# Geostrophic And Hydrostatic Balance

On the large-scale, water obeys the same fluid dynamics as air; so we have already derived the equations we will need: Eq. 6-44 applies just as well to the ocean as to the atmosphere. One simplification we can make in application of these equations to the ocean is to recognize that the density varies rather little in the ocean (by only a few %; see Fig. 9.2), so we can rewrite the horizontal momentum equations Eq. 6-44a,b thus (using our local Cartesian coordinate system; see Fig. 6.19)

Dt pref dx

Dv 1 dp

Dt Pref dy y

without incurring serious error, where pref is our constant reference density. We write the vertical equation, Eq. 6-44c (hydrostatic balance) in terms of the density anomaly, Eq. 9-2:

If we neglect the contribution from a, for a moment, Eq. 9-8 implies that the pressure increases linearly downward from its surface value (ps = 105 Pa, or one atmosphere) thus, p(Z) = ps - gPref (Z - n) , (9-9)

where the ocean's surface is at z = n, and z decreases into the interior of the ocean. This linear variation should be contrasted with the exponential pressure variation of an isothermal compressible atmosphere (see Eq. 3-7). The pressure at a depth of, say, 1 km in the ocean is thus about 107 Pa or 100 times atmospheric pressure. However, the gPrefZ part of the pressure field is dynamically inert, because it does not have any horizontal variations. The dynamical part of the hydrostatic pressure is associated with horizontal variations in the free surface height, n, and interior density anomalies, a, associated with T and S variations and connected to the flow field by geostrophic balance. Finally, it is important to realize that time-mean horizontal variations in surface atmospheric pressure, ps, turn out to be much less important in Eq. 9-9 than variations in n and a.9

In Section 9.2 we observed that the maximum horizontal current speeds in ocean gyres are found at the surface in the western boundary currents where, instantaneously, they can reach 1ms-1. Elsewhere, in the interior of ocean gyres, the currents are substantially weaker, typically 5-10 cms-1, except in the tropical and circumpolar belts. The N-S extent of middle-latitude ocean gyres is typically about 20° latitude « 2000 km (the E-W scale is greater). Thus setting U = 0.1m s-1 and L = 2 x 106 m with f = 10-4 s-1, we estimate a typical Rossby number, Eq. 7-1, of Ro = jL ~ 10-3. This is very small, much smaller, for example, than we found to be typical in the atmosphere where Ro ~ 10-1 (see estimates in Section 7.1 and Fig. 7.5). Thus the geostro-phic and thermal wind approximation is generally excellent for the interior of the ocean away from the equator10 and away

"Suppose, for example, that ps varies by 10 mbar in 1000 km (corresponding to a stiff surface wind of some 10 ms-1 in middle latitudes); (see, e.g., Fig. 7.25) then the surface geostrophic ocean current required to balance this surface pressure gradient is a factor patoms /pccean ~ 1/1000 times smaller, or 1 cm s-1, small relative to observed surface ocean currents. Mean surface atmospheric pressure gradients are typically considerably smaller than assumed in this estimate (see Fig. 7.27).

10Our estimate of Ro is applicable to the gyre as a whole. Within the western boundary currents, a more relevant estimate is Ubdy/f A, where A is the width of the boundary current and Ubdy is its speed. If A ~ 100 km and Ubdy reaches 2 ms-1, then the Rossby number within these boundary currents can approach, and indeed exceed, unity.

from surface, bottom, and side boundary layers. As discussed later in Section 9.3.4, this fact can be exploited to infer ocean currents from hydrographic observations of T and S.

Thus away from (surface, bottom, and side) boundary layers, the geostrophic equations derived in Chapter 7 will be valid: Eqs. 7-3 and 7-4 with, as in Eq. 9-8, pref substituted for p. The associated thermal wind equation, Eqs. 7-16 and 7-17, will also apply.

Using Eq. 7-16 one can immediately infer the sense of the thermal wind shear from the a field shown in, for example, Fig. 9.7: du/dz > 0 where a increases moving northward in the northern hemisphere (f> 0), implying that u at the surface is directed eastward in these regions if abyssal currents are weak. Inspection of Fig. 9.7 suggests that du/dz and the surface u are positive (negative) poleward (equatorward) of 25° N, more or less as observed in Figs. 9.14 and 9.15. Moreover, Eq. 7-16 suggests a mean surface geostrophic flow of magnitude:

usurface 