deletions | additions       diff --git a/Magnetic_fields_can_provide_the__.tex b/Magnetic_fields_can_provide_the__.tex  index 06d316d..8e027d4 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 buoyancy acceleration created by a wave displacement $\xi_r$ in the radial direction is $\sim \! \xi_r N^2$, whereas the magnetic tension acceleration due to a horizontal wave displacement $\xi_{\rm h}$ in a radial magnetic field of field strength $B$ $B_r$  is $\xi_{\rm h} B^2 B_r^2  k_r^2/(4 \pi \rho)$, where $k_r$ is radial wavenumber and $\rho$ is the density. Gravity waves are strongly altered by the magnetic fields when these two forces are equal, which for dipole waves occurs at a critical magnetic field strength (see supplementary online material)} \begin{equation}  \label{eqn:Bc}  B_c= \sqrt{\frac{\pi \rho}{2}} \, \frac{\omega^2 r}{N} \, . 
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