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where k is a drag coefficient representing the rubbing of the Taylor column over the bottom and 5 is the Ekman layer depth. Now, just as in the derivation of the T-P theorem in Section 7.2, we cross-differentiate the above to eliminate the pressure p, to obtain:

where for convenience we have introduced e = k/5. Now, since, by continuity, (du/dx + dv/dy) = -dw/dz, the previous equation can be written:

What is Eq. 11-10 saying physically? If the flow is frictionless (e = 0), the rhs is zero, and we recover a version of Eq. 7-15, the T-P theorem:

J dz

But the theorem is violated by the presence of friction, which allows columns to stretch or squash at a rate that depends on e and dv/dx - du/dy.

The quantity Z = (dv/dx - du/dy) is the vertical component of a vector called the vorticity, which measures the spin of a fluid parcel about a vertical axis relative to the rotating Earth (or table). Imagine that we place a miniature, weightless paddle wheel in to a flow which floats along with a fluid parcel and spins around on its vertical axis. It turns out that the vorticity Z is exactly twice the rate of rotation of the paddle wheel (see Problem 7, Chapter 6). For example, a paddle wheel placed in a swirling flow such as that sketched in Fig. 7.26 (left) will spin cyclonically, (dv/dx - du/dy) > 0. Thus Eq. 11-10 then implies dw/dz > 0: so, since w = 0 at the bottom, fluid will upwell away from the boundary and the column will stretch. Cross-isobaric flow at the bottom, where the column rubs over the base of the tank, leads to the requisite acquisition of mass. This is exactly the same process studied in GFD Lab X: fluid is driven into a low pressure system at its base where frictional affects are operative (see Fig. 7.23, top).

Now let us return to the problem of boundary currents. Consider the southward flowing western boundary current sketched in Fig. 11.20. A paddle wheel placed in it will turn cyclonically because flow on its inside flank is faster than on its outside flank. Hence (dv/dx - du/dy) > 0.5 Now using Eq. 11-10 we see that dw/dz > 0. But this is just what is required of a southward flowing boundary current (moving to the deeper end of the tank) because, from Eq. 11-10, it must stretch, dw/dz > 0. Thus the signs in Eq.11-10 are consistent. But what happens in the southward flowing eastern boundary current sketched on the right of Fig. 11.20? There, (dv/dx - du/dy) < 0 and so Eq. 11-10 tells us that dw/dz < 0. But this is the wrong sign if the column is to stretch. Thus we conclude that the southward flowing eastern boundary current cannot satisfy Eq. 11-10 and so is disallowed.

5This is obvious since, by definition, \v\ >> |u| and d/dx >> d/dy in a meridional boundary current.

FIGURE 11.20. Schematic diagram of boundary currents in the laboratory experiment shown in Fig. 11.18 and Fig. 11.22. The mass balance of Taylor columns moving in boundary currents on the western margin (whether flowing southwards as sketched here or northward) can be satisfied; the mass balance of Taylor columns moving in boundary currents on the eastern margin (whether running north or south) cannot.

FIGURE 11.20. Schematic diagram of boundary currents in the laboratory experiment shown in Fig. 11.18 and Fig. 11.22. The mass balance of Taylor columns moving in boundary currents on the western margin (whether flowing southwards as sketched here or northward) can be satisfied; the mass balance of Taylor columns moving in boundary currents on the eastern margin (whether running north or south) cannot.

FIGURE 11.21. Source-sink driven flow can be studied with the apparatus shown above. The pump and associated tubing can be seen, together with the intravenous device used to dye the fluid travelling toward the source. Fluid enters the tank through a submerged diffuser. The pumping rate is very slow, only about 20 cm3 min-1. An example of the subsequent evolution of the dyed fluid is shown in Fig. 11.22.

On consideration of the balance of terms in northward flowing boundary currents, we deduce that eastern boundary currents of both signs are prohibited, but western boundary currents of both signs are allowed.

11.3.4. GFD Lab XV: Source sink flow in a rotating basin

The preference for western as opposed to eastern boundary currents can be studied in our rotating tank by setting up a source sink flow using a pump, as sketched in Fig. 11.21. The pump gently draws fluid out of the tank (to create a sink) and pumps it back in through a diffuser (the source). On the way, the fluid is dyed (through an intravenous device) so that when it enters at the source its subsequent path can be followed. Experiments with different arrangements of source and sink are readily carried out. [See Problem 3 at the end of this Chapter.] One example is shown in Fig. 11.22. Fluid is sucked out toward the northern end of the eastern boundary (marked by the black circle) and pumped in on the southern end of the eastern boundary (marked by the white circle). Rather than flow due north along the eastern boundary, fluid tracks west, runs north along the western boundary and then turns eastwards at the "latitude" of the sink. And all because eastern boundary currents are disallowed!

FIGURE 11.21. Source-sink driven flow can be studied with the apparatus shown above. The pump and associated tubing can be seen, together with the intravenous device used to dye the fluid travelling toward the source. Fluid enters the tank through a submerged diffuser. The pumping rate is very slow, only about 20 cm3 min-1. An example of the subsequent evolution of the dyed fluid is shown in Fig. 11.22.

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