Lecture 14. Equations of Motion Currents With Friction Sverdrup, Stommel, and Munk Solutions Remember that Ekman's solution for wind-induced transport

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1 Lecture 14. Equations of Motion Currents With Friction Sverdrup, Stommel, and Munk Solutions Remember that Ekman's solution for wind-induced transport is which can also be written as (14.1) i.e., #Q x,y = M x,y. Equations (14.1) represent wind-induced transports per unit length of coastline 2 and have units of L /T. Now, instead of assuming spatially uniform wind as Ekman did, if we take into account horizontal shears in the wind field, then we will arrive at the solution obtained by Sverdrup. Consider then the following situations of an eastward wind stress that is sheared in the N-S direction in the northern hemisphere: ) Q ) Q x y x y N E Note that the situation on the left causes transport convergence, and that on the right yields transport divergence. These effects are elucidated from equations (14.1) by differentiating the first with respect to y to represent the N-S shear of an E-W wind, and by differentiating the y component of (14.1) with respect to x to represent the E-W shear of the N-S wind. This differentiating procedure yields 1

2 which, upon addition yield (14.2) Note that 3 is the change of the Coriolis parameter with latitude (,f /,y), which will be considered negligible for now. We will return to the full form of this equation, which is one form of Sverdrup's equation. Then, equation (14.2) with 3 = 0 describes transport convergences/ divergences (first term on the left hand side) as produced by the curl of the wind stress (term on the right hand side). From the diagram above, it is seen that,) y /,x = 0. Then, for the situation on the left,,) /,y is positive and (14.2) describes a negative divergence, i.e., x convergence. Similarly, for the situation on the right,,) /,y is negative and (14.2) describes x a divergence. Convergences and divergences cause sea level slopes, which in turn set up pressure gradients. Convergences tend to pile up water, while divergences set-down the sea surface. These sea level slopes will support the development of a geostrophic flow (u g) in the waters between the surface and bottom Ekman layers, as illustrated below z ) V Press.Grad. Force y Q D x E š u g Note that the geostrophic flow produced by the sea level slope is in the same direction as the wind stress, and that the net transport in the bottom Ekman layer is to the left of the geostrophic flow. Equation (14.2) can be simplified with the use of continuity (first term equals zero) to obtain 2

3 Sverdrup's meridional transport (parallel to meridians) Q : y (14.3) This is the familiar form of Sverdrup's equation. To obtain the zonal transport Q, Sverdrup used x the principle that allowed the development of the expression for Q (14.3), i.e., continuity: y With continuity and (14.3), Sverdrup obtained the following relationship: (14.4) by neglecting the change of meridional winds with latitude (,) /,y 0). Equation (14.4) says y that the zonal changes of the zonal transport are given by the curvature of the zonal wind with respect to latitude. To solve (14.4), Sverdrup integrated it from an eastern boundary where Q x equals zero, to a distance -x away from the boundary, or (14.5) which describes the zonal transport induced by wind stress. Equation (14.5) indicates that the transport is related to the meridional curvature of the zonal wind stress. This concept is illustrated in the figure presented in class. As seen, the transports predicted by (14.5) are qualitatively similar to those associated with the equatorial current system. Equation 14.5 also indicates that transports increase with distance from the coast. Sverdrup's circulation model allows the presence of a boundary and of horizontal shears in the wind stress (in contrast to Ekman's). However, Sverdrup's solution is limited to the eastern side of the oceans, i.e., nothing is known about the western parts. Also, Sverdrup's model depicts depth-integrated flows, i.e., nothing is known about the vertical structure of the currents. Stommel explained what happens in western portions. Stommel Solution 3

4 Stommel elucidated why currents are fast and narrow in the west and broad and relatively slow in the east. Stommel used Sverdrup's equation and included bottom friction. He calculated vertically integrated flow patterns over an idealized ocean (process-oriented study) with a prescribed wind forcing, which was a function of latitude, under a variety of conditions: no rotation (f = 0), constant Coriolis parameter f, and Coriolis parameter varied constantly as a function of latitude, i.e., 3 constant (3-plane approximation). His results were similar for the first two situations (f = 0, and f = constant). With the 3 effect, his results yielded a westward intensification of currents. Then, the change of f with latitude is the main responsible for westward intensification of oceanic currents as seen in the Figure presented in class. Stommel used wind patterns between 15 and 45 N. His addition of friction allowed a closed circulation across the entire basin, in contrast to Sverdrup's solution (restricted to eastern boundary). Stommel's result can be understood in terms of vorticity tendencies. We will digress to discuss vorticity and then we will come back to discuss Stommel's results in terms of vorticity. Vorticity Vorticity is the tendency for portions of a fluid to rotate. Vorticity is related to horizontal shears in the flow. We will consider the following types of vorticity: relative vorticity, planetary vorticity, absolute vorticity, and potential vorticity. Consider the following situations of flow horizontal shears that induce vorticity œ œ N E u v Relative vorticity is defined as the curl of the horizontal flow, i.e., (14.6) 4

5 Note from the above diagram that in the upper two cases, the relative vorticity is positive. On the upper left,,u/,y is negative and induces a counterclockwise or positive vorticity, which corresponds with (14.6). On the upper right,,v/,x is positive, thus inducing counterclockwise rotation. Similarly, the lower cases represent negative or clockwise vorticity as explained by (14.6). On the lower left,u/,y is positive, and on the lower right,v/,x is negative. Planetary vorticity is that arising from the Earth's rotation and equals the Coriolis parameter f. A stationary object on the surface of the Earth has planetary vorticity, which varies with latitude. Absolute vorticity is the addition of planetary plus relative vorticities. The changes in absolute vorticty are useful to help us understand the tendencies for parcels of water to rotate. To describe changes in absolute vorticity, we have to take the curl of the frictionless equations of motion The curl is obtained through cross-differentiation (u component with respect to y and v component with respect to x) and subtraction, to obtain (14.7) Equation (14.7) describes changes of absolute vorticity in time. These are related to convergences or divergences, as expressed in the right hand side of the equation. Imagine a column of water that is subject to stretching or squashing. The stretching situation is illustrated below 5

6 Gain of Absolute Vorticity Column Stretching f + œ f Plan View Side View In the stretching situation (probably due to increase in local depth), the particles at the perimeter of a given water column, will tend to move towards the center of the column (convergence) as illustrated in the plan view. Due to Coriolis accelerations, the particles will acquire cyclonic or positive relative vorticity and thus the gain of absolute vorticity. The divergence or column squashing situation is represented in the diagram below. The particles at the perimeter of the water column will now move outward. Due to Coriolis effects, the particles will acquire anticyclonic or negative relative vorticity. This negative relative vorticity results in the loss of absolute vorticity caused by the water column moving to shallow water. 6

7 Loss of Absolute Vorticity Column Squashing f f - Plan View Side View Now consider a layer of thickness D, and whose equation of continuity is (14.8) The changes of the layer thickness in time are given by convergences or divergences. This is a reiteration of the fact that convergences cause increases in D, and divergences cause decreases in D. If we combine this equation (14.8) with the equation for absolute vorticity (14.7), we get (14.9) This relationship says that the quantity (+f)/d =, which is called potential vorticity (absolute vorticity over layer thickness), remains constant. This is the statement of conservation of potential vorticity. This is a useful and powerful concept because it allows description of vorticity tendencies in the ocean. Consider the following situations in the northern hemisphere. 1) A column of water with constant thickness D. If the column moves zonally (along colatitude lines), then f remains constant and so does. If the water column moves meridionally, then if 7

8 f increases (column moves to the North), must decrease (column acquires anticyclonic motion) to keep constant. Under the same meridional motion, if the water column moves to the south, f decreases and must increase by acquiring cyclonic motion. 2) Increased water column thickness (moving towards deep regions). If the column moves zonally, f remains constant so must increase, i.e., the column acquires cyclonic vorticity (vortex stretching). Then, if the water column moves meridionally to the South, f decreases and must increase. If the column moves meridionally to the North, f increases and it is not obvious what happens to because it could increase or decrease, depending on the increases of both D and f. 3) Decreased water column thickness (moving towards shallow regions). must decrease in zonal motions (vortex squashing). For northward meridional motion, f increases and must decrease. It is not obvious what may happen in southward meridional motion. These situations derived from the concept of conservation of potential vorticity can be used to explain the westward intensification of the ocean currents. Now we return to explain why the oceanic currents are intensified to the west as Stommel proposed. Consider the area of the westerlies and the trade winds in the ocean. The distribution of the wind stress tends to cause anticyclonic (negative vorticity), as illustrated below ) Westerlies Easterlies Then, on the western part of the oceans, the wind-induced relative vorticity is the same as that in the eastern part. On the western side, however, the currents move northwards, which make f increase. To conserve, must decrease, i.e., northward motion on the western part of the oceans acquires anticyclonic (negative) vorticity. Analogously, southward motion in the eastern part of the oceans, produce cyclonic (positive) vorticity. Note that on the eastern part, the windinduced vorticity and the relative vorticity arising from southward flow are of opposite sign and tend to balance each other. On the other hand, over the western part, the vorticity induced by northward currents and the wind-induced vorticity are of the same sign. Then, to maintain vorticity balances at the east and west, there has to be a large source of positive vorticity on the 8

9 west. This source comes from strong lateral shears in the northward flow (flow increasing eastward), i.e., Western Part Wind Stress ( ) Wind Stress ( ) ) ) as f increases p Lateral Friction ( ) œ to balance... f œ as f decreases + c 0 ; ) p ) p f ) p Flow looks like p Eastern Part Lateral friction on the western part is the necessary ingredient to have vorticity balances in the ocean. That is why currents intensify to the west. Munk's Solution Munk extended the domain of Stommel to include a larger range of latitudes. He used more realistic wind patterns than those idealized by Stommel and also added horizontal shears (horizontal friction) to obtain his solution. As seen in the figure in class, Munk's solution depicts ocean gyres that approximate the actual circulation. 9