## 1. Introduction

It has been recognized that microphysics may play an important role in regulating dynamics of stratocumulus clouds and their impact on the weather from local to climate scales. These roles include the so-called “indirect aerosol effects” as discussed by Twomey (1977) and Albrecht (1989), as well as the precipitation effects that may change the stability in stratocumulus cloud-topped boundary layers (e.g., Paluch and Lenschow 1991; Wang and Wang 1994; Stevens et al. 1998b). For this reason, efforts have increasingly focused on representing detailed microphysics in a dynamic framework of stratocumulus clouds to understand physical processes and to predict the impacts on large-scale meteorological fields. Large eddy simulation (LES) and turbulence closure models are two commonly used dynamic frameworks for stratocumulus clouds. For example, Feingold et al. (1994) and Bott et al. (1996), respectively, implemented explicit bin cloud microphysics representations in an LES and turbulence closure model.

A successful dynamical–microphysical model must include some basic coupling between turbulence dynamics and microphysical processes. It is well known that vertical motion strongly affects cloud droplet activation, condensation, and liquid water transport (e.g., Stevens et al. 1998a). An excess of water vapor is produced at the cloud base by turbulent updrafts and initiates droplet activation. In the turbulent updrafts, the condensational latent heat released from rapid droplet growth further enhances upward motion; while in the downdrafts, evaporation in the subsaturated environment strengthens the downward motion. It is clear that in this process condensation/evaporation (CE) is strongly controlled by turbulent vertical motion and liquid water is not conserved during turbulent ascent and descent. In the LES framework, this coupling is relatively straightforward, because the stochastic large eddies that contain most of the energy are explicitly resolved. Therefore, a bin-resolved cloud microphysical model that generates supersaturation and activates and grows droplets based on resolved vertical motions can be directly implemented in an LES model without major parameterization of the turbulence–microphysics coupling (e.g., Feingold et al. 1994; Kogan et al. 1995).

Microphysical representation in a turbulence-parameterized model is significantly more complex; because the turbulence dynamics in such models is parameterized, the basic turbulence–microphysics coupling must be explicitly parameterized too. A key question in this parameterization is how the ensemble mean CE rate and the turbulent liquid water (and droplet number) flux should be related to the mean and turbulence fields. Stevens et al. (1998a) discussed this issue in detail and pointed out that the interaction between turbulence and microphysics is critical in developing an explicit bin microphysical model for a turbulence parameterization.

Turbulence–microphysics interaction has been a subject of many studies (e.g., Telford and Chai 1980; Cooper 1989). These studies have, however, primarily focused on turbulence effects on the broadening of the cloud droplet size spectra, rather than on the turbulence–microphysical statistical quantities and their parameterizations. Khvorostyanov and Curry (1999) developed a stochastic condensation theory, which includes a treatment of the covariance of velocity and liquid water variables based on the turbulence statistical theory and characteristics of the turbulence and the droplet size spectra.

A coupled LES–bin-microphysical (LES-BM) model explicitly resolves both the large eddy turbulence field and the associated cloud droplet spectrum, and thus can be used to provide detailed and quantitative information on some basic turbulence–microphysics coupling issues. Since our focus is on statistical quantities, the turbulence budget analysis is particularly useful because this type of analysis (e.g., Moeng and Wyngaard 1986) may provide not only the physical understanding, but also some implications for the parameterization approaches.

In this work, the budgets of four liquid water variables will be derived in terms of bulk cloud mean and turbulence fields. These variables are mean liquid water content, turbulent liquid water flux, mean cloud droplet number concentration, and the number density flux, denoted by *q*_{l}*w*′*q*^{′}_{l}*N**w*′*N*′

## 2. LES-BM model

The LES-BM model used in this study is very similar to that of Stevens et al. (1996a). A detailed description and a comprehensive evaluation and description of the LES code is given by Stevens et al. (1999). Of relevance to this study is the fact that monotone operators are used for scalar advection (following Zalesak 1979) and Deardorff's prognostic turbulent kinetic energy (TKE) technique is used for the subgrid-scale (SGS) model (Deardorff 1980). At the surface, Monin–Obukhov similarity theory provides the surface fluxes based on assumed sea surface temperature and the predicted winds, temperature, and moisture at the first level. At the top of the domain, a constant gradient condition is applied to all scalar variables and a free slip for momentum. The lateral boundary conditions are periodic.

The bin microphysical model was developed by Feingold et al. (1994), and was also described in detail in Stevens et al. (1996a). Briefly, the droplet size spectrum is divided into 25 size bins in which both the droplet mass and number concentration are predicted. The diagnostic activation scheme is based on the cumulative method discussed by Clark (1973) where the number of droplets in the first bin is incremented by the difference between the number of cloud condensation nuclei (CCN) that should be activated at the calculated supersaturation, and the local droplet number. The aerosol spectrum is assumed to follow a constant lognormal distribution with a total aerosol concentration *N*_{a}.

Our focus is on the basic interaction among turbulence, CE rates, and liquid water fluxes. Particularly, response of the turbulence to the different CE timescales will be studied. Therefore, the processes of droplet coalescence–collection and sedimentation (“drizzle processes”) are not represented in the first three simulations so as to isolate the targeted coupled system and facilitate comparisons. Drizzle is included in two other simulations for evaluation of our results.

The four-stream radiation parameterization developed by Fu et al. (1995) is used to calculate radiative cooling rates. To better focus on our objective, we purposely disconnect the link between the predicted droplet spectrum and the radiation by specifying the droplet number mixing ratio at 100 mg^{−1} for the longwave, and by not simulating shortwave radiation. For the stratocumulus case here, the model uses 60 × 60 grid points with uniform spacing Δ*x* = Δ*y* = 50 m in the horizontal; there are 76 grid points in the vertical with a minimum spacing of 5 m within the inversion and 25 m below the inversion to span a 3 km × 3 km × 2.1 km domain. The time step is 1.5 s.

## 3. Budget equations

*q*

_{l}and

*N*(instead of the droplet mass and number in each bin, or the radius), although the simulations are performed with the full bin microphysical model. The starting point for the budgets is the governing equation for the time rate of change of the droplet number density due to CE (e.g., Clark 1973):

*n*(

*r*) is the droplet number density with radius

*r,*

*n*(

*r*)

*dr*the droplet number per unit air mass in the interval (

*r,*

*r*+

*dr*),

*G*(

*T,*

*p*) a function of temperature and pressure, and

*S*the surplus vapor (

*S*=

*q*

_{υ}−

*q*

_{s}). By definition,

*N*=

^{∞}

_{0}

*n*(

*r*)

*dr.*The third moment integration gives the local instantaneous CE rate

*R*is the integrated radius [

^{∞}

_{0}

*rn*(

*r*)

*dr*], and

*ρ*

_{l}is the liquid water density. Applying the Reynolds averaging operation gives the following ensemble mean CE rate:

*G*

*T,*

*p*) ≅

*G*(

*T*

*p*

*G*and others are neglected due to its relatively weak dependence on

*T*and

*p,*which has been confirmed in the LES-BM model simulations (see appendix A). We need to express

*R*

*R*′ in terms of

*q*

_{l},

*N*

*q*

^{′}

_{l}

*N*′, which requires a specific distribution. Feingold et al. (1998) successfully used lognormal distribution functions to parameterize the droplet and rain spectra for stratocumulus modeling. Based on this distribution function, the integral radius can be determined by

*a*is the geometric standard deviation and is specified as a constant. If (4) is expanded at (

*q*

_{l},

*N*

*q*

_{l}

*b*≡ 4

*π*

*G*

*ρ*

_{l}, the double prime “ ″ ” represents SGS fluctuation, and angle brackets are a grid-volume averaging operator. The first two terms are the large-scale subsidence (

*L*) and divergence of the turbulent flux (

*T*), respectively. The four terms in the square brackets—that is, mean saturation, two turbulence covariance terms, and activation—are denoted by the numbers 1–4, respectively. The liquid water flux budget, derived in appendix B, is written as

*T*); mean gradient production (

*G*); buoyancy production (

*B*); pressure perturbation (

*P*); SGS term (

*SGS*); and microphysical terms (in the square brackets), which are water vapor surplus flux, second-order covariance, third-order covariance, and activation flux, denoted by the corresponding numbers under (7).

*N*

*w*′

*N*′

*A*

_{N}and

*E*

_{N}are droplet number activation and evaporation rates. Due to the complexity of the activation and evaporation formulations, no attempts have been made to explicitly express

*w*′

*A*

^{′}

_{N}

*w*′

*E*

^{′}

_{N}

*A*” and “

*E*” under the equations represent activation and evaporation related processes, respectively, in the budgets, while other letters have similar meaning to those for

*q*

_{l}

*w*′

*q*

^{′}

_{l}

## 4. Stratocumulus cloud case

The San Nicolas Island sounding composited by Albrecht et al. (1995) from observations taken during the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment (FIRE) is used as a basis for the model initialization. A LES simulation with the sounding by Wang and Stevens (2000) showed that the modeled liquid water mixing ratio is relatively large; the maximum is 0.61 g kg^{−1} compared with 0.1∼0.4 g kg^{−1} obtained by many other observations (e.g., Paluch and Lenschow 1991). Therefore, we intentionally reduce the moisture above the inversion from 6.5 g kg^{−1} to 4.5 g kg^{−1}, which is more in line with other observations (e.g., Paluch and Lenschow 1991; Lilly 1968), for the first four simulations, but still use 6.5 g kg^{−1} for the fifth simulation that includes drizzle. In addition, the constant sea surface temperature and the large-scale divergence is specified to be 288.8 K (Albrecht et al. 1995) and 6 × 10^{−6} s^{−1}, respectively. We emphasize that the specification of these large-scale conditions does not have any qualitative impact on our results as long as a radiatively driven stratocumulus-topped boundary layer is simulated. The model is first run for 3 h with the saturation adjustment procedure, then run with the bin-microphysical model for the next 3 h until a quasi steady state is achieved. All the turbulence and microphysics statistics are calculated and averaged over the last hour of the simulation with a sample interval of 9 s. Five simulations (S1–S5) are performed and the procedures are shown in Table 1.

## 5. Cloud water budgets

For the budget study, the results of S1 are analyzed. First, the general characteristics of simulated clouds are presented, and then the four cloud water budgets [(6)–(9)] are discussed. A classic well-mixed cloud-topped boundary layer is simulated with some mean and turbulence statistics displayed in Figs. 1a–c, which are consistent with our expectations and past experience. Figures 1d–f show profiles of the five cloud variables (*q*_{l}*w*′*q*^{′}_{l}*N**w*′*N*′*r*_{e}), which are in general agreement with other modeling studies (e.g., Stevens et al. 1996a) and observations (e.g., Nicholls 1984). Nonetheless, several important features are worth mentioning. The number concentration *N**q*_{l}*r*_{e} with *z*^{1/3}. The liquid water flux *w*′*q*^{′}_{l}*w*′*N*′*w*′*q*^{′}_{l}*w*′*N*′*q*_{l} and *N* in the downdrafts than those in the updrafts at those levels.

### a. The q_{l} budget

An excellent balance exists between the flux divergence (including both resolved and SGS fluxes) and the condensation terms, while the subsidence advection term is negligibly small (Fig. 2a). Between 500 and 580 m near the cloud base (∼530 m), the net evaporation is compensated by the flux convergence as shown in Fig. 2a. This evaporation and the negative liquid water flux (Fig. 1e) are largely a result of turbulent transport of liquid water in the downdrafts. The net condensation is balanced by the flux divergence between 580 and 640 m. In the middle of the cloud layer (700 m), both the flux divergence and CE terms are small, indicating that there are no source and sink terms at those levels. Just below the cloud top (between 720 and 780 m) both the divergence and the net condensation are significantly increased by radiative cooling. Finally, at the cloud top, the turbulent eddies transport the droplets to the warmer and drier inversion where significant evaporation occurs.

Figure 2b shows various terms of the CE term in (6). The mean saturation term is almost always negative from the cloud base up to 750 m except at the level of 600 m where the mean condition is saturated due to considerably supersaturated updrafts. The negative *S*

In contrast to the mean saturation term, contributions from the two turbulence covariance terms are generally positive; a larger *S* tends to result in more condensation/activation or less evaporation, leading to positive covariance *N*′*S*′*q*^{′}_{l}*S*′*N*′*S*′*R*′*S*′

Recall from (3) that these two turbulence terms represent the correlation between *R* and *S* and therefore, *R*′*S*′*R*′*S*′ is almost always positive. This means that the effects of the turbulence are to enhance the mean CE rate by increasing condensation in the updrafts (usually *S* > 0) as well as decreasing the evaporation in the downdrafts (usually *S* < 0). Therefore, without the turbulence covariance terms, the mean CE rate not only ignores the condensation in the updrafts, but also overestimates the evaporation in the downdrafts. This is particularly true at cloud base and cloud top, as shown in Fig. 2b. This analysis shows that the mean supersaturation should not be used as the *only* factor to drive the droplet growth and the covariance *R*′*S*′

### b. The w′q^{′}_{l} budget

^{′}

_{l}

The most striking feature in the *w*′*q*^{′}_{l}*G*) and the microphysical terms in (7). The mean gradient term represents the traditional downgradient transport (or the diffusive mixing). Its large negative values are a direct result of the large gradient of *q*_{l}*w* and CE perturbations. Therefore, it is condensation (evaporation) in the updrafts (downdrafts) that results in positive (negative) *q*_{l} fluctuations and leads to a mostly positive liquid water flux for both updrafts and downdrafts.

The buoyancy term is positive below 760 m in the cloud layer where the buoyancy fluctuations are positively correlated with vertical velocity, and it becomes strongly negative at the cloud top due to the entrainment of more buoyant and drier air. The pressure correlation term is negative below 760 m and becomes strongly positive in the inversion, therefore it tends to cancel the contribution of the buoyancy term. This is consistent with the general description of the corresponding terms in other scalar flux budgets (Moeng 1986).

The first three components of the microphysical part in (7) are presented in Fig. 3b. The fourth is negligibly small due to the small radius (∼1.5 *μ*m) at which droplets are activated, and thus is not included. The contribution of the *w*′*S*′*w*′*S*′*S* and *w* in the updrafts (downdrafts) at those levels as discussed by many authors (e.g., Stevens et al. 1996a). At the very top levels of the cloud layer, the warmer and drier inversion air (*S*′ < 0) is entrained into the downdrafts (*w*′ < 0), leading to the second maximum of the *w*′*S*′

*w*′

*S*′

*w*′

*S*′

*S*tendency equation (e.g., Clark 1973):

*γ*=

*L*

*c*

^{−1}

_{p}

*T*

*θ*

^{−1}∂

*q*

_{s}/∂

*T*and is considered a known function of the ensemble mean atmospheric pressure and temperature. Assuming

*R*in (10) remains the ensemble mean value, multiplying by

*w*′, and performing Reynolds averaging, we have

*dq*

_{sa}/

*dz*is the gradient of saturation water vapor mixing ratio following the mean moist adiabat. Note that (11) only applies to the levels where all the grid points are cloudy. Equation (11) states that

*w*′

*S*′

*w*and the moist adiabatic CE rate (−

*wd*

*q*

_{sa}/

*dz*). The estimated

*w*′

*S*′

*w*′

*S*′

*q*

_{l}

*w*′

*S*′

*w*′

*S*′

*R*

*N*

This analysis shows that the *w*′*S*′*w*′*q*^{′}_{l}*w*′*q*^{′}_{l}

### c. The N budget

As in the case of the *q*_{l}*T*) and microphysical terms (*A* + *E*) in (8) are closely balanced in the *N*

Figure 4b shows the activation (*A*) and the evaporation (*E*) components in the microphysical term. Below cloud base, the evaporation is likely to have resulted from the downward transport of small cloud droplets, which leads to the *w*′*N*′

### d. The w′N′ budget

As shown in Fig. 5a, the gradient and the microphysical terms again have opposite signs and are dominant in the budget in the lower part of the cloud. In the inversion layer, the buoyancy and pressure perturbation terms are in close balance. Figure 5b shows that both components of the microphysical term, the activation and evaporation fluxes as defined by “*A*” and “*E*” terms in (9), are generally positive, because the activation is most likely to occur in the updrafts and the evaporation in the downdrafts. Similar to the budget of *w*′*q*^{′}_{l}*w*′*N*′*w*′*N*′*N*

## 6. Parameterizations

The above results clearly demonstrate the importance of turbulence–microphysics interaction in defining the ensemble mean CE rate and various liquid water–related fluxes. Therefore, any attempt to parameterize these variables must include both turbulence and microphysical statistics.

### a. Liquid water flux w′q^{′}_{l}

^{′}

_{l}

*C*is cloud fraction,

*q*

_{t}total water mixing ratio, and

*θ*

_{l}liquid water potential temperature. Note that (13) is not related to the cloud droplet spectrum because liquid water is diagnosed by any excess above the saturation mixing ratio (saturation adjustment scheme); that is,

*S*= 0 at any cloudy grid point. Wang and Wang (1999) analyzed (13) and concluded that the turbulence-generated condensation is dominant in defining the flux profile in the parameterization. The second parameterization is the mass flux scheme in which

*w*′

*q*

^{′}

_{l}

*u*” and “

*d*” represent updraft and downdraft variables,

*σ*is the fractional coverage of either updrafts or downdrafts,

*w*

_{u}the average updraft velocity, and

*φ*any variable (

*φ*=

*q*

_{l}for liquid water flux), respectively.

*w*′

*q*

^{′}

_{l}

*w*′

*q*

^{′}

_{l}

*ρ*

_{0}×

*q*

^{′}

_{l}∂

*p*′/∂

*z*

*g*/

*θ*

_{0}×

*θ*

^{′}

_{υ}

*q*

^{′}

_{l}

*w*′

*q*

^{′}

_{l}

*τ*

_{R}, where

*τ*

_{R}is return-to-isotropy timescale. Furthermore, the transport term and all the third moments in the microphysical term are neglected. The two second-order turbulence–microphysics covariance terms of (7) are assumed to be related by

*q*

_{t}(

*q*

_{t}=

*q*

_{υ}+

*q*

_{l}) and

*S,*we have

*S*′ =

*q*

^{′}

_{t}

*q*

^{′}

_{l}

*q*

^{′}

_{s}

*T,*

*p*). Further expressing

*q*

^{′}

_{s}

*θ*

^{′}

_{l}

*q*

^{′}

_{l}

*S*′ as

*w*′

*S*′

*τ*

_{R}is of the same order as the large eddy turnover time (Moeng and Wyngaard, 1986), and

*τ*

_{CE}is the CE timescale defined by

*τ*

_{CE}is the same as the commonly used supersaturation absorption or phase relaxation timescale defined by an

*e*-folding decrease of the initial surplus water vapor discussed by many authors (e.g., Squires 1952; Kogan et al. 1995).

Parameterization (17) represents a balance among three dynamic and one microphysical terms in (7); these terms are the mean gradient, buoyancy, pressure correlation, and surplus vapor flux terms. An interesting feature of (17) is that *w*′*q*^{′}_{l}*τ*_{CE}, which is defined by the droplet spectrum through (18). A smaller *τ*_{CE} (or larger *R**effective* timescale over which the turbulence can affect *w*′*q*^{′}_{l}*τ*_{R} + 1/*τ*_{CE})^{−1} (rather than *τ*_{R}), as shown by (17). Therefore, when *τ*_{R}/*τ*_{CE} ≫ 1, the turbulence has a response timescale that is equivalent to *τ*_{CE} to counteract the microphysical change. Consequently, a smaller *τ*_{CE} would result in a smaller turbulence diffusion flux and a larger *w*′*q*^{′}_{l}*τ*_{R}/*τ*_{CE} → 0, *w*′*q*^{′}_{l}*τ*_{R}/*τ*_{CE} → ∞, *w*′*q*^{′}_{l}

For real clouds, both *τ*_{R} and *τ*_{CE} are finite. For example, given the generalized convective velocity scale (Deardorff 1980) *w*∗ ≅ 0.9 m s^{−1} and the boundary layer height *z*_{i} ≅ 800 m, we obtain *τ*_{R} ∼ *z*_{i}/*w*∗ ≅ 880 s. Figure 6a shows that *τ*_{CE} is between 3 and 10 s. Clearly, *τ*_{CE} ≪ *τ*_{R}, demonstrating that the microphysical processes are always important and cannot be ignored. Since the gradient of *q*_{l} is significant for stratocumulus clouds, the turbulence diffusion term is large and must also be included for a supersaturation-based cloud model as well.

*w*′

*q*

^{′}

_{l}

*q*

_{l}resulting from vertical motion includes both the condensation/evaporation and the gradient parts. However, a major weakness of (19) is that the microphysical contribution is independent of

*τ*

_{CE}due to the approximations made in deriving (11). Consequently, both the turbulence and CE contributions depend

*only*on the turbulence timescale

*τ*. [See Eq. (17) for comparison.] Note that (19) has been obtained by Khvorostyanov and Curry (1999) and Sigg (2000). The development in this paper differs from theirs mainly in that (17) and (19) are derived from the dynamic balance among the turbulence and microphysical processes.

_{R}Figures 6b and 6c show the comparison between the LES-BM resolved and the parameterized fluxes. The parameterized *w*′*q*^{′}_{l}*w*′*q*^{′}_{l}*w*′*q*^{′}_{l}*within* updraft or downdraft plumes, it understandably underestimates *w*′*q*^{′}_{l}*C* = 1 at certain levels (i.e., all the grid points are cloudy at that level), the value calculated from (13) should be almost equal to the LES resolved flux if the saturation adjustment were used in the model. The flux from (13) is, however, significantly larger than the LES-BM resolved flux. This implies that an LES coupled with a saturation adjustment (SA) cloud scheme gives larger *w*′*q*^{′}_{l}

### b. Droplet number density flux w′N′

*w*′

*N*′

*φ*=

*N*in (14). As shown in Fig. 7, the flux without the transport term derived from (20) does not even provide solutions of reasonable magnitude. However, the inclusion of the transport term significantly improves the parameterization, although its performance is still less satisfactory than (17) for

*w*′

*q*

^{′}

_{l}

*w*′

*N*′

*w*′

*q*

^{′}

_{l}

*N*

_{u}and

*N*

_{d}, respectively, while it considerably underestimates the negative values near the cloud base.

Although the parameterizations of *w*′*q*^{′}_{l}*w*′*N*′*q*_{t} and *θ*_{l}. However, the major difference between *q*_{l} and *q*_{t} is that the former typically increases significantly with height due to condensation, while the latter is conserved during condensation and is therefore usually well mixed in the boundary layers. For an LES, the subgrid-scale diffusion timescale *τ*_{R} can be estimated as Δ*z* (SGS TKE)^{−0.5} ∼ 100 s using the simulation parameters, and is still significantly larger than *τ*_{CE}. Therefore, the microphysical impact should be considered in formulating the SGS liquid water fluxes.

### c. Ensemble mean CE rate

At the heart of any microphysical parameterization is the mean CE rate, which must include the turbulence contribution as discussed in the last section. One possibility is to use a higher-order turbulence closure model to directly predict or diagnose *q*^{′}_{l}*S*′*N*′*S*′*q*^{′}_{l}*S*′*N*′*S*′

*w*and

*S*at those levels, which is the basis for the parameterization. The consistency does not, however, exist in the inversion layer because the circulation is ill-defined there due to strong influences of small-scale turbulence. Therefore, even though (21) can accurately describe the CE process in most of the cloud layer, a special treatment of the entrainment zone at the cloud top must be formulated to include the effects of small-scale turbulence. A similar framework was used by Considine (1997) to study diurnal variability in microphysical properties of stratocumulus clouds. Note that (21) has the same form as the mass flux representation of the turbulence statistics in many models (e.g., Wang and Albrecht 1986; Lappen and Randall 2001). Therefore, it may be particularly useful in a coupled mass flux–microphysics parameterization.

## 7. Discussion

### a. Dynamic feedback of the CE timescale

*w*′

*q*

^{′}

_{l}

*S*= 0 at a cloudy grid point for a LES-SA, while it can be greater or smaller than zero at a cloudy grid for a LES-BM. This can be expressed in the following relationship obtained from (16):

*w*′

*S*′

*w*′

*q*

^{′}

_{l}

*S*must be zero for a saturation adjustment when cloud is present. Consequently, the magnitudes of

*q*

^{′}

_{l}

*w*′ from LES-SA are larger than those from LES-BM.

The above results suggest that the CE timescale *τ*_{CE}, to some extent, regulates the turbulence field. Note that droplet number *N* is related to *τ*_{CE} through the integral radius *R.* When *N**R**τ*_{CE} as seen from (18). Then, larger CE fluctuations are produced and result in larger *w*′*q*^{′}_{l}*N* is, the smaller *τ*_{CE} is, and the closer the turbulence structure derived from LES-BM is to that derived from LES-SA. More specifically, when *τ*_{CE} decreases (i.e., *R* increases), *w*′*S*′*w*′*q*^{′}_{l}*w*′*q*^{′}_{l}*τ*_{CE}, *w*′*q*^{′}_{l}*N* and approaches (13) when *τ*_{CE} → 0. Because the extra *q*^{′}_{l}*N* (or smaller *τ*_{CE}) in up- and downdrafts are likely to cancel for the ensemble mean due to their different signs, and because instantaneous CE rate is strongly coupled to the vertical motion of turbulent eddies, the direct impact of this feedback is on the turbulence field.

To test these ideas, two more simulations are performed. One (S2) uses the same LES-BM model, but with a background CCN number of 1000 mg^{−1}. The other (S3) uses the same LES dynamic model but with a saturation adjustment scheme. The high CCN number concentration used in S2 is chosen to facilitate discussion, but it may not be unrealistic for a polluted air mass. All the simulation procedures are identical to those of the previous simulation (S1) and listed in Table 1.

Figure 9 shows the comparison of some statistics among three simulations over the last hour (hour 5–6). Both S2 and S3 result in significantly larger resolved *w*′*q*^{′}_{l}*w*′*q*^{′}_{l}*R**w*′*S*′*R**w*′*S*′*w*′*q*^{′}_{l}*R**τ*_{CE}), the cloud-top entrainment is enhanced, which is demonstrated by the larger total water flux near the cloud top, and consequently *q*_{l} and cloud fraction from S3 are reduced as shown in Fig. 9d. Because LES-SA results in the largest instantaneous local CE rate among the three simulations, the entrainment from S3 is strongest and *q*_{l} is the smallest, as can be seen in both Figs. 9c and 9d.

The above results suggest that different CCN number concentrations may lead to different turbulence fields due to the different CE timescales. Consequently, the impact of variable CCN number density is not only due to the drizzle process, but also due to the different CE timescale.

### b. Impact of drizzle on the parameterizations

The parameterizations (17) and (20) have been derived based on the formulation without drizzle processes. It is natural to ask how drizzle will affect these parameterizations. To answer this question, we perform two simulations (S4 and S5) that include the processes of droplet coalescence–collection and sedimentation as shown in Table 1. In S5, the higher water vapor mixing ratio above the cloud leads to a stronger drizzle rate as seen in Fig. 10a. The flux *w*′*q*^{′}_{l}*w*′*q*^{′}_{l}*τ*_{CE} is relatively large (Fig. 6a) and *S* has a positive maximum value in updrafts.

The inclusion of drizzle may affect parameterization (20) through the correlation between *w*′ and droplet collection, and sedimentation, respectively. We have found in particular that the contribution from the collection tendency flux is significant and highly variable, and thus the quasi-equilibrium condition is not met for the *w*′*N*′*w*′*q*^{′}_{l}*w*′*N*′

The mass flux representation of *w*′*q*^{′}_{l}*w*′*N*′

## 8. Summary and conclusions

The main objectives of this work are to understand how turbulence interacts with microphysics to produce an ensemble mean CE rate and liquid water fluxes, and to suggest methods to parameterize them. The approach is to simulate a case of a nonprecipitating stratocumulus cloud with a coupled large eddy simulation and an explicit bin-microphysical model, and then perform a budget analysis for four liquid water variables: *q*_{l}*w*′*q*^{′}_{l}*N**w*′*N*′

The ensemble mean CE rate can be decomposed into mean saturation and turbulence parts; the former is directly computed from the ensemble mean supersaturation and the latter comes from the covariance *R*′*S*′*S* > 0) and reducing evaporation in subsaturated regions (*S* < 0). The dominant component of the turbulence contribution is the covariance *N*′*S*′

For the liquid water flux budget, a close balance is reached between the negative mean gradient term and the positive *w*′*S*′*w*′*S*′*w*′*q*^{′}_{l}*w*′*q*^{′}_{l}*w*′*q*^{′}_{l}*w*′*N*′

It is shown that the CE timescale defined by (18) may regulate the turbulence dynamics, because a smaller CE timescale tends to result in larger condensation fluctuations, which enhance *w*′*q*^{′}_{l}*w*′*S*′*w*′*q*^{′}_{l}*N* may affect the turbulence dynamics through the CE timescale. The simulations show that the larger *N* is, the stronger the entrainment, and the smaller the mean liquid water content. Since an LES-SA *instantly* condenses (evaporates) all available water vapor (liquid water) surplus, it produces stronger entrainment rates and smaller liquid water contents than an LES-BM for nonprecipitating stratocumulus clouds.

Two possible methods of computing the ensemble mean CE rate are proposed. One is to derive budgets for *q*^{′}_{l}*S*′*N*′*S*′

Understanding the interaction between the turbulence and microphysics is crucial for a successful representation of cloud droplet spectrum in a boundary layer parameterization. This paper shows that parameterization of the condensation, turbulent fluxes of the cloud water mass, and the droplet number concentration should include both the turbulence statistics and cloud droplet spectrum information. The parameterizations developed in this paper are steps towards that goal. One interesting aspect of turbulence–microphysics coupling discussed in the paper is the dynamic feedback of the CE timescale on the coupled turbulence–microphysics field. Immediate questions are as follows. How much does this feedback contribute to the overall impact of changing the CCN number concentration? And how should one represent it in a coupled turbulence–microphysical parameterization? These issues should be addressed in order to fully understand the interaction between the microphysics and turbulence dynamics.

## Acknowledgments

We thank Marat Khairoutdinov for the stimulating discussion that lead to Eq. (11) and Bjorn Stevens for providing his LES code. Bjorn Stevens, Charlie Cohen, and Steve Burk are thanked for their comments on the manuscript. The constructive comments of the anonymous reviewers are greatly appreciated. This work was started when S. Wang was affiliated with Universities Space Research Association and supported by the NASA FIRE III and EOS programs. Most of the analyses and writing were done when S. Wang was at Naval Research Laboratory, and supported by the Office of Naval Research under PE 0602435N. Q. Wang was supported by NSF Grants ATM-9700845 and ATM-9900496. G. Feingold acknowledges support from the NSF/NOAA EPIC program.

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## APPENDIX A

### Components of the Ensemble Mean CE Rate

*G*and the other variables results from relatively weak dependence of

*G*on temperature. Therefore, it is justified to neglect the last three terms in Eq. (A1) to obtain (2), which is used in the budget analysis.

## APPENDIX B

### Derivation of Liquid Water Flux Budget

*w*′ on both sides and applying Reynolds averaging operation,

*R*′ from (5), the above equation becomes

*without*microphysics:

The *q*_{l} budget defined by (6). (a) Condensation (CE, solid), the flux divergence (T, dashed), the subsidence (L, dotted), net tendency (Net, dot–dash); (b) *R**S**q*^{′}_{l}*S*′*N*′*S*′*R*′*S*′*S* (solid) and for negative *S* (dotted)

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *q*_{l} budget defined by (6). (a) Condensation (CE, solid), the flux divergence (T, dashed), the subsidence (L, dotted), net tendency (Net, dot–dash); (b) *R**S**q*^{′}_{l}*S*′*N*′*S*′*R*′*S*′*S* (solid) and for negative *S* (dotted)

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *q*_{l} budget defined by (6). (a) Condensation (CE, solid), the flux divergence (T, dashed), the subsidence (L, dotted), net tendency (Net, dot–dash); (b) *R**S**q*^{′}_{l}*S*′*N*′*S*′*R*′*S*′*S* (solid) and for negative *S* (dotted)

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *w*′*q*^{′}_{l}*G*), transport (*T*), buoyancy correlation (*B,* dot–dash), and pressure correlation (P, long–short-dashed); (b) *w*′*S*′*w*′*q*^{′}_{l}*q*^{−1}_{l}*w*′*N*′*N*^{−1}) term (denoted by 2), and triple correlation term (denoted by 3) [numbers represent corresponding terms in the square bracket of (7)]; (c) LES-BM resolved *w*′*S*′

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *w*′*q*^{′}_{l}*G*), transport (*T*), buoyancy correlation (*B,* dot–dash), and pressure correlation (P, long–short-dashed); (b) *w*′*S*′*w*′*q*^{′}_{l}*q*^{−1}_{l}*w*′*N*′*N*^{−1}) term (denoted by 2), and triple correlation term (denoted by 3) [numbers represent corresponding terms in the square bracket of (7)]; (c) LES-BM resolved *w*′*S*′

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *w*′*q*^{′}_{l}*G*), transport (*T*), buoyancy correlation (*B,* dot–dash), and pressure correlation (P, long–short-dashed); (b) *w*′*S*′*w*′*q*^{′}_{l}*q*^{−1}_{l}*w*′*N*′*N*^{−1}) term (denoted by 2), and triple correlation term (denoted by 3) [numbers represent corresponding terms in the square bracket of (7)]; (c) LES-BM resolved *w*′*S*′

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *N**T,* long–short-dashed), microphysics [*M* = *A* + *E* in Eq. (8), solid], subsidence (*L,* dotted), and net tendency (dot–dash); (b) activation (*A*) and evaporation (*E*).

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *N**T,* long–short-dashed), microphysics [*M* = *A* + *E* in Eq. (8), solid], subsidence (*L,* dotted), and net tendency (dot–dash); (b) activation (*A*) and evaporation (*E*).

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *N**T,* long–short-dashed), microphysics [*M* = *A* + *E* in Eq. (8), solid], subsidence (*L,* dotted), and net tendency (dot–dash); (b) activation (*A*) and evaporation (*E*).

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *w*′*N*′*M* = *A* + *E* in Eq. (9), dot–dash), mean gradient (*G,* solid), transport (*T,* dashed), buoyancy correlation (*B,* dot–dot–dash), pressure correlation (*P,* dotted); (b) activation flux term (*A,* solid), evaporation flux term (*E,* dashed). The SGS term is significantly smaller than the other terms and is not shown

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *w*′*N*′*M* = *A* + *E* in Eq. (9), dot–dash), mean gradient (*G,* solid), transport (*T,* dashed), buoyancy correlation (*B,* dot–dot–dash), pressure correlation (*P,* dotted); (b) activation flux term (*A,* solid), evaporation flux term (*E,* dashed). The SGS term is significantly smaller than the other terms and is not shown

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *w*′*N*′*M* = *A* + *E* in Eq. (9), dot–dash), mean gradient (*G,* solid), transport (*T,* dashed), buoyancy correlation (*B,* dot–dot–dash), pressure correlation (*P,* dotted); (b) activation flux term (*A,* solid), evaporation flux term (*E,* dashed). The SGS term is significantly smaller than the other terms and is not shown

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *w*′*q*^{′}_{l}*R**w*′*q*^{′}_{l}*without* B term (dotted) and the calculated from (15) (dot–dash); (c) *w*′*q*^{′}_{l}*w*′*q*^{′}_{l}

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *w*′*q*^{′}_{l}*R**w*′*q*^{′}_{l}*without* B term (dotted) and the calculated from (15) (dot–dash); (c) *w*′*q*^{′}_{l}*w*′*q*^{′}_{l}

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *w*′*q*^{′}_{l}*R**w*′*q*^{′}_{l}*without* B term (dotted) and the calculated from (15) (dot–dash); (c) *w*′*q*^{′}_{l}*w*′*q*^{′}_{l}

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *w*′*N*′*without* the transport (dotted, top axis); Eq. (20) *with* the transport (dot–dash, bottom axis). (bottom) LES-resolved flux (solid), the mass flux parameterization (14) (dot–dash), Eq. (15) (dotted)

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *w*′*N*′*without* the transport (dotted, top axis); Eq. (20) *with* the transport (dot–dash, bottom axis). (bottom) LES-resolved flux (solid), the mass flux parameterization (14) (dot–dash), Eq. (15) (dotted)

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The *w*′*N*′*without* the transport (dotted, top axis); Eq. (20) *with* the transport (dot–dash, bottom axis). (bottom) LES-resolved flux (solid), the mass flux parameterization (14) (dot–dash), Eq. (15) (dotted)

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The CE parameterization defined by (21). (a) Average supersaturation for updrafts (U, solid) and downdrafts (D, solid), integral radius for updrafts (U, dashed) and downdrafts (D, dashed). (b) Updraft CE rate from the LES (U, solid) and parameterization (21) (U, dashed), downdraft CE rates from the LES (D, solid) and parameterization (21) (D, dashed)

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The CE parameterization defined by (21). (a) Average supersaturation for updrafts (U, solid) and downdrafts (D, solid), integral radius for updrafts (U, dashed) and downdrafts (D, dashed). (b) Updraft CE rate from the LES (U, solid) and parameterization (21) (U, dashed), downdraft CE rates from the LES (D, solid) and parameterization (21) (D, dashed)

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

The CE parameterization defined by (21). (a) Average supersaturation for updrafts (U, solid) and downdrafts (D, solid), integral radius for updrafts (U, dashed) and downdrafts (D, dashed). (b) Updraft CE rate from the LES (U, solid) and parameterization (21) (U, dashed), downdraft CE rates from the LES (D, solid) and parameterization (21) (D, dashed)

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

Impacts of *N* and the saturation adjustment. (a) Resolved *ρ*_{0}*L**w*′*q*^{′}_{l}*ρ*_{0}*L**w*′*S*′*ρ*_{0}*L**w*′*q*^{′}_{t}*q*_{l} and cloud fraction (C). Solid lines denote S1, long-dashed S2, and long–short-dashed S3

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

Impacts of *N* and the saturation adjustment. (a) Resolved *ρ*_{0}*L**w*′*q*^{′}_{l}*ρ*_{0}*L**w*′*S*′*ρ*_{0}*L**w*′*q*^{′}_{t}*q*_{l} and cloud fraction (C). Solid lines denote S1, long-dashed S2, and long–short-dashed S3

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

Impacts of *N* and the saturation adjustment. (a) Resolved *ρ*_{0}*L**w*′*q*^{′}_{l}*ρ*_{0}*L**w*′*S*′*ρ*_{0}*L**w*′*q*^{′}_{t}*q*_{l} and cloud fraction (C). Solid lines denote S1, long-dashed S2, and long–short-dashed S3

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

Impact of drizzle on the parameterizations. (top) Simulation S4, (bottom) S5. (left) Drizzle rate (dashed) and *q*_{t} (solid); (center) *ρ*_{0}*L**w*′*q*^{′}_{l}*w*′*N*′

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

Impact of drizzle on the parameterizations. (top) Simulation S4, (bottom) S5. (left) Drizzle rate (dashed) and *q*_{t} (solid); (center) *ρ*_{0}*L**w*′*q*^{′}_{l}*w*′*N*′

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

Impact of drizzle on the parameterizations. (top) Simulation S4, (bottom) S5. (left) Drizzle rate (dashed) and *q*_{t} (solid); (center) *ρ*_{0}*L**w*′*q*^{′}_{l}*w*′*N*′

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

Fig. A1. Comparison among various terms in Eq. (A1): (top) 780 m ≤ *z* ≤ 830 m; (bottom) 400 ≤ *z* ≤ 780 m. They are *R**S**G**G**R*′*S*′*S**R*′*G*′*R**S*′*G*′*G*′*R*′*S*′^{−2} (g kg^{−1} s^{−1}), and thus the lines for these terms cannot be differentiated

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

Fig. A1. Comparison among various terms in Eq. (A1): (top) 780 m ≤ *z* ≤ 830 m; (bottom) 400 ≤ *z* ≤ 780 m. They are *R**S**G**G**R*′*S*′*S**R*′*G*′*R**S*′*G*′*G*′*R*′*S*′^{−2} (g kg^{−1} s^{−1}), and thus the lines for these terms cannot be differentiated

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

Fig. A1. Comparison among various terms in Eq. (A1): (top) 780 m ≤ *z* ≤ 830 m; (bottom) 400 ≤ *z* ≤ 780 m. They are *R**S**G**G**R*′*S*′*S**R*′*G*′*R**S*′*G*′*G*′*R*′*S*′^{−2} (g kg^{−1} s^{−1}), and thus the lines for these terms cannot be differentiated

Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2

Simulation conditions and procedures (CE means that only activation, and condensation/evaporation are considered; drizzle means that all BM processes are included)