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. We perform a linear analysis of the evolution of perturbations in the shearing sheet model. 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. Differential rotation can transiently stabilize slow MHD waves with a vertical wavelength longer than the scale height of the disc, against the Parker instability. 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, i.e. 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. These analytical results, obtained in a WKB approximation, are found to be in excellent agreement with numerical solutions of the full set of linearized equations. Our results can be applied to both galactic and accretion discs.