The Parker-Shearing instability in azimuthaly magnetized discs

We describe the effects of both magnetic buoyancy and differential rotation on a disc of isothermal gas embedded in a purely azimuthal magnetic field, in order to study the evolution and interplay of Parker and shearing instabilities. Both instabilities occur on the slow MHD branch of the dispersion relation, and can affect the same waves. We put a stress on the natural polarization properties of the slow MHD waves to get a better understanding of the physics involved. The mechanism of the shearing instability is described in details. Waves with a vertical wavelength shorter than the scale height of the disc are subject to both the Parker and the transient shearing instabilities. They occur in different ranges of radial wavenumbers, \ie at different times in the shearing evolution ; these ranges can overlap or, on the contrary, be separated by a phase of wave-like oscillations, depending on the strength of differential rotation. Our results can be applied to both galactic and accretion discs.