A Hierarchy of Turbulence Closure Models for Planetary Boundary Layers

Turbulence models centered on hypotheses by Rotta and Kolmogoroff are complex. In the present paper we consider systematic simplifications based on the observation that parameters governing the degree of anisotropy are small. Hopefully, we shall discern a level of complexity which is intuitively attractive and which optimizes computational speed and convenience without unduly sacrificing accuracy. Discussion is focused on density stratified flow due to temperature. However, other dependent variables—such as water vapor and droplet density—can be treated in analogous fashion. It is, in fact, the anticipation of additional physical complexity in modeling turbulent flow fields that partially motivates the interest in an organized process of analytical simplification. For the problem of a planetary boundary layer subject to a diurnally varying surface heat flux or surface temperature, three models of varying complexity have been integrated for 10 days. All of the models incorporate identical empirical constants obtained from neutral flow data alone. The most complex of the three models requires simultaneous solution of 10 partial differential equations for turbulence moments in addition to the equations for the mean velocity components and temperature; the least complex eliminates all of the 10 differential equation whereas a “compromise” model retains two differential equations for total turbulent energy and temperature variance. We conclude that all of the models give nearly the same results. We find the two-differential-equation model particularly attractive.