deletions | additions       diff --git a/subsection_Joule_Damping_A_gravity__.tex b/subsection_Joule_Damping_A_gravity__.tex  index 1540cfa..d3d9cbe 100644  --- a/subsection_Joule_Damping_A_gravity__.tex  +++ b/subsection_Joule_Damping_A_gravity__.tex 
...
where $\eta$ is the magnetic diffusivity.  The joule damping rate of equation \ref{joule} can be compared with the damping rate from radiative diffusion, $\Gamma_r = k_r^2 \kappa$, where $\kappa$ is the thermal diffusivity. The ratio of joule damping to thermal damping is  \begin{equation} \begin{align}  \label{eqn:jouleratio}  \frac{\Gamma_B}{\Gamma_r} = &=  \frac{\eta}{\kappa} \frac{B^2 k_r^2}{(4 \pi)^2 \rho \omega^2} \nonumber \\ = &=  \frac{\eta}{\kappa} \frac{l(l+1) B^2 N^2}{(4 \pi)^2 \rho r^2 \omega^4} \, . \end{equation} \end{align}