Fig. \ref{fig:DipoleTime} shows the exact value of $t_{\rm leak}$ calculated from Eqn. \ref{eqn:tleak2}, and $t_{\rm leak}$ approximated from Eqn. \ref{eqn:tleak}, with $T$ calculated via Eqn.s \ref{eqn:integral2} and \ref{eqn:integral}. Clearly, evaluating $t_{\rm leak}$ via Eqn. \ref{eqn:tleak} with $T$ calculated from Eqn. \ref{eqn:integral} is a very good approximation, accurate to within $\sim \! 10 \%$ for our stellar models. However, using the approximation of Eqn. \ref{eqn:integral2} is not very accurate, and generally produces a value of $t_{\rm leak}$ too large by a factor of $\sim \! 2$. We conclude that we may accurately estimate mode visibilities using Eqn. \ref{eqn:vis}, so long as the value of $T$ is calculated with an integral over the evanescent region as in Eqn. \ref{eqn:integral}. The approximation of $T$ in Eqn. \ref{eqn:integral2} should not be used for visibility calculations, although it is still useful because it demonstrates the scaling of $T$ with wave angular degree $\ell$ and the size of the evanescent region.

\subsection{Magneto-Gravity Waves}

The properties of magnetohydrodynamic waves in red giant cores can be understood using a local (WKB) analysis for high wavenumbers ${\bf k}$, in which $k r \gg 1$ and $k H \gg 1$ (where $H$ is a pressure scale height). We show below the WKB limit is a good approximation in our stellar models. In what follows, we shall also use the adiabatic, anelastic, and ideal MHD approximations, which are all valid for the magneto-gravity waves we consider in red giant cores.

Using the approximations above, the dispersion relation for MHD waves is \cite{unno:89}

\begin{equation}
\label{eqn:disp}
\bigg( \omega^2 - \omega_A^2 \bigg) \bigg( \omega^2 - \frac{k_\perp^2}{k^2}N^2 - \omega_A^2 \bigg) = 0.
\end{equation}
Here, $\omega$ is the angular frequency of the wave, $k_\perp = \sqrt{l(l+1)}/r$ is the horizontal wavenumber, $N$ is the Brunt-V\"{a}is\"{a}l\"{a} (buoyancy) frequency, and the Alfv\'en frequency is

\begin{equation}
\label{eqn:alven}
\omega_A^2 = \frac{ \big( {\bf B} \cdot {\bf k} \big)^2}{4 \pi \rho},
\end{equation}
where ${\bf B}$ is the magnetic field and $\rho$ is the density. The Alfv\'en frequency can also be expressed as

\begin{equation}
\label{eqn:alven2}
\omega_A^2 = v_A^2 k^2 \mu^2,
\end{equation}
where $v_A$ is the Alfv\'en speed,

\begin{equation}
\label{eqn:valfven}
v_A^2 = \frac{B^2}{4 \pi \rho} \, .
\end{equation}
and $\mu= \cos \theta$ is the angle between the magnetic field and wave vector.

Eqn. \ref{eqn:disp} has two classes of solutions corresponding to each term in parentheses: Alfv\'en waves and magneto-gravity waves. Alfv\'en waves satisfy $\omega^2 = \omega_A^2$ and have wavenumber

\begin{equation}
\label{eqn:alvendisp}
k^2 = \frac{\omega^2}{\mu^2 v_A^2}.
\end{equation}
Alfv\'en waves have fluid velocity perpendicular to the field lines and group velocity $v_g = v_A$ parallel to magnetic field lines.

Magneto-gravity waves have $\omega^2 = k_\perp^2 N^2/k^2 + \omega_A^2$. A little algebra demonstrates that their wavenumber is

\begin{equation}
\label{eqn:magnetodisp}
k^2 = \frac{\omega^2}{2 v_A^2 \mu^2} \bigg[ 1 \pm \sqrt{1 - \frac{4 \mu^2 v_A^2 N^2 k_\perp^2}{\omega^4}} \bigg].
\end{equation}
The positive and negative roots correspond to the ``slow" and ``fast" magneto-gravity waves, respectively. In the limit of vanishing magnetic field or buoyancy ($v_A \rightarrow 0$ or $N \rightarrow 0$), the slow waves reduce to Alfv\'en waves,

\begin{equation}
\label{eqn:magnetodisp1}
k^2 \simeq \frac{\omega^2}{\mu^2 v_A^2} \, .
\end{equation}
The fast waves reduce to gravity waves,

\begin{equation}
\label{eqn:magnetodisp2}
k^2 \simeq \frac{N^2 k_\perp^2}{\omega^2} \, .
\end{equation}
Gravity waves have fluid velocity nearly perpendicular to the stratification (i.e., nearly horizontal). Their group velocity is primarily horizontal, with

\begin{equation}
\label{eqn:vgravperp}
v_{g,\perp} = \frac{\omega}{k_\perp} \, ,
\end{equation}
but with a small radial component of

\begin{equation}
\label{eqn:vgravr}
v_{g,r} = \frac{\omega^2}{N k_\perp} \, . 
\end{equation}

In the limit of very strong magnetic field or stratification (such that the second term in the square root of Eqn. \ref{eqn:magnetodisp} dominates), the wavenumber obtains a large imaginary component. Therefore, magneto-gravity waves become evanescent in regions of very strong magnetic field. Low frequency waves approaching regions of high field strength can reflect off the stiff field lines, similar to low frequency fluid waves reflecting off a solid boundary. The evanescent skin depth is small, with $H_{\rm ev} \sim \sqrt{v_{A}/(N k_\perp)} \ll H$ when the second term in the square root of Eqn. \ref{eqn:magnetodisp} dominates.

The transition from propagating to evanescent magneto-gravity waves occurs when

\begin{equation}
\label{eqn:magnetogravity}
2 \mu v_A = \frac{\omega^2}{N k_\perp} \, ,
\end{equation}
i.e., when

\begin{equation}
\label{eqn:magnetogravity2}
v_{A,r} \sim v_{g,r} \, .
\end{equation}
Here, we have used $\mu v_A \sim v_{A,r}$ because $k_r \gg k_\perp$ for gravity waves in the WKB limit, and therefore ${\bf B} \cdot {\bf k} \approx B_r k_r$, unless the field is almost completely horizontal. Hence, the \emph{radial} component of the field typically dominates the interaction between the magnetic field and gravity waves. The physical reason for this is that the large horizontal motions and vertical wavenumbers of gravity waves generate large magnetic tension restoring forces by bending radial magnetic field lines.

Fig. \ref{fig:Prop2} shows wave speeds and wavenumbers corresponding to the propagation diagram in Fig. \ref{fig:Prop}. We note that the Alfv\'en speed is always much less than the sound speed, i.e., the magnetic pressure is much smaller than the gas pressure and the magnetic field has a negligible effect on the background stellar structure. We also note that both Alfv\'en and magneto-gravity waves always have $k \gg 1/H$ and $k \gg 1/r$ in the inner core of our RGB models. Therefore, the WKB analysis used above is justified.

Several previous works (e.g., \cite{Mathis_2012} and references therein) have examined the propagation of magneto-gravity waves in stellar interiors, focusing primarily on the solar tachocline. However, nearly all of these works have considered a purely toroidal (horizontal) magnetic field configuration, because they were motivated by the strong toroidal field thought to exist due to the shear flows in the solar tachocline. Horizontal fields must be stronger by a factor $k_r/k_\perp \sim N/\omega \gg 1$ in order to strongly affect gravity waves. Consequently, these works did not examine the extremely important effect of radial magnetic fields on gravity wave dynamics.

Finally, many papers (such as \cite{Saio_2012} and references therein) have examined the effect of magnetic fields on the acoustic oscillations of rapidly oscillating Ap stars. In this case, the magnetic field strongly affects the acoustic waves only near the surface of the star where the magnetic pressure becomes comparable to the gas pressure. These authors reach similar conclusions to those discussed below: some wave energy can be lost by transmission into Alfv\'en waves, and the geometry of the magnetic field is important. However, the oscillation modes in these stars indicates that observable modes can still exist in the presence of strong magnetic fields, and future studies should further examine possible connections between the physics of oscillating Ap stars and red giants with magnetic cores.

\subsection{Reflection/Transmission}

We define the magneto-gravity radius, $r_{\rm MG}$, 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 Alfv\'en wave.

Incoming $\ell \! = \! 1$ magneto-gravity waves can transmit energy into a continuous spectrum \cite{Reese_2004,Levin_2006} of Alfv\'en waves with a broad spectrum of $\ell$ values \cite{Rincon_2003}. Reflected waves will also transfer energy to high $\ell$ waves (for reasons below), and because the location of $r_{\rm MG}$ is a function of latitude since the magnetic field cannot be spherically symmetric. Even in the simplest case of a purely dipolar magnetic field, any resulting oscillation modes will contain a broad spectrum of $\ell$ \cite{Lee_2007,Lee_2010}. In reality, the field will likely have a complex geometry containing both poloidal and toroidal components \cite{Braithwaite_2004,Braithwaite_2006,Duez_2010}, and dipole waves will inevitably scatter into higher $\ell$ waves in the presence of a strong magnetic field.

Wave reflection or transmission at $r_{\rm MG}$ is analogous to the propagation of light between materials of differing refractive indices. In the present case, magneto-gravity waves will likely be reflected due to the high effective refractive index at $r_{\rm MG}$ due to the differing speeds of magneto-gravity and Alfv\'en waves. Just above $r_{\rm MG}$, the group velocity of the incoming magneto-gravity waves is primarily horizontal and is approximately

\begin{equation}
\label{vgroup}
{\bf v}_g  \sim \bigg[ \frac{\omega}{k_\perp} {\bf \hat{n}}_\perp - \frac{\omega^2}{N k_\perp} {\bf \hat{r}} \bigg]
\sim \bigg[ \frac{\omega}{k_\perp} {\bf \hat {n}}_\perp - v_{A,r} {\bf \hat{r}} \bigg] \, .
\end{equation}
Below $r_{\rm MG}$, the group velocity of Alfv\'en 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 Alfv\'en waves, their horizontal group velocity is much larger than the Alfv\'en velocity. Except in the case of nearly horizontal fields, coupling to Alfv\'en waves 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 Alfv\'en waves.

In the solar atmosphere, an analogous process occurs where magneto-acoustic-gravity waves become magnetically dominated as they propagate upward. In general, the reflection or transmission of the wave depends on the geometry of the magnetic field \cite{Zhugzhda_1984}. Mostly radial fields tend to reflect waves downward at the effective value of $r_{\rm MG}$ in the solar atmosphere \cite{Newington_2009,Newington_2011}. Moreover, the waves are reflected onto the slow branch, i.e., they transition into Alfv\'en waves as they propagate downward. The same process may occur in stellar interiors: ingoing waves will mostly reflect at $r_{\rm MG}$ and will then transition into Alfv\'en waves as they propagate back outward. Sufficiently horizontal fields will allow more wave transmission into Alfv\'en waves in the core, however, stronger fields are required in this case.

The reflected waves will dissipate much faster than the incident dipole waves, preventing them from ever tunneling back to the surface. Waves reflected back onto the fast branch will have higher $\ell$, shorter wavelengths, and will damp out more quickly than dipole waves. Waves reflected onto the slow branch have wavenumbers orders of magnitude larger than the fast branch of magneto-gravity waves (see Fig. \ref{fig:Prop2}) as they propagate outward into weakly magnetized regions. Therefore, any wave energy reflected into slow magneto-gravity waves will be quickly dissipated via radiative diffusion.

For perfect wave trapping in the core, purely dipole modes only exist in the envelope, with part of their energy leaking into the core as running magneto-gravity waves. If some wave energy does escape the core, it may leave a signature in the form of mixed magneto-gravity acoustic modes, or by producing magnetic mode splitting, which could be used to constrain the internal magnetic field geometry.