As an initial investigation into the long-term evolution of protostellar disks, we explore the conditions required to explain the large outbursts of disk accretion seen in some young stellar objects. In Zhu et al. 2010a, we use one-dimensional time-dependent disk models with a phenomenological treatment of the magnetorotational instability (MRI) and gravitational torques to follow disk evolution over long timescales. Comparison with our previous two-dimensional disk model calculations (Zhu et al. 2009b) indicates that the neglect of radial effects and two-dimensional disk structure in the one-dimensional case makes only modest differences in the results; this allows us to use the simpler models to explore parameter space efficiently. We find that the mass infall rates typically estimated for low-mass protostars generally result in AU-scale disk accretion outbursts, as predicted by our previous analysis (Zhu et al 2009a). We also confirm quasi-steady accretion behavior for high mass infall rates if the values of a-parameter for the magnetorotational instability is small, while at this high accretion rate convection from the thermal instability may lead to some variations. We further constrain the combinations of the a-parameter and the MRI critical temperature, which can reproduce observed outburst behavior. Our results suggest that dust sublimation may be connected with full activation of the MRI. This is consistent with the idea that small dust captures ions and electrons to suppress the MRI. In a later paper we will explore both long-term outburst and disk evolution with this model, allowing for infall from protostellar envelopes with differing angular momenta.

In Zhu et al. 2010 b, we use one-dimensional two-zone time-dependent accretion disk models to study the long-term evolution of protostellar disks subject to mass addition from the collapse of a rotating cloud core. Our model consists of a constant surface density magnetically coupled active layer, with transport and dissipation in inactive regions only via gravitational instability. We start our simulations after a central protostar has formed, containing ~ 10% of the mass of the protostellar cloud. Subsequent evolution depends on the angular momentum of the accreting envelope. We find that disk accretion matches the infall rate early in the disk evolution because much of the inner disk is hot enough to couple to the magnetic field. Later infall reaches the disk beyond ~10 AU, and the disk undergoes outbursts of accretion in FU Ori-like events as described in Zhu et al. 2009c. If the initial cloud core is moderately rotating most of the central star's mass is built up by these outburst events. Our results suggest that the protostellar ``luminosity problem'' is eased by accretion during these FU Ori-like outbursts. After infall stops the disk enters the T Tauri phase. An outer, viscously evolving disk has structure that is in reasonable agreement with recent submillimeter studies and its surface density evolves from S~ R-1 to R-1.5. An inner, massive belt of material-- the ``dead zone'' -- would not have been observed yet but should be seen in future high angular resolution observations by EVLA and ALMA. This high surface density belt is a generic consequence of low angular momentum transport efficiency at radii where the disk is magnetically decoupled, and would strongly affect planet formation and migration.