1 dp

dy V Preff dxJ f where we have remembered that f varies with y and, noting that dy = adp, df 1 df 2Q ß = = — cos p ay aap a

is the meridional gradient in the Coriolis parameter, Eq. 6-42. The variation of f with latitude is known as the (!- effect.

We see, then, that because f varies with latitude, the geostrophic flow is horizontally divergent. Hitherto (in Eq. 10-6, for example) we have assumed that Vh ■ u = 0. However, on the planetary scales being considered here, we can no longer ignore variations in f and the resulting horizontal divergence of geostrophic flow is associated with vertical stretching of water columns, because:4

Combining Eqs. 10-9 and 10-11, we obtain the very useful expression:

r&w

which relates horizontal and vertical currents.

If vertical velocities in the abyss are much smaller than surface Ekman pumping velocities, then Eq. 10-12 tells us that ocean currents will have a southward component in regions where wEk < 0, and northward where wEk > 0. This is indeed observed in the interior regions of ocean gyres. Consider Figs. 9.14 and 9.15, for example, in the light of Eq. 10-12 and Fig. 10.11. Does Eq. 10-12 make any quantitative sense? Putting in numbers, f = 10-4 s-1, wEk = 30 my-1, H the depth of the thermocline ~ 1 km, we find that v = 1 cm s-1, typical of the gentle currents observed in the interior of the ocean on the large scales. Note that Fig. 9.22 (top) shows surface currents, whereas our estimate here is an average current over the depth of the thermocline. This is the basic mechanical drive of the ocean circulation: the pattern of Ekman pumping imposed on the ocean from above induces meridional motion through Eq. 10-12.

10.2.2. Wind-driven gyres and western boundary currents

We have seen that Ekman pumping, downward into the interior of the ocean, must drive an equatorward flow in the subtropical latitude belt between the mid-latitude westerlies and tropical easterlies. Although this flow is indeed observed (see the equatorward interior flow over the subtropics in Figs. 9.14 and 9.15), we have

4Eq. 10-11 tells us that Vh • Ug ~ wgt/H where H is the vertical scale of the thermocline, and so is 5/H smaller than Vh • uag = WEk/5 in Eq. 10-6, where 5 is the Ekman layer depth. Because 5/H < 0.1, we are thus justified in neglecting Vh • Ug in comparison with Vh • uag in the computation of w^t, as was assumed in deriving Eq. 10-7.

not yet explained the remainder of the circulation. If we take a very simple view of the geometry of the midlatitude ocean basins (Fig. 10.12), then the equatorward flow induced by Ekman pumping could be "fed" at its poleward side by eastward or westward flow, and in turn feed westward or eastward flow at the equa-torward edge. Either would be consistent with Eq. 10-12, which only dictates the N-S component of the current. However, the general sense of circulation must mirror the anticyclonic sense of the driving wind stress in Fig. 10.12. Hence the required poleward return flow must occur on the western margin, as sketched in Fig. 10.12.5 In the resulting intense western boundary current, our assumptions of geostrophy and/or of negligible friction break down (as will be discussed in more detail in Section 11.3.3), and therefore Eq. 10-12 is not applicable. This current is the counterpart in our simple model of the Gulf Stream or the Kuroshio, and other western boundary currents.

The preference for western as opposed to eastern boundary currents is strikingly evident in the surface drifter observations shown in Fig. 10.13, which displays the same data as in Fig. 9.15, but zooms in at high resolution on the surface circulation in the North Atlantic. We clearly observe the northward flowing Gulf Stream and the southward flowing Labrador Current hugging the western boundary. Here mean currents can exceed 40cms-1. Note how the path of the Gulf Stream and its interior extension, the North Atlantic Current, tends to follow the zero Ekman pumping line in Fig. 10.11, which marks the boundary between the subtropical and subpolar gyres and the region of enhanced temperature gradients.

Before going on in Section 10.3 to a fuller discussion of the implications of Eq. 10-12 on ocean circulation, we now discuss what it means physically. Equation 10-12 can be simply understood in terms of the attendant rotational and geometrical constraints on the fluid motion, i.e., the Taylor-Proudman

5Indeed, to sustain the circulation against frictional dissipation at the bottom and side boundaries, the wind stress must do work on the ocean. Since the rate of doing work is proportional to the product of wind stress and current velocity (u • F > 0) these two quantities must, on average, be in phase for the work done to be positive. This is the case if the sense of circulation is as depicted in Fig. 10.12, whereas the work done would be negative if the circulation were returned to the east.

Surface Current Components

cm/s

cm/s

FIGURE 10.13. (Top) Time-mean zonal velocity and (Bottom) the meridional velocity in cms-1 computedfrom surface drifters averaged over a 0.25° x 0.25° grid over the North Atlantic. Green colors denote positive values, brown colors negative. Data courtesy of Maximenko and Niiler (2003).

FIGURE 10.13. (Top) Time-mean zonal velocity and (Bottom) the meridional velocity in cms-1 computedfrom surface drifters averaged over a 0.25° x 0.25° grid over the North Atlantic. Green colors denote positive values, brown colors negative. Data courtesy of Maximenko and Niiler (2003).

FIGURE 10.14. In Section 10.2.3 we consider the possibility that a Taylor column subjected to Ekman pumping at its top might conserve mass by expanding its girth, as sketched on the left. We argue that such a scenario is not physically plausible. Instead the column maintains its cross-sectional area and increases its length, as sketched on the right.

FIGURE 10.14. In Section 10.2.3 we consider the possibility that a Taylor column subjected to Ekman pumping at its top might conserve mass by expanding its girth, as sketched on the left. We argue that such a scenario is not physically plausible. Instead the column maintains its cross-sectional area and increases its length, as sketched on the right.

theorem, on the sphere. This can be seen as follows.

Let us first consider rotational constraints on the possible motion. If the ocean were homogeneous then, as described in Section 7.2, the steady, inviscid, low-Rossby-number flow of such a fluid must obey the Taylor-Proudman theorem, Eq. 7-14. Thus the velocity vector cannot vary in the direction parallel to the rotation vector, and flow must be organized into columns parallel to Q, an expression of gyroscopic rigidity, as illustrated in GFD Lab VII, Section 7.2.1, and sketched in Fig. 7.8.

Now, consider what happens to such a column of fluid subjected to Ekman pumping at its top. According to Eq. 7-14, we might think that mass continuity would be satisfied by uniform flow sideways out of the column, as sketched in Fig. 10.14 left.

This satisfies the constraint that the flow be independent of height, but cannot be sustained in a steady flow. Why not? If the flow is axisymmetric about the circular column, it will conserve its angular momentum, Qr2 + vgr, where vg is the azimuthal component of flow around the column, and r is the column radius. The column must continuously expand as fluid is being pumped into it at its top; thus r must increase and so vg must change by an amount Svg ~ -2QSr (this, of course, is just Eq. 7-22) as the column increases its girth by an amount Sr. Thus vg must become increasingly negative as r increases. This is obviously inconsistent with our assumption of steady state and not physically reasonable.

So, what else can happen? Let's now introduce the geometrical constraint that our Taylor columns must move in a spherical shell, as sketched in Fig. 10.15a. (We have obviously exaggerated the depth h of the fluid layer in the figure—recall Fig. 1.1!) We see that the columns have greatest length, in the direction parallel to Q, near the equator. Therefore, if supplied by fluid from above by systematic Ekman pumping, a fluid column can expand in volume without expanding its girth (which we have seen is not allowed in steady state), by moving systematically equatorward in the spherical shell and hence increasing its length in the manner illustrated in Fig. 10.15b. The column will move equatorward at just the rate required to ensure that the "gap" created between it and the spherical shell is at all times filled by the pumping down of water from the surface. This, in essence, is how the wind, through the Ekman layers, drives the circulation in the interior of the ocean. The rate at which fluid is pumped down from the Ekman layer must be equal to the rate of change of the volume of the Taylor columns beneath.

Let's think about this process in more detail following Fig. 10.15. The Taylor columns are aligned parallel to the rotation axis; hence, since cos g = sin <p = h/d where g is co-latitude and p latitude (see Fig. 10.15a), their length is given by:

if the shell is thin (this is inaccurate within less than 1% of the equator). If the change in

FIGURE 10.15. (a) An illustration of Taylor-Proudman on a rotating sphere. We consider a spherical shell of homogeneous fluid of constant thickness h. Taylor columns line up parallel to Q with length d. The latitude is p and the co-latitude 6. (b) A Taylor column in a wedge. If the wedge narrows, or fluid is pumped down from the top at rate WEk, the Taylor column moves sideways to the thicker end of the wedge. This is just how one flicks a lemon seed. The downward motion between finger and thumb generates lateral (shooting) motion as the seed slips sideways. Modified from a discussion by Rhines (1993).

FIGURE 10.15. (a) An illustration of Taylor-Proudman on a rotating sphere. We consider a spherical shell of homogeneous fluid of constant thickness h. Taylor columns line up parallel to Q with length d. The latitude is p and the co-latitude 6. (b) A Taylor column in a wedge. If the wedge narrows, or fluid is pumped down from the top at rate WEk, the Taylor column moves sideways to the thicker end of the wedge. This is just how one flicks a lemon seed. The downward motion between finger and thumb generates lateral (shooting) motion as the seed slips sideways. Modified from a discussion by Rhines (1993).

the volume of the Taylor column is ADd/Dt, where (see Fig. 10.16) A is its cross-sectional

sectional area of a Taylor column, A, projected on to the surface of the sphere, where p is the latitude.

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