  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.
