Nonlinear Waves and Weak Turbulence

This book is an outgrowth of the NSF-CBMS conference Nonlinear Waves £3 Weak Turbulence held at Case Western Reserve University in May 1992. The principal speaker at the conference was Professor V. E. Zakharov who delivered a series of ten lectures outlining the historical and ongoing developments in the field. Some twenty other researchers also made presentations and it is their work which makes up the bulk of this text. Professor Zakharov's opening chapter serves as a general introduction to the other papers, which for the most part are concerned with the application of the theory in various fields. While the word "turbulence" is most often associated with f:l. uid dynamics it is in fact a dominant feature of most systems having a large or infinite number of degrees of freedom. For our purposes we might define turbulence as the chaotic behavior of systems having a large number of degrees of freedom and which are far from thermodynamic equilibrium. Work in field can be broadly divided into two areas: . The theory of the transition from smooth laminar motions to the disordered motions characteristic of turbulence. . Statistical studies of fully developed turbulent systems. In hydrodynamics, work on the transition question dates back to the end of the last century with pioneering contributions by Osborne Reynolds and Lord Rayleigh.

Contenu

I Hamiltonian Systems.- 1 Turbulence in Hamiltonian Systems.- 1.1 Introduction.- 1.2 Some examples.- 1.3 Classical hydrodynamic turbulence.- 1.4 Order from chaos.- 1.5 Bibliography.- 2 Revised Universality Concept in the Turbulence Theory.- 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.- 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.- 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.- 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 Schrödinger Equation.- 6.1 Introduction.- 6.2 Gravity waves: Hopf formulation.- 6.3 Statistical steady states.- 6.4 Cubic Schrödinger equation.- 6.5 Rossby waves: statistical steady states Ill.- 6.6 Readability.- 6.7 Conclusion.- 6.8 Bibliography.- 7 Hidden Symmetries of Hamiltonian Systems over Holomorphic Curves.- 7.1 Introduction.- 7.2 Hidden Hamiltonians.- 7.3 Linear 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.- 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.- 9.1 Introduction.- 9.2 Analysis of outer linear flow.- 9.3 Critical layer analysis.- 9.4 Mean flow change.- 9.5 Pure oblique mode interaction.- 9.6 Pure parametric resonance interaction.- 9.7 Parametric resonance.- 9.8 Fully interactive case.- 9.9 Bibliography.- 10 Modeling Turbulence by Systems of Coupled Gyrostats.- 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.- 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.- 12.1 Introduction.- 12.2 Hamiltonian structure.- 12.3 Soliton-like solutions.- 12.4 Fokker-Plank equation.- 12.5 Appendix.- 12.6 Bibliography.- 13 Collision Dynamics of Solitary Waves in Nematic Liquid Crystals.- 13.1 Introduction.- 13.2 Collision of walls.- 13.3 Nonlinear diffusion equation.- 13.4 Discussion.- 13.5 Bibliography.- IV Statistical Problems.- 14 Statistical Mechanics, Euler's Equation, and Jupiter's Red Spot.- 14.1 Introduction.- 14.2 Vorticity field and Hamiltonian.- 14.3 Thermodynamic formalism.- 14.4 Thermodynamics of the vorticity field.- 14.5 Examples and generalizations.- 14.6 Dressed vorticity corollary.- 14.7 Toy model for Euler equation.- 14.8 Bibliography.- 15 Stochastic Burgers' Flows.- 15.1 Nondispersive waves.- 15.2 Exact solutions.- 15.3 Propagation of chaos.- 15.4 Scaling limits.- 15.5 Maximum principle.- 15.6 Statistics of shocks.- 15.7 Gravitational instability.- 15.8 Bibliography.- 16 Long Range Prediction and Scaling Limit for Statistical Solutions of the Burgers' Equation.- 16.1 Introduction.- 16.2 Preliminaries.- 16.3 A general scaling limit result.- 16.4 Shot noise initial data.- 16.5 Non-Gaussian scaling limits.- 16.6 Bibliography.- 17 A Remark on Shocks in Inviscid Burgers' Turbulence.- 17.1 Introduction.- 17.2 Hausdorff dimension of shock points.- 17.3 Bibliography.