  We improve on our previous treatments of long-term evolution of protostellar
disks by explicitly solving disk self-gravity in two dimensions. The current
model is an extension of the one-dimensional layered accretion disk model of
Bae et al. We find that gravitational instability (GI)-induced spiral density
waves heat disks via compressional heating (i.e. $P\rm{d}V$ work), and can
trigger accretion outbursts by activating the magnetorotational instability
(MRI) in the magnetically inert disk dead-zone. The GI-induced spiral waves
propagate well inside of gravitationally unstable region before they trigger
outbursts at $R \lesssim 1$ AU where GI cannot be sustained. This long-range
propagation of waves cannot be reproduced with the previously used local
$\alpha$ treatments for GI. In our standard model where zero dead-zone residual
viscosity ($\alpha_{\rm rd}$) is assumed, the GI-induced stress measured at the
onset of outbursts is locally as large as $0.01$ in terms of the generic
$\alpha$ parameter. However, as suggested in our previous one-dimensional
calculations, we confirm that the presence of a small but finite $\alpha_{\rm
rd}$ triggers thermally-driven bursts of accretion instead of the GI +
MRI-driven outbursts that are observed when $\alpha_{\rm rd}=0$. The inclusion
of non-zero residual viscosity in the dead-zone decreases the importance of GI
soon after mass feeding from the envelope cloud ceases. During the infall phase
while the central protostar is still embedded, our models stay in a quiescent
accretion phase with $\dot{M}_{acc}\sim10^{-8}-10^{-7}~M_{\odot}~{\rm yr^{-1}}$
over $60~\%$ of the time and spend less than $15~\%$ of the infall phase in
accretion outbursts.

Solar and Stellar Astrophysics