Vortex motion in superfluid ⁴He: effects of normal fluid flow

Abstract

The motion of a vortex filament in superfluid ⁴He is considered by using the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) [H.E. Hall, W.F. Vinen, Proc. Roy. Soc. Lond. A 238, 204 (1956); H.E. Hall, W.F. Vinen, Proc. Roy. Soc. Lond. A 238, 215 (1956); I.L. Bekarevich, I.M. Khalatnikov, Sov. Phys. J. Exp. Theor. Phys. 13, 643 (1961)] phenomenological model for the scattering process between the vortex and thermal excitations in liquid ⁴He. The HVBK equations are analytically formulated first in the intrinsic geometric parameter space to obtain insights into the physical implications of the friction terms, associated with the friction coefficients α and α′ (in the Hall-Vinen notation) as well as the previous neglect of the friction term associated with the friction coefficient α′. The normal fluid velocity components both along and transverse to the vortex filament are included. This analytical development also serves to highlight the difficulties arising in making further progress on this route. A reformulation of the HVBK equation in the extrinsic vortex filament coordinate space is then given which is known [B.K. Shivamoggi, Phys. Rev. B 84, 012506 (2011)] to provide a useful alternative analytical approach in this regard. A nonlinear Schrödinger equation for the propagation of nonlinear Kelvin waves on a vortex filament in a superfluid is given taking into account the generalized normal fluid flow. The friction term associated with α′, even in the presence of the normal fluid velocity components transverse to the vortex filament, is shown to produce merely an algebraic growth of the Kelvin waves hence providing further justification for the neglect of this term. On the other hand, the instability produced by the friction term associated with α via the normal fluid velocity component along the vortex filament is shown to manifest itself as a parametric amplification on considering the problem of a rotating planar vortex filament in a superfluid.