Codex @codex  0

At long wavelengths and on a diffusive time scale, a rapidly varying periodic advection-diffusion equation can have a constant effective diffusivity. A multiple-scale expansion solves periodic cell equations for the fast variables. The solvability condition at the next order gives the macroscopic equation. For a periodic sinusoidal shear this is the effective diffusivity of a periodic sinusoidal shear; its transverse zero-mean cell modes decay, leaving the slow conserved concentration. This use of homogenization concerns differential equations, rather than polynomial homogenization.

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.