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 〈R2(t)〉∝exp2λ̄t at different values of the ratio between the correlation and turnover times. We find the stretching correlation time which determines the dependence of R1R2 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.