Magnetic and vertical shear instabilities in accretion discs

Abstract

The stability properties of magnetized discs rotating with angular velocity Ω = Ω(s, z), dependent on both the radial and the vertical coordinates s and z, are considered. Such a rotation law is adequate for many astrophysical discs (e.g., galactic and protoplanetary discs, as well as accretion discs in binaries). In general, the angular velocity depends on height, even in thin accretion discs. A linear stability analysis is performed in the Boussinesq approximation, and the dispersion relation is obtained for short‐wavelength perturbations. Any dependence of Ω on z can destabilize the flow. This concerns primarily small‐scale perturbations for which the stabilizing effect of buoyancy is strongly suppressed due to the energy exchange with the surrounding plasma. For a weak magnetic field, instability of discs is mainly associated with vertical shear, whilst for an intermediate magnetic field the magnetic shear instability, first considered by Chandrasekhar and Velikhov, is more efficient. This instability is caused by the radial shear which is typically much stronger than the vertical shear. Therefore the growth time for the magnetic shear instability is much shorter than for the vertical shear instability. A relatively strong magnetic field can suppress both these instabilities. The vertical shear instability could be the source of turbulence in protoplanetary discs, where the conductivity is low.

Turbulence and its parameterization in accretion discs

“…Without entering the discussion about the reality of non-magnetic turbulence in accretion discs (e.g., Balbus et al 1996), we note that under some circumstances (e.g. in protostellar discs where the electric conductivity is poor) the MRI is not likely to operate, so less efficient mechanisms such as the inflow into the disc during its formation and vertical shear (Urpin & Brandenburg 1998) cannot be excluded as possible agents facilitating turbulence. Also the possibility of nonlinear instabilities (Richard & Zahn 1999;Chagelishvili et al 2003;Afshordi et al 2005) should be mentioned.…”

A comparison study of the vertical and magnetic shear instabilities in accretion discs

“…In convectively stable Keplerian discs (ω 2 g > 0), we have & Brandenburg 1998;Rüdiger et al 2002) and, hence, only the first mode can be unstable if Q 2 < 0. The growth time of instability, τ L , is given by…”

“…However, small-scale motions in discs are likely non-adiabatic. The exchange of heat between perturbations and the surrounding medium substantially reduces the influence of the buoyancy force and decreases the stabilizing effect of stratification (Urpin & Brandenburg 1998). As a result, the stability properties of thermally conducting discs can well be different from those obtained in the adiabatic limit.…”

The stability of magnetized protostellar disks with the Hall effect and buoyancy

“…However, ω 2 g may also be negative if the temperature gradient is subadiabatic but ∆∇T is not parallel to the "effective gravity", G (Knobloch & Spruit 1986;Urpin & Brandenburg 1998;Balbus 2000). This obliqueness can be caused, in principle, either by the dependence of Ω on z or by radiative heat transport in the radial direction.…”