  We present a detailed analysis of the dynamics of proto-compact star (PCS)
convection and the core ${}^2\!g_1$-mode in core-collapse supernovae based on
general relativistic 2D and 3D neutrino hydrodynamics simulations. Based on 2D
simulations, we derive a mode relation for the core $g$-mode frequency in terms
of PCS and equation of state parameters, and discuss its limits of accuracy.
This relation may prove useful for parameter inference from future supernova
gravitational wave (GW) signals if the core $g$-mode or an emission gap at the
avoided crossing with the fundamental mode can be detected. The current 3D
simulation does not show GW emission from the core $g$-mode due to less power
in high-frequency convective motions to excite the mode, however. Analysing the
dynamics of PCS convection in 3D, we find that simple scaling laws for
convective velocity from mixing-length theory (MLT) do not apply. Energy and
lepton number transport is instead governed by a more complex balance between
neutrino fluxes and turbulent fluxes that results in roughly uniform rates of
change of entropy and lepton number in the PCS convection zone. Electron
fraction and enthalpy contrasts in PCS convection are not well captured by the
MLT gradient approximation. We find distinctly different spectra for the
turbulent kinetic energy and turbulent fluctuations in the electron fraction,
which scale approximately as $l^{-1}$ without a downturn at low $l$. We suggest
that the different turbulence spectrum of the electron fraction is naturally
expected for a passive scalar quantity.
