Universal dependences between turbulent and mean flow parameters instably and neutrally stratified Planetary Boundary Layers

Abstract. We consider the resistance law for the planetary boundary layer (PBL) from the point of view of the similarity theory. In other words, we select the set of the PBL governing parameters and search for an optimal way to express through these parameters the geostrophic drag coefficient C_g=u_* /U_g and the cross isobaric angle α (where u_* is the friction velocity and U_g is the geostrophic wind speed). By this example, we demonstrate how to determine the "parameter space" in the most convenient way, so that make independent the dimensionless numbers representing co-ordinates in the parameter space, and to avoid (or at least minimise) artificial self-correlations caused by the appearance of the same factors (such as u_*) in the examined dimensionless combinations (e.g. in C_g=u_* /U_g) and in dimensionless numbers composed of the governing parameters. We also discuss the "completeness" of the parameter space from the point of view of large-eddy simulation (LES) modeller creating a database for a specific physical problem. As recognised recently, very large scatter of data in prior empirical dependencies of C_g and α on the surface Rossby number Ro=U_g| fz₀|^-1 (where z₀ is the roughness length) and the stratification characterised by µ was to a large extent caused by incompactness of the set of the governing parameters. The most important parameter overlooked in the traditional approach is the typical value of the Brunt-Väisälä frequency N in the free atmosphere (immediately above the PBL), which involves, besides Ro and µ, one more dimensionless number: µ_N=N/ | f |. Accordingly, we consider C_g and α as dependent on the three (rather then two) basic dimensionless numbers (including µ_N) using LES database DATABASE64. By these means we determine the form of the dependencies under consideration in the part of the parameter space representing typical atmospheric PBLs, and provide analytical expressions for C_g and α.