Zeldovich planar flux balance ($B_0⟨w{a}^{\prime }⟩=-\eta ⟨|\nabla a^{\prime }{|}^2⟩-\eta ⟨{\bar{a}}_z^2⟩$)

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 $\overline{w{a}^{\prime }}=\eta {\bar{a}}_z+C$, with $C=⟨w{a}^{\prime }⟩$. Multiplying the fluctuation equation by $a^{\prime }$ and applying integration by parts gives $⟨\overline{w{a}^{\prime }}{\bar{a}}_z⟩+B_{0}C=-\eta ⟨|\nabla a^{\prime }{|}^2⟩$. 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.