\subsection{Mode Visibility}

Here we estimate the visibility of modes suppressed via the magnetic greenhouse effect. To do this, we consider the energy balance between driving and damping of a mode. Each mode receives a stochastic energy input $\dot{E}_{\rm in}$ \cite{Dupret_2009}. At its time-averaged equilibrium amplitude, the energy input and damping rates of the mode are equal, such that

\begin{equation}
\label{eqn:enorm}
\dot{E}_{\rm in} = \dot{E}_{\rm out} = E_{\alpha} \gamma_\alpha \, ,
\end{equation}
where $E_{\alpha}$ is the energy contained in the mode and $\gamma_\alpha$ is its damping rate.

Modes suppressed via the magnetic greenhouse effect have an extra source of damping determined by the rate at which energy leaks through the evanescent region separating the acoustic cavity from the g mode cavity. For suppressed modes, we assume that any mode energy which leaks into the g mode cavity is completely lost via the magnetic greenhouse effect. Similar calculations have been employed for more evolved red giants where waves entering the core are damped by radiative diffusion \cite{Dziembowski_1977,Osaki_1977,Dziembowski_2012}. Given some mode energy contained within the acoustic cavity, $E_{\rm ac}$, the rate at which mode energy leaks into the core is

\begin{equation}
\dot{E}_{\rm leak} = \frac{E_{\rm ac}}{t_{\rm leak}} ,
\end{equation}
where $t_{\rm leak}$ is the time scale on which mode energy leaks into the core. As explained in the text, the energy leakage time scale is

\begin{equation}
\label{eqn:tleak}
t_{\rm leak} = \frac{2 t_{\rm cross}}{T^2} \, ,
\end{equation}
where the transmission coefficient $T$ is

\begin{equation}
\label{eqn:integral}
T =  \exp{ \bigg[ \int^{r_2}_{r_1} i k_r dr} \bigg] \, .
\end{equation}
The value of $T$ is approximately the fractional decrease in wave amplitude across the evanescent region, whereas $T^2$ is the fractional decrease in wave energy. In the WKB approximation, the value of the radial wavenumber $k_r$ within the evanescent region is

\begin{equation}
\label{eqn:kr}
k_r^2 = \frac{ \big(L_\ell^2 - \omega^2 \big) \big(N^2 - \omega^2 \big) }{v_s^2 \omega^2} \, .
\end{equation}
Deep within an evanescent region where $N^2 \! \ll \! \omega \! \ll \! L_\ell^2$, Eqn. \ref{eqn:kr} evaluates to $k_r \! \sim \! i \sqrt{\ell (\ell +1)}/r$, and Eqn. \ref{eqn:integral} yields the expression for $T$ in Eqn. \ref{eqn:integral2}. The wave crossing time for acoustic waves is

\begin{equation}
t_{\rm cross} = \int^R_{r_2} \frac{dr}{v_s} \, .
\end{equation}

A suppressed mode is also damped by the same mechanisms as a normal mode. In the case of envelope modes for stars low on the RGB, this damping is created by convective motions near the surface of the star \cite{Houdek_1999,Dupret_2009}. The equilibrium energy of the suppressed mode is

\begin{equation}
\label{eqn:esup}
\dot{E}_{\rm in} = \dot{E}_{\rm out} = E_{\rm ac} \bigg[\gamma_{\rm ac} +  \frac{T^2}{2 t_{\rm cross}} \bigg],
\end{equation}
where $\gamma_{\rm ac}$ is the damping rate due to convective motions in the acoustic cavity.

Now, we assume that the suppression mechanism is localized to the core and that the energy input $\dot{E}_{\rm in}$ is unaltered. Then we can set Eqns. \ref{eqn:enorm} and \ref{eqn:esup} equal to each other to find

\begin{equation}
\label{edamp}
E_{\alpha} \gamma_\alpha =  E_{\rm ac} \bigg[\gamma_{\rm ac} +  \frac{T^2}{2 t_{\rm cross}} \bigg] \, .
\end{equation}
The damping of a normal mode is localized to the acoustic cavity, so its energy loss rate can be written

\begin{equation}
{E_\alpha} \gamma_\alpha \simeq E_{\alpha,{\rm ac}} \gamma_{\rm ac} \, ,
\end{equation}
where $E_{\alpha,{\rm ac}}$ is the mode energy contained in the acoustic cavity. Inserting this into Eqn. \ref{edamp}, we have

\begin{equation}
\label{eqn:ebalance}
E_{\alpha,{\rm ac}} \gamma_{\rm ac} =  E_{\rm ac} \bigg[\gamma_{\rm ac} +  \frac{T^2}{2 t_{\rm cross}} \bigg] \, .
\end{equation}

The energy of a mode within the envelope is proportional to its surface amplitude squared, hence, the visibility of a mode scales as $V_{\alpha}^2 \propto E_{\alpha,{\rm ac}}$. Then the ratio of the visibility of the suppressed mode to that of the normal mode is

\begin{equation}
\frac{V_{\rm sup}^2}{V_{\rm norm}^2} = \frac{E_{\rm ac}}{E_{\alpha,{\rm ac}}} \, .
\end{equation}
Then Eqn. \ref{eqn:ebalance} leads to

\begin{equation}
\frac{V_{\rm sup}^2}{V_{\rm norm}^2} = \frac{\gamma_{\rm ac}}{\gamma_{\rm ac} + T^2/(2 t_{\rm cross})} \, .
\end{equation}
Using the fact that the large frequency separation is $\Delta \nu \simeq (2 t_{\rm cross})^{-1}$ \cite{Chaplin_2013} and defining $\tau = \gamma_{\rm ac}^{-1}$, we have our final result:

\begin{equation}
\frac{V_{\rm sup}^2}{V_{\rm norm}^2} = \bigg[1 + \Delta \nu \tau T^2 \bigg]^{-1} \, .
\end{equation}
The damping time $\tau$ is the lifetime of wave energy located in the acoustic cavity. It is not equal to the lifetime of a normal dipole mode, because much of the dipole mode energy resides within the core. Instead, $\tau$ is approximately equal to the lifetime of a radial mode, because all of its energy is in the acoustic cavity. Thus, $\tau$ can be equated with observed/theoretical lifetimes of radial modes.

We emphasize that the magnetic greenhouse mechanism that operates in depressed dipole oscillators is not directly related to the suppression of solar-like oscillations in stars exhibiting surface magnetic activity \cite{Garcia_2010,Chaplin_2011,Gaulme_2014}. In the latter case, the depression likely arises from magnetic effects in the convective envelope quenching the amplitudes of all oscillation modes, not just the dipole modes.

\subsection{Wave Leakage Time}

The wave leakage time scale $t_{\rm leak}$ on which wave energy tunnels from the acoustic cavity into the stellar core can be estimated from Eqn. \ref{eqn:tleak}, with the value of $T$ calculated from the first part of Eqn. \ref{eqn:integral}, or approximated from Eqn. \ref{eqn:integral2}.

To compute a more precise estimate, we solve the forced adiabatic linearized hydrodynamic (non-magnetic) wave equations for our stellar models (using the Cowling approximation), assuming all wave energy that tunnels into the outer core is lost within the inner core. To do this, we place the inner boundary of our computational grid at a radius $r/R = 0.01$, which is always within the stably stratified regions of our red giant models. We then impose a radiative inner boundary condition \cite{Fuller_2012}. On the outer boundary, we impose a forcing/normalization condition on the real part of the wave displacement vector {\boldmath $\xi$}, i.e., we set ${\rm Re} \big( \xi_r \big) =1$, where $\xi_r$ is the radial component of the displacement vector. For the imaginary component, we adopt the standard reflective outer boundary condition, ${\rm Im} \big( \delta P \big) = \rho g {\rm Im} \big( \xi_r \big)$, where $\delta P$ is the Eulerian pressure perturbation, $\rho$ is the density, and $g$ is the gravitational acceleration. Physically, this scenario represents the forcing of the stellar surface at a given angular frequency $\omega$, and the eventual leakage of the wave energy into the core of the star.

After solving the wave equations, we compute the leakage time of the wave energy contained within the acoustic cavity at radii $r_2 \! < r \! < R$. The wave energy contained within the acoustic cavity is simply

\begin{equation}
\label{eqn:ecav}
E_{\rm ac} = \int^{R}_{r_2} dr \, \rho r^2 \omega^2 \bigg( | \xi_r |^2 + \ell(\ell+1) | \xi_\perp |^2 \bigg) \, ,
\end{equation}
where $\xi_\perp$ is the horizontal component of the wave displacement vector. The rate at which energy leaks through the inner boundary is

\begin{equation}
\label{eqn:eleak}
\dot{E}_{\rm leak} = \rho r^3 \omega^3 \bigg[ {\rm Re}\big( \xi_\perp \big) {\rm Im}\big( \xi_r \big) - {\rm Re}\big( \xi_r \big) {\rm Im} \big( \xi_\perp \big) \bigg] \, ,
\end{equation}
evaluated at the inner boundary of the grid. The wave energy leakage time is then

\begin{equation}
\label{eqn:tleak2}
t_{\rm leak} = \frac{ E_{\rm ac} }{\dot{E}_{\rm leak}} \, .
\end{equation}
We have calculated the leakage timescales for waves with frequencies near $\omega_{\rm max} \! = \! 2 \pi \nu_{\rm max}$ for stellar models on the RGB. For this computational technique, the energy $E_{\rm ac}$ contained within the acoustic cavity peaks at the p mode frequencies of the stellar model. The energy loss rate $\dot{E}_{\rm leak}$ also peaks at the mode frequencies, so that the value of $t_{\rm leak}$ is essentially independent of $\omega$.