On the global stability of magnetized accretion discs — III. Non-axisymmetric modes

We investigate the global stability of a differentially rotating, ideal MHD fluid shell to linear, non-axisymmetric perturbations. This system, which approximates an accretion disc near its mid-plane, is known to be unstable to both axisymmetric and non-axisymmetric local perturbations. We find two distinct classes of globally unstable modes. One of these is a magnetic analogue of the Papaloizou-Pringle (PP) instability of thick hydrodynamic tori. The other is a pure Alfvénic mode coupled to the rotation frequency of the shell. For shells of sufficient radial extent, these two modes merge with the consequence that enhanced dynamical growth occurs. Keplerian discs, which are entirely stable to the hydrodynamic PP mode, show the most rapid growth for radially extended shells. In general, the largest growth rates are obtained in the limit of high vertical and low azimuthal wavenumber, and do not exceed the local axisymmetric rates. In more slender systems, however, the magnetic PP instability can exhibit more rapid growth than in the axisymmetric case. By calculating the critical Alfvén speed for stability, we show that discs large in both radial and vertical extent should be more unstable to non-axisymmetric modes than to axisymmetric ones. Since the instability acts for all allowable angular momentum distributions, both thick (i.e., radiation-pressure-supported and ion tori) and thin discs should be equally affected.