Linear stability of a vortex ring revisited

We revisit the stability of an elliptically strained vortex and a thin axisymmetric vortex ring, embedded in an inviscid incompressible fluid, to three-dimensional disturbances of infinitesimal amplitude. The results of Tsai & Widnall (1976) for an elliptically strained vortex are simplified by providing an explicit expression for the disturbance flow field. A direct relation is established with the elliptical instability. For Kelvin's vortex ring, the primary perturbation to the Rankine vortex is a dipole field. We show that the dipole field causes a parametric resonance instability between axisymmetric and bending waves at intersection points of the dispersion curves. It is found that the dipole effect predominates over the straining effect for a very thin core. The mechanism is attributable to stretching of the disturbance vortex lines in the toroidal direction.