On the Hill's Spherical Vortex in Fluid and Plasma, its Generalization, and Stability

TL;DR

This paper revisits Hill's spherical vortex in fluid dynamics, simplifies its derivation, generalizes it to include azimuthal flow, and analyzes its stability, revealing linear instability under certain perturbations.

Contribution

The paper provides a simpler derivation of Hill's spherical vortex, introduces a generalized version with azimuthal flow, and investigates its linear stability in the MHD framework.

Findings

Hill's spherical vortex is linearly unstable under specific perturbations.

A new generalized vortex solution with azimuthal component is derived.

The derivation simplifies previous approaches using the Bragg-Hawthorne equation.

Abstract

In 1894 M.J.M. Hill published an article describing a spherical vortex moving through a stationary fluid. Using cylindrical coordinates and assuming the azimuthal velocity component zero, Hill found a simple solution that described this flow. A similar modern problem in the MHD framework was put forth in 1987 by A. A. Bobnev and in 1995 by R. Kaiser and D. Lortz who applied the setup to model a ball lighting. We present a much simpler derivation of Hill's spherical vortex using the Bragg-Hawthorne equation. In particular, by using the moving frame of reference, the Euler equations reduce to equilibrium flow which are equivalent to the static equilibrium MHD equations up to relabelling. A new generalized version of Hill's spherical vortex with a nonzero azimuthal component is derived. A physical solution to the static equilibrium MHD equations is computed by looking at a separated solution to the Grad-Shafranov equation in spherical coordinates. Finally, the stability of Hill's spherical vortex is examined by performing an axisymmetric perturbation described; it is shown that the Hill's spherical vortex is linearly unstable with respect to certain kinds of small perturbations.