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Cellular Automata for Clouds and Convection

  • Daan CrommelinEmail author
Chapter
Part of the Emergence, Complexity and Computation book series (ECC, volume 27)

Abstract

Numerical models of the global atmosphere have spatial resolutions that are much too coarse to resolve clouds and convection processes explicitly. Because these processes play an important role in the atmosphere and climate system, they are included in numerical models by means of simplified representations, so-called parameterizations. Traditional parameterization schemes for atmospheric convection are deterministic. To overcome the limitations of these deterministic schemes, stochastic parameterizations are being developed. The use of probabilistic cellular automata (PCA) for this application is very new and can provide a way to generate spatial patterns of convection as observed in the atmosphere. It is approached from two directions, both briefly reviewed here. In one approach, convection and other sub-grid-scale processes are represented with deterministic CA. In recent work, this is extended to PCA. In the other approach, convection is represented by means of discrete stochastic processes (finite state Markov chains) on a lattice. In most studies in this direction, there is no direct coupling between neighboring lattice nodes, however recently such couplings are considered as well. To illustrate the concept of parameterization, a frequently used test model (the L96 model) is discussed as well in this chapter. Parameterization of atmospheric convection and clouds with PCA has several interesting mathematical aspects. One is the interactive (two-way) coupling of the PCA to a partial differential equation for large-scale atmospheric flow. The state of the PCA couples to the time evolution of the flow, and in turn the PCA rules (transition probabilities) depend on the flow state. Furthermore, for convection it is natural to consider N-state PCAs with \(N > 2\) rather than a binary (\(N = 2\)) PCA. Finally, statistical inference can be a fruitful approach to construct the PCA rules or transition probabilities for convection. The PCA dependence on the time-evolving atmospheric flow and the large number of configurations for PCAs with \(N > 2\) provide interesting challenges for such inference.

Keywords

Markov chains Stochastic parameterization Atmospheric convection Statistical inference 

Notes

Acknowledgements

DC is financially supported by the Netherlands Organisation for Scientific Research (NWO) through the Vidi project Stochastic models for unresolved scales in geophysical flows.

References

  1. 1.
    Stephens, G.L.: Cloud feedbacks in the climate system: a critical review. J. Clim. 18, 237–273 (2005)CrossRefGoogle Scholar
  2. 2.
    Arakawa, A.: The cumulus parameterization problem: past, present, and future. J. Clim. 17, 2493–2525 (2004)CrossRefGoogle Scholar
  3. 3.
    Randall, D., Khairoutdinov, M., Arakawa, A., Grabowski, W.: Breaking the cloud parameterization deadlock. B. Am. Meteorol. Soc. 84, 1547–1564 (2003)CrossRefGoogle Scholar
  4. 4.
    Tan, J., Jakob, C., Lane, T.P.: The consequences of a local approach in statistical models of convection on its large-scale coherence. J. Geophys. Res. Atmospheres 120, 931–944 (2015)CrossRefGoogle Scholar
  5. 5.
    Palmer, T.N.: A nonlinear dynamical perspective on model error: a proposal for non-local stochastic-dynamic parameterization in weather and climate prediction models. Q. J. R. Meteorol. Soc. 127, 279–304 (2001)Google Scholar
  6. 6.
    Palmer, T.N.: On parametrizing scales that are only somewhat smaller than the smallest resolved scales, with application to convection and orography. In: Proceedings of the ECMWF Workshop on New Insights and Approaches to Convective Parametrization, pp. 328–337 (1997)Google Scholar
  7. 7.
    Berner, J., Doblas-Reyes, F.J., Palmer, T.N., Shutts, G., Weisheimer, A.: Impact of a quasi-stochastic cellular automaton backscatter scheme on the systematic error and seasonal prediction skill of a global climate model. Philos. Trans. R. Soc. A 366, 2561–2579 (2008)CrossRefzbMATHGoogle Scholar
  8. 8.
    Shutts, G.: A stochastic kinetic energy backscatter algorithm for use in ensemble prediction systems. Technical Memorandum 449, ECMWF (2004)Google Scholar
  9. 9.
    Shutts, G.: A kinetic energy backscatter algorithm for use in ensemble prediction systems. Q. J. R. Meteorol. Soc. 131, 3079–3102 (2005)CrossRefGoogle Scholar
  10. 10.
    Bengtsson, L., Körnich, H., Källén, E., Svensson, E.: Large-scale dynamical response to sub-grid-scale organization provided by cellular automata. J. Atmos. Sci. 68, 3132–3144 (2011)CrossRefGoogle Scholar
  11. 11.
    Bengtsson, L., Steinheimer, M., Bechtold, P., Geleyn, J.F.: A stochastic parametrization for deep convection using cellular automata. Q. J. R. Meteorol. Soc. 139, 1533–1543 (2013)CrossRefGoogle Scholar
  12. 12.
    Berner, J., Shutts, G., and Palmer, T.: Parameterising the multiscale structure of organised convection using a cellular automaton. In: ECMWF Workshop on Representation of Sub-grid Processes Using Stochastic-Dynamic Models, pp. 129–139 (2005)Google Scholar
  13. 13.
    Khouider, B., Biello, J., Majda, A.J.: A stochastic multicloud model for tropical convection. Commun. Math. Sci 8, 187–216 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dorrestijn, J., Crommelin, D.T., Siebesma, A.P., Jonker, H.J.J.: Stochastic parameterization of shallow cumulus convection estimated from high-resolution model data. Theor. Comput. Fluid Dyn. 27, 133–148 (2013)CrossRefGoogle Scholar
  15. 15.
    Dorrestijn, J., Crommelin, D.T., Biello, J.A., Böing, S.J.: A data-driven multicloud model for stochastic parameterization of deep convection. Philos. Trans. R. Soc. A 371(1991), 20120374 (2013)CrossRefGoogle Scholar
  16. 16.
    Gottwald, G.A., Peters, K., Davies, L.: A data-driven method for the stochastic parametrisation of subgrid-scale tropical convective area fraction. Q. J. R. Meteorol. Soc. 142, 349–359 (2016)CrossRefGoogle Scholar
  17. 17.
    Majda, A.J., Khouider, B.: Stochastic and mesoscopic models for tropical convection. Proc. Natl. Acad. Sci. 99, 1123–1128 (2002)CrossRefzbMATHGoogle Scholar
  18. 18.
    Dorrestijn, J., Crommelin, D.T., Siebesma, A.P., Jonker, H.J.J., Jakob, C.: Stochastic parameterization of convective area fractions with a multicloud model inferred from observational data. J. Atmos. Sci. 72, 854–869 (2015)CrossRefGoogle Scholar
  19. 19.
    Crommelin, D., Vanden-Eijnden, E.: Subgrid-scale parameterization with conditional Markov chains. J. Atmos. Sci. 65, 2661–2675 (2008)CrossRefGoogle Scholar
  20. 20.
    Lorenz, E.N.: Predictability - a problem partly solved. In: Proceedings of the 1995 ECMWF Seminar on Predictability, ECMWF, Reading, UK, 118 (1996)Google Scholar
  21. 21.
    Pavliotis, G., Stuart, A.: Multiscale Methods: Averaging and Homogenization. Springer (2008)Google Scholar
  22. 22.
    Khouider, B.: A coarse grained stochastic multi-type particle interacting model for tropical convection: nearest neighbour interactions. Comm. Math. Sci 12, 1379–1407 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Richards, F.C., Meyer, T.P., Packard, N.H.: Extracting cellular automaton rules directly from experimental data. Physica D 45, 189–202 (1990)CrossRefzbMATHGoogle Scholar
  24. 24.
    Adamatzky, A.I.: Identification of Cellular Automata. CRC Press (1994)Google Scholar
  25. 25.
    Billings, S.A., Yang, Y.: Identification of probabilistic cellular automata. IEEE Trans. Syst. Man Cybern. B Cybern. 33, 225–236 (2003)CrossRefGoogle Scholar
  26. 26.
    Sun, X., Rosin, P.L., Martin, R.R.: Fast rule identification and neighborhood selection for cellular automata. IEEE Trans. Syst. Man Cybern. B Cybern. 41, 749–760 (2011)CrossRefGoogle Scholar
  27. 27.
    Guo, Y., Billings, S.A., Coca, D.: Identification of N-state spatio-temporal dynamical systems using a polynomial model. Int. J. Bifurc. Chaos 18, 2049–2057 (2008)MathSciNetCrossRefGoogle Scholar
  28. 28.
    De La Chevrotiere, M., Khouider, B., Majda, A.J.: Calibration of the stochastic multicloud model using Bayesian inference. SIAM J. Sci. Comput. 36, B538–B560 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Centrum Wiskunde & Informatica (CWI)AmsterdamThe Netherlands
  2. 2.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

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