  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. 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, Z2009b) 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,Z2009a). We also confirm quasi-steady accretion behavior for high
mass infall rates if the values of $\alpha$-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 $\alpha$-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.
