Fast and slow resonant triads in the two layer rotating shallow water equations

TL;DR

This paper investigates triad resonances in a two-layer rotating shallow water system, revealing new interactions between fast barotropic and baroclinic waves and analyzing their dynamics in low Rossby and Froude number regimes.

Contribution

It introduces the first analysis of triad resonances involving multiple fast waves in a two-layer model, extending understanding beyond the one-layer system.

Findings

Triad resonances can occur between mixed barotropic and baroclinic fast waves.

The model reveals two branches of slow geostrophic modes with a repeated dispersion relation.

Derived interaction equations are validated through energy and enstrophy conservation.

Abstract

In this paper we examine triad resonances in a rotating shallow water system when there are two free interfaces. This allows for an examination in a relatively simple model of the interplay between baroclinic and barotropic dynamics in a context where there is also a geostrophic mode. In contrast to the much-studied one-layer rotating shallow water system, we find that as well as the usual slow geostrophic mode, there are now two fast waves, a barotropic mode and a baroclinic mode. This feature permits triad resonances to occur between three fast waves, with a mixture of barotropic and baroclinic modes, an aspect which cannot occur in the one-layer system. There are now also two branches of the slow geostrophic mode with a repeated branch of the dispersion relation. The consequences are explored in a derivation of the full set of triad interaction equations, using a multi-scale asymptotic expansion based on a small amplitude parameter. The derived nonlinear interaction coefficients are confirmed using energy and enstrophy conservation. These triad interaction equations are explored with an emphasis on the parameter regime with small Rossby and Froude numbers.