Nonlinear Waves and Weak Turbulence with Applications in Oceanography and Condensed Matter

The classical concept of turbulence is most often associated with fluid dynamics. However, it is in fact a dominant feature of most systems having a large or infinite number of degrees of freedom. In demonstration of this fact, the current volume covers topics such as acoustics, optics, and Jupiter's red spot, as wen as traditional hydrodynamics. The emphasis of the volume is on applications of the relatively new theory of weak turbulence. 'nis theory, which has been developed largely in the last twenty five years, anows for the existence of a multiplicity of linearly unstable modes interacting in a nonlinear "soup." It makes many intriguing connections to such topics as Hamiltonian mechanics, nonlinear parties, equations and integrable systems, stochastic analysis, and methods developed in quantum field theory. Most of the contributions in this book aim at finding and applying the proper mathematical and statistical tools to describe fully developed turbulence. These diverse applications serve to illustrate the power of a unified approach based for the most part on a Hamiltonian formulation. A few chapters address a class of stochastic nonlinear nondispersive waves known as Burger:e turbulence. Set into historical context by V. E. Zakharov's opening chapter, the contributions to this book will be of interest to research workers and graduate students in pure and applied mathematics, theoretical physics, fluid mechanics, oceanography, and various areas of engineering.

Table of Contents:
Introduction. Authors. I HAMIELTONIIAN SYSTEMS. 1 Turbulence in Hqmiltonian Systems by V. V. Zakharov. 1.1 Introduction. 1.2 Some exqmples. 1.3 Classical hydrodynamic turbulence. 1.4 Order from chaos. 1.5 Bibliography. 2 Revised Universality Concept in the Turbulence Theory by G.E. Falkovich. 2.1 Steady spectra and their instabilities. 2.2 Multi-flux spectra. 2.3 Four-wave case. 2.4 Turbulence of incompressible fluid. 2.5 Summary. 2.6 Bibliography. 3 Wave Spectra of Developed Seas by R.E.Glazman 3.1 Introduction. 3.2 Buoy observations of developed seas. 3.3 Shape of the wave spectrum. 3.4 Spatially inhomogeneous wave field. 3.5 Effect of energy and action advection. 3.6 Gravity wave turbulence. 3.7 Conclusions. 3.8 Bibliography. 4 Gravity Waves in the Large Scales of the Atmosphere by J. Herring 4.1 Introduction. 4.2 Stratified vs. 2-D turbulence. 4.3 Physics of 2-D turbulence. 4.4 Numerical experiments. 4.5 Concluding comments. 4.6 Bibliography. 5 Physical Applications of Wave Turbulence: Wind Waves and Classical Collective Modes by A. Larraza. 5.1 Introduction. 5.2 Scaling for wave turbulence. 5.3 Collective modes. 5.4 Experimental Perspectives. 5.5 Bibliography. 6 Strong and Weak Turbulence for Gravity Waves and the Cubic Schrodinger Equation by H.H. Shen. 6.1 Introduction. 6.2 Gravity waves: Hopf formulation. 6.3 Statistical steady states. 6.4 Cubic Schr6dinger equation. 6.5 Rossby waves: statistical steady states. 6.6 Realizability. 6.7 Conclusion. 6.8 Bibliography. 7 Hidden Synunetries of Hamiltonian Systems over Holomorphic Curves by S.J. Alber. 7.1 Introduction. 7.2 Hidden Hamiltonians. 7.3 Lineax Hamiltonian flows. 7.4 Linear collections of curves. 7.5 Triangular collections of curves. 7.6 Multiparameter and discrete systems. 7.7 Vector bundles of Hamiltonian algebras. 7.8 Bibliography. II FLOW STABILITY. 8 Chaotic Motion in Unsteady Vortical Flows by J. J. Li 8.1 Introduction. 8.2 Vortex triplet. 8.3 Resulting chaotic motion. 8.4 Concluding remarks. 8.5 Bibliography. 9. Oblique Instability Waves in Nearly Parallel Shear Flows by M.E. Goldstein and S.S. Lee. 9.1 Introduction. 9.2 Analysis of outer lineax flow. 9.3 Critical layer analysis. 9.4 Mean flow change. 9.5 Pure oblique mode interaction. 9.6 Pure paxametric resonance interaction. 9.7 Parametric resonance. 9.8 Fully interactive case. 9.9 Bibliography. 10 Modeling Turbulence by Systems of Coupled Gyrostats by A. Gluhovsky. 10.1 Introduction. 10.2 Volterra gyrostat. 10.3 Coupled gyrostats in GFD problems. 10.4 Cascade of gyrostats. 10.5 Conclusion. 10.6 Bibliography. III NONLINEAR WAVES IN CONDENSED - MATTER 11 Soliton Turbulence in Nonlinear Optical Phenomena by A.B. Aceves. 11.1 Introduction. 11.2 Governing equations and dynamics. 11.3 Soliton-like solutions. 11.4 Bibliography. 12 Solitons Propagation in Optical Fibers with Random Parameters by D. Gurarie and P. Mishnayevskly. 12.1 Introduction. 12.2 Hamiltonian structure. 12.3 Soliton-like solutions.