deletions | additions       diff --git a/section_Magnetic_Trapping_subsection_Magneto__.tex b/section_Magnetic_Trapping_subsection_Magneto__.tex  index 2f8bf87..ed24bb4 100644  --- a/section_Magnetic_Trapping_subsection_Magneto__.tex  +++ b/section_Magnetic_Trapping_subsection_Magneto__.tex 
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\label{eqn:Bc}  B_c= \sqrt{\frac{\pi \rho}{2}} \, \frac{\omega^2 r}{N} \, ,  \end{equation}  where $\rho$ is the density. This field strength approximately corresponds to the point at which thegroup velocity of  Alfven waves speed  becomes larger than the radial group velocity of gravity waves. %A simple estimate of the magnetic field strength needed to compete with the restoring force of buoyancy gives $v_{A,r} \approx v_{g,r}$, where $v_{A,r}$ is the radial component of the Alfven velocity and $v_{g,r}$ is the radial component of the gravity wave group velocity. The Alfven velocity is ${\bf v}_A = {\bf B}/\sqrt{4 \pi \rho}$, and the gravity waves propagate in the radial direction at $v_{g,r} = \omega^2/(k_\perp N)$. Here, $\omega$ is the wave angular frequency, $r$ is the radial position, $k_\perp = \sqrt{l(l+1)}/r$ is the horizontal wave number, and $N$ is the Brunt-Vaisala frequency. 
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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 waves are evanescent in regions where their 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:cartoon} illustrates the general properties of waves that produce solar-like oscillations in red giants with strongly magnetized cores, while Figure \ref{fig:Prop} shows a propagation diagram for a red giant model. model with a magnetic core.  In red giant cores, $B_c$ is typically minimized 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 depressed dipole modes thus provides a {\it lower limit} to the internal magnetic field strength, given by equation \ref{eqn:Bc}, evaluated in the H-burning shell. We refer to this field strength as $B_{c,{\rm min}}$, which corresponds to a minimum {\it radial} field strength for magnetic suppression. Magnetic suppression via horizontal fields can also occur, but in general requires much larger field strengths (see Supplementary Material).