
\section{Turbulent kinetic energy in the convection zone}
We define the turbulent kinetic energy
in terms of the turbulent velocity fluctuations
$\delta v$ and the density $\rho $ as
\begin{equation}
E_\text{kin}^\text{turb} = \frac{1}{2}\int_\text{PCS} \delta v^2 \rho \mathrm{d}V,
\end{equation}
where the volume element $\mathrm{d}V$ includes the appropriate relativistic metric factors\footnote{The deviation of the Lorentz factor from unity is negligible in the proto-compact star because the fluid velocities are small.}, i.e., $\mathrm{d} V=2\pi \phi^6 r^2 \sin\theta \,\mathrm{d}r\, \mathrm{d}\theta$ in axisymmetry with the lapse function $\phi$. The integral is performed over the entire proto-compact star ($\mathrm{PCS}$), i.e., for densities above $10^{11}\, \mathrm{g}\,\mathrm{cm}^{-3}$. The turbulent velocity fluctuations are given by
\begin{equation}
\delta v^2=(v_r-\langle v_r\rangle)^2+
v_\theta^2,
\end{equation}
where $\langle v_r\rangle$  is the spherical Favre average of the radial velocity.

\begin{figure}[H]
\centering 
	\includegraphics[width=0.8\columnwidth]{plots/ekin_turb.pdf}
    \caption{Turbulent kinetic energy of the PCS as function of post-bounce time for the progenitors \texttt{z35} (grey) and \texttt{z85} (black) using the \texttt{SFHx} EoS (dashed lines) and \texttt{CMF} EoS (solid lines).}
    \label{fig:ekin turb}
\end{figure}

We plot $E_\text{kin}^\text{turb}$ for all four models as function of time in Figure~\ref{fig:ekin turb}. The \texttt{CMF} models show higher turbulent kinetic energies compared to the \texttt{SFHx} models, especially for the lighter \texttt{z35} progenitor. In this case, the turbulent kinetic energy in the PCS convection zone is about an order of magnitude higher in the \texttt{CMF} EoS compared to \texttt{SFHx}. For \texttt{z85}, the difference is modest but robust, and the rise of the turbulent kinetic energy to the plateau at $\sim 5\times 10^{50}\, \mathrm{erg}$ occurs significantly earlier.

\section{Electron fraction gradient}
\begin{figure}[H]
\centering 
	\includegraphics[width=\columnwidth]{plots/dydr_evolv.pdf}
    \caption{Evolution of the gradient $\mathrm{d}Y_e/\mathrm{d}r$
    of the spherically-averaged electron fraction
    as function of post-bounce time for the \texttt{CMF} model (left), \texttt{SFHx} (right) for \texttt{z85} (upper panel) and \texttt{z35} (lower panel). The black circles approximately track the PCS convection zone. Red dashes show buoyantly stable regions where $\omega_\mathrm{BV}^2 > 0$. Those areas are susceptible to quadrupolar perturbations at the frequency of the $g^2_1$-mode. The black-dashed vertical line roughly corresponds to the onset of the $g^2_1$-signal in the GW spectrograms.}
    \label{fig:dydr}
\end{figure}

The differences in the Brunt-V\"ais\"al\"a frequency in the region below the PCS convection zone in the various models affect the frequency of the $g^2_1$-mode and may also have some bearing on its excitation by turbulent motions in the overlying convection zone. Aside from the different sound speed above nuclear saturation density, differences in the Brunt-V\"ais\"al\"a frequency between the \texttt{CMF} and \texttt{SFHx} models also arise because the electron fraction evolves differently during the post-bounce phase. To illustrate this effect we show
the gradient of the spherically-averaged electron fraction.

Initially, all models show a slight negative electron fraction gradient in the PCS core inside $\mathord{\sim}0.7\,\mathrm{M}_\odot$ (inner blue region in
Figure~\ref{fig:dydr}), and then a small ``hump'' that is visible as a red stripe and blue stripe further outside in Figure~\ref{fig:dydr}. With time, neutrino diffusion erases the hump, and the two blue regions merge in the \texttt{CMF} models. This process takes considerably longer in model \texttt{z85:SFHx}, and in model \texttt{z85:SFHx} the hump is still clearly present at the end of the simulation. The evolution of the electron fraction hump in the code is affected by a combination of factors that determine the neutrino opacities in this region, i.e., density, temperature, and neutron, proton, and neutrino chemical potentials. Hence there is no straightforward explanation for the faster disappearance of the hump in the \texttt{CMF} models at this time.
\newpage 
\section{Decomposition of Terms Contributing to the Brunt-V\"ais\"al\"a frequency}
To elucidate how differences in the entropy and electron fraction gradients and various thermodynamic derivatives affect the Brunt-V\"ais\"al\"a frequency at the inner boundary of the PCS convection zone, we rewrite the relativistic
 Brunt-V\"ais\"al\"a frequency as
\begin{align}\label{eq:bv}
    \omega_\text{BV}^2= \dv{\alpha}{r}\frac{\alpha}{\rho h \phi^4}\frac{1}{c_s^2}
    \left[\left(\pdv{ P}{s}\right)_{\!\!\tilde{\rho},Y_e}\dv{s}{r}+\left(\pdv{P}{Y_e}\right)_{\!\tilde{\rho},s}\dv{Y_e}{r}\right].
\end{align}
Here, $\alpha$ is the lapse function, $\phi$ is the conformal factor, $\rho$ is the baryonic mass density, $\tilde{\rho}$ is the total mass-energy density, $P$ is the pressure, $h=(\tilde{\rho}+P/c^2)/\rho$ is the relativistic enthalpy, $s$ is the specific entropy per baryon, $Y_\mathrm{e}$ is the lepton fraction, and $c_\mathrm{s}^2$ is the (adiabatic) sound speed\footnote{Note that the small effect of neutrino pressure and energy density is not included in Equation~(\ref{eq:bv}).}.

The individual terms in Equation~(\ref{eq:bv}) and their products are plotted in Figure~\ref{fig:z85} for the \texttt{z85} models and in Figure~\ref{fig:z35} for the \texttt{z35} models. Each figure shows the results for the \texttt{CMF} models (solid lines) and the \texttt{SFHx} models (dotted) at 4 different times. 

\begin{figure}[H]
\centering 
	\includegraphics[width=0.4\columnwidth]{plots/z35_0.pdf}
    \caption{Individual terms in Equation~(\ref{eq:bv}) for \texttt{z35:CMF} (solid lines) and \texttt{z35:SFHx} (dotted) at four different post-bounce times. The left panel in the fourth row shows the term in square brackets of Equation~\ref{eq:bv}, i.g. $\displaystyle{\sum \equiv \left(\pdv{ P}{s}\right)_{\!\!\tilde{\rho},Y_\mathrm{e}}\dv{s}{r}+\left(\pdv{P}{Y_\mathrm{e}}\right)_{\!\tilde{\rho},s}\dv{Y_\mathrm{e}}{r}}$.}
    \label{fig:z35}
\end{figure}
\begin{figure}[H]
\centering 
	\includegraphics[width=0.4\columnwidth]{plots/z85_0.pdf}
    \caption{Individual terms for \texttt{z85:CMF} (solid lines) and \texttt{z85:SFHx} (dotted) at four different post-bounce times. See Figure~\ref{fig:z35}.}
    \label{fig:z85}
\end{figure}

The figures show that 
$\displaystyle{\left(\pdv{ P}{s}\right)_{\!\!\tilde{\rho},Y_\mathrm{e}}}$ and 
$\displaystyle{\dv{s}{r}}$ are quite different for the \texttt{CMF} and \texttt{SFHx} models, but the product
$\displaystyle{\left(\pdv{ P}{s}\right)_{\!\!\tilde{\rho},Y_e} \dv{s}{r}}$ is rather similar.  The electron fraction gradient becomes steeper in the \texttt{CMF} models, which pushes the $\Sigma$-term in square brackets in Equation~(\ref{eq:bv}),
$$\Sigma=\left(\pdv{ P}{s}\right)_{\!\!\tilde{\rho},Y_e}\dv{s}{r}+\left(\pdv{P}{Y_e}\right)_{\!\tilde{\rho},s}\dv{Y_e}{r}$$
to slightly lower values, compared to \texttt{z85:SFHx} where the radial profile of $\left(\pdv{P}{Y_e}\right)_{\!\tilde{\rho},s}\dv{Y_e}{r}$ is flatter and largely positive. The effect is less pronounced for the more massive progenitor \texttt{z85:CMF}; the $\Sigma$-term here is dominated by a larger entropy gradient. For both progenitors, \texttt{z85} and \texttt{z35}, the sound speed in the region below the PCS convection zone is increasingly higher in the \texttt{CMF} models, which systematically pushes $\omega_\mathrm{BV}$ to lower values. In the case of model \texttt{z85}, this effect explains the difference in the Brunt-V\"ais\"al\"a frequency, as the $\Sigma$-term is similar for the
\texttt{SFHx} and \texttt{CMF} EoS.


\section{Radially resolved GW signals}
For completeness, we show plots analogously to Figure~2 in the main text. Figures~\ref{fig:rgwz85CMF}, \ref{fig:rgwz85SFHx}, and \ref{fig:rgwz35SFHx}
present the results for model \texttt{z85:CMF}, \texttt{z85:SFHx}, and \texttt{z35:SFHx} respectively.
The figures show the radius- and frequency-dependent amplitude $\tilde{q}(r,f)$ of quadrupolar perturbations (first two panels from the left) for two different time intervals $\Delta t_1$ and $\Delta t_2$, which are indicated in the third panel by black and red dash-dotted vertical lines.  The third panels also show the quark fraction (background color; only present in Figure~\ref{fig:rgwz85CMF}) and the adiabatic index (contour lines). The fourth panel shows the squared Brunt-V\"ais\"al\"a frequency and the spherically averaged specific entropy per baryon. 

\begin{figure}[H]
\centering \includegraphics[width=1\columnwidth]{plots/radial_prof_z85_cmf_gamma_cmf_NEW.pdf}
    \caption{Quadrupolar perturbations, quark fraction, adiabatic index, Brunt-V\"ais\"al\"a frequency and spherically averaged entropy for model
    \texttt{z85:CMF}; see detailed description in text.}
    \label{fig:rgwz85CMF}
\end{figure}

\begin{figure}[H]
\centering \includegraphics[width=1\columnwidth]{plots/radial_prof_z85_sfhx.pdf}
    \caption{Quadrupolar perturbations, adiabatic index, Brunt-V\"ais\"al\"a frequency and spherically averaged entropy for model \texttt{z85:SFHx}; see detailed description in text. Note that the \texttt{SFHx} EoS is purely hadronic.}
    \label{fig:rgwz85SFHx}
\end{figure}

\begin{figure}[H]
\centering \includegraphics[width=1\columnwidth]{plots/radial_prof_z35_sfhx.pdf}
    \caption{Analogously to Figure~\ref{fig:rgwz85SFHx} but for model \texttt{z35:SFHx} where no signal is observed.}
    \label{fig:rgwz35SFHx}
\end{figure}


\end{document}