Stability of compressible reduced magnetohydrodynamic equilibria - analogy with magnetorotational instability

TL;DR

This paper analyzes the stability of compressible reduced magnetohydrodynamic equilibria using the Energy-Casimir method, compares results with ideal MHD, and reveals a formal analogy with magnetorotational instability in rotating disks.

Contribution

It introduces a stability analysis of CRMHD equilibria, identifies destabilizing and stabilizing effects, and establishes an analogy with MRI in rotating disk models.

Findings

Destabilizing effects correspond to kink and interchange instabilities.

Stability conditions with flow are derived.

Analogy between CRMHD and MRI conditions is established.

Abstract

Stability analyses for equilibria of the compressible reduced magnetohydrodynamics (CRMHD) model are carried out by means of the Energy-Casimir (EC) method. Stability results are compared with those obtained for ideal magnetohydrodynamics (MHD) from the classical {\delta}W criterion. An identification of the terms in the second variation of the free energy functional for CRMHD with those of {\delta}W is made: two destabilizing effects present for CRMHD turn out to correspond to the kink and interchange instabilities in usual MHD, while the stabilizing roles of field line bending and compressibility are also identified in the reduced model. Also, using the EC method, stability conditions in the presence of toroidal flow are obtained. A formal analogy between CRMHD and a reduced incompressible model for magnetized rotating disks, due to Julien and Knobloch [EAS Pub. Series, 21, 81 (2006)], is discovered. In light of this analogy, energy stability analysis shows that the condition for magnetorotational instability (MRI) for the latter model, corresponds to the condition for interchange instability in CRMHD, with the Coriolis term and shear velocity playing the roles of the curvature term and pressure gradient, respectively. Using the EC method, stability conditions for the rotating disk model, for a large class of equilibria with possible non-uniform magnetic fields, are obtained. In particular, this shows it is possible for the MRI system to undergo, in addition to the MRI, another instability that is analogous to the kink instability. For vanishing magnetic field, the Rayleigh hydrodynamical stability condition is recovered.