An approximate analytical solution of flux-profile relationships for the atmospheric surface layer with different momentum and heat roughness lengths

An approximate method for calculating the relationship between z/L (z = reference height, L = Obukhov length) and the bulk Richardson number is presented. If this relationship is known, the momentum and heat fluxes can be computed easily without any iteration. The avoidance of iteration can speed up computations in large-scale models considerably (up to 10 times) and cases which do not converge or converge very slowly cannot occur. The proposed formulae take into account the difference between momentum (z(0M)) and heat roughness lengths (z(0H)). Because the roughness lengths are not neglected at any step of the derivation, the resulting analytical formulae can be used not only between the surface and the reference height but also between two finite levels z(1) and z(2) (by replacing z(0M) and z(0H) by z(1) and z by z(2)). The equations remain correct even in the limit z(1) --> z(2).The formulae are based upon the (partially modified) Businger-Dyer flux-profile relationships and, consequently, they are restricted to predominantly homogeneous terrain. These new approximations are an improvement over the existing solutions because they are simpler than most of the formulae in the literature and are able to match the numerical exact solution for different parameter sets (Businger, Dyer, Hogstrom) with an maximum error of about 2% for a wide range of z/L, z/z(0M) and z(0M)/z(0H). Furthermore, in stable conditions, schemes with and without a finite critical bulk Richardson number can be approximated. The possible ambiguity of the exact solution zeta = f(RIB) in (moderately) stable conditions is discussed briefly. The performance of the new formulae is compared to the exact numerical solution and to different formulae proposed in the literature.