Abstract
Two alternative depth scales have been proposed for the case of a stably stratified boundary layer (SBL) where static stability is due to the surface buoyancy flux B~s~. Kitaigorodskii in 1960 assumed that the Earth's rotation is no longer important as static stability becomes strong and that the SBL depth scales with the Obukhov length \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$L=-u_*^3/B_{\rm s}$ \end{document}, u~*~ being the surface‐friction velocity. Zilitinkevich in 1972 proposed an alternative scale, (u~*~L/|f|)^1/2^, which depends on the Coriolis parameter f no matter how strong the static stability. Similarly, two alternative depth scales have been proposed for the case of a SBL dominated by static stability at its outer edge with buoyancy frequency N. The depth scale u~*~/N introduced by Kitaigorodskii and Joffre in 1988 does not depend on the Coriolis parameter, whereas the Pollard, Rhines and Thompson scale u~*~/|Nf|^1/2^ introduced in 1973 does.
In the present article, the above formulations for the SBL depth are shown to be consistent with the budgets of momentum and of turbulence kinetic energy in the SBL. Furthermore, it is demonstrated that in the case of sufficiently strong static stability the alternative depth‐scale formulations represent particular cases of more general power‐law expressions. For a SBL dominated by the surface buoyancy flux, the generalized depth scale is given by L(|f|L/u~*~)^−γ^. For a SBL dominated by outer‐edge static stability, the generalized scale is (u~*~/N)(|f|/N)^−δ^. The exponents γ and δ lie in the range from 0 to 1/2. With γ = 1/2 and δ = 1/2, these expressions yield the Zilitinkevich scale and the Pollard et al. scale, respectively. In the limits γ = 0 and δ = 0, the SBL depth scales cease to depend on the Coriolis parameter in their explicit form and the formulations proposed by Kitaigorodskii and by Kitaigorodskii and Joffre, respectively, are recovered.
Simple dimensionality arguments are not sufficient to determine γ and δ. To do this would require an exact solution to equations governing the structure of mean fields and turbulence in the SBL. Since such a solution is not known, the exponents should be evaluated from experimental data. Available data from observations and from large‐eddy simulations are uncertain. They do not make it possible to evaluate γ and δ to adequate accuracy and to decide conclusively between the alternative formulations for the SBL depth. As regards practical applications, previously proposed multi‐limit formulations based on the above depth scales with γ and δ in the range from 0 to 1/2 are expected to give similar results for stability conditions typical of the atmospheric and oceanic SBLs, provided the disposable dimensionless coefficients in the multi‐limit formulations are appropriately tuned. Copyright © 2010 Royal Meteorological Society