Helicity of the toroidal vortex with swirl

TL;DR

This paper analyzes the helicity of thin toroidal vortices with swirl using solutions of the Bragg-Hawthorne equations, revealing how circulation relationships depend on azimuthal velocity distribution and differ from classical models.

Contribution

It introduces a new analysis of vortex helicity considering non-homogeneous swirl, extending the classical Moffat relationship for linked circulations.

Findings

Helicity depends on azimuthal velocity distribution in the vortex core.

Non-homogeneous swirl alters the classical circulation relationship.

Results applicable to planetary atmospheres and galactic vortex dynamics.

Abstract

On the basis of solutions of the Bragg-Hawthorne equations we discuss the helicity of thin toroidal vortices with the swirl - the orbital motion along the torus diretrix. It is shown that relationship of the helicity with circulations along the small and large linked circles - directrix and generatrix of the torus - depends on distribution of the azimuthal velocity in the core of the swirling vortex ring. In the case of non-homogeneous swirl this relationship differs from the well-known Moffat relationship - the doubled product of such circulations multiplied by the number of links. The results can be applied to vortices in planetary atmospheres and to vortex movements in the vicinity of active galactic nuclei.