#### 4.1. Entrainment Flux

[21] In response to diurnally varying solar forcing, the boundary layer grows each day through entrainment, which exchanges heat, water vapor, CO_{2} between the ABL and the free atmospheric layer. It is difficult to make direct measurement of entrainment fluxes. The entrainment buoyancy flux is typically assumed as a constant fraction of the surface flux in the ZOM [*Lilly*, 1968] or described as a function of the entrainment zone thickness in the FOM [*Betts*, 1974; *Sullivan et al.*, 1998]. In contrast, entrainment fluxes within the LES are explicitly calculated and are a result of the simulation.

[22] Figure 6 shows the time evolution of entrainment fluxes of heat, water vapor and CO_{2} calculated with the LES for four different geostrophic wind conditions. The values shown here represent the domain- and hourly-averaged fluxes. Based on the study of *Sullivan et al.* [1998], a 10 m vertical resolution is fine enough to resolve the turbulence organized structure within the entrainment zone. The entrainment flux of heat follows a similar time variation pattern of the surface flux. Both entrainment fluxes of water vapor and CO_{2} flux follow similar patterns but before 0915 LST are not sensitive to the surface fluxes. The geostrophic wind variations have negligible influence on the three entrainment fluxes (Table 1 and Figure 6). Therefore, for the atmospheric situations simulated here, the entrainment fluxes are more sensitive to the surface flux than to the geostrophic wind.

[23] We now use the LES results to evaluate several entrainment parameterizations. These parameterizations are based on the budget equation originally written for virtual potential temperature (neglecting radiation and mean vertical motion) [*Betts*, 1974, equation (2)] and are extended here to a general scalar ϕ (i.e., θ, *q*, and CO_{2} mixing ratio *c* in this study),

where 〈 〉 denotes spatial averaging in the *x* − *y* directions, overbar represents temporal averaging, double prime denotes fluctuation from the spatial average, _{i,g} represents the domain- and hourly-averaged entrainment flux at *z*_{i} and subscript *g* denotes GSM, *z*_{i} is the vertical position where buoyancy flux reaches the minimum value, *z*_{2} is the vertical location where the buoyancy flux first goes to zero above the ABL height *z*_{i}, the height of *z*_{2} is usually taken as the vertical location where the vertical flux reaches a prescribed fraction of the entrainment flux at z_{i} (Figure 7), ϕ_{i} and ϕ_{2} are the scalar values at *z*_{i} and *z*_{2}, respectively, _{i2} is the average of ϕ between *z*_{i} and *z*_{2}, is the derivative with respect to time (*t*). The readers are reminded that all the terms on the right side in equation (2a) represent horizontally averaged quantities and the angle brackets are omitted for simplicity. Our GSM is established in terms of the budget equation derived by *Betts* [1974, equation (2)]. As compared to the GSM of *Deardorff* [1979], the GSM discussed in this study has a relative simple form and is convenient for our discussion because we do not need to determine the integral shape factor in *Deardorff*'s [1979] formula (equation (30) in his study). In this study, our GSM framework assumes , which means that the entrainment flux can be written as

where *δ*ϕ_{i2} = ϕ_{2} − ϕ_{i}, *δz* is the inversion-layer thickness (*δz* = *z*_{2} − *z*_{1}) and . *Betts* [1974] neglected the time change of the inversion thickness to further simplify equation (2b) into the FOM as follows:

where _{i,f} denotes the entrainment flux at *z*_{i} calculated with the FOM and subscript *f* indicates FOM. If *δ*ϕ_{i2} is constant and *δ*ϕ_{12} = 2*δ*ϕ_{i2}, equation (3a) becomes

where Γ_{ϕ,1} denotes the vertical gradient of scalar ϕ between *z*_{i} and *z*_{2} (within entrainment zone), *δ*ϕ_{12} is the jump for the FOM across the entrainment zone and defined as the difference of ϕ between height *z*_{1} where the flux _{i,z} reaches zero below *z*_{i} and height *z*_{2} where the flux _{i,z} reaches zero above *z*_{i} (e.g., *δ*θ illustrated by Figure 7). The jumps *δ*ϕ_{12} are similar to the potential temperature jump *δ*θ defined by *Fedorovich et al.* [2004, Figure 1] and *Sun and Wang* [2008, Figure 1] but different from the one used by *Sullivan et al.* [1998]. In fact, our FOM (equation 3a) is very similar to equation (8) of *Sullivan et al.* [1998] if we consider *δ*ϕ_{12} = 2(ϕ_{2} − ϕ_{i}) and *δz* = 2(*z*_{2} − *z*_{i}).

[24] On the other hand, ZOM expresses the entrainment flux at *z*_{i} [*Fedorovich et al.*, 2004]

where Γ_{ϕ,2} is the vertical gradient of scalar in the free atmospheric layer (e.g., Γ illustrated by Figure 7) and *δh* = *z*_{2} − *z*_{i}. The ZOM can be rewritten in a simplified form

where _{i,z} denotes the entrainment flux at *z*_{i} and calculated with ZOM with subscript *z* indicating ZOM, Δϕ is the jump of ZOM across the entrainment zone and defined as the difference between the value of extrapolation of the scalar profile from the free atmosphere down to z_{i} and the value at the bottom of the entrainment zone (Figure 7), similar to the one defined by *Fedorovich et al.* [2004] and *Sun and Wang* [2008]. The bottom of the entrainment zone is defined as the height where the flux _{i,z} reaches zero below *z*_{i}.

[25] In Figure 8, the entrainment fluxes of heat, water vapor and CO_{2} simulated by the LES are compared with entrainment flux estimates based upon the ZOM, FOM and GSM. While the ZOM slightly underpredicts the magnitude of the entrainment fluxes at *z*_{i}, the FOM and GSM show very good agreement with the LES-resolved entrainment fluxes at *z*_{i}. It is worth noting that determining the zero-order scalar jump Δϕ is crucial to the parameterized entrainment fluxes. For example, the scalar jump calculated with Δϕ = ϕ_{i} − ϕ_{1} or Δϕ = ϕ_{2} − ϕ_{1} may cause significant underprediction (overprediction) as compared to the LES-resolved entrainment fluxes. Our calculations suggest the ZOM is a good approximation to the entrainment fluxes at *z*_{i} if Δϕ = ϕ_{2} − ϕ_{1} − Γ_{2,}_{ϕ}*δh* is utilized. The results are consistent with the findings of *Fedorovich et al.* [2004].

[26] Overall the FOM is a good approximation to parameterize the entrainment flux for the three scalars, and the ZOM is also a good approximation if the zero-order jump across the entrainment zone is calculated correctly. The performance of FOM suggests that the role of time change of the thickness of entrainment zone is negligible in the calculation of entrainment flux at *z*_{i}.

#### 4.2. Jumps of θ, *q*, and *c* Across the Capping Inversion

[27] The jumps of θ, *q* and *c* across the capping inversion, or differences in these scalar quantities between the free atmosphere and the mixed layer largely control the entrainment fluxes at the top of the ABL. The inversion jump of *c*, a critical parameter for inferring the surface CO_{2} flux at the landscape scale from the time series of the CO_{2} mixing ratios measured in the ABL [*Helliker et al.*, 2004; *Betts et al.*, 2004; *Lai et al.*, 2006; *Cleugh and Grimmond*, 2001], is rarely measured. Instead, it is sometimes taken as the difference between the tower-based measurement on land and the observation taken in the marine boundary layer, a proxy representing CO_{2} mixing ratio in the free atmosphere, for the regional CO_{2} flux calculations [*Chen et al.*, 2006], or is arbitrarily specified in snapshot LES studies [*Górska et al.*, 2008; *Huang et al.*, 2009]. In both snapshot and time-varying forcing simulations, the inversion jumps vary with the surface fluxes and the PBL growth. However, for the time-varying forcing cases, the surface fluxes evolve over time in response to the solar forcing and the entrainment. Thus the jump value in the current time-varying simulation is closer to the reality than the one calculated with the snapshot simulation.

[28] Figure 9 illustrates the evolution of the jumps of potential temperature, specific humidity and CO_{2} mixing ratio (*δ*θ, *δq*, *δc*) derived from the time varying LES-LSM simulations. Here *δ*θ, *δq*, *δc* are calculated as the difference between the values at z_{2} (the location where the vertical heat flux reaches zero above *z*_{i}; see Figure 7) and the value at z_{1} (the location where the vertical heat flux reaches zero below *z*_{i}; see Figure 7). The *δ*θ follows a similar temporal variation pattern of the surface heat flux, increasing quickly in the morning and then tending to slightly decrease in the afternoon, which indicates that surface flux plays a dominant role in the evolution of *δ*θ. In contrast, both −*δq* and *δc* follow linear increasing trends. The changes in *δ*θ and *δq* are consistent with the findings of *Vilà-Guerau de Arellano et al.* [2004], but *δc* is different from their result in which dilution process is more important than ours and negative jumps are reported for the morning hours which results from different initial CO_{2} profiles between the two investigations. The inversion jumps are insensitive to the geostrophic wind variations investigated here.

#### 4.3. Relative Entrainment Efficiency

[29] The entrainment process can be further understood through the examination of the relative entrainment efficiency for heat (*B*_{θ}), water vapor (*B*_{q}), and CO_{2} (*B*_{c}). In this study, we define the relative entrainment efficiency as

where *δ*θ, *δq*, and *δc* represent the jumps of θ, q, and c across the entire entrainment zone (as defined in section 4.2), and _{i}, _{i}, and _{i} represent the hourly and domain-wide averaged LES-resolved entrainment fluxes at *z*_{i}.

[30] These coefficients can also be interpreted as the normalized entrainment fluxes or apparent diffusivities of the entrainment process. In equations (5a)–(5c), all the quantities on the right hand side are calculated directly from the LES simulations. In general, the entrainment efficiency is highest for CO_{2} and lowest for temperature. In the time varying simulations, *B*_{q} and *B*_{θ} are higher in the morning and lower in the afternoon due to slower growth of the ABL in the afternoon, and *B*_{c} is relatively constant at 1 through the simulations (data not shown). The entrainment efficiency cannot be assumed equal among the three scalars.

[31] According to equation (4a), the entrainment efficiency of ϕ can be expressed as

where is used for the equation derivation. The entrainment efficiency is dependent on the ratio of the vertical gradient of scalar ϕ in the free atmospheric layer to the inversion jump layer.

[32] So generally, the efficiency is less than unity because the quantity is positive. Only when the vertical gradient in the free atmosphere is zero, a condition satisfied for CO_{2} in the current time-varying simulations (Figure 10) and for *q* and CO_{2} in the snapshot simulations [*Huang et al.*, 2009, Figure 3], does *B*_{ϕ} approach 1. CO_{2} is a well-mixed gas in the free atmosphere. The profile measurement above the ABL [*Lloyd et al.*, 2001; *Górska et al.*, 2008] shows that Γ_{ϕ,2} = 0 is indeed a good approximation to the real atmospheric flows.