Large-eddy simulations are conducted to investigate the impacts of the scale of chessboard-like heteroge- neous surface heating and the background wind on secondary circulations （SCs） in the convective boundary layer （CBL）. When the wind blows along the diagonal of the chessboard pattern, the cases with different heterogeneity length scales （λ = 1.2, 2.4, and 4.8 km） and weak background wind （U = 2.5 m s-1） suggest that there exists a threshold for the roll-like SCs, which is satisfied when the heterogeneity length scale is 1.6 times the boundary layer height （λ = 1.6zi）. During the CBL development, the SC intensity increases before this threshold is met, whereas it decreases thereafter. The cases with different background wind speeds （U = 2.5, 5.0, and 10.0 m s-1） and relatively large heterogeneity length scale （λ = 4.8 km） show that the SCs are strengthened by larger wind speeds when the heterogeneity length scale is so large that the threshold cannot be met during the CBL development. Another case with wind direction along neither the diagonal nor the side of the chessboard pattern shows that the roll-like SCs can still be triggered, but the roll axes are orientated along the diagonal of the chessboard pattern rather than along the wind direction.
The entrainment flux ratio Ae and the inversion layer （IL） thickness are two key parameters in a mixed layer model. Ae is defined as the ratio of the entrainment heat flux at the mixed layer top to the surface heat flux. The IL is the layer between the mixed layer and the free atmosphere. In this study, a parameterization of Ae is derived from the TKE budget in the first- order model for a well-developed CBL under the condition of linearly sheared geostrophic velocity with a zero value at the surface. It is also appropriate for a CBL under the condition of geostrophic velocity remaining constant with height. LESs are conducted under the above two conditions to determine the coefficients in the parameterization scheme. Results suggest that about 43% of the shear-produced TKE in the IL is available for entrainment, while the shear-produced TKE in the mixed layer and surface layer have little effect on entrainment. Based on this scheme, a new scale of convective turbulence velocity is proposed and applied to parameterize the IL thickness, The LES outputs for the CBLs under the condition of linearly sheared geostrophic velocity with a non-zero surface value are used to verify the performance of the parameterization scheme. It is found that the parameterized Ae and IL thickness agree well with the LES outputs.
Following the parameterization of sheared entrainment obtained in the companion paper, Liu et al. （2016）, the present study aims to further investigate the characteristics of entrainment, and develop a simple model for predicting the growth rate of a well-developed and sheared CBL. The relative stratification, defined as the ratio of the stratification in the free atmosphere to that in the entrainment zone, is found to be a function of entrainment flux ratio （Ae）. This leads to a simple expression of the entrainment rate, in which Ae needs to be parameterized. According to the results in Liu et al. （2016）, Ae can be simply expressed as the ratio of the convective velocity scale in the sheared CBL to that in the shear-free CBL. The parameterization of the convective velocity scale in the sheared CBL is obtained by analytically solving the bulk model with several assumptions and approximations. Results indicate that the entrainment process is influenced by the dynamic effect, the interaction between mean shear and environmental stratification, and one other term that includes the Coriolis effect. These three parameterizations constitute a simple model for predicting the growth rate of a well-developed and sheared CBL. This model is validated by outputs of LESs, and the results show that it performs satisfactorily. Compared with bulk models, this model does not need to solve a set of equations for the CBL. It is more convenient to apply in numerical models.
