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III. NUMERICAL ANALYSIS OF A THREE DIMENSIONAL PROBLEM
111.1 Introduction
The purpose of this work is to numerically approach by the Galerkin's method, the
solution of a three-dimensional general circulation model:
db
db dr-db)
-+u Vb +U3--- ,1,- = 0
at'
dz dz dz
divu + du3 = 0
dz
(1Il.1)
(Ill. 2)
(1Il.3 )
(1Il.4)
(1Il.5)
One of the main feature of this system is that it is not an evolution system due to the two
diagnostic equations, (ID.3) and (ID.4), involving U3 and q. Usually both diagnostic equations
are integrated according to the vertical and they yield an evolution system. The resulting
system then depends on the boundary conditions that we impose on the velocity field. A
natural condition to impose on the free surface can be written as :
(III. 6)
The other conditions on surface mainly express the continuity of the buoyancy flows and
momentum at the interface air-sea. We usually write:
and
for z = ~
where 1: s is a function of the wind, and P can be expressed according to the temperature and
salinity flows (Nihoul [1975]). On the bottom, we impose an impermeability condition:
(III. 7)