Instabilities of rotational flows in azimuthal magnetic fields of arbitrary radial dependence

Using the Wentzel–Kramers–Brillouin (WKB) approximation we perform a linear stability analysis for a rotational flow of a viscous and electrically conducting fluid in an external azimuthal magnetic field that has an arbitrary radial profile. In the inductionless approximation, we find the growth rate of the three-dimensional perturbation in a closed form and demonstrate in particular that it can be positive when the velocity profile is Keplerian and the magnetic field profile is slightly shallower than 1/R.