Ventilation and Homogenization: A Unified Theory
225
The Sverdrup relation, ( 4.3.15), written in terms of layer thicknesses is:
(4.9.18)
When (4.9.17) is used, (4.9.18) yields the solution for h2 and thus the total
depth in the pool region. Note that although the thickness of layer 3 is a
function only of latitude within the pool, the base of layer 3 is at a depth
h2 + h3 and is therefore a function of longitude as well. Thus the pressure field
in layer 3 in the pool is a function of both latitude and longitude, and the flow
has both meridional and zonal components. Figure 4.9.1 shows the solution as
determined by Pedlosky and Young (1983) for the case in which the outcrop
line is moved all the way to the southern edge of the gyre so that layer 2 covers
the whole gyre and thus the solutions (4.9.17) and (4.9.18) are valid in the
whole basin (where layer 3 is in motion). The thicknesses of the layers are
shown as well as the streamlines. The pool region can be easily discerned in the
lower layer thickness field as the region where the thickness is a function only
of latitude.
This solution is qualitatively similar to the quasi-geostrophic calculation of
Section 3.9.2 with the only difference being the relaxation of the constraint of
small isopycnal displacements. Qualitative effects of ventilation only become
obvious and interesting when more ventilated layers are added, for example,
when the outcrop line at e = 82 lies within the basin.
Consider the region south of the outcrop line where there are now three
layers in motion. Once again the solution domain is split into different zones in
which the solution differs. Figure 4.9.2 shows a schematic diagram, in plan
view, of the domains of the solution. Starting at the northern boundary of the
gyre, where fluid in layer 2 with density p 2 is exposed to Ekman pumping, there
are two regions of flow. In the east there is a region in which only layer 2 is in
motion, and layer 3 is at rest. Beyond the pool boundary, ¢ = p(8), is the
pool region in layer 3 labeled "pool3" in which the potential vorticity is
homogenized to fo/H3 and where the total solution is given by (4.9.18) while
the boundary of the region is given by (4.9.5).
After layer 2 subducts, all three layers are in motion in parts of the gyre.
East of the pool of uniform q3 (whose extension south of the outcrop line we
must find) layer 3 is at rest, and the solution is similar to that for the two-layer
model found in Section 4.4, although we must be more precise about the
domain of its validity. There are actually three critical curves, as shown in
Fig. 4.9.2, and the solution changes across them. The eastern most curve is the
boundary between the shadow zone boundary in layer 2 and the ventilated
zone in layer 2. East of this curve both layer 2 and layer 3 are at rest, and the
total Sverdrup transport is carried in the upper layer. The solution in the
shadow zone is exactly as found in Section 4.4 and is given by (4.4.23) while the
boundary itself is given by ( 4.4.22). Again, the solution in the eastern part of the
gyre is unaffected by alterations in the structure of the flow in the west.
225
The Sverdrup relation, ( 4.3.15), written in terms of layer thicknesses is:
(4.9.18)
When (4.9.17) is used, (4.9.18) yields the solution for h2 and thus the total
depth in the pool region. Note that although the thickness of layer 3 is a
function only of latitude within the pool, the base of layer 3 is at a depth
h2 + h3 and is therefore a function of longitude as well. Thus the pressure field
in layer 3 in the pool is a function of both latitude and longitude, and the flow
has both meridional and zonal components. Figure 4.9.1 shows the solution as
determined by Pedlosky and Young (1983) for the case in which the outcrop
line is moved all the way to the southern edge of the gyre so that layer 2 covers
the whole gyre and thus the solutions (4.9.17) and (4.9.18) are valid in the
whole basin (where layer 3 is in motion). The thicknesses of the layers are
shown as well as the streamlines. The pool region can be easily discerned in the
lower layer thickness field as the region where the thickness is a function only
of latitude.
This solution is qualitatively similar to the quasi-geostrophic calculation of
Section 3.9.2 with the only difference being the relaxation of the constraint of
small isopycnal displacements. Qualitative effects of ventilation only become
obvious and interesting when more ventilated layers are added, for example,
when the outcrop line at e = 82 lies within the basin.
Consider the region south of the outcrop line where there are now three
layers in motion. Once again the solution domain is split into different zones in
which the solution differs. Figure 4.9.2 shows a schematic diagram, in plan
view, of the domains of the solution. Starting at the northern boundary of the
gyre, where fluid in layer 2 with density p 2 is exposed to Ekman pumping, there
are two regions of flow. In the east there is a region in which only layer 2 is in
motion, and layer 3 is at rest. Beyond the pool boundary, ¢ = p(8), is the
pool region in layer 3 labeled "pool3" in which the potential vorticity is
homogenized to fo/H3 and where the total solution is given by (4.9.18) while
the boundary of the region is given by (4.9.5).
After layer 2 subducts, all three layers are in motion in parts of the gyre.
East of the pool of uniform q3 (whose extension south of the outcrop line we
must find) layer 3 is at rest, and the solution is similar to that for the two-layer
model found in Section 4.4, although we must be more precise about the
domain of its validity. There are actually three critical curves, as shown in
Fig. 4.9.2, and the solution changes across them. The eastern most curve is the
boundary between the shadow zone boundary in layer 2 and the ventilated
zone in layer 2. East of this curve both layer 2 and layer 3 are at rest, and the
total Sverdrup transport is carried in the upper layer. The solution in the
shadow zone is exactly as found in Section 4.4 and is given by (4.4.23) while the
boundary itself is given by ( 4.4.22). Again, the solution in the eastern part of the
gyre is unaffected by alterations in the structure of the flow in the west.