Context :

Aims : The stability of dissipative Taylor-Couette flows with an axial stable density stratification and a prescribed azimuthal magnetic field is considered .

Methods : Global nonaxisymmetric solutions of the linearized MHD equations with toroidal magnetic field , axial density stratification and differential rotation are found for both insulating and conducting cylinder walls .

Results : Flat rotation laws such as the quasi-Kepler law are unstable against the nonaxisymmetric stratorotational instability ( SRI ) .
The influence of a current-free toroidal magnetic field depends on the magnetic Prandtl number Pm : SRI is supported by Pm > 1 and it is suppressed by Pm \lower 1.72 pt \hbox { $ \buildrel < \over { \scriptstyle \sim } $ } 1 .
For too flat rotation laws a smooth transition exists to the instability which the toroidal magnetic field produces in combination with the differential rotation .
This nonaxisymmetric azimuthal magnetorotational instability ( AMRI ) has been computed under the presence of an axial density gradient .

If the magnetic field between the cylinders is not current-free then also the Tayler instability occurs and the transition from the hydrodynamic SRI to the magnetic Tayler instability proves to be rather complex .
Most spectacular is the ‘ ballooning ’ of the stability domain by the density stratification : already a rather small rotation stabilizes magnetic fields against the Tayler instability .

An azimuthal component of the resulting electromotive force only exists for density-stratified flows .
The related alpha-effect for magnetic SRI of Kepler rotation appears to be positive for negative d \rho / dz < 0 .

Conclusions :