1. Introduction

Individual cloud elements cannot generally be resolved by grids used in mesoscale (or larger) meteorological models, so subgrid parameterization of partly cloudy atmospheric boundary layers has been a long- standing subject for research. Regardless of the sophistication used in modeling the dynamics of partly cloudy layers, some representation of the horizontal mean buoyancy flux in terms of the fluxes of adiabatically conserved variables, for example, total water mixing ratio, q_t, and liquid potential temperature, θ_l, is usually required.

To a good approximation the buoyancy flux¹ may be written as

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where q_l is the liquid water mixing ratio, α and β are thermodynamic coefficients defined in appendix A, and the overbars indicate a horizontal average. Expressing the mean liquid water flux, w′q^′_l, in terms of the fluxes of θ_l and q_t is the chief difficulty, since this depends on how the clouds and vertical velocity are correlated locally. In the limit where a given vertical level is all clear or all cloudy this difficulty is avoided. We can define “dry” and “wet” buoyancy fluxes that, to a good approximation, equal the buoyancy flux in the limit of all- clear or all-cloudy conditions, respectively:

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Here s is the difference between the total-water and saturation mixing ratios, which may be approximated by linearizing the saturation formula,

sq_tq_saq^′_tbθ^′_lc,(5)

leading to the approximation in (4). Here q_s is the saturation mixing ratio and a, b, and c are thermodynamic coefficients given in appendix A.

In the general partly cloudy case we may formally write the buoyancy flux as an interpolation between the dry and wet limits,

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where, from (1) to (3), the interpolation variable R̂ is given by,

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Modeling of w′θ^′_υ in terms of w′q^′_t and w′θ^′_l amounts to finding a representation for R̂. One approach has been statistical formulations relating the variances and covariances of the relevant fluctuations. Sommeria and Deardorff (1977) and Mellor (1977) assumed joint- Gaussian distributions for the probability density functions (pdfs) of the fluctuations of w, q_t, and θ_l. In this limit (in which the vertical velocity and cloud water fluctuations are uncorrelated) R̂ is found to be identically equal to the mean partial cloud fraction, which in turn is determined as a function of the appropriate variances and covariances.

In partly cloudy conditions where latent heat release is very important to the dynamics, the vertical velocity and cloud water fluctuations tend to be strongly correlated and the actual R̂ can depart significantly from the mean cloud fraction. The pdfs in this case are not well represented by joint-Gaussians. Bougeault (1981) found that many simulated clouds produced from large- eddy simulations (LESs) showed positively skewed conserved variable distributions better fit with exponential or gamma distributions. Under highly skewed conditions with small cloud fractions and relatively strong updrafts such as may occur in shallow cumulus, Randall (1987) demonstrated that the phenomenological updraft–downdraft model developed for tropospheric cumulus parameterization provided a much better representation than the Gaussian pdf partial cloudiness models. Lewellen and Yoh (1993) introduced a binormal formulation that provided for a smooth transition between the Gaussian distribution and a double Gaussian. In the limit of high skewness, the double Gaussian approaches the two-delta functions of the simple updraft– downdraft model. This formulation represented distributions obtained from LES results well when analyzed values of the skewness were used, but did not provide a model for the skewness.

Bechtold and colleagues have tried to empirically determine relations between the fractional cloudiness, the cloud water content, and the liquid water flux as a function of s/σ_s, the mean saturation deficit normalized by its standard deviation. Bechtold et al. (1992) indicated that such relations based on the Gaussian distribution could produce reasonable boundary layer structure for layers with cloud fractions greater than approximately 20%. Bechtold et al. (1995) added a function to account for non-Gaussian behavior at small cloud fractions and improved upon this empirical function in Cuijpers and Bechtold (1995) and Bechtold and Siebesma (1998). In this approach R̂ [which is equal to the combination f_NN in Bechtold and Siebesma (1998)] is empirically determined as a function of s/σ_s from LES results.

Though the main focus in these works (and here) is partly cloudy layers (particularly shallow cumulus, stratocumulus over cumulus, and broken stratocumulus), it should be noted that even for solid stratocumulus determining R̂ is an issue at cloud top. This is particularly important for treatments of cloud-top entrainment that require parameterizing the buoyancy flux at the inver-sion, w′θ^′_υi. See, for example, Lilly (2002) for a recent treatment [R̂(z_i) here is his parameter α, which he estimates from LES results under the assumption that a particular parameterization for w′θ^′_υi holds].

Most of the preceding approaches aim to express R̂(z) as a function of thermodynamic moments defined locally at z. The underlying dynamics, however, is typically dominated by coherent circulations extending over much of the boundary layer depth, which, upon horizontal averaging, would naturally lead to a nonlocal result in z. Invoking higher-order moments (skewnesses, saturation variance, etc.) is done in part to try to circumvent this difficulty and provide a formulation local in z. An alternative is to actually provide a simple model for the underlying eddy dynamics. This is the case for mass flux models, where w′θ^′_υ is computed directly, but this requires a specification of the convective mass flux and assumptions about lateral entrainment and detrainment in and out of the updrafts.

In the following section we formulate expressions for R̂ incorporating nonlocality in z at a very simple level, inspired by the mass flux approach. This enables w′θ^′_υ to be expressed approximately using only the most basic of inputs, the q_t, θ_l, w′q^′_t, and w′θ^′_l profiles. This formulation was tested against LES predictions for a variety of boundary layers; the results are sampled and summarized in section 3. The LES cases include the large set of idealized layers driven by surface heat and humidity fluxes that was employed in Lewellen and Lewellen (2002, hereafter referred to as LL02), and a sample from recent LES intercomparisons organized by the Global Water and Energy Experiment (GEWEX) Cloud Systems Studies (GCSS) boundary layer clouds working group (WG1).

In section 4, R̂ is used to extend the results of LL02 on the relation between boundary layer entrainment and circulation structure for shallow cumulus conditions. For quasi-steady conditions this allows reasonably successful predictions of cumulus-coupled layer dynamics (e.g., the fluxes, q_t and θ_l within the cloud layer) to be made as a function of a basic set of input parameters.

2. Approximating R̂

A basic requirement of R̂ in (6) is that it interpolate between 0 and 1 for all-clear and all-cloudy conditions, respectively. For conditions with a relatively uniform cloud base (e.g., typical stratocumulus) the details of that interpolation are not so important. The more challenging case is shallow cumulus, where R̂ can be large even while the cloud fraction is small because the cloud elements and strong updrafts are highly correlated. Accordingly, we concentrate on developing an appropriate formulation of R̂ for shallow cumulus and will check its adequacy for other conditions a posteriori.

As in Bechtold and Siebesma (1998) we employ the mass flux model to approximate R̂, but do so in a different fashion. In general, for the flux of any scalar quantity Ψ, we have

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where M_c is the convective mass flux, ν is a proportionality constant, and the superscripts denote updraft and downdraft values. We apply this to (7) at what is a priori a very crude level: using the horizontal mean profiles q_t and θ_l to define downdraft properties, and constant values q^sc_t and θ^sc_l characteristic of the lower portion of the subcloud layer to define updraft properties, giving

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Here, s^sc(z) = q^sc_t − q_s(θ^sc_l, q^sc_t, z) is the saturation deficit of a parcel with values θ^sc_l and q^sc_t at height z; q^sc_l(z) = q_l(θ^sc_l, q^sc_t, z) = max[0, s^sc(z)] is the liquid water mixing ratio of such a parcel. The variables s^b(z) = q_t(z) − q_s[θ_l(z), q_t(z), z] and q^b_l(z) = q_l[θ_l(z), q_t(z), z] = max[0, s^b(z)] are the corresponding quantities for a parcel at height z possessing the values θ_l(z) and q_t(z). Note that q^b_l and s^b are not equal to the horizontal means q_l and s.

Alternatively we can apply (8) to (7) in the analogous way but using the approximation (4) for W arising from the linear approximation to the saturation formula, giving

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where a, b, α, β, and θ^b_υ are all computed using q_t and θ_l.

The expression (9) does not involve a linearization of the saturation formula and is conceptually simpler than (11) (all the terms of which generally contribute significantly, precluding further simplification). On the other hand, by using (4) for W the buoyancy flux is given by (6) in terms of w′θ^′_l and w′q^′_t alone without reference to w′s′, which is often convenient in practice. We use the forms (4), (10), and (11) later in comparing with our LES results, for example, because w′s′ was not included in the statistics stored for each run. In practice the predictions from (9) and (11) agree with each other to a within a few percent or better, with the largest differences occurring where liquid water levels are highest, at the tops of thick cloud layers.

Some further comments on these approximations to R̂ are in order. First, since q_l = max(0, s), it is clear that (9) is 0 below cloud base and 1 in a solid cloud, as expected; indeed (9) remains bounded between 0 and 1 regardless of what q_t and θ_l profiles (physical or unphysical) are used to evaluate it. Second, while the mass flux approximation (8) is sometimes rough and we have taken liberties with it, several sources of uncertainty drop out, at least partially, in the final expression, suggesting that the result might prove more successful than the derivation would indicate at first glance. In replacing q^u_t and θ^u_l with q^sc_t and θ^sc_l we have nominally ignored mixing between the updraft and its environment, which is not a good approximation for traditional mass flux applications to cumulus. In using (8) to get from (7) to (9) [or from (10) to (11)], however, the leading effect of this mixing—forcing updraft values toward the background ones—is common to the numerator and denominator and so at least partly cancels in the ratio. In this substitution the factors of ν and M_c that are introduced cancel exactly. Third, the resulting approximations to R̂ are nonlocal in z but in a simple way: through the dependence, at all levels, on the near-surface subcloud values q^sc_t and θ^sc_l. Finally, the result depends only on q_t and θ_l, unlike other expressions given in the literature that depend on the horizontal mean liquid water and cloud fraction [as in the mass-flux-derived expression in Bechtold and Siebesma (1998)], the saturation deficit variance (Cuijpers and Bechtold 1995), or w-skewness and s-skewness (Lewellen and Yoh 1993).

Our search for a form with very basic ingredients was motivated in part by an empirical finding from our LES results: the instantaneous horizontal mean fluxes measured in simulations with modest domain sizes exhibited a very large degree of variability in time, but in the R̂ measured from (10) most of this variability dropped out, suggesting the possibility of a formulation independent of higher moment quantities. To give a concrete, representative example: in simulation NHb from LL02 (cf. Figs. 2a and 6 later) the standard deviation in time of the horizontal average normalized by the time average for the heat, moisture, and buoyancy fluxes were σ_{〈w′θ^′l〉}/|w′θ^′_l| = 0.19, σ_{〈w′q^′t〉}/w′q^′_t = 0.20, and σ_{〈w′θ^′υ〉}/w′θ^′_υ = 0.26, respectively, at 600-m height, while the fluctuations of R̂ at that height were nearly an order of magnitude smaller, σ_〈R̂〉/R̂ = 0.027.

One limitation of (9) or (11) is straightforwardly improved upon. The transitions at cumulus or stratocumulus cloud base occur at fixed heights instead of distributed in z because no contribution from fluctuations away from the mean updraft or mean downdraft values has been included. Assuming a Gaussian distribution of fluctuations in properties of the subcloud parcels contributing to cumulus plumes (with means q^sc_t, θ^sc_l, and standard deviations σ^sc_qt, σ^sc_θl) and in fluctuations in downdraft/background properties (means q_t, θ_l, and standard deviations σ^b_qt, σ^b_θl), we can incorporate these effects to linear order in the fluctuations as described in Sommeria and Deardorff (1977) and Mellor (1977) (SDM hereafter). Note that we are considering the fluctuations about the updraft and downdraft means separately, not the contribution of the updraft–downdraft differences to the total variances (which would be poorly represented by a Gaussian for cumulus conditions). Averaging over the appropriate ensembles in the numerator and denominator of (9) leaves the denominator unchanged and replaces q^sc_l and q^b_l in the numerator with the appropriate SDM expressions,

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with

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and similarly for q^b_l,SDM. The corresponding change is effected in (11) by adding

βq^sc_l,SDMq^sc_lq^b_l,SDMq^b_l(16)

to the numerator.

We will see in comparing with LES results later that the effects of these fluctuations on R̂ are secondary, so rough estimates for the standard deviations of the saturation deficits, σ^sc_s and σ^b_s, suffice for these purposes. Here we use characteristic θ_l and q_t differences based on the surface fluxes to estimate these variances,

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Here κ is a proportionality constant, and w^*_sc and w* are convective velocity scales for the subcloud circulation [e.g., (g/θ₀ ∫^{\[mu4\]z_t\[md4\]}₀ w′θ^′_υ)^1/3 with z_t the top of the subcloud layer] and full layer, respectively. Arguably σ^b_s should include some contribution from the entrainment fluxes at cloud top as well as from the surface fluxes, but we have kept the given form for simplicity.

3. Comparison with LES results

The LES results we will compare to were all generated with the same LES model employed in Lewellen and Lewellen (1998, hereafter referred to as LL98), described there and in references therein. The simulations considered fall into two categories: an idealized set generated for the study in LL02, and a more “realistic” set representing our contributions to recent LES intercomparisons organized by the GCSS-WG1. For convenience, the conditions for the cases that are featured most prominently in the later figures are briefly summarized in the table in appendix B.

The first set includes results from more than 30 simulations described in LL02 of idealized boundary layers driven solely by prescribed heat and moisture fluxes (see appendix B in LL02 for a full summary). All the layers were “tightly capped,” that is the inversion layer thickness was much smaller than the boundary layer depth. This was achieved in some cases using true inversions with large temperature jumps, and in others through the artifice of a rigid lid placed at the top of the layer with a negative heat flux (and a zero or positive moisture flux) forced through it to build up a thin inversion layer immediately below the lid. The principal variations considered were in the magnitude of the surface fluxes, the cloud-top humidity and temperature jumps, and the relative cloud depth. The set was designed in particular to sample the transition from nearly horizontally uniform clouds to shallow cumulus.

The second set includes the four most recent GCSS- WG1 intercomparison cases, all based on particular sets of field observations: a low cloud fraction shallow cumulus case (Siebesma et al. 2003) based on data from the Barbados Oceanographic and Meteorological Experiment (BOMEX); a higher cloud fraction cumulus case (Stevens et al. 2001) based on data from the Atlantic Trade Wind Experiment (ATEX); a diurnal cycle of shallow cumulus over land (Brown et al. 2002) based on data taken at the Atmospheric Radiation Measurement (ARM) site; and, as part of the European Project on Cloud Systems in Climate Models (EUROCS), a diurnal cycle of marine stratocumulus (Duynkerke et al. 2003, manuscript submitted to Quart. J. Roy. Meteor. Soc.) based on observations taken during the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment (FIRE) off the California coast. In each case the LES intercomparisons showed good agreement between different LES models, providing a robust target for the testing of simpler models or relationships; the results from our LES were entirely representative.

Figures 1–4 show sample comparisons between model predictions and LES results. Each case considered is represented by a pair of figure panels. The left member shows the LES mean cloud fraction (thin solid line) and R̂ profile from (10) (thick solid line) compared with the predicted R̂ profile from (11) (thin short dash) and with the correction in (16) (thick long dash). The right member shows the corresponding LES and predicted buoyancy fluxes using the same dash patterns. The LES results are of 1- or 2-h averages about selected times. The R̂ comparison is based on the form in (10) rather than in (7) because the exact saturation deficit flux was not collected during the simulations, but w′θ^′_υ, w′θ^′_l, and w′q^′_t were.

The agreement between predictions and LES results is, in general, excellent. The results from the rest of the LES dataset of LL02 (in all three regimes identified there) are in comparably good agreement, as are the GCSS cases at different times than those shown. The conditions represented are quite varied, including: a convective boundary layer with a growing partial cloud layer (Fig. 2d); stratocumulus boundary layers driven by surface fluxes (Fig. 2c), driven by cloud-top radiative cooling (Fig. 4b), and exhibiting solar-induced afternoon decoupling (Fig. 4a); quasi-steady shallow cumulus underlying stratocumulus with different cloud depths, cloud fractions, surface fluxes, and cloud-top conditions (Figs. 1a–d and 2a,b); somewhat deeper cumulus for both quasi-steady (Figs. 3a,b) and strongly time-varying (growing and collapsing) conditions (Figs. 4c,d).

For the results shown in these and subsequent figures we have set q^sc_t and θ^sc_l equal to the average of the mean measured profiles over the bottom quarter of the subcloud layer; κ = 2 has been used in (17) for estimating the fluctuation contribution. The results prove to be fairly insensitive to the precise choice of subcloud parcel properties; the chief differences appear near cloud base due to the change in lifting condensation level. The improvement from the inclusion of fluctuations via (16) is modest (sometimes negligible; e.g., figs. 3a,b and 4c,d) but consistent, with the biggest effect around cloud base for shallower boundary layers. Again the results are relatively insensitive to the precise choice of fluctuation levels, justifying the rough treatment afforded in (17). In several cases the R̂ and w′θ^′_υ predictions with the (16) term included are nearly indistinguishable from the LES profiles.

We note in passing several other features apparent in the figures. 1) As advertised, R̂, which from (6) may be interpreted as an effective cloud fraction, can be much greater than the actual mean cloud fraction when cumulus columns are present. 2) Generally for cumulus- coupled layers R̂ rises approximately linearly with height in the region with column clouds, reaching the value 1 if and when the mean cloud fraction exceeds one-half. 3) In cases with thick solid cloud layers the predicted and measured R̂ often modestly exceed 1; this a consequence of the approximation made in going from (3) to (4) and, correspondingly, using (11) instead of (9). The excess above 1 observed in the figures in these cases represents the largest differences encountered between the alternative formulations (9) and (11). 4) Above z_i the fluxes approach 0, therefore the measured R̂, determined by a ratio of fluxes, becomes inaccurate and eventually undefined. 5) In the BOMEX and ARM GCSS examples the LES-measured R̂ is observed to significantly exceed 1 in the upper part of the layer. These are the only cases presented with very weakly capped cumulus layers, and the large values trace to regions with overshooting cumulus tops where there can be both clear and cloudy updrafts at the given height. The simple picture of section 2, which assumes all updrafts to be clear or all to be cloudy at a given height, does not attempt to include these effects. 6) The wet and dry buoyancy fluxes, W and D, are often of opposite sign and comparable magnitude, so from (6) a small relative change in R̂ often leads to a larger relative change in w′θ^′_υ. 7) The effects of neglecting lateral mixing into the updraft plumes may just be apparent for some of the deeper cumulus layers (particularly the BOMEX case). In the interesting regime where 0 < R̂ < 1, (9) reduces to R̂ = q^sc_l/(q^sc_l − s^b); the leading effect of lateral mixing is to reduce the updraft liquid water q^sc_l by some amount δq_l, a change which will appear in both numerator and denominator. For this to make a significant difference in the ratio we must have that δq_l is significant compared with q^sc_l and that s_b is significant compared with q^sc_l − δq_l. In other words we expect that not including mixing corrections may lead to an overestimate of R̂ for cases with tall cumulus columns and large saturation deficits in the background flow. 8) In Fig. 4b near cloud base the measured R̂ is less than the measured cloud fraction, while the predicted R̂ is greater (as in other cases). While the discrepancy is minor, this suggests that for layers such as this one driven dominantly by downdrafts (e.g., radiatively cloud-top cooled stratocumulus with little or no surface forcing) one should consider using downdraft parcels in the formulation rather than (or in addition to) the updraft parcels originating in the subcloud layer that were used.

4. Applications to entrainment and flux predictions for shallow cumulus

The results of the last section demonstrate a strong correlation between the mean θ_l and q_t profiles and R̂. It is tempting to employ this in the coupling of a fully predictive single-column model (e.g., a higher-order closure model), using the current θ_l and q_t at each time step to determine R̂ and thence the buoyancy flux. However, a good correlation between variables is insufficient to guarantee that a coupling will be well behaved in a model: cause and effect must be properly represented in the implementation or else small errors will grow and the model predictions wander away from physical reality. If the physics determining the θ_l and q_t profile shapes is not properly represented in the model then the suggested determination of R̂ will ultimately perform poorly.

It was argued in LL02 that the feedback between large-scale circulation structure and entrainment into the boundary layer from above is a critical ingredient in determining the quasi-steady properties a layer will equilibrate to, including the θ_l and q_t profile shapes. The remainder of this paper explores this issue further for cumulus-coupled layers: first, using R̂ to extend the entrainment parameterizations presented in LL98 and LL02 to partly cloudy layers; and second providing an indirect test of the postulated feedback by showing that it can lead to good a priori predictions for quasi-steady cumulus-coupled layer dynamics.

a. Entrainment efficiency for partly cloudy conditions

In LL98 we considered a large-eddy entrainment efficiency [closely related to the entrainment parameterization of Stage and Businger (1981)],

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where w′θ^′_υ is the actual buoyancy flux profile in the layer and w′θ^′_υNE is that buoyancy flux that would result if the entrainment process being considered were not present. The numerator represents an estimate of the energetic cost of entrainment incurred by the buoyantly driven layer-scale circulation; the denominator is an estimate of the energy input available to drive the circulation. The working assumption, supported by LES results (LL98; Lewellen and Lewellen 2000; LL02), is that for buoyantly driven quasi-steady layers the cloud- top entrainment rate is set by the upper limit that the large-scale circulation can energetically support.

Computing η for a given case, or using (18) to predict entrainment by assuming a given value for η, both re- quire knowledge of w′θ^′_υNE. As discussed in LL98, for quasi-steady conditions the “no entrainment” fluxes of the conserved variables θ_l and q_t can be estimated given the surface fluxes and the contributions (if any) from large-scale advection, radiation, or precipitation. If the latter effects are absent, the quasi-steady w′θ^′_lNE and w′q^′_tNE profiles are simply linear from their surface values to 0 at the inversion height, z_i. LL98 concentrated on layers with approximately horizontally uniform cloud layers so that w′θ^′_υNE follows from w′θ^′_lNE and w′q^′_tNE using (2) or (4). Employing R̂ these results can be extended to partly cloudy boundary layers, in particular cumulus-coupled layers, using (6) to relate the no-entrainment fluxes.

LL02 employed the large-eddy entrainment efficiency and a simple classification of circulation structures to study the “decoupling” transition leading to a cumulus- coupled boundary layer. Starting from three basic eddy scales (full layer, cloud layer, subcloud layer) we defined three idealized circulation regimes (I, II, III) depending on which eddy type was hypothesized to limit the entrainment flux transport. In regime I the flux transport is driven by full-layer-scale eddies (e.g., as in the dry convective boundary layer). In regimes II and III the entrained flux is transported across the full layer in two steps, first by cloud-scale, then by subcloud-scale eddies. In quasi-steady state (i.e., with the shapes of the θ_l and q_t profiles approximately constant in time) the flux transport in the upper and lower circulations equilibrates by weakening the stronger circulation or strengthening the weaker. In regime II (identified with a class of stratocumulus layers) the cloud deck is assumed solid and the entrained flux is the maximum that the upper circulation can support; the subcloud layer is only partially capped with stronger plumes passing through, but with the downward circulation still sufficient to mix the entrained fluxes across that layer. In regime III (identified with cumulus coupling), the subcloud layer transports the maximum entrainment flux it can support (it is effectively capped, behaving as a dry convective layer), while the upper circulation is weakened by a drop in cloud fraction over at least part of the depth of the layer (leading to the appearance of column clouds).

Beginning in regime III and increasing the surface sensible heat flux, for example, augments the lower circulation relative to the upper and a transitional period follows in which the subcloud layer heats and dries faster than the upper layer, reducing the column cloud depth until equilibrium is restored. A big enough change eliminates the column cloud and drives the system into II where the upper circulation has trouble keeping up with the lower; the lower circulation overentrains relative to the upper, weakening the capping of the lower layer until equilibrium is restored with the stronger subcloud plumes exiting the layer. With a large enough increase in the surface sensible heat flux the circulation will become organized more into a full-layer circulation and the regime I picture becomes more appropriate.

In LL02 the cloud-top entrainment in the quasi-steady cumulus regime (III) was predicted by applying (18) to the subcloud circulation, and found in good agreement with LES results. We now apply (18) to the upper circulation, to check whether using R̂ to determine w′θ^′_υNE for a partly cloudy circulation gives an entrainment efficiency consistent with our previous results. An advantage of the entrainment prediction based on the cloud circulation is that it could be used even if the upper and lower circulations had not yet equilibrated, in which case the prediction based on the subcloud circulation would not be correct. The predicted versus LES-measured results for the entrainment dry buoyancy flux, D_i, normalized by the surface flux, D₀, are shown in Fig. 5 for the entire regime II and III datasets from LL02. The details of the application of (18), assumed entrainment efficiency (η = 0.55), and plot format are all as in LL02; the R̂ used is that measured from the LES results. The agreement is generally quite good.

The data points with the poorest agreement are from a single LES run, R8, at different times. In this case a vertically uniform large-scale advective cooling term was included so that the cloud thickness grew in time even though the mean cloud-top conditions were close to the nominal buoyancy-reversal threshold. Within the bulk of the cloud layer there is little difference between w′θ^′_υ and w′θ^′_υNE as currently computed, so that the perceived entrainment cost is largely contributed from the regions around cloud base and cloud top. There are probably local regions of buoyancy reversal in this simulation (where entrainment would drive, rather than inhibit, the cloud-layer circulation) that should probably be included in the estimate of the total energy input available to drive the circulation, ∫ w′θ^′_υNE. This would tend to improve the predictions in this case, though we do not know how to include this contribution quantitatively; this effect would be less important or absent over the rest of the dataset where the upper cloud layer is thin or broken, or the cloud-top conditions are not close to the buoyancy-reversal threshold.

5. Concluding remarks

In this work an approximate formulation for the horizontal mean buoyancy flux within a partly cloudy boundary layer has been developed in terms of the liquid potential temperature and total water fluxes, requiring only the mean θ_l and q_t profiles as additional information. Comparison with a broad range of LES results has shown that the formulation does a good job representing the buoyancy flux, bridging between the limits of layers that are well characterized by joint-Gaussian pdfs and cumulus-coupled layers with strongly correlated clouds and updrafts and very low cloud fractions. The formulation has the advantage of not requiring knowledge of less accessible quantities such as the mean liquid water, cloud fraction, saturation variance, or vertical velocity skewness, as in related formulations (e.g., Bechtold and Siebesma 1998; Cuijpers and Bechtold 1995; Lewellen and Yoh 1993).

The formulation performs particularly well for transitional layers in between uniform stratocumulus and well-developed shallow cumulus. The behavior in such cases (e.g., cumulus under stratocumulus) differs from that predicted by the formulation of Bechtold and Siebesma (1998), which has been empirically fit to the limits of very high or very low cloud fraction. In the present formulation there is not a one-to-one correspondence between the effective cloud fraction R̂ and the mean cloud fraction; the underlying dynamical regime is important and is reflected in the structure of the θ_l and q_t profiles. This conclusion is supported by the LES results. For example, Fig. 9 shows a scatterplot of LES results for R̂ versus mean partial cloud fraction (pcf). The points are taken at each vertical grid level from mean profiles from the full LL02 simulation set plus the ATEX, BOMEX, EUROCS, and ARM cases (including a full sequence of 1-h averages through the diurnal cycles of the latter two cases). The solid line is the prediction from Bechtold and Siebesma (1998; adapted from their Figs. 4 and 6). The scatterplot does not support a one-to-one correspondence between R̂ and pcf. Both the cumulus (low pcf) and Gaussian (R̂ = pcf) branches considered by Bechtold and Siebesma (1998) are well populated, but so are other regions. In the binormal model of Lewellen and Yoh (1993) R̂ and pcf are not predicted to be in one-to-one correspondence; each is a function of s/σ_s [as in Bechtold and Siebesma (1998)] but also functions of skewness. The predicted R̂ versus pcf curve shifts up and to the left with increasing skewness. For example, the dashed line in Fig. 9 is the binormal model prediction using skewness (of both s and w) = 3, while the skewness = 0 prediction is the R̂ = pcf line. In the LES results, we find that there is a general trend toward increasing R̂ with increasing w-skewness (at fixed pcf), particularly within an individual simulation, but even for fixed w- skewness R̂ and pcf are not found to be in one-to-one correspondence. Besides s/σ_s and w-skewness other parameters, such as differing values of s-skewness, must come into play; in a given region of the R̂ versus pcf plot, the w-skewness can vary significantly. These results (following from a more detailed analysis of the LES results) can be seen in a rough way in Fig. 9, where we have used different symbols to indicate whether the w-skewness at that point is less than 1, between 1 and 3, or greater than 3.

Even though the comparison between LES results and the R̂ formulation of section 2 is generally very good, there are possibilities for improvements or additions to the basic formulation that may be worth exploring. In particular, motivated by a simple mass flux picture, the leading effects of the lateral mixing into updrafts could be included and/or a small number of different parcels could be sampled. This could lead to improved treatment of cumulus layers with weaker, more vertically smeared capping inversions.

This formulation for the buoyancy flux was used to extend our previous work on entrainment and quasi- steady structure in cloudy boundary layers (LL02): first using it to apply a large-eddy entrainment efficiency to predict entrainment rates for partly cloudy cumulus layers; and second, using this result to predict the equilibrium structure (particularly the θ_l, q_t, and flux profiles) for those layers. Again the predictions agree well with the LES results.

The success of these predictions for many different cases supports the underlying picture advocated in LL02 of the physical balances responsible for the quasi-steady behavior in shallow cumulus layers. We emphasize in particular how much is being predicted from a minimal set of physical inputs, without invoking a posteriori features of the LES results such as the buoyancy flux, cloud fraction, entrainment rate, or temperature and moisture differences between the cloud and subcloud layers. This evidence for a large-scale balance constraining the buoyancy flux may in part explain the results from LES studies (e.g., Brown 1999) showing an insensitivity to resolution and subgrid model of the mean flux profiles for quasi-steady cumulus simulations even though the structure of the individual cumulus plumes shows a clear sensitivity. Our primary focus on quasi-steady boundary layers here is based on the belief that an understanding of the physical balances involved in quasi-steady layers is a critical ingredient required to construct simple models that can more robustly treat the evolution between different boundary layer regimes.

Finally, we would caution in implementing the suggested formulation for the buoyancy flux within a boundary layer model that the appropriate physics must be included in that model to ensure a good representation of θ_l and q_t in different dynamical regimes or else the overall results may not be satisfactory. In particular, as has been emphasized in the literature (e.g., Bretherton and Wyant 1997; LL02) and as our quasi- steady predictions here demonstrate, cloud-top entrainment heating and drying plays a critical role in setting up the “decoupled” θ_l and q_t profiles characteristic of cumulus-coupled dynamics.