Based on multi-dimensional multi-group radiation 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). Conclusions of this work are threefold: First, as before, we do not see any large-scale overturn of the inner PNS material. Second, we see no evidence of doubly-diffusive instabilities in the PNS, expected to operate on diffusion timescales of at least a second, but instead observe the presence of convection, within a radius range of 10-20km, operating with a timescale of a few milliseconds. Third, we identify unambiguously the presence of gravity waves, predominantly at 200-300milliseconds (ms) past core bounce, in the region separating the convective zones inside the PNS and between the PNS surface and the shocked region. Our numerical study is an improvement over past work in a number of ways: we follow the evolution of the collapsing envelope from ∼200ms before bounce to ∼500ms after bounce; the spatial grid switches from Cartesian inside to spherical outside, permitting a handling of the inner PNS region at good spatial resolution, all the way inside to the center, and without severe Courant-time limitation; neutrino-transport is treated with a Multi-Group, Flux-Limited-Diffusion (MGFLD) approach, wel
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