Effect of the Earth's rotation on the equilibrium depth of a stably stratified barotropic planetary boundary layer

Formulations for the equilibrium depth of the stably stratified atmospheric planetary boundary layer (SBL) are examined from the standpoint of their consistency with the momentum budget and the turbulence kinetic energy (TKE) budget in the presence of the Earth’s rotation characterized by the Coriolis parameter f. Two SBL depth scales commonly discussed in the literature (see, e.g., Zilitinkevich et al. 2007) are considered. The first is the Zilitinkevich (1972) scale (u_*L/|f|)^1/2, which accounts for static stability due to the surface buoyancy flux B_s through the Obukhov length L = –u_*³/B_s, where u_* is the surface friction velocity. The second is the Pollard et al. (1973) scale u_*(N|f|)^–1/2 for the SBL affected by the static stability at its outer edge with the buoyancy frequency N>0.

Both SBL-depth scales are shown to be consistent with the momentum and TKE budgets. Furthermore, it is demonstrated that in case of a sufficiently strong static stability (L or/and u_*/N is small compared to u_*/|f|), the scales are particular cases of generalized SBL depth scales. For the SBL dominated by the surface buoyancy flux, the generalized depth scale is given by L (|f|L/u_*) ^–γ. For the SBL dominated by the outer static stability, the generalized scale is (u_*/N)(|f|/N) ^–d. The exponents γ and d lie in the range from 0 to 1. With γ=1/2 and d=1/2, the first expression yields the Zilitinkevich (1972) scale, whilst the second one yields the Pollard et al. (1973) scale. In the limit of γ=0 and d=0, the SBL depth scales cease to depend on the Coriolis parameter in their explicit form. In this case, the first generalized scale turns into L, which was first proposed as the SBL depth scale for the regime of strong stability by Kitaigorodskii (1960). The second generalized scale turns into the Kitaigorodskii and Joffre (1988) scale u_*/N. It remains to be seen, however, whether these depth scales are representative of equilibrium (steady) flow regimes in a horizontally homogeneous SBL.

Simple scaling arguments are not sufficient to fix γ and d. To do this would require an exact solution to equations governing the structure of mean fields and turbulence in the SBL. Since such solution is not known, γ and d should be evaluated from experimental or/and numerical simulation data. Available data on the SBL from both sources are uncertain and do not make it possible to evaluate γ and d to sufficient accuracy. It is, therefore, not possible at present to conclusively identify an optimal formulation for the SBL depth on purely empirical grounds. As regards practical applications, the multi-limit formulations (e.g. Zilitinkevich and Mironov 1996, Zilitinkevich et al. 2002, 2007) based on the above depth scales with γ and d in the range from 0 to 1/2 are expected to give similar results over a range of stability conditions typical of the atmospheric and oceanic SBLs once the disposable dimensionless coefficients in these formulations are appropriately tuned.