Effective diffusivity in non-isotropic gradient flows

We study the effective diffusivity tensor for particles in a random gradient flow that, statistically, lacks rotational symmetry. The effective diffusivity tensor is computed up to two-loop order in perturbation theory and the 'Ward identity' that relates this tensor to the effective coupling is verified to the same order. We re-examine the renormalization group calculation that produced a very accurate numerical value for the effective diffusivity in the rotationally symmetric case and formulate two versions based on distinct divisions of the random potential field that gives rise to the flow. Both types of renormalization group calculation give good results when compared with numerical simulations. However, at two-loop order in perturbation theory the two methods differ in detail from each other and from the exact perturbation calculation. This is in contrast to the corresponding results in the isotropic case.