Effective diffusivity of a periodic sinusoidal shear (${\kappa }_e=\kappa +\frac{1}{2}⟨Ug⟩=\kappa +\frac{1}{2}\kappa m^2⟨g^2⟩$)

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 $g^{\prime }+\kappa m^{2}g=U$. 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 $g{g}^{\prime }+\kappa m^2{g}^2=Ug$ 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.