  Based on multi-dimensional, multi-group, flux-limited-diffusion hydrodynamic
simulations of core-collapse supernovae with the VULCAN/2D code, we study the
physical conditions within and in the vicinity of the nascent protoneutron star
(PNS). Our numerical study follows the evolution of the collapsing envelope of
the 11-Msun model of Woosley & Weaver, from ~200 ms before bounce to ~300ms
after bounce, on a spatial grid that switches from Cartesian at the PNS center
to spherical above a 10km radius.
  As has been shown previously, we do not see any large-scale overturn of the
inner PNS material. Convection, directly connected to the PNS, is found to
occur in two distinct regions: between 10 and 20km, coincident with the region
of negative lepton gradient, and exterior to the PNS above 50km. Separating
these two regions, an interface, with no sizable outward or inward motion, is
the site of gravity waves, emerging at ~200-300ms after core bounce, excited by
the convection in the outer convective zone.
  In the PNS, convection is always confined within the neutrinospheric radii
for all neutrino energies above just a few MeV. We find that such convective
motions do not appreciably enhance the nu_e neutrino luminosity, and that they
enhance the anti-nu_e and "nu_mu" luminosities modestly, by ~15% and ~30%,
respectively, during the first post-bounce 100--200ms. Moreover, we see no
evidence of doubly-diffusive instabilities in the PNS, expected to operate on
diffusion timescales of at least a second, much longer than the millisecond
timescale associated with PNS convection.
  PNS convection is thus found to be a secondary feature of the core-collapse
phenomenon, rather than a decisive ingredient for a successful explosion.
