the cross-

sectional area of a Taylor column, A, projected on to the surface of the sphere, where p is the latitude.

area measured perpendicular to Q (which, as discussed above, does not change in time), and d is its length parallel to Q, then a, A A

sin p sin p

Dt Dt since A' _ A/sin p is the area of the Taylor column projected onto the surface of the sphere over which fluid is being pumped down from the Ekman layer at rate wEk. Note that the minus sign ensures that if wEk < 0 (pumping down into the ocean), then Dd/Dt > 0: so that following the Taylor column along its length increases.

Rewriting, wEk sin p

Dt h cos p sin 2p

where Eq. 10-13 has been used and v = aDp/Dt is the meridional velocity of y the column. Multiplying both sides by 2Q and rearranging, Eq. 10-14 may be written in the form:

where f is given by Eq. 6-42 and fi by Eq. 10-10. Note that, setting dw/dz = wEk/h, we have arrived at a version of Eq. 10-12.

The simple mechanism sketched in Fig. 10.15b is the basic drive of the wind-driven circulation; the gentle vertical motion induced by the prevailing winds, wEk, is amplified by a large geometrical factor, f = h tan p ~ radius of Earth/depth of ocean ~ 1000, to create horizontal currents with speeds that are 1000 times that of Ekman pumping rates, which is 1 cm s-1 compared to 30 m y-1. Thus we see that the stiffness imparted to the fluid by rotation results in strong lateral motion as the Taylor columns are squashed and stretched in the spherical shell.

There are two useful mechanical analogies:

1. 'pip' flicking: a lemon seed shoots out sideways on being squashed between finger and thumb (see Fig. 10.15b).

2. a child's spinning top: the ''pitch'' of the thread on the spin axis results in horizontal motion when the axis is pushed down (see Fig. 10.17). The tighter the pitch the more horizontal motion one creates (note, however, that in practice friction prevents use of a very tight pitch).

10.2.4. GFD Lab XIII: Wind-driven ocean gyres

The previous discussion motivates a laboratory demonstration of the wind-driven circulation. We need the following three essential ingredients: (i) geometrical and (ii) rotational constraints, and (iii) a representation of Ekman pumping. The apparatus, shown in Fig. 10.18, consists of a rotating

FIGURE 10.17. The mechanism of wind-driven ocean circulation can be likened to that of a child's spinning top. The tight pitch of the screw thread (analogous to rotational rigidity) translates weak vertical motion (Ekman pumping of order 30 my-1) into rapid horizontal swirling motion (ocean gyres circulating at speeds of cm s-1).

FIGURE 10.17. The mechanism of wind-driven ocean circulation can be likened to that of a child's spinning top. The tight pitch of the screw thread (analogous to rotational rigidity) translates weak vertical motion (Ekman pumping of order 30 my-1) into rapid horizontal swirling motion (ocean gyres circulating at speeds of cm s-1).

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