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rotation rate. This is the only component that matters (a consequence of the thinness of the atmosphere and ocean). For one thing this means that since f ^ 0 at the equator, rotational effects are negligible there. Furthermore, f < 0 in the southern hemisphere (see Fig. 6.17, right). Values of f at selected latitudes are set out in Table 6.1.

We can now write Eq. 6-29 as (rearranging slightly),

Dt p where 2Q has been replaced by fx. Writing this in component form for our local Cartesian system (see Fig. 6.19) and making the hydrostatic approximation for the vertical component, we have

Du Dt

Dv 1 dp

Dt pdy y

1 dp

where (Fx, Fy) are the (x, y) components of friction (and we have assumed the vertical component of F to be negligible compared with gravity).

The set, Eq. 6-44 is the starting point for discussions of the dynamics of a fluid in

FIGURE 6.20. A leveled cylinder is filled with water, covered by a lid and left standing for several days. Attached to the small hole at the center of the cylinder is a hose (also filled with water and stopped by a rubber bung) which hangs down in to a pail of water. On releasing the bung the water flows out and, according to theory, should acquire a spin which has the same sense as that of the Earth.

FIGURE 6.20. A leveled cylinder is filled with water, covered by a lid and left standing for several days. Attached to the small hole at the center of the cylinder is a hose (also filled with water and stopped by a rubber bung) which hangs down in to a pail of water. On releasing the bung the water flows out and, according to theory, should acquire a spin which has the same sense as that of the Earth.

a thin spherical shell on a rotating sphere, such as the atmosphere and ocean.

6.6.6. GFD Lab VI: An experiment on the Earth's rotation

A classic experiment on the Earth's rotation was carried out by Perrot in 1859.13 It is directly analogous to the radial inflow experiment, GFD Lab III, except that the Earth's spin is the source of rotation rather than a rotating table. Perrot filled a large cylinder with water (the cylinder had a hole in the middle of its base plugged with a cork, as sketched in Fig. 6.20) and left it standing for two days. He returned and released the plug. As fluid flowed in toward the drain hole, it conserved angular momentum, thus "concentrating" the rotation of the Earth, and acquired a "spin" that was cyclonic (in the same sense of rotation as the Earth).

According to the theory below, we expect to see the fluid spiral in the same sense of rotation as the Earth. The close analogue with the radial inflow experiment is clear when one realizes that the container

FIGURE 6.21. The Earth's rotation is magnified by the ratio (r1 /ro)2 » 1, if the drain hole has a radius ro, very much less than the tank itself, r1.

sketched in Fig. 6.20 is on the rotating Earth and experiences a rotation rate of Q x sin lat!

Theory We suppose that a particle of water initially on the outer rim of the cylinder at radius r1 moves inward, conserving angular momentum until it reaches the drain hole at radius ro (see Fig. 6.21). The Earth's rotation QEarth resolved in the

13Perrot's experiment can be regarded as the fluid-mechanical analogue of Foucault's 1851 experiment on the Earth's rotation using a pendulum.

direction of the local vertical is QEarth sin v where v is the latitude. Therefore a particle initially at rest relative to the cylinder at radius r1, has a speed of v1 = r1QEarth sin v in the inertial frame. Its angular momentum is A1 = v1 r1. At ro what is the rate of rotation of the particle?

If angular momentum is conserved, then Ao = Qor2 = A1, and so the rate of rotation of the ball at the radius ro is:

Thus if r1 /ro>> 1 the Earth's rotation can be ''amplified'' by a large amount. For example, at a latitude of 42° N, appropriate for Cambridge, Massachusetts, sin v = 0.67, QEarth = 7.3 x 10-5 s-1, and if the cylinder has a radius of r1 = 30 cm, and the inner hole has radius ro = 0.15 cm, we find that Qo = 1.96 rad s-1, or a complete rotation in only 3 seconds!

Perrot's experiment, although based on sound physical ideas, is rather tricky to carry out. The initial (background) velocity has to be very tiny (v <<fr) for the experiment to work, thus demanding great care in setup. Apparatus such as that shown in Fig. 6.20 can be used, but the experiment must be repeated many times. More often than not, the fluid does indeed acquire the spin of the Earth, swirling cyclonically as it exits the reservoir.

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