136
The important sizes of the domains and the difficulties linked to the adjustment of the
parameters make the Galerkin's method particularly well adapted. As a matter of fact, the
heavy calculi linked to the dimensions are carried out once and for all.
The plan of the study is the following:
We present first the models and notations. In the paragraph I we give an existence result
of the shallow water problem and in paragraph II we give the numerical analysis of this
problem. In the paragraph III we give the analysis of the three dimensional problem.
GENERALITIES
1.1 The Mathematical Models
The state variables used to describe an hydrodynamic model are the velocity field v, the
temperature T, the salinity S, the pressure p and the density of the seawater p. The equations
governing the behaviour of these quantities are the momentum equation, the continuity
equation, the thermodynamical equation, the equation of diffusion of the salinity and the
equation of state linking these quantities between themselves. These general equations are
then simplified in regard to some features of the studied system. One of the salient feature of
the geophysics fluids lies on the fact that the density p does not vary a lot compared to its
constant reference value po. Then the density differences can be neglected in the equations
except in the gravity force where, multiplicated by the acceleration of gravity g, they
contribute to a vertical force called buoyancy. Another simplification comes from the fact that
in the sea the horizontal scale is much larger than the vertical one. The scale analysis ensures
that the dominant forces in the vertical momentum equation come from the vertical
component of the pressure gradient and gravity. That is called the hydrostatic approximation.
The buoyancy is related to the stratification of the sea and is primarily a function of the
temperature and the salinity of the seawater. In the stratified seas, we usually replace the
equations of the temperature and salinity by the equation of the buoyancy b. The primitive
equations describing hydrodynamical processes, expressed in a Cartesian coordinates system
(which is restrictive regarding to the dimensions of the studied domains), can be written as :
Ju
Ju
d(_Ju)
-+ V.u + U3-+ IDAU - - v- = - Vq
dt
U
dz
dzdz
The important sizes of the domains and the difficulties linked to the adjustment of the
parameters make the Galerkin's method particularly well adapted. As a matter of fact, the
heavy calculi linked to the dimensions are carried out once and for all.
The plan of the study is the following:
We present first the models and notations. In the paragraph I we give an existence result
of the shallow water problem and in paragraph II we give the numerical analysis of this
problem. In the paragraph III we give the analysis of the three dimensional problem.
GENERALITIES
1.1 The Mathematical Models
The state variables used to describe an hydrodynamic model are the velocity field v, the
temperature T, the salinity S, the pressure p and the density of the seawater p. The equations
governing the behaviour of these quantities are the momentum equation, the continuity
equation, the thermodynamical equation, the equation of diffusion of the salinity and the
equation of state linking these quantities between themselves. These general equations are
then simplified in regard to some features of the studied system. One of the salient feature of
the geophysics fluids lies on the fact that the density p does not vary a lot compared to its
constant reference value po. Then the density differences can be neglected in the equations
except in the gravity force where, multiplicated by the acceleration of gravity g, they
contribute to a vertical force called buoyancy. Another simplification comes from the fact that
in the sea the horizontal scale is much larger than the vertical one. The scale analysis ensures
that the dominant forces in the vertical momentum equation come from the vertical
component of the pressure gradient and gravity. That is called the hydrostatic approximation.
The buoyancy is related to the stratification of the sea and is primarily a function of the
temperature and the salinity of the seawater. In the stratified seas, we usually replace the
equations of the temperature and salinity by the equation of the buoyancy b. The primitive
equations describing hydrodynamical processes, expressed in a Cartesian coordinates system
(which is restrictive regarding to the dimensions of the studied domains), can be written as :
Ju
Ju
d(_Ju)
-+ V.u + U3-+ IDAU - - v- = - Vq
dt
U
dz
dzdz