Magnetorotational instability in Keplerian discs: a nonlocal approach

We revisit the modal analysis of small perturbations in Keplerian ideal gas flows with a constant vertical magnetic field leading to magneto-rotational instability (MRI) using the nonlocal approach. In the general case, MRI modes are described by a Schro center dot 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 = 27t/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 Alfven velocity c(A) << (c(A))(cr), the MRI instability increment omega is suppressed compared to the value obtained in the local perturbation analysis, omega approximate to -root 3ic(A)k(z). We also investigate for the first time the case of a radially variable background magnetic field.