Five different formulations of the stability functions used forvertical turbulent transfer in atmospheric models are compared in a 1-Dmodel of the nocturnal boundary layer. One of them has a critical valueof the Richardson number around 0.2 and leads to the traditional log-linear profile, while other more empirical formulations maintain sometransfer at values of Ri around 1.0. Although no new observationalevidence is presented, it is suggested that the latter formulations aremore appropriate for use in atmospheric models because the unresolvedvariability inside a model grid box induces some turbulent transfer evenat super-critical values of the mean Ri. The study shows that themagnitude of the stability functions is principally important in theeffective range of Ri values found in the stable boundary layer and thattheir slopes near the origin are less important. This permits the use inatmospheric models of a simple explicit function of Ri containing asingle parameter, with results similar to those obtained with earlier morecomplex formulations.
The results of the simulation with the 1-D model are used toexamine the errors introduced by the relatively thick surface layers of most atmospheric models, in which, for the stable case, the traditionalassumption of constancy of the fluxes with height is often clearlyviolated. When a height variation of the fluxes is introduced in surface-layer formulations, the error in the magnitude of the surface fluxes isdecreased with some of the formulations but not all of them. This lackof sensitivity is explained by a compensation mechanism in which theassumed decrease of the fluxes with height implies a correspondingdecrease of the Obukhov length which acts in the oppositedirection, reducing, and sometimes eliminating, the adverse effect of theunrealistic specification of the fluxes. It may be argued that thiscompensation mechanism also explains the wide range of validity of theMoini-Obukhov similarity theory.