Wind-Driven Circulation: Stommel s gyre & Sverdrup s balance

Transcription

1 Wind-Driven Circulation: Stommel s gyre & Sverdrup s balance We begin by returning to our system o equations or low o a layer o uniorm density on a rotating earth. du dv h + [ u( H + h)] + [ v( H t y d h F v = g ru +, y h F + u = g rv + y + u + v t y + h)] =, where We recall that z = -H is the bottom o the luid layer and z = h is the ree surace. For simplicity, we will deine D to be the total depth o the luid: D = H+h. Because the bottom o the luid is not changing in time, we can replace the time rate o change o h in the third equation with the time rate o change o D. Net we deine vorticity. An important luid property associated with shearing and rotating elements o motion is vorticity, ζ. It is deined as ollows: ζ v It is a vector, but we will be considering here the vertical component o vorticity, which is z ζ v = ( u ) y This quantity depends on the shears in the low, not the low itsel. We will see that is an important quantity in a moment. But irst note in the above equations that momentum can change ollowing a luid particle because o eternal orces, riction and pressure gradients. The latter element o nonconservation can be removed by considering vorticity because i one dierentiates the second equation above by, the second equation above by

2 y and then subtracts them, the term associated with pressure gradients can be eliminated. We will not derive the ollowing equation here, but you can do this on your own [in act it may be a new homework eercise], but by using the above procedure and the deinition or vorticity, and all three o the above equations, we can obtain the ollowing without any assumptions about the Coriolis parameter being constant (ecept in time!): d z + ζ r 1 r 1 y [ ] = ζ + ( F Fy ) = ζ + ( h F) D D D D D In the absence o riction or eternal orces, the quantity [(+ζ)/d] is conserved, where we have now dropped the subscript v or vertical, which is assumed. This quantity is called potential vorticity and is one o the undamental properties used in physical oceanography. It can be changed by riction and by a curl o an eternal orce (e.g. wind stress) but otherwise is remains constant. It is so important, that it is even known by the cartoon character Dilbert (see igure shown in class). It states that, in the absence o eternal orces (or riction), luid will tend to ollow contours o constant depth in the ocean. Since depth variations occur much more rapidly that variations o the Coriolis parameter, luid parcels can cross depth contours only by developing a substantial relative vorticity. Concept o linearization: I the equations are eamined, not all terms are always o the same size. We have already recognized that overall, riction will be small and or many o the lows the basic balance will be geostrophic, or eample. Because the equations are non-linear, this makes solution very diicult. SO i some o the small terms are the non-linear ones, ignoring them can allow us to obtain a solution! Consider the quantity potential vorticity, or eample. It contains quantities which have dierent magnitudes. We will now look at some o these terms [this eercise is called scaling ]. We can denote the scale o a quantity by square brackets:

3 ζ [ u / L] h [ h] [ H [ ] H [ H ] + ζ ] [ + h H 1 + ( u / L) ][ ] [ 1 + ( h / H ) R [ u / L] << 1, and [ h / H H ] << 1 ], i The quantity R, is called the Rossby Number and is a ratio o the relative vorticity, ζ to the planetary vorticty,. For most velocities and or most horizontal length scales we will use in the course, this is a pretty good approimation (e.g. R<<1). I the total depth o the luid is much larger than the ree surace elevation changes, then the second approimation is good as well. In this case, potential vorticity simpliies to (/H) which, in the absence o other orces, must be conserved. What this means is that the luid must always low along lines o constant (/H). For an -plane, this means that luid cannot cross depth contours without some eternal orce operating. This is a pretty powerul statement! Now we will look at Stommel s (1948) eamination o wind-driven gyres. We have already derived everything we need. We will use the above linearizations to simpliy the potential vorticty (PV) equation. His model was or an ocean o constant depth, so this simpliies things even more. Since H is constant and varies with latitude (y-direction), this means that PV conservation requires that luid can low only east/west NOT north/south unless there are eternal orces. This is now d [ + ζ D z ] = r D 1 ζ + ( F D y F ), or ater scaling 1 y βv = r( v u y ) ( F Fy ) With constant depth, (u +v y =) and we can write the second equation above in terms o a stream unction ψ : (u,v) = ( ψ y,ψ ) y

4 1 y r( ψ + ψ yy ) + βψ = ( F Fy ) Stommel used a simpliied orm o the eternal orcing as well: is was a sinusoidally-varying body orce in the -direction, but varying only in the y direction. At this stage, one can associate this roughly with a meridionally-varying zonal wind stress. I this orce is given by (F,F y )=((-aπ/b)cos(πy/b,), using Stommel s notation, then the above becomes r ( ψ ψ ) + βψ = a sin( πy/b) + yy This is the equation Stommel solved which showed the dierent types o low depending on the rotation (or non-rotation) o the earth, and due to the earth s curvature. The solutions are shown in the tetbook on p. 94. We will discuss this in class. Because β is always positive, the intense western boundary low is always on the western boundary. I the wind stress is reversed, the sense o rotation o the low will be reversed but the strong boundary low will be to the south along the western boundary. This is what we would epect in the subpolar gyres (to the north o the subtropical gyre Stommel considered. This general result eplains a lot about the structure o the wind-driven circulation in both hemispheres. You might think through what happens in the southern hemisphere where is negative (but NOT β). Sverdrup s balance I we eamine Stommel s solution or the low in the case o rotation on a β- plane on p. 94, lower let, we see that ecept or the region near the western boundary o the basin, where the low is strong and to the north, the low everywhere else is zero or to the south. We can think o this interior region as somehow having dierent dynamics than the western boundary current (wbc) region. I one looks at the previous equation or the streamunction o the low, we see three terms: one proportional to riction, one having β, and a third with the orcing. The irst term becomes important near the western boundary where gradients in the streamunction are large. Elsewhere riction is less important because gradients o the streamunction (actually second derivatives!) are small. So one might imagine an interior region in which the second two terms are important and a boundary layer region in which the irst two terms are important, the orcing there having no strong variation.

5 The interior region is one that Sverdrup studied in 1947 and the balance o these two terms has become known by his name: The Sverdrup Balance. I we re-write this balance slightly, it s value will become more apparent. 1 βψ = ( h F), or H ddz v = d( F) β h I we integrate the irst equation across a basin rom east to west at any ied latitude (y=constant) and then multiply by the depth o the luid, we obtain an estimate (second equation) or the interior wind-driven transport across the basin in the meridional direction. The direction o this transport depends on the sign o the curl o the orcing, F. In the case o the Stommel orcing, the curl o the orcing is negative and thus the interior low is to the south. So we can then estimate, using knowledge about the orcing, what the winddriven circulation must look like. Whatever interior low is demanded by the Sverdrup balance must then be returned in the wbc as a boundary layer low. With knowledge about the wind stress and its curl, one can make direct estimates o the wind-driven low using the Sverdrup Balance. Below we have etracted two igures rom a manuscript written by Josey, Kent, and Taylor (JPO, submitted) that discusses wind stress rom a couple o wellknown climatologies (Hellermann and Rosenstein, JPO, 1983; and Josey, Kent & Taylor, 1998, rom Southampton Oceanogaphy Centre [SOC]).

6 Image removed due to copyright concerns. In the irst igure (above), we show some global maps o wind curl rom these two climatologies (based on ship observations over many years). Regions o negative (positive) curl are subtropical gyres in the northern (southern) hemisphere. We are not interested in the dierences in these two climatologies at this point: you can read all about this in the paper on the web at What we are interested in here are the patterns and in the zonal integration o the curl ield to produce a volume transport streamunction ollowing the Sverdrup Balance, which is shown below (also rom the same reerence).

7 Image removed due to copyright concerns. In case you are wondering about the units in the second igure, they are in Sverdrups (1Sv = 1 6 m 3 /s, also named ater guess who?). As a point o comparison, the Amazon river, which transports the most water o any river in the world, has a mean volume transport o.3 Sv. So the wind-driven low drives transports in the ocean that are a hundred times larger than the Amazon outlow. As we will see later, the actual transports in wbc s are substantially larger than what is anticipated by the Sverdrup balance. But this is not going to deter us & we will not throw out this valuable tool or understanding the wind-driven circulation.

8 The Sverdrup circulation is constructed by beginning the integration o the wind curl at the eastern boundary o all oceans and requiring that there be no low into the ocean through/rom the eastern boundary. You will see that there is a lot o low into and out o the boundary on the western sides o the ocean basins. This is because we need western boundary currents to return the wind-driven low in these regions and they are not part o the dynamical balance. Finally, note that while assumptions o geostrophic motion break down at the equator, there is nothing in the Sverdrup Balance that breaks down there. So unlike circulation maps inerred rom dynamic height, maps o the Sverdrup circulation can be used to iner wind-driven circulation on and across the equator. Compare the subtropical gyres in the N. Atlantic and N. Paciic. The winds are similar in the two basins and so are the wind stress curls. But because the N. Paciic is nearly twice as wide zonally as the Atlantic, the Sverdrup transport (which is the zonal integral o the wind curl) is nearly twice as large. The two wind climatologies used above are quite dierent in the Southern Ocean. Problems there are largely because there are so ew ship observations in that part o the ocean. Another problem is because there are no continental boundaries over the latitude zone deined by Drake Passage (between S. America and the Antarctic peninsula). Thereore we can t deine a zonally-integrated Sverdrup Balance there as there are no eastern or western boundaries to start rom and end at! Beore we return to the N. Atlantic subtropical gyre, one urther digression is needed. We have been rather vague about how the body orce, F, is actually related to the wind stress. We have also been rather vague about the nature o the rictional orce. As these are two o the three terms in the winddriven theory o Stommel, it is probably appropriate to spend some time on this now. They are both related to the way that stress is transerred between luid particles in the ocean. Once we have done this, we can also briely discuss Munk s (195) paper on the wind-driven circulation and how it diers rom Stommel s.