A theoretical study of the microphysical structure of mixed stratiform frontal clouds and their precipitation

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1 Ž. Atmospheric Research A theoretical study of the microphysical structure of mixed stratiform frontal clouds and their precipitation Svetlana V. Krakovskaia ), Anne M. Pirnach Ukrainian Hydrometeorological Research Institute, KieÕ, Ukraine Abstract One-dimensional numerical models with a detailed description of the evolution of cloud particles Ž drops, crystals, cloud nuclei, snowflakes, etc.. are used to study the microphysical processes in supercooled winter frontal clouds. A set of equations is used to simulate the evolution of the processes of condensation, nucleation, freezing, sedimentation, accretion, collection, aggregation, etc. The radius spectrum of liquid drops is divided into two parts: the diameter either smaller or greater than 20 mm. The spectrum of cloud droplets is formed from the condensation, turbulent diffusion and motion of drops. The size distribution function of the ice particles is assumed to form due to sublimation, turbulent diffusion, motion, riming, accretion and aggregation. Simulations of mixed supercooled clouds are examined to see what extent the different microphysical processes Ž such as collection, aggregation, freezing, accretion, riming, etc.. and thermodynamical conditions Ž such as surface temperature and updrafts. can impact on the development of cloud and precipitation. They have shown that the liquid, as well as solid, precipitation from supercooled mixed clouds may be significant, especially at surface temperatures greater than 08C. The influence of updraft values on liquid precipitation is significant, while the surface temperature affects liquid phase precipitation slightly. The opposite holds for solid precipitation; the temperature is a principal factor, while the updrafts affect solid precipitation only at temperatures above 08C. A study of the different mechanisms of cloud and precipitation formation shows that all such mechanisms are important. If one is absent, others compensate and can form the precipitation successfully. The obtained spectra of cloud droplets and ice crystals conform to a g-distribution and spectra of the raindrops correspond to the power distribution. Inclusion of new mechanisms of CCN originating within the cloud changes the droplets spectrum ) Corresponding author. Ukrainian Hydrometeorological Research Institute, 37 Nauki Avenue, Kiev , Ukraine. Tel.: q ; fax: q ; krasvet@ozsol.kiev.ua r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. Ž. PII S

2 492 ( ) S.V. KrakoÕskaia, A.M. PirnachrAtmospheric Research noticeably. Numerical experiments confirm that spectra of aggregates conform to an exponential law and indices of the distributions decrease with an increase of precipitation formation intensity and size of particles. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Cloud microphysics; Precipitation; Spectra of cloud particles; Numerical simulation 1. Introduction Microphysical and dynamical processes in winter stratiform frontal clouds have been the subject of continuous theoretical and experimental investigations for many years at the Ukrainian Hydrometeorological Research Institute. As a result, the fundamental theory of the stratiform frontal cloudiness and precipitation formation has been developed ŽBuikov and Dehtyar, 1968; Buikov and Pirnach, 1972, 1976, 1984; Pirnach, 1976; Khvorostyanov, 1987; Pirnach and Krakovskaya, 1994; etc... In the present research, one-dimensional numerical models with a detailed description of the evolution of cloud particles Ž droplets, raindrops, crystals, cloud nuclei, etc.. are used to study the microphysical processes in supercooled winter frontal clouds. The theoretical studies mixed stratiform clouds, with the aid of detailed microphysical numerical models, have shown that models are good tools in the study of the internal structure and microphysical processes of clouds, particularly frontal ones. The paper does not aim at a perfect correlation between modelled and observed data. To focus attention on the important processes is the ultimate goal in the creation of the microphysical models and their use as a tool in analysis; ultimately to explore to what extent the different microphysical processes Žsuch as collection, aggregation, freezing, accretion, riming, etc.. and thermodynamical conditions Žsuch as surface temperature and updrafts. can influence cloud and precipitation development. 2. A short description of the model Only the essential details for a full understanding of the presented results and the principles of model construction are described. The reader is referred to Buikov and Dehtyar Ž 1968., Buikov and Pirnach Ž 1972, 1976, 1984., Pirnach Ž 1976, 1979., Pirnach and Krakovskaya Ž and Akimov et al. Ž for a detailed presentation of the methodology, equations and numerics. The spectrum of liquid cloud drops is divided into two parts: cloud droplets Žwith radii smaller than 20 mm. and raindrops Ž radii greater than 20 mm.. The spectrum of cloud droplets is the result of condensation, turbulent diffusion and dynamical motion. Additionally, the spectrum of raindrops is determined by the collection of droplets. The size distribution function of the ice particles is assumed to result from sublimation, turbulent diffusion, motion, riming, glaciation Ž heterogeneous freezing., accretion and aggregation.

3 ( ) S.V. KrakoÕskaia, A.M. PirnachrAtmospheric Research The set of the basic model equations is as follows: d fi E Efi q Ž rf i i. yõ i sl a i "l fi yd f i qd f i, dt E r EZ d fj E E Efj q Ž rf j j. q Ž r c jfj. yõj s"l fjqd f j, dt E r E r EZ dt 4 syg a wq Ý a qdt k k, dt ks1 dq 4 sy Ý kqdq, akslkrc p, 1 dt ks1 where isks1 is for droplets, isks2 is for crystals, js1 and ks3 is for raindrops, j s 2 and k s 4 is for aggregates; f and f are cloud Ž CP. i j and precipitation particles Ž PP. size distribution functions; r i and r j are rates of CP s and PP s growths due to condensation Ž sublimation.; r c i is rate of continuous coagulation growth of PP; d fi describes the decrease of CP by collection; Ia i are values that describe generation of the cloud condensation and ice nuclei; If i and If j are values that describe freezing of droplets and raindrops; DS Ž S sf, f, T, q. i i i j describe turbulent transfer; Õi and Õj are the fall speeds of CP and PP; T and q are the temperature and specific humidity of air; are rates of water vapour condensation Ž sublimation. k ; w are updrafts; ga is the dry-adiabatic temperature gradient; L is the latent condensation Ž sublimation. k heats, and Cp is the specific heat at constant pressure of dry air. Updrafts in the model have an elliptical profile and are simulated in a layer 0-z -4 km, where z s2 km is a height of the maximum updraft Ž w. m m m. Updraft speeds, surface temperature and humidity are constant during a cloud evolution. The depth of the updraft layer is a target parameter. The top of the layer limits the vertical development of the cloud. A frontal cloud top at zs4 km is common in the Ukraine. The calculated values fi give a full description of the cloud microphysics and allow the derivation of the integrated characteristics, such as concentrations of the cloud particles Ž N., their averaged radii Ž r., the water Ž ice. contents Ž q. i i i, the total precipita- tion rate Ž j. etc. Ž. 3. Droplet collection by raindrops and ice particles The numerical runs presented in this section focus on the gravitational collection by large drops Ž raindrops. and ice particles of small drops Ž droplets.. The kernel of gravitational collection and coefficients of accretion for solid and liquid precipitation are calculated as in Shishkin Ž 1964.: 3 RR 0 Ec is 1y, R0 s14.5 mm 2< 2 2 4r R yr < 2

4 494 ( ) S.V. KrakoÕskaia, A.M. PirnachrAtmospheric Research Fig. 1. Diagrams of solid and liquid precipi-tation intensities averaged over 24 h and their differences depending on thermodynamic characteristics. Ž. a numbers in lines are solid and liquid Ž dashed line. precipitation rates Ž mmrh.; Ž b. differences of solid and liquid precipitation rates Ž mmrh.. where R, r and R are the respective raindrops Ž or ice crystals. 0, droplets and minimum collection radii Ži is as in Eq. Ž Interaction of thermodynamical characteristics with cloud microphysics To estimate the effect of thermodynamic characteristics Žsuch as surface temperature T and maximum updrafts w in the middle of the cloud. s m on the development and phases of precipi-tation, simulations of microphysical features under different thermodynamic conditions in the atmosphere are studied. The results of many numerical runs are presented in Fig. 1a. The diagram shows that while the maximum updraft in the middle of simulated cloud determines liquid precipitation intensity, the surface temperature Ts does not significantly affect the liquid phase precipitation. At the same time, the solid precipitation rate depends more on T s than on w m. For T s) 2738K, the influence of wm on solid precipita-tion is significant: for example at Ts s 2758K, the precipitation intensity increases from 0.1 at wm s 2 cmrs, to 0.8 mmrh atwm s10 cmrs. Fig. 1b shows the differences between the solid and liquid pre-cipitation rates. The zero isoline means equal contributions to the precipitation rate from the solid and liquid phases. From the diagram, the droplet collection by raindrops is important in the temperature range 268 to 275 K. For lower temperatures, the isolines are almost parallel to the ordinate Ž see Fig. 1b.. Liquid precipitation rates of more than 1 mmrh are less than that for solid ones, and therefore may be excluded from the simulation. It should be noted that wm of about 10 cmrs and winter stratiform frontal clouds of 4 km depth, especially at Ts lower than 2688K, are rare events for extratropical cyclones. Hence, precipitation intensities of more than 3 mmrh for each water phase Žin the upper left region of Fig. 1. are more appropriate for cumulus clouds or embedded convection in stratus The influence of freezing parameter and accretion processes on precipitation deõelopment In this section, the results of the freezing parameter effect on precipitation intensities and cloud microphysics are presented. The freezing term If i in Eq. Ž. 1 is as follows: I sa expž BT. r 3 f Q Ž T.,T s273.15yt, Ž 2. f i f f f i i f f

5 ( ) S.V. KrakoÕskaia, A.M. PirnachrAtmospheric Research Fig. 2. Precipitation intensities, j Ž mmrh., in dependence on surface temperature, T s, freezing coefficient, A f, and time of evolution with w s2 cmrs: Ž. a and Ž. d ice precipitation without accretion processes; Ž. b and Ž. m e ice precipitation with accretion processes; Ž. c and Ž. f liquid precipitation with collection processes. where A and B are the empirical constants Ž f f Fletcher, 1962; Vali, 1968, 1975; Buikov and Pirnach, 1976, 1984.; Q Ž T. s1 for T )0 and Q Ž T. f f f s0 for Tf-0. Fig. 2 shows the precipitation rate dependence on the cloud evolution and freezing coefficient, A f. The study is conducted for surface temperatures Ts s 268 K and Tss 273 K and wms 2 cmrs. The accretion processes in the simulation increases pre-cipitation rate by a factor of two at T s 2688K and accelerates the onset of s precipitation by 7 h at Ts s2738k. At the same time, the accounting of collection allows the estimation of the contribution of liquid precipitation to the total sum. Thus, for T s 268 K, the averaged Ž for all A. s f contribution of liquid precipitation is 17% of the total Ž with a maximum of 34%.. For T s273 K, the contribution increases to 49% Ž s with the maximum of 84%.. Thus, even with such weak dynamics Ž w s 2cmrs. m, the inclusion of accretion of ice particles and droplet collection by raindrops is necessary for all Af and allows the simulation of more realistic clouds Microphysics of the simulated stratiform cloud In this section, a more detailed example of stratiform cloud simulation is presented. The emphasis is on the integrated microphysical characteristics Žsee Eq. Ž 2.. and precipitation development with and without accretion. Fig. 3 presents an example of one of the runs used to construct the diagrams in Fig. 1. It shows the evolution of the microphysical characteristics of the cloud. The initial conditions correspond to the zero isoline in Fig. 1b with equal contributions to the

6 496 ( ) S.V. KrakoÕskaia, A.M. PirnachrAtmospheric Research Fig. 3. Vertical cross-sections of integral microphysical characteristics of the cloud at Tss271 K and wms6 cmrs: N_cr and N_dr are ice and drop concentration; q_cr and q_dr are ice and liquid water content; r_cr is mean radius of ice CP, and r_dr is mean radius of cloud drops. precipitation amount by the two water phases. They are Tss271 K, wms6 cmrs and y3 Ž 3 freezing coefficient A s2p10 1r cm s. t. Other numerics are described in Pirnach Ž and Pirnach and Krakovskaya Ž As illustrated in Fig. 3, the ice concentration, N cr, and the ice content, q cr, of the cloud achieve their maxima at the same time as the drop concentration, N dr. Thus, Ncr equals to 7.6P10 3 m y3 near the cloud top at zs3.4 km and Ndr equals to 0.9P10 6 m y3 at zs1.2 km at time ts9 h of the cloud evolution. This is the time of the maximum solid precipitation rate Ž see curve 3 in Fig. 4b.. The maximum water content q Ž Fig. 3. and liquid precipitation rate Ž see curve 4 in Fig. 4b. dr is delayed by 2 h from the drop concentration maximum. Diagrams for average sizes of ice CP, r cr, and drops, r dr, are shown in Fig. 3. They represent mean radii of CP and indicate intense widening of the cloud spectra and growth of drops due to collection of droplets, particularly during the initial 6 h of the cloud evolution. As described below, after that the drop spectra becomes steadier. It is interesting to make a comparison between concentrations of ice nuclei and ice CP. The relation of Hobbs et al. Ž is used in this investigation, and ice nuclei concentration is set to 3P10 y2, 1 and 30 g y1 when T equals to y10, y20 and y268c, respectively. In this research, these temperatures correspond to heights of 1, 2.4 and 3.7 Ž 3 y3 km and to ice concentrations of 2.9, 4.1 and m. at ts9 h. Therefore, in the

7 ( ) S.V. KrakoÕskaia, A.M. PirnachrAtmospheric Research Fig. 4. Time evolution of precipitation rates with and without accretion processes. Ž. a Summarised precipitation rates of solid and liquid phases: 1 only Findeisen Bergeron precipitation formation mechanism; 2 simulation with accretion; 3 simulation with collection raindrops for droplets without accretion accounting; 4 both accretion and collection. Ž. b Separated solid Ž 1, 3. and liquid Ž 2, 4. precipitation rates corresponded to 3 and 4 curves in Ž. a, respectively. lower cloud, Ncr is two orders of magnitude larger than the ice nucleus concentration, indicating that ice enhancement by seeding from upper part of the cloud, from «seeder zones», is occurring. Note that Ndr is two orders of magnitude greater than N cr. This agrees with the results obtained in the previous investigations ŽHobbs et al., 1975; Buikov and Pirnach, An investigation of precipitation development and its dependence on collection processes is presented in Fig. 4. Curve 1 in Fig. 4a presents the Findeisen Bergeron mechanism. In this case, the total 24-h precipitation amount is only 9.4 mm, and this is insufficient. Curve 2 in the same figure includes accretion of droplets by ice crystals. The total precipitation amount in this simulation is 34.0 mm but still insufficient. Curve 3 presents the case with the collection of cloud droplets by raindrops for droplets. It is the sum of the solid and the liquid phase precipitation presented in Fig. 4b Žcurves 1 and 2.. In this case, the 24-h precipitation amount is 46.7 mm. The curves 4 in Figs. 4 and 3 and 4 in Fig. 4b, correspond to the above described case including accretion processes for both ice crystals and raindrops. As above, this is an example of equal contributions by the liquid and solid phases to the precipitation rate. Hence, curves 3 and 4 in Fig. 4b are close to each other, and their sum is presented in Fig. 4a as curve 4. In this case, the total pre-cipitation amount for 24-h cloud development is 48.9 mm. The inclusion of latent heat of condensation, sublimation and freezing processes Žwith the absence of melting and evaporation since the temperature of the atmosphere is under 273 K. produces a temperature inversion in the low troposphere after this time. It is clear from Figs. 3 and 4, that under the above thermodynamic conditions and microphysical parameters, the raindrops collection of droplets is an important mechanism for precipitation formation and alone could be sufficient to produce the required cloud moisture. If this mechanism is included, ice crystal accretion could be excluded from the simulation Ž see Fig. 4a, cases 3 and 4..

8 498 ( ) S.V. KrakoÕskaia, A.M. PirnachrAtmospheric Research Influence on the winter precipitation of the collection of cloud particles by solid precipitation particles As an example, a case of frontal cloud observed in the field experiments of January 6, 1976 is more presented Žsee Akimov and Leskov, 1977; Pirnach, 1979; Akimov et al., For the numerical simulation the one-dimensional model described in Pirnach Ž is used. The constants and parameters are taken from Pirnach Ž The initial fields are constructed by interpolating the rawinsonde data. The temperature decreases from y108c toy368c at0-z-5 km and the humidity exceeds 90% at the initial time. The plate crystals and snowflakes are chiefly presented. These particle shapes are used in the model. The numerical simulation shows that the cloud is located at 1-z-3 km above a thin mixed layer. The presence of deep mixing cloud caused precipitation formation by sublimation and coagulation. The mixing layer disappears after a short time Ž about 1 h., following which the cloud becomes crystallized both in the observations and in the cloud simulation. Occasionally, mixed layers appear and force a brief but intense activation of aggregation processes. According to most research, the distribution function of snowflakes follows an exponential law. In Gunn and Marshall Ž 1958., size distribution curves for melted crystals with diameters Ž D. up to 4 mm were approximated by the expression: fsf0 e yl D where f0 is an initial parameter. For these sizes l changes from 1.8 to 3.8 mm y1. Rogers Ž have fulfilled an investigation of snowflakes distribution in framework from 1 to 15 mm for a large number of snowfalls and found that index of the Table 1 Precipitation intensity, j Ž mmrh., and 20-h total precipitation with different precipitation formation mechanisms No. t Ž. h S, Ž mm Ž. 1 Sublimation, riming, aggregation Žthe coagulation coefficient for riming, Ers1, for aggregation, Ea was computed as in Rogers, Ž. 2 Sublimation, riming. Ž. 3 As number 1 and Ea s1. Ž. 4 As 1 and nucleation occurred if ice supersaturation D s )0. Ž. 5 Sublimation, aggregation. Ž. 6 Sublimation. In Cases 1, 2, 3, 5, 6, nucleation occurred if liquid water supersaturation D 1)0.

9 ( ) S.V. KrakoÕskaia, A.M. PirnachrAtmospheric Research exponential distribution in the investigated size limits was roughly constant and equal to 0.296"0.145mm y1. For the considered field experiment the simulated distributions of aggregates with sizes of 1.2-r-4 mm confirm that spectra of such particles agree to an exponential law with the almost constant index of 4.5 mm y1. A small number of aggregates with r)4 mm is observed in this case. Thus, simulations were carried out for particles less than 4 mm. Curves of size distribution functions can be divided into 2 3 straight parts. The index l of the exponential distribution decreases with an increase in particle sizes and intensification of precipitation formation mechanisms. It changes from 4 5 to y mm Ž Pirnach, Table 1 shows the precipitation rates and amounts for different precipitation mechanisms showing a strong influence on precipitation rate by riming processes. The role of snowflakes is negligible if the riming mechanism is working. The appearance of the mixing layer stimulates the riming and aggregation processes and produces the highest precipitation rates. The precipitation amounts are defined by cloud dynamics chiefly if the precipitation mechanism is capable of realising the liquid water and all the free water vapour accumulated in the cloud. There is a question of which precipitation formation mechanism would operate most efficiently. In the absence of one particular mechanism, the others compensate and can form the precipita-tion successfully. 5. Spectra of cloud particles This section is devoted to spectra of cloud particles obtained in the above investigation considered in Section 3.3. These spectra are presented in Figs The ice and liquid water nucleation mechanisms used in this study are: Ž. 1 the continuous source of droplets into the cloud base with water supersaturation; Ž. 2 the generation of droplets in the clouds if there is further water supersaturation exceeding that at cloud base; Ž. 3 ice nucleation in layers with the water supersaturation; and Ž. 4 the freezing of drops. These processes are parameterised using theoretical and experimental data cited in Hallet and Mason Ž 1958., Latham and Mason Ž 1958., Mason Ž 1971., Fletcher Ž 1962., Vali Ž 1968, 1975., Auer Ž 1971., Buikov and Pirnach Ž 1972, 1984., Pruppacher and Klett Ž 1978., and Hobbs et al. Ž 1975, These relationships can be found in Pirnach and Krakovskaya Ž In Fig. 5, the spectra of ice CP corresponds to the g-distribution. At ts3 h, the size distribution functions become narrower with height, but at z s 4 km, the spectrum is wider. The latter could be explained by the assumption of nucleation at water supersaturation only. The best conditions for nucleation are at z s 3.2 km, where ice concentra- 3 y3 tion is 4.4P10 m. and Ts247 K. During cloud development, the spectra of ice CP becomes narrower, particularly at zs4 km and after ts12 h of the cloud simulation. The spectra of droplets presented in Fig. 6 also conform to the g-distribution. These results differ from those obtained in Buikov and Pirnach Ž 1972.; the difference is the inclusion in the present study of a new process of cloud condensation nuclei generation

10 500 ( ) S.V. KrakoÕskaia, A.M. PirnachrAtmospheric Research Fig. 5. Size distribution functions for ice cloud particles depending on time and height. Fig. 6. Size distribution functions for droplets depending on time and height.

11 ( ) S.V. KrakoÕskaia, A.M. PirnachrAtmospheric Research Fig. 7. Spectra of cloud drops depending on time and height. Fig. 8. The same as for Fig. 7.

12 502 ( ) S.V. KrakoÕskaia, A.M. PirnachrAtmospheric Research in the cloud. This occurs when water supersaturation Ž D. l in inner part exceeds D l at cloud base. The size distribution functions for droplets at z s 3 km are the narrowest, while at zs4 km they are the widest. Consequently, the most intensive process of CCN generation occurs at z s 3 km, where most suitable and constant conditions for the droplets formation exist. Figs. 7 and 8 show cloud spectra dependence on time and height. Particles with 20- r- 50 mm corresponds to the transition between droplets described by the g-distribution to raindrops with a power distribution. This is a characteristic feature for the start of the evolution and at the cloud edges. After 6 h, the cloud becomes more steady and spectra have a closer agreement with the power distribution. Indices for the distributions vary from 2 to 11 and correlate with data of Borovikov et al. Ž Conclusions One-dimensional numerical models with detailed description of the evolution of cloud particles Ž drops, crystals, cloud nuclei, snowflakes, etc.. are used to study the microphysical processes into supercooled winter frontal clouds. A number of simulations of mixed supercooled clouds have studied the impact on the development of cloud and precipitation by different microphysical processes Žsuch as collection, aggregation, freezing, accretion, riming, etc.. and thermodynamical conditions Žsuch as surface temperature and updrafts.. For all the studied parameters and selected thermodynamic conditions, the liquid precipitation of simulated supercooled mixed clouds has been significant, especially at surface temperatures greater than 2738K. The influence of updraft speeds on liquid precipitation is significant, while the surface temperature affects liquid phase precipitation only slightly. The opposite holds for solid precipitation where the temperature is a principal factor, and the updraft speed affects solid precipitation only at relatively high temperatures. The total amount of precipitation depends on both cloud dynamics and microphysics. The study of the different mechanisms of precipitation formation has shown that every mechanism is important. The absence of one particular mechanism results in others becoming more active and the precipitation can be successfully produced. The obtained spectra of cloud droplets and ice crystals conform to a g-distribution and the spectra of the raindrops corresponds to a power distribution. Numerical experiments confirm that spectra of aggregates conform to an exponential law and indices of the distributions decrease with increasing precipitation formation intensity and the size of particles. Acknowledgements This work was supported in part through Grant No. PSU of the International Soros Science Education Program Ž ISSEP.. Special thanks are due to Dr. Alan Gadian, guest editor, who helped edit the paper.

13 ( ) S.V. KrakoÕskaia, A.M. PirnachrAtmospheric Research References Akimov, N.M., Leskov, B.N., About periodicity of particles concentration changing and intensity of solid precipitation. Trudy UHRI 162, 91 96, Ž in Russian.. Akimov, N.M., Palamarchuk, L.V., Pirnach, A.M., Investigation of intensity and microstructure of winter frontal precipita-tion for real synoptic situation. Trudy UHRI 203, 56 69, Ž in Russian.. Auer, A.N., Observation of ice crystal nucleation by drop freezing in natural clouds. J. Atmos. Sci. 28 Ž. 2. Borovikov, A.M., Mazin, I.P., Nevsorov, A.N., Some features of large particles distributions in the different cloud types. Izv. AN USSR, Ser. Phys. Atmos. Oceans 1, , Ž in Russian.. Buikov, M.V., Dehtyar, M.I., Stratiform clouds theory. Trudy UHRI 70, 21 49, Ž in Russian.. Buikov, M.V., Pirnach, A.M., Investigation of the droplets size distribution function in a two-phases stratiform cloud. Numerical experiment. Trudy UHRI 114, 3 13, Ž in Russian.. Buikov, M.V., Pirnach, A.M., Influence on precipitation formation into mixing stratiform clouds of coagulation processes. Trudy UHRI 144, 3 13, Ž in Russian.. Buikov, M.V., Pirnach, A.M., Simulation of occluded frontal cloudiness features. Trudy UHRI 161, 26 35, Ž in Russian.. Buikov, M.V., Pirnach, A.M., Influence on the precipitation formation of the intensity of ice formation mechanisms. Trudy UHRI 199, 19 28, Ž in Russian.. Fletcher, N.H., The Physics of Rainclouds. Cambridge Univ. Press., London, p Gunn, K., Marshall, J., The distribution with size of aggregate snowflakes. J. Meteorol. 15 Ž. 5, Hallet, J., Mason, B.J., The influence of temperature and supersaturation on the habit of ice crystal growth from vapour. Proc. R. Soc. A 247, Hobbs, P.V., Houze, R.A., Mateika, T.I., The dynamical and microphysical structure of the occluded front and its modification by orography. J. Atmos. Sci. 32, Hobbs, P.V., Bowdle, D.A., Radke, L.F., Aerosol over Height Plains of the University of Washington. Research Report, XII, 143 pp. Khvorostyanov, V.I., A three-dimensional numerical model of cloud crystallization under seeding with solid carbon dioxide. Meteorol. Hydrol. 4, 29 38, Ž in Russian.. Latham, R., Mason, B.J., The heterogenous nucleation of super-cold water. Proc. R. Soc. A 247 Ž 1951., Mason, B.J., The Physics of Cloud. Oxford Univ. Press, London, 540 pp. Pirnach, A.M., Influence on mixing cloud microstructure of a collection by solid precipitation particles of cloud drops. Trudy UHRI 144, 20 24, Ž in Russian.. Pirnach, A.M., Study of aggregation in stratiform mixing clouds Ž numerical experiment.. Trudy UHRI 170, 32 43, Ž in Russian.. Pirnach, A.M., Krakovskaya, S.V., Numerical studies of dynamics and cloud microphysics of the frontal rainbands. Atmos. Res. 33, Pruppacher, H.R., Klett, J.D., Microphysics of Cloud and Precipitation. Reidel, Dordrecht, 714 pp. Rogers, The aggregation of natural ice crystals. Report No. AR 110, 35 pp. Shishkin, N.S., Clouds, Precipitation and Thunderstorm Electricity. Gidrometeoizdat, p. 280 Žin Russian.. Vali, G., Ice nucleation relevant to formation of hail. McGill Univ. Stormy Wea. Group Rep. MW-58, 51 pp. Vali, G., Remarks of the mechanisms of atmospheric ice nucleation. In: Proc. VIII ICN, Gidrometeoizdat, Moscow, pp