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FIGURE 10.4. The stress at the sea surface, t(0) = Twind, the wind stress, diminishes to zero at a depth z = -5. The layer directly affected by the stress is known as the Ekman layer.

FIGURE 10.5. The mass transport of the Ekman layer is directed to the right of the wind in the northern hemisphere (see Eq. 10-5). Theory suggests that horizontal currents, uag, within the Ekman layer spiral with depth as shown.

Since z is a unit vector pointing vertically upwards, we see that the mass transport of the Ekman layer is exactly to the right of the surface wind (in the northern hemisphere). Eq. 10-5 determines MEk, which depends only on rwind and f. But Eq. 10-5 does not predict typical velocities or boundary layer depths that depend on the details of the turbulent boundary layer. A more complete analysis (carried out by Ekman, 19051) shows that the horizontal velocity vectors within the layer trace out a spiral, as shown in Fig. 10.5. Typically 5 ~ 10 - 100 m. So the direct effects of the wind are confined to the very surface of the ocean.

10.1.2. Ekman pumping and suction and GFD Lab XII

Imagine a wind stress blowing over the northern hemisphere ocean with the general anticyclonic pattern sketched in Fig. 10.6

(left). We have just seen that the Ekman transport is directed to the right of the wind and hence there will be a mass flux directed inward (as marked by the broad open arrows in the figure), leading to convergence into the center. Since the mass flux across the sea surface is zero (neglecting evaporation and precipitation, which, as discussed in Chapter 11, are typically ± 1my-1), water cannot accumulate in the steady state, and so it must be driven down into the interior. Conversely, the cyclonic pattern of wind stress sketched in Fig. 10.6 (right) will result in a mass flux directed outward from the center, and therefore water will be drawn up from below.

If the wind stress pattern varies in space (or, more precisely, as we shall see, if the wind stress has some "curl") it will therefore result in vertical motion across the base of the Ekman layer. The flow within the Ekman layer is convergent in anti-cyclonic flow and divergent in cyclonic flow, as sketched in Fig. 10.6. The convergent flow drives downward vertical motion (called Ekman pumping); the divergent flow drives upward vertical motion from beneath (called Ekman suction). We will see that it is this pattern of vertical motion from the base of surface Ekman layers that brings the interior of the ocean into motion. But first let us study Ekman pumping and suction in isolation, in a simple laboratory experiment.

QFD Lab XII: Ekman pumping and suction

The mechanism by which the wind drives ocean circulation through the action of Ekman layers can be studied in a simple laboratory experiment in which the cyclonic

Vagn Walfrid Ekman (1874—1954), a Swedish physical oceanographer, is remembered for his studies of wind-driven ocean currents. The role of Coriolis forces in the wind-driven layers of the ocean was first suggested by the great Norwegian explorer Fridtjof Nansen, who observed that sea ice generally drifted to the right of the wind and proposed that this was a consequence of the Coriolis force. He suggested the problem to Ekman—at the time a student of Vilhelm Bjerknes—who, remarkably, worked out the mathematics behind what are now known as Ekman spirals in one evening of intense activity.

Ekman transport

FIGURE 10.6. The Ekman transport is directed perpendicular to the applied stress (to the right if Q > 0, to the left if Q < 0 ) driving (left) convergent flow if the stress is anticyclonic and (right) divergent flow if the stress is cyclonic. (The case Q > 0—the northern hemisphere—is shown.)

FIGURE 10.7. We rotate a disc at rate w on the surface of a cylindrical tank of water (the disc is just submerged beneath the surface). The tank of water and the disc driving it are then rotated at rate Q using a turntable; 10 rpm works well. We experiment with disc rotations of both signs, w = ±5 rpm. Low values of | w| are used to minimize the generation of shearing instabilities at the edge of the disc. The apparatus is left for about 20 minutes to come to equilibrium. Once equilibrium is reached, dye crystals are dropped into the water to trace the motions. The whole system is viewed from above in the rotating frame; a mirror can be used to capture a side view, as shown in the photograph on the right.

FIGURE 10.7. We rotate a disc at rate w on the surface of a cylindrical tank of water (the disc is just submerged beneath the surface). The tank of water and the disc driving it are then rotated at rate Q using a turntable; 10 rpm works well. We experiment with disc rotations of both signs, w = ±5 rpm. Low values of | w| are used to minimize the generation of shearing instabilities at the edge of the disc. The apparatus is left for about 20 minutes to come to equilibrium. Once equilibrium is reached, dye crystals are dropped into the water to trace the motions. The whole system is viewed from above in the rotating frame; a mirror can be used to capture a side view, as shown in the photograph on the right.

and anticyclonic stress patterns sketched in Fig. 10.6 are created by driving a disc around on the surface of a rotating tank of water (see Fig. 10.7 and legend). We apply a stress by rotating a disc at the surface of a tank of water that is itself rotating, as depicted in Fig. 10.7. If Q > 0, the stress imparted to the fluid below by the rotating disc will induce an ageostrophic flow to the right of the stress (Eq. 10-5): outward if the disc is rotating cyclonically relative to the rotating table (w/Q > 0), inward if the disc is rotating anticyclonically (w/Q < 0), as illustrated in Fig. 10.6. The Ekman layers and patterns of upwelling and downwelling can be made visible with dye crystals.

With Q, w > 0 (i.e., the disk is rotating cyclonically and faster than the table) the

FIGURE 10.8. Schematic of the ageostrophic flow driven by the cyclonic rotation of a surface disc relative to a homogeneous fluid that is itself rotating cycloni-cally below. Note that the flow in the bottom Ekman layer is in the same sense as Fig. 7.23, top panel, from GFD Lab X. The flow in the top Ekman layer is divergent. If the disc is rotated anticyclonically, the sense of circulation is reversed.

FIGURE 10.8. Schematic of the ageostrophic flow driven by the cyclonic rotation of a surface disc relative to a homogeneous fluid that is itself rotating cycloni-cally below. Note that the flow in the bottom Ekman layer is in the same sense as Fig. 7.23, top panel, from GFD Lab X. The flow in the top Ekman layer is divergent. If the disc is rotated anticyclonically, the sense of circulation is reversed.

column of fluid is brought into cyclonic circulation and rubs against the bottom of the tank. The flow in the bottom Ekman layer is then just as it was for the cyclonic vortex studied in GFD Lab X (Fig. 7.23, top panel) and is directed inward at the bottom as sketched in Fig. 10.8. Convergence in the bottom Ekman layer thus induces upwelling, drawing fluid up toward the rotating disc at the top, where it diverges in the Ekman layer just under the disk. This is clearly evident in the photographs of upwelling fluid shown in Fig. 10.9. The Ekman layer directly under the disc drives fluid outward to the periphery, drawing fluid up from below. This process is known as Ekman suction.

If the disc is rotated anticyclonically, convergence of fluid in the Ekman layer underneath the disc drives fluid downward into the interior of the fluid in a process known as Ekman pumping (in this case the sign of all the arrows in Fig. 10.8 is reversed).

Before going on to discuss how Ekman pumping and suction manifest themselves on the large scale in the ocean, it should be noted that in this experiment the Ekman layers are laminar (nonturbulent) and controlled by the viscosity of water.2 In the ocean the momentum of the wind is carried down into the interior by turbulent motions rather than by molecular processes. Nevertheless, the key result of Ekman theory, Eq. 10-5, still applies. We now go on to estimate typical Ekman pumping rates in the ocean.

10.1.3. Ekman pumping and suction induced by large-scale wind patterns

Figure 10.10 shows a schematic of the midlatitude westerlies (eastward wind stress) and the tropical easterlies (westward stress) blowing over the ocean, as suggested by Fig. 10.2. Because the Ekman transport is to the right of the wind in the northern hemisphere, there is convergence and downward Ekman pumping in the subtropics. This Ekman layer convergence also explains why the sea surface is higher in the subtropics than in subpolar regions (cf. Fig. 9.19): the water is ''piled up'' by the wind through the action of counter-posed Ekman layers, as sketched in Fig. 10.10. So the interior of the subtropical ocean ''feels'' the wind stress indirectly, through Ekman-induced downwelling. It is this downwelling that, for example, causes the a = 26.5 surface plotted in Fig. 9.8 to bow down in the subtropics. Over the subpolar oceans, by contrast, where the westward stress acting on the ocean diminishes in strength moving northward (see Fig. 10.2), Ekman suction is induced, drawing fluid from the interior

2Theory tells us (not derived here, but see Hide and Titman, 1967) that:

where V = 10-6 m2 s-1 is the kinematic viscosity of water (see Table 9.3). In the experiment of Fig. 10.7, typically ^ =

1 rads-1 (that is 10 rpm) and w = 0.5 rads-1, and so wEk — 1.5 x 10-4 m s-1. Thus in Fig. 10.9 , fluid is sucked up at a rate on the order of 9 cm every 10 minutes.

FIGURE 10.9. (Top) The anticlockwise rotation of the disc at the surface induces upwelling in the fluid beneath (Ekman suction), as can be clearly seen from the "dome" of dyed fluid being drawn up from below. (Bottom) We see the experiment from the top and, via the mirror, from the side (and slightly below). Now, sometime later, the dye has been drawn up in to a column reaching right up to the disk and is being expelled outward at the top. The white arrows indicate the general direction of flow. The yellow line marks the rotating disk.

FIGURE 10.9. (Top) The anticlockwise rotation of the disc at the surface induces upwelling in the fluid beneath (Ekman suction), as can be clearly seen from the "dome" of dyed fluid being drawn up from below. (Bottom) We see the experiment from the top and, via the mirror, from the side (and slightly below). Now, sometime later, the dye has been drawn up in to a column reaching right up to the disk and is being expelled outward at the top. The white arrows indicate the general direction of flow. The yellow line marks the rotating disk.

upward into the Ekman layer (as in GFD Lab XII, just described). Hence isopycnals are drawn up to the surface around latitudes of 60° N, S. This general pattern of Ekman pumping/suction imposed on the interior ocean is represented by the vertical arrows in the schematic Fig. 10.1.

FIGURE 10.10. A schematic showing midlatitude westerlies (eastward wind stress) and tropical easterlies (westward stress) blowing over the ocean. Because the Ekman transport is to the right of the wind in the northern hemisphere, there is convergence and downward Ekman pumping into the interior of the ocean. Note that the sea surface is high in regions of convergence.

FIGURE 10.10. A schematic showing midlatitude westerlies (eastward wind stress) and tropical easterlies (westward stress) blowing over the ocean. Because the Ekman transport is to the right of the wind in the northern hemisphere, there is convergence and downward Ekman pumping into the interior of the ocean. Note that the sea surface is high in regions of convergence.

We can obtain a simple expression for the pattern and magnitude of the Ekman pumping/suction field in terms of the applied wind stress as follows. Integrating the continuity equation, Eq. 6-11, across the Ekman layer, assuming that geostrophic flow is nondivergent (but see footnote in Section 10.2.1):

and noting w _ 0 at the sea surface, then the divergence of the Ekman layer transport results in a vertical velocity at the bottom of the Ekman layer that has magnitude, using Eq. 10-5

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