1. Introduction
An entrainment zone or a capping inversion is a transition layer between the turbulent convective boundary layer (CBL) and the stably stratified free atmosphere. Convective entrainment, a continual turbulent exchange of energy and substances between the mixed layer and the free atmosphere, is a crucial mechanism for the development of the CBL, as well as for distribution of heat, water vapor, CO2, and air pollutants in the CBL (Betts and Ball 1994; Barr and Betts 1997; Huang et al. 2011). Accurate quantification of entrainment is important for parameterization of turbulent exchange processes in regional numerical weather and air quality prediction models (Hu et al. 2010a,b; Nielsen-Gammon et al. 2010). Difficulties in entrainment modeling stem from the complexity of physical processes that control interactions between the CBL and the free atmosphere, and the scarcity of observational data on entrainment (Moeng et al. 1999; Huang et al. 2011). Thermal and dynamic properties of the CBL entrainment can be modified by the presence of atmospheric aerosols due to their shortwave radiative absorption effect (Yu et al. 2002; Barbaro et al. 2013). In particular, uncertainties of entrainment quantification under conditions of strong aerosol pollution tend to be larger as compared to the entrainment predictions for the CBL without aerosols or with a low load of aerosols.
Parameterizations of entrainment are vital to regional or global numerical models due to the unresolved scales of motions within the entrainment zone. Such parameterizations are usually constructed in terms of the so-called integral parameters of entrainment (Fedorovich et al. 2004). The commonly used integral parameters include entrainment zone thickness; entrainment heat flux ratio, defined as the negative of the entrainment to surface sensible heat fluxes; and entrainment rate 
Proposed entrainment parameterizations are usually evaluated with the so-called CBL bulk models. Typically, the bulk models are classified into three categories in terms of the degree of complexity in representing the entrainment zone structure: zero-order model (ZOM), first-order model (FOM), and general structure model (GSM). They show a similarity in describing the CBL properties (height-constant temperature and velocity) but are different with respect to representation of entrainment zone. The ZOM assumes an infinitesimally thin capping inversion layer with discontinuities in temperature and velocity profiles (e.g., Lilly 1968; Fedorovich 1995). The FOM, on the other hand, considers a linear change of temperature in entrainment zone with a finite depth (e.g., Betts 1974; Sullivan et al. 1998). The GSM uses a self-similar representation of the temperature profile within the entrainment layer (e.g., Deardorff 1979; Fedorovich and Mironov 1995). Although FOMs and GSMs in some respects are more realistic than ZOMs in representing the entrainment-zone structure, their added complexity has limited their applications in atmospheric modeling. In comparison, ZOMs are relatively simple but show a competitive performance in prediction of the basic entrainment relationships (Fedorovich 1995; Fedorovich et al. 2004).
Entrainment integral parameter relationships obtained within the conceptual framework of bulk models for the clear and shear-free CBL have been evaluated in Fedorovich et al. (2004) using data from laboratory experiments and numerical large-eddy simulations (LESs). Typically the integral parameters of entrainment are evaluated within the so-called equilibrium entrainment regime, when the entrainment heat (buoyancy) flux remains in a quasi-steady balance with the surface heat (buoyancy) flux and the dissipation rate of turbulence kinetic energy (TKE) integrated throughout the CBL (Liu et al. 2018). The laboratory experiments showed that, in the shear-free CBL, the entrainment rate normalized with the convective velocity scale 
The CBL entrainment flux ratio is variable and dependent on several factors such as surface heat flux, wind shear, and stratification in the free atmosphere. In the quasi-equilibrium entrainment regime, the ZOM-retrieved entrainment flux ratio 
Aerosols exert significant impact on the CBL flow structures and entrainment. They reduce surface heat flux, modify vertical profiles of potential temperature, and change stability of the CBL due to aerosol absorption of the shortwave (SW) atmospheric radiation (Yu et al. 2002; Kedia et al. 2010; Barbaro et al. 2013; Ding et al. 2016). The impact depends on the vertical distribution of aerosols (Raga et al. 2001; Barbaro et al. 2013). Following the methodology of Betts (1974) and Van Zanten et al. (1999), Barbaro et al. (2013) derived an equation for the normalized entrainment rate 
In the present study, an entrainment-rate equation for the CBL under aerosol-loaded conditions is derived within framework of the ZOM. A radiation transfer model is coupled with the National Center for Atmospheric Research (NCAR) LES (e.g., Moeng 1984; Sullivan et al. 1996; Huang et al. 2008, 2009, 2011) to validate the ZOM entrainment-rate equation. The specific objectives of this study are the following: 1) to investigate the response of vertical profiles of potential temperature and heat flux to the aerosol radiative heating, 2) to determine factors that control the entrainment flux ratio in presence of the radiative heating, and 3) to derive and validate, by means of LES, a ZOM entrainment-rate equation for the aerosol-polluted shear-free CBL.
2. Methodology
a. Coupling LES with radiative transfer model
The Santa Barbara Discrete Ordinates Radiative Transfer (DISORT) Atmospheric Radiative Transfer (SBDART) model (Ricchiazzi et al. 1998) is utilized in our study to calculate the radiative flux and heating rate at different heights. The SBDART model has been widely used for the calculation of aerosol radiative forcing (Kim et al. 2004; Tripathi et al. 2007; Gao et al. 2008; Kedia et al. 2010; Liu et al. 2012). The input parameters of SBDART include longitude, latitude, date, time, wavelength, height above the ground, surface albedo, and aerosol optical parameters such as AOD, single scattering albedo (SSA), and asymmetric factor gf. The definitions and specific values of these three parameters are given in section 2c. The surface albedo in our calculations is set to 0.2. It is assumed that all the aerosols are confined to the CBL interior. Given the constancy of AOD and other aerosol optical parameters (e.g., SSA and gf) adopted in the SBDART model during the whole simulation time period, the radiative flux R (or F) in each atmospheric layer may be calculated based on the radiative transfer equation (see Liou 2002). The corresponding heating rate due to aerosol absorption is given by 
In the coupled model system, the LES provides the CBL depth to the SBDART, the SBDART-calculated surface downward SW radiation is sent to the land surface module (LSM), and the LSM-predicted surface fluxes of sensible and latent heat are then used in the LES to drive the development of the CBL (Lee et al. 2012; Fig. 1). Meanwhile, the SBDART-predicted shortwave radiative heating rate is horizontally uniformly prescribed at each model level. Note that in this study the aerosol longwave (LW) radiative cooling is not included because it is assumed that LW cooling and the gaseous SW radiative heating nearly offset each other during the daytime CBL development (Barbaro et al. 2014).
Schematic diagram of the coupled SBDART–LES–LSM system: u is wind velocity, 
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0107.1
b. ZOM entrainment-rate equation under conditions of radiative heating
Profiles of (a) potential temperature, (b) turbulent heat flux, and (c) radiation flux in ZOM framework. Thin solid lines indicate the actual (LES) profiles and heavy dashed lines show their ZOM counterparts.
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0107.1
c. Simulated flow configurations
We used the LES domain with size of 5 km × 5 km × 1.92 km and grid spacing of 50 m, 50 m, and 20 m in the x, y, and z directions, respectively. A larger domain of 10 km × 10 km × 1.92 km with finer grid mesh of 25 m × 25 m × 10 m was also used to test the effects of grid spacing on the findings. The initial potential temperature profile included a height-constant section with 290 K beneath 640 m and a section with potential temperature growing with height above 640 m at three prescribed rates: dθ/dz = 3, 6, and 9 K km−1. A very dry and cloud-free boundary layer is assumed in all the simulations in order to remove the impact of water vapor and cloud effects. Geostrophic wind is set to zero (i.e., shear free) in all the numerical experiments due to weak winds observed on heavy aerosol pollution events (usually less than 5.0 m s−1), which has negligible impact on CBL development and entrainment (e.g., Pino and Vilà-Guerau de Arellano 2008; Liu et al. 2018).
The radiation-absorbing aerosol is assumed to be distributed uniformly within the CBL, which has been confirmed by many observational and modeling studies (e.g., Steyn et al. 1999; Barbaro et al. 2013). AOD is a commonly used dimensionless parameter representing the degree to which the aerosols impede the transmission of light through the atmosphere. A total of 18 simulations were conducted through combining six AOD values (i.e., AOD = 0, 0.3, 0.6, 0.9, 1.2, 1.5) and three potential temperature gradients (i.e., 3, 6, and 9 K km−1) in the free atmosphere. Information about simulations is summarized in Table 1, with the individual simulation parameters encoded in the case label. For example, the CTLN3 represents the control case without aerosols (AOD = 0) and with a free-atmosphere potential temperature vertical gradient of 3 K km−1, while A03N6 is the case with AOD = 0.3 and a potential temperature gradient of 6 K km−1. Two other optical parameters of aerosols (SSA and gf) are taken from the observations conducted in the Yangtze River Delta region, China (Liu et al. 2012). Here SSA is the ratio of scattering to the total extinction of radiation and gf represents the relative strength of forward scattering (Barbaro et al. 2013). Values of SSA = 0.9 and gf = 0.6 were used in all the simulations. Latitude and longitude are set to 32.21°N and 118.70°E, respectively. The surface heat flux in LES is calculated by a land surface module with surface energy balance equation described in Huang et al. (2009). The SW radiative flux is calculated by the SBDART model (Ricchiazzi et al. 1998).
Details of simulated cases: CTL denotes control case (no aerosol heating), AOD is aerosol optical depth, and dθ/dz is the temperature gradient in the free atmosphere.
Horizontal periodic boundary conditions were used in the simulations. All the simulations represented the CBLs at the noontime on 24 January 2015, and for each case the surface heat flux was kept practically unchanged during the simulation. The 200-time-step samples of LES data were postprocessed after the turbulence reached a quasi-steady state (spinup time is about 1 h with slight difference among the cases presented in this study) as described in Patton et al. (2005) and Huang et al. (2008). The time step of integration was dynamically determined as explained in Huang et al. (2008). In this study it typically varied between 1.6 and 2.0 s.
3. Results and discussion
a. Turbulence statistics
Table 2 presents hourly-averaged values of basic parameters for all simulated CBL cases. As presented data indicate, the surface heat flux diminishes greatly with increasing AOD. In fact, it drops by 63% as AOD increases from 0 to 1.5. Over the same AOD range, the boundary layer depth 
The hourly and domain-averaged surface heat flux 
The entrainment rate 
b. Vertical profiles of potential temperature and heat flux
The impact of aerosol radiative effects on the CBL thermal regime is readily illustrated by changes in the potential temperature field. Figure 3 shows the profiles of potential temperature averaged over 200 time steps (i.e., over about 6–7 min) and over the horizontal planes under different aerosol loading and stratification conditions. Figures 3a–c correspond to three different stratifications. In each panel, vertical profiles of 
Vertical profiles of potential temperature under three different stratification strengths in the free atmosphere: (a) 3, (b) 6, and (c) 9 K km−1. Black: CTL, blue: A03, red: A06, green: A09, magenta: A12, cyan: A15. The short horizontal dash represents the position of the zero-mean 
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0107.1
Another effect of the aerosol-related radiative heating is the modification of the static stability in the CBL. As indicated in Fig. 3, the height at which the potential temperature profile becomes stable (i.e., when dθ/dz becomes positive; this level is marked in the figure by a short horizontal dash) shifts downward as the AOD increases. For example, this height is 420 m in the CTL case, and it changes to 240 m in the A15 case. This effect of growing stability with increase of the aerosol loading becomes more prominent as the free-atmosphere stratification strengthens (Fig. 3c). Barbaro et al. (2013) found a similar earlier stable feature when the aerosols located at the top of the CBL. These observed tendencies are consistent with results from the previous modeling studies showing that the aerosol radiative effect enhances the stability in the upper portion of the CBL (e.g., Gao et al. 2015; Ding et al. 2016; Petäjä et al. 2016; Qiu et al. 2017; Li et al. 2017).
Figure 4 presents the profiles of simulated kinematic heat flux together with profiles of the SBDART-calculated SW radiation flux for the CBL cases with the free-atmospheric stratification of 6 K km−1. The observed profiles support our previous analyses, based on Eq. (3), predicting that the turbulent heat flux in the CBL with aerosol loading departs from a linear shape due to the aerosol SW radiation absorption. However, the total heat (thermal energy) flux (the sum of sensible heat flux and SW radiation flux affected by the aerosol-related absorption) remains linear, in agreement with well-mixed potential temperature field within the CBL (see Fig. S1 in the online supplemental material). Figure 4a demonstrates linearity of the sensible heat flux with height when the CBL does not contain aerosols (CTL case; black line). The flux-profile deviation from the linear behavior becomes more prominent with increase of presence of aerosols in the CBL (the increase of the AOD) for a given SSA. The nonlinearity degree appears to be less pronounced as SSA increases (see Fig. S2). For the outer stratification of 6 K km−1, the surface heat flux drops by 63% (Table 2) as the AOD increases from 0 to 1.5. At the same time, there is an overall decrease of the sensible heat flux over the entire CBL. The entrainment heat flux shows an overall decreasing trend as AOD increases. This is apparently a combined result of the reduction of the energy supply from the surface due to the weaker surface heat flux, and less free-atmospheric air engulfed into the CBL associated with the (slightly) stable stratification in the upper part of the CBL.
Vertical profiles of (a) turbulent heat flux and (b) radiation flux at different aerosol optical depth (AOD) conditions with the same free-atmospheric potential temperature gradient of 6 K km−1. The color scheme for the profiles is as in Fig. 3.
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0107.1
c. Entrainment flux ratio
Now we turn our attention to the entrainment flux ratio 
The changes of LES and ZOM entrainment flux ratios with AOD at the three free-atmospheric potential temperature gradients: (a) 3, (b) 6, and (c) 9 K km−1. The error bar shows plus and minus one standard deviation. Filled boxes represent LES ratio, and open boxes represent ZOM-calculated flux ratio. The list of simulated cases and color scheme for symbols are as in Fig. 3.
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0107.1
For given stratification, the ZOM flux ratio is reduced as AOD increases. On the other hand, the ZOM flux ratio remains constant (about 0.2) when no aerosols are present (Fedorovich et al. 2004; Liu et al. 2018). It follows from Eq. (7) that it is the aerosol radiative heating term [i.e., 
The LES and ZOM flux ratios for the same AOD are quite different. This is apparently because the LES ratio is evaluated locally, whereas the ZOM ratio is obtained from the interpolated profiles, so it is more integral in nature. The LES-predicted 
For a given AOD, the ZOM flux ratio is not sensitive to the free-atmospheric temperature gradient. However, the LES flux ratio is positively correlated with the stratification in the free atmosphere, in a manner that is similar to the case of the clean CBL (Fedorovich et al. 2004; Liu et al. 2018). Close examination of the LES heat flux profiles under different stratification (Fig. S3) suggests that, as the stratification becomes stronger, the negative entrainment flux is constrained to a shallower region below 
The entrainment flux ratio dependence on the surface heat flux, free-atmosphere stratification, and aerosol loading may have important implications for numerical weather prediction and air quality modeling. Typically, such ratios are assumed to be constant in these models. For example, the Yonsei University (YSU) planetary boundary layer parameterization scheme uses an entrainment flux ratio of 0.15 when applied in the Weather Research and Forecasting (WRF) Model (Hong et al. 2006; Hu et al. 2010a). Our study suggests that the entrainment flux ratio should be parameterized as a function of all abovementioned forcings (including aerosol optical depth) rather than kept constant.
d. Evaluation of entrainment equations
In the derivation of the ZOM entrainment-rate equation [Eq. (6)], entrainment flux is assumed to be proportional to the heat flux averaged over the entire CBL depth, following parameterization of Deardorff (1976). The proportionality coefficient Ah is taken constant and equal 0.5 for clean CBL case, in which Ah = 0.5 corresponds to the commonly used ZOM entrainment flux ratio of 0.2. It is not known a priori whether this proportionality assumption is suitable for the aerosol-loaded CBL. Figure 6 shows the calculated Ah for various AOD values and stratification strengths. To determine Ah, we first calculated the entrainment heat flux from LES as 
The changes of the ratios Ah defined by the Deardorff’s (1976) closure assumption with AOD for the three free-atmospheric potential temperature gradients of (a) 3, (b) 6, and (c) 9 K km−1. The error bar shows plus and minus one standard deviation. The list of simulated cases and color scheme for symbols are as in Fig. 3.
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0107.1
The entrainment flux equation [Eq. (7)] for an aerosol-loaded CBL is derived within the framework of ZOM using the closure assumption of Deardorff (1976), and with the aerosol radiative effect included. We now check the validity of this equation against the LES output. The term on the left-hand side of the equation is obtained from the LES data postprocessed with the ZOM representation of the CBL structure, while the term on the right-hand side is computed using the surface sensible heat flux and calculated radiation flux profiles assuming Ah = 0.5. The comparison of the terms is presented in Fig. 7. The overall agreement between the terms indicates that the assumptions involved in the derivation of Eq. (7) make sense, even though the uncertainties of 
A comparison of right- with left-hand sides of ZOM entrainment equation [Eq. (7)] at three different free-atmospheric potential temperature gradients of (a) 3, (b) 6, and (c) 9 K km−1. Squares represent the mean values and the error bars represent plus and minus one standard deviation. Color scheme for symbols is as in Fig. 3.
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0107.1
Equation (7) allows quantification of the relative contributions of aerosol radiative heating effect and surface heat flux to the entrainment flux. For instance, with free-atmosphere stratification of 6 K km−1, as AOD increases from 0 to 1.5, the entrainment flux is reduced by 63% due to the reduction in the surface heat flux, and by an additional 22% due to the aerosol radiative heating. In other words, although the diminishing surface heat flux is a main factor causing the reduction of entrainment flux under such conditions, the aerosol radiative effect is quite strong. Putting it differently, the traditional ZOM parameterization would overestimate the entrainment flux in the considered CBL case because it neglects the aerosol radiative heating contribution.
Figure 8 compares the ZOM-calculated entrainment flux with the flux predicted by LES at level 
A comparison of left-hand sides of ZOM entrainment equation [Eq. (7)] with LES minimum heat flux at the three free-atmospheric potential temperature gradients: (a) 3, (b) 6, and (c) 9 K km−1. The color scheme for the symbols is as in Fig. 3. Other notation is as in Fig. 7.
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0107.1
The modified Richardson number RiR in Fig. 9 shows a decreasing trend with increasing AOD; a similar decreasing trend is also seen in the traditional Ri (Table 2). This trend is primarily due to the decreasing temperature jump Δθ (Table 2) induced by the aerosol absorption of SW radiation. The Richardson number is a measure of static stability within the entrainment zone. Thus, one may conclude that aerosol loading weakens this stability. Similar results were obtained by Barbaro et al. (2013). In response to the less stable entrainment zone, the CBL is expected to grow faster, leading to a deeper CBL. However, the mean CBL depth is slightly decreased as AOD increases (Table 2). A similar contradiction is also found in Barbaro et al. (2013). Such attenuation effect of aerosol-related heating on the CBL growth may be associated with the fact that aerosol heating, besides reducing the temperature jump, stabilizes the flow, in terms of potential temperature gradient, within the upper portion of the CBL, beneath the entrainment zone (Fig. 3). On the other hand, the modified Richardson number RiR shows an increasing trend with the free-atmospheric stratification since 
Richardson number (RiR) for free-atmospheric potential temperature gradients: (a) 3, (b) 6, and (c) 9 K km−1. The color scheme for symbols is as in Fig. 3. Other notation is as in Fig. 7.
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0107.1
According to the study of Fedorovich et al. (2004), the normalized entrainment rate E in the CBL without aerosol loading follows the −1 power law as function of Ri at low Richardson numbers (Ri < 10). Figure 10 shows the relationship of the modified normalized entrainment rate 
Scatterplots of (a) the modified entrainment rate ER with the modified Richardson number (RiR) and (b) the traditional entrainment rate E with the standard Richardson number (Ri) for the free-atmospheric potential temperature gradients of 3 (circles), 6 (triangles), and 9 K km−1 (asterisks). The color scheme for symbols is as in Fig. 3. All points represent mean values.
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0107.1
4. Summary and conclusions
In this study, an entrainment equation has been derived within the framework of a zero-order model (ZOM) to account for the radiative effect of aerosols in the atmospheric convective boundary layer (CBL). A radiation transfer model (SBDART) has been coupled with LES to account for the impact of aerosol heating on thermodynamics of the CBL and entrainment. The new ZOM entrainment equation has been evaluated with the LES results for the dry CBL with different aerosol loadings and free-atmospheric stratifications.
In the studied CBL, the solar radiation reaching the surface is reduced due to the scattering and absorption effects of aerosols, whereas the CBL air is warmed by the aerosol radiative heating. The resulting redistribution of heat energy inside the CBL modifies the vertical profiles of potential temperature and heat flux. Because of aerosol radiative heating, a stably stratified flow region is formed in the upper portion of the CBL beneath the capping inversion. The profiles of heat flux in the aerosol-loaded CBL become more nonlinear in shape as compared to the case of the clean CBL, and the degree of nonlinearity depends on the AOD of the layer.
Both actual (LES) and ZOM-derived entrainment ratios show a decreasing trend with increasing AOD. The study hints that the entrainment flux ratio in atmospheric models needs to be parameterized as a function of aerosol concentration or aerosol optical depth in order to improve weather and air quality predictions associated with the heavy-air-pollution events.
The LES results show that the closure assumption for the entrainment heat flux proposed by Deardorff (1976) is appropriate for a ZOM of the CBL with the aerosol radiative heating being accounted for. The proposed ZOM entrainment equation quantitatively agrees with the LES output processed in terms of the ZOM. This ZOM equation works especially well for CBL cases with strong free-atmospheric stratification. The traditional ZOM entrainment-rate relationship between the dimensionless entrainment and Richardson number has been found no longer valid for heavy aerosol pollution conditions. However, the modified dimensionless entrainment rate closely follows the −1 power law with modified Richardson number.
In the study, uniform aerosol optical properties have been assumed throughout the entire CBL. In real life, these properties may vary within multiscale three-dimensional turbulent flow characteristic of the atmospheric CBL. In the future studies, the turbulent transport of aerosol will be incorporated in the LES along with aerosol sources and sinks to better represent the interactions of aerosol radiative effects with dynamics and thermodynamics of the aerosol-polluted CBL. Moreover, the aerosol radiative heating in current study is incorporated in the conducted LES through the bulk (slab) representation of the aerosol-loaded layer, which is in conformity with zero-order representation of the entrainment zone (i.e., a discontinuity interface; see Fig. 2). Thus we derived an equation within the ZOM framework and demonstrated that it could be successfully applied to parameterize entrainment rate. Nevertheless, once a feasible and physically consistent method is developed, derivation of a first- (or higher-) order model of entrainment in aerosol-loaded CBL accounting for entrainment-zone depth and local (as opposite to bulk) heating in LES is required to further provide new insights into entrainment affected by aerosol radiative heating. The impact of single scattering properties and vertical distribution of aerosols on the CBL dynamics and entrainment will be further investigated through numerical experiments.
Acknowledgments
The research was supported jointly by the National Natural Science Foundation of China (Grant 41575009), the National Key Research and Development Program of China (Grant 2017YFC0210102), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). The first author also acknowledges the support Postgraduate Research and Practice Innovation Program of Jiangsu Province (Grant KYLX16_0945). The computing for this project was performed at the OU Supercomputing Center for Education and Research (OSCER) at the University of Oklahoma (OU). We thank three anonymous reviewers for their helpful comments, which greatly improved the manuscript.
APPENDIX
Derivation of Eq. (4) from Eq. (2)
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