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.