Effects of finite scales of turbulence on dispersion estimates

The eddy-diffusion parameterization of turbulent transport is based on the assumption, often questionable for geophysical flows, of infinitesimal scales of turbulence. In this paper we quantitatively compute what are the consequences of this assumption on the estimates of dispersion for some simple flows with oceanographical scales. We compare the analytical results of two different models, one with infinitesimal turbulent time scale (corresponding to a model with eddy-diffusion) and the other with finite scale T. The difference between the two model results normalized with respect to the estimate of the eddy-diffusion model, called A, is computed and it is used as a measure of the accuracy of the eddy-diffusion parameterization. In the case of homogeneous turbulence, the eddy-diffusion model is accurate for asymptotic times t > T, as is known, whereas for small times it is characterized by an overestimate of the dispersion. A decreases asymptotically as T/t, with the overestimate less than 10% fort > lOT, so that the eddy-diffusion model is appropriate for the meso or global scales but it could be inadequate for short-term studies or for the interpretation of Lagrangian data. When a linear shear is superimposed on the turbulence, A still decreases as = T/t, which indicates that the range of applicability of the eddy-diffusion model is approximately the same for homogeneous and for linearly inhomogeneous flows. We then consider an oscillating linear shear with frequency o, and the results change drastically. A does not vanish asymptotically, but rather converges to an oscillating function with constant average and constant extrema. The values of A depend on the ratio between the turbulent time scale T and the period of oscillation l/w. When T and l/o are comparable, the eddy-diffusion model leads to an overestimate of order one even at asymptotic times. This result is a specific and analytically solvable example of a potentially much wider class of results valid for time-dependent flows with more complex spatial structure. A discussion of the possible generalizations and examples of applications to oceanographical flows are presented.