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We compare dynamical and energetical stability criteria for vortex rings. Itis argued that vortex rings will be intrinsically unstable againstperturbations with short wavelengths below a critical wavelength, because thecanonical vortex Hamiltonian is unbounded from below for these modes. Toexplicitly demonstrate this behaviour, we derive the oscillation spectrum ofvortex rings in incompressible, inviscid fluids, within a geometrical cutoffprocedure for the core. The spectrum develops an anomalous branch of negativegroup velocity, and approaches the zero of energy for wavelengths which areabout six times the core diameter. We show the consequences of this dispersionrelation for the thermodynamics of vortex rings in superfluid $^4$He at lowtemperatures.Comment: 7 pages, 4 figures, final version to appear in Phys. Rev.

Short wavelength spectrum and Hamiltonian stability of vortex rings