Diffusion and geometric effects in passive advection by random arrays of vortices

The Lagrangian transport of a passive scalar in a class of incompressible, random stationary velocity fields, termed "random-vortex" models, is studied. These fields generally consist of random distributions of finite-sized elementary vortices in space with zero mean velocity in the presence of molecular diffusion D. The effects of vortex density, vortex strength, and sign of the vorticity on the Lagrangian history of a fluid particle [i.e., mean-square displacement σ²(t) and velocity autocorrelation function script R(t)] on the specific random-vortex models which possess identical energy spectra but different higher-order statistics for a Péclet number of 100 are investigated. This is done by a combination of Monte Carlo simulations of the Langevin equations and analysis. It is found that the Lagrangian autocorrelation script R(t) and the mean-square displacement σ²(t) can be significantly different as the density of the vortices increases and when there are long-range correlations in the sign of the vorticity. A simple theory based on a model for script R(t) agrees strikingly well with the present simulations. It is found that D* increases with vortex density, suggesting that Gaussian fields are maximally dissipative among a wide class of vortex flows with given energy spectra.