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 t he fluid is assumed to be incompressible and differentially rotating with t he rotation law Omega = a+b/R-2. The model is fully global in all three spa tial directions with boundaries on each side; finite diffusivities are also imposed. The computations are carried out for several values of the azimut hal wavenumber m of the perturbations in order to analyze whether or not no n-axisymmetric modes are preferred, which in a nonlinear extension of the s tudy finally might lead to a dynamo-generated magnetic field. For magnetic Prandtl number of order unity we find that with a magnetic field the instab ility is generally easier to excite than without a magnetic field. The crit ical Reynolds number for Pm = 1 is of the order of 50, independent of wheth er 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 effe ct results which is negative (positive) for the upper (lower) part of the c ylinder. For magnetic Prandtl number smaller than unity the critical Reynol ds number appears to scale with Pm-0.65. If this was true even for very sma ll magnetic Prandtl numbers (e.g. for 10(-5), the magnetic Prandtl number o f 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 simul taneous existence of both sorts of instabilities in related experiments. Si milar phenomena are also discussed for cold accretion disks with their basi cally small magnetic Prandtl numbers.