Diffusion by a Random Velocity Field

Single-particle diffusion in a multivariate-normal, incompressible, stationary, isotropic velocity field is calculated in two and three dimensions by computer simulation and by the direct-interaction approximation. The computer simulations are carried out by storing the velocity field as a set of Fourier components and synthesizing in physical space only along the particle trajectories. The spectra taken for the velocity field are of the form E(k) ∝ δ(k − k0) and E(k) ∝ k4 exp (−2k2/k02) in three dimensions, and E(k) ∝ δ(k − k0) and E(k) ∝ k3 exp (−3k2/2k02) in two dimensions. Both frozen Eulerian fields and fields with Gaussian time correlation are treated. The simulation results agree with Taylor's picture of a classical diffusion process for times long compared with the eddy circulation time, except for the frozen-Eulerian-field runs in two dimensions, where strong trapping effects are found. The direct-interaction approximations for Lagrangian velocity correlation, eddy diffusivity, dispersion, and modal response functions agree well with the computer experiments except where there are trapping effects, which the approximation completely fails to represent.