A stochastic closure for two-moment bulk microphysics of warm clouds: part II, parameter constraint and validation


The representation of clouds and associated processes of rain and snow formation remain one of the major uncertainties in climate and weather prediction models. In a companion paper (part I), we systematically derived a two-moment bulk cloud microphysics model for warm rain based on the kinetic coalescence equation (KCE) and the use of stochastic approximations to close the high order moment terms, independently of the collision kernel. Conservation of mass and consistency of droplet number concentration of the evolving cloud distribution combined with numerical simulations are used as design principles to reduce the parametrization problem to three key parameters. Here, we further derive physical limits or region of validity for these three parameters based on the physics of collision and coalescence processes: “the stochastic region of validity”. More importantly, in this second part, we validate the stochastically derived bulk cloud microphysics model against detailed simulations based on the KCE and in comparison with a similar model by Seifert and Beheng (J Atmos Sci 59–60:265–281, 2001; hereafter SB01) who instead prescribed the shapes of the droplet distributions of rain and clouds in order to close the high-order moments and have done so specifically for one given kernel only. A thorough parameter exploration of the stochastic validity region is conducted, and parameter values that faithfully reproduce the detailed KCE results are identified. The results show that for typical parameter values, dependent on the environmental conditions, the new parameterization outperforms that of SB01 when compared to the KCE benchmark simulations. These results can be explored in the future to design a Markov jump process to randomly select adequate parameters within the validity region conditional on the environmental conditions and the age of the cloud. Furthermore, sensitivity tests indicate that the stochastically derived model can be used with a time step as large as 30 s without significantly compromising accuracy, which makes it very attractive to use in medium to long range weather prediction models.

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  1. 1.

    Devenish, B., Bartello, P.: Review article: droplet growth in warm turbulent clouds. J. Q. R. Meteorol. Soc. 138, 1401–1421 (2012)

    Article  Google Scholar 

  2. 2.

    Bony, S., Dufresne, J.: Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models. Geophys. Res. Lett. 32, 20806 (2005)

    Article  Google Scholar 

  3. 3.

    Benner, T.C., Curry, J.A.: Characteristics of small tropical cumulus clouds and their impact on the environment. J. Geophys. Res. 103(28), 28753–28767 (1998)

    Article  Google Scholar 

  4. 4.

    Devenish, B., Bartello, P.: Review article: droplet growth in warm turbulent clouds. J. Q. R. Meteorol. Soc. 138, 1401–1421 (2012)

    Article  Google Scholar 

  5. 5.

    Khouider, B.: Models for Tropical Climate Dynamics: Waves, Clouds, and Precipitation. Springer, Berlin (2019)

    Book  Google Scholar 

  6. 6.

    Pruppacher, H., Klett, J.: Microphysics of Clouds and Precipitation. Kluwer Academic Publishers, Boston (1997)

    Google Scholar 

  7. 7.

    Baker, M., Breidenthal, R., Choularton, T., Latham, J.: The effects of turbulent mixing in clouds. J. Atmos. Sci. 41(2), 299–304 (1984)

    Article  Google Scholar 

  8. 8.

    Franklin, C.: A warm rain microphysics parameterization that includes the effects of turbulence. J. Atmos. Sci. 65, 1795–1816 (2008)

    Article  Google Scholar 

  9. 9.

    Grabowski, W.W., Wang, L.: Growth of cloud droplets in a turbulent environment. Annu. Rev. Fluid Mech. 45, 293–324 (2013)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Shaw, R.: Particle turbulent interactions in atmospheric clouds. Annu. Rev. Fluid Mech. 35, 183–227 (2003)

    Article  Google Scholar 

  11. 11.

    Kessler, E.: On the distribution and continuity of water substance in atmospheric circulations. In: Kessler, E. (ed.) On the Distribution and Continuity of Water Substance in Atmospheric Circulations, Vol. 10 of Meteorological Monographs, pp. 1–84. American Meteorological Society, Providence (1969)

    Google Scholar 

  12. 12.

    Liu, Y., Daum, P.: Parameterization of the autoconversation process. Part I: analytical formulation of the Kessler-type parameterizations. J. Atmos. Sci. 61, 1539–1548 (2004)

    Article  Google Scholar 

  13. 13.

    Liu, Y., Daum, P., McGraw, R., Wood, R.: Parameterization of the autoconversation process. Part II: generalization of the Sundqvist-type parameterizations. J. Atmos. Sci. 63, 1103–1109 (2006)

    Article  Google Scholar 

  14. 14.

    Wood, R., Blossey, P.: Comments on parameterization of the autoconversion process. Part I: analytical formulation of the Kessler-type parameterizations. Am. Meteorol. Soc. 62, 3003–3006 (2005)

    Google Scholar 

  15. 15.

    Khairoutdinov, M., Kogan, Y.: A new cloud physics parameterization in a large eddy simulation model of marine stratocumulus. Mon. Wea. Rev. 128, 229–243 (2000)

    Article  Google Scholar 

  16. 16.

    Seifert, A., Beheng, K.D.: A double moment parameterization for simulating autoconversion, accretion, and self-collection. J. Atmos. Sci. 59–60, 265–281 (2001)

    Google Scholar 

  17. 17.

    Collins, D., Khouider, B.: A Stochastic closure for two-moment bulk microphysics of warm clouds: part I, derivations, Research in the Mathematical Sciences. In press (2021)

  18. 18.

    Sundqvist, H.: A parameterization scheme for non-convective condensation including prediction of cloud water content. Q. J. Met. Soc. 104, 667–690 (1978)

    Google Scholar 

  19. 19.

    Zawadzki, I., Fabry, F.: The development of drop size distribution is light rain. J. Atmos. Sci. 51(1100–1114)

  20. 20.

    Wood, R.: Drizzle in stratiform boundary layer clouds. Part II: microphysical aspects. J. Atmos. Sci. 62, 3034–3050 (2005)

    Article  Google Scholar 

  21. 21.

    Wood, R., Field, P., Cotton, W.R.: Autoconversion rate bias in stratiform boundary layer cloud parameterizations. Atmos. Res. 65, 109–128 (2002)

    Article  Google Scholar 

  22. 22.

    Franklin, C., Vaillancourt, P., Yau, M.K.: Statistics and parameterizations of the effect of turbulence on the geometric collision Kernel of cloud droplets. J. Atmos. Sci. 64, 938–954 (2007)

    Article  Google Scholar 

  23. 23.

    Khouider, B., Biello, J., Majda, A.J.: A stochastic multicloud model for tropical convection. Commun. Math. Sci. 8(1), 187–216 (2010)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Seifert, A., Stevens, B.: Microphysical scaling relations in a kinematic model of isolated shallow cumulus clouds. J. Atmos. Sci. 67(1575–1590)

  25. 25.

    Hall, W.D.: A detailed microphysical model within a two-dimensional dynamic framwork: model description and preliminary results. J. Atmos. Sci. 37, 2486–2507 (1980)

    Article  Google Scholar 

  26. 26.

    Bott, A.: A flux method for the numerical solution of the stochastic collection equation. J. Atmos. Sci. 55, 2284–2293 (1998)

    Article  Google Scholar 

Download references


This research is part of D. C.’s Ph.D. thesis. The research of B. K. is partly supported by a grant from the Natural Sciences and Engineering Research Council of Canada. D. C.’s fellowship is partly funded through this grant.

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Correspondence to Boualem Khouider.

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Collins, D., Khouider, B. A stochastic closure for two-moment bulk microphysics of warm clouds: part II, parameter constraint and validation. Res Math Sci 8, 15 (2021). https://doi.org/10.1007/s40687-021-00247-6

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  • Cloud microphysics
  • Bulk parameterizations
  • Stochastic differential equations
  • Kinetic collection equation
  • Collision and coalescence
  • Two moment closure