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b V2

Fig. 4.39 Structure of the western boundary currents: a meridional section of the inertial western boundary currents. Y represents the nondimensional meridional coordinates, thin curves are layer interfaces at the outer edge of the western boundary, heavy curves are interfaces at the western wall, and the dashed curves are the non-physical solutions; b boundary layer structure in the streamfunction coordinates (Huang, 1990c).

b V2

Fig. 4.39 Structure of the western boundary currents: a meridional section of the inertial western boundary currents. Y represents the nondimensional meridional coordinates, thin curves are layer interfaces at the outer edge of the western boundary, heavy curves are interfaces at the western wall, and the dashed curves are the non-physical solutions; b boundary layer structure in the streamfunction coordinates (Huang, 1990c).

Fig. 4.40 Dependence of the boundary layer structure on the lower layer thickness at the eastern wall. The solid line indicates the continuous solution, the long-dashed line represents the spurious solution, and the short-dashed lines indicate branches of the solutions (Huang, 1990c).

Fig. 4.40 Dependence of the boundary layer structure on the lower layer thickness at the eastern wall. The solid line indicates the continuous solution, the long-dashed line represents the spurious solution, and the short-dashed lines indicate branches of the solutions (Huang, 1990c).

In the classical theories of wind-driven circulation with a single moving layer, the western boundary layer has been assigned a passive role - closing the mass flux and dissipating the extra vorticity. The interior solution is uniquely determined by the wind-stress curl. For a stratified model, the solution also depends on the stratification and surface density distribution, as discussed in the previous sections.

It was surprising that Blandford (1965) failed to find continuous solutions for a two-moving-layer inertial western boundary layer model. Here we have shown that continuity of the western boundary layer imposes a constraint over the mid-ocean thermocline. Although there are many degrees of freedom for the interior thermocline, each time we add on a new piece of the circulation, the system loses some freedom; as more and more pieces are added on, there will be only few possible solutions remaining. These implicit constraints reflect the interaction between the interior flow and other parts of the circulation.

Remarks on closing the ventilated thermocline with western boundary currents Since the ventilated thermocline theory was first proposed, an inevitable question has been whether such beautiful solutions can be closed along the western boundary. It is, however, very clear that no ideal-fluid model can accomplish such a job because, to close the circulation, we need to include friction/dissipation so that the budget of potential vorticity and energy can be closed for any closed streamlines. Matching an inertial western boundary current with multiple moving layers represents an effort in pushing this issue to the theoretical limit. The circulation obtained from an ideal-fluid model without mixing/dissipation cannot be closed. It is more convenient to use a numerical model, in which the dynamical effect of small diapycnal mixing can be incorporated, to study the circulation in a closed basin. Samelson and Vallis (1997) gave a clear picture of the linkage between the nearly ideal-fluid thermocline in the ocean interior and other parts of the basin-scale circulation through the boundary layers. However, including mixing/dissipation will render the solution analytically rather difficult to solve, and the beauty of an analytical solution hard to achieve.

4.1.9 Thermocline theory applied to the world's oceans

Potential vorticity maps in the oceans

Talley (1985) applied the LPS model to the north subtropical Pacific and successfully explained the observed shallow salinity minimum. The circulation in the Pacific can be seen clearly through maps of potential vorticity on different isopycnal surfaces (Fig. 4.41). Among many other features, potential vorticity p-1f Ap/Az in the upper layers is non-homogenized, which is apparently due to the strong potential vorticity source/sink in these shallow layers; however, potential plateaus can be clearly identified for deeper isopycnal surfaces og = 26.0 kg/m3 and 26.2 kg/m3. These potential vorticity plateaus appear to be not inconsistent with the potential vorticity homogenization theory. However, the existence of such potential vorticity plateaus in the world's oceans may be due to quite different dynamical processes.

Potential vorticity homogenization diagnosed from numerical models Primitive equation numerical experiments have been carried out to simulate the circulation dynamics predicted by the ventilated and unventilated thermocline theories. Potential vorticity maps generated from the Geophysical Fluid Dynamics Laboratory's primitive equation model are shown in Figure 4.42. The dashed lines are the Bernoulli contours that can be used as the streamlines for the respective isopycnal surfaces, assuming the dissipation is relatively weak. Within the major part of the subtropical gyre interior, potential vorticity is nearly homogenized.

From Figures 4.41 and 4.42, one can identify all the dynamical regions, such as the pool, the ventilated region, and the shadow zone, proposed by Rhines and Young (1982b) and Luyten et al. (1983). On the other hand, we will notice that there are some important discrepancies between the theories and numerical experiments or observations. A common shortcoming of these two theories is that the dynamical effects of strong eddy mixing are ignored for the sake of analytical simplicity. For example, as seen from Figure 4.42, potential vorticity is advected downstream in the form of low/high potential vorticity tongues, consistent with the potential vorticity conservation used in the LPS model. However, potential vorticity is not exactly conserved, as assumed in the ventilated thermocline theory by LPS. In fact, low/high potential vorticity tongues gradually lose their identity owing to cross-stream eddy mixing, as seen from Figures 4.41 and 4.42. There are pools of almost uniform potential vorticity in both Figures 4.41 and 4.42; however, the reason for

Fig. 4.41 Maps of potential vorticity on selected isopycnals for the North Pacific (in 10-10/m/s). Shaded regions are seasonal; their southern edge is the winter sea-surface outcrop (Talley, 1988).

potential vorticity uniformity seems quite different from the original theory by Rhines and Young (1982b).

First, smearing vorticity gradients on the isopycnal surface at the base of the thermocline is relatively fast compared with the circulation time scale. In fact, potential vorticity gradients cause baroclinic instability that tends to homogenize potential vorticity on isopycnal surfaces within just part of the trajectory around a subtropical gyre.

The potential vorticity homogenization process was simulated by an eddy-resolving primitive equation model with idealized topography and forcing (Cox, 1985) (Fig. 4.43). The results from the non-eddy-resolving and eddy-resolving models gave rise to the same basic

Fig. 4.42 Potential vorticity (—f az) at z = -95 m (a), and on the 26.7, 27.0, and 27.3 a surfaces (b, c, d) in 10-10/m/s (solid lines). The Bernoulli function with contour interval 4 cm (equivalent vertical displacement) is shown as short dashed lines with arrows in b, c, d. The intersection of the sigma surfaces with z = -95 m is shown with long dashed lines (Cox and Bryan, 1984).

Fig. 4.42 Potential vorticity (—f az) at z = -95 m (a), and on the 26.7, 27.0, and 27.3 a surfaces (b, c, d) in 10-10/m/s (solid lines). The Bernoulli function with contour interval 4 cm (equivalent vertical displacement) is shown as short dashed lines with arrows in b, c, d. The intersection of the sigma surfaces with z = -95 m is shown with long dashed lines (Cox and Bryan, 1984).

flow pattern. (It would be more accurate to call such models "non-eddy-permitted" and "eddy-permitted", because a model with 1 degree resolution cannot really resolve eddies of higher baroclinic modes in the ocean.) However, the mixing process is quite different in these two cases. Mixing by eddies in the westward-flowing sector of the subtropical gyre is quite effective in homogenizing the potential vorticity. Whereas previous theory predicted homogenization of potential vorticity on long time scales only within a recirculating gyre, in an eddy-resolving model homogenization takes place on a much shorter time scale across recirculating/ventilating flow boundaries. In fact, anomalous potential vorticity that is advected into the thermocline from isopycnal outcrops by ventilated flow causes changes in the sign of the meridional gradient of local potential vorticity, which in turn gives rise to baroclinic instability.

Fig. 4.43 Potential vorticity distribution on the a = 26.0 surface obtained from a numerical model, in 10-10/m/s for a the case with 1° resolution, and b the case with 1 /3° resolution (Cox, 1985).

Fig. 4.43 Potential vorticity distribution on the a = 26.0 surface obtained from a numerical model, in 10-10/m/s for a the case with 1° resolution, and b the case with 1 /3° resolution (Cox, 1985).

Second, atmospheric cooling plays a crucial role in creating mode water that has nearly homogenized potential vorticity and other properties (McCartney, 1982). As shown much earlier in Figure 1.7, there is a lot of heat lost to the atmosphere near the recirculation region in both the North Atlantic and North Pacific. This strong cooling is closely associated with mode water formation, which takes place primarily south of the Gulf Stream system. A key index for mode water formation is the local maximum of mixed layer depth in late winter.

As discussed in detail in Section 5.1.5 about mode water formation, late-winter cooling leads to a deep mixed layer and a thick column of water with vertically almost homogenized properties. Springtime warming in the upper ocean induces a rapid shoaling of the mixed layer, and the newly formed mode water is sealed off. As a result, a large amount of mode water with very low potential vorticity is formed within the recirculation regime south of the Gulf Stream and the Kuroshio. As an example, mode water formed in the upper ocean can be seen clearly through a map of potential vorticity along the 65° W section in the North Atlantic Ocean (Fig. 4.44).

The same idea can be demonstrated by numerical models (Huang and Bryan, 1987). Their numerical experiments showed that at the outer edge of the western boundary outflow, cooling induces a strong mass flux from the warm and light upper layers to the cold and dense lower layers. In a four-layer model, there is a strong mass flux from the second layer to the third layer at the latitude corresponding to where the Gulf Stream leaves the western boundary (Fig. 4.45).

When water leaves the western boundary, it carries high potential vorticity; however, this high-potential vorticity tongue is rapidly transformed into quasi-uniform, low-potential vorticity water through convective overturning (Fig. 4.46). Thus, the thermodynamic processes in the mixed layer and the upper ocean could be the primary machinery generating these huge masses of quasi-homogenized potential vorticity.

Fig. 4.44 Meridional section of potential vorticity (in 10-11 /m/s) along the 65° WIGY (International Geophysical Year) section. The broad shallow minimum (q < 10) is the 18°C water formed by convective cooling at the surface and subsequently capped off by a seasonal pycnocline. The q maximum at 800 m is the permanent pycnocline (McDowell et al., 1982).

Fig. 4.44 Meridional section of potential vorticity (in 10-11 /m/s) along the 65° WIGY (International Geophysical Year) section. The broad shallow minimum (q < 10) is the 18°C water formed by convective cooling at the surface and subsequently capped off by a seasonal pycnocline. The q maximum at 800 m is the permanent pycnocline (McDowell et al., 1982).

4.2 Thermocline models with continuous stratification 4.2.1 Diffusive versus ideal-fluid thermocline

Diffusive thermocline

Even though multi-layer models of the wind-driven circulation were not very successful, people have tried to attack the continuously stratified model. They simply could not resist the temptation of trying to solve such a simple-looking equation set:

This equation set looks really simple because all the equations are linear except the density equation (4.255). This simple-looking equation, however, turns out to be very difficult to solve. Since these equations are very complicated nonlinear partial differential equations, it was not clear how to formulate suitable boundary value problems and how to solve them. For a long time, the only way that people could solve these equations was by using similarity solutions.

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