Point vortices on a rotating sphere

TL;DR

This paper investigates the dynamics and stability of point vortices on a rotating sphere, demonstrating the persistence of certain equilibria under rotation and relating findings to atmospheric circulation observations.

Contribution

It proves the stability of latitudinal vortex rings on a rotating sphere and analyzes vortex configurations on a rotating plane, extending understanding of vortex dynamics in geophysical contexts.

Findings

Latitudinal vortex rings persist under sphere rotation.

Polygon vortex stability studied on rotating plane.

Results relate to atmospheric circulation patterns.

Abstract

We study the dynamics of $N$ point vortices on a rotating sphere. The Hamiltonian system becomes infinite dimensional due to the non-uniform background vorticity coming from the Coriolis force. We prove that a relative equilibrium formed of latitudinal rings of identical vortices for the non-rotating sphere persists to be a relative equilibrium when the sphere rotates. We study the nonlinear stability of a polygon of planar point vortices on a rotating plane in order to approximate the corresponding relative equilibrium on the rotating sphere when the ring is close to the pole. We then perform the same study for geostrophic vortices. To end, we compare our results to the observations on the southern hemisphere atmospheric circulation.