Helical magnetorotational instability of Taylor-Couette flows in the Rayleigh limit and for quasi-Kepler rotation

The magnetorotational instability (MRI) of differential rotation under the simultaneous presence of axial and azimuthal components of the (current-free) magnetic field is considered. For rotation with uniform specific angular momentum the MHD equations for axisymmetric perturbations are solved in a local short-wave approximation. All the solutions are overstable for B_z \cdot B_\phi \neq 0 with eigenfrequencies approaching the viscous frequency. For more flat rotation laws the results of the local approximation do not comply with the results of a global calculation of the MHD instability of Taylor-Couette flows between rotating cylinders. -- With B_phi and B_z of the same order the traveling-mode solutions are also prefered for flat rotation laws such as the quasi-Kepler rotation. For magnetic Prandtl number Pm\to 0 they scale with the Reynolds number of rotation rather than with the magnetic Reynolds number (as for standard MRI) so that they can easily be realized in MHD laboratory experiments. -- Regarding the nonaxisymmetric modes one finds a remarkable influence of the ratio B_\phi /B_z only for the extrema. For B_\phi >> B_z and for not too small Pm the nonaxisymmetric modes dominate the traveling axisymmetric modes. For standard MRI with B_z >> B_\phi, however, the critical Reynolds numbers of the nonaxisymmetric modes exceed the values for the axisymmetric modes by many orders so that they are never prefered.