with the following constraints

B = Ba, Bp = Bap, at p = pb ( pb is unknown) (4.306)

' 2 rpb a2 2 g p f2 rxe B2dp + / Ba dp = wedx (4.307)

Although this approach seems to be a simple extension of the early work by Welander (1971a), there are subtle differences between the new formulation and Welander's old formulation. In Welander's formulation, it was assumed that potential vorticity Q(B, p) in Eqn. (4.304) is a given function of B and p, and this equation is subject to boundary conditions specified at the upper and lower boundaries. In particular, Welander did not consider the potential role of the mixed layer, i.e., he implicitly assumed h(x, y) = 0. In addition, he assumed that the lower boundary is fixed at p — p—ix>. Since the ordinary differential equation (4.304) with a given function Q(B, p) can satisfy only two boundary conditions, it was not clear how to find a solution satisfying additional boundary conditions, such as the Sverdrup constraint, Eqn. (4.307), which are essential for describing the wind-driven circulation in the ocean.

As discussed above, the major problem with Welander's original formulation is the assumption that potential vorticity is a given function for the entire thermocline. It took a long time and much effort before it was realized that the thermocline consists of many regions which are regulated by different dynamics, such as the ventilated zone, the unventi-lated thermocline, the shadow zone, and the pool zone. It was shown that potential vorticity in the unventilated thermocline is fairly well homogenized (Rhines and Young, 1982a; McDowell et al., 1982). Owing to strong wind forcing, potential vorticity in the ventilated zone is not homogenized in general. Thus, any a priori assumption about the form of the potential vorticity function in the ventilated zone is artificial, and the most important progress made in the early 1980s was the realization of such an essential limit in previous approaches and the creation of a new approach which allows us to calculate potential vorticity in the ventilated zone as a part of the solution, as demonstrated by Luyten et al. (1983).

Another major difference in the new approaches is that the base of moving water is no longer a constant-density surface; instead, the boundary between the moving part of the wind-driven gyre and the stagnant water below is a free boundary which is calculated as part of the solution. In the model of continuous stratification, the shadow zone next to the eastern boundary in the multi-layer ventilated thermocline discussed in the original LPS model is now replaced by the regime of stagnant water with continuous stratification. Thus, the technical difficulty in dealing with the exponentially increasing number of different shadow zones can easily be avoided in the model with continuous stratification.

In light of these new discoveries, the early model of Welander may be classified as some kind of similarity solution. The continuously stratified model has incorporated these new features. As one of the major differences from Welander's model, Q in Eqn. (4.304) is a given function only for the unventilated thermocline, and is unknown for the ventilated thermocline. The special nature of the boundary value, including the facts that the lower boundary is a free boundary and that function Q(B, p) is not completely specified, gives rise to a unique problem in which a second-order ordinary differential equation is subject to four constraints.

This free boundary value problem is solved with a shooting method by starting from a first guess of the bottom of the moving water ps. Integrating upward (toward lower density) to the base of the mixed layer, we can determine the potential vorticity of the uppermost ventilated layer as q = f Ap/Ah, where Ap is the density increment, and Ah is the thickness of the uppermost layer. The generalized Sverdrup relation is then checked. If it is not satisfied, the base of the moving water, pb, is adjusted until the integral constraint is met.

The overall structure of a continuously stratified model is shown in Figure 4.49. For the subtropical wind-driven gyre, the model integration starts from the inter-gyre (between the subpolar and subtropical gyres) boundary in the north. In the vertical direction there are four dynamical regimes. On the top there is the mixed layer where density is vertically constant. Both the mixed-layer density and depth are specified from late-winter properties. Below the mixed layer, there is the ventilated thermocline where potential vorticity is unknown and is calculated as part of the solution. Each ventilated layer is subducted at the next outcrop line. South of this new outcrop line, the newly subducted layer continues its southward motion underneath, keeping the potential vorticity formed during the subduction process. Below

Intergyre boundary

Intergyre boundary

the ventilated thermocline there is the unventilated thermocline where potential vorticity is specified. Although any reasonably chosen forms of potential vorticity can be used, it is more convenient to assume that potential vorticity in the unventilated thermocline is homogenized toward the planetary vorticity along the northern boundary of the model. The lowest part of the model is the stagnant water in the abyss, where stratification is specified. Since water there does not move, a constant stratification in each layer implies that potential vorticity there is a function of latitude, although the concept of potential vorticity for stagnant water does not have much dynamical meaning.

This model was applied to both the North Atlantic and the North Pacific, with both h and ps specified functions of geographical location, taken from the climatological mean density and depth datasets. The model ocean is divided into m x n grids, and the calculation of the three-dimensional structure of the wind-driven subtropical gyre is reduced to repeatedly solving this second-order ordinary differential equation at each station along the individual outcropping line. Typical isopycnal outcropping lines in late winter for the North Pacific are shown in Figure 4.50.

Along the northernmost outcrop line, starting from the first station next to the eastern boundary (or the northern boundary), we solve the free boundary value problem at each station. We assume that potential vorticity is a given function of density for the unventilated thermocline; thus, the solution at each station gives us the base of moving water at this station, and the Bernoulli function at the sea surface, Bs. After the completion of calculation along this outcrop line, we have a functional relation between potential vorticity and the Bernoulli function for this density p i, and this function is stored in the form of a data array

in the computer. For outcrop lines southward, we can use this functional relation from the computer's data storage to solve the free boundary value problem along the next outcrop line with density lighter than p1, and this process continues until we reach the southern boundary of the model basin.

This process produces the horizontal distribution of Bernoulli function on each isopycnal surface, and the application of the geostrophic condition gives rise to the horizontal velocity on each isopycnal surface. For example, streamlines on four isopycnal surfaces in the subtropical gyre of the North Pacific are shown in Figure 4.51. The shaded areas next to the eastern boundary depict the stagnant water on each isopycnal surface, which corresponds to the shadow zone in the multi-layered ventilated thermocline model. According to the model results, most of the wind-driven circulation in the North Pacific is ventilated.

The most prominent feature of the model is the strong ventilation due to the inclusion of a mixed layer of finite, horizontally varying depth. The southern shoaling of the late-winter mixed-layer depth gives rise to a strong lateral induction and subduction rate (Fig. 4.52c, d).

Three factors contribute to the volume flux in the ventilated thermocline: the vertical pumping from the Ekman layer convergence (Fig. 4.52a, b), the lateral induction (Fig. 4.52c), and the inter-gyre boundary outflow due to the northeast-southwest orientation of the zero-Ekman-pumping line (Fig. 4.53). In fact, these three are equal contributors. The southern shoaling of the mixed layer and the induced lateral induction from the mixed layer into the main thermocline have a very important impact on the wind-driven circulation

Fig. 4.51 Circulation on four different isopycnal surfaces of the wind-driven subtropical gyre in the North Pacific; solid lines with arrows for the layer-integrated streamlines and numbers for the layer-integrated volume flux (Sv), dotted lines for the water age in years since subduction; the shadow areas (shaded) for the stagnant water on each isopycnal surface (Huang and Russell, 1994).

Fig. 4.51 Circulation on four different isopycnal surfaces of the wind-driven subtropical gyre in the North Pacific; solid lines with arrows for the layer-integrated streamlines and numbers for the layer-integrated volume flux (Sv), dotted lines for the water age in years since subduction; the shadow areas (shaded) for the stagnant water on each isopycnal surface (Huang and Russell, 1994).

and climate; these issues will be discussed again in Section 5.1.5 specially dedicated to water mass formation through subduction. These fluxes can be identified from either a model of continuous stratification (Fig. 4.53a) or a diagnostic calculation from historical hydrographic data (Fig. 4.53b). The numbers on the left edge of each panel indicate the mass exchange with the western boundary or the inter-gyre boundary. Since the inter-gyre boundary has a large meridional distance, most of the influx from the western boundary actually comes from the inter-gyre boundary. As shown in Figure 4.53, inter-gyre mass flux is thus a major contributor to the flux on each isopycnal layer.

4.3 Structure of circulation in a subpolar gyre 4.3.1 Introduction

Wind-driven circulation is a dominant component of the current system in the upper ocean. The simplest model is the widely used reduced-gravity model. In such a model, the specification of boundary conditions is quite similar for both the subtropical and subpolar

120°E 140°E 160°E 180° 160°W 140°W 120°W 100°W 120°E 140°E 160°E 180° 160°W 140°W 120°W 100°W

C Lateral induction, in m/y d Subduction rate, in m/y

Fig. 4.52 Vertical pumping and subduction rate for the North Pacific, based on the ideal-fluid thermocline model (Huang and Russell, 1994).

120°E 140°E 160°E 180° 160°W 140°W 120°W 100°W 120°E 140°E 160°E 180° 160°W 140°W 120°W 100°W

C Lateral induction, in m/y d Subduction rate, in m/y

Fig. 4.52 Vertical pumping and subduction rate for the North Pacific, based on the ideal-fluid thermocline model (Huang and Russell, 1994).

gyres; therefore, the discussion in Section 4.1.2 about the reduced-gravity model with a single moving layer for the subtropical gyre also applies to the description of the subpolar gyre, with the only difference being that the Ekman pumping in the subpolar basin is positive. As a result, the geostrophic flow in the subpolar gyre interior flows poleward under the Ekman suction owing to the positive wind-stress curl, and isopycnal surfaces are dome-shaped. The corresponding western boundary current in the subpolar basin moves equatorward. The corresponding theories of the Stommel layer, the Munk layer, and the inertial western boundary layer all work in a way similar to those in the subtropical basin.

However, the wind-driven circulation in a subpolar basin is different from that in a subtropical basin in the following ways. First, Ekman pumping is upward. Although this does not seem to matter much for a model with a single moving layer, it changes the dynamics and formulation of models with multiple layers or continuous stratification. The ideal-fluid thermocline model belongs to the so-called hyperbolic system in mathematics. For a

Seasonal thermocline

Seasonal thermocline

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