Comparison of collision velocity differences of drops and graupel particles in a very turbulent cloud

Ž. Atmospheric Research 49 1998 99 113 Comparison of collision velocity differences of drops and graupel particles in a very turbulent cloud M. Pinsky ), A. Khain, D. Rosenfeld, A. Pokrovsky The Institute of Earth Science, The Hebrew UniÕersity of Jerusalem, GiÕat Ram, Jerusalem 91904, Israel Received 17 March 1997; accepted 24 April 1998 Abstract The motion of water drops and graupel particles within a turbulent medium is analyzed. The turbulence is assumed to be homogeneous and isotropic. It is demonstrated that the inertia of drops and graupel particles falling within a turbulent flow leads to the formation of significant velocity deviations from the surrounding air, as well as to the formation of substantial relative velocity between drops and graupel particles. The results of calculations of the continuous growth of raindrops and graupel particles moving within a cloud of small droplets are presented both in a non-turbulent medium and within turbulent flows of different turbulence intensity. Continuous growth of a drop-collector was calculated with the coalescence efficiency E s1, as well as using E values provided by Beard and Ochs w Beard, K.V., Ochs, H.T., 1984. Collection and coalescence efficiencies for accretion. J. Geophys. Res., 89: 7165 7169. x ranging from 0.5 to about 0.75 for different droplet sizes. In the case of graupel droplet interaction E was assumed equal to 1. It is shown that in the case E s1 in a non-turbulent medium, the growth rates of graupel and raindrops are close. Under turbulent conditions typical of mature convective clouds, graupel grows much faster than a raindrop. In the case E -1 the growth rate of a water drop slows down significantly, so that graupel grows faster than raindrops even under non-turbulent conditions. Turbulence greatly increases the difference between the growth rates of graupel and drop-collectors. Possible consequences of the faster growth of graupel in terms of cloud microphysics are discussed. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Inertia of atmospheric particles; Cloud turbulence; Riming; Graupel particles; Coalescence; Cloud seeding ) Corresponding author. 0169-8095r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. Ž. PII S0169-8095 98 00073-8

100 ( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113 1. Introduction The important role of ice processes in the formation of precipitation, especially in non-tropical clouds, was realized many years ago: it can be traced back to the theory of Bergeron Ž 1935., who emphasized the significance of the lower vapor pressure over ice in an environment in which ice crystals and water droplets coexist. Braham Ž 1964. concluded from observations of precipitation development in Missouri clouds that graupel grow faster by riming than do unfrozen drops due to coalescence. This conclusion suggests that freezing of supercooled drizzle drops increases the precipitation efficiency of summertime convective clouds in the central United States ŽBraham, 1964, 1968.. These results seem to be supported by the observations of Rosenfeld and Woodley Ž 1996. in seeded clouds, showing a very fast growth of graupel after seeding, which seems to substantially exceed the growth rate of raindrops under non-seeding conditions. For example, it takes graupel only about 6 min to grow from 0.5 mm to 5 8 mm in diameter. Johnson Ž 1987. carried out quantitative calculations comparing the continuous riming growth rates for graupel with those of unfrozen drops growing by coalescence. He showed that graupel had an advantage over unfrozen raindrops in the regions where cloud droplets were comparably large Ž e.g., 20 mm in diameter.. This difference was reached mainly owing to the utilization of a smaller coalescence efficiency of water drops, E Ž the fraction of collisions that results in coalescence., which was taken from the empirical formulas of Beard and Ochs Ž 1984.. For graupel particles the values of coalescence efficiency in the calculations were set equal to 1. The calculated difference in growth rates, however, was not large. For example, the growth rate Ž d Mrdt. of rough graupel with density of 0.3 g cm y3 and melted diameter of 1 mm was found to be greater than that of a corresponding raindrop by a factor of 1.2 for E s0.69, while for E s0.82 the growth rate of graupel was even lower than that of a water drop. This result can be attributed to a smaller terminal velocity of graupel as compared to rain drops and correspondingly, to a smaller swept volume of graupel. Thus, it appears that the fast growth of graupel observed by Rosenfeld and Woodley cannot be fully explained by the mechanism described by Johnson Ž 1987.. Note that faster collisional growth of graupel as compared to rain drops is one of the main points of conceptual model of Rosenfeld and Woodley Ž 1993. for glaciogenic seeding precipitation enhancement in clouds with active coalescence processes. As it is known, in convective clouds Žfor example, those which were the subject of seeding in. y1 Texas during the rain enhancement program cloud water content can reach 3 g kg within the layer, where temperature ranges from about y5 toy158c. In the supercooled, water-rich part of the cloud, small droplets are quickly lifted with the updrafts. If these small cloud droplets are collected within the region by larger raindrops or graupel particles, they contribute to precipitation. Otherwise, these small droplets are transported to the upper levels in clouds and freeze into small ice crystals that spread within the anvils, and many of them are lost eventually by sublimation. We suppose that one of the causes leading to the preferable growth of graupel within a cloud of small droplets is cloud turbulence.

( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113 101 As it is known, clouds are areas of enhanced turbulence. The intensity of turbulence varies widely depending on the type of clouds and their age. Thus, Mazin et al. Ž1984,. 1989 observed in stratus clouds a turbulent energy dissipation rate of 10 cm s as compared to of 100 cm s in small cumuli. Ackerman Ž 1967. found a median value of 207 cm s in well developed cumuli. In deep cumuli Panchev Ž 1971. quotes a Ž. 2 typical value of 700 cm s. Weil et al. 1989 observed values of up to 2000 cm s y3 in large cumulonimbi. In their studies, Khain and Pinsky Ž 1995, 1997. and Pinsky and Khain Ž 1996, 1997. demonstrated that the inertia of drops falling within a turbulent flow leads to an increase of relative velocities between drops and to an increase of the number of collisions per unit of time, which can be interpreted as an increase of the collision kernel. One can expect that turbulence effects are even more pronounced in the case of ice water drop and ice ice collisions, because the large inertia of ice particles is accompanied by a smaller terminal fall velocity. Let us consider a supercooled cloud with some warm rain process in which water drops can grow into precipitation particles in two ways: Ž. a precipitation particles remain supercooled raindrops, and Ž. b the droplets convert to graupel particles upon reaching radius of 200 mm. The aim of the study is to compare the growth rates of supercooled drop-collector and graupel particles colliding with small cloud droplets under non-turbulent conditions and different intensities of cloud turbulence. As a first step, the continuous growth of a single drop-collector and graupel will be analyzed. 2. Drop drop and drop graupel relative velocities in a turbulent flow Statistical properties of the relative velocities between graupel and water drops are calculated under the assumption that graupel particles move as effective spheres. The main consequence of the effective spheres assumption is that for the description of graupel particle motion we shall use the same motion equation as for water drops ŽKhain and Pinsky, 1997., with the exception of still air terminal velocities determined as Ž. 0.6 Ž y1 Rogers and Yau, 1989 : V s343pd in cm s. t, where D is the diameter of the sphere which just circumscribes the particle, in centimeter. Under the effective spheres assumption, D will be referred to as the graupel diameter. y3 The density of graupel is set as 0.3 g cm Žsee, e.g., Johnson, 1987; Takahashi and Kuhara, 1993.. To calculate the statistical characteristics of the relative velocity between drops, or between drops and graupel particles moving within a three-dimensional turbulent flow, an analytical study similar to that in Pinsky and Khain Ž 1997. has been carried out. The statistical properties of a turbulent flow velocity field were set by the latitudinal structure function Ž Batchelor, 1951., which describes the spectrum of homogeneous and isotropic turbulence both in the viscous and inertial subranges Žsee Pinsky and Khain, 1997; Khain and Pinsky, 1997 for details.. The contribution of turbulence effects, as compared to that of gravitational effects, to relative velocities between

102 ( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113

( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113 103 X 2 : 1r2 Fig. 1. Ž. a The dependence of ratio ² DV rdvt on the sizes of drops for dissipation rate of turbulence Ž. Ž. Ž. Ž. 2 energy s100 cm s. b Same as in a, but for s400 cm s. c Same as in a, but for s800 cm s y3. X particles about to collide can be characterized by the ratio ² DV 2 : 1r2 rdvt where X ² DV 2 : 1r2 is the RMS of the relative drop velocities induced by particle inertia and DV t is the relative drop velocity induced by gravity. Since the formation of graupel takes place during the maturing stage of cloud evolution, characterized by the maximum intensity of turbulence, we shall present the results of calculations under eddy dissipation rates up to 800 cm 2 s y3. Fig. 1a,b,c and X Fig. 2a,b,c show the values of ² DV 2 : 1r2 rdvt as a function of sizes of particles about to collide for the dissipation rates s100, 400 and 800 cm 2 s y3, for drop drop and drop graupel collisions. Analysis of Figs. 1 and 2 shows that: Ž. 1 Turbulence effects under comparably strong values of turbulence are significant. In cases of collisions of comparably small drops the X ratio ² DV 2 : 1r2 rdvt is greater than one. It means that for these drops the relative velocity between the drops is determined mainly by their differential inertia, and not by the difference in the velocity of sedimentation. Ž. 2 Relative velocities between graupel particles and water drops about to collide retain their high values for much larger sizes of graupel, than in the case of water drops. This is due to smaller terminal velocities for X graupel particles than water drops of the same mass. Ž. 3 The values of ² DV 2 : 1r2 rdv t in the case of graupel water drop collisions are significantly greater than for water

104 ( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113

( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113 105 X 2 : 1r2 Fig. 2. Ž. a The dependencies of ratio ² DV rdvt on the sizes of drops and graupel about to collide for a Ž. Ž. dissipation rate of turbulence energy s100 cm s. b Same as in a, but for s400 cm s. Ž. c Same Ž. as in a, but for s800 cm s. drops of the same size. It means that turbulence significantly increases the efficiency of riming. 3. Turbulence effects on the collisional growth of water drops and graupel As it was shown in Section 2, the increase of relative velocity leads to an increase of the swept volume and the collision kernel. Due to the drop and ice particles inertia the relative velocity between particles about to collide is greater than the difference in their terminal velocities. As a first approximation, the still air terminal velocity difference < DV < in the expression for the collision kernel KŽr, r X.Ž Pruppacher and Klett, 1978. t X X 2 Ž. Ž. Ž X. < < t Ž. K r,r sp rqr E r,r DV, 1 was simply replaced by the difference in the mean square relative velocities between 2 X particles about to collide DV ² 2 :4 t q DV 1r2, so that the expression for the collision kernel used was: 4 X X 2 X X2 1r2 < < ² : 2 Ž. Ž. Ž. t t Ž. K r,r sp rqr E r,r DV 1q DV rdv. 2

106 ( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113 X In Eqs. Ž. 1 and Ž. 2, r and r are the radii of colliding particles and Er,r Ž X. is their collection efficiency. The applicability of this replacement, in our opinion, can be explained by the fact that the spectrum of relative velocity between particles of different radii depends mainly on the particle with the greater terminal velocity Ž mass. Žsee Figs. 1 and 2.. The increase of the swept volume Žwhich in our case is close to the increase of the particle track length relative to the surrounding turbulent flow. depends mainly on the velocity of the larger particle, and the situation appears similar to the classical approach. X As can be seen from Eq. Ž. 2, if the values of the ratio ² DV 2 : 1r2 rdvt are significant, the collision kernel substantially increases. In Eq. Ž. 2 the collection efficiency can be written as the product of collision efficiency and coalescence efficiency Ž Beard and Ochs, 1984; Johnson, 1987.: EsEcol PE. Ž 3. According to Beard and Ochs Ž 1984., there exists a physical mechanism leading to E -1 in the case of drops collisions, which is related to the ability of water drops to deform; the possible deformation of the large drop should increase, with an associated reduction in the coalescence efficiency. As a graupel particle is unable to deform, this mechanism is not effective in the case of water graupel collisions Ž Johnson, 1987., so that in this case E s1. Note that the utilization of E -1 for drop drop interaction, Fig. 3. Size distributions of cloud droplets used in calculations. The cloud water content was assumed equal to 3gm y3.

( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113 107 which according to Beard and Ochs Ž 1984. can be as small as 0.5, decreases substantially the rate of droplet spectrum broadening and slows down the process of rain formation. The explanation of fast rain formation in warm clouds appears to be a difficult problem in cloud physics even under E s 1 Ž see discussion in Pinsky and Khain, 1996.. The utilization of E - 1 would definitely require an additional mechanism accelerating the size distribution broadening. Note, however, that there exist many uncertainties as regards to the values of the coalescence efficiency. In many studies, the coalescence efficiency of water drops is set equal to 1 Ž e.g., Hall, 1980; Seesselberg et al., 1996; Khain and Sednev, 1996.. We shall simulate the growth of a drop-collector in both cases: E s1 and E -1. Comparable turbulence effects on the collisional growth will be illustrated by calculations of the continuous growth of a water drop-collector and a graupel-collector moving within a cloud of small droplets. The initial masses of a drop-collector and graupel were set the same. This mass corresponds to the radius of a graupel particle equal to 200 mm. This size can be considered to be close to the minimum size of graupel Ž Pruppacher and Klett, 1978.. The calculations were carried out both under non-turbulent and turbulent conditions of different turbulent intensity Ž dissipation rate.. The droplet size distribution was assumed to follow a gamma-distribution. In order to check the sensitivity of drop- and graupel- Fig. 4. Continuous growth of a drop-collector and graupel with time under non-turbulent conditions. Ž. a drop-collector growth with the coalescence efficiencies presented by Beard and Ochs Ž 1984. and Ochs and Beard Ž 1984.; Ž b. growth of a drop-collector with E s1; Ž c. growth of graupel.

108 ( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113 collector growth to the shape of cloud droplet size, three distributions were used Ž Fig. 3.. The maxima in the size distributions are reached at about 7 mm, 8 mm and 10 mm, respectively. In all cases cloud water content was set equal to 3 g cm y3. This cloud water content corresponds to droplet concentrations of 280 cm y3, 180 cm y3 and 95 y3 cm, respectively. The first two values of droplet concentration Žas well as the value of cloud water content used. are typical of the range of values found in fresh growing convective elements of clouds in West Texas Ž Rosenfeld and Woodley, 1996.. The third distribution with a cloud drop concentration of 95 cm y3 corresponds to maritime cumulus clouds observed by Dr. Rosenfeld in Thailand. The sizes of drop-collector and graupel-collector are much greater than those of cloud droplets, and the equation for continuous growth provides good precision for particlecollector growth Ž Pruppacher and Klett, 1978.. The collision efficiencies for drop drop and graupel drop interaction were set as in Hall Ž 1980.. Note that the collision efficiencies of large collectors are only slightly dependent on the size of cloud droplets. As a result, when E s 1 drop-collector and graupel sizes are actually independent of the shape of droplet size spectrum and depend on cloud droplet content only. For example, in similar experiments, the maximum difference in graupel radii caused by utilization of different cloud drop distributions does not exceed 6%. Maximum difference in drop-collector radii was 10.1%. The fact that the difference in drop-collector size is greater than in the case of graupel can be attributed to the fact that the graupel particle is larger than Fig. 5. Same as in Fig. 4, but for the dissipation rate of turbulence energy s400 cm 2 s y3.

( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113 109 a water drop of the same mass. The larger the particle-collector, the sensitivity of the collision efficiency to the size of small droplets is less. In the case of E -1 the sensitivity of the drop-collector growth rate to the shape of cloud drop spectrum is greater because the coalescence efficiency decreases with the increase of cloud drop radius Ž Beard and Ochs, 1984.. The maximum difference in drop-collector sizes reaches 25% in this case. The lowest growth rate is observed for the spectrum with maximum at 10 mm. In the case of the use of other more continental cloud size distributions, the maximum difference in drop-collector radii was about 10%. Taking into account the low sensitivity of the results to the droplet size spectrum shape, the results obtained by utilization of only one-drop size distribution, corresponding to the concentration of 280 cm y3 will be presented. Figs. 4 6 show the time growths of rain drop and graupel particle radii in a non-turbulent flow and in flows with dissipation rates of turbulence 400 cm 2 s y3 and 800 cm 2 s y3, respectively. For graupel particles the growth of melted radii is presented. The growth of the bulk radius of graupel under these conditions is presented in Fig. 7. One can see that under non-turbulent conditions the growth of the bulk radius of a graupel particle by riming is actually equal to that of a rain drop under E s1: toward 700 s the bulk radius of the graupel and the drop radius are equal to 0.35 cm ŽFigs. 4 and 7.. However, taking into account the difference in the densities of a graupel particle and of a water drop, these nearly equal rates of growth of the particle sizes correspond to a faster growth of the drop-collector mass that can be seen by comparison to the growth Fig. 6. Same as in Fig. 4 but for the dissipation rate of turbulence energy s800 cm 2 s y3.

110 ( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113 Ž. Ž. Fig. 7. Continuous growth of graupel bulk radius under a non-turbulent conditions; b s400 cm s and Ž. c s800 cm s. of the graupel melted radius Ž Fig. 4.. Thus, under non-turbulent conditions graupel collects a smaller number of water drops than a drop-collector of the same initial mass Ž under E s1.. The rate of collisional growth of the graupel particle under non-turbulent conditions appears to be two to three times slower than that observed in seeded clouds in West Texas under the cloud water content used Ž Rosenfeld and Woodley, 1996.. To facilitate comparison with the study of Johnson Ž 1987., the growth rates of a drop-collector were calculated using the collection efficiencies E presented by Beard and Ochs Ž 1984. and Ochs and Beard Ž 1984.. The values of the coalescence efficiency ranged from 0.5 to about 0.8 for small droplets of different size. In case of E -1 the radius of a liquid collector grows at about half the rate as the radius of graupel even under non-turbulent conditions. This result agrees well with the findings of Johnson Ž 1987.. Figs. 5 and 6 indicate that turbulence increases the collisional growth rate of a graupel particle to a greater extent as compared to raindrops. The graupel particle diameter reaches 6 mm in 8 min under the dissipation rate of 400 cm 2 s y3 and in 6 min under the dissipation rate of 800 cm 2 s y3. Because the fast growth of graupel was observed in a mature stage cumulus clouds, the values of the dissipation rate of 800 cm 2 s y3 seem to be reasonable, and even higher turbulence intensity in these clouds may be suggested.

( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113 111 Turbulence increases this difference in the coalescence growth rate of graupel and rain drop mass significantly. The dominating role of graupel in turbulent clouds with regard to the collection of small droplets can be clearly seen. 4. Conclusions It is demonstrated that turbulence increases the rate of riming substantially. The terminal fall velocity of large graupel is about half the terminal fall velocity of raindrops of the same mass. Since the multiplication of the increased cross section of a graupel particle by the reduced fall velocity is smaller than 1, the swept volume of the graupel particle in a non-turbulent medium is close to that of a drop of the same mass. However, in a turbulent medium the increase of the swept volume for graupel is more pronounced than for a rain drop. The results of the calculations of continuous growth of raindrops and graupel particles moving within a supercooled cloud of small droplets are presented both in a non-turbulent medium and within turbulent flows of different turbulence intensity. It is shown that in the case of equal coalescence efficiencies for liquid collectors and graupel Ž E s 1. in a non-turbulent medium, the growth rates of graupel and rain drops radii are close, but the mass of liquid collectors grows faster. The rate of graupel growth under non-turbulent conditions seems too slow to explain the observed fast growth of graupel size in seeded clouds Ž Rosenfeld and Woodley, 1996.. Under turbulent conditions typical of mature convective clouds, the growth of graupel is faster than that of a rain drop of the same size. The calculations show that the observed growth rates of graupel can be achieved under turbulence intensities typical of moderate cumulus clouds at the mature stage. Thus, turbulence increases the rate of riming which seems to be the main mechanism of converting cloud droplets into precipitation particles at temperatures where graupel particles are generated in clouds. The increase in the rate of riming promotes the role of ice in precipitation formation. When the coalescence efficiency for liquid collectors and small cloud droplets ranges from 0.5 to about 0.8, the mass of the drop-collectors grows slower than that of graupel even under non-turbulent conditions. Turbulence increases this difference in growth rates substantially. Note, that the coalescence efficiency being less than one, the growth rate of water drops decreases significantly. This fact explains why water drops grow more slowly than graupel, but it cannot explain the observed fast growth of graupel without taking into account the dynamics of graupel motion within a turbulent flow. Note that the significant difference in the mass of graupel and water drops crossing the melting level found by Johnson Ž 1987. is attributed to the fact that due to the slower fall velocity it takes graupel particles a longer period to fall out. As one sees in Figs. 6 and 7, continuous growth rate by coalescence increases with time Žunder the assumption that cloud water content remains unchanged with time.. Thus, a comparably small Ž several minutes. difference in the duration of drop and graupel location within a cloud can lead to a substantial difference in their mass. In this paper we focus on another effect, where graupel wins in the competition with water drops during a short period of time by collecting a greater number of small cloud droplets.

112 ( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113 The finding of the effect that graupel particles collect cloud droplets more efficiently than unfrozen rain drops has far reaching implications: e.g., glaciogenic seeding for precipitation enhancement in clouds with an active warm rain processes. A cloud seeding method that will lead to the preferable freezing of larger drops, while leaving most of the smaller cloud droplets unfrozen, will lead to more effective collection of small droplets and possibly to precipitation enhancement. Note that the results were obtained using a simple model of the continuous growth of drop-collector and graupel within a homogeneous field of small cloud droplets. More detailed calculations using complicated cloud models with detailed microphysics and parameterized effects of turbulence are necessary to make definitive statements concerning this problem. Note that in this study the collisions between large collectors and small cloud droplets were considered. It was shown that cloud turbulence can change the relationship between growth rate of largest droplets at early stage of rain formation and growth rate of crystals by riming. This problem needs further investigation. Acknowledgements The authors are grateful to the anonymous referees for their valuable comments on the manuscript. This study was partially supported by the Germany Israel Science Foundation Ž grant 0407-008.08r95. and the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities Ž grant 572r97.. References Ackerman, B., 1967. The nature of the meteorological fluctuations in clouds. J. Appl. Meteorol. 6, 61 71. Batchelor, G.K., 1951. Pressure fluctuations in isotropic turbulence. Proc. Cambridge Philos. Soc. 47, 359 374. Beard, K.V., Ochs, H.T., 1984. Collection and coalescence efficiencies for accretion. J. Geophys. Res. 89, 7165 7169. Bergeron, T., 1935. On the physics of cloud and precipitation. Proc. 5th Assembly IUGG, Lisbon, pp. 156 178. Braham, R.R. Jr., 1964. What is the role of ice in summer rainshowers? J. Atmos. Sci. 21, 640 645. Braham, R.R. Jr., 1968. Meteorological bases for precipitation development. Bull. Am. Meteorol. Soc. 49, 343 353. Hall, W.D., 1980. A detailed microphysical model within a two-dimensional dynamic framework: model description and preliminary results. J. Atmos. Sci. 37, 2486 2507. Johnson, D.B., 1987. On the relative efficiency of coalescence and riming. J. Atmos. Sci. 44, 1671 1680. Khain, A.P., Pinsky, M.B., 1995. Drops inertia and its contribution to turbulent coalescence in convective clouds: Part 1. Drops fall in the flow with random horizontal velocity. J. Atmos. Sci. 52, 196 206. Khain, A.P., Pinsky, M.B., 1997. Turbulence effects on collision kernel: Part 2. Increase of the swept volume of colliding drops. Q. J. R. Meteorol. Soc. 123, 1543 1560. Khain, A.P., Sednev, I.L., 1996. Simulation of precipitation formation in the Eastern Mediterranean coastal zone using a spectral microphysics cloud ensemble model. Atmos. Res. 43 Ž. 1, 77 110. Mazin, I.P., Silaeva, V.I., Strunin, M.A., 1984. Turbulent fluctuations of horizontal and vertical wind velocity components in various cloud forms. Izvestia. Atmos. Oceanic Phys. 20, 6 11. Mazin, I.P., Khrgian, A.Kh., Imyanitov, I.M., 1989. Handbook of Clouds and Cloudy atmosphere. Gidrometeoizdat, 647 pp.

( ) M. Pinsky et al.ratmospheric Research 49 1998 99 113 113 Ochs, H.T., Beard, K.V., 1984. Laboratory measurements of collection efficiencies for accretion. J. Atmos. Sci. 41, 863 867. Panchev, S., 1971. Random Fluctuations in Turbulence. Pergamon, 256 pp. Pinsky, M.B., Khain, A.P., 1996. Simulations of drops fall in a homogeneous isotropic turbulence flow. Atmos Res. 40, 223 259. Pinsky, M.B., Khain, A.P., 1997. Turbulence effects on the collision kernel: Part 1. Formation of velocity deviations of drops falling within a turbulent three-dimensional flow. Q. J. R. Meteorol. Soc. 123, 1517 1542. Pruppacher, H.R., Klett, J.D., 1978. Microphysics of Clouds and Precipitation. Reidel, 714 pp. Rogers, R.R., Yau, M.K., 1989. A Short Course in Cloud Physics. Pergamon, 293 pp. Rosenfeld, D., Woodley, W.L., 1993. Effects of cloud seeding in West Texas: additional results and new insights. J. Appl. Meteorol. 32, 1848 1866. Rosenfeld, D., Woodley, W.L., 1996. Microphysical response of summer convective clouds in West Texas to AgI dynamic seeding. 12th International Cloud Physics Conference, Zurich, August, pp. 1325 1328. Seesselberg, M., Traumann, T., Thorn, M., 1996. Stochastic simulations as a benchmark for mathematical methods solving the coalescence equation. Atmos. Res. 40, 33 48. Takahashi, T., Kuhara, K., 1993. Precipitation mechanisms of cumulus clouds at Pohnpei, Micronesia. J. Meteorol. Soc. Jpn. 71, 21 31. Weil, J.C., Lawson, R.P., Rodi, A.R., 1989. Relative dispersion of ice crystals in seeded cumuli. J. Appl. Meteorol. 32, 1055 1073.