## S

FIGURE 11.17. Apparatus used to illustrate the driving of deep ocean circulation by localized sinking of fluid. A sloping base is used to represent the influence of sphericity on Taylor columns as in GFD XIII. The 50 cm square tank is filled with water and set rotating anticlockwise at a rate of Q = 5 rpm. (The sense of rotation is thus representative of the northern hemisphere). Dyed water, supplied via a funnel from an overhead bucket, flows slowly into the tank through a diffuser located in the ''northeast'' corner at a rate of typically 20 cm3 min"1 or so. The circulation of the dyed fluid is viewed from above using a camera.

the boundary currents, a feature of deep boundary currents in the real ocean.

### Application of T-P theory to the experiment

We now apply dynamical ideas to the experiment in a way that directly parallels that developed in Section 11.3.1 to infer abyssal circulation patterns in the ocean. If fluid is introduced to the tank of side L at rate S, then the depth of the fluid in the tank, h, (see Fig. 11.19) increases at a rate given by:

In the presence of rotation, columns of fluid in steady, slow, frictionless motion must, by the Taylor-Proudman theorem, remain of constant length. Hence, if the free surface rises, an interior column, marked by the thick vertical line in Fig. 11.19, must move toward the shallow end of the tank conserving its length. In a time At the free surface has risen by Ah and so the vertical velocity is w = Ah/At = S/L2, from above. Given that the displacement of the column must match the geometry of the wedge defined by the upper surface and the sloping bottom, we see that the vertical and horizontal velocities must be in the ratio w/v = dz/dy = a, where a is the slope of the bottom and we have used the definitions w = dz/dt, v = dy/dt. Thus v is given by:

exactly analogous to Eq. 11-8.4 The column moves northwards, towards the shallow end of the tank i.e.,''polewards''.

Typically we set S = 20 cm3 min"1, a = 0.2, L = 50 cm, and so we find that v = -L x 20 gf^"1 = 3.3 x 10"4 m s"1, or only 20 cm in 10 min. The boundary currents returning the water to the deep end of the tank are much swifter than this and are clearly evident in Fig. 11.18.

Our experiment confirms the preference for western boundary currents. But why are western boundary currents favored over eastern boundary currents? In Chapter 10 we explained the preference of wind-driven ocean gyres for western, as opposed to eastern boundary currents, by invoking an interior Sverdrup balance, Eq. 10-17, and arguing that the sense of the circulation must reflect that implied by the driving wind. But

4Comparing Eqs. 11-9 and 11-8, we see that a plays the role of ¡D/f ~ D/a (since, see Eq. 10-10, 1 ~ f /a) where D is a typical ocean depth and a is the radius of the Earth. It is interesting to note that Q does not appear in either Eq. 11-9 or 11-8. Neverthless it is important to realize that rotation is a crucial ingredient through imposition of the T-P constraint.

FIGURE 11.18. Three photographs, taken at 10min intervals, charting the evolution of dye slowly entering a rotating tank of water with a sloping bottom, as sketched in Fig. 11.17. The funnel in the centre carries fluid to the diffuser located at the top right hand corner of the tank. The shallow end of the tank is marked with the 'N' and represents polar latitudes. The dyed fluid enters at the top right, and creates a ''northern'' and then ''western'' boundary current.

FIGURE 11.18. Three photographs, taken at 10min intervals, charting the evolution of dye slowly entering a rotating tank of water with a sloping bottom, as sketched in Fig. 11.17. The funnel in the centre carries fluid to the diffuser located at the top right hand corner of the tank. The shallow end of the tank is marked with the 'N' and represents polar latitudes. The dyed fluid enters at the top right, and creates a ''northern'' and then ''western'' boundary current.

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