Oscillations of quantized vortices in rotating liquid helium II

Abstract

The vortex wave experiments in rotating liquid helium II have been usually analyzed by associating a Magnus force and a vortex tension with the vortices. Simple examples are advanced to show that these concepts are not proper in the hydrodynamics of free vortex motions. The dispersion relation for a vortex in a finite container is first calculated and it is proved to be incompatible with the ideas of Magnus force and vortex tension. The vibrations of a hollow columnar vortex in a rotating liquid are then derived. The connections to the limiting cases of the inertial oscillations of the rotating liquid and Kelvin's results on the vibrations of an isolated vortex are pointed out. The occasion is also taken to correct a slip in Rosenhead's kinematic analysis of the vibrating vortex. The oscillations of a random array of vortices, with a mean spacing b, are investigated next and the dominant frequency in the rotating frame is Ω = (2-J)ω 0+(4π) −1κk 2 [ ln( 1 /ka )+0.1159], where the function J( kb) varies from zero at kb = 0 to unity as kb → ∞. Hall's well-known experiments are consistent with this purely hydrodynamic analysis and give a core radius of ∼4 Å, compared with the earlier estimates of ∼30−7 Å. The studies of vortex motions in an atomistic framework are also discussed.