224
Theory of the Ventilated Thermocline
where Cr is the baroclinic Rossby wave speed. Thus the critical point occurs if
and where the averaged eastward Sverdrup flow on the northern boundary of the
gyre balances the westward propagation of the baroclinic Rossby waves. The
result ( 4.9 .14) is the generalization of the quasi-geostrophic result of Chapter 3
to a case in which Cr depends on the local values of the layer thicknesses and
not their constant, rest values. On the northern boundary of the gyre these
coincide.
The zonal gradient of q3, again using the solution (4.9.3) for the region of
resting fluid is:
1 8q3
hf 2
- - - - - - - - v
R cos 8 8¢ - y 2 h2h~ s
(4.9.15)
which vanishes at the northern boundary of the gyre. The slope of the q3
isolines in the horizontal plane is, collecting the above results, given by:
1 ( 88)
cos e 8¢ q 3
(4.9.16)
Thus at the critical point the slope becomes indeterminate, and the critical
value of the potential vorticity, fo/ H3, simultaneously corresponds to the value
of q3 all along the latitude circle of the northern boundary of the gyre as well as
the boundary of the pool region. This boundary enters the gyre and isolates the
pool in the northwest corner from the rest of the circulation. The splitting of
this isoline at the critical point is a consequence of the indeterminacy of the
direction of information propagation where both Vs and Us-Cr vanish.
Within the pool region layer 3 can be in motion. We argued in Chapter 3
that although it is consistent to have a solution in which layer 3 remains at rest,
the existence of closed q3 contours, which allow free geostrophic motion,
makes it likely that the solution without motion is unstable and is driven to an
0( 1) velocity by the slightest frictional or eddy coupling with the overlying
layer. Of course, the contours are really only "semiclosed" in the sense that
they must actually close through a western boundary layer in which the
dynamics may be much different than the interior adiabatic dynamics.
However, as in the previous chapter, we assume that the closure through the
western boundary current is dynamically equivalent to a closure in the interior.
To determine this motion we once again appeal heuristically to the notion
that the potential vorticity becomes homogenized in any region, as argued in
the quasi-geostrophic theory, once that region is encircled by a contour of
constant potential vorticity and after diffusion of potential vorticity spreads
that constant value through the girdled domain. If we take this as a working
hypothesis, it would imply that north of the outcrop line and west of the
boundary of the pool:
(4.9.17)
Theory of the Ventilated Thermocline
where Cr is the baroclinic Rossby wave speed. Thus the critical point occurs if
and where the averaged eastward Sverdrup flow on the northern boundary of the
gyre balances the westward propagation of the baroclinic Rossby waves. The
result ( 4.9 .14) is the generalization of the quasi-geostrophic result of Chapter 3
to a case in which Cr depends on the local values of the layer thicknesses and
not their constant, rest values. On the northern boundary of the gyre these
coincide.
The zonal gradient of q3, again using the solution (4.9.3) for the region of
resting fluid is:
1 8q3
hf 2
- - - - - - - - v
R cos 8 8¢ - y 2 h2h~ s
(4.9.15)
which vanishes at the northern boundary of the gyre. The slope of the q3
isolines in the horizontal plane is, collecting the above results, given by:
1 ( 88)
cos e 8¢ q 3
(4.9.16)
Thus at the critical point the slope becomes indeterminate, and the critical
value of the potential vorticity, fo/ H3, simultaneously corresponds to the value
of q3 all along the latitude circle of the northern boundary of the gyre as well as
the boundary of the pool region. This boundary enters the gyre and isolates the
pool in the northwest corner from the rest of the circulation. The splitting of
this isoline at the critical point is a consequence of the indeterminacy of the
direction of information propagation where both Vs and Us-Cr vanish.
Within the pool region layer 3 can be in motion. We argued in Chapter 3
that although it is consistent to have a solution in which layer 3 remains at rest,
the existence of closed q3 contours, which allow free geostrophic motion,
makes it likely that the solution without motion is unstable and is driven to an
0( 1) velocity by the slightest frictional or eddy coupling with the overlying
layer. Of course, the contours are really only "semiclosed" in the sense that
they must actually close through a western boundary layer in which the
dynamics may be much different than the interior adiabatic dynamics.
However, as in the previous chapter, we assume that the closure through the
western boundary current is dynamically equivalent to a closure in the interior.
To determine this motion we once again appeal heuristically to the notion
that the potential vorticity becomes homogenized in any region, as argued in
the quasi-geostrophic theory, once that region is encircled by a contour of
constant potential vorticity and after diffusion of potential vorticity spreads
that constant value through the girdled domain. If we take this as a working
hypothesis, it would imply that north of the outcrop line and west of the
boundary of the pool:
(4.9.17)