Effect of Finite Mixed Layer Depth
245
which have finite depths on the eastern boundary, and each of these layers has
a shadow zone. Shadow zones therefore generally occur in all layers. The
stratification on the eastern boundary of the ocean is finite rather than singular
since now it is not necessary that all the isopycnals (except the lowest ventilated
one) rise to the surface. The shadow zones spread from the wall into the eastern
region of the gyre, altering the structure of the circulation.
Figure 4.10.5 shows the streamlines of the three-layer flow as calculated by
Pedlosky and Robbins. Panel a shows the solution domains of the flow. South
of the first outcrop line the solution is split in layer 3 into a ventilated zone, V3 ,
and a shadow zone, SZ3. South of the next outcrop line the solution domain
fragments further. Most of the subdomains are discussed above in Section 4. 7.
The details of the solution alter in the presence of the variable mixed layer in
the western region, but the solution is similar to that found above. There is the
ventilated zone in layer 3, V3, as well as the hybrid zone in layer 2, M2, in which
layer 2 is ventilated on the outcrop line where layer 3 is in motion but then
passes into the hybrid region where layer 3 is at rest. East of this region is the
simple ventilated region in layer 2, V2, where layer 3 is at rest, and where the
potential vorticity in layer 2 is determined on the eastern portion of the outcrop
line in the shadow zone of layer 3. Then there is the new region, SZ2, which is
the shadow zone in layer 2, and in which only layer 1 is in motion beneath the
mixed layer. Panels b-e show the streamlines in layers 1, 2 and 3, respectively.
In particular the shadow zone in layer 2 is evident. Figure 4.10.5e, and f shows
latitude and zone cross sections of the solution in the region in which all three
layers exist. The shadow zone in layer 2 is so narrow that the sections do not
display it clearly.
If more layers are added to improve the vertical resolution of the model,
each new layer has a shadow zone. Each shadow zone boundary in turn
produces new subdomains of the flow when it encounters successive outcrop
lines further southward. It is not hard to see that the improvement of the model
in the direction of greater vertical resolution would soon lead to analytic
complexity great enough to vitiate the advantages normally associated with
analytic representations of the motion. The principal advantage of an
analytical solution is the help its compact representation of the physics gives
to develop our physical insight. An analytic solution fragmented into ever
smaller zones, requiring a map and a table to reference and locate each domain,
soon exhausts its instructive value as well as our patience.
The step to improved resolution requires going finally to a model with
continuous distributions of density and velocity. Since the solution can be
expected to differ physically and mathematically in the various subdomains, it
is unlikely that an analytical solution will emerge, and a numerical treatment is
in fact necessary. The formulation and the execution of such a model and its
algorithm is taken up in the next section. We see there that it closely follows the
essentials of our discussion of the layer models.
245
which have finite depths on the eastern boundary, and each of these layers has
a shadow zone. Shadow zones therefore generally occur in all layers. The
stratification on the eastern boundary of the ocean is finite rather than singular
since now it is not necessary that all the isopycnals (except the lowest ventilated
one) rise to the surface. The shadow zones spread from the wall into the eastern
region of the gyre, altering the structure of the circulation.
Figure 4.10.5 shows the streamlines of the three-layer flow as calculated by
Pedlosky and Robbins. Panel a shows the solution domains of the flow. South
of the first outcrop line the solution is split in layer 3 into a ventilated zone, V3 ,
and a shadow zone, SZ3. South of the next outcrop line the solution domain
fragments further. Most of the subdomains are discussed above in Section 4. 7.
The details of the solution alter in the presence of the variable mixed layer in
the western region, but the solution is similar to that found above. There is the
ventilated zone in layer 3, V3, as well as the hybrid zone in layer 2, M2, in which
layer 2 is ventilated on the outcrop line where layer 3 is in motion but then
passes into the hybrid region where layer 3 is at rest. East of this region is the
simple ventilated region in layer 2, V2, where layer 3 is at rest, and where the
potential vorticity in layer 2 is determined on the eastern portion of the outcrop
line in the shadow zone of layer 3. Then there is the new region, SZ2, which is
the shadow zone in layer 2, and in which only layer 1 is in motion beneath the
mixed layer. Panels b-e show the streamlines in layers 1, 2 and 3, respectively.
In particular the shadow zone in layer 2 is evident. Figure 4.10.5e, and f shows
latitude and zone cross sections of the solution in the region in which all three
layers exist. The shadow zone in layer 2 is so narrow that the sections do not
display it clearly.
If more layers are added to improve the vertical resolution of the model,
each new layer has a shadow zone. Each shadow zone boundary in turn
produces new subdomains of the flow when it encounters successive outcrop
lines further southward. It is not hard to see that the improvement of the model
in the direction of greater vertical resolution would soon lead to analytic
complexity great enough to vitiate the advantages normally associated with
analytic representations of the motion. The principal advantage of an
analytical solution is the help its compact representation of the physics gives
to develop our physical insight. An analytic solution fragmented into ever
smaller zones, requiring a map and a table to reference and locate each domain,
soon exhausts its instructive value as well as our patience.
The step to improved resolution requires going finally to a model with
continuous distributions of density and velocity. Since the solution can be
expected to differ physically and mathematically in the various subdomains, it
is unlikely that an analytical solution will emerge, and a numerical treatment is
in fact necessary. The formulation and the execution of such a model and its
algorithm is taken up in the next section. We see there that it closely follows the
essentials of our discussion of the layer models.