The Biot–Savart description of Kelvin waves on a quantum vortex filament in the presence of mutual friction and a driving fluid

Abstract

We study the dynamics of Kelvin waves along a quantum vortex filament in the presence of mutual friction and a driving fluid while taking into account non-local effects due to Biot-Savart integrals. The Schwarz model reduces to a nonlinear and nonlocal dynamical system of dimension three, the solutions of which determine the translational and rotational motion of the Kelvin waves, as well as the amplification or decay of such waves. We determine the possible qualitative behaviours of the resulting Kelvin waves. It is well known from experimental and theoretical studies that the Donnelly-Glaberson instability plays a role on the amplification or decay of Kelvin waves in the presence of a driving normal fluid velocity, and we obtain the relevant stability criterion for the non-local model. While the stability criterion is the same for local and non-local models when the wavenumber is sufficiently small, we show that large differences emerge for the large wavenumber case (tightly coiled helices). The results demonstrate that non-local effects have a stabilizing effect on the Kelvin waves, and hence larger normal fluid velocities are required for amplification of large wavenumber Kelvin waves. Additional qualitative differences between the local and non-local models are explored.

Localized nonlinear waves on quantized superfluid vortex filaments in the presence of mutual friction and a driving normal fluid flow

“…As might be expected from the known Donnelly-Glaberson instability results for Kelvin waves (which correspond to helical vortex filaments [29,30]), we observe a parametric amplification of the localized spatiotemporal waves along the quantized vortex filaments. However, while the helical filaments would exhibit a uniform amplification over the entire vortex filament length, the localized waves may exhibit parametric amplification only locally.…”

“…For the vortex filament, there will be more rapid dissipation of an initial disturbance or wave corresponding to α > 0 rather than in the classical case when α = 0. This is consistent with what has been observed in the literature for Kelvin waves along quantum vortex filaments [29,30] (corresponding to helical filaments). For those cases, the dissipation manifests as a factor which scales as e −αk 2 eff t , where k eff is an effective wave number [k eff = O(1) in α] for the helical filament.…”

“…This is essentially the statement of the Donnelly-Glaberson instability [24][25][26] for the model (11). Under the quantum LIA, results like this were shown for regular helical vortex filaments in [29], while recently such results were shown under the quantum form of the nonlocal Biot-Savart dynamics in [30]. What this means is that, for small U 3 , there should be no drastic changes in the solutions between the quantum Hasimoto map and the PDE (11), since both are essentially the dissipative limit.…”

Self-similar vortex filament motion under the non-local Biot–Savart model

“…While (1.2) defines a nonlinear vector partial differential equation, its solution is still simpler than that of the non-local and singular equation (1.1). The LIA is useful when the vortex filaments are very thin, which is true for instance in the case of quantized vortex filaments in superfluid helium, provided that they are not too tightly coiled (Van Gorder, 2014), while for more tightly coiled filaments the Biot-Savart dynamics are needed (Van Gorder, 2015a). For classical vortex filaments, the LIA has well-known limitations, hence solution to the full Biot-Savart dynamics (1.1) are desirable where possible.…”

Solitons and nonlinear waves along quantum vortex filaments under the low-temperature two-dimensional local induction approximation

“…The nonlinear dynamics of such Kelvin waves under the Schwarz model were recently studied in [41], where is was shown that the rate of amplification or decay of Kelvin waves along these quantum vortex filaments will depend strongly on the mutual friction parameter, with the rate becoming larger as the superfluid warms. Helical filaments in both classical and quantum fluids continue to be an active area of research interest [42][43][44][45].…”