Magnetic instability in a sheared azimuthal flow

We study the magneto-rotational instability of an incompressible flow which rotates with angular velocity Omega(r) = a + b = r(2) where r is the radius and a and b are constants. We find that an applied magnetic field destabilises the flow, in agreement with the results of Rudiger & Zhang (2001). We extend the investigation in the region of parameter space which is Rayleigh stable. We also study the instability at values of magnetic Prandtl number which are much larger and smaller than Rudiger & Zhang. Large magnetic Prandtl numbers are motivated by their possible relevance in the central region of galaxies (Kulsrud & Anderson 1992). In this regime we find that increasing the magnetic Prandtl number greatly enhances the instability; the stability boundary drops below the Rayleigh line and tends toward the solid body rotation line. Very small magnetic Prandtl numbers are motivated by the current MHD dynamo experiments performed using liquid sodium and gallium. Our finding in this regime confirms Rudiger & Zhang's conjecture that the linear magneto-rotational instability and the nonlinear hydrodynamical instability (Richard & Zahn 1999) take place at Reynolds numbers of the same order of magnitude.