## 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.

^{1}may be written as

*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}

*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:

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

*s*

*q*

_{t}

*q*

_{s}

*aq*

^{′}

_{t}

*bθ*

^{′}

_{l}

*c,*

*q*

_{s}is the saturation mixing ratio and

*a,*

*b,*and

*c*are thermodynamic coefficients given in appendix A.

*R̂*is given by,

Modeling of *w*′*θ*^{′}_{υ}*w*′*q*^{′}_{t}*w*′*θ*^{′}_{l}*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*_{N}*N* 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*^{′}_{υ}_{i}*R̂*(*z*_{i}) here is his parameter *α,* which he estimates from LES results under the assumption that a particular parameterization for *w*′*θ*^{′}_{υ}_{i}

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}

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}*q*_{t}*θ*_{l}*w*′*q*^{′}_{t}*w*′*θ*^{′}_{l}

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.

*R̂,*but do so in a different fashion. In general, for the flux of any scalar quantity Ψ, we have

*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}

*θ*

_{l}

*q*

^{sc}

_{t}

*θ*

^{sc}

_{l}

*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}

*q*

^{sc}

_{t}

*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}

*s*

^{b}are not equal to the horizontal means

*q*

_{l}

*s*

*W*arising from the linear approximation to the saturation formula, giving

*a,*

*b,*

*α,*

*β,*and

*θ*

^{b}

_{υ}

*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}*w*′*q*^{′}_{t}*w*′*s*′*w*′*s*′

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}*q*^{u}_{t}*θ*^{u}_{l}*q*^{sc}_{t}*θ*^{sc}_{l}*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}*θ*^{sc}_{l}*q*_{t} and *θ*_{l}*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}*σ*_{〈w′q′t〉}*w*′*q*^{′}_{t}*σ*_{〈w′θ′υ〉}*w*′*θ*^{′}_{υ}*R̂* at that height were nearly an order of magnitude smaller, *σ*_{〈R̂〉}/*R̂*

*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}

*σ*

^{sc}

_{qt}

*σ*

^{sc}

_{θl}

*q*

_{t},

*θ*

_{l}

*σ*

^{b}

_{qt}

*σ*

^{b}

_{θl}

*q*

^{sc}

_{l}

*q*

^{b}

_{l}

*q*

^{b}

_{l,SDM}

*β*

*q*

^{sc}

_{l,SDM}

*q*

^{sc}

_{l}

*q*

^{b}

_{l,SDM}

*q*

^{b}

_{l}

*R̂*are secondary, so rough estimates for the standard deviations of the saturation deficits,

*σ*

^{sc}

_{s}

*σ*

^{b}

_{s}

*θ*

_{l}and

*q*

_{t}differences based on the surface fluxes to estimate these variances,

*κ*is a proportionality constant, and

*w*

^{*}

_{sc}

*w** are convective velocity scales for the subcloud circulation [e.g., (

*g*/

*θ*

_{0}

^{\[mu4\]zt\[md4\]}

_{0}

*w*′

*θ*

^{′}

_{υ}

^{1/3}with

*z*

_{t}the top of the subcloud layer] and full layer, respectively. Arguably

*σ*

^{b}

_{s}

## 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}*w*′*q*^{′}_{t}

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}*θ*^{sc}_{l}*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}

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}*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}*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}*q*_{t}*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

*w*′

*θ*

^{′}

_{υ}

*w*′

*θ*

^{′}

_{υ}

_{NE}

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*^{′}_{υ}_{NE}*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*^{′}_{l}_{NE}*w*′*q*^{′}_{t}_{NE}*z*_{i}. LL98 concentrated on layers with approximately horizontally uniform cloud layers so that *w*′*θ*^{′}_{υ}_{NE}*w*′*θ*^{′}_{l}_{NE}*w*′*q*^{′}_{t}_{NE}*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*^{′}_{υ}_{NE}*D*_{i}, normalized by the surface flux, *D*_{0}, 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}*w*′*θ*^{′}_{υ}_{NE}*w*′*θ*^{′}_{υ}_{NE}

### b. Predicting the quasi-steady-state behavior of a shallow cumulus-coupled layer

Consider the problem of predicting the quasi-steady- state behavior of a cloudy boundary layer given only a minimal set of input parameters: the surface fluxes (*w*′*θ*^{′}_{l}_{0}*w*′*q*^{′}_{t}_{0}*θ*_{l}*q*_{t}*z*_{i}; mean subcloud *θ*_{l} and *q*_{t} values; and specified vertical profiles of sources of *θ*_{l} and *q*_{t} within the layer, if present (e.g., radiation or precipitation fluxes or large-scale advective forcing). In LL02, predictions were formulated for the cloud-top entrainment fluxes and basic circulation type as functions of these input parameters. In a quasi-steady boundary layer this suf-fices to determine the conserved fluxes *w*′*θ*^{′}_{l}*w*′*q*^{′}_{t}*θ*_{l} and *q*_{t} profiles within the layer are well mixed, the cloud layer horizontally uniform, and the buoyancy flux determined by the simple limits (2) or (4). For a cumulus-coupled layer, however, these approximations no longer hold; a critical component of the layer dynamics is not yet determined.

In LL02 it was argued that in the cumulus-coupled regime the *θ*_{l} and *q*_{t} profiles will naturally dynamically adjust, altering the cloud structure and hence the buoyancy flux, until the entrainment fluxes at cloud top driven by the upper circulation are in equilibrium with what the subcloud circulation can support. Consider, for example, an initially horizontally uniform cloud cover, with *θ*_{l} and *q*_{t} well mixed, in a boundary layer that the analysis of LL02 predicts will be cumulus coupled if in a quasi-steady state. In this case the upper, cloudy circulation can support a larger entrainment flux across cloud top than the lower, subcloud circulation can transport through the subcloud layer. Consequently, the upper layer warms and dries relative to the subcloud layer and the cloud base becomes nonuniform (a raised general base with narrow column clouds below). The change in cloud structure reduces the buoyancy flux driving the upper circulation and thereby the cloud-top entrainment flux. This development proceeds until the entrainment flux driven by the upper circulation is reduced enough to be in equilibrium with what the subcloud circulation can support.

This picture suggests that for a quasi-steady cumulus- coupled layer the equilibration of the upper- and lower- layer entrainment predictions should be used in predicting the buoyancy flux profile in terms of the chosen input parameters. Further assumptions are required since equating the two entrainment predictions provides only a single constraint, insufficient for determining a vertical profile. We proceed by postulating a one-parameter family of solutions within the cloud layer, then use the entrainment constraint to choose from among this family. The hope is that if the key dynamical feedbacks are as postulated, then the solution will perhaps be forgiving of the errors necessarily associated with this level of approximation.

As in LL02 the entrainment prediction from the subcloud circulation in the cumulus-coupled regime is used to predict the entrainment fluxes and thereby the quasi- steady *w*′*θ*^{′}_{l}*w*′*q*^{′}_{t}*R̂,* these could be used to obtain *w*′*θ*^{′}_{υ}*R̂* as suggested in section 2 requires *θ*_{l}*q*_{t}*z*_{t}, in the “transition layer”) and above *z*_{i}. Empirically, we find that across the bulk of the cloud layer *θ*_{l} is often approximately linear in *z* (though this admittedly ignores meaningful structure one might hope to incorporate). For simplicity, then, we consider a one-parameter family of *θ*_{l} profiles which is linear across the bulk of the cloud layer, with quadratic interpolations above and below (in the inversion and transition layers) to match onto the known profile above *z*_{i} and well below *z*_{t}. The *q*_{t} profile is then chosen to be consistent with the *θ*_{l} profile and predicted *w*′*θ*^{′}_{l}*w*′*q*^{′}_{t}*θ*_{l} and *q*_{t} profiles are given explicitly in appendix C.

The one-parameter family of solutions is sampled in Fig. 6 for case NHb from LL02 (cf. Fig. 2a) along with the member of that family satisfying the upper-circulation entrainment constraint, and the LES profiles. The successful prediction of *w*′*θ*^{′}_{l}*w*′*q*^{′}_{t}*θ*_{l} profiles chosen, the predictions for the *θ*_{l}*q*_{t}*R̂,* and *w*′*θ*^{′}_{υ}

Repeating this exercise for all of the cumulus-coupled cases in LL02 gives comparable results, with the exception of case R8 discussed above. Figure 7 shows the *R̂* and *w*′*θ*^{′}_{υ}

Figure 8 shows the analogous predictions for the ATEX and BOMEX GCSS-WG1 cases. In these simulations there were additional contributions from large- scale advective forcing, radiation, and subsidence. These effects were included in predicting *w*′*θ*^{′}_{l}*w*′*q*^{′}_{t}

Analogous predictions based on a piecewise linear one-parameter family of *R̂* profiles generally matched the LES results even better (particularly for the ATEX and BOMEX cases), but without providing any prediction for the *θ*_{l} and *q*_{t} profiles.

## 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}

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}*q*_{t}*θ*_{l} and *q*_{t} profiles characteristic of cumulus-coupled dynamics.

## Acknowledgments

This work was supported by Grant N00014-98-1-0595 from the Office of Naval Research with R. Ferek as technical monitor. We would also like to thank the GCSS Working Group 1 for its influence in stimulating this work, and for the comments of three anonymous reviewers that led to improvements in the presentation.

## REFERENCES

Bechtold, P., and A. P. Siebesma, 1998: Organization and representation of boundary layer clouds.

,*J. Atmos. Sci***55****,**888–895.Bechtold, P., C. Fravalo, and J. P. Pinty, 1992: A model of marine boundary layer cloudiness for mesoscale applications.

,*J. Atmos. Sci***49****,**1723–1744.Bechtold, P., J. W. M. Cuijpers, P. Mascart, and P. Trouilhet, 1995: Modeling of trade wind cumuli with a low-order turbulence model: Toward a unified description of Cu and Sc clouds in meteorological models.

,*J. Atmos. Sci***52****,**455–463.Bougeault, P., 1981: Modeling the trade-wind cumulus boundary layer. Part I: Testing the ensemble cloud relations against numerical data.

,*J. Atmos. Sci***38****,**2414–2428.Bretherton, C. S., and M. C. Wyant, 1997: Moisture transport, lower- tropospheric stability, and decoupling of cloud-topped boundary layers.

,*J. Atmos. Sci***54****,**148–167.Brown, A. R., 1999: The sensitivity of large-eddy simulations of shallow cumulus convection to resolution and subgrid model.

,*Quart. J. Roy. Meteor. Soc***125****,**469–482.Brown, A. R., and Coauthors, 2002: Large-eddy simulation of the diurnal cycle of shallow cumulus convection over land.

,*Quart. J. Roy. Meteor. Soc***128****,**1075–1093.Cuijpers, J. W. M., and P. Bechtold, 1995: A simple parameterization of cloud water related variables for use in boundary layer models.

,*J. Atmos. Sci***52****,**2486–2490.Lewellen, D. C., and W. S. Lewellen, 1998: Large-eddy boundary layer entrainment.

,*J. Atmos. Sci***55****,**2645–2665.Lewellen, D. C., and W. S. Lewellen, 2000: Boundary layer entrainment for different capping conditions. Preprints,

*14th Symp. on Boundary Layers and Turbulence,*Aspen, CO, Amer. Meteor. Soc., 80–83.Lewellen, D. C., and W. S. Lewellen, 2002: Entrainment and decoupling relations for cloudy boundary layers.

,*J. Atmos. Sci***59****,**2966–2986.Lewellen, W. S., and S. Yoh, 1993: Binormal model of ensemble partial cloudiness.

,*J. Atmos. Sci***50****,**1228–1237.Lilly, D. K., 2002: Entrainment into mixed layers. Part II: A new closure.

,*J. Atmos. Sci***59****,**3353–3361.Mellor, G. L., 1977: The Gaussian cloud model relations.

,*J. Atmos. Sci***34****,**3161–3169.Randall, D. A., 1987: Turbulent fluxes of liquid water and buoyancy in partly cloudy layers.

,*J. Atmos. Sci***44****,**850–858.Siebesma, A. P., and Coauthors, 2003: A large-eddy simulation intercomparison study of shallow cumulus convection.

,*J. Atmos. Sci***60****,**1201–1219.Sommeria, G., and J. W. Deardorff, 1977: Subgrid-scale condensation in models of nonprecipitating clouds.

,*J. Atmos. Sci***34****,**344–355.Stage, S., and J. Businger, 1981: A model for entrainment into a cloud-topped marine boundary layer. Part I: Model description and application to a cold-air outbreak episode.

, .*J. Atmos. Sci***38****,**2213–2229.Stevens, B., and Coauthors, 2001: Simulations of trade wind cumuli under a strong inversion.

,*J. Atmos. Sci***58****,**1870–1891.

## APPENDIX A

### Thermodynamic Coefficients

*q*

_{sl}=

*q*

_{s}(

*T*

_{l}) for

*T*

_{l}=

*θ*

_{l}(

*T*

*θ*

*q*

_{sl,T}= ∂

*q*

_{s}/∂

*T*|

_{T=Tl}

*Lq*

_{s}(

*T*

_{l})/(

*R*

_{υ}

*T*

^{2}

_{l}

*L*is the latent heat of vaporization,

*c*

_{p}the specific heat of air at constant pressure, and

*R*

_{υ}the gas constant of water vapor.

## APPENDIX C

### One-Parameter θl and qt Profiles for Quasi-Steady Cumulus Predictions

*θ*

_{l}vertical profiles are defined as

*λ*the single undetermined parameter. Above and below this region, the given form quadratically interpolates to the measured values at

*z*

_{i}and

*z*

_{1}, smoothly matching onto the slope

*λ*at

*z*

_{3}and

*z*

_{2}. The lower height,

*z*

_{1}, was chosen sufficiently below the top of the subcloud layer,

*z*

_{t}, as to be in the “mixed layer” where “universal” convective boundary layer behavior is found (cf. Fig. 6 in LL02).

*q*

_{t}profiles found from a crude mass-flux approximation is

*w*′

*q*

^{′}

_{t}

*z*)

_{p}and

*w*′

*θ*

^{′}

_{l}

*z*)

_{p}are the

*q*

_{t}and

*θ*

_{l}fluxes predicted in the quasi-steady approximation using the cloud-top entrainment prediction deduced from the subcloud circulation as described in the main text.

Mean vertical profiles from sample cloudy boundary layers considered in LL02, cases (a) NHb, (b) T0, (c) N2, and (d) T5. Lines and variables as in Fig. 1. Note changes in scales between plots

Citation: Journal of the Atmospheric Sciences 61, 10; 10.1175/1520-0469(2004)061<1147:BFMFCB>2.0.CO;2

Mean vertical profiles from sample cloudy boundary layers considered in LL02, cases (a) NHb, (b) T0, (c) N2, and (d) T5. Lines and variables as in Fig. 1. Note changes in scales between plots

Citation: Journal of the Atmospheric Sciences 61, 10; 10.1175/1520-0469(2004)061<1147:BFMFCB>2.0.CO;2

Mean vertical profiles from sample cloudy boundary layers considered in LL02, cases (a) NHb, (b) T0, (c) N2, and (d) T5. Lines and variables as in Fig. 1. Note changes in scales between plots

Citation: Journal of the Atmospheric Sciences 61, 10; 10.1175/1520-0469(2004)061<1147:BFMFCB>2.0.CO;2

Mean vertical profiles from (a) ATEX and (b) BOMEX GCSS-WG1 cases. Lines and variables as in Fig. 1.

Mean vertical profiles from (a) ATEX and (b) BOMEX GCSS-WG1 cases. Lines and variables as in Fig. 1.

Mean vertical profiles from (a) ATEX and (b) BOMEX GCSS-WG1 cases. Lines and variables as in Fig. 1.

Mean vertical profiles from GCSS-WG1 cases: (a) EUROCS diurnal simulation of stratocumulus at simulation hour 14 [1400 local time (LT)] and (b) at hour 26 (0200 LT); (c) ARM case at simulation hour 8 (1330 LT) and (d) at hour 12 (1730 LT). Lines and variables as in Fig. 1. Note changes in scales between plots

Mean vertical profiles from GCSS-WG1 cases: (a) EUROCS diurnal simulation of stratocumulus at simulation hour 14 [1400 local time (LT)] and (b) at hour 26 (0200 LT); (c) ARM case at simulation hour 8 (1330 LT) and (d) at hour 12 (1730 LT). Lines and variables as in Fig. 1. Note changes in scales between plots

Mean vertical profiles from GCSS-WG1 cases: (a) EUROCS diurnal simulation of stratocumulus at simulation hour 14 [1400 local time (LT)] and (b) at hour 26 (0200 LT); (c) ARM case at simulation hour 8 (1330 LT) and (d) at hour 12 (1730 LT). Lines and variables as in Fig. 1. Note changes in scales between plots

Normalized LES-measured entrainment flux vs that predicted based on the cloud-layer circulation dynamics. Points from the regime II (pluses) and regime III (large and small stars) datasets from LL02. Small stars are from case R8 discussed in the text.

Normalized LES-measured entrainment flux vs that predicted based on the cloud-layer circulation dynamics. Points from the regime II (pluses) and regime III (large and small stars) datasets from LL02. Small stars are from case R8 discussed in the text.

Normalized LES-measured entrainment flux vs that predicted based on the cloud-layer circulation dynamics. Points from the regime II (pluses) and regime III (large and small stars) datasets from LL02. Small stars are from case R8 discussed in the text.

Sample mean vertical profiles for case NHb from LL02. Thick solid lines are LES results; thin solid lines are sample members of the one-parameter family of approximate solutions discussed in the text; thick short dashed lines are the predicted members of this family satisfying the entrainment constraints, as discussed in the text.

Sample mean vertical profiles for case NHb from LL02. Thick solid lines are LES results; thin solid lines are sample members of the one-parameter family of approximate solutions discussed in the text; thick short dashed lines are the predicted members of this family satisfying the entrainment constraints, as discussed in the text.

Sample mean vertical profiles for case NHb from LL02. Thick solid lines are LES results; thin solid lines are sample members of the one-parameter family of approximate solutions discussed in the text; thick short dashed lines are the predicted members of this family satisfying the entrainment constraints, as discussed in the text.

As in Fig. 1 but with the predictions (long dashes) made using the entrainment constraints as discussed in the text.

As in Fig. 1 but with the predictions (long dashes) made using the entrainment constraints as discussed in the text.

As in Fig. 1 but with the predictions (long dashes) made using the entrainment constraints as discussed in the text.

As in Fig. 3 but with the predictions (long dashes) made using the entrainment constraints as discussed in the text

As in Fig. 3 but with the predictions (long dashes) made using the entrainment constraints as discussed in the text

As in Fig. 3 but with the predictions (long dashes) made using the entrainment constraints as discussed in the text

Scatterplot of LES results for *R̂* vs partial cloud fraction with the different symbols indicating the vertical velocity skewness interval (“−” if less than 1, “o” between 1 and 3, “+” if greater then 3). The solid line is the prediction from Bechtold and Siebesma (1998), the dashed line the prediction of Lewellen and Yoh (1993) for skewness = 3.

Scatterplot of LES results for *R̂* vs partial cloud fraction with the different symbols indicating the vertical velocity skewness interval (“−” if less than 1, “o” between 1 and 3, “+” if greater then 3). The solid line is the prediction from Bechtold and Siebesma (1998), the dashed line the prediction of Lewellen and Yoh (1993) for skewness = 3.

Scatterplot of LES results for *R̂* vs partial cloud fraction with the different symbols indicating the vertical velocity skewness interval (“−” if less than 1, “o” between 1 and 3, “+” if greater then 3). The solid line is the prediction from Bechtold and Siebesma (1998), the dashed line the prediction of Lewellen and Yoh (1993) for skewness = 3.

APPENDIX B Table of Simulations

^{1}

Throughout the text we refer to the virtual potential temperature flux, *w*′*θ*′_{υ}