Astrophysical and experimental implications from the magnetorotational instability of toroidal fields

TL;DR

This paper investigates the magnetorotational instability of toroidal magnetic fields in differentially rotating fluids, revealing conditions for instability, implications for laboratory experiments, and effects on angular momentum transport and mixing in stellar cores.

Contribution

It provides a detailed analysis of the azimuthal magnetorotational instability (AMRI) in various rotation laws, including laboratory experiment design and nonlinear energy growth predictions.

Findings

Nonaxisymmetric perturbations are unstable when rotation rate and Alfvén frequency are comparable.

Laboratory experiments with liquid metals can be designed based on the instability thresholds.

Magnetic energy increases monotonically with magnetic Reynolds number, affecting plasma mixing and angular momentum transport.

Abstract

The interaction of differential rotation and toroidal fields that are current-free in the gap between two corotating axially unbounded cylinders is considered. It is shown that nonaxisymmetric perturbations are unstable if the rotation rate and Alfv\'en frequency of the field are of the same order, almost independent of the magnetic Prandtl number Pm. For the very steep rotation law \Omega\propto R^{-2} (the Rayleigh limit) and for small Pm the threshold values of rotation and field for this Azimuthal MagnetoRotational Instability (AMRI) scale with the ordinary Reynolds number and the Hartmann number, resp. A laboratory experiment with liquid metals like sodium or gallium in a Taylor-Couette container has been designed on the basis of this finding. For fluids with more flat rotation laws the Reynolds number and the Hartmann number are no longer typical quantities for the instability. For the weakly nonlinear system the numerical values of the kinetic energy and the magnetic energy are derived for magnetic Prandtl numbers \leq 1. We find that the magnetic energy grows monotonically with the magnetic Reynolds number Rm, while the kinetic energy grows with Rm/\sqrt{Pm}. The resulting turbulent Schmidt number, as the ratio of the `eddy' viscosity and the diffusion coefficient of a passive scalar (such as lithium) is of order 20 for Pm=1, but for small Pm it drops to order unity. Hence, in a stellar core with fossil fields and steep rotation law the transport of angular momentum by AMRI is always accompanied by an intense mixing of the plasma, until the rotation becomes rigid.