Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
Chapitre 2
Derivation of a dynamic multilayer
shallow water model of
approximation of free surface
Navier-Stokes equations ; existence of
local in time strong solution
We propose a new simple approximation of the viscous primitive equations of the ocean
including Coriolis force (2.1.1), by a multilayer shallow water type model. Using a finite
volume type discretization in the vertical direction, we show that our system is a consistent
approximation of (2.1.1) and we compare it briefly with other multilayer shallow water type
existing models. Next, existence and uniqueness of local in time strong solution is proved
for the new model.
2.1 Introduction and Main Result
The main goal of this work is to propose a simple and numerically efficient model of
geophysical flows such as large-scale ocean circulations. Many of these flows are generally
described by the incompressible Navier-Stokes equations with a free surface [137]. Due
to the mathematical complexity of this system, different approximations are usually performed,
which aim in particular at finding a compromise between physical consistency
and reasonable computational cost. Going beyond the Boussinesq approximation [159] we
start our study by considering an homogeneous fluid (water), with density equal to one.
Moreover we use the so-called hydrostatic approximation, that is we assume the pressure
is hydrostatic and is not an unknown of the problem. Precisely, the departure model consists
in the primitive equations of the ocean, given in the conservative form below. We
use bold characters to indicate vector valued functions or variables. Hence the 3D velocity
of the fluid, for which we separate the horizontal component and the vertical one as
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