1rTir"i"T"illllq,"il"r"i 20 30

Longitude

I111!111!111!111!111!111!111!111!1 0 10 20 30 40 50 0 10 20 30 40 50

Fig. 4.62 Perturbations due to the anomalous Ekman pumping, shown in Fig. 4.61: a surface elevation (cm); b surface density (kg/m3); c depth of a = 27.0 kg/m3 isopycnal surface (m); d depth of the wind-driven gyre (in 100 m).

1rTir"i"T"illllq,"il"r"i 20 30

Longitude

I111!111!111!111!111!111!111!111!1 0 10 20 30 40 50 0 10 20 30 40 50

Fig. 4.62 Perturbations due to the anomalous Ekman pumping, shown in Fig. 4.61: a surface elevation (cm); b surface density (kg/m3); c depth of a = 27.0 kg/m3 isopycnal surface (m); d depth of the wind-driven gyre (in 100 m).

4.4 Recirculation

Our discussion has been limited to models based on the Sverdrup dynamics in the interior and western boundary currents matched to the interior solution along the western boundary, including the case with an inertial western boundary current matched to the interior solution for the southern half of the western boundary region. Within this theoretical framework, the maximal streamfunction is totally determined by a zonal integration of the Ekman pumping velocity, started from the eastern boundary of the basin. Observations, however, indicate that the maximal volume flux in the Gulf Stream is about 150 Sv, which is several times larger than the value calculated from the Sverdrup relation (Fig. 4.63).

The linear Sverdrup dynamics suggests that the volume transport of the gyre should reach the maximal value at the latitude of the Florida Strait, and then gradually decline for sections at higher latitude. On the other hand, the observed transport is much larger than that from the Sverdrup dynamics. Even at the latitude of the Florida Strait, the observed transport is larger than that of the Sverdrup dynamics. Near 40° N, the observed transport reaches the maximal value of 150 Sv (Gill, 1971), but the transport predicted by the Sverdrup dynamics is nearly zero because this is close to the zero wind-stress curl latitude. There is a similar phenomenon in the North Pacific Ocean, where the Kuroshio Extension and its dynamics and variability have been studied extensively (for a comprehensive review, see Qiu, 2002a).

The discrepancy between the linear theory and observations is due to two factors. First, the inertial terms in the horizontal momentum equations have been neglected for analytical simplicity in the commonly used linear theories. As discussed above, inertial terms are crucial within strong boundary currents, such as the Gulf Stream, the Kuroshio, and the ACC. There are recirculation regions in the northwestern (southwestern) corners of the subtropical gyre in the Northern (Southern) Hemisphere where the circulation is strongly nonlinear. Second, there is a strong interaction with stratified flow over topography, which is called bottom pressure torque or the JEBAR (joint effect of baroclinicity and bottom relief) term, and this is discussed in detail in subsection 4.4.5, "The role of bottom pressure torque."

The contribution due to the JEBAR term can be seen clearly from the western boundary current transport as diagnosed from ai°x 1° low-resolution model (SODA). Such a model cannot simulate the effect of eddies. Nevertheless, the transport of the anticyclonic gyre is more than 50 Sv around 34° N.

In this section we discuss preliminary theories of the recirculation. Most of the published theoretical work relating to the role of meso-scale eddies in the recirculation is based on the quasi-geostrophic theory. Furthermore, the role of bottom pressure torque can be explained without explicitly invoking eddies; thus, the meaning of bottom pressure torque will be explained using results from a simple non-eddy-resolving model. The potential shortcomings of such models will be commented on at the end of this section.

4.4.2 Fofonoff solution

For a model with a single moving layer, the quasi-geostrophic vorticity equation is

where R ^ O (1) is a frictional parameter. Fofonoff (1954) suggested looking for solutions free of forcing and friction. Therefore, we set both the Ekman pumping and friction at zero (R = 0); since potential vorticity is conserved, such a solution should satisfy the vorticity conservation law

where F(f ) is an arbitrary function of f.

There are many different solutions, as first discussed by Fofonoff (1954). However, many of these solutions are not linked to the circulation in the ocean interior, so it is desirable to find solutions which satisfy Eqn. (4.344), and are thus valid solutions for the whole basin.

Integrating Eqn. (4.344) over an area Af within a closed streamline leads to an integral constraint ffA f0Wfy) = R$C* u- dl (4.346)

This constraint can be used to find a solution for the problem; however, it is not an easy job. One way to find a solution for this problem is to assume some simple form of this function F(f ), and thus a solution consistent with this constraint (Niiler, 1966). As an example, we assume that this function is in the form

Interior solution

In the ocean interior, the inertial term, i.e., the relative vorticity, can be neglected; thus, from Eqn. (4.347) the solution is fi = y - y0 (4.348)

which represents a westward flow with a uniform velocity. For convenience we choose the southern boundary as y0 = 0, so there is no boundary layer along the southern boundary of the model basin.

Boundary layers along other boundaries of the model basin It is clear that along other boundaries of the basin, the streamfunction calculated from the interior solution is not zero, fI = 0. In order to satisfy the boundary condition of zero streamfunction, there must be boundary layers along all other boundaries. At the western boundary x = 0. Let us separate the streamfunction into two parts:

Substituting Eqns. (4.348,4.349) into Eqn. (4.347), we obtain

within the western boundary, the y-derivative term is much smaller than the x-derivative terms, so this equation is reduced to an ordinary differential equation in x d 2

subject to boundary conditions fw (0) = - fi, and fw ^ 0, at x ^<x> (4.352)

Boundary layer solutions along the boundary at x = 1 and y = 1 have a similar structure; therefore, the complete solution is f = y h - e-^x - e-^(1-x)] + e-VP(1-y) (4.354)

Two solutions are shown in Figure 4.64.

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