Taylor-Couette flow: MRI, SHI and SRI
  The linear stability theory of Taylor-Couette flows (unbounded in_z_) is described including magnetic fields, Hall effect or a density stratification in order to prepare laboratory experiments to probe the stability of differential rotation in astrophysics. If an axial field is present then the magnetorotational instability (MRI) is investigated also for small magnetic Prandtl numbers. For rotating outer cylinder beyond the Rayleigh line characteristic minima are found for magnetic Reynolds number of the order of 10 and for Lundquist numbers of order 1. The inclusion of extra toroidal current-free fields leads to new oscillating solutions with rather low Reynolds numbers and Hartmann numbers. The Hall effect establishes an unexpected `shear-Hall instability' (SHI) where shear and magnetic field have opposite signs. In this case even rotation laws increasing outwards may become unstable. Recently global solutions have been found for the Taylor-Couette flows with density stratified in axial directions (`stratorotational instability', SRI). They exist beyond the Rayleigh line for Froude numbers of moderate order but also not for too flat rotation laws with \hat\eta^2 < \hat\mu <\hat \eta. 
Guenther Ruediger