Dispersion of Lagrangian trajectories in a random large-scale velocity field

We study the distribution of the distance R(t) between two Lagrangian trajectories in a spatially smooth turbulent velocity field with an arbutrary correlation time and a non-Gaussian distribution. There are two dimensionless parameters, the degree of deviation from the Gaussian distribution alpha and beta = tau D, where tau is the velocity correlation time and D is a characteristic velocity gradient. Asymptotically, R(t) has a lognormal distribution characterized by the mean runaway velocity lambda and the dispersion Delta. We use the method of higher space dimensions d to estimate lambda and Delta for different values of alpha and beta. It was shown previously that lambda similar to D for beta much less than 1 and lambda similar to root D.tau for beta much greater than 1. The estimate of Delta is then nonuniversal and depends on details of the two-point velocity correlator. Higher-order velocity correlators give an additional contributions to Delta estimated as alpha D(2)tau for beta much less than 1 and alpha beta/tau for beta much greater than 1. For alpha above some critical value alpha(cr), the values of lambda and Delta are determined by higher irreducible correlators for the velocity gradient, and our approach loses its applicability. This critical value can be estimated as alpha(cr) similar to beta(-1) for beta much less than 1 and alpha(cr) similar to beta(-1/2) for beta much greater than 1.