1. Introduction
Description of the diffusional growth and evaporation of an ensemble of cloud particles is one of the fundamental tasks in cloud physics. The first analytical description of the condensation process of an ensemble of liquid droplets goes back to Squires (1952). A later, detailed theoretical analysis of the supersaturation equation was provided in the work of Kabanov et al. (1971). The behavior of the supersaturation equation was analyzed in many studies (e.g., Twomey 1959; Sedunov 1965; Rogers 1975; Fukuta 1993; Khvorostyanov and Curry 2009). Korolev and Mazin (2003) generalized this equation for a three-phase system consisting of liquid droplets, ice particles, and water vapor. The outcome of these works was an analytical description of supersaturation 
Substitution of Eq. (3) into Eq. (5) and successive integration leads to the linear dependence of droplet mass on height. According to this dependence, droplet mass depends only on the distance between initial and final levels and does not depend on the ascend velocity (e.g., Khain et al. 2000). The linear dependence between vertical velocity and supersaturation creates the major problem in explaining of droplet size distribution (DSD) broadening during the diffusion growth stage so that mechanisms allowing breaking such dependencies were looked for in many studies (e.g., Sedunov 1974; Khvorostyanov and Curry 1999) [see also surveys by Mazin and Merkulovich (2008) and Devenish et al. (2012)].
This type of equation has been used for the analysis of cloud condensation nuclei (CCN) activation near cloud base (e.g., Twomey 1959; Sedunov 1974; Ghan et al. 1993, 1995; Bedos et al. 1996; Cohard et al. 1998; Abdul-Razzak et al. 1998; Abdul-Razzak and Ghan 2000; Fountoukis and Nenes 2005; Khvorostyanov and Curry 2006, 2009; Shipway and Abel 2010). In these studies approximate solutions for supersaturation maximum near cloud base were proposed for different activation CCN spectra. Equation (6) was also used in analysis of supersaturation behavior inside clouds by Korolev and Mazin (2003).
In numerical models supersaturation and droplet sizes are calculated from a numerical integration of the relevant system of differential equations. If special precautions are not taken into account the errors in calculations of S and r may become overly large (Klaassen and Clark 1985; Stevens et al. 1996; Grabowski and Morrison 2008).
Closed equations for supersaturation and water content enabling its analytical treatment would be useful for the analysis of behavior of cloud microphysical variables and for development of parameterizations for numerical models. In the frame of this study we undertook efforts to derive such equations based on the water mass balance equation centered with respect to the adiabatic liquid water mixing ratio. The obtained equations allowed one (i) to estimate the range of droplet spectra broadening caused by fluctuations of supersaturation, (ii) to find analytical expression for the supersaturation maximum magnitude and the altitude above the cloud base, and (iii) to demonstrate universality of the vertical profiles of supersaturation and water content.
The rest of study is organized as follows. In section 2 the equation of water balance as well as closed equations for supersaturation and cloud water content are derived and analyzed. In section 3 the equation for supersaturation maximum near cloud base is derived and analyzed. In section 4 it is shown that equations for supersaturation and liquid water content can be represented in universal nondimensional form. In section 5 the applicability of the approach to real cloud conditions is discussed. Conclusions can be found in section 6.
2. Basic equations of the collective droplet growth
In the following sections we consider an ensemble of monodisperse droplets with concentration 
a. Water balance equation
The water mass balance equation derived in this section forms a basis for the entire analysis in the frame of this study.
The variable 
The ratio 
b. Supersaturation equation
This section presents derivation of a closed equation for supersaturation in a new form.
Equations (12) and (13) represent a new form of the supersaturation equation in comparison to its traditional form [e.g., Eq. (2)] introduced by Squires (1952) and its subsequent modifications. After several simple transformations, Eq. (12) can be reduced to Eq. (2).
The supersaturation equation in the form of Eq. (13) is a closed differential equation with just one dependent variable. Other forms of closed equations for 
Figure 1 shows comparisons of the supersaturation calculated from Eq. (13) and that deduced from a numerical integration of a full system of equations describing a collective droplet growth in adiabatic parcel. Equation (13) was integrated assuming that 
Vertical changes of supersaturation in a cloud parcel ascending with and descending with w = ±5 and ±1 m s−1. The droplet concentration is N = 100 cm−3. The supersaturation was calculated from numerical simulation of the collective droplet growth (solid gray), nonlinear Eq. (13) (solid black), linearized Eq. (14) (dashed–dotted), quasi-steady approximation by Eq. (3) (dashed), and limiting approximation by Eq. (15) (dotted). Initial conditions are 
Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-077.1
c. Limiting supersaturation
Here rad is the adiabatic droplet radius related to the adiabatic water content qad as in Eq. (10). Equation (17) coincides with the expression for the quasi-steady supersaturation in Eq. (3), with the only difference being that the droplet radius 
As seen from Eq. (16) for the case of uniform ascent, 
Figure 1 shows comparisons of the supersaturation calculated from the linearized Eq. (14) and the modeled one. As seen from Fig. 1 the linearized Eq. (14) approximates the exact solution well for z > 40 m at w = 1 m s−1 and z > 150 m at w = 5 m s−1 above the level of the supersaturation maximum. When these altitudes are translated into time required for the parcel to reach them, it turns out that this time remains approximately the same (i.e., 30 s for this specific case). Equation (14) also leads to a formation of a local supersaturation maximum near the cloud base, but this maximum is lower than that obtained from the numerical model.
Figure 1 also shows that 
Here, 
The horizontal, thick arrows on the right-hand side in Fig. 1 indicate the altitude below which 
d. Water content and droplet radius equations
This section presents derivation and analysis of a closed equation for water content.
Analysis of Eq. (19) shows that, when the height is large enough, then each of the two terms in the right-hand side become significantly larger than dqw/dz. The balance between these two large terms leads to a linear dependence of 
After substitution of Eq. (10) into Eq. (21), Eq. (21) turns into a trivial equality 
Figure 2 shows the changes of water content and droplet sizes computed in Eqs. (19) and (20), respectively, for two different vertical velocities. The initial conditions were kept the same as for the case in Fig. 1. As seen from Fig. 2 the integration of Eqs. (19) and (20) provides a good agreement with 
Vertical changes of (a) 
Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-077.1
It is worth noting that 
The ratio Δr/r can be used as a surrogate for the variation coefficient, which is usually referred to as droplet size spectrum relative dispersion and is equal to the ratio of DSD width 
The ratio Δr/r in Eq. (24) was used for estimation of the difference of potential supersaturation in cloud parcels required to obtain a variation coefficient typical for real clouds. The estimations presented in Table 1 were performed for Δr/r = 0.2. As seen from Table 1, the values of 
Difference in the supersaturation required to obtain the size spectrum relative dispersion equal to 0.2. The microphysical parameters used here obtained from the studies indicated in the table.
In case of vertical oscillations, Δr/r may serve as a measure of nonreversibility of microphysical parameters in adiabatic cloud parcels experiencing cycling ascents and descents. Numerical simulation shows that Δr/r reaches its maximum near the cloud base. This effect is clearly seen in Fig. 2b. However, above the level of supersaturation maximum at the cloud base, Δr/r typically does not exceed 0.1 and it asymptotically approaches zero with altitude.
For the constant 
e. Vertical changes of the phase relaxation time
The value of 
Fig. 3 shows changes of 
Changes of 
Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-077.1
3. Maximum of supersaturation
Dependences 
Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-077.1
Equations (32)–(34) show that all three variables 
In other words, 
Modeled dependences (a) 
Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-077.1
One of the interesting features of the initial stage of the cloud formation, which follows from Eq. (35), is that at the level of supersaturation maximum, regardless of N, w, T, and F, 45% of potentially condensed water exists in liquid phase, whereas the remaining 55% exists in a form of supersaturated vapor.
4. Universal profiles for supersaturation and water content
Here 
Normalized supersaturation and liquid mixing ratio vs normalized height.
Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-077.1
The existence of universal profiles of supersaturation and water content reflect, supposedly, the existence of deep laws of diffusional droplet growth.
5. Applicability of the approach to real cloud conditions
There are few simplifications used during derivation of the equations describing changes of supersaturation, water content, and droplet size. The purpose of this section is to consider consequences and limitations in use of the obtained equations related to three most significant of those simplifications.
The first simplification is related to the assumption that the coefficients A1, A2, and F in Eqs. (1) and (5) are constant and their dependences of T and P were neglected. As it was discussed in section 2b, this assumption provides an accurate solution for S, q, and r with a few percent accuracy for vertical motion within several hundred meters.
Using the balance equation (7), one can represent droplet radius and then equilibrium supersaturation as a function of 
Figure 7 shows vertical changes of supersaturation calculated for monodisperse CCN with radii of 0.02 and 0.1 μm. Vertical profiles of 
Vertical profiles of supersaturation and profiles of equilibrium supersaturation calculated using monodisperse CCN with radii rn of 0.02 and 0.1 μm. The profile of supersaturation calculated using Eq. (13) is also presented.
Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-077.1
The main purpose of this examination is to identify the limitations of the universality of the supersaturation presentation described by Eq. (31)—that is, the applicability of the approach in a more general case of polydisperse CCN.
Figure 8 (left) shows vertical profiles of supersaturation calculated using the bin model. One can see that the profiles resemble those plotted in Figs. 1 and 5a. For instance, 
(left) Vertical profiles of supersaturation calculated using the parcel model (Pinsky and Khain 2002) with explicit calculation of diffusion growth of aerosols. (right) As in (left), but plotted in the normalized form (see text).
Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-077.1
The closeness of the supersaturation profiles calculated for the monodisperse and polydisperse size distributions can be interpreted as follows. The contribution of CCN of different size in the CCN spectrum to 
Dependencies of minimal, mean, mean volume, and effective radii of aerosols on CCN concentration n2 in the second CCN mode at w = (left) 1 and (right) 3 m s−1. Size distribution of CCN is given as a sum of three lognormal distributions (modes) (Respondek et al. 1995; Pinsky and Khain 2002). Parameters of the distributions are the following. Concentrations in the first and the third modes are n1 = 50 and 1 cm−3, respectively. The modes are centered at radii 0.006, 0.03, and 0.5 μm, respectively. The values of 
Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-077.1
It should be noted that simplified equation for diffusional growth in Eq. (5) was used in many studies for calculation of supersaturation maximum (Twomey 1959; Ghan et al. 1993, 1995; Bedos et al. 1996; Cohard et al. 1998; Abdul-Razzak et al. 1998; Abdul-Razzak and Ghan 2000; Fountoukis and Nenes 2005; Shipway and Abel 2010).
6. Conclusions
The main outcomes of this study can be formulated in the following way:
- A new form of the closed equations for supersaturation, water content, and droplet radii were obtained [i.e., Eqs. (13), (19), and (20), respectively] based on the consideration of the water mass balance equation centered with respect to the adiabatic liquid water mixing ratio.
- An analytical expression for the supersaturation maximum
- The approach developed in the frame of this study enabled us to obtain universal solutions for supersaturation
The results obtained in this study provide a new look at the nature of supersaturation formation in liquid clouds. The findings of this work open the door for an entirely new way of parameterization of cloud microphysical processes and specifically for the parameterization of CCN nucleation in cloud models. In the recent study by Pinsky et al. (2012), the formula for the supersaturation maximum derived in the present work is used to calculate the droplet concentration under different vertical velocities and different types of the CCN spectra (wide maritime, narrow continental, urban, etc.), which are approximated by either single mode spectra or by multiple modal CCN spectra with different contributions of the modes. The dependencies of droplet concentration on vertical velocity and parameters of CCN spectra obtained in this study were compared with those obtained using approaches developed in different studies. It was shown that the approach proposed in the present study produces comparable and, in many cases, more accurate results than other approaches. We suppose also that the utilization of the equation for supersaturation [Eq. (12)] derived in the study will substantially improve the procedure for calculating supersaturation in the current cloud models. In a majority of the models, supersaturation is calculated using a splitting method, according to which calculation of supersaturation is performed in several substeps (e.g., Khain and Sednev 1996; Khain et al. 2008). At the first substep, supersaturation changes due to advection and mixing are calculated. At the “microphysical” time step, the process of drop growth/evaporation and corresponding change in supersaturation is considered as in adiabatic volume. There are many difficulties in the calculation of supersaturation at this microphysical substep. In some models, supersaturation is assumed to be constant during this substep. In other models the supersaturation changes under the assumption of constant phase relaxation time. In this sense the nonlinear Eq. (12) for the supersaturation derived in this study will be quite useful to improve the treatment of supersaturation calculations in cloud models.
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). Dr. Korolev was supported by Environment Canada.
APPENDIX A
APPENDIX B
Dependence of Coefficients in the Microphysical Equations on Temperature and Pressure
Figure B1 shows the temperature dependence of coefficients 
Temperature dependence of coefficients (a) 
Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-077.1
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