Short wavelength spectrum and Hamiltonian stability of vortex rings - art.no. 016306

We compare dynamical and energetical stability criteria for vortex rings. I t is argued that vortex rings will be intrinsically unstable against pertur bations with short wavelengths below a critical wavelength because the cano nical vortex Hamiltonian is unbounded from below for these modes. To explic itly demonstrate this behavior, we derive the oscillation spectrum of vorte x rings in incompressible, inviscid fluids within a geometrical cutoff proc edure for the core. The spectrum develops an anomalous branch of negative g roup velocity and approaches the zero of energy for wavelengths that are ab out six times the core diameter. We show the consequences of this dispersio n relation for the thermodynamics of vortex rings in superfluid He-4 at low temperatures.