Magnetorotational instability in Keplerian disks: a nonlocal approach

We revisit the modal analysis of small perturbations in Keplerian ideal gas flows with a constant vertical magnetic field leading to magnetorotational instability (MRI) using the nonlocal approach. In the general case, MRI modes are described by a Schr$\ddot {\rm o}$dinger-like differential equation with some effective potential, including ‘repulsive’ ($1/r^{2}$) and ‘attractive’ ($-1/r^{3}$) terms, and are quantized. In shallow potentials, there are no stationary ‘energy levels.’ In thin Keplerian accretion discs, the perturbation wavelengths $\lambda =2\pi /k_{z}$ are smaller than the disc semi-thickness $h$ only in ‘deep’ potential wells. We find that there is a critical magnetic field for the MRI to develop. The instability arises for magnetic fields below this critical value. In thin accretion discs, at low background Alfv$\acute {\rm e}$n velocity $c_{\rm A}\ll (c_{\rm A})_{\rm cr}$, the MRI instability increment $\omega $ is suppressed compared to the value obtained in the local perturbation analysis, $\omega \approx -\sqrt {3}{\rm i}c_{\rm A}k_{z}$. We also investigate for the first time the case of a radially variable background magnetic field.