deletions | additions       diff --git a/Magnetic_fields_can_provide_the__.tex b/Magnetic_fields_can_provide_the__.tex  index f0c01bc..4b28d1c 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. {\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 (see supplementary online material)}  \begin{equation}  \label{eqn:Bc} \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. 
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