The Tayler instability of toroidal magnetic fields in a columnar gallium experiment

TL;DR

This paper investigates the Tayler instability in a gallium experiment, analyzing how magnetic fields and rotation influence the onset and growth of nonaxisymmetric perturbations in a cylindrical setup.

Contribution

It provides a detailed linear stability analysis of the Tayler instability in a columnar gallium experiment, including effects of rotation and magnetic Prandtl number variations.

Findings

Critical Hartmann numbers are independent of magnetic Prandtl number without rotation.

Growth rates vary significantly with magnetic Prandtl number, being smaller for very low or high values.

A magnetic field of 1 kG in gallium leads to a 5-second growth time for the instability.

Abstract

The nonaxisymmetric Tayler instability of toroidal magnetic fields due to axial electric currents is studied for conducting incompressible fluids between two coaxial cylinders without endplates. The inner cylinder is considered as so thin that even the limit of R_in \to 0 can be computed. The magnetic Prandtl number is varied over many orders of magnitudes but the azimuthal mode number of the perturbations is fixed to m=1. In the linear approximation the critical magnetic field amplitudes and the growth rates of the instability are determined for both resting and rotating cylinders. Without rotation the critical Hartmann numbers do {\em not} depend on the magnetic Prandtl number but this is not true for the growth rates. For given product of viscosity and magnetic diffusivity the growth rates for small and large magnetic Prandtl number are much smaller than those for Pm=1. For gallium under the influence of a magnetic field at the outer cylinder of 1 kG the resulting growth time is 5 s. The minimum electric current through a container of 10 cm diameter to excite the kink-type instability is 3.20 kA. For a rotating container both the critical magnetic field and the related growth times are larger than for the resting column.