Vortex Pairs on the Triaxial Ellipsoid: Axis Equilibria Stability

We consider a pair of opposite vortices moving on the surface of the triaxial ellipsoid $\mathbb{E}(a,b,c):$ $x^2/a+y^2/b+z^2/c=1, \, a < b < c$. The equations of motion are transported to $S^2 \times S^2$ via a conformal map that combines confocal quadric coordinates for the ellipsoid and sphero-conical coordinates in the sphere. The antipodal pairs form an invariant submanifold for the dynamics. We characterize the linear stability of the equilibrium pairs at the three axis endpoints.