Geodesic
Dispersion of Lagrangian trajectories in a random large-scale velocity field
Teoretičeskaâ i matematičeskaâ fizika, Tome 122 (2000) no. 3, pp. 456-467 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\beta\gg1$. 
@article{TMF_2000_122_3_a11,
     author = {V. R. Kogan},
     title = {Dispersion of {Lagrangian} trajectories in a random large-scale velocity field},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {456--467},
     year = {2000},
     volume = {122},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2000_122_3_a11/}
}
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V. R. Kogan. Dispersion of Lagrangian trajectories in a random large-scale velocity field. Teoretičeskaâ i matematičeskaâ fizika, Tome 122 (2000) no. 3, pp. 456-467. http://geodesic.mathdoc.fr/item/TMF_2000_122_3_a11/

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