  Synthetic models of Eulerian turbulence like so called "Kinematic
Simulations" (KS) have been criticized by Thomson & Devenish (TD) (2005), who
argued that sweeping decorrelation effects suppress pair dispersion in such
models. We derive analytical results for Eulerian turbulence modeled by
Gaussian random fields for the case with zero mean velocity. Our starting point
is an exact integrodifferential equation for the particle pair separation
distribution. When memory times of particle locations are short, a Markovian
approximation leads to a Richardson-type diffusion model. We obtain a
diffusivity tensor of the form $K_{ij}(r,t)=S_{ij}(r)\tau(r,t)$ where
$S_{ij}(r)$ is the structure-function tensor and $\tau(r,t)$ is an effective
correlation time of velocity increments. This is found to be the minimum value
of three times: the intrinsic turnover time $\tau_{eddy}(r)$ 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 moderate inertial-ranges like those achieved in current KS, our model is
found to reproduce the $t^{9/2}$ power-law for pair dispersion predicted by TD
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 $t^1$ 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 TD for KS with large mean velocities, which we
argue hold also for KS with zero mean velocity. Our results support the basic
conclusion that sweeping effects make Lagrangian properties of KS fundamentally
different from hydrodynamic turbulence for very extended inertial-ranges.
