## 1. Introduction

Turbulence in (cloudy) convective boundary layers (CBLs) is characterized by a skewed vertical velocity distribution. In surface-driven (cumulus-topped) boundary layers, updrafts cover a relatively small area but have large vertical velocities, whereas the compensating downdrafts cover a larger area and have smaller vertical velocities. In boundary layers with stratocumulus, skewness can be either positive or negative depending on the strength of radiative cooling, entrainment processes, and the surface fluxes.

Conservative scalars introduced into the bottom of the CBL with no flux through the top (bottom-up diffusion) have a radically different eddy diffusivity profile than scalars introduced at the CBL top having no flux through the surface (top-down diffusion) in the same turbulence field. Wyngaard (1987) argues that a positive skewness of the vertical velocity and the time change in the mean concentration during scalar transport across the CBL could explain this difference. However, he also suggests that the vertical inhomogeneity of statisticalproperties of the velocity field may provide an alternative explanation. Weil (1990) concludes that both skewness and vertical inhomogeneity are necessary to explain the diffusion asymmetry in the CBL. Wyngaard and Weil (1991, hereafter WW91) suggest that the roots of the transport asymmetry lie in the interaction between the skewness of the transporting turbulence and the gradient of the flux of the transported scalar. As such, these authors propose an expression for scalar fluxes in CBLs that combines a downgradient diffusion term with a nonlocal flux term depending on the skewness of the vertical velocity and the flux gradient of the transported scalar.

An alternative approach is followed by Holtslag and Moeng (1991, hereafter HM91). Using the large-eddy simulation (LES) results of Moeng and Wyngaard (1989), they analyze the heat flux budget in the dry CBL. By introducing an empirical closure for the transport term in the heat flux budget, they arrive at an expression for the nonlocal term, which is related to bulk parameters of the dry CBL and the bottom-up flux. Wyngaard and Moeng (1993) discuss the similarity between the expressions for scalar fluxes in the dry CBL as proposed by HM91 and WW91 in the case of bottom-up transport. It is found that both expressions can describe the countergradient transport in a dry CBL (Deardorff 1966, 1972).

However, the similarity between the two expressions breaks down for top-down diffusion. Also it is not clear how the expressions behave in cloudy CBLs. Since thetwo expressions have different origins and reflect the dependence on different physical processes, the question remains: what is the impact of skewness and nonlocal effects on the fluxes in CBLs?

To answer this question, we use LES to create different types of CBLs. We simulate a dry CBL, driven from the bottom (by a surface heat flux), a boundary layer driven from the top (by radiative cooling), and a boundary layer driven by both surface heat flux and radiative cooling from the top. In this way, LES of boundary layers with positive, negative, and nearly zero skewness are constructed (some of these boundary layers can be considered as simplified versions of cloudy CBLs).

By analyzing the scalar and buoyancy flux budgets of the CBLs with positive and negative skewness, useful approximations for the budget terms are found. For this, a generalized convective scaling is introduced. This directly provides an expression for scalar and buoyancy fluxes, which combines downgradient and nonlocal effects on the fluxes. The nonlocal term is a function of vertical velocity variance and an integral form of the flux. As such our result can be seen as a generalization of the expression by HM91.

The new expression is compared with the one of WW91. For this purpose, we will stretch the application of the latter. The LES results are used to verify the flux expressions and determine the coefficients that appear in them. Furthermore, we discuss the calculation of length scales, one of the important ingredients of the expressions. This leads to a general but still simple expression for scalar fluxes. Simplicity is desirable since the final expression is aimed at use in mesoscale and global models.

The LES model is briefly described in section 2. In that section we also present the three CBLs and the smoke flux profiles, which are used to test the expressions. In section 3 we show heat and smoke flux budgets in two of these CBLs. The turbulent transport and pressure covariance terms in these budgets are discussed and scaling variables are introduced. Based on these results we derive an expression for fluxes in section 4. The characteristics of this expression and the one given by WW91 are discussed. Comparison of the two flux expressions with the LES data is made in section 5, where we present profiles for heat and smoke fluxes. A discussion and summary is given in section 6.

## 2. LES model and prototype boundary layers

The LES model used in this study is extensively described in Cuijpers and Duynkerke (1993) and Siebesma and Cuijpers (1995). Here we only discuss a modification we made in the code.

The CBLs we simulate have a rather strong inversion (Δ*θ* ≈ 7 K). Calculating the advection terms of temperature and scalars with the discretization method of Piacsek and Williams (1970), as was done before, leadsto nonphysical wiggles near the inversion. Therefore, we implemented a monotone scheme, which prevents the occurrence of nonphysical wiggles (the so-called limited *κ* = −;d1 scheme; Vreugdenhil and Koren 1993). This kappa scheme is applied to temperature and scalars only.

In the simulations of the three prototype CBLs, no large-scale forcings (as subsidence and the Coriolis forcing) nor mean wind are applied. So, all simulations are entirely buoyancy driven. We also omit a sponge layer at the top of the domain.

The numerical domain is 3200 m × 3200 m × 1250 m with 64 × 64 × 50 grid points. For each case the model is run for 3 h, so that we obtain a quasi-steady state with linear flux profiles. The results that will be presented are averages over the third hour.

In Fig. 1 the initial and final profiles are given for two of the three simulated cases. Below we will discuss these cases in detail. Figure 2 gives a quick and sketchy overview of the buoyancy and scalar flux profiles realized with the different simulations. The motivation for the selection of the different cases will become clear in section 5.

### a. The dry CBL

The first prototype boundary layer is a dry CBL with uniform potential temperature profile (Fig. 1a), driven by a surface heat flux of 60 W m^{−2}. In this simulation there is no radiative cooling. The smoke is a passive tracer that has no effect on the turbulence. In one dry CBL case, there is a uniform smoke concentration within the boundary layer and zero concentration above (Fig.1b). Since we apply a zero surface smoke flux (*wS*_{1}_{0} and because there is a positive entrainment flux (*wS*_{1}_{e} at the top of the CBL (as a result of the chosen smoke concentration profile), this smoke flux is top-down (Fig. 2a1).

In the second dry CBL case, the smoke concentration is taken as zero within the boundary layer and one above. As a result the entrainment flux (*wS*_{2}_{e} is negative. A positive surface smoke flux (*wS*_{2}_{0} ≈ −(*wS*_{2}_{e} is applied (Fig. 2a2).

### b. The smoke-cloud boundary layer

The second prototype boundary layer is the so-called smoke-cloud boundary layer, which was originally set up for the Second GCSS Boundary Layer Cloud Workshop (KNMI, 30 Aug–1 Sep 1995). The smoke-cloud case was designed not only as an intercomparison, but also to help shed light on the important problem of entrainment closures for marine stratocumulus clouds. However, in place of water substance, a radiatively active “smoke” tracer was advected. The smoke concentration is initially uniform in the lower layer and zero in the upper layer (Fig. 1b). The radiative cooling in each column is assumed to occur only in regions with smoke, with cooling (of 60 W m^{−2}) concentrated within 50 m of the top of the smoke layer (Fig. 2b). This is in analogy to the cloud-top radiative cooling that occurs in actual marine stratocumulus clouds.

In the first smoke-cloud case, the surface smoke flux is again zero, while there is a positive entrainment smoke flux at the top (Fig. 2b1). However, in contrast to *wS*_{1}*wS*_{1}*wS*_{1}

In the second smoke-cloud case, we add a surface smoke flux (*wS*_{2}_{0} such that the smoke flux gradient across the mixed layer is about zero (Fig. 2b2). Because of this extra smoke source, the evolution of the boundary layer is slightly different from the first smoke cloud case (Table 1). A (nearly) zero humidity flux gradient is often encountered in (cumulus-topped) boundary layers (e.g., Mahrt 1991). With this simulation we mimic this effect qualitatively.

### c. Boundary layer with zero skewness

The third prototype boundary layer is a combination of the dry CBL and the smoke-cloud case. To simulate a boundary layer that has zero skewness in a large part of the mixed layer, we apply radiative cooling as well as a surface heat flux (Fig. 2c). This simulation is a simplified version of a stratocumulus-topped boundary layer where both radiative cooling and surface heating can be present. The surface smoke flux is zero, while there is a positive entrainment flux of smoke at the top (Fig. 2c1).

## 3. Flux budgets

*wS*

_{1}

*wχ*

*w*and

*χ*are vertical velocity and scalar (concentration) fluctuations,

*X*is the mean value of the scalar (concentration),

*σ*

^{2}

_{w}

*g*is the gravitational acceleration,

*θ*

_{0}and

*ρ*

_{0}are a reference state potential temperature and density respectively, and

*p*is pressure fluctuation. The terms on the right-hand side (rhs) of (1) are denoted as the mean-gradient

*M,*the buoyancy production

*B,*the turbulent transport

*T,*and the pressure covariance

*P,*respectively.

*h,*and the generalized convective velocity scale (Deardorff 1976)

*w** for a dry CBL (assuming that the entrainment flux equals −0.2(

*wθ*

_{0}, with (

*wθ*

_{0}the surface heat flux). Further, we may define a convective scale for a scalar

*χ*as

*χ*is either potential temperature

*θ*or smoke concentration

*S,*in our case.

The results in Fig. 3 compare favorably with the top- down and bottom-up fluxes shown by Moeng and Wyngaard (1984). However, we note that in the heat flux budget for the dry CBL (Fig. 3a), the turbulent transport term *T* does not differ from the pressure covariance term*P* by a constant value. This assumption was made by HM91, utilizing the LES results of Moeng and Wyngaard (1989), in order to derive an expression for the nonlocal term in the heat flux parameterization. As a matter of fact, the results in Fig. 3 show that for none of the budgets such a simple relation between *T* and *P* exists.

*T,*which yields

*b*=

*b*(

*z*/

*h*) is a function of height in general.

*P*in the scalar flux equation. They show that

*P*in a dry CBL without significant wind shear is given by

*τ*is a return-to-isotropy timescale and

*a*≈ ½. This result is confirmed by our LES results for the dry CBL in both the heat and smoke flux budgets (not shown). Here we have calculated the timescale

*τ*directly from LES by

*τ*=

*c*

_{ε}

*σ*

^{2}

_{w}

*ε*(with

*ε*the dissipation rate) and initially

*c*

_{ε}= 1. However, it appears that varying

*c*

_{ε}by a factor of 2 significantly impacts on the value of

*a.*Moreover, in the smoke-cloud boundary layer, the LES results suggest that

*a*should be a function of height. Thus, in general,

*a*=

*a*(

*z*/

*h*).

The closures for the turbulent transport (4) and pressure covariance (5) can be used to simplify the budget equation for a scalar flux. This provides a useful flux expression, to be discussed in the next section.

## 4. Flux expressions

*wχ*

*t*≈ 0), yields

*X*is directly related to a downgradient diffusion term (first term on rhs) and quantities that represent buoyancy and transport effects. Note that a similar relation has been obtained by Abdella andMcFarlane (1997) using a mass flux decomposition for the higher-order terms.

*B*as well:

*c*=

*c*(

*z*/

*h*) a function of height. This results in

*β*

_{0}= [

*c*(1 −

*a*) +

*b*]. Note that for the fluxexpression (6b), only the combination of coefficients as given by

*β*

_{0}is relevant. Equation (6b) shows that the scalar flux depends on a local downgradient transport (first term on the rhs) and a nonlocal convective transport (second term). The latter is proportional to the integral of the scalar flux

For the case of bottom-up diffusion (and assuming a linear flux profile), (6b) is equivalent to the result of HM91. The latter authors obtained their findings by utilizing a different assumption for the transport term *T* in the heat flux budget (which is nearly a bottom-up flux in a dry CBL). Apparently, the format of (6b) is rather robust and not so much dependent on the basic assumptions for the higher-order terms in the flux budget. For top-down diffusion, (6b) is different in such a way that now the scalar flux in a dry CBL is also influenced by the top-down entrainment flux. As such (6b) can be seen as a generalization of Eq. (7) in HM91.

*X*in a skewed turbulent boundary layer driven from the surface. Using the constitutive equation derived by Lumley (1975), they found an alternative expression for a scalar flux, which can be written as

*T*

_{L}is the Lagrangian integral timescale and

*S*

_{w}=

*w*

^{3}

*σ*

^{3}

_{w}

*S*

_{w}and the scalar flux gradient. For a bottom-up flux in a dry CBL the latter is equivalent to −(

*wχ*

_{0}/

*h*, where (

*wχ*

_{0}is the surface scalar flux. Although (8) is developed for a passive conservative scalar in homogeneous skewed turbulence, we will stretch its application here to see how well we can describe heat and smoke fluxes with it in differently driven CBLs.

*L*∼

*σ*

_{w}

*τ*or

*L*∼

*σ*

_{w}

*T*

_{L}(here we assume that

*T*

_{L}and

*τ*are proportional to each other), we may generalize the two expressions given by (6b) and (8) into one form as

*K*∼

*σ*

_{w}

*L*(see below). We note that for heat,

*wχ*

_{NL}in (9) is equivalent to

*wχ*

_{NL}=

*Kγ,*where

*γ*is the countergradient term for temperature (Deardorff 1972; HM91).

*wχ*

_{NL}in (9) can be written as

*β*

_{1}a proportionality coefficient that includes

*β*

_{0}. Similarly we have

*β*

_{2}an other proportionality coefficient. The coefficients

*β*

_{1}and

*β*

_{2}may be a function of height in general and are therefore to be determined from the LES cases (section 5).

The differences in the formulations of *wχ*_{NL} are twofold: namely, we have *w**/*σ*_{w} versus *S*_{w} and *w***χ**/*h* versus ∂*wχ**z.* The first two terms are typically different and discriminate between vertical velocity variance or skewness. The latter two terms are often more similar (especially when the scalar flux profile is linear) but differ in the sense that *w***χ**/*h* is a bulk quantity, since it equals the integral of the scalar flux, while ∂*wχ**z* is the flux gradient. This will have a clear consequence as we will see in section 5.

*K*in (9) can generally be written as

*K*

*c*

_{k}

*σ*

_{w}

*L,*

*c*

_{k}a coefficient. In the absence of convection, we have that

*w** = 0 and

*S*

_{w}= 0, so

*wχ*

_{NL}= 0 and (9) reduces to the familiar downgradient relationship with a diffusivity given by (11). As such we have the proper limit for the neutral boundary layer. This provides us with a possibility to estimate

*c*

_{k}by matching (11) to the well-known similarity results for the neutral surface layer (Stull 1988),

*K*=

*ku**

*z.*Here,

*k*is the von Ká®án constant (

*k*= 0.4) and

*u** is the friction velocity. Equating this to (11) and utilizing the normal findings of

*σ*

_{w}= 1.3

*u** and

*L*=

*z*for the neutral surface layer, we arrive at

*c*

_{k}= 0.3. This value is used throughout the remainder of this paper.

*L*in (10)–(11) depends in general on stability and the depth over which the turbulence extends. To quantify

*L*in this context, we explored two different methods. First, the dissipation length scale, based on the vertical velocity variance

*σ*

_{w}and the dissipation

*ε,*yields

*c*

_{ε}= 0.4 (Hunt et al. 1985).

The second method follows Bougeault and Lacarrère (1989). As such, the length scale is calculated from two length scales *L*_{up} and *L*_{down}. These length scales are calculated in each column of the LES model as the distance between the initial level of an upward (downward) moving parcel and the level where it has lost all its kinetic energy. The turbulent kinetic energy at the initial level of the parcel is taken as the fueling energy. During its ascent (descent) the parcel may increase its kinetic energy as long as it is positively buoyant, while it looses kinetic energy when it is negatively buoyant. So, the parcel may overshoot its level of neutral buoyancy, but in the inversion layer the kinetic energy is generally rapidly consumed. In the downward direction, the length scale (*L*_{down}) is bounded by the distance to the surface. The length scale, denoted by *L*_{BL}, is then calculated as the square root of the product of the horizontally averaged *L*_{up} and *L*_{down}.

In Fig. 4 we show the results for the two length scales*L*_{ε} and *L*_{BL} for the dry CBL and the smoke-cloud case. For the dry CBL both length scales compare reasonably well, for the smoke-cloud simulation the dissipation length scale is larger in the bulk of the boundary layer. For the zero skewness case the agreement is intermediate (not shown).

*L*that is actually used in the flux expressions is taken as the minimum of either one of

*L*

_{ε}or

*L*

_{BL}and the buoyancy length scale

*L*

_{BV}:

*L*

_{BV}=

*σ*

_{w}/

*N*, where

*N*is the Brunt–Vä∩älä frequency given by

*L*

_{1}=

*L*

_{ε}and

*L*

_{2}=

*L*

_{BL}, but in the upper part of the boundary layer there is a smooth transition to

*L*

_{BV}.

## 5. Results

In this section we provide a comparison between the two alternative expressions, that is, (9) with either (10a) or (10b). They both rely on a proper turbulent length scale [see (10)–(11)]. In spite of the differences in *L*_{1} and *L*_{2} in the bulk of the boundary layer (Fig. 4), it was found that both length scales give qualitatively similar results for the flux expressions. Therefore, we will only show the results by using *L*_{1} (Fig. 5) since this is a more fundamental length scale. In larger scale models, however, *L*_{2} is preferred because it is easier to calculate.

In Fig. 5 we also show the vertical velocity skewness and the vertical velocity variance of the three prototype CBLs. The dry CBL has a positive skewness, while in the smoke cloud case it is mainly negative, except for a small layer near the inversion. Comparing the vertical velocity variance profile of the dry CBL (full line in Fig. 5, lower panel) with the profile of the smoke cloud case (dotted line) reveals that the smoke cloud case is not exactly an upside-down version of the dry CBL. While in the dry CBL the maximum is at *z*/*h* = 0.3, in the smoke cloud case the maximum occurs at about the middle of the boundary layer.

As discussed in section 4, the coefficients *β*_{1} and *β*_{2}, which appear in (10a) and (10b) respectively, may be a function of height in general. Overall quite satisfactory results, however, are obtained by utilizing *β*_{1} = 1.5 for all the cases below. Similarly, we take *β*_{2} = 1. We note that this simplifies the application of (9) in a larger scale context.

### a. The dry CBL

In Fig. 6 we show the different terms of the scalar flux equation (9) for the heat flux in the dry CBL. The full line gives the turbulent flux as calculated directly from the LES data (resolved and subgrid fluxes). The dotted line is the contribution of the local term [first term on the rhs of (9)] and the dashed line indicates the nonlocal term [second term on the rhs of (9)]. The sum of local and nonlocal terms is the total parameterized flux, given by the dash–dotted line. In the upper panel of Fig. 6 the results using (10a) as the nonlocal term are shown, while in the lower panel (10b) is used.

The local term contributes most near the surface and at the inversion where the potential temperature gradient is large, while the nonlocal term is important in the bulk of the boundary layer, as was found before (e.g., HM91). It is seen that (9) with either (10a) or (10b) describesthe heat flux (which is nearly a bottom-up flux) rather well, as already noted by Wyngaard and Moeng (1993).

In Fig. 7 the smoke flux *wS*_{1}*wS*_{1}

The smoke flux *wS*_{2}*β*_{2} that varies with height would give better results than shown here with a constant *β*_{2}).

Comparing the findings for the scalar fluxes *wS*_{1}*wS*_{2}

### b. The smoke-cloud boundary layer

For the smoke-cloud case, the local term fairly well describes the actual heat flux in the inversion, but below 600 m this term is no longer sufficient (Fig. 9). Due to a very slightly stable layer in the lower part of the mixed layer (Fig. 1a), the local term becomes even negative there. It reflects the upside-down character of this boundary layer with respect to the dry CBL, showing countergradient behavior in the lower part of the mixed layer. Again the nonlocal term contributes to the total flux mostly in the bulk of the mixed layer. In the lower panel of Fig. 9, we show the results using (10b). Also in this case the nonlocal term is necessary to obtain the proper total flux. However, this nonlocal term variesmore with height and becomes very small between 500 m and 600 m, where both the skewness and the flux gradient are small.

Comparable results are found for the smoke flux *wS*_{1}

The flux parameterization using (10a) for a smoke flux *wS*_{2}

### c. Boundary layer with zero skewness

For the boundary layer where both radiative cooling at the top and surface heating are applied, the heat flux is nearly constant with height (of about 0.01 K m s^{−1}; not shown). The expressions are evaluated for the smoke flux *wS*_{1}

## 6. Discussion and summary

In this paper we have presented an evaluation of scalar and buoyancy flux expressions in three prototype convective atmospheric boundary layers (CBLs). By applying a surface heat flux, radiative cooling at the boundary layer top or a combination of these two, CBLs with positive, negative, and zero vertical velocity skewness are simulated with an LES model. Furthermore, in these CBLs different smoke fluxes are applied.

The flux expressions consist of a local downgradient term and a nonlocal term. The expressions differ in the formulation of the nonlocal term, namely, in one case[Eq. (10a)] we have *w**/*σ*_{w} (vertical inhomogeneity) and an integral form of the scalar flux [Eq. (3)]. In the other case [Eq. (10b)], the skewness *S*_{w} and the (local) scalar flux gradient ∂*wχ**z* are used. Formulation (10a) is different from the expression by Holtslag and Moeng (1991, HM91), in the sense that we use an integral form of the scalar flux, while HM91 use either the surface flux (for bottom-up diffusion) or entrainment flux (for top-down diffusion). Expression (10b) is derived for a passive conservative scalar in homogeneous skewed turbulence (Wyngaard and Weil 1991). Here we stretched its application to evaluate its performance in differently driven CBLs.

All variables that we need for the flux expressions are derived from the LES data. In this way, a clean and consistent verification of the expressions is performed. It is shown that both expressions work well for the heat flux in the dry CBL and the smoke-cloud case. The nonlocal term contributes most in the bulk of the boundary layer, where the local downgradient diffusion term alone is insufficient to have a correct description of the turbulent flux. These findings support previous results (e.g., Deardorff 1972; HM91; Wyngaard and Moeng 1993).

Comparing the smoke flux *wS*_{1}*wS*_{1}

The smoke flux *wS*_{2}

The constant flux case (*wS*_{2}

In summary, we find that the vertical velocity variance and the integral form of the flux are sufficient to describe the nonlocal transports. Significant nonlocal transport is still found in the absence of skewness. Furthermore, it is shown that the coefficient *β*_{1} in (10a) can be taken constant (*β*_{1} = 1.5). Thus, the flux is very well described by (9) with (10a) for the cases examined. Provided that *σ*_{w} and *L* can be properly modeled, we directly achieve realistic flux profiles. This avoids the necessity of utilizing higher-order closure modeling.

The implications of the current findings for larger- scale atmospheric models have to be seen. In this study we found that a correct formulation of the length scale in the upper part of the CBL is very important to obtain the proper entrainment fluxes. This fact might become even more critical in coarse resolution global climate and weather forecast models. Therefore, as a next step we want to test these parameterizations in a 1D model for the same cases. This will be done for the high resolution used in this study and for a resolution typical for global models. We then hope to clarify whether the nonlocal term or a correct description of the length scale near the inversion is the more critical issue in flux parameterizations.

## Acknowledgments

We like to thank Peter Duynkerke and Ge Verver for discussions. We also like to thank them, and Arthur Petersen, Chin-Hoh Moeng, Dr. Canuto, and two unknown referees for useful comments on an earlier draft of this paper. Cees Beets is acknowledged for his help by implementing the kappa scheme. The first author acknowledges support by the Netherlands Geosciences Foundation (GOA) with financial aid from the Netherlands Organisation for Scientific Research (NWO) under Contract 750-194-13. This work was sponsored by the National Computing Facilities Foundation (NCF) for the use of supercomputer facilities, with financial support from NWO.

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Schematic of the heat fluxes (left panels) and smoke fluxes (middle and right panels) in the three prototype CBLs. (a)–(a2): The dry CBL driven by a surface heat flux. (b)–(b2): The smoke-cloud case driven by cloud-top radiative cooling. (c)–(c1): CBL driven by surface heating and cloud-top radiative cooling.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

Schematic of the heat fluxes (left panels) and smoke fluxes (middle and right panels) in the three prototype CBLs. (a)–(a2): The dry CBL driven by a surface heat flux. (b)–(b2): The smoke-cloud case driven by cloud-top radiative cooling. (c)–(c1): CBL driven by surface heating and cloud-top radiative cooling.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

Schematic of the heat fluxes (left panels) and smoke fluxes (middle and right panels) in the three prototype CBLs. (a)–(a2): The dry CBL driven by a surface heat flux. (b)–(b2): The smoke-cloud case driven by cloud-top radiative cooling. (c)–(c1): CBL driven by surface heating and cloud-top radiative cooling.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

The normalized budget terms of heat flux [(a) and (b)] and smoke flux (*wS*_{1}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

The normalized budget terms of heat flux [(a) and (b)] and smoke flux (*wS*_{1}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

The normalized budget terms of heat flux [(a) and (b)] and smoke flux (*wS*_{1}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

Comparison of length-scale formulations for the dry CBL (upper panel) and smoke-cloud case (lower panel). Dissipation length scale [Eq. (12); thin full line], the length scale following Bougeault and Lacarrère (1989; thin dotted line), buoyancy length scale (dashed line), and *L*_{1} (thick full line) and *L*_{2} (thick dotted line) given by (13).

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

Comparison of length-scale formulations for the dry CBL (upper panel) and smoke-cloud case (lower panel). Dissipation length scale [Eq. (12); thin full line], the length scale following Bougeault and Lacarrère (1989; thin dotted line), buoyancy length scale (dashed line), and *L*_{1} (thick full line) and *L*_{2} (thick dotted line) given by (13).

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

Comparison of length-scale formulations for the dry CBL (upper panel) and smoke-cloud case (lower panel). Dissipation length scale [Eq. (12); thin full line], the length scale following Bougeault and Lacarrère (1989; thin dotted line), buoyancy length scale (dashed line), and *L*_{1} (thick full line) and *L*_{2} (thick dotted line) given by (13).

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

The length scale *L*_{1} [Eq. (13); upper panel], vertical velocity skewness (middle panel), and vertical velocity variance (lower panel) for the three prototype CBLs.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

The length scale *L*_{1} [Eq. (13); upper panel], vertical velocity skewness (middle panel), and vertical velocity variance (lower panel) for the three prototype CBLs.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

The length scale *L*_{1} [Eq. (13); upper panel], vertical velocity skewness (middle panel), and vertical velocity variance (lower panel) for the three prototype CBLs.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

The variation with height (*z*) of the terms in Eq. (9) for the heat flux as calculated from LES data for a dry CBL. Upper panel is the result by using (10a) and lower panel by using (10b).

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

The variation with height (*z*) of the terms in Eq. (9) for the heat flux as calculated from LES data for a dry CBL. Upper panel is the result by using (10a) and lower panel by using (10b).

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

The variation with height (*z*) of the terms in Eq. (9) for the heat flux as calculated from LES data for a dry CBL. Upper panel is the result by using (10a) and lower panel by using (10b).

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{1}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{1}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{1}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{2}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{2}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{2}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for heat flux in the smoke-cloud case.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for heat flux in the smoke-cloud case.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for heat flux in the smoke-cloud case.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{1}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{1}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{1}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{2}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{2}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{2}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{1}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{1}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

As in Fig. 6 but for smoke flux *wS*_{1}

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0151:IOSANE>2.0.CO;2

Characteristics of the simulated cases. Shown are the inversion height *h,* the convective velocity scale *w*_{*}, the entrainment velocity*w _{e}*, and the sign of the vertical velocity skewness

*S*, respectively. We also indicate the integrated fluxes

_{w}*w*

_{*}

*θ*

_{*}and

*w*

_{*}

*S*

_{*}, respectively, as used in Eq. (10a) and the sign of the smoke flux gradient ∂

*wS*

*z.*