On the stability of an accretion disc containing a toroidal magnetic field

We study the stability of an accretion disc with an embedded toroidal magnetic field to general perturbations. Disc models are considered in which the equilibrium variables depend on both the radial and vertical coordinates. We consider the full global problem in which the disc may be in the form of a narrow annulus, or occupy a significant radial extent. Perturbations with azimuthal mode number m in the range zero up to the ratio of the radius to disc semithickness are considered. Discs containing a purely toroidal magnetic field are always found to be unstable. We find spectra of unstable modes using local techniques. In the absence of dissipation, these modes may occupy arbitrarily small scales in the radial and vertical directions. One class of modes is driven primarily by buoyancy, while the other is driven by shear independently of the equilibrium stratification. The first type of instability predominates if the field is large, while the second type predominates if the field is weak and the underlying medium is strongly stable to convection. We also investigate stability by solving the initial value problem for perturbations numerically. We find, for our disc models, that local instabilities predominate over any possible global instability. Their behaviour is in good accord with the local analysis. The associated growth rates become just less than the orbital frequency when the ratio of magnetic energy density to pressure reaches about 10 per cent. Instabilities of the kinds discussed here may provide a mechanism for limiting the growth of toroidal fields in dynamo models of accretion discs.