Mathematics in Atmospheric Sciences: An Overview

  • Pierre Gauthier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5810)


Several sectors of human activities rely on weather forecasts to plan in preparation of high impact weather events like snow storms, hurricanes or heat waves. Climate studies are needed to make decisions about the long-term development in agriculture, transport and land development. Observing and modeling the evolution of the atmosphere is needed to provide key reliable information for both weather prediction and climate scenarios. This paper gives an overview of the scientific research underlying the development and validation of numerical models of the atmosphere and the monitoring of the quality of the observations collected from several types of instruments. A particular emphasis will be given to data assimilation which establishes the bridge between numerical models and observations. The mathematical problems arising in atmospheric research are diverse as the problem is one of stochastic prediction for which errors in both the model and the observations need to be considered and estimated. Atmospheric predictability is concerned with the chaotic nature of the nonlinear equations that govern the atmosphere. Ensemble prediction is one area that has expanded significantly in the last decade. The interest stems from the necessity to evaluate more than just a forecast: it aims at giving an estimate of its accuracy as well. This brings up more questions than answers.


Tropical Cyclone Data Assimilation Weather Forecast Tropical Rainfall Measurement Mission Numerical Weather Prediction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [Barker et al., 2003]
    Barker, H.W., Pincus, R., Morcrette, J.-J.: The Monte-Carlo Independent Column Approximation: Application within large-scale models. In: Proceedings GCSS/ARM Workshop on the Representation of Cloud System in Large-Scale Models, Kananaskis, Al, Canada, 10 pp. (May 2002),
  2. [Candille and Talagrand, 2005]
    Candille, G., Talagrand, O.: Evaluation of probabilistic prediction systems for a scalar variable. Quart. J.R. Metor. Soc. 131, 2131–2150 (2005)CrossRefGoogle Scholar
  3. [Durran, 1998]
    Durran, D.R.: Numerical methods for wave equations in geophysical fluids. Texts in Applied Mathematics, vol. 32, 463 pages. Springer, Heidelberg (1998)Google Scholar
  4. [Evensen, 1994]
    Evensen, G.: Sequential data assimilation with a nonlinear quasigeostrophic model using Monte Carlo methods to forecast error statistics. em J. Geophys. Res. 99(C5), 10143–10162 (1994)CrossRefGoogle Scholar
  5. [Gauthier and Thépaut, 2001]
    Gauthier, P., Thépaut, J.-N.: Impact of the digital filter as a weak constraint in the preoperational 4DVAR assimilation system of Météo-France. Mon. Wea. Rev. 129, 2089–2102 (2001)CrossRefGoogle Scholar
  6. [Gauthier et al., 2007]
    Gauthier, P., Tanguay, M., Laroche, S., Pellerin, S., Morneau, J.: Extension of 3D-Var to 4D-Var: implementation of 4D-Var at the Meteorological Service of Canada. Mon. Wea. Rev. 135, 2339–2354 (2007)CrossRefGoogle Scholar
  7. [Guckenheimer and Holmes, 1983]
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of vector fields, 453 pages. Springer, Heidelberg (1983)CrossRefzbMATHGoogle Scholar
  8. [Houtekamer et al., 2005]
    Houtekamer, P.L., Mitchell, H.L., Pellerin, G., Buehner, M., Charron, M., Spacek, L., Hansen, B.: Atmospheric Data Assimilation with an Ensemble Kalman Filter: Results with Real Observations. Mon. Wea. Rev. 133, 604–620 (2005)CrossRefGoogle Scholar
  9. [Lorenz, 1963]
    Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)CrossRefGoogle Scholar
  10. [Lovejoy et al., 2001]
    Lovejoy, S., Schertzer, D., Tessier, Y., Gaonach, H.: Multifractals and Resolution independent remote sensing algorithms: the example of ocean colour. Inter. J. Remote Sensing 22, 1191–1234 (2001)CrossRefGoogle Scholar
  11. [Palmer and Hagedorn, 2006]
    Palmer, T., Hagedorn, R.: Predictability of Weather and Climate, 695 pages. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  12. [Rabier et al., 2000]
    Rabier, F., Jrvinen, H., Klinker, E., Mahfouf, J.-F., Simmons, A.: The ECMWF operational implementation of four dimensional variational assimilation. Part I: experimental results with simplified physics. Quart. J.R. Meteor. Soc. 126, 1143–1170 (2000)CrossRefGoogle Scholar
  13. [Rawlins et al., 2007]
    Rawlins, F., Ballard, S.P., Bovis, K.J., Clayton, A.M., Li, D., Inverarity, G.W., Lorenc, A.C., Payne, T.J.: The Met Office global four-dimensional variational data assimilation scheme. Quart. J.R. Meteor. Soc. 623(623), 347–362 (2007)CrossRefGoogle Scholar
  14. [Robert, 1982]
    Robert, A.: A semi-implicit and semi-Lagrangian numerical integration scheme for the primitive meteorological equations. J. Meteor. Soc. Japan 60, 319–325 (1982)Google Scholar
  15. [Schertzer and Lovejoy, 1987]
    Schertzer, D., Lovejoy, S.: Physical modeling and Analysis of Rain and Clouds by Anisotropic Scaling of Multiplicative Processes. Journal of Geophysical Research D8(8), 9693–9714 (1987)CrossRefGoogle Scholar
  16. [Trémolet, 2006]
    Trémolet, Y.: Accounting for an imperfect model in 4D-Var. Quart. J.R. Meteorol. Soc. 132, 2483–2504 (2006)CrossRefGoogle Scholar
  17. [Uppala et al., 2008]
    Ippala, S., Simmons, A., Dee, D., Källberg, P., Thépaut, J.N.: Atmospheric reanalyses and climate variations. In: Climate variability and Extremes during the past 100 years, 364 pages, pp. 103-118. Springer, Heidelberg (2008)Google Scholar
  18. [van Leeuwen, 2009]
    van Leeuwen, P.J.: Particle filters in geophysical systems. To appear in Mon Wea. Rev. (2009)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Pierre Gauthier
    • 1
  1. 1.Department of Earth and Atmospheric SciencesUniversité du Québec à MontréalCanada

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