The Pool of Constant Potential Vorticity
205
4.6 The Pool of Constant Potential Vorticity
Examining Fig. 4.4.4a carefully, we see that there is a streamline which issues
from the intersection of the outcrop line with the western boundary of the
basin. This streamline carves out a region on the western flank of the ventilated
layer in which all the streamlines within that region emanate from the western
boundary south of the outcrop line. Thus the fluid in layer 2 in this region does
not originate from the outcrop line. If we follow a streamline of the solution
given by ( 4.4.18), its origin on the outcrop line lies west of the oceanic western
boundary for fluid in the shaded region. Hence our solution, which depends on
the process of subduction to determine the potential vorticity of the fluid, is
inapplicable in the shaded zone. This is the same situation as above in
discussing the solution in the shadow zone on the eastern edge of the ventilated
region. There we argued that the flow in layer 2 should be at rest since the
shadow zone region is bounded on both its eastern and western edges by curves
across which there is no flow. The shaded region on the western flank of the
ventilated zone, on the other hand, receives fluid from the western boundary,
and in fact the fluid clearly recirculates through the western boundary current.
Luyten et al. (1983), following the ideas of Young and Rhines (1982), assumed
that the recirculation in this region, absent of the influence of any source of
potential vorticity, would lead to a final state of homogenized potential
vorticity in layer 2, and they called this region the pool of constant potential
vorticity.
The extent of the pool depends on the position of the outcrop line. Any of
the streamlines in layer 2 in Fig. 4.4.4a could be the boundary of the pool
region if the outcrop line coincides with the intersection of the streamline with
the western boundary. Thus the farther north the outcrop line is, the larger is
the area of layer 2 covered by the pool. For outcrop lines that are far enough
south there are no streamlines emanating from the western boundary, and
there is no pool for such outcrop lines. To find the boundary of the pool
consider the trajectory:
( 4.6.1)
which emanates from the intersection of the outcrop line and the western
boundary of the basin where = w. If we again use the streamline equation
(4.4.20), we obtain for the trajectory (4.6.1) the implicit relation:
n~(w, e) = n~( w, e2) { 1 + ~~ ( 1 - Izr}
+Hf 1 (1- L)
2
Y2
h
(4.6.2)