1 J4.6 A FORMULA FOR THE DEPTH OF THE STABLE BOUNDARY LAYER: EVALUATION AND DIMENSIONAL ANALYSIS G.J. Steeneveld 1 , B.J.H. van de Wiel and A.A.M. Holtslag Wageningen University, Wageningen, The Netherlands 1. INTRODUCTION 1 The stable boundary-layer (SBL) height (h) is important to understand SBL devel- opment and vertical structure. Clearly, h in- fluences the SBL mixing properties. Model formulations with an explicit prescription of the vertical profile of the eddy diffusion coëf- ficient, require an explicit expression for h (e.g. Troen and Mahrt, 1986, Holtslag and Boville, 1993). For stable conditions, many models of this type overestimate the vertical mixing and thus h. So there is a clear need for an alternative h formulation for Numerical Weather Prediction models. Furthermore, the dispersion of pollutants is strongly affected by h. Release of pollut- ants below h during periods of weak winds results in very high concentrations of pri- mary and secondary pollutants, which can cause serious consequences for the envi- ronment. This means that for meteorological preprocessors in air quality models, h is the most critical quantity to estimate (Lena and Desiato, 1999). Measuring h is not straightforward be- cause the turbulence is suppressed at night and can be of intermittent nature or ill- defined (e.g. Holtslag and Nieuwstadt, 1986). In addition to turbulence, radiation divergence, gravity waves, wave breaking and baroclinicity influence the SBL structure for the very stable case. In that case, prob- lems occur in measuring h, since no univer- sal relationship exists between the profiles of temperature, wind speed and turbulence variables. We evaluate the performance of two multi-limit equations (Zilitinkevich and Mi- ronov, 1996; henceforth ZM96) against four observational datasets and Large Eddy Simulation (LES). Secondly, we present an alternative, robust and practical formulation for h. Finally, we will show that the Coriolis 1 Corresponding author address: G.J. Steeneveld, Wageningen University, Meteorology and Air Quality Group, Duivendaal 2, 6701 AP Wageningen, The Netherlands. E-mail: Gert-Jan.Steeneveld@wur.nl parameter is not a priori a necessary quan- tity for h estimation. 2. BACKGROUND ZM96 identified rotation, surface buoy- ancy flux and free flow stability to be the key physical processes that govern h. Conse- quently, ZM96 derived a formula for h by inverse quadratic interpolation of the rele- vant boundary-layer height scales that rep- resent these three processes. The formula uses the friction velocity ( * u ), buoyancy flux ( s s w g B θ θ = ), Coriolis parameter (f) and free flow stability ( z g N ∂ θ ∂ θ = 2 ) (g is the gravity acceleration, θ the potential tem- perature and z the height above ground) and reads: 1 * * 2 * = + +         u C Nh L C h u C fh i s n (1) Herein s B u L 3 * * − = is the Obukhov length (without Von Kármán constant). The main advantage of Eq. (1) is its multi-limit behav- iour, i.e. both for f → 0, or N → 0 or L * → ∞, Eq. (1) remains defined. Based on Zilitinkevich (1972) and Pol- lard et al. (1973), ZM96 add two additional terms to “include the cross interactions” be- tween f, B s and N: 1 * 2 * * * 2 * = + + + +         u C h fN u C h fB u C Nh L C h u C fh ir sr s i s n (2) with C n = 0.5, C s = 10, C i = 20, C sr = 1, C ir = 1.7. Apart from the benefits discussed above, Eqs. (1) and (2) have several draw- backs. Firstly, a large amount of parameters is required in both equations. Several of these coefficients are hard to determine (ZM96, Joffre et al., 2002, Vickers and Mahrt, 2004). For example, C n ranges from 0.045 to 0.6 and C s ranges from 1.2-100 (ZM96) in the literature. Secondly, it is not a priori clear that the method of inverse quadratic interpolation gives the proper weight to the relevant