MHD instability in differentially-rotating cylindric flows

The possibility that the magnetic shear-flow instability (MRI, also Balbus-Hawley instability) might give rise to turbulence in a cylindrical Couette ow is investigated through numerical simulations. The study is linear and the fluid is assumed to be incompressible and differentially rotating with the rotation law Omega = a+b/R-2. The model is fully global in all three spatial directions with boundaries on each side; finite diffusivities are also imposed. The computations are carried out for several values of the azimuthal wavenumber m of the perturbations in order to analyze whether or not non-axisymmetric modes are preferred, which in a nonlinear extension of the study finally might lead to a dynamo-generated magnetic field. For magnetic Prandtl number of order unity we find that with a magnetic field the instability is generally easier to excite than without a magnetic field. The critical Reynolds number for Pm = 1 is of the order of 50, independent of whether or not the nonmagnetic ow is stable. We find that i) the magnetic field strongly reduces the number of Taylor vortices, ii) the angular momentum is transported outwards and iii) for finite cylinders a net dynamo-alpha effect results which is negative (positive) for the upper (lower) part of the cylinder. For magnetic Prandtl number smaller than unity the critical Reynolds number appears to scale with Pm-0.65. If this was true even for very small magnetic Prandtl numbers (e.g. for 10(-5), the magnetic Prandtl number of liquid sodium) the critical Reynolds number should reach the value of 10(5) which, however, is also characteristic of the nonlinear finite-amplitude hydrodynamic Taylor-Couette turbulence-so that we have to expect the simultaneous existence of both sorts of instabilities in related experiments. Similar phenomena are also discussed for cold accretion disks with their basically small magnetic Prandtl numbers.