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 } \gg B _ { z } and for not too small Pm the nonaxisymmetric modes dominate the traveling axisymmetric modes .
For standard MRI with B _ { z } \gg 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 .