  The source of viscosity in astrophysical accretion flows is still a hotly
debated issue. We investigate the contribution of convective turbulence to the
total viscosity in a self-consistent approach, where the strength of convection
is determined from the vertical disc structure itself. Additional sources of
viscosity are parametrized by a beta-viscosity prescription, which also allows
an investigation of self-gravitating effects. In the context of accretion discs
around stellar mass and intermediate mass black holes, we conclude that
convection alone cannot account for the total viscosity in the disc, but
significantly adds to it. For accretion rates up to 10% of the Eddington rate,
we find that differential rotation provides a sufficiently large underlying
viscosity. For higher accretion rates, further support is needed in the inner
disc region, which can be provided by an MRI-induced viscosity. We briefly
discuss the interplay of MRI, convection and differential rotation. We conduct
a detailed parameter study of the effects of central masses and accretion rates
on the disc models and find that the threshold value of the supporting
viscosity is determined mostly by the Eddington ratio with only little
influence from the central black hole mass.
