Point vortices on the sphere: stability of relative equilibria

TL;DR

This paper investigates the stability of symmetric point vortex configurations on the sphere, analyzing how their eigenvalues and bifurcations determine stability or instability in these fluid dynamics models.

Contribution

It introduces a symmetry-based method to analyze stability of vortex arrangements on the sphere, including configurations with rings and polar vortices, and describes bifurcation phenomena.

Findings

Certain symmetric configurations are linearly stable

Eigenvalue analysis reveals bifurcation points

Symmetry simplifies stability calculations

Abstract

We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortices. Such configurations have dihedral symmetry, and the symmetry is used to block diagonalize the relevant matrices, to distinguish the subspaces on which their eigenvalues need to be calculated, and also to describe the bifurcations that occur as eigenvalues pass through zero.