# Analysis of the hydrostatic approximation in oceanography with compression term

Tomás Chacón Rebollo; Roger Lewandowski; Eliseo Chacón Vera

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 34, Issue: 3, page 525-537
- ISSN: 0764-583X

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topRebollo, Tomás Chacón, Lewandowski, Roger, and Vera, Eliseo Chacón. "Analysis of the hydrostatic approximation in oceanography with compression term." ESAIM: Mathematical Modelling and Numerical Analysis 34.3 (2010): 525-537. <http://eudml.org/doc/197586>.

@article{Rebollo2010,

abstract = {
The hydrostatic approximation of the incompressible 3D 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 state equation 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.
},

author = {Rebollo, Tomás Chacón, Lewandowski, Roger, Vera, Eliseo Chacón},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Navier-Stokes equations;
Oceanography; Compression term.; hydrostatic approximation; oceanography; Coriolis force; additional pressure term; existence theorem; mixed variational formulation; finite element spaces; uniqueness; evolutionary linear problem},

language = {eng},

month = {3},

number = {3},

pages = {525-537},

publisher = {EDP Sciences},

title = {Analysis of the hydrostatic approximation in oceanography with compression term},

url = {http://eudml.org/doc/197586},

volume = {34},

year = {2010},

}

TY - JOUR

AU - Rebollo, Tomás Chacón

AU - Lewandowski, Roger

AU - Vera, Eliseo Chacón

TI - Analysis of the hydrostatic approximation in oceanography with compression term

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 34

IS - 3

SP - 525

EP - 537

AB -
The hydrostatic approximation of the incompressible 3D 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 state equation 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.

LA - eng

KW - Navier-Stokes equations;
Oceanography; Compression term.; hydrostatic approximation; oceanography; Coriolis force; additional pressure term; existence theorem; mixed variational formulation; finite element spaces; uniqueness; evolutionary linear problem

UR - http://eudml.org/doc/197586

ER -

## References

top- Azerat P. and Guillén F., Équations de Navier-Stokes en bassin peu profond: l'approximation hydrostatique. Submitted for publication.
- 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.
- 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).
- 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.
- 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).
- Ciarlet Ph., The Finite Element Method for Elliptic Problems. North-Holland (1978).
- Gill A.-E., Atmosphere-Ocean dynamics. Academic Press (1982).
- Lewandowski R., Analyse mathématique et océanographie. Masson (1997).
- 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.
- Lions J.-L., Teman S. and Wang S., On the equations of the large scale Ocean. Nonlinearity5 (1992) 1007-1053.
- Pedlosky J., Geophysical fluid dynamics. Springer-Verlag, New York (1987).
- Teman R., Sur la stabilité et la convergence de la méthode des pas fractionnaires. Ann. Mat. Pura ed ApplicataLXXIX (1968) 191-379.
- Teman R., Navier-Stokes Equations. North-Holland, Elsevier (1985).
- Zeidler E., Nonlinear functional analysis and its applications, II/A. Springer-Verlag (1990).
- Zuur E. and Dietrich D.E., The SOMS model and its application to Lake Neuchâtel. Aquatic Sci.52 (1990) 115-129.

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