  We construct models of core-collapse supernovae in one spatial dimension,
including rotation, angular momentum transport, and viscous dissipation
employing an alpha-prescription. We compare the evolution of a fiducial 11
M_sun non-rotating progenitor with its evolution including a wide range of
imposed initial rotation profiles (1.25<P_0<8 s, where P_0 is the initial,
approximately solid-body, rotation period of the iron core). This range of P_0
covers the region of parameter space from where rotation begins to modify the
dynamics (P_0~8 s) to where angular velocities at collapse approach Keplerian
(P_0~1 s). Assuming strict angular momentum conservation, all models in this
range leave behind neutron stars with spin periods <10 ms, shorter than those
of most radio pulsars, but similar to those expected theoretically for
magnetars at birth. A fraction of the gravitational binding energy of collapse
is stored in the free energy of differential rotation. This energy source may
be tapped by viscous processes, providing a mechanism for energy deposition
that is not strongly coupled to the mass accretion rate through the stalled
supernova shock. This effect yields qualitatively new dynamics in models of
supernovae. We explore several potential mechanisms for viscosity in the
core-collapse environment: neutrino viscosity, turbulent viscosity caused by
the magnetorotational instability (MRI), and turbulent viscosity by entropy-
and composition-gradient-driven convection. We argue that the MRI is the most
effective. We find that for rotation periods in the range P_0<~5 s, and a range
of viscous stresses, that the post-bounce dynamics is significantly effected by
the inclusion of this extra energy deposition mechanism; in several cases we
obtain strong supernova explosions.
