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Sphere

Rotating table

Sphere

FIGURE 6.17. In the rotating table used in the laboratory, Q and g are always parallel or (as sketched here) antiparallel to one another. This should be contrasted with the sphere.

6.6.5. Putting things on the sphere

Hitherto, our discussion of rotating dynamics has made use of a laboratory turntable in which Q and g are parallel or antiparallel to one another. But on the spherical Earth Q and g are not aligned and we must take into account these geometrical complications, illustrated in Fig. 6.17. We will now show that, by exploiting the thinness of the fluid shell (Fig. 1.1) and the overwhelming importance of gravity, the equations of motion that govern the fluid on the rotating spherical Earth are essentially the same as those that govern the motion of the fluid of our rotating table if 2Q is replaced by (what is known as) the Coriolis parameter, f = 2Q sin p. This is true because a fluid parcel on the rotating Earth "feels" a rotation rate of only 2Q sin p—2Q resolved in the direction of gravity, rather than the full 2Q.

The centrifugal force, modified gravity, and geopotential surfaces on the sphere

Just as on our rotating table, so on the sphere the centrifugal term on the right of Eq. 6-28 modifies gravity and hence hydrostatic balance. For an inviscid fluid at rest in the rotating frame, we have

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