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

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 σˆ of the electrical conductivities of the cylinders and the fluid, such that σˆ=O(10) provides a very good approximation to perfectly conducting cylinders, and σˆ=O(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 realisation of the super-rotating instability, with liquid sodium as the fluid and cylinders made of copper, would need an electric current of at least 33.5 kA running along the central axis.