  We present a sub-grid model that emulates the magnetic dynamo operating in
magnetized accretion disks. We have implemented this model in the general
relativisic radiation magnetohydrodynamic (GRRMHD) code \koral, using results
from local shearing sheet simulations of the magnetorotational instability to
fix the parameters of the dynamo. With the inclusion of this dynamo, we are
able to run 2D axisymmetric GRRMHD simulations of accretion disks for
arbitrarily long times. The simulated disks exhibit sustained turbulence, with
the poloidal and toroidal magnetic field components driven towards a state
similar to that seen in 3D studies. Using this dynamo code, we present a set of
long-duration global simulations of super-Eddington, optically-thick disks
around non-spinning and spinning black holes. Super-Eddington disks around
non-rotating black holes exhibit a surprisingly large efficiency,
$\eta\approx0.04$, independent of the accretion rate, where we measure
efficiency in terms of the total energy output, both radiation and mechanical,
flowing out to infinity. Super-Eddington disks around spinning black holes are
even more efficient, and appear to extract black hole rotational energy through
a process similar to the Blandford-Znajek mechanism. All the simulated models
are characterized by highly super-Eddington radiative fluxes collimated along
the rotation axis. We also present a set of simulations that were designed to
have Eddington or slightly sub-Eddington accretion rates ($\dot{M} \lesssim
2\dot M_{\rm Edd}$). None of these models reached a steady state. Instead, the
disks collapsed as a result of runaway cooling, presumably because of a thermal
instability.
