Velocity, energy and helicity of vortex knots and unknots

TL;DR

This study numerically analyzes the velocity, energy, and helicity of vortex knots and unknots in a perfect fluid, revealing how their complexity influences their dynamics and stability, with implications for superfluid vortices.

Contribution

It provides a detailed numerical investigation of vortex filament properties, including velocity, energy, and helicity, for torus knots and unknots, highlighting the effects of vortex complexity.

Findings

Vortex speed and energy depend on the winding number w.

Helicity is mainly due to writhe contribution.

Knots are structurally stable over several diameters.

Abstract

In this paper we determine the velocity, the energy and estimate writhe and twist helicity contributions of vortex filaments in the shape of torus knots and unknots (toroidal and poloidal coils) in a perfect fluid. Calculations are performed by numerical integration of the Biot-Savart law. Vortex complexity is parametrized by the winding number $w$, given by the ratio of the number of meridian wraps to that of the longitudinal wraps. We find that for $w<1$ vortex knots and toroidal coils move faster and carry more energy than a reference vortex ring of same size and circulation, whereas for $w>1$ knots and poloidal coils have approximately same speed and energy of the reference vortex ring. Helicity is dominated by the writhe contribution. Finally, we confirm the stabilizing effect of the Biot-Savart law for all knots and unknots tested, that are found to be structurally stable over a distance of several diameters. Our results also apply to quantized vortices in superfluid $^4$He.