  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 the 3D axisymmetric
geometry. This approach allows us to systematically account for the effects of
boundary deformations on the linearized evolution of the vortex ring. We
investigate the instantaneous amplification of perturbations assumed to have
the same the circulation as the vortex rings in their equilibrium
configuration. These 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 to axisymmetric perturbations,
they become linearly unstable to such perturbations when they are sufficiently
``fat''. Analysis of the structure of the eigenmodes demonstrates that they
approach the corresponding eigenmodes of Rankine's vortex and Hill's vortex in
the thin-vortex and fat-vortex limit, respectively. This study is a stepping
stone on the way towards a complete stability analysis of inviscid vortex rings
with respect to general perturbations.
