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Kolmogorov Spectra of Turbulence I: Wave Turbulence PDF

274 Pages·1992·14.825 MB·English
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I I I N D Springer Series in Nonlinear Dynamics I I I N D Springer Series in Nonlinear Dynamics Series Editors: F. Calogero, B. Fuchssteiner, G. Rowlands, H. Segur, M. Wadati and V. E. Zakharov Solitons - Introduction and Applications Editor: M. Lakshmanan What Is Integrability? Editor: V E. Zakharov Rossby Vortices and Spiral Structures By M. V Nezlin and E. N. Snezhkin Algebro-Geometrical Approach to Nonlinear Evolution Equations By E. D. Belokolos, A.!, Bobenko, V Z. Enolsky, A. R. Its andY B. Matveev Darboux Transformations and Solitons By V B. Matveev and M. A. Salle Optical Solitons By F. Abdullaev, S. Darmanyan and P. Khabibullaev Wave Turbulence Under Parametric Excitation Applications to Magnetics By V.S. L'vov Kolmogorov Spectra of Turbulence I Wave Turbulence By V E. Zakharov, V S. L'vov and G. Falkovich V.E. Zakharov V.S. ~vov G. Falkovich Kolmogorov Spectra of Turbulence I Wave Turbulence With 34 Figures Springer -Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Professor Dr. Vladimir E. Zakharov Landau Institute tor Theoretical Physics. Russian Academy of Sciences, ul. Kosygina 2, 117334 Moscow, Russia Professor Dr. Victor S. L'vov Dr. Gregory Falkovich Physics Department, Weizmann Institute of Science, 76100 Rehovot, Israel ISBN 978-3-642-50054-1 ISBN 978-3-642-50052-7 (eBook) DOI 10.1007/978-3-642-50052-7 l.ibrary of Congress Cataloging·in-Publication Data. Zakharov, Vladimir Evgen'evich. Kolmogorov spectra of turbulence 1 V. E. Zakharov, V.S. L'vov, G. Falkovich. p. cm. - (Springer series in nonlinear dynamics) Includes bibliographical references (v. I, p. ) and index. Contents: [1] Wave turbulence ISBN 3·540-54533-6 (Springer-Verlag Berlin Heidelberg New York: acid-free paper). - ISBN 0-387-54533-6 (Springer-Verlag New York Berlin Heidelberg: acid-free paper) I. Turbulence-Spectra. 2. Waves. 3. Spectrum analysis. 4. Nonlinear theories. I. L'vov, V. S. (Victor Sergeevich), 1942-. II. Falkovich, G. (Gregory), 1958-. III. Title. IV. Series. QC1S7.Z35 1992 532'.OS27-dc20 92-4801 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is con cerned, 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-Verlag. Violations are liable for prosecution under the German Copyright Law. o Springer-Verlag Berlin Heidelberg 1992 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. Typesetting: Data conversion by Springer-Verlag 57/3140· 5 43 210 - Printed on acid-free paper In memory of A. N. Kolmogorov and A. M. Obukhov Preface Since the human organism is itself an open system, we are naturally curious about the behavior of other open systems with fluxes of matter, energy or information. Of the possible open systems, it is those endowed with many degrees of freedom and strongly deviating from equilibrium that are most challenging. A simple but very significant example of such a system is given by developed turbulence in a continuous medium, where we can discern astonishing features of universality. This two-volume monograph deals with the theory of turbulence viewed as a general physical phenomenon. In addition to vortex hydrodynamic turbulence, it considers various cases of wave turbulence in plasmas, magnets, atmosphere, ocean and space. A sound basis for discussion is provided by the concept of cascade turbulence with relay energy transfer over different scales and modes. We shall show how the initial cascade hypothesis turns into an elegant theory yielding the Kolmogorov spectra of turbulence as exact solutions. We shall describe the further development of the theory discussing stability prob lems and modes of Kolmogorov spectra formation, as well as their matching with sources and sinks. This volume is dedicated to developed wave turbulence in different media. It contains a detailed exposition so that the reader can use it as an introductory textbook on wave turbulence theory. Moreover, it also provides an introduction to the general theory of developed turbulence, since wave turbulence at low ex citation level is closely related to the Richardson-Kolmogorov-Obukhov cascade picture. In the second volume developed turbulence of incompressible fluids will be described. This text is based on lecture courses given at the Arizona, Chicago and Novosibirsk Universities, at the Moscow Institute of Space Researches, and the Weizmann Institute of Science in Rehovot The book is useful for specialists in hydrodynamics, plasma and solid-state physics, meteorology, and astrophysics. We also hope it will prove instructive for students and young researchers starting their academic careers with studies of the problem of turbulence. June 1992 Vladimir Zakharov Victor L'vov Gregory Falkovich Acknowledgements We acknowledge the valuable remarks on Sects. 3.3 and 4.2 by Dr. A.M. Balk, and a number of useful recommendations by Dr. A.V. Shafarenko for improving the presentation of the material in this book. Contents O. Introduction 1 1. Equations of Motion and the Hamiltonian Formalism 9 1.1 The Hamiltonian Formalism for Waves in Continuous Media ............................... . 9 1.1.1 The Hamiltonian in Normal Variables ........... . 9 1.1.2 Interaction Hamiltonian for Weak Nonlinearity ... . 15 1.1.3 Dynamic Perturbation Theory. Elimination of Nonresonant Terms ............. . 18 1.1.4 Dimensional Analysis of the Hamiltonian Coefficients ................ . 21 1.2 The Hamiltonian Formalism in Hydrodynamics .......... . 25 1.2.1 Clebsh Variables for Ideal Hydrodynamics ....... . 25 1.2.2 Vortex Motion in Incompressible Fluids ......... . 29 1.2.3 Sound in Continuous Media ................... . 29 1.2.4 Interaction of Vortex and Potential Motions in Compressible Fluids ....................... . 31 1.2.5 Waves on Fluid Surfaces ..................... . 33 1.3 Hydrodynamic-Type Systems ........................ . 37 1.3.1 Langmuir and Ion-Sound Waves in Plasma ...... . 37 1.3.2 Atmospheric Rossby Waves and Drift Waves in Inhomogeneous Magnetized Plasmas ......... . 43 1.4 Spin Waves ....................................... . 51 1.4.1 Magnetic Order, Energy and Equations of Motion .. 51 1.4.2 Canonical Variables ......................... . 53 1.4.3 The Hamiltonian of a Heisenberg Ferromagnet ... . 54 1.4.4 The Hamiltonian of Antiferromagnets ........... . 56 1.5 Universal Models .................................. . 58 1.5.1 Nonlinear SchrOdinger Equation for Envelopes ... . 59 1.5.2 Kadomtsev-Petviashvili Equation for Weakly Dispersive Waves ................. . 60 1.5.3 Interaction of Three Wave Packets ............. . 61 2. Statistical Description of Weak Wave Thrbulence . . . . . . . . . . . 63 2.1 Kinetic Wave Equation .............................. 63 2.1.1 Equations of Motion .......................... 63 X Contents 2.1.2 Transition to the Statistical Description . . . . . . . . . . 64 2.1.3 The Three-Wave Kinetic Equation . . . . . . . . . . . . . . 66 2.1.4 Applicability Criterion of the Three-Wave Kinetic Equation (KE) ........ 67 2.1.5 The Four-Wave Kinetic Equation ............... 70 2.1.6 The Quantum Kinetic Equation ................. 72 2.2 General Properties of Kinetic Wave Equations ........... 75 2.2.1 Conservation Laws ........................... 75 2.2.2 Boltzmann's H-Theorem and Thennodynamic Equilibrium ............... 78 2.2.3 Stationary Nonequilibrium Distributions .......... 80 3. Stationary Spectra of Weak Wave Thrbulence ..... . . . . . . . . . 83 3.1 Kolmogorov Spectra of Weak Turbulence in Scale-Invariant Isotropic Media ..................... 83 3.1.1 Dimensional Estimations and Self-Similarity Analysis .. . . . . . . . . . . . . . . . . . 84 3.1.2 Exact Stationary Solutions of the Three-Wave Kinetic Equation ............. 86 3.1.3 Exact Stationary Solutions for the Four-Wave Kinetic Equations ............ 93 3.1.4 Exact Power Solutions of the Boltzmann Equation . 101 3.2 Kolmogorov Spectra of Weak Turbulence in Nearly Scale-Invariant Media ....................... 102 3.2.1 Weak Acoustic Turbulence ..................... 102 3.2.2 Media with Two Types of Interacting Waves ...... 108 3.3 Kolmogorov Spectra of Weak Turbulence in Anisotropic Media ................................ 117 3.3.1 Stationary Power Solutions .................... 117 3.3.2 Fluxes of Integrals of Motion and Families of Anisotropic Power Solutions ...... 120 3.4 Matching Kolmogorov Distributions with Pumping and Damping Regions ................... 123 3.4.1 Matching with the Wave Source ................ 124 3.4.2 Influence of Dissipation ....................... 135 4. The Stability Problem and Kolmogorov Spectra . . .. . . . . . . . . 145 4.1 The Linearized Kinetic Equation and Neutrally Stable Modes .......................... 145 4.1.1 The Linearized Collision Tenn . . . . . . . . . . . . . . . . . 145 4.1.2 General Stationary Solutions and Neutrally Stable Modes .................... 147 4.2 Stability Problem for Kolmogorov Spectra of Weak Turbulence ................................. 156 4.2.1 Perturbation of the Kolmogorov Spectrum ........ 160 Contents XI 4.2.2 Behavior of Kolmogorov-Like Turbulent Distributions. Stability Criterion ........ 173 4.2.3 Physical Examples ........................... 184 4.3 Nonstationary Processes and the Formation of Kolmogorov Spectra .............. 190 4.3.1 Analysis of Self-Similar Substitutions ............ 191 4.3.2 Method of Moments .......................... 197 4.3.3 Numerical Simulations ........................ 200 5. Physical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 5.1 Weak Acoustic Turbulence ........................... 207 5.1.1 Three-Dimensional Acoustics with Positive Dispersion: Magnetic Sound and Phonons in Helium 209 5.1.2 Two-Dimensional Acoustics with Positive Dispersion: Gravity-Capillary Waves on Shallow Water and Waves in Flaky Media .................... 218 5.1.3 Nondecay Acoustic Turbulence: Ion Sound, Gravity Waves on Shallow Water and Inertia-Gravity Waves ..................... 227 5.2 Wave Turbulence on Water Surfaces ................... 229 5.2.1 Capillary Waves on Deep Water ................ 229 5.2.2 Gravity Waves on Deep Water .................. 230 5.2.3 Capillary Waves on Shallow Fluids .............. 232 5.3 Turbulence Spectra in Plasmas, Solids, and the Atmosphere 233 5.3.1 Langmuir Turbulence in Isotropic Plasmas ........ 233 5.3.2 Optical Turbulence in Nonlinear Dielectrics and Turbulence of Envelopes ................... 236 5.3.3 Spin Wave Turbulence in Magnetic Dielectrics .... 237 5.3.4 Anisotropic Spectra in Plasmas ................. 239 5.3.5 Rossby Waves ............................... 242 6. Conclusion 245 A. Appendix .............................................. 249 A.1 Variational Derivatives ............................... 249 A.2 Canonicity Conditions of Transformations ............... 250 A.3 Elimination of Nonresonant Terms from the Interaction Hamiltonian ...................... 252 References ................................................. 257 Subject Index ............................................... 263

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