Effect of Finite Mixed Layer Depth
241
(4.10.20)
and:
(4.10.21)
With the Sverdrup relation (4.10.14), the solution can be completed.
Pedlosky and Robbins present first the circulation in this region for the case
where the latitude of the second outcrop line is moved to the equator so that
the solution presented here fills the gyre, except for the shadow zone. There is, of
course, a shadow zone near the eastern boundary, just as found in Section 4.4,
in which the fluid in layer 3 is at rest. The boundary· of the shadow zone is
found in just the same manner as in the earlier theory. That is, the streamline
emanating from the intersection of the outcrop line with the eastern wall, on
which h = H(/3), is followed using the solution described above. The solution
in the shadow zone has only layer 2 in motion (as well as the mixed layer) and
so in that region h = H(/3) and the Sverdrup relation (4.10.14) yields the
shadow zone solution directly:
h2 = y 3 Dij + (Hm + H2) 2 - H;,
Y2
(4.10.22)
where Hm + H2 is given by ( 4.1 0.16). Thus on the eastern wall h2 = H2 =I- 0.
Figure 4.1 0.4 shows the solution in the region south of the outcrop line for
a sinusoidal Ekman pumping. Panel c shows the variable part of the mixed
layer density and the variation of the mixed layer depth used in the calculation.
Panel a shows the streamlines in layer 3 in the case in which the mixed layer is
negligibly thin. Panel b shows the streamlines in the case in which the mixed
layer is variable. In this case the shadow zone broadens linearly, instead of
parabolically, with distance from the outcrop line, and in fact all the
streamlines are shifted westward. As a consequence of this shift the constant
potential vorticity pool region in the west is absent when the mixed layer is
variable, at least for this outcrop position. Outcrop positions farther
northward would again yield a western pool region, but it would be reduced
in size. In panels d and e we see the flow in layer 2. Panel d describes the flow
for a negligible mixed layer thickness while panel e shows the flow in the case
where the mixed layer depth varies. Note that only a very weak flow issues
from the eastern boundary. The variations in density and mixed layer depth
tend to offset each other, and therefore there is only a very weak zonal flux of
fluid in the mixed layer at the wall requiring compensation. Thus to a large
extent layer 2 develops a nonzero thickness at the wall due mostly to the
kinematic shrinking of the mixed layer and, to first order, the constancy of the
depth of the base of layer 2 as shown in Fig. 4.10.4f.
241
(4.10.20)
and:
(4.10.21)
With the Sverdrup relation (4.10.14), the solution can be completed.
Pedlosky and Robbins present first the circulation in this region for the case
where the latitude of the second outcrop line is moved to the equator so that
the solution presented here fills the gyre, except for the shadow zone. There is, of
course, a shadow zone near the eastern boundary, just as found in Section 4.4,
in which the fluid in layer 3 is at rest. The boundary· of the shadow zone is
found in just the same manner as in the earlier theory. That is, the streamline
emanating from the intersection of the outcrop line with the eastern wall, on
which h = H(/3), is followed using the solution described above. The solution
in the shadow zone has only layer 2 in motion (as well as the mixed layer) and
so in that region h = H(/3) and the Sverdrup relation (4.10.14) yields the
shadow zone solution directly:
h2 = y 3 Dij + (Hm + H2) 2 - H;,
Y2
(4.10.22)
where Hm + H2 is given by ( 4.1 0.16). Thus on the eastern wall h2 = H2 =I- 0.
Figure 4.1 0.4 shows the solution in the region south of the outcrop line for
a sinusoidal Ekman pumping. Panel c shows the variable part of the mixed
layer density and the variation of the mixed layer depth used in the calculation.
Panel a shows the streamlines in layer 3 in the case in which the mixed layer is
negligibly thin. Panel b shows the streamlines in the case in which the mixed
layer is variable. In this case the shadow zone broadens linearly, instead of
parabolically, with distance from the outcrop line, and in fact all the
streamlines are shifted westward. As a consequence of this shift the constant
potential vorticity pool region in the west is absent when the mixed layer is
variable, at least for this outcrop position. Outcrop positions farther
northward would again yield a western pool region, but it would be reduced
in size. In panels d and e we see the flow in layer 2. Panel d describes the flow
for a negligible mixed layer thickness while panel e shows the flow in the case
where the mixed layer depth varies. Note that only a very weak flow issues
from the eastern boundary. The variations in density and mixed layer depth
tend to offset each other, and therefore there is only a very weak zonal flux of
fluid in the mixed layer at the wall requiring compensation. Thus to a large
extent layer 2 develops a nonzero thickness at the wall due mostly to the
kinematic shrinking of the mixed layer and, to first order, the constancy of the
depth of the base of layer 2 as shown in Fig. 4.10.4f.