Diffusive MHD Instabilities: Beyond the Chandrasekhar Theorem

TL;DR

This paper investigates the stability of magnetized cylindrical flows with azimuthal magnetic fields, revealing a diffusive magnetorotational instability that occurs beyond classical ideal fluid stability criteria, especially at low magnetic Prandtl numbers.

Contribution

It demonstrates the existence of a diffusive azimuthal magnetorotational instability in flows with finite diffusivity, extending Chandrasekhar's ideal fluid stability results to more realistic conditions.

Findings

Instability occurs at lower fields and rotation rates than predicted by ideal theory.

The instability domain scales with Reynolds and Hartmann numbers for low magnetic Prandtl numbers.

The Tayler instability appears at small Hartmann numbers, displacing the magnetorotational instability.

Abstract

The magnetohydrodynamic stability of axially unbounded cylindrical flows is considered which contain a toroidal magnetic background field with the same radial profile as the linear azimuthal velocity. Chandrasekhar (1956) has shown for ideal fluids the stability of this configuration if the Alfven velocity of the field equals the velocity of the background flow. It is demonstrated for magnetized Taylor-Couette flows at the Rayleigh line, however, that for finite diffusivity such flows become unstable against nonaxisymmetric perturbations where the critical magnetic Reynolds number of the rotation rate does not depend on the magnetic Prandtl number Pm if Pm much << 1. In order to study this new diffusive azimuthal magnetorotational instability, flows and fields with the same radial profile but with different amplitudes are considered. For Pm << 1 the instability domain with the weakest fields and the slowest rotation rates lies below the Chandrasekhar line of equal amplitudes for Alfven velocity and rotation velocity. We find that then the lines of marginal instability scale with the Reynolds number and the Hartmann number. The minimum values of the field strength and the rotation rate which are needed for the instability (slightly) grow for more and more flat rotation. Finally, the corresponding electric current of the background field becomes so strong that the Tayler instability (which even exists without rotation) also appears in the bifurcation map at small Hartmann numbers displacing after all the azimuthal magnetorotational instability.