"Noisy" spectra, long correlations and intermittency in wave turbulence

TL;DR

This paper investigates the fluctuations and intermittency in wave turbulence spectra using a Random Phase Approximation model, revealing how long-range correlations influence the evolution of waveaction fluctuations.

Contribution

It introduces a minimal RPA-based model that captures the dynamics of waveaction fluctuations and intermittency in dispersive wave turbulence, even in extreme non-Gaussian cases.

Findings

Fluctuations depend on long-range correlations in the system.

Non-Gaussian initial conditions propagate without change to high amplitudes.

At large times, the probability distribution function becomes Gaussian at each fixed amplitude.

Abstract

We study the k-space fluctuations of the waveaction about its mean spectrum in the turbulence of dispersive waves. We use a minimal model based on the Random Phase Approximation (RPA) and derive evolution equations for the arbitrary-order one-point moments of the wave intensity in the wavenumber space. The first equation in this series is the familiar Kinetic Equation for the mean waveaction spectrum, whereas the second and higher equations describe the fluctuations about this mean spectrum. The fluctuations exhibit a nontrivial dynamics if some long coordinate-space correlations are present in the system, as it is the case in typical numerical and laboratory experiments. Without such long-range correlations, the fluctuations are trivially fixed at their Gaussian values and cannot evolve even if the wavefield itself is non-Gaussian in the coordinate space. Unlike the previous approaches based on smooth initial k-space cumulants, the RPA model works even for extreme cases where the k-space fluctuations are absent or very large and intermittent. We show that any initial non-Gaussianity at small amplitudes propagates without change toward the high amplitudes at each fixed wavenumber. At each fixed amplitude, however, the PDF becomes Gaussian at large time.