Ventilation and Homogenization: A Unified Theory
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where the sum is over all moving layers. We define the zonal and meridional
components of the transport as Us and Vs, respectively, and thus Us and Vs can
be thought of as the vertical average of the interior velocity or equivalently, its
barotropic component. The continuity equation summed over all the layers
yields:
V' · Us = -WE·
On the other hand, the Sverdrup relations allow us to write:
f
1'2
aD6
Vs = PWE = 2JR COS e 0 This allows us to use (4.9.9) to solve for Us:
1'2 aD6
Us = hus = - 2 /R ae .
(4.9.9)
(4.9.10)
(4.9.11)
In layer 3, where it is motionless, the northward gradient of potential vorticity
is, using (4.9.3):
_!_ aq3 _ __!__ { f3 + _L _!_ aD6
1
}
Rae - h3
2h3R ae Jn 6 +H}
_ __!__ {!3 + _f_ aD6}
- h3
2h2h3 Rae
_ 1 {{3 f
2
hus}
- h3
- h2h31'2 .
(4.9.12)
Along the northern boundary of the gyre where WE is zero and where h2
and h3 take on the constant values H2 and H3 :
(4.9.13)
If this result is used in (4.9.12) and compared with (4.9.7) it foiiows that the
existence of the critical point northern boundary of the gyre, is also a location where the northward gradient
of potential vorticity vanishes. Since the eastward component of the gradient of
f jh3 also vanishes everywhere along the latitude circle e = e0 , the point
( This point occurs where the bracket in (4.9.12) vanishes so that the critical
point also corresponds to the location on the northern boundary of the gyre
where:
(4.9.14)