Effects of different size distributions of cloud condensational nuclei (CCN) on the evolution of deep convective clouds under dry unstable continental thermodynamic conditions are investigated using the spectral microphysics Hebrew University Cloud Model (HUCM). In particular, high supercooled water content just below the level of homogeneous freezing, as well as an extremely high concentration of ice crystals above the level, observed recently by Rosenfeld and Woodley at the tops of growing clouds in Texas, were successfully reproduced.
Numerical experiments indicate a significant decrease in accumulated precipitation in smoky air. The fraction of warm rain in the total precipitation amount increases with a decrease in the CCN concentration. The fraction is low in smoky continental air and is dominating in clean maritime air. As warm rain is a smaller fraction of total precipitation, the decrease in the accumulated rain amount in smoky air results mainly from the reduction of melted precipitation.
It is shown that aerosols significantly influence cloud dynamics leading to the elevation of the level of precipitating particle formation. The falling down of these particles through dry air leads to a loss in precipitation. Thus, close coupling of microphysical and dynamical aerosol effects leads to the rain suppression from clouds arising in dry smoky air.
The roles of freezing, CCN penetration through lateral cloud boundaries, and turbulent effects on cloud particles collisions are evaluated.
Results, obtained using spectral microphysics, were compared with those obtained using two well-known schemes of bulk parameterization. The results indicate that the bulk parameterization schemes do not reproduce well the observed cloud microstructure.
The microphysical and dynamical features of deep convective clouds over continents and over oceans are quite different. According to Takahashi and Kuhara (1993), liquid water in tropical cloud bands at Pohnpei, Micronesia, is concentrated below the level of −15°C, with pure ice above that level. In an individual cumulonimbus no supercooled water has been found.
At the same time Rosenfeld and Woodley (2000) measured, in the tops of growing continental convective clouds, a significant content of supercooled cloud water (up to 1.8 g m−3). The mean volume diameter of droplets at temperatures as low as −37.5°C in summertime Texas clouds was evaluated to be as small as 17 μm. Freezing these droplets produces a huge amount of ice crystals with the concentration as high as 500–700 cm−3. Supercooled water with the content of 2.5 g m−3 at −32°C was observed in the tops of premonsoon clouds in Thailand, where clouds are, as a rule, polluted by smoke from agricultural burning (Rosenfeld and Woodley 1999). In vigorous convective clouds in Argentina the content of supercooled water was as high as 4 g m−3 at −38°C (Rosenfeld and Woodley 2000). Thus, the presence of a huge amount of liquid water up to heights of 9–10 km seems to be a common feature of vigorous convective clouds developing in smoky air.
The difference in the microstructure of maritime and continental clouds can be attributed to two main factors: different stabilities of the continental and maritime atmospheres and different aerosol properties. The main characteristics of continental clouds with high supercooled cloud water content (CWC) are strong updrafts exceeding 20 m s−1 and a high droplet concentration of about 1000 cm−3. According to the observations in Thailand, clouds arising under similar thermodynamic conditions, but under cloud condensation nuclei (CCN) concentrations typical of maritime air masses, contain no supercooled cloud water above the −25°C level (D. Rosenfeld 2003, personal communication).
A useful tool in the investigation of atmospheric aerosol effects on cloud developments and precipitation formation is numerical modeling using spectral microphysics cloud models, in which size distributions of cloud particles are calculated in the course of the model integration. Aerosol effects were investigated using mainly Lagrangian cloud parcel models (Warner 1973; Johnson 1982; Cooper et al. 1997; Chen and Lamb 1999; Pinsky and Khain 2002). Recently, advanced two-dimensional cloud models were developed. These models include the aerosol particle (AP) budget (e.g., Flossmann et al. 1985; Kogan 1991; Khain and Sednev 1996; Khain et al. 1999, 2000, 2001a,b; Yin et al. 2000a,b; Ovtchinnikov and Kogan 2000; Ovtchinnikov et al. 2000). The APs are described by a special size distribution function that changes as a result of advection and in-cloud scavenging. The values of supersaturation are used to determine the sizes of APs (playing the role of CCN) to be activated and the corresponding sizes of newly nucleated cloud droplets. Collisions are calculated by solving stochastic equations for size distribution functions.
Many spectral microphysics models do not include ice processes. At the same time, a significant fraction of precipitation (even in tropical clouds) is caused by cloud ice melting. Hall (1980), Khvorostyanov et al. (1989), Ovtchinnikov and Kogan (2000), and Ovtchinnikov et al. (2000) describe cloud ice using a single size distribution function. The categories with the smallest masses were regarded as ice crystals, while larger ice particles were regarded as graupel. There are only a few models of mixed-phase clouds containing several size distribution functions for different types of cloud ice (e.g., Khain and Sednev 1996; Khain et al. 1996, 1999, 2000, 2001b; Reisin et al. 1996a–c; ,Chen and Lamb 1999; Yin et al. 2000a,b). The advection of cloud particles is calculated in these studies by taking into account the dependence of sedimentation velocities on the particle mass and shape.
In spite of significant efforts, the formation and contribution of cloud ice to precipitation are concepts that are not well understood as yet. This can be attributed to the lack of sufficient knowledge of both ice–ice collision and collection efficiencies and the role of APs in the formation of cloud ice. No estimations of a comparable contribution of warm and melted precipitation under different AP size distributions were presented in these studies. Since the rates of freezing and some other ice-generating microphysical processes depend on the droplet size, APs must influence ice microphysics and ice precipitation as well.
The effect of another mechanism, namely, the effect of atmospheric turbulence on the collision of inertial particles and precipitation amount, is not yet understood. The turbulent effects on drop–drop collisions have been under investigation for several years, and some quantitative results have already been obtained (Khain et al. 2000; Pinsky et al. 2001; Pinsky and Khain 2002). At the same time, the turbulent effects on graupel–drop and ice–ice collisions remain essentially unknown. Pinsky and Khain (1998) and Pinsky et al. (1998) showed that cloud turbulence significantly increases the rate of ice–ice and ice–water collisions. The effects of turbulence on the collision efficiencies of nonspherical ice particles remain largely unknown. Taking into account significant inertia and low fall velocities of ice particles, one can expect that the role of turbulence for ice–ice and ice–water collisions is even more significant than is that for drop–drop collisions. Since the effects of turbulence on collisions are closely related to the size of the colliding particles, turbulence effects have to be closely related to aerosols effects on the droplet spectra.
We employ the Hebrew University Cloud Model (HUCM) herein to address the following problems:
What is the influence of AP size and spatial distributions on the microstructure, precipitation rate, and the accumulated precipitation amount from deep cumulonimbus clouds under continental thermodynamic conditions?
What is the relationship between warm and melted precipitation in deep cumulonimbus clouds under different aerosol concentrations?
What microphysical and dynamical processes are responsible for the transport of huge supercooled water content up to the levels of homogeneous freezing?
What is the possible contribution of turbulent-induced acceleration of particle collisions to the rain rate and the accumulated rain amount under different aerosol concentrations?
What is the influence of atmospheric aerosols on the dynamics of continental clouds?
What is the difference in the reproduction of cloud microphysics by spectral (bin) microphysics and bulk parameterization schemes?
2. Model description
The HUCM is described in detail in Part I of this paper (Khain et al. 2004). So, only a short description will be presented below. Eight size (number) distribution functions are used to describe water drops, ice crystals (columnar, platelike, and dendrites), snowflakes (aggregates), graupel, and hail/frozen drops, as well as cloud condensation nuclei (CCN). Each size distribution is represented by 33 mass doubling categories (bins), so mass mk in the category k is determined as mk = 2mk−1, where k = 2, … , 3. The size distribution for water drops includes all drop sizes from small cloud droplets to raindrops. The minimum mass in the hydrometeor mass grids (except aerosols) corresponds to that of a 2-μm-radius droplet. The mass grids used for hydrometeors of all types are similar, which simplifies the calculation of the interaction between hydrometeors of different bulk densities. The model microphysics is specifically designed to take into account the effect of atmospheric aerosols on cloud development and precipitation formation, as well as the effects of clouds on CCN concentration in the atmosphere.
Nucleation of droplets (CCN activation) is based on the utilization of a separate size distribution function for CCN. In the current model version, the initial size distribution of CCN is calculated using the dependence of concentration N of activated CCN on supersaturation with respect to water Sw. Such dependence can be taken either from specific observations or from climatological data indicating typical dependencies for a certain geographical region. The procedure for such calculations is described by Mazin and Shmeter (1983) and, in more details, by Khain et al. (2000). In particular, the empirical dependence (Pruppacher and Klett 1997) can be written as
where Sw (in %), No, and k are measured constants. Using supersaturation Sw as calculated in the course of the model integration, the value of the critical size of dry CCN rNcrit is determined at each model time step. Aerosol particles with the radii rN > rNcrit are activated and transformed into droplets. Corresponding bins of the CCN size distribution mass grid become empty. In case there are no aerosol particles with rN > rNcrit in the CCN spectra for a particular grid point, no new droplet nucleation takes place at that point. The size of fresh nucleated droplets is calculated as follows. In the case where the radii of CCN rN < 0.03 μm, the equilibrium assumption is used to calculate the radius of a nucleated droplet corresponding to rN (see Khain et al. 2000 for more details). In the case of rN > 0.03 μm, the radius of the nucleated water droplet is assumed to be equal to 5 times as much as 5rN (Kogan 1991; Khain et al. 1999a; Yin et al. 2000a,b). Since large CCN do not reach their equilibrium size at the cloud base, this approach prevents the formation of unrealistically large droplets and inhibits raindrop formation that is too fast.
Nucleation of ice crystals proceeds from the formula presented by Meyers et al. (1992) relating the number concentration of deposition and condensation–freezing ice nuclei (IN), Nd, to supersaturation with respect to ice, Sice:Nd = Ndo exp(ad + bdSice), where Ndo = 10−3 m−3, ad = 0.639, and bd = 12.96. No ice nucleation is assumed for temperatures over −5°C. The number of newly activated ice crystals at each time step in a certain grid point, dNo, is calculated as follows:
where dSice is calculated using the semi-Lagrangian approach (see Khain et al. 2000a for more details). The type of ice crystals nucleated depends on temperature. According to Takahashi et al. (1991), temperature-dependent nucleation proceeds as follows: platelike crystals form at −8° > Tc ≥ −14°C and −18° > Tc ≥ −22.4°C, columnar crystals arise at −4° > Tc ≥ −8°C, Tc < −22.4°C, and dendrites (branch-type crystals) form at −14° > Tc ≥ −18°C.
Secondary ice generation is described by the Hallett and Mossop (1974) mechanism, according to which at T = −5°C each 250 collisions of droplets with the diameter exceeding 24 μm with graupel particles leads to the formation of one ice splinter. According to the measurements, this process is assumed to occur within the −3° to −8°C temperature range. We suppose that the density of splinters is the same as that of pure ice (0.9 g cm−3) and, hence, the splinters are assigned to plate-type ice crystals.
The rate of immersion drop freezing is described following Vali (1975, 1994); the rate of homogeneous freezing is described by Pruppacher (1995). The rate of freezing is calculated using the semi-Lagrangian approach allowing calculation of changes in supersaturation and temperature in moving cloud parcels reaching model grid points (Khain et al. 2000).
Melting is treated in a simplified way by the instantaneous conversion of all ice particles into drops of equal mass at the grid points just below the freezing level. A more sophisticated procedure will be included if the model is to be used for simulations of vigorous hail-bearing clouds.
At each time step, supersaturations with respect to water and ice are calculated by solving a system of corresponding differential equations (Khain and Sednev 1996). Besides droplet and ice nucleation, these values of supersaturation are used for the calculation of diffusion growth/evaporation of water droplets and deposition/sublimation of ice particles. We take into account the shape of the ice crystals to calculate the diffusion growth of different ice crystals.
An efficient and precise method of solving the stochastic kinetic equation for droplet collisions (Bott 1998) was extended to a system of stochastic kinetic equations used to calculate water–water, water–ice, and ice–ice collisions. The model uses height-dependent drop–drop and drop–graupel collision kernels calculated using a hydrodynamic method valid within a wide range of drop and graupel sizes (Khain et al. 2001a; Pinsky et al. 2001). Ice–water and ice–ice collision kernels are calculated taking into account the shape of ice crystals. Ice–ice collision rates are assumed to be temperature dependent. An increase in the water–water and water– ice collision kernels by turbulence/inertia mechanisms was taken into account following Pinsky and Khain (1998) and Pinsky et al. (1998, 1999, 2000). As a result of riming, ice crystals and snowflakes can convert into graupel or hail depending on the temperature. At comparatively low temperatures, collisions of hail (frozen drops) with water drops lead to graupel formation. Collisions of ice crystals lead to snow (aggregates) formation. Khain and Sednev (1996) and Khain et al. (2000) describe in detail the procedure of hydrometeor types conversion as a result of collisions of different kinds.
3. Design of numerical experiments
The computational domain consists of 257 × 129 grid points, with 250-m resolution in the horizontal direction and 125-m resolution in the vertical direction. This rather high resolution was chosen based on the preliminary sensitivity experiments.
Below we will describe the results of simulations of deep cumulus clouds under thermodynamic conditions that occurred in Texas on 13 August 1999 (Rosenfeld and Woodley 1999). Sounding reveals a significant instability of the atmosphere. The surface temperature was as high as 36°C, and the surface relative humidity was about 35%. The freezing level was at 4.5 km above the surface. The cloud-base level was observed at the height of 3.5 km.
The effects of different CCN size distributions on cloud dynamics and microphysics are studied. The initial size distributions of aerosols (CCN) are calculated as was described in section 2. In simulations with continental-type aerosols, k in Eq. (1) was set equal to 0.308 (Khain et al. 2001b). The CCN size distribution was truncated by rN = 0.5 μm. This cutoff radius corresponds to the radius of nucleated droplets of about 2.5 μm. In a supplemental experiment the sensitivity of the results to the concentration of CCN with the “dry” radii ranging from 0.5 to 2 μm is studied.
In the case of maritime aerosols, k in Eq. (1) was set equal to 0.462. The CCN size distribution was truncated by the 2-μm radius, which corresponds to the maximum size of nucleated droplets of about 10 μm. Measurements reported by Hudson (1984, 1993) and Hudson and Frisbie (1991) showed no increase in the CCN concentration for supersaturation Sw exceeding about 0.6%, which indicates the lack of small CCN in extremely clean maritime air. According to Hudson and Xie (1999), maritime air can contain larger concentrations of small CCN. We assumed that the “maritime” aerosol spectrum did not contain CCN, which could be activated at Sw > 1.1%. This limitation allows us to obtain droplet concentrations typical of maritime clouds. To investigate the effects of aerosol concentration on cloud microphysics and dynamics, the values of amplitude A were varied from 10 to 1260 cm−3 in different simulations.
Note that the terms continental and maritime relate in this study to aerosol distributions. Thermodynamic conditions in all experiments were similar (dry air, high instability).
In most experiments, the initial concentration of APs was assumed to be height independent. The list of experiments is as follows, starting with the control experiments:
experiment C in which a deep continental cloud is simulated using the collision kernels calculated for a pure gravity case (no turbulence/inertia mechanisms are taken into account); the magnitude of A is set equal to 1260 cm−3; and
experiment M, which is similar to experiment C, except that the maritime CCN distribution with A = 100 cm−3 is used.
The sensitivity experiments are aimed at the investigation of different microphysical processes on the cloud evolution. The sensitivity experiments include the following:
experiment C-exp is similar to experiment C, except that the CCN concentration above the cloud-base level decreases exponentially with the characteristic 2-km spatial scale; and
experiment C-pf (probability freezing) aims at revealing the effects of the freezing parameterization of cloud microphysics. In experiment C-pf, Vali's (1975) formula for immersion freezing Nim = Nimo(−0.1Tc)γ (where Nim is the number of active immersion nuclei per unit volume of liquid water, Nimo = 107 m−3, γ = 4.4 for cumulus clouds; Tc is the air temperature in °C) was replaced by the formula of probability freezing: (1/fw)(∂fw/∂t) = −afrm exp(−bfrTc), where fw is the size distribution of water drops, m is the drop mass, afr = 10−4 s−1 g−1, and bfr = 0.66 (°C)−1 (Bigg 1953; Reisin et al. 1996a, 1998; see Khain et al. 2000 for more details).
Turbulence effects includes a special set of experiments aimed at investigating the possible turbulent effects on precipitation formation in cases of high and low CCN concentration. Theoretical and laboratory results indicate that the rate of collisions in turbulent flow is significantly higher than is that in calm air and depends on the dissipation rate ɛ. In deep cumuli, typical values of ɛ are about 500–700 cm2 s−3 (Panchev 1971; Mazin et al. 1989). Weil et al. (1993) observed heavy cumulonimbi values of ɛ up to 2 × 103 cm2 s−3. The factors showing an increase in the drop–drop and graupel–drop collision kernels in a turbulent flow as compared to those in calm air were chosen under the assumption that a typical value of ɛ in a simulated cloud is 600 cm2 s−3. These factors are approximate. We do not claim that they provide a precise parameterization of turbulent effects on drop–drop and, especially, drop–graupel collisions. The main question we address in these experiments is does the turbulence inertia mechanism lead only to an acceleration of rain formation, or does it affect the accumulated rain as well? Numerical experiments dedicated to the analysis of turbulence/inertia effects include the following:
experiment C-t (turbulence in) is similar to C, except that the effects of turbulence on particle collisions are taken into account;
experiment M-t (turbulence in) is similar to C-t, except that the maritime CCN (A = 100 cm−3) distribution is used;
experiments C-tdd (turbulence effects on water droplets collisions are included) and C-tgd (turbulence effects on graupel–drop collisions are included), whose goal is to reveal the effects of the turbulence/inertia mechanism in two cases: only drop–drop collision kernels are increased, and only drop–graupel collision kernels are increased, respectively; and
experiment C-t-tail, whose goal is to reveal the role of possible large CCN in the CCN spectra under continental unstable conditions. In C-t-tail, parameter k = 0.308 was used for dry aerosol particles with the radii up to 1 μm (instead of 0.5 μm as in experiment C-t and k = 2 was applied for CCN with the radii from 1 to 2 μm. The turbulence effects on collisions were taken into account in this experiment.
A special set of numerical simulations has been conducted to investigate the effects of aerosol concentration on cloud dynamics and the relationship between cloud dynamics and microphysics. In these experiments the CCN concentration varied within a wide range down to A = 10 cm−3. Other conditions of the experiments are similar to those used in M-t or C-t.
Finally, a comparison of the results obtained using spectral and bulk microphysics is presented.
In all simulations the development of a cumulus cloud is triggered by a 5-min-duration heating within a 5-km-wide region. The heating has a parabolic shape in the horizontal direction. The maximum heating rate is 3.0°C h−1. In the vertical direction the heating was distributed as exp[−2(z − 0.5)], where z is in kilometers. Such heating leads to the formation of updraft, with a vertical velocity of 3 m s−1 at the cloud base at t = 680 s (cloud formation).
a. Simulation of deep convective clouds in smoky and clear air (experiments C and M)
The following figures illustrate the differences in the microstructure of deep convective clouds in experiments C (cloud development in smoky air) and M (low aerosol concentration typical of maritime conditions).
Figure 1 shows the fields of cloud droplet concentration in experiments C (left) and M (right) at t = 30 min when clouds reach their maximum height of about 12.5 km (measured by the cloud ice content). The droplet concentration in experiment C remains 900–1000 cm−3 up to 8-km height indicating negligible droplet collision rates, and decreases to 500–600 cm−3 in the vicinity of the level of homogeneous freezing (−38°C or 9.8 km).
The maximum droplet concentration in experiment M is about 90 cm−3 near the cloud base, and it rapidly decreases with height to about 5–10 cm−3 at the 7-km level indicating efficient droplet collisions and raindrop formation in this case. The increase in droplet concentration near the lateral boundaries is caused by nucleation of new droplets as a result of CCN penetration.
Figure 2a shows the fields of cloud water content (CWC; droplet radii below 50 μm) obtained in these experiments at t = 30 min. In experiment C the CWC reaches its maximum of 3 g m−3 at the height of 7–8 km and decreases to 1.6–1.8 g m−3 in the vicinity of the homogeneous freezing level. The upper boundary of the CWC field coincides with the homogeneous freezing level. The values of both the droplet concentration and the CWC at the top of the growing cloud are consistent with the observations of Rosenfeld and Woodley (2000).
In experiment M the CWC maximum of 2.2 g m−3 reaches the 5-km level (2–3 km below that in experiment C) and decreases with height indicating efficient elimination of droplets above.
Figures 3a and 3b show the drop mass distribution functions at different heights in experiments C and M, respectively. The spectra in experiment C are quite narrow, with the maximum at 10 μm within the 8–9-km layer. This radius is a little larger as compared to 8.5 μm reported by Rosenfeld and Woodley (2000). In experiment M the droplet spectrum broadening is much faster as compared to that in experiment C. Effective collisions lead to raindrop formation already at the 3–5-km level and eliminate a significant number of cloud droplets, leading to a rapid decrease of CWC with height above the 6-km level. Fields of the rainwater content (RWC) at 45 min are presented in Fig. 4 for experiment C (left) and experiment M (right). One can see that in the case of low maritime CCN concentration warm rain develops faster and attains much larger values as compared to the case of high (continental) CCN concentration. This result clearly indicates that due to aerosol effects the continental cloud with a significantly larger CWC (3 g m−3 in experiment C versus 2.2 g m−3 in experiment M) produces much lower rain. This fact contradicts the prediction of a well-known Kessler (1969) formula (used in most bulk parameterization microphysics models), according to which the raindrop production is proportional to CWC.
In accordance with the evolution of the mass distribution spectra, the first radar echo is observed in experiment to be C much higher (8 versus 4 km), and 5– 10 min later as compared to experiment M (not shown). Therefore, thermodynamic conditions being similar, the height of the first echo can serve as an indicator of the existence of APs in the atmosphere: the higher the APs concentration, the weaker the first echo is and the higher the level where it is located.
As the rates of drop freezing and riming depend on the drop size, APs crucially influence cloud ice microphysics. Figure 5a shows the fields of plate ice crystal concentration in experiments C (left) and M (right). Most of these crystals are formed by droplet freezing near the level of homogeneous freezing. The maximum concentration reaches 500 cm−3, which agrees well with the measurements of Rosenfeld and Woodley (2000). The concentration of ice crystals in experiment M is 30 times lower than that in experiment C. The mass of ice crystals in experiment, C (with the maximum of 1.1 g m−3 at 40 min; see Fig. 5b) is higher than that in experiment M (the maximum is 0.6 g m−3). At the same time the size of these ice crystals in experiment M is larger. Since at low temperatures the coalescence efficiency of ice crystals is negligibly small, these crystals are spreading over a large area and sublimate, without contributing to precipitation. This is an important factor leading to the lower accumulated precipitation in experiment C as compared to that in experiment M.
The efficiency of ice precipitation formation depends on the ability of graupel to collect ascending liquid droplets. The efficiency increases with the size and concentration of graupel and droplets, as well as with the intensity of turbulence (see below). Graupel size is especially important because small graupel cannot collect small cloud droplets (Khain et al. 2001a). Graupel concentration and content are determined by drop freezing, whose rate increases with the droplet size and the decrease in temperature. Graupel concentration in experiment C (200–300 L−1) is significantly higher than that in experiment M. At the same time, the graupel mass content in experiment M is higher than that in experiment C throughout the whole simulation period (e.g., Fig. 6), which means that the graupel in experiment M are larger than those in experiment C. Graupel content in experiment C is higher as compared to that in experiment M only at the highest levels because of a lower sedimentation velocity of smaller graupel. Differences in the droplet size distributions and the dependence of drop freezing on the droplet size determine different shapes of graupel mass fields, as shown in Fig. 6. In experiment M the shape of the graupel content contours resembles that of a mushroom, with the root within the cloud updraft, indicating the existence of a significant amount of raindrops in the updraft that freeze easily. In experiment C graupel content at the cloud axis is much smaller than that in experiment M, indicating a comparatively low concentration of large drops in the cloud core. Figures 7a and 7b show mass distributions of graupel at different levels in experiments C and M at t = 35 min at the cloud axis (x = 23 km). Figure 7c shows the graupel mass distributions in the points of graupel mass content maximum (x = 19 km). While in experiment C the melted radius of graupel does not exceed about 1000 μm; a fraction of the large graupel in experiment M is significantly larger, as can be seen in Fig. 7c. Note that mass distributions of graupel at the cloud axis are bimodal. The formation of graupel bimodal spectra can be attributed to the fact that at low levels only the largest drops freeze, while at higher levels droplets of a smaller size can freeze as well.
Note that there is no bulk parameterization microphysics scheme that could reproduce the bimodality of cloud particles mass distributions.
Being of a larger mass, the graupel in experiment M produces melted rain earlier than in experiment C. Figures 8 and 9 show graupel and RWC in experiments C and M at the cloud dissipation stage. Precipitation at this stage of a cloud's evolution is entirely determined by ice melting, mainly in the form of graupel. The higher graupel content in experiment M near the freezing level results in a higher precipitation rate.
One of the principal characteristics of clouds is radar reflectivity. In the present study radar reflectivity (R in dBZ) is calculated as R = log(Z), where Z in the case of water drops is calculated in accordance with its definition as Z = 64 ∫∞0 f1(r)r6 dr. Radar reflectivity from dry ice particles is calculated under the assumption that ice particles of mass M can be regarded as a mixture of pure ice and air with masses Mi and Ma, respectively. According to Debye (Battan 1973, p. 40), (K/ρ)M = (Ki/ρi)Mi + (Ka/ρa)Ma, where ρ is the particle density; the subscripts i and a refer to ice and air, respectively. Here Ki/a = (mi/a − 1)/(m2i/a + 2), where mi is the complex index of ice refraction.
Since Ka approaches zero, the second term may be neglected. Since Mi ≅ M, it follows that K/ρ is nearly constant for ice–air mixtures. The backscattering cross-section σ of an ice particle with mass M is proportional to |K2|/ρ2 (Battan 1973, p. 38) and to M2. If the density is chosen to be 1 g cm−3, σ will be proportional to r6m, where rm is the melted radius. As is shown by Battan (1973, p. 39), in the case ρ = 1 g cm−3, |K|2 = 0.197. Thus, radar reflectivity from dry ice particles was calculated using |K|2 = 0.197 and the melted radii.
Following Battan (1973, p. 46), we assume that ice particles of high density (graupel and hail) develop skins of water whose average thickness increases with time as the particles move through the air. According to Fig. 5.1 in Battan (1973), the particles covered by the water skin with a mass over 20% of the total mass actually reflect as water drops.
In the present version of the model, immediate melting is assumed just at the level of the 0°C isotherm. So, below this level no ice is assumed. This is why we calculate reflectivity from the melting layer using the size distributions of ice particles just above the freezing level as input. Then we attribute these values of reflectivity to the layer containing the 0°C level.
According to Battan (1973), melting snowflakes may be treated as a mixture of ice and water. Since in the present version of the model we do not calculate the water fraction in melting particles, a simple parameterization of reflectivity from melting snowflakes was used. It was assumed that melting snowflakes contain a significant fraction of liquid water and are covered by a significant water film. At the same time it was assumed that the density of these melting particles is significantly higher compared to the density of dry snowflakes. The assumed density of 0.3 g cm−3 allowed us to obtain a bright band with the reflectivity about 6–8 dBZ higher than that above the freezing level. This value is consistent with observations. Note that the radar reflectivity is the model output that does not influence any model variables.
Figure 10 depicts the radar reflectivity fields at 80 min. The bright band reflecting the area of ice particles melting can be seen clearly. The radar reflectivity in experiment M is higher than that in experiment C, indicating the existence of larger ice particles above the freezing level and more intensive melted rain in experiment M.
The spatial–temporal distributions of precipitation rates in experiments C and M are shown in Figs. 11a and 11b, respectively. One can see that precipitation in “the continental aerosol case” begins 20 min later. Two modes of precipitation are clearly seen. During the first 20–30 min of rainfall, warm rain precipitation dominates; the succeeding precipitation is determined by ice melting. Melting precipitation covers a larger area and lasts longer. In spite of the fact that the intensity of the warm rain in experiment M is much higher, the contribution of melting rain to the total rain amount dominates in this case as well.
b. Sensitivity experiments
1) Effects of vertical distribution of aerosols
The distribution of APs with height is often quite inhomogeneous (Pruppacher and Klett 1997). One of the mechanisms that may influence the droplet spectrum evolution is the entrainment and mixing of cloud air with the environment. Such mixing can lead, on the one hand, to total or partial evaporation of cloud droplets and, on the other hand, to a new droplet nucleation on aerosols penetrating clouds through their boundaries.
To illustrate the effects of aerosol (CCN) entrainment above the cloud base in the case of a vigorous convective cloud, we conducted experiment C-exp, which differs from experiment C only in one aspect; namely, the initial concentration of CCN was assumed to decrease exponentially [as it was suggested by Mazin and Shmeter (1983)] with height above the cloud-base level with the spatial scale of 2 km. The results show that at the cloud-growing stage the droplet concentration near the cloud boundaries in experiment C-exp is about 500 cm−3 lower as compared to that in experiment C. Taking into account that the maximum droplet concentration is about 1000 cm−3, this decrease in the droplet concentration at the cloud periphery is significant.
The distance the environmental air penetrates a cloud updraft through the lateral cloud boundaries can be crudely evaluated as Dt, where D is the coefficient of turbulent diffusion and t is the characteristic time period, during which cloud air ascends from the cloud base to the cloud-top level. Assuming D and t to be 200 m2 s−1 and 10–15 min, respectively, one can estimate the distance of the environmental air penetration due to turbulent diffusion as 400–500 m. The larger distance of the environmental air penetration into the cloud (about 1 km) observed in the calculations is caused, possibly, by the presence of the mean radial velocity component.
Note that the environmental air does not penetrate into the cloud core of the simulated deep cloud. As a result, the main characteristic features of the cloud structure, such as the maximum droplet concentration and CWC, etc., in experiment C-exp are close to those in experiment C. Thus, for the deep and wide cloud simulated, the contribution of APs penetrating into the cloud through the cloud base turns out to be dominating.
The effect of CCN entrainment above the cloud base is, possibly, more important in cumulus clouds, which are smaller than those simulated in this study.
2) Sensitivity of droplet concentration to the parameterization of drop freezing
A comparison of the results of numerical experiments C and C-pf (probability freezing) shows that the main difference in droplet concentration is observed in the upper troposphere at temperatures below −30°C. The probability freezing formula (with generally accepted magnitudes of the coefficients) seems to overestimate the rate of droplet freezing at temperatures as low as −33° to −37°C leading to a 30% (by 250 cm−3) decrease in droplet concentration within this temperature zone as compared to that obtained in experiment C. In this sense the Vali (1975) formula provides results that are closer to the droplet concentration measured at high levels.
At temperatures over −30°C, no significant difference in droplet concentration was found in these experiments. We attribute this result to the fact that in the continental aerosol case droplets are small and the rate of their freezing is slow, no matter what formula is used. It is possible that the difference would be more significant in a less extreme continental case. Since droplets frozen above the homogeneous freezing level do not contribute to precipitation, the accumulated rain in experiment C-pf turned out to be about 7% larger than that in experiment C.
3) Turbulence–inertia effects on the microphysical structure and precipitation formation
An investigation was conducted comparing the results of experiments C-t (turbulence effects included) and M-t (turbulence effects included) with those obtained in experiments C and M. Figure 2b shows CWC fields in experiments C-t (left) and M-t (right), respectively. A comparison of the CWC fields with those in the corresponding nonturbulent experiments (Fig. 2a) shows that turbulence decreases the maximum values of the CWC, indicating more effective rain production in both cases. Note, however, that all characteristic features of deep Texas clouds observed by Rosenfeld and Woodley (2000; high values of droplet concentration and CWC just below the homogeneous freezing level, high concentration of ice crystals above this level, etc.) are reproduced well in the turbulent experiments. A comparison of Figs. 2b and 2a shows that turbulent effects lead to a significant decrease in the CWC above the −20°C level (8 km) in the case of low (maritime) CCN concentration. This effect is caused by the increase in the rates of accretion and riming in a turbulent flow. Larger cloud droplets arising in the case of low CCN concentration are collected by ice particles more efficiently than are the small cloud droplets that formed in the case of high CCN concentration. As the lack of any significant supercooled cloud water is repeatedly observed in deep maritime convective clouds at heights above the −20°C level (e.g., Takahashi and Kuhara 1993; D. Rosenfeld 2003, personal communication), turbulence effects are important with regard to the formation of a realistic profile of LWC in clouds, whose droplet concentration is not extremely high and the droplet size not extremely small. The maximum magnitudes of the CWC obtained in turbulent experiments also seem to be closer to the observations of Rosenfeld and Woodley (2000).
A comparison of the droplet spectra with those formed in nonturbulent experiments C and M indicates that in the case of high APs the concentration, the droplet size spectrum at the growing stage actually remains as narrow as in the nonturbulent experiment (not shown). This result can be attributed to the fact that the time period of droplets ascending along the cloud axis up to the level of homogeneous freezing is quite short, so that the largest droplets that formed due to turbulence effects appear at high levels and are rapidly eliminated by freezing. An increase in the rate of drop–drop and drop–ice collisions leads to a decrease in the maximum value of the CWC by only 15%–20%. Turbulence effects become more important at later stages of cloud evolution. In the case of maritime aerosols turbulence significantly accelerates raindrop formation. Large raindrops in experiment M-t form about 0.6 km lower than do those in experiment M. In both experiments droplet spectra are wide and contain a significant amount of raindrops at the 4-km level.
Turbulence effects decrease the graupel concentration by a factor of 2–3. Since the graupel concentration obtained in experiments C and M is possibly too high (200–300 L−1), turbulence effects, being taken into account, lead to a more realistic graupel concentration. The turbulence–inertia mechanism accelerates the formation of raindrops at lower levels. Freezing of these drops leads to the formation of graupel large enough to collect supercooled droplets (especially the largest ones) aloft. Hence, turbulent effects lead to a decrease in the graupel concentration and to an increase in the graupel size.
Figure 12 shows time dependence of rain accumulation in different experiments with and without turbulent effects taken into account. One can see that in the case of high APs concentration (experiment C) accumulated rain turns out to be smaller than in the case of low APs concentrations (experiment M) by a factor of 5. In the case simulated, the turbulence–inertia effect increases the accumulated rain by about 20% with each CCN concentration.
Transition of the warm rain to the melted rain period can be seen by a decrease in the slope of the curves in Fig. 12: being stratiform, the melted rain is less intensive. One can see that warm rain contributes significantly to the accumulated rain only in the case of low CCN concentration (experiments M and M-t). However, even in this case melted precipitation exceeds the warm rain by a factor of about 2.5.
Experiments C-tdd (turbulence effects on water droplets collisions included) and C-tgd (turbulence effects on graupel–drop collisions included) indicate a nearly equal contribution of drop–drop and graupel–drop collision kernels increase induced by turbulence–inertia effects to the accumulated rain amount.
Effects of large CCN with the radii ranging from 0.5 to 2 μm on cloud microphysics were studied in experiment C-t-tail, in which parameter k = 0.308 in Eq. (1) was used to calculate the initial CCN size distribution with the radii up to 1 μm (instead of 0.5 μm as in other “continental aerosol” experiments). At the larger end of the aerosol size distribution, k = 2 was applied. The magnitudes of A for the larger end of the aerosol size distribution were calculated so as to provide the continuity of the CCN size distribution. As a result, the concentration of larger APs was significantly increased in this experiment as compared with that in experiment C-t.
The result turned out to be not very sensitive to the tail of larger CCN in the CCN spectrum: the accumulated rain amount increased by about 30%. We attribute this comparatively low sensitivity of the precipitation amount to the additional large CCN to two effects: 1) high droplet concentration decreases supersaturation and slows down the growth rate of all droplets including the largest ones and 2) strong updrafts shorten the time of droplet collisions, so that raindrops form (if at all) at high levels where they rapidly freeze. As a result, the tail of the large CCN leads to a certain increase in graupel mass at high levels. Possible effects of ultragiant CCN were not analyzed in this study.
c. Effects of CCN concentration on cloud dynamics and precipitation
A special set of numerical simulations has been conducted to investigate the effects of aerosol concentration on cloud dynamics and the relationship of cloud dynamics and microphysics. In these experiments the CCN concentration was varied from 1260 cm−3 down to A = 10 cm−3. Other conditions were similar to those applied in experiments M-t and C-t.
The further decrease in CCN concentration down to A = 50 and 10 cm−3 leads to a more maritime cloud structure. Figure 2c shows CWC at t = 30 min in numerical experiments M-t-50 and M-t-20 (the numbers denote the concentration of activated CCN at 1% supersaturation). Figure 2a–c show that the magnitude of the CWC decreases with the decrease in the CCN concentration. Besides, the CWC decreases with height more rapidly, indicating earlier raindrop formation.
Figure 13 shows the dependence of accumulated rain on the CCN concentration (taken at 1% of supersaturation). One can see that maximum accumulated rain continuously increases with the decrease in CCN concentration, no matter what kind of aerosol (“continental” or “maritime”) is used. The possible mechanism of this behavior is assumed to be the following.
Figure 14 shows time dependencies of maximum updraft (Wmax) and downdraft (Wmin) velocities in the numerical experiments with different CCN concentrations. One can see that clouds that developed in more smoky air have larger vertical velocities. Thus, aerosols significantly affect cloud dynamics. We can suggest several mechanisms leading to this effect. First, a decrease in the CCN concentration leads to the formation of raindrops at lower levels. A significant loading arises at lower levels, decreasing the cloud updraft aloft. Besides, early precipitation also decreases the air temperature in the subcloud layer due to full or partial raindrop evaporation. These mechanisms slow down cloud development. Third, in smoky air droplets reach higher levels and diffusion growth continues for a longer period of time. Thus, latent heat of condensation (as well as latent heat of freezing) in the case of higher CCN concentration contributes to the increase in the updraft velocity and to the elevation of the raindrop formation level. As a result, precipitation particles in the case of smoky air fall down from the higher levels. The wind shear leads to the fact that these particles fall down through very dry air. Thus, we attribute the decrease in precipitation with the increase in the CCN concentration to the increase in the precipitation loss by evaporation (or sublimation) of precipitation within a dry sheared atmosphere.
d. Comparison with bulk parameterization schemes
In many studies bulk microphysical parameterizations are used for simulation of clouds or cloud-related phenomena. To verify the ability of bulk parameterization schemes to reproduce microphysical features of deep continental clouds, parameterizations by Lin et al. (1983) and Rutledge and Hobbs (1984) were chosen. Figure 15 shows the fields of the CWC at t = 35 min in experiments using these parameterizations. One can see that the CWC in both cases attains its maximum at 4.5 km and decreases with height aloft. The decrease with height is related to the freezing of droplets and the formation of graupel. Since the bulk parameterization schemes do not control droplet size, the rate of freezing does not depend on the droplet size in these schemes. In the case of the Lin et al. (1983) parameterization the freezing leads to a decrease in the CWC with height, which is as strong as in the case of maritime aerosols, with A between 50 and 100 cm−3. The decrease in the CWC in the case of the Rutledge and Hobbs (1984) parameterization is slower than in the case of the Lin et al. (1983) parameterization. Anyway, in both cases the CWC decreases with height significantly faster than was observed by Rosenfeld and Woodely (2000) and simulated using spectral microphysics (Figs. 2a,b; left panels). Intensive freezing leads to very strong updrafts (up to 30 m s−1 maximum). This leads to the transfer of graupel into the upper troposphere and to a delay in precipitation as compared to spectral microphysics. Figure 16 shows the spatiotemporal dependence of the precipitation rate in the experiment in which the Rutledge and Hobbs bulk parameterization has been used. Comparison with Fig. 11a shows a significant difference in the spatiotemporal distribution of the precipitation rates. Precipitation in the bulk parameterization experiment started about 50 min later than did that in the experiment with spectral microphysics C-t. The precipitation rate is higher and the rain lasts for a shorter time as compared to those in C-t. The maximum precipitation rate takes place at 140 min. The time dependencies of rain accumulation in the cases using the Rutledge and Hobbs parameterization and the spectral microphysics (experiment C-t) are presented in Fig. 17. One can see a significant difference in the time characteristics of rain accumulation with regard to the spectral and bulk microphysics. Note that the total rain amounts turn out to be quite similar in these simulations.
5. Discussions and conclusions
The main goal of the study was to investigate aerosol effects on the microphysics and dynamics of deep clouds arising under continental thermodynamic conditions. Comprehensive simulations of deep convective clouds under continental thermodynamic conditions are conducted using the Hebrew University Cloud Model (HUCM) with spectral microphysics. In the simulations of deep growing clouds in Texas observed by Rosenfeld and Woodley (2000), such typical features as 1) a high concentration of droplets up to the level of homogeneous freezing and 2) a high supercooled water content of 1.5–2 g m−3 just below this level, as well as 3) an extremely high concentration of ice crystals (up to 500 cm−3) above the level, were successfully reproduced. The high concentration of CCN along with the high vertical velocity in clouds lead to the nucleation of cloud droplets with the concentration of about 1000 cm−3. The high concentration of droplets decreases supersaturation and slows down the rate of droplet growth by diffusion. The rates of both the collisions and freezing are very low and cannot decrease CWC significantly. The collision efficiencies between ice crystals with sizes below 100 μm and small droplets, as well as between small graupel and small droplets, are close to zero. Thus, riming cannot deplete the CWC of small supercooled droplets in spite of the fact that the graupel concentration is high.
Sensitivity experiments show that in the continental deep clouds simulated APs penetrating into a cloud through the cloud base play the main role with regard to aerosol effects on cloud microphysics and dynamics. Since the simulated clouds have significant horizontal size, APs penetrating into the cloud through its lateral boundaries increase the droplet concentration in the vicinity of the cloud boundary, but do not affect the structure of the cloud core. As a result, the effects of the AP entrainment through the cloud boundaries are of little importance in the case simulated.
Experiments reveal the crucial effect of APs concentration and size distribution on both warm and melted precipitation amounts. When the CCN (and droplet) concentrations are small, diffusion growth leads to a rapid formation of large droplets triggering coalescence and raindrop formation. Raindrops either fall down or freeze, forming frozen drops that rapidly transform into graupel. The graupel particles that formed are larger, than those in the case of high aerosol concentration, and they collect droplets efficiently. As a result, both warm and melted precipitation amounts increase. Accumulated rain in the case of APs with characteristics typical of maritime CCN turns out to be 4–5 times higher than in the case of a high concentration of APs with characteristics typical of continental CCN.
Numerical experiments conducted within a wide range of CCN concentrations indicate a monotonic decrease in the accumulated rain amount with the increase in the CCN concentration. This conclusion is the main result of the study. We found that an increase in the CCN concentration leads to stronger cloud updrafts. Both dynamical and microphysical mechanisms lead to an elevation of the level of precipitating particle formation and to a decrease in their size. As a result, in smoky air precipitating particles start falling down from higher levels, being of smaller size. Their fall through very dry air is accompanied by evaporation (or/and sublimation), which decreases precipitation at the surface. Thus, the coupling of “microphysical” and “dynamical” aerosol effects leads to the rain suppression from clouds arising in dry smoky air.
Note that an addition of a “tail” of larger CCN to the CCN size spectrum (leading to the nucleation of droplets with radii up to 8 μm has a comparatively small effect on the accumulated rain (a 30% increase). This effect can be attributed to the fact that high droplet concentration decreases supersaturation and hinders the diffusion growth of the largest droplets as well.
Turbulent effects on the collision rate (as they were evaluated in the study) lead to a more realistic microphysical structure of clouds simulated and to a certain (about 30%) increase in the accumulated rain. This comparatively low increase in accumulated rain can also be attributed to the small droplet size and high vertical velocities in the cloud simulated. As a result, a turbulence-induced increase in the collision rate does not lead to a significant increase in the formation of precipitating particles. A change in the CCN concentration turns out to be the major factor influencing the precipitation amount. We suppose that turbulence effects might be more pronounced in cases where the thermodynamic conditions are not so unstable.
The results show a significant contribution of melted rain to the accumulated rain amount. This contribution is dominating in clouds developed in smoky air and it decreases with a decrease in the APs concentration. However, even in the case of low concentration maritime aerosols (about 100 cm−3), the fraction of melted precipitation exceeds 50%. As warm rain is a small fraction of total precipitation, the decrease in the accumulated rain amount in smoky air results mainly from the reduction of melted precipitation.
Simulations of cloud development using two bulk parameterization schemes showed that these schemes could not reproduce the observed cloud microphysical structure. Both schemes lead to a rapid decrease in the cloud mass content with height, caused by an overestimation of the freezing rate. The bulk parameterizations lead to the formation of clouds with a higher vertical velocity, a higher graupel content, and a significant delay in the commencement of precipitation, as well as a higher precipitation rate and a shorter lifetime as compared to those in the spectral microphysics experiments.
The experiments discussed in the study have been conducted assuming extremely dry and unstable thermodynamic conditions. In our future studies we are going to analyze aerosol effects on the development of clouds under maritime and “intermediate” thermodynamical conditions.
The authors are grateful to Prof. D. Rosenfeld and Dr. M. Pinsky for valuable comments and remarks. This study was partially supported by the Israel Ministry of Science (Grant WT 0403, German– Israeli Cooperation in Water Technology), the Binational U.S.–Israel Science Foundation (Grant 2000215), and the European Grant SMOCC.
Calculation of Supersaturation and Droplet Nucleation
The method used for the calculation of supersaturation values is similar to that used by Tzivion et al. (1989) and Khain and Sednev (1996) with some additional modifications. The calculation of supersaturations with respect to water, Sw(=e/ew − 1), and with respect to ice, Sice(=e/eice − 1) (where e, ew, and eice are water vapor pressure and its saturated values with respect to water and ice, respectively), are performed in two steps. First, the equations for the advection of potential temperature θ and of the mixing ratio q are integrated during a dynamical time step Δtdyn. As a result, the values of supersaturations S*w and S*ice, as well as the dynamical caused tendencies (δSw/δt)dyn = (S*w − St0w)/Δtdyn and (δSice/δt)dyn = (S*ice − St0ice)/Δtdyn, are calculated at each grid point. The dynamical time step is divided into several microphysical time steps, Δtdiff. The change of supersaturation at each microphysical time step is calculated as the sum of the dynamical tendency [e.g., (δSw/δt)dyn Δtdiff] and changes caused by diffusion growth/evaporation of drops or deposition/sublimation of ice.
The changes of supersaturation due to microphysical processes are calculated as follows. The rate of change of the mixing ratio and potential temperature can be written as
where i is the type of hydrometeor (i = 1, water drops; 2–4, ice crystals; 5, aggregates; 6, graupel; and 7, frozen drops/hail); ɛ1,2 denotes the rates of condensation/evaporation of drops or deposition/sublimation of ice particles; Lw/ice(Lw or Lice) is the specific latent heat with respect to water or ice; fi, is the size (number) distribution function and ∫∞0 fi dmi is the concentration (in cm−3) of hydrometeors of ith type; and ρa is the air density.
The mass grid for each type of hydrometeor is represented in the model by 33 mass doubling categories (bins): mi,k+1 = 2mi,k, where k is the mass bin number. Diffusion growth (evaporation) of liquid drops and deposition (sublimation) of ice particles of mass mik by water vapor deposition (sublimation) is expressed as (Pruppacher and Klett 1997)
where Gi = RυT/es,w/iceDυ + Lw/ice/kaT(Lw/ice/RυT − 1), Dυ, ka, and Rυ are the water and air diffusivity coefficients and the moist air gas constant, respectively; es,w/ice is the saturation water vapor pressure over water or ice. Expressions for the “electrostatic capacitance” of particles of different shape Cik are taken from Pruppacher and Klett (1997); see also Khain and Sednev (1996). Using Eqs. (A1)–(A3), expressions for the water vapor mixing ratio q = 0.622(ew/p), as well as the dependences of the saturation vapor pressure over water and ice es,w/ice = Aw/ice exp(−Bw/ice/T) (Rogers and Yau 1989), one can derive the following equations for Sw and Sice:
where coefficients P1, P2, R1, and R2 are
where Bw = 5.42 × 103 K and Bice = 6.13 × 103 K. If the microphysical time step Δtdiff is small enough, the coefficients of Eqs. (A4) can be considered as constants and the analytical solution of (A4) during the time τ ≤ Δtdiff can be written as
where α = [(P1 − R2)2 + 4R1P2]1/2, β = (1/2)(α + P1 + R2), and γ = (1/2)(α − P1 − R2).
Using (A3), one can evaluate the approximate magnitude of Δtdiff needed for particles belonging to the smallest nonempty bin to reach the next mass category in the case of condensation (deposition), or to be transferred to a neighboring smaller mass bin (or to be fully evaporated), in the case of evaporation (sublimation). Using this choice of Δtdiff the coefficients in the differential equations for supersaturations usually change by much less than 1% during Δtdiff. In rare cases the coefficients changed by more than 1%; the magnitude of Δtdiff was decreased until the required precision was achieved. At each nth time step, Δtdiff,n, the change of the droplet–ice particle mass due to microphysical processes, is calculated analytically as Fik Sw/ice dτ using solutions (A5) and (A6). The utilization of this approach takes into account the fact that supersaturation changes during one microphysical time step. The values of supersaturations obtained to the end of each Δtdiff,n are corrected using the dynamical tendency of supersaturation mentioned above. The final particle mass at t + Δtdyn is calculated as
where Sw/ice dτ = Σn Sw/ice dτ. To calculate new values of distribution functions, the new spectrum has to be remapped into the regular mass grid (e.g., Kovetz and Olund 1969) that leads to the artificial spectrum broadening. To decrease the number of the remappings, the calculation of new size distribution functions is performed only at the end of the dynamical time step Δtdyn which leads to a significant time delay in raindrop formation as compared to the case when the remapping is conducted at each step Δtdiff,n. Utilization of analytical solutions (A5) and (A6) in (A7) avoids the artificial (numerically induced) formation of negative supersaturations during diffusion growth/deposition or positive supersaturations during evaporation/sublimation. The utilization of small values of Δtdyn,n allows one to obtain a smoothed behavior of supersaturations during Δtdyn. Supplemental experiments showed that a further decrease in Δtdyn,n did not influence either particle size distributions or supersaturations to the end of Δtdyn. For droplet nucleation, at each dynamical time step Δtdyn maximum positive values of supersaturations Sw and Sice are calculated and used for the calculation of minimum (critical) radii of aerosol particles rNcrit, so that CCN with radii rN > rNcrit are activated, giving rise to the formation of new droplets. Using the Kohler theory, the critical radius of CCN rNcrit can be calculated as rNcrit = (A/3)(4/BS2)1/3 (Rogers and Yau 1989), where A = 2σ/ρwRυT and σ is the surface tension of the droplet. The “solution” term B = n(ρNMW/ρwMs), where n is the number of ions for each solvent molecule (assumed equal to 2); ρN is the density of dry aerosol particle; and Ms and Mw are the molecular masses of the solvent and water, respectively. In the case where the CCN size distribution at a given grid point does not contain aerosols with rN > rNcrit, no nucleation takes place.
According to simulations of aerosol particle growth below cloud base performed by Ivanova et al. (1977) and recently by Segal et al. (2004), in the case of rNcrit ≤ 0.03 μm, the radii of nucleated droplets equal, with high precision, to the equilibrium size rcrit = (2/3)A(1/Sw) (Rogers and Yau 1989). The time needed to achieve the equilibrium increases with CCN size. Thus, being applied to large CCN, the equilibrium assumption can lead to an overestimation of droplet size at cloud base. The calculations show that in the case of rNcrit > 0.03 μm the radius of a corresponding droplet at zero supersaturation can be determined as K × rNcrit, where 3 < K < 8. This expression was also used by Kogan et al. (1984), Kogan (1991), Khain and Sednev (1996), Yin et al. (2000a,b), and in other studies. Under the assumption that CCN particles are fully soluble, the latter expression determines the size distributions of nucleated droplets.
Corresponding author address: Prof. Alexander Khain, The Institute of Earth Sciences, The Hebrew University of Jerusalem, Givat Ram 91904, Israel. Email: firstname.lastname@example.org