Convection and the Core $g$-mode in Proto-Compact Stars -- A detailed analysis

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.