  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.
