Table Of ContentI
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
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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
