## 1. Introduction

The turbulence cloud parameterizations developed by Sommeria and Deardorff (1977) and Mellor (1977) have been widely used in various turbulence closure models of cloud-topped boundary layers. This type of parameterization describes clouds by specifying the distribution of fluctuating moisture departure from saturation. In essence, the method defines the condensation–evaporation (CE) rate produced by the mean as well as turbulence fields, which is at the heart of any cloud parameterization. However, the CE rate does not explicitly appear in the parameterizations because they are usually used in a diagnostic form. Some studies (e.g., Smith 1990; Sommeria and Deardorff 1977) used similar cloud parameterizations in a prognostic form to calculate the net CE rate, which, however, cannot identify each physical component that contributes to the CE.

In recent years, many prognostic approaches have been proposed to define clouds in large-scale or turbulence closure models. These methods include highly complicated explicit bin microphysical models, like that of Bott et al. (1996), as well as relatively simple ones like that of Tiedtke (1993). Despite the different complexity in the model cloud processes, an essential question in all the prognostic methods is how one should relate the CE processes to the mean and turbulence fields.

In this note, we do not intend to investigate the CE processes in real clouds. Rather, we will provide new insight into the CE processes in the turbulence cloud parameterization by Sommeria and Deardorff (1977) and Mellor (1977). Particularly, we will focus on how the CE is parameterized in terms of the mean and turbulence fields. We will first derive an analytical expression for the CE rate from the diagnostic cloud relations and then discuss individual components in the CE parameterization based on a numerical simulation. Finally, we will make further comments on liquid water flux and comparisons with other prognostic cloud schemes.

## 2. Derivation

*q*

_{l}is LWC,

*w*vertical velocity, and an overbar denotes an ensemble average. Before introducing the cloud parameterization, we define following variables. Saturation water vapor mixing ratio at mean liquid water temperature is

*q*

_{sl}

*q*

_{s}(

*T*

_{l}

*p*

*T*

_{l}=

*T*−

*q*

_{l}

*L*

*c*

^{−1}

_{p}

*C*is cloud fraction, and

*G*a probability density function (PDF) of the random variable

*u*essentially representing normalized fluctuating LWC and available vapor. It is important to notice that

*q*

_{l}

*C*may be defined with any specified PDF for

*G*in (4) and (5), while the turbulent flux by (6) is only strictly derived for a Gaussian PDF (Mellor 1977). Therefore, the derivation of the CE rate presented below is correct only for a Gaussian PDF. Although some studies have provided empirical and more generalized turbulence cloud models (e.g., Bougeault 1982; Bechtold and Siebesma 1998), no attempt is made in this note to include those parameterizations.

*q*

_{l}

*u*is an independent variable and

*G*(

*u*) is independent of

*t*and

*z.*In addition,

*Q*

_{1}(

*t, z*) defined by (2) is independent of

*u.*We thus differentiate (5) with respect to time by applying the generalized Leibniz rule (see appendix A) and obtain

*γ*is considered to be constant with time and the last term is, of course, zero. We use (2)–(3) to derive a ∂

*Q*

_{1}/∂

*t*and rewrite (7) as

*q*=

*q*

_{t}

*q*

_{sl}

*q*

_{t}

*θ*

_{l}

*z*and obtain the large-scale advection term:

*T*

_{l}

*T*

^{1}to obtain

*F*

_{R}is the radiative flux. Comparing the terms on the rhs of (10) with those of (1), one clearly sees that the {•} term in (10) represents the CE rate. We further derive the gradient of

*C*using (4), substitute it into (10), and finally obtain the CE rate in the following form:where

*d*/

*dt*≡ ∂/∂

*t*+

*w*

*z.*The term CE1 is related to the large-scale vertical motion and radiation and does not directly involve turbulence variables; CE2 is derived from the gradient of cloud fraction and is directly related to turbulence CE; CE3 represents the effects of the variance (

*σ*

^{2}

_{s}

## 3. Cloud processes

In this section, we present an example of the LWC budget defined by (10) and discuss how individual terms in (12)–(14) contribute to the net CE rate based on a simulation of stratocumulus-topped boundary layer. For this purpose, we include the cloud parameterization with a Gaussian PDF [i.e., *G* = (2*π*)^{−1} exp(−*u*^{2}) in (4) and (5)] in the third-order turbulence closure model used by Wang and Wang (1994). This model is coupled with the four-stream radiation parameterization developed by Fu et al. (1995). We choose the case of the stratocumulus-topped boundary layer used in the first Global Energy and Water Experiment Cloud System Study workshop. The complete model setup, large-scale fields, initial conditions, and discussion of results can be found in Moeng et al. (1996) and Bechtold et al. (1996). The only difference between our model setup and theirs is that we use 20 m for the grid spacing while they used 25 m. The model was run for 5 h and all the analyses were conducted over the last hour with the data sampled at each time step.

*q*

^{*}

_{l}

*t*), and then (5) is used to calculate

*q*

_{l}

*t*) based on the updated mean and turbulence structure. Clearly, the difference between

*q*

_{l}

*t*) and

*q*

^{*}

_{l}

*t*) is solely due to CE processes, and then the CE rate can be calculated as

*t*is the time step (5 s).

### a. The LWC budget

As shown in Fig. 1a, the three cloud variables (*q*_{l}*C,* and *w*′*q*^{′}_{l}*q*_{l}*q*_{l}*C* ≅ 1, the CE rate is small since it is related to the gradient of cloud fraction. Near the top of the cloud, CE becomes large and negative, reaching the minimum value of −40 g kg^{−1} day^{−1} at 750 m, indicating very strong evaporation there. In the meantime, the flux divergence term is negative below 720 m, implying that the liquid water produced by the CE term below is transported upward. At the cloud top, the convergence of the flux balances the evaporation. This picture of the LWC budget is basically consistent with our general understanding of the cloud generation and dissipation processes. Since the CE term represents only the net rate, we need to look into the individual terms in (12)–(14) to understand how it is related to the various mean and turbulence fields.

### b. The condensation and evaporation processes

We further decompose CE1, CE2, and CE3 into the individual terms, which are shown in Figs. 2a,b. Appendix B shows that *g*[*γ*/*L* − *q*_{sl}*R**T*_{l}*γ*)^{−1} is approximately equal to the gradient of saturation water vapor mixing ratio following a mean moist adiabat. Thus, the first term in CE1 defined by (12) describes condensation or evaporation by large-scale vertical motion following the mean moist adiabat. Clearly, this term is negative due to the subsidence in the simulation. The second term represents the effects of radiative process:the longwave cooling at the cloud top contributes significantly to condensation, while the effect of the cloud-base warming is limited. Thus, CE1 is the CE rate produced by the mean fields.

Both CE2 and CE3 are produced by turbulent processes, as they are directly associated to the turbulent fluxes and variances. As seen in (10), CE2 is related to vertical gradient of the cloud fraction. We have further expressed this gradient in terms of gradients of the mean and turbulence variables, which actually define an ensemble of turbulent eddies following different thermodynamic paths. Therefore, CE2 can be represented by the three terms inside the square bracket in (13), multiplied by the factor [*w*′*q*^{′}_{t}*c*_{p}*γ*/*L*)(*T**θ**w*′*θ*^{′}_{l}*γ*)^{−1}, which is always positive for the marine boundary layer. The first term describes the moist-adiabatic motion and makes a major positive contribution at the cloud base, as shown in Fig. 2b. This term decreases upward and has a second maximum just below the cloud top. The second term and the third terms describe impacts of nonadiabatic paths of turbulent eddies. Obviously, the decrease of *q*_{t}*θ*_{l}*q* = *q*_{t}*q*_{sl}*G*(−*Q*_{1}), which has maxima at Δ*q* = 0 for the Gaussian PDF. Thus, condensation starts where the mean condition is still below saturation (i.e., Δ*q* = *q*_{t}*q*_{sl}*just* saturated. The cloud-top entrainment strongly affect the gradients of the mean and the turbulence variables, and thus it also significantly affects these negative contributing terms. It is interesting to note that the adiabatic condensation term [i.e., term 1 in (13)] has a similar profile as those of positive supersaturation calculated in some coupled large eddy simulation (LES) and bin-microphysics models such as in Kogan et al. (1995).

The variance-generation term (*CE3*) defined by (14) is positive at the cloud top due to the production of *σ*_{s} there, because stronger turbulence under relatively dry mean conditions means that some extreme eddies may become moist and cool enough to get saturated. This condensation is important, since it represents the growth of the cloud top.

The above analysis clearly demonstrates that, at the cloud base, the adiabatic condensation by turbulent eddies offsets the negative contributions by the gradient terms to result in a net condensation there. Near the cloud top, the negative contributions by the gradients of the mean and the variance dominate among all the terms, leading to the net evaporation there. This analysis of the CE profile shown in Fig. 1b, is consistent with the classic condensation theory.

We can summarize the major cloud processes in the parameterization for this simulation as follows. At the cloud base, turbulent eddies following the mean moist adiabat initiate condensation to form the clouds; the turbulence transports liquid water upward to the cloud top where the mean radiative cooling significantly enhances the condensation. In the meantime, evaporation occurs as a result of the stratification of the cloud layer. In the middle of the cloud layer, both the condensation and evaporation rates reach a minimum because the air at the levels is almost everywhere saturated and no water vapor excess or deficit is present. At the cloud top, evaporation is significant due to the increase of the variance *σ*^{2}_{s}

## 4. Further comments

*z*to obtain

*w*′

*q*

^{′}

_{t}

*θ*

_{l}and

*q*

_{t}variances except near the cloud top. Therefore, the mean condensation term CE1 dominates in (17). This is why some one-dimensional models (e.g., Bott et al. 1996) that include only the mean condensation due to the radiation (not turbulence condensation processes) still produce reasonable

*q*

_{l}

Of many cloud prognostic parameterizations, Tiedtke’s (1993) scheme stands out as it explicitly parameterizes *each* individual process contributing to condensation and evaporation, and to the flux transport. Despite differences in the detailed formulation between the Tiedtke’s scheme and the one discussed in this note, the cloud processes considered in both schemes are similar. For condensation, Tiedtke includes the processes of moist-adiabatic convective upward motion, defined by convective updraft mass flux, and radiative cooling, which are equivalent to the two individual terms in CE1 and CE2. For evaporation, Tiedtke’s scheme includes moist-adiabatic subsidence warming and evaporation due to turbulence mixing in an unsaturated environment, which should correspond to the terms of (12) and (13). Tiedtke formulated the evaporation due to turbulence mixing in terms of the mean saturation deficit and an evaporation timescale, while the turbulence cloud scheme uses turbulent fluxes and the gradient of conserved variables and their variance and covariance. It suggests that the timescale in the Tiedtke’s scheme is related to the environmental turbulence structure. The detrainment of liquid water from the updrafts in the Tiedtke’s scheme should correspond to part of the liquid water flux divergence term in the budget (10).

*C*and

*q*

_{l}

*C*from (4). We first differentiate (4) with respect to time, then compare the result with the prognostic equation for

*q*

_{l}

^{2}

*G*can be any PDF, since only (4) and (5) are used in the derivation. Thus, in principle, (18) can be used for any type of clouds. This equation states that the tendency of

*C*is proportional to that of the normalized liquid water [

*q*

_{l}

*γ*)

*σ*

^{−1}

_{s}

*q*

_{l}

*C,*which is consistent with Tiedtke’s approach. In addition,

*C*is a decreasing function of

*σ*

_{s}. It is because for same

*q*

_{l}

*σ*

_{s}and (

*q*

_{t}

*q*

_{sl}

*σ*

_{s}means that more liquid water needs to be condensed in fewer turbulent eddies. Thus, a smaller cloud fraction results. Furthermore, (18) suggests that the cloud fraction tendency is directly related to the liquid water flux divergence being distributed by

*σ*

_{s}, while Tiedtke’s scheme only relates the tendency to the mass flux divergence. Although the derivation of (18) is specifically based on (4) and (5), it provides some useful ideas on how a prognostic equation for

*C*should be formulated in general.

*G*(−

*Q*

_{1}) → 0 for either

*Q*

_{1}→ −∞ (i.e.,

*C*= 0) or

*Q*

_{1}→ +∞ (i.e.,

*C*= 1); and it reaches maximum when

*Q*

_{1}= 0 (i.e.,

*C*= 0.5) for a symmetric PDF. Figure 3 demonstrates the similarity between the profiles of

*C*(1 −

*C*) and the various functions of

*G*(−

*Q*

_{1}) for the simulated structure. Thus, it is reasonable to express

*G*(−

*Q*

_{1}) in terms of

*C,*for example,

*G*(−

*Q*

_{1}) ≅

*kC*(1 −

*C*), where

*k*is an adjustable parameter. Substituting it to (18) and performing the integration, one has the following diagnostic relation between

*C*and

*q*

_{l}

*k.*This expression can be used with diagnostic or prognostic

*q*

_{l}

*q*

_{l}

*C*using (19). One advantage of the diagnostic relationship is that

*C*is directly linked to

*q*

_{l}

*k*are likely to be different for different types of clouds and

*σ*

_{s}is difficult to obtain in a large-scale meteorological model.

Finally, a comprehensive understanding of a cloud model is critical if it is to be used to study real physical processes in clouds. We believe that this study not only provides more understanding of the classic turbulence condensation theory, but also gives implications for future development of the new parameterizations.

## Acknowledgments

We appreciate the comments of Charlie Cohen and Bob Haney on the original manuscript. Bjorn Stevens provided constructive suggestions on the discussion of the condensation and evaporation processes. Three anonymous reviewers are thanked for their thorough reviews. Shouping Wang was supported by NASA/FIRE III and EOS programs. Qing Wang was supported by NSF Grant ATM 9700845.

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## APPENDIX A

### Leibniz’s Rule

## APPENDIX B

### Approximated Moist-Adiabatic Water Vapor Mixing Ratio Gradient

*q*

_{s}(

*T, p*) in terms of

*T*

_{l}. Thus, the saturated water vapor mixing ratio is approximated as

Condensation and evaporation components. (a) CE1 and CE3; solid line denotes the moist-adiabatic large-scale subsidence term (CE1), dashed the radiative heating (CE1), and dotted the variance generation (CE3) terms; (b) terms in CE2; solid line represents the turbulence adiabatic condensation rate, dashed the mean gradient term, and dotted the variance gradient term. The number “1” denotes the first term in (16).

Citation: Journal of the Atmospheric Sciences 56, 18; 10.1175/1520-0469(1999)056<3338:OCAEIT>2.0.CO;2

Condensation and evaporation components. (a) CE1 and CE3; solid line denotes the moist-adiabatic large-scale subsidence term (CE1), dashed the radiative heating (CE1), and dotted the variance generation (CE3) terms; (b) terms in CE2; solid line represents the turbulence adiabatic condensation rate, dashed the mean gradient term, and dotted the variance gradient term. The number “1” denotes the first term in (16).

Citation: Journal of the Atmospheric Sciences 56, 18; 10.1175/1520-0469(1999)056<3338:OCAEIT>2.0.CO;2

Condensation and evaporation components. (a) CE1 and CE3; solid line denotes the moist-adiabatic large-scale subsidence term (CE1), dashed the radiative heating (CE1), and dotted the variance generation (CE3) terms; (b) terms in CE2; solid line represents the turbulence adiabatic condensation rate, dashed the mean gradient term, and dotted the variance gradient term. The number “1” denotes the first term in (16).

Citation: Journal of the Atmospheric Sciences 56, 18; 10.1175/1520-0469(1999)056<3338:OCAEIT>2.0.CO;2

Profiles of *G*(−*Q*_{1}) and *C*(1 − *C*). The exponential PDF is defined by *G* = (1/*e*^{−√2|u|}*G* = *H*(*u* + 1)*e*^{−(u+1)}, where *H* is Heaviside function. The functions of *G*(−*Q*_{1}) are calculated with the same *Q*_{1} profile calculated by the model.

Profiles of *G*(−*Q*_{1}) and *C*(1 − *C*). The exponential PDF is defined by *G* = (1/*e*^{−√2|u|}*G* = *H*(*u* + 1)*e*^{−(u+1)}, where *H* is Heaviside function. The functions of *G*(−*Q*_{1}) are calculated with the same *Q*_{1} profile calculated by the model.

Profiles of *G*(−*Q*_{1}) and *C*(1 − *C*). The exponential PDF is defined by *G* = (1/*e*^{−√2|u|}*G* = *H*(*u* + 1)*e*^{−(u+1)}, where *H* is Heaviside function. The functions of *G*(−*Q*_{1}) are calculated with the same *Q*_{1} profile calculated by the model.