Cross-Gyre Flow
281
Note that in order to keep h2 real:
!!_ < 1 + 1'2 H2
< 1 + 1'2
{
2 }1/2
{
}1/2
H
1'3 H2
1'3
( 5.2.8)
The variation of h2 might be substantial with respect to H 2 , but the variation of
the total depth of the moving fluid on the intergyre boundary is limited by
(5.2.8).
For flow in layer 3 which crosses the intergyre boundary both (5.2.6) and
(5.2.7) must apply so that on 8 = 8o where f = fo and where:
( 5.2.9)
we have:
Q ( h)_
fo
3
-h-ii2(h)
(5.2.10)
Thus if the flow in layer 3 crosses the intergyre boundary, h3 must be related to
h by the unforced Sverdrup relation (5.2.7), which in turn determines the potential vorticity on the boundary. If potential vorticity is conserved, this relation is carried into the adjacent gyre. Thus on all streamlines emanating from
the intergyre boundary and on which potential vorticity is conserved in layer 3:
f
fo
h3 h- ii2(h)
(5.2.11)
or:
f
-
h3 = Jo{h- H2(h)}
h2 = h- h3
( 5.2.12)
= (1- L)h + Lii2(h)
fo
fo
where H2(h) is given by (5.2.7).
The Sverdrup balance (5.2.4) then becomes:
h 2 + ' )' 2 {h(l- L) + Lii2(h)}
2
= ')' 2 D6 + H 2 + ' )' 2 H~
1'3
fo
fo
1'3
1'3
(5.2.13)
which yields an equation for h( ¢, 8).
One of the most interesting questions that arises is the extent of the
longitudinal interval on the intergyre boundary on which cross-gyre flow can
arise. That is, how big is the "window" through which flow from one gyre can
enter and influence the other? We anticipate that this window cannot stretch all
the way across the gyre boundary. It certainly cannot reach the eastern
boundary apart from exceptional cases. This follows from the fact that if H 2