Application of the Theory to the Subtropical Gyre
145
pool region in which motion is allowed, also suggests the simultaneous
appearance of a mechanism of small-scale eddy driving of the deeper layers to
produce the motion that moves along the geostrophic contours.
For realistic values of the parameters Xr lies within the ocean basin. For
examplewithfo = IQ- 4 s- 1 ,p = IQ- 13 cm- 1 s-l,H=800m, andL~ = l/F = 25x
10 12 cm 2 and W0 = IQ- 4 cmjs with L = 1000 km, then Xe - Xn the distance from
the critical point to the eastern boundary is 1270 km for the Ekman pumping
of the form (3.9.6). This is close enough to the eastern boundary to fit the
critical point into all the major ocean basins.
In the pool region of constant potential vorticity in layer 2, the base of
layer 2 is depressed by the anticyclonic motion shown in Fig. 3.9.2. We
assumed layer 3 to be at rest, but now that layer 2 is in motion and the isolines
of potential vorticity in layer 3 are distorted from latitude circles by the flexing
of the base of layer 2, it is possible that within the pool region oflayer 2 these q 3
contours may now become closed, repeating the process that occurred in layer
2. Should they close they would be nested within the pool region of layer 2.
Within this subpool the motion can be recalculated by attributing to layer 3 a
constant potential vorticity equal to PL, since it too touches the northern
boundary of the gyre. The process can be repeated indefinitely with deeper
layers set into motion if the thickness field of each layer, presumed to be at rest,
is altered enough by the motion of the layers above. Since the intensity of the
motion decreases with depth, this becomes increasingly difficult to achieve, and
in most cases a layer is finally reached in which its critical point on the northern
boundary of the gyre lies outside of the basin, and this layer and all deeper
layers remain at rest. It is important to note here that the pools are nested in
the sense that each deeper layer in motion has its pool completely contained
within the region of the pool of the layer above it since it is only the motion of
the upper layer that can distort a layer's potential vorticity contours away from
latitude circles. The region of uniform potential vorticity would then form an
inverted bowl or cone with the vertex at depth in the northwest corner and then
opening outward to the south and east as one rises in the water column. Note
that each layer would have the potential vorticity PL within its pool so that the
potential vorticity is constant from layer to layer within the moving region as
well as being uniform within the moving part of each layer. At each depth the
horizontal plan form would resemble the pool region found in the 2! layer
model shown in Fig. 3.9.2.
The vertical resolution of the layer model can be improved by adding more
layers and making each layer thinner. In the limit, the layers become
infinitesimally thin, and the fluid has a continuous stratification of density.
In this limit the layer equations must be changed to equations governing a fluid
with a continuous variation of density, p. The limiting form of this bowl is
obtained as each of the layers has an infinitesimal thickness, i.e., when the fluid
has a continuous vertical stratification. Although the continuously stratified
problem is generally a very difficult one, it is tractable within the quasigeostrophic approximation.
145
pool region in which motion is allowed, also suggests the simultaneous
appearance of a mechanism of small-scale eddy driving of the deeper layers to
produce the motion that moves along the geostrophic contours.
For realistic values of the parameters Xr lies within the ocean basin. For
examplewithfo = IQ- 4 s- 1 ,p = IQ- 13 cm- 1 s-l,H=800m, andL~ = l/F = 25x
10 12 cm 2 and W0 = IQ- 4 cmjs with L = 1000 km, then Xe - Xn the distance from
the critical point to the eastern boundary is 1270 km for the Ekman pumping
of the form (3.9.6). This is close enough to the eastern boundary to fit the
critical point into all the major ocean basins.
In the pool region of constant potential vorticity in layer 2, the base of
layer 2 is depressed by the anticyclonic motion shown in Fig. 3.9.2. We
assumed layer 3 to be at rest, but now that layer 2 is in motion and the isolines
of potential vorticity in layer 3 are distorted from latitude circles by the flexing
of the base of layer 2, it is possible that within the pool region oflayer 2 these q 3
contours may now become closed, repeating the process that occurred in layer
2. Should they close they would be nested within the pool region of layer 2.
Within this subpool the motion can be recalculated by attributing to layer 3 a
constant potential vorticity equal to PL, since it too touches the northern
boundary of the gyre. The process can be repeated indefinitely with deeper
layers set into motion if the thickness field of each layer, presumed to be at rest,
is altered enough by the motion of the layers above. Since the intensity of the
motion decreases with depth, this becomes increasingly difficult to achieve, and
in most cases a layer is finally reached in which its critical point on the northern
boundary of the gyre lies outside of the basin, and this layer and all deeper
layers remain at rest. It is important to note here that the pools are nested in
the sense that each deeper layer in motion has its pool completely contained
within the region of the pool of the layer above it since it is only the motion of
the upper layer that can distort a layer's potential vorticity contours away from
latitude circles. The region of uniform potential vorticity would then form an
inverted bowl or cone with the vertex at depth in the northwest corner and then
opening outward to the south and east as one rises in the water column. Note
that each layer would have the potential vorticity PL within its pool so that the
potential vorticity is constant from layer to layer within the moving region as
well as being uniform within the moving part of each layer. At each depth the
horizontal plan form would resemble the pool region found in the 2! layer
model shown in Fig. 3.9.2.
The vertical resolution of the layer model can be improved by adding more
layers and making each layer thinner. In the limit, the layers become
infinitesimally thin, and the fluid has a continuous stratification of density.
In this limit the layer equations must be changed to equations governing a fluid
with a continuous variation of density, p. The limiting form of this bowl is
obtained as each of the layers has an infinitesimal thickness, i.e., when the fluid
has a continuous vertical stratification. Although the continuously stratified
problem is generally a very difficult one, it is tractable within the quasigeostrophic approximation.