!58
Vertical Structure: Baroclinic Quasi-Geostrophic Models
Only the first condition is clearly satisfied for the gyre circulation problem
that we have discussed. The dissipation of potential vorticity by small-scale
eddy motion may act as a diffusion, but it is hard to argue that the diffusion
coefficient is constant, and in some cases it may even be negative (Holland and
Rhines 1980).
Perhaps the most critical difficulty in applying the homogenization
argument in the case of the gyre circulation is the fact that the isolines of
potential vorticity, including the outermost one, must close in the western
boundary current. Normally, the western boundary current is a location of
substantial potential vorticity flux, and, as we saw in Chapter 2, it is difficult in
general to construct a circulation pattern in which this dissipation is not a
controlling factor in the circulation. Moreover, it is precisely the outermost
contour that is closest to the western boundary and on which it will be most
likely for dissipation to act. In the absence of adequate theories of the western
boundary current it is therefore not possible to give a completely convincing a
priori argument of whether conditions 3 and 4 are likely to be met.
Young and Rhines (1982) argue that for a quasi-geostrophic layer model
there are no substantial inputs of vorticity below the first layer, at least in the
region outside the western boundary current, and therefore there is no need for
the western boundary current at depth to diffuse substantial amounts of
potential vorticity through the side wall on the western boundary. They present
an interesting heuristic attempt to construct a dissipationless western boundary
current for the deeper layers to demonstrate the applicability of the PrandtlBatchelor theorem to the wind driven gyre. Unfortunately, an essential part of
their argument is based on the Moore scenario for the western boundary
current in the region of eastward interior flow, which as we saw in Chapter 2, is
fundamentally deficient. Indeed, Ierley and Young (1983) later presented a
counterexample in which the matching of the western boundary current to the
Sverdrup interior places a condition on the interior flow that blocks the process
of homogenization. However, the Ierley and Young argument, while rigorous,
is limited to a rather artificial parameter regime in which relative vorticity is
considered unimportant in the western boundary layer. From a theoretical
point of view the issue remains obscure.
The question of the existence of sources of potential vorticity for the flow
in the gyre is a complex one which we take up in greater detail in Chapters 4
and 5. However, as we have seen in Fig. 3.1.1, the interfaces between the
upper density strata rise to the surface within the latitude band of the
subtropical gyre where they are exposed directly to Ekman pumping. The
potential vorticity of fluid in the surface mixed layer is determined by a
complex interaction with heating and cooling by the atmosphere; then enters
the thermocline in these outcrop zones and carries the information of the
surface-determined potential vorticity to great depth within the thermocline.
For such density layers it seems problematic from the beginning whether we
can ignore such sources in determining the potential vorticity structure. It
Vertical Structure: Baroclinic Quasi-Geostrophic Models
Only the first condition is clearly satisfied for the gyre circulation problem
that we have discussed. The dissipation of potential vorticity by small-scale
eddy motion may act as a diffusion, but it is hard to argue that the diffusion
coefficient is constant, and in some cases it may even be negative (Holland and
Rhines 1980).
Perhaps the most critical difficulty in applying the homogenization
argument in the case of the gyre circulation is the fact that the isolines of
potential vorticity, including the outermost one, must close in the western
boundary current. Normally, the western boundary current is a location of
substantial potential vorticity flux, and, as we saw in Chapter 2, it is difficult in
general to construct a circulation pattern in which this dissipation is not a
controlling factor in the circulation. Moreover, it is precisely the outermost
contour that is closest to the western boundary and on which it will be most
likely for dissipation to act. In the absence of adequate theories of the western
boundary current it is therefore not possible to give a completely convincing a
priori argument of whether conditions 3 and 4 are likely to be met.
Young and Rhines (1982) argue that for a quasi-geostrophic layer model
there are no substantial inputs of vorticity below the first layer, at least in the
region outside the western boundary current, and therefore there is no need for
the western boundary current at depth to diffuse substantial amounts of
potential vorticity through the side wall on the western boundary. They present
an interesting heuristic attempt to construct a dissipationless western boundary
current for the deeper layers to demonstrate the applicability of the PrandtlBatchelor theorem to the wind driven gyre. Unfortunately, an essential part of
their argument is based on the Moore scenario for the western boundary
current in the region of eastward interior flow, which as we saw in Chapter 2, is
fundamentally deficient. Indeed, Ierley and Young (1983) later presented a
counterexample in which the matching of the western boundary current to the
Sverdrup interior places a condition on the interior flow that blocks the process
of homogenization. However, the Ierley and Young argument, while rigorous,
is limited to a rather artificial parameter regime in which relative vorticity is
considered unimportant in the western boundary layer. From a theoretical
point of view the issue remains obscure.
The question of the existence of sources of potential vorticity for the flow
in the gyre is a complex one which we take up in greater detail in Chapters 4
and 5. However, as we have seen in Fig. 3.1.1, the interfaces between the
upper density strata rise to the surface within the latitude band of the
subtropical gyre where they are exposed directly to Ekman pumping. The
potential vorticity of fluid in the surface mixed layer is determined by a
complex interaction with heating and cooling by the atmosphere; then enters
the thermocline in these outcrop zones and carries the information of the
surface-determined potential vorticity to great depth within the thermocline.
For such density layers it seems problematic from the beginning whether we
can ignore such sources in determining the potential vorticity structure. It