Fig. 1. Sketch of the winddriven circulation. Under the Ekman pumping (upwelling) water moves equatorward (poleward) in order to conserve f/h.


 Berenice McCoy
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1 Lecture 8. The physics of winddriven circulation 3/7/006 11:50 AM 1. Interior solution Our discussion about winddriven circulation in the previous lecture is based on detailed dynamic analysis; however, it is also very important to understanding the basic structure of the winddriven circulation without some elaborating mathematics. In this lecture, we will explain the basic structure of the circulation in terms of the fundamental physics. A) Meridional flow driven by Ekman pumping The winddriven circulation in the ocean interior can be explained by the following cartoon. In order to maintain a relatively steady rotation of the Earth, the globally integrated frictional torque exerted by the atmosphere on the solid earth should be zero; thus, both westerlies and easterlies are the necessary components of the atmospheric circulation. As shown in Fig. 1, there are prevailing westerlies at middle latitudes, and easterlies at low latitudes and polar regime. Fig. 1. Sketch of the winddriven circulation. Under the Ekman pumping (upwelling) water moves equatorward (poleward) in order to conserve f/h. This wind stress pattern drives a poleward Ekman flow in both low and high latitudes, but it drives an equatorward Ekman flow at middle latitudes. As a result, the meridional convergence of Ekman flux gives rise to the Ekman pumping and upwelling below the base of the Ekman layer. In the subtropical basin, a downward Ekman pumping induces a compression of the water column. In the basin interior, the relative vorticity is negligible, so potential vorticity for a water column is f/h. Ekman pumping compresses the water column height h. In order to conserve the potential vorticity f/h, individual water column moves toward the equator where the Coriolis parameter f is smaller. Thus, Ekman pumping drives an equatorward flow in the ocean interior. Similarly, the Ekman upwelling in the subpolar basin drives a poleward flow in the ocean interior. B) Close the circulation with the western/eastern boundary layers 1
2 The above argument applies to the direction of the meridional flow in the ocean interior only, and we still do not know how to close the circulation. Obviously, in order to have a steady circulation in the subtropical basin we must find out a way to transport the water mass poleward. Since our model is twodimensional, the poleward flow must be either in forms of a western boundary current or an eastern boundary current where some additional dynamics of higher order can play the role of completing the circulation. From the view of simple mass conservation, thus, either a western boundary current or an eastern boundary current are equally possible candidates, as shown in Fig.. However, as will be shown shortly, only the western boundary current can play the role of completing the vorticity balance in a closed basin. Fig.. Two possible circulation patterns driven by Ekman pumping.. An integral potential vorticity constraint for the steady circulation In the discussion about the interior solution, we do not pay much attention to the possible roles played by the friction because frictional force is negligible in the basin interior. However, for the circulation in a closed basin, the frictional force may play a vitally important role, and this can be seen clearly through the momentum torque balance integrated over the whole basin. For steady circulation, potential vorticity equation conservation law states f + vx uy hu = C (1) h where y x τ A κv τ A κu C = + ( hv) + ( hu) () ρ0h h h ρ x 0h h h y is the source of potential vorticity due to wind stress curl and the vorticity sinks due to the lateral and bottom friction. We integrate the potential vorticity equation (1) over area A ψ defined by a closed streamline C ψ, Fig. 1. Using the continuity equation ( hu ) = 0, the integration of the lefthand side gives f + vx u y I = hu dxdy = u ( f + vx uy) dxdy Aψ h A (3) ψ
3 Since the lateral boundary is a streamline. i.e., u n= 0, the divergence theorem leads to I = un( f + vx uy) ds = 0 (4) Cψ Fig. 3. A line integral going through the interior and western boundary of the model basin. Thus, the final balance is τ u ( hu) ds = κ ds A ds C ρ C C 0h h (5) h ψ ψ ψ Namely, the momentum torque balance for a closed streamline (or the whole basin) is between the frictional torque (mostly generated along the edge of the basin where the friction is nonnegligible) and the wind stress torque imposed over the basin. Remarks: 1) No matter how small is the frictional force, it is essential for the basinwide vorticity balance between wind stress input and frictional output. ) Purely inertial model is physically impossible! In the studies of the theory of inertial western boundary layer, it was shown that purely inertial boundary current could not exist in the northern part of the western boundary region where the currents are leaving the coast to join the interior. To close the circulation, an interesting solution was postulated where the subtropical gyre and subpolar gyre are joined together by an eastward internal jet going through basin at middle latitudes, Fig. 4. 3
4 Fig. 4 Midocean jets proposed to close the winddriven circulation with purely inertial boundary currents. However, a careful examination reveals that such a model is physically troublesome: 1) Vorticity is not balanced: there is only negative vorticity source from the wind stress, but no vorticity sink. Anticyclonic vorticity would constantly accumulate in the model, making a steady state impossible to reach. ) Similarly, energy in the model would grow unbounded. 3) Near the midocean jet strong source of positive vorticity is required. For example, the southern half of the midocean jet has strong negative relative vorticity; however, the relative vorticity is nearly zero in the ocean interior. Thus, in the transition from the midocean jet to the southward interior flow, a strong positive vorticity source is required in such a model. This argument also applies to the junction between the interior flow and the northern half of the midocean jet. There is no such vorticity sources in the oceans, so the model formulation is physically inconsistent. 3. Vorticity dynamics of boundary layers In Section 1 the interior solution was obtained by integrating from the eastern boundary, but why do not we start from the western boundary? As will be shown shortly, there is no eastern boundary current, so interior solution is valid all way up to the eastern wall; thus, solution in the interior can be obtained by integration starting from the eastern boundary. On the other hand, there is western boundary, so we could not obtain the interior solution by integration from the western boundary. If we do want to start this way, we would have to integrate the western boundary layer solution first, then add on the interior solution when we reach the interior. As discussed above, the western boundary solution itself depends on the interior solution. A) Vorticity transport within the boundary current Here we follow Stewart's argument. Let us discuss the vorticity balance for a narrow boundary current moving along the western boundary, and the right edge of the boundary current is defined as the point b where the meridional velocity is zero, Fig. 5. 4
5 Fig. 5. Sketch of the western boundary current. Using the boundary layer approximation, we can write ς = vx uy vx The total vorticity contained in the boundary layer and the total vorticity transport within the boundary layer are b b I = ς dx v dx = v( b) v(0) = v(0) b ς x b 1 I = v dx vv ( ) 0 0 xdx= v b Assume that noslip condition applies, then I1 = I = 0 Thus, the whole boundary layer contains no relative vorticity and does not transport vorticity in the meridional direction. However, if we divide the boundary layer into two branches, the right branch R and the left branch L, which separated at the place where the meridional velocity reaches local maximum, v M, we can show that I1R = vm < 0 1 IR = vm < 0 Thus, R does transport large amount of negative vorticity northward. I1L = vm > 0 1 IL = vm > 0 Therefore, the transports in the left branch are positive. This corresponds to the case of slip boundary condition, if we treat the Lbranch as a very thin boundary layer adhered to the wall. As the currents move poleward, the amount of vorticity contained in the boundary current and the vorticity transported by the boundary current change accordingly. Such changes are closely associated with the frictional torque produced within the boundary current, as discussed below. B) Vorticity balance in a closed basin. 5
6 Let us concentrate on a subtropical basin. Water continuously gains negative vorticity from the upper boundary. The circulation in the ocean interior is very slow, the relative vorticity is negligible; thus, water moves southward to a place where the planetary vorticity is smaller. To close the circulation water with low vorticity has to move northward along the eastern/western boundary and eventually rejoin the interior flow where the water should have high vorticity. That is to say water must give up the negative vorticity somewhere in order to maintain a steady circulation. Taking a control volume in the southern part of the western boundary between two zonal sections through the western boundary, there is an influx of negative vorticity by the incoming flow from the ocean interior. Since boundary layer does not carry vorticity north/southward through the zonal sections, in order to balance vorticity water must gain positive vorticity either through the lateral boundary or the lower boundary. Thus, no matter what kind of model we use, there should be always a place where positive vorticity is generated through either friction or intensifying shear to counterbalance the wind stress curl input vorticity in the interior. C) Possible source of positive vorticity for a subtropical basin: a) Lateral boundary layer Munk model As an example, we discussed the case with a noslip boundary condition. (Note that y y although the frictional stress τ = 0 at x=0 for a slip boundary condition. τ / x is maximal at x=0, and this is the term in the ymomentum equation and similarly for the vorticity balance!) The frictional torque reaches maximum near the western wall, so a positive frictional torque is produced that balances the negative vorticity input in the basin interior, as shown in the left panel of Fig. 6. Fig. 6. Vorticity balance for a western boundary current with lateral friction. On the other hand, an eastern boundary current would be unable to produce the positive momentum torque needed for a basinwise vorticity balance. Thus, western boundary current is an essential component of the vorticity balance in a closed basin, and the eastern boundary current is dynamically much less important. b) Bottom boundary layer  Stommel model 6
7 The frictional torque needed for balance the negative wind stress torque imposed on the basin interior is again generated along the western boundary where the strongest boundary current produced the strongest friction, and hence the positive momentum torque, Fig. 7. Fig. 7. Vorticity balance for a western boundary current with bottom friction. As an example, the structure of the Stommel boundary layer, based on the same parameters used in Lecture, is shown in Fig Thus, strong meridional boundary current produces a positive bottom fiction torque that is one of the essential ingredients for the vorticity balance in the model basin. The meridional velocity v is maximal at the western wall; however, the frictional torque Q maximum is slightly off the wall. Fig. 8. Structure of the western boundary based on the bottom friction, the solid line for v (in 3 cm/s ) and the dashed line for the frictional torque in 10 cm / s. D) Application of the potential vorticity integral 7
8 The potential vorticity constraint is one of the fundamental constraints for circulation in a closed basin. For example, in the Arctic, observations indicated the existence of a broad cyclonic boundary current system along the edge of the basin. From 14 numerical models, which are forced with the seemingly same boundary conditions and bottom topography, half of the models produced cyclonic boundary currents, but the other half produced anticyclonic boundary currents. This puzzle has not been fully understood. Yang (005) has recently posed a theory based on potential vorticity balance of the Arctic basin. According to his theory, the direction of the boundary currents along the rim of the basin is directly linked to the next exchange of potential vorticity between the Arctic Ocean and the environment, including the North Atlantic, the North Pacific (through the Bering Strait), and the river runoff into the basin. Yang (005) has demonstrated that: if the next exchange of PV with the environment is positive (negative), the boundary currents along the edge of the basin is anticlockwise (clockwise). As discussed above, only a cyclonic boundary current can generated negative PV to balance the positive source of PV from exchange with its environment. However, the whole story about what went wrong in these models remains illusive because the exchange of PV with the environments is related to the circulation dynamics itself. In fact, the most important source/sink of PV for the Arctic Ocean is associated with inflow of the saline water from the NorwegianGreenland Sea and the outflow of cold and freshwater through the Fram Strait; but, these exchanges depend on the basin scale circulation itself. As a result, finding out and fix this problem for Arctic Ocean circulation remains an exiting challenge. E) Inertial western boundary layer as a mean of a partial closing f + v Potential vorticity is conserved along a streamline, i.e., x = Gy ( ). In order to balance the h mass a northward boundary current is required. Since f increases northward, the potential vorticity equation requires a negative relative vorticity to balance the increase of planetary vorticity. This can be achieved by horizontal convergence in the southern part of the western boundary. Similar to the cases of frictional boundary currents, only an inertial western boundary current can produce the needed negative relative vorticity. Here again, the vorticity balance rules out the possibility of closing the gyre circulation with an eastern boundary current. In the northern part of the western boundary, flow is divergent and the relative vorticity declines; thus, a purely inertial boundary current in this part of the basin is unable to compensate the further increase of planetary vorticity f. Consequently, the purely inertial western boundary current does not work for the northern part of the western boundary. The structure of the western boundary layer is shown schematically in Fig. 9. 8
9 Fig. 9. Different dynamic zones within the western boundary regime. In addition, within the framework of purely inertial boundary current there is no communication of vorticity between the current and the solid boundary, such as the bottom and the lateral walls, so the model cannot get rid of the negative vorticity imposed in the interior. Thus, a purely inertial model cannot satisfy the vorticity balance in a closed basin. There must a place where other mechanisms of exporting negative vorticity works out and establish the vorticity balance in the whole basin. 4. Energetics of the winddriven circulation A) GPE of the thermocline in the reduced gravity model For a simple reduced gravity model, the free surface elevation η is link with the layer Δρ thickness h: η = h, Fig. 10. Therefore, the center of mass is at a position ( h/ δ h), where ρ0 η Δρ δ h= = h. GPE depends on the choice of a reference state, and in the present case we choose ρ a reference state in which the density is ρ everywhere, so the corresponding free surface elevation is zero. Using this as reference state, GPE for the warm water above the thermocline is 1 Eρ = δh ρgh= ρg' h, where we have used the relation ρh= ( ρ Δ ρ)( h+ η), which is the statement of zero pressure gradient below the thermocline. Therefore, in a reduced gravity model, increase of h means increase of GPE. Physically, increase of h indicates the increase of the free surface elevation η. 9
10 Fig. 10. Sketch for a reduced gravity model. B) Balance of GPE in a winddriven gyre Based on the Stommel model discussed in the previous section, the mechanical energy balance equation can be obtained by multiplying the x momentum equation by u and the y momentum equation by v and adding up the results: u Ep = u τ κρ( u + v ) a). Interior solution In the basin interior, wind stress does work on the geostrophic velocity, u τ > 0. Thus, u h> 0, i.e., h( η ) increases down stream; or wind stress input energy by pushing water toward high GPE regime, depicted by the arrows on the top of the surface in Fig. 10 and the arrows on the right panel of Fig. 11. Note the minor difference in the direction of the flow for the southern half of the subtropical gyre illustrated in Fig. 10 and 11. The streamlines shown in Fig. 11 are the Sverdrup streamfunction, so it includes both the Ekman flux and the geostrophic flow below the Ekman layer. Although these two components have similar flow direction for the northern half of the subtropical gyre, their directions are opposite in the equatorward half of the subtropical gyre. As indicated in the lower panel of Fig. 10, the Ekman flux flows toward the pole and the subsurface geostrophic current moves toward the equator. The surface velocity is the vector sum of these two components, and it receives energy from wind stress. Energy input into the Ekman layer is stored in the subtropical basin through the Ekman pumping and pushed down into the bowshaped main thermocline, as depicted by the downward arrow in Fig
11 Fig. 11. Schematic uphill and downhill flows in the Stommel model. b) In the western boundary The wind stress work term is negligible here, so the balance is now between the friction term that is negative definite. Thus, water parcels must loss their GPE and thus going toward lower surface elevation and shallow thermocline depth, as shown by the arrows in Fig. 4.a. c) Physical interpretation of the bottom friction The socalled bottom friction in the Stommel model is a crude parameterization of baroclinic instability. Therefore, GPE loss to bottom friction can be interpreted as loss to baroclinic instability within the western boundary. Furthermore, baroclinic instability in the oceans do not take place along the western boundary of the basin; instead, it takes place primarily within the outflow regime, such as the Gulf Stream Extension or the Recirculation. 4. Rossby waves and Western intensification A) Dispersion relation In quasigeostrophic theory we have derived the quasigeostrophic potential vorticity conservation f L ( ψ + βy Fψ) + J( ψ, ψ + βψ Fψ) = 0, F = (6) t gd Assume a single frequency wave ψ = Acos( kx+ ly σt+ φ) (7) The dispersion relation is β k σ = (8) k + l + F The phase velocity is β c x = < 0 (9) k + l + F 11
12 The group velocity is σ l + F k cgx = = β k ( k + l + F) σ kl cgy = = β l ( k + l + F) (10) (11) Thus, for long waves, which satisfy k < l + F, the wave energy or the wave packet moves westward; while the short waves, which satisfy k > l + F, the wave packet moves eastward. This can be shown geometrically with an energy propagation diagram. Fig. 1. The Rossby wave diagram. The dispersion relation can be rewritten as β β k+ + l = F σ 4σ β Thus, the tip of the wave vector should correspond to a circle at,0 σ β F. The wave vector is in the direction of 4σ g y kl tanθ = = gx k l F As shown in the figure, OW is the direction of group velocity. (1) and of radius (13) B) Reflection For wave energy, αo = αi. Thus, Rossby waves in the oceans behave like this: eastward short waves ==> westward long waves westward long waves ==> eastward short waves 1
13 C) Western intensification Wind field in the atmosphere is not steady, it consists of very broad spectrum in time and space. As discussed above, small scale wind stress patterns create Rossby waves of short scale, which propagate eastward; at the eastern boundary of the basin, Rossby waves reflected back as long Rossby waves which propagate westward; while large scale wind stress pattern in the atmosphere create long Rossby waves which propagate westward. At the western wall all these long waves reflect back as short waves. However, short waves are much vulnerable to friction and diffusion, thus they lose their energy within short distance of the western wall. In this model, the western intensification is seen as the local accumulation of small scale energy. D) Boundary layer scaling According to the above argument, the western boundary current is the site where the wave energy is dissipated. The dissipation time scale is 1 ψt AH ψ Td O Ak (14) H Since the group velocity is c = β k, the length scale is / gx 0 4 / gx d = β0 H (15) l c T A k Thus, energy is trapped within the scale of 1/3 1 A l k l H (16) β0 This is the scale of the Munk layer. Similarly, the scale of Stommel layer and inertial layer can be estimated. Fig. 13. Rossby wave reflection in a basin and the western boundary current. Reference Yang, J.Y., 005: The Arctic and Subarctic ocean flux of potential vorticity and the Arctic ocean circulation, J. Phys. Oceanogr., 35,
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