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\subsection{Local Analysis}  Many of the properties of magnetohydrodynamic waves can be understood from a local analysis for high wavenumbers ${\bf k}$, in which $k r \gg 1$ and $k H \gg 1$. This analysis is technically only valid for large horizontal wavenumbers, $k_\perp r \gg 1$, whereas dipole oscillation modes have $k_\perp r = \sqrt{2}$. We must therefore be careful in extrapolating to global scale waves, especially for arbitrary magnetic field geometries.   In the adiabatic, anelastic, and ideal MHD approximations, the local dispersion relation for MHD waves is \citep{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, $N$ is the Brunt-Vaisala frequency, and the Alfven 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 $B$ is the magnetic field and $\rho$ is the density. The Alfven 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 Alfven velocity,  \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.  Equation \ref{eqn:disp} has two classes of solutions: Alfven waves and magneto-gravity waves. Alfven 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}  Alfven 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 Alfven 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}  For non-negligible buoyancy or magnetic field, the slow and fast magneto-gravity waves have similar characteristics. In the limit of very strong magnetic field or stratification (such that the second term in the square root of equation \ref{eqn:magnetodisp} dominates), we have  \begin{equation}  \label{eqn:magnetodisp3}  k \simeq \pm \sqrt{\frac{N k_\perp}{2 \mu v_A}} \big( 1 \pm i \big).  \end{equation}  Therefore, magneto-gravity waves become evanescent in regions of very strong magnetic field. In essence, low frequency waves 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 $H_{\rm ev} \sim \sqrt{v_{A}/(N k_\perp)} \ll H$ for realistic field strengths.  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 \sim v_{g,r} \, .  \end{equation}  In other words, gravity waves will be strongly modified when the local Alfven velocity is comparable to the radial component of the gravity wave group velocity. Another way of expressing this is that gravity waves will be altered when $\omega \lesssim \omega_{MG}$, where the magneto-gravity wave frequency is defined as  \begin{equation}  \label{eqn:maggrav}  \omega_{MG} = \sqrt{2 v_A N k_\perp}.  \end{equation}  In a magnetized red giant core, the value of both $N$ and $v_A$ will increase from the exterior of the radiative zone inward. Incoming gravity waves become increasingly Alfvenic magneto-gravity waves as they propagate toward the helium core. If the magnetic field strength becomes large enough to satisfy equation \ref{eqn:magnetogravity2}, the waves are very strongly altered by the magnetic field. They may either reflect (i.e., they become evanescent as in equation \ref{eqn:magnetodisp3}), or they may transform into purely Alfven waves. Both processes are likely to occur for an arbitrary field geometry.   \subsection{Mode Visibility}  Here we estimate the visibility of modes suppressed via the magnetic greenhouse effect. In the absence of magnetic suppression, 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. 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} = E_{\rm ac} \frac{T^2}{2 t_{\rm cross}} ,  \end{equation}  where $T$ is the transmission coefficient from equation \ref{eqn:integral}, and  \begin{equation}  t_{\rm cross} = \int^R_{r_2} \frac{dr}{v_s} \, ,  \end{equation}  is the wave crossing time across the acoustic cavity. The 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 \citep{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 non-adiabatic effects 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 equations \ref{eqn:enorm} and \ref{eqn:esup} equal to each other to find  \begin{equation}  E_{\alpha} \gamma_\alpha = E_{\rm ac} \bigg[\gamma_{\rm ac} + \frac{T^2}{2 t_{\rm cross}} \bigg] \, .  \end{equation}  Since the damping is produced almost entirely in the acoustic cavity, the normal mode damping rate is  \begin{equation}  \gamma_\alpha \simeq \frac{E_{\alpha,{\rm ac}}}{E_\alpha} \gamma_{\rm ac} \, ,  \end{equation}  where $E_{\alpha,{\rm ac}}$ is the mode energy contained in the acoustic cavity. Then 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 $V_{\alpha} \propto E_{\alpha,{\rm ac}}$. 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_\alpha^2} = \frac{E_{\rm ac}}{E_{\alpha,{\rm ac}}} \, .  \end{equation}  Then equation \ref{eqn:ebalance} leads to  \begin{equation}  \frac{V_{\rm sup}^2}{V_\alpha^2} = \frac{\gamma_{\rm ac}}{\gamma_{\rm ac} + T^2/(2 t_{\rm cross})} \, .  \end{equation}  Using the fact that the large frequency spacing is $\Delta \nu \simeq (2 t_{\rm cross})^{-1}$ \citep{Chaplin_2013} and defining $\tau_{\rm ac} = \gamma_{\rm ac}^{-1}$, we have our final result:  \begin{equation}  \frac{V_{\rm sup}^2}{V_\alpha^2} = \bigg[1 + \Delta \nu \tau_{\rm ac} T^2 \bigg]^{-1} \, .  \end{equation}    \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.