What causes the divergences in local second-order closure models?

It has been known for three decades that in the case of buoyancy-driven flows the widely used second-order closure (SOC) level-2.5 turbulence models exhibit divergences that render them unphysical in certain domains. This occurs when the dimensionless temperature gradient G(h) (defined below) approaches a critical value G(h)(cr) of the order of 10; thus far, the divergerices have been treated with ad hoc limitations of the type G(h) <= G(h)(cr) similar to 10, G(h) equivalent to -tau(2) g alpha partial derivative T/partial derivative z, where tau is the eddy turnover time scale, g is the gravitational acceleration, g is the coefficient of thermal expansion, T is the mean potential temperature and z is the height. It must be noted that large eddy simulation (LES) data show no such limitation. The divergent results have the following implications. In most of the aT/az < 0 portion of a convective planetary boundary layer (PBL), a variety of data show that tau increases with z, -aT/az decreases with z, and G(h) decreases with G(h)(cr). As one approach's the surface laver from above, at some z(cr) (similar to 0.2H, H is the PBL height), G(h), approaches G(h)(cr) and the model results diverge. Below z(er), existing models assume the displayed equation above. Physically, this amounts to artificially making the eddy lifetime shorter than what it really is. Since short-lived eddies are small eddies, one is essentially changing-large eddies into small eddies. Since large eddies are the main contributors to bulk properties such as heat, momentum flux. etc., the artificial transformation of large eddies into small eddies is equivalent to underestimating the efficiency of turbulence (is a mixing process. In this paper the physical origin of the divergences is investigated. First, it is shown that it is due to the local nature of the level-2.5 models. Second. it is shown that once an appropriate nonlocal model is employed, all the divergences cancel out, yielding a finite result. An immediate implication of this result is the need for a reliable model for the third-order moments (TOMs) that represent nonlocality. The TOMs must not only compare well with LES data, but in addition, they must yield nondivergent second-order moments.