Enhanced diffusivity and intercell transition layers in 2-D models of passive advection

A class of two-dimensional, isotropic, divergence-free vector fields is introduced and the effective diffusivity of the corresponding advection-diffusion equations is studied. These examples are very idealized flows, but they can be solved exactly in the limit Pe ≫ 1. Scaling laws D* ∝ D₀(Pe)^α are obtained, where D ₀ = molecular diffusion, Pe = Peclet number, with exponents in the range 0 < α < 1, and examples of "stream functions" with logarithmic singularities for which D* ∝ D₀Pe. The exponent α is related by a simple formula to the shape of the stream function along cell boundaries, suggesting that similar scaling laws should hold for more general 2-D closed-cell flows.