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Now, consider Fig. 6.18.

FIGURE 6.18. The centrifugal vector q x q x r has magnitude q2r , directed outward normal to the rotation axis. Gravity, g, points radially inward to the center of the Earth. Over geological time the surface of the Earth adjusts to make itself an equipoten-tial surface—close to a reference ellipsoid—which is always perpendicular the the vector sum of Q x Q x r and g. This vector sum is ''measured'' gravity: g* = -gz - q x q x r.

FIGURE 6.18. The centrifugal vector q x q x r has magnitude q2r , directed outward normal to the rotation axis. Gravity, g, points radially inward to the center of the Earth. Over geological time the surface of the Earth adjusts to make itself an equipoten-tial surface—close to a reference ellipsoid—which is always perpendicular the the vector sum of Q x Q x r and g. This vector sum is ''measured'' gravity: g* = -gz - q x q x r.

The centrifugal vector Q x Q x r has magnitude Q2r, directed outward normal to the rotation axis, where r = (a + z) cos v ~ a cos v, a is the mean Earth radius, z is the altitude above the spherical surface with radius a, and v is latitude, and where the ''shallow atmosphere'' approximation has allowed us to write a + z ~ a. Hence on the sphere, Eq. 6-30 becomes:

O2a2cos2v 2 '

defining the modified gravitational potential on the Earth. At the axis of rotation the height of a geopotential surface is geometric height, z (because v = §). Elsewhere, geopotential surfaces are defined by:

O2a cos v 2g

We can see that Eq. 6-40 is exactly analogous to the form we derived in Eq. 6-33

for the free surface of a fluid in solid body rotation in our rotating table, when we realize that r = a cos v is the distance normal to the axis of rotation. A plumb line is always perpendicular to z* surfaces, and modified gravity is given by g* = -Vz*

Since (with Q = 7.27 x 10-5 s a = 6.37 x 106m) Q2a2/2g « 11 km, geopo-tential surfaces depart only very slightly from a sphere, being 11km higher at the equator than at the pole. Indeed, the figure of the Earth's surface—the geoid—adopts something like this shape, actually bulging more than this at the equator (by 21 km, relative to the poles).12 So, if we adopt these (very slightly) aspherical surfaces as our basic coordinate system, then relative to these coordinates the centrifugal force disappears (being subsumed into the coordinate system) and hydrostatic balance again is described (to a very good approximation) by Eq. 6-8. This is directly analogous, of course, to adopting the surface of our parabolic turntable as a coordinate reference system in GFD Lab V.

Components of the Coriolis force on the sphere: the Coriolis parameter

We noted in Chapter 1 that the thinness of the atmosphere allows us (for most purposes) to use a local Cartesian coordinate system, neglecting the Earth's curvature. First, however, we must figure out how to express the Coriolis force in such a system. Consider Fig. 6.19.

At latitude v, we define a local coordinate system such that the three coordinates in the (x, y, z) directions point (eastward, northward, upward), as shown. The components of Q in these coordinates are (0, Q cos v, Q sin v). Therefore, expressed in these coordinates,

12The discrepancy between the actual shape of the Earth and the prediction from Eq. 6-40 is due to the mass distribution of the equatorial bulge, which is not taken in the calculation presented here.

FIGURE 6.19. At latitude & and longitude X, we define a local coordinate system such that the three coordinates in the (x, y, z) directions point (eastward, northward, upward): dx = a cos &dX; dy = ad&; dz = dz, where a is the radius of the Earth. The velocity is u = (u, v, w) in the directions (x, y, z). See also Appendix A.2.3.

FIGURE 6.19. At latitude & and longitude X, we define a local coordinate system such that the three coordinates in the (x, y, z) directions point (eastward, northward, upward): dx = a cos &dX; dy = ad&; dz = dz, where a is the radius of the Earth. The velocity is u = (u, v, w) in the directions (x, y, z). See also Appendix A.2.3.

We can now make two (good) approximations. First, we note that the vertical component competes with gravity and so is negligible if Qu ^ g. Typically, in the atmosphere |u| ~ 10 ms-1, so Qu ~ 7 x 10-4ms-2, which is negligible compared with gravity (we will see in Chapter 9 that ocean currents are even weaker, and so Qu ^ g there too). Second, because of the thinness of the atmosphere and ocean, vertical velocities (typically < 1cm s-1) are much less than horizontal velocities; so we may neglect the term involving w in the x-component of Q x u. Hence we may write the Coriolis term as

2Q x u ~ (-2Q sin & v,2Q sin & u,0) (6-41) =fx x u , where f = 2Q sin & (6-42)

is known as the Coriolis parameter. Note that Q sin & is the vertical component of the Earth's

TABLE 6.1. Values of the Coriolis parameter, f = 2Q sin ty (Eq. 6-42), and its meridional gradient, P = df/dy = 2Q/a cos ty (Eq. 10-10), tabulated as a function of latitude. Here Q is the rotation rate of the Earth and a is the radius of the Earth.

Latitude

f (x10-4s-1)

ß (x10-11 s-1 m-1)

90°

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