Continuous Models of the Ventilated Thermocline
257
layers. The thickness of the uppermost layer and its potential vorticity are then
determined in the layer model in terms of the total depth of the thermocline by
the condition of outcropping. In the same way, q is determined at the final grid
point in the continuous model by ( 4.11.40), i.e., by the depth of the layer at the
grid point at Ps + Ap, which is in turn a function of Pn from the integration of
(4.11.22).
The integration of ( 4.11.22b) across the final grid interval then also
determines the surface pressure field from ( 4.11.27).
The final condition to be met is the Sverdrup integral condition ( 4.11.39).
The integral requires information about the fields, in particular the depth of the
thermocline in density space, Pn, only east of the point under consideration. It
is therefore necessary to start the process of calculation at the most eastward
point at the given latitude and work westward a step at a time. The solution
obtained by the vertical integration of ( 4.11.22) depends on the assumed
starting value of Pn, the density of the thermocline base. Unless that value has
been chosen properly, the Sverdrup condition is not satisfied. The value of Pn
must be iterated until the Sverdrup balance is satisfied. As we have noted, this
is exactly equivalent to the use of the Sverdrup relation in the layer models to
determine the base of the moving zone, h. Once the solution along a latitude
circle (or in the more general case the outcrop line) is determined by marching
westward along that line, the geostrophic velocities can be calculated, and the
fluid trajectories can be advanced to the position of the next outcrop line. Thus
at every depth in the new location the potential vorticity, carried with the fluid,
is known (except for the fluid that will be subducted at the new location), and
the process of the solution just outlined can be repeated until the whole gyre
has been covered. In advancing the potential vorticity from one point to the
next along streamlines, q is kept track of numerically in tabular form instead of
analytically, but the essential process is the same as in the layer model. Just as
in the layer models, the entire process is started at the northern edge of the gyre
where (except for the first layer to outcrop just south of the intergyre
boundary) the potential vorticity on each density surface is given by the
potential vorticity on the intergyre boundary or fixed by the flow out of the
western boundary. The direct relation to the layer model calculation is stressed
both to clarify the procedure for the continuous model and to emphasize the
physical continuity between the layer and continuous models.
Huang (1989b) has applied his model to the subtropical gyre of the North
Atlantic. He chose an Ekman pumping velocity that is a function of latitude
and longitude so that the zero value of WE occurs on a line that slopes
southwest-northeast in the basin (see Fig. 4.11.4a). The dynamical quantity
which enters the theory is the integral of the Ekman pumping ( 4.11.39) which
determines the Sverdrup transport streamfunction. This transport streamfunction is shown in Fig. 4.11.4b. Note that the zero line of the Sverdrup transport
slopes less sharply than that of the Ekman pumping and occurs further
northward. The boundary of the subtropical gyre could be plausibly associated
with either of these zero lines. These two lines coincide when the Ekman
257
layers. The thickness of the uppermost layer and its potential vorticity are then
determined in the layer model in terms of the total depth of the thermocline by
the condition of outcropping. In the same way, q is determined at the final grid
point in the continuous model by ( 4.11.40), i.e., by the depth of the layer at the
grid point at Ps + Ap, which is in turn a function of Pn from the integration of
(4.11.22).
The integration of ( 4.11.22b) across the final grid interval then also
determines the surface pressure field from ( 4.11.27).
The final condition to be met is the Sverdrup integral condition ( 4.11.39).
The integral requires information about the fields, in particular the depth of the
thermocline in density space, Pn, only east of the point under consideration. It
is therefore necessary to start the process of calculation at the most eastward
point at the given latitude and work westward a step at a time. The solution
obtained by the vertical integration of ( 4.11.22) depends on the assumed
starting value of Pn, the density of the thermocline base. Unless that value has
been chosen properly, the Sverdrup condition is not satisfied. The value of Pn
must be iterated until the Sverdrup balance is satisfied. As we have noted, this
is exactly equivalent to the use of the Sverdrup relation in the layer models to
determine the base of the moving zone, h. Once the solution along a latitude
circle (or in the more general case the outcrop line) is determined by marching
westward along that line, the geostrophic velocities can be calculated, and the
fluid trajectories can be advanced to the position of the next outcrop line. Thus
at every depth in the new location the potential vorticity, carried with the fluid,
is known (except for the fluid that will be subducted at the new location), and
the process of the solution just outlined can be repeated until the whole gyre
has been covered. In advancing the potential vorticity from one point to the
next along streamlines, q is kept track of numerically in tabular form instead of
analytically, but the essential process is the same as in the layer model. Just as
in the layer models, the entire process is started at the northern edge of the gyre
where (except for the first layer to outcrop just south of the intergyre
boundary) the potential vorticity on each density surface is given by the
potential vorticity on the intergyre boundary or fixed by the flow out of the
western boundary. The direct relation to the layer model calculation is stressed
both to clarify the procedure for the continuous model and to emphasize the
physical continuity between the layer and continuous models.
Huang (1989b) has applied his model to the subtropical gyre of the North
Atlantic. He chose an Ekman pumping velocity that is a function of latitude
and longitude so that the zero value of WE occurs on a line that slopes
southwest-northeast in the basin (see Fig. 4.11.4a). The dynamical quantity
which enters the theory is the integral of the Ekman pumping ( 4.11.39) which
determines the Sverdrup transport streamfunction. This transport streamfunction is shown in Fig. 4.11.4b. Note that the zero line of the Sverdrup transport
slopes less sharply than that of the Ekman pumping and occurs further
northward. The boundary of the subtropical gyre could be plausibly associated
with either of these zero lines. These two lines coincide when the Ekman