Kolmogorov Spectra of Turbulence I: v. 1 Wave Turbulence(Springer Series in Nonlinear Dynamics)

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.

Table of Contents:
0. Introduction.- 1. Equations of Motion and the Hamiltonian Formalism.- 1.1 The Hamiltonian Formalism for Waves in Continuous Media.- 1.1.1 The Hamiltonian in Normal Variables.- 1.1.2 Interaction Hamiltonian for Weak Nonlinearity.- 1.1.3 Dynamic Perturbation Theory. Elimination of Nonresonant Terms.- 1.1.4 Dimensional Analysis of the Hamiltonian Coefficients.- 1.2 The Hamiltonian Formalism in Hydrodynamics.- 1.2.1 Clebsh Variables for Ideal Hydrodynamics.- 1.2.2 Vortex Motion in Incompressible Fluids.- 1.2.3 Sound in Continuous Media.- 1.2.4 Interaction of Vortex and Potential Motions in Compressible Fluids.- 1.2.5 Waves on Fluid Surfaces.- 1.3 Hydrodynamic-Type Systems.- 1.3.1 Langmuir and Ion-Sound Waves in Plasma.- 1.3.2 Atmospheric Rossby Waves and Drift Waves in Inhomogeneous Magnetized Plasmas.- 1.4 Spin Waves.- 1.4.1 Magnetic Order, Energy and Equations of Motion.- 1.4.2 Canonical Variables.- 1.4.3 The Hamiltonian of a Heisenberg Ferromagnet.- 1.4.4 The Hamiltonian of Antiferromagnets.- 1.5 Universal Models.- 1.5.1 Nonlinear Schrodinger Equation for Envelopes.- 1.5.2 Kadomtsev-Petviashvili Equation for Weakly Dispersive Waves.- 1.5.3 Interaction of Three Wave Packets.- 2. Statistical Description of Weak Wave Turbulence.- 2.1 Kinetic Wave Equation.- 2.1.1 Equations of Motion.- 2.1.2 Transition to the Statistical Description.- 2.1.3 The Three-Wave Kinetic Equation.- 2.1.4 Applicability Criterion of the Three-Wave Kinetic Equation (KE).- 2.1.5 The Four-Wave Kinetic Equation.- 2.1.6 The Quantum Kinetic Equation.- 2.2 General Properties of Kinetic Wave Equations.- 2.2.1 Conservation Laws.- 2.2.2 Boltzmann's H-Theorem and Thermodynamic Equilibrium.- 2.2.3 Stationary Nonequilibrium Distributions.- 3. Stationary Spectra of Weak Wave Turbulence.- 3.1 Kolmogorov Spectra of Weak Turbulence in Scale-Invariant Isotropic Media.- 3.1.1 Dimensional Estimations and Self-Similarity Analysis.- 3.1.2 Exact Stationary Solutions of the Three-Wave Kinetic Equation.- 3.1.3 Exact Stationary Solutions for the Four-Wave Kinetic Equations.- 3.1.4 Exact Power Solutions of the Boltzmann Equation..- 3.2 Kolmogorov Spectra of Weak Turbulence in Nearly Scale-Invariant Media.- 3.2.1 Weak Acoustic Turbulence.- 3.2.2 Media with Two Types of Interacting Waves.- 3.3 Kolmogorov Spectra of Weak Turbulence in Anisotropic Media.- 3.3.1 Stationary Power Solutions.- 3.3.2 Fluxes of Integrals of Motion and Families of Anisotropic Power Solutions.- 3.4 Matching Kolmogorov Distributions with Pumping and Damping Regions.- 3.4.1 Matching with the Wave Source.- 3.4.2 Influence of Dissipation.- 4. The Stability Problem and Kolmogorov Spectra.- 4.1 The Linearized Kinetic Equation and Neutrally Stable Modes.- 4.1.1 The Linearized Collision Term.- 4.1.2 General Stationary Solutions and Neutrally Stable Modes.- 4.2 Stability Problem for Kolmogorov Spectra of Weak Turbulence.- 4.2.1 Perturbation of the Kolmogorov Spectrum.- 4.2.2 Behavior of Kolmogorov-Like Turbulent Distributions. Stability Criterion.- 4.2.3 Physical Examples.- 4.3 Nonstationary Processes and the Formation of Kolmogorov Spectra.- 4.3.1 Analysis of Self-Similar Substitutions.- 4.3.2 Method of Moments.- 4.3.3 Numerical Simulations.- 5. Physical Applications.- 5.1 Weak Acoustic Turbulence.- 5.1.1 Three-Dimensional Acoustics with Positive Dispersion: Magnetic Sound and Phonons in Helium.- 5.1.2 Two-Dimensional Acoustics with Positive Dispersion: Gravity-Capillary Waves on Shallow Water and Waves in Flaky Media.- 5.1.3 Nondecay Acoustic Turbulence: Ion Sound, Gravity Waves on Shallow Water and Inertio-Gravity Waves.- 5.2 Wave Turbulence on Water Surfaces.- 5.2.1 Capillary Waves on Deep Water.- 5.2.2 Gravity Waves on Deep Water.- 5.2.3 Capillary Waves on Shallow Fluids.- 5.3 Turbulence Spectra in Plasmas, Solids, and the Atmosphere.- 5.3.1 Langmuir Turbulence in Isotropic Plasmas.- 5.3.2 Optical Turbulence in Nonlinear Dielectrics and Turbulence of Envelopes.- 5.3.3 Spin Wave Turbulence in Magnetic Dielectrics.- 5.3.4 Anisotropic Spectra in Plasmas.- 5.3.5 Rossby Waves.- 6. Conclusion.- A. Appendix.- A.1 Variational Derivatives.- A.2 Canonicity Conditions of Transformations.- A.3 Elimination of Nonresonant Terms from the Interaction Hamiltonian.- References.