Short wavelength spectrum and Hamiltonian stability of vortex rings

We compare dynamical and energetical stability criteria for vortex rings. It is argued that vortex rings will be intrinsically unstable against perturbations with short wavelengths below a critical wavelength because the canonical vortex Hamiltonian is unbounded from below for these modes. To explicitly demonstrate this behavior, we derive the oscillation spectrum of vortex rings in incompressible, inviscid fluids within a geometrical cutoff procedure for the core. The spectrum develops an anomalous branch of negative group velocity and approaches the zero of energy for wavelengths that are about six times the core diameter. We show the consequences of this dispersion relation for the thermodynamics of vortex rings in superfluid ⁴He at low temperatures.