Linear and Weakly Nonlinear Analysis of the Magneto-Rotational-Instability in Thin Keplerian Discs

TL;DR

This paper performs a linear and weakly nonlinear analysis of the magnetorotational instability in thin Keplerian discs, deriving explicit growth rates and mode structures, and shows saturation by stable magnetoacoustic modes near the threshold.

Contribution

It provides explicit solutions for the MRI growth rates and mode structures in thin discs, including a weakly nonlinear analysis showing mode saturation.

Findings

Explicit expressions for MRI growth rates and unstable modes.

Decoupling of in-plane and vertical perturbations.

Saturation of MRI by stable magnetoacoustic modes near threshold.

Abstract

The linear instability of thin, vertically-isothermal Keplerian discs, under the influence of axial magnetic field is investigated. Solutions of the stability problem are found explicitly by asymptotic expansions in the small aspect ratio of the disc. It is shown that the perturbations are decoupled into in-plane and vertical modes. Exact expressions for the growth rates as well as the number of unstable modes are derived. Those are the discrete counterpart of the continuous infinite homogeneous cylinder magnetorotational (MRI) spectrum. In addition, a weakly nonlinear analysis of the MRI is performed. It is shown that near the instability threshold the latter is saturated by the stable magnetoacoustic modes.