Using the barotropic streamfunction f B, the nonlinear equations (4.154a) and (4.154b) can be rewritten as

These equations are linear because f B is a given function fromEqn. (4.155). These equations are first-order partial differential equations in the characteristic form, with a quantity that behaves like a "barotropic potential vorticity" q2 = py + F fB as the characteristics. Under the assumption of infinitesimal friction, potential vorticity in the second layer is conserved along streamlines. According to Eqn. (4.156b), potential vorticity contours in the second layer are the same as the "barotropic potential vorticity" contours.

To study the effect of strong forcing on the geostrophic contours in subsurface layers, Rhines and Young chose a very idealized Ekman pumping function wq = —ax, for r < r\\ w0 = 0, for r > r1 (4.157)

where r = ^x2 + y2. Therefore, the barotropic streamfunction is fs =

The q2 contours are circles or arcs of circles if r < r1 (Fig. 4.22); outside the circle of non-zero Ekman pumping rate they are just straight zonal lines q2 =

— [r2 + y2 — x2 — (y — y0)2l, if r < r1,

It can readily be seen that if forcing is weak, the barotropic potential vorticity is controlled by the planetary term jy, so vorticity contours are close to straight lines and there are no closed vorticity contours. For such cases, the eastern boundary blocks all possible flows in the second layer, as pointed out by Rooth et al. (1978). However, if forcing is strong enough, the second term, Ff B, dominates the value of q2, and there can be closed vorticity contours (Fig. 4.22).

For this given type of Ekman pumping rate distribution, closed contours will appear, if r1 > y0, or equivalently, ar1 > j2/F (4.160)

In general, the closed contours will appear near the northern boundary of the forcing field, where the negative meridional gradient of the barotropic streamfunction can cancel the positive meridional gradient of the planetary vorticity.

A crucial technique used in their model is to assume a special Ekman pumping pattern which satisfies wo (x') dxX = 0 (4.161)

By using such a pattern, they were able to avoid the complications associated with the western boundary layer and find dynamically consistent solutions. Note that the same technique was first employed by Goldsbrough for the evaporation/precipitation-driven circulation. As we discuss later, adding the western boundary current is not an easy exercise. In fact, due to the strong friction/dissipation within the western boundary layer, potential vorticity cannot be homogenized in the subsurface layer (Ierley and Young, 1983).

When closed geostrophic contours appear in the second layer, the number of possible solutions for an ideal fluid is infinite; these solutions are in the form of f 2 = A2(q2) (4.162)

where A2 is an arbitrary function. In order to find solutions that are physically meaningful, one has to include the next-order terms. Typically, by working on some integrals for the next-order terms, one can find constraints on the lowest-order dynamics, and this will eventually lead to unique solutions. A most important attribute of this solution is that it is stable to small perturbations, as demonstrated by Rhines and Young (1982a).

This example leads to an interesting observation. A potential vorticity field dominated by the strong planetary vorticity, combining with the eastern boundary, blocks the potential ideal-fluid flow in the subsurface interior ocean; however, strong Ekman pumping creates closed geostrophic contours in the oceanic interior, overcoming the blockage and making free solutions possible there.

Determination of the flow inside the closed geostrophic contours

For convenience, we assume that D = R. Integrating Eqn. (4.149b) along a closed contour, leads to

where u1 = k xVf 1 and U2 = k xVf2 are horizontal velocity in the upper and lower layers, and k is a unit vector in the vertical direction. Note that the Jacobian term vanishes identically! Equation (4.163) can be rewritten as fiu2 ■ dU = 1PtuB ■ dU (4.164)

where uB = k x VfB = u1 +u2 is the barotropic velocity. Using Eqn. (4.162), the term on the left-hand side of Eqn. (4.164) can be rewritten as fiu2 ■ dU = fiA'2(q2)(k x Vhv2) ■ dU = A'2(q2)fi(FuB - px) ■ ds (4.165)

where A2 = dA2/dq2. Thus, from Eqns. (4.164) and (4.165) we finally obtain a relation

Using Eqn. (4.162), the final solution is

1 1 1 py f 2 = — q2 + const = - fB + -— + const (4.167a)

For the case discussed above, w0 = -ax, r < r1, so that flow in the lower layer is

- — (x2 + (y - y0)2) + const. for closed q2 f2 =\ 6ß * (4.168)

0, elsewhere

The solution is shown in Figure 4.23. In the lower layer, flow is confined to the outermost closed geostrophic contour. Outside this boundary the lower layer is stagnant, so flow in this regime is entirely confined to the upper layer. Since the lower layer is in motion within the closed geostrophic contours, flow in the upper layer is reduced as the Sverdrup constraint applies to the sum of the volumetric transport in these two moving layers.

When D = R, the corresponding solution is q2 = (D/2R + D) q2 + const; thus, in the limit of D ^ R, potential vorticity in the second layer becomes homogenized within the closed streamlines. This is an example of the more generalized potential vorticity homog-enization theory discussed by Rhines and Young (1982a, b). This example of a two-layer model, satisfying the constraint D ^ R, shows that potential vorticity homogenization in the subsurface layer can be realized only if that layer is shielded from strong forcing or dissipation. Therefore, a natural choice to demonstrate the idea of potential vorticity homogenization is to use a three-layer model, in which the second layer is shielded from surface forcing and bottom friction.

For a model with three moving layers, an analysis similar to that presented above demonstrates that, in the second layer, potential vorticity within the closed geostrophic contours is homogenized toward its value along the northern boundary of the gyre. Since potential vorticity outside the big circle is controlled by the planetary vorticity alone, its contours are zonally oriented; thus, potential vorticity homogenization within the big circle implies an expulsion of potential vorticity contours toward the edge of the big circle. As a result, there is a sharp potential vorticity front adjacent to the outermost closed geostrophic contours, as illustrated in Figure 4.24.

Another important phenomenon related to the three-moving-layer model is that the center of the wind-driven gyre in each layer is gradually moved northward from the topmost layer to the deeper layers, as shown in Figure 4.25. This is called the northern intensification of the subtropical gyre, and this phenomenon can be identified from hydrographic data or a model with continuous stratification, as will be discussed later. Note that volume flux in the upper layer is reduced over the domain where the lower layers are in motion.

The potential vorticity homogenization theory discussed in this section represents a beautiful combination of many vital dynamic concepts related to the vertical structure of wind-driven gyres in the oceans.

Fig. 4.25 Streamfunction maps for a three-layer model for the case with y0 = r1/8. In the middle panel, the smallest dashed circle indicates the outermost closed geostrophic contour in the lower layer, and the middle-sized dashed circle indicates the outermost closed geostrophic contour in the middle layer, while the outermost dashed circle indicates the domain with non-zero Ekman pumping.

Fig. 4.25 Streamfunction maps for a three-layer model for the case with y0 = r1/8. In the middle panel, the smallest dashed circle indicates the outermost closed geostrophic contour in the lower layer, and the middle-sized dashed circle indicates the outermost closed geostrophic contour in the middle layer, while the outermost dashed circle indicates the domain with non-zero Ekman pumping.

• Strong surface forcing can induce closed geostrophic contours in the subsurface layers.

• There is an infinite number of possible solutions; however, unique solutions stable to small perturbations can be found.

• Potential vorticity is homogenized within the closed contours when the dissipation is parameterized as lateral potential vorticity diffusion.

• Potential vorticity is homogenized toward its value along the northern boundary.

These are very useful when we want to find a solution for the wind-driven circulation in the case of a multi-layer model or a model with continuous stratification.

4.1.7 The ventilated thermocline

There is a prominent layer of steep vertical temperature gradient in the world's oceans, the so-called main thermocline, the structure of which was discussed in Section 1.4. Since density is very closely related to temperature, the structure of the thermocline is very closely related to currents in the upper ocean. The vertical shear of the horizontal velocity field is related to the horizontal density gradient through the thermal wind relation; therefore, solving the density structure is equivalent to finding the structure of the wind-driven circulation. For historical reasons the relevant theory is called thermocline theory; however, it is somewhat equivalent to a theory for the wind-driven circulation in the upper ocean. The fundamental questions at the heart of the thermocline theory are as follows.

• First, the existence problem: Why is there the main thermocline in the ocean?

• Second, the direct problem for the thermocline: Given the surface forcing conditions, including wind stress, heat and freshwater fluxes, how is the stratification, or the potential vorticity, set up in the subsurface ocean?

From the very beginning of the development of the theory, there have been two approaches to this problem. In 1959, two papers were published side by side in Tellus. Robinson and Stommel (1959) proposed a theory of the thermocline in which the vertical diffusion plays a vital role; Welander (1959) proposed an ideal-fluid theory for the thermocline.

According to the theory of Robinson and Stommel (1959), the main thermocline is viewed as an internal density front or internal thermal boundary layer; thus, the vertical diffusion term should be retained as an essential component of the dynamical description. The latest advance along this line of thought is presented by Salmon (1990,1991). The most challenging difficulty associated with this approach is the fact that nobody knows how to formulate suitable boundary value problems for the corresponding nonlinear equation system, much less how to find the solutions to this problem. In order to overcome such a difficulty, similarity solutions have been sought. The early developments along this line of research were summed up in a comprehensive review by Veronis (1969).

Similarity solutions which satisfy a given differential equation can be found through a systematic approach based on group-invariant solutions under the infinitesimal group transformation, i.e., the Lie group. Filippov (1968) applied the Lie group theory to the thermohaline equations and discussed the group-invariant solutions. More up-to-date mathematical tools can be found in books by Oliver (1986) and Rogers and Ames (1989). Similarity solutions to the oceanic thermocline have been discussed by Salmon and Hollerbach (1991) and Hood and Williams (1996); time-varying similarity solutions have been discussed by Edwards (1996).

The major disadvantage of the similarity solutions is as follows. Although they do satisfy the thermocline equations, they do not satisfy some essential boundary conditions, such as the Sverdrup constraint. A solution that does not satisfying the Sverdrup constraint cannot describe the global structure of the wind-driven circulation accurately. More importantly, the function relation for the potential vorticity in the thermocline is prescribed a priori; thus, such solutions cannot provide a clear answer to the direct problem of the thermocline: how potential vorticity in the thermocline is set up for given surface forcing conditions.

In the other Tellus paper, Welander (1959) proposed a quite different approach; he argued that the main thermocline can be treated in terms of an ideal-fluid theory. Accordingly, the structure of the main thermocline can be interpreted without the vertical and horizontal diffusion terms.

These two approaches are based on different simplifications of the same set of dynamical equations. From the beginning, the basic equations for the thermocline seemed so naively simple that many people believed they could be solved easily, so most people did not want to spend time working on the apparently incomplete ideal-fluid thermocline theory, except for Welander, who also published another very influential paper on the ideal-fluid thermocline (Welander, 1971a).

The most critical point in the ideal-fluid thermocline theory is the smallness of the vertical diffusivity in the upper ocean (below the mixed layer). Recent field observations confirm that diapycnal diffusivity above the main thermocline is indeed quite weak, on the order of 10-5 m2/s (e.g., Ledwell et al., 1993). Thus, flow associated with the main thermocline can be treated in terms of an ideal fluid. As explained in Section 5.4.5, the vertical diffusion term is not the most crucial ingredient in formulating the main thermocline. As a matter of fact, the main thermocline in the subtropical ocean is primarily due to the downward squeeze of the Ekman pumping induced by the negative wind-stress curl.

A convenient way to study the ideal-fluid thermocline is to formulate the problem in terms of density coordinates and the Bernoulli function B = p + pgz, wherep is pressure, p is density, g is gravity, and z is the vertical coordinate. Assuming that the potential vorticity Q = f Pz is a linear function of both the density and the Bernoulli function, Welander (1971a) was able to find the first analytical solution for the thermocline.

Although this is the first elegant solution for the ideal-fluid thermocline, and thus has been cited in many textbooks, it has some major defects: the solution satisfies neither the Sverdrup relation nor the eastern boundary condition, and the lower boundary is set at z = —cc Since the Sverdrup constraint is the most important constraint, thermocline solutions that do not satisfy the Sverdrup relation are less meaningful dynamically.

During the 1960s, similarity solutions remained the mainstream in thermocline theory. However, the limitations of the similarity approach became clear. Beginning in the early 1980s there were major breakthroughs in the theory of the dynamical structure of the wind-driven gyre, including the potential vorticity homogenization theory by Rhines and Young (1982a) and the ventilated thermocline by Luyten et al. (1983). The historic events that led to the discovery of the new thermocline theory have been vividly described by Pedlosky (2006).

According to these new theories, the wind-driven gyre includes several regions with distinctively different dynamics: the ventilated thermocline where potential vorticity is set at the surface; the unventilated thermocline where potential vorticity is homogenized; the shadow zone near the eastern boundary; and the pool region near the western boundary. These new theories were combined and extended to form a theory of the wind-driven gyre in the continuously stratified oceans by Huang (1988a, b). The dynamical role of the mixed layer, excluded from the early theories, was incorporated into the thermocline theory (e.g., Huang, 1990a; Pedlosky and Robbins, 1991; Williams, 1989, 1991). The progress made during this period was reviewed by Huang (1991a) and Pedlosky (1996).

A major defect of thermocline theories is the lack of the western boundary layer and recirculation. It is clear that in order to explain the structure of the wind-driven circulation, numerical models must be used in which the dynamical effects of vertical/horizontal mixing, the western boundary layer, and the recirculation are explicitly included. From the physical point of view, mixing must play a critical role in setting up the global structure of the thermocline; thus, the ocean can be classified into regimes of different dynamics. In particular, as Welander (1971b) pointed out, "the thermocline may not be a diffusive boundary layer, but rather an ideal-fluid regime imbedded between diffusive regimes."

More recent studies based on numerical simulation provide a more dynamically complete picture. For example, Samelson and Vallis (1997) studied the thermocline structure in a closed basin, and showed that by using a small diapycnal mixing rate in the ocean interior, the thermocline does appear in two dynamical regimes, i.e., the ventilated thermo-cline for the water entering from the surface layer in the subtropical basin due to Ekman pumping, and the diffusive thermocline over the density range corresponding to the subpolar basin. Vallis (2000) went through a series of carefully designed numerical experiments based on a primitive equation model and showed that stratification below the ventilated thermocline is a result of global dynamics, involving the effect of wind forcing, geometry of the world's oceans, and diffusion. In particular, his study indicated that the geometry of the ACC (Antarctic Circumpolar Current) plays a subtle but important role in setting up the stratification at the mid depth of the world's oceans.

The modern theory of the ventilated thermocline is based on two cornerstones, i.e., the ventilation, as postulated by Iselin, and the "Stommel demon." These two concepts are discussed here before we introduce the formulation of the layered ventilated thermocline.

Iselin's conceptual model

A major conceptual difficulty in getting subsurface layers in motion is that they are not directly in contact with the atmospheric forcing. One way to get these subsurface layers in motion is the close geostrophic contours induced by strong forcing upon the surface layer. There is also another way, called ventilation, through which the subsurface layer can be put into motion. Many isopycnals outcrop in poleward parts of the oceans. When a layer outcrops, it is exposed to the atmospheric forcing directly. Thus, the outcropping layer is in motion under the wind stress forcing, and it should continue its motion even after it has been subducted under the other layers.

Iselin (1939) made a link between the T-S relation found in a vertical section and the wintertime mixed layer at higher latitudes. His schematic picture for this ventilation process is shown in Figure 4.26. The speculated motion is indicated by arrows. Iselin's model was the first prototype for water mass formation in the oceans; however, it is surprising that such a simple and important dynamical idea was not pursued further in the ensuing decades.

In modern terminology the basic idea is that, within the subtropical gyre, water is pushed downward into the thermocline by Ekman pumping and then downwells along isopycnals as it moves southward, induced by the Sverdrup dynamics. The motion of the particles after their rejection from the base of the mixed layer is confined within the corresponding isopycnal surfaces, because mixing is relatively weak below the mixed layer and above the rough bottom topography. The weakness of mixing in the upper ocean and below the mixed layer has been confirmed by observations, such as the recent tracer-release experiments. The process described by Iselin is now called "ventilation by Ekman pumping." Thus,

4.1 Simple layered models Region of convergence

4.1 Simple layered models Region of convergence

Fig. 4.26 Schematic representation of water mass formation due to water sinking along isopycnal surfaces (Iselin, 1939).

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