Self-similar vortex dynamics in superfluid 4He under the Cartesian representation of the Hall-Vinen model including superfluid friction

Abstract

We determine conditions under which the fully nonlinear form of the local induction approximation (LIA) governing the motion of a vortex filament in superfluid 4He (that is, the Hall-Vinen model) in the Cartesian frame of reference permits the existence of self-similar solutions, even in the presence of superfluid friction parameters. Writing the Cartesian Hall-Vinen LIA in potential form for the motion of a vortex filament, we find that a necessary condition for self-similarity is that the normal-fluid component vanishes (which makes sense in the low temperature limit), and we reduce the potential form of the Hall-Vinen LIA to a complex nonlinear ordinary differential equation governing the behavior of a similarity solution. In the limit where superfluid friction parameters are negligible, we provide some analytical and asymptotic results for various regimes. While such analytical results are useful for determining the qualitative behavior of the vortex filament in the limit where superfluid friction parameters vanish, numerical simulations are needed to determine the true behavior of the filaments in the case of non-zero superfluid friction parameters. While the superfluid friction parameters are small, the numerical results demonstrate that the influence of the superfluid friction parameters on the self-similar vortex structures can be strong. We classify two types of filaments from the numerical results: singular filaments (which demonstrate growing oscillations and hold kink-type solutions as a special case) and bounded filaments (whose behavior is a bounded function of x). We also comment on how to include the case where there is a non-zero normal fluid, and we find a transformation of the self-similar solutions into non-similar solutions that can account for this.