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\subsection{Reflection/Transmission}
We can now understand the behavior of waves and oscillation modes in red giants with magnetic cores. Waves with angular frequency $\omega \sim \omega_{\rm max}$ are excited via convective motions near the surface of the star. As in other red giants, the waves propagate downward as acoustic waves until their frequency is less than the local Lamb frequency for waves of angular degree $\ell$, i.e., until $\omega < L_{\ell}$. At this boundary, part of the wave flux is reflected, and part of it tunnels through.
The degree of reflection is determined by the tunneling integral through the evanescent region. The transmission coefficient is
\begin{equation}\label{eqn:integral}
T = \exp{\int^{r_2}_{r_1} i k_r dr} \simeq \exp{\int^{r_2}_{r_1} - \frac{\sqrt{\ell(\ell+1)}}{r} dr } \, ,
\end{equation}
where $r_1$ and $r_2$ are the boundaries of the evanescent region (in this case, the upper boundary occurs where $\omega
Waves that tunnel through the evanescent region into the radiative core continue to propagate inward as gravity waves, since magnetic effects are likely negligible near the top of the radiative zone. As the waves propagate inward, however, the background magnetic field strength likely increases ($B \propto r^{-3}$ in the simple case of a dipole field). Moreover, the Brunt-Vaisala frequency also increases sharply toward the H-burning shell (see Figure \ref{Fig:Struc}). Consequently, the magneto-gravity frequency $\omega_{\rm MG}$ (equation \ref{eqn:maggrav}) increases inward, such that magnetic restoring forces begin to dominate the wave dynamics.
For strong enough magnetic fields, there exists a critical magneto-gravity radius, $r_{\rm MG}$, defined as the radius where $\omega=\omega_{\rm MG}$. At this location, magneto-gravity waves become evanescent and can no longer propagate inward. An incoming wave must either reflect or propagate inward as a pure Alfven wave.
The reflection or transition at $r_{\rm MG}$ is analogous to reflection of light between materials of differing refractive indices. It is well known that light at small incidence angles, with ${\bf {\hat k}} \cdot {\bf {\hat n}} = \cos \theta $ (where ${\bf {\hat k}}$ is the direction of the wave vector and ${\bf {\hat n}}$ is the surface normal) is transmitted. Light at large incidence angles is reflected.
In our case, the location of $r_{\rm MG}$ is similar to such an interface. Just above $r_{\rm MG}$, incoming magneto-gravity waves have $k = \omega/(\sqrt{2} v_A \mu)$, whereas just below the interface an Alfven wave has $k = \omega/(v_A \mu)$. Across the location of $r_{\rm MG}$ a transmitted wave has a sudden jump in wave number and group velocity by a factor of $\sqrt{2}$. Thus, the interface has an effective index of refraction of $n = \sqrt{2}$, similar to that of many common liquids (water has $n=1.33$).
Despite a relatively low effective index of refraction, it is not clear how well the waves will be transmitted across $r_{\rm MG}$. Just above $r_{\rm MG}$, the group velocity of the incoming magneto-gravity waves is primarily horizontal and is approximately
\begin{align}
{\bf v}_g & \sim \bigg[ \frac{\omega}{k_\perp} {\bf \hat{n}}_\perp + \frac{\omega^2}{N k_\perp} {\bf \hat{r}} \bigg] \nonumber \\
& \sim \bigg[ \frac{\omega}{k_\perp} {\bf \hat {n}}_\perp + v_A {\bf \hat{r}} \bigg] \, .
\end{align}
Below $r_{\rm MG}$, the group velocity of Alfven waves is $v_A {\bf \hat{B}}$, in the direction of the magnetic field. Thus, although the radial group velocity of the incoming magneto-gravity waves is comparable to that of Alven waves, their horizontal group velocity is much larger than the Alfven velocity. Coupling to Alfven waves therefore requires a large change in both direction and magnitude of the group velocity. The same is true for the phase velocity. This may cause most of the gravity wave energy to reflect at $r_{\rm MG}$ rather than being transmitted into Alfven waves.
An additional possibility is that some of the wave energy will be converted into the slow branch of magneto-gravity waves, since the fast and slow branches have the same wavenumber at $r_{\rm MG}$ (equation \ref{eqn:magnetodisp}). The slow branch of magneto-gravity waves behave like Alfven waves in the weakly magnetized radiative region above $r_{\rm MG}$, and have wavenumbers orders of magnitude larger than the fast branch of magneto-gravity waves. Therefore, any wave energy reflected into slow magneto-gravity waves will likely be quickly dissipated via radiative diffusion.
\subsection{Magnetic Greenhouse Effect}
\label{greenhouse}
Regardless of whether magneto-gravity waves are transmitted or reflected at $r_{\rm MG}$, we show here that strong magnetic fields will effectively trap waves in the radiative core of a red giant star. We refer to this process as the magnetic greenhouse effect, because it acts as a filter that allows gravity waves to enter the radiative core, but prevents them from escaping. However, the magnetic greenhouse effect arises not from the alteration of incoming wave frequencies, but rather due to modification of the wave angular structure. Such angular modification originates from the inherently non-spherical structure of even the simplest field configurations.
Let's first consider waves that are transmitted at $r_{\rm MG}$. The incoming magneto-gravity waves couple to Alfven waves in the cavity $r
The wave energy transmitted to Alfven waves is not necessarily quickly damped. The wavenumber $k$ of the Alfven waves is smaller than that of gravity waves in the absence of a magnetic of a magnetic field (see bottom panel of Figure \ref{fig:Prop}, therefore damping via radiative diffusion will occur slower for Alfven waves than for g-modes. Furthermore, in Appendix \ref{Bdamp} we show that damping of Joule damping of currents created by Alfven waves is negligible compared to radiative diffusion. The Alfven waves will therefore travel along field lines and may re-emerge at $r_{\rm MG}$ and transmit their energy back into magneto-gravity waves. However, even if this occurs, the energy will be spread over a large number of $\ell$ values. The higher multipole magneto-gravity waves have shorter wavelengths and damp out more quickly than dipole waves.
Most importantly, the tunneling integral of equation \ref{eqn:integral} is much larger for waves of larger $\ell$, due to both the larger value of the integrand and the larger width of the evanescent region (because of a higher Lamb frequency for high $\ell$ modes). Therefore, any wave energy with $\ell \gtrsim 3$ will be completely trapped within the radiative core. The initially dipolar wave will be trapped in the core until it dissipates, unable to tunnel back toward the surface to create an observable signature. This is the essence of the magnetic greenhouse effect.
A similar effect occurs for magneto-gravity waves that are reflected at $r_{\rm MG}$ rather than coupling with Alfven waves. The location of $r_{\rm MG}$ is a function of latitude, because the magnetic field cannot be spherically symmetric (since $\nabla \cdot \textbf{B} = 0$). Even in the simplest case of a purely dipolar field, the waves will scatter into a broad spectrum of $\ell$ (see \cite{Rincon_2003,Reese_2004}). In reality, purely poloidal fields are unstable, and the field will likely have a complex geometry containing both poloidal and toroidal components. The incoming $\ell=1$ wave is thus inevitably scattered into higher $\ell$ waves. As described above, these waves cannot couple back to acoustic modes in the envelope, and remain trapped within the radiative core until they dissipate.
We have shown that, due to the magnetic greenhouse effect, waves that tunnel from the envelope into the radiative region cannot escape from a strongly magnetized stellar core. Their degree of suppression is therefore controlled by the degree of reflection at the bottom of the acoustic cavity (as determined by the tunneling integral through the evanescent region, see Eq.~\ref{eqn:integral}). This quantity can be used, as a first approximation, to predict the visibility of the $\ell=1,2$ modes (Fig.~\ref{visibility}).
\end{appendix}
