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 In the present study a new method of calculating droplet concentration near cloud base is proposed. The ratio of maximum supersaturation Smax to the liquid water mixing ratio when Smax is reached near cloud base is found to be universal, and it does not depend on the vertical velocity w and droplet number concentration N. It is found that Smax depends on vertical velocity as Smax ∝ w3/4 and on droplet concentration as Smax ∝ N−1/2. The droplet concentration calculated using the simple approach agrees well with exact solutions obtained numerically using high precision parcel models. Comparison with the results of other parameterizations is presented. It is demonstrated that the approach proposed in the study can be applied to an arbitrary form of activation spectra or any CCN size distribution given either analytically or by tables. Moreover, it can be applied for the cases when the CCN size spectrum changes with time. Temperature dependencies of Smax and related quantities are analyzed.
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 Droplet concentration in clouds is largely determined by the value of maximum supersaturation near cloud base Smax and CCN activity spectrum NCCN(S). Droplets nucleated near cloud base form the first mode of the largest droplets within the droplet size distribution (DSD) which plays the dominating role in precipitation formation and radiation transfer. For typical vertical velocity w and NCCN(S) the level of formation of Smaxabove the cloud base varies from tens of centimeters to tens of meters and it is quite narrow. To resolve the supersaturation maximum in the cloud models, the grid spacing in the vertical direction should be less than 1 to 10 m. This maximum cannot be resolved in large scale models or in most cloud models with the exception of 1-D parcel models. The droplet concentration determines major microphysical cloud properties such as height of precipitation onset, type of precipitation (warm versus cold), as well as radiative cloud properties. The influence of the atmospheric aerosols on clouds represents itself first of all via the effects of aerosols on droplet concentration. Thus, droplet concentration is one of the most important parameters to be determined in cloud models.
 The value of Smax depends on the vertical velocity at cloud base and the size distribution of cloud condensational nuclei (CCN) that can be represented by the activity spectra. When droplet concentration is known, for instance, from in situ measurements, the relationship between Smax and droplet concentration can provide useful information about CCN under different atmospheric conditions.
 Several studies were aimed at calculating Smax and estimating the droplet concentration [e.g., von der Emde and Wacker, 1993; Ghan et al., 1993, 1997, 2011; Cohard et al., 1998; Abdul-Razzak and Ghan, 2000; Abdul-Razzak et al., 1998; Shipway and Abel, 2010]. All these studies are based on the classical work by Twomey , and they analyze the process of new droplet nucleation within an air volume ascending from cloud base at a given vertical velocity. The growth rate of the newly nucleated droplets affects both supersaturation and supersaturation maximum. Thus, the process considered is actually a complicated process of continuously increasing droplet concentration within the narrow layer between the cloud base and the level of the supersaturation maximum. Mathematically, a system of transcendental equations for Smaxand droplet concentration expressed in terms of beta- and hypergeometric functions is solved in the studies. The complexity of the equations increases with the increasing complexity of expressions for CCN activity spectraNCCN(S) or of size distributions of CCN.
 In this study we present analytical estimations of Smax based on a novel equation for supersaturation, and apply the results for calculation of droplet concentration at cloud base.
2. Analytical Estimation of the Maximum Supersaturation Near Cloud Base
 In a study by M. Pinsky et al. (On the theory of cloud droplet diffusion growth. Part 1: Monodisperse spectra, submitted to Journal of the Atmospheric Sciences, 2012) a new closed equation for supersaturation was obtained for an adiabatic air volume ascending from cloud base and containing particles of a uniform size with concentration N (see Appendix A)
where w is the vertical velocity at cloud base and z is the height above the cloud base. All other notations are presented in the Notation section. This equation is based on the balance equation (see equation (A2) in Appendix A)
as well as on the equation for diffusion growth, while the curvature and the chemical terms are not taken into account. It is assumed that at cloud base S(z = 0) = 0 and liquid water mixing ratio qw (z = 0) = 0. Comparisons with exact numerical solutions show that equation (1) describes the vertical profile S(z) with high accuracy at distances exceeding a few tens of centimeters above cloud base (Pinsky et al., submitted manuscript, 2012). It is assumed also that the coefficients A1, A2 and B1 in equations (1) and (2) are constant, which is valid for distances of a few hundred meters above cloud base.
Equation (1) is a closed nonlinear differential equation. It has a simpler form as compared to that suggested by Korolev and Mazin  and hence allows further analytical advances. An important feature of equation (1) is that it allows estimation of Smax and the height zmax where Smaxis reached. For the following analysis we introduce two new variables: the non-dimensional altitudeh = A1zand the non-dimensional parameter
where F is the coefficient slightly dependent on T and P (see Appendix A). Typically in water clouds R ≫ 1. Using the new variables, equation (1) can be rewritten in a dimensionless form
 An eligible initial condition for this equation is S|h = 0 = 0 at cloud base. By inspection, equation (4) shows that the maximum of supersaturation solution Smax depends on the sole parameter R. The condition in equation (4) yields an expression relating Smax and hmax as:
where constants C1 and C2 are related by the equation
 The relationship (7) follows from equations (5) and (6). Coefficients C1 and C2 are universal, i.e., they do not depend on the environmental or thermodynamical characteristics of the cloud parcel. In order to find the coefficients C1 and C2 differential equation (4) was numerically integrated for a wide range of values R. For each R the values of Smax(R) and hmax(R) were determined.
 As it was demonstrated by Pinsky et al. (submitted manuscript, 2012), the power law dependencies (6) agree with numerical integrations nearly perfectly. The values C1 and C2 were determined using these dependencies. These coefficients are C1 = 1.058 and C2 = 1.904 and satisfy equation (7) with an accuracy of 10−4.
Equation (6) shows that the value Smax and height of this maximum zmax are linearly related as:
 For instance, for the cloud base temperature equal to 10°C, Smax [%] = 3.06∙10−2zmax [m].
Equations (3) and (6) allow establishing the relationships between droplet concentration, vertical velocity and supersaturation maximum as:
where coefficient depends on the thermodynamical parameters.
Equations (2) and (8) show that a split of water vapor between the condensed liquid water and supersaturated fraction of water vapor at the level zmax is universal, namely:
 Since is the adiabatic liquid water mixing ratio corresponding to the full condensation of supersaturated water vapor, equation (11) shows that at z = zmax the liquid water mixing ratio is a universal fraction (=0.44) of the adiabatic value.
where . Under TC = 10°C and w = 1 ms−1, equation (12) yields rzmax = 3.28 μm and rzmax = 1.04 μm for particle concentrations of 100 cm−3 and 1000 cm−3, respectively. The values of rzmax can be considered as the characteristic values of droplet size forming at the level of supersaturation maximum. Equation (12) reflects the result of diffusion growth of particles just after nucleation. Accordingly, it shows that drop size increases with an increase in cloud base velocity and decreases with increasing drop concentration. Equations (8)–(12) can be useful in cloud models as a description of the process of diffusion growth in the narrow layer in the vicinity of cloud base.
 It is widely accepted in cloud physics to relate the concentration of nucleated drops (CCN activity) with supersaturation measured in %, using the following expression for activity spectra [Pruppacher and Klett, 1997]
where the parameter N0 and the slope parameter k vary within a wide range depending on geographical locations, meteorological conditions and even the time of day. The typical values of N0 are about 100 cm−3 for maritime clean conditions and it ranges from 500 cm−3 to several thousand cm−3 for continental clouds at different levels of aerosol loading [Pruppacher and Klett, 1997; Andreae et al., 2004; Prabha et al., 2011]. The values of the slope parameter k vary from about 0.3 to 1 in clear and polluted air, respectively. Since above the level of zmax supersaturation decreases, drop concentration does not change, so droplet concentration above the supersaturation maximum can be found from the expression N = N0Smaxk. Substituting this expression in equation (9) leads to two expressions for supersaturation maximum and drop concentration, respectively:
3. Comparison With Previous Results
3.1. Parameterizations Using CCN Activity Spectra
Twomey  was the first to obtain expressions for supersaturation maximum and droplet concentration (the amount of activated CCN) at the level of this maximum. Twomey derived an equation for supersaturation as a function of height, describing the process of droplet nucleation in a parcel ascending at a given vertical velocity from cloud base up to the level of supersaturation maximum. The nucleated droplets grow affecting supersaturation and decreasing supersaturation maximum. Twomey used activation spectra in the form (13)to describe the continuous nucleation and solved the integro-differential equation for supersaturation using the semi-geometrical approach. The final expression derived by Twomey for the supersaturation maximum is the following:
where B(3/2, k/2) is the beta function and . The expression for drop concentration was obtained by Twomey in the form:
Equations (16) and (17) demonstrate the same dependencies of droplet concentration on N0 and w as those in equations (14) and (15) obtained in the present study. The formulas differ only by coefficients CT and C3, the former being dependent on parameter k. The dependencies of ratios and on k at TC = 0°C and 10°C are presented in Figure 1 for comparison. While calculating the dependencies shown in Figure 1, we took into account that Twomey measures supersaturation in °C. The units °C and % are related as . One can see that the ratio N/NT changes only slightly with variation of the slope parameter k. At TC = 10°C and k < 0.8 Twomey's formula (1959) yields about 95–98% of the drop concentration calculated using equation (15) obtained in this study. At higher values of the slope parameter k the difference between concentrations predicted by Twomey and that obtained from the proposed method increases.
Equation (13) for activity spectra is often used in cloud models to calculate droplet concentration on cloud base. Strictly speaking, while this formula is valid within a certain range of supersaturation values, it becomes invalid at very high supersaturations since it yields infinitely large CCN concentration at S→∞. Since concentration of small CCN is limited, the slope parameter should tend to zero when S→∞. Equation (13) should also be corrected for very small supersaturations in order to take into account the lack of very large CCN. Accordingly, the expression for activity spectra is often written in the following form [e.g., Cohard et al., 1998; Shipway and Abel, 2010]:
where the hypergeometric function 2F1(Smax, k, μ, β) plays the role of a limiting factor preventing nucleation at high supersaturations. Parameters μ and β allow to tune the asymptotic behavior of N(S) for large and small supersaturations in order to match actual observations. Note that in the study by Shipway and Abel  parameters k, μ and β are functions of the underlying aerosol physicochemistry and are not chosen to match observations.
 The expression for supersaturation maximum (9) can be combined with any modifications of the CCN activity spectrum formula, for instance, with equation (18). In order to test the validity of the parameterization based on equation (9), calculations were performed for the activity spectra in the forms (13) and (18) for parameters used by Cohard et al.  for clean and polluted air which are presented in Table 1. Figure 2 shows the dependencies of droplet concentration on the vertical velocity for polluted air and clean air at T = 283 K and P = 800 hPa using these activity spectra. One can see that the curves obtained using equation (9) actually coincide with those obtained using Twomey's and Cohard et al.  parameterizations with the exception for the case with k = 1.57, where our approach predicts droplet concentration slightly higher than that obtained by the Twomey's solution (17). This result is in agreement with the analysis presented above (see Figure 1). These results show that equation (9) can be used for reproduction of the dependence of droplet concentration on the vertical velocity for different activity spectra.
Table 1. Values of the Adjusted Parameters for Activity Spectra Described by Twomey's Power Law and by Equation (18) for the Polluted and Clean Aira
 The CCN size distribution often has a complicated form. It is often represented using a lognormal distribution [Ghan et al., 2011] or as a sum of three lognormal distributions (such a sum is often referred to as 3-mode lognormal distribution) with parameters varying within wide ranges [Respondek et al., 1995; Segal and Khain, 2006]. In spectral bin microphysics models the CCN distribution is represented on a mass grid [e.g., Khain et al., 2004]. The shape of CCN size distribution changes with time as a result of cloud-aerosol interaction accompanied by the washout of CCN, mixing, penetration of air with lower CCN concentration from higher levels, nucleation of largest CCN, etc. Accordingly, in the course of cloud systems development, the parameters of the activity spectrum change with time. In this case, it is quite complicated to calculateSmax and droplet concentration using the approach based on the CCN nucleation spectra in the form (18). Instead, it is preferable to use the CCN size distribution as the basis.
 Since the droplet concentration is equal to the concentration of CCN activated at S = Smax, it can be calculated as
where f(rn) is a given size distribution of dry aerosol particles and rn_cr is the critical radius of aerosol activated under Smax. This radius relates to Smax as [e.g., Khain et al., 2000]
where coefficients A and B are the coefficients of the Kohler equation , presented in the Notation section. From equations (9), (19), and (20) one obtains a closed equation for Smax
Equation (21) is the equation with respect to Smax that can be easily solved numerically using simple iteration method. The r.h.s. in equation (21) is assumed to be known. Note that the l.h.s. of equation (21) is a monotonically increasing function of Smax. Simply increasing Smax from the first guess at which the l.h.s. is smaller than r.h.s. leads to a value at which the l.h.s. becomes equal to the r.h.s. with a given precision. As soon as Smax is found, equation (9) can be used to calculate the drop concentration N and other parameters.
 Let the CCN distribution fn be given as a sum of I lognormal distributions:
where niis the number concentration in the i-th mode,Ri and σiare the mean radius and the width of the i-th aerosol mode, respectively.
 Let us first consider a single mode lognormal distribution with parameters used by Ghan et al. .
Figure 3 shows maximum supersaturation both parameterized and simulated using a numerical parcel model and number fraction activated as functions of updraft velocity for a single lognormal aerosol mode with n = 1000 cm−3, the mode radius R = 0.05 μm and log σ = 0.3 and composition of ammonium sulfate. The figure is plotted for different parameterization schemes: ARG is the Abdul-Razzak and Ghan  modal parameterization. Nenes is the Fountoukis and Nenes  scheme. Ming is the Ming et al.  scheme. Shipway is the Shipway and Abel  scheme. The curves corresponding to the approach presented in this study are denoted as “This study.”
Figure 4 shows parameterized and simulated (using a numerical parcel model [Ghan et al., 2011]) maximum supersaturation and number fraction activated as a function of aerosol number concentration for a fixed updraft velocity of 0.5 m/s. The aerosol number concentration is equal to 1000 cm−3. Figure 5 shows parameterized and simulated number fraction activated as a function of geometric standard deviation of the lognormal size distribution, for a fixed updraft velocity of 0.5 m/s and the geometric standard deviation σ = 2 (i.e., log σ = 0.3). Finally, Figure 6 shows parameterized and simulated maximum supersaturation and number fraction activated as a function of mean radius of the aerosol size distribution.
 1. The simple method based on solving equation (21) produces the dependencies of Smax and number fraction of activated CCN on vertical velocity and on CCN concentration that are very close to the exact solution. The agreement is especially good for aerosol concentration less than 1000 cm−3.
 2. Similarly to parameterizations proposed by Shipway and Abel  and Fountoukis and Nenes , the proposed method does not well predict the slope of the dependence of number fraction of activated CCN on the width of the CCN spectrum. At the same time, within the range of typically used values of σ from 1.9 to 2.2, the method predicts the number fraction of activated CCN with high accuracy. Note that variation of drop concentration with the change of σ is substantially less than that with the variation of CCN concentration and vertical velocity.
 3. The proposed method underestimates somehow the value of the supersaturation maximum at the low mean radius (∼0.01–0.03 μm), but gives reasonable agreement with the numerical solution at higher mean radii. As regards to the fraction of CCN nucleated as a function of the mean radius, the method agrees well with the results of Shipway and Abel . Again, for typical CCN mean radii a reasonable agreement with numerical solution is seen.
 Analysis and interpretation of these results is given in the discussion and conclusion section.
 The calculations are performed also using the CCN spectrum given as a three mode lognormal CCN distribution (22). Following Ghan et al.  four cases of aerosol spectra were considered: “marine,” “clean continental,” “background” and “urban.” The parameters of these distributions are given in Ghan et al.  and are presented in Table 2. The first mode represents very small aerosols (nuclei mode), the second mode, which plays dominating role in formation of droplet concentration, is the accumulated mode and the third mode (coarse mode) represents a small number of the largest CCN. Figure 7 compares the dependencies of supersaturation maximum on vertical velocity obtained for these aerosol distributions using different parameterization schemes including the approach presented here (denoted as “This study”). Figure 8 shows the dependencies of the fraction of activated CCN to the entire aerosol concentration.
Table 2. Aerosol Distribution Parameters for Four Tested Casesa
From Ghan et al. . Aerosol radii and the width of aerosol modes are given in μm, concentrations in the modes are in cm−3.
 Analysis of the results shows that the approach proposed in the present study shows that in most cases (except the “urban” case) droplet concentrations predicted by the new approach are quite close to the exact values within the whole range of vertical velocities. Even in the extreme urban case the agreement with exact solution is quite reasonable. In many cases the results obtained using this simple approach turn out to me closer to the exact solutions than those obtained by previous methods.
 Now we compare the results obtained using equation (21) with those presented in lookup tables calculated by Segal and Khain  using a detailed parcel model [Pinsky and Khain, 2002; Segal et al., 2004]. The main feature of the model is an accurate description of diffusional growth of wetted aerosols and droplets. To describe the DSD of particles (non-activated aerosols and droplets) 2000 mass bins are used within the range from 0.01μm to 2000 μm. The grid spacing was 0.001 μm for small particles and it gradually increases up to 8 μm for large particles. Such resolution is sufficient for explicit description of the process of separation of all particles into growing droplets and non-activated wetted aerosols. Accordingly, the process of droplet nucleation is treated directly without using any parameterization procedures. To describe the diffusion growth, a non-regular grid with a variable set of masses is used. The masses related to corresponding bins are shifted with time according to the equation of diffusion growth. Correspondingly, no remapping is applied, i.e., no artificial spectrum broadening is introduced when droplet growth by diffusion is calculated. The time step of 0.005 s was used to calculate diffusion growth of drops and aerosol particles.
 The calculations are performed using the CCN spectrum given as a three mode lognormal CCN distribution (22). The values of these parameters used by Segal et al.  and Segal and Khain  are given in Table 3. The first mode describes the smallest CCN, the second mode is the mode of small CCN which give rise to most cloud droplets, and the third mode represents the giant CCN. Figure 9 shows the dependencies of Smax and droplet concentration on the CCN concentration in the second mode. The CCN concentration in the second mode varies from 50 cm−3 which is typical of clean air to 2000 cm−3 typical of polluted air. The results shown in Figure 9 agree well with those obtained from lookup tables presented by Segal and Khain . The drop concentration/CCN concentration ratio is close to one at low droplet concentration (typical of maritime clouds) and decreases with increasing CCN concentration. Such dependence qualitatively agrees with that reported by Ramanathan et al. .
Table 3. Parameters of Aerosol Spectra, Used in the Example for Comparison With Results by Segal and Khain 
4. Temperature Dependencies
 The classical equation (17) was derived by Twomey  for certain temperature and pressure values at cloud base. To our knowledge, the sensitivity of Smax and of droplet concentration to thermodynamical parameters was not investigated earlier. However, the cloud base temperatures can be quite different in different geographical zones and seasons. The temperature dependence comes from the dependence of the coefficient C3 on temperature. This dependence is shown in Figure 10. The dependence is quite strong which indicates a possible significant influence of temperature on Smax and drop concentration at cloud base.
Figures 11a–11f show dependencies of Smax(TC) (a, b), zmax(TC) (c, d) and rmax(TC)(e, f) at different droplet concentrations for w = 1 m/s (left panels) and at different w for N = 103 cm−3 (right panels). One can see that Smax(TC) significantly depends on the temperature of cloud base. Accordingly, the changes in zmax(TC) calculated using equation (8) are also significant, as zmax(TC) increases by factor of two with a decrease in the temperature at cloud base. The vertical velocity also varies within a significant range along cloud base. It means that fluctuations of wat cloud base lead to formation of supersaturation maximums at different levels, and, accordingly, to different droplet concentrations. Even high-resolution LES models in many cases do not allow an explicit resolution of this maximum and its fluctuations.
 Spectral bin microphysics models, as well as two-moment bulk parameterization schemes, require the data on the size of droplets formed at the levelzmax. Dependencies rzmax(TC) of drop radii formed at height zmax (calculated using equation (12)) on temperature are shown in Figures 11e and 11f. In the given examples, rzmax(TC) varies up to ∼20–30% with temperature. In a spectral bin microphysics model [e.g., Khain et al., 2004], droplets with such a difference in radii belong to different mass bins. Neglecting this fact can affect the model representation of cloud microstructure at higher levels.
 The dependence of supersaturation at cloud base on the temperature leads to the corresponding dependence of droplet concentration on the temperature. Figure 12 illustrates this dependence at different pressure values for clean and polluted air masses for parameters of CCN presented in Table 1. The dependencies are calculated using equation (15). One can see that changes in the cloud base temperature can lead to significant changes in drop concentration.
5. Discussion and Conclusions
 The value of supersaturation maximum and droplet concentration near cloud base were calculated in many studies [Ghan et al., 2011]. All these studies are based on the approach developed by Twomey , where a complex integro-differential equation for supersaturation is solved taking into account the process of droplet nucleation within an air parcel ascending from cloud base. The growth of newly nucleated droplets is calculated by means of a simplified equation for diffusional growth, the curvature and the chemical terms being omitted. The drop growth affects supersaturation and determines the value of supersaturation maximum. Thus, the supersaturation maximum in the above mentioned studies is calculated by considering aerosols and droplets moving from cloud base upward.
 In the present study, a simpler approach is used where the relationship between Smax, N and w is obtained without explicit treating the nucleation process. It has been found that under certain assumptions the relationship between Smax and qmax at the level zmax is universal and it does not depend on the environmental conditions and thermodynamical properties of the cloud parcel. The supersaturation maximum is related to vertical velocity and droplet concentration as Smax ∝ w3/4N−1/2 (equation (9)). The relationship (9) can be utilized for an arbitrary form of activation spectra or to any CCN size distribution given either analytically or by tables. Moreover, it can be equally applied for the cases, when the CCN size spectrum changes with time due to various processes, such as nucleation of some fraction of CCN, mixing, penetration of clean air from above, washout of aerosols, etc.
 The approach allows calculation of additional parameters related to supersaturation maximum, among them: the height of the maximum above cloud base and droplet size. It is shown that the relationship between supersaturation maximum and liquid water mixing ratio at this level is universal.
 Comparison with other parameterization schemes shows that this simple approach provides results comparable or even closer to exact solutions in case of both single and multimodal aerosol distributions. The good agreement with exact solutions obtained using the simplified assumptions about the monodisperse particle size distribution and omitting the curvature and the chemical terms in the equation of diffusion growth is somehow surprising and requires physical explanation. It should be noted that larger CCN have higher contribution to the formation of Smax as compared to smaller ones for the following reasons. First, since due to the Kohler theory the equilibrium radii of haze particles are proportional to rN3/2, and therefore larger CCN absorb larger amount of water than the smaller ones. Second, more important reason is that the larger CCN are activated at smaller distances above cloud base (∼1 m) and then they continue rapidly growing as droplets, whereas the non-activated CCN (haze particles) grow slowly remaining in equilibrium with the ambient water vapor. The difference in the masses of the activated droplets and haze near the cloud base can reach several orders of magnitude. There exists competition between large and small CCN: droplets forming on large CCN absorb water vapor and decrease supersaturation preventing nucleation of the smallest CCN in the CCN size distribution. The competition tends to narrow the DSD at the level of supersaturation maximum.
 The closeness of the supersaturation profiles calculated for the monodisperse and polydisperse CCN size distributions can be interpreted in the way that the supersaturation profile is determined by “equivalent” CCN which has comparatively large size. The equivalent CCN size of a monodisperse CCN can be defined in such a way that the effect of CCN of this size on supersaturation and drop concentration will be equivalent to that of the polydisperse CCN spectrum. It is reasonable to assume that the equivalent CCN size is close to the mean volume radius of activated CCN. Estimations show that for CCN distributions in equation (22) with mean the radius of the second mode (the most important as regards to effect on droplet concentration) of 0.03 μm and log σ ∼ 0.3, the mean volume radius of activated CCN is about of 0.2 μm. Detailed calculations presented in Pinsky et al. (submitted manuscript, 2012) show that for CCN with rn ∼ 0.2 μm the effect of the corrections on supersaturation due to curvature and chemical terms are cancelling each other and that for polydisperse CCN the supersaturation can be approximated using the approach developed in the present study.
 The exception is the case when CCN size distribution contains only small CCN (less than 0.02 μm in radius). In this case the proposed method overestimates the number of activated droplets as seen in Figure 4 (bottom) at CCN concentrations exceeding 2000–3000 cm−3 in the single mode case. Note that such cases are not typical in the atmosphere, especially at large scales. More often the CCN concentration is less than 2000–3000 cm−3 or aerosol spectrum contains also larger CCN. In this case the proposed method based on equation (21) provides results very close to exact ones.
 The supersaturation maximum and concentration of activated droplets have been found to be sensitive to the cloud base temperature. Since cloud base temperatures vary considerably depending on geographical regions, seasons and cloud type, the temperature dependence should be taken into account in different cloud models, and especially large-scale ones. The proposed method also allows accounting for the solubility of aerosols.
 In the present study it has been demonstrated the advantage of the proposed method and its applicability for any CCN size spectra including cases when the CCN change in space and in time as a result of cloud-aerosol interaction (e.g., nucleation scavenging or wet scavenging). This method can be effectively used in spectral bin microphysics models, where the size distribution of CCN is calculated numerically and has a shape that cannot be approximated using lognormal or gamma distribution.
Appendix A:: Derivation of the Expression for Superstauration
 The equation describing changes of supersaturation S in an adiabatic cloud air parcel can be written in the following form [e.g., Warner, 1969]:
where qwis liquid water mixing ratio. On the right-hand side, the first term describes an increase in supersaturation due to adiabatic air cooling during ascent, whereas the second term describes the supersaturation decrease caused by condensation of water vapor on droplets. CoefficientsA1 and A2 are presented in the Notation section. Integration of equation (A1) yields:
where C = A2q0 + S0 is determined by the values at cloud base (z = 0). Since at cloud base S0 = 0 and the mass of particles is negligible, we use the condition C = 0.
 Consider a cloud parcel with monodisperse DSD, containing droplets of radii r and liquid water mixing ratio
where N is the droplet concentration. Following Twomey  and other studies where Smax, is calculated, the equation for diffusional growth can be written is the form where the curvature term and the chemical term are omitted [Rogers and Yau, 1996]:
 The expression for coefficient F is presented in the Notation section. Utilization of equation (A4) can be justified by the fact that wet CCN particles (haze particles) are significantly larger than dry particles. For wet particles with radii exceeding ∼1 μm, equation (A4) provides a quite accurate solution. Coefficients A1, A2 and F slightly depend on temperature and can be assumed constant in the analysis. Substitution of (A3) into (A4) leads to the following expression for the liquid water mixing ratio:
where . Excluding qw from equations (A2) and (A5) leads to the following equation for supersaturation:
which can be rewritten for the independent variable z as:
specific heat capacity of moist air at constant pressure (J kg−1K−1).
coefficient of water vapor diffusion in the air (m2 s−1).
water vapor pressure (N m−2).
saturation vapor pressure above the flat surface of water (N m−2).
acceleration of gravity (m s−2).
h = A1z.
parameter of activity spectra.
coefficient of air heat conductivity (J m−1s−1K−1).
latent heat for liquid water (J kg−1).
molecular weight of aerosol salt (kg mol−1).
molecular weight of water (kg mol−1).
concentration of liquid droplets (m−3).
parameter of activity spectra.
pressure of moist air (N m−2).
water vapor mixing ratio (mass of water vapor per 1 kg of dry air).
liquid water mixing ratio (mass of liquid water per 1kg of dry air).
liquid droplet radius (m).
drop radius at z = zmax.
S = e/ew−1.
maximum of supersaturation.
absolute temperature (°K).
temperature at cloud base (°C).
vertical velocity (m s−1).
height over the condensation level (m).
height of supersaturation maximum (m).
parameter of activity spectra.
density of the air (kg m−3).
density of a dry aerosol particle (kg m−3).
density of liquid water (kg m−3).
surface tension of water-air interface (Nm−1).
parameter of activity spectra.
Van't Hoff factor.
molal osmotic coefficient.
 The authors express our deep gratitude to S. J. Ghan for his interest to the study, an important contribution to the improvement of the article, and for providing the data for comparison and testing the method. This research was supported by the Office of Science (BER), U.S. Department of Energy Award DE-SC0006788 and the Binational U.S.-Israel Science Foundation (grant 2010446). A. Korolev is supported by Environment Canada.