Discreteness and its effect on the water-wave turbulence

TL;DR

This study investigates how wave resonance discreteness affects water-wave turbulence, revealing that certain WT predictions hold while new phenomena like threshold effects and bursty cascades emerge due to finite basin size.

Contribution

It demonstrates the impact of wave resonance discreteness on turbulence properties, including the existence of a threshold for cascade development and the bursty energy transfer dynamics.

Findings

Wave spectrum agrees with Zakharov-Filonenko predictions.

Probability density function shows large, non-Gaussian strong waves.

Identifies a threshold wave intensity for cascade onset due to discreteness.

Abstract

We perform numerical simulations of the dynamical equations for free water surface in finite basin in presence of gravity. Wave Turbulence (WT) is a theory derived for describing statistics of weakly nonlinear waves in the infinite basin limit. Its formal applicability condition on the minimal size of the computational basin is impossible to satisfy in present numerical simulations, and the number of wave resonances is significantly depleted due to the wavenumber discreteness. The goal of this paper will be to examine which WT predictions survive in such discrete systems with depleted resonances and which properties arise specifically due to the discreteness effects. As in \cite{DKZ,onorato,naoto}, our results for the wave spectrum agree with the Zakharov-Filonenko spectrum predicted within WT. We also go beyond finding the spectra and compute probability density function (PDF) of the wave amplitudes and observe an anomalously large, with respect to Gaussian, probability of strong waves which is consistent with recent theory \cite{clnp,cln}. Using a simple model for quasi-resonances we predict an effect arising purely due to discreteness: existence of a threshold wave intensity above which turbulent cascade develops and proceeds to arbitrarily small scales. Numerically, we observe that the energy cascade is very ``bursty'' in time and is somewhat similar to sporadic sandpile avalanches. We explain this as a cycle: a cascade arrest due to discreteness leads to accumulation of energy near the forcing scale which, in turn, leads to widening of the nonlinear resonance and, therefore, triggering of the cascade draining the turbulence levels and returning the system to the beginning of the cycle. ~