Wave Turbulence

TL;DR

This paper reviews recent advances in the statistical theory of wave turbulence, introducing a generalized approach that considers both phase and amplitude randomness, deriving new equations for multi-mode probability densities, and exploring non-Gaussianity and intermittency.

Contribution

It extends wave turbulence theory by developing a generalized random phase and amplitude approach and deriving multi-mode PDF equations for four-wave systems.

Findings

Derived a new PBP equation for four-wave systems

Validated statistical assumptions about phase and amplitude randomness

Linked intermittency to probability flux in wave amplitude space

Abstract

In this paper we review recent developments in the statistical theory of weakly nonlinear dispersive waves, the subject known as Wave Turbulence (WT). We revise WT theory using a generalisation of the random phase approximation (RPA). This generalisation takes into account that not only the phases but also the amplitudes of the wave Fourier modes are random quantities and it is called the ``Random Phase and Amplitude'' approach. This approach allows to systematically derive the kinetic equation for the energy spectrum from the the Peierls-Brout-Prigogine (PBP) equation for the multi-mode probability density function (PDF). The PBP equation was originally derived for the three-wave systems and in the present paper we derive a similar equation for the four-wave case. Equation for the multi-mode PDF will be used to validate the statistical assumptions about the phase and the amplitude randomness used for WT closures. Further, the multi-mode PDF contains a detailed statistical information, beyond spectra, and it finally allows to study non-Gaussianity and intermittency in WT, as it will be described in the present paper. In particular, we will show that intermittency of stochastic nonlinear waves is related to a flux of probability in the space of wave amplitudes.