Finally, let us consider the effect of Earth's sphericity on the Coriolis force. The rotation axis of Earth is a line between the North Pole and the South Pole, and the Co-riolis force always acts in a direction perpendicular to this line. We are primarily interested in the horizontal deflection of fluid parcels by the Coriolis force because in the vertical direction, gravitational effects dominate and the Coriolis force is relatively small. The horizontal deflection is caused mainly by the component of the rotation that points in the local vertical direction acting on the horizontal fluid flow. (There is also a horizontal force caused by the horizontal component of the rotation acting on the vertical fluid motion, but because vertical velocities are small, this force is small also.) If Earth is rotating with an angular velocity X, then at a latitude j the component of rotation in the local vertical is equal to X sin j (figure 3.2). This angle is equal to X at the North Pole but diminishes as one moves equatorward, falling to zero at the equator itself. In the Southern Hemisphere, the component in the local vertical direction decreases from zero at the equator to —Q at the South Pole. So here the magnitude of the Coriolis force still increases with latitude, but the force acts in the opposite direction and tends to deflect objects to the left.

At the poles, Earth's surface is perpendicular to the axis of rotation; horizontal motion is therefore also perpendicular to the axis of rotation and so experiences the full effects of Earth's rotation, just like flow on a rotating disk. At the equator, a horizontal motion does not involve any motion toward or away from the axis of rotation, and the Coriolis force must involve vertical motions, which are small for large-scale flow. That is, at the equator the only components of the Coriolis force that are nonzero either involve the vertical velocity (which is small) or act in the vertical direction (and in this direction, the gravitational force swamps the Coriolis force).

The "effective" rotation of Earth thus increases as we move poleward, and thus Earth's atmosphere and ocean are said to be in differential rotation. We take this effect into account by allowing the Coriolis parameter, f, to equal twice the vertical component of the rotation. Thus, f = 2X sin j, and it increases from the South Pole (where f = —2X) to the equator (where f = 0) and to the North Pole (where f = 2X). Equations involving the Coriolis force, such as equation 3.4a, all hold with this new definition off. The variation of the Coriolis parameter turns out to be crucial in the production of western boundary currents like the Gulf Stream, as we see in the next chapter.

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