The wind-driven models of Stommel and Munk employed a linearization involving a small parameter, the Rossby number, which we need to reconsider.

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1 Equatorial twists to mid-latitude dnamics As we saw or Stommel s or Munk s wind-driven gres and or Sverdrup s balance, there was no particular problem with the equator. In act, Stommel solved his gre or no rotation o the earth at all. But we do need to return to the barotropic potential vorticit balance and discuss one aspect (involving lineariation) that is dierent in equatorial dnamics. The wind-driven models o Stommel and Munk emploed a lineariation involving a small parameter, the Rossb number, which we need to reconsider. Recall ζ [ u / L] h [ h] [ H [ ] H [ H ] + ζ ] [ + h H 1 + ( u / L) ][ ] [ 1 + ( h / H ) H R [ u / L] << 1, and [ h / H ] << 1 ], i Clearl near the equator, becomes arbitraril small and this deinition o the Rossb number leads one to suspect that the low cannot be linear. However, we can i this b recogniing that or stead lows: ζ = β ; [ β ], [ u/ L ] d 2 [ + ς] = vβ + ( uς + vς ) [ βu][1 + ( u/ βl )] dt 2 d + ζ βu 1 + ( u/ βl ) βu i [ ] [ ][ ] [ ], dt H + h H 1 + ( h / H ) H E R u L << and h H << 2 [ / β ] 1, [ / ] 1 For equatorial dnamics a new parameter R E must be small or the low to be linear in the same sense as or mid-latitudes. This is somewhat less stringent than or mid-latitudes. Tr putting some values into the two dierent

2 Rossb numbers and see which one is larger. Recall the deinition o β. At the equator it has a value o approimatel (ms) -1. The equations o motion near the equator can be written using the equatorial β-plane as ollows: du dt dv dt = g, ρ u v w + + =, d dt βv = + βu = ru +, ρ ρ rv + ρ ρ + u + v t + w Stead, linear low near the equator demands that terms with (d/dt) are set to ero. The irst two equations then reduce to the ollowing: βv = ru + ρ ρ + βu = rv + ρ ρ At the equator, the let hand sides o both o the above must vanish. This results in a balance in which ver near the surace, where the pressure gradient is small, the currents are down-wind. Where the pressure gradient is larger than the wind orcing, the currents are then down-pressure gradient. While Stommel (196, DSR) seems to have been the onl one to eplore this or the case in which the wind stress is onal and uniorm with latitude, this balance lies at the heart o the description o surace and subsurace lows at the equator, and the simpliied eplanation or the equatorial undercurrent (EUC) in Ocean Circulation. We will look urther along these lines shortl.

3 The EUC occurs at a depth where horiontal pressure gradients (higher pressure on western sides o the basins due to westward wind stress at the surace piling up water) drive an eastward low at depth. It also can appl to the meridional (north/south) lows: at the surace, where wind stress is blowing to the north across the equator, there is a surace low in the direction o the wind (South Equatorial Current, SEC), but at depth, pressure gradients ma drive an opposing low to the south. Indeed, there is evidence or mean meridional slopes o the sea surace across the equator, with generall higher sea surace height to the north as a consequence o positive meridional wind stress at the equator. Recall that the ITCZ is, in the mean, located to the north o the equator. The second equation above has been used to calculate a geostrophic onal velocit AT the equator. (Jerlov, 1953, Tellus) was the irst eplore this balance. It can be most simpl stated b neglecting the eects o riction (proportional to r ) as ollows. Let u = u g + u. In the second momentum equation above, we now have + ) g ρβ ( u + u = p + I we take the derivative o both sides o the above and evaluate it at the equator (=), this equation becomes + ρβu + ρβu g = p = + This balance assumes that = p + and that u is inite. The irst o these three equations is oten used to calculate geostrophic low at the equator, but the additional two constraints are usuall ignored. This is discussed urther b Joce (right, that s me!, 1988, JPO). A recent (as et unpublished) stud o Sverdrup dnamics in the equatorial Paciic b Kessler can be ound on the web at

4 Equatorial Dnamics 11 To eamine the onal momentum balances at the equator, we will use the simpliied dnamics contained in = p + Z=h Z= Consider the igure at the right, which we alread have used in previous notes. I we think o ρ 1 ourselves looking at the equatorial Paciic Ocean (or eample) rom a point in the northern hemisphere, the sea surace slopes upwards to the west under the trades and the pcnocline underneath slopes oppositel so as to make or no ρ 2 Z= -H Z= -H or reduced pressure gradients at depth. In the upper laer, which can be in motion, we have p =ρ 1 gh. This is constant throughout the upper laer since the densit is assumed constant. I we integrate the above equation rom the ree surace to a depth within the upper laer, we obtain = ρ g( h ) h + ( = h) ( ), but 1 ( ) = r( ) u( ), where r( ) is the riction parameter, thus ru ( ) ( ) = ( = h) ρ gh ( h ) 1 I we knew something about r(), we could solve or the onal current variation with depth using the last o the equations above. Near the surace (=h), the second term on the right o the last equation is small and the low is in the direction o the surace stress (to the west since (h) is negative on the equator). This is the westward SEC. In this simple model, the vertical integral o the onal low must vanish since there is no source or inlow rom the sides or thru the bottom o the upper laer. Near the bottom o the upper laer, r() becomes small, approaches H, the second term on the right is larger than the irst and the low becomes positive (and possibl large). This is the EUC. The depth o no onal low (between the SEC and

5 EUC), called (= ) can be written down without knowledge o the riction parameter. = r( ) u( ) = ( = ) ρ1g( h ) h, or It is merel the depth where the surace stress eactl balances the horiontal pressure gradient. This can be solved or and estimated readil rom data. As ou will recall rom an earlier discussion, the variation o surace elevation h is much smaller than that o the upper laer depth H b a actor o ρ/ρ. So we could also simpli the above even more b replacing ρ 1 h b ρh and using a igure such as ig. 5.4 (p. 148 o 2 nd ed. tetbook). Let s tr putting in some numbers and seeing what we get. From the above, we can solve or. ( = )/ gh, or h/ << 1, 2.4 nt / m, 6 h.5 m / 9km =.56 1, thus 74 m, which is indeed much larger in magnitude than the sea surace elevation. This depth separates the westward lowing South Equatorial Current on the equator rom the eastward lowing Equatorial Undercurrent. It is not a bad estimate looking at the upper panel o ig. 5.5 in the tet. Observations o the Equatorial Ocean Because o the legac o the Tropical Ocean, Global Atmosphere (TOGA ) Program, we have learned a great deal about equatorial dnamics and the reasons or changes in the equatorial circulation. One such change, in which the trades weaken signiicantl across the Paciic Ocean, the thermocline slope is greatl reduced rom west to east, and the upwelling o cold water in the eastern Paciic Ocean is terminated, is something called El Niño. This is discussed in the tet. Because o the demonstrated human dimension o these changes, mainl due to changes in tropical precipitation, a real time monitoring sstem has been maintained in the tropical Paciic with data emploed in some predictive models o equatorial dnamics than can be

6 used to orecast the occurrence o El Niño and its opposite etreme, La Niña. Access to these data can be ound on the web at: An arra o moorings is in place in the Paciic as shown below: All o the blue dots represent surace moorings with atmospheric boundar laer and subsurace temperature and velocit measurement. On selected locations on the equator, moored ADCP (acoustic Doppler current proilers) collect current proiles in the upper ocean. All o the data are telemetered back and can be accessed in realtime. For eample, one can see the wind anomalies rom the mean seasonal ccle and SST (sea surace temperature) rom 8 October 22 (below). Winds are blowing towards a SST maimum in the western tropical Paciic and cold SSTs can be seen in the east. This is close to the normal condition since the anomalies are weak.

7 Other data can be seen showing the changes in SST and in surace dnamic height on the equator, the latter coming rom the moorings. In addition to the seasonal ccle o SST on the let, one can see interannual changes in the dnamic topograph at the sea surace (relative to 5 db) showing a large reduction in the sea surace height across the Paciic between, sa, March 21 and Sept. 22. A similar but less etensive arra is in place in the tropical Atlantic, called PIRATA ( ). The mooring arra is shown net.

8 Data are somewhat more diicult to access, and oten have gaps due to mooring ailure or damage rom vandalism. The arra is still considered a pilot arra, and there are ongoing discussions to make additions/changes. But compared to the rest o the ocean, the Paciic and Atlantic tropics are in MUCH better condition or data availabilit rom in situ instrumentation.