Effect of Turbulence on the Collision Rate between Settling Ice Crystals and Droplets

M. Z. Sheikh aDepartment of Mechanical Engineering, University of Engineering and Technology Lahore, Lahore, Pakistan

Search for other papers by M. Z. Sheikh in
Current site
Google Scholar
PubMed
Close
,
K. Gustavsson bDepartment of Physics, Gothenburg University, Gothenburg, Sweden

Search for other papers by K. Gustavsson in
Current site
Google Scholar
PubMed
Close
,
E. Lévêque cCNRS, Ecole Centrale de Lyon, INSA Lyon, Universite Claude Bernard Lyon 1, Laboratoire de Mécanique des Fluides et d’Acoustique, UMR5509, Ecully, France

Search for other papers by E. Lévêque in
Current site
Google Scholar
PubMed
Close
,
B. Mehlig bDepartment of Physics, Gothenburg University, Gothenburg, Sweden

Search for other papers by B. Mehlig in
Current site
Google Scholar
PubMed
Close
,
A. Pumir dUniv Lyon, ENSL, CNRS, Laboratoire de Physique, Lyon, France
eMax Planck Institute for Dynamics and Self-Organization, Göttingen, Germany

Search for other papers by A. Pumir in
Current site
Google Scholar
PubMed
Close
, and
A. Naso cCNRS, Ecole Centrale de Lyon, INSA Lyon, Universite Claude Bernard Lyon 1, Laboratoire de Mécanique des Fluides et d’Acoustique, UMR5509, Ecully, France

Search for other papers by A. Naso in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

In mixed-phase clouds, graupel forms by riming, a process whereby ice crystals and supercooled water droplets settling through a turbulent flow collide and aggregate. We consider here the early stage of the collision process of small ice crystals with water droplets and determine numerically the geometric collision kernel in turbulent flows (therefore neglecting all interactions between the particles and assuming a collision efficiency equal to unity), over a range of energy dissipation rate 1–250 cm2 s−3 relevant to cloud microphysics. We take into account the effect of small, but nonzero fluid inertia, which is essential since it favors a biased orientation of the crystals with their broad side down. Since water droplets and ice crystals have different masses and shapes, they generally settle with different velocities. Turbulence does not play any significant role on the collision kernel when the difference between the settling velocities of the two sets of particles is larger than a few millimeters per second. The situation is completely different when the settling speeds of droplets and crystals are comparable, in which case turbulence is the main cause of collisions. Our results are compatible with those of recent experiments according to which turbulence does not clearly increase the growth rate of tethered graupel in a flow transporting water droplets.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding authors: A. Pumir, alain.pumir@ens-lyon.fr; A. Naso, aurore.naso@cnrs.fr

Abstract

In mixed-phase clouds, graupel forms by riming, a process whereby ice crystals and supercooled water droplets settling through a turbulent flow collide and aggregate. We consider here the early stage of the collision process of small ice crystals with water droplets and determine numerically the geometric collision kernel in turbulent flows (therefore neglecting all interactions between the particles and assuming a collision efficiency equal to unity), over a range of energy dissipation rate 1–250 cm2 s−3 relevant to cloud microphysics. We take into account the effect of small, but nonzero fluid inertia, which is essential since it favors a biased orientation of the crystals with their broad side down. Since water droplets and ice crystals have different masses and shapes, they generally settle with different velocities. Turbulence does not play any significant role on the collision kernel when the difference between the settling velocities of the two sets of particles is larger than a few millimeters per second. The situation is completely different when the settling speeds of droplets and crystals are comparable, in which case turbulence is the main cause of collisions. Our results are compatible with those of recent experiments according to which turbulence does not clearly increase the growth rate of tethered graupel in a flow transporting water droplets.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding authors: A. Pumir, alain.pumir@ens-lyon.fr; A. Naso, aurore.naso@cnrs.fr
Save
  • Anand, P., S. S. Ray, and G. Subramanian, 2020: Orientation dynamics of sedimenting anisotropic particles in turbulence. Phys. Rev. Lett., 125, 034501, https://doi.org/10.1103/PhysRevLett.125.034501.

    • Search Google Scholar
    • Export Citation
  • Ayala, O., B. Rosa, and L.-P. Wang, 2008a: Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 2. Theory and parametrization. New J. Phys., 10, 075016, https://doi.org/10.1088/1367-2630/10/7/075016.

    • Search Google Scholar
    • Export Citation
  • Ayala, O., B. Rosa, L.-P. Wang, and W. W. Grabowski, 2008b: Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 1. Results from direct numerical simulations. New J. Phys., 10 (7), 075015, https://doi.org/10.1088/1367-2630/10/7/075015.

    • Search Google Scholar
    • Export Citation
  • Bec, J., K. Gustavsson, and B. Mehlig, 2023: Statistical models for the dynamics of heavy particles in turbulence. Annu. Rev. Fluid Mech., 56, 189213, https://doi.org/10.1146/annurev-fluid-032822-014140.

    • Search Google Scholar
    • Export Citation
  • Bhowmick, T., J. Seesing, K. Gustavsson, J. Guettler, Y. Wang, A. Pumir, B. Mehlig, and G. Bagheri, 2024: Inertia induces strong orientation fluctuations of nonspherical atmospheric particles. Phys. Rev. Lett., 132, 034101, https://doi.org/10.1103/PhysRevLett.132.034101.

    • Search Google Scholar
    • Export Citation
  • Brenner, H., 1961: The Oseen resistance of a particle of arbitrary shape. J. Fluid Mech., 11, 604610, https://doi.org/10.1017/S0022112061000755.

    • Search Google Scholar
    • Export Citation
  • Cabrera, F., M. Z. Sheikh, B. Mehlig, N. Plihon, M. Bourgoin, A. Pumir, and A. Naso, 2022: Experimental validation of fluid inertia models for a cylinder settling in a quiescent flow. Phys. Rev. Fluids, 7, 024301, https://doi.org/10.1103/PhysRevFluids.7.024301.

    • Search Google Scholar
    • Export Citation
  • Chen, J.-P., and D. Lamb, 1994: The theoretical basis for the parameterization of ice crystal habits: Growth by vapor deposition. J. Atmos. Sci., 51, 12061222, https://doi.org/10.1175/1520-0469(1994)051<1206:TTBFTP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Cheng, K.-Y., P. K. Wang, and T. Hashino, 2015: A numerical study on the attitudes and aerodynamics of freely falling hexagonal ice plates. J. Atmos. Sci., 72, 36853698, https://doi.org/10.1175/JAS-D-15-0059.1.

    • Search Google Scholar
    • Export Citation
  • Choi, Y.-K., J.-W. Chang, W. Wang, M.-S. Kim, and G. Elber, 2009: Continuous collision detection for ellipsoids. IEEE Trans. Visualization Comput. Graphics, 15, 311325, https://doi.org/10.1109/TVCG.2008.80.

    • Search Google Scholar
    • Export Citation
  • Cox, R. G., 1965: The steady motion of a particle of arbitrary shape at small Reynolds numbers. J. Fluid Mech., 23, 625643, https://doi.org/10.1017/S0022112065001593.

    • Search Google Scholar
    • Export Citation
  • Dabade, V., N. K. Marath, and G. Subramanian, 2015: Effects of inertia and viscoelasticity on sedimenting anisotropic particles. J. Fluid Mech., 778, 133188, https://doi.org/10.1017/jfm.2015.360.

    • Search Google Scholar
    • Export Citation
  • Devenish, B. J., and Coauthors, 2012: Droplet growth in warm turbulent clouds. Quart. J. Roy. Meteor. Soc., 138, 14011429, https://doi.org/10.1002/qj.1897.

    • Search Google Scholar
    • Export Citation
  • Dhanasekaran, J., A. Roy, and D. L. Koch, 2021: Collision rate of bidisperse, hydrodynamically interacting spheres settling in a turbulent flow. J. Fluid Mech., 912, A5, https://doi.org/10.1017/jfm.2020.1113.

    • Search Google Scholar
    • Export Citation
  • Dubey, A., G. P. Bewley, K. Gustavsson, and B. Mehlig, 2022: Critical charges in droplet collisions. arXiv, 2209.05427v1, https://doi.org/10.48550/arXiv.2209.05427.

  • Falkovich, G., A. Fouxon, and M. G. Stepanov, 2002: Acceleration of rain initiation by cloud turbulence. Nature, 419, 151154, https://doi.org/10.1038/nature00983.

    • Search Google Scholar
    • Export Citation
  • Falkovich, G., S. Musacchio, L. Piterbarg, and M. Vucelja, 2007: Inertial particles driven by a telegraph noise. Phys. Rev., 76E, 026313, https://doi.org/10.1103/PhysRevE.76.026313.

    • Search Google Scholar
    • Export Citation
  • Fitch, K. E., and T. J. Garrett, 2022a: Graupel precipitating from thin arctic clouds with liquid water paths less than 50 g m−2. Geophys. Res. Lett., 49, e2021GL094075, https://doi.org/10.1029/2021GL094075.

    • Search Google Scholar
    • Export Citation
  • Fitch, K. E., and T. J. Garrett, 2022b: Measurement and analysis of the microphysical properties of arctic precipitation showing frequent occurrence of riming. J. Geophys. Res. Atmos., 127, e2021JD035980, https://doi.org/10.1029/2021JD035980.

    • Search Google Scholar
    • Export Citation
  • Fröhlich, K., M. Meinke, and W. Schröder, 2020: Correlations for inclined prolates based on highly resolved simulations. J. Fluid Mech., 901, A5, https://doi.org/10.1017/jfm.2020.482.

    • Search Google Scholar
    • Export Citation
  • Garrett, T. J., S. E. Yuter, C. Fallgatter, K. Shkurko, S. R. Rhodes, and J. L. Endries, 2015: Orientations and aspect ratios of falling snow. Geophys. Res. Lett., 42, 46174622, https://doi.org/10.1002/2015GL064040.

    • Search Google Scholar
    • Export Citation
  • Gavze, E., and A. Khain, 2022: Gravitational collision of small nonspherical particles: Swept volumes of prolate and oblate spheroids in calm air. J. Atmos. Sci., 79, 14931514, https://doi.org/10.1175/JAS-D-20-0336.1.

    • Search Google Scholar
    • Export Citation
  • Good, G. H., P. J. Ireland, G. P. Bewley, E. Bodenschatz, L. R. Collins, and Z. Warhaft, 2014: Settling regimes of inertial particles in isotropic turbulence. J. Fluid Mech., 759, R3, https://doi.org/10.1017/jfm.2014.602.

    • Search Google Scholar
    • Export Citation
  • Grabowski, W. W., and L.-P. Wang, 2013: Growth of cloud droplets in a turbulent environment. Annu. Rev. Fluid Mech., 45, 293324, https://doi.org/10.1146/annurev-fluid-011212-140750.

    • Search Google Scholar
    • Export Citation
  • Grazioli, J., G. Lloyd, L. Panziera, C. R. Hoyle, P. J. Connolly, J. Henneberger, and A. Berne, 2015: Polarimetric radar and in situ observations of riming and snowfall microphysics during CLACE 2014. Atmos. Chem. Phys., 15, 13 78713 802, https://doi.org/10.5194/acp-15-13787-2015.

    • Search Google Scholar
    • Export Citation
  • Gunn, R., and G. D. Kinzer, 1949: The terminal velocity of fall for water droplets in stagnant air. J. Meteor., 6, 243248, https://doi.org/10.1175/1520-0469(1949)006<0243:TTVOFF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Gustavsson, K., B. Mehlig, and M. Wilkinson, 2008: Collisions of particles advected in random flows. New J. Phys., 10, 075014, https://doi.org/10.1088/1367-2630/10/7/075014.

    • Search Google Scholar
    • Export Citation
  • Gustavsson, K., M. Z. Sheikh, D. Lopez, A. Naso, A. Pumir, and B. Mehlig, 2019: Effect of fluid inertia on the orientation of a small spheroid settling in turbulence. New J. Phys., 21, 083008, https://doi.org/10.1088/1367-2630/ab3062.

    • Search Google Scholar
    • Export Citation
  • Gustavsson, K., M. Z. Sheikh, A. Naso, A. Pumir, and B. Mehlig, 2021: Effect of particle inertia on the alignment of small ice crystals in turbulent clouds. J. Atmos. Sci., 78, 25732587, https://doi.org/10.1175/JAS-D-20-0221.1.

    • Search Google Scholar
    • Export Citation
  • Happel, J., and H. Brenner, 1983: Low Reynolds Number Hydrodynamics. Springer, 553 pp.

  • Jeffery, G. B., 1922: The motion of ellipsoidal particles immersed in a viscous fluid. Proc. Roy. Soc. London, A102, 161179, https://doi.org/10.1098/rspa.1922.0078.

    • Search Google Scholar
    • Export Citation
  • Jiang, F., L. Zhao, H. I. Andersson, K. Gustavsson, A. Pumir, and B. Mehlig, 2021: Inertial torque on a small spheroid in a stationary uniform flow. Phys. Rev. Fluids, 6, 024302, https://doi.org/10.1103/PhysRevFluids.6.024302.

    • Search Google Scholar
    • Export Citation
  • Jost, A., M. Szakáll, K. Diehl, S. K. Mitra, A. Hudertmark, B. S. Klug, and S. Borrmann, 2019: The effect of turbulence on the accretional growth of graupels. J. Atmos. Sci., 76, 30473061, https://doi.org/10.1175/JAS-D-18-0200.1.

    • Search Google Scholar
    • Export Citation
  • Jucha, J., A. Naso, E. Lévêque, and A. Pumir, 2018: Settling and collision between small ice crystals in turbulent flows. Phys. Rev. Fluids, 3, 014604, https://doi.org/10.1103/PhysRevFluids.3.014604.

    • Search Google Scholar
    • Export Citation
  • Kajikawa, M., 1972: Measurement of falling velocity of individual snow crystals. J. Meteor. Soc. Japan, 50, 577584, https://doi.org/10.2151/jmsj1965.50.6_577.

    • Search Google Scholar
    • Export Citation
  • Khayat, R. E., and R. Cox, 1989: Inertia effects on the motion of long slender bodies. J. Fluid Mech., 209, 435462, https://doi.org/10.1017/S0022112089003174.

    • Search Google Scholar
    • Export Citation
  • Kim, S., and S. J. Karrila, 1991: Microhydrodynamics: Principles and Selected Applications. Elsevier Science & Technology Books, 507 pp.

  • Klett, J. D., 1995: Orientation model for particles in turbulence. J. Atmos. Sci., 52, 22762285, https://doi.org/10.1175/1520-0469(1995)052<2276:OMFPIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Klett, J. D., and M. H. Davis, 1973: Theoretical collision efficiencies of cloud droplets at small Reynolds numbers. J. Atmos. Sci., 30, 107117, https://doi.org/10.1175/1520-0469(1973)030<0107:TCEOCD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Korolev, A., and T. Leisner, 2020: Review of experimental studies of secondary ice production. Atmos. Chem. Phys., 20, 11 76711 797, https://doi.org/10.5194/acp-20-11767-2020.

    • Search Google Scholar
    • Export Citation
  • Kramel, S., 2017: Non-spherical particle dynamics in turbulence. Ph.D. thesis, Wesleyan University, 165 pp., https://doi.org/10.14418/wes01.3.78.

  • Li Sing How, M., D. L. Koch, and L. R. Collins, 2021: Non-continuum tangential lubrication gas flow between two spheres. J. Fluid Mech., 920, A2, https://doi.org/10.1017/jfm.2021.308.

    • Search Google Scholar
    • Export Citation
  • Lopez, D., and E. Guazzelli, 2017: Inertial effects on fibers settling in a vortical flow. Phys. Rev. Fluids, 2, 024306, https://doi.org/10.1103/PhysRevFluids.2.024306.

    • Search Google Scholar
    • Export Citation
  • Martin, S. J., P. K. Wang, and H. R. Pruppacher, 1980: A theoretical determination of the efficiency with which aerosol particles are collected by simple ice crystal plates. J. Atmos. Sci., 37, 16281638, https://doi.org/10.1175/1520-0469(1980)037<1628:ATDOTE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Maxey, M. R., 1987: The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J. Fluid Mech., 174, 441465, https://doi.org/10.1017/S0022112087000193.

    • Search Google Scholar
    • Export Citation
  • Meng, X., and E.-W. Saw, 2023: Sharp depletion of radial distribution function of particles due to collision and coagulation inside turbulent flow: A systematic study. Phys. Rev. Fluids, 8, 084304, https://doi.org/10.1103/PhysRevFluids.8.084304.

    • Search Google Scholar
    • Export Citation
  • Moisseev, D., A. von Lerber, and J. Tiira, 2017: Quantifying the effect of riming on snowfall using ground-based observations. J. Geophys. Res. Atmos., 122, 40194037, https://doi.org/10.1002/2016JD026272.

    • Search Google Scholar
    • Export Citation
  • Montero-Martínez, G., A. B. Kostinski, R. A. Shaw, and F. García-García, 2009: Do all raindrops fall at terminal speed? Geophys. Res. Lett., 36, L11818, https://doi.org/10.1029/2008GL037111.

    • Search Google Scholar
    • Export Citation
  • Morrison, H., and Coauthors, 2020: Confronting the challenge of modeling cloud and precipitation microphysics. J. Adv. Model. Earth Syst., 12, e2019MS001689, https://doi.org/10.1029/2019MS001689.

    • Search Google Scholar
    • Export Citation
  • Naso, A., J. Jucha, E. Lévêque, and A. Pumir, 2018: Collision rate of ice crystals with water droplets in turbulent flows. J. Fluid Mech., 845, 615641, https://doi.org/10.1017/jfm.2018.238.

    • Search Google Scholar
    • Export Citation
  • Ouchene, R., 2020: Numerical simulation and modeling of the hydrodynamic forces and torque acting on individual oblate spheroids. Phys. Fluids, 32, 073303, https://doi.org/10.1063/5.0011618.

    • Search Google Scholar
    • Export Citation
  • Patra, P., D. L. Koch, and A. Roy, 2022: Collision efficiency of non-Brownian spheres in a simple shear flow—The role of non-continuum hydrodynamic interactions. J. Fluid Mech., 950, A18, https://doi.org/10.1017/jfm.2022.817.

    • Search Google Scholar
    • Export Citation
  • Pinsky, M. B., and A. P. Khain, 1998: Some effects of cloud turbulence on water-ice and ice-ice collisions. Atmos. Res., 47–48, 6986, https://doi.org/10.1016/S0169-8095(98)00041-6.

    • Search Google Scholar
    • Export Citation
  • Pinsky, M. B., A. P. Khain, D. Rosenfeld, and A. Pokrovsky, 1998: Comparison of collision velocity differences of drops and graupel particles in a very turbulent cloud. Atmos. Res., 49, 99113, https://doi.org/10.1016/S0169-8095(98)00073-8.

    • Search Google Scholar
    • Export Citation
  • Pinsky, M. B., A. P. Khain, and M. Shapiro, 1999: Collisions of small drops in a turbulent flow. Part I: Collision efficiency. Problem formulation and preliminary results. J. Atmos. Sci., 56, 25852600, https://doi.org/10.1175/1520-0469(1999)056<2585:COSDIA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Pinsky, M. B., A. P. Khain, and M. Shapiro, 2007: Collisions of cloud droplets in a turbulent flow. Part IV: Droplet hydrodynamic interaction. J. Atmos. Sci., 64, 24622482, https://doi.org/10.1175/JAS3952.1.

    • Search Google Scholar
    • Export Citation
  • Pitter, R. L., H. R. Pruppacher, and A. E. Hamielec, 1973: A numerical study of viscous flow past a thin oblate spheroid at low and intermediate Reynolds numbers. J. Atmos. Sci., 30, 125134, https://doi.org/10.1175/1520-0469(1973)030<0125:ANSOVF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Pruppacher, H. R., and J. D. Klett, 1997: Microphysics of Clouds and Precipitation. 2nd ed. Springer, 954 pp.

  • Pumir, A., and M. Wilkinson, 2016: Collision aggregation due to turbulence. Annu. Rev. Condens. Matter Phys., 7, 141170, https://doi.org/10.1146/annurev-conmatphys-031115-011538.

    • Search Google Scholar
    • Export Citation
  • Roy, A., R. J. Hamati, L. Tierney, D. L. Koch, and G. A. Voth, 2019: Inertial torques and a symmetry breaking orientational transition in the sedimentation of slender fibres. J. Fluid Mech., 875, 576596, https://doi.org/10.1017/jfm.2019.492.

    • Search Google Scholar
    • Export Citation
  • Saffman, P. G., and J. S. Turner, 1956: On the collision of drops in turbulent clouds. J. Fluid Mech., 1, 1630, https://doi.org/10.1017/S0022112056000020.

    • Search Google Scholar
    • Export Citation
  • Sheikh, M. Z., K. Gustavsson, D. Lopez, E. Lévêque, B. Mehlig, A. Pumir, and A. Naso, 2020: Importance of fluid inertia for the orientation of spheroids settling in a turbulent flow. J. Fluid Mech., 886, A9, https://doi.org/10.1017/jfm.2019.1041.

    • Search Google Scholar
    • Export Citation
  • Sheikh, M. Z., K. Gustavsson, E. Lévêque, B. Mehlig, A. Pumir, and A. Naso, 2022: Colliding ice crystals in turbulent clouds. J. Atmos. Sci., 79, 22052218, https://doi.org/10.1175/JAS-D-21-0305.1.

    • Search Google Scholar
    • Export Citation
  • Siewert, C., R. P. J. Kunnen, and W. Schrőder, 2014: Collision rates of small ellipsoids settling in turbulence. J. Fluid Mech., 758, 686701, https://doi.org/10.1017/jfm.2014.554.

    • Search Google Scholar
    • Export Citation
  • Sundarajakumar, R. R., and D. L. Koch, 1996: Non-continuum lubrication flows between particles colliding in a gas. J. Fluid Mech., 313, 283308, https://doi.org/10.1017/S0022112096002212.

    • Search Google Scholar
    • Export Citation
  • Um, J., G. M. McFarquhar, Y. P. Hong, S.-S. Lee, C. H. Jung, R. P. Lawson, and Q. Mo, 2015: Dimensions and aspect ratios of natural ice crystals. Atmos. Chem. Phys., 15, 39333956, https://doi.org/10.5194/acp-15-3933-2015.

    • Search Google Scholar
    • Export Citation
  • Voßkuhle, M., E. Lévêque, M. Wilkinson, and A. Pumir, 2013: Multiple collisions in turbulent flows. Phys. Rev., 88E, 063008, https://doi.org/10.1103/PhysRevE.88.063008.

    • Search Google Scholar
    • Export Citation
  • Voßkuhle, M., A. Pumir, E. Lévêque, and M. Wilkinson, 2014: Prevalence of the sling effect for enhancing collision rates in turbulent suspensions. J. Fluid Mech., 749, 841852, https://doi.org/10.1017/jfm.2014.259.

    • Search Google Scholar
    • Export Citation
  • Wallace, J. M., and P. V. Hobbs, 2006: Atmospheric Science: An Introductory Survey. 2nd ed. Academic Press, 484 pp.

  • Wang, L.-P., A. S. Wexler, and Y. Zhou, 1998: On the collision rate of small particles in isotropic turbulence. I. Zero inertia case. Phys. Fluids, 10, 266276, https://doi.org/10.1063/1.869565.

    • Search Google Scholar
    • Export Citation
  • Wang, L.-P., O. Ayala, S. E. Kasprzak, and W. W. Grabowski, 2005: Theoretical formulation of collision rate and collision efficiency of hydrodynamically interacting cloud droplets in turbulent atmosphere. J. Atmos. Sci., 62, 24332450, https://doi.org/10.1175/JAS3492.1.

    • Search Google Scholar
    • Export Citation
  • Wang, P. K., 2013: Physics and Dynamics of Clouds and Precipitation. Cambridge University Press, 452 pp.

  • Wang, P. K., and W. Ji, 1997: Numerical simulation of three-dimensional unsteady flow past ice crystals. J. Atmos. Sci., 54, 22612274, https://doi.org/10.1175/1520-0469(1997)054<2261:NSOTDU>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Wang, P. K., and W. S. Ji, 2000: Collision efficiencies of ice crystals at low–Intermediate Reynolds numbers colliding with supercooled cloud droplets: A numerical study. J. Atmos. Sci., 57, 10011009, https://doi.org/10.1175/1520-0469(2000)057<1001:CEOICA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Wilkinson, M., and B. Mehlig, 2005: Caustics in turbulent aerosols. Europhys. Lett., 71, 186, https://doi.org/10.1209/epl/i2004-10532-7.

    • Search Google Scholar
    • Export Citation
  • Wilkinson, M., B. Mehlig, and V. Bezuglyy, 2006: Caustic activation of rain showers. Phys. Rev. Lett., 97, 048501, https://doi.org/10.1103/PhysRevLett.97.048501.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 498 498 151
Full Text Views 99 99 30
PDF Downloads 137 137 43