Abstract

Using a nonhydrostatic model with a double-moment bulk cloud microphysics scheme, the authors introduce an aerosol effect on a convective cloud system by accelerating the condensation and evaporation processes (the aerosol condensational effect). To evaluate this effect, the authors use an explicit condensation scheme rather than the saturation adjustment method and propose a method to isolate the aerosol condensational effect. This study shows that the aerosol condensational effect not only accelerates growth rates but also increases cloud water, even though the degree of the acceleration of evaporation exceeds that of condensation. In the early developing stage of the convective system, increased cloud water is, in turn, linked to ice-phase processes and modifies the ice water path of anvil clouds and the ice cloud fraction. In the mature stage, although the aerosol condensational effect has a secondary role in dynamical feedbacks when combined with other aerosol effects, the degree of modulation of the cloud microphysical parameters by the aerosol condensational effect continues to be nonnegligible. These findings indicate that feedback mechanisms, such as latent heat release and the interaction of various aerosol effects, are important in convective cloud systems that involve ice-phase processes.

1. Introduction

In climate systems, the aerosol indirect effects play an important role through the radiation budget. For example, when a certain amount of cloud water is present, the scattering cross section of a cloud increases as the number concentration of cloud condensation nuclei (CCN) increases (the aerosol albedo effect; Twomey 1977). In addition, it has been suggested that CCN loading increases the lifetime of a cloud by preventing cloud droplets from growing through collision and coalescence. As a result, the additional suspended cloud water increases the level of the cloud radiative forcing (the aerosol lifetime effect; Albrecht 1989). Although many researchers have attempted to implement these aerosol indirect effects in general circulation models (GCMs), many uncertainties remain owing to the limitation of GCMs (Lohmann and Feichter 2005). One difficulty associated with the estimation of the aerosol indirect effects is due to the insufficient understanding of cloud microphysics and feedbacks to cloud dynamics, although observational data have shown evidence of a linkage between convective clouds and aerosols (e.g., Koren et al. 2005; Lin et al. 2006). A more precise treatment of the effects of aerosols on convective clouds has proven challenging in climate studies. For this purpose, higher-resolution cloud-resolving models (CRMs) can be used instead of GCMs, which cannot explicitly resolve convective cloud systems owing to their coarse spatial resolution and larger temporal step length.

Recently, CRMs with more advanced cloud microphysics schemes have been used to investigate the mechanisms of aerosol effects on convective clouds (e.g., Khain et al. 2004; Khain and Pokrovsky 2004; Wang 2005; Seifert and Beheng 2006b; Tao et al. 2007; Khain et al. 2008; Lee 2012). These studies have discussed the role of the warm-phase collisional processes in feedback mechanisms in convective clouds and have provided insights in secondary cloud development, which involves ice-phase processes. The basic idea of these feedback mechanisms is similar to the delayed-precipitation mechanism that was proposed by Khain et al. (2004) and Khain and Pokrovsky (2004). Under low CCN conditions, cloud droplets are likely to be converted into rain droplets in the lower troposphere. The resulting increase in precipitation induces a stronger downdraft via evaporative cooling that enhances the transport of low potential temperature air from the middle levels to low levels. These effects result in a decrease in convective updraft just below the initial convection. In contrast, under high-CCN conditions, a number of small droplets are transported upward and water vapor is condensed into cloud droplets for a long time with a small amount of precipitation. As a result, CCN loading increases latent heat release in the convective core, and additional latent heat release is obtained owing to freezing if the top of the cloud reaches above the freezing level. Hence, it has been suggested that higher CCN conditions are more favorable than lower CCN conditions to the invigoration of convective clouds that involve ice-phase processes. In this respect, the delayed-precipitation mechanism through which convective clouds are enhanced by higher CCN loading is regarded as a type of the aerosol lifetime effect. Although many researchers have explored this concept, the contributions of aerosol effects via other microphysical processes have not been discussed as intensively.

In addition to the hypothesis based on the microphysical aspect, numerous studies using numerical simulations have revealed that the effects of aerosols on the invigoration of convective clouds are easily changed by the environmental conditions (Seifert and Beheng 2006b; Khain et al. 2008; Lee et al. 2008; Fan et al. 2009; Khain 2009). Seifert and Beheng (2006b) used a double-moment bulk cloud microphysics scheme to show that the degree and even the sign of the relative changes in the accumulated precipitation amount and the maximum updraft velocity differ by the environmental vertical wind shear and the convective available potential energy (CAPE) in an isolated convective cloud system. In general, an increase in aerosols decreases (increases) the strength of convection under lower (higher) CAPE conditions, and an increase in the vertical wind shear decreases the strength of the invigoration of convection by high aerosol loading under lower CAPE conditions. This feature is supported by the results reported by Fan et al. (2009), who simulated various vertical shear cases using a spectral bin cloud microphysical scheme. In addition, it is suggested that, depending on the strength of the wind shear, there is an optimal CCN amount that gives a local maximal value of precipitation amount to deep convective clouds. This means that the aerosol effects could differ depending on the CCN amount even under the same environmental conditions. Further, Lee et al. (2008) showed that the interaction between low-level convergence and evaporative cooling becomes strong under high CCN conditions and that the secondary organization of convective clouds is supported by aerosol effects. Khain et al. (2008) attempted to classify preceding studies on the sensitivity of aerosols to the precipitation amount according to the balance between the changing gain and loss of condensate by aerosol loading: aerosol loading provides more precipitation when the gain exceeds the loss at each environmental condition.

In this study, to isolate the effects of aerosols on convective clouds, we focused on the condensation and evaporation processes. Condensation and evaporation of cloud droplets in the bulk cloud microphysics schemes of CRMs are generally solved using the saturation adjustment method (e.g., Kessler 1969; Lin et al. 1983): if an air parcel becomes supersaturated, it is instantaneously adjusted to the saturated value. However, it is known that the saturation adjustment method sometimes causes a nonnegligible error in the condensational growth. Kogan and Martin (1994) showed that the saturation adjustment method can overestimate the amount of condensation and evaporation, and this error increases as the CCN amount decreases. In other words, the dependence of condensational growth on the CCN amount indicates the possibility of an additional aerosol indirect effect (denoted the aerosol condensational effect). In particular, it has been shown that the CCN loading modifies the cumulative condensation to induce further invigoration through the freezing of cloud water (Khain et al. 2008; Lee 2012). In addition, it is important to investigate the role of the aerosol condensational effect, particularly in terms of the modification of the condensate through the condensation and evaporation processes. The investigation of the aerosol condensational effect could be a key to understanding the linkage between aerosols and dynamical feedback mechanisms. As a first step, we attempted to reveal the aerosol condensational effect on the invigoration of a squall-line system, as an example of deep convective clouds, although the aerosol effects on a convective cloud system is not monotonic and could be strongly affected by environmental conditions and feedbacks from cloud dynamics.

In section 2, we introduce the dependence of the condensation rate on the CCN. In section 3, we introduce a numerical model to investigate the aerosol condensational effect and the numerical settings. Section 4 shows the results and section 5 summarizes and discusses the results.

2. Dependence of the condensation rate on the CCN

In this section, we introduce the dependence of the condensation rate on the CCN. The growth rate of a cloud droplet with droplet mass x due to condensation is obtained by simultaneously solving the thermal diffusion and the vapor diffusion equations:

 
formula

where D is the maximum dimension of the particle; Gliq is a thermodynamic coefficient, which is a function of temperature T and pressure p; Fυ is the ventilation factor; and δliq is the supersaturation of liquid water, which is defined by the deviation of the specific humidity qυ from the saturated specific humidity qυs,liq. By integrating (1) over the particle mass distribution (PMD) fc(x), we derive the condensation rate of the cloud water mixing ratio qc:

 
formula

where ρ is air density, Nc is the number concentration of cloud droplets, Dm,c ≡ 0.124(ρqc/Nc)1/3 is the mean mass diameter, and CPMD is a weighting coefficient in the integration of the PMD. In this study, we assume that Fυ is approximately 1 for cloud water because the ventilation effect becomes quite small when the droplets fall slowly. Thus, the condensation rate is formulated as the consumption of the supersaturation according to the characteristic time scale of condensation τcnd,c.

Figure 1 shows the dependence of τcnd,c on Nc at various values of qc under the assumption that CPMD equals 1 [e.g., the value of CPMD can vary by up to approximately 10% with the typical PMDs of cloud droplets, as suggested by Berry and Reinhardt (1974)]. Under typical maritime conditions (Nc < 100 cm−3), the time scale is several tens of seconds. In contrast, the time scale is less than 10 s in typical urban areas, where Nc > 1000 cm−3. Thus, the supersaturation is not likely to be spontaneously consumed under maritime conditions. Similarly, the evaporation rate could be relatively slow near the cloud edge, where qc and Nc are small.

Fig. 1.

The dependence of the characteristic time scale of condensation on the cloud droplet number concentration with qc = 1 (circle), 0.1 (rectangle), and 0.01 g kg−1 (triangle) under the conditions of T = 293 K and p = 1000 hPa.

Fig. 1.

The dependence of the characteristic time scale of condensation on the cloud droplet number concentration with qc = 1 (circle), 0.1 (rectangle), and 0.01 g kg−1 (triangle) under the conditions of T = 293 K and p = 1000 hPa.

Figure 2 shows the sensitivities of qc and the vertical velocity to various values of fixed Nc determined using a moist adiabatic parcel model with a condensation process that was calculated using a spectral bin method (Bott et al. 1990; Bott 1992). The vertical profile of the environmental conditions was obtained from observations made by the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) on 22 February 1993 (Jorgensen et al. 1997; Fig. 4b), and the parcel rises from the level of neutral buoyancy with vertical velocity w = 0, δliq = 0, and various values of fixed Nc. The vertical velocity is accelerated by the buoyancy B defined as

 
formula

where g is the gravity constant, θ is the potential temperature, and the bar over some of the symbols indicates the horizontal-averaged values of the corresponding variables. In the moist adiabatic process, the amount of qc is potentially determined by the initial condition. However, the timing of the condensation process accelerates as Nc increases and, consequently, the updraft rapidly becomes strong. In addition, the maximum vertical velocity increases with an increase in Nc. The aerosol condensational effect accelerates the hydrological cycle to modify the characteristic time scale of condensation and the vertical motion in the condensation and evaporation processes, although the effect is not sufficiently strong to drastically change the cloud system. These effects would almost saturate at a value of Nc of more than 160 cm−3 in the ascending moist adiabatic parcel model. In three-dimensional simulations, the dynamical effect through buoyancy would generally be reduced by the counteraction owing to the associated vertical perturbation pressure gradient force (Eastin et al. 2005). In addition, the entrainment associated with the positive vertical gradient of buoyancy also reduces the dynamical effect.

Fig. 2.

The time evolution of (a) the mixing ratio of cloud water and (b) vertical velocity in the ascending parcel for different values of Nc.

Fig. 2.

The time evolution of (a) the mixing ratio of cloud water and (b) vertical velocity in the ascending parcel for different values of Nc.

3. Experimental design

a. Nonhydrostatic atmospheric model

Numerical experiments were performed using the Nonhydrostatic Icosahedral Atmospheric Model (NICAM), which was originally designed as a global cloud-resolving model based on an icosahedral grid system (Tomita and Satoh 2004; Satoh et al. 2008). The NICAM can also be used as a three-dimensional nonhydrostatic cloud-resolving model by reducing the radius of the applied model sphere (Satoh and Matsuda 2009). In addition to reducing the radius, the stretched-grid system proposed by Tomita (2008) provides a locally high-resolution domain to simulate an isolated convective cloud system. Figure 3 shows the center of the stretched-grid system at the sixth grid division level and using a stretching factor of β = ⅛ and a radius of approximately 570 km. In this study, we utilized the same physical packages as those used in the global simulations performed by Satoh et al. (2011) and Dirmeyer et al. (2012), with the exception of the cloud microphysics scheme. A double-moment bulk cloud microphysics scheme, the NICAM double-moment bulk scheme with six water categories (NDW6), which predicts both the mixing ratio q and the number concentration N of cloud water, rain, cloud ice, snow, and graupel (denoted by the subscripts c, r, i, s, and g, respectively) was developed to investigate the aerosol effects on cloud microphysics as shown in section 2 (see  appendix A for detail). The turbulent mixing was calculated based on the second-order closure model with moist processes (Nakanishi and Niino 2004; Noda et al. 2010), and the bulk surface fluxes were calculated using the method developed by Louis (1979). The radiative transfer model was a broadband model with 29 spectral bands (Sekiguchi and Nakajima 2008).

Fig. 3.

The horizontal grid resolution in the analyzed model domain.

Fig. 3.

The horizontal grid resolution in the analyzed model domain.

b. Initial conditions

To examine the aerosol condensational effect, we chose the case of a tropical squall line system proposed by the Global Energy and Water Cycle Experiment (GEWEX) Cloud System Study (GCSS) model intercomparison (Redelsperger et al. 2000), which is based on TOGA COARE, as described in section 2. This study used the stretched-grid system to capture the squall-line system in a finer grid region, the finest horizontal resolution of which was approximately 1 km (see Fig. 3), and a model time step of 6 s was used. The vertical grid spacing, the thermodynamical profiles, and the horizontal velocity profiles are shown in Fig. 4. The top of the model domain reaches an altitude of 20 km and covers the tropopause at an altitude of approximately 15 km. Humid air at an altitude below 1 km generates a conditionally unstable atmosphere, and, as a result, the deep convective system reaches the tropopause after the initial perturbations (Fig. 4b). This study used the initial perturbations to induce convective clouds in the horizontally uniform initial conditions during the first 20 min of the simulations, as described by Redelsperger et al. (2000), when four dry cold bubbles with a horizontal radius of 6.5 km are linearly located north and south at an interval of 15 km:

 
formula

where , xc,i = 60 000 m, and yc,i = 60 000 + (I − 2.5) × 15 000 (i = 1, 2, 3, 4) m. The cold bubbles are uniform within the circle of normalized radius rp = 1 and at an altitude of less than 2.5 km. The system of the squall line moves rapidly owing to the low-level jet (Fig. 4c). To maintain the system centered on the model domain, we used a moving frame to shift the horizontal velocities uniformly as u = u – 12 m s−1 and υ = υ + 2 m s−1. This moving frame is active in the dynamics but inactive in the boundary layer scheme because the surface friction decreases the absolute value of the wind speed. The lower boundary condition is nonslip, the upper boundary condition is free slip, and Rayleigh damping was applied to the top layer of model. The bulk surface fluxes were calculated assuming the sea surface in Louis’s scheme, and the sea surface temperature was fixed at 300 K. The solar incidence was fixed to 1365 W m−2 at all location in the nadir. The Coriolis force was omitted.

Fig. 4.

(a) Vertical grid spacing, (b) the vertical profiles of initial potential temperature and equivalent potential temperature, and (c) the horizontal components of wind speed.

Fig. 4.

(a) Vertical grid spacing, (b) the vertical profiles of initial potential temperature and equivalent potential temperature, and (c) the horizontal components of wind speed.

c. Sensitivity experiments

In this study, we considered aerosols only as precursors of cloud droplets and ignored other aerosol processes. We assumed a traditional formulation of the aerosol activity spectrum as follows:

 
formula

where Cccn and κccn are measured constants, ssw = (qυ/qυs,liq – 1) × 100% is equal to the supersaturation of liquid water, and Nccn (cm−3) is the number concentration of CCN at the supersaturation of ssw (Twomey 1959). Because the aerosol activity spectrum is a function of the supersaturation and unbounded by the total aerosol number concentration, we chose an upper limit of activated aerosols of 1.5 × Cccn, similarly to that chosen by Seifert and Beheng (2006a, hereafter SB06): the maximum activated aerosol number concentration is 1.5-fold higher than the activated aerosol number concentration at ssw = 1%. With the aerosol activity spectrum, we applied the approximated aerosol nucleation scheme (Rogers and Yau 1989) including the effect of turbulence (Lohmann 2002):

 
formula
 
formula
 
formula

where weff is the effective vertical velocity, TKE is the subgrid turbulent kinetic energy, and cpa is the specific heat of the atmosphere at constant pressure. Here, we note that (4), (6), and (7) are described in cgs units following their original formulations, and (5) is then calculated in mks. The nucleation scheme only works below the layer in which the first local maximum of Nc,new from the ground appears in each column, assuming that nucleation mainly occurs near the cloud base. This simple approach ignores the spatial distribution of aerosols and in-cloud activation.

We set the control experiment (CTL) as the typical maritime CCN condition with Cccn = 100 cm−3 and κccn = 0.462, following the protocol described by Khain et al. (2001). We performed the sensitivity experiments using the same settings as those used for the CTL, except that we set Cccn = 200, 400, and 800 cm−3 (CCN2, CCN4, and CCN8 experiments, respectively) to estimate the total aerosol effects including the warm-phase collisional processes (see  appendix B). In this study, we propose a simple approach for the evaluation of the aerosol condensational effect. The condensational growth rate contains the dependence on CCN through the cloud number concentration and the mean mass diameter [Dm,c = 0.124(ρqc/Nc)1/3]. Assuming that the number concentration of cloud water is proportional to the number concentration of CCN, we can define a factor Accn that accelerates the condensation process by modifying the CCN through the factor Fccn to approximately isolate the aerosol condensational effect:

 
formula

We then performed additional sensitivity experiments using the same settings as the CTL, except for setting Fccn = 2, 4, and 8 (TAU2, TAU4, and TAU8, respectively) to estimate the aerosol condensational effect alone. Similarly, we performed sensitivity experiments in which the total aerosol effects, with the exception of the aerosol condensational effect, were considered by choosing Cccn = 200 with Fccn = ½, CCN = 400 with Fccn = ¼, and Cccn = 800 with Fccn = ⅛ (MTAU2, MTAU4, and MTAU8, respectively). In addition, we performed an experiment using a single-moment bulk cloud microphysics scheme, as described by Lin et al. (1983), as a reference for the CTL (LIN). All of the experimental settings are summarized in Table 1.

Table 1.

Microphysical settings of each experiment.

Microphysical settings of each experiment.
Microphysical settings of each experiment.

4. Results

a. Overview of the CTL experiment

The development of the simulated squall line system in the CTL experiment is shown in Fig. 5. During the first 20 min, the initial perturbations by four cold bubbles induce a cold pool, as shown by the isoline of the virtual potential temperature depression. Subsequently, moist warm inflow in the lower troposphere lifts up at the upwind edge of the cold pool to reach the tropopause after a few hours, as shown in Fig. 5b. The anvil cloud formation provides further precipitation along the squall line, and the air mass behind the convective cells is cooled by evaporation. Thus, the cold pool is enhanced by precipitation, and the convective system is self-sustained and can develop for several hours. These features are known as typical long-lived squall lines (Rotunno et al. 1988).

Fig. 5.

(a) The time series of surface precipitation (mm h−1; shading), the horizontal wind velocity (m s−1; vectors) at the level of z = 35 m, and the isoline of the virtual potential temperature of 1 K below the domain-averaged value at the level of z = 35 m (contour) for (left to right) t =1, 3, and 5 h. (b) The (left to right) time series of total hydrometeor content (kg kg−1; shading) averaged along the distance from the leading edge of the squall line and vertical wind (vectors). The vectors show the composition of an anomaly of the zonal wind (m s−1) and the vertical wind (m s−1) scaled by multiplying the vertical wind speed by 10 from the horizontal mean.

Fig. 5.

(a) The time series of surface precipitation (mm h−1; shading), the horizontal wind velocity (m s−1; vectors) at the level of z = 35 m, and the isoline of the virtual potential temperature of 1 K below the domain-averaged value at the level of z = 35 m (contour) for (left to right) t =1, 3, and 5 h. (b) The (left to right) time series of total hydrometeor content (kg kg−1; shading) averaged along the distance from the leading edge of the squall line and vertical wind (vectors). The vectors show the composition of an anomaly of the zonal wind (m s−1) and the vertical wind (m s−1) scaled by multiplying the vertical wind speed by 10 from the horizontal mean.

The time series of the domain-averaged surface precipitation in the CTL and LIN experiments are shown in Fig. 6. Hereafter, the results were analyzed within a 400-km radius from the center of the domain. Following the method described by Redelsperger et al. (2000), surface precipitation was categorized into convective precipitation and stratiform precipitation: the convective grid points exhibit precipitation rate more than 20 mm h−1, and the convective grid points are the eight surrounding grid points of the convective cores where precipitation rates are more than 4 mm h−1 and twofold higher than the precipitation rate averaged over the 24 surrounding grid points. At the initiation of the squall line system, the contribution of the surface precipitation originates mostly from convective cores. The total precipitation in the CTL experiment is less than that in the LIN experiment, and the difference between the systems gradually increases as the squall-line system develops. Correspondingly, after two hours, the ratio of the convective precipitation to the total precipitation in the CTL experiment gradually decreases. In contrast, the convective precipitation in the LIN experiment almost predominates in the total precipitation. The ratio of the convective precipitation to the total precipitation in the CTL experiment is approximately 70% after 3 h. This feature agrees well with the observations reported by the TOGA COARE (Short et al. 1997) and the previous work performed by Redelsperger et al. (2000).

Fig. 6.

The time series of the domain-averaged total precipitation (lines without symbols) and 0.1 × the ratio convective precipitation/total precipitation (lines with symbols) in the CTL (solid lines) and LIN (dashed lines) experiments.

Fig. 6.

The time series of the domain-averaged total precipitation (lines without symbols) and 0.1 × the ratio convective precipitation/total precipitation (lines with symbols) in the CTL (solid lines) and LIN (dashed lines) experiments.

b. Verification of the method to isolate the aerosol condensational effect

Figure 7 shows the instantaneous results of the vertical profiles of Nc, τcnd, and δliq under various CCN conditions. Many cloud droplets are activated and transported upward by strong updraft; correspondingly, τcnd decreases with height. In contrast, near the cloud bottom, the rain droplets increase in size by consuming cloud droplets, and τcnd becomes larger than 1 min. The comparison of the results of the CCN8 experiment with the results of the CTL experiment revealed that the aerosol effect on τcnd is apparent along convective cores where aerosols are easily activated. Corresponding to the amount of Cccn in the nucleation scheme, Nc in the CCN8 experiment is approximately eightfold higher than Nc in the CTL experiment, and the supersaturation in the CCN8 experiment is smaller than that in the CTL experiment. In contrast, there is less modification of τcnd near the cloud bottom. Interestingly, there are negligible differences in the number concentration of rain between the CTL and the CCN8 experiments.

Fig. 7.

Vertical cross sections of (a),(b) the number concentrations of cloud droplets (cm−3; shading), rain droplets (L−1; contours), and wind vectors for (left) the CTL and (right) CCN8 experiments along y = 70 km at t = 1 h. (c)–(f) As in (a),(b), but for the time scales of condensation (s) and liquid supersaturation (%), respectively The dashed lines show the isoline of qc = 10−6 kg kg−1 as an indicator of cloud boundary, and the vectors show the composition of the zonal wind (m s−1) and the vertical wind scaled by multiplying by 10 (m s−1).

Fig. 7.

Vertical cross sections of (a),(b) the number concentrations of cloud droplets (cm−3; shading), rain droplets (L−1; contours), and wind vectors for (left) the CTL and (right) CCN8 experiments along y = 70 km at t = 1 h. (c)–(f) As in (a),(b), but for the time scales of condensation (s) and liquid supersaturation (%), respectively The dashed lines show the isoline of qc = 10−6 kg kg−1 as an indicator of cloud boundary, and the vectors show the composition of the zonal wind (m s−1) and the vertical wind scaled by multiplying by 10 (m s−1).

We then examined the strength of the aerosol condensational effect on the growth rate and verified the method to isolate the aerosol condensational effect given by (8). The results were analyzed statistically within a 400-km radius from the center of the domain (see Fig. 3). Figures 8a and 8b show the domain-averaged condensation and evaporation rates sorted by τcnd in the CTL, CCN2, CCN4, and CCN8 experiments. The condensation and evaporation rates of cloud water exhibit distinct peaks at τcnd values of less than 10 s. The aerosol condensational effect is evident in the decrease in the mode values of the time scales of the condensation and evaporation of cloud water, which decrease as Cccn increases. In contrast, the mode values of the time scales of the condensation and evaporation of rain have almost no dependence on Cccn. The collisional processes relax the aerosol condensational effect on precipitating particles by consuming the population of droplets, as shown in Figs. 8a and 8b. Furthermore, the total amount of evaporated rain mass decreases slightly as Cccn increases (Fig. 8b). This is explained as follows: aerosols affect the collisional processes to prevent cloud droplets from growing into rain, and the total amount of rain itself decreases as Cccn increases (as will be shown later). The deposition rates of cloud ice, snow, and graupel are affected by the aerosol condensational effect through the Bergeron–Findeisen process, although the extent of the impact is limited in the convective cores, and the total amount of their growth rates are relatively small compared with those of cloud water (results are not shown). Figures 8c and 8d show the averaged condensation and evaporation rates of cloud water and rain in the CTL, TAU2, TAU4, and TAU8 experiments. The simple estimation of the aerosol condensational effect through the acceleration of the condensation and evaporation rates in the TAU2, TAU4, and TAU8 experiments presented here leads to the modification of the mode value of the condensation and evaporation rates by CCN loading. In addition, there exist differences between the results in the series of CCN experiments and TAU experiments. An increase in cloud water by the aerosol effect on the collisional processes leads to an increase in the total amount of evaporated cloud water. In addition, the increased cloud water, which can undergo gradual cloud growth, results in a wide range in the time scale distribution of the condensation and evaporation rates near its mode value. Thus, the shape of the time scale distribution of the condensation and evaporation rates in the series of CCN experiments are not exactly reproduced in the series of TAU experiments owing to other aerosol effects. In the MTAU2, MTAU4, and MTAU8 experiments, the change in the mode of the condensation and evaporation rates are quite small, and the total evaporation rates of rain decreases as Cccn increases (Figs. 8e and 8f).

Fig. 8.

The time-domain-averaged (left) condensation and (right) evaporation rates vs the time scale of condensation within a 400-km radius from the center of the domain over the entire simulation period for CTL and (a),(b) CCN2, 4, and 8; (c),(d) TAU2, 4, and 8 (e),(f) MTAU2, 4, and 8 experiments. Solid lines represent the contributions of cloud water and dashed lines represent the contributions of rain.

Fig. 8.

The time-domain-averaged (left) condensation and (right) evaporation rates vs the time scale of condensation within a 400-km radius from the center of the domain over the entire simulation period for CTL and (a),(b) CCN2, 4, and 8; (c),(d) TAU2, 4, and 8 (e),(f) MTAU2, 4, and 8 experiments. Solid lines represent the contributions of cloud water and dashed lines represent the contributions of rain.

c. The aerosol condensational effect on the convective system

Figure 9 shows the time series of the cumulative precipitation (CP) in the analysis domain in the CTL, TAU8, CCN8, and MTAU8 experiments. The aerosol condensational effect increases the precipitation, and the intensity increases as the cloud system develops. Interestingly, in the CCN8 experiment, the total aerosol effects decrease the precipitation slightly, even though the increase in the CP in the TAU8 experiment is larger than the decrease in the CP observed in the MTAU8 experiment. The total aerosol effects consist of two opposing effects on the precipitation, and their behaviors are nonlinear. Figure 10 shows the time series of the liquid water path (LWP) and the ice water path (IWP). There exists distinct diversity in the IWP compared with the LWP among the experiments, particularly after 2 h, when the diversity in the CP becomes large. This finding suggests that the hydrological cycle is affected by the amount of aerosols involved in ice-phase processes. Figure 11 shows the time series of the domain-averaged cumulative production rate of the IWP. The diversity among experiments is mainly caused by riming of graupel. The aerosol condensational effect induces additional precipitation through the production of graupel, which efficiently removes hydrometeors from the atmosphere. The ice water path in the CCN8 and MTAU8 experiments increases such that it becomes larger than the IWP in the CTL and TAU8 experiments after 4 h. This result indicates that the aerosol effect via the collisional process is effective later and longer compared with the aerosol condensational effect because the time scale of the collisional processes is approximately thousands of seconds or more and the time scale strongly depends on Nc in the collisional processes (see  appendix B). We defined the cloud system before 2 h as the early developing stage and the one after 5 h as the mature stage and then investigated the vertical profiles of hydrometeors at each stage.

Fig. 9.

(a) The time series of the cumulative precipitation in the CTL experiment and (b) deviation of the cumulative precipitation in TAU8, CCN8, and MTAU8 from the CTL experiments.

Fig. 9.

(a) The time series of the cumulative precipitation in the CTL experiment and (b) deviation of the cumulative precipitation in TAU8, CCN8, and MTAU8 from the CTL experiments.

Fig. 10.

The time series of (a) the liquid water path and (b) the ice water path averaged over the domain for the CTL, CCN8, TAU8, and MTAU8 experiments.

Fig. 10.

The time series of (a) the liquid water path and (b) the ice water path averaged over the domain for the CTL, CCN8, TAU8, and MTAU8 experiments.

Fig. 11.

(a) The time series of the column-integrated production term of the ice-phase hydrometeors by riming (thin solid lines), vapor deposition (thick solid lines), and freezing (lines with symbols) averaged over the domain for the CTL, CCN8, TAU8, and MTAU8 experiments. (b) As in (a), but for graupel (thin solid lines), snow (thick solid lines), and cloud ice (lines with symbols) by riming averaged over the domain.

Fig. 11.

(a) The time series of the column-integrated production term of the ice-phase hydrometeors by riming (thin solid lines), vapor deposition (thick solid lines), and freezing (lines with symbols) averaged over the domain for the CTL, CCN8, TAU8, and MTAU8 experiments. (b) As in (a), but for graupel (thin solid lines), snow (thick solid lines), and cloud ice (lines with symbols) by riming averaged over the domain.

Figure 12 shows a comparison of the vertical profiles of hydrometeors averaged for the first 2 h, during which time the convective system is not well organized and precipitation originates mainly from convective cores (Figs. 5 and 6). We found that the aerosol condensational effect increases the mixing ratio of rain rather than the mixing ratio of cloud water below the freezing level (near an altitude of 5 km; Figs. 12a and 12b). In terms of cloud microphysics, increased condensation means an increase in the mixing ratio and also in the mean mass radius rc = [10−3(3/4π)(ρqc/Nc)]1/3, which is important for the initiation of autoconversion ( appendix B). Figure 13 shows the vertical profile of the autoconversion rate (Paut), the sum of the autoconversion and the accretion rates (Pacc), and the number-weighted mean mass radius of cloud water. The aerosol condensational effect increases the autoconversion rate in the TAU8 experiment (Fig. 13a). In the CTL and TAU8 experiments, the mean mass radius of cloud water in most cloudy layers is higher than 15 μm (Fig. 13c), which indicates that the clouds are likely to begin precipitating (Rosenfeld and Gutman 1994; Masunaga et al. 2002; Suzuki et al. 2010). The aerosol condensational effect accelerates Paut by first increasing qc and rc. Subsequently, the higher produced qr enhances accretion, and the accelerated production of qc is balanced with a reduction in qc through autoconversion and accretion. This is opposite to the logic used in the aerosol lifetime effect: CCN loading prevents the autoconversion of cloud droplets by reducing the particle radii as shown in Figs. 13a and 13c. In other words, an increase in the particle radii increases autoconversion and the subsequent processes. In addition, the increased qr results in an increase in qg by freezing and riming above the freezing level in the TAU8 experiment. In contrast, there is a negligible difference in the Paut value between the MTAU8 and the CCN8 experiments. Based on the definition, the mean mass radius becomes insensitive to an increase in qc when Nc is large:

 
formula

Under high-CCN conditions, the aerosol condensational effect produces more rain by increasing qc in accretion rather than rc in autoconversion (Fig. 13). This leads to a small difference in the mixing ratio of graupel between the MTAU8 and the CCN8 experiments (Fig. 12d).

Fig. 12.

Vertical profiles of the mixing ratio of hydrometeors averaged over the domain for the first 2 h for the CTL, CCN8, TAU8, and MTAU8 experiments: (a) qc, (b) qr, (c) qi + qs, and (d) qg.

Fig. 12.

Vertical profiles of the mixing ratio of hydrometeors averaged over the domain for the first 2 h for the CTL, CCN8, TAU8, and MTAU8 experiments: (a) qc, (b) qr, (c) qi + qs, and (d) qg.

Fig. 13.

Vertical profiles of the (a) domain-averaged autoconversion rate, (b) accretion rate, and the (c) number-weighted mean mass radius of cloud water for the first 2 h for the CTL, CCN8, TAU8, and MTAU8 experiments.

Fig. 13.

Vertical profiles of the (a) domain-averaged autoconversion rate, (b) accretion rate, and the (c) number-weighted mean mass radius of cloud water for the first 2 h for the CTL, CCN8, TAU8, and MTAU8 experiments.

In addition to these microphysical effects, there exists the associated dynamical effect, which was introduced in section 2. Figure 14 shows the departure of the heating rates obtained in the condensation and evaporation of cloud water in each experiment from those in the CTL experiment. In contrast to the results from the moist adiabatic parcel model described in section 2, the amount of condensates differs through the aerosol condensational effect. Furthermore, the departure of the net heating rates demonstrates that the aerosol condensational effect reduces cloud water, particularly in the lower troposphere, by enhancing evaporation more than enhancing condensation (Figs. 14b and 14c). Nevertheless, cloud water remains at the almost same amount in the TAU8 and CCN8 experiments compared with the CTL and MTAU8 experiments, respectively (Fig. 12a). This finding indicates the existence of a feedback mechanism involving cloud dynamics through latent heat release. Figure 15 shows the vertical profile of the domain-averaged upward mass fluxes in the early developing stage. Accelerated condensation enhances convection through positive buoyancy, and the effect propagates over the freezing level to impact the ice-phase cloud. Thus, the aerosol condensational effect increases the mixing ratio of the ice-phase hydrometeors, as shown in Fig. 12c, and produces more precipitation involving the ice-phase processes, as shown in Figs. 9 and 10b.

Fig. 14.

(a) The vertical profile of the heating rates (K h−1) due to condensation and evaporation of cloud water in the CTL experiment, and deviation of the heating rates (K h−1) due to (b) condensation and (c) evaporation of cloud water in the CCN8, TAU8, and MTAU8 experiments from the CTL experiment averaged over the domain for the first 2 h.

Fig. 14.

(a) The vertical profile of the heating rates (K h−1) due to condensation and evaporation of cloud water in the CTL experiment, and deviation of the heating rates (K h−1) due to (b) condensation and (c) evaporation of cloud water in the CCN8, TAU8, and MTAU8 experiments from the CTL experiment averaged over the domain for the first 2 h.

Fig. 15.

Vertical profiles of the upward mass flux averaged over the domain for the first 2 h for the CTL, CCN8, TAU8, and MTAU8 experiments.

Fig. 15.

Vertical profiles of the upward mass flux averaged over the domain for the first 2 h for the CTL, CCN8, TAU8, and MTAU8 experiments.

Figure 16 shows a comparison of the vertical profiles of hydrometeors averaged for the last 2 h, at which point the convective system is well organized, and stratiform precipitation increases with the expansion of anvil clouds (Figs. 5 and 6). In the mature stage, the aerosol condensational effect continues to produce more conversion between rain and graupel (Figs. 16b and 16d). However, there is a large difference in the mixing ratio of cloud ice and snow in the upper troposphere, where the aerosol condensational effect has a secondary role. In the CTL and TAU8 experiments, the production of rain and graupel by the aerosol condensational effect removes cloud ice and snow through mixed-phase collection with graupel and rain (Fig. 17). However, under high-Nc conditions as in the CCN8 experiment, the reduction in cloud ice and snow by graupel is less effective because the aerosol condensational effect increases the mixing ratio of cloud water by keeping the mean mass radius small. This could lead to an increase in cloud ice and snow by the freezing of cloud water and the Bergeron–Findeisen process (Fig. 16c). In addition to the microphysical feedback, the upward mass flux just above the altitude of 5 km (near the freezing level) in the TAU8 and CCN8 experiments becomes smaller compared with that in the CTL and MTAU8 experiments (Fig. 18). The aerosol condensational effect on the cloud dynamics in the mature stage becomes different from that in the early developing stage after complicated feedbacks.

Fig. 16.

As in Fig. 12, except averaged for the last 2 h.

Fig. 16.

As in Fig. 12, except averaged for the last 2 h.

Fig. 17.

Vertical profiles of the reduction term of the mixing ratio of cloud ice and snow to produce graupel by the mixed-phase collection processes for the last 2 h for the CTL, CCN8, TAU8, and MTAU8 experiments.

Fig. 17.

Vertical profiles of the reduction term of the mixing ratio of cloud ice and snow to produce graupel by the mixed-phase collection processes for the last 2 h for the CTL, CCN8, TAU8, and MTAU8 experiments.

Fig. 18.

As in Fig. 15, except averaged for the last 2 h.

Fig. 18.

As in Fig. 15, except averaged for the last 2 h.

Figure 19 shows the time-domain-averaged liquid water path of cloud water below the freezing level (LWPc), the ice water path of cloud ice with snow (IWPi+s), the liquid water path of rain plus the ice water path of graupel (LWPr + IWPg), and the ice cloud fraction, which is defined by the cloudy area where the visible ice optical thickness is larger than 0.1 (the typical threshold that can be detected by satellite observations; e.g., Rossow and Schiffer 1999). In the early developing stage, the aerosol condensational effect on the cloud system consistently increases the ice-phase cloud through the intensification of convection and induces additional conversion between rain and graupel. As a result, the ice-phase cloud grows thick and the ice cloud fraction increases to increase ice cloud radiative forcing. When the total aerosol effects are included, the impact of the aerosol condensational effect becomes slightly small and saturates under conditions in which Cccn is higher than 200 cm−3. In contrast, in the mature stage, an increase in the LWPc by the aerosol condensational effect leads to an increase in the LWPr + IWPg and a decrease in the IWPi + IWPs in the series of TAU experiments (Fig. 20). The modulation of these microphysical parameters by the aerosol condensational effect is systematic, although the dynamical feedbacks that occur during the mature stage differ from those in the early developing stage (Figs. 15 and 18). In the series of CCN experiments, the degree of modulation of the cloud microphysical parameters by the aerosol condensational effect is not monotonic to the amount of Cccn, and even the signs change. These complicated results are not explained based on the aerosol condensational effect, as discussed in section 2. In contrast to the role of graupel in the removal of cloud ice and snow in the series of TAU experiments, the modification of the amount of graupel is not directly related with the amount of cloud ice and snow as discussed in the above paragraph. The comparison of the series of CCN and MTAU experiments revealed that a change in cloud water coincides with a change in cloud ice and snow, in rain and graupel, and in the ice cloud fraction. This finding indicates that the feedbacks of the aerosol effects on cloud dynamics promote the whole cloud system to strengthen (weaken) in order to increase (decrease) the hydrometeors in the mature stage.

Fig. 19.

The time-domain-averaged LWPc (g m−2) where (a) T > 273 K, (b) IWPi + IWPs (g m−2), (c) IWPg (g m−2), and (d) ice cloud fraction for the first 2 h. Solid lines show the results from the CTL, CCN2, CCN4, and CCN8 experiments; solid lines with symbols show the results from the TAU2, TAU4, and TAU8 experiments; and dashed lines show the results from the MTAU2, MTAU4, and MTAU8 experiments.

Fig. 19.

The time-domain-averaged LWPc (g m−2) where (a) T > 273 K, (b) IWPi + IWPs (g m−2), (c) IWPg (g m−2), and (d) ice cloud fraction for the first 2 h. Solid lines show the results from the CTL, CCN2, CCN4, and CCN8 experiments; solid lines with symbols show the results from the TAU2, TAU4, and TAU8 experiments; and dashed lines show the results from the MTAU2, MTAU4, and MTAU8 experiments.

Fig. 20.

As in Fig. 19, except averaged for the last 2 h.

Fig. 20.

As in Fig. 19, except averaged for the last 2 h.

5. Summary and discussion

The new insights into the aerosol indirect effects provided by this study concern the condensation and evaporation processes in a tropical squall-line system as a particular case of convective clouds. The aerosol condensational effect originates from the dependence of the characteristic time scale of condensation τcnd on the cloud droplet number concentration Nc. We proposed a method to approximately isolate the aerosol condensational effect by increasing Nc by the same factor as the CCN loading while calculating the condensation rate, although the approach ignores the contribution of other processes to the value of Nc (e.g., collision and coalescence). This approach captures various mode values of the time scales through the modification of the amount of CCN. It should be noted that the aerosol condensational effect is not detected when applying the saturation adjustment instead of the explicit condensation scheme (e.g., Milbrandt and Yau 2005; SB06). Since the approach to evaluate the aerosol condensational effect is easily applicable to GCMs, which use the explicit condensation scheme, the aerosol condensational effect will be evaluated on the globe as the aerosol lifetime effects (Lohmann et al. 2000). We ignored the role of aerosols as ice nuclei in the numerical simulation to isolate the aerosol effects. Moreover, additional aerosol indirect effects through ice nucleation are beyond the scope of this study.

Two aspects of the aerosol condensational effect are relevant in cloud microphysics. First, the size of each cloud droplet increases to induce more collision and coalescence during updraft. Second, qc is increased or decreased with latent heat release. In cases in which the CCN loading is small, additional qr is produced owing to the increase in qc that results from the aerosol condensational effect. In contrast, when the CCN loading is large, qr is insensitive to increased values of qc because the modification of qc does not effectively increase rc. Thus, in heavily polluted cases, the aerosol condensational effect acts only to increase qc, and subsequently increase qi by the freezing of qc.

In previous studies, the basic idea underlying the feedback mechanism provided by the aerosol indirect effects on convective cloud systems was similar to the delayed-precipitation mechanism proposed by Khain et al. (2004) and Khain and Pokrovsky (2004), which did not consider the contribution of the condensation and evaporation processes. Based on the statistical analysis, Wang (2005) noted that the intensification of convection under high CCN conditions is also due to accelerated heating by the aerosol condensational effect. He showed that the cumulative cloud condensation, mean core updraft, and maximum vertical velocity increase as the initial CCN number concentration increases. The response of the vertical velocity to aerosols is identified using the moist adiabatic parcel model, and the feature appears in the three-dimensional simulations, particularly in the early developing stage. In contrast, the modification of the cumulative cloud condensation appears in the three-dimensional simulations. We further revealed that accelerated condensation and evaporation results in increased cloud water, although the net acceleration of condensation and evaporation decreases cloud water. Increased cloud water induces more conversion between rain and graupel through latent heat release to intensify the convection. As a result, the aerosol condensational effect produces more precipitation.

The abovementioned effects, which originate from the condensational process, play different roles in the developing stage of convective systems. In the early developing stage, the aerosol condensational effect invigorates cumuli to produce more the IWP and increase the ice cloud radiative forcing. These effects could weaken when the total aerosol effects are included owing to the nonlinearity of cloud microphysics and dynamics. In addition, the aerosol condensational effect saturates under relatively low CCN conditions, such as Cccn = 200 cm−3, compared with the aerosol effect on the collisional processes owing to the limitation of the available supersaturation in convective cores. As a result, the aerosol condensational effect would be effective under moderate CCN conditions in the early developing stage. In contrast, in the mature stage, cloud microphysical and dynamical feedbacks cancel the aerosol condensational effect to intensify convection as shown in the early developing stage. Although the aerosol condensational effect has a secondary role in the convective system, the degree of the deviation in the microphysical parameters by the aerosol condensational process is nonnegligible when combined with other aerosol effects. In particular, the aerosol condensational effect on cloud radiative forcing is enhanced in ice clouds because the modulated IWP is larger than the modulated LWP. In other words, anvil clouds act as reservoirs of the aerosol condensational effect when the total aerosol effects are included because the aerosol effect on the collisional processes significantly increases the cloud ice and snow in the upper troposphere, as discussed in the delayed-precipitation mechanism.

In our simulation, it is difficult to explain the role of the aerosol condensational effect in cloud dynamics during the mature stage. One of the reasons is the simplification of the approach that was used to isolate the aerosol condensational effect. In the mature stage, cloudy parcels experience several pathways of cloud microphysics, in which the errors due to this simplification will grow. Hence, it is difficult to strictly isolate an aerosol effect in convective cloud systems. Another reason is that this case may just be a piece of various deep convective cloud systems. In different simulation cases, the aerosol condensational effect would play a more distinct role in feedback mechanisms. For example, the aerosol condensational effect could affect the dynamical feedback mechanism suggested by Lee et al. (2008) in a different case: an increase in evaporative cooling by the CCN loading enhances low-level convergence, which induces the secondary organization of convective clouds, particularly in strong shear cases. In contrast, Tao et al. (2007) reported that the CCN loading is less sensitive to evaporative cooling in the Preliminary Regional Experiment for STORM-Central (PRESTORM) (dry environment and strong vertical wind shear). In such cases, the aerosol condensational effect would be minor. Further investigations under many environmental conditions are required to achieve a more in-depth understanding of the aerosol condensational effects.

As discussed by Stevens and Feingold (2009), the aerosol effects involve several feedbacks as counterworks in nature. To understand this complicated buffered system, further studies are required to understand the dependences of aerosols on the cloud microphysics and their influence on the dominant dynamics of each convective system. In addition, we need to statistically evaluate the degree of modification of the cloud microphysical parameters by aerosols to achieve robust understanding of the system, as discussed by Morrison and Grabowski (2011).

Acknowledgments

The authors wish to thank Dr. Masaki Satoh, Dr. Hirohumi Tomita, and the NICAM development team for helpful discussions. The numerical experiments were performed using the HITACHI SR11000 at the Information Technology Center at the University of Tokyo. This study was partly supported by the Core Research for Evolutional Science and Technology program (CREST) of the Japan Science and Technology Agency and the Innovation Program of Climate Projection for the 21st Century of the Ministry of Education, Culture, Sports, Science, and Technology (MEXT). One of the authors (T. Nakajima) was supported by the Research Program on Climate Change Adaptation of MEXT.

APPENDIX A

Cloud Microphysics Scheme

The NDW6 scheme is designed to maintain the self-consistency of the assumptions regarding PMD and the shapes of ice particles throughout cloud microphysical processes. Following the method described by SB06, NDW6 predicts the moments of the PMDs of each hydrometeor under the assumption of a generalized gamma distribution to analytically formulate the cloud microphysics as follows:

 
formula

For a given PMD, the kth moment of a PMD Ma(k) can be defined as follows:

 
formula

The evolutions of PMDs are represented by updating αa and λa using Na and qa with two fixed parameters: νa and μa, respectively. The diagnostic parameters αa and λa are calculated as follows:

 
formula

where the mean particle mass ρqa/Na. We chose to use constant parameters in the representations of PMDs for cloud water, cloud ice, snow, and graupel based on SB06 and for rain (assuming collisional–breakup equilibrium conditions) based on the protocol described by Seifert (2008). We maintained the self-consistency of the shape of the ice particles by assuming power-law relationships 1) between the particle mass and the maximum dimension D and 2) between the particle mass and the projected area to the flow A as follows:

 
formula
 
formula

where am, bm, aA, and bA are constant coefficients. The shapes of the ice particles are those provided by Mitchell (1996) and we assumed cloud ice as hexagonal plates, snow as assemblages of planer polycrystals in cirrus clouds, and graupel as lump graupel. The abovementioned constant parameters for each hydrometeor are summarized in Table A1. According to the theory proposed by Mitchell (1996), the terminal velocities of ice particles depend on the projected areas and maximum dimensions. We fitted the curves of the terminal velocities to the theoretical values for ice, snow, and graupel. The terminal velocity of cloud water is given by Stokes’ law, and the terminal velocity of rain is given by the analytical formulation developed by Rogers et al. (1993). In the advection of q and N, conservation, consistency with continuity, monotonicity, and non-negativity are fully retained (Miura 2007; Satoh et al. 2008; Niwa et al. 2011).

Table A1.

Constant parameters chosen for the generalized gamma distribution; power-law coefficients used for maximum dimensions and the projected area.

Constant parameters chosen for the generalized gamma distribution; power-law coefficients used for maximum dimensions and the projected area.
Constant parameters chosen for the generalized gamma distribution; power-law coefficients used for maximum dimensions and the projected area.

The cloud microphysical processes used in this study were based on SB06 with several modifications. Here, we describe the modification to SB06, and the original paper should be referred to for details. We solved for the condensation, evaporation, deposition, and sublimation processes using a semi-analytical method that considers the time variation of the supersaturation within a model time step (Khvorostyanov and Sassen 1998; Morrison et al. 2005; Phillips et al. 2007) using the exact moist thermodynamics (Satoh et al. 2008). The warm-phase collisional processes were calculated in the same way as SB06, but we used the modified parameters reported by Seifert (2008) (see  appendix B). We used the mixed-phase collisional processes following SB06, but these were slightly altered from the original scheme. We assumed that the collisional cross section is approximately given by the projected area (A5) instead of circumscribing a circle using the maximum dimension. In addition, the category of hydrometeors that result from binary collision changes by temperature as summarized in Table A2. The Hallett–Mossop process was formulated following the method described by Cotton et al. (1986).

Table A2.

Hydrometeors that result from binary collision. Collecting hydrometeors are written in the first row and collected hydrometeors are written in the first column.

Hydrometeors that result from binary collision. Collecting hydrometeors are written in the first row and collected hydrometeors are written in the first column.
Hydrometeors that result from binary collision. Collecting hydrometeors are written in the first row and collected hydrometeors are written in the first column.

APPENDIX B

Autoconversion and Accretion

We used autoconversion and accretion as proposed by Seifert and Beheng (2001, 2006a) and Seifert (2008). These researchers consider that the warm-phase collisional processes can be parameterized using the dimensionless time scale τcolqr/(qc + qr) based on the theoretical similarity in the processes, in which the liquid water content Lρ(qc + qr) is conserved. Autoconversion and accretion are parameterized through a theoretical formulation with correction functions as follows:

 
formula
 
formula
 
formula

where xaut = 2.68 × 10−10 kg, kcc = 4.44 m3 kg s−1, kcr = 5.78 m3 kg s−1, ρ0 = 1.28 kg m−3, and ϕaut and ϕacc are correction functions of the dimensionless time scale that match the results from a bin scheme. The lifetime of qc in the warm-phase collisional processes, τALE, depends on τcol and is estimated as follows:

 
formula

Figure B1 illustrates the dependence of τALE on τcol with various mean mass radii under typical conditions. In the initiation of collision and coalescence (τcol ~ 10−5), τALE is dominated by autoconversion and is inversely proportional to the sixth power of the mean mass radius of cloud water as shown in (B3) with (B1) and (A4).

Fig. B1.

Dependence of τALE on τcol with LWC = 10−3 kg kg−1, ρ = ρ0, and rc = 8 (solid line) or 16 μm (dashed line). Lines with cross symbols represent the contribution of autoconversion and the line with square symbols represents the contribution of accretion.

Fig. B1.

Dependence of τALE on τcol with LWC = 10−3 kg kg−1, ρ = ρ0, and rc = 8 (solid line) or 16 μm (dashed line). Lines with cross symbols represent the contribution of autoconversion and the line with square symbols represents the contribution of accretion.

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