Vortex Knots Dynamics in Euler Fluids

Abstract

In this paper we examine certain geometric and topological aspects of the dynamics and energetics of vortex torus knots and un- knots. The knots are given by small-amplitude torus knot solutions in the local induction approximation (LIA). Vortex evolution is then studied in the context of the Euler equations by direct numerical integration of the Biot-Savart law and the velocity, helicity and kinetic energy of different vortex knots and unknots are presented for comparison. Vortex complexity is parametrized by the winding number w given by the ratio of the number of meridian wraps to 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 the same size and circulation, whereas for w > 1, knots and poloidal coils have approximately the same speed and energy as the reference vortex ring. Kinetic helicity is dominated by writhe contributions and increases with knot complexity. All torus knots and unknots tested under Biot-Savart show much stronger permanence than under LIA.