deletions | additions       diff --git a/Magnetic_fields_can_provide_the__.tex b/Magnetic_fields_can_provide_the__.tex  index 8601b00..ce706ab 100644  --- a/Magnetic_fields_can_provide_the__.tex  +++ b/Magnetic_fields_can_provide_the__.tex 
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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. The acceleration required to restore a wave of angular frequency $\omega$ and horizontal displacement $\xi_{\rm h}$ is $\xi_{\rm h} \omega^2$, whereas the magnetic tension acceleration due to a radial magnetic field of strength $B_r$ is $\xi_{\rm h} B_r^2 k_r^2/(4 \pi \rho)$, where $k_r$ is the radial wavenumber and $\rho$ is the density. Gravity waves are strongly altered by the magnetic fields when the magnetic tension force dominates, which for dipole waves occurs at a critical magnetic field strength \cite{supplementary}  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. {\bf The acceleration required to restore a wave of angular frequency $\omega$ and horizontal displacement $\xi_{\rm h}$ is $\xi_{\rm h} \omega^2$, whereas the magnetic tension acceleration due to a radial magnetic field of strength $B_r$ is $\xi_{\rm h} B_r^2 k_r^2/(4 \pi \rho)$, where $k_r$ is the radial wavenumber and $\rho$ is the density. Gravity waves are strongly altered by the magnetic fields when the magnetic tension force dominates, which for dipole waves occurs at a critical magnetic field strength \cite{supplementary}}  \begin{equation}\label{eqn:Bc}  B_c= \sqrt{\frac{\pi \rho}{2}} \, \frac{\omega^2 r}{N} \, .  \end{equation}  This field strength approximately corresponds to the point at which the Alfv\'en speed becomes larger than the radial group velocity of gravity waves.  %Magneto-gravity waves become evanescent in regions where the radial magnetic field strength exceeds the critical value of equation \ref{eqn:Bc}.   Magneto-gravity waves cannot exist in regions with $B_r>B_c$ where magnetic tension overwhelms the buoyancy force, i.e., the stiff field lines cannot be bent by the placid gravity wave motion. {\bf Dipole magneto-gravity waves become evanescent when $\omega < \omega_{\rm MG}$, where the magneto-gravity frequency $\omega_{\rm MG}$ is defined as}  \begin{equation}  \label{eqn:maggrav}  \omega_{\rm MG} = \bigg[ \frac{2}{\pi} \frac{B_r^2 N^2}{\rho r^2} \bigg]^{1/4} \label{eqn:Bc}  B_c= \sqrt{\frac{\pi \rho}{2}} \, \frac{\omega^2 r}{N}  \, . \end{equation}  {\bf Fig. \ref{fig:Prop} shows a wave propagation diagram in which a strong internal magnetic This  field prevents magneto-gravity wave propagation in the 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. Therefore, gravity waves are most susceptible strength approximately corresponds  tomagnetic alteration in  the H-burning shell, and point at which  the observation of a star with {\bf depressed} dipole modes thus provides a {\it lower limit} to Alfv\'en speed becomes larger than  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. group velocity of gravity waves.