Averaged dynamics of time-periodic advection diffusion equations in the limit of small diffusivity

TL;DR

This paper analyzes how small diffusion affects passive tracers in time-periodic flows, deriving an averaged model using canonical transforms and validating it through numerical simulations.

Contribution

It introduces a novel averaged equation for tracer motion in time-periodic flows with small diffusion, including explicit formulas and validity estimates.

Findings

First-order approximation valid for extended times

Explicit formulas for averaged coefficients in vortical flows

Numerical simulations confirm theoretical validity range

Abstract

We study the effect of advection and small diffusion on passive tracers. The advecting velocity field is assumed to have mean zero and to possess time-periodic stream lines. Using a canonical transform to action-angle variables followed by a Lie-transform, we derive an averaged equation describing the effective motion of the tracers. An estimate for the time validity of the first-order approximation is established. For particular cases of a regularized vortical flow we present explicit formulas for the coefficients of the averaged equation both at first and at second order. Numerical simulations indicate that the validity of the above first-order estimate extends to the second order.