Helicity within the vortex filament model

TL;DR

This paper introduces a method to define and compute helicity in superfluid vortex filaments using a Seifert framing, restoring its key properties and linking it to classical vortex behavior.

Contribution

It presents a novel approach to incorporate twist into vortex filament helicity calculations, clarifying its invariance and connection to classical vortex helicity.

Findings

Helicity can be recovered in superfluids by defining a spanwise vector with Seifert framing.

Internal twist contributes significantly to the total helicity of vortex filaments.

The approach links superfluid vortex helicity to classical vortex helicity in the quasi-classical limit.

Abstract

Kinetic helicity is one of the invariants of the Euler equations that is associated with the topology of vortex lines within the fluid. In superfluids, the vorticity is concentrated along vortex filaments. In this setting, helicity would be expected to acquire its simplest form. However, the lack of a core structure for vortex filaments appears to result in a helicity that does not retain its key attribute as a quadratic invariant. By defining a spanwise vector to the vortex through the use of a Seifert framing, we are able to introduce twist and henceforth recover the key properties of helicity. We present several examples for calculating internal twist to illustrate why the centreline helicity alone will lead to ambiguous results if a twist contribution is not introduced. Our choice of the spanwise vector can be expressed in terms of the tangential component of velocity along the filament. Since the tangential velocity does not alter the configuration of the vortex at later times, we are able to recover a similar equation for the internal twist angle for classical vortex tubes. Our results allow us to explain how a quasi-classical limit of helicity emerges from helicity considerations for individual superfluid vortex filaments.