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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  · B ϕ   ≠ 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 ϕ   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  →  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 ϕ /B z   only for the extrema. For  B ϕ  ≫ B z   and for not too small Pm the nonaxisymmetric modes dominate the traveling axisymmetric modes. For standard MRI with  B z  ≫ B ϕ  , 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. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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