Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematics of Climate and Weather Forecasting

Ed. by Khouider, Boualem

Covered by MathSciNet and Zentralblatt Math (zbMATH)

Open Access
See all formats and pricing
More options …

Stochastic Galerkin method for cloud simulation

A. Chertock / A. Kurganov
  • Department of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ M. Lukáčová-Medvid’ová / P. Spichtinger / B. Wiebe
Published Online: 2019-11-20 | DOI: https://doi.org/10.1515/mcwf-2019-0005


We develop a stochastic Galerkin method for a coupled Navier-Stokes-cloud system that models dynamics of warm clouds. Our goal is to explicitly describe the evolution of uncertainties that arise due to unknown input data, such as model parameters and initial or boundary conditions. The developed stochastic Galerkin method combines the space-time approximation obtained by a suitable finite volume method with a spectral-type approximation based on the generalized polynomial chaos expansion in the stochastic space. The resulting numerical scheme yields a second-order accurate approximation in both space and time and exponential convergence in the stochastic space. Our numerical results demonstrate the reliability and robustness of the stochastic Galerkin method. We also use the proposed method to study the behavior of clouds in certain perturbed scenarios, for examples, the ones leading to changes in macroscopic cloud pattern as a shift from hexagonal to rectangular structures.


  • [1] Alboussière, T., and Ricard, Y. Rayleigh-Bénard stability and the validity of quasi-Boussinesq or quasi-anelastic liquid approximations. J. Fluid Mech. 817 (2017), 264–305.Web of ScienceGoogle Scholar

  • [2] Ascher, U. M., Ruuth, S. J., and Spiteri, R. J. Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 2-3 (1997), 151–167. Special issue on time integration (Amsterdam, 1996).CrossrefGoogle Scholar

  • [3] Baldauf, M., Förstner, J., Klink, S., Reinhardt, T., Schraff, C., Seifert, A., and Stephan, K. Kurze modell- und datenbankbeschreibung COSMO-DE (LMK).Google Scholar

  • [4] Beheng, K. D. The Evolution of Raindrop Spectra: A Review of Microphysical Essentials. In Rainfall: State of the Science (2010), F. Y. Testik and M. Gebremichael, Eds., Geophysical Monograph Series, pp. 29–48.Google Scholar

  • [5] Bispen, G. IMEX finite volume schemes for the shallow water equations, PhD thesis. Johannes Gutenberg-University, Mainz, 2015.Google Scholar

  • [6] Bispen, G., Luká£ová-Medvid’ová, M., and Yelash, L. Asymptotic preserving IMEX finite volume schemes for low Mach number Euler equations with gravitation. J. Comput. Phys. (2017).CrossrefGoogle Scholar

  • [7] Bodenschatz, E., Pesch, W., and Ahlers, G. Recent developments in rayleigh-benard convection. Annual Review of Fluid Mechanics 32 (2000), 709–778.Google Scholar

  • [8] Bryan, G. H., and Fritsch, J. M. A benchmark simulation for moist nonhydrostatic numerical models. Mont. Weather Rev. 130 (2002), 2917–2928.Google Scholar

  • [9] Canuto, C. Spectral Methods - Fundamentals in Single Domains. Springer, 2006.Google Scholar

  • [10] Davies, R. M., and Taylor, F. J. The mechanism of large bubbles rising through extended liquids and through liquids in tubes. Proc. Royal Soc. Lond. A 200 (1950), 375–390.Google Scholar

  • [11] Després, B., Poëtte, G., and Lucor, D. Robust uncertainty propagation in systems of conservation laws with the entropy closure method. In Uncertainty Quantification in Computational Fluid Dynamics. Springer, 2013, pp. 105–149.Google Scholar

  • [12] Dixon, J. The Shock Absorber Handbook. Wiley, 2007.Google Scholar

  • [13] Elman, H. C., Miller, C. W., Phipps, E. T., and Tuminaro, R. S. Assessment of collocation and Galerkin approaches to linear diffusion equations with random data. Int. J. Uncertain. Quantif. 1, 1 (2011), 19–33.Google Scholar

  • [14] Getling, A. V. Rayleigh-Bénard convection, Structure and Dynamics. World Sci. Publ., Singpore, 2001.Google Scholar

  • [15] Grabowski, W. W. Untangling microphysical impacts on deep convection applying a novel modeling methodology. J. Atmos. Sci. 72, 6 (2015), 2446–2464.Web of ScienceGoogle Scholar

  • [16] Igel, A. L., and van den Heever, S. C. The role of latent heating in warm frontogenesis. Quart. J. Roy. Met. Soc. 140, 678, A (2014), 139–150.Google Scholar

  • [17] Kessler, E. On the distribution and continuity of water substance in atmospheric circulations., vol. 32 of Meteorol. Monographs. American Meteorological Society, Boston, 1969.Google Scholar

  • [18] Khain, A. P., Ovtchinnikov, M., Pinsky, M., Pokrovsky, A., and Krugliak, H. Notes on the state-of-the-art numerical modeling of cloud microphysics. Atmos. Res. 55, 3-4 (2000), 159 – 224.Google Scholar

  • [19] Khvorostyanov, V. I. Mesoscale processes of cloud formation, cloud-radiation interaction, and their modeling with explicit cloud microphysics. Atmos. Res. 39, 1-3 (1995), 1–67.Google Scholar

  • [20] Köhler, H. The nucleus in and the growth of hygroscopic droplets. T. Faraday Soc. 32, 2 (1936), 1152–1161.CrossrefGoogle Scholar

  • [21] Kurganov, A. Finite-volume schemes for shallow-water equations. Acta Numer. 27 (2018), 289–351.Web of ScienceGoogle Scholar

  • [22] Lamb, D., and Verlinde, J. Physics and chemistry of clouds. Cambridge University Press, 2011.Google Scholar

  • [23] Lukáčová-Medvid’ová, M., Rosemeier, J., Spichtinger, P., and Wiebe, B. IMEX finite volume methods for cloud simulation. In Finite volumes for complex applications VIII—hyperbolic, elliptic and parabolic problems, vol. 200 of Springer Proc. Math. Stat. Springer, Cham, 2017, pp. 179–187.Google Scholar

  • [24] Ma, X., and Zabaras, N. An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228, 8 (2009), 3084–3113.Web of ScienceGoogle Scholar

  • [25] Marinescu, P. J., van den Heever, S. C., Saleeby, S. M., Kreidenweis, S. M., and DeMott, P. J. The microphysical roles of lower-tropospheric versus midtropospheric aerosol particles in mature-stage mcs precipitation. J. Atmos. Sci. 74, 11 (2017), 3657–3678.CrossrefGoogle Scholar

  • [26] Marshall, J. S., and Palmer, W. M. The distributions of raindrops with size. J. Meteorol. 5 (1948), 165–166.CrossrefGoogle Scholar

  • [27] Medovikov, A. A. Dumka 3 code, available at http://dumkaland.org/.

  • [28] Medovikov, A. A. High order explicit methods for parabolic equations. BIT 38, 2 (1998), 372–390.CrossrefGoogle Scholar

  • [29] Mishra, S., and Schwab, C. Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data. Math. Comp. 81, 280 (2012), 1979–2018.Web of ScienceGoogle Scholar

  • [30] Mishra, S., Schwab, C., and Šukys, J. Multi-level Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws. In Uncertainty quantification in computational fluid dynamics, vol. 92 of Lect. Notes Comput. Sci. Eng. Springer, Heidelberg, 2013, pp. 225–294.CrossrefGoogle Scholar

  • [31] Murphy, D., and Koop, T. Review of the vapour pressure of ice and supercooled water for atmospheric applications. Quarterly Journal of the Royal Meterorological Society 131 (2005), 1539–1565.Google Scholar

  • [32] Pauluis, O., and Schumacher, J. Idealized moist Rayleigh-Bénard convection with piecewise linear equation of state. Comm. Math. Sci. 8 (2010), 295–319.Google Scholar

  • [33] Petters, M. D., and Kreidenweis, S. M. A single parameter representation of hygroscopic growth and cloud condensation nucleus activity. Atmos. Chem. Phys. 7, 8 (2007), 1961–1971.Web of ScienceCrossrefGoogle Scholar

  • [34] Poëtte, G., Després, B., and Lucor, D. Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228, 7 (2009), 2443–2467.Web of ScienceGoogle Scholar

  • [35] Porz, N., Hanke, M., Baumgartner, M., and Spichtinger, P. A consistent model for liquid clouds. Math. Clim. Weather Forecast. 4, 1 (2018), 50–78.Google Scholar

  • [36] Pruppacher, H. R., and Klett, J. D. Microphysics of Clouds and Precipitation. Springer, 2010.Google Scholar

  • [37] Schuster, D., Brdar, S., Baldauf, M., Dedner, A., Klöfkorn, R., and Kröner, D. On discontinuous Galerkin approach for atmospheric flow in the mesoscale with and without moisture. Meteorol. Z. 23, 4 (2011), 449–464.Web of ScienceGoogle Scholar

  • [38] Šukys, J., Mishra, S., and Schwab, C. Multi-level Monte Carlo finite difference and finite volume methods for stochastic linear hyperbolic systems. In Monte Carlo and quasi-Monte Carlo methods 2012, vol. 65 of Springer Proc. Math. Stat. Springer, Heidelberg, 2013, pp. 649–666.Google Scholar

  • [39] Szakall, M., Diehl, K., Mitra, S. K., and 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. Journalof the Atmospheric Sciences 66, 3 (2009), 755–765.Google Scholar

  • [40] Szakall, M., Mitra, S. K., Diehl, K., and Borrmann, S. Shapes and oscillations of falling raindrops - a review. Atmospheric Research 97, 4, SI (2010), 416–425.CrossrefGoogle Scholar

  • [41] Tryoen, J., Le Maître, O., Ndjinga, M., and Ern, A. Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems. J. Comput. Phys. 229, 18 (2010), 6485–6511.Web of ScienceGoogle Scholar

  • [42] Wan, X., and Karniadakis, G. E. Long-term behavior of polynomial chaos in stochastic flow simulations. Comput. Methods Appl. Mech. Engrg. 195, 41-43 (2006), 5582–5596.Google Scholar

  • [43] Warner, J. The microstructure of cumulus cloud. part i. general features of the droplet spectrum. J. Atmos. Sci. 26, 5 (1969), 1049–1059.CrossrefGoogle Scholar

  • [44] Weidauer, T., Pauluis, O., and Schumacher, J. Cloud patterns and mixing properties in shallow moist rayleigh-benard convection. New Journal of Physics 12 (OCT 7 2010).CrossrefWeb of ScienceGoogle Scholar

  • [45] Weidauer, T., and Schumacher, J. Toward a mode reduction strategy in shallow moist convection. New Journal of Physics 15 (2013), 125025–125249.Web of ScienceGoogle Scholar

  • [46] Witteveen, J. A. S., Loeven, A., and Bijl, H. An adaptive stochastic finite elements approach based on Newton-Cotes quadrature in simplex elements. Comput. & Fluids 38, 6 (2009), 1270–1288.CrossrefWeb of ScienceGoogle Scholar

  • [47] Xiu, D., and Hesthaven, J. S. High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 3 (2005), 1118–1139 (electronic).CrossrefGoogle Scholar

  • [48] Xiu, D., and Karniadakis, G. E. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 2 (2002), 619–644 (electronic).CrossrefGoogle Scholar

About the article

Received: 2018-11-29

Accepted: 2019-11-01

Published Online: 2019-11-20

Citation Information: Mathematics of Climate and Weather Forecasting, Volume 5, Issue 1, Pages 65–106, ISSN (Online) 2353-6438, DOI: https://doi.org/10.1515/mcwf-2019-0005.

Export Citation

© 2019 A. Chertock et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Manuel Baumgartner, Max Sagebaum, Nicolas R. Gauger, Peter Spichtinger, and André Brinkmann
Geoscientific Model Development, 2019, Volume 12, Number 12, Page 5197

Comments (0)

Please log in or register to comment.
Log in