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Use a planar flux function $F=B_{0}z+a$ and writeThis automatically satisfies the solenoidal magnetic-field constraint. There is a sign convention to settle: with the usual definition $\mathbf{B}=\nabla \times \mathbf{A}$, the magnetic vector potential is $\mathbf{A}=-F\hat{\mathbf{y}}$. The positive sign printed for $A_y$ instead gives the opposite field, or corresponds to defining $\mathbf{B}=-\nabla \times \mathbf{A}$. We use the standard curl convention and the flux $F$ oriented along the prescribed positive $B_0$; the flux evolution and all transport identities below are unchanged by reversing the overall potential convention.

For planar incompressible flow, substitute this representation into the resistive induction equation. The two in-plane components implyAn additive time-dependent constant in $F$ leaves the field unchanged and removes $C(t)$. Since ${\nabla }^2(B_{0}z)=0$, this givesThis is the scalar advection-diffusion equation underlying the planar anti-dynamo theorem, with the imposed-field source retained.

Put $M(z)=\bar{a}(z)$, $a=M+a^{\prime }$ and $F_c(z)=\overline{w{a}^{\prime }}$. Stationarity of the horizontal averages, $\bar{w}=0$, and incompressibility giveHere the horizontal derivative of the advective flux has zero average. Integration gives $F_c=\eta M^{\prime }+C$. Averaging once more, with $⟨M^{\prime }⟩=0$, identifies $C=⟨w{a}^{\prime }⟩$. ThusDecay or periodic averaging removes the endpoint contribution from the mean derivative. The periodic assumption used later replaces decay at vertical infinity; a nonzero periodic function cannot also decay there.

Subtract the averaged equation from the scalar equation and multiply by $a^{\prime }$. The average of the time derivative is zero by stationarity; the advective term becomes a boundary flux and vanishes by incompressibility and the stated boundary averaging. Terms depending only on $z$ multiply $\overline{a^{\prime }}=0$. Integration by parts in the diffusion term therefore givesSubstituting $F_c=\eta M^{\prime }+C$ and using $⟨M^{\prime }⟩=0$ yields the Zeldovich planar flux balanceBoth terms on the right are dissipative. In a nontrivial stationary forced problem this requires transport opposing the imposed flux gradient.

For finite-energy localized fields, enforcing decay at both vertical infinities literally gives $C=0$ from the mean-flux boundary values; the corresponding integral balance then forces both dissipation terms to vanish. The later nonzero-transport, periodic case therefore replaces that localized boundary setting rather than satisfying both simultaneously.

Now assume $B_0\ne 0$ and a positive turbulent magnetic diffusivity ${\eta }_1$. DefineSince $F_c=-{\eta }_1{B}_{0}f$ and $⟨f⟩=1$, its spatial mean is $C=-{\eta }_1{B}_0$. HenceThe Zeldovich planar flux balance becomesBecause $C\ne 0$, the fluctuation cannot vanish, and a nontrivial fluctuation has $G>0$ under the stated boundary or zero-mean periodic conditions. ThereforeIf ${\eta }_1\gg \eta$, then $\Vert f-1{\Vert }_{L^2}^2<\eta /{\eta }_1\ll 1$: the transport profile must be nearly constant in mean square. This does not, without additional regularity, imply a uniform pointwise bound on narrow spikes.

For the final estimate use the supplied Poincaré inequality in the periodic setting and the Cauchy-Schwarz inequality:Insert the expression for $G$, cancel $B_0^2$, and multiply by $V$. Setting $y=r_{t}V\in [0,1)$ givesConsequently the variance bound for planar turbulent magnetic transport isThe factor $1/4$ comes from maximizing $y(1-y)$, so the weaker supplied Poincare constant already suffices. Under a zero vertical mean at each horizontal location, the usual period-$d$ Fourier bound is even stronger, with $4{\pi }^2/d^2$ in place of ${\pi }^2/d^2$.