Quartet, higher order and near resonant interactions in nonlinear wave equations

TL;DR

This paper investigates resonant and near-resonant wave interactions in nonlinear wave equations, focusing on geophysical fluid dynamics, and develops a multi-scale asymptotic framework to analyze higher order interactions and their timescales.

Contribution

It introduces a multi-scale asymptotic expansion method to derive higher order nonlinear interaction coefficients and analyzes how near-resonances influence interaction timescales in geophysical systems.

Findings

Higher order interaction coefficients are derived from first order triad interactions.

Near-resonances reduce the complexity of higher order expansions by focusing on dominant contributions.

Interactions involving slow modes are pushed to slower timescales due to conserved quantities.

Abstract

Motivated by problems arising in geophysical fluid dynamics, we investigate resonant and near resonant wave interactions in nonlinear wave equations with quadratic nonlinearity, We place a special focus on interactions between slow wave modes, with zero frequency in the linear limit, and fast modes. These regularly occur in geophysical fluid systems with conserved potential vorticity or similar conserved quantities. A general multi-scale asymptotic expansion is used to show how the higher order nonlinear interaction coefficients are derived as a combination of the first order terms arising at the triad interaction level. From the general nth-order interaction coefficient we present a proof by induction how the limiting effect for particular combinations of slow and fast modes pushes their interactions to a slower timescale, and we show how this is linked to the form of the conserved quantities. We compare near resonant expansions with exact resonant expansions, and show how near-resonances allow higher order expansions to be reduced to just the most dominant contributions. These contributions then occur at one order higher in the expansion when compared to the analogous exact resonance expansion.