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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 $\delta J \sim c k_r \delta B/(4 \pi)$, where $c$ is the speed of light. The volumetric energy dissipation rate is $\delta J^2/\sigma$, where $\sigma$ is the electrical conductivity. The wave energy density is $\rho \omega^2 \xi_\perp^2$, therefore the local damping rate is  \begin{equation \begin{equation}  \label{eqn:joule}  \Gamma_B = \frac{\eta B^2 k_r^4}{(4 \pi)^2 \rho \omega^2} \, ,  \end{equation}