Local stability analysis of the azimuthal magnetorotational instability of ideal MHD flows

A local stability analysis is carried out of axisymmetric rotating flows of a perfectly conducting fluid subjected to an external azimuthal magnetic field Bθ to non-axisymmetric as well as axisymmetric perturbations. We use the Hain-Lüst equation, capable of dealing with perturbations over a wide range of the axial wavenumber k, which supplements the previous results of |k| → ∞ [G. I. Ogilvie and J. E. Pringle, Mon. Not. R. Astron. Soc. 279, 152 (1996)]. When the magnetic field is sufficiently weak, instability occurs for Ro = rΩ′/(2Ω) < Ro_c with Ro_c close to zero, with the maximum growth rate given by the Oort A-value |Ro|, where Ω(r) is the angular velocity of the rotating flow as a function only of r, the distance from the axis of symmetry, and the prime designates the derivative in r. As the magnetic field is increased the flow becomes unstable to waves of long axial wavelength for a finite range of Ro around 0 when Rb = r²(Bθ/r)′/(2Bθ) < -1/4 with growth rate proportional to |Bθ|, giving unstable rotating flows under the current-free background magnetic field.