Imperfect bifurcation for the quasi-geostrophic shallow-water equations

TL;DR

This paper investigates the bifurcation structure of rotating vortex patches in the quasi-geostrophic shallow-water equations, revealing complex disconnections in the bifurcation diagram influenced by the Rossby deformation length.

Contribution

It provides the first analytical proof of bifurcation persistence near circular vortices for all Rossby deformation lengths and uncovers the disconnection of the Kirchhoff ellipse branch.

Findings

Bifurcation diagram persists near circular vortices for all deformation lengths.

The Kirchhoff ellipse branch is disconnected and split into multiple disjoint branches.

Numerical analysis confirms the complex global bifurcation structure.

Abstract

We study analytical and numerical aspects of the bifurcation diagram of simply-connected rotating vortex patch equilibria for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations are a generalisation of the Euler equations and contain an additional parameter, the Rossby deformation length $\EE^{-1}$, which enters in the relation between streamfunction and (potential) vorticity. The Euler equations are recovered in the limit $\EE\to 0$. We prove, close to circular (Rankine) vortices, the persistence of the bifurcation diagram for arbitrary Rossby deformation length. However we show that the two-fold branch, corresponding to Kirchhoff ellipses for the Euler equations, is never connected even for small values $\EE$, and indeed is split into a countable set of disjoint connected branches. Accurate numerical calculations of the global structure of the bifurcation diagram and of the limiting equilibrium states are also presented to complement the mathematical analysis.