We approach certain problems of statistical fluid mechanics from a kinematic point of view. As an alternative to the customary Gaussian fields of similar stochastic kinematic descriptions, we consider a Poisson velocity field ansatz constructed as a sum of eddies. We compute its spectrum and show that it can exhibit some features of fully developed turbulence à la Kolmogorov. We define a related field with a Poisson initial condition and a more physical evolution. Next, we look at advection-diffusion on these fields. We address the question of long time asymptotics which is related to Central Limit Theorems under weak dependence. With the aid of molecular diffusivity, bounded and toroidal versions of the Poisson field are shown to have homogenization.; Particle trajectories are simulated and the intermediate and long-time scaling behaviour of the mean square displacement is measured. Depending on the density of the eddies and the power law of the energy spectral density, both diffusive and subdiffusive limits are obtained. Based on comparison with real float data, a homogeneous model for the large scales of the ocean surface is proposed.
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