Resonant interactions of nonlinear water waves in a finite basin

TL;DR

This paper investigates exact four-wave resonances in nonlinear water waves within a finite square basin, revealing how these resonances form clusters that influence energy transfer and wavefield evolution.

Contribution

It identifies and classifies resonant quartets and their clusters in a finite basin, elucidating their roles in energy redistribution and transfer among wave scales.

Findings

Resonant quartets form interconnected clusters.

Angle-resonances redistribute energy without cascading.

Scale-resonances transfer energy across scales.

Abstract

We study exact four-wave resonances among gravity water waves in a square box with periodic boundary conditions. We show that these resonant quartets are linked with each other by shared Fourier modes in such a way that they form independent clusters. These clusters can be formed by two types of quartets: (1) {\it angle-resonances} which cannot directly cascade energy but which can redistribute it among the initially excited modes and (2) {\it scale-resonances} which are much more rare but which are the only ones that can transfer energy between different scales. We find such resonant quartets and their clusters numerically on the set of 1000 x 1000 modes, classify and quantify them and discuss consequences of the obtained cluster structure for the wavefield evolution. Finite box effects and associated resonant interaction among discrete wave modes appear to be important in most numerical and laboratory experiments on the deep water gravity waves, and our work is aimed at aiding the interpretation of the experimental and numerical data.