2009MNRAS.393..822V

We add the effect of turbulent viscosity via the α-prescription to models of the self-consistent formation and evolution of protostellar discs. Our models are non-axisymmetric and are carried out using the thin-disc approximation. Self-gravity plays an important role in the early evolution of a disc, and the later evolution is determined by the relative importance of gravitational and viscous torques. In the absence of viscous torques, a protostellar disc evolves into a self-regulated state with the Toomre parameter Q ∼ 1.5 - 2.0, non-axisymmetric structure diminishing with time and maximum disc-to-star mass ratio ξ = 0.14. We estimate an effective viscosity parameter α_eff associated with gravitational torques at the inner boundary of our simulation to be in the range 10^–4-10^–3 during the late evolution. The addition of viscous torques with a low value α = 10^–4 has little effect on the evolution, structure and accretion properties of the disc, and the self-regulated state is largely preserved. A sequence of increasing values of α results in the discs becoming more axisymmetric in structure, being more gravitationally stable, having greater accretion rates, larger sizes, shorter lifetimes and lower disc-to-star mass ratios. For α = 10^–2, the model is viscous-dominated, and the self-regulated state largely disappears by late times. The axisymmetry and low surface density of this model may contrast with observations and pose problems for planet formation models. The use of α = 0.1 leads to very high disc accretion rates and rapid (within 2 Myr) depletion of the disc, and seems even less viable observationally. Furthermore, only the non-viscous-dominated models with low values of α = 10^–4-10^–3 can account for an early phase of quiescent low accretion rate (interspersed with accretion bursts) that can explain the recently observed Very Low luminosity Objects (VeLLOs). We also find that a modest increase in disc temperature caused by a stiffer barotropic equation of state (γ = 1.67) has little effect on the disc accretion properties averaged over many disc orbital periods (∼10⁴ yr), but can substantially influence the instantaneous mass accretion rates, particularly in the early embedded phase of disc evolution.