## 1. Introduction

Shallow low-level clouds within the marine boundary layer (MBL) are known to be an important regulator of the earth’s radiative budget because of their large spatial extent and omnipresence (Hartmann et al. 1992). Because of their proximity to the surface, changes in the cloud properties and their lifetime alter the albedo effect disproportionately compared to their emissivity. The strong susceptibility of marine stratiform clouds to changes in environmental conditions increases the difficulty in physically describing their properties and evolution (e.g., Stevens and Brenguier 2009; Wood et al. 2009). Consequently, uncertainties in estimates of climate sensitivity have found to be correlated to the predicted changes of marine stratiform cloud amount, suggesting a causal relation (Stephens 2005; Bony and Dufresne 2005; Soden and Vecchi 2011). It is therefore crucial to be able to appropriately represent the physical processes associated with these types of clouds in atmospheric models.

One key process is the formation of precipitation (here only warm-phase precipitation is considered), which, although small in amount and referred to as drizzle in shallow marine clouds, is dependent on the state of the cloud. Following Stevens and Brenguier (2009), the latter is mainly determined by the dynamics and the atmospheric aerosol, where in principle the dynamics (i.e., large-scale dynamics and thermodynamic state) governs the macrostructure and aerosols (i.e., cloud condensation nuclei) govern the microstructure of the cloud. Precipitation-size drops are characterized by their terminal fall velocity being large enough to sediment below cloud base without them necessarily reaching the surface. Collision and coalescence are the driving processes for the formation of drizzle and the limitation in its onset lies in the production of sufficiently large drops around 20 *μ*m in radius (Pruppacher and Klett 1997). Without having to follow the evolution of every single drop, liquid water is divided into two classes following Kessler (1969), where one accounts for cloud and the other for rainwater. This bulk approach is what most conventional atmospheric models adopt (for both one- and two-moment schemes, where for the latter traditionally the additional information on drop number concentration is used). It implies discrete and artificial processes describing the transition from one class into the next by the collection of drop pairs; self-collection is the term used for the collection of a pair of drops within the same class and resulting in a drop remaining in that class, autoconversion is when two cloud droplets form a raindrop, and accretion is when a cloud droplet is collected by a raindrop (Beheng 2010). Depending on the environmental conditions, precipitation does not necessarily need to occur. The case of “no precipitation” is very difficult for certain models to represent, where the state of the cloud is often correlated with low liquid water content and high cloud droplet number concentrations (CDNC; Suzuki et al. 2011; Wood et al. 2009). Many observations (e.g., Pawlowska and Brenguier 2003; Comstock et al. 2004; van Zanten et al. 2005; Kubar et al. 2009) and model studies (e.g., Savic-Jovcic and Stevens 2008; Wang and Feingold 2009; Wood et al. 2009) confirm the dependence of the precipitation rate in low clouds on CDNC and that the feedback on the dynamics and structure of the MBL can be significant if drizzle sets in (Wood et al. 2009). The following effects can be listed:

- The simple presence of drizzle size drops (
*r*> 20–25*μ*m) enhances the collision–coalescence growth rates and as a result depletes the cloud more rapidly. This is in agreement with observations, which show that accretion is essential for the production of drizzle (Wood 2005b; Pruppacher and Klett 1997; Rogers and Yau 1989). - The cloud (and subcloud) layer is cleansed by scavenging of aerosol particles by drizzle drops leading to a positive feedback in keeping cloud condensation nuclei (CCN) concentrations low. Alternatively, the absence of drizzle is often correlated with a high amount of CCN. This bistability in CCN concentrations correlated with the presence or lack of drizzle has been hypothesized by Baker and Charlson (1990).
- The formation of precipitation is suggested to be correlated with coherent and long-lived regions of open-cellular structures (e.g., Stevens et al. 2005), which form in overcast marine stratocumulus and are a good example of the Baker and Charlson bistability. It has been recognized that whether a cloud precipitates or not is largely dependent on the interplay between aerosols, microphysics, and dynamics with a potentially significant radiative feedback (Wang and Feingold 2009).
- Drizzle may stabilize or destabilize the boundary layer (Albrecht 1989), such that in the case of a stabilization a decoupling of the cloud and the subcloud layer can occur, leading to the transition to trade wind cumulus (Savic-Jovcic and Stevens 2008). For those clouds the dependence of precipitation formation on CCN concentrations is still debated. For example, modeling work by Xue and Feingold (2006) suggests sensitivity whereas observations by Nuijens et al. (2009) show no significant impact.
- The presence of drizzle increases the effective radius of low clouds (Wood 2000). If this is not considered, the effective radius is underestimated and the cloud albedo overestimated. For a no-precipitation case hardly any drizzle is present and the change in effective radius is negligible.

The appropriate representation of shallow clouds in the MBL depends on many environmental conditions on different spatial as well as temporal scales as shown in many field campaigns (e.g., Albrecht et al. 1988; Lenschow et al. 1988; Stevens et al. 2003; Albrecht et al. 1995; Bretherton et al. 2004). Under the assumption that the macro- and microstructure of the cloud are well captured, precipitation can be considered as a proxy for the state of the cloud, the boundary layer, and its radiative properties. Hence, describing the formation of precipitation in marine stratiform clouds correctly is paramount in capturing the evolution of the cloud, its boundary layer, and, therefore, its radiative feedback.

Many bulk parameterizations of the collision–coalescence process (e.g., Kessler 1969; Sundqvist et al. 1989; Khairoutdinov and Kogan 2000) are based on empirical relations of the total drop mass density (and number density for two-moment schemes) of each class, where important information, especially on the shape of the drop size distribution (DSD), is lost. Others, such as Berry and Reinhardt (1974b), Lüpkes et al. (1989), Beheng (1994), or Seifert and Beheng (2001) derived parameterizations based on the stochastic collection equation (SCE). This approach boils down to the assumptions of a number density function *f*(*x*) describing the DSD and a collection kernel *K*(*x*, *y*) of two drops with masses *x* and *y*, which results in a physical representation of the collection process.

*c*(cloud),

*d*(drizzle), and

*r*(rain) the drop spectrum is divided such thatwhere

*x*

_{1}is the drop mass separating the cloud and the drizzle class and

*x*

_{2}is the mass separating the drizzle from the rain class. Based on existing modeling studies (e.g., Khairoutdinov and Kogan 2000; Mechem and Kogan 2008; Savic-Jovcic and Stevens 2008) and many observations (cf. Fig. 1),

*x*

_{1}was chosen to correspond to a drop radius of 25

*μ*m, which is argued to be the relevant size threshold for stratocumulus, because drops larger than this size have significant fall velocities. The observations shown in Fig. 1 show a distinct shoulder and minimum recognizable in the number (Fig. 1a) and mass density distribution (Fig. 1b), respectively, which underlines the choice of

*x*

_{1}. Determining

*x*

_{2}is much less evident, especially from observations. However, defining drizzle drops as drops that are too large for condensation to determine their growth and fall sufficiently fast to overcome typical updrafts in marine stratocumulus, but that remain small enough such that a large fraction of their mass evaporates below cloud base, gives a physically based size range. From calculations of drop evaporation due to sedimentation into subsaturated areas below cloud base using observed relative humidity profiles yielded that drops with a radius larger than 100

*μ*m experience a very small amount of mass loss (Sant 2012), which corresponds to the conventionally accepted definition of precipitating drops reaching the ground (e.g., Rogers and Yau 1989; Pruppacher and Klett 1997). Thus,

*x*

_{2}was chosen to correspond to a radius of 100

*μ*m.

Lüpkes et al. (1989) have been the only other authors that we are aware of to derive a bulk parameterization for three liquid water classes, focusing, however, on the partitioning of the cloud class. Consistent with the findings by Beheng and Doms (1990), their idea was to capture the broadening of the cloud droplet spectrum due to self-collection, which led to an improved representation of the DSD and collection processes. Similarly, Saleeby and Cotton (2004) investigated the introduction of a large cloud water class to improve the bimodal structure of the cloud DSD often observed in nature and to take the activation of giant CCN into account separately. However, relying on bin microphysics descriptions there was no need of a new bulk description. Conceptually, the idea of splitting the liquid water into three classes is picked up here, but with the focus laid on the separation of the rain class and different assumptions on *f* and *K*. Offering an alternative approach, but different from the empirical schemes and the other existing SCE methods, the aim is to provide an alternative method for applications that benefit from an explicit drizzle class. The main consequence of introducing an additional liquid water class is that several more collection pathways need to be considered as is graphically illustrated in Fig. 2, where the acronyms of the different processes are given in Table 1. An additional advantage of the suggested triclass decomposition is a better representation of sedimentation of drizzle. For a two-class scheme, even if it does well in representing the collision–coalescence processes like autoconversion and accretion, the parameterization of the sedimentation flux of drizzle remains challenging. This is because the drizzle in such schemes is part of the tail of either the cloud droplet or the raindrop distribution. This may lead to an under- or overestimation, respectively, of the drizzle water flux at cloud base. The details will, of course, depend on the actual implementation of the sedimentation in the two-class two-moment scheme.

Description of the different acronyms used in Fig. 2 and Eqs. (10) to describe the different collection processes.

In view of evaluating the performance of a new two-moment bulk parameterization of the collision–coalescence process for three liquid water classes, the one-dimensional kinematic cloud model of Seifert and Stevens (2010, hereafter SS10) is used and described in section 2. The concept and the derivation of the new parameterization for the collection process is presented in section 3, where in addition observations and spectral model results are used to determine the free shape parameters of the number density functions *f _{i}* (

*i*∈ {

*c*,

*d*,

*r*}). Section 4 presents results of idealized simulations of the new parameterization and its performance is compared to a spectral and the bulk microphysics formulations from Khairoutdinov and Kogan (2000, hereafter KK00) and Seifert and Beheng (2001, hereafter SB01) within the 1D kinematic cloud model. The last section discusses and gives a further outlook of our findings.

## 2. Model description

For examining the performance of the newly developed parameterization, the 1D kinematic cloud model used in SS10 provides an adequate framework in which new microphysics schemes can be tested [a similar model is the kinematic driver model (KiD); cf. Shipway and Hill 2012]. Extended from a traditional 1D rain shaft model (Hu and Srivastava 1995), it investigates the evolution of the whole size spectrum of liquid water within a single vertical column and for a prescribed flow. Owing to the simplified thermodynamic treatment, the kinematic cloud model is dependent on only a few input parameters out of which the essential ones are the in-cloud lapse rate Γ_{*}, the cloud lifetime *τ _{w}*, the ambient CCN concentration

*N*

_{CCN}, and the maximum updraft velocity

*w*

_{0}. The vertical velocity follows a sinusoid in time with its maximum amplitude given by

*w*

_{0}. Furthermore, the in-cloud lapse rate and cloud lifetime are fixed to Γ

_{*}= 1.5 K km

^{−1}and

*τ*= 60 min. Other parameters are regarded as secondary and are held constant. The cloud-base height, for example, is assumed to be at 500 m at a pressure of 950 hPa and with a relative humidity of 100%. Below the cloud base the boundary layer is assumed to be well mixed with a potential temperature of

_{w}*θ*

_{0}= 297.2 K. Hence, all simulations presented here, for which a time step of 1 s is used, focus on the variations of

*N*

_{CCN}and

*w*

_{0}.

The 1D kinematic cloud model features a spectral or bin model description, where the collection growth of drops is based on Bott (1998) with mass doubling every four bins (i.e., 142 bins between 1 *μ*m and 3.4 mm in radius). Thus, the model offers an ideal test bed for comparing bin and bulk models. For the processes of activation, sedimentation, and condensation both bin and bulk schemes make use of the same formulation as described in SS10. The evaporation of raindrops is treated following Seifert (2008).

*ρ*

_{0}= 1.065 kg m

^{−3}(SS10), the empirical autoconversion and accretion rates, which they derived from large-eddy simulations of marine stratocumulus, are given byandwhere

*q*and

_{c}*q*are the cloud and rainwater mass densities (g cm

_{r}^{−3}) and

*N*is the cloud droplet number concentration (cm

_{c}^{−3}). For the collection rates of the number densities the expressions by SB01 are used, as no expression for self-collection is offered by KK00. Note that the collection rates of SB01 are based on a separation radius of 40

*μ*m, which differs from the cloud and rainwater partitioning of KK00 at 25

*μ*m; that is, KK00 include drizzle in the rainwater class.

## 3. Drizzle as a third liquid water class

The introduction of a third bulk liquid water class (i.e., drizzle) implies using an extended collision–coalescence parameterization for three classes (Lüpkes et al. 1989), which in this case will be derived using the SCE for partial moments.

### a. Collection for three drop classes

*f*(

*x*,

*t*) be a droplet number density function (i.e., DSD), where

*x*is the drop mass and

*t*is time. The evolution of

*f*due to the kinetic effect of collection (i.e., collision with subsequent coalescence) can be expressed by the SCE:where

*f*(

*x*,

*t*)

*dx*corresponds to the number of drops in the mass range [

*x*,

*x*+

*dx*] at time

*t*,

*x*and

*y*correspond to the pair of drops subject to collection, and

*K*(

*x*,

*y*) is the so-called collision–coalescence kernel, which is a measure of how fast drops of mass

*x*and

*y*collect each other. The first term describes the rate of generation of drops of mass

*x*—the factor ½ avoids counting collections twice, and the second term accounts for the reduction of drops with mass

*x*due to collision–coalescence with all other drops. Solving the SCE implies finding

*f*for an analytically suitable approximation of

*K*. In the following, the time dependence of

*f*will be omitted for simplicity.

*f*is related to bulk quantities by so-called moments of

*f*:where

*k*∈ ℕ and (with

*x*being the drop mass) the first three moments (

*k*= {0, 1, 2}) of

*f*are the number density

*N*, the mass density

*q*, and a value proportional to the radar reflectivity

^{1}

*Z*, respectively. Now, instead of solving the SCE for

*f*, the SCE can be expressed in terms of the moments

*M*following Drake (1972) as it is done in SB01.

^{k}*f*comprises the entire drop spectrum, but in view of the separation of the liquid water into two or (in this case) three bulk classes, a central assumption is whether a DSD of a class

*i*∈ {

*c*,

*d*,

*r*},

*f*, is complete or not. Here, completeness refers to the property of

_{i}*f*having a significant minimum at the separation size(s) of the individual classes (Berry and Reinhardt 1974a), such that a moment of class

_{i}*i*,

*M*, can be expressed following Eq. (5) (SB01). Note that the minimum referred to appears only if—instead of

_{i}*f*(

*x*)—the modified spectrum

*g*{ln(

*r*)} {=

*r*·

*x*·

*f*(

*r*)} is plotted. Dropping the assumption of completeness the following notation for partial moments is introduced:where

*l*and

*u*are the lower and upper integration limits of class

*i*, respectively. Taking the partial derivative with respect to

*t*of Eq. (6) and following Drake (1972) yields the following expression for the SCE for partial moments:

*ν*,

_{i}*λ*, and

_{i}*μ*control the shape of the distribution and the amplitude

_{i}*A*controls the overall scaling. Here,

_{i}*ν*and

_{i}*μ*are chosen as free parameters to be defined (see below), whereas

_{i}*λ*and

_{i}*A*will be related to the moments of number and mass density. Note that assuming partial moments results in the use of the incomplete gamma function (Abramowitz and Stegun 1965). As a result,

_{i}*λ*needs to be calculated iteratively. More details are given in appendix A.

_{i}*K*(

*x*,

*y*) Long (1974) put forward the idea of a piecewise polynomial approximation, which until today is being used with some success (e.g., Lüpkes et al. 1989; Cohard and Pinty 2000; SB01). However, all existing approximations of the collection kernel are given for two drop classes, meaning that a new approximation of the kernel needs to be found for a triclass spectrum with the proposed size ranges. To this end, a nonlinear least squares fit using a Levenberg–Marquardt algorithm (Press et al. 1992) was used to derive a new piecewise approximation for three classes resulting in the following form:where the values of the collection kernel parameters areThis new collection kernel was derived using the collision efficiencies of Pinsky et al. (2001), the coalescence efficiencies of Low and List (1982a,b) and Beard and Ochs (1995), and the terminal fall velocity by Beard (1976). Note that for reasons of simplicity the presented kernel does not take effects of turbulence on the collision frequency into account. Furthermore, as turbulence is small in stratocumulus clouds its influence on the kernel is assumed to be small too. Nevertheless, Pinsky and Khain (2002) and Seifert et al. (2010) point out that taking turbulence into account can enhance

*K*(

*x*,

*y*) by a factor of up to 6 for certain drop combinations, although values larger than 2 are for the more convective type clouds. Thus, omitting the effects of turbulence may lead to an underestimation of the autoconversion process and the subsequent onset of precipitation.

#### Rate equations

_{cd+}and ac

_{cd−}denoting the source and sink term of the accretion process with respect to the drizzle class (cf. Fig. 2 and Table 1). Note that in a standard two-class bulk scheme accretion does not change the number of raindrops, but in a three-class scheme accretion of cloud water by drizzle can lead to a growth of drizzle into rain leading to a change of

*N*. The new rate equations for the bulk number and mass mixing ratios of each class

_{d}*i*consequently have the following dependencies:The detailed results are derived by defining the lower (

*l*) and upper (

*u*) bounds in Eq. (7) according to the class of interest and are presented in appendix B. Owing to the nature of some of the collection processes, for example, aut

_{c}or ac

_{cd+}, cannot be solved analytically, and, therefore, are solved numerically.

*i*and

*j*at a given separation mass

*x** is assumed, which under the assumption of partial moments is more adequate leading to a natural transition from one class to the next and is consistent with the fact that observed spectra are smooth. Consequently, the assumption of continuity replaces the use of a moment to determine the coefficients

*A*and

_{i}*λ*(

_{i}*i*∈ {

*d*,

*r*}), but introduces a dependence on class

*j*∈ {

*c*,

*d*}.

Since the condition of continuity is needed and applicable for the beginning stages of cloud evolution, continuity between the cloud and drizzle class is assumed as long as no rain is present. If both drizzle and rainwater are present, continuity between these two precipitation classes is assumed. Conceptually, this approach may run into problems as the relation between adjacent classes may lose its causality. The effect of sedimentation illustrates this nicely: let a cloud be vertically extended, and the drizzle and raindrops produced in the upper part of the cloud will sediment into the lower part, where they have lost their causal relation to the cloud droplets. Hence, the assumption of continuity does not hold anymore. As such, assuming continuity only between the drizzle and rain class when both are present seems to be the best approach.

Because of the use of partial moments certain limitations are imposed on this new scheme. On one hand, the mean radius *i* has to be within the size range of the class for the iterative determination of *λ _{i}* to converge. On the other hand, partial-moment calculations can result in unnaturally small values for autoconversion in the beginning stages. Therefore, if no drizzle or rain mass is present, threshold values

_{c}or aut

_{d}to set in, respectively. Note that these values are well below the values observed in a DSD (cf. Fig. 3). The mass ranges and thresholds are summarized in Table 2.

Mean radius ranges (*μ*m) and threshold values (cm^{−3} g^{−1}) for all classes.

The computational costs of the triclass scheme compared to the two-class schemes by KK00 and SB01, which are very similar, is approximately a factor of 4 higher, mainly because of the iterative calculations of *λ _{i}*. Replacing these and the integral expressions for given collection processes (cf. appendix B) by fitting functions would largely reduce the costs. As a comparison, the computational cost of the bin model with respect to the two-class schemes is approximately a factor of 20 higher.

### b. Closing the set of equations

Having chosen the MGD to describe the three classes of cloud, drizzle, and rainwater, the free shape parameters *ν _{i}* and

*μ*need to be determined to close the set of equations. As they have a strong influence on the development of the collision–coalescence process (e.g., Milbrandt and Yau 2005; Seifert 2008; Posselt and Lohmann 2009) they ought to be chosen carefully. To this end, observed and bin model DSDs are analyzed.

_{i}#### 1) Observational perspective

The observational DSDs offer a realistic picture of the parameter space of *ν _{i}* and

*μ*, but only for a given instant and place in time. The more data is available, the more trustworthy the choice of shape parameters becomes. However, as variability occurs in four different dimensions (one temporal and three spatial) and sampling or averaging errors need to be accounted for, the uncertainty nevertheless remains high.

_{i}There are numerous datasets of DSDs for different types of clouds and produced precipitation. Since the focus is on marine stratiform clouds, the dataset of 12 flights of stratiform boundary layer clouds over the Atlantic from Wood (2005a) is used. Focusing on the formation of drizzle, these observations offer an ideal basis on which the DSDs can be fitted, thus determining the parameter space of *ν _{i}* and

*μ*. By calculating the bulk number concentration and mass mixing ratio from the average measurements and separating them into each (artificially determined) class enables to fit the measured DSD with the best

_{i}*ν*and

_{i}*μ*—that is, the fit with the smallest error in log space, varying the values between [−0.8; 1.0] and [0.1; 1.0], respectively. The dataset provides DSDs at different heights within the cloud for all flights such that the data can be divided into cloud-top, in-cloud, and cloud-base observations. As two of the flights {A644 and A648 in Wood (2005a)} observed clouds with multiple layers and, hence, the vertical extent of the cloud was not clearly distinguishable, these were not taken into consideration. Out of the remaining 10 flights, 8 can be regarded as clean and 2 as polluted (flights A049 and A641, owing to the high CDNC observed). Table 3 summarizes the best-fitting shape parameters for both cases and for each height, where in addition the best fit for the sum over all heights is shown too. Note that since only two flights were available for the polluted case, the deduced parameters are not as reliable. In addition, the observed in-cloud DSDs of the clean cases and the result of fitting the averaged data with a MGD are shown in Figs. 3a and b, respectively.

_{i}Results of fitting the observations with an MGD to DSDs from Wood (2005a), where *μ*m.

By dividing the data into different heights within the cloud, the evolution of the DSD can be observed as the average cloud droplet size becomes larger from cloud base to top and the precipitation drops become larger from the top to the bottom of the cloud (Rogers and Yau 1989; Wood 2005a). However, no clear dependence of the shape parameters can be deduced, at least for the cloud class and more importantly for the difference between the clean and the polluted case. Even though a significant difference is seen in the CDNC between the two regimes, there is no significant response in the shape parameters. For the drizzle class the shape parameters seem rather stable and do not vary much. Hence, they seem reliable for both clean and polluted cases. The rain class seems to act as an extension of the drizzle class as long as the average size of the former stays small. As soon as the average size becomes larger, the rain class seems to exhibit a more parabolic shape; *ν _{r}* going toward positive values and

*μ*becoming larger. One may state that the precipitating classes do not seem to exhibit a dependence on the background state of the cloud, and, therefore, can be parameterized independently.

_{r}#### 2) Bin model perspective

Another possibility is to rely on bin or cloud resolving models to determine the parameter space of *ν _{i}* and

*μ*, thus implicitly assuming that these models capture the observed variability of DSDs. Though not linked directly to what is observed in reality, this enables and/or restrains a functional description of the shape parameters.

_{i}To this end, the 1D kinematic cloud model with bin microphysics is used and offers the possibility of analyzing the evolution of the DSD in a very detailed way. Varying *w*_{0} and *N*_{CCN}, the DSD can be evaluated at each height and at every instant in time. A set of simulations were carried out for *N*_{CCN} = {100, 500} cm^{−3} and *w*_{0} = {0.75, 1.0, 1.5} m s^{−1} all within the range of relevance for shallow clouds (e.g., Ditas et al. 2012). Figure 4 illustrates the evolution of the complete DSD at each height by snapshots at intervals of 10 min between *t* = 20 and 40 min for the simulations with *N*_{CCN} = 100 cm^{−3} and *w*_{0} = {0.75, 1.0, 1.5} m s^{−1}. Within the first 20 min of all simulations, a cloud is created and the mass is carried aloft. For *w*_{0} = 0.75 m s^{−1} (cf. Figs. 4a–c) the precipitation formed remains exclusively in the drizzle class, resulting in little surface precipitation as a large fraction evaporates below cloud base. For *w*_{0} = 1.0 m s^{−1} (cf. Figs. 4d–f) both drizzle and rain are present in significant amount in terms of precipitation, whereas for *w*_{0} = 1.5 m s^{−1} (cf. Figs. 4g–i) drizzle water seems to play an intermediary role, transferring the mass from the cloud to the rain class without having a large effect on the produced precipitation. Consequently, this suggests a change in regime when going from low to high values of *w*_{0} with respect to the importance of drizzle, which is to be expected when going from stratocumulus to trade wind cumulus clouds. This will be explored further in section 4.

For this large dataset the same fitting procedure as for the observational data is applied, yielding the best-fitting shape parameters *ν _{i}* and

*μ*for every height and time. By plotting them as a function of the mean radius

_{i}^{2}(cf. Fig. 5) or the drop number concentration, unfortunately, no particular functional dependencies could be found. Similar to the fitting of the observational data,

*ν*and

_{i}*μ*were varied between [−0.8; 2.0] and [0.1; 1.0], respectively, where for a given height and time the best-fitting pair is plotted.

_{i}Concentrating on the findings for the cloud class (cf. Figs. 5a and d), a large scatter without any noticeable dependence, especially for *μ _{c}*, can be observed. The dependence of

*ν*on

_{c}*ν*is found. In general, there are no significant differences between the clean and the polluted cases in terms of the shape parameters, confirming what is found with the observed DSD dataset. Statistically the best-paired shape parameters are

_{c}*ν*= 2.0 and

_{c}*μ*= 1.0, in both single and paired terms (cf. Table 4), corresponding to what others have used to describe a DSD with a single maximum (Pruppacher and Klett 1997). Negative values for

_{c}*ν*are very unrealistic, as this would mean having a monotonically decreasing distribution, which typically is not observed for cloud drops (cf. Fig. 3). For comparison, Seifert and Beheng (2006) chose

_{c}*ν*=

_{c}*μ*= 1.0, which corresponds to a somewhat wider distribution. However, as they consider the cloud droplet range to extend up to

_{c}*r*= 40

*μ*m their parameterization slightly differs from the one presented here. The higher value of

*ν*= 2.0 might be due to the smaller cloud droplet range but might also lead to an underestimation of the smallest cloud droplets.

_{c}Statistics from the fitting of the shape parameters for the cloud water. The percentages refer to the total number of fits (Tot No.) produced.

For the drizzle class there is a slight tendency for both shape parameters to follow a parabolic behavior with a minimum around *ν _{d}* in that range,

*μ*generally seems to compensate by being rather low. For large values of

_{d}The rain class exhibits the largest scatter and except for the tendency to form a parabolic shape with a maximum, nothing else is revealed (cf. Figs. 5c and f). Monotonically decreasing distributions occur only for a very small amount of cases (<30%) and small

#### 3) Summary

*r*

_{d}_{,mid}= 40

*μ*m,

*r*

_{d}_{,max}= 90

*μ*m, and

*r*

_{1}denoting the lower limit of the drizzle class at 25

*μ*m. The functional dependence of

*ν*on

_{d}For the drizzle class it is inevitable to ascribe a dependence of either *ν _{d}* or

*μ*on

_{d}*ν*is assigned a functional dependence on

_{d}*r*

_{d}_{,mid}is chosen such that the value of

*ν*reflects the shape of the DSD in the observations for

_{d}*r*

_{d}_{,mid}= 40

*μ*m will influence the temporal evolution of the DSD such that rain is produced earlier (later).

From the analysis above it is reasonable to assume constant values for the shape parameters of the cloud and rain class, since no functional dependence is found, neither on the mean radius nor on the background conditions.

Based on other studies (e.g., Seifert and Beheng 2006) and the bin model results, *μ _{c}* is set to 1, having the advantage that the MGD reduces to the three-parameter gamma distribution (Petty and Huang 2011). One reason why the observations exhibit a wider distribution than the bin model results (i.e., lower values in both

*ν*and

_{c}*μ*) might be due to the presence of a higher number of small cloud droplets, which might be underestimated in the bin model. As the bin model results show a preference for

_{c}*ν*to be 2.0, for

_{c}Confronted with many uncertainties it is very difficult to find constant shape parameters for the rain class. Both observations and bin model results presented here underestimate the amount of large raindrops [the underestimation in the bin model has been recognized when comparing the radar reflectivity—i.e., the second moment with *k* = 2 in Eq. (5)—with radar measurements] especially when reverting to rain distribution measurements as in Marshall and Palmer (1948), who found an exponentially decreasing distribution in diameter. The so-called Marshall–Palmer distribution is widely accepted and many parameterizations for evaporation (e.g., Rotstayn 1997; Seifert 2008) and sedimentation (e.g., Seifert and Beheng 2006) use it with slight modifications. Converting the Marshall–Palmer distribution for diameter to a distribution in mass, the shape parameters adopt the values *ν _{r}* = −⅔ and

*μ*= ⅓. The value of

_{r}*μ*is hence set lower than what is found from the fit to the observations by Wood (2005a) to take the higher number of large raindrops into account. For

_{r}*ν*, however, a negative value leads to an overestimation of “small” raindrops. Consequently, the value chosen is positive and set to ⅓, owing to the fact that for large

_{r}*ν*should not be chosen negative.

_{r}## 4. Results

### a. Idealized simulations

Idealized simulations were carried out by only considering the process of collection—that is, using the solutions of the SCE for partial moments for three drop classes. The temporal evolution of the DSD can then be simulated by prescribing number and mass densities (*N _{i}* and

*q*) for each class mimicking the procedure followed in a bulk model. For simplicity and clarity these box model results are shown only for the new parameterization to explain the properties of that scheme. A comparison with bin microphysics is shown below. Moreover, the invariance of the new triclass scheme under time-stretching transformations can be tested.

_{i}A number of simulations were performed initializing cloud water with typical bulk values for the marine boundary layer (e.g., Wood 2005b). Here, simulations initialized with the pairs (*N _{c}*,

*q*) = {(50, 0.5), (75, 0.75), (100, 1.0)} (cm

_{c}^{−3}, g m

^{−3}) corresponding to a mean radius of

*N*and

_{c}*q*(and, consequently,

_{c}The detailed results of (*N _{c}*,

*q*) = (75, 0.75) (cm

_{c}^{−3}, g m

^{−3}) are shown in Figs. 7 and 8. Figure 7 illustrates how the DSD evolves in time and Fig. 8 presents the temporal evolution of the most important conversion rates. Initializing the simulation exclusively with cloud water number and mass density, drizzle drops are produced after the first time step. After around 15 min, rain is produced, which corresponds well to what other studies found (e.g., Berry and Reinhardt 1974b; Lüpkes et al. 1989). However, the subsequent evolution is slightly slower than what other studies show, which is due to the fact that no drizzle number or mass was initialized. The overall evolution shows that only when rain comes into play does the cloud DSD show a significant decrease in its mass or number density (cf. Fig. 6). This is underlined by the fact that the accretion rate of cloud and raindrops dominates as soon as enough rain is present and leads to the decrease of the other conversion rates as seen in Fig. 8. The evolution of the individual

*i*∈ {

*c*,

*d*,

*r*}) also shown in Fig. 7 is reasonable, except that

Further simulations showed that the higher the cloud number and mass, the earlier drizzle and, subsequently, rain is produced and, more importantly, the faster the transfer occurs. This is to be expected as all mass conversion rates (i.e., autoconversion and accretion) are proportional to *q _{i}q_{j}*|

_{i≠j}({

*i*,

*j*} ∈ {

*c*,

*d*,

*r*}), which can be seen in appendix B. As the accretion between cloud and rainwater dominates ultimately, the transfer occurs a lot faster if more mass is involved. Changing the mean radius of the cloud droplets by changing the number concentration, but keeping the mass mixing ratio constant, showed that the larger

*q*, the larger the amount of mass in the drizzle class.

_{c}The amount of drizzle water is at least an order of magnitude smaller than the amount of cloud and rainwater (cf. Fig. 6). This shows that for high cloud liquid water contents, which are found in more convective clouds, the drizzle class acts as pure intermediate to the rain class as should be the case. For lower cloud liquid water densities as in stratocumulus, drizzle water is present a lot longer and can contribute to the amount of produced precipitation. Furthermore, as the accretion of cloud and rainwater dominates the conversion rates, especially when large masses are involved, the parameterization shows an aptitude to differentiate between stratiform and convective clouds. For stronger convection beyond trade wind cumulus, this parameterization would not be apt, as a drizzle class would be superfluous and most probably slow down the transfer to rain size drops as was found by Saleeby and Cotton (2004).

*c*∈ ℝ. Every solution

*f*(

*x*,

*t*) possesses similarity solutionsBy integrating this relation over the size range of interest, the same is found for the momentswhere again

*i*∈ {

*c*,

*d*,

*r*}. For

*k*= 1, this means that

*c*corresponds to the ratio of the mass densities, which means that the speed at which the collection process occurs is proportional to the mass, as was mentioned above. Therefore, because of the invariance under time-stretching transformations, the relation in Eq. (15) governs the simulations and acts as a measure of quality. Hence, all moments or even the DSD should correspond to each other under corresponding transformation.

Using the simulation with (*N _{c}*,

*q*) = (100, 1.0) (cm

_{c}^{−3}, g m

^{−3}) as reference, the other simulations were transformed with respect to it and mapped onto each other (Sant 2012). For (

*N*,

_{c}*q*) = (50, 0.5) (cm

_{c}^{−3}, g m

^{−3}) for instance, the constant

*c*would take on the value 2. This acts as a validation of the new parameterization with respect to Eq. (15).

### b. 1D kinematic cloud model

To assess the new triclass parameterization’s performance within an atmospheric relevant framework, the 1D kinematic cloud model is used. A large set of simulations is performed for the ranges of *N*_{CCN} = 50–500 cm^{−3} and *w*_{0} = 0.5–1.5 m s^{−1}.

The simulation with *N*_{CCN} = 100 cm^{−3} and *w*_{0} = 1.5 m s^{−1}, though representing a more convective case, is the most complex as larger mass densities are present in all classes and the interaction between them is greatest. Figure 9 presents the liquid water contents of the corresponding three classes for the bin and triclass model in time–height diagrams. The general evolution and the magnitudes are well captured, except for the rain class, where the amount of liquid water is largely underestimated as compared to the bin scheme. The amount of drizzle water is well represented, although the triclass model produces drizzle water a little faster, resulting in an earlier depletion of the cloud. Since rain gains most of its mass through the accretion of cloud droplets, the earlier depletion of the cloud might be the reason for the underestimation in rain mass. Furthermore, the rainwater in the triclass model reaches the surface earlier, hence sedimenting faster, which points toward larger drop sizes.

For a clearer comparison, the liquid water paths including evaporation and surface precipitation are shown together with an additional polluted simulation with *N*_{CCN} = 500 cm^{−3} in Fig. 10. For the clean case, the earlier and faster creation of drizzle water in the triclass model, leading to the earlier depletion of cloud water, is more evident. This is the main difference between the two models. Ultimately, rainwater is produced earlier and is correlated with the onset of surface precipitation, but not necessarily with the amount of precipitation, which is overestimated. Nevertheless, in both clean and polluted cases the triclass scheme performs and compares reasonably well to the bin model, especially in terms of its response to a fivefold increase in CCN. In this case, the amount of surface precipitation is very similar, whereas the amount of evaporation is slightly underestimated compared to the bin model.

The cause for the higher amount of surface precipitation in the triclass model is due to different mean volume radii (Sant 2012). The cloud and drizzle drops mean volume radii of the two model runs agree well with each other, where *μ*m at cloud top and *μ*m sedimenting fastest. Both precipitating classes show that because of size sorting, the largest drops reach the surface first, but as was suspected, the raindrop mean radius is larger in the triclass scheme, such that raindrops up to 0.6 mm in radius are created, compared to 0.35 mm for the bin model. It was found that the higher mean volume radii, especially for the precipitating classes, is due to a smaller drop number concentration, the reason for which has not been identified yet. As the main rain shaft in the triclass model consists of such large drops, their residence time and evaporation time is low, therefore leading to the underestimation in evaporation and, consequently, overestimation in surface precipitation.

A very important feature are the conversion rates, of which the cloud droplet autoconversion and accretion rates are shown in Fig. 11 again for *N*_{CCN} = 100 cm^{−3}. The overall comparison is satisfying, since the autoconversion reaches its maximum values at the upper edge of the cloud where the cloud droplets are largest and only when large-enough drops are created does accretion set in. The maximum accretion rates in the bin model are not reached in the bulk scheme, which is because of the lower mass densities in the triclass model. Altogether, the triclass parameterization performs well despite the simple expression of the approximated collection kernel [cf. Eq. (9)].

Again, for clarity and to investigate the response of the conversion rates to an enhanced aerosol concentration, the conversion rates are averaged over height and plotted with respect to time for the clean and polluted case in Fig. 12. Autoconversion and hence the accretion set in a little earlier for the triclass model, causing the faster buildup of drizzle, but the overall comparison to the bin model is reasonable in terms of magnitude. The sensitivity to the background aerosol condition is captured particularly well by the triclass scheme, as a strong decrease in the autoconversion is observed. The decrease is around an order of magnitude, similar to the sensitivity of the bin model. The influence of an increased aerosol concentration on the accretion rates is not as strong as that of the autoconversion rate in both models. The fact that the accretion rate of the bulk model goes from having similar values in Fig. 12a to an underestimation in Fig. 12b is mainly due to the vertical averaging. However, in general the accretion rate is underestimated, meaning that the peak values of the bin model are not reached in the triclass model, which is mainly due to the underestimated rain mass mixing ratio discussed before. Even though drizzle and rain are produced at earlier stages in the triclass model, the transfer to the precipitating classes occurs at a slower rate.

Additional simulations for the parameterizations by KK00 and SB01 covering the same ranges of *N*_{CCN} and *w*_{0} were done to characterize the amount of surface precipitation over the whole simulation period more thoroughly. They are summarized in Fig. 13, once showing the surface precipitation as a function of *w*_{0} (Fig. 13a) and as a function of *N*_{CCN} (Fig. 13b). The figure confirms that the triclass model overestimates the amount of surface precipitation with respect to the bin model, but that the lower the vertical velocity, the better the comparison. Especially, the results for *w*_{0} ≤ 1.0 m s^{−1} are in reasonable agreement, such that for stratocumulus regimes the parameterization performs well. In addition, the response of the new parameterization to the change in aerosol concentration is reasonable, even though the response from *N*_{CCN} = 50 to 100 cm^{−3} is not as rigorous as in the bin model, the decreases in surface precipitation from the clean to the most polluted situations are on the same order of magnitude. The bin model tends to have an exponential decrease in surface precipitation with respect to *N*_{CCN}, whereas the bulk models tend to have a more linear response.

Taking into account the other bulk models, the parameterization by SB01 compares best to the bin model results, which was to be expected as the collection rates are based on the functional dependence of the ratio of cloud and rain mass densities derived from bin model calculations. Note that the formulation by SB01 uses constant shape parameters, which here is set arbitrarily to *ν* = 3. In comparison to the parameterization by KK00, the triclass model reveals a clear improvement, even for *w*_{0} ≤ 1.0 m s^{−1}. Most evident is the large overestimation of surface precipitation for polluted cases in the KK00 scheme, which may be the main weakness of such empirical parameterizations. It points toward the fact that the proportionality of the autoconversion to

To underline the benefit of a triclass scheme including drizzle, Fig. 14 presents the precipitation at cloud base over the whole simulation period as a function of *w*_{0} for all schemes for *N*_{CCN} = 100 cm^{−3}. Here, two points need to be made. First, for small *w*_{0} (≤0.8–1.0 m s^{−1}) the contribution of drizzle water is largest, which is the case in stratocumulus clouds. For larger *w*_{0} there is a clear regime change, where the amount of rainwater clearly dominates, as expected. These features are well captured by the triclass scheme although a positive bias compared to the bin model is present, as was recognized earlier for the surface precipitation (cf. Fig. 13). The second point concerns the comparison with the two-class schemes by KK00 and SB01. As KK00 separates cloud droplets and precipitating drops at the same drop radius (25 *μ*m) as the triclass and bin model, the cloud-base precipitation by the KK00 scheme needs to be compared to the total water (i.e., sum of drizzle and rainwater). For this case it behaves similar to the triclass scheme exhibiting the same positive bias as compared to the bin model. Concerning the SB01 parameterization it is important to stress that at these low precipitation rates the separation of cloud droplets and precipitating drops at a larger drop radius (40 *μ*m) has important implications. Foremost, the amount of cloud-base precipitation from the SB01 scheme needs to be compared to the gray area shown in Fig. 14, which is between the amount of rainwater and total water produced by the bin model. The SB01 scheme performs well as the amount of precipitation lies nicely within that area. Nevertheless, for regimes with low *w*_{0} and low precipitation rates a substantial amount of drizzle is formed, such that the SB01 scheme, which does not account for precipitation-size drops below 40 *μ*m, would lead to an underestimation of the produced precipitation, liquid water flux, and evaporation below cloud base.

## 5. Discussion and conclusions

A new two-moment bulk parameterization for the collision–coalescence process with three liquid water classes adapted for the description of drizzle formation in marine stratiform clouds is presented. To improve the representation of the DSD, the derived rate equations are based on the SCE for partial moments and the use of the MGD as number density function. This involves using the incomplete gamma function (Abramowitz and Stegun 1965; Jawitz 2004) and calculating one of the shape parameters iteratively (Lüpkes et al. 1989). As expected, this makes the triclass scheme more complex and expensive than conventional schemes. In the attempt to physically restrain the free shape parameters *ν _{i}* and

*μ*of each class

_{i}*i*, observed DSDs (Wood 2005a) and bin model studies performed with the 1D kinematic cloud model (SS10) are used. As the analysis hardly shows any dependence of

*ν*and

_{i}*μ*on mean drop radius or number concentration, all but the parameter

_{i}*ν*are set constant. Other studies such as Milbrandt and Yau (2005), Seifert (2008), or Posselt and Lohmann (2009) have shown that relating the shape-parameter

_{d}*ν*(for the precipitating classes) to the mean diameter of the DSD considerably improves results for the sedimentation and evaporation of rain. To improve the understanding of the importance of the DSD shape and, in particular, to validate the theoretical findings, further model studies and observations are needed. As such, the values of

*ν*and

_{i}*μ*given here are not considered to be the only possibilities.

_{i}With the introduction of an intermediate drizzle class, separating the three classes at a drop radius of 25 and 100 *μ*m, a new collection kernel was approximated for the relevant size ranges following Long (1974) and Seifert (2002) omitting the effects of turbulence. Overall, the triclass parameterization works and performs well. Idealized studies and simulations with the 1D kinematic cloud model show a realistic development of the DSD, where the conversion rates are on the order of magnitude expected from previous studies and bin model results, which is satisfactory considering that the collection kernel is approximated by a simple polynomial expression. The latter is very helpful as it simplifies the integration of the moments, although an improved approximation, which yields better collection efficiencies, but retains a simplicity of solving the moment’s integration, is still lacking. Main differences between the triclass and bin model arise from the earlier production of drizzle water and larger raindrop sizes in the triclass model. The larger raindrops lead to a faster sedimentation because of the corresponding higher terminal fall velocities and accordingly a slight overestimation in surface precipitation with a corresponding underestimation in evaporation.

The response in surface precipitation to changes in CCN concentration and maximum vertical velocity is very similar between the triclass scheme and the bin model, as the lower *w*_{0} becomes, the more similar is the amount of surface precipitation. Comparing the bulk parameterizations of KK00 and SB01 to the bin model, the scheme by SB01, where the collection rates are driven by functions based on bin model calculations, performs best, as expected. Nevertheless, the new parameterization is an improvement compared to the scheme by KK00, especially for high CCN concentrations. It is suggested that this may be the main weakness of such empirical collection equations, as the KK00 scheme has large difficulties representing the no-precipitation case. Thus, it is put forward that for a good representation of the collection processes and the subsequent formation of precipitation, the information on the shape of the DSD is paramount.

By investigating the precipitation at cloud base, a transition in regime is observed. For *w*_{0} < 0.8–1.0 m s^{−1} drizzle dominates the contribution to the formed precipitation, whereas for larger *w*_{0} rain does. This is to be expected when going from stratocumulus to trade wind cumulus–type clouds and shows that the triclass scheme can mimic such a transition although a positive bias compared to the bin model is present. Furthermore, the analysis points out that when comparing different parameterizations, the radius at which cloud and rainwater are separated is crucial and hints toward a preferred regime.

An important aspect that future work will need to address is the optimization of the computational cost of the triclass parameterization. As analytically precise expressions were favored for this study, replacing the iterative procedure for calculating the parameters *λ _{i}* (cf. appendix A) and the numerically solved integrals (cf. appendix B) by fitting functions would largely improve the efficiency of the calculations. As a result, the triclass scheme could be used in LES models, studying the evolution of the DSD in a more realistic framework potentially accounting for the effect of turbulence, or larger-scale models, improving the collection process for the formation of drizzle with respect to the influence of different CCN and giant CCN concentrations (e.g., Saleeby and Cotton 2004; Posselt and Lohmann 2008; Mechem and Kogan 2008). On a further note, following the findings by SS10, the time in which and the amount of precipitation formed is mainly given by the amount of cloud liquid water and the cloud droplet number concentration. Hence, the microphysics scheme could be further optimized by using the triclass parameterization only for selected cases, which involves identifying situations where the DSD should be split into three classes or where two suffice. As was put forward, in a first instance the aim is not to replace existing formulations, but to provide an alternative method for applications that would benefit from an explicit drizzle class, as for studying the effect of giant CCN or the more physical representation of the DSD. Accordingly, the validation of the scheme with respect to observations will also be sought.

Note that other physical parameterizations such as sedimentation or evaporation of hydrometeors can influence the results significantly in terms of amount of surface precipitation and/or what resides in the atmosphere. A coupling of these often separate parameterizations—for instance, by introducing an upper truncation of the rain class to improve the sedimentation (Ziemer and Wacker 2012)—would improve the physical representation of precipitation but may also lead to significant changes in the results. A last point to be noted is the transfer between the classes due to evaporation, which has not been taken into account. Most parameterizations of evaporation assume that the average size of a bulk quantity stays the same. However, the details of the evaporation process are difficult to capture, especially for drizzling cases (e.g., Seifert 2008). With the triclass model the transfer of rain to drizzle due to evaporation could be considered, especially in the early stages of precipitation formation.

We thank Robert Wood for providing the data of Wood (2005a), which is shown and used in Figures 1 and 3. The authors further wish to thank the staff of the Meteorological Research Flight and the C-130 aircrew and ground crew for their dedication to collecting the data presented in this study. We also thank the reviewers for their very helpful and constructive comments on improving the manuscript.

# APPENDIX A

## Partial Moments Using the MGD

*A*and

*λ*in dependence of the first two moments (drop number and mass density) become nonlinear. For

*A*the zeroth moment yieldswhere

*a*

_{1}= (

*ν*+ 1)/

*μ*and

*z*(

_{g}*x*) =

*λx*. It is evident from the expression of

^{μ}*A*that a linear expression for

*λ*cannot be found for partial moments and will have to be solved iteratively, where the bisection method was used. For completeness, the first moment yields the expressionwhere

*a*

_{2}= (

*ν*+ 2)/

*μ*.

Note that the incomplete gamma function is computed using the numerical algorithm by Press et al. (1992, their chapter 6), but there are alternatives such as the approximation by Blahak (2010). For the use of other number density functions for partial moments the reader is referred to Jawitz (2004).

# APPENDIX B

## Formal Solution of the Rate Equations

Based on the derived SCE for partial moments given in Eq. (7) the explicit rate equations for each class can be derived using the according lower (*l*) and upper (*u*) bounds, where the dependencies have already been summarized in Eqs. (10).

*l*= 0 and

*u*=

*x*

_{1}. Deriving the individual terms of the change in number concentration and mass mixing ratio yieldsandNote that

*l*=

*x*

_{1}and

*u*=

*x*

_{2}the following was derived:andwhereAs remarked in section 3, because of the separation of liquid water into three classes the accretion of cloud droplets by drizzle drops (ac

_{cd}) can lead to the formation of raindrops (cf. Fig. 2). Consequently, for this process

*N*(which would not change under a two-class scheme) and the mass gained for

_{d}*q*is reduced by the process ac

_{d}_{cd−}. Note that rather than calculating the integrals for ac

_{cd−}[in Eqs. (B9) and (B15)] these can be replaced by their complement ac

_{cd+}, which yield larger values and, therefore, are numerically easier to determine.

*l*=

*x*

_{2}and

*u*= ∞, the derived rate equation for the according number concentration and mass mixing ratio areand

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