The time-dependent motion of a vortex ring obeying the Hall-Vinen equations in a superfluid He counterflow is shown to be locally stable. Both the real and imaginary parts of the normal modes of oscillation scale as wave number squared.
- INTRODUCTION J. J. Thomson's calculation of the normal modes of a ring vortex has proved of more enduring value than W. Thomson's theory of vortex atoms, which was its motivation, l't Vortex rings in an ideal fluid are neutrally stable and superimposed on their translational motion are a discrete set of oscillatory modes with a frequency proportional to m (m 2_ 1)1/2 for integer m. The m = 0, 1 modes correspond respectively to a uniform change in radius R and a rigid motion of the plane of the loop. Now in a counterflow experiment in superfluid 4He as a consequence of the core-normal fluid drag, an isolated ring will retain its shape while its orientation and radius vary. 3"4 Its radius can increase no faster than linearly in time. The latter motion can be likened to a weak instability of the m = 0 mode and one is lead to ask, in view of the close correspondence between vortex motion in an ideal fluid and a superfluid at sufficiently low temperatures, whether in a counterflow any of the m > 1 modes might also be weakly unstable. We have addressed a somewhat more comprehensive problem, namely the stability of the time-dependent motion of an isolated vortex ring in a counterflow. The analysis in particular shows that the coupling between the m = 0, 1 and higher modes does not induce any new instabilities.