Vortex Pairs on the Triaxial Ellipsoid: Axis Equilibria Stability

Abstract

We consider a pair of opposite vortices moving on the surface of the triaxial ellipsoid E(a, b, c): x²/a+ y²/b+ z²/c = 1, a < b < c. The equations of motion are transported to S² ×S² 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.