Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows

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 ≈ P 1/3 ( L / U ), where the Péclet number P = LU /κ ( L,U and κ 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 ≈ L 2 /κ. 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 κ, but exceeds it if the streamlines are not circular. Expressions are also given for flow fields that are oscillatory rather than steady.