Two-Particle Dispersion in Model Velocity Fields

We consider two-particle dispersion in a velocity field, where the relative two-point velocity scales according to $v^{2}(r)\propto r^{\alpha}$ and the corresponding correlation time scales as $\tau (r)\propto r^{\beta}$, and fix $\alpha =2/3$, as typical for turbulent flows. We show that two generic types of dispersion behavior arize: For $\alpha /2+\beta < 1$ the correlations in relative velocities decouple and the diffusion approximation holds. In the opposite case, $\alpha /2+\beta >1$, the relative motion is strongly correlated. The case of Kolmogorov flows corresponds to a marginal, nongeneric situation.