The Effect of Helicity on the Effective Diffusivity for Incompressible Random Flows

Abstract

The advection of a passive scalar by a quenched (frozen) incompressible velocity field is studied by extensive high precision numerical simulation and various approximation schemes. We show that second order self consistent perturbation theory, in the absence of helicity, perfectly predicts the effective diffusivity of a tracer particle in such a field. In the presence of helicity in the flow simulations reveal an unexpectedly strong enhancement of the effective diffusivity which is highly nonperturbative and is most visible when the bare molecular diffusivity of the particle is small. We develop and analyse a series of approximation schemes which indicate that this enhancement of the diffusivity is due to a novel second order effect whereby the helical component of the field, which does not directly renormalize the effective diffusivity, enhances the strength of the non helical part of the flow, which in turn renormalizes the molecular diffusivity. We show that this renormalization is most important at low bare molecular diffusivity, in agreement with the numerical simulations.