Nonlocal convective PBL model based on new third- and fourth-order moments

The standard approach to studying the planetary boundary layer (PBL) via turbulence models begins with the first-moment equations for temperature, moisture, and mean velocity. These equations entail second-order moments that are solutions of dynamic equations, which in turn entail third-order moments, and so on. How and where to terminate (close) the moments equations has not been a generally agreed upon procedure and a variety of models differ precisely in the way they terminate the sequence. This can be viewed as a bottom-up approach. In this paper, a top-down procedure is suggested, worked out, and justified, in which a new closure model is proposed for the fourth-order moments (FOMs). The key reason for this consideration is the availability of new aircraft data that provide for the first time the z profile of several FOMs. The new FOM expressions have nonzero cumulants that the model relates to the z integrals of the third-order moments (TOMS), giving rise to a nonlocal model for the FOMs. The new FOM model is based on an analysis of the TOM equations with the aid of large-eddy simulation (LES) data, and is verified by comparison with the aircraft data. Use of the new FOMs in the equations for the TOMS yields a new TOM model, in which the TOMS are damped more realistically than in previous models. Surprisingly, the new FOMs with nonzero cumulants simplify, rather than complicate, the TOM model as compared with the quasi-normal (QN) approximation, since the resulting analytic expressions for the TOMS are considerably simpler than those of previous models and are free of algebraic singularities. The new TOMS are employed in a second-order moment (SOM) model, a numerical simulation of a convective PBL is run, and the resulting mean potential temperature T, the SOMs, and the TOMS are compared with several LES data. As a final consistency check, T, SOMs, and TOMs are substituted from the PBL run back into the FOMs, which are again compared with the aircraft data.