Kolmogorov Spectra of Turbulence I Wave Turbulence

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.

0. Introduction.- 1. Equations of Motion and the Hamiltonian Formalism.- 1.1 The Hamiltonian Formalism for Waves in Continuous Media.- 1.2 The Hamiltonian Formalism in Hydrodynamics.- 1.3 Hydrodynamic-Type Systems.- 1.4 Spin Waves.- 1.5 Universal Models.- 2. Statistical Description of Weak Wave Turbulence.- 2.1 Kinetic Wave Equation.- 2.2 General Properties of Kinetic Wave Equations.- 3. Stationary Spectra of Weak Wave Turbulence.- 3.1 Kolmogorov Spectra of Weak Turbulence in Scale-Invariant Isotropic Media.- 3.2 Kolmogorov Spectra of Weak Turbulence in Nearly Scale-Invariant Media.- 3.3 Kolmogorov Spectra of Weak Turbulence in Anisotropic Media.- 3.4 Matching Kolmogorov Distributions with Pumping and Damping Regions.- 4. The Stability Problem and Kolmogorov Spectra.- 4.1 The Linearized Kinetic Equation and Neutrally Stable Modes.- 4.2 Stability Problem for Kolmogorov Spectra of Weak Turbulence.- 4.3 Nonstationary Processes and the Formation of Kolmogorov Spectra.- 5. Physical Applications.- 5.1 Weak Acoustic Turbulence.- 5.2 Wave Turbulence on Water Surfaces.- 5.3 Turbulence Spectra in Plasmas, Solids, and the Atmosphere.- 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.