## 1. Introduction

Parameterization of clouds in numerical models is complicated because of the need to account for processes on a wide range of scales. Large-eddy simulation (LES) models employ high spatial resolution and, therefore, are capable of accurate description of turbulent dynamics that, in turn, is the foundation for physically grounded representation of cloud microphysics. The latter can be implemented in LES models in two ways. The first approach, referred to as explicit microphysics, employs many bins to discretize drop size distributions (DSDs), which evolve unconstrained according to microphysical processes of condensation/evaporation, coalescence, gravitational sedimentation, etc. The explicit microphysics approach is computationally expensive, but it is also limited to high-resolution models, as the coarse spatial grid of mesoscale and large-scale models does not allow prediction of the local supersaturation needed for calculation of drop condensational growth.

The computationally less expensive approach is to predict several moments of DSD rather than DSD itself. These moments represent bulk cloud characteristics such as liquid water content and cloud drop concentration. Bulk microphysical parameterizations, being relatively simple and computationally efficient, are extensively used in models of a wide range of scales. Currently many bulk parameterizations of warm rain cloud microphysics (Tripoli and Cotton 1980; Beheng 1994; Khairoutdinov and Kogan 2000; Cohard and Pinty 2000; Seifert and Beheng 2001; Liu and Daum 2004; Liu et al. 2004, 2006), as well as more general liquid and ice phase microphysics (Lin et al. 1983; Ferrier 1994; Ferrier et al. 1995; Morrison et al. 2005; Milbrandt and Yau 2005a,b), have been developed and employed mostly in numerical cloud models (i.e., with scales capable of resolving individual cloud updrafts).

The majority of bulk parameterizations are based on the partial moments approach where cloud liquid water is divided into two categories: cloud and rain (drizzle) water (e.g., Kessler 1969). The amount of cloud water can be diagnosed from temperature and water vapor content, assuming that all excess water vapor above the saturation value instantly converts into the cloud water. The initial rain drops form as a result of the autoconversion process. Further growth of the rain drops occurs by the accretion of cloud water droplets. The original Kessler’s formulation assumed that rain drop population followed the Marshall–Palmer drop size distribution (Marshall and Palmer 1948) and considered only one moment of the DSD, liquid water content. Current bulk parameterizations use two- and three-moment formulations and allow more accurate descriptions of microphysical conversion and sedimentation rates.

The threshold radius between the cloud and rain drops is cloud type dependent and not well defined. For example, Berry and Reinhardt (1974) assumed it to be about 50 *μ*m for convective type clouds, while Cohard and Pinty (2000) defined it at 40 *μ*m. In marine stratocumulus, where precipitation is mostly in the form of drizzle, the threshold is defined at 25 *μ*m (e.g., Khairoutdinov and Kogan 2000). The need to prescribe the threshold radius as an empirical constant may be eliminated following the approach developed by Y. Liu and collaborators (Liu and Daum 2004; Liu et al. 2004, 2006). They derived an analytical expression where the threshold radius is theoretically defined based on the cloud liquid water content and the droplet number concentration.

A different approach is to use full statistical moments of the particle size distribution function where the division of total water into two parts is avoided altogether. A general formulation of such approach was suggested by Hulburt and Katz (1964) for problems in chemical engineering. A cloud physics formulation was offered by Clark (1974), who proposed a parameterization based on a single mode gamma distribution function defined for the whole range of spectra. The parameters of the gamma distributions were considered as model variables and used for calculations of cloud characteristics at the nucleation and condensation stage. In Clark (1976) the approach was further expanded to include two-mode distribution functions that allowed formulation of the coagulation process as well. The comparisons with bin microphysics formulation done in a Lagrangian parcel model framework showed that the use of double lognormal rather than gamma distributions resulted in more accurate calculations of both condensation and coagulation parameters. Szyrmer et al. (2005) used the particle size distribution functions to develop a bulk microphysical scheme for optimal calculations of rain parameters in the context of data assimilation into numerical models.

In this study we describe an approach for cloud-scale model parameterization of microphysics based on the full moments of the DSD—which, however, does not assume a prescribed form of a drop size distribution function. We are using full moments of the DSD that have a clear physical meaning, such as drop size concentration and surface area, liquid water content, precipitation flux, and radar reflectivity. As some of these variables can be obtained directly from observations, there is no need to introduce another level of uncertainty associated with microphysical retrievals. The determination of the parameterization parameters is done using datasets from simulations of the LES model with explicit (spectral) microphysics. The explicit LES model also serves as a benchmark for comparison and assessment of the accuracy of the derived bulk formulation.

The paper is organized as follows. In section 2 we describe the general design of the parameterization. Parameterizations of individual microphysical processes are described in section 3. Comparison of the parameterization predictions with results produced by explicit microphysics model are shown in section 4. The study is summarized in section 5.

## 2. Formulation of the parameterization

### a. Evolution equations

*f*(

*x*,

_{i}*r*,

*t*), which describes the number of drops in the size range from

*r*to

*r + dr*in a unit volume with coordinates

*x*at the time

_{i}*t*. In general, the equation describing the evolution of the cloud drop population can be written as

*ν*(

*r*). The second term describes the transformation of the DSD due to condensation/evaporation of droplets with the rate

*M*

_{0}is the concentration of cloud droplets. The first and the second moments,

*M*

_{1}and

*M*

_{2}, when normalized by

*M*

_{0}, are proportional, respectively, to the mean radius and the mean cross-sectional area of the drop population. The third moment

*M*

_{3}is proportional to the total liquid water content. For the case of stratocumulus clouds where the precipitation is limited to drizzle size drops, the terminal velocities in the 40–600-

*μ*m radius range are well approximated by linear function of drop radius

*r*,

*ν*(

*r*) ~

*r*, and the fourth DSD moment

*M*

_{4}may be interpreted as a parameter proportional to the local drizzle rate. Finally, the sixth moment

*M*

_{6}is proportional to the radar reflectivity factor

*Z*, defined aswhere

*D*is a drop diameter.

*n*th DSD moment tendency by multiplying Eq. (2) by

*r*and performing the integration from zero to infinity, assuming the simplified formula for the rate of change of cloud drop radius due to condensation (Rogers and Yau 1989):

^{n}*S*is the ambient supersaturation and

*G*(

*T*,

*p*) is a function of temperature

*T*and pressure

*p*only and is defined aswhere

*L*is the latent heat of vaporization of water,

*R*is the gas constant for water vapor,

_{υ}*K*and

*D*are, respectively, the coefficients of thermal conductivity and molecular diffusivity of air,

*ρ*is the density of water, and

_{w}*e*(

_{s}*T*) is the saturated water vapor pressure.

*n*th moment is defined as

*M*

_{0}, as long as the drop radius remains larger than some threshold radius

*r*

_{act}. When the drop radius falls below this threshold, the change in drop concentration is accounted for by the regeneration process described below. Thus, the second term in the Eq. (1) for the DSD evolution yields zero after the integration and is not present in Eq. (6). Note that the

*n*th moment tendency given by Eq. (7) includes a term proportional to

*M*

_{n}_{−2}. Therefore, the evolution equations for the odd moments bring to play other odd moments of the lower orders, making the problem in consideration somewhat similar to the problem of turbulence closure. The solution is to parameterize the lowest odd moment in the set in terms of other model variables. In our case such a moment is

*M*

_{1}. So, the resulting system of evolution equations for the chosen set of model variables is given by

The subscripts “activ” and “regen” represent the rate of change of the moment due to CCN activation and regeneration; “coag” refers to the effect of droplet coagulation. To close the system (9) and (10) the sink and source terms on the right-hand sides, as well as the sedimentation velocities in Eq. (8) and the first moment, need to be parameterized in terms of predictive variables as described below.

### b. Parameterization approach

*X*is used to denote either the rate of change of the moment

*M*due to coagulation or the sedimentation velocity. A similar expression is used to find the parameterization for the first moment. We found that, in general, the two terms are needed to accurately parameterize the coagulation and sedimentation rates; however, for parameterization of the first moment a sufficiently accurate parameterization can be found using only one term. The constants

_{n}*k*is done over the entire DSD dataset. The minimization of the cost function given by expression (12) was performed using the modified Levenberg–Marquardt method (Dennis and Schnabel 1983) (see the appendix).

Similar to Khairoutdinov and Kogan (2000) the drop spectra database was obtained from four simulations of a marine drizzling cloud layer in a wide range of cloud liquid water and drop concentrations. The simulations initialized with data from Atlantic Stratocumulus Experiment (ASTEX), Monterey Area Ship Track (MAST), and Aerosol Characterization Experiment (ACE) were performed using the Cooperative Institute for Mesoscale Meteorological Studies (CIMMS) LES model (Kogan et al. 1995; Khairoutdinov and Kogan 1999). The dataset included spectra characteristic of light and heavy drizzling conditions that correspond to well-mixed solid stratocumulus boundary layer clouds, as well as decoupled broken cloud layers.

More than 30 000 cloud drop spectra with liquid water content greater than 0.1 g m^{−3} were selected from the dataset. Table 1 shows squared correlation coefficients *r*^{2} for the chosen model variables. One can see that *r*^{2} values are significantly lower than 1 for almost all pairs of moments, which indicates mutual independence for most of the model variables. The notable exceptions are pairs (*M*_{2}, *M*_{3}) and (*M*_{3}, *M*_{4}), which show high degrees of correlation. Correlation between *M*_{2} and *M*_{4} is somewhat lower but not negligible. The observed strong interdependence between *M*_{3} and other moments suggests that *M*_{3} can be omitted from the parameterized expressions. In other words, *M*_{3}, as well as *M*_{1}, will be present in the parameterization implicitly.

Correlation coefficient squared for the selected DSD moments.

## 3. Parameterization of individual microphysical processes

### a. Drop activation

*n*

_{ccn}rather than the CCN spectrum as defined in the explicit microphysics formulation. The evolution equation for the CCN particle concentration is written in the form

*n*

_{ccn}varies in the model domain due to processes of drop activation and CCN regeneration; however, it cannot exceed the background CCN concentration

*n*

_{bg}specified initially from observations. The new drop activation is determined similar to the approach of Khairoutdinov and Kogan (2000), which is based on the Twomey relationship between the number of activated drops

*N*and supersaturation

*S*(Twomey 1959):where

*C*and

*k*are empirical constants.

*S*

_{thr}as the threshold supersaturation defined aswhere

*n*

_{ccn}and

*N*are the current time step values of CCN and cloud drop particles, respectively. If the supersaturation on the current time step

*S*exceeds

*S*

_{thr}, then the additional CCN are activated according to the following expression:This number is added to

*N*and subtracted from

*n*

_{ccn}. In addition to drop concentration, other moments of the DSD are also modified during CCN activation. A simple assumption is made that all newly activated drops have the same size. Thus, the following expression is adopted:where

### b. Cloud drop regeneration

*r*

_{act}. We consider these drops as regenerated CCN and add their concentration to the CCN concentration. The concentration of regenerated CCNs is estimated by assuming that the size spectrum of drops before evaporation can be approximated by a unimodal gamma distribution:where

*N*is the drop concentration,

*α*and

*β*are constants, and Γ(

*x*) is the gamma function. The parameters

*N*,

*α*, and

*β*can be expressed through the moments

*M*

_{0},

*M*

_{1}, and

*M*

_{2}, according to the formulas

*dt*is (

*r*

_{act},

*r*

_{reg}), then the size of the largest drop in this range can be estimated by integration of Eq. (4):

Unlike activation, other moments do not change during regeneration, since this change is taken into account in the evaporation process.

### c. Condensation and evaporation processes

*n*th moment due to condensation/evaporation:where

*δ*is the Kronecker symbol. Note, that

_{nl}*M*

_{0}] = cm

^{−3}, [

*M*

_{1}] = cm cm

^{−3}, [

*M*

_{2}] = cm

^{2}cm

^{−3}, and [

*M*

_{3}] = cm

^{3}cm

^{−3}. Precision

*ϵ*is defined aswhere

*N*is the number of spectra in the dataset. This definition is used for estimation of the accuracy of other parameterizations as well.

Using the nonlinear regression analysis we found that the choice of constants *C* = 0.98, *γ* = 0.47, *μ* = 0.73, and *ν* =−0.14 approximated the first moment with an error of 5.4%. The scatterplots in Fig. 1 show the comparison of the parameterized moment *M*_{1} with the benchmark values from the explicit model.

### d. Drop sedimentation

*n*th moment, defined by expression (8), can be approximated asFall velocity of the zero moment

*V*

_{0}is on the order of 1 cm s

^{−1}and thus can be safely neglected. The parameterizations for fall velocities of higher moments are presented in Table 2 and, on average, have an accuracy of 10% or better. Figure 2 shows the scatterplots of parameterized fall velocities against the explicit model values.

Parameterization of weighted terminal fall velocity *V*_{n} and the approximation error *ϵ* defined according to Eq. (24).

### e. Drop coagulation

*M*

_{0}and

*M*

_{2}describing the small cloud drop portion of the spectra. The second term mimics the accretion process (i.e., collection between small/cloud and large/rain drops) and therefore depends on all moments:Here

*M*due to the coagulation process

_{i}*i*= 0, 2, 4, 6. The constants

*A*,

_{i}*B*,

_{i}*γ*, … . are determined by the multiple regression analysis; their values, together with the approximation errors, are given in Table 3. Figure 3 shows examples of the scatterplots of parameterized coagulation rates. Parameterizations for all moments have precisions better or around 20%.

_{i}Parameterization of the coagulation rates and the approximation error *ϵ*.

## 4. Testing of the parameterization

The bulk parameterization described in the previous section was incorporated into the same dynamical framework as used in the CIMMS LES explicit microphysics model. The detailed formulation of the dynamical framework of the CIMMS LES is given in Khairoutdinov and Kogan (1999). In the explicit version of the model, the cloud physics processes are formulated based on prediction equations for cloud condensation nuclei and cloud/drizzle drops (19 and 25 bins, respectively) as described in Kogan (1991) and Kogan et al. (1995). For this study we simulated several cases of stratocumulus clouds observed during ASTEX (Albrecht et al. 1995). The setup of the model and the initial thermodynamic profiles were similar to the ones used in Khairoutdinov and Kogan (1999). The initial aerosol size distributions were based on the ASTEX A209 case and transformed to activation spectra as detailed in Kogan (1991).

First, we test processes of activation, nucleation, regeneration, condensation, and evaporation. For this we simulate nonprecipitating stratocumulus by turning off the coagulation process in both explicit and bulk models. The thermodynamic profiles in both explicit and bulk models (not shown) are identical; the microphysical parameters shown in Fig. 4 have various degrees of deviation but in general are represented quite accurately. The largest deviations are at cloud bottom and top boundaries. The deviations are especially large for the fourth moment, when maximum at cloud top is about 20% less that in the explicit model; however, for all parameters the deviation, on average, is less than 10%.

Our main focus is on parameterization of the precipitation process, as nonprecipitating clouds for a number of important cases can be rather well represented even by a simple moist saturation adjustment scheme. The results of the test of the full parameterization in a drizzling case simulation with coagulation turned on are shown in Figs. 5 and 6, where we display the 1-h time-averaged vertical profiles of various parameters. In general, the thermodynamic profiles, such as the virtual liquid water potential temperature, total water content, radiative cooling rate, and cloud cover profile, are quite well reproduced by the bulk simulation (Fig. 5) and are very close to the ones reproduced by the two-moment partial moment scheme of Khairoutdinov and Kogan (2000).

The microphysical characteristics (Fig. 6), such as CCN count and the DSD moments in the cloud layer, are reproduced by the bulk parameterization with an error better than 20% for all moments, except for the zero moment (drop concentration). The approximation of the drop concentration in the subcloud area is less accurate; the most likely reason for this, in our opinion, is the assumption (18) of the Gaussian distribution for the shape of the evaporated drops. Other shape distributions may prove to be more accurate, and for analysis of the observed forward scattering spectrometer probe (FSSP) spectra at the left, small drop tails of the DSDs would be helpful; however, for the purpose of this study we considered the obtained accuracy acceptable.

For comparison we also show in Fig. 6 profiles of microphysical variables that are directly predicted in the partial moment Khairoutdinov and Kogan (2000) parameterization; the latter include CCN, total drop concentration, and liquid water content. The second, fourth, and sixth moments can be retrieved from the partial moments predicted in the Khairoutdinov and Kogan (2000) parameterization only indirectly; therefore, they will be subject to errors and assumptions made in the retrieval algorithms. In Fig. 6 the apparent differences between simulations using full and partial moment schemes as revealed by profiles of CCN, cloud drop concentration, and liquid water content are rather insignificant and fall well within the errors of approximations of both schemes.

The most important part of a microphysical parameterization is, of course, an accurate approximation of precipitation, which in turn depends on correct approximation of the coagulation rates. The latter are represented in our parameterization by rather complex mathematical expressions given by Eq. (30). As our experience showed, the specific choice of the mathematical form of these expressions directly affected the accuracy of coagulation rate approximations. This sensitivity of the regression analysis method suggests that the described approximations may not represent the best possible accuracy and, therefore, not reveal the full potential of the integral moments approach. The specification of various thresholds used in the parameterization (e.g., the threshold radius below which drops completely evaporate) may also be important. The need to use a priori prescribed mathematical form for an approximation is a limitation of the regression analysis method; incidentally, it is inherent to partial moment approach as well.

## 5. Conclusions

We propose an approach for a bulk microphysical parameterization based on full integral moments of the droplet size distribution (DSD) unconstrained by assumptions about the shape of the DSD. Among the advantages of the full moments approach is the use of model variables that can be directly measured. As a result, the observational data can be assimilated into the model forecast cycle without intermediate steps requiring retrievals with additional assumptions and approximations (Szyrmer et al. 2005). Another advantage is avoiding artificial division of the drop spectra into cloud and rain drops. This eliminates the problem of defining the threshold between these two categories and splitting the physical parameters, such as liquid water into cloud and precipitable water or concentration into cloud and rain drop concentrations. The use of full moments also allows the application of the physical grounded process of coagulation without the need to introduce artificial subprocesses, such as autoconversion, accretion, and self-collection.

The proposed parameterization is based on five moments of the drop size distribution that have a clear physical interpretation. These include drop concentration (the zeroth moment), drop mean geometrical cross section (the second moment), liquid water content (the third moment), drizzle flux, and the radar reflectivity (the sixth moment). The drizzle flux in the case of boundary layer stratocumulus is directly related to the fourth moment, but for more vigorous clouds it can be approximated with a high degree of accuracy through the fourth and the sixth moments. The sixth variable, the mean drop radius, is parameterized in terms of other moments and therefore serves as a diagnostic parameter.

The development and testing of the parameterization was made using the CIMMS LES explicit warm rain microphysical model. However, we would like to note that the full moment approach is not is limited to warm rain processes and can be applied to ice-phase parameters as well; in the latter case the dataset needs to include conversion and sedimentation rates both for ice- and liquid-phase parameters.

We focused specifically on boundary layer marine stratocumulus clouds that were simulated using integrated measurements obtained from the research flight A209 made during Atlantic Stratocumulus Transition Experiment (ASTEX; Albrecht et al. 1995). The cloud drop spectra dataset was used to calculate the rates of change of each moment due to coagulation and sedimentation. The parameterized expressions of sedimentation rates were sought in the form of a product of power functions using nonlinear regression analysis approach to determine the exponents of the approximated expressions. This functional form proved to provide a rather accurate parameterization of sedimentation rates, as well as the first moment. For example, the first moment was parameterized within 5% accuracy; the expressions for the weighted terminal fall velocities of most moments have been approximated with a precision of about 6%–10%.

Finding an approximation for the coagulation rates is always the most difficult and challenging part of any parameterization effort. This is why different parameterization approaches exhibit significant variation in approximation of autoconversion and accretion rates (see, e.g., Wood 2005; Wood et al. 2002). In our study, we sought the parameterization of coagulation rates as a sum of two terms, each representing a product of power functions of model variables. The two terms were intended to mimic the separation of the coagulation process into two processes used in the conventional partial moment approach. The first term represented the autoconversion process (coagulation between cloud drops only) and therefore depended only on the lower two moments of the DSD function, while the second term represented the accretion process (coagulation between rain and cloud drops) and therefore depended on all four moments of the DSD function. This approach provided reasonably accurate approximations of coagulation rates with precision of about 7% for the lower two moments; however, the higher moments are approximated with a larger error of 20%–25%.

The predictions of the LES bulk microphysics model were compared with the predictions of the explicit microphysics LES for two cases: nondrizzling and drizzling stratocumulus-topped boundary layer (STBL). In the first case we tested the part of parameterization responsible for CCN activation/regeneration and the cloud formation due to condensation/evaporation processes. The representation of these processes in the parameterization allowed reasonably accurate reproduction of all dynamically significant parameters of STBL such as concentration, mean geometric cross section of a drop, liquid water content, and precipitation flux in the cloud. The testing of the coagulation formulation was done by comparing results of the explicit and bulk microphysics simulations in the case of a drizzling STBL. This comparison also demonstrated reasonably good prediction of both thermodynamic and microphysical parameters of STBL.

The specific numerical implementation of the parameterization proposed in this study revealed certain weaknesses and problems that are outlined below. Use of multiple regression analysis resulted in mathematical formulas that are rather cumbersome and quite likely do not represent the most accurate approximation of the implicit functional dependencies. We were not able to justify from physical principles why the particular dependencies, especially for coagulation rates, have to be, for example, in the form of a product of power functions. It is quite possible that other functional dependences could provide a more accurate representation of coagulation and sedimentation rates. Studying the asymptotic behavior of the coagulation rates and using known analytical solutions (e.g., Golovin 1963; Scott 1968) may be helpful; however, these tasks are no less challenging. Better approximations may also be achieved by using other approximation methods, such as a neural network approach; the latter are computationally more efficient as well (V. Krasnopolsky 2011, personal communication). During the testing of the parameterization we found that its accuracy may be improved if the whole dataset is stratified by precipitation intensity, which indicates that constants of the parameterized expressions may actually be functions of the precipitation rate.

The mathematical expressions that constitute the parameterization describe a complex nonlinear dynamical system with many variables. In testing various functional forms during the implementation of the full moment approach we identified an additional challenge, which is the need to investigate the stability of the derived system of equations as a whole. Evidently the system has to be robust and stable in the sense that small errors in regression parameters will not lead with time to positive feedbacks and the increase in approximation errors of other variables.

We leave the challenge of finding more robust, physically based functional forms to others researchers and see the main goal of this study as the description of the parameterization design and demonstration of one possible approach to its implementation, thus providing the proof of concept of the full integral moment approach.

## Acknowledgments

We thank two anonymous reviewers for constructive comments. This investigation was supported by NOAA/Office of Oceanic and Atmospheric Research under NOAA-University of Oklahoma Cooperative Agreements NA17RJ1227 and NA08OAR4320904, the U.S. Department of Commerce by the ONR Grants N00014-08-10681 and N00014-11-10439, and by the Office of Science (BER), U.S. Department of Energy, Grant DE-FG02-05ER64062, as part of the Atmospheric Radiation Measurement Program, U.S. Department of Energy. The computing for this project was performed at the OU Supercomputing Center for Education and Research (OSCER) at the University of Oklahoma.

## APPENDIX

### Nonlinear Regression and its Software Implementation

The minimization of the cost function given by expression (12) was performed using the modified Levenberg–Marquardt method (Dennis and Schnabel 1983). To find an approximation to the global minimum of the cost function *σ* the minimization procedure was run with variable initial values of the fit parameters *α _{i}* were chosen separately for each parameterized process depending on the order of magnitude of

*X*.

_{k}The regression analysis software was implemented as a multiprocess client/server program. Each subprocess of the server was responsible for fitting a dataset with a particular formula, while its main process controlled the execution of the subprocesses and gathered information about their progress. The client allowed an operator to obtain progress data from the server and control its execution. This design approach allowed us to facilitate the management of concurrently running data fitters and to efficiently share memory among them. The regression analysis software used the implementation of the Levenberg–Marquardt algorithm was provided by the International Mathematics and Statistics Libraries (IMSL) subroutine library. A technical problem that arose from it should be noted here. The regression analysis subroutine RNLIN would not return the execution control to the calling program if it was provided with some sets of input parameters and data. An “educated guess” that the subroutine should not run for more than 5 min was made in order to overcome this deficiency. For this particular situation the respective subprocess would be terminated and restarted with new values of the initial parameters of the fit.

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