The Buoyancy- and Wind-Driven Subtropical Gyre: Analytical Solutions
311
(5.4.17)
which is the solution (5.4.2) already found in Section 4.4.
In the presence of heating (w. > 0) and with the negative Ekman pumping
in the subtropical gyre (wE < 0), it follows that b < 0, and thus ( 5.4.16) implies
that increases with Jl more rapidly than in the adiabatic case. Figure 5.4.1
shows the solution for for the case in which:
(5.4.18)
so that the cross-isopycnal velocity vanishes at the outcrop line and goes to
zero at the equator. The line in the figure with a slope of -1 is the solution for
in the absence of heating. The line above it is the solution for b given as in
(5.4.18). We see that is, as expected, everywhere greater than the value which
it would have in the absence of heating. This has several important consequences.
The trajectory for any streamline in the ventilated region is given by the
condition that h be constant, or:
(5.4.19)
where ¢' is the longitudinal position of the trajectory at its starting location on
the outcrop line. The equation for the eastern edge of the ventilated region
(5.4.14) is a special case of (5.4.19) as¢' __, ¢e where D6 vanishes. Since heating
increases the magnitude of at each latitude, the right side of (5.4.19) becomes
larger than would be the case for adiabatic motion. Since D6 is an increasing
function of distance from the eastern boundary, the only way in which the left
side of (5.4.19) can increase at each latitude is if the trajectory of the streamline
is shifted further westward. Heating, then, moves all the ventilated streamlines
westward of their adiabatic positions. In particular, the eastern boundary of
the ventilated regime is also moved westward. The eastern unventilated region
then increases in size while at the same time the size of any unventilated pool in
the western part of the circulation decreases in longitudinal extent.
When(f) = 1, the thickness of the lower layer goes to zero. For adiabatic motion this occurs only at the equator although we recognize that the
solution based on the geostrophic approximation may not be valid there.
Figure 5.4.1 shows that equals 1 at a value off> 0 when there is heating. If
this occurs, it means that the flux of fluid from layer 2 into the upper layer and
the general thinning of the layer as fluid moves westward into the western
boundary layer, combine to completely exhaust the volume of layer 2 at the
latitude in which = 1. In Fig. 5.4.1 this occurs atf/h = 0.14. The lower layer
then pinches off at that latitude, and layer 1 comes into direct contact with layer
3. Our solution is not valid of course south of this point, but one can easily
imagine that if the heating persists, it would then lead to a cross-isopycnal flux
311
(5.4.17)
which is the solution (5.4.2) already found in Section 4.4.
In the presence of heating (w. > 0) and with the negative Ekman pumping
in the subtropical gyre (wE < 0), it follows that b < 0, and thus ( 5.4.16) implies
that increases with Jl more rapidly than in the adiabatic case. Figure 5.4.1
shows the solution for for the case in which:
(5.4.18)
so that the cross-isopycnal velocity vanishes at the outcrop line and goes to
zero at the equator. The line in the figure with a slope of -1 is the solution for
in the absence of heating. The line above it is the solution for b given as in
(5.4.18). We see that is, as expected, everywhere greater than the value which
it would have in the absence of heating. This has several important consequences.
The trajectory for any streamline in the ventilated region is given by the
condition that h be constant, or:
(5.4.19)
where ¢' is the longitudinal position of the trajectory at its starting location on
the outcrop line. The equation for the eastern edge of the ventilated region
(5.4.14) is a special case of (5.4.19) as¢' __, ¢e where D6 vanishes. Since heating
increases the magnitude of at each latitude, the right side of (5.4.19) becomes
larger than would be the case for adiabatic motion. Since D6 is an increasing
function of distance from the eastern boundary, the only way in which the left
side of (5.4.19) can increase at each latitude is if the trajectory of the streamline
is shifted further westward. Heating, then, moves all the ventilated streamlines
westward of their adiabatic positions. In particular, the eastern boundary of
the ventilated regime is also moved westward. The eastern unventilated region
then increases in size while at the same time the size of any unventilated pool in
the western part of the circulation decreases in longitudinal extent.
When
solution based on the geostrophic approximation may not be valid there.
Figure 5.4.1 shows that equals 1 at a value off> 0 when there is heating. If
this occurs, it means that the flux of fluid from layer 2 into the upper layer and
the general thinning of the layer as fluid moves westward into the western
boundary layer, combine to completely exhaust the volume of layer 2 at the
latitude in which = 1. In Fig. 5.4.1 this occurs atf/h = 0.14. The lower layer
then pinches off at that latitude, and layer 1 comes into direct contact with layer
3. Our solution is not valid of course south of this point, but one can easily
imagine that if the heating persists, it would then lead to a cross-isopycnal flux