Nonlinear Stability of a Latitudinal Ring of Point-Vortices on a Nonrotating Sphere

We study the nonlinear stability of relative equilibria of configurations of identical point-vortices on the surface of a sphere. In particular, we study how the stability changes as a function of the colatitude θ and of the number of vortices N. By using the integrals of motion, we view the system in asuitable corotating frame where the polygonal vortex configuration is at rest. Then after a sufficient criterion due to Dirichlet, the stability ranges are the θ-intervals for which the Hessian of the Hamiltonian---evaluated at the equilibrium configuration---is positive or negative definite. We find that the stability intervals coincide with those for linear stability determined by Polvani and Dritschel [J. Fluid Mech., 255 (1993), pp. 35--64]. For $N=3$ we recover the result previously established by Pekarsky and Marsden [J. Math. Phys., 39 (1998), pp. 5894--5907].