Abbreviated Journal Title
ROTATING HELIUM-II; SUPERFLUID HE-4; DYNAMICS; MOTION; INSTABILITY; TURBULENCE; VORTICES; SOUND; Mechanics; Physics, Fluids & Plasmas
We study the dynamics of helical Kelvin waves moving along a quantum vortex filament driven by a normal fluid flow. We employ the vector form of the quantum local induction approximation (LIA) due to Schwarz. For an isolated filament, this is an adequate approximation to the full Hall-Vinen-Bekarevich-Khalatnikov dynamics. The motion of such Kelvin waves is both translational (along the quantum vortex filament) and rotational (in the plane orthogonal to the reference axis). We first present an exact closed form solution for the motion of these Kelvin waves in the case of a constant amplitude helix. Such solutions exist for a critical wave number and correspond exactly to the Donnelly-Glaberson instability, so perturbations of such solutions either decay to line filaments or blow-up. This leads us to consider helical Kelvin waves which decay to line filaments. Unlike in the case of constant amplitude helical solutions, the dynamics are much more complicated for the decaying helical waves, owing to the fact that the rate of decay of the helical perturbations along the vortex filament is not constant in time. We give an analytical and numerical description of the motion of decaying helical Kelvin waves, from which we are able to ascertain the influence of the physical parameters on the decay, translational motion along the filament, and rotational motion, of these waves (all of which depend nonlinearly on time). One interesting finding is that the helical Kelvin waves do not decay uniformly. Rather, suchwaves decay slowly for small time scales, and more rapidly for large time scales. The rotational and translational velocity of the Kelvin waves depend strongly on this rate of decay, and we find that the speed of propagation of a helical Kelvin wave along a quantum filament is large for small time while the wave asymptotically slows as it decays. The rotational velocity of such Kelvin waves along the filament will increase over time, asymptotically reaching a finite value. These decaying Kelvin waves correspond to wave number below the critical value for the Donnelly-Glaberson instability, and hence our results on the Schwarz quantum LIA correspond exactly to what one would expect from prior work on the Donnelly-Glaberson instability.
Physics of Fluids
Van Gorder, Robert A., "Decay of helical Kelvin waves on a quantum vortex filament" (2014). Faculty Bibliography 2010s. 6215.