Role of triad interactions in spectral evolution of surface gravity waves in deep water

TL;DR

This paper investigates how triad and quartet interactions influence the spectral evolution of deep-water surface gravity waves, revealing that non-resonant triad interactions significantly contribute to energy transfer, especially at low wavenumbers.

Contribution

It demonstrates that non-resonant triad interactions can form quartet resonances and rapidly transfer energy to low wavenumbers, expanding understanding of wave turbulence dynamics.

Findings

Triad interactions follow quartet trends for most wavenumbers.

Triad interactions rapidly energize low wavenumber spectrum regions.

Analytical models describe energy distribution between free and bound modes.

Abstract

It is generally accepted that the evolution of deep-water surface gravity wave spectrum is governed by quartet resonant and quasi-resonant interactions. However, it has also been reported in both experimental and computational studies that non-resonant triad interactions can play a role, e.g., generation of bound waves. In this study, we investigate the effects of triad and quartet interactions on the spectral evolution, by numerically tracking the contributions from quadratic and cubic terms in the dynamical equation. In a finite time interval, we find that the contribution from triad interactions follows the trend of that from quartet resonances (with comparable magnitude) for most wavenumbers, except that it peaks at low wavenumbers with very low initial energy. This result reveals two effects of triad interactions: (1) the non-resonant triad interactions can be connected to form quartet resonant interactions (hence exhibiting the comparable trend), which is a reflection of the normal form transformation applied in wave turbulence theory of surface gravity waves. (2) the triad interactions can fill energy into the low energy portion of the spectrum (low wavenumber part in this case) on a very fast time scale, with energy distributed in both bound and free modes at the same wavenumber. We further analyze the latter mechanism using a simple model with two initially active modes in the wavenumber domain. Analytical formulae are provided to describe the distribution of energy in free and bound modes with numerical validations.