Short-wavelength analysis of magnetorotational instability of resistive MHD flows (Mathematical Analysis of Viscous Incompressible Fluid)

Abstract

Local stability analysis is made of axisymmetric rotating flows of a perfectly conducting fluid and resistive flows with viscosity, subjected to external azimuthal magnetic field B_{$theta$} to non-axisymmetric as well as axisymmetric perturbations. For perfectly conducting fluid (ideal MHD), we use the Hain-Lüst equation, capable of dealing with perturbations over a wide range of the axial wavenumber k to take short wavelength approximation. When the magnetic field is sufficiently weak, the maximum growth rate is given by the Oort A-value |Ro|, where $Omega$(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 keplerian flow becomes unstable to waves of short axial wavelength when Rb=r^{2}(B_{$theta$}/r)'/(2B_{$theta$})>-3/4 with growth rate proportional to |B_{ $theta$}|. We also incorporate the effect of the viscosity and the electric resistivity and apply the WKB method in the same way as we do to the perfectly conducting fluid. In the inductionless limit, i.e. when the magnetic diffusivity is much larger than the viscosity, Keplerian-rotation flow of arbitrary distributions of the magnetic field, including the Liu limit, becomes unstable.