= Effective diffusivity of a periodic sinusoidal shear
{title2=$\kappa_e=\kappa+\tfrac12\langle Ug\rangle=\kappa+\tfrac12\kappa m^2\langle g^2\rangle$}
For a <passive scalar> transported by $U(t)\sin(my)$ with transverse period $2\pi/m$, positive <diffusivity> $\kappa$ and periodic $U$, let $g$ be the unique periodic solution of $g'+\kappa m^2g=U$. A <multiple-scale expansion> has first corrector $-g(t)\sin(my)\partial_X\overline\chi$. Averaging the next-order equation over the space-time cell gives the displayed longitudinal <diffusion coefficient>. Averaging $gg'+\kappa m^2g^2=Ug$ proves the second form. This is a long-time, long-wavelength effective equation, after transverse and periodic transients have decayed, rather than an exact finite-time closure for arbitrary concentrations.
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