# Analysis of the hydrostatic approximation in oceanography with compression term

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

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## Abstract

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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.

## How to cite

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Rebollo, 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

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