Particle dispersion in a multidimensional random flow with arbitrary temporal correlations

We study the statistics of relative distances R(t) between fluid particles in a spatially smooth random flow with arbitrary temporal correlations. Using the space dimensionality d as a large parameter we develop an effective description of Lagrangian dispersion. We describe the exponential growth of relative distances at different values of the ratio between the correlation and turnover times. We find the stretching correlation time which determines the dependence of on the difference t1−t2. The calculation of the next cumulant of R2 shows that statistics of R2 is nearly Gaussian at small times (as long as d 1) and becomes log-normal at large times when large-d approach fails for high-order moments. The crossover time between the regimes is the stretching correlation time which surprisingly appears to depend on the details of the velocity statistics at t τ. We establish the dispersion of the ln(R2) in the log-normal statistics.