Dispersive Waves In Ocean Systems And FPU Chains

We study the dispersive waves in two nonlinear wave systems,one is about the coupling effects between internal waves and surface waves in the ocean,and another is about ?-Fermi-Pasta-Ulam(FPU)chains.Our intension is to estimate the properties of internal waves,such as height and wavelength of internal waves,from SAR images of their surface signature.However,the mechanism for the coupling effects between the internal waves and the free surfaces waves is yet to be understood.By applying numerical simulations and theoretical analysis,we focus on the complicated dynamics of the free surface when an internal soliton passes beneath.For the nonlinear FPU chains,how to characterize the turbulent state of strongly nonlinear dispersive waves is a long-standing challenging scientific problem,and we are dedicated to developing a universal theory for strongly nonlinear wave turbulence.In our study,we effectively combine methods of partial differential equations,direct numerical simulation,stochastic processes,statistical mechanics,and asymptotic analysis to investigate various aspects of turbulent wave phenomena.To study the spatiotemporal manifestation and the dynamical behavior of the coupling effects between internal waves and surface waves in a two-layer density-stratified fluid,we develop a weakly-nonlinear model by applying an asymptotic method.The model can well capture the ubiquitous broadening of large-amplitude internal waves in the ocean.However,the traditional KdV equation describing the motion of internal waves cannot capture the broadening of internal waves.Our numerical results show the asymmetric behavior of surface waves in the vicinity of an internal soliton,i.e.,surface waves become short in wavelength and high in amplitude at the leading edge,and long in wavelength and low in amplitude at the trailing edge of an internal soliton.From the spectrum of surface waves,we can observe that both frequency and wavenumber are discrete in a bounded domain.We demonstrate that these surface waves are characterized not only by modulation but by resonance as well.Our numerical results are quite different from others.The previous researches focus more on the derivation of multi-scale models,but do not focus on the spatiotemporal manifestation and dynamical behavior of the coupling waves.It has been shown in many references that the surface waves are controlled by the Schr¨odinger equation,which is different from our weakly nonlinear dispersive model.Our results may provide a new possible mechanism for the narrow bands of roughness at the leading edge and millpond effects at the trailing edge of an internal soliton in various experimental observations.How to characterize the turbulent state of strongly nonlinear dispersive waves is a long-standing challenging scientific problem in many scientific disciplines from nonlinear optics,plasma physics to atmosphere-ocean science.Using the FPU nonlinear system as a prototypical example,we develop a strong turbulence theory for understanding dispersive wave turbulence.We demonstrate that though the FPU chain is a nonlinear chaos system,in thermal equilibrium and non-equilibrium steady state the turbulent state even in the strongly nonlinear regime possesses an effective linear stochastic structure with a renormalized dispersion relation in renormalized normal variables.In this framework,we can well characterize the spatiotemporal dynamics,which are dominated by long-wavelength renormalized waves.By applying a multiple scale analysis to the FPU chain,we show that the contributions of the trivial and nontrivial resonance are both significant to the renormalization of the dispersion relation.The scenario of the stochastic linearization of the turbulence dynamics provides a new perspective to study strongly nonlinear wave turbulence and can be extended to study turbulent states in many other nonlinear wave systems.