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\section{Propagation diagrams for B16 models}

The essence of the B16 envelope models is that the enthalpy flux is
not just given by a term proportional to the super-adiabatic gradient,
but there is an additional term, $-\tau\bra{s^2}\grav\rho T/c_p$, also
called the Deardorff term, which carries enthalpy outward, even in layers
that are (weakly) Schwarzschild-stable.

The deeper layers of the convection zone can indeed be weakly
Schwarzschild-stable when the Deardorff term is driven by mechanical
stirring through descending plumes.
This phenomenon is superficially reminiscent of overshoot, but,
unlike standard overshoot where the enthalpy flux is inward,
the Deardorff flux is always outward.

Given the presence of Schwarzschild-stable layers, there is the
possibility of trapping g-modes closer to the surface.
The question is therefore what their properties are and whether
they could be observable.
In the following, we compute wave propagation diagrams for such models.
When comparing with earlier propagation diagrams that employed
standard mixing length models, it should be noted that
the B16 envelope models employed simplistic prescriptions for the
opacity and ignored partial ionization effects.
They also ignored the spherical geometry, i.e., gravity and total flux
were constant, instead of varying like $r^{-2}$.
(In reality, $4\pi r^2 F_{\rm tot}$ and $4\pi r^2 g_{\rm r}$ would
be constant.)
The depth of the convection zone was in all cases less than
$200\Mm$, so the bottom of the convection zone was at a fractional
radius $r/R$ that was larger than 0.7.
Furthermore, the propagation diagrams will be unrealistic in the deeper
parts due to the neglect of spherical effects, but they may still give
an idea about the anticipated effects.

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pmodels_nabD1_xi1p5_kfHp00}
\end{center}\caption[]{
Case I, beta=tbeta=0, fs0=0.3, Fig.4, green line,
\url{pmodels_nabD1_xi1p5_kfHp00}
}\label{pmodels_nabD1_xi1p5_kfHp00}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pmodels_nabD2_xi1p5_kfHp01}
\end{center}\caption[]{
Case II, beta=0, tbeta=1, fs0=0.1, Fig.5, blue line,
\url{pmodels_nabD2_xi1p5_kfHp01}
}\label{pmodels_nabD2_xi1p5_kfHp01}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pmodels_nabD2_xi1p5_kfHp}
\end{center}\caption[]{
Case III, beta=tbeta=1, fs0=0.1, Fig.6, blue line,
\url{pmodels_nabD2_xi1p5_kfHp}
}\label{pmodels_nabD2_xi1p5_kfHp}\end{figure}

\Figss{pmodels_nabD1_xi1p5_kfHp00}{pmodels_nabD2_xi1p5_kfHp} show
propagation diagrams similar to those in Appourchaux+10.
The frequencies $\omega^\pm$ are the zeros of the function
\begin{equation}
K(r)=\frac{1}{\cs^2}\left[\omega^2-\omega_{\rm c}^2-S_\ell^2\left(1-\frac{N^2}{\omega^2}\right)\right],
\end{equation}
where $\omega$ is the eigenfrequency,
$\omega_{\rm c}=(\cs/2H_{\rm p})\sqrt{1-2\dd H_{\rm p}/\dd r}$
is the acoustic cutoff frequency, $S_\ell=\sqrt{\ell(\ell+1)}\cs/r$
is the Lamb frequency, and $N=\sqrt{g(\dd(s/c_p)/\dd r}$ is the
Brunt-V\"ais\"al\"a frequency.
The term $K(r)$ enters in the equation
\begin{equation}
\frac{\partial^2\Psi}{\partial r^2}+K(r)\Psi=0,
\end{equation}
where $\Psi$ is the eigenfunction, which is proportional to the divergence
of the displacement $\xxi$ of the relevant mode.

%\Figss{pmodels_ell8_nabD1_xi1p5_kfHp00}{pmodels_nabD2_ell8_xi1p5_kfHp}
%give the corresponding three cases for $\ell=8$ and
%\Figss{pmodels_ell1_nabD1_xi1p5_kfHp00}{pmodels_nabD2_ell1_xi1p5_kfHp}
%for $\ell=1$.

We see that both $\omega^+$ and $\omega^-$ increase with $\ell$,
but $\omega^+$ increases faster, so that the gap between them
increases with increasing $\ell$.

%\begin{table}[htb]\caption{
%}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
%$R$ & $T$ & $\kf$ & $\omf$ & $\EEGW$ & $\hrms$ & $k_{\rm peak}$ \\
%\hline
%\label{Ttimescale}\end{tabular}}\end{table}

%r e f
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%\bibitem[Biskamp \& M\"uller(1999)]{BM99}
%Biskamp, D., \& M\"uller, W.-C.\yprl{1999}{83}{2195}

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