## 1. Introduction

A stably stratified boundary layer (SBL) is a matter of keen interest in geophysical fluid dynamics. Although a lot of studies have been devoted to the SBL, there are still many controversial issues that need to be discussed further and eventually resolved. One such issue is the equilibrium SBL depth, that is the depth in a quasi-steady state to which the SBL evolves in response to external forcing. In the present article, this issue is reconsidered with an emphasis on the effect of the Earth's rotation on the equilibrium depth of a stably stratified barotropic boundary layer.

It should be stressed that the discussion in the present article is limited to the depth-scale formulations for a quasi-steady-state barotropic SBL. Other aspects of the real-world SBL, such as the effect of the horizontal components of the angular velocity of the Earth's rotation and the Ekman-layer rectification phenomenon (Zikanov *et al.*, 2003; McWilliams and Huckle, 2006; McWilliams *et al.*, 2009), although very important, are beyond the scope of the present study. However, even highly idealized SBL archetypes, such as a barotropic quasi-steady-state SBL, are of great significance. Apart from their academic utility, they are widely used in applications. For example, the equilibrium SBL depth is one of the key parameters in pollution dispersion studies.

A number of formulations for the equilibrium SBL depth *h* have been proposed to date (see discussions in Zilitinkevich and Mironov, 1996, hereafter ZM96; Zilitinkevich *et al.*, 2002, hereafter ZBRSLC02; Zilitinkevich and Esau, 2003; Hess, 2004; Zilitinkevich *et al.*, 2007). The major formulations are summarized in Table I, where *u*_{*} is the surface friction velocity, *B*_{s} is the surface buoyancy flux, *f* is the Coriolis parameter, *N* is the buoyancy frequency at the SBL outer edge and is the Obukhov (1946) length scale (we omit the von Kármán constant *κ* from the expression for *L*).

Truly neutral boundary layer | u_{*}/|f| | |

(Rossby and Montgomery, 1935) | ||

Surface-flux-dominated SBL | ||

(Kitaigorodskii, 1960) | (Zilitinkevich, 1972) | |

Imposed-stability-dominated SBL | u_{*}/N | u_{*}/|Nf|^{1/2} |

(Kitaigorodskii | (Pollard, Rhines | |

and Joffre, 1988) | and Thompson, 1973) |

As seen from Table I, there are two alternative depth scales for an SBL dominated by a stabilizing surface buoyancy flux. Kitaigorodskii (1960) held the viewpoint that in the case of strong static stability the Earth's rotation is no longer important and the SBL depth scales with the Obukhov length. Zilitinkevich (1972) proposed an alternative formulation, where the SBL depth depends on the Coriolis parameter no matter how strong the static stability. Similarly, the Kitaigorodskii and Joffre (1988) depth scale for an SBL affected by static stability at its outer edge (we will also refer to such an SBL as the imposed-stability-dominated SBL) does not depend on the Coriolis parameter, whereas the Pollard *et al.* (1973) depth scale does.

The results from earlier studies were summarized by ZM96, who concluded that the Rossby and Montgomery (1935), Kitaigorodskii (1960) and Kitaigorodskii and Joffre (1988) scales hold in the limiting cases of a truly neutral rotating boundary layer, a surface-flux-dominated SBL and an imposed-stability-dominated SBL, respectively. The Zilitinkevich (1972) and Pollard *et al.* (1973) scales were found to describe the intermediate regimes, where the effects of rotation and stratification are roughly equally important. A multi-limit SBL-depth formulation was proposed in ZM96 that accounts for the combined effects of rotation, surface buoyancy flux and static stability at the SBL outer edge. It reads

where *C*_{n}, *C*_{s}, *C*_{i}, *C*_{sr} and *C*_{ir} are dimensionless constants (we use the original ZM96 notation). Equation (1) shows that *h* ceases to depend on *f* in the case of strong static stability, i.e. when *L* or/and *u*_{*}*/N* is small compared with *u*_{*}/|*f*|. A simplified version of Eq. (1) that does not incorporate the intermediate scales, represented by the last two terms on the left-hand side (l.h.s.), was favourably tested against data from large-eddy simulations (LES) and from measurements in the atmospheric and benthic SBLs.

The problem of the equilibrium SBL depth was reconsidered by ZBRSLC02. They concluded that the appropriate depth scales for boundary layers dominated by the surface buoyancy flux and by static stability at the outer edge of the boundary layer are the Zilitinkevich (1972) scale and the Pollard *et al.* (1973) scale, respectively, and proposed the following multi-limit formulation for the equilibrium SBL depth:

where *C*_{R}, *C*_{s} and *C*_{uN} are dimensionless constants (the original ZBRSLC02 notation is retained). Equation (2), which was favourably tested against observational and LES data in ZBRSLC02, shows that *h* depends on *f* no matter how strong the static stability.

Worthy of mention is the depth scale obtained by McWilliams et al. (2009) by considering the oceanic SBL affected by both the surface buoyancy flux and the stable density stratification beneath the boundary layer. However, as the authors of op. cit. state, this scale ‘seems unlikely to be a physically common or important regime except in extreme conditions of heating and stratification’.

In the subsequent text, we examine the alternative formulations for the SBL depth given in Table I from the standpoint of their consistency with the budgets of turbulence kinetic energy (TKE) and of momentum in the boundary layer (sections 2 and 3, respectively). We demonstrate (section 4) that the alternative formulations can all be derived on the basis of TKE-budget and momentum-budget considerations, although different assumptions should be made on the way. In section 5, we propose generalized power-law formulations that incorporate the alternative SBL depth scales as particular cases. Then (section 6) we test Eq. (1) (more specifically, its reduced form without the last two terms on the l.h.s.) and Eq. (2) against numerical and observational data. Results of the study are summarized in section 7.