= Variance bound for planar turbulent magnetic transport
{title2=$\eta_t^2\operatorname{Var}(f)\leq\langle w^2\rangle d^2/(4\pi^2)$}
For $\overline{wa'}=-\eta_tB_0f$, $\langle f\rangle=1$ and positive diffusivities, the <Zeldovich planar flux balance> gives $G=(\eta_t/\eta)B_0^2[1-(\eta_t/\eta)\operatorname{Var}(f)]$, where $G=\langle|\nabla a'|^2\rangle$. The <Cauchy-Schwarz inequality> and a <Poincaré inequality> $\langle a'^2\rangle\leq d^2G/\pi^2$ give $\eta_t^2\operatorname{Var}(f)\leq\langle w^2\rangle d^2s(1-s)/\pi^2$, with $s=(\eta_t/\eta)\operatorname{Var}(f)\in[0,1]$. Maximizing $s(1-s)$ gives the displayed bound. Large turbulent-to-microscopic diffusivity therefore requires $f$ to be close to one in mean square, without guaranteeing pointwise uniformity.
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