analogously. Another definition of the transport exponent dW , which is usually called the internal dimensionality [16, 190, 191, 192, 193] is often used: t ∝ N ∝ RdW , 1 dW = . H
(10.2.7) (10.2.8)
3 Thus, the fractal dimension of self-avoiding random walks H (d) = 2+d in threedimensional space is 3/5, which corresponds to the internal dimensionality dW = 5/3. We can relate the Hurst exponent H and the internal dimensionality exponent dW to the properties of the fractal environment where transport occurs. A simple estimate that implies that fractal diffusion is a slower process than the conventional Brownian diffusion is widely used to investigate random walks in random environments: 1 R2 (t) ≈ θ . (10.2.9) Deff ≈ t RF Then, simple calculations yield the scaling: 1
R ∝ t (2+θF ) ,
(10.2.10)
and hence the internal dimensionality is given by: dW = 2 + θF .
(10.2.11)
10.3 Turbulence Scalings and Fractality
153
This example demonstrates the method for obtaining the relationships between the exponents that describe different physical properties of the system. In this case, the value dW describes transport processes and θF characterizes the complexity of a fractal structure. Fractal ideas have wide applicability to turbulent transport. Not only the walking particle trajectory, but also percolation streamlines, diffusive fronts, etc. appear to be fractal objects.
10.3 Turbulence Scalings and Fractality There is a good example of the relationship between fractal concepts and turbulence description. Even simple observation shows the fractal nature of the cloud boundary. In this connection, great interest has arisen in the experimental measurement of a perimeter of a rain area P and a cloud area A, determined from radar and satellite data. Lovejoy [198] obtained the simple scaling that relates the fractal perimeter P of the two-dimensional projections of the cloud region to its area A: P1/dF ∝ A1/2 .
(10.3.1)
From the fractal aspect, this expression is not surprising. Let us consider the formula for the length of the fractal curve that bounds the nonfractal area:
d F
1−dF λ δ ≈λ . (10.3.2) δ λ √ Here, λ is the characteristic spatial scale. If we suppose λ ∝ A, then the expression for L takes the form: (10.3.3) L (δ , A) ≈ δ 1−dF AdF /2 . L (δ ) ≈ δ N(δ ) ≈ δ
Now it is easy to obtain the relationship between the perimeter and the area: P1/dF ≈ L1/dF ≈ δ 1−dF A1/2 ,
(10.3.4)
where the small value δ is the parameter. The correctness of this result in a wide spectrum of parameters has led Hentschel and Procaccia [76] to the idea of the relationship between the fractal dimensionality of the cloud perimeter and the universal properties of the Kolmogorov model of isotropic turbulence. The simple method to obtain the relationship between the fractal dimensionality of the perimeter and the exponent that describes the turbulence spectrum is based on the calculation of the effective rate of the area increase of the cloud cross section due to the turbulent pulsation of velocity: dA ≈ lkV (lk )N(lk ). dt
(10.3.5)
154
10 Fractal and Percolation Concepts
Here, lk ≈ 1k is the characteristic spatial scale, V (lk ) is the characteristic velocity scale, and N(lk ) is the number of sections that approximate the fractal perimeter of the cloud. If we use the scaling expressions for the spectrum of energy with an arbitrary exponent ςk V2 E(k) ∝ k ∝ k−ςk , (10.3.6) k and for the number of sites: (10.3.7) N(lk ) ∝ lk dF , then substitution of (10.3.6) and (2.3.7) in (10.3.5) yields: 1−ςk dA ≈ lk 1−dF − 2 . dt
(10.3.8)
However, it is obvious that the rate of the area increase of the cloud cross section is independent of lk . Therefore, we obtain the following relationship: dF =
1 + ςk . 2
(10.3.9)
Then, in the Kolmogorov case with ςk = 53 , we obtain dF = 43 . This result is in good agreement with the value defined by Lovejoy: dF = 1.35. He studied shapes of clouds and showed that they have a fractal structure over more than four orders of magnitude (from 3 to 3 × 104 km) (see Fig. 10.6). The measurements were made from radar and satellite images. Even this simple estimate demonstrates the efficiency of using fractal ideas. Often the fractal representation more adequately mirrors the essence of the problem under consideration. Thus, an experiment has confirmed that the energy dissipation region of isotropic turbulence in three-dimensional space has a fractal structure. The correlation of energy dissipation rate, ε (x), at x is given by the often-studied function [28, 80, 83]: (10.3.10) Kεε (r) = ε (x + r,t) ε (x,t) , which decays as the power law: ε (x) ε (x + r) ∝ r−μi ,
(10.3.11)
where the intermittency exponent is μi ≈ 0.2. In the above discussions, we have implicitly assumed that εK is spatially homogeneous and not fractal. When the energy dissipation region forms a fractal structure with dimension dF , we cannot neglect the k-dependence of εK . Indeed, at an early stage of development of Kolmogorov and Obuchov’s phenomenology it was impossible to explain effects connected with intermittency. Energy dissipation means that kinetic energy of fluid is irreversibly transformed into thermal energy, and the rate is proportional to the square of the curl of the velocity field. The above scaling indicates that the velocity field has fractal properties with the dimensionality:
10.3 Turbulence Scalings and Fractality
155
Fig. 10.6 Area A versus perimeter P for cloud and rain areas. (After S. Lovejoy [198], with permission)
dF = 3 − μi .
(10.3.12)
Many researchers tried to correct “the law 5/3” for the energy spectrum E(k) [64, 65]. One of the most elegant models is related to the fractal representation of the energy dissipation regions. Mandelbrot [199] and then Fricsh, Sulem, and Nelkin [75] renormalized these expressions using the fractal representation of energy dissipation regions. The fraction of the volume corresponding to “one dissipation center” can be represented in the form:
156
10 Fractal and Percolation Concepts
QF ≈
lk d Wd ≈ d ≈ lk d−dF . N(lk ) lk F
(10.3.13)
Here, N(lk ) is the number of “dissipation centers” in the region of size lk , Wd ≈ lk d is the volume of this region, d is the dimensionality of Euclidean space, and dF is the fractal dimensionality of the “cluster” consisting of “dissipation centers.” If we assume that the dissipation rate is outside the fractal energy dissipation region then the classical Kolmogorov-Obukhov expression for εK and E(k) can be rewritten in the renormalized form: const = εF ≈ Vk3 kQF ≈ EF (k) ∝
Vl3 QF (l), l
(10.3.14)
Vk 2 QF (k) . k
(10.3.15)
Then, on performing calculations, we arrive at the expression: EF (k) ∝ EK (k)k−
d−dF 3
≈
1
1
k5/3
k(d−dF )/3
.
(10.3.16)
The last factor is the correction factor caused by the fractal nature of energy dissipation regions. Experiments are satisfactorily described by the value dF ≈ 2.8 [64, 65]. However, there are papers that discuss dissipation on eddy surfaces with dF = 2 or dissipation on eddy filaments, which can be considered in terms of self-avoiding random walks with dF = (2 + d)/3. For example, in the case of dF = 2 one obtains the energy spectrum: 1 (10.3.17) EF (k) ≈ 2 . k 2−dF
This spectrum yields the smooth velocity field Vk ∝ k 3 ∝ k0 , which corresponds to the famous Taunsend suggestion [28]. A similar analysis has led to modification of the Richardson scaling. Thus, based on the dimensional estimate: Kεε (k) ∝ Vk3 k ∝ k μi ,
(10.3.18)
one obtains the modified Richardson scaling: DF (l) ∝ Vl l QF (l) ∝ Vl (l)ll μi ∝ l 3 + 3 μi . 4
2
(10.3.19)
It is obvious that the Kolmogorov idea partially loses its initial universality after we introduce the new parameter dF . However, at the same time, such a correction essentially increases the possibilities to fit theory with experiment.
10.4 Percolation Transition and Correlations
157
10.4 Percolation Transition and Correlations Percolation was introduced in a paper by Broadbent and Hammersley [200]. The word “percolation” originated from an analogy with a fluid crossing a porous medium. The invasion of a porous medium by a fluid is in fact very complex and we will discuss the percolation mechanism in terms of the phase transition theory. It is a geometrical phase transition where the critical concentration pc separates a phase of finite clusters (p < pc ) from a phase where an infinite cluster is present (p > pc ). The structure of matter at a critical point is a typical fractal. In the vicinity of the critical point of a phase transition, many macroscopic quantities, such as magnetization and specific heat, are known to follow scaling laws. Just at the critical point, these quantities are divergent. From a microscopic viewpoint, such behavior is due to the divergence of correlation length. Since the correlation length is a length that characterizes the system, the divergence indicates that the system becomes scale-invariant. Let us consider an experiment on the percolation problem that exhibits a phase transition with a very simple mechanism. The problem of percolation occurs when fine metal particles are located at random on an insulator. If the covering of a metal is rare, it is not conductive. It becomes a conductor if the ratio is nearly equal to 1. Indeed, if the ratio equals 1, the surface of the insulator is completely covered by metal, and then it becomes a conductor. There is a critical ratio, pc , such that the system behaves as a conductor if p > pc and as an insulator if p < pc . This is a kind of phase transition from insulator phase to conductor phase and is called a percolation transition. In percolation, the concentration p of occupied sites plays the same role as the temperature in thermal phase transitions. Similar to thermal transition, long-range correlations control the percolation transition and the relevant quantities near pc are described by power laws and critical exponents. The percolation transition is characterized by the geometrical properties of the cluster near pc . An important quantity is the probability P∞ that a site belongs to the infinite cluster. For p < pc , there exist only finite clusters, and P∞ = 0, whereas for p > pc , P∞ increases with p by a power law (see Fig. 10.7): P∞ ∝ (p − pc )β = ε β .
(10.4.1)
Here, ε = p − pc is the small percolation parameter. The value P∞ describes the order in the percolation system and can be identified as the order parameter in terms of the phase transition theory. The linear size of the finite clusters is characterized by the correlation length a. Formally, the correlation length is defined as the mean distance between two sites on the same finite cluster. When p approaches pc , a increases as shown in Fig. 10.8: a (ε ) ∝
λ λ = ν, ε |p − pc |ν
(10.4.2)
158 Fig. 10.7 A typical plot of the dependence of the probability of a site being in the infinite cluster
10 Fractal and Percolation Concepts
P∞
1
0
pc
Fig. 10.8 Schematic illustration of the correlation length near a percolation threshold
1
p
a(p)
pc
p
with the same exponent ν below and above the threshold, where λ is the characteristic spatial scale. We will assume that we know the exponents ν and β . For the two-dimensional case, the exponents ν and β are rational numbers: they can be obtained in the model that has an exact solution: ν = 4/3; β = 5/36. In the threedimensional case ν = 0.875 and β = 0.417. We shall now consider the problems of percolation and calculate the fractal dimensionality of a percolation cluster (see Fig. 10.9). This dimensionality is easily expressed in terms of thermodynamic critical exponents. Near a critical point, a system can be regarded as fractal and self-similar on a scale λ0 << λ < a, where a is the correlation length, and as homogeneous on a larger scale. An idea of how such a system looks can be gained if a plane is covered with Sierpinski carpets (see Fig. 10.10) with a side of length a: on a scale less than a the system is self-similar, whereas on a larger scale it is homogeneous. If a system has a minimum scale, the
10.4 Percolation Transition and Correlations
159
Fig. 10.9 The percolation cluster. For scales less than a, the cluster is infinite fractal. For scales greater than a, the cluster is a weakly disordered lattice of mesh size a
Fig. 10.10 The Sierpinski carpet
dependence of its mass on the scale λ has the fractal character, M ∝ λ dF , where dF is the fractal dimensionality, for λ < a. For λ > a we have the conventional dependence, M ∝ λ d . Introducing the density ρ , one obtains: M = λ dF −d , for λ < a, λd M ρ = d = const, for λ > a. λ
ρ=
(10.4.3) (10.4.4)
Here, a is the scale at which the density becomes constant. In Fig. 10.11 we see this behavior of the density on a double logarithmic scale. The key problem in the framework of the scaling approach is the establishment of the relationships among the exponents describing geometric, correlation, and transport properties of a percolation cluster.
160 Fig. 10.11 Schematic illustration of the behavior of the density of a homogeneous fractal
10 Fractal and Percolation Concepts
ρ ∝ ldF − d
ln ρ(l)
ρ = const
Fractal
Homogeneous
ln l
The fraction of the sites or the volume belonging to a cluster varies with the concentration as P∞ (ε ) ∝ ε β . This quantity is the density of a cluster. The scale for which this quantity is defined is as follows: an infinite cluster is a fractal object if λ < a ∝ ε −ν and a homogeneous one for larger scales. When the characteristic scale a (ε ) is altered, i.e., when the concentration x is modified, the density varies as follows: β ρ (a) ∝ a− ν . (10.4.5) Bearing in mind that the fractal representation for the density is given by
ρ (a) ∝ adF −d ,
(10.4.6)
we find the relationship
β (10.4.7) ν between the thermodynamic and geometric exponents [201, 202, 203, 204, 205]. dF = d −
10.5 Continuum Percolation and Transport We discuss here the appearance of anomalous transport in two-dimensional flows in the framework of the percolation threshold. The percolation approach looks very attractive because it gives a simple and, at the same time, universal model related to both long-range correlation effects and complex topology. Kadomtsev and Pogutse [157] explained increasing turbulent diffusion for Ku >> 1 (or Rm >> 1) by the percolation character of the streamline that contributes most to transport near the threshold (see Fig. 10.12). In the context of this approach, the streamlines Ψ = Ψ(x, y) are considered as the coastlines in a hilly landscape flooded by water. It is expected that there is a sharp transition from separated lakes on a boundless land to individual islands in the infinite ocean.
10.5 Continuum Percolation and Transport
161
Fig. 10.12 The continuum percolation: the lake-ocean transition as the water level rises
Recall that incompressibility implies that the velocity field is related to the stream function Ψ: dx ∂Ψ = Vx = − , dt ∂y dy ∂Ψ = Vy = . dt ∂x
(10.5.1) (10.5.2)
Here, Ψ (x, y) is, at the same time, the Hamiltonian function. The percolation theory requires the existence of at least one coastline of infinite length, which is given by the scaling law [189, 201]: L(ε ) ∝
1 . ε Dp
(10.5.3)
Here, D p ≈ 2.4 is the coastline exponent and ε is a small dimensionless quantity that characterizes the degree of deviation of the system from the critical state (the percolation threshold): h ε≈ , (10.5.4) λ V0 where h is the value of the streamline function Ψ = Ψ(x, y) near the percolation threshold, λ is the characteristic scale, and V0 is the characteristic velocity of the flow. The expression for L(ε ) corresponds exactly to the fractal representation of the curve length. There are advantages and disadvantages of such a representation. One of the advantages is the possibility of describing the longest and tortuous curves and long range correlations [91, 157]. To describe effects related to the considerable increase in transport coefficients, it is not sufficient to consider the fractal presentation of a streamline alone. Moreover, the fractal character of the particle path leads to slower diffusion (subdiffusion). Therefore, it is necessary to introduce another important value. In percolation theory
162
10 Fractal and Percolation Concepts
the correlation length a(ε ) is the main magnitude characterizing spatial scales of the system that is located near the percolation threshold ε → 0: a(ε ) =
λ . |ε |ν
(10.5.5)
Here, λ is the geometric characteristic scale (see Fig. 10.8). Thus, the idea of longrange correlations was accomplished in the framework of the percolation approach. However, there is a problem, since the diffusion coefficient is directly related to the 2 expression for the correlation length: D ≈ ΔCOR τ . Here, τ is the correlation time. In the case under consideration estimates yield: ΔCOR ≈ a(ε )ε →0 → ∞.
(10.5.6)
Finally, for this reason Kadomtsev and Pogutse [157] based their consideration of long-range correlation effects on the “diffusion renormalization” of the quasilinear equations. However, in this approach the percolation character of correlation effects was lost. To develop the percolation approach, it is necessary to take into account the fact that the percolation cluster occupies only a small fraction of the space. Therefore the value (10.5.7) Deff (ε ) ≈ DC P∞ (ε ) ∝ ε μ can be the estimate of the effective diffusion coefficient. Here, DC is the diffusion coefficient, which corresponds to transport on the percolation cluster DC (a) ∝
1 a (ε )θF
(10.5.8)
where θF and μ are the transport exponents. The value P∞ (ε ) defines the fraction of the space that is occupied by the percolation cluster: P∞ (ε ) ∝ ε β .
(10.5.9)
On substituting DC (ε ) and P∞ (ε ) into the expression for Deff , we obtain the formula: Deff (ε ) ∝ (ε ν )θF ε β .
(10.5.10)
Now, the relationship among these percolation exponents is given by:
μ = β + νθF .
(10.5.11)
In the subsequent consideration, we will use these results to analyze two-dimensional random flows. Therefore, it is worth pointing out that there is an exactly calculated value ν for two-dimensional percolation problems: ν = 4/3 [206, 207, 208]. To characterize transport one needs to define the values θF and β , which depend on a flow topology.
10.6 Finite Size Renormalization and Scaling
163
To conclude our discussions, we note that besides the potential model considered above, there exist other continuum percolation models. These are the problem of voids, or the “Swiss-cheese” model, and the problem of spheres, or the “inverted Swiss-cheese” model, also known as the problem of random sites. The best general references here are [209, 210, 211], where the reader will find the application of the percolation method to transport and elastic properties of a porous medium. It is important to note that the scaling of the transport properties of the Swiss-cheese and the inverted Swiss-cheese models near their percolation thresholds is very different from those of a random network. For further details we refer the reader to the original papers [212, 213].
10.6 Finite Size Renormalization and Scaling The case when a small percolation parameter tends to zero ε → 0 looks rather abstract for solving real physical problems. We consider here a simple and effective method that permits us to estimate the percolation parameters by using the finite value of the percolation parameter ε∗ instead of ε → 0. The correlation length is one of the most important values describing transport. However, in a system of finite size L0 we cannot consider the infinite value a(ε ) |ε →0 → ∞.
(10.6.1)
Here, it is relevant to introduce a new small “renormalization” parameter ε∗ [209] as the value that provides the condition a(ε∗ ) ≈ L0 .
(10.6.2)
The simplest calculations yield:
ε∗ ≈
λ L0
1 ν.
(10.6.3)
This result can be interpreted in the framework of percolation experiments with finite size samples. Under these conditions, the percolation threshold arises when the value of ε∗ differs slightly from zero and is situated in some interval Δε . The estimate obtained for ε∗ can be considered as the characteristic width of this interval (see Fig. 10.13): (10.6.4) Δ ε ≈ ε∗ . Actually, we are starting from the initial small parameter
ε0 ≈
λ << 1, L0
(10.6.5)
164 Fig. 10.13 Schematic illustration of the finite size renormalization
10 Fractal and Percolation Concepts
a( p)
L
Pc
ε *(L)
p
which describes the real physical system with the characteristic scales L0 and λ . On renormalization, we obtain a new percolation parameter: 1
Δε ≈ ε∗ ≈ ε0ν .
(10.6.6)
It is natural that the value Δε decreases if the system size L0 increases. Now we can calculate the diffusion coefficient that is based on the estimate of the finite correlation length a (ε∗ ): a2 (ε∗ ) Deff ∝ . (10.6.7) τ In the percolation models of turbulent diffusion the key problem is to determine a small parameter ε0 and to find an adequate renormalization condition for the finite value of ε∗ . As a typical example of small-parameter renormalization, let us consider the Trugman approach to percolation in graded systems [214]. The graded character of the model corresponds to the assumption that the system is subject to a small external influence, which does not in general destroy the percolation character of system’s behavior, but it can essentially change its properties. First, we consider this method from the formal point of view. Let us introduce a parameter ε0 characterizing the smallness of an influence. In contrast to the renormalization, which uses the dependence of the percolation parameter on the system size, (10.6.8) Δε ≈ Δε (L0 ) ≈ ε∗ , here we will deal with the spatial dependence, which is related to the graded character of the problem ε ≈ ε (x). From the dimensional consideration we can obtain the expression that characterizes the uncertainty of the choice of the small parameter for these conditions:
10.6 Finite Size Renormalization and Scaling
165
∂ ε (x) a(ε∗ ). ∂x
(10.6.9)
1 λ ≈ . v ε∗ [ε (x)a(ε∗ )]v
(10.6.10)
ε∗ ≈ Δε ≈ Then, simple calculations yield a(ε∗ ) ≈ λ
After the dimensional estimate in the form
ε (x) ≈
ε0 , λ
(10.6.11)
we obtain the “graded” renormalization condition (the Trugman renormalization) for the correlation scale: 1 λ (10.6.12) a(ε∗ ) ≈ λ v ≈ v . ε∗ ε 1+v 0
Here, the value 1
ε∗ = ε01+v >> ε0
(10.6.13)
is the new renormalized small parameter. Note that the direct use of the value ε0 as the parameter in the percolation dependences is not correct, since it characterizes a destructive influence of a superimposed perturbation and not the degree of deviation of the system from the percolation threshold. This method looks quite formal, but renormalization (10.6.10) was repeatedly used to obtain the critical exponents that describe the hull of a percolation cluster, to analyze transport in a system with shear flows, and to consider the models of multiscale percolation. Using the graded percolation method allows one to describe the structure of the diffusion front, which has a fractal nature [189, 201]. In the context of the problem, the characteristic spatial scale is the average distance between the source and the “front”: √ (10.6.14) lF (t) ∝ Dt. Here, D is the diffusion coefficient. Now, the problem under analysis can be interpreted in terms of the graded percolation, where ε = ε (x). We choose the value
ε0 ≈
λ lF
(10.6.15)
as the initial dimensionless parameter of the problem. In accordance with the graded percolation scaling we obtain the relation: a v+1 v . lF (a) ≈ λ λ
(10.6.16)
Here, a is the correlation size that characterizes the structure of the diffusive front. Numerical and theoretical research on the external perimeter of the percolation
166
10 Fractal and Percolation Concepts
cluster has concluded that in the two-dimensional case the hull L(a) is equivalent to the value lF (a) [91, 189, 201]. Hence, the fractal dimensionality of the hull Dh is defined by the formula: Dh = 1 +
1 = 7/4, ν
where ν = 4/3.
(10.6.17)
Indeed, simulations and rigorous mathematical consideration confirms this result [206, 207, 208]. For a deeper discussion of this topic we refer the reader to [91, 190, 191, 192, 193]. In the two-dimensional percolation models of turbulent diffusion [91, 157, 215], the hull L(a) is considered as a percolation streamline. The characteristic scale of the flow velocity V0 is usually a known parameter; then, the dimensional estimates of different physical values can be obtained by combinations of the values D0 , λ , V0 , and L. From this standpoint the relationship between Dh and ν allows all the characteristics of the system to be described in terms of the universal exponent ν that reflects the correlation character of the percolation models.
Further Reading Fractals and Diffusion Ben-Avraham, D. and Havlin, S. (1996). Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge, U.K. Bunde, A. and Havlin, S., eds. (1996). Fractals in Science. Springer-Verlag, Berlin. Bunde, A. and Havlin, S., eds. (1995). Fractals and Disordered Systems. Springer-Verlag, Berlin. Feder, J. (1988). Fractals. Plenum Press, New York. Gouyet, J.-F. (1996). Physics and Fractal Structure. Springer-Verlag, Berlin. Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman, San Francisco. Schroeder, M. (2001). Fractals, Chaos, Power Laws. Minutes from an Infinite Paradise. W.H. Freeman, New York. Pietronero, L. (1988). Fractals’ Physical Origin and Properties. Plenum Press, New York.
Fractals and Turbulence Biferale, L. and Procaccia, I. (2005). Physics Reports, 254, 1–41. 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. Falkovich, G., Gawedzki, K., and Vergassola, M. (2001). Reviews of Modern Physics, 73, 913. Frisch, U. (1995). Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge, U.K. Gouyet, J.-F. (1996). Physics and Fractal Structure. Springer-Verlag, Berlin. Kida, S. and Takaoka, M. (1994). Annual Reviews of Fluid Mechanics, 26, 169.
Further Reading
167
Oberlack, M. and Busse, F.H., eds. (2002). Theories of Turbulence. Springer-Verlag, Vienna. Peinke, J., Kittel, A., Barth, S., and Oberlack, M., eds. (2005). Progress in Turbulence. SpringerVerlag, Berlin. Siggia, E.D. (1994). Annual Reviews of Fluid Mechanics, 26, 137. Sreenivasan, K.R. (1997). Annual Reviews of Fluid Mechanics, 29, 435. Sreenivasan, K.R. (1999). Reviews of Modern Physics, 71, S 383. Tsinober, A. (2004). An Informal Introduction to Turbulence. Kluwer Academic, Dordrecht. Vassilicos, J.C., ed. (2001). Intermittency in Turbulent Flows. Cambridge University Press, Cambridge, U.K.
Percolation Bunde, A. and Havlin, S., Eds. (1995). Fractals and Disordered Systems. Springer-Verlag, Berlin. Isichenko, M.B. (1992). Reviews of Modern Physics, 64, 961. Stauffer, D. (1979). Physics Reports, 2, 3. Stauffer, D. (1985). Introduction to Percolation Theory. Taylor and Francis, London. Stanley, H.E. (1971). Introduction to Phase Transitions and Critical Phenomena. Clarendon Press, Oxford. Sokolov, I.M. (1986). Soviet Physics Uspekhi, 29, 924.
Chapter 11
Percolation and Turbulent Transport
11.1 Random Steady Flows and Seed Diffusivity Isichenko et al. [216] developed the potentiality of the renormalization method to describe turbulent transport in two-dimensional random flows in the framework of the percolation method. In such an approach, the percolation streamline contributes most to turbulent transport near the threshold. The percolation scalings of the turbulent diffusion coefficient obtained by this method differ significantly from the quasilinear one. This considerably increases the range of problems that can be treated in terms of transport scaling. Let us consider a two-dimensional zero-average-velocity steady flow specified by the bounded “common position” stream function Ψ(x, y) (see Fig. 10.12). We also imply an isotropic-on-average oscillating function with a quasi-random location of saddle points along its height. To treat the random flow velocity field, we introduce the following scales: Ψ . (11.1.1) Ψ0 ≈ λ V0 , λ ≈ ∇Ψ Here, V0 is the characteristic velocity scale and λ is the spatial scale. The value h p = ελ V0 is the percolation scale of the stream function near the threshold, where ε is the small percolation parameter. To calculate the effective diffusion coefficient, we will consider the relation obtained above (10.5.7): Deff (ε ) ≈ DC (ε ) P∞ (ε ) ≈
a2 (ε ) P∞ (ε ) . τ
(11.1.2)
Here, a is the spatial correlation scale, τ is the correlation time, DC is the correlation contribution to the effective diffusion coefficient, and P∞ is the fraction of a space occupied by the percolation streamline. Effects of “long-range correlations” enter into the expression for the diffusion coefficient precisely through a(ε ). Following the ideas of the convective nature of the flow along the percolation streamline, we estimate P∞ in terms of the length of the percolation streamline L(ε ) and the stochastic layer width Δ (ε ) (see Fig. 11.1):
O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008
169
170
11 Percolation and Turbulent Transport
L(ε)
Δ(ε)
a(ε)
Fig. 11.1 Schematic illustration of a two-dimensional random flow
L(ε )Δ(ε ) . a2 (ε )
(11.1.3)
a2 L(ε )Δ(ε ) L (ε ) ≈ Δ (ε ) , τ a2 τ
(11.1.4)
P∞ (ε ) ≈ The expression derived, Deff (ε ) ≈
is similar to the formula for the effective diffusivity in the convective cell system: Deff ≈ V0 Δ ≈ D0 Pe1/2 ,
(11.1.5)
but here we deal with the percolation type of dependence Δ = Δ(ε ). Here, D0 is the seed diffusivity and Pe = λ V0 /D0 is the Peclet number. Moreover, in line with ideas in the literature on percolation theory, we are basing the expression on the renormalization, i.e., a method of calculating the universal value of the small parameter ε . Let us identify the small “width” of a percolation streamline with the small parameter of percolation theory: (11.1.6) Δ(ε ) = λ ε , which provides the universal small parameter that must be found as a solution of this equation. Such a renormalization is the main step in the framework of the percolation description of turbulent transport and we will actively use a similar technique to solve other percolation problems. To define the value Δ we consider the particle balance by analogy with the problem of convective cells (9.1.12) where the number of particles escaping from a convective cell per unit time ND ∝ D0 Δn λ is equal to the number of particles carried away by convective flow along the boundary layer NC ∝ nV0 Δ, where n is the scalar density. However, in the percolation case, the correlation effects should be taken
11.1 Random Steady Flows and Seed Diffusivity
171
into account by using the percolation streamline length L(ε ), instead of the geometric scale λ : 7 D0 L(ε ) Δ(ε ) ≈ . (11.1.7) V0 This approach creates a dependence of Δ on the parameter ε . Now, we obtain the equation for the determination of the “universal” value of ε∗ = λhV∗ as a function of 0 the flow parameters D0 ,V0 , λ : 7 D0 L(ε ) = ελ . (11.1.8) V0 The specific calculations can be completed by using the rigorous scaling results of percolation theory obtained for the correlation scale a:
λ , |ε |ν
a(ε ) =
(11.1.9)
and the length of the fractal streamline L:
L(ε ) = λ
a (ε ) λ
Dh ,
(11.1.10)
as functions of ε for the two-dimensional case, where the correlation exponent ν and the Hull exponent Dh are given by (10.6.17):
ν = 4/3,
1 Dh = 1 + . ν
(11.1.11)
The functional form of these dependencies reflects the fractal and percolation behavior of streamlines. The solution of renormalization equation (11.1.8) in terms of the Peclet number Pe = λ V0 /D0 leads to the scalings:
ε∗ =
1 Pe
1 3+ν
=
1 Pe
3
13
,
h∗ = λ V0 ε∗ = λ V0 Pe−3/13 ∝ V0 ,
1 1 3+ν 10/13 Deff ≈ V0 Δ(ε∗ ) ≈ V0 λ ε∗ ≈ V0 λ = D0 Pe10/13 ∝ V0 . Pe 10/13
(11.1.12) (11.1.13) (11.1.14)
In the framework of the percolation theory, where effective diffusivity is given by the relation Deff (ε ) ∝ ε μ , this corresponds to μ = 1. From the point of view of the renormalization of the initial small parameter 1 ε0 = Pe , the expression for the percolation small parameter can be obtained: 1
ε∗ = (ε0 ) 3+ν >> ε0 .
(11.1.15)
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11 Percolation and Turbulent Transport
Some “arbitrariness” in the expression Δ(ε ) = λ ε in the selection of the value λ ε , rather than λ ε 2 or λ ε 3 , can be interpreted as a desire to have a universal small parameter, which is analogous to a single characteristic scale—the correlation length in the theory of phase transitions. Here, it should be particularly emphasized that the length of the fractal percolation line is not infinitely large, because the small parameter ε∗ does not tend to zero, but has a finite value,
ε∗ =
h∗ , λ V0
(11.1.16)
for all types of flows with the characteristics D0 ,V0 , λ . Therein lies the universality of the formula: Δ(ε ) = λ ε . Note that the simplicity of the percolation estimate of turbulent transport is elusive. It suffices to recall in this connection the “hierarchy” of scales used here:
λ
a Dh
λ
≈L≈
a h∗ λ >> a ≈ ν >> λ >> ≈ Δ ≈ λ ε∗ . ε∗ ε∗ V0
(11.1.17)
Here, the length of the percolation streamline, L(ε∗ ) ≈ λ
1 , ε∗ 7/3
a(ε∗ ) = λ
1
(11.1.18)
and the correlation scale,
ε∗ 4/3
,
(11.1.19)
which are responsible for the percolation behavior, are not infinitely large, since ε∗ has a precise value. The fraction of volume occupied by the percolation streamlines is now estimated as: P∞ =
L(ε∗ )Δ(ε∗ ) λ ≈ ε∗ 4/3 . ≈ 2 a(ε∗ ) a (ε∗ )
(11.1.20)
Besides these scaling laws, we can also obtain some additional information that will be useful for the subsequent analysis. Note that in terms of the Peclet number the percolation regime is intermediate between the regime of convective cells Deff ≈ V0 Δ and the purely convective regime Deff ≈ λ V0 . We can also estimate the range of the percolation scaling applicability in term of spatial scales. It is necessary to take into account the finite size L0 of a real system. By analogy with the system size renormalization (10.6.2) we can consider the estimate: ν (11.1.21) a(ε∗ ) = λ Pe ν +3 ≤ L0 . Then, calculations yield the inequality for the Peclet number in the form
(ν +3)/ν L . 1 < Pe < λ
(11.1.22)
11.1 Random Steady Flows and Seed Diffusivity
173
There is also a correlation meaning of the percolation scaling. In the steady case the correlation time τ is described by the scaling law:
τ≈
L(ε ) λ 1 λ ≈ >> . 7/3 V0 V0 ε∗ V0
(11.1.23)
On the other hand, the correlation scale a can be represented in the form: a ≈ ε∗ L ≈ ε∗V0
Δ2 ≈ ε∗V0 τD << V0 τD . D0
(11.1.24)
From the formal point of view one can expect that there exists an additional spatial scale lS : L(ε )lS ≈ 1. λ2
(11.1.25)
Indeed, the scale lS (ε∗ ) ≈
λ 2 ε∗ ≈ λ ε∗ν +1 ≈ ε∗ν Δ(ε∗ ) a(ε∗ )
(11.1.26)
plays a significant role in the analysis of the flow topology reconstruction (reconnection process) [217]. This problem is discussed in the next section. Now the entire spatial hierarchy of scales is given by (see Fig. 11.2): a Dh
λ a >> a ≈ ν >> λ >> Δ ≈ λ ε∗ >> lS ≈ λ ε∗ν +1 ≈ ε∗ν Δ. ε∗ ε∗ (11.1.27) This five-level spatial hierarchy of scales opens wide possibilities to obtain new scalings in the framework of the percolation method. λ
λ
≈L≈
L(ε) = a(ε) Dh a (ε) = λ / εν Δ(ε) = λε
Fig. 11.2 A typical plot of the percolation spatial scales
0
1
ε
174
11 Percolation and Turbulent Transport
11.2 Reorganization of Flow Topology and Percolation Scalings The time dependence of complex flow topology, where the percolation streamlines play an important role, is a factor that significantly impacts on transport processes (see Fig. 11.3). To describe such regimes we must introduce a new dimensionless parameter that includes the characteristic time of flow topology reorganization T0 ≈ 1/ω . Thus, in the case of high frequencies ω , the path of a tested particle can be estimated ballistically: V0 lω ≈ V0 T0 ≈ , (11.2.1) ω and used as the correlation length to define the diffusion coefficient: D≈
lω 2 V0 2 1 ≈ 2 . τ ω τ
(11.2.2)
Then, applying 1/ω as the correlation time leads to the quasilinear limit: Deff = λ 2 ω Ku2 ∝ V0 2 .
Fig. 11.3 The reconstruction of the two-dimensional flow. (After S.D. Danilov et al. [218], with permission)
(11.2.3)
11.2 Reorganization of Flow Topology and Percolation Scalings
175
Here, Ku = V0 /λ ω is the Kubo number. However, in the low-frequency case, ω << V0 /λ or λ << V0 T0 , the percolation correlation scale a is much less than lω . Following the renormalization ideas, which were successfully applied in the description of steady percolation regimes with seed diffusivity, Grusinov, Isichenko, and Kalda [217] considered the percolation limit of the turbulent diffusion of a passive scalar in a time-dependent, incompressible, two-dimensional flow. In estimating the time it takes the flow pattern to change completely as T0 ≈ ω1 , we consider the low-frequency case, where: V0 or λ << V0 T0 . λ
ω <<
(11.2.4)
In this formulation of the problem, the lifetime of the individual percolation streamline τ is the main parameter. The standard expression can be used for the diffusion coefficient: a2 (ε ) . (11.2.5) DC (ε ) ≈ τ In the context of this problem the relation
τ ≈ε
1 ≈ ε T0 ω
(11.2.6)
is used as the renormalization equation to obtain the small parameter of the problem ε . This is analogous to the description of the steady percolation regime. In the time-dependent flow under consideration, one would also expect a universal result for a specific “universal” value of the small percolation parameter ε∗ . For this purpose, one can use a simple expression accounting for the convective nature of motion along the percolation streamline during the lifetime of this streamline: 1 L(ε ) ε = = τ (ε ) . (11.2.7) ω V0 This equation enables one to find the small percolation parameter ε∗ in terms of the time-dependent flow parameters: ω , V0 , λ . Substituting percolation scalings a(ε ) = λ ε −ν ,
L(ε ) = λ
a Dh
λ
ν = 4/3,
,
1 Dh = 1 + , ν
(11.2.8)
we can complete our calculations. Now, one easily obtains: h∗ ε∗ = = λ V0
λω V0
1 2+ν
= Ku− 10 ∝ ω 10 . 3
3
(11.2.9)
Here, we introduce the Kubo number Ku = λVω0 instead of the Peclet number, which was used for the analysis of a steady flow. From the standpoint of the renormalization of the initial small parameter ε0 ≈ 1/Ku we obtain the expression 1
ε∗ = (ε0 ) 2+ν >> ε0 .
(11.2.10)
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11 Percolation and Turbulent Transport
On direct substitution of the expression for ε∗ we arrive at the relation: DC (ε∗ ) ≈
a(ε∗ )2 , τ (ε∗ )
(11.2.11)
where the dependence on T0 appears quite Unexpected 1/10
DC ∝ T0
≈
1 . ω 1/10
(11.2.12)
The slow “restructuring” of the flow is unlikely to result in a significant growth in turbulent diffusion (see Fig. 11.4). The reason lies in the fact that we did not take into account the fraction P∞ of percolation streamlines in the total flow: P∞ =
L(ε )Δ(ε ) . a(ε )2
(11.2.13)
It is now evident that we need additional information on the value of Δ(ε ), despite the fact that we have calculated ε∗ . Unlike the steady percolation case, we are forced to make an additional assumption here other than the renormalization relation τ (ε ) ≈ ε T0 . One can set Δ(ε ), similar to the percolation approach for a steady random flow. In fact, use is made of the formulas: Δ ≈ ε∗ λ ,
P∞ ≈ ε∗ 4/3 ≈
λ . a(ε∗ )
(11.2.14)
Calculations now lead to the final expression for Deff :
Deff = DC (ε∗ )P∞ (ε∗ ) = V0 Δ (ε∗ ) = λ V0
1 Ku
1 ν +2
7
3
∝ V010 ω 10 .
(11.2.15)
Deff
Deff ∝1/ ω
V0λ
Fig. 11.4 A typical plot of the percolation dependence of the effective diffusivity on the frequency
Deff ∝ ω 3/10 Ku >> 1
V0 / λ
Ku << 1
ω
11.2 Reorganization of Flow Topology and Percolation Scalings Fig. 11.5 A typical plot of the dependence of the effective diffusivity on the Kubo number
177
Deff (Ku) Ku<<1
Ku>>1
Deff ∝ Ku 7/10 Deff ∝Ku2
0
Ku=1
KuMax Ku
Here, Deff is the renormalized effective diffusivity expressed in terms of the Kubo number (see Fig. 11.5). The formula obtained mirrors the universal character of percolation diffusion in time-dependent flows, since the value ε∗ depends only on the flow parameters ω ,V0 , λ . Moreover, the correlation scale a(ε∗ ) is really much less than the frequency pass lω for ε∗ < 1: (11.2.16) a(ε∗ ) ≈ ε∗2V0 /ω ≈ ε∗2 lω << lω , which demonstrates the correctness of the assumptions made. Note that the condition (11.2.17) a (ε∗ ) ≈ ε∗ lω leads to the renormalization
1 (11.2.18) ω that does not correspond to the low-frequency regime. 1 1 ν +2 Assuming the small percolation parameter ε∗ = Ku < 1 and accounting for the finite size of the system:
τB ≈ T0 ≈
a(ε∗ ) =
ν λ ν +2 ≤ L 0 ν = λ Ku |ε∗ (Ku)|
(11.2.19)
we can find an inequality for the Kubo number, which corresponds to the timedependent percolation (see Fig. 11.5):
ν +2 L ν = Kumax . 1 < Ku < λ By extensive numerical computation, the percolation prediction,
(11.2.20)
178
11 Percolation and Turbulent Transport
Deff (V0 ) ≈ λ 2 ω
V0 λω
7/10
7
7
∝ V010 ∝ Ku 10 ,
(11.2.21)
has been checked to hold the low-frequency domain for guiding centers in a k−3 turbulent spectrum simulated by a large number of randomly phased waves, and this has recently been confirmed at very high amplitudes [91, 219]. This has also been checked with a different spectrum involving a low-k Gaussian part and an asymptotic k−3 decreasing part [220]. Percolation scaling in the asymptotic highamplitude limit thus seems to be very well confirmed.
11.3 Spatial and Temporal Hierarchy of Scales In a manner similar to the above-considered hierarchy of spatial scales in steady percolation, we analyze characteristic times allowing us to distinguish the regimes of flow, where the “long-range” correlation effects become the principal ones. Besides characteristic time T0 and correlation time τ = τB = ε∗ T0 , we must estimate the effect of the diffusive escape of particles from the streamlines. In fact, it is necessary to calculate the time of escape of particles from the streamlines τD . The time-dependent percolation regime was considered in the framework of the condition τ < τD . To obtain estimates, we make use of the diffusion coefficient of streamlines Dψ and relate it to the coefficient of “seed” diffusion D0 :
τD ≈
h2 , Dψ
Dψ = V02 D0 .
(11.3.1)
The applicability condition for the low-frequency regime takes the form:
τ ≈ τB ≈
h h2 < 2 ≈ τD . λ V0 ω V0 D0
(11.3.2)
This is in fact a limitation on the value of the “seed” diffusion coefficient D0 , D0 < hω (h)
λ < hν +3 . V0
(11.3.3)
To conclude the discussion of this issue, we give the hierarchy of characteristic times in the framework of the percolation approach: 1 ω
h λ V0
1+ν
= τS << τ ≈
h h2 1 << ≈ τD << ≈ T0 , ωλ V0 ω D0V02
(11.3.4)
where τS = VlS0 is the characteristic time that describes the stochastic instability of streamlines. This means that one deals with the temporal interval
11.3 Spatial and Temporal Hierarchy of Scales
179
τS << τ < τD .
(11.3.5)
The interpretation of renormalization conditions in terms of the characteristic times is a very important aspect of the problem. For example, the condition for the steady case (11.1.6) can be represented by the balance of the ballistic and diffusive times: L(ε∗ ) Δ2 (ε∗ ) τ≈ ≈ . (11.3.6) V0 D0 It is easy to note that there are several characteristic times that do not play essential roles at this stage: h h << << τ . ω L(h)V0 ω a(h)V0
(11.3.7)
The mean-field estimates considered above were obtained in the framework of the percolation character of streamlines. Their quasi-steady topology is characterized by the percolation exponents ν and Dh . Let us consider now the evolutionary aspects of percolation structure growth, where the initial stage of the evolution of the correlation length a(t) and the stochastic layer width Δ (t) plays a significant role. Thus, simultaneously with the increase in the correlation scale,
aI (t) ≈
L (t) λ
1/Dh ,
(11.3.8)
it is necessary to take into account the increase in the stochastic layer width Δ = Δ(t), which, in the framework of percolation models of turbulent diffusion, is closely related to the value of the small parameter ε∗ ≈ Δ/λ , and hence to the definition of the correlation scale a ≈ λ /ε ν . Trivial calculations allow one to obtain the expression describing the increase in correlation scale aD (t) due to the increase in the stochastic layer width: λ (11.3.9) aD (t) ≈ ν . Δ(t) λ
In the framework of the mean field theory the consideration of the balance between aD (t) and aI (t) enables us to estimate the characteristic time t0 that has to be used to define the effective diffusion coefficient Deff . Thus, in the case of the diffusive increasing the stochastic layer width Δ2 ≈ D0t, one obtains:
aI (t) ≈ λ
V0t0 λ
1 Dh
Calculations lead to the estimate
λ ≈ √ ν ≈ aD (t) . D0t0 λ
(11.3.10)
180
11 Percolation and Turbulent Transport
t0 ≈
λ2 D0
1 Pe
1 ν +3
=
λ2 D0
D0 λ V0
1 ν +3
,
(11.3.11)
which after substitution in the formula for the stochastic layer width yields:
1 2
Δ ≈ (D0t0 ) ≈ λ
1 Pe
1 ν +3
,
(11.3.12)
It is easy to see that this expression coincides exactly with the expression for the steady case (11.1.4), and the corresponding estimate of the effective diffusion coefficient is given by: (11.3.13) Deff ≈ V0 Δ. Naturally, other estimates can also be used to describe the growth of the stochastic layer width. Thus, in the Hamiltonian dynamics the linear estimate Δ ∝ t is widely used. In the context of the description of time-dependence effects it is easy to represent this expression in the form: Δ(t) = (λ ω )t,
(11.3.14)
where ω is the characteristic frequency. Then, the correlation scales balance,
λ
V0t0 λ
1 Dh
≈
λ , (ω t0 )ν
(11.3.15)
allows the estimate of the characteristic time t0 to be obtained 1 t0 ≈ ω
λω V0
1 ν +2
1 ≈ ω
1 Ku
1 ν +2
.
(11.3.16)
The expression for the stochastic layer width then takes the form corresponding to the percolation model (11.2.15):
Δ = λ ε∗ = λ
1 Ku
1 ν +2
,
(11.3.17)
and hence the estimate of the turbulent diffusion coefficient is given by expression (11.2.15). Note that the approach under analysis makes it possible to use the correlation scale balance as the basis for constructing new percolation transport models based on approximations Δ(t). In addition, one can consider the dynamics of growth of two-dimensional smallscale structures in terms of λ (t). Formally, we could describe the structure evolution by the following expression: λ (t) ≈ ε ν a (t) , (11.3.18) where ballistic scaling is used:
11.4 Percolation in Drift Flows
181
L (t) ≈ λ
a (t) λ
Dh
≈ V0t.
(11.3.19)
Then, simple calculations yield:
λ (t) ≈ ε∗ ν a (t) ≈ ε∗ (λ )ν (V0t)1/Dh ,
(11.3.20)
where ε∗ is the renormalized percolation parameter, which could be taken from known percolation models. For instance, in the presence of seed diffusivity, we find the relation:
1
1 1 ν +3 1 ν +3 ε∗ ≈ ∝ . (11.3.21) Pe λ After substitution one obtains: 1 ν +3 λ ≈ V02 Dν0 +1 2ν +4 t 2ν +4 ∝ t 13/20 ≈ t 0.65 ,
(11.3.22)
where ν = 4/3. Formula (11.3.22) leads to the simplest percolation estimate of the growth of the characteristic scale of small structures in the system under analysis. Such regimes were observed in astrophysical turbulence, where the value of the exponent describing the structure evolution was found to be 0.7 [221].
11.4 Percolation in Drift Flows The presence of a small drift velocity can significantly changes turbulent transport. The influence of the small external perturbation on the percolation system has been considered in terms of the graded approach, where the renormalization of the small percolation parameter was used. The natural approach to analyze the influence of a small drift velocity Ud on the fractal topology of streamlines (see Fig. 11.6) can be written as: (11.4.1) V = V0 +Ud , Ud << V0 .
Fig. 11.6 Schematic illustration of an unclosed percolation streamline
182
11 Percolation and Turbulent Transport
Here, V0 is the mean flow velocity. The simplest way to introduce the small parameter is by using the value: Ud ε0 = . (11.4.2) V0 However, in this approach the fractal character of percolation streamlines is completely lost. Yushmanov suggested the use of the following dimensional estimate of the drift velocity [222, 223]: Ud =
a(ε ) P∞ (a), τ (ε )
(11.4.3)
where the correlation time and correlation scale are given by:
τ (ε ) ≈
L(ε ) , V0
a(ε ) ≈ λ |ε |−ν .
(11.4.4)
For P∞ we use an additional expression, which was suggested for the steady case, P∞ ≈ λ /a. Simple calculations lead to the parametric dependence for the renormalized small parameter ε∗ on the flow parameters V0 and Ud :
ε∗ =
Ud V0
1 1+ν
,
where
ν = 4/3.
(11.4.5)
It is easy to see that this expression completely coincides with the graded renormalization result (10.6.13) and can be interpreted in terms of the streamline function (see Fig. 11.7):
ψ(x,y)
Fig. 11.7 A drift percolation landscape
11.4 Percolation in Drift Flows
183
Ud ≈
Ψ1 ε Ψ0 ≈ . a(ε ) a(ε )
(11.4.6)
Indeed, this expression relates the drift velocity to both the amplitude of the mean stream function Ψ0 and the percolation correlation scale a (ε ). Here, we are dealing with the conditions: Ψ1 << Ψ0 ≈ λ V0
and
a(ε ) >> λ .
(11.4.7)
One obtains the estimate that is related to the finite size of a system a(ε∗ ) ≤ L0 :
V0
λ L0
(ν +1)/ν < Ud < V0 .
(11.4.8)
Moreover, we can note how the spatial hierarchy of scales is included in the description of perturbations [224]: Ud ≈ ε
λ λ Δ V0 ≈ V0 ≈ V0 . a(ε ) L(ε ) a(ε )
(11.4.9)
We see that parameter λ does not enter into the expression for the renormalized value ε∗ . One can determine some velocity U, which will be much less than Ud : U≈
Δ V0 << Ud << V0 . L(ε )
(11.4.10)
This problem was considered in connection with the study of compressibility or dissipation effects [225]. Indeed, the presence of even small compressibility or dissipation can considerably change the character of streamlines behavior. In principle, there are also alternative possibilities of the renormalization. For example, we can consider the following expression: Ud = ε
a(ε ) ε a(ε ) = V0 . τ (ε ) L(ε )
(11.4.11)
This case corresponds to setting P∞ (ε ) ≈ ε (compare with the steady case where P∞ ≈ λ /a(ε )). However, in contrast to the previous case (11.4.5), these estimates lead to the expression: (11.4.12) Ud ≈ ε 2V0 , which does not contain the percolation exponent ν . Undoubtedly, it is an important drawback. Therefore, taking into account the transport effects, we will follow the graded renormalization (11.4.5). Now we can obtain the estimate of the diffusion coefficient in terms of the stochastic layer width Δ ≈ λ ε∗ (see Fig. 11.8): 1
Deff ≈ V0 Δ(ε∗ ) ≈ λ V0 (ε0 ) 1+v ≈ λ V0
Ud V0
3 7
4
∝ V07 .
(11.4.13)
184
11 Percolation and Turbulent Transport
Fig. 11.8 The diffusion coefficient as a function of the relative fluctuation amplitude in the presence of drift. (After A.I. Smolyakov and P. Yushmanov [223], with permission)
A more complex situation arises when both the drift effects and the time dependence characterized by the time T0 ≈ 1/ω exist simultaneously in a system.
11.5 Drift and Low-Frequency Regimes The analysis of drift effects on the basis of the graded renormalization performed in the previous section does not take into account processes related to the temporal fluctuations of the velocity field. Moreover, in the framework of the “graded” approach (11.4.5), the correlation time τ plays a formal role because to define the effective diffusion coefficient Deff = V0 Δ, the conjecture Δ = λ ε∗ was applied. Nevertheless, the scaling approach allows us to consider percolation transport in the presence of both drift effects and fluctuations. The main parameter that characterizes the temporal dependence of the stream function is the characteristic time:
τD ≈
Ψ ∂Ψ ∂t
.
(11.5.1)
11.5 Drift and Low-Frequency Regimes
185
The percolation character of transport is manifested in the estimate Ψ ≈ ε∗ Ψ0 ≈ ε∗V0 λ .
(11.5.2)
Here, ε∗ is the small parameter of the percolation model. On the other hand, it is necessary to take into account the “graded” character of perturbations (11.4.6): ΔΨ ≈ Ud a(ε∗ ),
1
ε∗ = ε01+ν .
(11.5.3)
Then, estimates yield:
∂Ψ ≈ Ud a(ε∗ )ω . (11.5.4) ∂t Here, Ud is the drift velocity, a is the correlation scale, and ω is the characteristic frequency of the stream function alteration. Calculations yield the correlation time: τ=
1 1+ν ν ε . ω 0
(11.5.5)
The simplest quasilinear dependence is given by the relation
τ (ω ) ≈
1 . ω
(11.5.6)
One can calculate the effective diffusion coefficient in the renormalized quasilinear form: a2 a(ε∗ ) . (11.5.7) Deff ≈ P∞ ≈ Ud2 τ τ λ Here, P∞ ≈ λ /a(ε∗ ) << 1, which corresponds to the drift model [222] with the small percolation parameter:
ε∗ =
Ud V0
1 1+ν
,
(11.5.8)
which corresponds to the graded renormalization. On substituting the expression for τ = 1/ω into (11.5.7), we obtain the scaling: Deff ≈
Ud 2 ω
1 ε0
ν 1+ν
≈
Ud 2 ω
V0 Ud
4 7
10
∝ Ud7 .
(11.5.9)
This dependence shows that there is an essential difference between transport in the presence of velocity field fluctuations and the steady case (11.4.2). The quasilinear method of calculating the effective diffusion coefficient Deff only corrects the percolation expression (11.5.5), whereas the relevant approach would be to include the frequency ω into the renormalization condition that characterizes transport in the drift flow in the presence of the time-dependent perturbation of the stream function. Thus, in the steady case [216] the balance of characteristic times is
186
11 Percolation and Turbulent Transport
the main assumption to obtain the small percolation parameter:
τ≈
Δ(ε )2 L(ε ) ≈ . D0 V0
(11.5.10)
In considering the fluctuations of the stream function, the diffusion estimate of the characteristic time is given by (11.3.1):
τD ≈
h2 (ελ V0 )2 ≈ , where DΨ ≈ V02 D0 . DΨ DΨ
(11.5.11)
However, when considering the drift effects we must use the correlation estimate [222]: (11.5.12) ΔΨ ≈ Ud a(ε∗ ). The important aspect is that the streamline diffusivity DΨ is responsible for the physical mechanism of streamline distortion. Let us approximate the value DΨ in the form that mirrors the certain character of the external influences (drift flow and time dependent perturbation): DΨ ≈ (ΔΨ)2 ω ≈ Ud2 a(ε )2 ω .
(11.5.13)
The new equation for the small percolation parameter ε∗ takes the form [226]: (ελ V0 )2 L(ε ) = . V0 Ud2 a(ε )2 ω
(11.5.14)
In fact, we have renormalized the streamline diffusivity, DΨ ≈ V02 D0 , in accordance with mechanisms distorting streamlines (the drift flow with the characteristic velocity Ud and temporal fluctuations with the frequency ω ). Solving Eq. (11.5.14), we obtain a new small parameter for the problem of percolation transport in the presence of both the drift flow Ud and the fluctuations with the characteristic frequency ω :
ε∗ ≈
Ud V0
2 3(1+ν )
1 Ku
1 3(ν +1)
2
−3
1
∝ Ud7 V0 7 ω 7 .
(11.5.15)
Note that the characteristic size λ is included in this definition of the small parameter, in contrast to the quasilinear case. The expression for the effective diffusion coefficient can be written as:
Deff ≈ V0 λ ε∗ ≈
Ud V0
2 3(ν +1)
λω V0
1 3(ν +1)
2
1
∝ Ud7 ω 7 .
(11.5.16)
This result differs significantly from the quasilinear representation (11.5.9), Deff ∝ 10/7
Ud ω
. This new scaling (11.5.16) is in accord with other low-frequency regimes (see Fig. 11.5). For example, in the case Ud = λ ω we obtain:
11.6 Renormalization and the Stochastic Instability Increment
ε∗ ≈
Ud V0
2 3(1+ν )
1 Ku
1 3(ν +1)
187
.
(11.5.17)
This leads to the characteristic time balance:
τB =
L(ε ) 1 = = T0 , V0 ω
(11.5.18)
which shows the absence of the intermediate characteristic time τD in this case. It is possible also to obtain other estimates based on the conditions Ud << λ ω or Ud >> λ ω , though we do not do so here because we shall not make any further use of these results.
11.6 Renormalization and the Stochastic Instability Increment The consideration of time-dependent two-dimensional random flow is of particular interest because there is no exponential separation of initially nearby streamlines in the steady case in a bounded two-dimensional area. Actually, the steady case corresponds to the Hamiltonian one-dimensional problem, whereas when the characteristic frequency ω is not zero we deal with the stochastic behavior. The percolation approach to the consideration of the hierarchy of spatial and temporal scales allows us to treat long-range correlation effects in terms of simple scalings. One of the important problems here is to estimate the stochastic instability increment. Analyzing the spatial scale hierarchy in a two-dimensional random time-dependent velocity field, the authors of [217] estimated the square corresponding to the stochastic layer Δ as: L (ε∗ ) Δ (ε∗ ) ≈
a (ε∗ ) λ ε∗ ≈ aλ >> λ 2 . ε∗
(11.6.1)
This mirrors the existence of many streamlines in the stochastic layer (see Fig. 11.9). At the same time, this means that there is a spatial scale lS that characterizes a single streamline. A relevant estimate of lS is the expression: lS (ε ) ≈
λ2 ≈ λ ε ν +1 ≈ ε Δ << Δ ≈ λ ε . L (ε )
(11.6.2)
The characteristic reconnection time of a pair of nearby separatrixes can be evaluated as: λω VS L (ε ) ω 1 ≈ ≈ ≈ γS ≈ , (11.6.3) τS (ε ) lS (ε ) lS (ε ) λ where VS = λ ω is the dimensional estimate of separatrix motion. On the other hand, the balance of characteristic times τS ≈ τB , which has the form
188
11 Percolation and Turbulent Transport
Δ(ε) ≈ λε
Fig. 11.9 Schematic illustration of a stochastic layer
τS =
L ( ε∗ ) λ ≈ = τB , L ( ε∗ ) ω V0
(11.6.4)
allows the small percolation parameter ε∗ to be defined:
ε∗ ≈
λω V0
1 2(ν +1)
≈
1 Ku
1 2(ν +1)
≈
1 Ku
3
14
,
(11.6.5)
where for the two-dimensional case ν = 4/3. The final expression for the stochastic instability increment γS then takes the following form:
γS ≈
√ L (ε∗ ) 1 ≈ω ≈ ω Ku. τS λ
(11.6.6)
It is easy to take into consideration here the logarithmic factor (8.1.8), which leads to the result: √ Ku for Ku > 1. (11.6.7) γS ≈ ω ln Ku The Kubo number Ku can be easily interpreted in terms of the magnetic Kubo number Rm ≈ b0 LZ /Δ⊥ and hence for the stochastic magnetic field one finds: √ 1 Rm γZ ≈ . (11.6.8) LZ ln Rm Here, b0 is the perturbation amplitude of stochastic magnetic field, Δ⊥ is the perpendicular correlation scale, and LZ is the longitudinal characteristic scale. It is interesting to compare this expression with the quasilinear result (8.2.10) discussed above (see Fig. 11.10):
11.7 Stochastic Magnetic Field and Percolation
189 γZ ( Rm )
Fig. 11.10 A typical plot of the dependence of the stochastic instability increment on the magnetic Kubo number
Rm << 1
Rm << 1
γ
Rm = 1
0
γZ ≈
γ ∝ R 1/ 2 Z m
2 Z ∝ Rm
R2m . LZ
RMax
Rm
(11.6.9)
We see that the stochastic instability increment in the percolation case (long-range correlations) is characterized by a smooth scaling, which is analogous to the dependence of the effective diffusion coefficient on the Kubo number. The simulations [227, 228] confirm this conclusion.
11.7 Stochastic Magnetic Field and Percolation The percolation approach allows us to obtain the diffusion coefficient of magnetic streamlines Dm [15, 45, 50, 91, 229]. The chaotic behavior of magnetic field streamlines creates a complex topological pattern. In this nontrivial situation, the use of the fractality and percolation ideas is very fruitful, since the notion of correlations and characteristic spatial scales of structures under analysis can be related to topological characteristics. The percolation model describes stochastic magnetic field on the basis of the analogy between the Kubo number Ku = V0λT0 and the magnetic Kubo number Rm ≈ bΔ0 LZ , where regimes with ⊥
Rm ≈
b0 LZ >1 Δ⊥
(11.7.1)
are considered. Therefore, the result of the time-dependent percolation case can be used to analyze a “braided” magnetic field. Isichenko [230, 231] applied this analogy directly. Consider the expression for Dm in the form (11.2.15): − 1
Dm = b0 Δ⊥ Rm ν +2 .
(11.7.2)
190
11 Percolation and Turbulent Transport
Here, the value of the relative amplitude of the stochastic magnetic field b0 is used instead of the velocity V0 , Δ⊥ corresponds to the spatial scale λ , and the value LZ that enters into the quasilinear definition of the longitudinal correlation length corresponds to T0 that was used in defining the Kubo number Ku = V0λT0 . Many authors conducted numerical simulations [219, 228, 232], which allow us to consider that the analytical result (for the monoscale model with the exponent 7/10): 7/10 (11.7.3) Dm ∝ b0 is in qualitative agreement with the simulation results: 0.6–0.8. Percolation effects in the stochastic magnetic field can also be considered in the context of the well-known Kadomtsev-Pogutse transport scaling (8.3.15): 2 (11.7.4) Deff ≈ D// D⊥ . Here, D// is the longitudinal diffusivity and D⊥ is the perpendicular diffusion coefficient. The model suggested in [157] is based on the assumption of the presence of a strong longitudinal magnetic field. In these conditions, the longitudinal diffusion coefficient is much greater than the transverse diffusion coefficient: D// >> D⊥ and the choice of the value
ε0 =
D⊥ D//
(11.7.5)
(11.7.6)
as an initial small parameter is fairly natural. The renormalization condition, which allows the true small parameter to be obtained, needs to mirror the competition between the longitudinal and transverse decorrelation mechanisms. It is convenient to reformulate the particle balance condition in the stochastic layer with allowance for the diffusive character of longitudinal motions: n n D⊥ L ≈ D// Δ. Δ L
(11.7.7)
Here, n is the density of the particle number, L = L(ε ) is the length of the fractal streamline, and Δ = Δ(ε ) is the width of the stochastic layer. We can rewrite this condition in terms of the equivalence of decorrelation times, by analogy with the convective cell case: L(ε )2 Δ(ε )2 τ≈ ≈ . (11.7.8) D// D⊥ Equations (11.7.7) and (11.7.8) allow one to determine the small parameter of the percolation problem:
D⊥ . (11.7.9) ε∗ = ε∗ (ε0 ) = ε∗ D//
Further Reading
191
To solve these equations we need to find the value Δ(ε ). The percolation definition of the effective diffusion coefficient is given by the relation: a2 a2 LΔ LΔ P∞ ≈ ≈ . τ τ a2 τ
(11.7.10)
2 Δ L D// ≈ D⊥ ≈ D// D⊥ , L Δ
(11.7.11)
Deff ≈ It is easy to see that in our case Deff ≈ where
Δ L
≈
2
D⊥ D// .
Using the percolation expression for L(ε ) and the conjecture
Δ = ελ , one can obtain the expression for the correlation scale:
a=λ
D// D⊥
ν 2ν +4
.
(11.7.12)
Hence, the percolation interpretation of the Kadomtsev-Pogutse scaling is related to the renormalization of the initial small parameter in the following form: 1
ε∗ = ε02ν +4 .
(11.7.13)
Now it is possible to express all the values by means of the small parameter ε0 and the percolation exponent ν . The percolation interpretation for scaling (11.7.4) is only an example of the percolation phenomenology. However, this result enables us to complete a new kind of hierarchy for the small percolation parameter: 1
1
1
1
1
ε0 , ε0ν , ε0ν +1 , ε0ν +2 , ε0ν +3 , ε02ν +4 , . . . .
(11.7.14)
This hierarchy is obtained from the analysis of size limitation, graded (drift) renormalization, low-frequency regime, steady case, and transport in a stochastic magnetic field.
Further Reading Percolation and Turbulent Transport Balescu, R. (1997). Statistical Dynamics. Imperial College Press, London. Balescu, R. (2005). Aspects of Anomalous Transport in Plasmas. IOP, Bristol and Philadelphia. Bakunin, O.G. (2004). Report on Progress in Physics, 67, 965. Chorin, A.J. (1994). Vorticity and Turbulence. Springer-Verlag, New York. Gouyet, J.-F. (1996). Physics and Fractal Structure. Springer-Verlag, Berlin.
192
11 Percolation and Turbulent Transport
Horton, W. and Ichikawa, Y.-H. (1994). Chaos and Structures in Nonlinear Plasmas. World Scientific, Singapore. Isichenko, M.B. (1992). Reviews of Modern Physics, 64, 961. Stauffer, D. (1985). Introduction to Percolation Theory. Taylor and Francis, London. Ziman, J.M. (1979). Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems. Cambridge University Press, Cambridge, U.K.
Chapter 12
Multiscale Approach and Scalings
12.1 The Nested Hierarchy of Scales and Drift Effects It was stated in the previous chapter that to describe correlation effects in the framework of the percolation approach one needs to built a scale hierarchy. The interrelation between the spatial and temporal scale hierarchies in terms of the percolation model is given by:
λ ε ν ≈ lS << Δ ≈ λ ε << λ << a ≈ τS ≈
λ a << L ≈ , ν ε ε
lS 1 λ ε ε ≈ << << T0 ≈ , VS aω ω ω
(12.1.1) (12.1.2)
which is based on the monoscale representation of a flow with the parameters V0 , λ , ω . Recall here the basic definitions from the previous chapter: λ is the spatial scale, ε is the small percolation parameter, lS is the scale corresponding to a single streamline, Δ ≈ λ ε is the stochastic layer width, a ≈ λ /ε ν is the spatial correlation scale, ν = 4/3 is the correlation exponent, L(ε ) is the length of the percolation streamline, VS = λ ω is the dimensional estimate of separatrix motion, ω is the characteristic frequency, and τS ≈ lS /VS is the characteristic reconnection time. We generalize here the monoscale percolation model by analysis of multiscale flows with the hierarchy of spatial scales, which is related to the velocity hierarchy. The main difference of this multiscale approach from the Kolmogorov hierarchical model of turbulence is the assumption of the percolation character of the streamline behavior. Such a model has been considered by Isichenko and Kalda in [233, 234]. The expression for the streamline function Ψλ as a function of the parameter λ was presented in the scaling form with the stream function exponent M:
M λ . (12.1.3) Ψλ ≈ Ψ0 λ0
O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008
193
194
12 Multiscale Approach and Scalings
Here, Ψ0 is the scale of the streamline function that corresponds to the characteristic spatial scale λ0 and the characteristic velocity V0 . In agreement with the dimensional estimates, we can obtain the expression for the velocity Vλ that corresponds to the scale λ :
M−1 λ Ψ Vλ ≈ λ ≈ V0 . (12.1.4) λ λ0 In the framework of the percolation approach, the most intensive streamlines will contribute most to transport. Two different situations occur with respect to the value of M. Let us consider the first case, M > 1. Here, the characteristic flow pattern is determined by the maximum value of λ ≈ λm for such a system. The second case of interest arises when M < 1 and appears to be more complex. Let us consider the hierarchy of spatial scales,
λ0 << λ1 << λ2 << . . . ,
(12.1.5)
and the corresponding hierarchy of velocities, V0 >> V1 >> V2 >> . . . .
(12.1.6)
V = V0 +V1 +V2 + . . .
(12.1.7)
Here, the total velocity is the superposition of flows entering in the hierarchy. Formally, this flow pattern is similar to the hierarchy used by Kolmogorov to describe isotropic turbulence [28, 64]. In that case, the scaling law for the spectrum of energy E(k) is one of the important characteristics E(k) ∝
Vk2 k
∝
1 . Here, k is the wave number. The k5/3 1/3 λ . A comparison with the multiscale
1 ≈ estimate of the velocity gives Vk ∝ k1/3 expression yields M = 4/3 > 1. An analogous situation arises when considering the spectrum for the two-dimensional model E(k) ≈ 1/k3 , which is more relevant for the two-dimensional percolation case under consideration. We see that the Kolmogorov theory differs significantly from the multiscale percolation approach where M < 1. We will analyze the percolation hierarchy basing on the ideas of “graded” percolation. This approach is based upon the concept of the distortion of a small scale related flow, by way of superimposition of a weaker drift flow related to a large scale over V = V0 +V1 . In the case of the scale hierarchy, the small parameter renormalization [222] can be represented in the form: 1
ε∗ (i) = ε0 (i) 1+ν >> ε0 (i) ≈
Vi+1 . Vi
(12.1.8)
Following the ideas of monoscale percolation approach, it is convenient to introduce ai (ε∗ ) = λi ε∗
−ν
, Li (ε∗ ) = λi
ai λi
Dh ,
P∞ (i) ∝
1 . ai
(12.1.9)
12.1 The Nested Hierarchy of Scales and Drift Effects
195
Here, the values a0 , Δ0 , L0 correspond to the scale λ0 , and Dh is the Hull exponent, Dh = 7/4. The values P∞ (i) and Δi are interdependent: P∞ (i) =
Li Δi . ai 2
(12.1.10)
Therefore, we obtain a relationship that is similar to the case of monoscale percolation, Δi (ε∗ ) ≈ ε∗ λi . An assumption was made about the hierarchical flow pattern based on the system of nested scales (see Fig. 12.1): ai (ε∗ ) ≤ λi+1 .
(12.1.11)
This condition plays a significant role in the multiscale approach, since the value ε∗ can be expressed as a function of λi+1 and λ1i : Vi+1 ε0 (i) ≈ ≈ Vi
λi+1 λi
M−1 .
(12.1.12)
Then, simple calculations yield:
λi λi+1
λi+1 λi
ν (1−M) 1+ν
< 1.
(12.1.13)
This condition means that the assumption of nested scales is correct only for −
1 < M < 1. ν
(12.1.14)
In fact, we have a double inequality for the value M that describes the case of interest.
λ0 Fig. 12.1 Schematic illustration of the nested hierarchy of scales
λ1
196
12 Multiscale Approach and Scalings
12.2 The Brownian Landscape and Percolation The multiscale approach considered above is the “percolation version” of the rough surface conception (see Fig. 12.2). As a result of the importance of rough surfaces in many areas of science and technology, considerable effort has been devoted to the development of ways of classifying them and describing them in quantitative terms. For the most part, this effort has led to complicated empirical equations with a large number of parameters. However, in our case we will construct the turbulent transport scaling only on the streamline exponent M, and the scaling representation of closed fractal loops will be a key to this complex problem. The simplest interpretation of such an approach [188, 189] is based on the representation of a “rough” 1D + 1D landscape as a graph of one-dimensional random walks in the x −t axes, where the t-axis can be interpreted as a horizontal coordinate and the x-axis can be a vertical one. Then, different values Hurst exponent cor of the 2 respond to different types of landscape “roughness,” (Δx) ∝ t 2H . This implies that the “rough landscape” is a statistically self-affine fractal over a corresponding range of length scales with the characteristic Hurst exponent equal to the roughness 2 exponent H (see Fig. 12.3). For such landscapes the mean height difference (Δh)2 between the pairs of points separated by a “horizontal” distance Δr is given by: 2 (Δh)2 ∝ (Δr)H . (12.2.1)
It is easy to generalize this representation for the case of a rough surface with another dimensionality. In the framework of turbulent transport description, a similar model was analyzed, where the streamline function Ψλ is used as the “height” characteristic of the two-dimensional random field:
M λ . (12.2.2) Ψλ ≈ Ψ0 λ0 This is analogous to the model of continuum percolation; the loops arise when considering a coastline that was formed under the conditions of a “landscape flooded by water.” In this connection, there is a problem in obtaining the relationship between
λ//
Fig. 12.2 A rough surface. The inset shows part of the surface on an enlarged scale to visualize the self-similar properties of the surface
λ⊥
12.2 The Brownian Landscape and Percolation
197
Fig. 12.3 Schematic illustration of the Brownian landscape
Δh
Δr
the fractal dimensionality characterizing a single loop D˜ h and the Hurst exponent H (or the stream function exponent M). Here, we will use another symbol D˜ h instead of the hull percolation exponent Dh , since in order to describe the Brown surface the percolation ideas were used as approximations only. The probabilistic approximation is, as usual, the simplest method. The authors of [235, 236] used the model of self-avoiding random walks to describe the single loop character. The functional form to treat the characteristic probability (6.1.7) could be an adequate model. However, to describe cases with different Hurst’s exponents it is necessary to use the probability density function with the arbitrary values H : R (N) ∝ N H , instead of the Brownian case, where H = 1/2. Then, the expression for the probability of self–avoiding Brownian motion takes the form: ∞
PS (t) = −∞
1 1 R2 (dR)d . exp − d (N)2 exp − 2H d H N R (N )
(12.2.3)
We assume that the main contribution to the integral comes from the extremum of the integrand:
1 R2 2 . (12.2.4) (N) + min N 2H Rd Performing calculations, we can write the scaling: d+2
N ∝ R 2(1+H)
(12.2.5)
Now one can consider the fractal dimensionality: dF (H, d) =
d +2 1+H
(12.2.6)
for the two-dimensional case (d = 2) as a fractal dimensionality of the single contour loop (coastline) of a self-affine surface with the Hurst exponent H: % h (H) = D
2 . 1+H
(12.2.7)
198
12 Multiscale Approach and Scalings
We can now make several simple estimates. The value of H = 1 yields result, which corresponds to the linear type of behavior with D˜ h = 1. The random walk with H = 1/2 corresponds to D˜ h = 4/3. However, approximation (12.2.7) is not correct in the region of small H since the following condition must be realized: % h ≤ Dh = 1 + 1 = 7 . D ν 4
(12.2.8)
This condition has a clear physical meaning. The fractal dimensionality of the hull that has a percolation nature has to be larger than the fractal dimensionality of the coastline of the self-affine surface. Here we consider a different version of the approximation D˜ h based on the hierarchy of nested scales (12.1.5). In the framework of this approach the correct description of the regimes, where D˜ h tends to the hull exponent Dh , is possible, since the multiscale approach is based on “graded” percolation. Under these new conditions the expression for the value Dh , with a0 → λ1 should be reconsidered from the standpoint of the interaction of nearby scales of the hierarchy. Recall that Dh = 1 + 1/ν ; therefore, in the monoscale approach we have: L(a) ≈ λ
a Dh
λ
≈
a . ε∗
(12.2.9)
Then, using the assumption a0 → λ1 , the authors of refs. [233, 234] suggested the simple approximation: λi . (12.2.10) L(λi ) ∝ ε∗ (λi ) In the framework of this approach, the calculations yield the renormalized value D˜ h :
ν 1 + νM D˜ h = Dh (1 − M) + . 1+ν 1+ν
(12.2.11)
For the two-dimensional case, the following simple expression was derived (see Fig. 12.4): 10 − 3M , (12.2.12) D˜ h (M) = 7 where D˜ h (−1/ν ) = Dh and D˜ h (1) = 1. However, this scaling has not passed recent tests [237, 238, 239, 240, 241], because there is a rigorous result that gives the value D˜ h (0) = 3/2. A new approximation for 0 < M < 1 was suggested, namely: 3−M , (12.2.13) D˜ h (M) = 2 which is in agreement with the rigorous result for D˜ h (0) [237, 238, 239, 240, 241]. Unfortunately, this new formula does not give any information about the behavior of D˜ h in the region of negative M. This is not surprising, since in the framework of the Brownian theory of “rough” surface we cannot interpret the values H < 0.
12.3 Correlations and Transport Scalings
199 Dh(M)
2 7/4 3/2
1
−1/ν
−1
1 M
Fig. 12.4 The Hull exponent approximations
However, the “graded” percolation method allows these values to be analyzed. The physical interpretation of the exponent M is treated below from the correlation aspect.
12.3 Correlations and Transport Scalings Completing the consideration of the basic assumptions of the multiscale approach, let us focus on the physical meaning of the parameter M. Obviously, this parameter governs the character of a flow. We can relate it to the correlation properties of a flow. The simple estimate for the spatial correlation function in the scaling law form is given by: 1 C(λ ) = V (x + λ ,t)V (x) ∝ V (λ )2 ∝ λ 2(M−1) ∝ α . λ E
(12.3.1)
Here, αE is the correlation exponent that describes the rate of decay of the correlation function: αE = 2(M − 1). (12.3.2) The cases where M ≈ 1 correspond to C(λ ) ≈ const. In the cases M ≥ −1/ν the correlation function decays faster. The condition of applicability of the hierarchical model takes the following form [242]: 7 0 < αE < . 2
(12.3.3)
200
12 Multiscale Approach and Scalings
In subsequent considerations, to describe transport properties we will use the exponent αE together with the landscape exponent M. The above analysis permits us to interpret transport scalings since there are many cases in which the relationships between the Hurst exponent and the correlation exponent can be applied. Thus, on the basis of the ballistic character of percolation transport, a scaling for the calculation of the Hurst exponent H in the multiscale case was suggested in [91]:
λ (t) ∝ V (λ )t ≈ V0
λ (t) λ0
M−1 t.
(12.3.4)
Simple calculations lead to the expression:
λ (t) ≈ λ0
V0t λ0
1 M−2
∝ tH .
(12.3.5)
2 1 = . 2−M 2 + αE
(12.3.6)
Hence, the Hurst exponent is given by: H(αE ) =
This ballistic estimate (12.3.4) looks fairly rough (compare with (4.2.7)). However, in terms of the correlation exponent αE , one obtains the rigorous result for incompressible flows with the power correlation function [117] where αE < 2. We see that the steeper correlation functions correspond to the lower values of the Hurst exponent. The correlation function C(λ ) ∝ 1/λ 2 corresponds to the case of classical diffusion with H = 1/2. The efficiency of similar estimates allows us to consider more complex flows, where anisotropy effects of a medium play an important role together with the multiscale effects: λ⊥ (t) ∝ V⊥ (λ⊥ , λ// ) t. (12.3.7) Formula (12.3.6) does not take into account the anisotropy effects; however, an analogous approach can be applied to the simplest anisotropic case with separated spatial scales [91, 135]: λ⊥ (t) ∝ V⊥ (λ ) t. (12.3.8) // Here, λ⊥ is the perpendicular displacement and λ// is the longitudinal displacement. If the case under consideration has a diffusive character of the longitudinal motion √ (double diffusion, random shear flows, etc.), then we obtain λ// ≈ 2D0t. On substitution of this estimate into (12.3.8) we find the scaling:
λ⊥ (t) ∝ V0
λ02 D0
1−M 2 t
1+M 2
.
The expression for the Hurst exponent takes the form [242, 243]:
(12.3.9)
12.4 Diffusive Approximation and the Multiscale Model
H=
201
αE 1+M = 1− , 2 4
(12.3.10)
where 0 < αE < 7/2. Comparison between (12.3.10) and (12.3.6) shows the coincidence of both these dependences at two points αE = 0 (H = 1 is the ballistic case) and αE = 2 (H = 1/2 is the classical diffusion equation). The characters of both these dependences H(αE ) are similar. The landscape exponent M is fairly efficient for the description of the physical properties of a flow. Thus, in the anisotropic case (12.3.10), for M = 1/2 we obtain the scaling for the random shear flow, H = 3/4. In the isotropic case (12.3.6), for M = 1/2 we find H = 3/2, which corresponds to the “Manhattan grid” flow [91] (the generalization of the shear flows model [135]). Note that for αE > 2 scaling (12.3.10) yields the subdiffusive regime, which contradicts the initial assumptions about the incompressibility of the flow and using the streamline concept.
12.4 Diffusive Approximation and the Multiscale Model Besides the ballistic approach (12.3.4), it is interesting to consider the diffusion approximations: λ 2 ≈ D(λ )t. (12.4.1) Here, D(λ ) is the coefficient corresponding to the scale λ . However, the expressions for D(λ ) are usually based on the approximation of the exponent D˜ h . This is not surprising, since in the monoscale approach the expression for Dh = 1 + ν1 also plays a key role. The diffusion coefficient D(λ ) can be expressed in the form that is analogous to the monoscale case: D(λ ) ≈
λ2 P∞ (λ ), τ
where
P∞ (λ ) ≈
L(λ )Δ(λ ) . λ2
(12.4.2)
Let us consider the approximation of P∞ (λ ) in the form corresponding to the monoscale approach (11.2.14): P∞ (i) ≈
λi . ai (λi )
(12.4.3)
Then, calculations yield the expression:
P∞ (λ ) ≈
λ0 λ
4(1−M) 7
.
(12.4.4)
This result allows the width of the percolation layer Δ(λ ) to be defined: Δ(λ ) ≈ P∞ (λ )
λ2 ≈ λ0 L(λ )
λ λ0
M .
(12.4.5)
202
12 Multiscale Approach and Scalings
Introducing the correlation time τ in the ballistic form τ ≈ VL0 in the expression for D(λ ) leads to the Koch-Brady result (12.3.6). In this case, the conditions M > 0 (αE < 2) are automatically satisfied, since D(λ ) increases with λ :
D(λ ) ≈ V0 Δ(λ ) ≈ V0 λ0
λ λ0
M .
(12.4.6)
A new regime arises when the correlation time is considered in the diffusive form, 2 τ = ΔD(x) , 0 L(λ ) t. (12.4.7) λ 2 ≈ D0 Δ(λ ) Simple calculations [233, 234] lead to the scaling for the Hurst exponent: H (M) =
7 . 10M + 4
(12.4.8)
Here, −2/5 < M < 1. It easy to see that the scaling obtained depends essentially on the choice of the expression for L(λ ). Thus, for the approximation of L(λ ), the substitution of ai for λi was used, whereas for the approximation of F(λ ) the conventional form of ai was retained. It is easy to explain the choice made in [233, 234], since the expression in the form L(λi ) ≈ aεi leads to the value D˜ h (1) = 0, which is absolutely unacceptable. On the other hand, the graded approximation (12.2.10) provides the correct limit for negative values of M : D˜ h (− ν1 ) = Dh and the realization of Mandelbrot’s condition D˜ h < 2−M with M < 1. From the modern standpoint correct calculations could be based on the recent results by Kondev et al. [237, 238, 239] and Kalda [240, 241]: 3−M . (12.4.9) D˜ h = 2 Then, on carrying out calculations we obtain: L(λ ) L2 (λ ) ≈ D0 2 D ≈ D0 ≈ D0 Δ(λ ) λ P∞
λ λ0
11(1−M) 7
.
(12.4.10)
The diffusion coefficient D(λ ) increases with λ for the cases M < 1, providing the superdiffusion character of the behavior in the region 0 < M < 1: H=
7 . 3 + 11M
(12.4.11)
Note that the use of new approximation (12.4.9) leads to alteration in the formula for the diffusion coefficient D(λ ) that is based on the ballistic expression for τ , since 1 . (12.4.12) D(λ ) ≈ V0 Δ(λ ) ∝ L(λ )
12.5 Stochastic Instability and the Temporal Hierarchy of Scales
203
In this case, the new scaling for the Hurst exponent does not coincide with the expressions derived by Koch and Brady (12.3.6). However, there are also other possibilities, since the approximation of P∞ (λ ) in the form (12.4.3) is not universal. Thus, the estimate
P∞ ≈ ε0 ≈
λ λ0
M−1 (12.4.13)
can be used in the interval M ≤ 1. We can also use the united approximation formula on the basis of three characteristic values of D˜ h (M) at the points M = 1, M = 0, and M = −1/ν , for the whole interval of parameters − ν1 < M < 1. The estimates considered show that diffusive approximation (12.4.1) has several degrees of freedom, which leads to considerable uncertainties of the results. Nevertheless, the multiscale approach essentially increases the possibilities of applying percolation ideas for the description of turbulent diffusion.
12.5 Stochastic Instability and the Temporal Hierarchy of Scales To consider stochastic instability in the framework of the multiscale approach it is convenient to introduce the hierarchy of characteristic times (frequency scales): tλ ≈
1 ≈ ωλ
λ λ0
G p
λ0 , V0
(12.5.1)
where G p is the frequency exponent, which permits modeling both ballistic G p = 1 and diffusive regimes G p = 2. Based on the multiscale results obtained by the multiscale approximation, we can consider the stochastic instability increment in terms of the monoscale definition:
γ˜S (λ ) ≈
VS L(λ )λ ωλ ˜ ≈ ∝ λ Dh −G p −1 . lS λ2
(12.5.2)
Introducing the local Kubo number Kuλ corresponding to the selected scale λ yields: V Kuλ ≈ λ ∝ λ G p +M−2 . (12.5.3) λ ωλ Then, in terms of the Kubo number the expression for the stochastic instability increment takes the form: ˜ 1+G p −Dh (M) 2−G p −M
γ˜S (λ ) ∝ Kuλ
.
(12.5.4)
As previously mentioned, there is still no correct approximation for D˜ h (M) in the whole interval: (12.5.5) −1/ν < M < 1.
204
12 Multiscale Approach and Scalings
However, the exact value of D˜ h is calculated for M = 0: D˜ h (0) = 3/2,
(12.5.6)
which in combination with the assumption of the ballistic character of particle motion along percolation streamlines could yield a scaling similar to the monoscale estimate. Indeed, if (12.5.7) M = 0, D˜ h = 3/2, and G p = 1, we obtain [243, 244]:
7
γ˜S (λ ) ∝
1 Vλ ∝ Kuλ . λ ωλ
(12.5.8)
Note that the passage from the monoscale percolation model to the multiscale one makes it possible to describe essentially nondiffusive transport regimes in both isotropic and anisotropic models. Therefore, it is natural to apply the multiscale description to other problems where long-range correlation effects dominate.
12.6 Isotropic and Anisotropic Magnetohydrodynamic Turbulence The multiscale model of percolation transport considered in the previous sections is based on the scaling representation for velocity. However, the consideration of the system of percolation transport channels requires the separation of characteristic scales in the framework of the hierarchy:
λ0 ;
λ1 = μ p λ0 ;
λ2 = μ p 2 λ0 ; . . . λm ;
μ p >> 1.
(12.6.1)
Analysis shows that the limit of applicability of the multiscale model is μ p ≈ 2. On the other hand, the multiscale percolation model is closely related to drift effects, which, naturally, lead to a considerable difference between the percolation scaling for velocity Vk ∝ (1/k)M−1 , where −1/ν < M < 1, and the expression for the twodimensional isotropic turbulence [64, 65]: Vk ≈
1
1 E (k) k ≈ , k
(12.6.2)
1 . k3
(12.6.3)
where the energy spectrum is given by: E (k) ∝
The consideration of the multiscale transport model in the framework of magnetohydrodynamic (MHD) turbulence is also closely related to the energetic spectrum form. Here it is convenient to apply the Kolmogorov phenomenology to plasma wave turbulence. Formally, the time of nonlinear interaction of waves is defined as:
12.6 Isotropic and Anisotropic Magnetohydrodynamic Turbulence
τCASC ≈
Vp Vk
mW −1
1 , ωp
205
(12.6.4)
where ω p = Vp k, Vp is the phase velocity, and mW characterizes the type of nonlinear interaction (mW = 3 for a three-wave interaction, mW = 4 for a four-wave interaction, etc.). The MHD turbulence model of Iroshnikov and Kraichnan [245, 246] for Alfven’s waves is widely used now:
τCASC ≈
VA Vk
2
1 , VA k
where
mW = 3,
τA << τCASC ,
B0 >> Vk . Vp ≡ VA = 1 4π n p
(12.6.5) (12.6.6)
Here, VA is the Alfven velocity, B0 is the magnetic field amplitude, and n p is the plasma density. Calculations based on the constancy of the Kolmogorov dissipation rate, V2 εK ≈ k , (12.6.7) τCASC yield the scaling for the turbulence spectrum: E (k) ∝
1 . k3/2
(12.6.8)
In terms of energy dissipation,
δ Edis ≈ εK τA ≈
Vk3 , VA
(12.6.9)
the model from [245, 246] describes the diffusive character of energy transfer in 1/2 nonlinear wave interaction Vk2 ∝ τCASC : Vk2 ≈ δ Edis
VA Vk
0
2 ≈
τCASC . τA
(12.6.10)
Moreover, Kraichnan noted [246] that in the framework of the isotropic model of MHD turbulence there is a linear dependence of the dissipation rate on the characteristic time εK ∝ τA , whereas in the classical Kolmogorov approach εK ∝ 1/τCASC . Naturally, the isotropic model is incorrect for describing turbulent transport processes in a strong magnetic field. Indeed, in the case of strong MHD turbulence, the separation of longitudinal and transverse scales plays an important role (see Fig. 12.5). The scaling approach to the description of essentially nonisotropic MHD turbulence was suggested by Goldreich and Sridhar [247]. It was based on the balance of characteristic times in Alfven’s MHD turbulence 1 1 ≈ k// VA ≈ k⊥V⊥ (k⊥ ) ≈ , τA τ⊥
(12.6.11)
206
12 Multiscale Approach and Scalings
Fig. 12.5 Schematic illustration of Alfven’s waves propagating along folded fields
B
lS
where ωA = 1/τA is Alfven’s frequency, VA is Alfven’s velocity, k// ≈ 1/l// is the longitudinal wave number, k⊥ ≈ 1/l⊥ is the transverse wave number, V⊥ (k⊥ ) is the scale of transverse velocity related to the spatial scale k⊥ , and τ⊥ is the dimensional estimate of the time of nonlinear interaction that characterizes the turbulent cascade in the transverse direction to the magnetic field with the constant dissipation rate:
ε⊥ =
V⊥3 = const. l⊥
(12.6.12)
This considerably differs from the conventional isotropic representations. The new relation (2.6.11) is only an approximation analogous to the percolation renormalizations, and its efficiency has been confirmed repeatedly by simulations, which demonstrate the correctness of the scaling: l// ≈
1 VA l⊥ VA l⊥ VA 2/3 ≈ ≈ ≈ 1/3 l⊥ . 1/3 k// V⊥ (ε⊥ l⊥ ) ε⊥
(12.6.13)
The relationship between the longitudinal and transverse scales is expressed as α l// ∝ l⊥G , where αG = 2/3 corresponds to the strong MHD turbulence. The method of separation of longitudinal and transverse spatial scales is fairly effective in analyzing transport in a stochastic magnetic field. Thus, the authors of [248] obtained an estimate of electron heat conductivity in galaxy clusters; the assumption was made that the characteristic spatial scale lB in the model used in [173] must be calculated by taking into account the anisotropic MHD turbulence spectrum. The phenomenological relation (12.6.11) was used, α where the longitudinal and transverse scales are related by l// ∝ l⊥G , where αG = 2/3 corresponds to strong MHD turbulence, and αG = 4/3 corresponds to the intermediate turbulence regime. This allows us to assume that the correlation scales characterizing transport in a stochastic magnetic field have the form:
αG Δ⊥ LCOR ≈ . (12.6.14) lB lB
12.6 Isotropic and Anisotropic Magnetohydrodynamic Turbulence
207
where Δ⊥ is the perpendicular correlation scale and lB is the longitudinal scale. This estimate differs significantly from the isotropic estimate considered in Chap. 8, LCOR ≈ 30lB . Thus, for transverse displacements Δ⊥ ≈ lB it leads to the value of longitudinal correlation length: LCOR ≈ Δ⊥ ≈ lB << 30lB .
(12.6.15)
Then, the electron heat-conductivity estimate has the form:
χe f f ≈ D//
χSp lB , ≈ LCOR 3
(12.6.16)
which agrees well with the data of astrophysical observations. Here, D// is the longitudinal diffusion coefficient. To substantiate obtained scalings, let us consider the model equation describing the separation of initially close force lines. The approximation equation describing the exponential divergence of force lines Δm takes the form: Δm 2 d Δm 2 ∝ . (12.6.17) dl lB However, the exponential regime for Δm 2 > lB2 has to move into the diffusive one, l⊥ 2 dΔm 2 ≈ Dm ≈ dl lB
.
(12.6.18)
Here, Dm is the magnetic diffusivity. From this standpoint, it is natural to describe intermediate regimes by the modification of the factor 1/lB in this equation, taking into account the increasing role of large scales. Considering the scale hierarchy corresponding to the model of a quasi-isotropic stochastic magnetic field, one can use the scaling from the model of strong Alfven’s turbulence:
α l// l⊥ ≈ , (12.6.19) lB lB with the hierarchy of scales: lmin < l⊥ < lB ≤ LCOR . Then, in terms of the wave numbers k⊥ ≈
1 l⊥
and k// ≈
(12.6.20) 1 l// , it is convenient to rewrite
model equation (12.6.17) in a form that takes into account the contribution of different scales of hierarchy in the representation of LK ≈ lB : 1/Δ⊥ 1/l min k (k ) 2 d Δm 2 ⊥ // ≈ Δm k// (k⊥ )d ln k⊥ + d ln k⊥ , 2 dl k⊥ 1/lB
1/α
(12.6.21)
1/Δ⊥
where k// ∝ k⊥ ; hence, this equation describes the transition to regimes with
208
12 Multiscale Approach and Scalings α Δ⊥ α ≈ l⊥ ≈ l// .
(12.6.22)
It is important to note that in spite of using the multiscale approach relating longitudinal and transverse scales in the strong Alfven’s turbulence, the characteristic spatial scale lB of a tangled magnetic field appears to be the universal parameter of model: (12.6.23) lB ≈ LCOR ≈ LK ≈ Δ⊥ , and the characteristic correlation scale does not enter the final expression for the effective heat conductivity χe f f . The generation and evolution of stochastic magnetic fields occurs largely through the action of turbulence. In many situations, the magnetic field is strong enough to influence many important properties of turbulence itself. Indeed, magnetohydrodynamic turbulence problems are common to many fields of astrophysics and physics of plasma, such as turbulent mixing, turbulent transport, dynamo problem, etc.
Further Reading Fractal Landscape Feder, J. (1988). Fractals. Plenum Press, New York. Gouyet, J.-F. (1996). Physics and Fractal Structure. Springer-Verlag, Berlin. Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman, San Francisco. Stauffer, D. (1979). Physics Reports, 2, 3. Stauffer, D. (1985). Introduction to Percolation Theory. Taylor and Francis, London.
Multiscale Approach and Structures Bohr, T., Jensen, M.H., Giovanni, P., and Vulpiani, A. (2003). Dynamical Systems Approach to Turbulence. Cambridge University Press, Cambridge, U.K. Frisch, U. (1995). Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge, U.K. 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. Pismen, L.M. (2006), Patterns and Interfaces in Dissipative Dynamics. Springer-Verlag, Berlin.
Turbulence and Magnetic Fields Balescu, R. (1997). Statistical Dynamics. Imperial College Press, London. Balescu, R. (2005). Aspects of Anomalous Transport in Plasmas. IOP Bristol and Philadelphia.
Further Reading
209
Falgarone, E. and Passot, T., eds. (2003). Turbulence and Magnetic Fields in Astrophysics. Springer-Verlag, Berlin. Horton, W. and Ichikawa, Y.-H. (1994). Chaos and Structures in Nonlinear Plasmas. World Scientific, Singapore. Isichenko, M.B. (1992). Reviews of Modern Physics, 64, 961. Kadomtsev, B.B. (1991). Tokamak Plasma: A Complex System. IOP, Bristol. Krommes, J.A. (2002). Physics Reports, 360, 1–352. Rosenbluth, M.N. and Sagdeev, R.Z., eds. (1984). Handbook of Plasma Physics. North-Holland, Amsterdam. Wesson, J.A. (1987). Tokamaks. Oxford University Press, Oxford.
Part IV
Trapping and the Escape Probability Formalism
Chapter 13
Subdiffusion and Trapping
13.1 Diffusion in the Presence of Traps Subdiffusive scalar transport can occur in regular steady flows as well as in random velocity fields. The basic mechanism responsible for this slow diffusion is trapping caused by the existence of complex vortex structures. Thus, the subdiffusive motion of a tracer particle in the array of convective rolls is related to both convection along streamlines and molecular diffusion, which allows “jumps” between streamlines. Note that without compressibility in two-dimensional flows the subdiffusion mechanism cannot be realized. First, we analyze a particular physical model of diffusion in a medium with traps [249, 250]. Using probabilistic estimates, one can derive transport scaling relations, which can be interpreted in terms of stretched relaxation functions. On long time scales, the diffusion of particles in a medium with traps is related to the fluctuating character of the appearance and disappearance of regions free of traps. Let us introduce a probability to avoid trapping in terms of the Poisson distribution:
t . (13.1.1) Pc ∝ exp − τD 2
Here, τD ≈ RD is the characteristic time scale on which the particle diffuses through the medium until it reaches the boundary of trap-free region of the radius R, and D is the seed diffusion coefficient. We assume that trap-free regions obey the Poisson distribution: & ' R d , (13.1.2) PT ∝ exp − R0 where R0 is the mean radius of the trap-free regions in the space of dimensionality d. Now, we can estimate how the radius R(t) of the trap-free region should change in time in order for the survival probability to be the highest: ' & R d t . (13.1.3) − Pef f = Pc PT ∝ exp − R0 τD (R)
O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008
213
214
13 Subdiffusion and Trapping
Calculating the time derivative of the argument of the exponential functions in this expression, we obtain: 1 (13.1.4) R(t) ∝ t 2+d . For d > 0, the diffusion described by this scaling is obviously slower than that described by the classical diffusion scaling: H(d) =
1 1 < . 2+d 2
(13.1.5)
Note that the final estimate for the escape probability is given by the relation: d−1 Peff ∝ exp −t d+2 , (13.1.6) which is a stretched exponential distribution. In the two-dimensional case, one ob tains the well-known Kohlrausch relaxation law Peff ∝ exp −t 1/2 . Though the models are motivated by solid-state applications, there have been many parallel pure mathematical and interdisciplinary (anomalous diffusion, phase space kinetics, etc.) developments. The ideas associated with traps are often used in the study of particle trapping by vortices in order to describe the behavior of a passive scalar in a turbulent field [3, 14, 18, 20, 34, 91, 251, 252] or in particle diffusion in a magnetic field with “braided” force √ lines [17, 45, 253, 254]. Thus the diffusion approximation L// ∝ 2D0t of longitudinal motion in the double diffusion model (7.4.7) allows us to expect that the subdiffusive character of transport can be also interpreted by means of the effective time renormalization t → t δNN , where δ N/N is the renormalization factor and D0 is the diffusion coefficient. Indeed, the diffusive character of motion in the one-dimensional case leads to numerous “returns”; hence the particle takes part in the transverse diffusion process only a fraction of the total time t. Let us assume a fraction of the time during which the particle takes part in the transverse diffusion has a fractal character with the dimensionality dF . One can use the representation for transverse displacement: Δ⊥ 2 ≈ D⊥t where
δN ∝
t τ//
dF
δN , N
, N∝
(13.1.7)
t . τ//
(13.1.8)
Here, τ// is the longitudinal correlation time and D⊥ is the transverse diffusion coefficient. Simple calculations yield the expression:
D⊥ 2 t dF . (13.1.9) Δ⊥ ≈ (τ// )dF −1 Changing to the symbols used above,
13.2 Trapping and Strong Turbulence
215
D⊥ ≈ Dm
L2 L// , D0 ≈ // , τ⊥ τ//
(13.1.10)
we rewrite the above expression as: √ Δ2⊥ ≈ Dm
D0 τ// τ⊥
t τ//
d F .
(13.1.11)
Here, Dm is the magnetic diffusivity and D0 is the seed (collisional) diffusivity. In the case of dF = 1/2, we obtain the transverse transport scaling for double diffusion Δ2⊥ ∝ t 1/2 . The rigorous theory for such problems is based on continuous time random walks (CTRW) [3, 4, 6, 7]. An important assumption in that approach is the consideration of the probability Φ(t) for a particle remaining in a trap during the time t. The following function in the scaling form is usually used: Φ(t) ∝
1 , tγ
where
γ = dF
for γ < 1,
(13.1.12)
where γ is the parameter of the probability function. A simple estimate for Φ(t) based on the probability of the return (6.1.3) is given by: Δ . Φ(t) ∝ ρ (0,t)Δ ∝ √ 4π D0t
(13.1.13)
Here, Δ is the correlation scale. We obtain the relationship between the probability of finding the particle in the trap and the probability density of one-dimensional random walks. We can imagine the “traps” as teeth in the “comb-like structure” as shown in Fig. 13.1. If the teeth length tends to infinity, then our diffusive approximation is satisfactory. The models with teeth of a limited length and the models with teeth, which have a fractal structure, appear to be the natural generalization of the “comb-like” model [3, 4, 10, 16].
Fig. 13.1 A comb-like structure
13.2 Trapping and Strong Turbulence To discuss trapping in the limit of strong turbulence we first consider an intermediate percolation regime [91], where particle transport at the initial stage (see Chap. 11),
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13 Subdiffusion and Trapping
t < τB ≈
L , V0
(13.2.1)
is defined by the fractal structure of the percolation streamline. Here, L is the length of the percolation streamline, V0 is the characteristic velocity, and τB is the ballistic time. Indeed, this case differs significantly from the percolation model of turbulent diffusion where t ≥ τB . The estimate of correlation scale is given by:
1/Dh L(t) 1/Dh V0t ≈λ , (13.2.2) aI (t) ≈ λ λ λ where λ is the spatial characteristic scale and Dh = 7/4 is the Hull exponent. Here, the supposition was made that the test particle path at this stage is approximately ballistic L(t) ≈ V0t. To describe the initial stage of percolation t < τB in the two-dimensional random flow, we write an expression (as suggested in [19]):
1/Dh
1/Dh L V0t ≈ λ2 . (13.2.3) R2 (t) ≈ a2 P∞ ≈ λ 2 λ λ Here, P∞ is the fraction of a space that is occupied by the percolation streamline, a is the spatial correlation scale, Dh = 1 + ν1 and ν = 4/3 are the percolation exponents, and R is the mean free path of the particle. This leads to the anomalous transport regime: (13.2.4) R ∝ t 1/2Dh = t 2/7 with the Hurst exponent H=
1 = 2/7. 2Dh
(13.2.5)
This corresponds to the subdiffusive character and describes the initial transport stage when the particle moves along the fractal streamline. Naturally, trapping influences not only the initial stage. Thus, in the framework of the time-dependent percolation model the correlation scale ΔCOR is much less than the frequency path lω [255, 256]: ΔCOR ≈ a (ε∗ ) ≈ ε∗2
V0 V0 << lω ≈ ω ω
(13.2.6)
and corresponding characteristic time τ∗ is much less than the correlation time:
τ∗ =
L ΔCOR L ≈ ε∗ << τCOR ≈ ≈ τB . V0 V0 V0
(13.2.7)
Here, ε∗ is the renormalized small percolation parameter and ω is the characteristic frequency. This mirrors the character of the particle motion trough the correlation zone along fractal streamlines. The deviation of τCOR from τ∗ can be related with the trapping mechanism:
13.2 Trapping and Strong Turbulence
τCOR = τ∗ + τTrap =
217
ΔCOR (Ku) + τTrap (Ku), V0
(13.2.8)
where τTrap = τTrap (Ku) is the characteristic trapping time, which in the case of strong turbulence depends on the Kubo number. Let us consider a simple heuristic argument to show that trajectory trapping generally has an effect of decreasing the diffusion coefficient and of changing the scaling exponent that describes the dependence of the effective diffusivity on the Kubo number to subunitary values σT < 1, D ∝ KuσT . The parameters of the stochastic field are combined in the Kubo number Ku, which represents the ratio of the distance covered by the particle in the characteristic time 1/ω to the correlation length 0 . Here, V0 is the amplitude of particle of the stochastic field ΔCOR : Ku = ω ΔVCOR velocity. When Ku is small, the particles cannot “feel” the structure of the potential, and move only a small distance before the stochastic field is changed. This is a temporal decorrelation that is characterized by the equality of the time step of the random walk with the correlation time: τ = 1/ω . The space step is the distance covered in 1/ω : lω ≈ Vω0 ≈ ΔCOR . The diffusion coefficient is estimated as: Deff (Ku) ≈
ΔCOR V02 2 ≈ ≈ ΔCOR ω Ku2 ∝ Ku2 . τ ω
(13.2.9)
This is the well-known quasilinear regime (σT = 2). When the Kubo number Ku is not small, the process is nonlinear and the diffusion coefficient is much more difficult to estimate. In these conditions, the particle can explore the structure of the potential. Considering that particles move freely with velocity V0 , the time necessary to cover this distance is τ∗ ≈ ΔCOR V0 , which is smaller than the characteristic time 1/ω (for large values of Ku). The estimation of the diffusion coefficient gives: Deff (Ku) ≈
2 Δ2COR ≈ V0 ΔCOR ≈ ΔCOR ω Ku ∝ Ku, τ∗
(13.2.10)
which is the well-known Bohm scaling σT = 1.The trajectories are much simplified in this estimation: they correspond to a free motion transit of the particle through the correlation zone. In reality, the motion is much more complicated. Various mechanisms of particle trapping occur, leading to a longer average time spent by the particle in the correlated zone (see Fig. 13.2). One of them is the percolation mechanism with an alternative scaling σT = 0.7 in this low-frequency domain. However, simple three-wave Hamiltonian dynamical systems in which particles remain confined most of the time on closed trajectories may exhibit large scale diffusion because of many separatrix crossings [18, 91, 257, 258]. The latter may be responsible for the appearance of a “flat” scaling σT = 0, instead of the Bohm result. In other words, the trapping process can greatly reduce the scaling exponent σT to subunitary values.
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13 Subdiffusion and Trapping
Fig. 13.2 Schematic illustration of particle trapping. The trajectory of a single particle shows a sequence of long flights interrupted by trapping events, in which the particle circles a trap (vortex). The vortex trapping regions are indicated
To treat the trapping mechanism we introduce the time scale in the estimation of the diffusion coefficient:
τCOR =
ΔCOR ΔCOR + τTrap >> , V0 V0
(13.2.11)
where τTrap is the average trapping time. Ignoring a weak dependence of the correlation length on the Kubo number we see that in the case of strong turbulence limit Ku >> 1 the trapping effects play the main role:
τCOR (Ku) =
ω + τTrap (Ku) → τTrap (Ku). Ku
(13.2.12)
Consequently, the diffusion coefficient is: D≈
2 ΔCOR ≈ τCOR
2 ΔCOR
ΔCOR V0
+ τTrap
2 ≈ ΔCOR ω
For Ku >> 1, one obtains: DE f f ≈
Ku . 1 + (τTrap ω ) Ku
2 ΔCOR . τTrap (Ku)
(13.2.13)
(13.2.14)
The trapping time (which is a function of Ku) can have values between zero (no trapping) and 1/ω (total trapping). The evaluation of τTrap (Ku) is very difficult task and generally not solved. The analysis of such a model [45, 259, 260] was carried out using the power form of correlation functions where the simple trapping approximation of the streamline function was applied. The special trajectory ensemble in the framework of the generalized Corrsin conjecture was used and is discussed in Chap. 15 in relation to phase-space problems.
13.3 Comb Structures and Transport Models of anomalous transport on comb structures are widely applied because of the fairly universal kind of trap topology. Here we treat a complex comb-structure
13.3 Comb Structures and Transport
219
Fig. 13.3 A complex comb structure where the length of a “teeth” is distributed in accordance with a power law
li li+1
Δ
model, where the length of the “teeth” is distributed in accordance with power laws (see Fig. 13.3). Let us consider a structure that consists of a backbone directed along the z-axis and orthogonal teeth connected with this backbone. The distances Δ between the teeth are identical and the tooth distribution along the length is given by the scaling: γ −1 lc (13.3.1) f (l) = 0 . l Here, f (l) is the probability density of finding a tooth with length l, l0 is the tooth characteristic length, and γc is the characteristic exponent. In the model under analysis, the minimum tooth length l0 must be of the order of the characteristic longitudinal scale Δ: (13.3.2) l0 ≈ Δ. On the other hand, it is necessary to take into account the renormalization condition of the distribution function f (l): ∞ l0 ∞
f (l) dl = 1,
(13.3.3)
f (l) l n dl = ∞,
(13.3.4)
l0
where n = 1, 2, 3 . . . In such a formulation of the problem, the description of anomalous transport on comb structures was repeatedly discussed in [3, 4, 14, 16, 91]. Therefore, here we develop only qualitative estimates. Let us define the mean length l that describes the spatial scale corresponding to diffusive walks along teeth with the diffusion coefficient D0 [261]: l(t)
√ D0 t
l0
D0t l f (l) dl ≈ l0
l √0 D0t
γc
∝
1 t γc /2
.
(13.3.5)
The mean time T that the particle diffuses along a tooth can be estimated as:
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13 Subdiffusion and Trapping
T ∝
Δl0 Δ t∝ l D0
D0 t l02
γc /2
∝ t γc /2 .
(13.3.6)
Then, the mean square displacement is estimated by the scaling: R2 = Δx2 (t) ≈ D0 T ≈ Δl0
D0t l02
γc /2
∝ t γc /2 .
(13.3.7)
On the other hand, one can estimate the probability of avoiding a return to the backbone Φ(t): T 1 ∝ γ /2 . (13.3.8) Φ(t) ≈ t tc In the case of comb structures with the power form of the length distribution, the 1 . expression for Φ(t) differs from the simple diffusive estimate Φ(t) ∝ t 1/2 The subsequent consideration of the probability Φ(t) in the framework of the more rigorous approach (continuous time random walk) makes it possible to obtain not only scalings for the mean square displacement but also fractional differential equations to describe particle density. This approach is developed in the next chapter.
13.4 Double Diffusion and Return Effects In the case of moving charged particles in a braided magnetic field, the constant return of charges to the initial point, which causes their one-dimensional diffusive motion along force lines (7.4.7), prevents significant transverse displacements. Therefore, in the model of transport in a braided magnetic field it is natural to use the estimate for the probability: Φ(t) ∝ P(0,t) ∝ ρ (0,t)Δ ∝ √
Δ 4π D0t
(13.4.1)
to establish the direct relationship between the transport scaling and the probability of returns to the initial point. Here, ρ is the Gaussian distribution and D0 is the seed diffusivity. In the presence of strong anisotropy, several spatial and temporal correlation scales describe the stochastic magnetic field, and they differ from correlation scales that correspond to particle transport. Therefore, scaling arguments (7.4.7) look oversimplified. Nevertheless, the development of methods describing anomalous transport in stochastic magnetic and electric fields in the conditions of anisotropy allows one to obtain visual estimates [45, 91, 262]. To describe transport in comb-like structures, use is made of independent estimates of characteristic times of particles staying in traps or on a bone. Then, using approximation (13.3.6) we can introduce the effective time of diffusion in the transverse direction:
13.4 Double Diffusion and Return Effects
te f f ≈
221
Δ⊥ t, LTrap
(13.4.2)
where Δ⊥ is the perpendicular correlation scale related to the stochastic magnetic field and LTrap is the correlation scale related to trapping mechanisms due to particle returns. The analysis of correlation effects leads to a simple estimate of the value LTrap through the effective time of particles staying in traps: LTrap ∝ tTrap .
(13.4.3)
Based on this approximation, it is easy to obtain the expression for teff in terms of the probability of charged particles returning: Δ⊥ . Φ(t) ∝ √ 4π D0t
(13.4.4)
Since the trapping time is given by: tTrap ≈ Φ (t)t, one can write teff ≈
Δ⊥ Δ⊥ ≈ 2l// t≈ LTrap D0 Φ (t)
(13.4.5) 0
πt , D0
(13.4.6)
where l// is the longitudinal correlation length related to longitudinal diffusion. The expression for the transverse displacements in a static braided magnetic field is given by:
1 D0 Δ⊥ 2 ≈ Dm 4π D0t, teff ≈ Dm (13.4.7) r⊥ ≈ Deffteff ≈ Dm l// Φ (t) which corresponds to the scaling: 2 r⊥ (t) ∝ t 1/2 ,
1 H= , 4
(13.4.8)
for double diffusion of charges in the braded magnetic field. Here, Dm is the magnetic diffusion coefficient. These trapping models are informative and simple in both the initial formulation of the problem and the ability to go far toward its solution. From the above considerations, one realizes that it is not particularly difficult to build up ad hoc probabilistic models exhibiting anomalous transport in systems where trapping effects are essential. Indeed, different power laws can be obtained with a naive dimensional analysis. On the contrary, it is much more difficult to treat anomalous diffusion in complex systems like transport by incompressible velocity fields where the explicit probabilistic features (given, for instance, by an experimental noise related to molecular diffusion) do not play a relevant role. Thus, for a time-independent potential, the charged particles in turbulent plasma will move according to the equipotential curves. However, the latter oscillate in time with the characteristic frequency ω . As a consequence, a particle that is moving according
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13 Subdiffusion and Trapping
to a given equipotential curve has a finite probability of being “trapped” in a finite time by another equipotential curve. Below, we analyze complex correlation effects in the framework of both the phase-space representation and the modified Corrsin conjecture.
Further Reading Diffusion and Trapping Effects Ben-Avraham, D. and Havlin, S. (1996). Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge, U.K. Bouchaud, G.P. and Georges, A. (1990). Physics Reports, 195, 132–292. Hanggi, P. and Talkner, P., eds. (1995). New Trends in Kramer’s Reaction Rate Theory. Kluwer Academic, Boston. Haus, J.W. and Kehr, K.W. (1987). Physics Reports, 150, 263. Metzler, R. and Klafter, J. (2000). Physics Reports, 339, 1. 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..
Trapping and Structures Balescu, R. (1997). Statistical Dynamics. Imperial College Press, London. Shiesinger, M.F. and Zaslavsky, G.M. (1995). Levy Flights and Related Topics in Physics. SpringerVerlag, Berlin. Zaslavsky, G.M. (2002). Physics Reports, 371, 461–580.
Chapter 14
Continuous Time Random Walks and Transport Scalings
14.1 The Montroll and Weiss Approach and Memory Effects Discoveries of anomalous diffusion in numerous phenomena have stimulated the search for transport equations that differ significantly from the conventional diffusive representation. An elegant integral equation corresponding to this problem was suggested by Einstein and Smoluchowski [89, 99]. However, trapping and memory effects were not included in their equation. To describe trapping and subdiffusive regimes, the continuous time random walk model was introduced in [263, 264]. A careful analysis of the problems involving random-walk processes shows that the transition probability density plays a fundamentally important role. In Markov’s approach, the transition probability density is assumed to depend on the spatial variable, W (Δ), where Δ is a spatial step. Montroll and Weiss [263] used a fundamentally different dependence: they assumed that the transition probability density depends on time, ψ (t). They also introduced a physically clear quantity: the probability of not undergoing a transition from a point y to any other points during time t: ΦY (t) = 1 −
t
ψY (t)dt.
(14.1.1)
0
The subscript y in the functions Φ and ψ served merely to mark an arbitrarily chosen point. The function Φ(t) reflects the relaxation properties of the system. In the simplest case, the function Φ(t) is represented in the form of the Poisson distribution:
t , (14.1.2) Φ(t) = exp − τ0 where τ0 is the mean time between transition events. The function Φ(t) can also be represented in some other forms capable of reflecting the characteristic behavior of relaxing systems:
O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008
223
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14 Continuous Time Random Walks and Transport Scalings
√ Kohlrausch relaxation function Φ(t) = exp − τ0 t , algebraic relaxation function Φ(t) = (τ0 t)−γ , Montroll function Φ(t) = exp − lnβm [−τ0 t] .
(14.1.3) (14.1.4) (14.1.5)
Here, βm and γ are the characteristic parameters of the problem. In this theory, the function Φ(t) plays a governing role. The authors of [263] succeeded in writing an elegant chain equation for the probability of a randomly walking particle to be at the point x at time t: t
P(x,t) =
R p (x, τ )Φ(t − τ )d τ .
(14.1.6)
0
Here, R p (x,t) is the probability of transitions from other points to the point x during the time interval (t;t + dt). The functions on the right-hand side of this equation have an essentially similar physical meaning as the function ψ [see definition (14.1.1)]. Consider the point x1 and let the function ψX1 be represented as a sum of the probability densities for transitions from the point x1 to all allowed points x. Then, we have: ψX1 (t) = ∑ ψ (x1 → x,t) (14.1.7) X1
and, consequently R p (x,t) = ∑ X1
t
R p (x1 , τ )ψ (x1 → x,t − τ )d τ + P(x, 0)δ (t),
(14.1.8)
0
/t
where ψ (x1 → x,t) is the probability density for a transition from the point x1 to 0
the point x at time t (see Fig. 14.1). Note that the function ψ depends not only on the time t but also on the relative spatial positions of the points x1 and x, in which case we have: ∞
t
dx −∞
ψ (x1 → x,t)dt = 1.
(14.1.9)
0
The probability density P(x,t) is related to the particle density by: n(x,t) =
N p P(x,t) , δx
(14.1.10)
where Np is the total number of particles and δ x is a volume element. Applying the Laplace transformation in time and the convolution theorem, we obtain from expressions (14.1.6) and (14.1.8) the equation: % s) − P(x, 0) = ∑[R%p (x1 , s) − R%p (x, s)]ψ˜ (x1 → x, s), sP(x, X1
(14.1.11)
14.1 The Montroll and Weiss Approach and Memory Effects Fig. 14.1 Schematic illustration of the two-dimensional continuous time random walk. The time at which the step is made is ti . The time differences, Δti = ti+1 − ti , are chosen randomly
225
r5
r4 t5
t4
t6 t8 r6 r3
r7 r1
t7
t2 r2
t1 = 0
t3
where, in accordance with expression (14.1.6), P% and R%p are related by: % s) % s) sP(x, P(x, = . R%p (x, s) = % 1 − ∑ ψ˜ (x1 → x, s) Φ(s)
(14.1.12)
X1
Then, returning to the physical variables, we arrive at the functional equation: t
∂ P(x,t) = ∑ [P(x1 , τ ) − P(x, τ )]F(x1 → x,t − τ )d τ , ∂t X1
(14.1.13)
0
where the memory function F is defined in terms of its Laplace transform: % 1 → x, s) = F(x
sψ˜ (x1 → x, s) . 1 − ∑ ψ˜ (x1 → x, s)
(14.1.14)
X1
In what follows, we are interested in the functions that depend only on the difference between x and x1 ; this corresponds to the case of a uniform medium: F(x1 → x, s) = F(x − x1 , s).
(14.1.15)
Assuming that the variable x takes on continuous values, we can generalize Eq. (14.1.13) to a sort of the Smoluchowski-Chapman-Kolmogorov equation (3.1.13) with memory effects:
∂ P(x,t) = ∂t
∞
t
dx1 −∞
d τ P(x1 , τ )F(x − x1 ,t − τ ) + Qm (x,t) ,
0
where Qm (x,t) is expressed in terms of Laplace transforms as follows:
(14.1.16)
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14 Continuous Time Random Walks and Transport Scalings
%m (x, s) = P(x, 0) − P(x, % s)F(x % − x1 , s). Q
(14.1.17)
The assumption that the memory function is of a multiplicative nature yields: F(x − x1 ,t − τ ) = G(x − x1 )Mm (t − τ ).
(14.1.18)
Here, G corresponds to the kernel of the nonlocal functional and Mm is the memory function. Switching now to Fourier transforms in x and Laplace transforms in t, we arrive at the following equation for the particle density: ˜ s) − n(k, sn(k, ˜ 0) = −
sψ˜ (s) % ˜ s). (1 − G(k)) n(k, 1 − ψ˜ (s)
(14.1.19)
˜ s) is both the Fourier and Laplace transformation of the density n(x,t). It is Here n(k, easy to draw an analogy between this equation and the Einstein functional equation. Obviously, under the conditions: Mm (t) ∝ δ (t),
%m (s) = const M
(14.1.20)
the Montroll-Weiss equation turns into the Smoluchowski-Chapman-Kolmogorov equation. In fact, choosing the Poisson distribution for the function Φ(t) ensures the required limiting transition for the equation with memory effects. The telegraph equation can be derived for an exponential memory function: t Mm (t) = exp(− ) τ
(14.1.21)
and for a Gaussian kernel function with % = −Dk2 . 1 − G(k)
(14.1.22)
It is of interest to note that, although the equation considered above and the memory function Mm (t), both have simple form, the expression for Φ(t) is fairly complicated in its structure [6, 7]. Hence, we see that it is necessary to select different model functions for different physical situations. There are numerous investigations of the continuous time random walk models. Fortunately, several detail reviews [3, 4, 6, 7, 11, 36, 45] have been published recently. Therefore, in the next sections we consider a few examples closely related to the renormalization of quasilinear equations in terms of the continuous time random walk approach.
14.2 Fractional Differential Equations A highly significant physical quantity in the description of random walk processes with memory is the mean waiting time t until an event occurs:
14.2 Fractional Differential Equations
227
t =
∞
t ψ (t)dt.
(14.2.1)
0
In the continuous time random walk approach, this time plays a role analogous to the mean length of a jump in the theory of Markovian processes. This is not surprising because, in the approach based on memory effects, the transition probability density Φ(t) is an analogue of the Einstein functional kernel WE (y). For the Poisson distribution (14.1.2), the mean waiting time is equal to the characteristic time t = τ0 . An important particular case of relaxation functions is represented by those that decrease according to a power law: τ γ 0 , 0 < γ < 1. (14.2.2) Φ(t) ∝ t In this model, the mean waiting time until an event occurs tends to infinity: t =
∞
t ψ (t)dt → ∞.
(14.2.3)
0
The power relaxation functions were found to provide an efficient tool for analysis of transport processes [3, 4, 6, 7, 14, 16, 34, 45]. For long times t, simple manipulations %m (s): yield the following expression for M %m (s) = M
% sΦ(s) % = Γ(1 − γ )sγ , ≈ sΦ(s) % 1 − sΦ(s)
(14.2.4)
where Γ(z) is Euler’s gamma function. The equation describing memory effects takes the form: % % n(k, % ˜ s) = (1 − G(k)) ˜ s) + n(k, sΦ(s) n(k, ˜ 0)Φ(s).
(14.2.5)
% n(k, ˜ s) ˜ s) ≈ sγ n(k, sΦ(s)
(14.2.6)
The expression
can be interpreted as a time derivative of order γ [263, 264]: % n(k, ˜ s) ≈ sγ n(k, ˜ s), sΦ(s)
(14.2.7)
which leads to the relation
∂ γ n(k,t) ˜ ∂ = ∂ tγ ∂t
t 0
n(k, ˜ τ) dτ . (t − τ )γ
(14.2.8)
Representing the results in such a manner facilitates interpretation of the scaling relations of the form:
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14 Continuous Time Random Walks and Transport Scalings γ
R(t) ∝ t αL ,
(14.2.9)
which has increasingly wider application in the analysis of fractional derivatives:
∂ γ n(x,t) ∂ αL n(x,t) = + QI (n, x,t). ∂ tγ ∂ x αL
(14.2.10)
Here, αL and γ are the parameters, and QI is the additional term related to the initial conditions. This is a formal approach and we will establish the relationship between the continuous time random walk theory and turbulent transport models.
14.3 Correlation Function and Waiting Time Distribution The approximations of the function ψ (x1 → x;t) used in the previous sections allow one to investigate the models in which it is easy to use decoupling:
ψ (x,t) = ϕ (x)ψ (t).
(14.3.1)
However, many important results can be obtained in the framework of a more general analysis. Let us consider the system of the Montroll-Weiss equations in the form: P(x,t) = ∑ x1
where
t
P(x1 , τ )ψ (x − x1 ,t − τ )d τ + Φ(t)δ (x),
(14.3.2)
ψ (t) = ∑ ψ (x;t) = ψ (k = o;t).
(14.3.3)
0
x
Then, on applying both the Laplace and Fourier transformations one obtains: ˜ s) = P(k, ˜ s)ψ˜ (k, s) + Φ(s), P(k,
(14.3.4)
where
1 − ψ˜ (s) . s ˜ s) in the form: Formal calculations yield the expression for P(k, Φ(s) =
1 ˜ s) = 1 − ψ˜ (s) . P(k, % s % (k, s) 1−ψ
(14.3.5)
(14.3.6)
Following the approach developed in [265] it is easy to obtain a formal expression for the mean square displacement: ˜ ∂ 2 P(k,t) |k=0 . R2 ≡ Δx2 (t) = x2 P(x,t)dx = − ∂ k2
(14.3.7)
14.3 Correlation Function and Waiting Time Distribution
229
On the other hand, the correlation definition of the turbulent diffusion coefficient suggested by Taylor (1.4.10) relates the mean square displacement to the correlation function: t t dR2 = 2 C(t )dt = 2 V (0)V t dt . (14.3.8) dt 0
0
The authors of Ref. [265] used the Fourier transformation of this equation: sR%2 (s) = 2
˜ C(s) s
(14.3.9)
to establish the relationship between the Lagrangian correlation function and the memory function. Comparison between expressions (14.3.7) and (14.3.9) yields this relationship in terms of the Laplace transform: % = ζm 2 s ψ˜ (s) , C(s) 1 − ψ˜ (s)
(14.3.10)
where we used the Gaussian approximation for the step length distribution:
ψ˜ (k, s) = ψ˜ (s) exp(−ζm 2 k2 ).
(14.3.11)
It is well known that the exponential correlation function corresponds to the conventional Brownian motion. Therefore, the correctness of this formula must be first proved for ψ (t) = δ (t − τ0 ). Then, substituting ψ˜ (s) = exp(−sτ0 ), it is easy to find: % = ζm 2 C(s)
s . esτ0 − 1
(14.3.12)
To compute the Laplace inverse of this expression, one keeps the three lowest orders of the exponential as sτ0 → 0(the observation time is much lower than the microscopic waiting time: t >> τ0 ) and obtains: & '
2ζm 2 t . (14.3.13) C(t) = exp −2 τ0 2 τ0 This result corresponds exactly to the stochastic Langevin equation:
ζm 2 ξ (t), V˙ (t) = − V + τ0 τ0 2
(14.3.14)
where V (t) is the Brownian particle velocity and ξ (t) is the white noise random function. Now we are able to consider more complex models, where the memory function is represented, for example, by the stable low : 9 (14.3.15) ψ˜ (s) = exp − (sτ0 )γm
230
14 Continuous Time Random Walks and Transport Scalings
with the parameter γm ; 0 < γm < 1. Simple calculations in the framework of continuous time random walk allow the following expression to be obtained: R2 = 2
ζm 2 t γm , σ τ0 m Γ(1 + γm )
(14.3.16)
where Γ is the gamma function. The expression for the Laplace transform of the velocity correlation function takes the form: 2 s1−γm % = ζm s1−γm # γ γ $ , C(s) m m τ0 1 + τ0 s
(14.3.17)
2
: 9 where three lowest orders in sτ0 were kept to compute exp (sτ0 )γm . Formal calculations yield the final expression for the correlation function in the form: γm t 2 ζm 2 2γm −2 , t Eγm ,2γm −1 −2 C(t) = γ m τ0 τ0 τ0
(14.3.18)
where 12 < γm < 1 since we restrict the parameter γm in order to express C(t) in terms of the Mittag-Leffler functions Eγm ,2γm −1 . This expression leads to the power tail asymptotic: 1 1 (14.3.19) C(t) ∝ 2−γ ∝ α . t m t C This result agrees well with the simple estimates based on the continuous time random walk transport scaling R2 ∝ t γ , since the dimensional representation of the Lagrangian correlation function is given by: C(t) ∝ V 2 ∝
R2 1 ∝ 2−γ . t2 t
(14.3.20)
In the case under analysis (14.3.19) we find: R2 (t) ∝ t 2H ∝ C(t)t 2 ∝
t2 t 2−γm
= t γm .
(14.3.21)
Then, one obtains the relation for the Hurst exponent H = γ2m and the correlation exponent αc = 2 − γm . The method considered is effective for analyzing the correlation properties of the complex system in which memory effects dominate.
14.4 The Klafter Blumen and Shlesinger Approximation Fractional differential equations are an especially effective tool for investigating anomalous transport. These equations allow us to obtain scalar probability density functions based on the scaling representation of waiting time distributions.
14.4 The Klafter Blumen and Shlesinger Approximation
231
Thus applying the continuous time random approach makes it possible to consider both nonlocality and memory effects by the approximation of the model function ψ (x,t). Blumen, Klafter, and Shlesinger [266] suggested using the advantages of this method to describe the Richardson relative diffusion R2 ∝ t 3 . Recall that in the Monin model [106] only the dimensional estimate of nonlocality effects was used by the power approximation of the kernel of the nonlocal functional (3.4.2): 1/3 ˜ (14.4.1) G(k) ∝ εK k2/3 . The authors of Ref. [266, 267] suggested a fruitful dynamical interpretation of nonlocality and memory effects by using the model function ψ (x,t) in the form:
x , (14.4.2) ψ (x,t) = ϕ (x)ψ (t/x) = ϕ (x)δ t − V (x) x where, through the function δ t − V (x) , x and t are coupled. Such a representation corresponds to the Levy walk (see Fig. 14.2). It includes the same set of points as the Levy flight plus the trail it takes connecting these points. Here, the functions ϕ (x) and V (x) are represented by the following scalings:
ϕ (x) ∝
1 x1+βR
V (x) ∝ xγR .
Fig. 14.2 Realization of a Levy walk, obtained from the distribution probability where βR = 2.5, γR = 1/3. The situation depicted evolved after 1000 time units. The starting point of the walk is marked by a dot. (After G. Zumofen et al. [268], with permission)
,
(14.4.3) (14.4.4)
232
14 Continuous Time Random Walks and Transport Scalings
The case γR = 0 corresponds to the ballistic model and the case γR = to the Kolmogorov scaling: Vk2 ∝ E(k)k ∝
1 ∝ l 2/3 . k2/3
1 3
corresponds
(14.4.5)
Here, E(k) is the energy spectrum [28]. Then for γR = 13 , using the expression for mean square displacement in the form:
∂2 ˜ P(k,t) |k=0 , ∂ k2
R2 = −
(14.4.6)
˜ where the expression for P(k,t) is given by formula (14.3.2), one obtains the following relationships: R2 ∝ t 3
for
2
R2 ∝ t 2+ 3 (1−βR ) R2 ∝ t
for for
1 βR ≤ ; 3 1 1 ≤ βR ≤ ; 3 2 1 βR ≥ . 2
(14.4.7) (14.4.8) (14.4.9)
It is easily seen that for βR ≤ 13 expression (14.4.6) reduces to the Richardson law R2 ∝ t 3 , as anticipated. Moreover, in [266] the modified model was considered, where the intermittency effects are included: V (x) ∝ xγR ; γR =
1 1 d − dF μi + = 1+ . 3 6 3 2
(14.4.10)
Then, the mean square separation of two particles is given by: 12
βR ≤
R2 ∝ t 4−μi ; R2 ∝ t
1−β 2+6 4−μR i
1 − μi ; 3
1 − μi 10 − μi ≤ βR ≤ ; 3 6
;
R2 ∝ t;
10 − μi ≤ βR . 6
(14.4.11) (14.4.12) (14.4.13)
Here, the scaling exponents depend on the index βR as well as the fractal dimension dF = d − μi . Note, that the scaling for
βR ≤
1 − μi 3
(14.4.14)
was obtained independently by Hentschel and Procaccia [76], who used a much different approach.
14.5 Stochastic Magnetic Field and Balescu Approach
233
Hundreds of research papers have been written on the application of continuous time random walk approach to turbulent diffusion. For a fuller treatment of this exciting subject, we refer the reader to [3, 6, 7, 34, 35, 36, 45, 49].
14.5 Stochastic Magnetic Field and Balescu Approach The double diffusion of charged particles in the stochastic magnetic field has the subdiffusive character of transport with the Hurst exponent H = 1/4. The corresponding transport equation for the particle density must be coordinated with the continuous time random walk approach. Such a model was considered by Balescu [259]. Here, we represent this approach in the simplified form, which is related to the earlier discussions of the renormalization of quasilinear equations (5.4.4) and 2 (5.4.5). Instead of introducing the additional diffusion term D ∂∂ xn21 that describes
transverse correlation effects, we will keep one of the usually omitted terms, v1 ∂∂nx1 , which allows the memory effects to be described. Here, D is the diffusion coefficient, n1 is the scalar density perturbation, and v1 is the velocity field perturbation. As a result, the transformations place Eq. (5.4.5) for n1 into the form:
∂ n1 ∂ n1 ∂ n0 + v1 = −v1 . ∂t ∂x ∂x
(14.5.1)
Here, n0 is the mean scalar density. This equation can be considered as a first-order linear hyperbolic equation with the source term: I(x,t) = −v1
∂ n0 , ∂x
(14.5.2)
where the derivative ∂∂nx0 is the parameter of the problem. We also supplement the equation with the uniform initial condition n1 (x, 0) = 0. This formulation is similar to (5.4.9) but here we deal with the characteristic, z = x − v1 (t − t1 ),
(14.5.3)
instead of the characteristic of the conventional quasilinear approach z = x − v0 (t − t1 ): n1 (x,t) = −
t
v1 (t1 ) 0
∂ n0 (z,t) dt1 . ∂z
(14.5.4)
Substituting this expression for n1 into the equation for the mean density n0 (5.4.4): ∂ n0 ∂ n1 = − v1 . (14.5.5) ∂t ∂x
234
14 Continuous Time Random Walks and Transport Scalings
and performing simple manipulations yields:
∂ n0 = ∂t
t
v1 (t)v1 (t1 )
0
∂ 2 n0 (z,t1 ) dt1 . ∂ z∂ x
(14.5.6)
Now one can see that the use of the correlation function in the power form: τ αC 0 (14.5.7) C(t) = MB (t) ∝ t leads to the continuous time random walk representation for the transport equation:
t 2 ∂ 2 n0 (x,t) ∂ τ0 αC D ∂ =− n0 (x,t ) dt . (14.5.8) ∂ t2 τ0 ∂ t ∂ x 2 t −t 0
Here, D0 is the seed diffusivity, τ0 is the characteristic time, and αC is the correlation exponent. The renormalized quasilinear equations together with the approximation of the correlation function make it possible to obtain the fractional differential equation:
D0 ∂ αC ∂ 2 n ∂ 2n , (14.5.9) = − ∂ t2 τ0 ∂ t αC ∂ x2 which corresponds to a continuous time random walk with the scaling for the Hurst exponent: R2 (t) ∝ t 2−αC , αC , αC < 2. H (αC ) = 1 − 2
(14.5.10) (14.5.11)
In this simplified approach, we do not consider anisotropy effects and longitudinal collisional diffusion, which are essential for describing the anomalous transport in the stochastic magnetic field. For a deeper discussion of these aspects, we refer the reader to Balescu et al. [259, 269]. The final equation for the transverse transport with the modified memory function, MB ≈ D0 δ (t) +
D0 τ0 αC , τ0 t
(14.5.12)
takes the form:
∂ n0 (x,t) ∂ 2 n0 D0 = −D0 2 − 2 ∂t ∂x τ0
t 0
∂2 τ0 αC n (x,t ) dt , 0 ∂ x2 t − t
(14.5.13)
which allows one to investigate both the conventional diffusion and the subdiffusive behavior, where the case with αC = 3/2 corresponds to the double diffusion.
14.6 Longitudinal Correlations and the Diffusive Approximation
235
14.6 Longitudinal Correlations and the Diffusive Approximation The diffusive renormalization of quasilinear equations [157] is based on the ballistic representation of longitudinal (z-axis) motion and the diffusive approximation Def f ∇2⊥ n1 of transverse correlation effects. In fact, the opposite case corresponds to the random shear flow model and √ double diffusion where longitudinal motions have the diffusive character λ// ≈ 2D0t. From this standpoint, it is easy to obtain an equation for the passive tracer density under conditions when longitudinal correla2 tion effects can be approximated by the longitudinal diffusive term D0 ∂∂ zn21 . Thus, in the two-dimensional case the corresponding renormalized equations have the form: ∂ n0 ∂ n1 = − VX (z) ; (14.6.1) ∂t ∂x
∂ n1 ∂ 2 n1 ∂ n0 = D0 2 −VX (z) . ∂t ∂z ∂x
(14.6.2)
Here, in contrast to [157], the diffusion coefficient D0 characterizes the seed diffusion. The dependences n0 = n0 (x,t) and n1 = n1 (x, z,t) were used to describe the two-dimensional case. Indeed, a similar set of equations was derived in [269] by averaging the diffusion equation with the random convective term:
∂n ∂n = D0 Δn −VX (z) . ∂t ∂x
(14.6.3)
Using the Laplace transformation over t and the Fourier transformation over z, one obtains from (14.6.1) and (14.6.2): 2 ˜ ∂ n˜ 0 , sn˜ 0 (s, x) − n0 (x, 0) = D(s) (14.6.4) 2
2 ∂x ⎫ ⎧ ⎪ ⎪ |2 /D ⎪ ⎪ |z s − z exp − L0 ∞ 0 ⎬ ⎨ 1 ˜ √ dz dz VX (z)VX (z ) . D(s) = limL0 →∞ ⎪ ⎪ 2L0 D0 s ⎪ ⎪ ⎭ ⎩ −∞ −L0
(14.6.5) Then, one can write a diffusion equation for the model of random drift flows [135]. This corresponds to the condition z → z (the condition of “return” of the particle to the initial point). A fractional differential equation was found:
∂ 3/2 n0 (t, x) ∂2 = ∂ t2 ∂ t 3/2
t 0
V 2 a ∂ 2 n0 (t, x) n0 (0, x) n0 (t , x)dt 1 = √0 − √ 3/2 . 2 ∂ x 2 πt 2D0 π (t − t )
(14.6.6)
Indeed, the “renormalization” of the quasilinear equations allows us to obtain the transport equations, which differ significantly from the classical diffusion equation [8, 9]. Now, we consider using the correlation properties of a system of
236
14 Continuous Time Random Walks and Transport Scalings
random flows. The correlation function can be represented in the power form. We can change the form of the equation for n0 by means of a more detailed consideration of the function KC (|z − z |) = VX (z)VX (z ) in (14.6.5). This function describes the correlation properties. The correlation nature of this function was discussed in detail in [268]. However, in that treatment, the form of the function K was not important because only return effects (z → z ) were taken into account. Let us consider the power approximation of the function KC (w): KC (w) = KC z − z ∝
V02 . 1 + wαE
(14.6.7)
Such power approximations of the correlation function are often used for obtaining a transport scaling. Using the symmetry of integral (14.6.5), we can easily simplify Eq. (14.6.4). In terms of the Laplace transformation, the renormalized transport equation takes the form: V2 s sn˜ 0 (s, x) − n0 (x, 0) = √ 0 2D0 2
αE 2
−1
∂ 2 n˜ 0 . ∂ x2
(14.6.8)
Changing to the dependence in time, we obtain the fractional differential equation [270, 271]:
∂ γ n0 ∂ 2 n0 = De f f − Q(t, x), γ ∂t ∂ x2 V02 aαE , De f f = (2D0 )αE /2 n0 (0, x) Q= √ γ . 2 πt
(14.6.9) (14.6.10) (14.6.11)
Here, the order of the derivative with respect to time γ depends on the parameter αE : αE , 0 < αE < 4, γ (αE ) = 2H = 2 − (14.6.12) 2 which describes correlation properties in the longitudinal direction. In the case of an anisotropic medium this relationship can be related to the Corrsin functional in the form: C(t) ≈
∞ −∞
V02 dz z2 √ . αE exp − 4D0t 1 + (z/z0 ) 4π D0t
(14.6.13)
Here, V0 and z0 are the dimensional parameters of the correlation function. Moreover, it is possible to recognize the Corrsin integral in the expression for the Laplace transformation (14.6.5). 2 Using the dimensionless variable 4Dz 0 t to calculate the integral, we obtain the simple estimate in the scaling form:
14.6 Longitudinal Correlations and the Diffusive Approximation
C(t) ≈ and hence
λ⊥ 2 const ≈ α /2 t2 t E
λ⊥ ∝ t 1−
αE 4
.
237
(14.6.14)
(14.6.15)
This is the scaling law (14.6.12). Note that the random shear flow regime with αE = 1 (H = 3/4) has clear physical interpretation in terms of the spectrum Sc (k) [148]: (14.6.16) V˜x (k)V˜x (k ) = Sc (k)δ (k − k ). Here, V˜x (k) is the Fourier representation of Vx (z), and δ (k −k ) is the Dirac function. For k << 1 and the power form Sc (k) ≈ kαE −1 ,
(14.6.17)
the regime with H = 3/4 corresponds to a random velocity field, which is white noise. On the other hand, because of the character of anisotropy, the scaling H = 1 − α4E can be obtained by the modification of the Taylor dispersion model: if instead of a laminar velocity profile one considers the system of random flows with the Eulerian correlation function C (z − z ) = VX (z) VX (z ). Indeed, applying the method of Green’s functions to the equation for the density perturbation n1 (14.6.2) yields: 3 4 ∂ n0 (x,t ) dk ik(z−z ) −D k2 (t−t ) n1 = − dz dt VX z e e 0 . (14.6.18) ∂x 2π The substitution of this expression in the formula for a flux leads to the relation in terms of the memory function:
∂ n1 qX ≈ −VX (z) ∂x
t
= 0
dt
∂ 2 n (x,t ) M t − t . 2 ∂x
(14.6.19)
Here, M(t − t ) is the memory function represented as 3 4 dk 2 e−D0 k (t−t ) d z − z VX (z)VX z e−ik(z −z) . M t − t = 2π (14.6.20) Using the Fourier transform C% (k) of the function C (z − z ) = VX (z)VX (z ), we can rewrite the expression for the flux qx in the form: qX ≈
t
dt 0
dk 2π
5 6 3 4 % (k) ∂ 2 n0 (x,t) 2 t−t ∂ n0 (x,t ) C dk −D k ( ) C% (k) e 0 ≈ ∂ x2 2π D0 k2 ∂ x2 (14.6.21)
for the case of a smooth profile n0 (x,t). The effective diffusion coefficient is given by the expression that coincides with the Howells form [125] but for an anisotropic model:
238
14 Continuous Time Random Walks and Transport Scalings
Def f ≈
% C (k) dk
D0 k2 2π
.
(14.6.22)
The case under consideration can be interpreted in terms of scaling representation of the spectrum C% (k) ≈ kαE −1 and the assumption about the diffusive character of the wave numbers, which make a major contribution to transport: k∝
1 1 ≈ √ . λ// D0t
(14.6.23)
Then simple calculations yield a scaling:
λ⊥2 ≈ Def f t ≈
% αE C(k)t ≈ λ// 1−αE λ//t ∝ t 2− 2 , k
(14.6.24)
which relates the Hurst exponent H = 1 − αE /4 describing anomalous transport in the transverse direction to the exponent αE characterizing the spatial correlation properties of a system of random shear flows.
14.7 Vortex Structures and Trapping The continuous time random walk approach is effective to investigate trapping effects in turbulent flows with vortex structures (see Fig. 14.3). Thus, Cardoso et al. [271, 272] conducted experiments on anomalous diffusion of scalar in the field of several vortices. They studied trapping effects related to the capture of test particles by vortices. In such a formulation of the problem, the continuous time random walks approach is an adequate theoretical model. From the formal standpoint we must introduce a scaling to describe the waiting time distribution: 1 ψ (t) ∝ γ −1 . (14.7.1) t Then, the mean waiting time between two “jumps” is estimated by the expression:
Fig. 14.3 Schematic illustration of tracer particle trapping by a coherent vortex structure
14.7 Vortex Structures and Trapping
τT ≈
239 tmax
3−γ
t ψ (t)dt ∝ tmax ≈ t 3−γ .
(14.7.2)
0
On the other hand, the complete walking time can be estimated through τT and the number of pauses N: (14.7.3) t ≈ N τT ≈ t 3−γ N. Hence, one obtains the scaling for N in the form: N (t) ∝ t γ −2 .
(14.7.4)
Since the mean square displacement is related to N by the estimate R2 ∝ N, one arrives at the relationship: γ √ −1 (14.7.5) R (N) ∝ N ≈ t 2 . The experimental results from [272, 273] allow one to define a scaling for the mean square displacement in the following form: 1
R (t) ∝ t 3 ,
(14.7.6)
where the Hurst exponent H = 13 and the escape probability exponent γ = 83 . On the other hand, the probability density function ψ (t) corresponding to these scalings was successfully measured [271, 272], which permits us to consider the continuous time random walk prediction as correct (see Fig. 14.4). The experimentally determined subdiffusive regime is not the only one possible. In the presence of vortex structures, there are both trapping effects and flights. The self-organized vortex structures (see Fig. 14.5) were obtained experimentally in axially symmetric zonal counter-flows in a rotating parabolic “shallow water” layer [273, 274]. Similar experiments became an important step to investigate turbulent transport and complex patterns in the presence of regular vortex structures. Thus, the authors of refs. [275, 276, 277] experimentally investigated test particle transport in almost two-dimensional flow in annular tank (“Texas experiments”). The tank was rotating at about 1 or 2 Hz and the bottom is sloped to simulate β -effects. Because
Fig. 14.4 Experimental results for the diffusion front compared with the theoretical prediction. (After O. Cardoso and P. Tabeling [273], with permission)
240
14 Continuous Time Random Walks and Transport Scalings
Fig. 14.5 Vortex structures in a rotating parabolic “shallow water” layer. (After M. Nezlin et al. [274], with permission)
Fig. 14.6 Trajectory of a tracer particle in a vortex flow
of the rapid rotation, the flow was quasi-two-dimensional. Two types of particle trajectories were found. Particles within a vortex remain trapped for very a long time (stick) (see Fig. 14.6). Particles in the azimuthal jet experience prolonged flights around the circumference of the tank (see Fig. 14.7). Because the vortex pattern is not perfectly stationary, particles alternate, apparently randomly, between flaying in jets and sticking in vortices. These experiments lead to the superdiffusive scaling for the angular displacement: 2 γ (14.7.7) Δθ ∝ t 0 , where γ0 ≈ 1.4 ÷ 1.7. It was also possible to observe conventional diffusion with γ0 = 1 by breaking the azimuthal symmetry of forcing the flow. While much of this work has been concerned with the development of scaling models and attempts to understand the consequences of these models, the main objective of turbulent transport theory is to obtain a better understanding of natural phenomena. Experiments are usually more expensive and require more planning and preparation. However, even simple experiments have made an important
Further Reading
241 Θ
Fig. 14.7 Schematic illustration of the angular displacement, Θ(t), for the trajectory in the figure
T
contribution to the development of turbulence and have stimulated additional simulations and theoretical work.
Further Reading Continuous Time Random Walk and Scaling Ben-Avraham, D. and Havlin, S. (1996). Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge, U.K. Bendler, J.T., Fontanella, J.J., and Shlesinger, M.F. (2004). Physica, D 48, 67. Metzler, R. and Klafter, J. (2000). Phys. Rep., 339, 1. 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.
Fractional Differential Equations and Turbulent Transport Balescu, R. (1997). Statistical Dynamics. Imperial College Press, London. Shiesinger, M.F. and Zaslavsky, G.M. (1995). Levy Flights and Related Topics in Physics. SpringerVerlag, Berlin. West, B.J., Bologna, M., and Grigolini, P. (2003). Physics of Fractal Operators. Springer-Verlag, New York.
Chapter 15
Correlation and Phase-Space
15.1 Kinetics and the Diffusion Equation There are certainly deep connections between the conventional approach to the transport equation in the configuration space and the phase-space representation. The Hamiltonian theory offers the advantage of using additional degrees of freedom to treat nonlocality and memory effects in the framework of phase-space. The kinetic model provides the possibility of describing ballistic modes and establishing the relationship between different exponents and distributions. As early as 1940, Kramers [278] pointed out the difficulties encountered in an attempt to obtain the diffusion equation in ordinary coordinate space,
∂n ∂ 2n ∂ =D 2 − (V0 n) , ∂t ∂x ∂x from the simplest kinetic equation which includes spatial nonuniformity:
kTp ∂ f 1 ∂ ∂f ∂f ∂f +V − KF (x) = Vf+ . ∂t ∂x ∂V τ0 ∂ V m ∂V
(15.1.1)
(15.1.2)
Here, f (t,V, x) is the particle distribution function, KF (x) is the acceleration, V is the velocity, Tp is the temperature, τ0 is the characteristic time, and m is the mass of the particle. Even here a demand arose for a nontrivial approach with integration over a simplified trajectory, r = r0 + V τ0 , in lieu of “conventional averaging” with the fixed value r0 . Here, r0 is an arbitrary initial point. This corresponds to the system of characteristic lines: V dV =− (15.1.3) dt τ0 and
dr = V. dt
(15.1.4)
From this point of view, the spatial nonuniformity of the distribution function f at scales λ ≤ V0 τ0 can be ignored: f (t,V, x) ≈ f (t,V, x + λ ). This means that only local effects are described by this kinetic equation (see Fig. 15.1). However, this argument was not effective enough for the introduction of corrections to the kinetic equation O.G. Bakunin, Turbulence and Diffusion. Springer Series in Synergetics c Springer-Verlag Berlin Heidelberg 2008
243
244
15 Correlation and Phase-Space
Fig. 15.1 Phase-space representation of a particle trajectory
V V0τ V0
x0
x
at that time. Kramers in fact pointed out the conventional character of the diffusion equation and its close relationship to the correlation function behavior. Indeed, a formal integration of the kinetic equation [279, 280, 281, 282],
kTp ∂ f V ∂f ∂f ∂ 1 ∂ +V − Vf+ , (15.1.5) − KF (x) f = ∂t ∂ x ∂V τ0 τ0 ∂ V m ∂V taking into account the following conditions: f (|V | → ∞) → 0 and
∂ f (M → ∞) , ∂V
(15.1.6)
leads only to the continuity equation:
∂n ∂ + U∗ n = 0, ∂t ∂x
(15.1.7)
where the mean values are given by: ∞
n (x,t) =
f (x,V,t) dV ,
(15.1.8)
V f (x,V,t) dV.
(15.1.9)
−∞
1 U∗ = n
∞ ∞
However, Davies [283] suggested using an additional expression to derive the correct transport equation. Multiplying the kinetic equation by v and integrating over the velocity space yields: U∗ ∂ ∂ 2 (U∗ n) + nU + n = KF (x) n, ∂t ∂x t
(15.1.10)
15.2 Phase Space and Transport Scaling
245
where U 2 is given by the integral: 1 U = n
∞
2
V 2 f (x,V,t) dV .
(15.1.11)
−∞
Then, neglecting the term nU∗ we find the well-known telegraph equation:
τ0
∂ 2n ∂ n ∂ ∂2 + [KF (x) τ0 n] = 2 U 2 τ0 n . + 2 ∂t ∂t ∂x ∂x
(15.1.12)
For t >> τ0 the telegraph equation is transformed into the classical diffusive one with the diffusion coefficient: D = U 2 τ0 = const.
(15.1.13)
The value KF (x) τ0 characterizes the drift velocity: Udri f t = KF (x) τ0 .
(15.1.14)
The analysis shows that the kinetic description enables one to incorporate the shape of the particle velocity distribution function when describing transport equations in the configuration space.
15.2 Phase Space and Transport Scaling At an early stage of turbulence research, it was clear that using both the phase space and scaling offered great promise for the description of turbulent transport. Thus, Yaglom [40] demonstrated that the classical Richardson scaling for the relative diffusion, DR (t) ≈ const · εK t 2 ,
(15.2.1)
can be interpreted in terms of conventional diffusion in a velocity space. Indeed, the correlation function of Lagrangian accelerations Ca (t) is given by the expression: Ca (t) = a (0) a (t) = εK δ (t) =
∞
εK cos (ω t) dt,
(15.2.2)
0
which allows the value εK to be considered as the constant diffusion coefficient in the velocity space:
εK ∝
DR (t) 1 1 ≈ (ΔV )2 ≈ t2 t t
ΔY t
2 .
(15.2.3)
246
15 Correlation and Phase-Space
Unfortunately, the scaling approach in kinetics was recognized only recently. In contrast to diffusion equations for a configuration space, the kinetic equations open wider possibilities for the description of complex systems such as suprathermal particles, plasma turbulence, etc. There are papers of interest in which fractional differentials are recommended for use in the description of the nondiffusive kinetic effects [37, 38, 39, 284, 285, 286]. Relationships among kinetic models, diffusion equations, and probabilistic estimates of course have to exist. On the one hand, the kinetic approach makes it possible to correctly take into account ballistic effects that are related to the convective fraction of hydrodynamic flows. On the other hand, the increase in the number of degrees of freedom, which is related to the velocity incorporation as an independent variable, permits the description of nonlocal effects in the presence of spatial nonuniformity and stochastic layers. Thus, recently Zaslavsky and Edelman [287] suggested a fruitful method to relate phase-space properties of the sticky island boundary to continuous time random walk scalings. A regime close to the ballistic motion was considered on the basis of the fairly universal Hamiltonian function: He f f = b1 (Δφ )2 + b2 Δg − b3 (Δg)3 .
(15.2.4)
Here, Δφ and Δg are the deviations of the phase φ and the energy g values on the trajectory from their special quantities φ∗ , and g∗ on the ballistic trajectory, which corresponds to an initial stage when an island is born. Ballistic behavior is related to long-range trapping near the boundary of the sticky island [288] (see Fig. 15.2). It is possible to estimate the escape probability distribution ψ (t) from the boundary layer in terms of the phase volume ΔΓ:
ψ (t) ∝
1 1 ≈ . ΔΓ Δφ Δg
(15.2.5)
One can consider a steady estimate for the relationship between Δφ and Δg that is based on the Hamiltonian function (15.2.4): Δφ ∝ Δg3/2 .
(15.2.6)
Then, we can express the phase volume in the scaling form: ΔΓ ∝ Δg5/2 .
(15.2.7)
This simple estimate mirrors the advantages of using additional degrees of freedom, which arise in the framework of the Hamiltonian approach. Escaping from the boundary layers implies that during an initial period Δg scales with time in accordance with Δg (t) ∝ t. Simple calculations yield (see Fig. 15.3):
ψ (t) ∝
1 1 1 ≈ ≈ . ΔΓ Δg5/2 t 5/2
(15.2.8)
15.3 The One-Flight Model and Anomalous Diffusion Fig. 15.2 The sticking system of nested islands
247
τout
τin
This means that the relaxation function exponent is given by (14.1.4): γ = 3/2. Now the Hurst exponent that describes transport effects (14.2.9) can be obtained as follows: 3 γ (15.2.9) H= = . 2 4 This approach appears very attractive due to a universal assumption about the relationship between ψ and ΔΓ (15.2.8). Indeed, different models give exponents [34, 35, 287, 288] that are very close to H = 3/4.
15.3 The One-Flight Model and Anomalous Diffusion A large variety of anomalous transport models leads to a search for “hyper-scalings” and relationships between exponents. This is not surprising since an analogous situation arose in considering phase transitions and percolation. The investigation of relationships between the kinetic and correlation approaches plays a significant role here. Thus, the simplest correlation scaling is the power approximation of the Lagrangian correlation function: C(t) = V (0)V (t) ∝
1 , t αC
(15.3.1)
which, in accordance with the Taylor correlation definition of the turbulent diffusion coefficient DT (1.4.10), yields the scaling for the mean displacement: R (t) ∝ t 1−αC /2 ,
0 < αC < 2.
(15.3.2)
248
15 Correlation and Phase-Space
This means that the Hurst exponent is expressed in terms of the correlation exponent αC . Note that relationships between the Lagrangian correlation function and the probability density ψ (t) could also exist. Indeed, Zaslavsky [34] considered a ballistic model in which the velocities V (0) and V (t) that enter the Lagrangian correlation function C(t) = V (0)V (t) belong to the same flight. That analysis was based on the approximation of the anomalous diffusion in a map model [287, 288] where the relationship between the Hurst exponent H and escape probability was proposed: 3−γ 2
H=
(15.3.3)
Formal calculations enable one to obtain from (15.3.3) and (15.3.2) the relation:
αC = 1 − γ .
(15.3.4)
Zaslavsky suggested an interesting interpretation of this expression in terms of the probability density: 1 ψ = Φ (t) ∝ γ +1 . (15.3.5) t In the framework of the “one-flight” model, the correlation function C(t) is proportional to the probability Pesc (t) that a particle will stay in the same flight at least during the time interval t. From the formal standpoint it can be rewritten in the functional form: C(t) ∝ Pesc (t) ≈
∞
∞
dt1 t
dt2 ψ (t2 ).
(15.3.6)
t1
One can see that the scaling interpretation of this expression leads to αC = 1 − γ . It is natural to consider other approximations of relaxation functions Φ(t) as well. The Kohlrausch relaxation law (14.1.3) is an interesting example:
0 t . (15.3.7) Φ(t) ∝ exp − τ0 Here, τ0 is the characteristic time. Of particular interest is the fact that the Kohlrausch slowed relaxation law is related to the Laplace transformation of familiar Levy’s law for jumps with the exponent αL = 1/2: ∞
√ exp − w s = exp(−sx) fαL (x) dx, 0
f aL
1 (x) = 3/2 2x
0
w w . exp − π 4x
(15.3.8)
(15.3.9)
This simple formula clearly shows a close relationship between the memory and nonlocality effects. Physically, this relationship is not surprising. A particle that
15.4 Correlations and Nonlocal Velocity Distribution
249
stays in a trap in phase space is not involved in events (does not undergo collisions). However, in conventional coordinate space, such a collisionless particle is transported over a large distance during the time it stays within the phase-space trap. In this sense, collisionless particles cannot be regarded as being involved in a conventional diffusion process, in contrast to the particles that undergo collisions [289]. The physical meaning of formal relationship (15.3.8) can easily be understood by treating its integral part as the averaging procedure for the Poisson law:
x . (15.3.10) exp (−sx) = exp − x0 As an example, let us consider the case x = V, s = 1/x0 ≈ t/L0 , w = V0 , where V is the particle velocity, V0 is the characteristic velocity, and L0 is the size of the region over which the averaging is performed. In this case, we have:
0 ∞
V0 Vt exp − f (V ) dV. t = exp − L0 L0
(15.3.11)
0
As a result, we see that the Kohlrausch relaxation law describes the Poisson probability for a particle not to undergo collision in a region of size L0 during the time t, averaged by means of a Levy distribution with αL = 1/2:
1 V0 . (15.3.12) f (V ) = f1/2 (V ) ∝ 3/2 exp − 4V V Note that power tails often appear in the kinetic consideration of suprathermal electrons anomalous transport in plasma [284, 285]. For strongly nonequilibrium systems in the presence of accelerating and trapping mechanisms, it is necessary to take into account nondiffusive effects, which lead to long-tail distribution in the phase space.
15.4 Correlations and Nonlocal Velocity Distribution From the common standpoint, the considerable distortion of the Maxwellian particle velocity distribution function leads to the difference between the anomalous transport equation and the classical diffusion equation. Thus, the ballistic particle motion can be interpreted as trapping in phase space, since if collisions (interactions) are absent, then the particle has constant velocity and, hence, does not change its position in the velocity space. The effectiveness of the one-flight approximation [34, 287] indicates the possibility of using the ballistic character of motion for obtaining simple scaling estimates. The probability Φ(t) of avoiding an “event” (capture by a trap, for example) during time t can become the basis for building a simple kinetic model. It is possible to represent this probability as a result of averaging over the aggregate of particles with the velocity distribution function f (V ).
250
15 Correlation and Phase-Space
Indeed, if the characteristic distance between traps is R0 , then the probability of avoiding trapping for a particle with the velocity V could be the estimate in the form: w(V,t) ≈ 1 −
Vt t ≈ 1− . T R0
(15.4.1)
Here, T = R0 /V is the characteristic time for the particle with the velocity V to reach a trap. This linear estimate can be interpreted as decomposition over the small parameter V t/R0 << 1 of the conventional Poisson probability of avoiding an “event” during time t:
Vt Vt ≈ 1− . (15.4.2) exp − R0 R0 In the framework of the randomization method, we can obtain the integral: Φ(t) =
V t
w(V,t) f (V ) dV .
(15.4.3)
0
Note that the upper limit of the integral is not constant: V < R0 /t. Then, the simple transformations of this expression yield the functional equation: Φ(t) =
R 0 /t
1−
0
Vt R0
f (V ) dV .
(15.4.4)
Following the Montroll-Weiss ideas [6, 7, 10], we can assume that the function Φ(t) is known or can be approximated by one of the characteristic probabilistic distributions. Then, it is possible to solve this integral equation. Using a double differentiation over t, we obtain the equation for the definition of the distribution f (V ). Thus, after the first differentiation of the functional, we obtain: d Φ(t) = −ψ (t) = − dt
R 0 /t 0
V f (V ) dV . R0
(15.4.5)
Here, we take into account that the upper limit of the integral under consideration depends on time: V = R0 /t. The next differentiation yields: f (V ) =
R0 2 d 2 R0 2 d Φ(t) |t= R0 = − 3 ψ (t) |t= R0 . 3 2 V dt V dt V V
(15.4.6)
It is a very simple and at the same time nontrivial relationship that connects kinetic and probabilistic characteristic functions. One can apply different characteristic functions for Φ(t). In the first case, we assume that Φ(t) has the exponential Poisson form, which corresponds to the absence of memory effects [3, 4]:
15.4 Correlations and Nonlocal Velocity Distribution
251
t . Φ(t) = exp − τ0
(15.4.7)
Here, τ is the characteristic time that is the parameter of the problem. Conventional averaging yields the mean waiting time: t =
∞
t ψ (t) dt = τ0 ,
(15.4.8)
0
where
ψ (t) = −
1 d t Φ(t) = exp − . dt τ0 τ0
(15.4.9)
As a result of simple calculations for the Poisson case, we obtain the expression for the velocity distribution function f (V ), which depends on two parameters, R0 and τ : 1 f (V ) = 3 V
R0 τ0
2
R0 . exp − V τ0
(15.4.10)
If values of V are larger, one deals with the scaling: f (V ) ∝
1 . V3
(15.4.11)
The exponential factor of this function is nonanalytical with V → 0 and therefore it cannot be obtained by the asymptotic technique from the diffusive phase-space Fokker-Planck equation. In the framework of the scaling approach, we consider the representation of the probability Φ(t) in the power form [6, 7]: 1
Φ(t) =
1 + τt0
γ .
(15.4.12)
Here, γ < 1 is the characteristic exponent and τ0 is the characteristic time. In the continuous time random walk approach, this corresponds to the consideration of memory effects. However, in our case it is important that the mean waiting time t is an infinite value [6, 7]. On substituting (15.4.12) into (15.4.6), we obtain the velocity distribution function [290]:
γ 1 τ0 1 ∝ β . (15.4.13) f (V ) ≈ γ (γ + 1) 1− γ R0 V V K This yields the relationship between both the kinetic exponent βK and the waiting time exponent γ : βK = 1− γ . Note that this distribution leads to the fractal character of collisionless particle velocity space:
δ Np (V ) ∝ f (V )δ V ∝ V βK
(15.4.14)
252
15 Correlation and Phase-Space
with the dimensionality dF = βK . Here, δ Np is the number of collisionless particles in a small element δ V of velocity space. These relations characterize systems under the unconventional condition t = ∞, which differs significantly from the conventional Poisson model, where the mean waiting time t is finite.
15.5 The Corrsin Conjecture and Phase-Space Vlad, Spineanu, Misguich, and Balescu analyzed the trapping of test particles using both the model representation of streamlines and the modification of the Corrsin conjecture [291]. In principle, it is possible to keep the Corrsin factorization (4.1.6) but to modify considerably the trajectory ensemble under consideration. Indeed, the subaggregate of trajectories in which the resulting displacement corresponds to some fixed value λ could be described as follows: V (x(0), 0)V (x(t),t) |X(t)=λ .
(15.5.1)
Let us consider a system of subaggregates, where the value of the initial velocity is fixed in each such system: CV (V0 ,t) = V (x(0), 0)V (x(t),t) |X (t)=V0 = V0 V (x(t),t) |X (t)=V0 .
(15.5.2)
On the one hand, this model leads to averaging of subaggregates over the velocity with the kinetic distribution f (V0 ,t): ∞
f (V0 ,t)CV (V0 ,t)dV0 .
C(t) =
(15.5.3)
−∞
On the other hand, one can analyze in more detail the streamline behavior. This is not surprising because for the new conditions the approximation of expression (15.5.2) by means of the specially chosen function VC (x,t) permits us to investigate the specific system of trajectories in the framework of the equation: dx = VC (x,t) = V (x(t),t) |V0 . dt
(15.5.4)
The option of an approximating function defines the character of trajectories and enables one to “visualize” correlation effects, which in this formulation of the problem are also determined by the expression for VC (x,t). In this approach the value V0 VC (x,t) replaces the expression for the Eulerian correlation function CE (x,t). The simplest example of the approximation VC (x,t) was examined in [292, 293] in the form that allows one to easily solve Eq. (15.5.4) and at the same time to satisfy the trapping character of transport:
x t exp − . (15.5.5) VC (x,t) = V02 exp − λ τ0
15.5 The Corrsin Conjecture and Phase-Space
253
Here, λ is the characteristic spatial scale and τ0 is the characteristic time. Simple transformations of Eq. (15.5.4) yield the expression for the displacement:
|V0 | τ0 t 1 − exp − . (15.5.6) x(V0 ,t) = sign(V0 )λ ln 1 + λ τ0 The trapping effects are described by the finite value of the tracer displacement: # τ0 $ ≈ λ ln (1 + Ku) . (15.5.7) x(t → ∞) ≈ λ ln 1 + |V | λ One can calculate the correlation function (15.5.3) and the diffusion coefficient using the one-dimensional Maxwellian distribution:
1 V0 2 (15.5.8) exp − 2 . f (V0 ) = √ π VT VT Here, VT is the characteristic velocity of the one-dimensional Maxwellian distribution. Calculations yield the turbulent diffusion coefficient:
τ0 DT = √ π VT
∞
dV0 −∞
V0 2 . exp − VT2 (1 + |V0 | τ0 /λ )2 V02
(15.5.9)
The results of the calculations allow us to obtain the quasilinear expression DT ≈ V02 τ0 ∝ Ku2 for the case Ku << 1, and “flat” scaling DT ∝ KuσT with σT ≈ 0 for the long-range correlation case Ku >> 1. Here, Ku = V0λτ0 is the Kubo number. The approach considered was subsequently developed in a number of papers [291, 292, 293] where the two-dimensional model is analyzed in the framework of the special Hamiltonian function Ψ (the stream function) characterizing the trapping in the flow under consideration: → → dx → → → = −∇Ψ( x,t) × e Z = V ( x(t),t) |V0 ,Ψ0 . dt
(15.5.10)
Here, Ψ0 is the initial value of the streamline function. Streamlines obtained in this approach are closed curves, except the single streamline, which is a straight line along the initial velocity V0 (see Fig. 15.3). In other words, the particles moving
Fig. 15.3 Model streamline function that consists of closed curves, except the single streamline, which is a straight line along the initial velocity
254
15 Correlation and Phase-Space
along closed streamlines do not make an essential contribution to the transport, if considered trajectories are localized. This case is opposite to the percolation one in which a percolation streamline embraces almost the whole plane. The universal character of the dependence of the effective diffusion coefficient DT on the turbulence strength (the Kubo number) for Ku >> 1, DT ≈ D0 KuσT was found. Here, σT ≈ 0.62 and D0 is the conventional (seed) coefficient of diffusion. This differs from the percolation prediction σT = 7/10. The majority of results in the decorrelation trajectory approach were obtained by means of simulations, which is natural for turbulent transport modeling the complex vortex structures. Scaling theory offers an interpretive framework for simulation, while numerical experiments test turbulence theory and can inspire new developments. In this last chapter, an attempt is made to provide a quantitative explanation of anomalous transport in terms of kinetic description. Of course, some of the conclusions are speculative and further analysis is necessary to validate or disprove the claims made, since the correspondence with the experimental results may occur for the wrong reasons, as happens from time to time in the field of turbulence.
Further Reading The Fokker-Planck Equation Chandrasekhar, S. (1943). Reviews of Modern Physics, 15, 1. 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., Borkovec, M., and Talkner, P. (1990). Reviews of Modern Physics, 62, 251. 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.
Phase-Space Approach and Transport Eyik, G.L. and Sreenivasan, K.R. (2006). Reviews of Modern Physics, 78, 87. 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. Pekalski, A. and Sznajd-Weron, K., eds. (1999). Anomalous Diffusion. From Basics to Applications. Springer-Verlag, Berlin. Zaslavsky, G.M. (2002). Physics Reports, 371, 461–580.
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Index
Advection, 27, 34–36, 75–77 Advection-diffusion equation, 34–36 Advection subrange, 35 Alfven, 205–208 Alfven’s turbulence, 207–208 Anisotropy, 7, 92, 200, 220, 234, 237 Anomalous diffusion, 7, 47–51, 53, 60–62, 92, 98, 110–111, 147, 151, 214, 221–222, 223, 238, 247–249 Atmosphere, 22, 24, 75 Batchelor, 32, 33, 34, 38, 51, 52 Blasius, 27 Bohm, 139, 141, 217 Bohm scaling, 138–141, 217 Boltzmann, 3 Boundary-layer theory, 27 Braded magnetic fields, 10 Broadbent, 157 Brown, 3, 197 Brownian landscape, 196–199 Brownian motion, 3–7, 47, 93, 197, 229 Cantor dust, 147, 148 set, 147, 149–150 Cascade phenomenology, 34–38 Chapman, 43, 225, 226 Continuous time random walk, 223–241 comb structures, 218–220 Complex structures, 10, 77, 135–136, 140 Complex systems, 87, 118, 131, 221, 230, 246 Convective cells, 10, 66, 115, 131–143, 170, 172, 190 Correlation length, 6, 13, 18, 77, 92, 108–109, 111–114, 120, 122, 124, 126, 127, 140,
151, 157–158, 162, 163–164, 172, 174, 179, 190, 207, 217, 218, 221 Correlation mechanisms, 10, 108, 117, 118, 190 Correlation scale, 3–18, 62, 63, 65, 69, 77, 108, 109, 122, 126, 127, 139–140, 165, 169, 171–173, 175, 177, 180–181, 183, 185, 188, 191, 193, 206–208, 215–216, 220–221 Correlation time, 6, 11, 16–18, 23, 64–65, 77, 88, 96, 109, 112, 122, 124–126, 131, 133–134, 139–142, 151, 162, 169, 173, 174, 178, 182, 184–185, 190, 202, 214, 216–217 Corrsin conjecture, 58–59, 60–62, 66, 218, 222, 252–254 independence hypothesis, 57–69 Critical exponent, 157, 158, 165 Davydov, 11 Diffusion coefficient, 5–8, 13, 16–18, 21, 24, 31–33, 42, 44, 57–58, 63–67, 71, 74, 76–77, 79, 81, 83, 87, 90–91, 105, 107–109, 111, 114–115, 121–122, 124–127, 132, 133, 135, 139–140, 142, 151, 162, 164–165, 169, 174–175, 178–182, 183, 184–186, 189–191, 201–202, 207, 213–214, 217–219, 221, 229, 233, 235, 237, 245, 247, 253–254 equation, 8–13, 18, 26, 33–35, 38, 41–53, 58, 71–86, 104, 201, 235, 243–245 Diffusive motion, 3, 117, 213 Dissipation range, 29 Dissipation rate, 27–28, 34–35, 38, 51, 154, 156, 205–206
263
264 Double diffusion, 112, 117, 124–125, 200, 214–215, 220–222, 233–234, 235 Einstein, 14, 41, 43, 44, 51, 223, 226–227 Energy-containing scales, 28 Energy spectrum, 30–31, 36–37, 58, 64, 155–156, 204, 232 Ensemble, 63, 81, 94, 98, 106, 218, 252 Escape probability, 213–222, 239, 246, 248 Euclidean dimension, 148 Eulerian correlation, 59, 60–61, 237 Fick law, 8 Field-line random walk, 107 Fluctuation-dissipative relation, 136–138 Fluctuations, 10, 13, 15, 34, 82, 90–91, 101–102, 106, 108–109, 136, 138–142, 185–186 Fluid turbulence, 104, 129 Flux-gradient relation, 11 Fractal, 34, 47, 48, 50, 99, 136, 147–166, 171–172, 182–183, 190, 196–198, 214–216, 232, 251 Fractal dimension, 147–151, 152, 153–154, 156, 158–159, 166, 197–198, 232 Fractional derivatives, 13, 47–51, 52 Fractional differential equation, 220, 226–228, 230–231, 234–236 Fully developed turbulence, 38 Gaussian distribution, 9, 45, 52–53, 58, 62, 220 Graded percolation, 165–166, 194, 198, 199 Hamiltonian function, 161, 246, 253 Hammersley, 157 Hausdorff dimension, 148 Hierarchy of scales, 33, 128, 172–173, 178–181, 183–184, 193–195, 203–204, 207 Howells, 64–65, 66–68, 79, 115, 237 Hurst exponent, 7, 22–23, 46, 61–62, 89–91, 96–98, 112, 152, 196–197, 200, 202–203, 216, 230, 233–234, 238, 239, 247, 248 Inertial range, 28, 30, 36, 38, 47 Ingenhousz, 3 Intermittency, 34, 154, 232 Internal-convective subrange, 35 Isotropic turbulence, 22, 31, 51, 54, 153–154, 194, 204 Kinetic equation, 13, 243–244, 246 Kohlrausch, 214, 224, 248–249
Index Kohlrausch relaxation function, 224 Kolmogorov, 204–205, 225–226, 232 Kolmogorov microscale, 36 Kolmogorov phenomenology, 137, 204–205 Kolmogorov spectrum, 38 Kraichnan, 58, 205 Kramers, 243–244 Kubo number, 77–80, 110, 119, 121–122, 125, 139, 141, 175, 177, 188–189, 203, 217–218, 253–254 Lagrangian correlation, 15–18, 21, 23, 57–59, 61, 64, 68, 229–230, 247–248 Lagrangian correlation functions, 21, 58, 68 Lagrangian description, 13–18 Lagrangian particle, 13–15, 17–18 Lagrangian statistics, 33 Lagrangian velocity, 14, 22, 59 Langevin, 14, 21, 229 Levy, 34, 41, 45, 50, 53, 248, 249 Levy flight, 46, 53, 221 Levy-Khintchine distribution, 41 Levy-stable distribution, 44–46 Levy walk, 34, 231 Liouville derivatives, 48 Longitudinal correlations, 235–238 Long-range correlations, 48, 62, 83–84, 110, 157, 169, 189 Long tails, 63 Lyapunov’s exponent, 118 Magnetic diffusion coefficient, 105, 107–108, 111, 114–115, 122, 221 Magnetohydrodynamics, 106, 129, 204–208 Magnetohydrodynamic turbulence, 129, 204–208 Mandelbrot, 147, 150, 155, 202 Markovian processes, 227 Mean displacement, 4, 94, 247 Memory effects, 12–13, 38, 44, 48, 75, 223, 225–227, 230–231, 233, 243, 251 Moffat, 64, 68 Molecular diffusivity, 18, 34 Monin, 51–53, 231 Navier-Stokes, 26–27, 38 Nested hierarchy, 193–195 Newtonian fluid, 26–27 Nondiffusive kinetic effects, 246 Nonlocal effects, 41–53 Obukhov, 26, 30, 31, 33, 35–36, 156 One-flight model, 247–249 Organized structures, 129
Index Passive scalar transport, 10, 109, 113 Peclet number, 64–66, 73, 74, 76, 132–143, 136–138, 170–172, 175 Percolation regime, 124, 172, 175, 178, 215 Percolation threshold, 158, 160–163, 165 Percolation transition, 157–160 Percolation transport, 95, 180, 184, 186, 200, 204 Perrin, 3 Phase space, 81, 214, 218, 222, 243–254 Phase transition, 157, 172, 247 Plasma, 81, 101–108, 111, 117, 119, 123, 127, 129, 131, 138–141, 143, 204, 205, 208, 221, 246, 249 Plasma turbulence, 104, 106, 139, 246 Poiseuille flow, 71 Poisson distribution, 213, 223, 226–227 Prandtl number, 36 Random walks, 3–18, 24, 87–89, 99, 107, 108, 111–112, 120, 122, 151–152, 156, 196–197, 215, 223–241 Rayleigh-Benard convective rolls, 134 Reconnection of streamlines, 7 Relative diffusion, 23, 31–33, 41, 52, 231, 245 Renormalization, 57–69, 74–76, 90, 92, 112–115, 141–143, 162–165, 169–172, 175–177, 179, 182–185, 187–189, 190–191, 194, 206, 214, 219, 226, 233, 235–236 Return effects, 87–99 Reynolds number, 27, 28, 30, 33, 36, 65, 104–106, 137 Richardson, 23–26, 31–34, 51–52, 156, 231–232, 245 Root-mean-square displacement, 3, 5, 6, 88 Scalar spectrum, 34–38 Seed diffusion, 7, 57, 68, 75, 81, 90, 122, 132, 178, 213, 235 Self-avoiding random walks, 89, 152, 156, 197 Self-similar, 52–53, 147–151, 158, 196 Shear flow, 37, 62, 71, 75, 87–98, 125, 165, 200–201, 235, 237–238 Sierpinski carpet, 158, 159 Smoluchowski, 43, 51, 223, 225–226 Sticky island, 246
265 Sticky island boundary, 246 Stochastic instability, 7, 112, 117–129, 178, 187–189, 203–204 Stochastic instability increment, 119–122, 187–189, 203 Stochastic magnetic field, 68, 101, 104, 106–112, 117, 119, 122, 124, 127–128, 189–191, 206–208, 220–221, 233–235 Strong turbulence, 18, 137–138, 215, 217–218 Subdiffusion, 161, 213–222 Superdiffusion, 7, 90–92, 96–97, 202 Suprathermal particles, 246 System of zonal flows, 10 Taylor, 14–16, 21–22, 26, 57, 66, 68, 71–77, 81, 83–84, 92, 109, 142, 229, 237, 247 Taylor dispersion, 71–75, 237 Telegraph equation, 11–13, 42, 44, 52, 84–86, 226, 245 Temperature spectrum, 36 Transport models, 7, 180, 218–220 Transverse correlations, 112, 115 Trapping, 213–222, 223, 238–239, 246, 249–250, 252–253 Turbulent diffusion, 6–7, 11–13, 15–18, 21–38, 44, 51, 53, 57, 64–65, 77, 92, 101, 107, 109, 114, 117, 122, 131–135, 138, 142, 160, 164, 166, 169, 175–176, 180–181, 203, 216, 229, 233, 247, 253 Turbulent mixing, 11, 24, 136–137, 140, 208 Turbulent transport, 7, 13, 16, 18, 21, 23, 32, 38, 51, 58, 62, 64, 66, 68, 74, 77, 80, 81, 84, 95–98, 101, 104, 106, 107, 115, 122, 138, 151, 153, 169–191, 196, 205, 208, 228, 239, 245, 255 Velocity distribution, 93, 98, 245, 249–252 Velocity fluctuations, 11, 13, 15, 136 Velocity spectrum, 37 Viscous-convective subrange, 37–38 Vlasov equation, 102–103 Vortex structures, 213, 238–240, 254 Waiting time distribution, 228–230, 238 Well-developed turbulence, 26 Wiener process, 47
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