Abstract

The process of collective diffusional growth of droplets in an adiabatic parcel ascending or descending with the constant vertical velocity is analyzed in the frame of the regular condensation approach. Closed equations for the evolution of liquid water content, droplet radius, and supersaturation are derived from the mass balance equation centered with respect to the adiabatic water content. The analytical expression for the maximum supersaturation formed near the cloud base is obtained here. Similar analytical expressions for the height and liquid water mixing ratio corresponding to the level where occurs have also been obtained. It is shown that all three variables , , and are linearly related to each other and all are proportional to , where w is the vertical velocity and N is the droplet number concentration. Universal solutions for supersaturation and liquid water mixing ratio are found here, which incorporates the dependence on vertical velocity, droplet concentration, temperature, and pressure into one dimensionless parameter. The actual solutions for and can be obtained from the universal solutions with the help of appropriate scaling factors described in this study. The results obtained in the frame of this study provide a new look at the nature of supersaturation formation in liquid clouds. Despite the fact that the study does not include a detailed treatment of the activation process, it is shown that this work can be useful for the parameterization of cloud microphysical processes in cloud models, especially for the parameterization of cloud condensation nuclei (CCN) activation.

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 in a vertically moving adiabatic cloud parcel.

The equation for water vapor supersaturation can be written in the form (e.g., Pruppacher and Klett 1997)

 
formula

where is the liquid water mixing ratio. For the sake of brevity in the forthcoming consideration, we will refer to as water content. The first term in the right-hand side of Eq. (1) describes changes of supersaturation of the moist air due to its adiabatic cooling or heating, whereas the second term describes the supersaturation changes caused by condensation or evaporation of water vapor by droplets.

Equation (1) can be rewritten as (Korolev and Mazin 2003)

 
formula

where is the vertical velocity, is the average droplet radius, and N is the droplet number concentration. In Eqs. (1) and (2), , , and are coefficients dependent on temperature and pressure (for variable notations see  appendix A). For the case when the temperature and pressure dependences of , , and are neglected and the droplet radii are assumed to be constant (), Eq. (2) can be integrated analytically. In this case the solution tends toward supersaturation , usually referred to as quasi-steady approximation:

 
formula

The characteristic time of approaching of to is determined by the time constant

 
formula

known as phase relaxation time. Supersaturation approaches the quasi-steady value exponentially with a characteristic time scale given by Eq. (4). In cases when is small and does not exceed a few seconds, the quasi-steady approximation can be effectively used for the estimation of the actual supersaturation in clouds, when the vertical velocity, droplet radii, and concentration are known from measurements (e.g., Warner 1968; Paluch and Knight 1984; Politovich and Cooper 1988; Prabha et al. 2011).

If the surface tension, salinity corrections, and dependence of F on Z are neglected, the rate of the droplet growth is described by the equation

 
formula

where coefficient depends on temperature and pressure.

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)].

Equation (2) relates two time-dependent variables and . Strictly speaking, the assumption that is not valid, since nonzero values of supersaturation result in changes in droplet size. The basis for this assumption is that, when the droplets are large enough, the characteristic time of changing of supersaturation, determined by the time of phase relaxation, is much smaller than the characteristic time of changing of the droplet radius. This assumption is not justifiable if droplets are small (e.g., in the vicinity of cloud base). In such cases a closed equation for supersaturation accounting droplet changes should be used. Integrating the equation of the droplet growth and then substituting it into Eq. (2) yields a closed integro–differential equation with just one dependent variable:

 
formula

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 and radii in a vertically moving adiabatic parcel. It is assumed that the cloud droplets move with the air and stay inside the parcel beginning from cloud base. No sedimentation and coalescence is allowed. The collective droplet growth and evaporation will be considered in the frame of regular condensation; that is, the water vapor pressure and temperature fields at large distance from cloud droplets are assumed to be uniform, and all droplets grow or evaporate under the same conditions.

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.

Assuming that and are constant, the integration of Eq. (1) yields

 
formula

where is the height above cloud base. The term is determined by the initial and at . In cases when the mass of water associated with the wetted CCN at the cloud base is negligibly small, with the high accuracy.

In a moist adiabatic process, when the supersaturation adjusts to zero, , Eq. (7) yields changes of water content as

 
formula

The variable in the subsequent discussion will be called “adiabatic” water content. Strictly speaking, all microphysical variables considered in the frame of this work are adiabatic because of the main assumption made at the beginning of this section. In order not to confuse with , the term adiabatic will be applied only for the special case, when , and omitted for the cases when .

The ratio in Eq. (8) is the adiabatic gradient of liquid mixing water ratio (e.g., Khrgian 1969). As shown in  appendix B, and are slow-changing functions of and , and the changes of remain small when varies within a few hundred meters. Therefore, the adiabatic water content with a high accuracy can be considered linearly related to altitude if changes of 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 , taking into account the temperature and pressure dependencies (e.g., Korolev and Mazin 1993).

In essence, Eq. (7) represents a water mass balance centered with respect to the adiabatic water content; that is,

 
formula

where is the mixing ratio of the supersaturated fraction of water vapor. In its traditional form, the equation of integral water balance in a vertically moving parcel is usually presented as . As it is shown in the following section the mass balance equation in the form of Eq. (7) enables deducing a set of equations describing microphysical parameters in a new form.

b. Supersaturation equation

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

For an ensemble of monodisperse droplets with concentration and radii r, water content can be written as

 
formula

Substituting Eq. (10) into Eq. (5) yields

 
formula

where . Then, substituting from the balance equation (7) into Eq. (11) leads to the equation for supersaturation

 
formula

Equation (12) can be rewritten for the independent variable as

 
formula

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 were considered in Sedunov (1974) and Korolev and Mazin (2003) [see Eq. (6)]. However, the earlier representations of the closed supersaturation equations have integro–differential form, which are essentially more complex and more difficult for analysis, in comparison to Eq. (13).

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 , , and remain constant, whereas in the numerical model, the dependences of , , and on and were accounted for. As seen from Fig. 1, Eq. (13) accurately depicts the changes of supersaturation and it agrees well with the numerically modeled supersaturation. In this particular case the difference between the modeled supersaturation and that calculated from Eq. (13) does not exceed a few percent. It should be noted that neglecting the dependences of , , and on and gives quite accurate solutions for within the vertical scale on the order of few hundred meters. However, for the displacements beyond 1 km, these dependences should be accounted for.

Fig. 1.

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 = 10°C, = 950 mb, , = 0.01 μm. The horizontal, thick arrows on the right-hand side indicate the levels above which the condition in Eq. (18) is valid, and the quasi-steady approximation is justified.

Fig. 1.

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 = 10°C, = 950 mb, , = 0.01 μm. The horizontal, thick arrows on the right-hand side indicate the levels above which the condition in Eq. (18) is valid, and the quasi-steady approximation is justified.

c. Limiting supersaturation

As seen from Fig. 1 and Eq. (13) that after passing its maximum, the supersaturation is monotonically decreasing with altitude toward zero. Therefore, above some level, the supersaturation becomes . At that point, can be neglected inside the brackets on the right-hand side in Eq. (13). In this case, Eq. (13) can be linearized, resulting in

 
formula

Since tends to zero with increasing of the distance above cloud base, the left-hand term becomes much smaller than any of two terms on the right-hand side. In this case, after neglecting the term dS/dz in Eq. (14), it yields

 
formula

Taking into account Eq. (7) and that , at Eq. (15) can be rewritten in a form

 
formula

Equation (16) represents a limiting (or asymptotic) supersaturation forming in adiabatic cloud parcel. After substitution of Eq. (10) into Eq. (16), it can be transformed into a form

 
formula

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 in Eq. (3) is replaced by rad 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, decreases with height as . Since approximates , then changes as , as well. The same dependence was obtained in Sedunov (1974) and Fukuta (1993), but in a much more challenging way.

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 [Eq. (16)] asymptotically approaches with altitude. Comparisons between [Eq. (3)] and [Eq. (16)] in Fig. 1 show that approaches faster as compared to . Such behavior is a result of using adiabatic values for liquid water or droplet radii in Eqs. (16) and (17), respectively.

The large deviation of from near the cloud base in Fig. 1 is a consequence of the limitations of the quasi-steady approximation. Indeed, according to Eq. (3), tends to infinity at cloud base since droplet size tends to zero. The condition of applicability of for estimation of supersaturation was presented by Korolev and Mazin (2003):

 
formula

Here, is the coefficient dependent on and (see  appendix A). Inequality (18) was derived from the condition that droplet radius should not change significantly during time changes of supersaturation.

The horizontal, thick arrows on the right-hand side in Fig. 1 indicate the altitude below which —that is, the altitude below which condition (18) is not satisfied. In other words, the quasi-steady approximation is justified only above the levels indicated by the horizontal, thick arrows corresponding to each velocity. As seen from Fig. 1, below the indicated levels, the deviation of and from is significant, whereas above these levels the agreement among , , and improves and the difference between them does not exceed 10%.

d. Water content and droplet radius equations

This section presents derivation and analysis of a closed equation for water content.

After substituting the mass balance equation (7) into Eq. (11), a closed equation for is obtained:

 
formula

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 on height, so that for large the solution of Eq. (19) is reduced to Eq. (8)—that is, when .

Equations (10) and (19) yield differential equations for the changes of r:

 
formula

At large-enough heights, when r only slowly changes with height, solutions of Eq. (20) can be approximated by the adiabatic dependence:

 
formula

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 and 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 . One can see that the deviation of from the adiabatic value is largest near the cloud base, where the supersaturation is maximal and it decreases with increasing altitude. As it is seen from Fig. 2 the deviation of and from and , respectively, increases with the increase of vertical velocities.

Fig. 2.

Vertical changes of (a) from Eq. (19) (solid lines) and (dashed lines) and (b) droplet radius calculated from Eq. (20). The calculations were performed from the vertical velocity w = ±1 and ±5 m s−1. Arrows indicate calculations corresponding to ascending and descending parcels. The initial conditions are as in Fig. 1.

Fig. 2.

Vertical changes of (a) from Eq. (19) (solid lines) and (dashed lines) and (b) droplet radius calculated from Eq. (20). The calculations were performed from the vertical velocity w = ±1 and ±5 m s−1. Arrows indicate calculations corresponding to ascending and descending parcels. The initial conditions are as in Fig. 1.

It is worth noting that , , and are irreversible in ascending and descending parcels. In ascending parcels supersaturation is positive (), whereas in descending parcels it is negative (). Substituting these inequalities into Eq. (7) yields the inequality for ascending parcels and for descending parcels . It should be noted that the inequalities obtained above are valid at the time scales . At shorter time scales they may be reversed. For example, in Fig. 1, when the cloud parcel changed its direction from ascent to descent at = 800 m, for some time () the supersaturation remained positive during its descent. From balance Eq. (7) it follows that . It means that for the same , liquid water in ascending parcels is always lower than that in the descending ones. The same relationship refers to the droplet radii as well. The last inequality also follows from two inequalities presented above. This type of behavior is a result of the finite rate of the diffusional processes during the equilibration water vapor supersaturation. If the release and absorption of the water vapor by the liquid droplets, in order to compensate supersaturation, were to occur instantly (i.e., ), then the condensational processes in ascending and descending parcels would be reversible and the above inequalities would turn into equalities.

The condensational inertia of the diffusional processes results in spatial inhomogeneities of the microphysical parameters. The horizontal fluctuations of supersaturation can be estimated as the difference between forming at the same altitude in the ascending and descending parcels—that is, . Substituting Eqs. (16) and (17) into this expression, and assuming that , yields

 
formula

The spatial fluctuation of droplet radii can be estimated from the difference in the droplet radii cubes in ascending and descending parcels at the same altitude. Thus, substituting Eq. (7) into Eq. (10) with the following differentiating gives

 
formula

Assuming that , Eq. (23) can be rewritten as

 
formula

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 to the mean radius . In clouds the variation coefficient changes from approximately 0.1 to 0.6. In situ observations suggest no significant changes of the variation coefficient with height (e.g., Politovich 1993; Martin et al. 1994; Prabha et al. 2011).

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 required to get such variation coefficient are much higher than the characteristic values of supersaturation existing in clouds. Thus, for the vertically moving adiabatic parcels having the same initial conditions, the diffusional growth and evaporation of droplets cannot explain formation of broad DSD usually observed in cloud. Earlier the same conclusion was obtained by Mazin and Smirnov (1969) and Bartlett and Jonas (1972), who considered droplet growth within a random turbulent velocity field.

Table 1.

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.

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.
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.

Substituting Eq. (22) into Eq. (24) yields an estimate of the droplet radii fluctuations

 
formula

For the constant , Eq. (9) yields the relationship between fluctuations of liquid water and mixing ratio of the supersaturated vapor as . Therefore, fluctuations can be estimated as (Korolev 1995).

e. Vertical changes of the phase relaxation time

The coefficient in front of in the right-hand side of Eq. (12) has a meaning of inverse phase relaxation time [see Eq. (4)]. Bringing up the similarity of Eq. (2) and Eq. (12), one can conclude that the expression in Eq. (12) has the same meaning as in Eq. (2)—that is, an inverse phase relaxation time:

 
formula

The value of in Eq. (26) coincides with from Eq. (4), for the case of monodisperse droplets—that is, when , where is the mean volume radius of droplets.

Assuming and substituting Eq. (16) into Eq. (26) yields

 
formula

For the altitudes , when the second term in the square brackets can be neglected, Eq. (27) turns into

 
formula

Fig. 3 shows changes of versus the height of cloud parcels with different droplet concentrations ascending through a cloud base with 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 law as in Eq. (28).

Fig. 3.

Changes of calculated from Eq. (27) for cloud parcels ascending with w = 1 m s−1 and having different concentrations. The initial conditions are as in Fig. 1.

Fig. 3.

Changes of calculated from Eq. (27) for cloud parcels ascending with w = 1 m s−1 and having different concentrations. The initial conditions are as in Fig. 1.

3. Maximum of supersaturation

An important feature of Eq. (13) is that it allows for the estimation of the supersaturation maximum and the height corresponding to the level where occurs. For simplicity, assume . Then introduce a nondimensional altitude and a nondimensional parameter . Using new variables, Eq. (13) can be rewritten in the form

 
formula

with the initial condition at the cloud base.

An inspection of Eq. (29) suggests that the solution depends on the sole parameter . The condition in Eq. (29) yields an expression relating and :

 
formula

Figure 4 shows the dependences and calculated from the numerical solutions of Eq. (29) for different R. Analyses of the results of the calculations shows that and obey the power law with high accuracy. Therefore, looking for a solution to Eq. (30) in a form and , one can get

 
formula

where coefficients and were obtained from the diagram in Fig. 4. It should be noted that the value of and are linked just to the type of the differential equation (29) and they are not related to any physical variables (i.e., , , , , etc.).

Fig. 4.

Dependences and .

Fig. 4.

Dependences and .

Substituting and Eq. (7) into Eq. (31) enables deriving the dependences of , , and versus droplet concentration and vertical velocity:

 
formula
 
formula
 
formula

Equations (32)(34) show that all three variables , , and are proportional to . The conclusion that was also obtained by Fukuta (1993), although in his study unjustified assumptions have been applied when deriving this relationship. As a result, in Fukuta (1993) the coefficients C1 and C2 are missed in formulations for and (equivalent of ). Proportionality was also found by Khvorostyanov and Curry (2009) in so-called diffusion growth regime.

Equations (32)(34) allow important conclusion about the universal nature of the following ratios:

 
formula

In other words, , , and are linearly related to one another. This finding was verified with the help of a numerical simulation of the droplet growth in the ascending adiabatic parcel. The diagrams in Fig. 5 show that the modeled relationships between , , and follow Eq. (35) with high accuracy for a wide range of and , which occur in the tropospheric liquid clouds.

Fig. 5.

Modeled dependences (a) and (b) calculated for various combinations of vertical velocities of ascending parcel and droplet number concentrations from the respective ranges 0.01 ≤ w ≤ 10 m s−1 and 10 ≤ N ≤ 1000 cm−3. The circles indicate the locations of . The calculations were performed for the initial conditions: = 10°C, = 950 mb, , and = 0.01 μm.

Fig. 5.

Modeled dependences (a) and (b) calculated for various combinations of vertical velocities of ascending parcel and droplet number concentrations from the respective ranges 0.01 ≤ w ≤ 10 m s−1 and 10 ≤ N ≤ 1000 cm−3. The circles indicate the locations of . The calculations were performed for the initial conditions: = 10°C, = 950 mb, , and = 0.01 μm.

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

Substitution of Eq. (7) into Eq. (29) yields a dimensionless equation for water content:

 
formula

Here is the dimensionless water content. The dimensionless equations for and [Eqs. (29) and (36), respectively] in a form dependent on just one coefficient, along with the linear relationships of Eqs. (35), are suggestive of the existence of universal equations for and .

Thus, introducing new variables

 
formula

and then substituting them into Eqs. (29) and (36) yields normalized equations for and , which do not contain any parameters:

 
formula
 
formula

The solutions and are universal and valid for any droplet concentration, vertical velocity, temperature, and pressure. Figure 6 shows the behavior of universal and . As seen from Fig. 6, the maximum of supersaturation and altitude remain constant. To obtain an actual , for specific , , , and , the normalized solutions and should be scaled using , , and from Eqs. (31)(34) and the corresponding definitions of and . For the new variables, normalized adiabatic water content is

 
formula
Fig. 6.

Normalized supersaturation and liquid mixing ratio vs normalized height.

Fig. 6.

Normalized supersaturation and liquid mixing ratio vs normalized height.

It should be noted that Eqs. (38) and (39) are dependent, and each can be derived from the other using the mass balance equation (7), written in the normalized from as

 
formula

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.

The second simplification was related to neglecting corrections in the droplet curvature and salinity in the droplet growth in Eq. (5). A more accurate treatment of the diffusional droplet growth requires accounting for this correction and it yields an equation (e.g., Pruppacher and Klett 1997)

 
formula

where is the dry radius of soluble fraction of aerosol that plays the role of a CCN and , are the coefficients (see  appendix A). In Eq. (42), is the equilibrium supersaturation (Kohler equation):

 
formula

Using Eq. (42) instead of Eq. (5) and substituting it in Eq. (13) yields

 
formula

Using the balance equation (7), one can represent droplet radius and then equilibrium supersaturation as a function of and ; that is, and thus, closing Eq. (44).

Figure 7 shows vertical changes of supersaturation calculated for monodisperse CCN with radii of 0.02 and 0.1 μm. Vertical profiles of for these cases are presented as well in Fig. 7. For large CCN the correction of the salinity cancels the correction of the curvature, which result in that and vertical profiles of supersaturation calculated using Eqs. (13) and (44) turn out to be close to each other. For small CCN, the correction on curvature dominates over the correction on salinity, which results in large maximum of near the cloud base and more significant difference in solutions of Eqs. (13) and (44) at the level of . Above the level of maximum supersaturation (z > zmax) the difference between the solutions of Eqs. (13) and (44) decreases. Since the chemical term in is proportional to , and the curvature term is proportional to , for the corrections on salinity become small and the changes of will be mainly determined by the curvature term. In other words, for S(z) becomes insensitive to CCN and its changes can be accurately described by Eq. (13).

Fig. 7.

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.

Fig. 7.

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.

To estimate the effect of curvature and salinity on and we introduce the residual supersaturation and nondimensional height . Similar to Eq. (30), the equation for supersaturation maximum can be written in the form

 
formula

The solution of Eq. (45) has the same form as that for Eq. (31); that is,

 
formula

Equation (46) yields

 
formula

where is the value of at . Thus, for a more accurate estimation of and zmax, one should introduce corrections in Eqs. (31) and (32) represented by the second terms in the right-hand side of Eq. (47) calculated for . Note that this correction becomes significant when CCN size spectrum consists only of very small CCN with .

The third simplification is related to utilizing the monodisperse model of size distribution. To estimate the limitations of this simplification a set of numerical simulations with the parcel model described by Pinsky and Khain (2002) were conducted. The main feature of the model is an accurate description of diffusional growth of wetted aerosols and droplets. To describe the size distributions (SD) of nonactivated aerosols and droplets, the mass grid containing 2000 mass bins are used. The grid covers the range of particle sizes from 0.01 to 2000 μm. The grid has resolution 0.001 μm for small particles and it gradually decreases down 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 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):

 
formula

where is the number concentration in the ith mode and and are the mean radius and the width of the ith aerosol mode, respectively. In simulations the mean radii of the modes were 0.006, 0.03, and 0.5 μm, respectively. The values of were 0.3, 0.3, and 0.396 for the first, second, and third modes, respectively. The CCN of the second mode have a major effect on and droplet concentration. Simulations were performed for the vertical velocities of 1, 3, and 5 m s−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, and are related linearly (dashed line). The normalized profiles are plotted in Fig. 8 (right). One can see that the separation of the normalized profiles turned out to be low, justifying the proposed approach. The difference between the normalized values of supersaturation at the level of supersaturation maximum is less than 5%–10% for most simulations. The exception is the case with low vertical velocity of 1 m s−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.

Fig. 8.

(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).

Fig. 8.

(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).

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 is different. The effect of large CCN in the CCN size distribution on the values of is substantially larger than small CCN because large CCN activate first in close vicinity to cloud base and then grow by diffusional growth, while CCN of smaller size remain nonactivated to higher levels. Moreover, there is a well-known competition effect, when the growth of droplets activated on the largest CCN decreases and prevents nucleation of smallest CCN. One can assume that there exists an “equivalent” CCN size of a monodisperse CCN such that the effect of CCN of this size on supersaturation and drop concentration is similar to that of polydisperse CCN distribution. It is reasonable to expect that this equivalent CCN size is close to the mean volume radius of activated CCN. Figure 9 shows the dependencies of minimal, mean, mean volume, and effective radii of aerosols on the concentration of CCN activated at 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 μm. 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 . Such cases are quite rare in real clouds. The case with vertical velocity of 1 m s−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.

Fig. 9.

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 are 0.3, 0.3, and 0.396, respectively.

Fig. 9.

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 are 0.3, 0.3, and 0.396, respectively.

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:

  1. 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.

  2. 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 .

  3. 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 solutions and 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.

Acknowledgments

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

List of Symbols

Table A1 shows the symbols that are used in this study.

Table A1.

List of symbols.

List of symbols.
List of symbols.

APPENDIX B

Dependence of Coefficients in the Microphysical Equations on Temperature and Pressure

Figure B1 shows the temperature dependence of coefficients , , , and for three different air pressures. As seen, the changes of and are relatively slow at positive temperatures and they change relatively fast below −20°C. The coefficient is nearly insensitive to pressure and its slope remains approximately constant in the entire temperature range of −30° < T < 30°C. The analysis of the diagrams in Fig. B1 suggests that the temperature and pressure dependencies of , , , and with an accuracy higher than 10% can be neglected for vertical displacements on the order of a few hundred meters at T > 0°C and a few tens of meters at T < −20°C, assuming moist adiabatic changes of temperature.

Fig. B1.

Temperature dependence of coefficients (a) , (b) , (c) , and (d) βad calculated for different air pressures.

Fig. B1.

Temperature dependence of coefficients (a) , (b) , (c) , and (d) βad calculated for different air pressures.

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