Resonant fast–slow interactions and breakdown of quasi-geostrophy in rotating shallow water

Abstract

In this paper we investigate the possibility of fast waves affecting the evolution of slow balanced dynamics in the regime $Ro\sim Fr\ll 1$ of a rotating shallow water system, where $Ro$ and $Fr$ are the Rossby and Froude numbers respectively. The problem is set up as an initial value problem with unbalanced initial data. The method of multiple time scale asymptotic analysis is used to derive an evolution equation for the slow dynamics that holds for $t\lesssim 1/(fRo^{2})$, $f$ being the inertial frequency. This slow evolution equation is affected by the fast waves and thus does not form a closed system. Furthermore, it is shown that energy and enstrophy exchange can take place between the slow and fast dynamics. As a consequence, the quasi-geostrophic ideology of describing the slow dynamics of the balanced flow without any information on the fast modes breaks down. Further analysis is carried out in a doubly periodic domain for a few geostrophic and wave modes. A simple set of slowly evolving amplitude equations is then derived using resonant wave interaction theory to demonstrate that significant wave-balanced flow interactions can take place in the long-time limit. In this reduced system consisting of two geostrophic modes and two wave modes, the presence of waves considerably affects the interactions between the geostrophic modes, the waves acting as a catalyst in promoting energetic interactions among geostrophic modes.