Stability functions correct at the free convection limit and consistent for for both the surface and Ekman layers

For the thermal stability function _h used to calculate heat and moisture fluxes in the surface layer, we choose a formulation which has the theoretically correct free convection limit % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeikaiabgk% HiTGqaciaa-PhacaqGVaGaamitaiaabMcadaahaaWcbeqaaiabgkHi% TiaaigdacaGGVaGaaG4maaaaaaa!3DFE![{rm{(}} - z{rm{/}}L{rm{)}}^{ - 1/3} ]. We then use the experimental result that z/L Ri to deduce a formulation with an exponent -1/6 for the momentum stability function _m. This formulation also resolves the matching problem at the interface between the surface and Ekman layers. The proposed functions are found to remain reasonably close to another formulation that is well supported by observations and has exponents -1/2 for _h and -1/4 for _m. The intent of the proposals is mainly to clarify and simplify the parameterization of the convective boundary layer in present day atmospheric models, without significantly altering the results.