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.

When velocity consists of impulses $U_1$ spaced by $T$, the periodic corrector jumps by $U_1$ and decays between kicks. With $a=\kappa m^2$, $g(t)=U_1{e}^{-at}/(1-e^{-aT})$ for $0<t<T$. Integrating $g^2$ in the effective diffusivity of a periodic sinusoidal shear formula yields the displayed result. For $aT\gg 1$, independent transverse phases give longitudinal variance $U_1^2/2$ per kick and enhancement $U_1^2/(4T)$. For $aT\ll 1$, the enhancement is $U_1^2/(2\kappa m^2{T}^2)$ to leading order, the steady-shear result for average speed $U_1/T$. The zero-diffusivity limit does not commute with the long-time limit: without transverse diffusion the displacement is ballistic.