Closure Approximations for Passive Scalar Turbulence: A Comparative Study on an Exactly Solvable Model with Complex Features

Some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep. This model has a general geometric structure of a jet flow with transverse disturbances, which occur in a number of contexts, and it encompasses a wide variety of possible spatio-temporal statistical structures for the velocity field, including strong long-range correlations. Even though the Eulerian and Lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader-than-Gaussian intermittency, this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations. Our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly. In particular it is shown that the direct interaction approximation (DIA), one of the most sophisticated closure approximations available, fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions, even though the equations it produces are rather complicated and difficult to analyze. Two alternative closure approximations, the Modified DIA (MDIA) and the Renormalized Lagrangian Approximation (RLA), with different levels of sophistication, both are simpler to use than the DIA and perform better. In particular, it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the MDIA is even able to capture intermittency effects.