## Effects Of Stratification And Topography

Our analysis of the wind-driven circulation in Section 10.2 assumed the ocean to have constant density, whereas (see, e.g., Fig. 9.7) the density of the ocean varies horizontally and with depth. In fact, the variation of density with depth helps us out of a conceptual difficulty with our physical interpretation in terms of the Taylor-Proudman theorem on the sphere presented in Section 10.2.3. We described how the Taylor columns of fluid must lengthen in the subtropical gyres to accommodate Ekman pumping from above; where there is no Ekman pumping, they must maintain constant length. But the bottom of the ocean is far from flat. In the Atlantic Ocean, for example, there is a mid-ocean ridge that runs almost the whole length of the ocean rising about 2 km above the ocean bottom (see Fig. 9.1). If the ocean were really homogeneous, water columns simply could not cross this ridge: there would be an enormous, elongated, stagnant Taylor column above it. And yet we have seen that homogeneous theory accounts qualitatively for the observed circulation, including the observation that, for example, the Atlantic subtropical gyre does indeed involve water flowing over the ridge. How does this happen?

8The discrepancy is due to the fact that the transport of the Gulf Stream can considerably exceed the prediction based on Sverdrup theory, because a portion of the fluid that flows in it recirculates in closed loops that do not extend far into the interior of the ocean.

In most regions the mean circulation, in fact, does not extend all the way to the bottom because, as discussed in Section 9.3.2, the interior stratification of the ocean largely cancels out surface pressure gradients. The thermal wind relation tells us that there can be no vertical shear in the flow of a homogeneous fluid since there are no horizontal density gradients. In the presence of density gradients the constraint of vertical coherence is weakened. Consider Fig. 10.22. We suppose for simplicity that the ocean has two layers of different density p1, p2 (with p1 > p2, of course). The density difference produces a stable interface (somewhat like an atmospheric inversion described in Section 4.4) which effectively decouples the two layers. Thus Taylor columns in the upper layer, driven by Ekman pumping/suction from the surface, "feel" the interface rather than the ocean bottom. As long as the interface is above the topography, they will be uninfluenced by its presence. Thus the density stratification "buffers" the flow from control by bottom topography. If we look at the density stratification in the real ocean (Fig. 9.7), we see that most of the density stratification is found in the main thermocline, within a few hundred meters of the surface. Thus the mean wind-driven circulation is largely confined to these upper layers.

### 10.4.1. Taylor-Proudman in a layered ocean

If we suppose that the ocean is made up of many layers of fluid with slightly differing densities, Ap = pi — p2, and so on then we can imagine miniature Taylor columns within each homogeneous layer, and because each layer is homogeneous, T-P applies. Let us suppose that each Taylor column has a length d, measured parallel to the axis of rotation within each layer, as sketched in Fig. 10.23. An interior column will "try" to maintain its length. Thus d is constant, as will be the quantity

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