Analysis of the hydrostatic approximation in oceanography with compression term. (English) Zbl 0978.76101

M2AN, Math. Model. Numer. Anal. 34, No. 3, 525-537 (2000); erratum ibid. 34, No. 5, 1117 (2000).
Summary: The hydrostatic approximation of the incompressible three-dimensional stationary Navier-Stokes equations 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 ocean), and it is usually studied as such. We consider in this paper an equivalent formulation to this hydrostatic approximation that includes Coriolis force and an additional pressure term that comes from taking into account the pressure in the equation of state 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 problem coupled with a limit process that yields the solution. We finish this paper with an existence and uniqueness result for the evolutionary linear problem associated to this model. This problem includes the same additional pressure term and Coriolis force.


76U05 General theory of rotating fluids
76M10 Finite element methods applied to problems in fluid mechanics
86A05 Hydrology, hydrography, oceanography
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI EuDML


[1] Azerat P. and Guillén F., Équations de Navier-Stokes en bassin peu profond: l’approximation hydrostatique. Submitted for publication. Zbl0938.35122 · Zbl 0938.35122
[2] Besson O. and Laydi M.R., Some Estimates for the Anisotropic Navier-Stokes Equations and for the Hydrostatic Approximation. RAIRO Modél. Math. Anal. Numér.26 (1992) 855-865. · Zbl 0765.76017
[3] Bresch D. and Simon J., Modèles stationnaires de lacs et mers. Équations aux dérivées partielles et applications. Articles dédiés à J.-L. Lions. Elsevier, Paris (1998). · Zbl 0926.35112
[4] Bresch D., Lemoine J. and Simon J., Écoulement engendré par le vent et la force de Coriolis dans un domain mince: II cas d’évolution. C. R. Acad. Sci. Paris Sér. I327 (1998) 329-334. · Zbl 0918.35105
[5] Bravo de Mansilla A., Chacón Rebollo T. and Lewandowski R., Observaciones sobre dos aplicaciones diversas del Método de los Elementos Finitos: Controlabilidad Exacta de la Ecuación Discreta del Calor y Ecuaciones Primitivas en Oceanografía, Actas de la Jornada Científica en Homenaje al Prof. Antonio Valle Sánchez (1997).
[6] Ciarlet Ph., The Finite Element Method for Elliptic Problems. North-Holland (1978). · Zbl 0383.65058
[7] Gill A.-E., Atmosphere-Ocean dynamics. Academic Press (1982).
[8] Lewandowski R., Analyse mathématique et océanographie. Masson (1997).
[9] Lewandowski R., Étude d’un système stationnaire linéarisé d’équations primitives avec des termes de pression additionnels. C. R. Acad. Sci. Paris Sér. I324 (1997) 173-178. Zbl0878.76062 · Zbl 0878.76062
[10] Lions J.-L., Teman S. and Wang S., On the equations of the large scale Ocean. Nonlinearity5 (1992) 1007-1053. · Zbl 0766.35039
[11] Pedlosky J., Geophysical fluid dynamics. Springer-Verlag, New York (1987). Zbl0713.76005 · Zbl 0713.76005
[12] Teman R., Sur la stabilité et la convergence de la méthode des pas fractionnaires. Ann. Mat. Pura ed ApplicataLXXIX (1968) 191-379. Zbl0174.45804 · Zbl 0174.45804
[13] Teman R., Navier-Stokes Equations. North-Holland, Elsevier (1985).
[14] Zeidler E., Nonlinear functional analysis and its applications, II/A. Springer-Verlag (1990). · Zbl 0684.47029
[15] Zuur E. and Dietrich D.E., The SOMS model and its application to Lake Neuchâtel. Aquatic Sci.52 (1990) 115-129.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.