## 1. Introduction

The development of numerical models has always stimulated progress to include, so far as possible, more detailed representation of physical processes in clouds. In general circulation models (GCMs), the critical impact of the clouds on solar radiative transfer at the climate scale (Slingo 1990) has called for studies on how to estimate the effective droplet radius *r*_{e} (ratio of the third to the second moment of the droplet size distribution) with additional information on the subgrid-scale velocity from the large-scale fields (Feingold and Heymsfield 1992; Ghan et al. 1997). At the mesoscale where subcloud scales can be resolved explicitly, bulk microphysical schemes are used routinely to predict the evolution of the mixing ratio of cloud droplets (or of total water) and of raindrops (Kessler 1969; Flatau et al. 1989), mostly for simulations of storms and other precipitating events. At even smaller scales, the coupling of large eddy simulation (LES) dynamics with evolving bin-size-resolved distribution of liquid water and aerosols in some circumstances (Clark 1973; Hall 1980; Kogan 1991; Brenguier and Grabowski 1993; Feingold et al. 1994) is of great promise for investigating a variety of physical and chemical processes at fine scales inside and in the vicinity of the clouds. Such complex models are essential to support some conclusions on the climatic role of variable CCN concentrations (Baker and Charlson 1990; Baker 1993).

More specific about bulk warm precipitation schemes is that only a few of them carry a prognostic equation for the cloud droplet number concentration *N*_{c}. To the authors’ knowledge, only Ziegler (1985) and Chaumerliac et al. (1987) developed independently a complete scheme but for different purposes. This can be partially explained by the inherent complexity of incorporating with enough efficiency the nucleation process by which cloud droplets form on CCN during the very early stage of cloud development. However, as the cloud-water mixing ratio *r*_{c} can be assumed to be in thermodynamical equilibrium with a good approximation (adjustment to the saturation of water vapor at all cloudy grid points), *N*_{c} and *r*_{c} are independent variables because it is obvious that nucleation has a strong effect on the budget of *N*_{c} while condensation affects *r*_{c} but not *N*_{c}. Consequently both *N*_{c} and *r*_{c} determine certain cloud properties such as their optical thickness and *r*_{e}, and also the efficiency of the clouds to precipitate by droplet conversion into precipitating raindrops (Richard and Chaumerliac 1989). In the simulation of strong convective clouds, the exact knowledge of *N*_{c} is of minor importance (except for physicochemical studies), because precipitation is likely to occur and the solar radiative effects can be neglected for the lifetime of a storm. This is not true for stratocumulus clouds, where the cloud number concentration could influence their energetics in several ways as hypothesized by Albrecht (1989), Ackerman et al. (1994), and Pincus and Baker (1994).

Thus, future progress in parameterized microphysics of warm clouds depends crucially on the way the nucleation of cloud droplets is going to be handled. Classically, the explicit modeling of the CCN activation is described in, for example, Lee et al. (1980) or Flossmann et al. (1985) where an initial condensation nuclei (CN) spectrum is grown beyond the saturation and then a diagnostic relation for the critical saturation of the Köhler theory is solved to partition between the activated CN (now CCN and hence newly formed cloud droplets) from the remaining unactivated interstitial CN. It is easy to see that such a reference technique is accurate provided the characteristics of the CN (dimensional size distribution, chemical composition, and solubility) are known at the stage of the activation (Pruppacher and Klett 1997). This appears to be rather compelling because during the early condensational growth of the aerosols or CN interchangeably, one needs to take into account the nonideal character of the salted particles [wet aerosols are very concentrated solutions whereas cloud droplets can be considered as infinitely diluted ones; see Young and Warren (1992)] as well as other complex cloud processing of the aerosols (Feingold et al. 1996; Kogan et al. 1994).

Starting from another point of view, Twomey (1959) showed that the nucleation problem could be solved analytically by making strong assumptions. His formula, which has received much interest indeed because of its simplicity, gives the total activable CCN as a function mainly of the vertical velocity. In addition to approximations made on the CN growth equation (see Johnson 1981; Feingold and Heymsfield 1992) in order to facilitate an integration, most serious criticisms were addressed later because of the deficiency of using a single power law like *N*_{CCN} = *Cs*^{k}_{υ,w}*s*_{υ,w}. Many field observations (Hudson 1984; Hudson and Frisbie 1991), laboratory experiments (Jiusto and Lala 1981), and theoretical studies (Ahr et al. 1989; Feingold et al. 1994) explain that a more universal activation spectrum should exhibit a concave curvature in log–log scale (or in other words, a decrease of the *k* coefficient) due to the limited availability of CN as *s*_{υ,w} is increased. So using Twomey’s original formula can induce important errors in predicting the nucleated cloud droplet number concentration unless the *C* and *k* parameters are not “adjusted” to simulate a specific nucleation regime (Herbert 1986; Ahr et al. 1989).

Motivated by the simplicity of Twomey’s formula, we show in this paper that it is possible to keep the same mathematical framework but for an improved description of the activation spectrum. Compared to the classic power law *N*_{CCN} = *Cs*^{k}_{υ,w}*N*_{CCN} function with four unknown parameters can be adjusted automatically to fit synthetic or experimental data representing clean/polluted maritime and continental air masses. Our study is a simple generalization of Twomey’s approach that offers a practical and powerful solution to most of the microphysical problems where the complex tracking of the CCN is unnecessary.

## 2. About the shape of the activation spectra

Most of the shape of the activation spectra can be interpreted by considering the size distribution of the CN close to saturation (Junge and McLaren 1971). For instance, starting from the Köhler theory and assuming a Junge power-function distribution for the CN in the form *N*_{CN} ∝ *r*^{−α}, where *r* is the CN radius and *α* ∼ 3, Jiusto and Lala (1981) identified the activation spectrum as a power law but with a *k* coefficient given by the simple relation *k* = 2*α*/3. Unfortunately, such situation is strongly idealized because the Junge-like distribution of CN cannot clearly describe the broad range of real cases; more complex size distributions of CN should be preferred. For that purpose lognormal distribution laws are often used in cloud physics to fit experimentally determined size distribution data. Within that framework the computation of *N*_{CCN} as a function of *s*_{υ,w} is still straightforward (see, e.g., von der Emde and Wacker 1993) but further analytical manipulations, such as the search for solutions to Twomey’s mathematical model of activation, is not feasable with this set of functions. So having admitted that the functional dependence of *N*_{CCN} to *s*_{υ,w} is complex by both the nature of the CN and the activation process acting on a population of CN, only a well-chosen approximation to *N*_{CCN}(*s*_{υ,w}) will allow progress toward a realistic solution to the generation of cloud droplets by nucleation. This was tentatively done by Ghan et al. (1993) but outside Twomey’s analysis.

Another point of concern is the concavity of true activation spectra. As simulated by von der Emde and Wacker (1993), this concavity is mostly expected in the range 0.02% < *s*_{υ,w} < 0.2% for typical maritime CN, depending on their solubility. For more typical continental CN, the slope variation of the *N*_{CCN}(*s*_{υ,w}) curves is smoother but spans a larger range (0.02% < *s*_{υ,w} < 1.0% approximately) because the lognormal distribution of these CN is characterized by a larger variance. The more the *N*_{CCN}(*s*_{υ,w}) curve is concave, the more it departs from the classic power law *N*_{CCN} = *Cs*^{k}_{υ,w}*s*_{υ,w} < 0.02%. Looking now to reference simulations of the condensation/nucleation processes made in a 1D model by Lee et al. (1980) with a moderate updaft speed of 1 m s^{−1}, it was shown that the maximal supersaturation *s*_{υ,wmax}*N*_{CCN}(*s*_{υ,w}) curve around *s*_{υ,w} ∼ 0.4% because it is this maximal value of the supersaturation that bounds the cloud droplet concentration. According to von der Emde and Wacker (1993), *s*_{υ,wmax}*s*_{υ,wmax}

Inspection of these two extreme situations, maritime and continental sources of CN, reveals that the respective nucleation regimes at *s*_{υ,w} ∼ *s*_{υ,wmax}*N*_{CCN} = *Cs*^{k}_{υ,w}*C* and *k* parameters to match locally the number of activated CN for *s*_{υ,w} ∼ *s*_{υ,wmax}*N*_{CCN} = *Cs*^{k}_{υ,w}*N*_{CCN}(*s*_{υ,w}) between zero and *s*_{υ,w} (which stands for *s*_{υ,wmax}*N*_{CCN}(*s*_{υ,w}) curve is misleading. Also, any suggestion based upon an adjustment of the *N*_{CCN}(*s*_{υ,w}) curve by means of partial power laws is arbitrary and not realistic because Twomey’s integration rapidly becomes untractable.

To summarize, it is important to consider complete activation spectra with sufficient accuracy for the wide range of possible supersaturations in clouds. A power law behavior seems satisfactory for *s*_{υ,w} < 0.02% and for *s*_{υ,w} > 1% in most academic cases of CN size distributions (von der Emde and Wacker 1993). This leads to the conclusion that one must seek a function, *N*_{CCN}(*s*_{υ,w}) with the following properties. To apply to a broader range of supersaturations, we have selected a continuous differentiable function *N*_{CCN}(*s*_{υ,w}) with four tunable parameters. As we shall show in the following section, this function behaves as a power law at very low and very high supersaturations.

## 3. Reexamination of the nucleated cloud droplet concentration estimate with Twomey’s model

As stressed in the introduction, the knowledge of the activation spectrum is imperative to simulate the explicit formation of the cloud droplets if no details about the CN population and its dynamics are foreseen. With this in mind, there are two possible ways to compute the nucleated cloud droplet concentration. The first one requires the prediction of the supersaturation field (Chen 1994) and the temporal tracking of *s*_{υ,wmax}*s*_{υ,wmax}*s*_{υ,wmax}

*w*(Pruppacher and Klett 1997). The rate of change of supersaturation is increased by the adiabatic lifting of the moist air in proportion to

*w*and is decreased by the condensational growth of water vapor on the cloud droplets. A second equation expresses the growth of a single droplet by diffusion of water vapor as given by the Köhler theory after simplifications, so we start fromwith symbols defined in appendix A. Notice that in case of precipitating cloud, the coalescence effects can deplete dramatically the cloud water mixing ratio

*r*

_{c}by autoconversion and accretion, leading to an unbalanced supersaturation rise until additional CCN are activated (see Ochs 1978; Song and Marwitz 1989). This effect is not accounted for in Eq. (1) because we are here only concerned with the early stage of cloud formation. Also in Eq. (2), the term

*y*(

*D, T*), which contains the Kelvin and Raoult effects, is seen as a correction so we refer to Feingold and Heymsfield (1992) for a possible handling of

*y*(

*D, T*) in the context of Twomey’s activation spectrum power law to improve our results.

*r*

_{c}may be approximated bywhere the concentration number of nuclei

*n*(

*s*) active in the interval

*s*and

*s*+

*ds*is given by

*s*

_{υ,wmax}

*n*(

*s*′) =

*kCs*′

^{k−1}, as in Twomey (1959), yields a rather concise expression for

*s*

_{υ,wmax}

*n*

*s*

*kCs*

^{k−1}

*βs*

^{2}

^{−μ}

*C, k, β,*and

*μ*) has a different power law asymptotic limit for large and small supersaturations

*s*′ when

*μ*> (

*k*− 1)/2. The

*β*parameter with dimension of an inverse squared supersaturation is related to the position of the maximum of the

*n*(

*s*′) function and hence to the concavity of the

*N*

_{CCN}(

*s*

_{υ,w}) spectrum.

*x*= (

*s*′/

*s*

_{υ,w})

^{2}allows for formal integration of Eq. (6) using Eq. (7):where

*B*(

*x, y*) is the beta function and

*F*(

*a, b, c*;

*x*) is the hypergeometric function (Gradshteyn and Ryzhik 1965).

*s*

_{υ,wmax}

*ds*

_{υ,w}/

*dt*= 0. With Eq. (8) inserted into Eq. (1), a solution of the upper limit of

*s*

_{υ,wmax}

*N*

_{CCN}is obtained from Eqs. (4) and (7), that is,This last expression can be viewed as a simple generalization of the classic CCN activation spectrum because, as will be shown later, it is applicable to more realistic CCN populations for supersaturations greater than 0.01%. The hypergeometric function involved in (9) and (10) can be easily tabulated (see appendix B for details).

Some properties of (10) are illustrated in Fig. 1 with various configurations for the set of parameters. Cases with *μ* = 0 correspond to the power law relation *N*_{CCN} = *Cs*^{k}, which remains valid for asymptotically small supersaturations. For *k* = 2, the solid line curves with variations of *μ* between 0.7 and 1.5 show that the asymptotic slope of *N*_{CCN} can be adjusted for large supersaturations (greater than 0.1%). The case *k* = 0.5 with *μ* = 2 and drawn with dotted lines shows the effect of the parameter *β* = *s*^{−2}_{break}*μ* parameter is strong in the [0, 2] range because the *N*_{CCN} curves flatten rapidly as *μ* reaches 2. As *N*_{CCN}(*s*) is a monotonic function, a flat curve means a total consumption of the CN reservoir.

## 4. A first validation of the formulation

In the last section, a parametric law with four parameters has been selected to describe the shape of realistic activation spectra but with the constraint that it enables an analytical integration in the Twomey nucleation scheme. Although the function defined by (10) offers greater tuning compared to the classic power law, it is necessary to make sure that it is meaningful when compared to true activation spectra. So for that purpose we choose to compare the new activation law against two synthetic quasi-continuous datasets computed from a complex physicochemical aerosol model coupled with an activation scheme, and against sparse experimental data published by Hudson and Li (1995).

### a. Synthetic datasets

The data are generated by running a model of aerosol growth for two different cases as suggested by von der Emde and Wacker (1993). A corresponding set of four coefficients is then determined to get the most accurate representation of the activation spectrum of each population of aerosols.

Concisely, the physicochemical model we use is composed of an aerosol model coupled to an activation scheme based on a simplified form of the Köhler theory (Bedos et al. 1996). The model computes the microphysical and chemical evolution of a single distribution of aerosol whose growth is given by finding the thermodynamical equilibrium of various gases over aerosols of complex chemical composition. The various equilibria are resolved in the SEQUILIB module (Pilinis et al. 1987), which takes into account the departure from nonideality of the aqueous mixture. The initial parameters of the distribution are chosen to select the accumulation mode of a population of natural aerosols (i.e., the CN) taken well below their point of deliquescence (i.e., under dry enough conditions). For the purpose of this test, the relative humidity is increased linearly up to 99%, so the physicochemical model of aerosol integrates the evolution of the aerosol distribution due to water condensation.

*r*

*σ*the geometric standard deviation, and

*N*the total particle number. An important factor controlling the growth of the aerosols is their chemical composition in NaCl, H

_{2}SO

_{4}, NH

_{3}, HNO

_{3}, HCl, and water; all these compounds need to be initialized to structure the chemical signature of the considered aerosol type. Furthermore, the aerosols are assumed to be completely soluble, chemically homogeneous, and internally mixed with respect to their size. Inclusion of inert species, to simulate the presence of a solid insoluble core of the aerosols, is not considered here because varying the solubility preserves, with a good approximation, the shape of the activation spectrum that undergoes a simple shift on the supersaturation axis (Fitzgerald 1973; von der Emde and Wacker 1993). The evolution of the distribution parameters of the aerosols (a mass increase due to water condensation, as diagnosed by SEQUILIB) is obtained by the method of the moments. Here

*σ*and

*N*are kept constant because we do not consider other processes such as the coagulation of the particles. Once humidity has reached 99%, the activation scheme is applied to convert the resulting size distribution of the CN,

*n*

_{99}, into an activation spectrum. This is done as in Bedos et al. (1996) [see Eqs. (8), (9), and (15) of their paper], where, given a series of supersaturations, the critical radius

*r*

^{crit}(

*s*

_{υ,w}) is computed by the Köhler theory and converted into the corresponding radius

*r*

^{crit}

_{99}

*s*

_{υ,w}) of the CN distribution, at 99% of humidity. The determination of total number of activable CCN is straightforward and results from a partial integration of

*n*

_{99}assuming (11), namely,

We chose to validate the new CCN activation spectrum function (10) for typical maritime and continental air parcels. The initial configurations of the CCN generating model are listed in Table 1 and results of the two computations are plotted in Fig. 2 with solid lines. In both cases, these results are consistent with those of von der Emde and Wacker (1993), although they have considered a multimodal distribution of CCN.

The problem now is to determine the sets of four parameters (*C, k, μ,* and *β*) for the purpose of fitting, after discretization, both solid curves of Fig. 2 by the function defined in (10). A rough estimation can be done by considering that *k* is the slope for a small supersaturation regime in a log–log plot (see Fig. 1) and that *C* is a scaling factor. Then *β* and *μ* can be adjusted after examination of the location of the slope break and of the flat aspect of the curves at high supersaturation, respectively. However, in order to proceed an objective nonlinear adjustment, we have tested the Levenberg–Marquardt algorithm (Press et al. 1992) to generate automatically the same set of parameters. This nonlinear least squares method needs the evaluation of the gradient of the expression in (10) with respect to *C, k, μ,* and *β.* This has been done explicitly for the *C* and *β* parameters while we found it was more efficient to integrate (7) to get a numerical estimate of the gradients to the “exponent” parameters *k* and *μ.* Because of the wide range of possible values for both *s*_{υ,w} (around four orders of magnitude) and *N*_{CCN} (a maximum of three orders of magnitude in clouds), the method must be adapted to these constraints. Therefore, we use the minimization algorithm in the log–log scale in the 0.005% ⩽ *s*_{υ,w} ⩽ 10% region. The results of the two independent adjustments (maritime and continental cases) are plotted with dotted lines in Fig. 2. These curves are barely distinguishable from their respective “reference” counterparts, so the method is able to produce very accurate adjustments. Moreover, a series of tests have shown that the method is robust enough because the convergence was always successful, even if the initial set of parameters was very different from the final one. This is probably due to the fact that *n*_{99} is a three-parameter function like (11), whereas a four-parameter function (10) is used to fit (12).

The computed coefficients are *C* = 3270 cm^{−3}, *k* = 1.56, *μ* = 0.70, and *β* = 136 for the continental air parcel and *C* = 1.93 × 10^{8} cm^{−3}, *k* = 4.16, *μ* = 2.76, and *β* = 1370 for the maritime air parcel. It can be noted that large differences exist between our *C* and *k* coefficients and the recommended values given in Pruppacher and Klett (1997, 226) for Twomey’s power law. This discrepancy arises because as *k* may be two to four times as large as the corresponding values given for the Twomey activation spectrum, *C* has to counteract this power effect and so increases accordingly. That *k* can reach larger values than expected in the literature was already observed by Jiusto and Lala (1981) from measurements with a thermal gradient diffusion chamber.

### b. Experimental case

This test is an example of fitting our new expression (10) with real case data of representative activation spectra. It is important to recall that activation spectra measurements are difficult to make on board research aircraft due to the fine control and calibration of the supersaturation field in the cloud chamber of the instrument. As a consequence of the delicate experimental procedure, which may require significant extra time to get several stabilized supersaturated conditions, only a few data points are generally available for a continuous air sample.

The data we use were collected during the Atlantic Stratocumulus Transition Experiment (ASTEX) and reported by Hudson and Li (1995). Details concerning the CCN measurements and their accuracy are given in Hudson (1989). The analyzed dataset illustrates the marked differences between clean and polluted air masses in the mid-Atlantic atmospheric boundary layer. Each activation spectrum is an average taken along a sounding path. The experimental *N*_{CCN}(*s*_{υ,w}) curves are given at *s*_{s,w} = 0.04%, 0.1%, 0.4%, and 0.7% and the total CCN number concentrations (*N*^{total}_{CCN}

The results are presented in Figs. 3a and 3b for the polluted and the clean air parcels, respectively. The solid line corresponds to the nonlinear fit of expression (10) while the dashed line corresponds to the linear fit (in log–log scale) of a power law, excluding the *N*^{total}_{CCN}*N*^{total}_{CCN}*s*_{s,w} = 20%, 10%, or 5% yields quite identical results for the adjustment of (10) so we arbitrarily consider an upper limit of *s*_{s,w} = 10% for the *N*^{total}_{CCN}*s*_{s,w} = 0.04% and 0.1%. We found it consistent with the fact that the CCN spectrometer estimate is poor when the air sample contains very few activable nuclei at these low supersaturations. The adjusted coefficients are given in Table 2.

Looking at Figs. 3a and 3b, we can see that the adjustments of (10) to the CCN measurements perform better than they do for Twomey’s power law as confirmed by the *χ*^{2} test (not reported here). Furthermore, the new expression (10) is able to integrate the *N*^{total}_{CCN}

In order to estimate the impact of these different adjustments, the cloud droplet number concentration *N*_{droplet} has been computed as a function of *w* using the standard Twomey scheme (1959) and its generalization [i.e., Eqs. (9) and (10)] with data from Table 2 and for *T* = 283 K and *P* = 800 hPa. The results, plotted in Figs. 4a and 4b for the polluted and the clean cases, respectively, indicate that *N*_{droplet} coming from our new activation spectra increases less with *w.* In Fig. 4a, the deviation between the two estimates of droplet concentration is moderate for *w* < 300 cm s^{−1} (Δ*N*_{droplet} < 70 cm^{−3} at *w* ∼ 100 cm s^{−1}). Beyond this limit the standard Twomey scheme dramatically overpredicts *N*_{droplet} compared to the present scheme. In Fig. 4b, the situation is worse as, for instance, Δ*N*_{droplet}/*N*_{droplet} may reach 20% for *w* ∼ 30 cm s^{−1} and considerably more for *w* > 100 cm s^{−1}. This greater sensitivity of clean maritime CCN to nucleation scheme, due to the large value of the *k* parameter, adds some uncertainties to the climatic response of marine low-level clouds to environmental parameters.

In conclusion, adjusting experimental activation spectra to simple power laws or using more sophisticated functions such as Eq. (10) leads to significant differences in *N*_{droplet}. We show that introducing a CCN number limitation at large supersaturations is important and also has noticeable impact at low supersaturation regimes.

## 5. Conclusions

The need for an accurate and economical droplet nucleation parameterization is of great interest for cloud-resolving mesoscale/LES models with bulk advanced microphysical schemes where both cloud droplet and raindrop mixing ratio and number concentration could be predicted. In this context, droplet nucleation, which is actually one of the most challenging processes to model in warm clouds, is of prime importance because it determines the droplet concentration and hence the onset of drizzle formation, the effective scattering of the (solar) radiative fluxes, and the scavenging efficiency of interstitial aerosols and gases.

There are several ways to represent the nucleation processes in bulk models. If one assumes a simple power law for the droplet activation, Twomey’s analytical formula is very powerful for computing the cloud droplet number concentration. However, there are numerous observations that show such a simple power law form is inadequate under certain circumstances and can give a misrepresentation of the real cloud droplet number concentration if used. To remedy this deficiency one might try to fit the activation curve by a series of power functions, but this requires the accurate prediction of the supersaturation field in the whole domain. This is a difficult task, especially at cloud edges where nucleation is most likely to occur (see Stevens et al. 1996 for a discussion). Furthermore, such explicit techniques seem questionable as the maximal supersaturation, which has a clear definition in the Köhler theory (see Pruppacher and Klett 1997), is not readily available in such computations because of the very short timescale of the physical process.

In this study, we have shown that Twomey’s approach, based on reasonable assumptions about the behavior of the supersaturation field at the microscale, can be easily extended to more realistic shapes of activation spectra, thus preserving the original strength of the analytical derivation. The new method only needs a slightly more complex function to represent a broad variety of activation spectra with four parameters instead of two as for the classic power-law dependence. We showed that such a mathematical function is a good candidate because it provides a close fit to realistic activation spectra produced by a single lognormal mode of aerosol population. Moreover, a first attempt to match sparse data of activation spectra was successful and appealing for numerous tests against other experimental data. Note that the solution given in (10) can also be extended to a multimodal shape of CN distribution (Quinn et al. 1990; Hoppel et al. 1990). By applying the superposition principle, each mode can be approximated separately by a set of four coefficients in order to build up a complex shape of the activation spectrum after summation. This is not possible using Twomey’s power law. However, as it is true that only the “accumulation mode” is crucial for droplet nucleation because of the typical maximal supersaturation range in clouds, many details of the activation spectrum curves (or on the corresponding CN size distributions) outside that range can be disregarded. Finally, it is worth noting that the scheme can also benefit from some improvements discussed by Feingold and Heymsfield (1992), who retain the effects of kinetic diffusion for heat and water vapor (Fukuta and Xu 1996) together with the curvature and a solute term in the droplet growth equation.

The present scheme has been incorporated in a newly developed bulk scheme to study warm precipitating clouds. It is currently being tested to examine the sensitivity of the four parameters that shape the activation spectrum.

## Acknowledgments

We are pleased to acknowledge our colleague K. Suhre for his patient correction and remarks made on an earlier draft. We thank the reviewers for helping to improve and clarify the manuscript.

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## APPENDIX A

### List of Symbols

*B*(*a, b*) beta function*C*activation spectrum coefficient*D*droplet diameter*D*_{υ}diffusivity of water vapor in the air*e*_{s}saturation vapor pressure*F*(*a, b, c*;*x*) hypergeometric function*g*gravitational acceleration*G*(*D, T, P*)*k*activation spectrum coefficient*k*_{a}heat conductivity of air*L*_{υ}latent heat of vaporization*n*(*s*) nuclei concentration number between*s*and*s*+*ds**n*_{99}CN size distribution at 99% of humidity*N*_{c}cloud droplet number concentration*N*_{CCN}total activable CCN concentration*P*pressure*r*radius of the aerosol median radius of the aerosol size distribution*r**r*_{c}cloud water mixing ratio critical radius at the critical supersaturation (*r*^{crit}(*s*_{υ,w})*s*_{υ,w}) (from Köhler’s theory) corresponding radius at 99% of humidity*r*^{crit}_{99}(*s*_{υ,w})*r*_{e}effective droplet radius*R*_{d}gas constant for dry air*R*_{υ}gas constant for water vapor*s*_{υ,w}supersaturation (%) maximum supersaturation (%)*s*_{υ,wmax}*T*temperature*w*updraft velocity*y*(*D, T*) Kelvin’s and Raoult’s effects correction term*α*Junge’s power law coefficient*β*activation spectrum coefficient*γ*Junge power coefficient*ϵ*molecular weight ratio of water vapor to dry air*μ*activation spectrum coefficient*ψ*_{1}(*T, P*)*ψ*_{2}(*T, P*)*ρ*_{a}air density*ρ*_{w}liquid water density*σ*geometric standard deviation of aerosol spectrum.

## APPENDIX B

### Computation of the Hypergeometric Function

*F*(

*a, b, c*;

*x*) is computed by the series defined below:

*x*| < 1, so out of this interval it is necessary to use the following transformation:Notice that the above formula is applied for

*x*= −

*βs*

^{2}

_{υ,wmax}

*x*away from 1; the singularity for |

*x*| = 1 is overcome by interpolating numerical values of

*F*(

*a, b, c*;

*x*) on both sides of |

*x*| = 1.

Initial values of the lognormal aerosol distribution parameters used to produce the two aerosol types (continental and maritime).

Values of the adjusted parameters for activation spectra described by Twomey’s power law and by expression (10) for the polluted and clean air cases of Hudson and Li (1995).