Abstract
The problem of stirring a passive scalar by a random velocity field is considered. By the short-correlation approximation one means the assumption that the decorrelation time of the velocity field is infinitely small. There is a widespread misunderstanding that such an approach readily yields the Fokker-Planck equation for the mean field of passive scalar. Indeed, the resulting equation strongly depends on the order of a “hidden” time scale appearing in this problem. This time scale called the turnover time is defined as the ratio of the correlation radius and the mean square velocity fluctuation. We show that the effective diffusivity is different under different assumptions on the order of turnover time. As a consequence we have different physical effects for different forms of the scale separation. Also some new rigorous results related to the regime of superdiffusion are obtained.
This work was supported by ONR Grant No. N00014-91-J-1526.
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Piterbarg, L. (1997). Short-Correlation Approximation in Models of Turbulent Diffusion. In: Molchanov, S.A., Woyczynski, W.A. (eds) Stochastic Models in Geosystems. The IMA Volumes in Mathematics and its Applications, vol 85. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8500-4_15
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