deletions | additions       diff --git a/subsection_Magneto_Gravity_Waves_The__.tex b/subsection_Magneto_Gravity_Waves_The__.tex  index b836adf..be9dc83 100644  --- a/subsection_Magneto_Gravity_Waves_The__.tex  +++ b/subsection_Magneto_Gravity_Waves_The__.tex 
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\label{eqn:alven2}  \omega_A^2 = v_A^2 k^2 \mu^2,  \end{equation}  where $v_A$ is the Alfven Alfv\'en  speed, \begin{equation}  \label{eqn:valfven}  v_A^2 = \frac{B^2}{4 \pi \rho} \, . 
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\label{eqn:alvendisp}  k^2 = \frac{\omega^2}{\mu^2 v_A^2}.  \end{equation}  Alfven 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} 
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\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.   Figure \ref{fig:Prop2} shows wave speeds and wavenumbers corresponding to the propagation diagram in Figure \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 Alfven Alfv\'en  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. Several previous works \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 radial magnetic fields on gravity wave dynamics.  {\bf Finally, many papers \cite{Campbell_2006,Dziembowski_1996,Cunha_2000,Cunha_2006,Sousa_2008,Saio_2012,Saio_2014} 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 Alfven 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. }