deletions | additions       diff --git a/section_Magnetic_Greenhouse_Effect_Magnetic__.tex b/section_Magnetic_Greenhouse_Effect_Magnetic__.tex  index a6cef6a..f73d823 100644  --- a/section_Magnetic_Greenhouse_Effect_Magnetic__.tex  +++ b/section_Magnetic_Greenhouse_Effect_Magnetic__.tex 
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\section{Magnetic Greenhouse Effect}  Magnetic fields can provide the mechanism for trapping oscillation mode energy in the core by altering gravity wave propagation. The nearly horizontal motions and short radial wavelengths of gravity waves in RGB cores will bend radial magnetic field lines, creating strong magnetic tension forces. A simple estimate of the magnetic field strength required to compete with the restoring force of buoyancy for dipole gravity waves gives  \begin{equation}  \label{eqn:Bc} 
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\end{equation}  where $\rho$ is the density. This field strength approximately corresponds to the point at which the Alfven speed becomes larger than the radial group velocity of gravity waves.  A more careful calculation (supplementary online text) shows that the wavenumber of magneto-gravity waves is   \begin{equation}  \label{eqn:k} 
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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 dipole  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. 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} 
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\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 stellar model 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.  In red giants, $B_c$ is typically smallest at the peak in $N$ corresponding to the sharp density gradient within the hydrogen burning (H-burning) shell (see Figure \ref{fig:Prop}). Therefore, gravity waves are most susceptible to magnetic alteration in the H-burning shell. The observation of a star with suppressed dipole modes thus provides a {\it lower limit} to the radial field strength (equation \ref{eqn:Bc}) evaluated in the H-burning shell. We refer to this field strength as $B_{c,{\rm min}}$. Magnetic suppression via horizontal fields can also occur, but in general requires much larger field strengths.