Thus, Eqn. (4.263) can be rewritten as u ■ V(f pz) = 0 (4.265)

This is the potential vorticity conservation law, following a streamline.

Multiplying the momentum equations by v and u, and adding together, we obtain the mechanical energy balance, noting that the Coriolis force does no work,

0 = upx + vpy = u ■ Vp - wpz = u ■ Vp + wpg (4.266)

Since w = u ■ Vz, we obtain the Bernoulli conservation law u- V(p + pgz) = 0 (4.267)

Thus, density, potential vorticity, and the Bernoulli function are all conserved along streamlines. In the oceans these quantities are not exactly conserved; however, deviations from the conservation laws are small enough, so the theory of ideal-fluid thermocline has been rather successful.

Simple solutions

Reducing to a single ordinary differential equation

Welander (1971a) made another very important contribution to the ideal-fluid thermocline theory by applying the vorticity conservation and Bernoulli conservation laws to the thermocline problem. His original formulation was based on the z-coordinate. It is more convenient to formulate the problem in density coordinates and to use the Bernoulli function

as the dependent variable in the density coordinates. The first and second derivatives of the Bernoulli function with respect to the density are

is the potential vorticity. As stated above, both the Bernoulli function B and the potential vorticity q are conserved along streamlines.

Welander made a vital step in observing that if the functional form of q(B, p) is given, the equation can be solved either analytically or numerically. In particular, he discussed several cases.

If potential vorticity is a function of the density alone The equation can be solved easily (Welander, 1959). In fact, a double integration leads to an analytical solution

Assume a linear function q(B, p) = f pz = ap + bB + c This equation can be easily integrated. First, differentiating this linear function with respect to z and applying the hydrostatic relation leads to f pzz = (a + bgz)pz (4.273) Integrating Eqn. (4.273) twice leads to az+0.5bgz2

Welander added the following boundary conditions: density at the sea surface should match the observed annual-mean surface density ps(X, 0 ), and at great depth it should approach a uniform value p—œ.

Welander also discussed the general case where q is a function that depends on a linear combination of B and p, say q(B, p) = F(ap + bB + c). Although Welander laid the foundation for solving the thermocline equation, some very challenging difficulties remained to be overcome.

The most difficult issue had again been the conceptual difficulty associated with the boundary conditions. Welander's formulation reduces the thermocline problem to a second-order ordinary differential equation in density coordinates. A second-order ordinary differential equation can normally satisfy two boundary conditions only; however, a solution for the thermocline structure in a basin has to satisfy many boundary conditions. For example, the Ekman pumping condition requires the solution to fit a two-dimensional array, which seemed to be almost impossible within the original approach proposed by Welander. It is essential to satisfy the Ekman pumping condition, and the model can be modified by including such a constraint. In addition, specifying potential vorticity for the whole water column turns out to be unnecessary. As will be discussed in the following sections, the new theory shows that the problem is overdetermined, so it cannot satisfy other dynamically important boundary conditions.

Today, as we review the advances in the past decade, we realize that progress in solving the thermocline equation has been made step by step through the deepening of our understanding of the thermocline structure. The two contributions of Welander came primarily from some physical insights: ignoring diffusion and introducing conservation of the Bernoulli function and potential vorticity. The improvement of this solution requires additional physical insights. It was only after the innovative study of potential vorticity homogenization by Rhines and Young and the ventilated thermocline by Luyten, Pedlosky, and Stommel that we arrived at a better understanding of the physics of the thermocline. As we discuss later, similar to the situation for the multi-layered model of the thermocline, potential vorticity has to be calculated as part of the solution; one can only specify the potential vorticity for the unventilated thermocline. Therefore, Welander's suggestion of specifying the form of the potential vorticity function for all moving water requires too much information, and actually gives rise to an overdetermined system. As a result, the solution cannot satisfy more than two boundary conditions.

An analytical solution for the ideal-fluid thermocline

An analytical solution for the ideal-fluid thermocline can be found by using assumptions slightly different from Welander's. Instead of assuming a linear function of the potential vorticity, we assume that:

a) In the ventilated thermocline the potential thickness (the inverse of the potential vorticity) is a linear function of the Bernoulli function, i.e., D = —Zp /f = a2B, where a is a given parameter;

Fig. 4.47 Structure of an analytical solution for the ideal-fluid thermocline. Solid lines are ventilated thermocline, dashed lines are unventilated thermocline, and the thick lines are the velocity contours in 10-2/m/s.

Thus, the basic equation for the ventilated thermocline is

b) In the unventilated thermocline, potential thickness is constant.

The solution of Eqn. (4.276) is in the form of B = a cos b(p - ps), where b = ^Jfga, and a = a(x, y) can be determined by the boundary conditions, including the Sverdrup constraint. After simple manipulations, the Sverdrup constraint is reduced to a single transcendental equation (Huang, 2001). An example for a model basin covering (0°-60° E, 15°-45° N), mimicking the North Atlantic, is shown in Figure 4.47. This solution includes the ventilated thermocline, the unventilated thermocline, and the shadow zone (as indicated by the flat isopycnals near the southern and eastern boundaries). Most importantly, the solution satisfies the Sverdrup constraint, so it provides a rather complete dynamical picture of the thermocline and the associated three-dimensional wind-driven gyre.

It is clear that the solution resembles the subtropical gyres observed in the oceans. In some previous studies it was speculated that the main thermocline may appear in the form of a density discontinuity in a truly continuous model. Such a density front is, however, not necessary. In fact, the simple solution shown in Figure 4.47 has a truly continuous structure in three-dimensional space, including weak discontinuities in potential vorticity. In some sense, this solution is also a similarity solution because the potential vorticity function is specified a priori. However, this solution is quite different from previously studied similarity solutions in two aspects. First, the solution satisfies all essential boundary conditions, and in particular the Sverdrup relation. Second, the solution includes weak discontinuity in potential vorticity, but is continuous in density structure.

This model belongs to the category of solutions in which potential vorticity in the ventilated thermocline is set as a constant a priori, and the surface density distribution that matches the solution is found as part of the solution. In the oceans, potential vorticity of the ventilated thermocline is set by the basin-wide circulation, including the upper boundary conditions, such as the Ekman pumping rate, the surface density, and mixed layer depth. The mixed layer depth is assumed to be zero in the case presented here; however, it is easy to modify the formulation to include a mixed layer with a non-zero depth.

4.2.2 Models with continuous stratification

How to improve the ventilated thermocline models The ventilated thermocline theory by Luyten et al. (1983) laid the foundation for the ideal-fluid thermocline. In principle, their theory can be extended into a model with many more moving layers. Thus, such a model may be able to provide useful information about the thermocline structure in the ocean with continuous stratification.

However, such a multiple-layer model may encounter some difficulties when the number of layers increases. For example, from the work by Luyten et al. (1983), it is clear that the number of shadow zones with different dynamics increases exponentially as the number of layers is increased. In addition, when the number of layers is very large, it is a rather tedious job to derive and calculate the analytical solution. Thus, it is desirable to formulate the continuous model in the spirit of the ventilated thermocline.

The singularity in the models

One of the most critical problems in the early ventilated thermocline models is due to the lack of a mixed layer. In fact, the upper surface of the models is set to z = 0. This choice is apparently made to simplify the models. There are several problems associated with such an upper boundary condition.

First, the mixed layer is the major buffer between the atmosphere and the permanent thermocline. Mixed-layer depth reaches the annual maximum in late winter, to the order of 200-400 m. Volume flux in the mixed layer constitutes a substantial part of the total volume flux in the wind-driven circulation. Since density is almost vertically uniform in the mixed layer, its dynamics is quite different from that in the ocean interior. Including the mixed layer is a vital step toward a more realistic wind-driven circulation.

Second, this upper boundary forces all isopycnals to outcrop at the same depth, z = 0. As a result, all these models have singularity along the eastern, northern, and southern boundaries.

Third, this boundary condition excludes the contribution to the subduction rate due to the mixed-layer depth gradient. It will be shown in Section 5.1.5 that including a mixed layer with horizontally varying depth can give rise to a subduction rate that is substantially larger than the rate of Ekman pumping. Thus, including the mixed layer is an essential step in bringing the thermocline models closer to reality.

As will be shown in this section, including the mixed layer is actually not very difficult at all, and it has substantially improved the thermocline models. For example, coupling the mixed layer helps to overcome one of the major problems in thermocline models, i.e., the singularity along the eastern, northern, and southern boundaries.

The eastern boundary condition

The eastern boundary has not received enough attention. In earlier theoretical models of the wind-driven circulation with one moving layer, the eastern boundary is simply a place to start the integration. The importance of suitable eastern boundary conditions for a stratified model was first encountered in the Luyten-Pedlosky-Stommel model. A major feature of this model is that all ventilated layers have zero thickness along the eastern boundary. This is apparently inconsistent with observations that all layers have a finite thickness along the eastern boundary. Furthermore, since the calculations have to be started from the eastern boundary, it seems clear that if ventilated layers have non-zero thickness along the eastern boundary, the entire solution might change.

The special nature of the eastern boundary condition can be appreciated through an interesting argument by Killworth (1983b). Assuming that the flow can be described by the ideal-fluid thermocline equation, then the suitable kinematical condition is: at the eastern boundary u = 0, at x = 0 (4.277)

so that uz = 0. The geostrophy equations (4.257) and (4.258) imply

Along the eastern boundary, the density conservation equation (4.261) is reduced to wpz = 0, at x = 0 (4.279)

Now, if pz = 0 at the wall, i.e., the fluid is stratified, then w = 0, at x = 0 (4.280)

From the Sverdrup relation ¡3v = f wz, we obtain v = 0, at x = 0 (4.281)

By the thermal wind relation, this leads to px =, at x = 0

Differentiating the density conservation equation and the Sverdrup relation repeatedly, we can show that for variables u, v, w, px, py, pz, the first, second, and all higher-order x-derivatives are zero at the eastern boundary.

If the thermocline solution can be expanded in Taylor series from the eastern boundary, then the solution should be zero everywhere in the basin. Although water in the vicinity of the eastern boundary is stagnant and thus has homogeneous properties, water in the basin interior can have quite different properties. As we discussed in the previous section about the ventilated thermocline, water subducted from the basin interior can have dynamical properties completely different from the stagnant water adjacent to the eastern boundary. In fact, the system is of a hyperbolic nature; thus, along the interfaces which separate water with different dynamical properties, these quantities may be non-differentiable, and the expansion in Taylor series is not valid.

To explore suitable eastern boundary conditions for stratified models, Pedlosky (1983) studied a model of two moving layers. A new eastern boundary condition was used, which required only the vertically integrated zonal volume flux to be zero and allowed ventilated layers to have non-zero thickness along the eastern wall. Since the model has only two moving layers, the stratification along the eastern wall could be specified ad hoc and the solution in the interior can be calculated accordingly. This eastern boundary condition gives rise to the eastern boundary ventilated thermocline and alters the global structure of the gyre circulation.

The generalized eastern boundary conditions have been extended to the case of a continuously stratified model by Huang (1989a), where it is shown that the stratification along the eastern boundary can no longer be specified ad hoc; instead, it should be calculated as a part of the unified gyre-scale circulation. Thus, the eastern boundary conditions are closely tied to the gyre-scale circulation and cannot be specified arbitrarily.

There are some difficulties involved in using the generalized eastern boundary conditions. First, the model requires an unknown eastern boundary layer that can transfer water vertically. Second, the stratification along the eastern boundary implies some extra freedom of the system, so the system becomes highly underdetermined. The question is how to find a physically meaningful solution.

In searching for an answer to the general questions concerning the suitable boundary conditions at the eastern wall for continuously stratified models, Young and Ierley (1986) studied a thermohaline circulation model with vertical diffusion. They used a family of similarity solutions to explore the physical meaning of suitable eastern boundary conditions for the ideal-fluid thermocline. By examining the solutions obtained when the vertical diffusiv-ity approached zero, they came to the conclusion that the ideal-fluid thermocline equation has weak solutions, i.e., solutions that have a density discontinuity, which they interpreted as a thermocline. However, Huang (1988a, 2001) showed that truly continuously stratified solutions of the equations do exist, although potential vorticity would be discontinuous across the base of the moving water and there would be some singularity along the boundaries of a basin. Thus, it is possible to construct a solution that has a smooth density field in the interior ocean, although the eastern boundary always involves some kind of singularity.

In a model including a mixed layer of horizontally varying density and depth, Huang (1990a) returned to the old eastern boundary condition of zero zonal velocity below the base of mixed layer. An implicit assumption of the model is that the onshore geostrophic flow in the seasonal thermocline is exactly balanced by the offshore Ekman flux due to the southward along-shore wind stress. This new formulation successfully overcomes the artificial singularities existing in many previous theoretical models. Due to the finite depth of the mixed layer, meridional velocity is finite everywhere. The application of this eastern boundary condition eliminates the potential vorticity singularity along the eastern, northern, and southern boundaries in previous models, and gives rise to shadow zones in the ventilated thermocline.

It is fair to say that the problem with the eastern boundary condition has not been completely solved. Since the local offshore Ekman flux may not always exactly balance the onshore flux in the seasonal thermocline, there is again some singularity involved that requires further study. In fact, there is a gap between coastal oceanography and basin-scale oceanography. In coastal oceanography, stratification in the interior ocean is assumed given, and for most cases this stratification is taken to be independent of latitude; while for basin-scale oceanography, the stratification along the eastern boundary is assumed given, presumably determined by some coastal circulation processes. These two parts should be related through the general circulation in a closed basin.

Coupling with a mixed layer of variable depth In the early stages of development, the mixed layer was neglected in most ideal-fluid thermocline models for simplicity, and the upper boundary conditions of these models were that of specifying we and ps at z = 0. Neglecting the mixed layer leads to many problems in the models. A close examination reveals that a model with a mixed layer is a fairly easy extension of the previous models whose upper surface was artificially put at the sea surface.

The mixed layer

Our concern is primarily the dynamics of the main thermocline below the mixed layer. In addition, it is very difficult to formulate a simple analytical model to incorporate the mixed-layer thermodynamics with the dynamics in the main thermocline. A major challenge involved in such a goal is the handling of the seasonal cycle in the mixed layer, with the Rossby waves interacting with each other and the mean currents. Thus, we will exclude the thermodynamics of the mixed layer; instead, we will prescribe the thermodynamic parameters of the mixed layer; the velocity in the mixed layer is, however, part of the solution. In addition, late-winter mixed-layer properties are chosen as the forcing functions, according to Stommel's (1979) suggestion.

In our idealization, the Ekman layer is treated as an infinitely thin layer on the surface of the ocean, where water is collected horizontally by Ekman drift and from which it descends owing to Ekman pumping. Since density is assumed to be vertically homogenized within the whole depth of the mixed layer, the Bernoulli function

is vertically constant within the mixed layer. At the sea surface, the Bernoulli function is the same as the pressure, so the horizontal pressure gradient in the mixed layer is

where the superscript s indicates the sea surface. The first term on the right-hand side is a barotropic term; the second term is a baroclinic term due to the horizontal density gradient in the mixed layer.

The horizontal pressure gradient induces a geostrophic flow in the mixed layer ug = -^L, wg = PJL (4.285)

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