deletions | additions       diff --git a/section_Magnetic_Greenhouse_Effect_Magnetic__.tex b/section_Magnetic_Greenhouse_Effect_Magnetic__.tex  index b81f7cc..dcc3636 100644  --- a/section_Magnetic_Greenhouse_Effect_Magnetic__.tex  +++ b/section_Magnetic_Greenhouse_Effect_Magnetic__.tex 
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\end{equation}  where $v_A = B/\sqrt{4 \pi \rho}$ is the Alfven speed, $k$ is the wavenumber, $k_\perp = \sqrt{l(l+1)/r}$ is its horizontal component, and $\mu = \cos \theta$ is the angle between ${\bf B}$ and ${\bf k}$. The positive and negative roots of equation \ref{eqn:k} correspond to the ``slow" and ``fast" magneto-gravity waves, respectively, which correspond to Alfven waves and gravity waves in the limit of vanishing magnetic field. Pure Alfven waves can also exist.  The key feature of equation \ref{eqn:k} is that dipolar magneto-gravity waves become evanescent in regions where the radial magnetic field strength exceeds the critical value of equation \ref{eqn:Bc}. Low-frequency gravity waves cannot exist in these regions because magnetic tension overwhelms the buoyancy force, i.e., the stiff field lines cannot be bent by the placid gravity wave motion. In other words, gravity Figure \ref{fig:cartoon} illustrates the basic properties of waves propagating in a red giant with a strongly magnetized core.  Magneto-gravity  waves are evanescent in regions where their angular frequency $\omega$ is less than the magneto-gravity frequency $\omega_{\rm MG}$ defined as \begin{equation}  \label{eqn:maggrav}  \omega_{MG} = \sqrt{2 v_{A,r} N k_\perp} \, ,  \end{equation}  and $v_{A,r}$ is the radial component of the Alfven velocity. Figure \ref{fig:Prop} shows a wave propagation diagram for a red giant in which a strong internal magnetic field prevents magneto-gravity wave propagation in the core.  %Figure \ref{fig:cartoon} illustrates the basic properties of waves in red giants with strongly magnetized cores, while Figure \ref{fig:Prop} shows a propagation diagram for a red giant model %with a magnetic core.