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Analysis of the hydrostatic approximation in oceanography with compression term

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ESAIM: M

ODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

T OMÁS C HACÓN R EBOLLO

R OGER L EWANDOWSKI

E LISEO C HACÓN V ERA

Analysis of the hydrostatic approximation in oceanography with compression term

ESAIM: Modélisation mathématique et analyse numérique, tome 34, n

o5 (2000), p. 1107

<http://www.numdam.org/item?id=M2AN_2000__34_5_1107_0>

© SMAI, EDP Sciences, 2000, tous droits réservés.

L’accès aux archives de la revue « ESAIM: Modélisation mathématique et analyse numérique » (http://www.esaim-m2an.org/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation com- merciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

(2)

M a t h e m a t i c a l M o d e l l i n g a n d N u m e r i c a l Analysis ESAIM: M2AN Modélisation Mathématique et Analyse Numérique Vol. 34, N° 5, 2000, p. 1117

Erratum

ESAIM: M2AN, Vol 34, N°3, 2000, pp. 525-537.

ANALYSIS OF THE HYDROSTATIC APPROXIMATION IN OCEANOGRAPHY WITH COMPRESSION TERM*

TOMÂS CHACUN REBOLLO1, ROGER LEWANDOWSKI2 AND ELISEO CHACÓN V E R A1

Abstract. The hy drost atic approximation of the incompressible 3D stationary Navier-S tokes équa- tions is widely used in oceanography and other applied sciences. It appears through a limit process due to the anisotropy of the domain in use, an océan, and it is usually studied as such. We consider in this paper an equivalent formulation to this hy drost atic approximation that includes Coriolis force and an additional pressure term that comes from taking into account the pressure in the state équation for the density. It therefore models a slight dependence of the density upon compression terms. We study this model as an independent mathematical object and prove an existence theorem by means of a mixed variational formulation. The proof uses a family of finite element spaces to discretize the prob- lem coupled with a limit process that yields the solution. We finish this paper with an existence and uniqueness resuit for the evolutionary linear problem associâted to this model. This problem includes the same additional pressure term and Coriolis force.

Mathematics Subject Classification. 35Q30, 76D05.

Received: November 5, 1998. Revised: October 7, 1999.

The following aknowledgements have been omitted:

Acknowledgements. This work has been partially funded by the Project: CYTMAR MAR97-1055-C02-02 of Ministerio de Educación y Ciencia of Spain.

Keywords and phrases. Navier-Stokes équations, Oceanography, Compression term.

* Dedicated to J.L. Lions %n honor to his 70th Btrthday.

1 Departamento de Ecuaciones Diferenciales y Anâlisis Numérico Universidad de Sevilla, 41.080-Sevilla, Spain, e-mail: [email protected]; [email protected]

2 Modal-X, Bât. G, Université Paris X, 200 avenue de la République, 92001 Nanterre, France, e-mail: lewandowQmodalx.u-paxisl0.fr

(c) EDP Sciences, SMAI 2000

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