A Vortex Filament Tracking Method for the Gross-Pitaevskii Model of a Superfluid

TL;DR

This paper introduces a spectral numerical method for accurately tracking quantized vortex lines in superfluids modeled by the Gross-Pitaevskii equation, enabling detailed analysis of vortex dynamics and configurations.

Contribution

The method utilizes the superfluid's complex scalar order parameter and spectral techniques to precisely identify vortex lines without assuming their geometry.

Findings

Validated the method with vortex knot curvature and torsion calculations.

Measured Kelvin wave spectra on vortex lines and rings.

Applicable to dense vortex tangles in turbulent superfluids.

Abstract

We present an accurate and robust numerical method to track quantized vortex lines in a superfluid described by the Gross-Pitaevskii equation. By utilizing the pseudo-vorticity field of the associated complex scalar order parameter of the superfluid, we are able to track the topological defects of the superfluid and reconstruct the vortex lines which correspond to zeros of the field. Throughout, we assume our field is periodic to allow us to make extensive use of the Fourier representation of the field and its derivatives in order to retain spectral accuracy. We present several case studies to test the precision of the method which include the evaluation of the curvature and torsion of a torus vortex knot, and the measurement of the Kelvin wave spectrum of a vortex line and a vortex ring. The method we present makes no a-priori assumptions on the geometry of the vortices and is therefore applicable to a wide range of systems such as a superfluid in a turbulent state that is characterised by many vortex rings coexisting with sound waves. This allows us to track the positions of the vortex filaments in a dense turbulent vortex tangle and extract statistical information about the distribution of the size of the vortex rings and the inter-vortex separations. In principle, the method can be extended to track similar topological defects arising in other physical systems.