Weak versus strong wave turbulence in the MMT model

TL;DR

This study investigates the transition from weak to strong wave turbulence in the MMT model, revealing how increased nonlinearity causes deviations from weak turbulence theory and introduces intermittency similar to 3D fluid turbulence.

Contribution

The paper provides a detailed numerical analysis of the MMT model, demonstrating the breakdown of weak wave turbulence predictions and the emergence of intermittency with increased nonlinearity.

Findings

Spectral slopes match WWT predictions at small nonlinearity.

Deviations from WWT occur as nonlinearity increases.

Intermittency phenomena emerge with higher nonlinearity.

Abstract

Within the spirit of fluid turbulence, we consider the one-dimensional Majda-McLaughlin-Tabak (MMT) model that describes the interactions of nonlinear dispersive waves. We perform a detailed numerical study of the direct energy cascade in the defocusing regime. In particular, we consider a configuration with large-scale forcing and small scale dissipation, and we introduce three non- dimensional parameters: the ratio between nonlinearity and dispersion, {\epsilon}, and the analogues of the Reynolds number, Re, i.e. the ratio between the nonlinear and dissipative time-scales, both at large and small scales. Our numerical experiments show that (i) in the limit of small {\epsilon} the spectral slope observed in the statistical steady regime corresponds to the one predicted by the Weak Wave Turbulence (WWT) theory. (ii) As the nonlinearity is increased, the WWT theory breaks down and deviations from its predictions are observed. (iii) It is shown that such departures from the WWT theoretical predictions are accompanied by the phenomenon of intermittency, typical of three dimensional fluid turbulence. We calculate the structure-function as well as the probability density function of the wave field at each scale and show that the degree of intermittency depends on {\epsilon}.