deletions | additions       diff --git a/Magnetic Trapping.tex b/Magnetic Trapping.tex  index a46e4ad..ec6023e 100644  --- a/Magnetic Trapping.tex  +++ b/Magnetic Trapping.tex 
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{\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. \subsection{Magnetic Greenhouse Effect}