Stochastic linearization of turbulent dynamics of dispersive waves in equilibrium and non-equilibrium state

Shixiao W Jiang,Haihao Lu,Douglas Zhou,David Cai

doi:10.1088/1367-2630/18/8/083028

Abstract

Characterizing dispersive wave turbulence in the long time dynamics is central to understanding of many natural phenomena, e.g., in atmosphere ocean dynamics, nonlinear optics, and plasma physics. Using the β-Fermi–Pasta–Ulam nonlinear system as a prototypical example, we show that in thermal equilibrium and non-equilibrium steady state the turbulent state even in the strongly nonlinear regime possesses an effective linear stochastic structure in renormalized normal variables. In this framework, we can well characterize the spatiotemporal dynamics, which are dominated by long-wavelength renormalized waves. We further demonstrate that the energy flux is nearly saturated by the long-wavelength renormalized waves in non-equilibrium steady state. The scenario of such effective linear stochastic dynamics can be extended to study turbulent states in other nonlinear wave systems.

Highlights

Dispersion relations play a controlling role in characterizing turbulence of weakly nonlinear dispersive waves [1]
By examining the spatiotemporal characteristics and dynamical behaviors of renormalized waves, we demonstrate that the long time behavior of the momentum correlation functions is dominated by the long-wavelength renormalized waves and the linear stochastic structure results in a power-law decay of the amplitude of momentum correlation functions [17,18,19,20]
We demonstrate that the energy carriers in nonequilibrium steady state are long-wavelength renormalized waves in the linear stochastic dynamic structure, which nearly saturates the total energy flux in the turbulence

Summary

August 2016

Using the β-Fermi–Pasta–Ulam nonlinear system as a prototypical example, we show that in thermal equilibrium and non-equilibrium steady state the turbulent state even in the strongly nonlinear regime possesses an effective linear stochastic structure in renormalized normal variables. In this framework, we can well characterize the spatiotemporal dynamics, which are dominated by longwavelength renormalized waves. We further demonstrate that the energy flux is nearly saturated by the long-wavelength renormalized waves in non-equilibrium steady state The scenario of such effective linear stochastic dynamics can be extended to study turbulent states in other nonlinear wave systems

Introduction

The FPU model

N åkÎKL

Effective linear stochastic structure

Nonequilibrium steady state turbulence

Conclusion and discussion