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 characterized 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 α-parametrization, provided that the disc aspect ratio is small (H/r≤ 0.1) which is generally the case when the disc-to-star mass ratio q≲ 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.