Linear Stability of Inviscid Vortex Rings to Axisymmetric Perturbations

We consider the linear stability to axisymmetric perturbations of the family of inviscid vortex rings discovered by Norbury (1973). Since these vortex rings are obtained as solutions to a free-boundary problem, their stability analysis is performed using recently-developed methods of shape differentiation applied to the contour-dynamics formulation of the problem in a 3D axisymmetric geometry. This approach allows us to systematically account for the effect of boundary deformations on the linearized evolution of the vortex ring. Stability properties are then determined by the spectrum of a singular integro-differential operator defined on the vortex boundary in the meridional plane. The resulting generalized eigenvalue problem is solved numerically with a spectrally-accurate discretization. Our results reveal that while thin vortex rings remain neutrally stable, they become linearly unstable when they are sufficiently "fat". Analysis of the structure of the eigenmodes demonstrates that they approach the corresponding eigenmodes of Rankine's vortex when the vortex ring is thin and the eigenmodes of Hill's vortex when the vortex ring is fat. This study is a stepping stone leading to a complete stability analysis of inviscid vortex rings with respect to general perturbations.