Spectral and algebraic instabilities in thin Keplerian discs under poloidal and toroidal magnetic fields

TL;DR

This paper analyzes the linear stability of thin Keplerian discs with poloidal or toroidal magnetic fields, revealing how magnetic field configuration influences the types of instabilities and growth mechanisms present.

Contribution

It provides explicit solutions for the stability problem using asymptotic expansions, and shows the suppression of axisymmetric MRI by toroidal fields, highlighting non-axisymmetric and algebraic growth mechanisms.

Findings

Axisymmetric MRI is suppressed by dominant toroidal magnetic fields.

Non-axisymmetric MRI or algebraic growth mechanisms can occur in these discs.

Algebraic growth is driven by rotation shear and involves non-resonant or resonant mode coupling.

Abstract

Linear instability of two equilibrium configurations with either poloidal (I) or toroidal (II) dominant magnetic field components are studied in thin vertically-isothermal Keplerian discs. Solutions of the stability problem are found explicitly by asymptotic expansions in the small aspect ratio of the disc. In both equilibrium configurations the perturbations are decoupled into in-plane and vertical modes. For equilibria of type I those two modes are the Alfv\'en-Coriolis and sound waves, while for equilibria of type II they are the inertia-Coriolis and magnetosonic waves. Exact expressions for the growth rates as well as the number of unstable modes for type I equilibria are derived. Those are the discrete counterpart of the continuous infinite homogeneous cylinder magnetorotational (MRI) spectrum. It is further shown that the axisymmetric MRI is completely suppressed by dominant toroidal magnetic fields (i.e. equilibria of type II). This renders the system prone to either non-axisymmetric MRI or non-modal algebraic growth mechanisms. The algebraic growth mechanism investigated in the present study occurs exclusively due to the rotation shear, generates the inertia-Coriolis driven magnetosonic modes due to non-resonant or resonant coupling that induces, respectively, linear or quadratic temporal growth of the perturbations.