= Periodic two-coordinate anti-dynamo energy bound
{title2=$\Delta\langle B_z^2\rangle\leq Q^2\langle A_0^2\rangle/(4\eta^2k^2)$}
For a solenoidal three-component velocity depending on two periodic coordinates and a zero-mean magnetic field with the same dependence, the in-plane flux potential satisfies a homogeneous <advection-diffusion equation>. The <Poincaré inequality> gives $\langle A^2\rangle\leq\langle A_0^2\rangle e^{-2\eta k^2t}$. Integrating the vertical stretching term by parts bounds it by $Q\sqrt{\langle A^2\rangle\langle|\nabla B_z|^2\rangle}$, where $Q=\|\nabla u_z\|_\infty$. Maximizing against resistive dissipation and integrating gives the displayed transient bound. Splitting off half the dissipation instead gives $E'+\eta k^2E\leq(Q^2/\eta)\langle A_0^2\rangle e^{-2\eta k^2t}$, proving vertical energy decays. This permits transient generation by a nonzero vertical velocity while excluding sustained fields in this zero-mean, z-independent periodic class.
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