6. Sverdrup Balance A

6. Sverdrup balance a. Introduction b-1. Step 1 b-2. Step 2 c. Answer a. Introduction It is widely known that the western boundaries in the mid-latitudes of the Pacific Ocean, the Atlantic Ocean and the Indian Ocean are occupied by strong currents flowing to higher latitudes at speeds up to 1 m s-1, while no strong current flows along the eastern boundaries. The western boundary currents are named the Kuroshio in the North Pacific, and the Gulf Stream in the North Atlantic. This east-west asymmetry had been a mystery, until Stommel explained it by the latitudinal gradient of the Coriolis coefficient in late 1940s. The western boundary currents are actually returning sea water, which flows to lower latitudes in the wide regions of the oceans east to the western boundary currents. Thus, the key players are not the western boundary currents but wide and slow ocean flows in the rest of the regions. We consider the North Pacific and Atlantic, over which the mean winds are westerly in higher latitudes (around 40oN) and easterly in lower latitudes (around 20oN). The western boundary currents are northward and compensate the southward transport in the wide regions. We have already the basis for the first step needed to solve this problem: i.e., it is Ekman transport in section 3. The steady case in a finite depth layer, which is presented in Problem A, is the most useful. You are asked to review this part first and show that sea water converges in Ekman layer between the westerly and the easterly. The other necessary step is Rossby wave theory in section 5. Potential vorticity receives the effect of vertical compression caused by Ekman transport, which is compensated by the effect of meridional velocity, so that a steady circulation is achieved in the wide region east to the western boundary current. The platform used in this section is the reduced gravity configuration, which was introduced in section 2 for the geostrophic adjustment problem and also in section 5: i.e., an upper layer is active on an infinitely deep and motionless lower layer. b-1. Step 1: Ekman convergence Let's consider the region in Fig.6-1 with the meridional length to be 2Ly, and the wind stresses to be + τx and − τx at the northern and southern boundaries, respectively. The wind stresses are assumed to have uniform meridional gradients. We neglect the particular effects associated with Ekman transport interacting with the northern and southern boundaries, and consider that Ekman trasnport is continuous there. The Coriolis coefficient is taken as f0 at the mid-point with the meridional gradient to be β. What are the meridional Ekman transports on the northern and southern boundaries? What is the convergence in Ekman layer over the wide region? You can refer to (3.1) and show the transports independent of a thickness of Ekman layer. The transports are southward − τx /[ρ0(f0+ βLy)] at the northern boundary and northward + τx /[ρ0(f0− βLy)] at the southern boundary. The term βLy associated with the gradient is not necessarily smaller than the mean value f0, whereas we omit the gradient term for simplicity. The convergence is the difference in the transport divided by the length to be a vertical velocity, τx /(ρ0 f0 Ly). Once you show the convergence, the next step is to obtain vorticity input by using the conservation of potential vorticity in Appendix A. b-2. Step 2: Potential vorticity input and balance Underneath the Ekman layer, the upper layer receives water from the Ekman layer and increases thickness, as shown in Fig.6-2. Then, a water column expands vertically and receives geostrophic adjustment. As the system approaches a steady state, the water column expands horizontally and is equivalently compressed vertically. Thus, potential vorticity is reduced in the northern hemispere. By referring to (5.2b) in section 5, we now evaluate a rate of change in potential vorticity. The left-hand-side is vertical velocity multiplied by −1 and divided by the upper-layer thickness, H, and hence, it results in −τx /(ρ0f0Ly). The other input is induced by meridional velocity v to be –βv/f0, as shown in section 5. Since no net input should occur at a steady state, the potential vorticity balances as /( f L ) v x 0 0 y 0 (6.1) H f0 The meridional velocity is derived from (6.1) to be v x (6.2) 0Ly H which represents a southward flow in the wide region of the basin. The interface structure is related to the meridional velocity through (5.1) in section 5 and shown as f 0 x (6.3) x gLy H Thus, the upper layer is thinner (thicker) in the more eastern (western) region. It is also possible to derive this relationship from (5.5) in section 5, by adding a wind forcing term in the right-hand-side. You are asked to work on this formulation and then check Answer. c. Answer Once we include the wind forcing effect into (5.5), the right-hand-side receives a negative term due to Ekman convergence in the northern hemisphere, x C (6.4) t x 0 f0Ly At a steady state, the first term vanishes, and we have a zonal gradient of the interface depth, x (6.5) x C 0 f0Ly 2 Using the relationship derived in section 5, we can substitute Cβ to be −βR , where 1/2 R=(gΔρH/ρ0) /f0 , and confirm the relationship in (6.3). .

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