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 accretion disc is investigated through numerical simulations. The study is linear and the fluid disc is supposed to be incompressible and differentially rotating with a simple velocity profile with Ω∝ R ^{− q} . The simplicity of the model is counterbalanced by the fact that the study is fully global in all three spatial directions with boundaries on each side; finite diffusivities are also allowed. The investigation is also carried out for several values of the azimuthal wavenumber of the perturbations in order to analyse whether non-axisymmetric modes might be preferred, which may produce, in a non-linear extension of the study, a self-sustained magnetic field.
  We find the final pattern steady, with similar kinetic and magnetic energies and the angular momentum always transported outwards. Despite the differential rotation, there are only small differences for the eigenvalues for various non-axisymmetric eigensolutions. Axisymmetric instabilities are by no means preferred; in fact for Prandtl numbers between 0.1 and 1, the azimuthal wavenumbers m =0,1,2(10¹⁶ g s^-1). All three quantities appear to be equally readily excited. The equatorial symmetry is quadrupolar for the magnetic field and dipolar for the flow field system. The maximal magnetic field strength required to cause the instability is almost independent of the magnetic Prandtl number. With typical white dwarf values, a magnetic amplitude of 10⁵ G is estimated.