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 T₂ is defined as the time at which the radius r_gr of the predominant rain mass of the developing S2 distribution first reaches 50 μm in their bin model (BR74b, their Table 1¹). The following are some other important properties valid at time T₂ (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 μm; 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 T₂, 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 T₂ 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 (∼1.1T₂) at which the developing S2 mass distribution forms a “hump” and T (∼1.25T₂) at which the radius of the predominant mass of the joint S1 + S2 (bimodal) distribution first reaches 50 μm in radius.

For the purposes of this review, we will focus on the earliest time scale presented in BR74b because it is the one for which both mass and number concentration are defined. That earliest characteristic time is represented in the first curve-fit equation of BR74b’s Fig. 8 (in this and subsequent equations, units are presented in curly braces):

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where the variables on the right-hand side represent initial values of the cloud distribution (as indicated by the “0” superscripts) and T̂₂ represents the approximation of T₂ from this regression. (All other variables herein with “hats” refer to the regression of the bin-model data.) It is very interesting that the T̂₂ is completely predicted by the initial cloud conditions: L⁰{g m⁻³} = ρ{g m⁻³}q_w{g g⁻¹} is the initial cloud content and r⁰_b is the initial radius corresponding to the initial cloud mass relative variance varM about an initial cloud mean-mass radius

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Because M_b = M_f(varM)^1/2 and r³ ∼ M, then

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The mass relative variance,

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can be expressed either in terms of a ratio of the predominant (M_g) and mean (M_f) masses,

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(e.g., Carrió and Nicolini 2002), or as a shape/breadth parameter ν_w. In the latter case, one substitutes 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 varM = (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 T̂₂ printed on BR74b’s Fig. 8 (and herein) is correct but that BR74b’s Eq. (16) has incorrect typesetting. Also, note that L⁰ in BR74b’s Fig. 8 has units of grams per meter cubed and not the grams per centimeter cubed used elsewhere in their paper.

After converting to SI units, converting radius to diameter, and simplifying, the result is

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The second equation involved in the BR74b scheme is shown in their Eq. (18) and their Fig. 9 as follows (except that we include units that were omitted by BR74b):

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where L̂′₂ is the approximate amount of cloud water mass converted to rainwater after T₂ given a starting/input cloud water content L⁰. The r⁰_b and r⁰_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

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Pruppacher and Klett (1978) were probably the first to suggest that Eqs. (5) and (7) can be combined to obtain an average rate of change in rain mixing ratio during T̂₂ for a bulk microphysics model:

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where T̂_2inv = T̂⁻¹₂ 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.”

BR74b also fit a curve between their bin-model results of total rain mass and total rain number concentration at time T₂ (from their Table 1). The result of that regression with a zero intercept assumption is shown in their Eq. (14):

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We confirmed their calculation with a regression line fit to the bin-model data shown in their Table 1 and similarly obtained

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From there, VC93 (the first to use this scheme in a double-moment model) defined the average rain number concentration rate accounting for the net autoconversion and rain self-collection over T₂:

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After a conversion to SI units, substitution, and noting that the slope parameter in this equation is also the average rain mass at time T₂, M_fr2{kg} = (3.5 × 10⁹)⁻¹, then Eq. (9) can be written

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By inspection of Eq. (10), the mean mass M_fr2 of newly introduced particles corresponds to an average rain mean-mass diameter of

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which is the same value as what BR74b report in their Eq. (12). One may also arrive at nearly this same rain diameter by combining Eqs. (3) and (4) above (varM_r = M_gr/M_fr − 1) along with the BR74b findings that both varM_r ≈ 1 [their Eq. (13)] and D_gr = 100 μm, by definition, at characteristic time T₂. After substitution and simplifying, D_fr2 ≈ 100 μm/2^1/3 ≈ 79 μm 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 T₂, 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 ≤ D⁰_f ≤ 36 μm and 0.25 ≤ varM ≤ 1 (corresponding to Golovin distribution shape parameter limits 0 ≤ ν_wO ≤ 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.”

It is the diagnostic solutions for L̂′₂, N̂′₂, and T̂₂ 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 D⁰_b and D⁰_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^1/2r for each of these schemes, for example, one obtains different varM 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 (varM = 1.0) as the N86 lognormal distribution with σ = 0.277 518 2 or the MY05 generalized gamma distribution with c = 3 and ν_wΓ = 1. In a similar way, shape parameters can also be chosen in most cases so that varM = 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_wΓ can be set differently elsewhere, VC93 has a hardwired ν_wΓ = 2 for the BR74b scheme giving an invariant cloud varM = 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_tw and varM. This largely explains T07’s poor performance in the upcoming comparison plots.

Before showing comparison plots, a few comments should be made regarding the very different appearance of the BR74b equations presented in VC93 and N86. By substituting respective values of D⁰_f and varM (from Table 2) into T̂₂ and L̂′₂, one may obtain an alternative form of Eq. (8) for VC93,

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where a₁ = 9.74 × 10¹⁶{m⁻⁴}, a₂ = 10.8 × 10⁻³{nondimensional}, a₃ = 1.124 836 × 10⁵{m² s⁻¹ kg⁻¹}, and a₄ = 2.027{m³ s⁻¹ kg⁻¹}, or for N86,

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where

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We have included units and corrected terms in both cases. One interesting assumption made by N86, which is made by no other version for the BR74b scheme, is that the cloud’s M_f (or D⁰_f )is constant. They calculate M_f by using

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(see definitions in appendix A). N86 allow alteration of σ and D_G, and we did so herein to match the varM used in the original BR74b (see Table 3).

After equalizing the initial conditions as much as possible, T̂₂ 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 D⁰_f is set to 20, 24, 28, 32, or 36 μm (set A), all with a varM = 1.0, and a D⁰_f of 28, 32, or 36 μm (set B), all with a varM = 0.25. Each experiment also has initial values of L⁰ = 1 g m⁻³ and

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following BR74b. It is immediately apparent that most of the schemes diagnose T̂₂ and L̂′₂ [from Eqs. (5) and (7)] very close to T₂ 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/T̂₂ for most versions of the scheme is a good approximation to the average bin-model rate L′₂/T₂ (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” varM (T07 for all cases and VC93, M97, W95, and W95v2 in some cases) result in L̂′₂ that is too small and T̂₂ 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 varM 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

BR74b unfortunately do not provide the bin model’s reduced net cloud number concentration at time T₂ 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):

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where α_w = (ν_wO + 2)/(ν_wO + 1) is for a generalized Golovin distribution and α_w = Γ(ν_wΓ + 6/c)Γ(ν_wΓ)/[Γ(ν_wΓ + 3/c)]² is for the generalized gamma distribution³ (also see Table 4). The combination of shape parameters that result in the same varM = 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_t rate.

The CN99 scheme (Table 5) assumes that the newly formed predominant mass M*_gr of the rain distribution at the end of the model time step Δt is a weighted average between the predominant rain mass that had existed at the beginning of Δt,

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and the predominant mass S2 produced owing to the BR74b process during Δt,

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giving

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where the new rain mixing ratio is q*_r = [q_r + Δt(dq_r/dt)]. In Eq. (16) the V_S2 is computed using the BR74b result that the mean mass of the newly introduced S2 over T̂₂ is M(D_fr2) = (3.5 × 10⁹)⁻¹ kg (corresponding to mean-mass diameter D_fr2 = 82 μm) 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

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The number concentration rate can then be simply written

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These equations can be combined into the condensed form shown in Table 5.

CP00 present a simpler method to address the problem of excessive S2 “seeding” a preexisting rain distribution. There, the mean rain mass rather than the predominant rain mass is preserved. The S2 particles that are introduced have the larger of the following two masses: the mass associated with the mean rain diameter at time T₂ (D_fr2 = 82 μm), or the mass associated with the mean existing rain diameter D_fr, that is,

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In the second case in which rain already exists, new S2 is literally “reshaped” into larger raindrops to preserve the mean mass of the droplets in the rain distribution, thereby resulting in a correspondingly smaller number concentration rate. In that case, Eq. (18) may also be rewritten as

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which readers may recognize as the general slope-preserving N_t rate equation advocated by Ferrier (1994). Similar to CP00, MY05 use Eq. (18) except that instead of D_fr2, the diameter corresponding to the hump in S2 mass (found at later time T_H) is used:

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Note that this has been converted to SI units along with a factor-of-10 correction to BR74b’s typeset equation. The main result of using D̂_H instead of D_fr2 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_0r for P87, W95v2, and T04; herein set to the Marshall–Palmer default of 8 × 10⁶ 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 varM_r within the BR74b bin model increases to unity between times 0 and T₂. However, the varM_r 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.

The simulations are conducted with a very simple water budget within a bulk microphysics framework to focus on the BR74b process for a single grid cell in space. This equation set is most valid for integrations from 0 to T₂ 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:

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The time step used in the simulations herein is 1 s. Solutions are integrated until time T₂ when BR74b’s bin-model solution has a predominant radius of 50 μm.

We now diagnose the evolution of joint predominant radius for the bulk-model solutions by calculating it using BR74b’s definition of the joint predominant mass M_g:

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where M_gr = α_rM_fr and M_gw = α_wM_fw. We place the additional criteria that if N_tw (or N_tr) does not meet an arbitrary minimum threshold of 1 × 10⁻³ m⁻³, then M_gw (or M_gr) is set to 0. These evolutions are then compared with the detailed bin-model evolution (Fig. 3), which is the reference solution.

Perhaps it is not too surprising that those bulk cases that evaluate the incorrect L̂′₂ and T̂₂ (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 T₂—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 L⁰ 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 T₂ (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 μm at T₂ (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 T₂ (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 μm (as it should be, by definition, at time T₂) 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 (Δt ≪ T₂). 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 T₂.

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 varM_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_r and q_w) 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.

We speculated that the rain might be better represented in the CN99, CP00, and MY05 schemes (after time T₂) 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 T₂ 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 varM 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 varM values smaller than 1/4) may in part be because the BR74b scheme cannot be reliably extrapolated to those varMs. Third, it is important to confirm that the same curve fits for L̂′₂ and T̂₂ 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).