172
The equation system that we then get is over determinated. We therefore usually
introduce a new unknown qs that represents the pressure at the surface of the ocean, and that is
only function of x and y, such as :
q(t;x,y,z) = qlt;x,y) - r b(t;x,y,~) d~
(Jll.36)
To write the weak formulation of the rigid lid equations, we use, following Lions & al.:
Vu =( cpEL2(D)2, divCPE £l(D), curlcpE £l(D), : E £l(Q), div f: cpdz = O,cp.n = 0 on T/nr,j
and we define Hu as the closure of Vu for the of L2 -norm, and:
H=Hux £l(D)
and
The weak formulation of the rigid-lid equations can then be written as :
For (UO,bO)E H given, find (u,b) such as :
(u,b) E U(O,T;V)nL""(O,T;H) , VJ>O
+ ji ( divudivcp + ji ( curlucurlcp = ( rscp - ( b r divcp
JQ
JQ
J~
JQ Jz
(Jll.37)
(Jll.38)
(u,b)t=O = (ua,ba)
(Jll.39)
for each (cp, "') belonging to VnH 2 , and for U3 given by (III.33).
The last term of the left member of the equation (III.37) comes from the integration of
Vqcp. We have:
=- ( ~ r divcp + ( qfO divcp n3+ ( q cp.n = - ( bfO divcp + ( q ((0 divcp n3+ cp.n)
J!l J z
Jr z
J!
JQ z
J! J-h