**Intro**

Different to most engineering flows, planetary boundary layers (BLs) are typical influenced by both shear and buoyancy with a wide range of varying intensities: a meteorologically significant feature. However, no consensus exists about how these forcing in varying combinations (atmospheric stability) determine the BL dynamics, particularly in terms of entrainment and BL growth rate (entrainment velocity).

Progress has been made in the representation of entrainment for the case of sheared convective boundary layers (CBLs). However, we found that even relatively recent improvements, such as the model of Conzemius and Fedorovich (2007), show systematic errors when compared to data.

**Analysis**

Aiming at an improved formulation for entrainment, we therefore systematically re-examined the budget of turbulent kinetic energy (**A**) and the bulk-model approximations (**B**) of sheared CBLs using a set of twenty-five Large Eddy Simulations (LESs), which cover a wide range of atmospheric stability. Different from previous studies, both aspects A and B are tested independently using appropriate dimensionless quantities. This approach reveals systematic shortcomings in previous concepts for both the turbulent kinetic energy (TKE) budget and the bulk model approximations:

**A. **Contrary to what was previously assumed for CBLs, our analysis indicates that TKE drainage into the stably stratified free atmosphere, via internal gravity waves, is systematically enhanced by shear in the entrainment zone (EZ). This significantly affects entrainment. This is illustrated by Figure 1, which displays the spatial distribution of normalized vertical velocities within the LES domain (cross section). One can recognize systematically tilted fronts of internal gravity waves, which cause significant drainage of fluctuation energy into the free atmosphere. As a consequence, the relationship between the production terms (shear and buoyancy) on one hand, and both entrainment (buoyancy consumption) and dissipation on the other hand, deviates in a systematic manner from the traditionally assumed linearity. This deviation is particularly significant for moderate levels of shear. Here the previously overlooked wave-driven drainage of TKE well explains the frequently observed overestimation of entrainment.

**B. **Based on simple considerations and strong empirical evidence, we demonstrate that the traditionally used convective length scale (also known as Deardorff'-length), defined as the height of minimum buoyancy flux, results in a systematically biased estimation of the buoyancy production, entrainment and dissipation for cases with strong shear. This bias is related to the increase of the entrainment zone thickness with increasing shear. These deviations well explain the previously unsolved problem of erratic behaviour of bulk model entrainment for cases with strong shear.

**Figure 1: Cross section trough the domain of an LES of a strongly seared CBL, approximately in flow direction. Shown are the local vertical velocity fluctuation ***w ***normalized by its standard deviation ***σ**w*** throughout the domain. Grid size of the LES is 1024 x 1024 x 512, representing a domain of 25.6 km x 25.6 km x 6.4 km. CBL: convective boundary layer. FA: free atmosphere. DL: damping layer.**

**Improvement**

In order to address the observed shortcomings, we finally suggest two improvements:

**A. **First we re-define the traditional linear scaling between shear production of TKE and entrainment as an idealized reference (without TKE losses due to waves radiated to the free atmosphere (FA)). We find that this reference can be empirically approximated by academic CBLs without a stably stratified free atmosphere. Based on this linear reference we determine the non-linear deviations that account for the influence of wave-induced drainage of TKE on both entrainment and dissipation. Finally we suggest a parametrization that accounts for these non-linear influences.

**B. **Based on data and simple consideration we suggest an alternative convective length scale for the mixed-layer TKE buoyancy production and a separate length-scale for TKE consumption in the entrainment zone.

To finally demonstrate the significance of the suggested measures, we use the improvement for A and B to derive a new formulation for a 0^{th}-order bulk entrainment model and demonstrate its potential in reproducing our set of LES.

**Figure 2: Dimensionless entrainment velocity ***W**e*** as function of dimensionless EZ shear ***D**U*** for various LES CBLs, which differ in pressure forcing, resolution and domain size. The traditional entrainment model by Conzemius and Fedorovich (2007) is shown by the thin dashed black line. The improved entrainment model shown by the thick black line. **

The outcome is shown in Figure 2. It displays the relationship between the dimensionless entrainment velocity *W**e* and the dimensionless entrainment zone shear* D**U* for various LES. For comparison, the 0^{th}-order model of Conzemius and Fedorovich (2007) as a typical representative of previous entrainment models is displayed (dashed thin black line). The latter features systematic deviation from LES for levels of moderate shear, which is caused by the mentioned energy leakage by gravity waves (A) and for strong shear, which is caused by the overestimation of the buoyancy length scales (B). Both deficiencies are corrected by our improved 0^{th}-order model (thick black line), which matches the LES very well.

**Practical relevance**

As indicated by our large set of various LES, based on two different independent computer codes, wave leakage of TKE (A) seems to be a significant and persistent feature of CBLs with moderate shear. Inclusion of this effect should therefore be considered as improvement for any TKE-based entrainment model.

The use of the Deardorff-length scales (B) is not restricted to 0-order models, but is also a basic element of more complex 1^{st}-order models and other types of TKE models. We therefore expect that an improved length scale formulation, analogously to what we demonstrated here for a 0^{th}-order model, can lead to comparable improvements in the determination of the entrainment rate.

**References**

Conzemius RJ, Fedorovich E. 2007. Bulk models of the sheared convective boundary layer: Evaluation through large eddy simulations. J. Atmos. Sci. 64: 786807.