deletions | additions       diff --git a/subsection_Magneto_Gravity_Waves_The__.tex b/subsection_Magneto_Gravity_Waves_The__.tex  index 0a6abf1..5906d9e 100644  --- a/subsection_Magneto_Gravity_Waves_The__.tex  +++ b/subsection_Magneto_Gravity_Waves_The__.tex 
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\omega_{\rm MG} = \sqrt{2 v_{A,r} N k_\perp}.  \end{equation}  We note that 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.  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, 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 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 (e.g., \cite{Saio_2014,Saio_2012,Dziembowski_1996,Cunha_2000,Cunha_2006} have examined the effect }