A stochastic closure for two-moment bulk microphysics of warm clouds: part I, derivations


We propose a mathematical methodology to two-moment parameterization of bulk warm cloud microphysics based on a stochastic representation of the high-order moment terms. Unlike previous bulk parameterizations, the stochastic closure approach proposed here does not assume any particular droplet size distribution, all parameters have physical meanings which are recoverable from data, and the resultant parameterization has the flexibility to utilize arbitrary collision kernels. Our strategy is a new twofold approach of approximating the kinetic collection equation (KCE). First, by partitioning the droplet spectrum into two large bins representing cloud and rain aggregate particles, we are able to represent droplet densities as the sum of mean and random fluctuation terms. Second, we use a Taylor approximation for the collision kernel around the centres of masses of bulk cloud and rain aggregates which allows the derivation of bulk rate equations for the cloud and rain aggregate droplet numbers and mixing ratios that are independent of the collision kernel. Detailed numerical simulations of the KCE demonstrate that the high-order (quadratic and cubic) stochastic fluctuation terms can be neglected which results in a closed set of equations for the mean bulk cloud and rain aggregates with only five parameters. Considerations of consistency of cloud number concentration and mass conservation constraints further reduce the parameter set to three key entities representing, respectively, the strength of cloud self-collection, the strength of auto-conversion, and the strength of rain self-collection, relative to auto-conversion. In an accompanying paper, bounds on the parameters’ space are derived and the mean bulk two-moment parameterization is validated against direct simulation of the KCE and compared to an existing competitor parameterization.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. 1.

    Beheng, K.D.: The evolution of raindrop spectra: a review of microphysical essentials. In: Testik, F.Y., Gebremichael, M. (eds.) Rainfall: State of the Science, Geophysical Monograph Series, pp. 29–48 (2010)

  2. 2.

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

    Article  Google Scholar 

  3. 3.

    David, C.: A stochastic bulk model for turbulent collision and coalescence of cloud droplets, Ph.D. thesis, University of Victoria, p. 7. https://dspace.library.uvic.ca/handle/1828/7413 (2016)

  4. 4.

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

    Article  Google Scholar 

  5. 5.

    Dohms, H., Beheng, K.D.: Mathematical formulation of self-collection, autoconversion and accretion rates of cloud and raindrops. Met. Rundschau 39, 98–102 (1986)

    Google Scholar 

  6. 6.

    Drake, R., Wright, T.J.: The scalar transport equation of coalescence theory: new families of exact solutions. J. Atmos. Sci. 29, 548–556 (1972)

    MathSciNet  Article  Google Scholar 

  7. 7.

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

    Article  Google Scholar 

  8. 8.

    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 

  9. 9.

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

    MathSciNet  Article  Google Scholar 

  10. 10.

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

    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, Volume 10 of Meteorological Monographs. American Meteorological Society, pp. 1–84 (1969)

  12. 12.

    Khain, A.P., Beheng, K.D., Heymsfield, A., Korolev, A., Krichak, S.O., Levin, Z., Pinsky, M., Phillips, V., Prabhakaran, T., Teller, A., van den Heever, S.C., Yano, J.-I.: Representation of microphysical processes in cloud-resolving models: spectral (bin) microphysics versus bulk parameterization. Rev. Geophys. 53(2), 247–322 (2015)

    Article  Google Scholar 

  13. 13.

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

    Article  Google Scholar 

  14. 14.

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

    Google Scholar 

  15. 15.

    Krueger, Steven: Linear eddy modeling of entrainment and mixing in stratus clouds. J. Atmos. Sci. 50(18), 3078–3090 (1993)

    Article  Google Scholar 

  16. 16.

    Krueger, S.K., Su, C.W., McMurtry, P.A.: Modeling entrainment and finescale mixing in cumulus clouds. J. Atmos. Sci. 54(23), 2697–2712 (1997)

    Article  Google Scholar 

  17. 17.

    Liu, Y., Daum, P.H.: Parameterization of the autoconversion process. Part I: analytical formulation of the Kessler-type parameterizations. J. Atmos. Sci. 61(13), 1539–1548 (2004)

    Article  Google Scholar 

  18. 18.

    Liu, Y., Daum, P.H., McGraw, R., Wood, R.: Parameterization of the autoconversion process. Part II: generalization of Sundqvist-type parameterizations. J. Atmos. Sci. 63(3), 1103–1109 (2006)

    Article  Google Scholar 

  19. 19.

    Liu, Y., Daum, P.H.: Parameterization of the autoconversion process. Part I: analytical formulation of the Kessler-type parameterizations. J. Atmos. Sci. 61(13), 1539–1548 (2004)

    Article  Google Scholar 

  20. 20.

    Lynch, P.: The origins of computer weather prediction and climate modeling. J. Comput. Phys. 227(7), 3431–3444 (2008)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Moncrieff, M.W.: The multiscale organization of moist convection and the intersection of weather and climate. Clim. Dyn. Why Does Clim. Vary 189, 3–26 (2010)

    Article  Google Scholar 

  22. 22.

    Oreskes, Naomi, Shrader-Frechette, Kristin, Belitz, Kenneth: Verification, validation, and confirmation of numerical models in the earth sciences. Science 263(5147), 641–646 (1994)

    Article  Google Scholar 

  23. 23.

    Pinsky, M., Khain, A., Krugliak, H.: Collisions of cloud droplets in a turbulent flow. Part v: applications of detailed tables of turbulent collision rate enhancement to simulation of droplet spectra evolution. J. Atmos. Sci. 65, 357–374 (2008)

    Article  Google Scholar 

  24. 24.

    Posselt, D.J., Vukicevic, T.: Robust characterization of model physics uncertainty for simulations of deep moist convection. Mon. Weather Rev. 138, 1513–1535 (2010)

    Article  Google Scholar 

  25. 25.

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

    Google Scholar 

  26. 26.

    Rauber, R.M., Stevens, B., Ochs, H.T., Knight, C., Albrecht, B.A., Blyth, A.M., Fairall, C.W., Jensen, J.B., Lasher-Trapp, S.G., Mayol-Bracero, O.L., et al.: Rain in shallow cumulus over the ocean: the RICO campaign. Bull. Am. Meteor. Soc. 88(12), 1912–1928 (2007)

    Article  Google Scholar 

  27. 27.

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

    Google Scholar 

  28. 28.

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

    Article  Google Scholar 

  29. 29.

    Simmel, M., Trautmann, T., Tetzlaff, G.: Numerical solution of the stochastic collection equation: comparison of the linear discrete method with other methods. Atmos. Res. 61, 135–148 (2002)

    Article  Google Scholar 

  30. 30.

    Staniforth, Andrew, Thuburn, John: Horizontal grids for global weather and climate prediction models: a review. Q. J. R. Meteor. Soc. 138(662), 1–26 (2012)

    Article  Google Scholar 

  31. 31.

    Stensrud, David J.: Parametrization Schemes: Keys to Understanding Numerical Weather Prediction Models. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  32. 32.

    Su, C.W., Krueger, S.K., McMurtry, P.A., Austin, P.H.: Linear eddy modeling of droplet spectral evolution during entrainment and mixing in cumulus clouds. Atmos. Res. 47, 41–58 (1998)

    Article  Google Scholar 

  33. 33.

    Szakáll, M., Diehl, K., Mitra, S.K., Borrmann, S.: A wind tunnel study on the shape, oscillation, and internal circulation of large raindrops with sizes between 2.5 and 7.5 mm. J. Atmos. Sci. 66(3), 755–765 (2009)

    Article  Google Scholar 

  34. 34.

    van Lier-Walqui, M., Vukicevic, T., Posselt, D.J.: Quantification of cloud microphysical parameterization uncertainty using radar reflectivity. Mon. Weather Rev. 140, 3442–3466 (2012)

    Article  Google Scholar 

  35. 35.

    Wang, L.-P., Xue, Y., Grabowski, W.: A bin integral method for solving the kinetic collection equation. J. Comp. Phys. 226, 59–88 (2007)

    MathSciNet  Article  Google Scholar 

  36. 36.

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

    Article  Google Scholar 

  37. 37.

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

    Article  Google Scholar 

  38. 38.

    Zawadzki, I., Fabry, F.: The development of drop size distribution is light rain. J. Atmos. Sci. 51, 1100–1114 (1994)

    Article  Google Scholar 

Download references


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

Author information



Corresponding author

Correspondence to Boualem Khouider.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Collins, D., Khouider, B. A stochastic closure for two-moment bulk microphysics of warm clouds: part I, derivations. Res Math Sci 8, 11 (2021). https://doi.org/10.1007/s40687-021-00246-7

Download citation


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