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if Aa ~ 1.5kgm-3 is the change in a between, say, 20° N and 40° N, a distance L ~ 2000 km, in the top H ~ 1000 m of the ocean seen in Fig. 9.7. This is in good accord with direct observations shown, for example, from surface drifters in Fig. 9.14.

9.3.1. Ocean surface structure and geostrophic flow

### Near-surface geostrophic flow

To the extent that the surface currents shown in Figs. 9.14 to 9.16 are in geostrophic balance, there must be a pressure gradient force balancing Coriolis forces acting on them. Consider, for example, the eastward flowing Gulf Stream of the Atlantic evident in Fig. 9.15. To balance the southward-directed Coriolis force acting on it, there must be a pressure gradient force directed northward. This is provided by a tilt in the free surface of the ocean: we shall see that the sea surface is higher11 to the south of the Gulf Stream than to the north of it.

Consider Fig. 9.18. Integrating the hydrostatic relation, Eq. 3-3, from some horizontal surface at constant z up to the free surface at z = n (where p = ps, atmospheric pressure) we obtain:

where (p) = 1/ (n - z) gp dz is the mean density in the water column of depth n - z. If we are interested in the near-surface region (z = z0, say, in Fig. 9.18), fractional variations in column depth are much greater

"Here "high" means measured relative to the position that the ocean surface would take up if there were no surface geostrophic flow. This equipotential reference surface is the "geoid," which is everywhere perpendicular to the local plumb line. Recall that the "geoid" of our rotating tank of water considered in Section 6.6 is the parabolic surface shown in Fig. 6.11. We saw in Fig. 6.18 that the geoid surface of the rotating Earth approximates a reference ellipsoid. Geoid height variations about this reference ellipsoid are of the order ± 100 m, 100 times larger than height variations associated with geostrophic flow.

than those of density, so we can neglect the latter, setting p = pref in Eq. 9-10 and leaving p(z0) = ps + gPref (n - z0) .

We see that horizontal variations in pressure in the near-surface region thus depend on variations in atmospheric pressure and in free-surface height. Since here we are interested in the mean ocean circulation, we neglect day-to-day variations of atmospheric pressure and equate horizontal components of the near-surface pressure gradient with gradients in surface elevation: (dp/dx, dp/y = gpref (dn/dx, dn/dy). Thus the geostrophic flow just beneath the surface is, from Eq. 7-4, (Ugsurface, %„face) =

g (-dtj/dy, drj/dx) or, in vector form ugsfce = gi x Vrr (9-11)

Note how Eq. 9-11 exactly parallels the equivalent relationship Eq. 7-7 for geos-trophic flow on an atmospheric pressure surface.

In Section 7.1.1 we saw that geostrophic winds of 15 m s-1 were associated with tilts of pressure surfaces by about 800 m over a distance of 5000 km. But because oceanic flow is weaker than atmospheric flow, we expect to see much gentler tilts of pressure surfaces in the ocean. We can estimate the size of n variations by using Eq. 9-11 along with observations of surface currents. If U is the eastward speed of the surface current, then n must drop by an amount An in a distance L given by:

An = -, g or 1m in 1000 km if U = 10-1ms-1and f = 10-4s-1. Can we see evidence of this in the observations?

### Observations of surface elevation

Maps of the height of the sea surface, n, give us the same information as do maps of the height of atmospheric pressure surfaces (and, of course, because ps variations are so slight, the ocean surface is, to a very good approximation, a surface of constant pressure). If we could observe the n field of the ocean then, just as in the use of geopoten-tial height maps in synoptic meteorology, we could deduce the surface geostrophic flow in the ocean. Amazingly, variations in ocean topography, even though only a few centimeters to a meter in magnitude, can indeed be measured from satellite altimeters and are mapped routinely over the globe every week or so. Orbiting at a height of about 1000 km above the Earth's surface, altimeters measure their height above the sea surface to a precision of 1-2 cm. And, tracked by lasers, their distance from the center of the Earth can also be determined to high accuracy, permitting n to be found by subtraction.

The 10y-mean surface elevation (relative to the mean geoid) is shown in Fig. 9.19. Consistent with Figs. 9.14 and 9.15, the highest elevations are in the anticyclonic subtropical gyres (where the surface is about 40 cm higher than near the eastern boundary at the same latitude), and there are strong gradients of height at the western boundary currents and near the circumpolar current of the Southern Ocean, where surface height changes by about 1 m across these currents.

### 9.3.2. Geostrophic flow at depth

At depths much greater than variations of n (at z = z1, say, in Fig. 9.18), we can no longer neglect variations of density in Eq. 9-10 compared with those of column depth. Again neglecting atmospheric pressure variations, horizontal pressure variations at depth are therefore given by, using Eq. 9-10, z xVp = g (p) z xVn + g(n - z) z xV(p).

Thus the deep water geostrophic flow is given by

FIGURE 9.19. Top: The 10-year mean height of the sea surface relative to the geoid, n, (contoured every 20 cm) as measured by satellite altimeter. The pressure gradient force associated with the tilted free surface is balanced by Coriolis forces acting on the geostrophic flow of the ocean at the surface. Note that the equatorial current systems very evident in the drifter data, Figs. 9.14 and 9.22, are only hinted at in the sea surface height. Near the equator,

FIGURE 9.19. Top: The 10-year mean height of the sea surface relative to the geoid, n, (contoured every 20 cm) as measured by satellite altimeter. The pressure gradient force associated with the tilted free surface is balanced by Coriolis forces acting on the geostrophic flow of the ocean at the surface. Note that the equatorial current systems very evident in the drifter data, Figs. 9.14 and 9.22, are only hinted at in the sea surface height. Near the equator, where f is small, geostrophic balance no longer holds. Bottom: The variance of the sea surface height, an = Eq. 9-17, contoured every 5 cm.

J Pref g

= f^ [<p> z xVn + (n - z) z xv<p>] - gz xVn + g(n - z)Z xV(p>, (9-12)

f fPref since we can approximate (p) ~ pref in the first term, because we are not taking its gradient. We see that u has two contributions: that associated with free-surface height variations, and that associated with interior ocean density gradients. Note that if the ocean density were to be uniform, the second term would vanish and the deep-water geostrophic flow would be the same as that at the surface: geostrophic flow in an ocean of uniform density is independent of depth. This, of course, is a manifestation of the Taylor-Proudman theorem, discussed at length in Section 7.2. However, observed currents and pressure gradients at depth are smaller than at the surface (cf. Figs. 9.15 and 9.17) suggesting that the two terms on the rhs of Eq. 9-12 tend to balance one another.

The second term in Eq. 9-12 is the ''thermal wind'' term (cf. Section 7.3), telling us that u will vary with depth if there are horizontal gradients of density. Thus the presence of horizontal variations in surface height, manifested in surface geostrophic currents given by Eq. 9-11, does not guarantee a geostrophic flow at depth. Indeed, as already noted, the flow becomes much weaker at depth.

How much would interior density surfaces need to tilt to cancel out deep pressure gradients and hence geostrophic flow? If u

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