Five different formulations of the stability functions used for vertical turbulent transfer in atmospheric models are compared in a 1-D model of the nocturnal boundary layer. One of them has a critical value of the Richardson number around 0.2 and leads to the traditional log- linear profile, while other more empirical formulations maintain some transfer at values of Ri around 1.0. Although no new observational evidence is presented, it is suggested that the latter formulations are more appropriate for use in atmospheric models because the unresolved variability inside a model grid box induces some turbulent transfer even at super-critical values of the mean Ri. The study shows that the magnitude of the stability functions is principally important in the effective range of Ri values found in the stable boundary layer and that their slopes near the origin are less important. This permits the use in atmospheric models of a simple explicit function of Ri containing a single parameter, with results similar to those obtained with earlier more complex formulations. The results of the simulation with the 1-D model are used to examine the errors introduced by the relatively thick surface layers of most atmospheric models, in which, for the stable case, the traditional assumption of constancy of the fluxes with height is often clearly violated. When a height variation of the fluxes is introduced in surface-layer formulations, the error in the magnitude of the surface fluxes is decreased with some of the formulations but not all of them. This lack of sensitivity is explained by a compensation mechanism in which the assumed decrease of the fluxes with height implies a corresponding decrease of the Obukhov length which acts in the opposite direction, reducing, and sometimes eliminating, the adverse effect of the unrealistic specification of the fluxes. It may be argued that this compensation mechanism also explains the wide range of validity of the Monin-Obukhov similarity theory.
