= Solution
Interpret the two-dimensional dependence as independence of $z$, allowing the velocity to have a vertical component $U=u_z(x,y)$. Write $u_h=(u_x,u_y)$ and $B_h=(A_y,-A_x)$. The <resistive induction equation>, with a gauge fixing the potential's zero mean, gives
$$
A_t+u_h\cdot\nabla A=\eta\nabla^2A,\qquad
B_t+u_h\cdot\nabla B=B_h\cdot\nabla U+\eta\nabla^2B.
$$
The first is a homogeneous <advection-diffusion equation>. Multiply it by $A$, average over the periodic box and use <integration by parts>. Advection contributes zero because $\nabla\cdot u_h=0$; there are no surface terms. Thus
$$
\frac12\frac{d}{dt}\langle A^2\rangle=-\eta\langle|\nabla A|^2\rangle
\leq-\eta k^2\langle A^2\rangle.
$$
Multiplying by $e^{2\eta k^2t}$ makes its derivative nonpositive. With the initial norm convention $\langle A^2(0)\rangle=A_0^2$, this proves
$$
\boxed{\langle A^2(t)\rangle\leq A_0^2e^{-2\eta k^2t}.}
$$
The <Poincaré inequality> uses the zero-mean assumption; a spatially constant flux potential would otherwise have no gradient. Zero mean remains preserved by this incompressible periodic equation.
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