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.