154
II. NUMERICAL ANALYSIS OF A SHALLOW WATER
PROBLEM OCCURING IN OCEANOGRAPHY
11.1. Introduction
This study was carried out with the aim of solving Physics Oceanography problems
dealing with the determination of mean major streams in the occidental Mediterranean sea or
concerning ponds such as the one of Thau where a preliminary study can be done in
dimension 2.
The difficulties of the numerical resolution of these models are essentially due to the
domain dimensions, the problem of initial conditions and the parameters adjustment. These
difficulties lead to prohibitive calculation times. Spectal methods are well adapted to this kind
of problems as the heavy calculi of the base functions and of the differential equations
coefficients are carried out once and for all. The adjutsment is done during the resolution of
these differential equations and the difficulty raised by the important size of the domains does
not intervene in this resolution.
We present the numerical analysis of the shallow water problem in depth velocity (h,
u) formulation with slip boundary conditions that is to say normal velocity fixed on the
boudary and not u fixed on the boundary. A dissipation condition is added to this slip
condition and that renders the problem well posed. In Oceanography, the use of this kind of
condition is preferred so the inextricable boundary layer calculi, that appear when the alongshore velocity is fixed, are avoided.
To solve this problem by the Galerkin's method, we used the eigen-base associated
with the Laplace's operator with conditions at the homogeneous boundary conditions above
mentioned, that is to say:
(PI)
-~u = A.U into n
u. n = 0 ; curlu = 0 on y
This basis does not present the same difficulties as the one obtained with Dirichet's
boundary conditions (no slip condition). The properties of this basis that we introduce
herafter allow an important adaptability of the use of the Galerkin's method.
In what follows,
II. NUMERICAL ANALYSIS OF A SHALLOW WATER
PROBLEM OCCURING IN OCEANOGRAPHY
11.1. Introduction
This study was carried out with the aim of solving Physics Oceanography problems
dealing with the determination of mean major streams in the occidental Mediterranean sea or
concerning ponds such as the one of Thau where a preliminary study can be done in
dimension 2.
The difficulties of the numerical resolution of these models are essentially due to the
domain dimensions, the problem of initial conditions and the parameters adjustment. These
difficulties lead to prohibitive calculation times. Spectal methods are well adapted to this kind
of problems as the heavy calculi of the base functions and of the differential equations
coefficients are carried out once and for all. The adjutsment is done during the resolution of
these differential equations and the difficulty raised by the important size of the domains does
not intervene in this resolution.
We present the numerical analysis of the shallow water problem in depth velocity (h,
u) formulation with slip boundary conditions that is to say normal velocity fixed on the
boudary and not u fixed on the boundary. A dissipation condition is added to this slip
condition and that renders the problem well posed. In Oceanography, the use of this kind of
condition is preferred so the inextricable boundary layer calculi, that appear when the alongshore velocity is fixed, are avoided.
To solve this problem by the Galerkin's method, we used the eigen-base associated
with the Laplace's operator with conditions at the homogeneous boundary conditions above
mentioned, that is to say:
(PI)
-~u = A.U into n
u. n = 0 ; curlu = 0 on y
This basis does not present the same difficulties as the one obtained with Dirichet's
boundary conditions (no slip condition). The properties of this basis that we introduce
herafter allow an important adaptability of the use of the Galerkin's method.
In what follows,