# A new two-dimensional shallow water model including pressure effects and slow varying bottom topography

• Volume: 38, Issue: 2, page 211-234
• ISSN: 0764-583X

top

## Abstract

top
The motion of an incompressible fluid confined to a shallow basin with a slightly varying bottom topography is considered. Coriolis force, surface wind and pressure stresses, together with bottom and lateral friction stresses are taken into account. We introduce appropriate scalings into a three-dimensional anisotropic eddy viscosity model; after averaging on the vertical direction and considering some asymptotic assumptions, we obtain a two-dimensional model, which approximates the three-dimensional model at the second order with respect to the ratio between the vertical scale and the longitudinal scale. The derived model is shown to be symmetrizable through a suitable change of variables. Finally, we propose some numerical tests with the aim to validate the proposed model.

## How to cite

top

Ferrari, Stefania, and Saleri, Fausto. "A new two-dimensional shallow water model including pressure effects and slow varying bottom topography." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.2 (2004): 211-234. <http://eudml.org/doc/245994>.

@article{Ferrari2004,
abstract = {The motion of an incompressible fluid confined to a shallow basin with a slightly varying bottom topography is considered. Coriolis force, surface wind and pressure stresses, together with bottom and lateral friction stresses are taken into account. We introduce appropriate scalings into a three-dimensional anisotropic eddy viscosity model; after averaging on the vertical direction and considering some asymptotic assumptions, we obtain a two-dimensional model, which approximates the three-dimensional model at the second order with respect to the ratio between the vertical scale and the longitudinal scale. The derived model is shown to be symmetrizable through a suitable change of variables. Finally, we propose some numerical tests with the aim to validate the proposed model.},
author = {Ferrari, Stefania, Saleri, Fausto},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Navier-Stokes equations; Saint Venant equations; free surface flows},
language = {eng},
number = {2},
pages = {211-234},
publisher = {EDP-Sciences},
title = {A new two-dimensional shallow water model including pressure effects and slow varying bottom topography},
url = {http://eudml.org/doc/245994},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Ferrari, Stefania
AU - Saleri, Fausto
TI - A new two-dimensional shallow water model including pressure effects and slow varying bottom topography
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 2
SP - 211
EP - 234
AB - The motion of an incompressible fluid confined to a shallow basin with a slightly varying bottom topography is considered. Coriolis force, surface wind and pressure stresses, together with bottom and lateral friction stresses are taken into account. We introduce appropriate scalings into a three-dimensional anisotropic eddy viscosity model; after averaging on the vertical direction and considering some asymptotic assumptions, we obtain a two-dimensional model, which approximates the three-dimensional model at the second order with respect to the ratio between the vertical scale and the longitudinal scale. The derived model is shown to be symmetrizable through a suitable change of variables. Finally, we propose some numerical tests with the aim to validate the proposed model.
LA - eng
KW - Navier-Stokes equations; Saint Venant equations; free surface flows
UR - http://eudml.org/doc/245994
ER -

## References

top
1. [1] V.I. Agoshkov, D. Ambrosi, V. Pennati, A. Quarteroni and F. Saleri, Mathematical and numerical modelling of shallow water flow. Comput. Mech. 11 (1993) 280–299. Zbl0771.76032
2. [2] V.I. Agoshkov, A. Quarteroni and F. Saleri, Recent developments in the numerical simulation of shallow water equations. Boundary conditions. Appl. Numer. Math. 15 (1994) 175–200. Zbl0833.76008
3. [3] J.P. Benque, J.A. Cunge, J. Feuillet, A. Hauguel and F.M. Holly, New method for tidal current computation. J. Waterway, Port, Coastal and Ocean Division, ASCE 108 (1982) 396–417.
4. [4] J.P. Benque, A. Haugel and P.L. Viollet, Numerical methods in environmental fluid mechanics. M.B. Abbot and J.A. Cunge Eds., Eng. Appl. Comput. Hydraulics II (1982) 1–10.
5. [5] S. Ferrari, A new two-dimensional Shallow Water model: physical, mathematical and numerical aspects Ph.D. Thesis, a.a. 2002/2003, Dottorato M.A.C.R.O., Università degli Studi di Milano.
6. [6] S. Ferrari, Convergence analysis of a space-time approximation to a two-dimensional system of Shallow Water equations. Internat. J. Appl. Analysis (to appear). Zbl1122.76053MR2072307
7. [7] 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) 89–102. Zbl0997.76023
8. [8] R.H. Goodman, A.J. Majda and D.W. Mclaughlin, Modulations in the leading edges of midlatitude storm tracks. SIAM J. Appl. Math. 62 (2002) 746–776. Zbl0989.86007
9. [9] E. Grenier, Boundary layers for parabolic regularizations of totally characteristic quasilinear parabolic equations. J. Math. Pures Appl. 76 (1997) 965–990. Zbl0914.35032
10. [10] E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Differential Equations 143 (1998) 110–146. Zbl0896.35078
11. [11] O. Guès, Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites. Grenoble Ann. Inst. Fourier 45 (1995) 973–1006. Zbl0831.34023
12. [12] M.E. Gurtin, An introduction to continuum mechanics. Academic Press, New York (1981). Zbl0559.73001MR636255
13. [13] F. Hecht and O. Pironneau, FreeFem++:Manual version 1.23, 13-05-2002. FreeFem++ is a free software available at: http://www-rocq.inria.fr/Frederic.Hecht/freefem++.htm
14. [14] J.M. Hervouet and A. Watrin, Code TELEMAC (système ULYSSE) : Résolution et mise en œuvre des équations de Saint-Venant bidimensionnelles, Théorie et mise en œuvre informatique, Rapport EDF HE43/87.37 (1987).
15. [15] S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN 35 (2001) 631–645. Zbl1001.35083
16. [16] A. Kurganov and L. Doron, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397–425. Zbl1137.65398
17. [17] O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and quasilinear equations of parabolic type. Providence, Rhode Island. Amer. Math. Soc. (1968).
18. [18] D. Levermore and M. Sammartino, A shallow water model with eddy viscosity for basins with varying bottom topography. Nonlinearity 14 (2001) 1493–1515. Zbl0999.76033
19. [19] E. Miglio, A. Quarteroni and F. Saleri, Finite element approximation of a quasi–3D shallow water equation. Comput. Methods Appl. Mech. Engrg. 174 (1999) 355–369. Zbl0958.76046
20. [20] J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial-boundary value problems. Trans. Amer. Math. Soc. 189 (1974) 303–318. Zbl0282.35014
21. [21] M. Sammartino and R.E. Caflisch, Zero viscosity limit for analytic solutions of the Navier–Stokes equations on a half-space. I. Existence for Euler and Prandtl Equations; II. Construction of the Navier–Stokes solution. Comm. Math. Physics 192 (1998) 433–461 and 463–491. Zbl0913.35102
22. [22] D. Serre, Sytems of conservation laws. I and II, Cambridge University Press, Cambridge (1996). Zbl0930.35001
23. [23] G.B. Whitham, Linear and nonlinear waves. John Wiley & Sons, New York (1974). Zbl0373.76001MR483954

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.