On the relationship between the gradient and the bulk Richardson number for the atmospheric surface layer

Summary Semi-empirical formulations which have been proposed to describe the wind and potential temperature profiles are used to derive relationships between the gradient Richardson number, Ri, the finite-difference layer Richardson number, Ri_b, the surface layer Richardson number, Ri_s, and the bulk Richardson number,B, through the atmospheric surface layer. The theoretical analysis for stable conditions indicates that Ri (z ₃)=Ri_b, wherez ₃=(z ₂−z ₁)/ln(z ₂/z ₁), andz ₂;z ₁=upper and lower levels at which temperature and wind speed are specified. It is also found that, during stable conditions, the wind profile power law exponent,p, is computed at the heightz ₃, instead of the widely used geometric mean height,z _m, between top (z ₂) and bottom (z ₁) of the layer considered.