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Fig_ref_fig_DipoleTime_shows__.tex  figures/DipoleEvolProp/DipoleEvolWave.png  subsection_Ray_Tracing_label_ray__.tex  subsection_Joule_Damping_A_gravity__.tex  subsection_Measurements_Uncertainties_and_varepsilon__.tex  subsection_MESA_Inlist_label_inlist__.tex  diff --git a/subsection_Joule_Damping_A_gravity__.tex b/subsection_Joule_Damping_A_gravity__.tex  deleted file mode 100644  index bed8335..0000000  --- a/subsection_Joule_Damping_A_gravity__.tex  +++ /dev/null 
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\subsection{Joule Damping}  A gravity wave propagating through a magnetized fluid induces currents which dissipate in a non-perfectly conducting fluid, causing the wave to damp. For gravity waves in the WKB limit which are not strongly altered by magnetic tension forces, the perturbed radial magnetic field is $\delta B \approx \xi_\perp k_r B$, where $\xi_\perp$ is the horizontal wave displacement. The perturbed current density is $\delta J \approx c k_r \delta B/(4 \pi)$, where $c$ is the speed of light. The volumetric energy dissipation rate is $\dot{\varepsilon} \approx (\delta J)^2/\sigma$, where $\sigma$ is the electrical conductivity. The gravity wave energy density is $\varepsilon \approx \rho \omega^2 \xi_\perp^2$, so the local damping rate is  \begin{equation}  \label{eqn:joule}  \Gamma_B = \frac{\dot{\varepsilon}}{\varepsilon} \approx \frac{\eta B^2 k_r^4}{(4 \pi)^2 \rho \omega^2} \, ,  \end{equation}  where $\eta = c^2/\sigma$ is the magnetic diffusivity.  The joule damping rate of equation \ref{eqn:joule} can be compared with the damping rate from radiative diffusion (in the absence of composition gradients), $\Gamma_r = k_r^2 \kappa$, where $\kappa$ is the thermal diffusivity. The ratio of joule damping to thermal damping is  \begin{equation}  \label{eqn:jouleratio}  \frac{\Gamma_B}{\Gamma_r} = \frac{\eta}{\kappa} \frac{B^2 k_r^2}{(4 \pi)^2 \rho \omega^2} = \frac{\eta}{\kappa} \frac{l(l+1) B^2 N^2}{(4 \pi)^2 \rho r^2 \omega^4} \, ,  \end{equation}  and the second equality follows from using the gravity wave dispersion relation. The maximum magnetic field possible before Lorentz forces strongly alter gravity waves is $B_c$ (equation \ref{eqn:Bc}), and putting this value into equation \ref{eqn:jouleratio} we find  \begin{equation}  \label{eqn:jouleratio2}  \frac{\Gamma_B}{\Gamma_r} = \frac{1}{16 \pi} \frac{\eta}{\kappa} \, .  \end{equation}  Therefore, for gravity waves, joule damping cannot exceed thermal damping unless the magnetic diffusivity is significantly larger than the thermal diffusivity. In stellar interiors (and our RGB models), the magnetic diffusivity is typically orders of magnitude smaller than the thermal diffusivity. Therefore joule damping can safely be ignored. We note that the same result occurs if we use the Alfv\'en wave dispersion relation in equation \ref{eqn:jouleratio}, so joule damping is also unimportant for Alfv\'en waves.