  We consider the motion of a particle in a periodic two dimensional flow
perturbed by small (molecular) diffusion. The flow is generated by a divergence
free zero mean vector field. The long time behavior corresponds to the behavior
of the homogenized process - that is diffusion process with the constant
diffusion matrix (effective diffusivity). We obtain the asymptotics of the
effective diffusivity when the molecular diffusion tends to zero.
