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

Journal article

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 non-local 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.

Authors: Van Gorder, R

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Royal Society
• Journal: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
• Publication date: 2015-07-08
• Acceptance date: 2015-06-03
• DOI: 10.1098/rspa.2015.0149
• EISSN: 1471-2946
• ISSN: 1364-5021
• Keywords: superfluid helium; Donnelly-Glaberson instability; mutual friction; quantum vortex dynamics; Kelvin waves; SBTMR; non-locality
• Subjects: low temperature physics; wave motion; mathematical physics