Nonlinear waves and weak turbulence with applications in oceanography and condensed matter physics

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Nonlinear waves and weak turbulence with applications in oceanography and condensed matter physics

N. Fitzmaurice ... [et al.], editors

（Progress in nonlinear differential equations and their applications / editor, Haim Brezis, v. 11）

Birkhäuser, c1993

  • : us
  • : gw

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Based on a conference held at Case Western Reserve University in May 1992

Includes bibliographical references

Description and Table of Contents

Volume

: us ISBN 9780817636678

Description

This book is an outgrowth of the NSF-CBMS conference Nonlinear Waves GBP3 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.

Table of Contents

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 Schroedinger Equation.- 6.1 Introduction.- 6.2 Gravity waves: Hopf formulation.- 6.3 Statistical steady states.- 6.4 Cubic Schroedinger 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.
Volume

: gw ISBN 9783764336677

Description

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.

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