Magnetorotational instability in Taylor-Couette flows between cylinders with finite electrical conductivity

TL;DR

This study investigates the magnetorotational instability in Taylor-Couette flows with finite electrical conductivity, revealing how boundary conditions and electrical conductivity ratios influence the onset of instability, especially in super-rotating flows.

Contribution

It provides new insights into how electrical conductivity ratios and boundary conditions affect the magnetorotational instability thresholds in finite-conductivity Taylor-Couette systems.

Findings

Perfectly conducting cylinders become unstable at lower magnetic fields for small Pm.

Electrical conductivity ratio of about 10 approximates perfect conductors; 0.1 approximates insulators.

Super-rotating flows show significant differences in instability thresholds based on boundary conditions.

Abstract

The nonaxisymmetric azimuthal magnetorotational instability is studied for hydromagnetic Taylor-Couette flows between cylinders of finite electrical conductivity. We find that the magnetic Prandtl number Pm determines whether perfectly conducting or insulating boundary conditions lead to lower Hartmann numbers for the onset of instability. Regardless of the imposed rotation profile, for small Pm the solutions for perfectly conducting cylinders become unstable for weaker magnetic fields than the solutions for insulating cylinders. The critical Hartmann and Reynolds numbers form monotonic functions of the ratio of the electrical conductivities of the cylinders and the fluid, such that a ratio of about 10 provides a very good approximation to perfectly conducting cylinders, and a ratio of about 0.1 a very good approximation to insulating cylinders. These results are of particular relevance for the super-rotating case where the outer cylinder rotates faster than the inner one; in this case the critical onset values are substantially different for perfectly conducting versus insulating boundary conditions. An experimental realization of the super-rotating instability, with liquid sodium as the fluid and cylinders made of copper, would require an electric current of at least 33.5 kAmp running along the central axis.