What Causes the Divergences in Local Second-Order Closure Models?

Abstract

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 Gh (defined below) approaches a critical value Gh(cr) of the order of 10; thus far, the divergences have been treated with ad hoc limitations of the type where τ is the eddy turnover time scale, g is the gravitational acceleration, α 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 ?T/?z < 0 portion of a convective planetary boundary layer (PBL), a variety of data show that τ increases with z, ??T/?z decreases with z, and Gh decreases with z. As one approaches the surface layer from above, at some zcr (?0.2H, H is the PBL height), Gh approaches Gh(cr) and the model results diverge. Below zcr, 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 as 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.