  We couple two-dimensional hydrodynamics to detailed one-dimensional
multigroup flux-limited diffusion neutrino transport to investigate prompt
convection in core collapse supernovae. Our initial conditions, time-dependent
boundary conditions, and neutrino distributions for computing neutrino heating,
cooling, and deleptonization rates are obtained from one-dimensional
simulations that implement multigroup flux-limited diffusion neutrino transport
and one-dimensional hydrodynamics. The development and evolution of prompt
convection and its ramifications for the shock dynamics are investigated for
both 15 and 25 solar mass models, representative of the two classes of stars
with compact and extended iron cores, respectively. In the absence of neutrino
transport, prompt convection develops and dissipates on a time scale $\sim$15
ms for both models. Prompt convection seeds convection behind the shock, which
causes distortions in the shock's sphericity, but on the average, the shock
radius is not boosted significantly. In the presence of neutrino transport,
prompt convection velocities are too small relative to bulk inflow velocities
to result in any significant convective transport of entropy and leptons. A
simple analytical model supports our numerical results, indicating that the
inclusion of transport reduces the convection growth rates and asymptotic
velocities by factors of 4--250.
