Semi-analytic models of self-gravitating discs often approximate the angular momentum transport generated by the gravitational instability using the phenomenology of viscosity .
This allows the employment of the standard viscous evolution equations , and gives promising results .
It is , however , still not clear when such an approximation is appropriate .

This paper tests this approximation using high resolution 3D smoothed particle hydrodynamics ( SPH ) simulations of self-gravitating protostellar discs with radiative transfer .
The nature of angular momentum transport associated with the gravitational instability is characterised as a function of both the stellar mass and the disc-to-star mass ratio .
The effective viscosity is calculated from the Reynolds and gravitational stresses in the disc .
This is then compared to what would be expected if the effective viscosity were determined by assuming local thermodynamic equilibrium or , equivalently , that the local dissipation rate matches the local cooling rate .

In general , all the discs considered here settle into a self-regulated state where the heating generated by the gravitational instability is modulated by the local radiative cooling .
It is found that low-mass discs can indeed be represented by a local \alpha -parametrisation , provided that the disc aspect ratio is small ( H / R \leq 0.1 ) which is generally the case when the disc-to-star mass ratio q \lesssim 0.5 .
However , this result does not extend to discs with masses approaching that of the central object .
These are subject to transient burst events and global wave transport , and the effective viscosity is not well modelled by assuming local thermodynamic equilibrium .
In spite of these effects , it is shown that massive ( compact ) discs can remain stable and not fragment , evolving rapidly to reduce their disc-to-star mass ratios through stellar accretion and radial spreading .