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\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 isalways  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.  In Using  the adiabatic, anelastic, and ideal MHD approximations, approximations above,  thelocal  dispersion relation for MHD waves is \citep{unno:89} \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. 
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\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: solutions corresponding to each term in parentheses:  Alfven waves and magneto-gravity waves. Alfven waves satisfy $\omega^2 = \omega_A^2$, \omega_A^2$  and have wavenumber \begin{equation}  \label{eqn:alvendisp}  k^2 = \frac{\omega^2}{\mu^2 v_A^2}. 
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\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}. 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}. 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} 
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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 equation \ref{eqn:magnetodisp} dominates), the wavenumber obtains a large imaginary component. Therefore, magneto-gravity waves become evanescent in regions of very strong magnetic field. In essence, low 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 equation \ref{eqn:magnetodisp} dominates. The transition from propagating to evanescent magneto-gravity waves occurs when  \begin{equation} 
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\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. Therefore, limit, and therefore ${\bf B} \cdot {\bf k} \approx B_r k_r$, unless  the radial 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.  Equation \ref{eqn:magnetogravity2} shows that gravity waves will be strongly modified when the radial component of the 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 $\omega \lesssim \omega_{\rm MG}$, where the magneto-gravity wave frequency is defined as \begin{equation}  \label{eqn:maggrav2}  \omega_{\rm MG} = \sqrt{2 v_{A,r} 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. We note that several previous works \citep{Barnes_1998,Schecter_2001,MacGregor_2011,Mathis_2010,Mathis_2012,Rogers_2010,Rogers_2011} \cite{Barnes_1998,Schecter_2001,MacGregor_2011,Mathis_2010,Mathis_2012,Rogers_2010,Rogers_2011}  have examined the propagation of magneto-gravity waves in stellar interiors, focusing primarily on the solar tachocline. However, 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 {\it radial} radial  magnetic fields on gravity wave dynamics.Figure \ref{fig:Prop2} shows wave speeds and wavenumbers corresponding to the propagation diagram in Figure \ref{fig:Prop}. We note that the Alfven speed is always much less than the sound speed, therefore the magnetic field has a negligible effect on the background stellar structure. We also note that both Alfven and magneto-gravity waves always have $k \gg 1/H$ and $k \gg 1/r$ near $r_{\rm MG}$. Therefore, the WKB analysis used above is justified.  \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 Alfven wave.  The reflection or transition at $r_{\rm MG}$ is analogous to reflection of light between materials of differing refractive indices. Waves 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) are transmitted, while waves at large incidence angles are reflected. 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 Figure \ref{fig:Prop2} shows wave speeds  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,r} {\bf \hat{r}} \bigg] \, .  \end{align}  Below $r_{\rm MG}$, wavenumbers corresponding to the propagation diagram in Figure \ref{fig:Prop}. We note that  thegroup velocity of  Alfven waves speed  is $v_A {\bf \hat{B}}$, in always much less than  the direction of sound speed, i.e.,  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 pressure  is much larger smaller  than the Alfven velocity. Except in gas pressure and  the case of nearly horizontal fields, coupling to Alfven waves requires magnetic field has  a large change in negligible effect on the background stellar structure. We also note that  both direction Alfven  and magnitude of magneto-gravity waves always have $k \gg 1/H$ and $k \gg 1/r$ near $r_{\rm MG}$. Therefore,  the group velocity. The same WKB analysis used above  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. justified.  In the solar atmosphere, the fate of magneto-gravity-acoustic waves propagating upward into the chromosphere is largely determined by the geometry of the magnetic field. \cite{Newington_2009} and \cite{Newington_2011} demonstrate that mostly radial fields tend to reflect waves downward at the effective value of $r_{\rm MG}$ in the solar atmosphere. Moreover, the waves are reflected onto the slow branch, i.e., they transition into Alfven waves as they propagate downward. We suspect the same process will occur in stellar interiors: ingoing waves will mostly reflect at $r_{\rm MG}$ and will then transition into Alfven waves as they propagate back outward. Sufficiently horizontal fields will allow more wave transmission into Alfven 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 Figure \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.