On the Linear Stability of a Vortex Pair Equilibrium on a Riemann Surface of Genus Zero

We present a simple procedure to perform the linear stability analysis of a vortex pair equilibrium on a genus zero surface with an arbitrary metric. It consists of transferring the calculations to the round sphere in $\mathbb{R}^3$, with a conformal factor, and exploring the Möbius invariance of the conformal structure, so that the equilibria, seen on the representing sphere, appear in the north/south poles. Three example problems are analyzed: $i)$ For a surface of revolution of genus zero, a vortex pair located at the poles is nonlinearly stable due to integrability. We compute the two frequencies of the linearization. One is for the reduced system, the other is related to the reconstruction. Exceptionally, one of them can vanish. The calculation requires only the local profile at the poles and one piece of global information (given by a quadrature). $ii)$ A vortex pair on a double faced elliptical region, limiting case of the triaxial ellipsoid when the smaller axis goes to zero. We compute the frequencies of the pair placed at the centers of the faces. $iii)$ The stability, to a restricted set of perturbations, of a vortex equilateral triangle located in the equatorial plane of a spheroid, with polar vortices added so that the total vorticity vanishes.