## Abstract

Glaciation in mixed-phase adiabatic cloudy parcels is investigated analytically using two new equations: the equation for coexistence of liquid water and ice and the mass balance equation. The analysis of glaciation time is performed for a vertically moving adiabatic mixed-phase cloud parcel. The effects of vertical velocity, liquid water content, and concentrations of ice particles, liquid droplets, temperature, and other parameters on the glaciation process are discussed. It is shown analytically that, for a certain envelope of vertical velocities, the glaciation time depends only on the vertical displacement of the parcel and does not depend on the trajectory along which the cloud parcel travels toward the glaciation point. Analytical dependencies of the glaciation time and of the altitude of glaciation on vertical velocity are presented. The results demonstrate a good agreement with those obtained using the corresponding parcel model. The limitations of the newly proposed approach are discussed as well, and it is shown that implementation of a simple correction factor allows one to calculate the glaciation time within a wide range of temperatures, from 0° down to −30°C.

## 1. Introduction

At temperatures below the freezing point, liquid droplets may stay in metastable condition. Therefore, cold clouds may consist of a mixture of liquid droplets and ice particles. The rate of precipitation formation and the life cycle of mixed-phase clouds in general depend on the liquid and ice partitioning (e.g., Tremblay et al. 1996; Jiang et al. 2000; Khain et al. 2011). Knowledge of the phase composition of clouds is important for development of remote sensing retrieval algorithms (e.g., Schols et al. 1999) and validation of radar and lidar measurements (e.g., Lohmeier et al. 1997; Iguchi et al. 2012; Young 1974). The phase composition determines to a large extent the radiation balance (e.g., Oshchepkov and Isaka 1997) and is important for climate modeling (e.g., Sun and Shine 1995; Wilson 2000).

Mixed-phase clouds are colloidally unstable, which leads to conversion of the liquid phase into ice. This process is usually called glaciation of mixed-phase clouds. There are three main mechanisms of the phase transformation. The first mechanism is freezing, either homogeneous or heterogeneous, of liquid droplets (Pruppacher and Klett 1997). The second mechanism is freezing of liquid droplets after collisions with ice particles, which results in ice particle riming. This mechanism is relevant for comparatively large particles, since ice particles with sizes lower than about 100 *μ*m are unable to collect droplets (Pruppacher and Klett 1997; Khain et al. 2001). The third mechanism is the interaction of ice particles and liquid droplets through the gaseous phase. Because of the difference between the saturation vapor pressure over ice and that over liquid, cloud droplets under certain conditions may evaporate in the presence of ice particles, resulting in expanded growth of ice. The enhanced growth of ice particles at the expense of evaporating liquid droplets is known as the Wegener–Bergeron–Findeisen (WBF) mechanism (Wegener 1911; Bergeron 1935; Findeisen 1938). The WBF mechanism is considered one of the major glaciation mechanisms in supercooled stratiform clouds.

Numerous in situ observations show that the mixed-phase is quite frequent in stratiform clouds (e.g., Tremblay et al. 1996; Zawadzki et al. 2000; Korolev et al. 2003) and in deep convective clouds (e.g., Heymsfield et al. 2009). Mixed-phase clouds can exist at temperatures as low as −37°C. At lower temperatures, the conversion of water into ice occurs rapidly through homogeneous freezing of droplets. This process may result in the high concentration of small ice crystals at the tops of convective storms (Rosenfeld and Woodley 2000).

Comparison of simulations of mixed-phase clouds (e.g., Klein et al. 2009; Morrison et al. 2009, 2011) shows a large diversity in prediction of liquid–ice partitioning under the same environmental conditions. Ovchinnikov et al. (2014) performed a detailed intercomparison of seven large-eddy simulation (LES) models simulating the same stratiform mixed-phase cloud and showed that the uncertainty in ice nucleation had the most pronounced effect on the evolution of the phase composition of the simulated cloud. However, introducing constant ice concentration values did not eliminate the discrepancies in the values of liquid–ice partitioning, ice water path, liquid water path, and so on, simulated by different models. These results made it clear to the cloud physics community that a better theoretical understanding of the fundamental processes determining liquid–ice interaction is required.

Glaciation via the WBF mechanism was theoretically studied by Korolev and Isaac (2003), Korolev and Mazin (2003), Korolev (2008), and Korolev and Field (2008). In these works, the existence of different regimes of interaction between the liquid phase and the ice phase in adiabatic parcels was established, and the glaciation time was estimated either numerically for an arbitrary vertical motion or analytically for zero vertical velocity.

In this study, we consider the processes of glaciation based on diffusional interaction between ice and liquid via water vapor. The glaciation time is determined here as the time of complete evaporation of liquid droplets in the presence of ice particles. A new set of analytical equations describing the glaciation time for a more general case of a vertically moving adiabatic mixed-phase parcel is obtained. This allowed us to achieve a better understanding of the phase transformation process in mixed-phase clouds.

## 2. Basic equations for analyzing glaciation time

In this study, the analysis of glaciation time is performed under the following simplifying assumptions:

the cloud parcel is assumed to be adiabatic;

the liquid droplets and ice particles are uniformly mixed;

the concentrations of droplets and of ice particles remain constant (i.e., riming, aggregation of ice particles, collision, and coalescence of droplets are disregarded);

size distributions of droplets and ice particles are assumed to be monodisperse; and

particle sedimentation is disregarded.

Results of LESs indicate that assumption about the uniformity in the liquid and ice microphysics is reasonable. We also believe that accurate evaluations of glaciation time can be obtained by analyzing the monodisperse spectra of droplets and ice particles, as it was done in several earlier studies (e.g., Korolev and Isaac 2003; Korolev 2008). Utilization of a monodisperse droplet size distribution in a warm parcel model allowed us to correctly reproduce the vertical profiles of supersaturation and droplet concentration (Pinsky et al. 2013). This suggests, though indirectly, that proposed simplifications are suitable for analytical investigation of the phase transformation process. Using the assumptions, we derived the two basic equations presented below.

### a. Equation describing the evolution of liquid water mixing ratio and ice water mixing ratio

Under the assumptions presented above, the equations of diffusion growth of liquid droplets and ice particles can be written in the form

where supersaturations with respect to water and to ice are defined as and , respectively, where *e* is the water vapor pressure, is the saturation vapor pressure over water, is the saturation vapor pressure over ice, and *T* is the air temperature. The expressions for coefficients and are given in appendix A. The evaluations of glaciation time are performed for pristine crystals that are modeled by the “effective” spheres of radius with ice density . The effect of nonsphericity can be included in Eq. (2) by introducing a normalized capacitance, depending on the aspect ratio of a crystal (see appendix B). Since the glaciation time is inversely proportional to the normalized capacitance, the spherical approximation of the ice particle shape leads to some overestimation of glaciation time in this study. In addition to the effects of capacitance, the gravity sedimentation of ice particles also has to decrease the glaciation time because of the ventilation effect. This effect can be taken into account exactly in the same way as the effect of nonsphericity by introducing the ventilation coefficient into the equations for diffusion growth of crystals.

For monodisperse spectra, the equations for the liquid water mixing ratio and ice water mixing ratio can be written as

where and are number concentrations of droplets and ice particles, with radii and , respectively, and and are densities of the air and the water, respectively. Using Eqs. (3) and (4), Eqs. (1) and (2) can be rewritten as

The expressions for coefficients and are given in appendix A.

The relationship between supersaturations over water and over ice can be written as (Korolev and Mazin 2003)

where is a function of temperature only. At temperatures below 0°C, the value of changes by less than 10% for the temperature changes (Murphy and Koop 2005). Thus, our analysis is performed under an assumption that remains constant during a parcel’s motion.

Excluding the supersaturation terms from Eqs. (5)–(7) yields the following differential equation describing the relationship between and :

This equation is valid if both liquid droplets and ice particles coexist in an adiabatic volume. Assuming the coefficients in Eq. (8) to be constant allows the integration of Eq. (8) with the initial conditions and . The integration results in

Equation (9) is the first basic equation derived in this study. It describes the relationship between the mass contents of water and ice in the process of mutual transformations and will be referred to as the equation of the coexistence of liquid water and ice.

While this equation is valid under a wide range of conditions, it is applicable only for mixed-phase volumes when both liquid and ice phase coexist, that is, and .

### b. Balance equation and parcel states

Equation (9) relates two variables and . To determine the behavior of each variable, we used the balance equation derived from the equation for supersaturation. The equation for supersaturation evolution in an adiabatic air volume can be written in the form (Korolev and Mazin 2003)

where is the vertical velocity of the air, and , , and are coefficients that are dependent on temperature and pressure ( appendix A). As shown in appendix C, for the entire temperature interval. It is also shown in appendix C (Fig. C1) that *A*_{1} and *A*_{2} depend on *T* and *p* only slightly, and so for vertical changes , the coefficients *A*_{1} and *A*_{2} change by less than 20%. These changes can be neglected; thus, the values of coefficients *A*_{1} and *A*_{2} in the following analysis are assumed to be constant during a parcel’s movement. The first term on the right-hand side of Eq. (10) determines the generation of supersaturation caused by adiabatic changes of temperature in a vertically moving parcel. The other terms on the right-hand side describe the decrease in the supersaturation due to depletion and due to the release of water vapor by droplets (second term) and ice particles (third term). Since in most mixed-phase cloud volumes (Korolev and Mazin 2003), we assume on the left-hand side of Eq. (10). Assuming coefficients , , and are constants, integrating Eq. (10) yields

where is a constant determining the initial conditions at . Equation (11) is the second basic equation derived in this study. It presents the mass balance for an adiabatic mixed-phase cloud parcel. This equation represents a generalization of the mass balance equation for a three-phase system. A similar equation for a two-phase system was obtained earlier by Pinsky et al. (2013).

The balance equation [Eq. (11)] is used for determining the initial states and final states of an adiabatic air volume. Different initial states determine different glaciation times . In our analysis, we use the following conditions for the initial state:

At the moment of glaciation, . Korolev and Mazin (2003) showed that in most mixed-phase clouds, supersaturation over water remains close to zero as . Actually, it means that on the left-hand side of Eq. (11) is typically much smaller than the terms and on the right-hand side. Indeed, a simple evaluation of the characteristic values of the terms on the right-hand side of Eq. (11) shows that can be neglected if or , which is the case for most mixed-phase clouds.

Thus, Eq. (11) can be simplified as

## 3. Different glaciation scenarios

### a. Scenario 1: Zero vertical velocity ()

The first scenario is for an idealized case when the vertical velocity of the cloud volume is equal to zero. Assuming that , one obtains from Eq. (13)

Korolev and Mazin (2003) also analyzed this case, using the condition instead of Eq. (14) and assuming that the final mass of ice is equal to the sum of the initial masses of ice and liquid water. Equation (14) shows that, at the moment of glaciation, the mass of ice is not exactly equal to the sum of the initial masses of liquid water and ice because some fraction of water passes into water vapor. However, because ( appendix C), the assumption made by Korolev and Mazin (2003) does not lead to errors.

This equation is the third basic equation derived in this study. For the case when and , that is, for the initial stage of glaciation, Eq. (15a) can be rewritten as

Equation (15b) shows that the initial water mass is distributed between ice particles whose initial size is negligibly small. In this case, glaciation means disappearance of droplets with concentration and emergence of ice particles with concentration .

Figure 1 shows the dependencies of the glaciation time on temperature. These dependencies were calculated using Eq. (15a) for different initial liquid water mixing ratios and different concentrations of ice particles. Figure 1 shows a good agreement between the values of *t*_{gl} calculated by Eq. (15a), the results of the parcel model developed by Korolev and Isaac (2003) and those obtained by Korolev and Mazin (2003). The glaciation time calculated by the parcel model is considered as benchmark in our study. One can see that the glaciation time is minimum at temperatures from about *T* = −12° to −14°C at which the difference in the values of equilibrium vapor pressure over water and ice reaches its maximum. One can also see that the glaciation time is highly sensitive to the initial water amount and to the ice concentration. The increase in the ice concentration from 1.0 to 100 L^{−1} decreases the glaciation time from tens of hours to a few tens of minutes.

Glaciation times for different initial liquid water mixing ratios and ice water mixing ratios are presented in Table 1. The glaciation time varies from about 0.5 h for a high initial mass of ice and a low mass of water to 9 h for a low amount of ice and a significant initial mass of water. As follows from the calculations, the concentration of droplets has a weak effect on the glaciation time. This weak dependence is explained by the fact that droplet concentration is typically much higher than that of ice particles. It means that the second term on the right-hand side of Eq. (15a) is two orders of magnitude lower than the first term.

Korolev and Mazin (2003) obtained an expression for glaciation time similar to Eq. (15a). Despite the similarity, there are two differences between the two expressions: Eq. (15a) contains the ratio and the term . Both terms are not used by Korolev and Mazin (2003). Although Eq. (15a) is more accurate, the difference between the results arises only when the droplet concentrations drop below 100 cm^{−3}. Under these conditions, the results obtained using Eq. (15a) show a better agreement with those of the parcel model, as can be seen in Fig. 2. At the same time, the difference does not exceed 5%–10%.

Glaciation time also depends on capacitance and ice density. As shown in appendix B, glaciation time is inversely proportional to normalized capacitance and directly proportional to . Thus, glaciation time for nonspherical particles may be substantially (by several times) lower than that for spherical ones.

### b. Scenario 2: Vertical displacement of an adiabatic parcel

A mixed-phase cloud parcel may experience complex vertical trajectories before reaching the level of glaciation, which raises the following question: if the level at which a parcel glaciates is known, does the glaciation time depend on the parcel trajectory? Or, in other words, what is the difference between the glaciation time for a parcel with zero velocity (i.e., staying at all the time) and a parcel that moves first upward by and then downward by , its summary displacement being zero? We do not impose strict conditions on the vertical velocity of parcels. For instance, a parcel can reach the level and then stop ascending. However, the process of glaciation continues at this height. Thus, the glaciation time considered in this section is equal to or larger than the time needed for a displacement of a cloud parcel by .

Figure 3 schematically shows three examples (A, B, and C) of a cloud parcel movement that differs by the summary displacement occurring before the parcel arrives at its glaciation height. In A, the glaciation takes place at the height of above the initial parcel location. In B, the glaciation takes place at the initial height. In C, the glaciation takes place at the height of below the initial parcel location. In each example, two parcel trajectories having the same final displacement value are designated by arrows. The question is what the difference in glaciation times is for parcels moving along trajectories 1 and 2 in all the three examples and how this difference depends on .

From the balance equation [Eq. (13)], it follows that the final state of a parcel’s location is characterized by the following equalities:

Displacement by in Eq. (16) can be either positive or negative. Equating Eqs. (9) and (16) yields the following expression for glaciation time:

where the parcel’s displacement is positive in case of an upward movement and negative in case of a downward movement. Equation (17) is the fourth basic equation derived in this study.

A few comments are required in regards to Eq. (17):

Equation (17) describes both the processes of condensation and of glaciation (i.e., droplet evaporation in the presence of ice). In case of a positive displacement, the condensation of water vapor on water droplets may take place within some segment of the parcel’s trajectory, slowing down the glaciation. In case of a negative displacement, droplets evaporate faster than in a quiescent parcel, thus decreasing the glaciation time. Vertical oscillations of a cloud parcel can be accompanied by deceleration of the glaciation time (due to condensation of droplets) or its acceleration (due to evaporation of droplets).

- In case the value of is given a priori, Eq. (17) allows the calculation of . Here Eq. (17) is applicable if liquid water exists in a parcel till it reaches the final height level . This condition of applicability of Eq. (17) implies the following restriction on glaciation time: , where is the mean parcel velocity during its movement until a total glaciation takes place. This inequality means the following restriction should be imposed on the mean vertical velocity: If , is small and the glaciation time can be evaluated from Eq. (15a) for a motionless parcel. Here is the minimum vertical velocity needed to maintain a mixed-phase state over the given value of.

In case of a downward displacement ( is negative), the supersaturation over water decreases both because of the increase in the temperature in an adiabatic downdraft, and because of the absorption of water vapor by ice particles. This requires the second necessary condition restricting the downward displacement, which can be derived from condition . Omitting the negligibly small last term in Eq. (17), this condition can be written as . The inequality yields

This expression is similar to Eq. (8) in the study by Korolev and Isaac (2003). The physical meaning of inequality Eq. (19) is that a downward displacement should be lower than that needed to evaporate droplets during an adiabatic descent, if the effects of ice are not taken into account. Condition (18) can be applied for both ascending and descending parcels, whereas Eq. (19) refers only to descending parcels.

Analysis of Eq. (17) leads to the following important conclusions:

- Glaciation time does not depend on a parcel’s trajectory, but is determined solely by the initial conditions and the vertical displacement up to values of where total glaciation takes place. This conclusion is illustrated in Fig. 4, where the glaciation time is calculated for different trajectories of parcels arriving at the same final height. Calculations were performed with the help of the parcel model developed and used by Korolev and Isaac (2003). It can be seen that parcels moving along different trajectories are glaciated at approximately the same time. These time values agree well with those calculated from Eq. (17) (black dot in Fig. 4a).Fig. 4.
It should be noted that under condition , the WBF mechanism may be not necessarily active during all the glaciation period. As shown by Korolev and Mazin (2003) and Korolev (2007), the WBF mechanism in a mixed-phase parcel “turns on” only when , whereas both droplets and ice particles grow simultaneously at and evaporate simultaneously at . According to Korolev and Mazin (2003), and , where and are the mean radii of liquid droplets and ice particles, respectively. For the typical values of , the characteristic threshold velocity is of the order of a few to tens of centimeters per second. The low value of separating the WBF and non-WBF types of interaction between ice particles and liquid droplets means that the condition is a regular situation in the natural clouds. In case both droplets and ice particles grow simultaneously in an ascending parcel, the value grows accordingly. As soon as growing ice particles reach such size that , the WBF mechanism is activated, and liquid droplets start evaporating, thus expanding the growth of ice crystals. In Fig. 4, green and black colors of the curve highlight time periods when the WBF mechanism is enabled and disabled, respectively. Trajectory 3 in Fig. 4 shows that an increase of ice particle sizes leads to turning on the WBF mechanism at the same updraft velocity.

If the summary displacement is equal to zero, Eq. (17) coincides with Eq. (15a). So, provided the initial conditions are the same, the glaciation time within a moving parcel whose average vertical velocity, and consequently, the summary vertical displacement are equal to zero, is the same as the glaciation time of a motionless parcel. The glaciation time averaged over many oscillating parcels is not equal to the glaciation time of a motionless parcel because of the nonlinear dependence of .

There is a substantial difference in glaciation times for positive and negative displacements. For negative displacements, the glaciation time is shorter than that for positive displacements. This result is the direct consequence of the additional droplet evaporation in downdrafts caused by the temperature increase.

For the case and , Eq. (17) turns into

One can consider the case when, at , the initial liquid water mixing ratio is equal to zero and so activates at and evaporates at . This case can be analyzed using the initial condition (however, is concentration of droplets) in Eq. (17). The resulting equation has the form

In case both the initial liquid water mixing ratio and the ice water mixing ratio are equal to zero, this equation is simplified to

Figure 5 shows dependencies of glaciation time and of the minimal vertical parcel velocity on the displacement value at different ice particle concentrations. Analysis of Fig. 5 shows that the glaciation time is highly asymmetric with respect to the sign of . In case , the minimum velocity value is comparatively small, so Eq. (17) can be applied within a wide range of updraft velocities. In case , the glaciation time dramatically decreases as compared to the case of positive displacement. Under conditions of the simulations for which the results are presented in Fig. 5, the downward displacement is limited by the value of approximately 100 m, regardless of the initial concentration of ice particles. This maximum negative displacement corresponds to the case in which water droplets just evaporate in a downdraft without any influence of ice [see Eq. (19)].

### c. Scenario 3: Glaciation in case the vertical displacement is not preset

In case at which glaciation takes place is not predetermined, Eq. (17) describes a relationship between two unknown values, namely, the glaciation time and at which the full glaciation takes place. An additional relationship between these two variables can be derived from the equation of cloud parcel motion. For instance, in case a cloud parcel oscillates around a certain height level so (, , and are known parameters of parcel oscillations), Eq. (17) can be rewritten in the form of a closed transcendental equation with respect to glaciation time:

As soon as the value *t*_{gl} is calculated, the parcel’s location at the time instance of full glaciation is determined as . Figure 6 illustrates the results of such calculations. The left-hand side of Eq. (21) reaches zero practically at the same time (middle panel) as calculated using the parcel model (top panel). This time instance corresponds to the time of full glaciation. The glaciation time determines the height of the parcel location *z*_{gl} (bottom panel). Figure 6 also shows the zones where the WBF mechanism is active.

### d. Accuracy of theoretical results and corrections

We define as the ratio of glaciation time calculated from Eq. (17) and that obtained from the parcel model. The deviation of from 1 characterizes the relative error in the estimation of *t*_{gl} from Eq. (17). The dependences of on temperature under different values are shown in Fig. 7. One can see that the maximum deviation of from 1 (i.e., the maximum error) is reached at temperatures from −5° to 0°C. At lower temperatures the relative error does not exceed 25%. Errors in cases of positive and negative displacement are of opposite signs. At , Eq. (17) provides a good accuracy in estimation of *t*_{gl} for the entire temperature range, whereas at , Eq. (17) gives the best estimation of *t*_{gl} for the temperature range −20° < *T* < −15°C.

The dependences shown in Fig. 7 can be used to calculate the correction factors for Eq. (17). We propose the following expression for this correction factor:

where , *T* is measured in degrees Celsius, and is measured in meters. The dependences of the correction factor on temperature under different values of are shown by solid lines. The approximation of the correction factor given by Eq. (22) is shown by asterisks in Fig. 7. Using both Eq. (17) and the correction factor expressed by Eq. (22) allows an accurate estimation of the glaciation time within a wide range of temperatures from −30° to almost 0°C.

### e. Dependence of glaciation time on vertical velocity

In the previous section, glaciation time was analyzed as a function of and thus had to satisfy the condition . In the following discussion, glaciation of a mixed-phase cloud is analyzed for constant vertical velocity . In this case, . Substituting into Eq. (17) yields an equation for glaciation time as a function of vertical velocity, that is, :

where is positive in updrafts and negative in downdrafts. This equation is the fifth basic equation derived in this study. It can be reduced to a cubic equation with respect to glaciation time. For that purpose, we introduced two new variables and , that is,

and then defined the normalized time that is linearly related to glaciation time *t*_{gl}:

This equation has three roots and discriminant . There are two physically relevant roots:

and

The first root should be chosen if , and the second root should be chosen in the opposite case. The conditions are related to the discriminant sign. Solving Eq. (27) and substituting the solution into Eq. (26) yields glaciation time value.

In the case and , and . In this case, Eq. (27) yields a solution coinciding with that of Eq. (15a). In the case and the initial liquid water ratio and ice water mixing ratio are equal to zero (i.e., ), then , and the glaciation time is proportional to the square of the vertical velocity:

Figure 8 shows the dependence of glaciation time on vertical velocity under different number concentrations of ice particles. One can see that the glaciation time in updrafts rapidly increases with the increasing vertical velocity and the decreasing concentration of ice particles. According to Eq. (30), the glaciation time, being inversely proportional to the squared concentration of ice crystals, decreases rapidly with the increase of the ice particle concentration. Figure 8 shows that, for high ice particle concentration of 1000 L^{−1}, glaciation may take place during a few tens of minutes even at high updraft velocities of a few meters per second.

At vertical velocities of and ice particle concentrations of , glaciation time becomes larger than the characteristic lifetime of clouds. Therefore, any glaciation that might take place at these conditions would be caused by other mechanisms such as droplet freezing and/or riming. At vertical velocities of 0 < *w* < 0.25 m s^{−1}, which are typical of stratiform clouds, glaciation may occur within a layer a few hundred meters deep.

As was shown by Korolev (2008), the WBF mechanism, manifesting itself in the mass transfer from evaporating droplets to growing ice crystals, is most pronounced only at very low vertical velocities.

Glaciation time values calculated using Eqs. (26) and (27) show a rather good agreement with those obtained using the parcel model. The maximal deviation of the solution of Eqs. (26) and (27) from the parcel model solution takes place at high vertical velocities and/or low ice concentrations. This leads to a long glaciation time and therefore to a large . For large values, the assumption regarding the constancy of the coefficients in Eqs. (9) and (11) and their independence on changes of and in Eqs. (26) and (27) becomes invalid, which may result in significant errors.

Analysis shows that Eqs. (28) and (29) are highly applicable within the temperature range from −7° to −30°C. We believe that the analytical formulas can be useful for the evaluation of glaciation time in mixed-phase stratocumulus clouds at different vertical velocities and aerosol concentrations. This was demonstrated in the study by Korolev and Isaac (2003), where the dependences of the glaciation time on vertical velocity obtained using a parcel model are presented within a wide range of parameters.

## 4. Conclusions

In the present study, the glaciation of the mixed-phase cloud parcel is investigated analytically. The analysis is conducted with the help of two new equations: the equation for the coexistence of liquid water and ice [Eq. (9)] and the equation of mass balance [Eq. (11)]. These equations allowed for the derivation of analytical expressions that provide good estimations of glaciation time *t*_{gl} within a wide range of vertical velocities and parcel displacements.

The main new results obtained in the study are the following.

First, we derived a new Eq. (15) that presents a refined version of the glaciation time equation for proposed by Korolev and Mazin (2003). The new equation takes into account the effect of droplet concentration and yields more accurate *t*_{gl} values in case droplet concentrations and ice particle concentrations are comparable.

Second, it has been shown for the first time that glaciation time can be considered a function of the vertical displacement of parcel . If the conditions in Eqs. (18) and (19) are satisfied, then the glaciation time does not depend on the trajectory along which the cloud parcel underwent before it reaches the level of glaciation. It is also demonstrated that there exists a possibility of calculating the glaciation time using the new equation for the case when dependence of is known but the glaciation level is unknown.

Third, a new cubic equation [Eq. (27)] was derived that describes the dependence of glaciation time on vertical velocity. The analytical equations for *t*_{gl} obtained here are useful for the analysis of factors affecting the glaciation process.

The analytical derivations have been performed under simplified assumptions that coefficients , , , , and do not change during parcel movement. These assumptions may result in some errors in estimating the glaciation time, which are high (up to 50%) at temperatures higher than −4°C. However, the proposed correction coefficients allow us to compensate these errors and to obtain an accurate estimation of *t*_{gl}.

Using the equations derived in the present study, it is shown that as long as the droplet concentration is much higher than that of the ice particles, the effects of droplet concentration on the glaciation time is negligible. The concentration of ice particles turned out to be one of the strongest factors affecting the glaciation time. Among the other important parameters affecting the time of glaciation are the initial liquid water content, the vertical velocity, and the temperature. These conclusions agree well with those obtained in the studies by Korolev and Isaac (2003) and Ovchinnikov et al. (2011).

The analytical dependencies show the crucial effect of the vertical velocity on glaciation time. At , the supersaturation over water can become negative because of water vapor absorption on ice particles. In this case, the WBF mechanism turns on during the parcel ascent (see Fig. 4, top). At large and low temperatures, the parcel can reach the level of homogeneous freezing, and glaciation takes place as a result of this freezing.

We consider this study as a further development of the analysis of the phase transformation in mixed-phase clouds presented in the pioneering studies by Korolev and Isaac (2003), Korolev and Mazin (2003), and Korolev and Field (2008). The analytical approach used in the study allowed us to explain why the glaciation time does not depend on the parcel track. Also the approach allowed us to determine a set of main parameters that controlled the glaciation time.

Although the proposed approach is a significant idealization of the processes because many processes such as riming and freezing are not considered in this study, we believe that the results obtained in this study reflect the properties of real clouds. For instance, our estimations of the effects of capacitance on glaciation rate qualitatively agree with the parcel model results reported by Sulia and Harrington (2011). Our results concerning the effects of ice particle concentration on glaciation time qualitatively and even quantitatively agree with those obtained by the LES 3D model of stratiform clouds presented by Ovchinnikov et al. (2011). Since glaciation time is determined largely by processes in cloud updrafts (total glaciation occurs rapidly in downdrafts), we suppose that a parcel’s displacement factor introduced in the study is closely related to the cloud depth or more precisely to the depth of the mixed-phase layer. Estimations of *t*_{gl} that we obtained show that, under typical ice particle concentrations and the displacement of about 200 m, the glaciation time is on the order of a few hours, which is in agreement with the evaluations made by Ovchinnikov et al. (2011). A more detailed comparison with results of LESs and observations requires a separate study. We believe, however, that the results obtained in this study may prove to be a useful tool for future theoretical analysis and in situ studies of mixed-phase clouds.

## 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 and Transport Canada.

### APPENDIX A

#### List of Symbols

Table A1 shows a list of symbols used in this paper.

### APPENDIX B

#### Effects of Particle Capacitance and Density on Glaciation Time

In case the growth of nonspherical ice particles is taken into account, Eq. (6) can be rewritten in the following form:

where , is the particle capacitance, and is the normalized by mean volume radius. Here is the aspect ratio of the particle. Figure B1 shows several simplified particle shapes used for the approximation of shapes of ice crystals. Table B1 presents the expressions for the capacitance of particles of the simplified shapes. For a spherical particle, its capacitance is equal to its radius.

Figure B2 shows the dependence of the normalized capacitance on the aspect ratio of a spheroid particle. The spheroid particles have minimum values of the normalized capacitance (equal to unity) at the aspect ratio equal to one, that is, for the spherical shape. The capacitance grows as the aspect ratio deviates from unity. In clouds, crystal shapes deviate from the spherical one as their mass increases. For plates, the value increases to about 3 with the aspect ratio decreasing to . For columnar crystals, increases to about 4 with the increase in the aspect ratio to 100.

Thus, the approximation of ice crystals to effective spheres, which has been widely used in numerous studies, is in fact suitable for comparatively small particles with the aspect ratio values close to unity. However, for large ice crystals, diffusion growth should be calculated taking into account the actual shape of a crystal.

Taking into account the nonsphericity of ice particles, the formulas for glaciation time change in such a way that the normalized capacitance is to be multiplied by coefficient . For instance, Eq. (17) is replaced by the following expression:

This equation shows that the glaciation time is inversely proportional to the normalized capacitance, so the glaciation time for nonspherical particles is lower than for spherical ones. It can be accounted for by the fact that nonspherical ice particles grow by deposition and decrease the supersaturation over water faster than spherical particles. Since , a decrease in the particle density leads to a decrease in the glaciation time.

### APPENDIX C

#### Temperature Dependence of Coefficients

Figure C1 shows the temperature dependences of coefficients , , , and . The changes of coefficients , , , and are relatively slow when the temperatures are not extremely negative but are relatively rapid below −20°C. Coefficient is almost insensitive to pressure, and its slope remains approximately constant within the entire temperature range up to −30°C. Coefficients and are approximately equal () within the temperature interval from −20° to 0°C. Coefficients and have close values within the entire range of negative temperatures.

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