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 r₁ < r₂ < ··· < r_K. Let us consider a distribution of which the cloud droplet number is N^*_i in bin i. As condensation occurs, the drop radius in ith category grows from r_i to r^*_i. Our goal is to find the new drop numbers in each stationary bin.

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.

Now, we define the cost function in the form

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subject to the strong constraints

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where 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), which represent the total droplet number, mean droplet radius, total droplet surface area, and the total liquid water content (hence the effective radius is also conserved). The zero (C), first (R), second (S), and the third moments (Q), of the distribution (r^*_i, N^*_i) are defined as

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By requiring the minimum of the cost function through the use of Lagrangian function L and Lagrangian multipliers λ₁, λ₂, λ₃, and λ₄,

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one can obtain

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Solving for N_i from Eq. (6),

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and substituting Eq. (7) into Eq. (2), (3), (4), and (5), we obtain four equations for four unknowns λ₁, λ₂, λ₃, and λ₄ that give

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where

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and

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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 λ₁, λ₂, λ₃, and λ₄ 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, where 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 μm. Our tests showed that this weight function works equally well in the case of nonprecipitating clouds.

3. Description of the Lagrangian air parcel model

The Lagrangian air parcel model used in our tests follows that of Kornfeld (1970). We consider an air parcel that contains dry air, water vapor, and cloud drops and is rising adiabatically. The heat balance equation during the moist-adiabatic process (see Byers 1965) is

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Here 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_vwR_a

where R_a is the specific gas constant for dry air and w = G_v/G_a is the water vapor mixing ratio.

Combing the heat balance equation, the equation of state for the moist air p = ρR_vT, and the hydrostatic equation dp/dz = −ρ(g + du/dt), one obtains

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where u = dz/dt is the vertical velocity of the air parcel.

Since G_v + G_w is conserved during the moist adiabatic process, we have

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where 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)

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Here 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)

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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 = r₁ exp[(i − 1)/I₀], (i = 1, 2, . . . , 25) with I₀ = 3/ln2. The bin sizes cover the range from r₁ = 1 to 256 μm 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).

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.

The tests of the method so far used a rather limited set of spectra. In the second set of experiments, we used a much wider variety of spectra obtained under a broad range of dynamic conditions typical for stratocumulus clouds. These spectra were produced by a three-dimensional LES model initialized with the dataset obtained by Nicholls during the field experiment conducted in the Northern Atlantic (Nicholls 1984). The case study based on this dataset is described in Kogan et al. (1995) and showed reasonably good agreement between the LES model-predicted microphysics and observations. The model simulation provided us with more than 15000 spectra, which comprised the dataset with LWC in the range of 0.1–0.5 g m⁻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

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The latter parameter allows us to separate the dataset into groups with different values of LWC in the drizzle mode size range. The classification of the spectra is summarized in Table 1. The average spectra and the standard deviation of the spectra in each group are shown in Figs. 6a, 6b, 7a, and 7b.

Each of the 15300 spectra was tested in the Lagrangian parcel model described in section 3. The vertical velocity of the air parcel was specified according to (9) with w₁ = 0.25 m s⁻¹, w₂ = 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 and the corresponding exact Lagrangian solution Ñ^j_i as

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where 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.

Another way to evaluate the numerical diffusion of the VO method is to compare the relative dispersion of the spectrum that is defined as σ_r = σ/r, where r is the mean radius and σ is the standard variance of the spectrum,

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Table 2 lists the relative dispersion averaged over all spectra in each of the four groups. The results show that small numerical dispersion still exists, and, in general, it increases with the broadness of the spectrum. However, even for the broadest spectra in our experiments (groups III and IV), the dispersion error is less than 10%. We would like also to note that the present tests were made using a rather coarse resolution in the drop-size coordinate where drop mass doubled every second category. It is our believe that the current generation of computers makes it quite practical to use the finer resolutions of drop spectra with mass doubling every third category. The accuracy of the VO method in this case can be increased even further.

6. Conclusions

A variational optimization method¹ 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 μm, 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 μm) 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 μm, 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.