It is demonstrated that the azimuthal magnetorotational instability (AMRI)also works with radially increasing rotation rates contrary to the standardmagnetorotational instability for axial fields which requires negative shear.The stability against nonaxisymmetric perturbations of a conductingTaylor-Couette flow with positive shear under the influence of a toroidalmagnetic field is considered if the background field between the cylinders iscurrent-free. For small magnetic Prandtl number Pm-->0 the curves of neutralstability converge in the Hartmann number/Reynolds number plane approximatingthe stability curve obtained in the inductionless limit Pm=0. The numericalsolutions for Pm=0 indicate the existence of a lower limit of the shear rate.For large Pm the curves scale with the magnetic Reynolds number of the outercylinder but the flow is always stable for magnetic Prandtl number unity as istypical for double-diffusive instabilities. We are particularly interested toknow the minimum Hartmann number for neutral stability. For models with restingor almost resting inner cylinder and with perfect-conducting cylinder materialthe minimum Hartmann number occurs for a radius ratio of 0.9. The correspondingcritical Reynolds numbers are smaller than 10.000.
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