1. Introduction
Determining how aerosols affect warm clouds challenges the cloud and climate modeling community (IPCC 2013). In particular, one subset of aerosols, cloud condensation nuclei (CCN), influence cloud drop spectra; in turn, changes to cloud droplet spectra affect rain initiation (Howell 1949; Mordy 1959). However, CCN effect on rain initiation is not represented well in current weather and climate models, bringing about a bias of “too dense clouds” (or the mixing ratio of cloud water is too high) in the models (Nam et al. 2012).
Weather and climate models usually use the one-moment scheme of Kessler (1969) (or its variation) to parameterize warm rain initiation. Why do they still use a simple parameterization developed 50 years ago? The question implies that our knowledge on rain initiation is insufficient, which motivates the present study with two threads: 1) the models are missing an important process, and 2) thus, the parameterization framework of rain initiation is ill-posed.
a. Processes of rain initiation
The classic question of rain initiation come from the radar observations of Squires (1958) and others: “rain can initiate within half an hour or less in cumulus clouds.” Because no rain initiation model with a single process can explain the observations (Blyth and Latham 1993; Blyth et al. 2013), many scientists began to question the classic modeling framework of Mordy (1959) and built complicated models with additional processes or factors, such as giant salt particles (Johnson 1982; Blyth et al. 2003; Jensen and Nugent 2017), radiative effect on microphysics [Roach 1976; Harrington et al. 2000; Zeng 2008, 2018; see Zeng et al. (2022) for review], cloud mixing (Latham and Reed 1977; Telford et al. 1984), and small-scale turbulence (Jonas 1996; Shaw 2003).
While applauding the insight introduced by these complicated models, here we return to the classic modeling framework of Mordy (1959) except for introducing drop collection. After representing CCN activation and drop collection properly with this simple framework (see section 2), the present model replicates the fast rain initiation observed. The fast rain initiation is attributed to the effective cooperation in rain initiation between CCN activation and drop collection, which is studied herein to lay a basis for its parameterization.
b. Parameterization framework
The current parameterization framework of rain initiation cannot accommodate the cooperation between CCN activation and drop collection properly because it usually treats CCN activation and drop collection separately. The parameterization of drop collection in models has evolved over time. The earliest framework for the parameterization, one-moment schemes (Kessler 1969; Cotton 1972), was first proposed where the drops with radius r > 100 μm are classified as raindrops and others as cloud drops. Since the one-moment schemes use only one variable (i.e., cloud water content) and thus have no information on drop number, they cannot represent CCN properly.
Inheriting this framework, many two-moment (or higher-order moment) schemes were developed (Berry and Reinhardt 1974; Tzivion et al. 1987; Khairoutdinov and Kogan 2000; Liu and Daum 2004; Seifert and Beheng 2006; Morrison et al. 2005; Lee and Baik 2017; Zeng and Li 2020). Since the schemes employ two variables (i.e., cloud drop number concentration and cloud water content), they can parameterize CCN explicitly. To better represent the effect of drop collection on rain initiation, the separation radius between cloud drops and raindrops is decreased from 100 μm (Kessler 1969), to 40 μm (Berry and Reinhardt 1974; Seifert and Beheng 2001; Lee and Baik 2017), and further to ∼25 μm (Khairoutdinov and Kogan 2000; Zeng and Li 2020).
Although the schemes differ in the details of their formulations, most schemes can usually be described by two typical parameterizations, lying at opposite ends of model variability. Figure 1 illustrates these typical parameterizations, where the left-hand ovals represent cloud water, the right-hand ovals represent rainwater, and the red arrows denote processes in rain initiation. In Fig. 1, the autoconversion of cloud water to rainwater represents the increase of rainwater from collision–coalescence growth of cloud droplets. The accretion is the growth of existing rainwater by collecting cloud water (Morrison et al. 2020). The first involves only cloud droplets; the second includes rain drops interacting with cloud droplets. Clearly the definition of the autoconversion and accretion vary with the separation radius.
Contrast in framework between two typical parameterizations: (top) the scheme of Kessler (1969) and (bottom) the one used here, where left and right black ovals identify cloud water and rainwater, respectively. The cloud water–rainwater separation is defined at a drop radius (top) 100 and (bottom) 28 μm. Red arrows between black circles represent processes occurring between CCN, cloud water, and rainwater explicitly, and gray arrows in black circles represent processes occurring implicitly. The two typical parameterizations group CCI as a part of cloud water and rainwater differently, where cloud drops are defined as drops with r < 28 μm, CCIs as drops with 28 < r < 100 μm, and raindrops with r > 100 μm. In contrast to the first parameterization, the second parameterization explicitly represents not only autoconversion and accretion but also the condensation-induced conversion of cloud water to rainwater via CCIs.
Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0017.1
Collision–coalescence initiators (CCIs), defined as the drops with 28 < r < 100 μm, can initiate warm rain effectively (Mason 1971; Johnson 1993; Small and Chuang 2008). The two typical parameterizations in Fig. 1 differ in how they group CCIs. The first classifies CCIs as cloud water, while the second considers CCIs to be rainwater. The first parameterization includes Kessler (1969), and the second one follows Khairoutdinov and Kogan (2000) and Zeng and Li (2020) and will be used in this study. Other schemes with separation radius of 40 μm (e.g., Berry and Reinhardt 1974; Seifert and Beheng 2001; Lee and Baik 2017) split CCIs into cloud water and rainwater and thus lie between the two typical parameterizations.
The second parameterization in Fig. 1, in contrast to the first one, treats CCIs as rainwater because CCIs can become raindrops quickly via drop collection once they form (Mason 1971; Johnson 1993). Furthermore, its autoconversion (or the collision-induced conversion of cloud drops to CCIs) can be parameterized accurately (Zeng and Li 2020) and its condensational growth of cloud droplets to CCIs be parameterized explicitly. In addition, the second parameterization can express the sedimentation of rainwater properly in terms of the average radius of raindrops (including CCIs). For example, if an air parcel contains only CCIs, its rainwater sediments with the small terminal velocity of CCIs.
The second parameterization illustrated in Fig. 1 will be studied in this study. In contrast to other two-moment schemes (e.g., Seifert and Beheng 2001; Khairoutdinov and Kogan 2000; Lee and Baik 2017), the scheme introduces a new process, the condensational conversion of cloud water to rainwater via CCIs. The new process is independent of the autoconversion of cloud water to rainwater. It consists of two steps: cloud droplets grow to CCIs via condensation first and then CCIs grow to raindrops quickly by accreting cloud drops.
To test the new process, a bin model is developed that includes drop collection and thus can simulate the growth of small CCN to large raindrops. The model and its simulations are introduced in this paper with six sections. The model structure is introduced in section 2 with a CCN representation. Simulations are presented in sections 3 and 4 to show 1) the model replicates the Twomey relationship and 2) rain initiation is sensitive to CCN, respectively. Section 5 outlines how to apply the simulation results to rain initiation parameterization, and section 6 summarizes.
2. Model description
A bin model is developed based on Mordy (1959) and Zeng (2018) to explicitly simulate CCN activation and drop collection. It is introduced in this section with numerical schemes, CCN representation, and model structure.
a. Representing drop collection and condensation growth in the Lagrangian system
The model adopts the framework of the bin model of Zeng (2018) that is based on the method of Kovetz and Olund (1969). Specifically, it simulates drop growth in the Lagrangian system by tracking drops with thousands of bins. Let mj denote drop mass of a bin. Then mj is increased (or decreased) to represent the condensation growth (or evaporation shrinkage) of drops, avoiding the spurious spectrum broadening that often happens in the bin models in the Eulerian system (Khain et al. 2008, 2010, 2012; Flossmann and Wobrock 2010; Iguchi et al. 2012).
The present model is like the model of Pinsky and Khain (2002), to some extent, for both the models use thousands of bins to explicitly simulate CCN activation. On the other hand, the models differ in CCN representation because the present model incorporates the effects of not only CCN size but also composition via the Twomey relationship (see section 2b).
In addition, the two models differ in drop collection simulation [or Eq. (2)]. Pinsky and Khain (2002) used the two-step method of Bott (1998) to simulate drop collection. In the first step drops with mass m′ newly formed in a collision process is added to bin j if mj ≤ m′ < mj+1. In the second step a fraction of drops is transported from bin j to j + 1, where the fraction depends on an imposed pattern of drop spectrum between mj and mj+1. The pattern (or fraction) is adjusted so that the modeled drop spectrum fits the analytic solution of SCE in specific cases (e.g., those with the Golovin collection kernel; Bott 1998), which is referred to as “hard adjustment.” In the method the “hard adjustment” satisfies (1b) but violates (1a).
The “hard adjustment” was often used in the models with bin number ∼ 40 (Berry and Reinhardt 1974; Zeng and Zhang 1989; Bott 1998). Since the “hard adjustment” was tested only in some specific cases, whether it works in other cases (e.g., those with condensational growth) is still unknown. In contrast, the method of Kovetz and Olund [or Eq. (1)] is general and consequently can work in any cases given sufficient bin resolution. Hence the present model employs the method of Kovetz and Olund with 4096 bins.
b. CCN representation
The model simulates CCN activation explicitly just as the other Lagrangian models (Mordy 1959; Pinsky and Khain 2002; Jensen and Nugent 2017). Different from the other models, it uses a spectrum expression of condensation nuclei (CN) so that it can replicate the Twomey relationship between active CCN and supersaturation (Twomey 1959).
The bin model uses (6) to represent CN. Its two parameters—k and NCCN1—are available from the CCN observations (Twomey 1959; Braham 1976; Hudson 1993). In this paper, two extreme cases are discussed: k = 0.4 and NCCN1 = 100 cm−3 for maritime clouds, and k = 0.9 and NCCN1 = 500 cm−3 for continental clouds. Other actual cases usually lie between these two extreme cases (Jiusto 1967).
Given k and NCCN1, (6) still needs the information of CN composition. For the sake of simplicity, CN are assumed to be composed of sea salt (NaCl). In fact, the present CN representation is suitable for other salt or multicomposition particles after replacing the particles with their equivalent CN of NaCl, where the equivalent CN of NaCl are defined as the virtual NaCl particles whose Twomey relationship equals that of real aerosol particles.
In summary, (6) is equivalent to the Twomey relationship if all drops are close to equilibrium with water vapor, which is illustrated in Fig. 2. Since the bin model uses (6) as an input, it generates the Twomey relationship as an output (when its initial drops are at equilibrium with water vapor), which supports the equivalence between (6) and the Twomey relationship (see section 3 for simulation details or the circles in Fig. 2 for simulation results).
Two equivalent CCN representations: (left) the CN spectrum dNCN(ms)/dlnms vs salt mass ms [or Eq. (6)], and (right) the number concentration of activated CCN vs supersaturation (or the Twomey relationship). Blue and red lines represent marine and continental clouds, respectively. A circle denotes the number concentration of activated CCN vs the maximum supersaturation in a bin model simulation.
Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0017.1
On the other hand, drops with large CCN in real clouds are rarely at equilibrium with water vapor because they reach the equilibrium in hours to days which depends on the mass of large CCN (Kogan 1991). Hence the Twomey relationship represents large CCN in real clouds inaccurately. If a model used the Twomey relationship as an input to represent CCN in clouds (Khain et al. 2008), it introduced a representation error of large CCN.
Different from the Twomey relationship, (6) can be used to represent CCN in real clouds. It is connected to the Twomey relationship as follows. Suppose we sample aerosols in a cloud and then move the sample of the aerosols into a cloud chamber for CCN measurement. Since the aerosols are at equilibrium with water vapor in the cloud chamber, they satisfy the Twomey relationship and thus their k and NCCN1 are obtained. After obtaining k and NCCN1, we assume that the aerosols besides the sampled ones satisfy (6) with the same k and NCCN1, no matter whether they satisfy the Twomey relationship in real clouds.
c. Air parcel model
The bin model is used to simulate an air parcel at an altitude of pressure p = 800 hPa and temperature T = 15°C. It uses (6) to represent CN spectrum and sets the maximum salt mass msmax = 10−11 g except for specification.
Equations (7) and (10) are integrated numerically, following Zeng (2018). For simplicity, T and p are assumed to be constant. In addition, (10) is integrated implicitly with a small time step of 0.05 s so that CCN activation is simulated accurately (see appendix).
3. Replication of the Twomey relationship
The Twomey relationship was often misused to represent CCN in cloud modeling by overlooking its condition (or the environment for drops in cloud chamber) (e.g., Khain et al. 2008, 2010, 2012; Iguchi et al. 2012), where the condition involves the equilibrium between drops and water vapor and thus does not exist in all clouds. To avoid the misuse of the Twomey relationship, the present model employs (6) to represent CN instead. As a result, the model satisfies the Twomey relationship when its initial drops are at equilibrium with water vapor, which is tested in this section via bin model simulations.
Experiment M1I is designed to simulate a marine cloud with w = 1 m s−1, p = 800 hPa, and T = 15°C. It chooses k = 0.4, NCCN1 = 100 cm−3, and msmax = 10−11 g to represent marine CCN. It incorporates vapor condensation but no drop collection. Its modeled drop spectrum is displayed in Fig. 3, showing CCIs but no raindrops form. For comparison, another experiment of C1I is carried out to obtain the sensitivity of CCIs to CCN. It takes the same setup as M1I except for continental CCN. It uses k = 0.9, NCCN1 = 500 cm−3 and msmax = 10−11 g to represent continental CCN. Its modeled drop spectrum resembles that of M1I except for the water content of CCIs (figure omitted).
Evolution of mass density dM(lnr)/dlnr vs drop radius r in Exp. M1I with marine CCN, where M(lnr) denotes the mass of drops with radius less than r. Black line denotes the initial spectrum, and the time interval between lines is 10 min.
Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0017.1
Since the two experiments incorporate no drop collection, their CCIs form via the condensation growth of cloud drops. To be specific, since their relative humidity Sw > 1, water vapor deposits on drops, bringing about an increase of liquid water content (LWC; see Fig. 4). As a result, a part of cloud drops grows to CCIs via condensation.
Time series of (left) liquid water content and (right) supersaturation in Exp. M1I with marine CCN (blue) and Exp. C1I with continental CCN (red).
Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0017.1
The modeled number concentration of activated CCN (or the number concentration of cloud drops and CCIs) satisfies the Twomey relationship. Using the same setup as M1I and C1I, five additional sets of experiments are carried out that take a vertical velocity of 0.1, 0.2, 0.5, 3, or 10 m s−1, respectively. Their maximum number concentrations of activated CCN are displayed against the maximum supersaturation by circles in Fig. 2. Clearly, the modeled concentration of activated CCN is close to the Twomey relationship, supporting the equivalence between (6) and the Twomey relationship.
4. Sensitivity of rain initiation to CCN
Since M1I was designed to test whether the model satisfies the Twomey relationship, its initial drops were assumed to be at equilibrium at Sw = 1. However, large drops need hours or days to reach their equilibrium radius at Sw = 1 (Kogan 1991). Hence, M1I implicitly introduced spurious large initial droplets if it addressed a real cloud. To avoid the spurious initial large droplets, experiments with a new initial drop spectrum are carried out in this section (see Table 1 for experiment summary), focusing on how CCN impact the initiation of CCI and raindrops given the air vertical velocity w and the maximum salt mass msmax.
Parameters of the control experiments. Experiment naming convention starts with a letter M or C for a marine or continental cloud, followed by a number of vertical velocity, and ends with a letter I for special initial drop spectrum or C for an experiment with drop collection.
a. Condensational formation of CCIs
A default experiment, M3, is designed to test the condensational formation of CCIs. It takes the same setup as M1I except for w = 3 m s−1 and a new initial drop spectrum. Its initial drop spectrum is similar to that of Mordy (1959).
Mordy (1959) assumed that at cloud base drops formed on nuclei smaller than 0.12 μm are at equilibrium at 100% relative humidity, while the drops formed on nuclei larger than 0.12 μm are at equilibrium at 90% relative humidity. However, the discontinuity of relative humidity at nuclei 0.12 μm brings about an abnormal phenomenon: some drops formed on large nuclei are smaller than those formed on small nuclei. To avoid the abnormal phenomenon, the relative humidity for drop equilibrium is assumed to decrease linearly with nuclei mass from 100% at ms = 0% to 90% at ms = 2.5 × 10−14 g (or dry sea-salt particle radius 0.14 μm) and maintain 90% at ms > 2.5 × 10−14 g. The drop spectrum at cloud base is then obtained with (6) and (12).
Experiment M3 uses the preceding drop spectrum at cloud base as initial drop spectrum. Like C1I, it uses 4096 bins to discretize the drop spectrum, one-half of which are used to represent solution drops whose salt mass ranges from 10−18 g to msmax (i.e., 10−11 g) and the other half are reserved to represent large drops that will form via drop collection with a = 246.2. Its modeled drop spectrum is displayed in Fig. 5. At 30 min, most of the drops are located with radius between 28.5 and 31.2 μm, showing that condensation does bring about CCIs even after the new initial drop spectrum is used.
As in Fig. 3, but for Exp. M3 where initial large drops are not at equilibrium with water vapor.
Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0017.1
For comparison, another default experiment of C3 is carried out to obtain the sensitivity of CCIs to CCN. C3 takes the same setup as M3 except for continental CCN (i.e., k = 0.9, NCCN1 = 500 cm−3, and msmax = 10−11 g). Its CCIs, as shown in Fig. 6, form more slowly than the counterpart of M3. If CCI initiation is defined as the water content of CCIs above 10−3 g m−3, CCIs initiate at 21.1 and 68.7 min in M3 and C3, respectively.
Time series of CCI water content in Exp. M3 with marine CCN (blue) and Exp. C3 with continental CCN (red).
Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0017.1
To study the condensational formation of CCIs extensively, 3 × 9 × 2 experiments are carried out that take the same setup as M3 and C3 except for w and msmax. They use the following parameters: w = 1, 3, or 5 m s−1; msmax = 10−13, 3 × 10−13, 10−12, 3 × 10−12, 10−11, 3 × 10−11, 10−10, 3 × 10−10, or 10−9 g; k = 0.4/NCCN1 = 100 cm−3, or k = 0.9/NCCN1 = 500 cm−3. Their modeled time of CCI initiation is displayed in Fig. 7, showing that the CCI initiation is sensitive to CCN and w. To be specific, given w and msmax, the difference in CCI initiation time between the two panels (or the difference between two lines with the same color) indicates that the marine clouds with k = 0.4/NCCN1 = 100 cm−3 initiate CCIs faster than their continental counterparts with k = 0.9/NCCN1 = 500 cm−3. In addition, each line in a panel indicates that a cloud, given w, initiates CCIs faster with increasing msmax.
Time of CCI initiation vs the maximum salt mass msmax at vertical velocity w = 1 (blue), 3 (red), and 5 m s−1 (green) with (left) marine and (right) continental CCN. Symbols (circle, cross, and plus) correspond to numerical experiments.
Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0017.1
b. Collision-induced conversion of cloud drops to raindrops
Once CCIs form via condensation growth, they grow to raindrops quickly via drop collection. Thus, the condensational conversion of cloud drops to raindrops via CCIs is independent of the collision-induced conversion of cloud drops to raindrops, which is discussed in this subsection.
Experiment C3C, in contrast to C3, is designed to show the formation of raindrops by drop collection. C3C takes the same setup as C3 except for incorporating drop collection, where the collection kernel is computed with the drop terminal velocity of Beard (1976) and the collection efficiency of De Almeida (1977) complemented by Mason (1971).
Its modeled drop spectrum is displayed in Fig. 8, showing that raindrops form in ∼15 min.1 Since the condensational formation of CCIs is weak during the rain initiation (as shown in Fig. 6), the CCIs that grow to raindrops originate in the self-collection of cloud drops, indicating that the collision-induced conversion of cloud drops to raindrops is more important than the condensational conversion in the experiment.
As in Fig. 5, but for Exp. C3C that incorporates drop collection.
Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0017.1
c. Sensitivity to CCN at high vertical velocity
To study the sensitivity of rain initiation to CCN and w, another 3 × 9 × 2 experiments are carried out that take the same setup as C3C except for w and CCN. Their parameters are the same as those in Fig. 7. That is, w = 1, 3, or 5 m s−1; msmax = 10−13, 3 × 10−13, 10−12, 3 × 10−12, 10−11, 3 × 10−11, 10−10, 3 × 10−10, or 10−9 g; k = 0.4/NCCN1 = 100 cm−3, or k = 0.9/NCCN1 = 500 cm−3. Their modeled time of rain initiation is displayed in Fig. 9, where rain initiation is defined as the rainwater content (or the water content of raindrops with r > 100 μm) above 10−3 g m−3.
Time of rain initiation vs the maximum salt mass msmax at vertical velocity w = 1 (blue), 3 (red), and 5 m s−1 (green) with (left) marine and (right) continental CCN. Symbols (circle, cross, and plus) correspond to numerical experiments. All the experiments take the same setup as their counterparts in Fig. 7 except for incorporating drop collection. For comparison, the dashed lines are copied from Fig. 7 to approximately represent the time of condensation-induced CCI initiation.
Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0017.1
Figure 9 shows that rain initiates earlier with increasing w and msmax. In addition, rain with marine CCN initiates earlier than that with continental CCN. In general, rain can initiate within half an hour or less, which agrees with the radar observations of rain initiation in cumulus clouds (Squires 1958; Saunders 1965).
The sensitivity of rain initiation to CCN is attributed mainly to the sensitivity of CCI initiation to CCN because CCIs collect cloud drops and subsequently become raindrops quickly. The sensitivity is explained by comparing two kinds of CCI formations: the condensational conversion and the collision-induced conversion of cloud drops to CCIs. In the experiments in Fig. 9, the condensational conversion is more sensitive to CCN than the collision-induced conversion, although they are mixed. To separate the two conversions, the experiments in Fig. 7 are revisited that exclude drop collection and consequently possess only the condensational conversion. Owing to the similarity in the experiments between Figs. 7 and 9 except for drop collection, the condensational conversion in the experiments in Fig. 7 can be substituted for its counterpart in the experiments in Fig. 9 approximately.
Suppose that the CCI initiation time in Fig. 7 represents the condensational CCI formation in Fig. 9. Thus, the (positive) difference between the CCI initiation time in Fig. 7 (or the dashed lines in Fig. 9) and the rain initiation time in Fig. 9 can be used to infer the collision-induced CCI formation in the experiments in Fig. 9. When the CCI initiation time in Fig. 7 is much shorter (or longer) than the rain initiation time in Fig. 9, CCIs form mainly by the condensational growth of cloud drops (or the collection between cloud drops). Hence the difference between the CCI initiation time (dashed lines) and the raindrop initiation time (solid lines) in Fig. 9 indicates that the condensational conversion of cloud drops to CCIs is more important than the collision-induced conversion when the maximum salt mass msmax > ∼1.5 × 10−10 g (or dry sea-salt particle radius >2.5 μm).
In Fig. 9, the rain initiation is not as sensitive to CCN as the condensation-induced CCI initiation because of coexistence of the collision-induced conversion and the condensational conversion of cloud drops to CCIs. Since the collision-induced conversion “dilutes” the condensational conversion, the sensitivity of rain initiation to CCN is reduced, which is understood from two sides. On one side, condensation not only brings about CCIs but also accelerates the collision-induced conversion via collision efficiency. The collision efficiency between two drops increases markedly as the size of the collector drop (with radius ∼ 20 μm) increases (Mason 1971; De Almeida 1977). Since condensation increases the size of drops and thus the collision efficiency between drops, condensation accelerates the collision-induced conversion via collision efficiency and subsequently contributes to the sensitivity of rain initiation to CCN.2 On the other side, the collision-induced conversion is proportional to the square of LWC (Srivastava 1971; Zeng and Li 2020) and LWC increases mainly with w. As a result, high w brings about strong collision-induced conversion so that the sensitivity of rain initiation to CCN looks small relative to the sensitivity of condensation-induced CCI initiation to CCN.
d. Sensitivity to CCN at low vertical velocity
To reveal the sensitivity of rain initiation to CCN at low w, 9 × 2 experiments are carried out that take the same setup as C3C except for w = 0.1 m s−1 and CCN. They use the same CCN parameters as those in Fig. 9 except with different vertical velocity. Their modeled time of rain initiation is displayed in Fig. 10, show that rain initiation is quite sensitive to CCN.
Time of rain initiation vs the maximum salt mass msmax at vertical velocity w = 0.1 m s−1 with marine (blue) and continental (red) CCN. Symbols (circle and plus signs) correspond to numerical experiments. The two dashed lines represent the time of CCI initiation with marine (blue) and continental (red) CCN, respectively, which are obtained from the numerical experiments with no drop collection.
Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0017.1
To show whether the condensational conversion plays an important role in rain initiation, another 9 × 2 experiments are carried out that take the same setup as C3C and the other experiments in Fig. 10 except for no drop collection. Their time of CCI initiation is displayed by the dashed lines in Fig. 10, approximately representing the time of condensation-induced CCI initiation in the counterparts with drop collection. The difference in time between dashed and solid lines with the same color in Fig. 10 indicates that the condensational conversion is more important than the collision-induced conversion when the maximum salt mass msmax > ∼4 × 10−11 g (or dry sea-salt particle radius > 1.4 μm).
At msmax < 4 × 10−11 g, the condensational conversion still contributes to rain initiation and consequently brings about the high sensitivity of rain initiation to CCN. To be specific, the condensation-induced shifting of cloud drop spectrum peak toward raindrops (see Fig. 5) accelerates the collision-induced conversion via collision efficiency (see footnote 2 for explanation). This effect of condensation on rain initiation is relatively important at low w because low w brings about low LWC that in turn brings about relatively low collision-induced conversion. In addition, the effect is embodied via another form of the condensational conversion of cloud drops to CCIs: two small cloud drops collide to form a large new cloud drop that in turn grows to a CCI via condensation. Obviously, this form of the condensational conversion is different from the form that a cloud drop grows directly to a CCI via condensation. Although this form of the condensational conversion is not represented by the dashed lines in Fig. 10, it does contribute to the sensitivity of rain initiation to CCN at ms < 4 × 10−11 g in Fig. 10, explaining the sensitivity of rain initiation to msmax at msmax < 4 × 10−11 g.
5. Discussion
The modeled sensitivities of rain initiation to CCN provide clue to parameterize warm rain initiation. After CCIs are grouped as a part of rainwater (Fig. 1), the bulk conversion of cloud water to rainwater is expressed in terms of three parameterizations: the autoconversion, the accretion, and the condensational conversion of cloud water to rainwater (see the three red arrows between cloud water and rainwater in Fig. 1). Since the accretion works only after rain initiation, the autoconversion and condensational conversion during rain initiation are compared with the aid of Figs. 9 and 10. Roughly speaking, the dashed lines in Figs. 9 and 10 provide information on the condensational conversion in Fig. 1; the difference in time between the dashed and solid lines provides information on the autoconversion. The two conversions, as shown in Figs. 9 and 10, have different importance in different cases. The autoconversion and condensational conversion are dominant when the maximum salt mass msmax are small and large, respectively.
The separation between the small and large msmax depends on vertical velocity and the drop spectrum at cloud base (or initial drop spectrum). The separation shifts to lower msmax with decreasing w (see Figs. 9 and 10). It also shifts to lower msmax with increasing the relative humidity (RH) of the equilibrium of initial large drops (see the difference in initial drop spectrum between Figs. 3 and 5 whose initial large drops are at equilibrium with water vapor RH = 100% and 90%, respectively; the initial drop spectrum in Fig. 5 is too low to be displayed and thus only the drop spectrum at 1 min is displayed instead). Specifically, when initial large drops are at equilibrium with water vapor at RH = 90% (see section 4), the small and large msmax is separated at salt particle mass ∼1.5 × 10−10 g (or dry sea-salt particle radius ∼ 2.5 μm) with vertical velocity around 1 m s−1 and salt particle mass ∼ 4 × 10−11 g (or dry sea-salt particle radius ∼ 1.4 μm) with a vertical velocity of 0.1 m s−1.
In contrast, when initial large drops are at equilibrium with water vapor at RH = 100% (see section 3), CCIs are initiated at cloud base (see Fig. 3) and consequently the small and large msmax is separated at salt particle mass < 6 × 10−12 g (or dry sea-salt particle radius < 0.86 μm), because a drop with radius 28 μm and salt mass 6 × 10−12 g is at equilibrium with water vapor at RH = 100%. Roughly speaking, the small and large msmax is separated at salt particle radius 0.5 μm, which is estimated after taking account of another process: two small cloud droplets collide to form a large new droplet that in turn grows to a CCI via condensation (please note that the process contributes to the solid lines but not the dashed lines in Figs. 9 and 10).
When msmax is large, the condensational conversion includes the contribution of giant CCN (GCCN), where the contribution is embodied via a series of processes: GCCN grow via condensation to CCIs that in turn collect cloud drops to become raindrops. Obviously, the condensational conversion is much more effective than the process that GCCN grow directly to raindrops via condensation (Jensen and Nugent 2017).
The sensitivity of rain initiation to the maximum salt mass msmax in Figs. 9 and 10 makes sense. Consider the tail of drop spectrum at radius > 28 μm and at 20 min in Fig. 8. Since the vertical axis of Fig. 8 is scaled logarithmically, the tail is small. However, it is the small tail that leads to rain initiation. If additional salt particles with mass ∼ msmax and low number concentration were introduced, the tail would become large and thus rain be initiated faster. In short, the tail of drop spectrum for rain initiation is affected by the tail of salt particle spectrum at mass ∼ msmax.
The maximum salt mass msmax corresponds to the minimum Sw for CCN activation in the Twomey relationship. When Sw is below the minimum Sw, no drops grow from CCN in a cloud chamber, indicating no salt particles with mass > msmax. The CCN observations show that both the minimum Sw and msmax vary greatly from case to case (Woodcock 1953; Jiusto 1967).
The introduction of msmax does not violate the Twomey relationship. When msmax is large, (6) can be used to describe the existence of GCCN (Woodcock 1953; Jensen and Nugent 2017), small, nonexistence of GCCN. Since msmax is an independent variable to describe large CCN in the framework of the Twomey relationship, it can be introduced into the process parameterizations to better represent the effect of CCN on rain initiation.
6. Conclusions
A bin model is developed to accurately simulate the sensitivities of rain initiation to CCN. It uses Eq. (6) to represent CCN whose parameters come from the observed Twomey relationship or CCN measurements. It also uses thousands of bins to seamlessly integrate CCN activation and drop collection, providing a benchmark to test the process parameterizations in rain initiation.
The model is used to simulate two extreme cases with CCN parameters of maritime and continental clouds (i.e., k = 0.4/NCCN1 = 100 cm−3, and k = 0.9/NCCN1 = 500 cm−3), respectively. Since other actual cases usually lie between these two extreme cases (Jiusto 1967), it is inferred that other actual cases possess rain initiation time between those of the two extreme cases. Hence, the modeling results of the two extreme cases provide climatological information on rain initiation.
The model results (or Fig. 9) show that rain can initiate within half an hour or less as observed in cumulus clouds (Squires 1958; Saunders 1965). They (or Fig. 10) also show that rain initiation is sensitive to CCN. This fast rain initiation modeled and the sensitivity of rain initiation to CCN come from the effective cooperation between CCN activation and drop collection, and the effective cooperation is embodied mainly by a new process: the condensational conversion of cloud drops to raindrops via CCIs.
The condensational conversion of cloud drops to raindrops via CCIs is independent of the autoconversion of cloud water to rainwater (see Fig. 1). It consists of two steps: cloud drops grow to CCIs by condensation first and then CCIs grow quickly to raindrops by accreting cloud drops. The condensational conversion and autoconversion, as shown by the bin model simulations, have different importance in different cases. Roughly speaking, the condensational conversion is more important than the autoconversion when large CCN exists (or the dry salt particle radius of the largest CCN with mass msmax > ∼0.5 μm); otherwise, the autoconversion is more important (see Figs. 6 and 8).
Since the condensational conversion has not been represented by the current parameterization schemes, its parameterization will be developed based on the bin model simulations. Theoretically, the parameterization can benefit the weather and climate models for better representing CCN especially large CCN. To be specific, the current weather and climate models do not represent clouds well. They have a bias of “too dense clouds” (Nam et al. 2012). If a new process was introduced to convert the excessive cloud water to rainwater, the bias would be mitigated because rainwater can deposit to the ground quickly. A candidate of the new process is the condensational conversion of cloud drops to raindrops via CCIs (see Fig. 1), which is tested herein via a simple case. The present bin model, in contrast to Berry and Reinhardt (1974), represents the condensational conversion of cloud drops to raindrops via CCIs. As a result, it replicates the fast rain initiation observed (i.e., rain can initiate within half an hour or less; see Fig. 9). The difference in rain initiation between the present bin model and Berry and Reinhardt (1974) suggests that, if a weather/climate model properly represents the condensational conversion of cloud drops to raindrops via CCIs, its bias of “too dense clouds” will be mitigated effectively.
The rapid fluctuation of drop spectrum near 25 μm at 20 min in Fig. 8 comes from the discontinuous growth of drops as well as the discontinuity of initial CN spectrum at ms = msmax. A drop with mass 10, for example, collects another drop with mass 9, forming a new drop with mass 19. As a result, its mass jumps from 10 to 19, manifesting a discontinuous increase in drop mass. The discontinuous drop growth brings about the rapid fluctuation near 25 μm via the discontinuity of initial drop spectrum at ms = msmax or bin 2048 that is caused by the observed discontinuity of CN spectrum at ms = msmax (Woodcock 1953; Jiusto 1967).
The effect of collision efficiency on rain initiation can be understood with the similarity between the collection kernel and time in the Eulerian system of SCE (Srivastava 1988). When the collection kernel is doubled, for example, the time for rain initiation is halved. Since the collection kernel is proportional to the collision efficiency, an increase in collision efficiency, caused by the condensation-induced shifting of drop spectrum peak toward raindrops (see Fig. 5), can accelerate the accretion of cloud droplets by relatively large droplets and CCIs and subsequently accelerate rain initiation.
Acknowledgments.
The authors are grateful to Drs. Gail Skofronick Jackson and David Considine at NASA headquarters for their support of this research. They also thanked the editor, Dr. Lorraine Remer, and three anonymous reviewers for their constructive comments and Mark Gatlin for reading the manuscript. The research was supported by the NASA Grants NNX16AE24G, NNX16AM06G, 80HQTR18T0100, and 80NSSC19K0738.
Data availability statement.
The datasets of modeled rain and CCI initiation time are available at https://osf.io/4tsn3/.
APPENDIX
Implicit Integration of the Condensational Growth Equation
When r is small, the surface and solute terms in (A2) are large and thus their r is represented on time level n + 1. Even though so, if the time step Δt is too large, (A2) may have two solutions: one for haze droplet and the other for cloud drop. To avoid an artificial jump from a haze droplet to a cloud drop, a small value of Δt (e.g., 0.05 s) is used so that CCN activation is simulated accurately.
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