## 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

*N*is the droplet number concentration. In Eqs. (1) and (2),

*F*on

*Z*are neglected, the rate of the droplet growth is described by the equationwhere coefficient

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 *z* do not exceed few hundred meters. A more accurate equation for the adiabatic water content requires integration of the last term in Eq. (8) over

### b. Supersaturation equation

This section presents derivation of a closed equation for supersaturation in a new form.

*r*, water content can be written as

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

### c. Limiting supersaturation

*dS*/

*dz*in Eq. (14), it yields

Here *r*_{ad} is the adiabatic droplet radius related to the adiabatic water content *q*_{ad} 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 *r*_{ad} in Eq. (17). The analysis of derivations of Eqs. (3) and (15) shows that the aforementioned difference is a result of linearization applied in Eq. (14).

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 *dq _{w}*/

*dz*. The balance between these two large terms leads to a linear dependence of

*r*only slowly changes with height, solutions of Eq. (20) can be approximated by the adiabatic dependence:

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 *r*, respectively, calculated from the numerical model. Figure 2a also shows the changes of the mixing ratio of the supersaturated or undersaturated fraction of water vapor

It is worth noting that

The ratio Δ*r*/*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 Δ*r*/*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 Δ*r*/*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, Δ*r*/*r* may serve as a measure of nonreversibility of microphysical parameters in adiabatic cloud parcels experiencing cycling ascents and descents. Numerical simulation shows that Δ*r*/*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, Δ*r*/*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 *w* = 1 m s^{−1} calculated from Eq. (27). The dashed portions of the curves in Fig. 3 indicate the regions where the condition in Eq. (18) for the quasi-steady approximation is not satisfied. The solid portion of the curves with a good approximation follows

## 3. Maximum of supersaturation

*R*. Analyses of the results of the calculations shows that

Equations (32)–(34) show that all three variables *C*_{1} and *C*_{2} are missed in formulations for

In other words,

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

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 *A*_{1}, *A*_{2}, 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 *μ*m. Vertical profiles of *z* > *z*_{max}) the difference between the solutions of Eqs. (13) and (44) decreases. Since the chemical term in *S*(*z*) becomes insensitive to CCN and its changes can be accurately described by Eq. (13).

*z*

_{max}, one should introduce corrections in Eqs. (31) and (32) represented by the second terms in the right-hand side of Eq. (47) calculated for

*μ*m. The grid has resolution 0.001

*μ*m for small particles and it gradually decreases down to 8

*μ*m for large particles. Such resolution is sufficient for explicit description of the process of separation of all particles into growing droplets and nonactivated wetted aerosols. Accordingly, the process of droplet nucleation is treated directly without using any parameterization procedures. To describe the diffusion growth, a nonregular 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. Such a scheme was not affected by the artificial size spectrum broadening and therefore no remapping has been applied during the diffusional growth calculations. The time step of 0.005 s was used to calculate diffusion growth of drops and aerosol particles. The size distribution CCN is a sum of three lognormal modes, representing small, intermediate, and larger aerosol particles (Respondek et al. 1995; Pinsky and Khain 2002):where

*i*th mode and

*i*th aerosol mode, respectively. In simulations the mean radii of the modes were 0.006, 0.03, and 0.5

*μ*m, respectively. The values of

^{−1}. The total CCN concentration was selected this way in order to produce droplet concentrations of approximately 100, 200, and 500 cm

^{−3}.

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, ^{−1} and high droplet concentration of 500 cm^{−3}. However, even in the latter case the maximum of the normalized supersaturation is located at nearly the same normalized height, which was obtained using Eq. (31) for the monodisperse case. Note that the case with low vertical velocity and high droplet concentration is not typical of real clouds.

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 *w* = 1 (left) and 3 m s^{−1} (right). These dependencies were obtained in simulations using the parcel model by Pinsky and Khain (2002), with the CCN spectrum given by Eq. (48) and the parameters of the CCN modes mentioned above. One can see that along with the large CCN, the small CCN are also activated. However, the mean volume radius of activated CCN is equal approximately to 0.2 *μ*m. For CCN of such size the effect of curvature and chemical terms is not substantial, and the effect of polydisperse CCN on supersaturation can be approximated using the approach developed in the present study. The exception is the case when CCN size distribution consists of only small CCN with ^{−1} and high droplet concentration of 500 cm^{−3} is close to this rare situation, when many small CCN are activated and, consequently, the effect of curvature and chemical terms, as well as the effect of polydispersity of CCN distribution, becomes more substantial. Note, however, that the comprehensive numerous test performed by Pinsky et al. (2012) showed that the application of Eq. (30) to the determination of the droplet concentration leads to quite realistic results under very different CCN spectra despite these simplified assumptions being used. Thus, the proposed method and corresponding simplifications are applicable to a wide range of conditions in real clouds.

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
forming in the vicinity of the cloud base has been obtained here. Similar expressions for the height and liquid water mixing ratio corresponding to this level of were obtained here as well. It is shown that the values of , , and are linearly related to each other and are all proportional to . - The approach developed in the frame of this study enabled us to obtain universal solutions for supersaturation
and liquid water mixing ratio . These solutions are independent of *w*,*N*,*T*, and*P.*The actual solutionsand can be obtained from and using scaling coefficient from Eqs. (37).

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 *T* < 30°C. The analysis of the diagrams in Fig. B1 suggests that the temperature and pressure dependencies of *T* > 0°C and a few tens of meters at *T* < −20°C, assuming moist adiabatic changes of temperature.

## REFERENCES

Abdul-Razzak, H., , and S. J. Ghan, 2000: A parameterization of aerosol activation: 2. Multiple aerosol types.

,*J. Geophys. Res.***105**(D5), 6837–6844.Abdul-Razzak, H., , S. J. Ghan, , and C. Rivera-Carpio, 1998: A parameterization of aerosol activation: 1. Single aerosol type.

,*J. Geophys. Res.***103**(D6), 6123–6131.Andreae, M. O., , D. Rosenfeld, , P. Artaxo, , A. A. Costa, , G. P. Frank, , K. M. Longlo, , and M. A. F. Silva-Dias, 2004: Smoking rain clouds over the Amazon.

,*Science***303**, 1337–1342.Bartlett, J. T., , and P. R. Jonas, 1972: On the dispersion of the sizes of droplets growing by condensation in turbulent clouds.

,*Quart. J. Roy. Meteor. Soc.***98**, 150–164.Bedos, C., , K. Suhre, , and R. Rosset, 1996: Adaptation of a cloud activation scheme to a spectral-chemical aerosol model.

,*Atmos. Res.***41**, 267–279.Cohard, J. M., , J. P. Pinty, , and C. Bedos, 1998: Extending Twomey's analytical estimate of nucleated cloud droplet concentrations from CCN spectra.

,*J. Atmos. Sci.***55**, 3348–3357.Devenish, B. J., and Coauthors, 2012: Droplet growth in warm turbulent clouds.

,*Quart. J. Roy. Meteor. Soc.***138,**1401–1429.Fountoukis, C., , and A. Nenes, 2005: Continued development of a cloud droplet formation parameterization for global climate models.

,*J. Geophys. Res.***110**, D11212, doi:10.1029/2004JD005591.Fukuta, N., 1993: Water supersaturation in convective clouds.

,*Atmos. Res.***30**, 105–126.Ghan, S. J., , C. C. Chuang, , and J. E. Penner, 1993: A parameterization of cloud droplet nucleation. Part I: Single aerosol type.

,*Atmos. Res.***30**, 198–221.Ghan, S. J., , C. C. Chuang, , R. C. Easter, , and J. E. Penner, 1995: A parameterization of cloud droplet nucleation. Part II: Multiple aerosol types.

,*Atmos. Res.***36**, 39–54.Grabowski, W. W., , and H. Morrison, 2008: Toward the mitigation of spurious cloud-edge supersaturation in cloud models.

,*Mon. Wea. Rev.***136**, 1224–1234.Kabanov, A. S., , I. P. Mazin, , and V. I. Smirnov, 1971: Supersaturation of water vapor in clouds (in Russian).

,*Proc. Cent. Aerol. Obs.***95**, 50–61.Khain, A. P., , and I. Sednev, 1996: Simulation of precipitation formation in the Eastern Mediterranean coastal zone using a spectral microphysics cloud ensemble model.

,*Atmos. Res.***43**, 77–110.Khain, A. P., , M. Ovtchinnikov, , M. Pinsky, , A. Pokrovsky, , and H. Krugliak, 2000: Notes on the state-of-the-art numerical modeling of cloud microphysics.

,*Atmos. Res.***55**, 159–224.Khain, A. P., , N. BenMoshe, , and A. Pokrovsky, 2008: Factors determining the impact of aerosols on surface precipitation from clouds: An attempt of classification.

,*J. Atmos. Sci.***65**, 1721–1748.Khrgian, A. Kh., 1969:

*Physics of the Atmosphere*(in Russian). Gidrometeoizdat, 647 pp.Khvorostyanov, V. I., , and J. A. Curry, 1999: Toward the theory of stochastic condensation in clouds. Part I: A general kinetic equation.

,*J. Atmos. Sci.***56**, 3985–3996.Khvorostyanov, V. I., , and J. A. Curry, 2006: Aerosol size spectra and CCN activity spectra: Reconciling the lognormal, algebraic, and power laws.

,*J. Geophys. Res.***111**, D12202, doi:10.1029/2005JD006532.Khvorostyanov, V. I., , and J. A. Curry, 2009: Parameterization of cloud drop activation based on analytical asymptotic solutions to the supersaturation equation.

,*J. Atmos. Sci.***56**, 1905–1925.Klaassen, G. P., , and T. L. Clark, 1985: Dynamics of the cloud environment interface and entrainment in small cumuli: Two dimensional simulations in the absence of ambient shear.

,*J. Atmos. Sci.***42**, 2621–2642.Korolev, A. V., 1995: The influence of supersaturation fluctuations on droplet spectra formation.

,*J. Atmos. Sci.***52**, 3620–3634.Korolev, A. V., , and I. P. Mazin, 1993: Zones of increased and decreased droplet concentration in stratiform clouds.

,*J. Appl. Meteor.***32**, 760–773.Korolev, A. V., , and I. P. Mazin, 2003: Supersaturation of water vapor in clouds.

,*J. Atmos. Sci.***60**, 2957–2974.Magaritz, L., , M. Pinsky, , O. Krasnov, , and A. Khain, 2009: Investigation of droplet size distributions and drizzle formation using a new trajectory ensemble model. Part II: Lucky parcels.

,*J. Atmos. Sci.***66**, 781–805.Mazin, I. P., , and V. I. Smirnov, 1969: On the theory of cloud drop size spectrum formation by stochastic condensation (in Russian).

,*Proc. CAO***89**, 92–94.Mazin, I. P., , and V. M. Merkulovich, 2008: Stochastic condensation and its possible role in liquid cloud microstructure formation (in Russian).

*Some Problems of Cloud Physics: Collected Papers,*A. A. Ivanov et al., Eds., National Geophysical Committee, Russian Academy of Science, 263–295.Paluch, I. R., , and Ch. A. Knight, 1984: Mixing and evolution of cloud droplet size spectra in vigorous continental cumulus.

,*J. Atmos. Sci.***41**, 1801–1815.Pinsky, M., , and A. P. Khain, 2002: Effects of in-cloud nucleation and turbulence on droplet spectrum formation in cumulus clouds.

,*Quart. J. Roy. Meteor. Soc.***128**, 501–533.Pinsky, M., , A. Khain, , I. Mazin, , and A. Korolev, 2012: Analytical estimation of droplet concentration at cloud base.

,*J. Geophys. Res.***117**, D18211, doi:10.1029/2012JD017753.Politovich, M. K., 1993: A study of the broadening of droplet size distribution in cumuli.

,*J. Atmos. Sci.***50**, 2230–2244.Politovich, M. K., , and W. A. Cooper, 1988: Variability of the supersaturation in cumulus clouds.

,*J. Atmos. Sci.***45**, 1651–1664.Prabha, T., , A. P. Khain, , B. N. Goswami, , G. Pandithurai, , R. S. Maheshkumar, , and J. R. Kulkarni, 2011: Microphysics of premonsoon and monsoon clouds as seen from in situ measurements during the Cloud Aerosol Interaction and Precipitation Enhancement Experiment (CAIPEEX).

,*J. Atmos. Sci.***68**, 1882–1901.Pruppacher, H. R., , and J. D. Klett, 1997:

*Microphysics of Clouds and Precipitation.*Kluwer Academic, 976 pp.Respondek, P. S., , A. I. Flossmann, , R. R. Alheit, , and H. R. Pruppacher, 1995: A theoretical study of the wet removal of atmospheric pollutants. Part V: The uptake, redistribution, and deposition of (NH4)2SO4 by a convective cloud containing ice.

,*J. Atmos. Sci.***52**, 2121–2132.Rogers, R. R., 1975: An elementary parcel model with explicit condensation and supersaturation.

,*Atmosphere***13**, 192–204.Sedunov, Yu. S., 1965: Fine cloud structure and its role in the formation of the cloud spectrum.

,*Atmos. Oceanic Phys.***1**, 416–421.Sedunov, Yu. S., 1974:

*Physics of Drop Formation in the Atmosphere.*John Wiley & Sons, 234 pp.Shipway, B. J., , and S. J. Abel, 2010: Analytical estimation of cloud droplet nucleation based on an underlying aerosol population.

,*Atmos. Res.***96**, 344–355.Squires, P., 1952: The growth of cloud drops by condensation.

,*Aust. J. Sci. Res.***5**, 66–86.Stevens, B., , R. L. Walko, , W. R. Cotton, , and G. Feingold, 1996: The spurious production of cloud-edge supersaturations by Eulerian models.

,*Mon. Wea. Rev.***124**, 1034–1041.Stevens, B., and Coauthors, 2003: On entrainment rates in nocturnal maritime stratocumulus.

,*Quart. J. Roy. Meteor. Soc.***129**, 3469–3492.Stevens, B., and Coauthors, 2005: Evaluation of large-eddy simulations via observations of nocturnal marine stratocumulus.

,*Mon. Wea. Rev.***133**, 1443–1462.Twomey, S., 1959: The nuclei of natural cloud formation part II: The supersaturation in natural clouds and the variation of cloud droplet concentrations.

,*Geofis. Pura Appl.***43**, 243–249.Warner, J., 1968: The supersaturation in natural clouds.

,*J. Rech. Atmos.***3**, 233–237.