1. tangential stresses at the ocean s surface due to the prevailing wind systems - the wind-driven circulation and

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1 Chapter 9 The wind-driven circulation [Hartmann, Ch. 7 ( )] There are two causes of the circulation of the ocean: 1. tangential stresses at the ocean s surface due to the prevailing wind systems - the wind-driven circulation and 2. thermohaline e ects associated with changes in the density of water due to insolation, exchange of heat with the atmosphere, evaporation and precipitation - the thermohaline circulation. In this chapter we discuss the wind-driven circulation. 9.1 The wind stress and Ekman layers One cannot escape noticing the similarity between the pattern of surface currents in the ocean - see g and that of the low-level winds in the atmosphere where, as discussed in Chapter 7, there are low-level westerly (eastward) winds in middle latitudes and easterly (westward) winds in the tropics. These winds, through the e ects of friction in the atmospheric boundary layer, exert a stress on the ocean s surface; the annual average of this stress isshowninfig.(9.1). Although there is some similarity between the pattern of surface currents and the pattern of surface winds, as we are about the see, the way in which the ocean responds to this wind stress is fascinating and rather subtle. 213

2 214 CHAPTER 9. THE WIND-DRIVEN CIRCULATION Figure 9.1: Annual mean wind stress on the ocean. A contour of 1 represents a wind-stress of magnitude 0.1 Nm 2. Stresses reach values of 0.1 to 0.2 Nm 2 under the middle-latitude westerlies, and are particularly strong in the southern hemisphere.

3 9.1. THE WIND STRESS AND EKMAN LAYERS 215 Figure 9.2: Balance of forces in the Ekman layer Consider the horizontal components of (8.2), which are, in +(u r) u + fbz u = 1 rp + F ; ½ 0 where F is the imposed body force per unit mass. To give the simplest possible conceptual model, we will assume steadiness, thus neglecting uctuations in the circulation. Since we are considering a steady ow, the rst term is zero, and we neglect the second (compared with the Coriolis term) because the Rossby number is small. Before proceeding, we need to express the body force F in terms of the wind stress. Consider Fig The stress component of interest here, x (z), isthe x component of force acting at depth z, per unit horizontal area on the layer beneath. Theslabofthickness±z at level z is subjected to a force per unit horizontal area x (z + ±z) at its upper surface, but also subjects the layers beneath it to a force x (z) per unit horizontal area. Therefore the net force per unit horizontal area felt by this layer is x (z + ±z) x (z). Since the slab has thickness ±z, ithasvolume ±z per unit horizontal area, and if the slab has uniform density ½ 0,ithasmass½ 0 ±z per unit horizontal area. Therefore the force per unit mass, F x,feltbytheslabis F x = force per unit area mass per unit area = x(z + ±z) x (z) ½ 0 ±z = 1 ½ ; for small slab thickness. We can obtain a similar relationship for F y.hence our momentum equation for the steady circulation becomes fbz u = 1 ½ 0 rp + 1 : (9.1)

4 216 CHAPTER 9. THE WIND-DRIVEN CIRCULATION Eq. (9.1) describes the balance of forces for the wind-driven circulation, but it does not yet tell us what the circulation is. The stress at the surface is known - it is the wind stress plotted in g.(9.1) - but we do not know the vertical distribution of stress beneath the surface. The wind stress will be communicated below the surface by turbulent motions con ned to the near-surface layers of the ocean. The direct in uence of wind-forcing decays (rather rapidly - in a few 10 s of meters or so, depending on wind strength) with depth so that by the time a depth z = ± has been reached, = 0. Typically ±. 100m. Conveniently, we shall see that we can bypass the need to know the detailed vertical distribution of by focusing on the vertical integral across the turbulent layer. The mechanically-driven turbulent layer is called the Ekman layer Wind-driven Ekman pumping Let s think about what is happening in the surface Ekman layer. Now, in Figure 9.3: Section and Fig.(6.26 and see GFDlab X)) we saw how frictional drag in a bottom Ekman layer leads to non-geostrophic ow and, in particular, to Ekman layer convergence in a cyclonic eddy, and Ekman layer divergence in an anticyclonic eddy. In the present case we have the opposite situation, because the frictional e ects communicating the wind stress from above, through the Ekman layer, are driving the ow, rather than damping

5 9.1. THE WIND STRESS AND EKMAN LAYERS 217 Figure 9.4: it. In this case, the ow within the Ekman layer is convergent in anticyclonic ow, and divergent in cyclonic ow. The convergent ow drives downward vertical motion (called Ekman pumping); the divergent ow drives upward vertical motion from beneath (called Ekman suction). This will be illustrated below in GFDlab XII. In the region of the subtropical oceans between the midlatitude westerlies (eastward wind stress) and the tropical easterlies (westward stress), the wind stress curl is anticyclonic (clockwise in the northern hemisphere), as shown in Fig.9.4. This induces downward Ekman pumping. So the deep, frictionless ocean feels the wind stress indirectly through Ekman-induced downwelling. We can obtain a simple expression for the pattern and magnitude of the Ekman pumping eld in terms of the applied wind-stress as follows. The ageostrophic component of (9.1) is: fbz u ag = ½ Multiplying by ½ o and integrating across the layer from the surface to a depth z = ±, where =0, we obtain: fbz Z 0 ½ o u ag dz = (z =0)= wind or ± M Ek = wind bz f (9.2)

6 218 CHAPTER 9. THE WIND-DRIVEN CIRCULATION where Z 0 M Ek = ½ o u ag dz ± is the mass transport of the Ekman layers. Since bz 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. Eq(9.2) determines M Ek independently of the details of the turbulent boundary layer, but not typical velocities or boundary layer depths. We can estimate a boundary layer depth from the external parameters wind and f giving: r wind ±» 1 f ½ o Putting in some numbers wind =0.1Nm 2, typical of middle latitudes, ½ o = 1000kg m 3, f =10 4 s 1,we ndthat±» 100m. So the direct e ects of the wind are con ned to the top few tens of meters of the ocean. Now if wind varies in the horizontal because the surface winds vary (westerlies in middle latitudes, easterly in the tropics, for example) then, since we must conserve volume, the resulting divergence of Ekman transport results in a vertical velocity directed in to or out of the interior of the ocean - see g.(9.4). More precisely, integrating the continuity equation across the Ekman layer: r h u ag =0; and noting w =0at the sea surface, then the divergence of the Ekman layer transport results in a vertical velocity w Ek at the bottom of the Ekman layer which has magnitude: w Ek = 1 r h M Ek = 1 µ wind bz:r (9.4) ½ o ½ o f The vertical velocity w Ek is called Ekman pumping - it depends on the curl of the wind-stress. Typically w Ek has a magnitude of 30 myr 1,some30 times the annual-mean precipitation rate, and is directed downward in the subtropics and upwards in subpolar regions.

7 9.1. THE WIND STRESS AND EKMAN LAYERS 219 Figure 9.5: We rotate a disc at rate! on the surface of a cylindrical tank of water and the whole apparatus is then rotated at rate on our turntable. GFD Lab XII: Ekman pumping and suction Here we study the mechanism by which the wind stress drives ocean circulation. We induce circulation by rotating a disc at the surface of a tank of water which is itself rotating. The laboratory set-up is as follows. We rotate a disc at rate! on the surface of a cylindrical tank of water (in fact the disc is just submerged beneath the surface). The tank of water and the disc driving it is then rotated at rate using our turntable and left for about 30 minutes to come to equilibrium. Once equilibrium is reached, dye crystals are dropped in to the water to trace the motions. The whole system is viewed from above in the rotating frame; mirrors can be used to capture a side view, as shown in g.(9.5). 1. In the interior (away from the bottom boundary) the ow is independent of height. Why? Since the water has uniform density, there is no thermal wind shear - r½ =0, whence, =0. 2. Near the bottom boundary, there is in ow when! has the same sign as (cyclonic ow) and out ow when! has the opposite sign (anticyclonic). Think about our Ekman layer experiment, GFD IX. 3. There is also an Ekman layer at the top (beneath the rotating lid), in which the radial component of the ow is opposite to that at the bot-

8 220 CHAPTER 9. THE WIND-DRIVEN CIRCULATION Figure 9.6: The cyclonic rotation of the disc at the surface induces upwelling in the uid beneath (Ekman suction) as can be seen from the dome of dyed uid being drawn up from below. The whole apparatus is rotating cyclonically. tom boundary. [Can you gure out why? Can you picture the overall meridional (radial/vertical) circulation in the tank? - see g.(9.6)]. 9.2 Response of the interior ocean to Ekman pumping Interior balances Beneath the Ekman layer the ow is in geostrophic balance, eq.(8.3). How does this geostrophic ow respond to an imposed pattern of vertical velocity from the Ekman layer above? To see the e ect of w on the interior ocean we make use of eq(8.3) along with the continuity equation for the geostrophic ow: r h u =0: where we are allowing for the fact that the geostrophic ow is (slightly)

9 9.2. RESPONSE OF THE INTERIOR OCEAN TO EKMAN PUMPING221 divergent because of the meridional variation in using the above continuity equation we where = df dy v (9.5) is the meridional gradient in f which takes on a value of =10 11 s 1 m 1 in middle latitudes. If vertical velocities in the abyss are much smaller than surface Ekman pumping velocities, then Eq.(9.5) tells us that the ocean currents will have a southward component in regions where w Ek < 0 and northward where w Ek > 0, much as is observed in g Does eq.(9.5) make any quantitative sense? Putting in numbers: f =10 4 s 1, w Ek =30myr 1, h the depth of the thermocline» 1km, we nd that v =1cm s 1, typical of the gentle currents observed in the interior of the ocean on the large-scales. Before going on to a fuller discussion of the implications of (9.5) on ocean circulation, we will discuss, and illustrate in a laboratory experiment, its physical content Taylor-Proudman on the sphere Eq.(9.5) is in fact nothing more than a consequence of the Taylor-Proudman theorem on the sphere. If the ocean were homogeneous, we know - see section that steady, inviscid, low-rossby-number ow of such a uid obeys the Taylor-Proudman theorem: (2 r) u =0: (9.6) Thus, the velocity vector does not vary in the direction parallel to the rotation vector and ow must be organized into columns parallel to, an expression of gyroscopic rigidity. Now, consider what happens to a such a column of uid subjected to Ekman pumping at its top. According to (9.6), we might think that mass continuity would be satis ed by uniform ow out of the column, as sketched in Fig.(9.7). This satis es the constraint that the ow be independent of height, but cannot be sustained in a steady ow. Why not? If the ow is axisymmetric about the circular column, it will conserve its angular momentum density m = r 2 + v 0 r,wherev 0 is the azimuthal component of ow

10 222 CHAPTER 9. THE WIND-DRIVEN CIRCULATION Figure 9.7: around the column, and r is the column radius. The column must continuously expand as uid is being pumped into it at its top; thus, r must increase with time, and so v 0 must change as v 0 = m r2 : r After a long time, v 0 ' r, sov 0 must become increasingly negative as r increases. This is obviously inconsistent with our assumption of steady state. So what else can happen? Consider Fig.(9.8)a. which shows our Taylor Column on the sphere. (We have obviously exaggerated the depth h of our ocean in this gure!) The Taylor columns are aligned parallel to the rotation axis, as shown; they have length d = h (9.7) cos # if the shell is thin (this is inaccurate very close to the equator). Here # is the colatitude. As shown in the gure, the columns have greatest length near the equator. Therefore, if subjected to systematic Ekman pumping, a uid column can expand in volume, without expanding horizontally (which we have seen is not allowed in steady state), by systematically moving equatorward. 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 lled by the pumping down of water from the surface; this is how the wind, through the Ekman layers, drives the circulation in the interior of the ocean - see g.9.7b.

11 9.2. RESPONSE OF THE INTERIOR OCEAN TO EKMAN PUMPING223 Figure 9.8: The pumping velocity (projected on to ) is exactly equal to the change in the length of the column, following the uid meridionally. Let s think about this process in more detail. If the Ekman velocity (projected on to ) just lls the gap created by the change in d following the column, then - see g.(9.8): w Ek cos # = Dd Dt = = D# Dt µ h sin # = v cos 2 # a µ h sin # cos 2 # (9.8) where (9.7) has been used. Here v = a D# is the meridional velocity of the Dt column (note that v is positive if the column moves poleward and #, the colatitude, increases equatorward). Rearranging (9.8), multiplying both sides by 2, wemaywriteitinthe form: where v = f w Ek h f =2 cos # =2 sin ' (9.9) is the Coriolis parameter and

12 224 CHAPTER 9. THE WIND-DRIVEN CIRCULATION Figure 9.9: The mechanism of wind-driven ocean circulation can be likened to child s spinning top. The tight pitch of the screw thread (analogous to rotational rigidity) translates weak vertical motion (Ekman pumping) in to rapid horizontal swirling motion (ocean gyres). df dy = 1 df a d' = 2 cos ' ; (9.10) a is the meridional gradient of the Coriolis parameter. Here y is our coordinate increasing northwards given by dy = ad#. Note that eq(9.19) is just eq.(9.5) = w Ek h This simple mechanism is the basic drive of the wind-driven circulation; the gentle vertical motion induced by the prevailing winds, w Ek, is ampli- ed by a large geometrical factor f = a tan ', to create very much larger h h horizontal currents v, given by (9.19). The sti ness imparted to the uid by rotation results in strong swirling motion as the Taylor Columns are squashed and stretched. There are two useful mechanical analogies: 1. pip icking: a lemon seed shoots out sideways on being squashed between nger and thumb - see g.(9.8)b. 2. a child s spinning top: the pitch of the thread on the spin axis is very tight so rapid turning motion results when it is pushed down - see g.(9.9).