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 arbitrary 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 $\bar\lambda$ and the dispersion $\Delta$. We use the method of higher space dimensions $d$ to estimate $\bar\lambda$ and $\Delta$ for different values of $\alpha$ and $\beta$. It was shown previously that $\bar\lambda\sim D$ for $\beta\ll1$ and $\bar\lambda\sim\sqrt{D/\tau}$ for $\beta\gg1$. The estimate of $\Delta$ is then nonuniversal and depends on details of the two-point velocity correlator. Higher-order velocity correlators give an additional contribution to $\Delta$ estimated as $\alpha D^2\tau$ for $\beta\ll1$ and $\alpha\beta/\tau$ for $\beta\gg1$. For $\alpha$ above some critical value $\alpha_\mathrm{cr}$, the values of $\bar\lambda$ and $\Delta$ are determined by higher irreducible correlators of the velocity gradient, and our approach loses its applicability. This critical value can be estimated as $\alpha_\mathrm{cr}\sim\beta^{-1}$ for $\beta\ll1$ and $\alpha_\mathrm{cr}\sim \beta^{-1/2}$ for $\beta\gg1$.