Cosine effect on shallow water equations and mathematical properties
Author:
Carine Lucas
Journal:
Quart. Appl. Math. 67 (2009), 283-310
MSC (2000):
Primary 76M45, 76U05; Secondary 35B40, 35Q35, 46E35
DOI:
https://doi.org/10.1090/S0033-569X-09-01113-0
Published electronically:
March 20, 2009
MathSciNet review:
2514636
Full-text PDF Free Access
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Additional Information
Abstract:
This paper presents a viscous Shallow Water type model with new Coriolis terms, and some limits according to the values of the Rossby and Froude numbers. We prove that the extension to the bidimensional case of the unidimensional results given by [J.–F. Gerbeau, B. Perthame. Discrete Continuous Dynamical Systems, (2001)] including the Coriolis force has to add new terms, omitted up to now, depending on the latitude cosine, when the viscosity is assumed to be of the order of the aspect ratio.
We show that the expressions for the waves are modified, particularly at the equator, as well as the Quasi-Geostrophic and the Lake equations. To conclude, we also study the mathematical properties of these equations.
References
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- Didier Bresch and Benoît Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl. (9) 86 (2006), no. 4, 362–368 (English, with English and French summaries). MR 2257849, DOI https://doi.org/10.1016/j.matpur.2006.06.005
- Didier Bresch and Guy Métivier, Global existence and uniqueness for the lake equations with vanishing topography: elliptic estimates for degenerate equations, Nonlinearity 19 (2006), no. 3, 591–610. MR 2209290, DOI https://doi.org/10.1088/0951-7715/19/3/004
- J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation, Discrete Contin. Dyn. Syst. Ser. B 1 (2001), no. 1, 89–102. MR 1821555, DOI https://doi.org/10.3934/dcdsb.2001.1.89
- C. David Levermore, Marcel Oliver, and Edriss S. Titi, Global well-posedness for models of shallow water in a basin with a varying bottom, Indiana Univ. Math. J. 45 (1996), no. 2, 479–510. MR 1414339, DOI https://doi.org/10.1512/iumj.1996.45.1199
- Carine Lucas, Effet cosinus sur un modèle visqueux de type Saint-Venant et ses équations limites de type quasi-géostrophique et lacs, C. R. Math. Acad. Sci. Paris 345 (2007), no. 6, 313–318 (French, with English and French summaries). MR 2359088, DOI https://doi.org/10.1016/j.crma.2007.07.013
- C. Lucas and A. Rousseau. New Developments and Cosine Effect in the Viscous Shallow Water and Quasi-Geostrophic Equations, SIAM Multiscale Modeling and Simulation, 7 2 (2008), 796–813.
- Andrew Majda, Introduction to PDEs and waves for the atmosphere and ocean, Courant Lecture Notes in Mathematics, vol. 9, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 1965452
- F. Marche. Theoretical and numerical study of shallow water models; applications to nearshore hydrodynamics, Ph.D., Université Bordeaux (2005).
- F. Marche and P. Fabrie. Another proof of stability for global weak solutions of 2D degenerated Shallow Water models, Journal of Mathematical Fluid Mechanics, (2008).
- J. Pedlosky. Geophysical fluid dynamics, 2d edition, Springer (1987).
- A.J.-C. de Saint-Venant. Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leur lit, Comptes Rendus de l’Académie des Sciences 73 (1871), 147–154.
- Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI https://doi.org/10.1007/BF01762360
References
- D. Bresch and B. Desjardins. Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 1-2 (2003), 211–223. MR 1989675 (2004d:76026)
- D. Bresch and B. Desjardins. On the construction of approximate solutions for the 2D viscous Shallow Water model and for compressible Navier-Stokes models, J. Math. Pure Appl., 86 (2006), 362–368. MR 2257849 (2007j:35161)
- D. Bresch and G. Métivier. Global existence and uniqueness for the lake equations with vanishing topography: elliptic estimates for degenerate equations, Nonlinearity, 19 3 (2006), 591–610. MR 2209290 (2007b:35261)
- J.-F. Gerbeau and B. Perthame. Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation, Discrete and Continuous Dynamical Systems-series B., 1 1 (2001), 89–102. MR 1821555 (2001m:76029)
- C. D. Levermore, M. Oliver and E. S. Titi. Global well-posedness for models of shallow water in a basin with a varying bottom, Indiana Univ. Math. J, 45 (1996), 479–510. MR 1414339 (97m:35214)
- C. Lucas. Effet cosinus sur un modèle visqueux de type Saint-Venant et ses équations limites de type quasi-géostrophique et lacs, C. R. Acad. Sci. Paris, Ser. I, 345 6 (2007), 313–318. MR 2359088 (2008h:35294)
- C. Lucas and A. Rousseau. New Developments and Cosine Effect in the Viscous Shallow Water and Quasi-Geostrophic Equations, SIAM Multiscale Modeling and Simulation, 7 2 (2008), 796–813.
- A. Majda. Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Math. 9, American Mathematical Society (2003). MR 1965452 (2004b:76152)
- F. Marche. Theoretical and numerical study of shallow water models; applications to nearshore hydrodynamics, Ph.D., Université Bordeaux (2005).
- F. Marche and P. Fabrie. Another proof of stability for global weak solutions of 2D degenerated Shallow Water models, Journal of Mathematical Fluid Mechanics, (2008).
- J. Pedlosky. Geophysical fluid dynamics, 2d edition, Springer (1987).
- A.J.-C. de Saint-Venant. Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leur lit, Comptes Rendus de l’Académie des Sciences 73 (1871), 147–154.
- J. Simon. Compact Sets in the Space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 4 (1987), 65–96. MR 916688 (89c:46055)
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Additional Information
Carine Lucas
Affiliation:
Laboratoire MAPMO, Université d’Orléans–UFR Sciences, Bât. de Mathématiques–Route de Chartres, BP. 6759, 45067 Orléans cedex 2, France
Email:
Carine.Lucas@univ-orleans.fr
Keywords:
Shallow Water equations,
viscosity,
Coriolis force,
asymptotics,
waves,
a priori estimates,
existence of solutions.
Received by editor(s):
November 1, 2007
Published electronically:
March 20, 2009
Article copyright:
© Copyright 2009
Brown University
The copyright for this article reverts to public domain 28 years after publication.