A vortex filament tracking method for the Gross–Pitaevskii model of a superfluid

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 pseudovorticity 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.

Evolution of a superfluid vortex filament tangle driven by the Gross-Pitaevskii equation

“…In [10] we checked that this procedure is able to capture well the KWs superimposed on a ring by performing a series of different tests (different α and amplitude of KWs). In this work we use a value of α = 0.1, the variation of this fraction only slightly modifies (as expected) the largescale values of the spectrum, but values in the inertial range remain unchanged.…”

“…Low-density regions corresponding to vortex lines are plotted in red, while density fluctuations (sound) are rendered in light blue. We track the vortex lines forming the tangle with a recently-developed algorithm [10]. Vortex lines are followed using the pseudovorticity field [14] and the exact vortex position is obtained by a Newton-Raphson method.…”

“…In this Letter we apply a novel numerical algorithm [10] to track accurately the configuration of a turbulent vortex tangle evolving accordingly to the GP model. We focus on the evolution and decay of the tangle.…”

“…We have recently developed a robust and accurate algorithm to track vortex lines in the Gross-Pitaevskii equa-tion (GP) with arbitrary geometries in a periodic domain. The full details of the algorithm and the case studies to check its validity can be found in [10]. We recall here briefly the basic ideas.…”

“…Once the line is completely tracked the process is repeated with another line until the whole domain has been fully explored. The algorithm has been extensively described and validated in [10] using different case tests. As a final remark, we underline that the algorithm takes full advantage of the spectral resolution when a field is generated integrating the Gross-Pitaevskii with a pseudo-spectral method.…”