We study the statistics of relative distances R(t) between fluid parti
cles in a spatially smooth random flow with arbitrary temporal correla
tions. Using the space dimensionality d as a large parameter we develo
p an effective description of Lagrangian dispersion. We describe the e
xponential growth of relative distances [R-2(t)] proportional to exp<2
(lambda)over bar t> at different values of the ratio between the corre
lation and turnover rimes. We find the stretching correlation time whi
ch determines the dependence of [R1R2] on the difference t(1)-t(2). Th
e calculation of the nest cumulant of R-2 shows that statistics of R-2
is nearly Gaussian at small times (as long as d much greater than 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 stre
tching correlation time which surprisingly appears to depend on the de
tails of the velocity statistics at t much less than tau. We establish
the dispersion of the In(R-2) in the log-normal statistics. (C) 1998
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