381
Hence,
apl
aT/I
-
= Pig-, i = 1,2
aXi
ax,
then the following integral equality is deduced,
l
b+ h z+h 1 1 apl
l b +h2+ h l ar/1
aT/1
--a dJ:;l = 9
-a> (XI,X2,t)d.L:J = gh l - a '
b+h2
PI Xi
b+h2
Ji
Xi
In order to obtain a maximal monotonous operator of T/I (we recall that monotony
plays an important role when solving numerically the equations), thp last term in the
former equality is rewritten under the form:
where 8 is the unit tensor.
Therefore, for the first layer, the following equations are obtained:
where
ill is the vertically integrated velocity in the first layer;
QI = (Qi, QD = illhl' is the water flux;
FI is the Coriolis force;
T w is the surface wind stress;
Ti is the drag stress between the two layers:
As can be observed, these equations are formally analogous to the shallow water
equations (21), (22), but now the equality
does not hold anymore, instead
ah l
at
ah l
ar!1
at
&'
ar/!
aT/2
- - -
at at .
( 41)
Hence, the second layer unknowns appear in the continuity equation through the timp
derivative of hi and in the momentum equatioll through the friction stress Tj.
Let us consider the second layer. On the interface