Analytical Solutions to the Stochastic Kinetic Equation for Liquid and Ice Particle Size Spectra. Part I: Small-Size Fraction

Vitaly I. Khvorostyanov Central Aerological Observatory, Dolgoprudny, Moscow, Russia

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Judith A. Curry School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, Georgia

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Abstract

The kinetic equation of stochastic condensation for cloud drop size spectra is extended to account for crystalline clouds and also to include the accretion–aggregation process. The size spectra are separated into small and large size fractions that correspond to cloud drops (ice) and rain (snow). In Part I of this two-part paper, analytical solutions are derived for the small-size fractions of the spectra that correspond to cloud drops and cloud ice particles that can be identified with cloud liquid water or cloud ice water content, and used in bulk microphysical schemes employed in cloud and climate models. Solutions for the small-size fraction have the form of generalized gamma distributions. Simple analytical expressions are found for parameters of the gamma distributions that are functions of quantities that are available in cloud and climate models: liquid or ice water content and its vertical gradient, mean particle radius or concentration, and supersaturation or vertical velocities. Equations for the gamma distribution parameters provide an explanation of the dependence of the observed spectra on atmospheric dynamics, cloud temperature, and cloud liquid water or ice water content. The results are illustrated with example calculations for a crystalline cloud. The analytical solutions and expressions for the parameters presented here can be used for parameterization of the small-size fraction size spectra in liquid and crystalline clouds and related quantities (e.g., optical properties, lidar, and radar reflectivities).

Corresponding author address: Dr. J. A. Curry, School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, GA 30332. Email: curryja@eas.gatech.edu

Abstract

The kinetic equation of stochastic condensation for cloud drop size spectra is extended to account for crystalline clouds and also to include the accretion–aggregation process. The size spectra are separated into small and large size fractions that correspond to cloud drops (ice) and rain (snow). In Part I of this two-part paper, analytical solutions are derived for the small-size fractions of the spectra that correspond to cloud drops and cloud ice particles that can be identified with cloud liquid water or cloud ice water content, and used in bulk microphysical schemes employed in cloud and climate models. Solutions for the small-size fraction have the form of generalized gamma distributions. Simple analytical expressions are found for parameters of the gamma distributions that are functions of quantities that are available in cloud and climate models: liquid or ice water content and its vertical gradient, mean particle radius or concentration, and supersaturation or vertical velocities. Equations for the gamma distribution parameters provide an explanation of the dependence of the observed spectra on atmospheric dynamics, cloud temperature, and cloud liquid water or ice water content. The results are illustrated with example calculations for a crystalline cloud. The analytical solutions and expressions for the parameters presented here can be used for parameterization of the small-size fraction size spectra in liquid and crystalline clouds and related quantities (e.g., optical properties, lidar, and radar reflectivities).

Corresponding author address: Dr. J. A. Curry, School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, GA 30332. Email: curryja@eas.gatech.edu

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  • Ackerman, S. A., and G. L. Stephens, 1987: The absorption of solar radiation by cloud droplets: An application of anomalous diffraction theory. J. Atmos. Sci., 44 , 15741588.

    • Search Google Scholar
    • Export Citation
  • Austin, P. H., M. B. Baker, A. M. Blyth, and J. B. Jensen, 1985: Small-scale variability in warm continental cumulus clouds. J. Atmos. Sci., 42 , 11231138.

    • Search Google Scholar
    • Export Citation
  • Borovikov, A. M., I. P. Mazin, and A. N. Nevzorov, 1965: Some features of distribution of the large particles in clouds of various forms. Izv. Acad. Sci. USSR, Atmos. Oceanic Phys., 1 , 357369.

    • Search Google Scholar
    • Export Citation
  • Brenguier, J-L., and L. Chaumat, 2001: Droplet spectra broadening in cumulus clouds. Part I: Broadening in adiabatic cores. J. Atmos. Sci., 58 , 628641.

    • Search Google Scholar
    • Export Citation
  • Buikov, M. V., 1961: Kinetics of distillation in a polydisperse fog. Izv. Acad. Sci. USSR, Ser. Geophys., 7 , 10581065.

  • Buikov, M. V., 1963: A method of the kinetic equations in the theory of clouds. Proc. All-Union Meteor. Conf., Vol. 5, Leningrad, Russia, USSR State Committee for Hydrometeorology, 122–128.

    • Search Google Scholar
    • Export Citation
  • Clark, T., 1974: A study in cloud phase parameterization using the gamma distribution. J. Atmos. Sci., 31 , 142155.

  • Cohard, J-M., and J-P. Pinty, 2000: A comprehensive two-moment warm microphysical bulk scheme. I: Description and tests. Quart. J. Roy. Meteor. Soc., 126 , 18151842.

    • Search Google Scholar
    • Export Citation
  • Considine, G., and J. A. Curry, 1996: A statistical model of drop size spectra for stratocumulus clouds. Quart. J. Roy. Meteor. Soc., 122 , 611634.

    • Search Google Scholar
    • Export Citation
  • Cotton, W. R., and R. A. Anthes, 1989: Storm and Cloud Dynamics. Academic Press, 883 pp.

  • Cotton, W. R., G. J. Tripoli, R. M. Rauber, and E. A. Mulvihill, 1986: Numerical simulation of the effects of varying ice nucleation rates and aggregation process on orographic snowfall. J. Climate Appl. Meteor., 25 , 16581680.

    • Search Google Scholar
    • Export Citation
  • Curry, J. A., 1986: Interactions among turbulence, radiation and microphysics in Arctic stratus clouds. J. Atmos. Sci., 43 , 525538.

  • Curry, J. A., and P. J. Webster, 1999: Thermodynamics of Atmospheres and Oceans. Academic Press, 467 pp.

  • Curry, J. A., E. E. Ebert, and G. F. Herman, 1988: Mean and turbulent structure of the summertime Arctic cloudy boundary layer. Quart. J. Roy. Meteor. Soc., 114 , 715746.

    • Search Google Scholar
    • Export Citation
  • Dyachenko, P. V., 1959: Experience of application of the methods of mathematical statistics to the studies of clouds and fogs microstructure. Proc. Main Geophys. Obs., Leningrad, 101 , 3245.

    • Search Google Scholar
    • Export Citation
  • Ferrier, B. S., 1994: A double-moment multiple-phase four-class bulk ice scheme. Part I: Description. J. Atmos. Sci., 51 , 249280.

  • Fowler, L. D., D. A. Randall, and S. A. Rutledge, 1996: Liquid and ice cloud microphysics in the CSU general circulation model. Part I: Model description and simulated microphysical processes. J. Climate, 9 , 489529.

    • Search Google Scholar
    • Export Citation
  • Gettelman, A., and D. E. Kinnison, 2007: The global impact of supersaturation in a coupled chemistry–climate model. Atmos. Chem. Phys., 7 , 16291643.

    • Search Google Scholar
    • Export Citation
  • Ghan, S., L. Leung, R. Easter, and H. Abdul-Razzak, 1997: Prediction of cloud droplet number in a general circulation model. J. Geophys. Res., 102 , 777794.

    • Search Google Scholar
    • Export Citation
  • Girard, E., and J. A. Curry, 2001: Simulation of Arctic low-level clouds observed during the FIRE Arctic Clouds Experiment using a new bulk microphysics scheme. J. Geophys. Res., 106 , 1513915154.

    • Search Google Scholar
    • Export Citation
  • Gradshteyn, I. S., and I. M. Ryzhik, 1994: Table of Integrals, Series, and Products. 5th ed. Academic Press, 1204 pp.

  • Hall, W. D., and H. R. Pruppacher, 1976: The survival of ice particles falling from cirrus clouds in subsaturated air. J. Atmos. Sci., 33 , 19952006.

    • Search Google Scholar
    • Export Citation
  • Harrington, J. Y., M. P. Meyers, R. L. Walko, and W. R. Cotton, 1995: Parameterization of ice crystal conversion process due to vapor deposition for mesoscale models using double-moment basis functions. Part I: Basic formulation and parcel model results. J. Atmos. Sci., 52 , 43444366.

    • Search Google Scholar
    • Export Citation
  • Herman, G. F., and J. A. Curry, 1984: Observational and theoretical studies of solar radiation in Arctic stratus clouds. J. Climate Appl. Meteor., 23 , 524.

    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., and C. M. R. Platt, 1984: A parameterization of the particle size spectrum of ice clouds in terms of the ambient temperature and the ice water content. J. Atmos. Sci., 41 , 846855.

    • Search Google Scholar
    • Export Citation
  • Jeffery, C. A., J. M. Reisner, and M. Andrejczuk, 2007: Another look at stochastic condensation for subgrid cloud modeling: Adiabatic evolution and effects. J. Atmos. Sci., 64 , 39533973.

    • Search Google Scholar
    • Export Citation
  • Jensen, E. J., O. B. Toon, D. L. Westphal, S. Kinne, and A. J. Heymsfield, 1994: Microphysical modeling of cirrus, 1. Comparison with 1986 FIRE IFO measurements. J. Geophys. Res., 99 , 1042110442.

    • Search Google Scholar
    • Export Citation
  • Jensen, E. J., and Coauthors, 2001: Prevalence of ice-supersaturated regions in the upper troposphere: Implications for optically thin ice cloud formation. J. Geophys. Res., 106 , 1725317266.

    • Search Google Scholar
    • Export Citation
  • Kessler, E., 1969: On the Distribution and Continuity of Water Substance in Atmospheric Circulation. Meteor. Monogr., No. 32, Amer. Meteor. Soc., 84 pp.

    • Search Google Scholar
    • Export Citation
  • Khvorostyanov, V. I., and K. Sassen, 1998: Cirrus cloud simulation using explicit microphysics and radiation. Part II: Microphysics, vapor and ice mass budgets, and optical and radiative properties. J. Atmos. Sci., 55 , 18221845.

    • Search Google Scholar
    • Export Citation
  • Khvorostyanov, V. I., and J. A. Curry, 1999a: Toward the theory of stochastic condensation in clouds. Part I: A general kinetic equation. J. Atmos. Sci., 56 , 39853996.

    • Search Google Scholar
    • Export Citation
  • Khvorostyanov, V. I., and J. A. Curry, 1999b: Toward the theory of stochastic condensation in clouds. Part II: Analytical solutions of gamma distribution type. J. Atmos. Sci., 56 , 39974013.

    • Search Google Scholar
    • Export Citation
  • Khvorostyanov, V. I., and J. A. Curry, 2002: Terminal velocities of droplets and crystals: Power laws with continuous parameters over the size spectrum. J. Atmos. Sci., 59 , 18721884.

    • Search Google Scholar
    • Export Citation
  • Khvorostyanov, V. I., and K. Sassen, 2002: Microphysical processes in cirrus and their impact on radiation, a mesoscale modeling perspective. Cirrus, D. Lynch et al., Eds., Oxford University Press, 397–432.

    • Search Google Scholar
    • Export Citation
  • Khvorostyanov, V. I., and J. A. Curry, 2005: Fall velocities of hydrometeors in the atmosphere: Refinements to a continuous analytical power law. J. Atmos. Sci., 62 , 43434357.

    • Search Google Scholar
    • Export Citation
  • Khvorostyanov, V. I., and J. A. Curry, 2008: Analytical solutions to the stochastic kinetic equation for liquid and ice particle size spectra. Part II: Large-size fraction in precipitating clouds. J. Atmos. Sci., 65 , 20442063.

    • Search Google Scholar
    • Export Citation
  • Khvorostyanov, V. I., J. A. Curry, J. O. Pinto, M. Shupe, B. A. Baker, and K. Sassen, 2001: Modeling with explicit spectral water and ice microphysics of a two-layer cloud system of altostratus and cirrus observed during the FIRE Arctic Clouds Experiment. J. Geophys. Res., 106 , 1509915112.

    • Search Google Scholar
    • Export Citation
  • Korolev, A. V., 1995: The influence of supersaturation fluctuations on droplet size spectra formation. J. Atmos. Sci., 52 , 36203634.

  • Landau, L. D., and E. M. Lifshitz, 1958: Quantum Mechanics, Non-Relativistic Theory. Vol. 3, Course of Theoretical Physics, Addison-Wesley, 526 pp.

    • Search Google Scholar
    • Export Citation
  • Lawson, R. P., B. Baker, B. Pilson, and Q. Mo, 2006: In situ observations of the microphysical properties of wave, cirrus, and anvil clouds. Part II: Cirrus clouds. J. Atmos. Sci., 63 , 31863203.

    • Search Google Scholar
    • Export Citation
  • Levin, L. M., 1954: On the size distributions functions of the cloud and rain droplets. Dokl. Acad. Sci. USSR, 44 , 10451049.

  • Levin, L. M., and Y. S. Sedunov, 1966: Stochastic condensation of drops and kinetics of cloud spectrum formation. J. Rech. Atmos., 2 , 425432.

    • Search Google Scholar
    • Export Citation
  • Lin, Y. L., R. D. Farley, and H. D. Orville, 1983: Bulk parameterization of the snow field in a cloud model. J. Climate Appl. Meteor., 22 , 10651092.

    • Search Google Scholar
    • Export Citation
  • Liu, Y., 1995: On the generalized theory of atmospheric particles systems. Adv. Atmos. Sci., 12 , 419438.

  • Liu, Y., and J. Hallett, 1998: On size distributions of cloud drops growing by condensation: A new conceptual model. J. Atmos. Sci., 55 , 527536.

    • Search Google Scholar
    • Export Citation
  • Liu, Y., Y. Laiguang, Y. Weinong, and L. Feng, 1995: On the size distribution of cloud droplets. Atmos. Res., 35 , 201216.

  • Ludwig, F. L., and E. Robinson, 1970: Observations of aerosols and droplets in California stratus. Tellus, 22 , 7889.

  • Manton, M. J., 1979: On the broadening of a droplet distribution by turbulence near cloud base. Quart. J. Roy. Meteor. Soc., 105 , 899914.

    • Search Google Scholar
    • Export Citation
  • Matrosov, S. Y., B. W. Orr, R. A. Kropfli, and J. B. Snider, 1994: Retrieval of vertical profiles of cirrus clouds microphysical parameters from Doppler radar and infrared radiometer measurements. J. Appl. Meteor., 33 , 617626.

    • Search Google Scholar
    • Export Citation
  • McGraw, R., and Y. Liu, 2006: Brownian drift-diffusion model for evolution of droplet size distributions in turbulent clouds. Geophys. Res. Lett., 33 .L03802, doi:10.1029/2005GL023545.

    • Search Google Scholar
    • Export Citation
  • Meyers, M. P., R. L. Walko, J. Y. Harrington, and W. R. Cotton, 1997: New RAMS cloud microphysics parameterization. Part II: The two-moment scheme. Atmos. Res., 45 , 339.

    • Search Google Scholar
    • Export Citation
  • Milbrandt, J. A., and M. K. Yau, 2005a: A multimoment bulk microphysics parameterization. Part I: Analysis of the role of the spectral shape parameter. J. Atmos. Sci., 62 , 30513064.

    • Search Google Scholar
    • Export Citation
  • Milbrandt, J. A., and M. K. Yau, 2005b: A multimoment bulk microphysics parameterization. Part II: A proposed three-moment closure and scheme description. J. Atmos. Sci., 62 , 30653081.

    • Search Google Scholar
    • Export Citation
  • Mitchell, D. L., 1994: A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part I: Microphysics. J. Atmos. Sci., 51 , 797816.

    • Search Google Scholar
    • Export Citation
  • Mitchell, D. L., S. K. Chai, Y. Liu, A. J. Heymsfield, and Y. Dong, 1996: Modeling cirrus clouds. Part I: Treatment of bimodal spectra and case study analysis. J. Atmos. Sci., 53 , 29522966.

    • Search Google Scholar
    • Export Citation
  • Monin, A. S., and A. M. Yaglom, 2007: Statistical Fluid Mechanics: Mechanics of Turbulence. Vol. 2. Dover, 896 pp.

  • Morrison, H., and J. Pinto, 2005: Mesoscale modeling of springtime Arctic mixed-phase stratiform clouds using a new two-moment bulk microphysics scheme. J. Atmos. Sci., 62 , 36833704.

    • Search Google Scholar
    • Export Citation
  • Morrison, H., J. A. Curry, and V. I. Khvorostyanov, 2005a: A new double-moment microphysics parameterization for application in cloud and climate models. Part I: Description. J. Atmos. Sci., 62 , 16651677.

    • Search Google Scholar
    • Export Citation
  • Morrison, H., J. A. Curry, M. D. Shupe, and P. Zuidema, 2005b: A new double-moment microphysics parameterization for application in cloud and climate models. Part II: Single-column modeling of arctic clouds. J. Atmos. Sci., 62 , 16781693.

    • Search Google Scholar
    • Export Citation
  • Nevzorov, A. N., 1967: Distribution of large drops in liquid stratiform clouds. Proc. Centr. Aerolog. Observ., 79 , 5769.

  • Noonkester, V. R., 1984: Droplet spectra observed in marine stratus cloud layers. J. Atmos. Sci., 41 , 829845.

  • Okita, T., 1961: Size distribution of large droplets in precipitating clouds. Tellus, 13 , 456467.

  • Paoli, R., and K. Shariff, 2003: Particle size distributions in atmospheric clouds. Annual Research Briefs, Center for Turbulence Research, NASA Ames Research Center, Ames, IA, 39–47. [Available online at http://www.stanford.edu/group/ctr/publications.html.].

  • Pinto, J. O., and J. A. Curry, 1995: Atmospheric convective plumes emanating from the leads. 2. Microphysical and radiative processes. J. Geophys. Res., 100 , 46334642.

    • Search Google Scholar
    • Export Citation
  • Platt, C. M. R., 1997: A parameterization of the visible extinction coefficient in terms of the ice/water content. J. Atmos. Sci., 54 , 20832098.

    • Search Google Scholar
    • Export Citation
  • Poellot, M. R., K. A. Hilburn, W. P. Arnott, and K. Sassen, 1999: In situ observations of cirrus clouds from the 1994 ARM Southern Great Plains Intensive Observational Period. Proc. Ninth ARM Science Team Meeting, San Antonio, TX, U.S. Department of Energy.

  • Pruppacher, H. R., and J. D. Klett, 1997: Microphysics of Clouds and Precipitation. 2nd ed. Kluwer, 997 pp.

  • Quante, M., and D. O’C. Starr, 2002: Dynamical processes in cirrus clouds. Cirrus, D. Lynch et al., Eds., Oxford University Press, 346–374.

    • Search Google Scholar
    • Export Citation
  • Rogers, R. R., 1979: A. Short Course in Cloud Physics. 2nd ed. Pergamon Press, 235 pp.

  • Rutledge, S. A., and P. V. Hobbs, 1983: The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones. VIII: A model for the “seeder-feeder” process in warm frontal rainbands. J. Atmos. Sci., 40 , 11851206.

    • Search Google Scholar
    • Export Citation
  • Ryan, B. F., 2000: A bulk parameterization of the ice particles size distribution and the optical properties in ice clouds. J. Atmos. Sci., 57 , 14361451.

    • Search Google Scholar
    • Export Citation
  • Sassen, K., and L. Liao, 1996: Estimation of cloud content by W-band radar. J. Appl. Meteor., 35 , 932938.

  • Sassen, K., and V. I. Khvorostyanov, 2007: Microphysical and radiative properties of mixed-phase altocumulus: A model evaluation of glaciation effects. Atmos. Res., 84 , 390398.

    • Search Google Scholar
    • Export Citation
  • Sassen, K., D. O’C. Starr, and T. Uttal, 1989: Mesoscale and microscale structure of cirrus clouds: Three case studies. J. Atmos. Sci., 46 , 371396.

    • Search Google Scholar
    • Export Citation
  • Sedunov, Y. S., 1974: Physics of Drop Formation in the Atmosphere. Wiley, 234 pp.

  • Seinfeld, J. H., and S. N. Pandis, 1998: Atmospheric Chemistry and Physics. Wiley, 1326 pp.

  • Shaw, R. A., 2003: Particle-turbulence interactions in atmospheric clouds. Annu. Rev. Fluid. Mech., 35 , 183227.

  • Srivastava, R. C., 1969: Note on the theory of growth of cloud drops by condensation. J. Atmos. Sci., 26 , 776780.

  • Starr, D. O’C., and S. K. Cox, 1985: Cirrus clouds. Part I: A cirrus cloud model. J. Atmos. Sci., 42 , 26632681.

  • Tisler, P., E. Zapadinsky, and M. Kulmala, 2005: Initiation of rain by turbulence-induced condensational growth of cloud droplets. Geophys. Res. Lett., 32 .L06806, doi:10.1029/2004GL021969.

    • Search Google Scholar
    • Export Citation
  • Vaillancourt, P. A., M. K. Yau, P. Bartello, and W. W. Grabowski, 2002: Microscopic approach to cloud droplet growth by condensation. Part II: Turbulence, clustering, and condensational growth. J. Atmos. Sci., 59 , 34213435.

    • Search Google Scholar
    • Export Citation
  • Voloshchuk, V. M., and Y. S. Sedunov, 1977: A kinetic equation for the evolution of the droplet spectrum in a turbulent medium at the condensation stage of cloud development. Sov. Meteor. Hydrol., 3 , 314.

    • Search Google Scholar
    • Export Citation
  • Warner, J., 1969: The microstructure of cumulus cloud. Part I: General features of the droplet spectrum. J. Atmos. Sci., 26 , 10491059.

    • Search Google Scholar
    • Export Citation
  • Young, K., 1993: Microphysical Processes in Clouds. Oxford University Press, 427 pp.

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