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  • View in gallery

    The vertical profiles of potential temperature (upper panel) for the initial state (thin line) and the final states. Broken line refers to the dry CBL and the thick line to the smoke case and for a smoke tracer (lower panel).

  • View in gallery

    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.

  • View in gallery

    The normalized budget terms of heat flux [(a) and (b)] and smoke flux (wS1) [(c) and (d)] for the dry CBL [left] and smoke-cloud boundary layer [right]. The terms are defined in the text in section 3.

  • View in gallery

    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 L1 (thick full line) and L2 (thick dotted line) given by (13).

  • View in gallery

    The length scale L1 [Eq. (13); upper panel], vertical velocity skewness (middle panel), and vertical velocity variance (lower panel) for the three prototype CBLs.

  • View in gallery

    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).

  • View in gallery

    As in Fig. 6 but for smoke flux wS1 in a dry CBL.

  • View in gallery

    As in Fig. 6 but for smoke flux wS2 in a dry CBL.

  • View in gallery

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

  • View in gallery

    As in Fig. 6 but for smoke flux wS1 in the smoke-cloud case.

  • View in gallery

    As in Fig. 6 but for smoke flux wS2 in the smoke-cloud case using (10a).

  • View in gallery

    As in Fig. 6 but for smoke flux wS1 in a CBL with almost zero skewness using (10a).

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Impact of Skewness and Nonlocal Effects on Scalar and Buoyancy Fluxes in Convective Boundary Layers

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  • 1 Institute for Marine and Atmospheric Research Utrecht, Utrecht University, Utrecht, the Netherlands
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Abstract

Large-eddy simulation (LES) results of three prototype convective boundary layers (CBL) are used to study flux budgets and to test simple expressions for scalar and buoyancy fluxes. The simulated CBLs differ in their characteristics of skewness of the vertical velocity field and the scalar flux gradients. This is accomplished by applying a surface heat flux or radiative cooling at the boundary layer top. In this way boundary layers with positive, zero, and negative skewness are simulated.

Useful approximations for the transport and buoyancy terms in the flux budgets are achieved by using a generalized convective scaling. This directly provides an expression for scalar and buoyancy fluxes that combines downgradient and nonlocal effects on the fluxes. The nonlocal effects are due to vertical velocity variance and the integrated flux over the boundary layer. This result is compared with an alternative expression in which the nonlocal flux is related to the skewness of the vertical velocity field and the scalar flux gradient.

Overall it appears that the nonlocal flux is mostly related to the vertical velocity variance and the integrated flux and not so much to the skewness. The final result is a simple expression for scalar and buoyancy fluxes, that can be used in mesoscale and global models.

Additional affiliation: Royal Netherlands Meteorological Institute, De Bilt, the Netherlands.

Corresponding author address: J. W. M. Cuijpers, KNMI, P.O. Box 201, NL-3730 AE De Bilt, the Netherlands.

Email: cuijpersh@knmi.nl

Abstract

Large-eddy simulation (LES) results of three prototype convective boundary layers (CBL) are used to study flux budgets and to test simple expressions for scalar and buoyancy fluxes. The simulated CBLs differ in their characteristics of skewness of the vertical velocity field and the scalar flux gradients. This is accomplished by applying a surface heat flux or radiative cooling at the boundary layer top. In this way boundary layers with positive, zero, and negative skewness are simulated.

Useful approximations for the transport and buoyancy terms in the flux budgets are achieved by using a generalized convective scaling. This directly provides an expression for scalar and buoyancy fluxes that combines downgradient and nonlocal effects on the fluxes. The nonlocal effects are due to vertical velocity variance and the integrated flux over the boundary layer. This result is compared with an alternative expression in which the nonlocal flux is related to the skewness of the vertical velocity field and the scalar flux gradient.

Overall it appears that the nonlocal flux is mostly related to the vertical velocity variance and the integrated flux and not so much to the skewness. The final result is a simple expression for scalar and buoyancy fluxes, that can be used in mesoscale and global models.

Additional affiliation: Royal Netherlands Meteorological Institute, De Bilt, the Netherlands.

Corresponding author address: J. W. M. Cuijpers, KNMI, P.O. Box 201, NL-3730 AE De Bilt, the Netherlands.

Email: cuijpersh@knmi.nl

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 (Δθ ≈ 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 (wS1)0 and because there is a positive entrainment flux (wS1)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 (wS2)e is negative. A positive surface smoke flux (wS2)0 ≈ −(wS2)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 wS1 in the dry CBL case, in this smoke-cloud boundary layer wS1 is not a top-down flux. Since the convection is driven from the top, the smoke-cloud boundary layer is a dry CBL driven “upside down” and wS1 should be interpreted as a “bottom-up” flux.

In the second smoke-cloud case, we add a surface smoke flux (wS2)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

In this section we provide the heat flux and smoke flux (wS1) budgets for the dry CBL and the smoke-cloud boundary layer and discuss the scaling of the various budget terms. Under horizontally homogeneous conditions, the budget equation for a scalar flux is given by
i1520-0469-55-2-151-e1
Here, w and χ are vertical velocity and scalar (concentration) fluctuations, X is the mean value of the scalar (concentration), σ2w is the vertical velocity variance, g is the gravitational acceleration, θ0 and ρ0 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.
Figure 3 shows the vertical profiles of these terms normalized with the height of the CBL, h, and the generalized convective velocity scale (Deardorff 1976)
i1520-0469-55-2-151-e2
The constant 2.5 ensures that this relation reduces to the usual definition of w* for a dry CBL (assuming that the entrainment flux equals −0.2()0, with ()0 the surface heat flux). Further, we may define a convective scale for a scalar χ as
i1520-0469-55-2-151-e3
where 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 termP 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.

The modeling of the turbulent transport term is rather difficult in general. Also second-order closure models have difficulty in representing third-order terms (see, e.g., Canuto et al. 1994). In addition, second-order closure modeling is too computationally expensive for many applications. As a simple alternative, we may apply convective scaling for T, which yields
i1520-0469-55-2-151-e4
Note that b = b(z/h) is a function of height in general.
Moeng and Wyngaard (1986) studied the closure of the pressure term P in the scalar flux equation. They show that P in a dry CBL without significant wind shear is given by
i1520-0469-55-2-151-e5
where 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 = cεσ2w/ε (with ε the dissipation rate) and initially cε = 1. However, it appears that varying cε 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

On the basis of the findings in the previous section we can now derive a general expression for buoyancy and scalar fluxes. Substituting (5) and (4) into (1), and assuming quasi-steady state (i.e., ∂/∂t ≈ 0), yields
i1520-0469-55-2-151-e6a
This result shows that in quasi-steady state, the vertical flux of a quantity 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.
Equation (6a) can further be simplified by applying convective scaling to the buoyancy term B as well:
i1520-0469-55-2-151-e7
with c = c(z/h) a function of height. This results in
i1520-0469-55-2-151-e6b
where β0 = [c(1 − a) + b]. Note that for the fluxexpression (6b), only the combination of coefficients as given by β0 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
i1520-0469-55-2-151-eq1
[see Eq. (3)], or in other words, to the averaged flux across the CBL. As such the fluxes at the boundaries of a CBL (both at the surface and the top) have impacton the flux at a particular height in the CBL. This demonstrates clearly the nonlocal character of (6b).

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.

Let us now compare relation (6b) with the proposal by WW91. These authors considered the turbulent flux of a passive, conservative scalar 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
i1520-0469-55-2-151-e8
where TL is the Lagrangian integral timescale and Sw = w3/σ3w is the skewness of the vertical velocity field. Again an additional term appears besides the local downgradient transport term. This additional term is proportional to the skewness Sw and the scalar flux gradient. For a bottom-up flux in a dry CBL the latter is equivalent to −()0/h, where ()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.
Introducing a turbulent length scale Lσwτ or LσwTL (here we assume that TL and τ are proportional to each other), we may generalize the two expressions given by (6b) and (8) into one form as
i1520-0469-55-2-151-e9
where KσwL (see below). We note that for heat, NL in (9) is equivalent to NL = Kγ, where γ is the countergradient term for temperature (Deardorff 1972; HM91).
The nonlocal flux NL in (9) can be written as
i1520-0469-55-2-151-e10a
on the basis of (6b), with β1 a proportionality coefficient that includes β0. Similarly we have
i1520-0469-55-2-151-e10b
in the case of (8) with β2 an other proportionality coefficient. The coefficients β1 and β2 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 NL are twofold: namely, we have w*/σw versus Sw and w*χ*/h versus ∂/∂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 ∂/∂z is the flux gradient. This will have a clear consequence as we will see in section 5.

The diffusivity K in (9) can generally be written as
KckσwL,
with ck a coefficient. In the absence of convection, we have that w* = 0 and Sw = 0, so 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 ck 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 σw = 1.3u* and L = z for the neutral surface layer, we arrive at ck = 0.3. This value is used throughout the remainder of this paper.
The length scale 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 σw and the dissipation ε, yields
i1520-0469-55-2-151-e12
with cε = 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 Lup and Ldown. 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 (Ldown) is bounded by the distance to the surface. The length scale, denoted by LBL, is then calculated as the square root of the product of the horizontally averaged Lup and Ldown.

In Fig. 4 we show the results for the two length scalesLε and LBL 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).

However, it appears that both methods produce length scales that are still so large near the top of the CBL that excessive entrainment fluxes result [as found previously in many studies, e.g., Canuto et al. (1994)]. Therefore, the length scale L that is actually used in the flux expressions is taken as the minimum of either one of Lε or LBL and the buoyancy length scale LBV:
i1520-0469-55-2-151-e13
Here LBV = σw/N, where N is the Brunt–Vä∩älä frequency given by
i1520-0469-55-2-151-e14
In most parts of the boundary layer L1 = Lε and L2 = LBL, but in the upper part of the boundary layer there is a smooth transition to LBV.

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 L1 and L2 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 L1 (Fig. 5) since this is a more fundamental length scale. In larger scale models, however, L2 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 β1 and β2, which appear in (10a) and (10b) respectively, may be a function of height in general. Overall quite satisfactory results, however, are obtained by utilizing β1 = 1.5 for all the cases below. Similarly, we take β2 = 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 wS1 in the dry CBL is presented. Since this boundary layer is driven from the surface, wS1 is a top-down flux. The nonlocal term (10a) enhances the local downgradient term in the middle of the boundary layer, giving a reasonably good description of the turbulent flux. In contrast, the positive skewness and smoke flux gradient result in a negative nonlocal term as given by (10b). So, although the skewness clearly can effect the scalar transport, here the nonlocal term deteriorates the representation of the turbulent flux!

The smoke flux wS2 is chosen such that it is about zero when integrated over the whole boundary layer (Table 1). This results in a negligible nonlocal term when using (10a), but the local downgradient term describes the turbulent flux fairly well (Fig. 8). Thus, here we have a particular example that, even in a dry CBL, local diffusion can be sufficient to describe a scalar flux! Due to the opposite signs of the vertical velocity skewness (positive) and the scalar flux gradient (negative), the nonlocal term (10b) gives a positive contributionhere (we realize that a β2 that varies with height would give better results than shown here with a constant β2).

Comparing the findings for the scalar fluxes wS1 and wS2, we conclude that skewness enhances the total flux when the relative signs of skewness and scalar flux gradient are opposite, while similar signs of skewness and scalar flux gradient diminish the total flux.

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 wS1 (Fig. 10). Since in this boundary layer the convection is driven from the top, the smoke flux represents a “bottom-up” flux (but upside down). The local term alone is not able to describe the actual flux correctly, particularly in the bulk of the mixed layer. Due to the opposite signs of skewness and smoke flux gradient (Table 1), the nonlocal term (10b) enhances the diffusivity as does (10a). However, due to the small value of the skewness above about 600 m, the nonlocal term (10b) vanishes at a lower height than with (10a).

The flux parameterization using (10a) for a smoke flux wS2 with (nearly) zero gradient is presented in Fig. 11. There is a finite contribution from the nonlocal term (10a), which sufficiently contributes to the flux in the upper part of the CBL. However, the total parameterized flux underestimates the flux calculated directly from LES in the lower part of the boundary layer. The nonlocal term given by (10b) vanishes in this special case because of the zero flux gradient.

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 wS1. The nonlocal term (10a) shows again a finite contribution in the bulk of the boundary layer. This expression gives indeed a good performance for the total parameterized flux (Fig. 12). Because the skewness is about zero in this boundary layer (Fig. 5) and the nonlocal term given by (10b) is derived for skewed turbulence, it obviously has to fail in this case. The downgradient term alone substantially underestimates the smoke flux.

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*/σw (vertical inhomogeneity) and an integral form of the scalar flux [Eq. (3)]. In the other case [Eq. (10b)], the skewness Sw and the (local) scalar flux gradient ∂/∂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 wS1 in the dry CBL (a top-down flux) with wS1 in the smoke-cloud case (a“bottom-up” flux in upside-down sense) shows that the skewness has an effect on the total flux. However, in the dry CBL the nonlocal term (10b) is negative, due to the similar signs of skewness and scalar flux gradient, thus deteriorating the flux description. In the smoke- cloud case, (10b) is positive because of opposite signs of skewness and flux gradient, thereby improving the flux parameterization.

The smoke flux wS2 in the dry CBL is very well described with a local diffusion term only. In this special case the nonlocal flux term by (10a) is small because the integral flux tends to zero. The signs of skewness and scalar flux gradient are opposite, leading to an enhancement of the total flux by the nonlocal term (10b). Thus, skewness enhances the total flux when the relative signs of skewness and scalar flux gradient are opposite, while similar signs of skewness and scalar flux gradient diminish the total flux. The latter case, however, causes an underestimation of the parameterized flux.

The constant flux case (wS2 in the smoke cloud case) was set up to mimic a situation encountered in (cumulus- topped) boundary layers, where the humidity flux gradient is often found to be (nearly) zero. The zero skewness case can be considered a stratocumulus-topped boundary layer with both radiative cooling at the cloud top and a surface heat flux. The nonlocal term (10a) fairly well describes these smoke fluxes. However, these cases cannot be described on the basis of a flux expression in which the nonlocal effects depend on either skewness or a flux gradient [Eq. (10b)]. The expression then reduces to the usual downgradient diffusion formulation, which can be seen to substantially underestimate the smoke fluxes.

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 β1 in (10a) can be taken constant (β1 = 1.5). Thus, the flux is very well described by (9) with (10a) for the cases examined. Provided that σw 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.

REFERENCES

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Fig. 1.
Fig. 1.

The vertical profiles of potential temperature (upper panel) for the initial state (thin line) and the final states. Broken line refers to the dry CBL and the thick line to the smoke case and for a smoke tracer (lower panel).

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

Fig. 2.
Fig. 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

Fig. 3.
Fig. 3.

The normalized budget terms of heat flux [(a) and (b)] and smoke flux (wS1) [(c) and (d)] for the dry CBL [left] and smoke-cloud boundary layer [right]. The terms are defined in the text in section 3.

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

Fig. 4.
Fig. 4.

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 L1 (thick full line) and L2 (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

 Fig. 5.
Fig. 5.

The length scale L1 [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

Fig. 6.
Fig. 6.

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

Fig. 7.
Fig. 7.

As in Fig. 6 but for smoke flux wS1 in a dry CBL.

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

Fig. 8.
Fig. 8.

As in Fig. 6 but for smoke flux wS2 in a dry CBL.

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

Fig. 9.
Fig. 9.

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

Fig. 10.
Fig. 10.

As in Fig. 6 but for smoke flux wS1 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

Fig. 11.
Fig. 11.

As in Fig. 6 but for smoke flux wS2 in the smoke-cloud case using (10a).

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

Fig. 12.
Fig. 12.

As in Fig. 6 but for smoke flux wS1 in a CBL with almost zero skewness using (10a).

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

Table 1.

Characteristics of the simulated cases. Shown are the inversion height h, the convective velocity scale w*, the entrainment velocitywe, and the sign of the vertical velocity skewness Sw, respectively. We also indicate the integrated fluxes w*θ* and w*S*, respectively, as used in Eq. (10a) and the sign of the smoke flux gradient ∂wS/∂z.

Table 1.
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