## 1. Introduction

Clouds are of much interest from a climate perspective because of their ability to alter radiative fluxes. For example, low, optically thick clouds increase the albedo relative to the surface and thus exert a net shortwave cooling. On the other hand, high, optically thin clouds (e.g., cirrus) exert a net longwave warming by emitting at temperatures much colder than the surface. These radiative impacts are inherently tied to the microscale cloud properties, which in turn depend on the ambient background aerosol physical and chemical properties. This is particularly important for low, warm marine boundary layer clouds that remain a key uncertainty in global simulations of climate change (e.g., Bony and Dufresne 2005; Webb et al. 2006; Alley et al. 2007; Medeiros et al. 2008).

Various studies have closely examined aerosol–cloud–precipitation interactions in marine stratocumulus. Commonly, the effects of aerosols on marine stratocumulus are categorized into the Twomey effect (albedo enhancement; Twomey 1977), Albrecht effect (drizzle suppression; Albrecht 1989), drizzle-entrainment effect (e.g., Lu and Seinfeld 2005; Wood 2007), and sedimentation-entrainment effect (e.g., Ackerman et al. 2004; Bretherton et al. 2007; Hill et al. 2009). Chen et al. (2011) provides a complete review of the potential effects of aerosol perturbations on marine stratocumulus, providing a comparison of both large-eddy simulation (LES) results and an analytical formulation of the aerosol indirect effects. However, a potential shortcoming of a large fraction of aerosol–cloud–precipitation studies is the lack of a physically consistent aerosol processing scheme. Most cloud models are capable of directly analyzing the effect of aerosol perturbations on cloud properties, but not necessarily the effect of clouds on aerosol properties and the inherent link between this aerosol–cloud cycle and cloud dynamics. Specifically, recent work (e.g., Wang and Feingold 2009a,b; Wood et al. 2011; Kazil et al. 2011) suggests the importance of aerosol–cloud feedbacks on the transition from closed to open boundary layer clouds—in other words, the formation of pockets of open cells (POCs). A complete understanding of the dynamical transition between these states requires a comprehensive and physically accurate model of the complete aerosol–cloud feedback system.

By definition, the indirect effects of aerosols are the changes to cloud properties and precipitation amount that are linked to the changes in background aerosol number concentration. Modeling studies that aim to tell how aerosols indirectly affect clouds systematically address the problem by looking at the linkage between aerosols, activation, and condensation/evaporation. But, cloud processing can result in potentially large changes to the background aerosol size spectrum (via collision–coalescence and subsequent evaporation that cause the regeneration of haze aerosols), wet deposition (due to surface precipitation), and aerosol scavenging (both from in-cloud diffusion of interstitial aerosols impacting onto cloud droplets and below cloud from impact scavenging of aerosols onto raindrops). Only recently have we been able to explicitly simulate aerosol chemical (Ovchinnikov and Easter 2010) and physical (Lebo and Seinfeld 2011) processing in marine stratocumulus in LES simulations using fully two-dimensional (2D) bin microphysics (i.e., a microphysical scheme that predicts both the liquid and solute amounts for given droplet categories or bins). While these models provide an elegant way to understand not only the indirect effects of aerosol perturbations on bulk cloud properties but also how the clouds themselves affect the aerosol population, they are computationally very expensive. As a result, large suites of simulations are impractical and we are forced to parameterize the aerosol processing in order to accurately represent the aerosol–cloud–precipitation interactions in high-resolution numerical studies.

Flossman et al. (1985) provided a complete set of integral–differential equations representing the in-cloud processing of aerosols by cloud droplets. While complete, the algorithm is complex and thus simpler, efficient approaches are warranted for most applications. Of particular attention over the last 20 years has been the regeneration of aerosol upon droplet evaporation. Mitra et al. (1992) showed that a single evaporating droplet containing soluble and insoluble compounds produces exactly one aerosol particle during the droplet–particle conversion. This concept is the basis for the subsequent regeneration parameterizations and assumptions, but there are large differences in the treatment of aerosol regeneration among schemes. Ackerman et al. (1995) proposed using a breath parameter for the aerosol size distribution within droplets to predict the size of the regenerated aerosols. Moreover, Kogan et al. (1995) assumed that the regenerated aerosols ought to fill the smallest bins of the aerosol size distribution first, up until the bin reaches its initialized number concentration. It has also been proposed that the regenerated aerosols should be first put in the largest bins upon regeneration (e.g., Feingold et al. 1996). The former method assumes that the smallest aerosols form the smallest and most readily evaporated cloud droplets, while the latter is based on the fact that aerosols ought to grow in size because of collisions between cloud droplets and thus the smaller aerosols will be diminished. Other proposed methods have included both scaling the mean aerosol size to the decrease in-cloud number mixing ratio upon evaporation (e.g., Feingold et al. 1996) and using the collision–coalescence rate to predict the regenerated aerosol size (e.g., Xue et al. 2010). More recently, Kazil et al. (2011) used the method of Feingold et al. (1996), in which the regenerated mass was scaled with the decrease in droplet number mixing ratio but also included a parameterization of the processing and partitioning of aerosols within cloud droplets and rain drops by scaling the cloud process rates.

The previous approaches are highly uncertain and large differences exist among them. For example, assuming that the regenerated aerosols fit into the smallest aerosol bins, implies that there is little to no collision–coalescence occurring. This is likely only true in relatively shallow clouds or in very polluted regimes, where cloud droplets remain quite small on average. On the other hand, assuming that the regenerated aerosols fit into the largest aerosol bins implies that collision–coalescence is very efficient, likely only true in deeper clouds with relatively large water contents. Moreover, scaling the processes may result in biases, since the underlying assumption here is that the cloud droplet size spectrum and the aerosol spectrum within the droplets match precisely. However, modeling studies (given the absence of substantial observational evidence of aerosols contained within cloud droplets) suggest otherwise (e.g., Ovchinnikov and Easter 2010; Lebo and Seinfeld 2011). Aerosols are often prescribed to follow lognormal size distributions, whereas the cloud droplet spectrum is frequently defined to follow a gamma function in bulk cloud microphysics schemes. As shown below, these biases in the scaling assumption can be quite large.

The aforementioned aerosol processing parameterizations have largely been untested against more detailed aerosol–cloud models, for example, 2D bin models (Ovchinnikov and Easter 2010; Lebo and Seinfeld 2011) and/or Lagrangian schemes (Shima et al. 2009). The purpose of this study is to use a detailed 2D bin scheme to improve the treatment of aerosol processing by using it to develop a new bulk parameterization and to test previous methods (e.g., scaled process rates). Bulk schemes are amenable to such a parameterization for several reasons. First, they are computationally inexpensive in comparison to a 2D bin scheme and even more traditional 1D bin microphysics models. Moreover, a simple parameterization of a single-mode aerosol distribution implemented in a bulk scheme requires only the addition of five scalars (and just three if the model already includes aerosol mass and number) using the approach proposed here. In the case of a traditional 1D bin scheme, the aerosol mass mixing ratio would have to be accounted for in each bin, thus greatly increasing the computation expense of an already computationally expensive model. In addition to the aerosol processing parameterization, a new treatment of aerosol activation is presented that predicts number, mass, and modal aerosol size, thus providing more accurate predictions of droplet activation.

The remainder of the paper is organized as follows. Section 2 describes the aerosol processing parameterization and the scaling assumption (discussed above as well) in detail. An analysis of the model results comparing the new aerosol processing parameterization to both the scaling assumption and 2D bin model results are presented in section 3. Last, a summary and review of key results are included in section 4.

## 2. Parameterization

The aerosol processing model described herein relies on a few fundamental assumptions regarding both initial conditions and the relationship between the number of aerosols regenerated and the number of cloud droplets and raindrops evaporated. For illustrative purposes, we address the former issue by assuming that the initial aerosol size distribution [*n ^{d}*(

*D*)] is lognormal. This assumption can be relaxed for a generic size distribution, but for conciseness, this is beyond the scope of the current study. The latter issue is addressed by assuming that each cloud droplet and raindrop, upon complete evaporation, will produce exactly one regenerated aerosol particle (Mitra et al. 1992). As a result, the number of aerosol particles contained within each of the two hydrometeor categories is identical to the number of hydrometeors in the corresponding category. This limits the number of additional variables required for the scheme and thus only the aerosol mass mixing ratios contained in cloud droplets (

_{p}*Q*

_{a}_{,c}) and in raindrops (

*Q*

_{a}_{,r}) are required to accurately represent the aerosol contained within the hydrometeors.

*Q*

_{a}_{,d}), (ii) the bulk number mixing ratio of unactivated aerosol particles (

*N*

_{a}_{,d}), and (iii) the bulk number mixing ratio of previously activated aerosols (

*N*

_{a}_{,c}). It should be noted that the

*N*

_{a}_{,c}is not necessarily equivalent to the sum of the cloud droplet and raindrop number mixing ratios (

*N*and

_{c}*N*, respectively) since

_{r}*N*

_{a}_{,c}corresponds to the number mixing ratio of aerosols that activated at some point prior to the current time step, while

*N*+

_{c}*N*represents the total number mixing ratio of hydrometeors, after microphysical processing has occurred (especially collection processes). The necessity of this additional variable will become clear below. In total, the scheme described herein requires five additional scalars and prognostic equations for these scalars can be defined as follows:

_{r}*V*

_{q}_{,c}and

*V*

_{q}_{,r}correspond to the mass-weighted terminal fall speeds of the aerosol mass contained with cloud droplets and raindrops (identical to the sedimentation rate of

*Q*and

_{c}*Q*, since the aerosol is contained within the cloud droplets and raindrops), respectively; and

_{r}*V*

_{n}_{,c}represents the number-weighted fall speed of cloud droplets. The scaling factor, (

*N*+

_{c}*N*)/

_{r}*N*

_{a}_{,c}, is incorporated to account for the fact that, after collection occurs,

*N*

_{a}_{,c}≠

*N*+

_{c}*N*. Furthermore,

_{r}**u**is the 3D wind vector,

**k**is the unit vector in the vertical,

*ρ*is air density, and

*t*is time.

The process rates are described in detail in the subsequent subsections. Of these processes, only the aerosol mass and number activation rates are calculated using a semi-analytic technique. The remaining processes are calculated using empirical fits of the explicit process rates computed using the detailed 2D bin microphysics model of Lebo and Seinfeld (2011). No explicit formulation of aerosol regeneration from evaporating raindrops is included in the current parameterization, and it is instead represented using a simple scaling representation (the evaporation of raindrops is treated in a continuous fashion in the 2D bin model, whereas traditional bulk models evaporate raindrops completely in a time step). Since the evaporation rate of raindrops in the simulations presented here is quite small, this assumption ought to have a negligible effect on the results.

### a. Activation

*N*

_{a}_{,t}is the total aerosol number mixing ratio (the sum of

*N*

_{a}_{,d}and

*N*

_{a}_{,c});

*σ*and

*D*are the standard deviation and geometric mean diameter, respectively; and

_{g}*D*is the aerosol particle diameter. On the first time step,

_{p}*N*

_{a}_{,t}=

*N*

_{a}_{,d}, since the model is initially void of cloud droplets. Once droplets activate, however,

*N*

_{a}_{,c}is required to complete the lognormal size distribution and to predict the number of particles that will activate on subsequent time steps (Fig. 1). To explicitly compute the activation rate, (∂

*N*

_{a}_{,c}/∂

*t*)

_{act}, we compute the critical diameter (

*D*) that corresponds to a particle in which its critical supersaturation is equivalent to the ambient supersaturation. Then, using the additional prognostic variable

_{c}*N*

_{a}_{,c}(the number of aerosols that activated previously, assumed to advect as a passive scalar), we have

*N*

_{a}_{,t}can be thought of as a potential aerosol number concentration where only

*N*

_{a}_{,d}is available for activation.

*or*large aerosols are advected into a supersaturated region, then activation is likely to occur. However, by simply assuming that the total number of aerosols is just equivalent to the dry aerosol number concentration, errors would be induced as this assumption would artificially add larger, readily activated aerosols (Fig. 1, top). Thus, by preserving the number of previously activated aerosols (

*N*

_{a}_{,c}) as a prognostic variable, we can define a system of equations:

*ρ*and

*ρ*

_{aer}are the air and aerosol densities, respectively; and

*D*

_{cut}corresponds to the size demarcating the dry aerosol from the previously activated aerosol (Fig. 1, bottom right). Since

*N*

_{a}_{,t}is a known quantity [see Eq. (7)], Eqs. (8) and (9) represents a closed system of equations with exactly two unknowns, namely,

*D*and

_{g}*D*

_{cut}(it is assumed that the change in

*σ*is negligible). Initially,

*D*corresponds to the geometric mean diameter of the initial aerosol size distribution. However, after collection,

_{g}*D*will change (shifting to larger values because of collision–coalescence) and is thus not necessarily a known quantity. Unfortunately, an analytic solution to the system does not exist. The choice of using the equation for

_{g}*N*

_{a}_{,c}instead of

*N*

_{a}_{,d}comes from the fact that when the latter is solved for

*D*

_{cut}, there is no guarantee that there is a single unique solution, while solving the former for

*D*

_{cut}does. To solve for

*D*

_{cut}and

*D*, we first solve Eq. (9) for

_{g}*D*

_{cut}to get

*D*instead of ln

_{p}*D*to get

_{p}*D*

_{cut}to −1 and 1, respectively, using a transformation function to get

*w*is a function of Legendre polynomials (

_{i}*P*; Abramowitz and Stegun 1972) of order

_{K}*K*and can be computed following Hildebrand (1956) as

*z*) are the roots of

_{i}*P*. The roots

_{K}*z*and weights

_{i}*w*are computed and placed in a lookup table to reduce the computational expense of the method. Sensitivity analysis showed that

_{i}*K*= 20 provided numerically accurate solutions without the additional computational expense of increasing the order of the polynomial.

*D*using the bisection method. Once the solution has converged, Eq. (10) is used to compute

_{g}*D*

_{cut}and the corresponding critical supersaturation is calculated (

*S*

_{c}_{,cut}) by finding the critical point in the Köhler equation (e.g., Rogers and Yau 1989; Pruppacher and Klett 1997). If the ambient supersaturation (

*S*) is greater than

*S*

_{c}_{,cut}, then activation of particles with critical supersaturations between

*S*(corresponding to a critical diameter of

*D*) and

_{c}*S*

_{c}_{,cut}will occur (diagonal region in the bottom right of Fig. 1) such that

*K*= 20). This procedure is applicable for all times during a simulation, since even if

*D*

_{cut}is unknown (as would be the case at the initial time step), one can simply pick an arbitrarily large value for

*D*

_{cut}and the solution will still converge. Since a single aerosol particle will form just one cloud droplet, the number and mass sink of dry aerosols due to activation—that is, (

*dN*

_{a}_{,d}/

*dt*)

_{act}and (

*dq*

_{a}_{,d}/

*dt*)

_{act}, respectively—are equal in magnitude but opposite in sign to the source of aerosol mass and number in cloud droplets due to activation—that is, (

*dN*

_{a}_{,c}/

*dt*)

_{act}and (

*dQ*

_{a}_{,c}/

*dt*)

_{act}.

### b. Parameterized process rates

To formulate the parameterizations for the remaining process rates (i.e., regeneration, autoconversion, and accretion), we use power-law relationships to fit the 2D bin microphysics model output data of Lebo and Seinfeld (2011). The 2D bin microphysics model consists of 51 and 35 mass-doubling bins for the droplet mass (number) and aerosol/solute mass (number). The smallest wet aerosol, or haze particle, size is set to 0.01 *μ*m. Condensation/evaporation is predicted using a forward Eulerian method, while the collection of particles is formulating using a variation of Bott (1998) to encompass the full 2D collection problem. For the purposes of these calculations, the Long (1974) kernel is utilized. Sedimentation of hydrometeors is predicted following Beard (1976). Eulerian bin models with mass-doubling bins have recently been shown to have biases compared with higher-bin-resolution models (e.g., Grabowski and Wang 2009; Dinh and Durran 2012). However, as suggested by Liu et al. (1997), conserving more than two moments of the drop size distribution greatly limits the numerical biases in such a scheme. As a result, the 2D bin model has been updated to conserve three moments of the size distribution during the remapping process to reduce biases that may arise as a result of numerics.

The 2D bin microphysics scheme is coupled to the 2D kinematic framework of Szumowski et al. (1998) [applied in Morrison and Grabowski (2007) and Rasinski et al. (2011)]. The kinematic modeling framework applies a specified flow field to the domain that is constant in time. This permits the testing of microphysics schemes and formulation of parameterizations in a realistic flow field, including sedimentation and advective transport without complications due to feedbacks between the microphysics and dynamics. The background flow consists of an updraft in the left half of the domain (red) and a downdraft in the right half (blue) as shown in Fig. 2. Advection of scalars is computed using the multidimensional positive-definite advection transport algorithm (MPDATA) scheme (Smolarkiewicz 1984; Smolarkiewicz and Margolin 1998). The kinematic modeling framework employs periodical lateral boundary conditions and prognoses both potential temperature and water vapor mixing ratio. In the current study, we exclude surface heat and moisture fluxes to limit the potential feedback on the microphysics and instead solely focus on the microphysical processing of aerosols in a very controlled state.

Background flow field (steamlines). Vertical velocities (m s^{−1}) are contoured, showing updrafts (red) and downdrafts (blue).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

Background flow field (steamlines). Vertical velocities (m s^{−1}) are contoured, showing updrafts (red) and downdrafts (blue).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

Background flow field (steamlines). Vertical velocities (m s^{−1}) are contoured, showing updrafts (red) and downdrafts (blue).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

The domain initialization is loosely based on the Variability of the American Monsoon Systems (VAMOS) Ocean–Climate–Atmosphere–Land Study (VOCALS) Regional Experiment (VOCALS-REx) observations of subtropical drizzling stratocumulus (Wood et al. 2011; Allen et al. 2011). The initial water vapor mixing ratio (*q _{υ}*) is assumed to be 7.5 g kg

^{−1}throughout the domain and the potential temperature (

*θ*) is 289 K (Figs. 3a,b), with no liquid water. The surface pressure is fixed at 1015 hPa. The model is allowed to spin up for 120 min to allow the cloud to form in the absence of collision–coalescence and sedimentation. The resulting profiles of

*θ*,

*q*, and

_{υ}*q*are shown in Fig. 3. Here, we see that the spinup phase produces an adiabatic cloud layer with a base at about 900 m above ground level. The cloud is thus about 600 m thick, producing a cloud water mixing ratio (

_{c}*Q*) at cloud top of between 0.95 and 1 g kg

_{c}^{−1}. The difference in

*θ*from cloud base to cloud top is 2.5 K.

Initial (black) and after spinup (blue): (a) potential temperature and (b) water vapor mixing ratio. (c) The equilibrium cloud water mixing ratio using the 2D bin (black) and bulk model, including aerosol processing (red).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

Initial (black) and after spinup (blue): (a) potential temperature and (b) water vapor mixing ratio. (c) The equilibrium cloud water mixing ratio using the 2D bin (black) and bulk model, including aerosol processing (red).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

Initial (black) and after spinup (blue): (a) potential temperature and (b) water vapor mixing ratio. (c) The equilibrium cloud water mixing ratio using the 2D bin (black) and bulk model, including aerosol processing (red).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

The domain is set to be 1500 × 1500 m^{2} in both the *x* and *z* directions, such that the model top corresponds to the inversion layer. For the purpose of formulating the parameterized process rates, the horizontal and vertical grid spacing is set to 40 m. The relatively coarse domain is sufficient for the formulation of the parameterization, as there are no dynamical feedbacks on the cloud in the kinematic modeling framework. The model time step is 2 s to maintain dynamical stability. Data are output every minute over the course of the 6-h simulation (including the initial 2-h spinup phase). This setup provides more than 5 million points in space and time in which to use to formulate the parameterization, encompassing a wide range of environmental and microphysical conditions (Table 1).

Environmental conditions present in the kinematic model.

The aerosols are assumed to be ammonium sulfate with a single mode. We perform a suite of 15 simulations by varying the aerosol number concentration (*N _{a}*), the geometric mean diameter (

*D*), and the standard deviation (

_{g}*σ*) of the initial lognormal aerosol size distribution. Note that we use aerosol number concentration for convention when distinguishing between simulations and cloud droplet number concentration in the discussion of results, but the model formulation is in mixing ratio units since this quantity is conserved during advection. The simulations are run with bounds on these three parameters, such that

*N*

_{a}**∈**[100, 500] cm

^{−3},

*D*

_{g}**∈**[0.05, 0.1]

*μ*m, and

*σ*

**∈**[1.2, 1.6]. The details are given in Table 2.

Simulation descriptions.

*n*corresponds to the number of parameters used to fit the data. The parameters

*β*are computed by first taking the natural logarithm of Eq. (24) to get

_{i}*P*. We reserve the remaining details for the subsequent subsections, since

*n*is allowed to vary from process to process so as to provide the best fit to the data for each physical process. Unless otherwise noted, the process rates are given in kilograms per kilograms per second.

#### 1) Regeneration

*completely*evaporate over the course of a time step), aerosol mass mixing ratio contained within the cloud droplets (

*Q*

_{a}_{,c}), and cloud droplet number mixing ratio (

*N*). Thus,

_{c}*n*= 3 in Eq. (24). Using least squares analysis to determine the parameters

*β*, we obtain

_{i}*R*

^{2}value of 0.74 and is shown in Fig. 4. Moreover, 92% of the parameterized regeneration rates lie within a factor of 2 of the explicit rates computed using the 2D bin model (only every 5000th point is shown for clarity). The color scheme in Fig. 4—that is, black for

*N*= 100 cm

_{a}^{−3}, red for

*N*= 200 cm

_{a}^{−3}, and blue for

*N*= 500 cm

_{a}^{−3}—aids in showing that the parameterization does not contain a bias toward the background aerosol number concentration. Figure 4 also suggests that there is no bias toward the shape of the background aerosol distribution. To explicitly demonstrate that the parameterized rate does not contain a bias, Fig. 5 compares the explicit 2D model output to the parameterized regeneration rate for all 15 simulations shown in Table 2. The

*R*

^{2}value tends to be slightly lower for

*N*= 100 and 200 cm

_{a}^{−3}in comparison to

*N*= 500 cm

_{a}^{−3}. This is because the collision–coalescence rate decreases with increasing aerosol number concentration and thus the regenerated aerosol tends to be more like the background aerosol for higher

*N*. Regardless, the parameterization does quite well at capturing the vast range of the bulk aerosol mass regeneration rate.

_{a}Explicit aerosol mass regeneration rate as computed using the explicit 2D bin microphysics model and the parameterized aerosol mass regeneration rate using Eq. (26). Symbols are defined in Table 2.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

Explicit aerosol mass regeneration rate as computed using the explicit 2D bin microphysics model and the parameterized aerosol mass regeneration rate using Eq. (26). Symbols are defined in Table 2.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

Explicit aerosol mass regeneration rate as computed using the explicit 2D bin microphysics model and the parameterized aerosol mass regeneration rate using Eq. (26). Symbols are defined in Table 2.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

Explicit aerosol mass regeneration rate as computed using the explicit 2D bin microphysics model and the parameterized aerosol mass regeneration rate using Eq. (26) for each of the 15 simulations defined in Table 2. Columns correspond to the background aerosol number concentration (*N _{a}*) and rows correspond to the shape of the distribution. These fits are used in the EXP model configuration.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

Explicit aerosol mass regeneration rate as computed using the explicit 2D bin microphysics model and the parameterized aerosol mass regeneration rate using Eq. (26) for each of the 15 simulations defined in Table 2. Columns correspond to the background aerosol number concentration (*N _{a}*) and rows correspond to the shape of the distribution. These fits are used in the EXP model configuration.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

Explicit aerosol mass regeneration rate as computed using the explicit 2D bin microphysics model and the parameterized aerosol mass regeneration rate using Eq. (26) for each of the 15 simulations defined in Table 2. Columns correspond to the background aerosol number concentration (*N _{a}*) and rows correspond to the shape of the distribution. These fits are used in the EXP model configuration.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

#### 2) Autoconversion

*μ*m in diameter but little sensitivity was found for thresholds 15

*μ*m above or below this value). Natural choices for the parameterized power-law relation are the magnitude of the autoconversion rate itself

*Q*

_{a}_{,c}, and

*N*. The parameterized aerosol mass autoconversion rate was found to be

_{c}*R*

^{2}value for the fit is 0.84, and 96% of the data points are within a factor of 2 of the explicit rates. We again extract the points from Fig. 6 for each of the 15 model configurations and present them in Fig. 7. There appears to be a slight decrease in the

*R*

^{2}value for

*N*= 200 and 500 cm

_{a}^{−3}compared to

*N*= 100 cm

_{a}^{−3}; however, this is simply a result of the fact that as

*N*increases, the number of points with autoconversion decreases and so we have fewer points with which to infer the statistics. Thus,

_{a}*R*

^{2}and the number of points within a factor of 2 remain almost constant across all model configurations (i.e., >95%).

As in Fig. 4, but for the aerosol mass autoconversion rate in Eq. (27).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Fig. 4, but for the aerosol mass autoconversion rate in Eq. (27).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Fig. 4, but for the aerosol mass autoconversion rate in Eq. (27).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Fig. 5, but for the aerosol mass autoconversion rate in Eq. (27).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Fig. 5, but for the aerosol mass autoconversion rate in Eq. (27).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Fig. 5, but for the aerosol mass autoconversion rate in Eq. (27).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

#### 3) Accretion

*Q*

_{a}_{,c}, and

*N*. We found that a good fit could be attained with just these three variables. The least squares analysis provided the following parameterization for the aerosol mass accretion rate:

_{c}*R*

^{2}value is 0.83 with 80% of the data points within a factor of 2 of the actual rates predicted by the 2D bin model. This result is corroborated by again decomposing the data into each model setup in Fig. 9, in which we see no distinct and significant bias in the data to size distribution shape and/or background aerosol number concentration.

As in Fig. 4, but for the aerosol mass accretion rate in Eq. (28).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Fig. 4, but for the aerosol mass accretion rate in Eq. (28).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Fig. 4, but for the aerosol mass accretion rate in Eq. (28).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Fig. 5, but for the aerosol mass accretion rate in Eq. (28).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Fig. 5, but for the aerosol mass accretion rate in Eq. (28).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Fig. 5, but for the aerosol mass accretion rate in Eq. (28).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

#### 4) Brownian coagulation, impact scavenging, and dry deposition

Unlike autoconversion and accretion, the effect of Brownian coagulation is to transfer interstitial aerosol to the aqueous phase in cloud droplets. However, because of the large difference in sizes (i.e., collisions between particles on the order of 10s of microns in diameter for autoconversion, and collisions between submicron particles and cloud droplets on the order of tens of microns), it is no surprise that the loss rate of interstitial aerosol is quite small in comparison to the other processing rates. In fact, in nondrizzling clouds, the lifetime of interstitial aerosols for particles with diameters greater than 0.05 *μ*m is greater than 2.8 h (Seinfeld and Pandis 2006, see their Table 17.5). These particles are below the minimum aerosol size permitted in the model because they are often too small to activate. Impact scavenging rates are several orders of magnitude smaller than the loss rates because of activation, and collection processes. Thus, we neglect Brownian coagulation and impact scavenging in the current model setup. Last, since the dry (or haze) aerosols are treated as passive scalars (i.e., no terminal fall speed) and *w* = 0 at the surface, the deposition of dry aerosols is not permitted in the model. Given that in reality, the fall speeds of such particles are minuscule, especially in comparison to the speed at which large cloud droplets and raindrops fall, the effect on the modeled bulk quantities is negligible.

### c. Scaling parameterization

*Q*

_{a}_{,c}would be 1, the power for

*N*would be −1, and the power for

_{c}To statistically demonstrate the potential significance of the errors induced by simply scaling the aerosol process rates, we present in Fig. 10 the scaled process rates in comparison to the explicit rates computed by the 2D bin microphysics model. In Fig. 10a, for regeneration, we see that the scaling assumption provides a reasonably good fit (*R*^{2} = 0.83, slightly higher than that using the parameterization above). However, even though the fit appears good, we see that for relatively low regeneration rates (i.e., <1 ng^{−1} kg s^{−1}), the scaling underestimates the regeneration rate. For larger regeneration rates, a slight overestimate occurs. These differences are small though in comparison to the gross underestimate of the aerosol mass autoconversion rate as shown in Fig. 10b. Here, we see that for almost all points, the scaling assumption for the aerosol mass autoconversion rate underestimates the explicit rate by at least a factor of 2. This leads to a very small *R*^{2} value (i.e., 0.30).

As in Figs. 4, 6, and 8, but for the (a) scaled regeneration rate, (b) scaled aerosol mass autoconversion rate, and (c) scaled aerosol mass accretion rate. These correspond to the parameterization used in the SC simulations.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Figs. 4, 6, and 8, but for the (a) scaled regeneration rate, (b) scaled aerosol mass autoconversion rate, and (c) scaled aerosol mass accretion rate. These correspond to the parameterization used in the SC simulations.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Figs. 4, 6, and 8, but for the (a) scaled regeneration rate, (b) scaled aerosol mass autoconversion rate, and (c) scaled aerosol mass accretion rate. These correspond to the parameterization used in the SC simulations.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

Scaling the aerosol mass accretion rate introduces additional biases, as shown in Fig. 10c. Here, we see that the scaling assumption overpredicts the aerosol mass accretion rate for small rates (i.e., <10^{−5} ng^{−1} kg s^{−1}), especially for low aerosol concentrations (black). As the aerosol mass accretion rate is increased, the scaling model tends to underpredict the process rate for moderate aerosol concentrations (red and blue), but it continues to grossly overpredict the rate for low aerosol concentrations (black) by a factor of 5 or more. In section 3 we analyze the potential effects of these biases, if any, on the simulated bulk cloud and aerosol properties.

## 3. Parameterization testing and validation

We systematically analyze the processes important in the processing of aerosols by clouds so as to determine which processes the bulk model(s) can represent with confidence and those that may produce errors in either the bulk cloud or aerosol properties. To do so, we first examine the spinup period in which only activation and condensation–evaporation are permitted to occur in all of the models. We then delve into the effects of collision–coalescence and sedimentation by analyzing the model output in 1-h increments after the initial spinup period. The two-moment bulk microphysics scheme of Morrison and Grabowski (2007) is employed for this study. In this version, the scheme predicts both number and mass mixing ratios for cloud droplets and raindrops, including autoconversion and accretion following Seifert and Beheng (2001), and includes sedimentation of cloud droplets and rain. Three configurations of the two-moment bulk microphysics scheme are implemented that include aerosol processing:

*Explicit aerosol treatment*—This version includes the newly formulated aerosol activation scheme and power-law relations for all cloud processes relevant to aerosol processing as presented in section 2. Hereafter, this configuration of the bulk model is referred to as EXP followed by the aerosol number concentration. For example, EXP100 corresponds to simulations performed with the entire newly formulated explicit aerosol treatment for*N*= 100 cm_{a}^{−3}.*Scaling method with explicit activation*—Here all aerosol processes are scaled to the cloud processing rates predicted by the bulk model except for activation. Here, activation of new cloud droplets is predicted following the method described in section 2a. For brevity, this model configuration will be referred to as SCEXP and followed by the initial aerosol number concentration.*Scaling method*—All aerosol processes, including the aerosol mass activation rate, are predicted by scaling the cloud process rates with the aerosol mass and number mixing ratios. We will refer to the scaling method simulations as SC followed by the aerosol number concentration.

### a. Aerosol activation and regeneration

Simulations using the domain setup described in section 2b are performed for the control simulations at each aerosol number concentration, as defined in Table 2. For the validation segment of this study, simulations are carried out for a total of 4 h, with the initial 2 h reserved for spinup. Understandably, even with the implementation of an aerosol processing scheme in the bulk microphysics model, there still remains other differences between the bulk and bin schemes that will introduce differences in the predicted cloud properties—most significantly, the representations of sedimentation and collision–coalescence. Because of these differences, we restrict the simulations to just 4 h to limit the effects of the differences in representing these processes on the aerosol processing itself. Beyond 4 h, determining causality in the differences among the schemes becomes highly difficult.

In Figs. 11a–c, horizontally averaged vertical profiles of cloud droplet number concentration after the model spinup period are shown. We focus first solely on the 120-min spinup period so as to isolate the activation and regeneration components of the aerosol processing parameterizations (note that sedimentation and collision–coalescence are turned off during the initial spinup phase). Here, we see the potential effects of not accurately predicting both the number *and* mass of activated aerosols. While the differences between *N _{c}* for EXP (blue) and SC (red) are quite small for

*N*= 100 cm

_{a}^{−3}(Fig. 11a), the differences increases substantially as

*N*increases (Figs. 11b,c). On the other hand, the agreement between the 2D bin (black) and EXP (blue) is quite good for all aerosol number concentrations simulated. Both bulk models do well at capturing the effect of increasing

_{a}*N*on

_{a}*N*, suggesting that more aerosols implies more interstitial aerosol below cloud base but that most particles activate (if not all) by the time rising parcels reach cloud top. By scaling the aerosol mass activation rate to that of the number activation rate, we see that the model grossly underpredicts

_{c}*N*, especially at cloud top, and overpredicts the interstitial aerosol number concentration. This is shown explicitly in Figs. 11d–f, where weighted probability distribution functions (PDFs) of

_{c}*Q*

_{a}_{,d}for the entire spinup period are depicted. For all aerosol number concentrations, Table 3 shows very good agreement in the first four moments of the aerosol mass mixing ratio PDF—that is, mean (

*μ*

_{a}_{,d}), variance

*Q*

_{a}_{,d}. However, SC, in conjunction with its overprediction of the interstitial aerosol number concentration erroneously predicts a second peak in the PDF of

*Q*

_{a}_{,d}at about 23, 45, and 115 ng kg

^{−1}for

*N*= 100, 200, and 500 cm

_{a}^{−3}, respectively. Moreover, the SC model configuration statistics do not match well with those of the 2D bin model, especially in comparison to the goodness of fit for the EXP and SCEXP models (Table 3).

(a)–(c) Horizontally averaged vertical profiles of cloud droplet number concentration, (d)–(f) haze aerosol mass mixing ratio weighted PDF (i.e., PDF × *Q _{a}*

_{,d}) and (g)–(i) in-cloud droplet aerosol mass mixing ratio weighted PDF for the entire spinup period:

*N*= (left) 100, (middle) 200, and (right) 500 cm

_{a}^{−3}. Simulations are presented for the control realization only using the 2D bin model (black), EXP (blue), SC (red), and SCEXP (green). The axes are adjusted for each figure in order to accurately represent the complete range of values for each aerosol number concentration. Note that the curves for EXP and SCECP (blue and green, respectively, overlap).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

(a)–(c) Horizontally averaged vertical profiles of cloud droplet number concentration, (d)–(f) haze aerosol mass mixing ratio weighted PDF (i.e., PDF × *Q _{a}*

_{,d}) and (g)–(i) in-cloud droplet aerosol mass mixing ratio weighted PDF for the entire spinup period:

*N*= (left) 100, (middle) 200, and (right) 500 cm

_{a}^{−3}. Simulations are presented for the control realization only using the 2D bin model (black), EXP (blue), SC (red), and SCEXP (green). The axes are adjusted for each figure in order to accurately represent the complete range of values for each aerosol number concentration. Note that the curves for EXP and SCECP (blue and green, respectively, overlap).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

(a)–(c) Horizontally averaged vertical profiles of cloud droplet number concentration, (d)–(f) haze aerosol mass mixing ratio weighted PDF (i.e., PDF × *Q _{a}*

_{,d}) and (g)–(i) in-cloud droplet aerosol mass mixing ratio weighted PDF for the entire spinup period:

*N*= (left) 100, (middle) 200, and (right) 500 cm

_{a}^{−3}. Simulations are presented for the control realization only using the 2D bin model (black), EXP (blue), SC (red), and SCEXP (green). The axes are adjusted for each figure in order to accurately represent the complete range of values for each aerosol number concentration. Note that the curves for EXP and SCECP (blue and green, respectively, overlap).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

First four moments of the aerosol PDFs shown in Fig. 11 (i.e., during spinup). The mean (*μ*, ng kg^{−1}), variance (*σ*^{2}, ng^{2} kg^{−2}), skewness (*γ*_{1}, ng^{3} kg^{−3}), and kurtosis (*γ*_{2}, ng^{4} kg^{−4}) are shown for dry–haze aerosols (subscript *a*, *d*) and aerosol contained in cloud droplets (subscript *a*, *c*).

Errors are also evident in the prediction of *Q _{a}*

_{,c}. In Figs. 11g–i, PDFs of

*Q*

_{a}_{,c}are shown, demonstrating the large errors in the predicted aerosol mass contained in the cloud for SC (red). These errors are also reflected in the first four moments of the PDFs for the SC simulations compared with the other three model configurations as shown in Table 3 (note that the subscript

*a*,

*c*for each moment corresponds to the statistic for the aerosol mass mixing ratio in cloud droplets). The SC simulations predict a peak in

*Q*

_{a}_{,c}at about half the mass mixing ratio predicted by the 2D bin model and EXP. We also see that the SC simulations underpredict the mean (

*μ*

_{a}_{,c}). The reason for this large difference is that in the scaling assumption, if half the aerosols activate, then half the mass activates, whereas in the explicit models (i.e., 2D bin and EXP), the mass distribution is integrated to determine the exact aerosol mass that activates.

If we simply replace the scaled activation assumption in the scaling parameterization with the explicit activation method described in section 2a (SCEXP), and already included in EXP, then very small differences between the two bulk aerosol schemes exist (Fig. 11, comparing blue and green curves). For the spinup period shown in Fig. 11, the only difference between EXP (blue) and SCEXP (red) is the formulation for regenerating aerosol mass upon the complete evaporation of cloud droplets, since droplet collision–coalescence is neglected during this period. Regardless, both EXP and SCEXP predict nearly the same values for *N _{c}* during the spinup period. This should come as no surprise given the very small biases present in the scaling assumption shown previously for aerosol regeneration (Fig. 10).

The bulk models, including the explicit treatment of aerosol number and mass transfer during activation, regardless of representation of aerosol regeneration (i.e., EXP and SCEXP), also predict very similar PDFs for both dry aerosol mass and aerosol mass contained within the cloud droplets. Reassuringly, the simulated *N _{c}* and weighted PDFs of

*Q*

_{a}_{,d}and

*Q*

_{a}_{,c}for the bulk models reflect the same qualitative and quantitative features elicited by the 2D bin model (and as corroborated by the statistics displayed in Table 3). This provides two key results: (i) scaling the aerosol mass regeneration rate to the droplet evaporation rate is a very accurate assumption and (ii) implementing a physically accurate aerosol activation scheme for both number

*and*mass is crucial to accurately predicting both bulk cloud properties (i.e.,

*N*) and bulk aerosol properties (i.e.,

_{c}*Q*

_{a}_{,d}and

*Q*

_{a}_{,c}). The latter is important for accurately representing aerosol scavenging and cloud processing in bulk cloud models.

### b. Collision–coalescence and sedimentation

To evaluate the bulk aerosol parameterizations under more physically realistic conditions (i.e., including all physically relevant warm-cloud processes), we show the model output for only the 2-h period following the initiation of collision–coalescence and sedimentation, that is, from 120 to 240 min. As *t* gets further from 120 min, the differences between the 2D bin model and bulk schemes become more dependent on the representation of droplet collision–coalescence and not necessarily on the aerosol processing scheme itself. This is especially true for low aerosol number concentrations, since, given the rather large liquid water contents in these simulations, collision–coalescence is very efficient at reducing the cloud droplet number concentration (especially given the lack of an aerosol source term to help buffer the background aerosol number concentration in these idealized simulations). The 2D bin model, in general, predicts a much quicker drop in *N _{c}* in comparison to the bulk models, especially in relatively pristine conditions (Fig. 12a). However, even with these differences in

*N*, the aerosol processing parameterization (EXP) suggests a rather good agreement in the weighted PDFs of

_{c}*Q*

_{a}_{,d},

*Q*

_{a}_{,c}. This agreement is confirmed using the first four moments of the aerosol mass mixing ratio PDFs after the spinup period (Table 4).

As in Fig. 11, but for 120 < *t* ≤ 240 min, i.e, including collision–coalescence and sedimentation for the explicit 2D bin (black), EXP (blue), SC (red), and SCEXP (green).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Fig. 11, but for 120 < *t* ≤ 240 min, i.e, including collision–coalescence and sedimentation for the explicit 2D bin (black), EXP (blue), SC (red), and SCEXP (green).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As in Fig. 11, but for 120 < *t* ≤ 240 min, i.e, including collision–coalescence and sedimentation for the explicit 2D bin (black), EXP (blue), SC (red), and SCEXP (green).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

As *N _{a}* is increased to 200 and 500 cm

^{−3}, including the effects of collision–coalescence and sedimentation on the bulk cloud and aerosol processes, we again see very good agreement between the 2D bin model and EXP for the 2 h after spinup (Fig. 12). The SC continues to predict a bimodal PDF for

*Q*

_{a}_{,d}and

*Q*

_{a}_{,c}, again because of the errors induced during activation. On the other hand, EXP predicts a single-mode weighted PDF with a peak very close to that which is predicted by the 2D bin model. This demonstrates the ability of EXP to accurately reflect the processing of aerosol particles by clouds but in a much simpler and efficient modeling framework. The agreement between EXP and SCEXP demonstrates the insignificance of the bias in autoconversion rate in SC (Fig. 10b). The large differences in the autoconversion parameterizations for the aerosol do not elicit large differences in the bulk aerosol weighted PDFs because, on average, the aerosol mass autoconversion rate is two to three orders of magnitude smaller than the aerosol mass accretion rate.

For all three simulations with an explicit treatment of aerosol activation—that is, 2D bin, EXP, and SCEXP (black, blue, and green, respectively, in Fig. 12)—after the initiation of collision–coalescence and sedimentation, the differences in the predicted *N _{c}* and the PDFs of the bulk aerosol properties are negligible (Fig. 12). There are small differences in

*N*near cloud top, especially for

_{c}*N*= 100 cm

_{a}^{−3}as a result of the more efficient collision–coalescence process in the 2D bin model. The overall excellent agreement (see statistics in Table 4) in the bulk aerosol properties between the SCEXP and EXP demonstrates the ability of such a simple scaling parameterization to accurately represent aerosol processing in a warm cloud as long as it includes a realistic treatment of the impact on both mass and number from aerosol activation. The agreement between the shapes (as defined by the second, third, and fourth moments shown in Table 4) of the weighted PDFs of

*Q*

_{a}_{,d}for the 2D bin model, EXP, and SCEXP are quite good. All three models predict a negative skew in the weighted PDFs of

*Q*

_{a}_{,d}, largest for

*N*= 100 cm

_{a}^{−3}since collision–coalescence is more effective at lower aerosol loadings.

Of particular interest is how the models partition the aerosol mass between cloud and rainwater. The partitioning of aerosol between cloud and rainwater is important for wet deposition. Since collision–coalescence is most efficient for lower aerosol number concentrations (assuming all else is equal), we restrict our analysis of the partitioning of aerosol between cloud and rainwater to just the case where *N _{a}* = 100 cm

^{−3}(Fig. 13). The absolute differences in the weighted PDFs of

*Q*

_{a}_{,r}are quite small, especially in comparison to

*Q*

_{a}_{,d}and

*Q*

_{a}_{,c}. Table 5 provides the first four moments of the PDFs for

*Q*

_{a}_{,r}. Overall, EXP and SCEXP predict

*μ*

_{a}_{,r},

Weighted PDF of the aerosol mass mixing ratio contained with raindrops (i.e., *Q _{a}*

_{,r}) for the explicit 2D bin (black), EXP (blue), SC (red), and SCEXP (green). The data are filtered to include only points for 120 <

*t*≤ 240 min and where

*N*= 100 cm

_{a}^{−3}.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

Weighted PDF of the aerosol mass mixing ratio contained with raindrops (i.e., *Q _{a}*

_{,r}) for the explicit 2D bin (black), EXP (blue), SC (red), and SCEXP (green). The data are filtered to include only points for 120 <

*t*≤ 240 min and where

*N*= 100 cm

_{a}^{−3}.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

Weighted PDF of the aerosol mass mixing ratio contained with raindrops (i.e., *Q _{a}*

_{,r}) for the explicit 2D bin (black), EXP (blue), SC (red), and SCEXP (green). The data are filtered to include only points for 120 <

*t*≤ 240 min and where

*N*= 100 cm

_{a}^{−3}.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-13-045.1

## 4. Discussion and conclusions

A kinematic model was used both to develop power-law relationships for in-cloud aerosol processing mechanisms and to compare with previously used simple scaling arguments. A new, nonbiased aerosol activation scheme was formulated that is capable of predicting the number, mass, and modal size of the unactivated aerosol size distribution. It was shown that using this newly formulated activation scheme, a two-moment bulk microphysics model, can more accurately predict bulk cloud properties like the cloud droplet number concentration and interstitial aerosol number concentration as well as bulk aerosol properties (EXP and SCEXP model configurations). Moreover, a comparison between the simple scaling parameterization for aerosol processing (SC) and the new explicit aerosol scheme (EXP) showed that scaling the activation rate produced significant errors in both the bulk cloud properties *and* the bulk aerosol properties. These errors were quite large, especially in regard to the predicted PDFs of *Q _{a}*

_{,d}and

*Q*

_{a}_{,c}. However, by simply replacing the overly simplified activation scheme with the newly formulated scheme presented herein (SCEXP), very good agreement between the two versions of the bulk microphysics model was shown. More importantly, these models also agreed quite well with the 2D bin model; thus solidifying the argument for using a scaling parameterization for regeneration of aerosols.

The new aerosol processing scheme (EXP) that relies on power-law relations of bulk variables was shown statistically to have no biases across a wide range of aerosol number concentrations, shapes of aerosol size distributions, cloud water mixing ratios, cloud droplet number concentrations, updraft velocities, and temperature using a kinematic model. However, scaling the processing of aerosols due to collision–coalescence produced some biases, especially in the case of autoconversion. Nonetheless, these biases, although quite large in terms of the aerosol mass autoconversion rate, were shown to produce negligible differences in the bulk aerosol properties in comparison to EXP and the 2D bin model. We related this finding to the fact that even with the bias in the aerosol mass autoconversion rate, the rate itself was still several orders of magnitude less than the aerosol mass accretion rate and thus was a second-order effect.

Given the good agreement between the 2D bin model (as measured by computing the first four moments of the PDFs), EXP, and SCEXP, we have provided solid evidence for the use of scaling parameterizations for aerosol processing, with one caveat: The activation of aerosol number *and* mass must both be predicted with high accuracy in order to provide a reasonably good prediction of bulk cloud and aerosol properties. Any errors in the activation of aerosols, whether they be in number or mass, will be exacerbated through the continuous cycle of parcels entering and leaving a cloudy region.

There are several avenues for further advancement and testing of the formulations presented here in the future. First, the parameterizations have been tested in a kinematic model with a specified flow field. This allowed us to develop and test the parameterizations without added complications due to feedbacks between the microphysics and dynamics. Nonetheless, it will be valuable to test the scheme in a fully coupled dynamical model in future work, allowing for testing against detailed observations of cloud and aerosol properties and studies of feedbacks between microphysics and dynamics, including a transition from closed-cell to open-cell dynamics. Second, one can imagine that as the collision–coalescence process evolves, the formation of a bimodal aerosol distribution will occur and does occur in the 2D bin simulations (even if the background aerosol distribution contains only a single mode). In the current formulation, the regeneration and activation schemes only permit a single mode. This may introduce errors, especially when a parcel reenters the cloud, since the single mode will most likely underestimate large and small aerosols and overestimate aerosols that have sizes that lie in the middle of the distribution. More work needs to be done to expand both the scaling model and the power-law formulations to handle additional modes. However, given the agreement between the single mode, single particle type formulated with the 2D bin model and demonstrated here, adding additional modes are suspected to have second-order effects on the simulated bulk cloud and aerosol properties. Finally, while the present work provides a useful and physically consistent aerosol processing formulation encompassing a wide array of environmental conditions, these conditions are not necessarily independent of one another, since the simulations are performed for a single background (or initial) environmental scenario. This point will be addressed in a future study in which the parameterization is expanded to other cloud regimes–conditions, for example, shallow convection.

Last, while the model was formulated for a single aerosol particle type (or chemical composition), since it relies on just the bulk aerosol and cloud properties and process rates, it can easily be expanded to other particle types and/or expanded such that the model can predict the processing of more than one type of aerosol at a time. With that said, the model provides prognostic process rates for the physical processing of aerosol by clouds only at this time and thus adding the effects of chemistry to the model would be a valuable next step.

## Acknowledgments

The authors thank Wojciech Grabowski for his help in setting up the case and providing the kinematic model. The case presented here stems from an aerosol processing case study that was part of the Eighth International Cloud Modeling Workshop held in Warsaw, Poland, and sponsored by the World Meteorological Organization. Funding for this work was provided by the Advanced Study Program at the National Center for Atmospheric Research. Hugh Morrison was partially supported by the U.S. DOE ASR DE-SC0008648, U.S. DOE ASR DE-SC0005336, subawarded through NASA NNX12AH90G, and the NSF Science and Technology Center for Multiscale Modeling of Atmospheric Processes (CMMAP), managed by Colorado State University under Cooperative Agreement ATM-0425247.

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