## 1. Introduction

The purpose of this paper is to compare and explain various formulations of the popular Berry and Reinhardt (1974b, hereinafter BR74b) bulk microphysical parameterization for initiating growth of rain from cloud droplets and to investigate how those differences in formulation impact initial rain development. It will be shown that differences among versions result from typesetting errors (some from the original BR74b paper), derivation errors, and methods. These differences are potentially important because precipitation development has previously been shown to be very sensitive to the degree or particular choice of how rain is initiated (e.g., Cotton 1972; Lin et al. 1983; Richard and Chaumerliac 1989; Chaumerliac et al. 1991; Farley et al. 2004).

The BR74b scheme (used herein to refer to the bulk parameterization derived from BR74b’s bin model output) is one of many schemes that can be used to initiate rain in cloud models with bulk microphysics. Although the purpose of this paper is not to compare the detailed results of BR74b schemes with other rain-initiation schemes, some general comparisons can be made. [The reader can see Berry and Reinhardt (1974c, hereinafter BR74c) for other comparisons.] The BR74b scheme is moderately sophisticated because it includes two important details about the cloud droplet distribution known to affect collision–coalescence: mean cloud droplet size and dispersion. Simpler schemes like Kessler (1969) only include a cloud mixing-ratio threshold and do not account for cloud dispersion, giving a poor performance when compared with bin models (e.g., Beheng and Doms 1986). Numerous studies have revealed that heavy model tuning is required when using these Kessler-like schemes to match the observed onset of rain and proportions of cloud and rain [e.g., see review by Seifert and Beheng (2001)]. Although it will be shown that some tuning of the cloud dispersion may still be required in some implementations of the BR74b scheme, the user has more sophisticated control. The BR74b scheme also provides at least some guidance on rain number concentration rates, whereas some newer schemes (e.g., Liu and Daum 2004) do not. Furthermore, the BR74b bulk scheme should remain a popular, reasonably accurate, and always less expensive solution than the stochastic collection equation solutions that are used in bin or spectral models (e.g., Berry and Reinhardt 1974a, hereinafter BR74a; Feingold et al. 1988; Beheng 1994; Bott 1998; Tzivion et al. 1999; Simmel et al. 2002; Lynn et al. 2005).

The BR74b scheme that has been widely adopted is a parameterization of net rain development that is derived from their more sophisticated bin model described in BR74a. It is common knowledge that the scheme incorporates autoconversion: rain production via collision–coalescence of cloud droplets (e.g., Glickman 2000; BR74a). Although Pruppacher and Klett (1978) and most subsequent authors loosely referred to this entire parameterization as “autoconversion,” the scheme also includes the further accretion of cloud by those growing small raindrops and the collision–coalescence (or self-collection) of the growing small raindrops (BR74b).

The BR74b parameterization has been around for over 30 years, but it is still the mainstay of many models because of its moderate sophistication yet small relative cost. The BR74b scheme is used within the Weather Research and Forecasting Model (Thompson et al. 2004) and the Regional Atmospheric Modeling System (RAMS, version 3b). Versions of the BR74b parameterization have been presented and used by Nickerson et al. (1986, hereinafter N86), Proctor (1987, hereinafter P87), Verlinde and Cotton (1993, hereinafter VC93), Walko et al. (1995, hereinafter W95), Meyers et al. (1997, hereinafter M97), Carrió and Nicolini (1999, hereinafter CN99), Cohard and Pinty (2000, hereinafter CP00), G. Thompson et al. [2004, hereinafter T04, 2006, 2007, unpublished manuscript—the version described in Thompson et al. (2006) and in the Thompson et al. paper under preparation that describes the BR74b scheme is hereinafter collectively referred to as T07], and Milbrandt and Yau (2005, hereinafter MY05). These are the citations providing the versions of the BR74b scheme that are analyzed herein.

We begin our analysis by starting with the equations in BR74b, and we modify the units to follow the International System of Units (SI) convention (section 2). Different versions of the BR74b scheme, as presented in the literature, are derived and compared. Average rates are illustrated at the end of section 2, and time-integrated solutions are presented and compared in section 3.

## 2. Equations and derivations of the BR74b scheme

Berry and Reinhardt studied, in a series of four papers (BR74a,b,c; Berry and Reinhardt 1974d, hereinafter BR74d), how cloud droplets (their distribution “S1”) form larger hydrometeor droplets (distribution “S2”). They used a bin model initialized with S1 only or both S1 and S2. They applied the stochastic collection equation and included three processes: S1 self-collection (i.e., autoconversion), S2 accretion of S1, and S2 self-collection (BR74a). The droplet spectra were initialized as a Golovin distribution (also known as the Pearson type III gamma distribution), and condensation was omitted. BR74b found that the S2 growth was more rapid with increasing initial S1 cloud dispersion and/or mean mass.

What Berry and Reinhardt intended as their parameterization is not what is in use today. BR74c and BR74d parameterized sophisticated rate equations that can be evaluated at any arbitrary time and captured the detailed changes in S1 and S2 spectra (including an evolving mass variance). However, the BR74b bulk parameterization that has since been widely adopted by subsequent modelers is a simpler form of the rate equations [probably first suggested by Pruppacher and Klett (1978, p. 536)] based upon BR74b curve fits to the bin-model data. BR74b presented curve fits that related the mean mass and mass relative variance of an initial S1 distribution to the mean mass and number concentration of an S2 distribution resulting from all accretion and self-collection processes during a characteristic time scale. This characteristic time scale *T*_{2} is defined as the time at which the radius *r*_{gr} of the predominant rain mass of the developing S2 distribution first reaches 50 *μ*m in their bin model (BR74b, their Table 1^{1}). The following are some other important properties valid at time *T*_{2} (from BR74b):

*L*′_{2}and*N*′_{2}are S2’s total mass and number concentration, respectively;S2 first attains a Golovin shape;

S2’s mass relative variance is 1 (increasing from smaller values prior);

S2 obtains a mean-mass radius of ∼41

*μ*m; and*r*_{m}is the threshold radius corresponding to the minimum in the mass between the two modes of the total liquid spectrum.^{2}

Before *T*_{2}, cloud self-collection and rain accretion of cloud are dominant. Thereafter, self-collection of S2 becomes more important to growing and broadening the rain distribution (BR74c). Much later, the rain spectrum becomes narrower (BR74c). Time scale *T*_{2} is important to modelers because that was the only time for which BR74b reported/tabulated *both* the rain mass and rain number concentration from the bin model. The other time scales defined by BR74b are *T _{H}* (∼1.1

*T*

_{2}) at which the developing S2 mass distribution forms a “hump” and

*T*(∼1.25

*T*

_{2}) at which the radius of the predominant mass of the joint S1 + S2 (bimodal) distribution first reaches 50

*μ*m in radius.

*T̂*

_{2}represents the approximation of

*T*

_{2}from this regression. (All other variables herein with “hats” refer to the regression of the bin-model data.) It is very interesting that the

*T̂*

_{2}is completely predicted by the initial cloud conditions:

*L*

^{0}{g m

^{−3}} =

*ρ*{g m

^{−3}}

*q*{g g

_{w}^{−1}} is the initial cloud content and

*r*

^{0}

_{b}is the initial radius corresponding to the initial cloud mass relative variance var

*M*about an initial cloud mean-mass radius

*M*=

_{b}*M*(var

_{f}*M*)

^{1/2}and

*r*

^{3}∼

*M*, then

*M*) and mean (

_{g}*M*) masses,

_{f}*ν*. In the latter case, one substitutes

_{w}*M*=

_{g}*M*

^{k}^{=2}/

*M*

^{k}^{=1}and

*M*=

_{f}*M*

^{k}^{=1}/

*M*

^{k}^{=0}into Eq. (4) using the first three moments (

*k*= 0, 1, and 2) of the mass distribution [e.g., see Eq. (2) of Carrió and Nicolini (2002) or BR74b]. Substitution of

*α*into Eq. (3) results in var

*M*= (1 +

*ν*)

_{wO}^{−1}for the case of a Golovin (1963) distribution of particles [e.g., definition in Eq. (A3)], and one sees that this form of the mass variance is in terms of the shape/breadth parameter.

We have plotted Eq. (1) to confirm that it reproduces their curve fit (their Fig. 8) of their Table 1 bin-model results (not shown). Note that the regression equation for *T̂*_{2} printed on BR74b’s Fig. 8 (and herein) is correct but that BR74b’s Eq. (16) has incorrect typesetting. Also, note that *L*^{0} in BR74b’s Fig. 8 has units of grams per meter cubed and not the grams per centimeter cubed used elsewhere in their paper.

*L̂*′

_{2}is the approximate amount of cloud water mass converted to rainwater after

*T*

_{2}given a starting/input cloud water content

*L*

^{0}. The

*r*

^{0}

_{b}and

*r*

^{0}

_{f}are as previously defined. It was verified that this equation reproduces BR74b’s Fig. 9 curve fit to BR74b’s Table 1 bin-model data (not shown). Converting to SI units, rewriting radius in terms of diameter, and simplifying gives

*average*rate of change in rain mixing ratio during

*T̂*

_{2}for a bulk microphysics model:

*T̂*

_{2inv}=

*T̂*

^{−1}

_{2}and where we have limited the result to a positive quantity. BR74b do not suggest this average mixing ratio rate shown in Eq. (8); this is probably because they only consider those curve fits as “an intermediate step” (BR74b, p. 1825) to their parameterization and because BR74c and BR74d present a way to evaluate precise rates at any arbitrary time (rather than average rates by means of a characteristic time scale). Nevertheless, the simple form in Eq. (8) is what all subsequent bulk microphysics modelers have used and what is herein designated as the “BR74b parameterization” or “BR74b scheme.”

*T*

_{2}(from their Table 1). The result of that regression with a zero intercept assumption is shown in their Eq. (14):

*T*

_{2}:

*T*

_{2},

*M*

_{fr}_{2}{kg} = (3.5 × 10

^{9})

^{−1}, then Eq. (9) can be written

*M*

_{fr}_{2}of newly introduced particles corresponds to an average rain mean-mass diameter of

*M*=

_{r}*M*/

_{gr}*M*− 1) along with the BR74b findings that both var

_{fr}*M*≈ 1 [their Eq. (13)] and

_{r}*D*= 100

_{gr}*μ*m, by definition, at characteristic time

*T*

_{2}. After substitution and simplifying,

*D*

_{fr2}≈ 100

*μ*m/2

^{1/3}≈ 79

*μ*m for all BR74b bin-model simulations. Changes made by later authors to Eq. (10) will be discussed in section 2c.

There are several limitations to the BR74b scheme that will now be discussed. BR74b unfortunately do not give the remaining cloud water number concentration *N _{tw}* at

*T*

_{2}, and therefore an average −

*dN*/

_{tw}*dt*(owing to S1 self-collection and S1 accretion by S2) cannot be derived. Furthermore, accurate mass and number rates are difficult to define because the S1 and S2 distributions overlap. Also, CP00 have noted that cloud accretion by rain and rain self-collection both appear twice: once implicitly within the BR74b scheme and a second time explicitly parameterized. This results in double counting. BR74b also do not advise the user on how to adjust the rain number concentration rate, if at all, when there is only a single category to represent both preexisting large raindrops and the newly introduced S2 droplets (further discussion on this point will come later). Last, BR74b note that their regressions are based upon specific initial values in the range from 20 ≤

*D*

^{0}

_{f}≤ 36

*μ*m and 0.25 ≤ var

*M*≤ 1 (corresponding to Golovin distribution shape parameter limits 0 ≤

*ν*≤ 3) with collision efficiencies that do not include the effects of turbulence. BR74b (p. 1831) warned that “until further calculations are completed, extrapolations of the [BR74b] equations given here beyond these limits should be made with care.”

_{wO}It is the diagnostic solutions for *L̂*′_{2}, *N̂*′_{2}, and *T̂*_{2} and the prognostic solutions for *dq _{r}*/

*dt*,

*dN*/

_{tr}*dt*, and

*dN*/

_{tw}*dt*that will now be compared between BR74b scheme implementations and the bin-model solutions (below).

### a. Differences among BR74b scheme implementations

There are 10 manuscripts of which we are aware in the literature that present versions of the BR74b scheme in their studies: N86, P87, VC93, W95, M97, CN99, CP00, T04, T07, and MY05. Two of these papers, W95 and T04, have two variations presented (e.g., see Table 1), thereby giving a total of 12 versions of the bulk scheme for comparison with the BR74b bin-model results. There are two versions of the BR74b scheme by Cohard and Pinty (J.-P. Pinty 2005, personal communication) that are referenced herein. The first version is described in CP00, and the second/later version appears in both the Meso-NH model of Cohard and Pinty and in MY05 and is referred to herein by the latter reference. Note that although the M97 version did appear in print, it was not used for very long in practice and was upgraded to a more sophisticated scheme (C. Finley, S. van den Heever, and W. Cotton 2006, personal communications).

These 12 versions are grouped by their common characteristics in Tables 1 and 2 herein. Three of these schemes represent both cloud and rain with at least two moments while others predict one moment for cloud and either one or two moments for rain (Table 1). These schemes simulate cloud and rainwater with a monodispersed, generalized gamma, generalized Golovin, or lognormal distribution function (and not necessarily the same function for both cloud and rain). In the cases where the simulated cloud is monodispersed, another distribution function is assumed or a tabular value is used in order to have a nonzero mass relative variance. The differences in the choice of the distribution function, mean diameter, and how to represent dispersion affects how each author chose to represent *D*^{0}_{b} and *D*^{0}_{f} .

One can see from comparing the various forms of the equations (Table 2) that each scheme uses information regarding a mean size of the cloud water distribution. The original BR74b intended a mean-volume (or mean-mass) diameter, and most authors used that (Table 1). The mean-volume or mean-mass diameter has the same shape-independent formula for any of these four distributions: Golovin, generalized gamma, lognormal, or monodispersed (e.g., also see appendix A). However, some authors departed from BR74b and chose to use the mean diameter (VC93; Walko et al. 1995; Meyers et al. 1997), and Table 2 shows that that is dependent upon distribution shape (also see derivation in appendix A). It will be shown that the use of a different diameter has important consequences on the rain mass (and number concentration) growth rates.

In addition to the issue of mean-mass diameter versus mean diameter, one can also see from comparing the various forms of the equations (Table 2) that each scheme uses a representation of the cloud distribution’s mass relative variance. One may think of this mass relative variance in terms of the ratio between the predominant and mean masses, Eq. (3), or as a function of the shape/breadth of the cloud water distribution for the Golovin, generalized gamma, and lognormal distributions (e.g., see Table 2 and appendix B). By using the default values of *ν _{w}*,

*c*,

*σ*, or var

^{1/2}

*r*for each of these schemes, for example, one obtains different var

*M*and therefore different BR74b scheme mass growth rates even if everything else were equal (not shown). This largely explains the orders-of-magnitude differences in rates that we initially found between different implementations of the BR74b scheme.

To remove this difference among implementations, we attempted to equalize the mass relative variance by choosing appropriate shape parameters in each case (Table 3). For instance, BR74b’s *ν _{wO}* = 0 Golovin-distribution case gives the same cloud mass relative variance (var

*M*= 1.0) as the N86 lognormal distribution with

*σ*= 0.277 518 2 or the MY05 generalized gamma distribution with

*c*= 3 and

*ν*

_{w}_{Γ}= 1. In a similar way, shape parameters can also be chosen in most cases so that var

*M*= 1/4, which was true for the other set of experiments performed by BR74b. One will note that model limitations, in some cases, prevented the choosing of shape parameters that would match BR74b’s initial conditions. For instance, although

*ν*

_{w}_{Γ}can be set differently elsewhere, VC93 has a hardwired

*ν*

_{w}_{Γ}= 2 for the BR74b scheme giving an invariant cloud var

*M*= 1/3. Also, in T07, the shape parameter is diagnosed from the cloud number concentration, which makes it impossible to set BR74’s initial

*N*and var

_{tw}*M*. This largely explains T07’s poor performance in the upcoming comparison plots.

*D*

^{0}

_{f}and var

*M*(from Table 2) into

*T̂*

_{2}and

*L̂*′

_{2}, one may obtain an alternative form of Eq. (8) for VC93,

*a*

_{1}= 9.74 × 10

^{16}{m

^{−4}},

*a*

_{2}= 10.8 × 10

^{−3}{nondimensional},

*a*

_{3}= 1.124 836 × 10

^{5}{m

^{2}s

^{−1}kg

^{−1}}, and

*a*

_{4}= 2.027{m

^{3}s

^{−1}kg

^{−1}}, or for N86,

*M*(or

_{f}*D*

^{0}

_{f})is constant. They calculate

*M*by using

_{f}*σ*and

*D*, and we did so herein to match the var

_{G}*M*used in the original BR74b (see Table 3).

*T̂*

_{2}and

*L̂*′

_{2}for each version of scheme can be fairly compared with the original BR74 bin-model solution and each other (Fig. 1). The initial conditions for eight experiments follow Table 1 of BR74b whereby an initial cloud

*D*

^{0}

_{f}is set to 20, 24, 28, 32, or 36

*μ*m (set

**A**), all with a var

*M*= 1.0, and a

*D*

^{0}

_{f}of 28, 32, or 36

*μ*m (set

**B**), all with a var

*M*= 0.25. Each experiment also has initial values of

*L*

^{0}= 1 g m

^{−3}and

*T̂*

_{2}and

*L̂*′

_{2}[from Eqs. (5) and (7)] very close to

*T*

_{2}and

*L*′

_{2}of the bin model, respectively (Fig. 1a), confirming that the linear regressions are sufficient. It is similarly true that the average rate from

*L̂*′

_{2}/

*T̂*

_{2}for most versions of the scheme is a good approximation to the average bin-model rate

*L*′

_{2}/

*T*

_{2}(Fig. 1b). However, those schemes shown in Table 3 having incorrect mean diameters that are too small (e.g., VC93, M97, W95, and W95v2) or “too small” var

*M*(T07 for all cases and VC93, M97, W95, and W95v2 in some cases) result in

*L̂*′

_{2}that is too small and

*T̂*

_{2}that is too large (Fig. 1a) with resulting average rates that are too small (Fig. 1b). This is consistent with the experience of at least two users of the W95 version of the scheme who noted that it did not seem to produce enough rain in convective storms (C. Cohen and C. Finley 2006, personal communications). In the VC93 case for set

**B**, the competing influence of a smaller diameter is more important than the slightly larger var

*M*and gives an

*L̂*′

_{2}and average rate smaller than the bin model. These schemes with the smallest mass rates require higher cloud water contents before the BR74b scheme “switches on.”

Although it is both interesting and important to analyze these diagnostics, it would be considered more interesting to investigate their impact during a time evolution. However, several more prognostic equations that have become a part of the double-moment version of the BR74b scheme need to first be defined below.

### b. Implementations of cloud number concentration rate associated with the BR74b scheme

*T*

_{2}resulting from self-collection and accretion. Such would be needed for deriving a consistent cloud number concentration sink rate for those microphysics schemes that predict total number concentration for cloud water. CP00, MY05, and CN99 have attempted to remedy this omission by

*approximating*the total cloud number sink with the self-collection kernel by Long (1974):

*α*= (

_{w}*ν*+ 2)/(

_{wO}*ν*+ 1) is for a generalized Golovin distribution and

_{wO}*α*= Γ(

_{w}*ν*

_{w}_{Γ}+ 6/

*c*)Γ(

*ν*

_{w}_{Γ})/[Γ(

*ν*

_{w}_{Γ}+ 3/

*c*)]

^{2}is for the generalized gamma distribution

^{3}(also see Table 4). The combination of shape parameters that result in the same var

*M*=

*α*− 1 for different distributions (Table 3) also necessarily give the same Long (1974) self-collection rates through Eq. (14). A discussion of the interpretation of

*α*is provided in the introduction and appendix B (or see CN99). Although not all of these authors claim that the inclusion of Eq. (14) is to make up for a lack of

*dN*/

_{tw}*dt*within the BR74b scheme, it is our view that the

*impact*of such an inclusion is just that.

### c. Implementations of rain number concentration rate associated with the BR74b scheme

Another large variation between double-moment versions of BR74b’s scheme is in how authors have implemented *dN _{tr}*/

*dt*(see Table 5). Many authors simply use Eq. (10) as it is presented in BR74b (Table 5). However, because the focus of the BR74b scheme is only on early rain production, the continuous production of small raindrops at later times can overwhelm the larger raindrops that may already exist and artificially reduce their size. Discussions of this problem and potential solutions are found in CN99, CP00, and Carrió and Nicolini (2002).

It is our view that this problem may have more to do with a limitation in representing drizzle and rain within a single unimodal “rain” category as nearly all bulk microphysics schemes do, than it does with fundamental limitations within BR74b. Figure 2 shows observations of a bimodal drizzle and rain distribution [from Beard et al. (1986) and Willis and Tattelman (1989)] and Fig. 2b additionally includes a unimodal gamma function “best fitted” to the bimodal distribution. The fitting procedure results in reducing the mean diameter of the distribution by reducing (increasing) the number of large (small) droplets. Likewise, separate rain and drizzle modes cannot be well represented with a single gamma, Golovin, or lognormal distribution function as are used in cloud models. Regardless, below is how various authors have chosen to preserve the large raindrops when adding the S2 distribution.

CN99 and Carrió and Nicolini (2002) present a new method that favors preserving the predominant mass of an existing distribution (such as larger raindrops) when new smaller particles are added to that distribution (such as small S2 droplets). This results in an adjustment of the average rain number concentration source term in Eq. (10) to a lower value; however, the result is that the predominant mass is preserved and the model-diagnosed radar reflectivities are a closer match to the observed storm (CN99). Others such as Ferrier (1994) have taken a similar view, favoring the preservation of radar reflectivity with an *N _{t}* rate that preserves slope rather than strictly following an analytic formula for the

*N*rate.

_{t}*M**

_{gr}of the rain distribution at the end of the model time step Δ

*t*is a weighted average between the predominant rain mass that had existed at the beginning of Δ

*t*,

*t*,

*q**

_{r}= [

*q*

_{r}+ Δ

*t*(

*dq*

_{r}/

*dt*)]. In Eq. (16) the

*V*

_{S}_{2}is computed using the BR74b result that the mean mass of the newly introduced S2 over

*T̂*

_{2}is

*M*(

*D*

_{fr}_{2}) = (3.5 × 10

^{9})

^{−1}kg (corresponding to mean-mass diameter

*D*

_{fr}_{2}= 82

*μ*m) with approximately unity mass relative variance (in other words,

*α*

_{S2}≈ 2). Using a similar form as Eq. (15), one may rearrange and solve for the new number concentration at the end of the time step, which is consistent with the new predominant mass, giving

*T*

_{2}(

*D*

_{fr}_{2}= 82

*μ*m), or the mass associated with the mean existing rain diameter

*D*, that is,

_{fr}*N*rate equation advocated by Ferrier (1994). Similar to CP00, MY05 use Eq. (18) except that instead of

_{t}*D*

_{fr}_{2}, the diameter corresponding to the hump in S2 mass (found at later time

*T*) is used:

_{H}*D̂*

_{H}instead of

*D*

_{fr}_{2}is to produce fewer rain particles (slower rain number concentration rate).

The single-moment versions have a diagnostic *dN _{tr}*/

*dt*equation (Table 5), and it is there that one must choose from one of the following: a typical constant intercept parameter (

*n*

_{0}

*for P87, W95v2, and T04; herein set to the Marshall–Palmer default of 8 × 10*

_{r}^{6}m

^{−4}); a varying intercept parameter (T04v2 and T07); or a typical rain mean-volume diameter (W95; herein set to their default of 1 mm).

## 3. Time-dependent analysis

In this section, time-integrated solutions to the various BR74b scheme implementations are compared with the original BR74b bin-model solutions. The experiments herein have the same initial cloud distribution conditions as described in Table 3 (or Table 1 of BR74b) and use the rate equations presented herein (summarized in Tables 2, 4 and 5). A constant rain mass relative variance is also required for these simulations, and we choose what we think is a representative “average” value of 0.75 to represent the fact that the var*M _{r}* within the BR74b bin model increases to unity between times 0 and

*T*

_{2}. However, the var

*M*is necessarily much larger than 0.75 for some of the single-moment models (Table 6) because of limitations in the choices of shape parameters for some of those models.

_{r}*T*

_{2}when rain self-collection is small (as is the case for no preexisting rain distribution). Because the air remains saturated, evaporation is neglected. Although condensation source term could have been included, only the nonreplenished cloud case is studied here following BR74b. The simple budget for predicting two moments from previously presented equations herein can be summarized as follows:

*T*

_{2}when BR74b’s bin-model solution has a predominant radius of 50

*μ*m.

*M*:

_{g}*M*=

_{gr}*α*and

_{r}M_{fr}*M*=

_{gw}*α*. We place the additional criteria that if

_{w}M_{fw}*N*(or

_{tw}*N*) does not meet an arbitrary minimum threshold of 1 × 10

_{tr}^{−3}m

^{−3}, then

*M*(or

_{gw}*M*) is set to 0. These evolutions are then compared with the detailed bin-model evolution (Fig. 3), which is the reference solution.

_{gr}Perhaps it is not too surprising that those bulk cases that evaluate the incorrect *L̂*′_{2} and *T̂*_{2} (e.g., VC93; W95; M97 in Fig. 1) perform poorly in the time-integrated version of joint predominant diameter (Fig. 3). However, even some of those single-moment cases that give the correct *L̂*′_{2} at *t* = 0 result in skyrocketing predominant diameter evolutions (e.g., T04 and P87) and this is mostly a result of choosing a rain intercept parameter representative of mature rain rather than drizzle. By choosing a much larger intercept parameter consistent with drizzle (but unlikely to be used in a forecast model), those predominant diameter evolutions can be improved relative to the bin model (not shown). Furthermore, those *double-moment* cases giving a very accurate diagnosis of *L*′_{2} using *L̂*′_{2} (Fig. 1; N86; CN99; MY05; CP00) tend to give much better time-integrated forecasts of initial rain than the single-moment versions (Fig. 3). Perhaps the CN99 case performs best because it and the BR74 bin model are both designed to preserve the predominant diameter.

Inspection of the detailed evolution in Fig. 3 reveals that even the best-performing cases all develop the joint predominant diameter initially too rapidly and later too slowly. The bulk model solution in every case tends to develop more quickly earlier and more slowly approaching time *T*_{2}—precisely the opposite behavior of all of the bin-model solutions! This happens because the *L̂*′_{2} represents the change over one large *T̂*′_{2} step and thus the rate cannot vary with time like it should. Another oddity is that because autoconversion and accretion are both represented in the parameterization, accretion of cloud by rain effectively begins the first time step even before any rain has formed! Also, *L̂*′_{2} decreases each model time step since *L*^{0} is continually being reset after each time step. Therefore, the rate *L̂*′_{2}/*T̂*′_{2} starts off too large and ends too small. The net result, perhaps coincidentally, is bulk-model solutions in some cases that are not too far off from the bin model by time *T*_{2} (e.g., see CN99 and MY05 cases in Fig. 3).

Another way to evaluate the schemes is to compare the bulk model integrations of rain mass, rain number concentration, and predominant rain radius, with their respective bin-model values of *L*′_{2} and *N*′_{2} when *r*_{gr} = 50 *μ*m at *T*_{2} (Fig. 4). Even those cases that otherwise give good predictions of joint predominant radius sometimes (for large cloud mass) do not produce enough total rain mass by bin-model time *T*_{2} (MY05; CP00; CN99; Fig. 4b). The cases that produce a total number of S2 drops closest to the bin model are consistently CN99 and CP00 (Fig. 4a). Last, the S2 predominant radius is near *r*_{gr} = 50 *μ*m (as it should be, by definition, at time *T*_{2}) for many of the bulk cases including CN99 and CP00 (Fig. 4c). Of interest is that the S2 predominant radius in VC93 and M97 ends up close to the bin-model solution (Fig. 4c) even though their S2 masses and number concentrations are far off the mark (Figs. 4a,b). A summary of how each scheme ranks, on average, in each comparison with the bin model is presented in Table 7. CN99, CP00, and N86 consistently perform better than the others.

Experiments with preexisting mature rain distributions using the equation sets herein (not shown) indicate that the CN99, CP00, and MY05 cases may be superior relative to N86 because they can preserve the predominant or mean mass of existing large raindrops as new smaller S2 drops are added to the rain distribution. In contrast, the mean and predominant rain diameter in N86 (which performs well in the case of no preexisting rain) rapidly decreases as it is “seeded” with numerous S2. To *fairly* compare these remaining bulk models in simulating *mature* rain distributions, one would need to develop *consistent* formula describing rain self-collection, rain accretion of cloud, and raindrop breakup for each probability distribution function within each model and compare the results with a BR74a bin-model reference solution that includes drop breakup. We suggest that the inability of representing both drizzle and large raindrops within a unimodal distribution function might be bypassed in the future by simply including a separate species category for drizzle.^{4}

## 4. Conclusions

The Berry and Reinhardt (1974b) autoconversion parameterization was presented and compared with 12 versions of itself from the literature (with any typesetting errors corrected but any intentional changes or additions retained). It was found that large differences in average growth rates for rain mass and rain number concentration result among versions for 1) the use of mean diameter instead of the mean-volume diameter and 2) model constraints that limit the range of simulated cloud dispersions. However, it was found that even those implementations that give perfect agreement with the bin model in the average rate formulas sometimes do not give good agreement when those rate formulas are integrated with a model time step that is much smaller than the BR74b scheme’s characteristic time scale (Δ*t* ≪ *T*_{2}). None of the bulk implementations reproduce the detailed bin-model S2 growth pattern of slow initial growth and faster later growth; rather, they do the exact opposite. Nevertheless, these evolution errors offset each other so that some implementations actually still give a good approximation to the bin model by time *T*_{2}.

Three schemes use the wrong mean diameter in the BR74b formula (VC93, W95, and M97), which causes reduced S2 mass and number production rates, and, because of that, the BR74b scheme “shuts down” under some conditions when it should not. The T04 and P87 implementations are single-moment schemes that perform poorly, and this is related to using a small rain intercept parameter (more representative of mature rain than small drizzle) as well as to using excessive var*M _{r}* (resulting from a lack of versatility in those model’s shape parameters). The best performing one-moment scheme is T04v2, and its better performance owes to use of a variable rain intercept parameter.

The schemes that additionally predict number concentration for S2 are even better in tracking the bin model’s evolution: N86, CN99, and CP00. Comparison of the results between N86 (two moment in *q _{r}* only) and CN99 and CP00 (two moment in

*q*and

_{r}*q*) suggests that predicting the second moment for cloud is apparently not essential in obtaining a good match against the bin model for the N86 case.

_{w}We speculated that the rain might be better represented in the CN99, CP00, and MY05 schemes (after time *T*_{2}) than in the N86 scheme. This is because those three double-moment implementations include modifications to the original BR74b rain number concentration rates in an effort to better preserve the maturing rain distribution characteristics [also see discussion in those papers and Carrió and Nicolini (2002)]. It is argued that such preservation is necessary because otherwise existing large raindrops could unrealistically decrease in size/mass as the rain distribution number concentration increases with the addition of copious S2 resulting from the BR74b scheme.

The idealized experiments herein ignore the observational uncertainties that would be present when using a real-time single-moment or double-moment forecast model (such as distribution shape parameters, cloud condensation nuclei amount, nucleation rate, etc). Such uncertainties could likely cause differences in solution larger than those seen among these different implementations of BR74b. What are needed in the future are reliable formulas that relate the cloud distribution characteristics to other prognosed environmental variables; T07 have taken important steps in that direction.

Some may ask why modelers do not use the more sophisticated parameterization presented in BR74c and BR74d. After all, it was what Berry and Reinhardt had intended for modelers to use [and not the simpler BR74b implementation first advocated by Pruppacher and Klett (1978)]. This is a good question that is perhaps related to the computer power available when the schemes were first implemented in the late 1970s. Regardless of which parameterization is used, we see several reasons why the BR74 experiments need to be repeated in the future.

First, rather than approximating the *dN _{tw}*/

*dt*during

*T*

_{2}using the Long kernel for self-collection, the actual

*dN*/

_{tw}*dt*from the bin model could be parameterized so that it also includes both the self-collection and implicit accretion experienced. Second, BR74b note that the range of var

*M*that they originally tested is small (indeed much smaller than the range of observed values; e.g., see Table B1) and needs to be expanded to improve its versatility. The poor performance of the T07 scheme (which uses observed var

*M*values smaller than 1/4) may in part be because the BR74b scheme cannot be reliably extrapolated to those var

*M*s. Third, it is important to confirm that the same curve fits for

*L̂*′

_{2}and

*T̂*

_{2}can be derived regardless of the distribution function that is simulated.

^{5}Fourth, the impact of turbulence and other factors on the collection efficiencies could be incorporated as additional parameters. Fifth, one might consider separating the rate of autoconversion from the rate of accretion, and this might help to solve the double counting problems that result when the BR74b scheme is used in conjunction with separate schemes of rain self-collection and rain accretion of cloud. Sixth, one might investigate whether evaluating the mass and number of S1 and S2 with a fixed size threshold within the bin-model equations [following Beheng and Doms (1986)] gives a significantly different result than using a “floating” size threshold and evaluating S1 and S2 from the bin output (following BR74b). All of these reasons suggest that the bin-model experiments should be repeated to rederive the average rate relationships as presented in BR74b. Because the BR74 model and other bin models like it are known to have artificial spreading that may influence any parameterizations derived from them, these bin-model experiments might also be reperformed with one or more schemes known to have greater accuracy such as the moment-conserving scheme of Tzivion et al. (1987).

## Acknowledgments

This work was supported in part by National Science Foundation Grants ATM-0446509, ATM-0339519, and ATM-0449753. We thank Greg Thompson and three anonymous reviewers who helped to improve an earlier draft. We thank Drs. G. Carrió, C. Cohen, W. Cotton, C. Finley, M. Meyers, J.-P. Pinty, R. Walko, and S. van den Heever for helpful discussions and/or for providing source code.

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## APPENDIX A

### Derivation of Df

There are two types of “mean diameters” that have been implemented into the BR74b scheme in the past when representing the separate cloud and rain distributions: the mean-volume or mean-mass diameter (the diameter corresponding to the mean of all of the drop masses or volumes) and the mean diameter (the diameter corresponding to the mean of all the drop diameters). Those are now derived for the different distribution functions used in the different implementations. It will be shown that the mean-volume/mean-mass diameter is the same for all schemes, which makes it particularly useful for comparing different implementations of the BR74b scheme.

*ρ*is the density of liquid water. The mean mass can then be defined using

_{w}*ν*> −1. Here,

_{O}*N*is total number concentration,

_{t}*ν*is the shape of the distribution,

_{O}*n*(

*m*)

*δm*is the number of drops per unit volume between mass

*m*and

*m*+

*δm*, and

*M*is the characteristic mass of the distribution. Noting that the

_{f}*x*th moment is given by

*x*> −1 and

*a*> 0, one may substitute Eq. (A3) into Eq. (A2) and integrate Eq. (A2), resulting in

*m*

_{mean}=

*M*, then

_{f}*D*

_{meanmass}=

*D*. Furthermore, with a similar derivation as above (or a simple substitution of variables), it is trivial to show that the mean-mass diameter is equivalent to the mean-volume diameter used by CN99, who also used a Golovin number density function. It is also trivial to show that the diameter of the mean mass/volume for a monodispersed distribution is the same as Eq. (A6).

_{f}*c*and

*ν*

_{Γ}are two separate shape parameters (not to be confused with the shape parameters from the Golovin distribution) that are both greater than 0 and where

*D*is the characteristic diameter (identical to CP00’s 1/

_{n}*λ*term) that can be solved for after integrating

*σ*is the standard deviation of the log-transformed diameters and

*D*is the geometric mean diameter, which is related to the mean of the log-transformed diameters

_{G}*μ*through

*D*= exp(

_{G}*μ*). Nickerson et al. (1986) also give the following relationship to solve for the mean-volume diameter:

*D*is used; in the case of cloud water,

_{G}*D*is assumed to be constant, which implies a constant

_{G}*D*

_{meanvol}and

*D*

^{0}

_{f}regardless of cloud mass. To preserve

*D*

_{meanvol}, cloud

*N*must adjust along with

_{t}*q*. To find

*D*for the rain case, one may substitute Eq. (A10) into

_{G}*D*:

_{G}*D*gives

_{n}*D*

_{mean}differs from

*D*

_{meanvol}by a factor involving the distribution shape parameters. Verlinde and Cotton (1993), Walko et al. (1995), and Meyers et al. (1997) all assume that

*c*= 1 so that Eq. (A14) may also be written

*D*

_{mean}=

*D*

_{n}ν_{Γ}. Last, Verlinde and Cotton additionally assume that

*ν*

_{Γ}= 2 always.

## APPENDIX B

### Derivation of varM

All versions of BR74b use the same definition of *D*^{0}_{b} = *D*^{0}_{f}(var*M*)^{1/6} for cloud water droplets. However, the appearance of var*M* differs among cases.

#### The generalized gamma distribution implementation

*M*used by CP00 and MY05, var

*M*= (

*M*/

_{g}*M*) − 1 =

_{f}*α*− 1 is calculated for a diameter-distributed generalized gamma distribution function shown in Eq. (A7) herein. In this case,

*M*and

_{g}*M*are defined by the first three moments of length (

_{f}*k*= 0, 3, and 6). By use of the moment-generating function [e.g., Eq. (3) of CP00],

*M*may not be immediately clear for readers of CP00 because they show a different form of the

*D*

^{0}

_{b}equation:

*D*(equivalent to CP00’s 1/

_{n}*λ*) is the characteristic diameter [shown in Eq. (A8) herein]. Through a few substitutions, one may see that the CP00 form indeed follows the same general definition of

*D*

^{0}

_{b}=

*D*

^{0}

_{f}(var

*M*)

^{1/6}. Substitution of

*D*[shown in Eq. (A8) herein] into Eq. (B2) results in

_{n}*D*as was derived herein [Eq. (A6)] for the first group of terms and rewrite the exponents for the second group of terms, resulting in

_{f}*D*

^{0}

_{b}is the same as BR74b’s

*D*

^{0}

_{b}=

*D*

^{0}

_{f}(var

*M*)

^{1/6}.

#### The generalized Golovin mass distribution implementation

*M*used in the case of the Golovin mass distribution [Eq. (A3) herein], we start with

*M*=

_{g}*M*

^{k}^{=2}/

*M*

^{k}^{=1}and

*M*=

_{f}*M*

^{k}^{=1}/

*M*

^{k}^{=0}using the first three moments (

*k*= 0, 1, and 2) and the moment-generating function (e.g., see Carrió and Nicolini 2002), resulting in

*M*=

*α*− 1, and simplifying, results in

*O*is used to indicate that this shape parameter is for a Golovin distribution.

#### The lognormal distribution implementation

*M*for a lognormal distribution [Eq. (A10) herein], we start with

*M*=

_{g}*M*

^{k}^{=6}/

*M*

^{k}^{=3}and

*M*=

_{f}*M*

^{k}^{=3}/

*M*

^{k}^{=0}using three moments of length (

*k*= 0, 3, and 6) and the moment-generating function,

*M*/

_{g}*M*= exp(9

_{f}*σ*

^{2}) =

*α*. Substituting that into var

*M*=

*α*− 1, and simplifying, results in

#### The implementation when using observed spectra

In the Proctor (1987) version, observed values of *N _{tc}* and cloud relative size dispersion var

^{1/2}

*r*are used (see Table B1). The approximate relationship var

*M*≈ (var

^{1/2}

*r*/0.38)

^{2}is used [e.g., Eq. (16b) of BR74a], resulting in the required mass relative variance.

In summary, all versions of BR74b use the same definition of *D*^{0}_{b} = *D*^{0}_{f}(var*M*)^{1/6} but the var*M* differs depending upon the assumed or actual cloud mass relative variance. Note that although in this context it is used for cloud droplets, these definitions can also be used for other types of species distributions.

**APPENDIX C**

### List of Symbols

Table C1 contains a list of the symbols used in this paper, along with their definitions, values, and associated units.

Example drop size distributions showing raindrops coexisting with more numerous drizzle drops as (a) observed with the 2D precipitation probe just above cloud base (temperature of 20°C) in a rainband near Cape Kumukahi on the island of Hilo, HI, (from Beard et al. 1986) and (b) for all 10-s optical spectrometer samples of all hurricane and tropical storm cases from 1975 to 1982 having rain rates > 225 mm h^{−1} at 450-m altitude, bin averaged with a 200-*μ*m-diameter interval, as compared with a corresponding best-fit gamma function (from Willis and Tattelman 1989).

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1573.1

Example drop size distributions showing raindrops coexisting with more numerous drizzle drops as (a) observed with the 2D precipitation probe just above cloud base (temperature of 20°C) in a rainband near Cape Kumukahi on the island of Hilo, HI, (from Beard et al. 1986) and (b) for all 10-s optical spectrometer samples of all hurricane and tropical storm cases from 1975 to 1982 having rain rates > 225 mm h^{−1} at 450-m altitude, bin averaged with a 200-*μ*m-diameter interval, as compared with a corresponding best-fit gamma function (from Willis and Tattelman 1989).

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1573.1

Example drop size distributions showing raindrops coexisting with more numerous drizzle drops as (a) observed with the 2D precipitation probe just above cloud base (temperature of 20°C) in a rainband near Cape Kumukahi on the island of Hilo, HI, (from Beard et al. 1986) and (b) for all 10-s optical spectrometer samples of all hurricane and tropical storm cases from 1975 to 1982 having rain rates > 225 mm h^{−1} at 450-m altitude, bin averaged with a 200-*μ*m-diameter interval, as compared with a corresponding best-fit gamma function (from Willis and Tattelman 1989).

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1573.1

Evolution of the predominant radius of the joint (rain plus cloud) distribution for each version of the BR74b parameterization (black lines) in comparison with the original BR74a bin-model results (dashed lines; from Fig. 7 of BR74b). Panels here are labeled by the initial cloud conditions of mean volume radius and mass relative variance. The experiments include (a)–(d) cloud mass relative variance var*M* = 1 and initial mean cloud radius *r*^{0}_{f} of 18, 14, 12, and 10 *μ*m, respectively, and (e), (f) cloud var*M* = 0.25 and *r*^{0}_{f} of 18 and 14 *μ*m, respectively. Other details about the initial conditions are in Table 3. Only those experiments that have a corresponding bin-model trace (from Fig. 7 of BR74b) are shown here.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1573.1

Evolution of the predominant radius of the joint (rain plus cloud) distribution for each version of the BR74b parameterization (black lines) in comparison with the original BR74a bin-model results (dashed lines; from Fig. 7 of BR74b). Panels here are labeled by the initial cloud conditions of mean volume radius and mass relative variance. The experiments include (a)–(d) cloud mass relative variance var*M* = 1 and initial mean cloud radius *r*^{0}_{f} of 18, 14, 12, and 10 *μ*m, respectively, and (e), (f) cloud var*M* = 0.25 and *r*^{0}_{f} of 18 and 14 *μ*m, respectively. Other details about the initial conditions are in Table 3. Only those experiments that have a corresponding bin-model trace (from Fig. 7 of BR74b) are shown here.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1573.1

Evolution of the predominant radius of the joint (rain plus cloud) distribution for each version of the BR74b parameterization (black lines) in comparison with the original BR74a bin-model results (dashed lines; from Fig. 7 of BR74b). Panels here are labeled by the initial cloud conditions of mean volume radius and mass relative variance. The experiments include (a)–(d) cloud mass relative variance var*M* = 1 and initial mean cloud radius *r*^{0}_{f} of 18, 14, 12, and 10 *μ*m, respectively, and (e), (f) cloud var*M* = 0.25 and *r*^{0}_{f} of 18 and 14 *μ*m, respectively. Other details about the initial conditions are in Table 3. Only those experiments that have a corresponding bin-model trace (from Fig. 7 of BR74b) are shown here.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1573.1

Values for the rain distribution’s (a) total number concentration (No. m^{−3}), (b) total mass concentration (g m^{−3}), and (c) predominant radius (*μ*m) after integrating each BR74b parameterization (labels) and bin model (symbols) to time *T*_{2} for each set of initial conditions shown in Table 3 and the key in (a). Abbreviations are given in the (b) and (c) keys for those BR74b scheme solutions that can be grouped by similar or identical values.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1573.1

Values for the rain distribution’s (a) total number concentration (No. m^{−3}), (b) total mass concentration (g m^{−3}), and (c) predominant radius (*μ*m) after integrating each BR74b parameterization (labels) and bin model (symbols) to time *T*_{2} for each set of initial conditions shown in Table 3 and the key in (a). Abbreviations are given in the (b) and (c) keys for those BR74b scheme solutions that can be grouped by similar or identical values.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1573.1

Values for the rain distribution’s (a) total number concentration (No. m^{−3}), (b) total mass concentration (g m^{−3}), and (c) predominant radius (*μ*m) after integrating each BR74b parameterization (labels) and bin model (symbols) to time *T*_{2} for each set of initial conditions shown in Table 3 and the key in (a). Abbreviations are given in the (b) and (c) keys for those BR74b scheme solutions that can be grouped by similar or identical values.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1573.1

Distribution representations of cloud and rain, distribution assumptions for the cloud relative dispersion, and number of predicted moments for cloud and rain used in cited implementations of BR74b’s autoconversion parameterization. The abbreviation that refers to each implementation is also shown along with any distinguishing characteristics in parentheses. One (two) moment(s) implies prediction of *q* (and *N _{t}*). The symbols are defined as follows:

*O*= generalized Golovin distribution,

*M*= monodisperse distribution,

*L*= lognormal distribution, and Γ = generalized gamma distribution.

Presentations of the components of the *L̂*′_{2} and *T̂*_{2} terms of the BR74b scheme in the literature. All terms in the table apply to cloud water and are written here in terms of *D* and SI units, and with any typographical errors from the original manuscripts corrected herein. Derivations of *D*^{0}_{f} and var*M* are in appendixes A and B, respectively. All versions use *L*^{0} = *ρq _{w}*,

*D*

^{0}

_{b}=

*D*

^{0}

_{f}(var

*M*)

^{1/6},

*L̂*′

_{2}= 2.7 × 10

^{−2}[(10

^{20}/16)(

*D*

^{0}

_{b})

^{3}(

*D*

^{0}

_{f})− 0.4]

*L*

^{0}, and

*T̂*

_{2}= 3.72〈[0.5 × 10

^{6}(

*D*

^{0}

_{b}) − 7.5]

*L*

^{0}〉

^{−1}, and these equations are valid over a range 20 ≤

*D*

^{0}

_{f}≤ 36

*μ*m and var

*M*of 1 to 1/4 (or 0 ≤

*ν*≤ 3, where

_{wO}*ν*is the Golovin distribution shape parameter). Note that

_{wO}*D*

_{meanvol}= [6

*ρq*/(

_{w}*πρ*)]

_{w}N_{tw}^{1/3}and that

*ν*

_{w}_{Γ},

*c*,

*ν*,

_{wO}*D*,

_{Gw}*σ*, and var

^{1/2}

*r*are other miscellaneous distribution constants.

The various BR74b-parameterization initial cloud distribution conditions set to match (as closely as possible) the initial conditions of the BR74b bin model. BR74b tested an initial cloud mean-volume diameter *D*^{0}_{f} of 20, 24, 28, 32, and 36 *μ*m (set **A** below) for a cloud var*M* = 1.0 (*ν _{wO}* = 0) and a

*D*

^{0}

_{f}of 28, 32, and 36

*μ*m (set

**B**below) for a var

*M*= 0.25 (

*ν*= 3). These two sets,

_{wO}**A**and

**B**, total eight experiments. Their total initial cloud water mass content

*L*

^{0}and number concentration

*N*were set to 1 kg m

_{tw}^{−3}and 6

*L*

^{0}/[

*π*

*ρ*

_{w}(

*D*

^{0}

_{f})

^{3}] m

^{−3}, respectively.

Cloud water number concentration rates for the BR74b schemes. All terms are written here in terms of *D* and SI units, and with any errors present in the original references corrected.

Rain number concentration rate for the various BR74b schemes (written here in terms of *D* and SI units). In T04, *n _{or}*

_{max}= 1 × 10

^{10},

*n*

_{or}_{min}= 8 × 10

^{6}, and

*z*= 4. In T07,

*n*

_{or}_{max}= 2 × 10

^{9},

*n*

_{or}_{min}= 2 × 10

^{6}, and

*z*= 0.25.

The BR74b distribution constraints that give var*M _{r}* = 0.75. BR74a found that var

*M*grew from small values and approached unity at time

_{r}*T*

_{2}. The var

*M*was evaluated using formulas presented in appendix B.

_{r}Rankings for each of the schemes averaged over all experiments (totaling six for joint predominant radius and eight for all others) for important rain distribution properties. Before calculating the average, the ranks for each scheme were assigned in ascending order of absolute difference between the bulk and bin-model solutions. Also, schemes with equal solutions were assigned the same rank, and inactive schemes were automatically assigned the last/largest ranking before calculating the average.

Table B1. Observed cloud droplet number concentrations, relative size dispersions, and calculated mass relative variance [var*M* ≈ (var^{1/2}*r*/0.38)^{2}] for different environments used in the Proctor (1987) implementation of BR74b. In boldface are those var*M* that fall outside the range simulated by BR74a and parameterized by BR74b. (Adapted from Proctor 1987)

Table C1. List of symbols.

^{1}

That table provides raw bin-model results (and the units/magnitudes they show are correct).

^{2}

This definition was necessary so that BR74b could evaluate the mass and number concentration for the S1 and S2 distributions.

^{3}

The appearance of the generalized gamma form of Eq. (14) can be transformed into that shown by CP00 after a substitution of *D _{n}* and some algebraic manipulations.

^{4}

Although they do not have a separate drizzle category, Saleeby and Cotton (2004) have demonstrated the impact of including two modes for a cloud distribution that was previously only represented by one mode.

^{5}

We speculate that different distribution functions integrated explicitly with the stochastic collection equation will reveal different rain production rates even with the same var*M*, initial cloud mass, and cloud number concentration.