Summary: The homogenization of a passive ”tracer” in a flow with closed mean streamlines occurs in two stages: first, a rapid phase dominated by shear-augmented diffusion over a time approximately \(P^{1/3}(L/U)\), where the Peclet number \(P=LU(\kappa\) (L, U and \(\kappa\) are lengthscale, velocity scale and diffusivity), in which initial values of the tracer are replaced by their (generalized) average about a streamline; second, a slow phase requiring the full diffusion time approximately \(L^ 2/\kappa\). The diffusion problem for the second phase, where tracer isopleths are held to streamlines by shear diffusion, involves a generalized diffusivity which is proportional to kappa, but exceeds it if the streamlines are not circular. Expressions are also given for flow fields that are oscillatory rather than steady.