Suppression of particle dispersion by sweeping effects in synthetic turbulence

Synthetic models of Eulerian turbulence like so-called kinematic simulations (KS) are often used as computational shortcuts for studying Lagrangian properties of turbulence. These models have been criticized by Thomson and Devenish (2005), who argued on physical grounds that sweeping decorrelation effects suppress pair dispersion in such models. We derive analytical results for Eulerian turbulence modeled by Gaussian random fields, in particular for the case with zero mean velocity. Our starting point is an exact integrodifferential equation for the particle pair separation distribution obtained from the Gaussian integration-by-parts identity. When memory times of particle locations are short, a Markovian approximation leads to a Richardson-type diffusion model. We obtain a time-dependent pair diffusivity tensor of the form K(ij)(r,t)=S(ij)(r)τ(r,t), where S(ij)(r) is the structure-function tensor and τ(r,t) is an effective correlation time of velocity increments. Crucially, this is found to be the minimum value of three times: the intrinsic turnover time τ(eddy)(r) at separation r, the overall evolution time t, and the sweeping time r/v(0) with v(0) the rms velocity. We study the diffusion model numerically by a Monte Carlo method. With inertial ranges like the largest achieved in most current KS (about 6 decades long), our model is found to reproduce the t(9/2) power law for pair dispersion predicted by Thomson and Devenish and observed in the KS. However, for much longer ranges, our model exhibits three distinct pair-dispersion laws in the inertial range: a Batchelor t(2) regime, followed by a Kraichnan-model-like t(1) diffusive regime, and then a t(6) regime. Finally, outside the inertial range, there is another t(1) regime with particles undergoing independent Taylor diffusion. These scalings are exactly the same as those predicted by Thomson and Devenish for KS with large mean velocities, which we argue hold also for KS with zero mean velocity. Our results support the basic conclusion of Thomson and Devenish (2005) that sweeping effects make Lagrangian properties of KS fundamentally differ from those of hydrodynamic turbulence for very extended inertial ranges.