## Abstract

In subgrid-scale condensation schemes of cloud models, the majority of previous authors have relied on results presented in a paper by Bougeault. In the present paper, second-order liquid water correlations are restated that differ from the former paper but are coherent with the corrigendum of Mellor. These differences are explained here through two different underlying definitions of cloud water content fluctuations; they can be summarized by whether or not unsaturated air within a grid box contributes to the eddy flux of the cloud water content. Taking into account the issue mentioned above, the “buoyancy flux” is also derived. Although the full impact of these changes has not been evaluated here, it may become important for future global cloud-resolving climate models.

## 1. Introduction

Early condensation schemes of cloud models contained the assumption that a computational grid box volume element was either entirely saturated or entirely unsaturated. To improve upon this crude approximation, Sommeria and Deardorff (1977) introduced the concept of statistical distribution for variables such as total water specific humidity *q _{w}* and liquid water potential temperature

*θ*, allowing the computation of partial cloudiness

_{l}*R*and liquid water specific humidity

*q*, while taking into account subgrid-scale fluctuations. Working on the same issue, Mellor (1977a, b) proposed a mathematical method to obtain similar expressions instead of the approximations and the empirical derivations made by Sommeria and Deardorff (1977). Later, Bougeault (1981) presented a paper where he synthesized the main equations for the description of the PBL. Different statistical distributions were tested for computing partial cloudiness, liquid water content, and higher-order correlations.

_{l}In this paper, the computation of second-order liquid water correlations (of the form , for any given quantity *x*) is restated. Indeed, two different formulations have appeared in the literature. The formulation presented here is consistent with the corrigendum by Mellor (1977b) but differs from Bougeault (1981) and the initial Mellor (1977a) formulation. As we will show in this paper, the difference comes from the underlying mathematical definition of the cloud water content fluctuations.

The second-order liquid water correlations are used to compute the buoyancy flux in turbulence closure schemes (e.g., in the prognostic equations for turbulent kinetic energy and its dissipation when such closure is used) and are also used in the expression of vertical fluxes of specific humidity in the prognostic equation of liquid water content. So a fully analytical formulation of the buoyancy flux for Gaussian cloud is briefly proposed, which takes into account the issue mentioned above. The formulation of this flux is based to some extent on Bechtold et al. (1995) and Cuijpers and Bechtold (1995) and uses Gaussian distribution theorems restated by Chen (1991). More detailed computations can be found in Bouzereau (2004).

## 2. Analysis

### a. Subgrid-scale condensation scheme

#### 1) Preliminary definitions

The pseudoconservative variables used here are the liquid water potential temperature *θ _{l}* and the nonprecipitating water specific humidity

*q*=

_{n}*q*+

*q*used, for example, in the Meso-NH meteorological model (Bougeault and Mascart 2000, chapter 20). Neglecting the pressure fluctuation, the variable

_{c}*θ*is defined by

_{l}where *T* is the absolute temperature, *θ* is the potential temperature, *q _{c}* is the specific humidity of the cloud liquid water,

*q*is the water vapor specific humidity,

*L*is the latent heat of vaporization, and

*C*is the specific heat at constant pressure. The ensemble mean of each quantity is denoted by an overbar, whereas primes denote fluctuating quantities (e.g.,

_{p}*q*=

*q*+

*q*′). Instead of the specific humidity of total liquid water

*q*=

_{w}*q*+

*q*(where

_{l}*q*is the liquid water specific humidity) and

_{l}*θ*based on

_{l}*q*, which is a more classical choice, we use these variables

_{l}*q*and

_{n}*θ*based on

_{l}*q*to allow for the possibility of taking cloud and rain into account separately in the model.

_{c}As for the definition of *q _{c}*, taking into account subgrid-scale fluctuations of

*q*and

_{n}*θ*when

_{l}*q*≥

_{n}*q*, we use the following (Mellor 1977a):

_{s}where *q _{s}* is the saturation specific humidity, =

*q*(), = (

_{s}*T*/

*θ*),

*R*is the specific gas constant for water vapor, and

_{υ}When *q _{n}* ≤

*q*, we can write (Bougeault 1981)

_{s}Note that *q _{c}* = 0 simplifies the computation of

*q*−

_{n}*q*so that it appears that

_{s}*A*is reduced to unity, which is not the case.

_{l}#### 2) Partial cloudiness, mean cloud liquid water content, and second-order liquid water correlation

We can now compute the partial cloudiness *R* and the ensemble-mean content of cloud liquid water . We assume a bivariate normal distribution function *G̃*(*q _{n}*,

*θ*) in order to represent the subgrid-scale fluctuations of the variables

_{l}*θ*and

_{l}*q*(Mellor 1977a):

_{n}Classically, we reduce the integration to a single variable and we use the available variables [e.g., *q _{sl}* =

*q*(

_{s}*T*) instead of

_{l}*q*]. We follow the computation of Mellor (1977a) and Bougeault (1981) to introduce the following variables:

_{s}We define the normalized variable *t* = *s*′/*σ _{s}*

_{′}with

*σ*

_{s′}= (

*A*/2)( +

_{l}*α*

^{2}

_{l}− 2

*α*)

_{l}^{1/2}, the standard deviation of

*s*′. Let

*G*(

*t*) be its probability density and

*Q*

_{1}a dimensionless measure of the departure of the mean state from saturation

*Q*

_{1}=

*s*/

*σ*

_{s′}. Equation (9) becomes

Equations (9)–(12) are identical to those presented in the works of Mellor (1977a, b) and Bougeault (1981).

For the second-order liquid water correlations, we found two different approaches. To explain the differences between these two, let us compute, for example, , where *w* is the vertical velocity.^{1} The approach of Bougeault (1981), used by authors such as Chen (1991) or Bechtold et al. (1995) and identical to the initial approach of Mellor (1977a), can be written as

or, equivalently,

Here we have used a different expression, which is coherent with Mellor (1977b):

The difference between these two computations can be seen as a difference in the definition of the cloud water content fluctuations *q*′_{c} or *q*′*_{c}. On the one hand, Eq. (15) is equivalent to the definition

Note that the definition of Eq. (17) implies that *q*′*_{c} = − when *q _{n}* <

*q*. This latter expression is not physically realistic and leads to an erroneous contribution to the eddy flux of cloud water content. According to Eq. (16) and using the normalized variable

_{s}*t*, we can write

which becomes after some algebra

This latter equation can be seen as a defining equation for the second-order liquid water correlation. We have to be cautious here because the overbar does not verify the mathematical property of a mean since we do the integration only on the cloudy fraction of the grid mesh [Eq. (16); e.g., *c* = ∫^{∞}_{−Q1 }*cG*(*t*) *dt* = *cR*, where *c* is any constant]. Hereafter, we will introduce the notation *N* ≡ /2*σ*^{2}_{s′}.

#### 3) Subgrid-scale distributions

To integrate these previous expressions, one has to postulate the form of the density *G*(*t*). The forms studied here are the “all or nothing” scheme, the Gaussian law (Mellor 1977a; Sommeria and Deardorff 1977; Bougeault 1981), and a positively skewed distribution of interest for the trade wind cumulus layer (Bougeault 1981). The three proposed forms are presented in Table 1. As previously discussed, the values of *N* differ from Bougeault (1981), in which *N* = *R* for the Gaussian law, whereas for the skewed distribution, *N* = *R* = 1 when *Q*_{1} ≥ 1 and *N* = *R*(2 − *Q*_{1}) when *Q*_{1} ≤ 1. These differences for *N*(*Q*_{1}) are shown in Fig. 1. Note that the differences between the “corrected” *N*(*Q*_{1}) and “noncorrected” start to be sensitive and nonnegligible for *Q*_{1} > −1; for values of *Q*_{1} smaller than −2.5, the corrected *N*(*Q*_{1}) and the *N*(*Q*_{1}) according to Bougeault (1981) are superposed.

### b. Buoyancy flux

The virtual potential temperature flux (henceforth “buoyancy flux”) can be computed for partially cloudy grid volume without the linear interpolation between the entirely saturated case and the entirely unsaturated one proposed by Sommeria and Deardorff (1977), Redelsperger and Sommeria (1981), Bechtold et al. (1992), and Cuijpers and Duynkerke (1993).

Similar to Bechtold et al. (1995) and Cuijpers and Bechtold (1995) but with less simplification, we can use the following for nonprecipitating cloud without any assumption on the degree of saturation of the grid volume:

where *τ* = 1 + 0.608 − 1.608, *β* = *Lθ*/*C _{p}T* and the virtual potential temperature being defined by

*θ*≡ (1 + 0.608

_{υ}*q*−

*q*)

_{c}*θ*. Let us now write as a function of and . Assuming a subgrid Gaussian distribution for the variables

*w*and

*t*, this will be done using a mathematical relation restated by Chen (1991; particularly theorem 4). This theorem states that for

*X*and

*Y*joint-Gaussian distribution variables, the expected value of

*Y*is given by

where *E*() is the ensemble mean. Rewriting Eq. (15) with a change of variable gives

where *G̃*(*w*′, *t*) is a bivariate Gaussian distribution, and using the theorem from Eq. (21) with *Y* = *w*′[*E*(*w*′) = 0] and *X* = *t*[*E*(*t*) = 0 and *σ _{t}* = 1], we obtain

This latter relation is similar to the equation given by Mellor (1977b, p. 1484):

where

where *D _{q}* =

*βτ*− 1.608

*θ*and

*N*≡ /2

*σ*

^{2}

_{s′}.

## 3. Discussion

Two different computations for the second-order liquid water correlation *N* ≡ /2*σ*^{2}_{s′} are found in the literature. We have shown in this paper that the origin of this difference comes from two different underlying definitions of the cloud liquid water content fluctuation *q*′_{c}. The definition chosen here [Eq. (16)] implies that unsaturated air within a grid box gives zero contribution to the eddy flux of cloud water content, which is the physically realistic definition. This statement is not true with the other definition [Eq. (17)]. This paper points out this issue and shows the deviation induced on *N* as a function of *Q*_{1}, a dimensionless measure of the departure of the mean state from saturation. The deviation becomes nonnegligible for values of *Q*_{1} > −1, that is, for cloud cover higher than 20%. We then restate the computation of the buoyancy flux with an expression that does not need any assumption to be made on the degree of saturation of the grid volume.

Because some authors still currently use the expression of *N* given in Bougeault (1981), the results shown in this paper may bring new perspectives to their work. Indeed, the deviation observed here on *N*(*Q*_{1}) (Fig. 1) for a given distribution is as important as in Bougeault (1981) for deviations on *N*(*Q*_{1}) between different distributions. We can expect then that these differences will have an impact on the behavior of low-level cloud development and particularly on stratocumulus, whose description is important in climate modeling. With the next generation of climate models likely to be either global cloud-resolving models or GCMs with a cloud-resolving model (CRM) embedded within each grid box, considerations of the subgrid-scale variability of microphysical quantities in cloud models may become important in future climate research.

## Acknowledgments

The authors thank Dr. J.-L. Redelsperger, Prof. C. Frankignoul, and D. Wendum for their helpful discussions; and M. Milliez for her kind review of the text. We also thank the two anonymous reviewers who helped with very useful comments on the manuscript. This work was performed for the Ph.D. thesis of E. Bouzereau and supported by Electricité de France.

## REFERENCES

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## Footnotes

*Corresponding author address*: Dr. Emmanuel Bouzereau, CEREA, Laboratoire Commun ENPC-EDF R&D, 6-8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne la Vallée CEDEX 2, France. Email: emmanuel.bouzereau@ensta.org

^{1}

Note that the total flux of cloud water content can be derived as follows: = + .