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FIGURE 10.18. A tank with a false sloping bottom is filled with water so that the water depth varies between about 5 cm at the shallow end and 15 cm at the deep end. The slope of the bottom represents the spherical ''beta-effect'': the shallow end of the tank corresponds to high latitudes. (The labels N/S correspond to Q > 0, appropriate to the northern hemisphere.) A disc is rotated clockwise very slowly at the surface of the water; a rate of 1 rpm works well. To minimize irregularities at the surface, the disc can be submerged so that its upper surface is a millimeter or so underneath the surface. The whole apparatus is then rotated anticlockwise on a turntable at a speed of Q = 10 rpm. It is left to settle down for 20 minutes or so. Holes bored in the rotating disc can be used to inject dye and visualize the circulation beneath, as was done in Fig. 10.19.

Plexiglas disc (to represent the action of the wind as in GFD Lab XII) on the surface of a anticlockwise-rotating square tank of water with a sloping bottom (to represent, as we shall see, spherical geometry). The stress applied by the rotating lid to the underlying water is analogous to the wind stress at the ocean surface. With clockwise differential rotation of the disc (Fig. 10.6, left), fluid is drawn inwards in the Ekman layer just under the lid and pumped downward in to the interior, mimicking the pumping down of water in subtropical gyres by the action of the winds, as sketched in Fig. 10.10. The varying depth of the tank mimics the variation in the depth of the spherical shell measured in the direction parallel to the rotation vector on the sphere (Fig. 10.15). The shallow end of the tank is thus analogous to the poleward flank of the ocean basin (the 'N' in Figs. 10.18 and 10.19) and the deep end to the tropics.

On introduction of dye (through bore holes in the Plexiglas disc) to help visualize the flow, we observe a clockwise (anti-cyclonic) gyre with interior flow moving toward the deep end of the tank (''equa-torward'') as charted in Fig. 10.19. This flow (except near the lid and the bottom) will be independent of depth, because the interior flow obeys the Taylor-Proudman theorem. Consistent with the discussion (see Section 10.2.2), a strong ''poleward'' return flow forms at the ''western'' boundary; this is the tank's equivalent of the Gulf Stream in the Atlantic or the Kuroshio in the Pacific.

In a manner directly analogous to that described in Section 10.2.3, we can relate the strength of the north-south flow to the Ekman pumping from under the disc and the slope of the bottom, as follows (compare with Eq. 10-14):

Dt Dt dy dy where d is the depth of the water in the tank, y represents the upslope (''poleward'') coordinate, and v is the velocity in that direc tion. In our experiment, wEk

FIGURE 10.19. A time sequence (every 7 min) showing the evolution of red dye injected through a hole in the rotating disc. The label 'N' marks the shallow end of the tank. The plume of dye drifts ''equatorward'' in the ''Sverdrup'' interior where Eq. 10-15 holds. In the bottom picture we see the dye being returned poleward in a western boundary current, the laboratory model's analogue of the Gulf Stream or the Kuroshio. Equatorward flow is broad and gentle, poleward flow much swifter and confined to a western boundary current.

FIGURE 10.19. A time sequence (every 7 min) showing the evolution of red dye injected through a hole in the rotating disc. The label 'N' marks the shallow end of the tank. The plume of dye drifts ''equatorward'' in the ''Sverdrup'' interior where Eq. 10-15 holds. In the bottom picture we see the dye being returned poleward in a western boundary current, the laboratory model's analogue of the Gulf Stream or the Kuroshio. Equatorward flow is broad and gentle, poleward flow much swifter and confined to a western boundary current.

and the bottom has a slope dd/dy _ -0.2. Thus v should reach speeds of 2.5 x 10-4 ms-1, or 15 cm in 10 minutes, directed equatorward (toward the deep end). This is broadly in accord with observed flow speeds.

The above relation cannot hold, however, over the whole domain, because it implies that the flow is southward everywhere, draining fluid from the northern end of the tank. As can be seen in Fig. 10.19, the water returns in a poleward flowing western boundary current, just as sketched in Fig. 10.12.

10.3. THE DEPTH-INTEGRATED CIRCULATION: SVERDRUP _THEORY_

One might wonder about the relevance of the previously discussed homogeneous model of ocean circulation to the real ocean. The ocean is not homogeneous and it circulates in basins with complicated geometry. It is far from the homogeneous spherical shell of fluid sketched in Fig. 10.15. And yet, as we will now see, the depth-integrated circulation of the ocean is indeed governed by essentially the same dynamics as a homogeneous fluid in a shallow spherical shell.

We begin by, just as in Section 10.2.1, eliminating (by cross-differentiation) the pressure gradient terms between the horizontal momentum balances, Eq. 10-3, to obtain, using continuity, Eq. 6-11:

rdw 1 d dTy dTx

dz pref dz \ dx dy which is Eq. 10-12 modified by wind stress curl terms. Now, integrating up from the bottom of the ocean at z _ -D (where we imagine w _ 0 and t _ 0) to the very top, where w is again zero but t _ twind, we obtain:

1 / dTwindy drwindx

Pref V dx dy

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