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

Abstract

The 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 the equilibrium configurations the perturbations are decoupled into in‐plane and vertical modes. For equilibria of type I those two modes are the Alfvén–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.

Linear and weakly nonlinear analysis of the magneto-rotational-instability in thin keplerian discs

“…Here however, we focus on the equilibrium that is characterized by comparable magnitudes of the axial and toroidal components of the magnetic field, and in particular will consider the case of a pure axial equilibrium magnetic field. As is shown in [6], the results may be easily extended to the more general case of a comparable toroidal component. All equilibrium variables are written in the leading order in ε, and depend on the radial variable only.…”

“…The two equilibria are distinguished by different scaling of the physical variables with ε. A detailed description of those two equilibria may be found in [6]. Here however, we focus on the equilibrium that is characterized by comparable magnitudes of the axial and toroidal components of the magnetic field, and in particular will consider the case of a pure axial equilibrium magnetic field.…”

“…z/r * ∼ ε (|z| < ∼ H * ), and consequently the following rescaled quantities may be introduced in order to further apply the asymptotic expansions in ε [(similar to [6]; [7]; [9]):…”

“…As a first step, all physical variables are transformed to non-dimensional quantities by using the following characteristic values ( [6]; [7]):…”

“…Traditionally, results of the analytical study of magneto-rotational instability (MRI) in an infinitely long cylinder ( [1]; [2]) have been adopted for thin disks in order to derive criteria for the spectral stability under various conditions ( [3]). In addition, it has recently been shown that if the thin disk geometry is taken into account both the growth rates as well as the number of unstable MRI modes are greatly reduced and are decreasing functions of the disk thickness ( [4]; [5]; [6]). The aim of the current work is twofold:1. to provide a complete description of the stable as well as of the unstable spectrum of thin disks under the influence of an axial magnetic field, and 2. to carry out a weakly nonlinear analysis of the MRI in order to investigate its saturation mechanism.…”