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Quasi-hydrostatic primitive equations for ocean global circulation models

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Abstract

Global existence of weak and strong solutions to the quasi-hydrostatic primitive equations is studied in this paper. This model, that derives from the full non-hydrostatic model for geophysical fluid dynamics in the zero-limit of the aspect ratio, is more realistic than the classical hydrostatic model, since the traditional approximation that consists in neglecting a part of the Coriolis force is relaxed. After justifying the derivation of the model, the authors provide a rigorous proof of global existence of weak solutions, and well-posedness for strong solutions in dimension three.

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References

  1. Burger, A. P., The potential vorticity equation: from planetary to small scale, Tellus, 43A, 1991, 191–197.

    Google Scholar 

  2. Benoit Cushman-Roisin, Introduction to Geophysical Fluid Dynamics, Prentice-Hall, Indiana, 1994.

    Google Scholar 

  3. Cao, C. S. and Titi, E. S., Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math., 166(2), 2007, 245–267.

    Article  MATH  MathSciNet  Google Scholar 

  4. Dellar, P. J. and Salmon, R., Shallow water equations with a complete coriolis force and topography, Physics of Fluids, 17(10), 2005, 106601.

    Article  MathSciNet  Google Scholar 

  5. Eckart, C., Hydrodynamics of Oceans and Atmospheres, Pergamon Press, New York, 1960.

    MATH  Google Scholar 

  6. Gill, A. E., Atmosphere-Ocean Dynamics, Academic Press, New York, 1982.

    Google Scholar 

  7. Kobelkov, G. M., Existence of a solution “in the large” for ocean dynamics equations, J. Math. Fluid Mech., 9(4), 2007, 588–610.

    Article  MATH  MathSciNet  Google Scholar 

  8. Kukavica, I. and Ziane, M., On the regularity of the primitive equations of the ocean, Nonlinearity, 20, 2007, 2739–2753.

    Article  MATH  MathSciNet  Google Scholar 

  9. Lucas, C. and Rousseau, A., New developments and cosine effect in the viscous shallow water and quasigeostrophic equations, Multiscale Modeling and Simulations, 7(2), 2008, 793–813.

    MathSciNet  Google Scholar 

  10. Lucas, C. and Rousseau, A., Cosine effect in ocean models, Discrete Contin. Dyn. Syst. Series B, 13(4), 2010, 841–857.

    Article  MATH  MathSciNet  Google Scholar 

  11. Lions, J. L., Temam, R. and Wang, S. H., New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5(2), 1992, 237–288.

    Article  MATH  MathSciNet  Google Scholar 

  12. Lions, J. L., Temam, R. and Wang, S. H., On the equations of the large-scale ocean, Nonlinearity, 5(5), 1992, 1007–1053.

    Article  MATH  MathSciNet  Google Scholar 

  13. Madec, G., NEMO ocean engine, Note du Pole de Modélisation, 27, Institut Pierre-Simon Laplace (IPSL), France, 2008.

    Google Scholar 

  14. Madec, G., Delecluse, P, Imbard, M. and Lévy, C., OPA 8.1 Ocean General Circulation Model reference manual, Note du Pole de Modélisation, Institut Pierre-Simon Laplace (IPSL), France, 1999.

    Google Scholar 

  15. Pedlosky, J., Geophysical Fluid Dynamics, 2nd ed., Springer-Verlag, New York, 1987.

    MATH  Google Scholar 

  16. Phillips, N. A., The equations of motion for a shallow rotating atmosphere and the “traditional approximation”, J. Atmospheric Sciences, 23, 1966, 626–628.

    Article  Google Scholar 

  17. Phillips, N. A., Reply (to George Veronis), J. Atmospheric sciences, 25, 1968, 1155–1157.

    Article  Google Scholar 

  18. Petcu, M., Temam, R. and Ziane, M., Some mathematical problems in geophysical fluid dynamics, Handbook of Numerical Analysis, Special Volume on Computational Methods for the Oceans and the Atmosphere, P. G. Ciarlet (ed.), Vol. XIV. Elsevier, Amsterdam, 2008.

    Google Scholar 

  19. Shchepetkin, A. F. and McWilliams, J. C., The regional oceanic modeling system (ROMS): a split-explicit, free-surface, topography-following-coordinate oceanic model, Ocean Modelling, 9(4), 2005, 347–404.

    Article  Google Scholar 

  20. Veronis, G., Comments on phillips proposed simplification of the equations of motion for a shallow rotating atmosphere, J. Atmospheric Sciences, 25, 1968, 1154–1155.

    Article  Google Scholar 

  21. Wangsness, R. K., Comments on “the equations of motion for a shallow rotating atmosphere and the ‘traditionnal approximation’ ”, J. Atmospheric Sciences, 27, 1970, 504–506.

    Article  Google Scholar 

  22. White, A. A. and Bromley, R. A., Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the coriolis force, Q. J. R. Meteorol. Soc., 121, 1995, 399–418.

    Article  Google Scholar 

  23. White, A. A., Hoskins, B. J., Roulstone, I. and Staniforth, A., Consistent approximate models of the global atmosphere: shallow, deep, hydrostatic, quasi-hydrostatic and non-hydrostatic, Q. J. R. Meteorol. Soc., 131, 2005, 2081–2107.

    Article  Google Scholar 

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Authors and Affiliations

  1. Laboratoire MAPMO (UMR 6628), Fédération Denis Poisson (FDP-FR2964), Université d’Orléans, Bâtiment de Mathématiques — Route de Chartres, B. P. 6759, 45067, Orléans Cedex 2, France

    Carine Lucas

  2. Laboratoire de Mathématiques et Applications, Téléport 2, B. P. 30179, Boulevard Marie et Pierre Curie, 86962, Futuroscope Chasseneuil Cedex, France

    Madalina Petcu

  3. Laboratoire Jean Kuntzmann, INRIA, 51 rue des Mathématiques, B. P. 53, 38041, Grenoble Cedex 9, France

    Antoine Rousseau

Authors
  1. Carine Lucas
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  2. Madalina Petcu
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  3. Antoine Rousseau
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Corresponding author

Correspondence to Carine Lucas.

Additional information

Dedicated to Professor Roger Temam on the Occasion of his 70th Birthday

Project supported by the ANR (No. ANR-06-BLAN0306-01), the National Science Foundation (No. NSF-DMS-0906440) and the Research Fund of Indiana University.

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Lucas, C., Petcu, M. & Rousseau, A. Quasi-hydrostatic primitive equations for ocean global circulation models. Chin. Ann. Math. Ser. B 31, 939–952 (2010). https://doi.org/10.1007/s11401-010-0611-6

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  • DOI: https://doi.org/10.1007/s11401-010-0611-6

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