## 1. Introduction

The current generation of cloud microphysical models that combine three-dimensional dynamics and an explicit formulation of microphysics requires an Eulerian drop-size framework in which drop sizes are fixed. The evolution of the spectra is described by varying the number concentration and/or mass within each size category. As a result of the drop growth processes, such as condensation, evaporation, coagulation, etc., new size categories will form that need to be remapped to the Eulerian fixed bin sizes at the end of each time step. It is during the remapping procedure that the added mass in the case of condensation (or number concentration in the case of coagulation) is usually spread over the entire bin size interval, resulting in numerical dispersion of the spectra. The numerical dispersion, to a certain degree unavoidable in any Eulerian formulation, if excessive, can result in broadening of the cloud drop spectrum and acceleration of the collection process, thus leading to an earlier development of precipitation. In a similar manner, it may accelerate the evaporation of cloud drops in the descending branches of cloud circulation.

The development of numerical methods for accurate calculation of cloud drop growth in an Eulerian drop-size framework has been the subject of many studies (e.g., Kovetz and Olund 1969; Bleck 1970; Egan and Mahoney 1972; Berry and Reinhardt 1974; Young 1974; Ochs and Yao 1978; Tzivion et al. 1987). A simple and computationally efficient method that conserves both the mass and number concentration has been proposed by Kovetz and Olund (1969) (hereafter referred to as KO). In essence, the KO method represents the first-order upwind advection algorithm and results in large numerical diffusion. However, due to its simplicity, the method has been applied both to condensational and coagulational growth calculations. In coagulation calculation, as shown by Ochs and Yao (1978), the large numerical dispersion of the method results in artificial production of precipitation. As we will show later in this paper, the KO method gives also quite inaccurate solutions for droplet condensational growth under conditions typical for stratocumulus cloud layers.

A much more accurate method has been developed by Egan and Mahoney (1972) (referred hereafter as EM). This method conserves the zero (drop concentration), first (liquid water content), and second (radar reflectivity) moments of a drop mass distribution. Later, Ochs and Yao (1978) extended the technique to a nonuniform exponential mass coordinate and applied it for collection and breakup, as well as for condensation calculations. Young (1974) proposed a numerical method using separate number and mass conservation equations, thus allowing subbin resolution. Both EM and Young’s methods provide much more accurate solutions than single-moment schemes and have been successfully used in models with simplified dynamic frameworks. It has to be noted, however, that the use of higher moments of the distribution function increases the number of microphysical variables that need to be retained in the EM method by a factor of 3 and, in Young’s method, by a factor of 2. In multidimensional models, the computational burden of these methods is quite significant.

Liu et al. (1995) described a variational optimization (VO) method that requires specification of only one variable in each bin size. The method significantly limits the numerical dispersion and can be formulated to conserve arbitrary number of moments of the drop-size distribution. It is also computationally inexpensive and can be easily incorporated in multidimensional cloud models. In the present paper we provide a detailed description of the method and describe modifications of the algorithm for the case of precipitating clouds. The accuracy of the method is tested using more than 15000 cloud drop spectra generated by the three-dimensional large eddy simulation (LES) cloud model with explicit microphysics (Kogan et al. 1995).

## 2. Formulation of the method

The variational methods, first applied in meteorology by Sasaki (1958), have become powerful tools in optimization, numerical analysis, and data assimilation (e.g., Lewis 1972; Stephens 1970; Ritchile 1975). In this paper, we apply a variational method for remapping the drop-size distribution function during calculations of condensational growth in an Eulerian drop-size framework. In the latter, the evolution of a drop spectrum is represented by the changing concentration of drops within the fixed size bins. The variational method controls the numerical dispersion by imposing the moment-conserving constraints. In the following discussion, we denote the drop radius in each stationary bin as *r*_{i} (*i* = 1, 2, . . . , *K*) assuming for convenience *r*_{1} < *r*_{2} < ··· < *r*_{K}. Let us consider a distribution of which the cloud droplet number is *N*^{*}_{i}*i.* As condensation occurs, the drop radius in *i*th category grows from *r*_{i} to *r*^{*}_{i}

We start with the first guess of a spectrum, which is obtained by the KO method. The first guess conserves the total number of cloud drops and the total liquid water content.

*w*

_{i}are the weights (

*w*

_{i}> 0) defined later in this section. Here,

*Ñ*

_{i}is the cloud drop number in category

*i*of the first guess and

*N*

_{i}is the sought-after new cloud drop number concentration. The new spectrum (

*r*

_{i},

*N*

_{i}) conserves the zero, first, second, and third moments of the distribution (

*r*

^{*}

_{i}

*N*

^{*}

_{i}

*C*), first (

*R*), second (

*S*), and the third moments (

*Q*), of the distribution (

*r*

^{*}

_{i}

*N*

^{*}

_{i}

*L*and Lagrangian multipliers

*λ*

_{1},

*λ*

_{2},

*λ*

_{3}, and

*λ*

_{4},

*N*

_{i}from Eq. (6),

*λ*

_{1},

*λ*

_{2},

*λ*

_{3}, and

*λ*

_{4}that give

In a very few cases, the determinant Δ may become very small (<10^{−7} ∼ 10^{−8}), for example, when the spectrum is very narrow (say cloud drops only spread over less than four bins). In this case, we simply use the KO method without the variational adjustment (see appendix A). Otherwise, we can solve for *λ*_{1}, *λ*_{2}, *λ*_{3}, and *λ*_{4} and then calculate the adjusted cloud droplet number for each bin *i* using Eq. (7).

Variational optimization technique does not impose constraints on the determination of weights (*w*_{i}) in the cost function. In most variational optimization problems the weights are chosen empirically based on the specifics of the problem. Liu et al. (1995) give the weights as *w*_{i} = *Ñ*^{−γ}_{i}*γ* = 1.0. A series of numerical tests using a Lagrangian air parcel model that provides an exact solution of the condensation problem showed that this weight function works very well in the case of nonprecipitating clouds. However, in the case of precipitating stratocumulus or convective clouds, where the cloud drop size range covers hundreds and even thousands microns, much better results are obtained by the weights in the form *w*_{i} = *Ñ*_{i}/(1 + (*r*_{i}/*β*)^{3}), where *β* = 10 *μ*m. Our tests showed that this weight function works equally well in the case of nonprecipitating clouds.

## 3. Description of the Lagrangian air parcel model

*L*is the latent heat of vaporization and

*s*is the specific heat of water. The symbols

*G*

_{a},

*G*

_{v}, and

*G*

_{w}are the mass of dry air, water vapor, and liquid water in the air parcel, respectively. Here

*T*and

*p*are the temperature and pressure of the air parcel, respectively. Also,

*C*

_{p}is the specific heat of dry air at constant pressure. The specific gas constant for moist air,

*R*

_{v}, is given by

*R*

_{v}

*w*

*R*

_{a}

*R*

_{a}is the specific gas constant for dry air and

*w*=

*G*

_{v}/

*G*

_{a}is the water vapor mixing ratio.

*p*=

*ρR*

_{v}

*T,*and the hydrostatic equation

*dp*/

*dz*= −

*ρ*(

*g*+

*du*/

*dt*), one obtains

*u*=

*dz*/

*dt*is the vertical velocity of the air parcel.

*G*

_{v}+

*G*

_{w}is conserved during the moist adiabatic process, we have

*dm*

_{i}/

*dt*is the condensational diffusion growth rate for a cloud drop with the mass

*m*

_{i}in bin

*i,*which, after neglecting both the curvature and the salt factor, can be written as (Houze 1993)

*F*

_{K}and

*F*

_{D}represent the effects of heat conductivity and the vapor diffusivity, respectively;

*f*

_{v}is the ventilation factor;

*S̃*is the supersaturation, which is defined as

*S̃*≡

*e*/

*e*

_{s}(

*T*) − 1, where

*e*is the ambient vapor pressure and can be calculated as

*e*=

*pw*/(0.622 +

*w*); and

*e*

_{s}(

*T*) is the saturation vapor pressure given as (Murray 1967)

The set of Eqs. (9)–(12) can be solved very accurately numerically in a Lagrangian drop-size framework and provides the benchmark (exact) solution of the condensation process. The solutions of the KO, EM, and VO schemes are obtained by remapping the drop spectrum to the Eulerian drop size framework at every time step, therefore they are subject to numerical diffusion errors.

## 4. Verification of the method in a Lagrangian model

We first show results of the test of the VO method in a Lagrangian air parcel model. A Gamma-type distribution is specified initially (Berry 1967) with the liquid water content of 0.2 g m^{−3} and drop concentration of 50 cm^{−3} (Fig. 1). The stationary bin sizes are defined as *r*_{i} = *r*_{1} exp[(*i* − 1)/*I*_{0}], (*i* = 1, 2, . . . , 25) with *I*_{0} = 3/ln2. The bin sizes cover the range from *r*_{1} = 1 to 256 *μ*m and are the same as in the Cooperative Institute for Mesoscale Meteorological Studies (CIMMS) large eddy simulation (LES) cloud model (Kogan et al. 1995).

The accuracy of the VO method was evaluated in two sets of experiments, in which only condensation and evaporation processes were considered. In the first test, the air parcel ascended with a constant vertical velocity of 1.0 m s^{−1}. The drop spectra were computed using three different methods: Kovetz and Olund (1969), Egan and Mahoney (1972), and the present variational optimization method. All three methods employ an Eulerian drop-size framework. The initial supersaturation is set to be 0.2%. Figures 2a and 2b show the resulting size distribution of drop number and mass at the 200-m height. The exact solution obtained in a Lagrangian model is plotted as the solid line. As evident from Figs. 2a, 2b, and 2f, the KO method has a significant numerical dispersion, while the EM scheme has a relatively small numerical dispersion error. Since all three schemes conserve drop number and mass, the liquid water content (Fig. 2c) is determined rather accurately in all three methods. The other moments of the distribution function, such as the mean and effective radius and standard deviation, are more accurately calculated by the EM and VO schemes. The VO method produces the smallest error not only in the prediction of the integral parameters of the spectrum, such as liquid water content, mean radius, effective radius, and the relative standard deviation of the spectrum (Figs. 2c–f), but also for the spectrum itself (Figs. 2a and 2b).

*u*

*u*

_{1}

*u*

_{2}

*ωt*

*u*

_{1}= 0,

*u*

_{2}= 1 m s

^{−1}and

*ω*= (2

*π*/600) s

^{−1}. The integration is made for 600 s during which time the air parcel first moves upward, reaching a height of about 191 m, and then moves downward to the same starting point, thus completing a cycle. Figures 3a and 3b show the size distribution of drop concentration and liquid water content (LWC) in the air parcel at the end of the cycle. The results are remarkably good for both the EM and VO schemes, which both preserve well the spectrum shape. The EM scheme does exhibit a moderate dispersion, while the VO method has a small phase error. The KO method shows a very significant dispersion (largest in both diffusion and phase error). Due to the strong dispersion, the KO method results in evaporation of a significant number of cloud drops (58%). Only a small fraction (less than 4%) of cloud drops evaporates in the VO scheme (Fig. 3c).

## 5. Modification of the VO method for precipitating clouds

The VO method has been implemented in the CIMMS LES model with explicit microphysics (Kogan et al. 1995). The three-dimensional experiments showed that the method works very well in simulations of nonprecipitating stratocumulus clouds. Since coalescence is weak in these clouds, the cloud drop spectra are mostly unimodal and occupy a rather limited size range from 1 to 50 microns in radius. The situation is more complex in the case of precipitating clouds where bimodal drop-size distributions are quite common. In this case, the constraints to conserve four moments of the drop distribution imposed by the VO method may lead to artificial reduction in the drop concentration at the tails of the spectrum. The right “large-droplet” tail of the spectrum is especially important as it determines the onset of coagulation and also rain and drizzle formation rates.

The problem can be effectively solved and the concentration of drops at the spectrum tails preserved by applying the VO method to the central part of the drop spectrum and to each of its tails separately. The decomposition of the spectrum into three subspectra produces the smallest error when the subspectra are smooth and do not have sharp discontinuities. This can be achieved by using Gamma distribution function in the decomposition procedure described in appendix B and illustrated conceptually in Fig. 4.

The VO method modified for precipitation clouds was tested in a two different sets of experiments. The first set used two typical spectra produced as a result of the coalescence process and characterized by the long large-drop tail and double peaks in the LWC distribution. The spectra were obtained using Berry and Reinhardt (1974) coalescence model that was initialized with a Gamma-type drop distribution with LWC of 1.0 g m^{−3} and cloud drop concentration of 50 cm^{−3}. The coalescence model was run for 316 and 476 s, respectively, to obtain the spectra 1 and 2 shown in Figs. 5a and 5b. Figures 5a and 5b also show the decomposition of the two spectra into three parts. The spectra 1 and 2 shown in Figs. 5a and 5b were then used as the input spectra for the Lagrangian condensation parcel model and run for another 500 time steps with a constant vertical velocity of 1.0 m s^{−1}. As in the experiments described in section 3, the exact solution of the Lagrangian condensation model is then compared to the solution given by the VO method in the Eulerian framework. The present experiments, however, use spectral 1 and 2 that have a significant drizzle mode compared to initial spectrum shown in Fig. 1. The results of the experiments are shown in Figs. 5c and 5d (note the difference in scales in Fig. 5 due to the increase in LWC as a result of condensation). The modified VO method that uses the decomposition procedure provides more accurate solution than the old version of the VO method, especially for the large-drop tail of the spectra. The increased accuracy in this size range is especially important for drizzle and rain prediction in precipitating clouds.

^{−3}and drop concentrations in the range of 30–90 cm

^{−3}. The whole dataset was divided into four groups based on the radar reflectivity parameter

*w*

_{1}= 0.25 m s

^{−1},

*w*

_{2}= 0.5 m s

^{−1}, and

*ω*= (

*π*/20) s

^{−1}. The resulting spectra of the modified VO method are then compared to the exact solutions after 600 time steps. We calculated the averaged difference (error)

*ε*

_{i}between the solution given by the VO method

*N*

^{j}

_{i}

*Ñ*

^{j}

_{i}

*K*is the total number of spectra in each of the four groups,

*i*denotes the spectrum bin size, and

*j*denotes the individual spectrum in each of the four groups described in Table 1.

Figures 6c, 6d, 7c, and 7d show the exact and the VO method solutions, as well as the error averaged over all spectra in each of the four groups. One can see that the errors are quite small and the VO method provides very satisfactory results; the improvement is especially noticeable in the drizzle size range.

*σ*

_{r}=

*σ*/

*r*

*,*where

*r*

*σ*is the standard variance of the spectrum,

## 6. Conclusions

A variational optimization method^{1} for condensation and evaporation calculations in an Eulerian drop-size framework has been proposed and tested against the exact solution given by the Lagrangian air parcel model. The variational method not only conserves the integral parameters of the spectrum, such as drop number, mean radius, mass, and the effective radius, but also provides an accurate calculation of the spectrum itself. The accuracy of the variational method is comparable to the accuracy of the Egan and Mahoney (1972) scheme. The variational method has, however, an important advantage compared to the latter method. It requires specification of only one variable in each bin size, while the EM scheme needs retaining of the three moments of the spectrum, thus tripling the memory requirements in the model. For multidimensional models, this presents a serious limitation and makes the application of EM method impractical.

The estimate of computation time showed that the VO method by itself is about 3.1 times slower than the KO method. However, one has to bear in mind that 1) the remapping in a full multidimensional model needs to be done only once during the dynamical time step, and 2) the cost of the remapping is only a small fraction compared to the computational cost of other processes, such as the advection of microphysical variables, etc. Tests with the CIMMS LES model in a 64 × 64 × 60 integration domain showed that the old VO scheme for nonprecipitating clouds (Liu et al. 1995) increased the total CPU time by 1.1%, while the present version of the VO scheme for precipitating clouds increased the total CPU time by 4.5%. Evidently this CPU time expense is justified given the significant increase in accuracy.

We would like to note that the errors associated with condensational remapping are most significant in the size range of 1–200 *μ*m, which is typical for drops in stratiform clouds. The application of the VO method for this type of cloud is recommended. In the convective clouds where the drop-size range is much wider and the drop spectra sometimes exhibit very complex multimodal shapes, the performance of the VO method has not been thoroughly tested. However, we do expect a good performance of the VO method in convective cloud models as well for two reasons. First, for large cloud drops (r ≥ 200 *μ*m) the growth by condensation is very small (Δr ≪ r), therefore the condensational remapping is rather insensitive to a particular scheme. In addition, for drops larger than 100–200 *μ*m, the condensational growth is much smaller than the coalescence growth, consequently, the errors of the condensational remapping are also smaller than the errors associated with the coagulation calculations. It is our experience that for drops larger than 100 microns, the use of a simpler method, such as Kovetz and Olund’s (1969) method, is quite warranted.

## Acknowledgments

The first author wishes to thank Prof. C. Qiou and Dr. Z. Wang for helpful discussions. Support for this research was provided by the DOE ARM Project 144880-A91, the DOD-ONR Grants N00014-96-1-1112 and N00014-96-0687, the NOAA’s Climate and Global Change Program Grants NA37RJ0203 and NA67RJ0150, and the NSF Grant ATM88-09862 to CAPS. Computations were performed on the DEC Alfa workstation funded by the NASA Earth Science and Application Division, Radiation Dynamic and Hydrology Branch, through Task 460-23-54-20.

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## APPENDIX A

### Adjustment of Negative Values

As noted in Liu et al. (1995), the method may produce some small negative values in cloud drop concentration that are normally associated with very sharp narrow spectra. We adjust these small negative values in the following way. Let us denote the summation of negative *N*_{i} as *C*^{−} and positive *N*_{i} as *C*^{+} (*C* = *C*^{−} + *C*^{+}). After setting the negative *N*_{i} to be zero, the new *N*^{new}_{i}*N*^{new}_{i}*N*_{i}(1 + *C*^{−}/*C*^{+}). By doing this, we still have ^{K}_{i=1}*N*^{new}_{i}*C,* but LWC is no longer conserved. Our tests show that the LWC error is very small in most cases, exceptions are associated with very narrow spectra occupying only 2–3 bins. Figure A1 shows the percentage of the negative value occurrence in the CIMMS LES model during the two dynamic time steps that produce 142985 calls to the VO method subroutine. The calls that produced negative values were counted whenever |*C*^{−}/*C*| > *δ* or |*Q*^{−}/*Q*| > *δ.* (Here, *Q*^{−} and *Q* are the total negative LWC and total LWC, respectively). The percentage of the calls that produce negative values are plotted in Fig. A1 as a function of *δ.* Of the 142985 calls, there were no cases with negative values for *δ* = 0.09 and only nine cases for *δ* = 0.05. All the nine cases show that the spectra were at the early stage of activation in which cloud condensation nuclei were just activated and have not grown large enough and the drops concentrated in the first two or three bins. As *δ* decreases, the percentage of the negative values increases. Even for *δ* = 10^{−5} (0.001%), the occurrence of the negatives values is less than 1%. In fact, if the negative value is large, we use the KO method (which broadens the spectra) for one time step. At the next time step, the VO method is automatically selected, and the error again will be checked. If the error exceeds our criteria, the KO method will be used. This procedure can be repeated until all errors meet our criteria.

## APPENDIX B

### Description of the Decomposition Procedure

This appendix describes how the mass distribution function *f*(*x*) is decomposed into three subspectra prior to condensation calculations (Fig. 4).

1) Find the maximum (B) of the spectrum, and the smallest *x*_{a} (A) and the largest *x*_{c} (*C*) bins that have liquid water content exceeding the minimum threshold (10^{−3} g m^{−3}).

2) Find the drop bins (*x*_{k} and *x*_{l}) at which the differences between *f*(*x*) and their corresponding values on lines AB and BC is at their maximum.

*G*

^{(1)}(

*G*

^{(2)}) that pass through the points

*x*

_{k}+1/2 (

*x*

_{l}−1/2) and have the same slopes as

*f*(

*x*) at these points. The Gamma function is defined here as a two-parameter function,

*G*

*x*

*c*

_{1}

*x*

^{2}

*c*

_{2}

*x*

*c*

_{1}and

*c*

_{2}are constants and can be calculated as

*N̂*

_{i}is the drop concentration in bin

*i*and Γ

_{i}(=

*G*

_{i}/

*m*

_{i}) is the drop concentration for the fitted Gamma function.

5) Use the VO method for each part separately, then sum the outputs together.

Classification of the cloud drop spectra.

Averaged standard deviation.

^{1}

The FORTRAN code of the variational optimization method can be obtained by sending a request to the leading author by e-mail at qliutornado.ou.edu.