Magnetic shear-flow instability in thin accretion discs

The possibility that the magnetic shear-flow instability (also known as the 'Balbus-Hawley' instability) might give rise to turbulence in a thin accre tion disc is investigated through numerical simulations. The study is Linea r and the fluid disc is supposed to be incompressible and differentially ro tating with a simple velocity profile with Omega proportional to R-q. The s implicity of the model is counterbalanced by the fact that the study is ful ly global in all three spatial directions with boundaries on each side; fin ite diffusivities are also allowed. The investigation is also carried out f or several values of the azimuthal wavenumber of the perturbations in order to analyse whether non-axisymmetric modes might be preferred, which may pr oduce, in a non-linear extension of the study, a self-sustained magnetic fi eld. We find the final pattern steady, with similar kinetic and magnetic energie s and the angular momentum always transported outwards. Despite the differe ntial rotation, there are only small differences for the eigenvalues for va rious non-axisymmetric eigensolutions. Axisymmetric instabilities are by no means preferred; in fact for Prandtl numbers between 0.1 and 1, the azimut hal wavenumbers m = 0, 1, 2 appear to be equally readily excited. The equat orial symmetry is quadrupolar for the magnetic field and dipolar for the no w field system. The maximal magnetic field strength required to cause the i nstability is almost independent of the magnetic Prandtl number. With typic al white dwarf values, a magnetic amplitude of 10(5) G is estimated.