= Zeldovich planar flux balance
{c}
{title2=$B_0\langle wa'\rangle=-\eta\langle|\nabla a'|^2\rangle-\eta\langle\bar a_z^2\rangle$}
For a stationary planar advected magnetic flux potential, average its <advection-diffusion equation> horizontally and split it into mean and fluctuation. The mean transport is $\overline{wa'}=\eta\bar a_z+C$, with $C=\langle wa'\rangle$. Multiplying the fluctuation equation by $a'$ and applying <integration by parts> gives $\langle\overline{wa'}\bar a_z\rangle+B_0C=-\eta\langle|\nabla a'|^2\rangle$. A vanishing mean derivative removes the constant term in the first correlation, yielding the displayed dissipative identity. The boundary assumptions must eliminate surface terms; a periodic layer replaces, rather than accompanies, decay at vertical infinity.
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