= Solution
The <advection-diffusion equation> is
$$
c_t+u(y)c_x=D(c_{xx}+c_{yy}),
\qquad
u(y)=\frac Uh y.
$$
Write
$$
c=\bar c(x,t)+c'(x,y,t),
\qquad
\overline{c'}=0,
$$
where the bar is the cross-gap average. In the long, late-time Taylor regime, $c'$ adjusts rapidly across the gap while $\bar c$ varies slowly along the cell. The leading fluctuation balance is
$$
\boxed{
u(y)\bar c_x=Dc'_{yy}}.
$$
Its scaling is
$$
\frac{U\bar c}{L}\sim\frac{Dc'}{h^2},
\qquad
\boxed{
\frac{c'}{\bar c}
\sim\frac{Uh^2}{DL}
=\operatorname{Pe}_h\frac hL\ll1}.
$$
This final inequality is precisely the transverse-equilibration condition from part i.
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