Analysis of some oceanography physics problems by the
Galerkin's method
Pierre ORENGA, Fran~ois Joseph CHATELON and Charles FLUIXA
Centre de Mathematiques et de Calcul Scientifique
Universite de Corse
Quartier Grossetti - B.P. 52
20 250 CORTE
ABSTRACT.
The essential of this work is the determination of the velocity fields of the marine
streams with a view of solving biologic models, pollution transport ... The studied models
concern phenomena specific to the seas where the main forcing is the wind. We present the
study of two models : a three dimensionnal model and a shallow water model in depth-mean
velocity formulation. This last, that is obtained by integration on the depth of the three
dimensionnal model, allows a preliminary study that can be useful to obtain for instance initial
conditions (or boundary) and even to solve the three dimensional model when the hypothesis
of the rigid-lid is not done.
First of all, we present an existence result of the shallow water problem, result that we
obtained thank to the help of Pr. P.L. Lions. We will not give all the details of the
demonstration but the main ideas.
The equations of this model differ from classical equations by the boundary conditions
that are of the kind u.n and curl u fixed on the boundary. This difference allowed us to solve
the equations by the Galerkin's method using the eigen basis of the operator I::. with u.n = 0
and curl u = 0 on the boundary. These boundary conditions give to this base particular
properties that allow to smoothly solve the equations of the Shallow water model herafter
presented.
This basis is also used to solve the three dimensionnal model.
NATO ASI Series. Vol. I 48
The Mathematics of Models for Climatology
and Environment
Edited by Jesus I1defonso Diaz
© Springer. Verlag Berlin Heidelberg 1997
Galerkin's method
Pierre ORENGA, Fran~ois Joseph CHATELON and Charles FLUIXA
Centre de Mathematiques et de Calcul Scientifique
Universite de Corse
Quartier Grossetti - B.P. 52
20 250 CORTE
ABSTRACT.
The essential of this work is the determination of the velocity fields of the marine
streams with a view of solving biologic models, pollution transport ... The studied models
concern phenomena specific to the seas where the main forcing is the wind. We present the
study of two models : a three dimensionnal model and a shallow water model in depth-mean
velocity formulation. This last, that is obtained by integration on the depth of the three
dimensionnal model, allows a preliminary study that can be useful to obtain for instance initial
conditions (or boundary) and even to solve the three dimensional model when the hypothesis
of the rigid-lid is not done.
First of all, we present an existence result of the shallow water problem, result that we
obtained thank to the help of Pr. P.L. Lions. We will not give all the details of the
demonstration but the main ideas.
The equations of this model differ from classical equations by the boundary conditions
that are of the kind u.n and curl u fixed on the boundary. This difference allowed us to solve
the equations by the Galerkin's method using the eigen basis of the operator I::. with u.n = 0
and curl u = 0 on the boundary. These boundary conditions give to this base particular
properties that allow to smoothly solve the equations of the Shallow water model herafter
presented.
This basis is also used to solve the three dimensionnal model.
NATO ASI Series. Vol. I 48
The Mathematics of Models for Climatology
and Environment
Edited by Jesus I1defonso Diaz
© Springer. Verlag Berlin Heidelberg 1997