Implications of the Co-rotation Theorem on the MRI in Axial Symmetry

TL;DR

This paper investigates how the co-rotation theorem influences the Magneto-Rotational Instability in axially symmetric plasma disks, revealing that it simplifies the vertical angular velocity dependence and affects the mode structure.

Contribution

It demonstrates the impact of the co-rotation theorem on the linear stability analysis of MRI, clarifying mode morphology and vertical dependence in stratified disks.

Findings

Co-rotation theorem cancels vertical angular velocity derivatives.

Unstable modes exhibit thin disk morphology.

Vertical dependence is parametrically simple.

Abstract

We analyze the linear stability of an axially symmetric ideal plasma disk, embedded in a magnetic field and endowed with a differential rotation. This study is performed by adopting the magnetic flux function as the fundamental dynamical variable, in order to outline the role played by the co-rotation theorem on the linear mode structure. Using some specific assumptions (e.g. plasma incompressibility and propagation of the perturbations along the background magnetic field), we select the Alfvenic nature of the Magneto-Rotational Instability and, in the geometric optics limit, we determine the dispersion relation describing the linear spectrum. We show how the implementation of the co-rotation theorem (valid for the background configuration) on the linear dynamics produces the cancellation of the vertical derivative of the disk angular velocity (we check such a feature also in the standard vector formalism to facilitate comparison with previous literature, both in the axisymmetric and three-dimensional case). As a result, we clarify that the unstable modes have, for a stratified disk, the same morphology, proper of a thin disk profile, and the $z$ dependence has a simple parametric role.