Existence and Stability of the Lamb Dipoles for the Quasi-Geostrophic Shallow-Water Equations

TL;DR

This paper proves the nonlinear orbital stability of vortex dipoles in the quasi-geostrophic shallow-water equations, providing a variational framework that extends classical vortex stability results to a more complex geophysical fluid model.

Contribution

It introduces the first stability proof for vortex dipoles in QGSW equations using a variational characterization approach.

Findings

Vortex dipoles are proven to be nonlinearly orbitally stable.

A variational characterization of vortex poles is established.

The results extend classical vortex stability to geophysical fluid models.

Abstract

In this paper, we prove the nonlinear orbital stability of vortex dipoles for the quasi-geostrophic shallow-water (QGSW) equations. The vortex dipoles are explicit travelling wave solutions to the QGSW equations, which are analogues of the classical circular vortex of Lamb and Chaplygin for the steady planar Euler equations. We establish a variational characterization of these vortex poles, which provides a basis for the stability result.