A Finite Element Algorithm of a Nonlinear Diffusive Climate Energy Balance Model

  • R. Bermejo
  • J. Carpio
  • J. I. Díaz
  • P. Galán del Sastre
Chapter
Part of the Pageoph Topical Volumes book series (PTV)

Abstract

We present a finite element algorithm of a climate diagnostic model that takes as a climate indicator the atmospheric sea-level temperature. This model belongs to the category of energy balance models introduced independently by the climatologists M.I. Budyko and W.D. Sellers in 1969 to study the influence of certain geophysical mechanisms on the Earth climate. The energy balance model we are dealing with consists of a two-dimensional nonlinear parabolic problem on the 2-sphere with the albedo terms formulated according to Budyko as a bounded maximal monotone graph in ℝ2. The numerical model combines the first-order Euler implicit time discretization scheme with linear finite elements for space discretization, the latter is carried out for the special case of a spherical Earth and uses quasi-uniform spherical triangles as finite elements. The numerical formulation yields a nonlinear problem that is solved by an iterative procedure. We performed different numerical simulations starting with an initial datum consisting of a monthly average temperature field, calculated from the temperature field obtained from 50 years of simulations, corresponding to the period 1950–2000, carried out by the Atmosphere General Circulation Model HIRLAM.

Key Words

Climate nonlinear energy balance finite elements 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bermejo, R., Carpio, J., Díaz, J. I, and Tello, L. (2007), Mathematical and numerical analysis of a nonlinear diffusive climate energy balance model (submitted).Google Scholar
  2. Badii, M. and Díaz, J. I. (1999), Time Periodic Solutions for a Diffusive Energy Balance Model in Climatology, J. Math. Anal. AppL. 233, 717–724.CrossRefGoogle Scholar
  3. Barrett, J. W. and Liu, W. B. (1994), Finite element approximation of the parabolic p-Laplacian, SIAM J. Numer. Anal. 31, 413–428.CrossRefGoogle Scholar
  4. Baumgardner, J. R. and Frederickson, P. O. (1985), Icosahedral discretization of the two-sphere, SIAM J. Numer. Anal. 22, 1107–1115.CrossRefGoogle Scholar
  5. Budyko, M. I. (1969), The effects of solar radiation variations on the climate of the Earth, Tellus 21, 611–619.Google Scholar
  6. Carl, S. (1992), A combined variational-monotone iterative method for elliptic boundary value problems with discontinuous nonlinearities, Applicable Analysis 43, 21–45.CrossRefGoogle Scholar
  7. Díaz, J. I., Mathematical analysis of some diffusive energy balance climate models. In Mathematics, Climate and Environment (eds. Bíaz, J. I., and Lions, J. L.) (Masson, Paris 1993) pp. 28–56.Google Scholar
  8. Díaz, J. I. (Ed.) The Mathematics of Models in Climatology and Environment, ASI NATO Global Change Series I, no. 48 (Springer-Verlag, Heidelberg, 1996).Google Scholar
  9. Díaz, J. I., Hernandez, J., and Tello, L. (1997), On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in Climatology, J. Math. Anal. Appl. 216, 593–613.CrossRefGoogle Scholar
  10. Díaz, J. I. and Hetzer, G. A quasilinear functional reaction-diffusion equation arising in Climatology, In Equations aux derivees partielles et applications: Articles dedies a Jacques Louis Lions (Gautier Villards, Paris 1998) pp. 461–480.Google Scholar
  11. Díaz, J. I. and Tello, L. (1999), A nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica 50, 19–51.Google Scholar
  12. Dziuk, G., Finite element for the Beltrami operator on arbitrary surfaces. In Partial Differential Equations and Calculus of Variations, Lectures Notes in Mathematics, vol. 1357 (Springer, Heidelberg 1988) pp. 142–155.CrossRefGoogle Scholar
  13. Giraldo, F. X. and Warburton, T. (2005), A nodal triangle-based spectral element method for the shallow water equations on the sphere, J. Comput. Phys. 207, 129–150.CrossRefGoogle Scholar
  14. Graves, C. E., Lee, W.-H. and North, G. R. (1993), New parameterizations and sensitivities for simple climate models, J. Geophys. Res. 98, 5025–5036.CrossRefGoogle Scholar
  15. Hairer, E., Norsett, S. P. and Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems (Springer-Verlag, Berlin, Heidelberg, 1993).Google Scholar
  16. Heinze, T. and Hense, A. (2002), The shallow water equations on the sphere and their Lagrange-Galerkin solution, Meterol. Atmos. Phys. 81, 129–137.CrossRefGoogle Scholar
  17. Hetzer, G. (1990), The structure of the principal component for semilinear diffusion equations from energy balance climate models, Houston Journal of Math. 16, 203–216.Google Scholar
  18. Hetzer, G., Jarausch, H. and Mackens, W. (1989), A multiparameter sensitivity analysis of a 2D diffusive climate model, Impact and Computing in Science and Engineering 1, 327–393.CrossRefGoogle Scholar
  19. Hyde, W. T., Kim, K.-Y., Crowley, T. J. and North, G. R. (1990), On the relation between polar continentality and climate: Studies with a nonlinear seasonal energy balance model, J. Geophys. Res. 95(D11), 18.653–18.668.CrossRefGoogle Scholar
  20. Ju, N. (2000), Numerical analysis of parabolic p-Laplacian. Approximation of trajectories, SIAM J. Numer. Anal. 37, 1861–1884.CrossRefGoogle Scholar
  21. Myhre, G., Highwood, E. J., Shine, K., and Stordal, F. (1998), New estimates of radiative forcing due to well mixed Greenhouse gases, Geophys. Res. Lett. 25, 2715–2718.CrossRefGoogle Scholar
  22. North, G. R., Multiple solutions in energy balance climate models. In Paleogeography, Paleoclimatology, Paleoecology 82 (Elsevier Science Publishers B.V., Amsterdam, 1990) pp. 225–235.Google Scholar
  23. North, G. R. and Coakley, J. A. (1979), Differences between seasonal and mean annual energy balance model calculations of climate and climate sensitivity, J. Atmos. Sci. 41, 1189–1204.CrossRefGoogle Scholar
  24. Sadourny, R., Arakawa, A. and Mintz, Y. (1968), Integration of the nondivergent barotropic vorticity equation with an icosahedral hexagonal grid for the sphere, Mon. Wea. Rev. 96, 351–356.CrossRefGoogle Scholar
  25. Sellers, W. D., Physical Climatology (The University of Chicago Press, Chicago, Ill. 1965).Google Scholar
  26. Sellers, W. D. (1969), A global climatic model based on the energy balance of the earth-atmosphere system, J. AppL. Meteorol. 8, 392–400.CrossRefGoogle Scholar
  27. Williamson, D. L. (1968), Integration of the barotropic vorticity equation on a spherical geodesic grid, Tellus 20, 642–653.CrossRefGoogle Scholar
  28. Xu, X. (1991), Existence and regularity theorems for a free boundary problem governing a simple climate model, Applicable Anal. 42, 33–59.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 2008

Authors and Affiliations

  • R. Bermejo
    • 1
  • J. Carpio
    • 1
  • J. I. Díaz
    • 2
  • P. Galán del Sastre
    • 3
  1. 1.Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingenieros IndustrialesUniversidad Politécnica de MadridMadridSpain
  2. 2.Dept. Matemática AplicadaUniversidad Complutense de MadridMadridSpain
  3. 3.Dept. Matemática AplicadaUniversidad Politécnica de MadridMadridSpain

Personalised recommendations