Analysis of azimuthal magnetorotational instability and Tayler instability of rotating MHD flows via an extended Hain-Lust equation

Short-wavelength local analysis and numerical global analysis are made of the azimuthal magnetorotational instability of rotating flows of an electrically conducting fluid, subject to an arbitrary but axisymmetric azimuthal magnetic field, to three-dimensional disturbances. We extend the Hain-L\"ust equation for the radial Lagrangian displacements by incorporating the effect of the viscosity and electrical resistivity and apply the Wentzel-Kramers-Brillouin method to it. We confirm that in the inductionless limit, in which the magnetic diffusivity is much larger than the viscosity, rotating flows with radial distributions of the angular velocity beyond the Liu limit, become unstable subject to a wide variety of the azimuthal magnetic fields, and so is the Keplerian flow. Our results compare well with the numerical global stability analysis.