The “separation formula” and its application to the Pacific Ocean

Deep-Sea Research I 45 (1998) 2011—2033

The ‘‘separation formula’’ and its application to the Pacific Ocean Doron Nof* Department of Oceanography 4320 and the Geophysical Fluid Dynamics Institute, The Florida State University, Tallahassee, FL 32306-4320, USA Received 31 January 1997; received in revised form 29 December 1997; accepted 27 March 1998

Abstract The ‘‘separation formula’’, a new method for computing the adiabatic inter-hemispheric meridional transport, is applied to the Pacific Ocean. The method involves an integration of the wind stress along a ‘‘horseshoe’’ path. It begins at the separation point of the East Australian Current, continues eastward across the ocean, progresses northward along the continental boundary, and then turns back westward across the ocean to the separation point of the Kuroshio. Since the Pacific is closed on the northern side, such an integration gives the winddriven Indonesian throughflow. The analytical formulas show that, in order for the adiabatic wind-driven throughflow to exist, it is necessary that there be an asymmetry in the winds associated with the two zonal cross-sections connecting the (northern and southern) separation points in the west to the continents in the east. It turns out that these asymmetries in the Pacific are relatively small and, consequently, do not allow for a significant (i.e. more than one Sverdrup) Indonesian transport. Specifically, in the Pacific, this wind-driven transport is directed to the south, implying a very small net Indian-to-Pacific transport rather than a Pacific-to-Indian transport. The adiabatic model fails, therefore, to explain the observed Pacific-to-Indian throughflow of 5—6 Sv. When an upwelling is added to the model (to simulate diabatic processes), then one obtains the result that all the water upwelled in the Pacific must exit the Pacific via the Indonesian seas, i.e. the wind field is effectively blocking the oceanic region between Australia and South America, forcing the upwelled water into the Indian Ocean. This model suggests, therefore, that the observed Pacific-to-Indian throughflow is a measure of the upwelling in the Pacific.  1998 Elsevier Science Ltd. All rights reserved.

*Fax: 001 904 644 2581; e-mail: [email protected] 0967-0637/98/$—see front matter  1998 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 3 7 ( 9 8 ) 0 0 0 5 2 - 1

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1. Introduction The question of how much water flows (from the Pacific to the Indian Ocean) through the Indonesian Passages and where is it coming from is important because of its relevance to the so-called ‘‘great global conveyor’’. The region’s geography is very complex, and the distribution of the islands reminds one of a ‘‘colander’’ (Fig. 1). There are numerous islands and passages and, consequently, answering the above question is difficult from both the observational and theoretical point of view. The observational difficulty stems simply from the relatively large number of passages that need to be studied, whereas the numerical difficulty is due to the fact that global models do not have sufficient resolution to describe the flow through narrow straits (typical to the region). For more details on these aspects, the reader is referred to the review articles of Lukas et al. (1996) and Godfrey (1996), and to other articles in the same special Journal of Geophysical Research issue devoted to the throughflow. Because of the above observational and numerical difficulties, analytical models are of particular importance to the Indonesian throughflow problem. The present article represents a new approach for analytically estimating the adiabatic transport and the origin of the throughflow. As we shall see, it relies on the presence of two western boundary currents that separate in mid-latitude. 1.1. Previous methods The first attempt to analytically compute the transport of the Indonesian throughflow was made by Godfrey (1989), who derived an ingeniously simple relationship for

Fig. 1. The observed flow within the Indonesian Archipelago (adapted from Gordon and Fine, 1996).

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the mass transport between an island (Australia) and a continent to the east (America). The problem involves two eastern oceanic boundaries, one along the western boundary of the South American continent and the other along the western boundary of the Australian continent. This allows one to integrate the momentum equations along a closed path (containing the island), eliminate all the pressure terms, and derive an expression for the desired transport. It turns out that the transport computed this way depends solely on the wind stress. Noting that the Pacific is closed on the northern side, one immediately concludes that this computed wind-driven transport, entering the Pacific between Australia and South America, can exit the Pacific only via the Indonesian Passages. Using the above relationship, a depth of no-motion of 1500 m, and Hellerman and Rothstein’s (1983) winds, Godfrey calculated a mean wind-driven transport of 16 Sv for the resulting Indonesian throughflow. Three comments should be made with regard to this estimate. First, it appears to be three times larger than the throughflow amount presently agreed upon (5—6 Sv according to Wijffels et al., 1996). This might be due to frictional effects within the Indonesian seas, which are neglected in the island rule calculations. After all, Australia is not truly an island in the sense that it is not situated much farther than an equatorial Munk layer [(t/b), where t is the horizontal eddy viscosity and b is the familiar linear variation of the Coriolis parameter with latitude] away from the Indonesian Islands, which separate Australia from Asia. Specifically, for t &10 cm s\, the equatorial Munk layer is roughly 20 km, a distance comparable to the gaps within the Indonesian Archipelago. Furthermore, the length of the Indonesian seas is comparable to the length scale of most western boundary currents, a distance that allows a dissipation of the entire wind input to the associated ocean basin. These aspects are consistent with recent theoretical calculations of Pedlosky et al. (1997) showing a reduction of 80—90% in the transport associated with a (rectangular) island situated a Munk layer away from a western boundary. The reader is also referred to Wajsowitcz (1993, 1994, 1996) for a discussion of related issues. The second comment is with regard to the island rule’s combination with (i) the observed condition of zero wind-stress curl a few degrees north of the equator (implying zero Sverdrup transport) and (ii) linear theory implying a boundary current with a transport equal and opposite to the interior transport (i.e. no boundary current transport at the latitude of zero curl). A simple (and erroneous) interpretation of this combination suggests that, since no water can cross the line of zero curl, all the water within the Indonesian throughflow originates from the South Pacific and enters the Indonesian seas in a zonal manner without ever seeing the North Pacific. This clearly contradicts the observations (e.g. Gordon, 1995). To resolve this difficulty, it is necessary to realize that the condition of no boundary current transport along the latitude of zero wind stress curl (corresponding to no interior transport) implies merely no net transport. It does not, however, prohibit the existence of two opposing boundary currents side by side (Fig. 2). This resolution of the conflict implies

 This vanishing of the wind stress curl should not be confused with the additional vanishing along the northern separation latitude, which will be discussed later.

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Fig. 2. Schematic diagram of a flow pattern consistent with Godfrey’s (1989) ‘‘island rule’’ and the condition of zero wind stress curl (implying zero transport) a few degrees north of the equator. Since the basin is closed from the north, the flow across AB (the latitude of zero curl) has no transport. It contains two equal and opposite boundary currents side by side. One corresponds to a fraction of the South Equatorial Current, which orignates from the South Pacific and turns eastward while crossing the line of zero wind stress curl. The other, the Mindanao Current, flows southward and enters the Indonesian seas. No upwelling is allowed in this scenario.

that the South Equatorial Current must retroflect into the North Equatorial Counter Current a few degrees north of the equator, a condition that was recognized by Godfrey et al. (1993). The third comment is related to the question whether or not diabatic processes are implicitly included in Godfrey’s (1989) island rule calculations. To answer this question we first note that the island rule calculations do not distinguish between barotropic and baroclinic fields, because the stratification does not enter the final result. This means that when an application of the barotropic island rule is considered (i.e. the resulting transport is interpreted as the barotropic transport) then thermodynamics do not enter the problem. Such a barotropic application to the Pacific would, therefore, yield the appropriate Indonesian throughflow even if there are significant diabatic processes in the actual flow surrounding Australia. This barotropic application is attractive, but it suffers from the neglect of bottom topography, which can be very important. To overcome this problem, Godfrey (1989) has used an application of the baroclinic island rule asssuming a depth of no motion above the highest sills (i.e. he interpreted the resulting transport to be the baroclinic transport in the upper 1500 m). In contrast to the barotropic application, such a baroclinic application does not allow for diabatic processes and hence is a measure of the wind-driven flow alone. The answer to the question posed in the beginning of this paragraph is then that diabatic processes are not included in Godfrey’s (1989) baroclinic calculations. This completes our introductory discussion of the island rule and its implications. In what follows we shall show that some progress can be made by considering a new

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model that does not involve specification of the flow conditions within the passages or an assumption that the passages are completely open water. This is the purpose of our present effort. 1.2. Present approach As is done in the baroclinic island rule calculation, we shall first consider the adiabatic wind-driven flow from the Pacific into the Indian Ocean. However, instead of considering Australia to be an island, we shall take advantage of the known theoretical and observational fact that western boundary currents (WBC) separate from the coast at mid-latitude. We shall begin by considering a reduced-gravity layer-and-a-half model (Fig. 3). This simple model (Section 2) is sufficient to demonstrate the essence of the problem in the sense that it illustrates the essential elements of our procedure. It employs familiar general circulation equations allowing for a Sverdrup balance in the interior and frictional boundary currents along western boundaries. Cross-equatorial flow is allowed in the model (via the western boundary current), because the friction erases the ‘‘memory’’ of the fluid (regarding its original vorticity), allowing a free inter-hemispheric exchange. Integration of the momentum equation in a ‘‘horseshoe’’ manner [along ABCD (Fig. 3)], consideration of the separation condition (i.e. surfacing thermocline) and the eastern boundary condition, give a simple and straightforward relationship for the transport ¹ in terms of the wind stress alone. It shows that there cannot be any  wind-driven Indonesian throughflow unless there is some sort of asymmetry in the wind field above the two zonal cross-sections connecting the separation points in the west to the boundary in the east. After presenting a few examples, we shall proceed and analyze the more realistic case where the ocean is subject to both zonal and meridional winds and is continuously stratified in the upper water. Also, we shall consider a boundary that is a better approximation of the actual geography as well as spherical coordinates (Section 3). Using COAD’s winds and this ‘‘realistic’’ adiabatic model we shall show that the Indonesian throughflow forced by the wind alone is less than 1 Sv and is directed from the Indian into the Pacific. This is very different from the observed amount (5—6 Sv from the Pacific to the Indian) and the amount computed using the island rule (16 Sv from the Pacific to the Indian). To simulate diabatic processes, we shall then add upwelling a` la Goldsbrough (1933) to the model (Section 4) and show that any upwelling in the Pacific cannot exit the basin via the southern cross-section connecting Australia and South America. Instead, it is always diverted by the wind into the Indonesian Passages no matter where it occurs. The results are discussed and summarized in Sections 5 and 6. 2. Derivation of the ‘‘separation formula’’ 2.1. Governing equations Consider again the model shown in Fig. 3. The familiar vertically integrated equations of motion are

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Fig. 3. (a) Schematic diagram of the (preliminary) Indonsian throughflow model. The upper layer (shaded region corresponding to p (26.20) is driven northward and into the passages solely by zonal winds. The F basin has a trapezoidal shape; hatched area denotes the boundary currents region. This simple model is considered merely for the clarity of the presentation. Actual computations will be done using a more realistic model that takes into account the appropriate geography, the sphericity of the earth, meridional winds and a continuous stratification in the upper water (Fig. 4). Two coordinate systems will be used. The x, y system (with x pointing eastward and y pointing northward) is used to describe the field equations; then n and s system (with n normal to the coast and s along the coast) is used to describe the boundary condition along the eastern boundary. The angle a measures the tilt of the eastern boundary. (b) A cross-section of the model shown in Fig. 3a. (c) A three-dimensional view of the throughflow model shown in Fig. 3a. ‘‘Wiggly’’ arrows denote wind and thick arrows denote the transport.

1 * qV !f»"! g (h)# 2 *x o

(2.1)

1 * fº"! g (h)!R» 2 *y

(2.2)

º #» "0 V W

(2.3)

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where º and » are the vertically integrated transports in the x and y direction (here, x is pointing eastward and y is pointing northward), f is the Coriolis parameter (varying linearly with y), g is the ‘‘reduced gravity’’ g *o/o (where *o is the density difference between the upper and the intermediate layer and o is the density of the upper layer), qV is the wind stress in the x direction, and R is an interfacial friction coefficient which need not be specified for the purpose of our analysis. (For convenience, all of the variables are defined in the text and in the Appendix.) Note that, in this simple case, there are no meridional winds. Five comments should be made with regard to Eqs. (2.1)—(2.3). First, in this adiabatic model, energy is supplied by the wind over the entire ocean and is dissipated through interfacial friction [i.e. the R» term in Eq. (2.2)] within the limits of the western boundary current system. Namely, since the velocities are small in the ocean interior the frictional term is negligible there. Also, the interfacial friction is not present in Eq. (2.1) because Rº is small and negligible within the boundary current. Second, Eq. (2.1) holds both in the sluggish interior away from the boundaries and in the intense boundary current where the flow is geostrophic in the cross-stream direction. Within the western boundary current the balance is between the Coriolis and pressure terms with the wind stress playing a secondary role. In the interior, on the other hand, the velocities are small and, consequently, all three terms are of the same order. Third, the inertial terms are excluded from the model but some nonlinearity is included through the pressure term because it contains h in a nonlinear form; the system is linear, however, in º, » and h. Since the model does not include relative vorticity within the boundary currents, it implicitly assumes that all the relative vorticity generated by moving the fluid meridionally (within the boundary layer) is dissipated immediately after its creation. This implies that the adiabatic cross-equatorial flow within the boundary current is allowed because the fluid does not ‘‘remember’’ its original vorticity. Fourth, Eqs. (2.1)—(2.2) are not valid within the throughflow and its immediate vicinity. This, however, has no consequence for our analysis, because the equations are later applied to regions situated away from the throughflow. Finally, it is important to realize that, even though the lower layer velocities are neglected in this ‘‘reduced gravity’’ layer-and-a-half model, the transports in the lower layer are not required to be small [and can be of O(1)]. This results from the condition that, due to the large depth, even small speeds (with negligible contributions to the upper layer pressure terms) correspond to relatively large transports. Elimination of the pressure terms between Eqs. (2.1) and (2.2) and consideration of Eq. (2.3) gives 1 *qV *» b»"! !R o *y *x

(2.4)

which, for the inviscid ocean interior, reduces to the familiar Sverdrup relationship, 1 *qV b»"! . o *y

(2.5)

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2.2. Boundary conditions The boundary condition along the solid boundaries is, of course, the familiar no-normal flow into the western and eastern walls (save the entrance to the Indonesian Archipelago). For the specification of the eastern boundary condition, it is convenient to use a tilted coordinate system n and s where n is normal to the eastern boundary and s is along the boundary. In terms of this tilted coordinate system Eqs. (2.1) and (2.2) (with the frictional term neglected) are, g * cos a ![ f #b (s cos a#n sin a) »"! (h)#qV  2 *n o

(2.6)

g * cos a [ f #b (s cos a#n sin a) º"! (h)#qV  o 2 *s

(2.7)

where a is the tilt of the boundary (see Fig. 3), and º and » are now the integrated transports in the n and s directions. Since º"0 along the boundary (n"0) it follows immediately from Eq. (2.7) that,



g 1 ! qV sin a ds [h!h]" ! 2 o

(2.8)

where h and h are the upper layer thicknesses at C and B, respectively. Relation (2.8) ! implies that along the eastern coast the sea level responds to the wind stress in the usual wind set-up manner (i.e. in the same manner that it would in a long and narrow lake). For a purely meridional boundary (a"0), Eq. (2.8) reduces to the familiar condition of constant depth along the eastern boundary (i.e. h "h ) as should be the ! case. This completes our discussion of the eastern boundary condition. As mentioned, along the western boundary the zonal velocity component also vanishes. However, in contrast to the eastern boundary, the upper layer thickness is not uniform along this boundary due to the frictional term. Nevertheless, there is an additional boundary condition that is crucial to our computation. It is the condition that both the southern and northern boundary currents separate from the coast at some points (A and D). This means that h "h "0 (2.9) "  where, as before, the subscripts ‘‘D’’ and ‘‘A’’ denote that the variable in question is associated with points D and A (see Fig. 3). 2.3. Integration of the momentum equation To derive the separation formula, we integrate Eq. (2.1) from the western to the eastern boundary along the separation latitudes (i.e. Sections 1 and 2 shown in Fig. 3a),




 

g ! * ! qV 0" (h) dx! dx 2 *x o " " g * qV f ¹ " (h) dx! dx   2 *x o  



(2.10a) (2.10b)

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where ¹ is the net wind-driven transport entering the Pacific and f is the Coriolis   parameter along Section 2 (Fig. 3a). Eqs. (2.10a) and (2.10b) give



g ! q6 (h!h )" dx " 2 ! o "

(2.11a)



g f ¹ " (h!h )!    2



q6 dx. o

(2.11b)

By subtracting Eq. (2.11b) from Eq. (2.11a) and taking into account the eastern boundary condition Eq. (2.8) and the separation condition Eq. (2.9) we find the surprisingly simple relationship, !1 ¹ "  f 







qV ! qV sin a " qV dx# ds# dx o o o  !



(2.12)

which is our desired formula. It should now be clear why we start and finish our integration at the separation points; only these two locations correspond to zero pressure along the western boundary. Relation (2.12) gives the adiabatic transport in terms of the wind field and the separation latitudes alone. It states that there is an intimate relationship between the wind field, the Indonesian wind-drive transport ¹ ,  and the position of the WBC separation. Hence, a change in one of these variables (such as ¹ ) will force a change in one of the other variables (e.g. the separation  latitudes). To understand what Eq. (2.12) means we shall consider three examples. First, we shall consider the simplest possible case where there is no asymmetry in either the wind field or the continents, i.e. the only asymmetry is in the (given) WBC separation latitudes. In this special case the second term in Eq. (2.12) drops out. To illustrate the properties of this case, we chose an ocean 15,000 km broad driven by the Hellerman and Rosenstein (1983) winds (Figs. 4a and b). The throughflow as a function of the separation latitude is shown in the lower panel of Fig. 4a. Fig. 4b shows the results for four hypothetical separation latitudes: one corresponds to a Pacific-to-Indian transport of 21 Sv (upper left panel), one corresponds to a net transport of 13 Sv (upper right panel), one to 9 Sv (lower right panel) and one to no transport (lower left panel). For the second example we add an asymmetry in the wind field but retain the symmetrical geometry. Namely, the eastern boundary is still meridional so that there is no asymmetry in the shape of the continents. For this special case the second term still drops out and there is no throughflow (i.e. ¹,0) unless there is some asymmetry in either the wind field or the position of the separation latitude (or both). For instance, consider a hypothetical rectangular ocean 15,000 km broad with boundary currents detaching from the western boundary at 40°N and 40°S. There is no wind stress along 40°N and an eastward wind corresponding to a stress of 1 dyne cm\ is blowing along 40°S. According to Eq. (2.12), such an ocean will have an Indonesian throughflow of 15 Sv.

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For the third example we now add an asymmetry in the geography. In this case the wind along the southern section (AB) is acting over a different distance than the wind along the northern section (DC) so that even identical winds cause a transport [provided that the wind stress along the eastern boundary (BC) is not identical to those winds]. For instance, consider now a trapezoidal ocean extending 15,000 km

Fig. 4a. (ºpper panel) The simplified zonal wind stress [adapted from Hellerman and Rosenstein (1983)] used to examine the dependency of the net Indonesian transport on the separation latitudes in a rectangular ocean 15,000 km broad. (¸ower panel) The net northward transport ¹ [computed from Eq. (2.12) and the  symmetrical wind stress shown in (a)] as a function of the two separation latitudes. The ocean is 15,000 km broad. When the separation latitudes in the two hemispheres are identical (i.e., the problem is completely symmetrical no net transport is possible. However, when the separation latitudes are not symmetrical, a net throughflow transport of as much as 30 Sv is possible.

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Fig. 4b. Four situations corresponding to a wind that forces 21 Sv (upper left), 13 Sv (upper right), 0 Sv (lower left) and 9 Sv (lower right) into the Indonesian Sea. All four situations correspond to (2.12) and the wind pattern shown in Fig. 4a.

along 40°N and 30,000 km along 40°S with currents separating again at 40°N and 40°S. As in the second example, the ocean is subject to no zonal wind stress along the northern boundary and an eastward wind corresponding to 1 dyne cm\ along the southern boundary. We also specify here a linear wind stress dependence along the eastern boundary. According to Eq. (2.12), such an ocean will have a net northward transport of 24 Sv, which is 60% greater than the net transport in the rectangular ocean. All three examples illustrate that only severe inter-hemispheric asymmetries generate a significant throughflow; mild asymmetries generate very weak meridional flow. It is important to realize that, in this model, the separation of the western boundary currents does not necessarily occur along the latitude of vanishing wind stress curl. Instead, it occurs where the thermocline depth along the wall vanishes, i.e. in the Charney (1955) manner where, due to the increase of the Coriolis parameter with y, a poleward flowing western boundary current ultimately reaches a point where the thermocline surfaces along the coast. Charney’s separation includes the

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inertial nonlinear terms, which are absent from this model, but the essence of the separation is identical in the two cases. Alternatively, one can think of our separation as Parsons’ (1969) separation without the condition of no net meridional flow across the ocean. To see this more clearly, recall that the idea of western boundary current separation occurring at the latitude of vanishing wind stress curl originates in the assumption of a boundary current transport being equal and opposite to the interior Sverdrup transport. The theory is that, in this scenario, when the interior transport vanishes (due to zero wind stress curl) then the boundary current meridional transport must vanish as well, implying a separation. A vanishing WBC transport is expected in our northern hemisphere separation latitude (i.e. the WBC associated with cross-section DC) because there is no net meridional transport there. However, the southern boundary current transport (i.e. the boundary current associated with section AB) is not necessarily equal and opposite to the interior transport because there is a net meridional transport. It is also appropriate to comment on the use of geostrophy at the separation point. Although it is true that any one-dimensional boundary current is geostrophic in the cross-stream direction (i.e. the east—west momentum balance is geostrophic), when the current separates (due to a vanishing thermocline depth), it is no longer one-dimensional and, consequently, may no longer be geostrophic (see, e.g. Moore and Niiler, (1974), where the separation of a fully nonlinear flow is discussed). This means that the applicability of Eq. (2.2) to the separation points is questionable. It turns out, however, that this is not an issue to be concerned about, because there is no question that a few deformation radii upstream of the separation point the current is geostrophic. Application of Eq. (2.2) to such an upstream cross-section (instead of the chosen one) will introduce extremely small errors to the computed transport [&O+(gH)/ f ) (R /a), where H is the undisturbed depth,  R is the Rossby radius and a is the radius of the earth]. For additional aspects  of boundary currents separation the reader is referred to the review article of Ierley (1990), and to Kamenkovich and Reznik (1972), Huang (1984, 1986), Cessi (1990, 1991), Cessi et al. (1990), Chassignet and Bleck (1993), Chassignet (1995) and Pedlosky (1996).

3. Application of the ‘‘separation formula’’ to an adiabatic continuously stratified upper layer on a spherical earth In this section we shall extend the results of the simple ‘‘reduced-gravity’’ model with one active layer to a continuously stratified upper ocean overlying again a stagnant lower layer with constant density (Fig. 5). The upper layer is again permitted to flow meridionally beyond the southern separation point, which is defined here to be the intersection of the surfacing o isopycnal [i.e. the densest water  within the upper layer (see Fig. 5)] with the coast. As before, all the topographic features are assumed to lie below the boundary separating the lower and the stratified upper layers.

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Fig. 5. A schematic north—south section of the ‘‘realistic’’ separating boundary current. The adiabatic current is bounded from the below and the sides by the surfacing isopycnal corresponding to o , i.e. the  active light water (shaded) is continuously stratified with a density smaller than o , whereas the inert lower  water has a constant density, o . The level of no motion is taken to be the depth of the o isopycnal,   implying that all the water below it is at rest.

With the Boussinesque approximation, the vertically integrated equations in spherical coordinates are 1 *pL qV !f »"! # !Rº o *x o   1 *pL qW fº"! # !R» o *y o   *º *» tan

# "» *y a *x

(3.1) (3.2)

(3.3)

where dx ("a cos dj) and dy ("a d ) correspond to increments in the eastward (x) and northward (y) directions, and j are latitude and longitude, o is a reference  density (taken to be the density of the isopycnal separating the upper and the lower water) and the vertically integrated pressure pL is defined by



pL "

E



p dz+



p dz. (3.4) \" \" Here, g is the free surface vertical displacement [much smaller than the maximum upper layer thickness D, which coincides with the maximum depth of the o isopyc nal], p, the total pressure, is given by the sum of the pressure associated with a state of rest pN (z) and the departure from this motionless pressure p, p"pN (z)#p(x, y, z).

(3.5)

For convenience, we choose the state of rest to be a no-upper-layer state, i.e. the stagnant reference ocean is a barotropic ocean with density o . The separation curve  is identified as the southwesternmost or northwesternmost curve that separates inert,

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lower water from active surface water, i.e. it is the intersection of the o isopycnal with  the surface. The last terms on the right-hand side of Eqs. (3.1) and (3.2), Rº and R», are the familiar linear interfacial frictional terms, whose detailed structure is not important for our present analysis. Here, both equations include frictional terms because the western boundary is not necessarily meridional. As before, dissipation occurs only under the western boundary current system so that the terms Rº and R» are neglected in the interior and along the eastern boundary. The integration will be performed counterclockwise along the ‘‘horseshoe’’ path shown in Fig. 6. Across the boundary currents the integration is done in a direction normal to the coast (and the boundary current axis) so that frictional terms do not enter the integration. Across the ocean the integration is done in the zonal direction, because otherwise the Coriolis parameter cannot be taken outside the integral of the transport. Along the eastern boundary the integration is done along the coast (i.e., in the ‘‘s’’ direction). For convenience, we use the normal and tangential coordinates n and s for both the integration across the boundary currents (AB and FE) and the integration along the eastern boundary. Within the boundary currents the balances are, 1 *pL qL !f »"! # (3.6a) o *n o   1 *pL qQ # !R» (3.6b) f º"! o *s o   where, because of the relatively small cross-current scale, the sphericity has been neglected and R, the frictional coefficient, need not be specified (to perform the calculations). Integration of Eq. (3.6a) across the two narrow currents (FE and AB) yields,

 

!f ¹ "!(pL !pL )/o #  $# # $ 

#

$

!f ¹ "!(pL !pL )/o #    

(qLo ) dn 

(3.7a)

(qLo ) dn 

(3.7b)

 where the small variations in f across AB and FE have been neglected. Here, ¹ and $# ¹ denote the transports across FE and AB and, as before, f and f are the Coriolis    parameters at cross-sections 1 and 2. We now note that the integrated pressure pL is zero at the separation points (pL "pL ,0) and integrate Eq. (3.1) with R"0 across $  ED and BC, " 0"!(pL !pL )/o # (qVo ) dx (3.8a) " #   # ! !f ¹ "!(pL !pL )/o # (q l/o ) dl (3.8b)  ! !  



where ¹

!



is the transport across cross-section BC.

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Fig. 6. The horseshoe integration path ABCDEF. Sections AB and EF are perpendicular to the boundary currents, whereas BC and ED are zonal sections. The integration of the COAD wind stress was done in 2° squares. The results are very insensitive to the choice of the separation latitudes; even variations of 10° or more do not change the results significantly.

It is now recalled that along the eastern boundary there is no flow into the wall (º"0) and friction is small and negligible. Under such conditions, the equation analogous to Eq. (3.6b) yields,



*pL " "qQ or pL !oL " qQ ds. (3.9) " ! *s # By combining Eqs. (3.7a)—(3.9) and using the conditions pL "pL "0 we finally arrive $  at our desired formula,



ql dl



f o (3.10)    !"#$ where l indicates the direction of the integration path (which is not necessarily identical to either x, y, n or s), and ¹ is the total transport due to the wind. Note that the  integration is done in a horseshoe manner in the sense that it is not an integration over a closed path but rather an integration over an open path which does not contain section FA. ¹ "! 

4. The adiabatic wind-driven throughflow We used COAD’s monthly mean winds (Woodruff et al., 1987) with a drag coefficient, C , of 1.6;10\ [which is the appropriate coefficient for winds (such as  ours) with a speed of less than 6.7 m s\ (see, e.g. Hellerman and Rosenstein, 1983)]

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and identified the separation points to be the annual mean intersection of the observed p "26.20 isopycnal [taken from Levitus’ (1982) climatology] with the coast F at a depth of 100 m (see Fig. 5). The choice of the p "26.20 isopycnal was made F because it corresponds to the maximum gradient in the thermocline. Much of the throughflow occurs in the upper 200 m (see, e.g. Wyrtki, 1987), which is lighter than this, suggesting that the choice is appropriate. It essentially implies that both North Pacific Intermediate Water (p '26.20) and South Pacific Intermediate Water F (p '26.20) do not participate in the process that we are addressing. This does not F mean, of course, that there are no actual contributions from these intermediate sources (see, e.g., Hautala et al., 1996; Molcard et al., 1996) but rather that such contributions are not included in our model. The choice of 100 m depth for the isopycnal separation was made in order to avoid mixed layer effects, i.e. it is assumed here that the intersection of the p "26.20 F isopycnal with the free surface in an ocean without a mixed layer would be similar to its intersection with the 100 m level. This is a very reasonable choice because the speed within the boundary currents is high [O(1 m s\)] so that the intersection of the 26.20 p isopycnal with the free surface in an ocean without a mixed layer would be a disF tance only of O(10 km) away from our chosen separation point. With the aid of (3.10) one finds a surprising result—a negligible net (adiabatic) Indonesian transport of less than one Sverdrup directed from the Indian to the Pacific. This implies that there is virtually no net wind-driven flow into the Indonesian seas of water lighter than p "26.20 gr cm\. How can this be reconciled with the observations that there are F 5—6 Sv of warm water that flow from the upper Pacific to the Indian Ocean? Furthermore, how can this be reconciled with calculations based on the baroclinic island rule which suggest that about 16 Sv of upper water are (adiabatically) driven by the wind into the Indonesian seas? We shall answer these two questions in the following sections.

5. Upwelling in the Pacific An aspect that is neglected in our calculations is heat exchange with the atmosphere above and heat exchange with the lower layer underneath. It is difficult to examine such diabatic processes in detail [because of the three-dimensionality of the problem (see, e.g. Nof, 1983)], but some (limited) progress can be made by looking at the heat exchange process without examining the details of the cooling process itself. To do so, we shall examine the influence of specified upwelling (which cools the upper water). Upwelling a` la Goldsbrough (1933) can be easily added to our two-layer model (Section 2). Goldsbrough’s approach (originally used to describe evaporation and precipitation) is to add a source or a sink only in the continuity equation, leaving the momentum equations unaltered. Furthermore, even though the sink or source may

 This estimate is easily obtained using the geostrophic condition.

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correspond to water with a density different than that of the upper layer, the Goldsbrough approach is to ignore these differences to first order. This approach is valid as long as (i) the upper water volume is so large that injection of foreign fluid will not have much influence on its density, and (ii) the density anomaly associated with the injected water is immediately diffused after the injection. The former condition is clearly satisfied because it would take 10,000 yr or so for upwelled water in the Pacific to significantly alter the upper layer density. It is not so easy to say whether or not the latter condition is also always satisfied, because this would depend on the horizontal and vertical diffusion coefficients whose values are, by and large, unknown. With the Goldsbrough approach the only alteration to our Eqs. (2.1)—(2.3) is that, instead of Eq. (2.3), we now have, *º *» # !w"0, *y *x

(5.1)

where w is the specified upwelling at the base of the thermocline. Eqs. (2.1) and (2.2) remain unaltered, but the Sverdrup transport now takes the modified form 1 *qV b»"! !fw. o *y

(5.2)

Since Eqs. (2.10a) and (2.10b) are unaltered, the previously derived expression for the wind-driven meridional transport ¹ [i.e. relation (2.12) corresponding to no upwell ing] is still relevant. This means that the transport entering the Pacific from the south (through section AB shown in Fig. 3) remains unaltered so that the total transport entering the Indonesian seas is





ql dl# w dx dy (5.3) o  !"  !" where the second term corresponds to upwelling occurring in the region confined by the separation latitude in the north and the separation latitude in the south. Clearly, relation (5.3) is also valid in spherical coordinates. However, in this upwelling case, it does not make any sense to speak about a stratified upper layer, because it is hard to imagine how upwelled water can immediately accept the surrounding stratified density as it is injected into the fluid. Consequently, we consider here an upper layer with uniform density, an outcropping interface and a stagnant lower layer. In this scenario, the wind-driven transport forced through section ABC (Fig. 5) is the same as before, i.e., a negligible transport. According to Eq. (5.3) and the observed wind pattern (which gives the value of  (ql/o) dl), the upwelled water can leave the Pacific basin only through the Indonesian seas. This implies that the observed 5—6 Sv is actually a measure of the amount of water upwelled (through the base of the thermocline) into the upper Pacific layer. In other words, the wind field over the Pacific essentially blocks the southern exit (i.e. section ABC, Fig. 5) forcing the upwelled water into the Indonesian seas. 1 ¹ "!  f 

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How the upwelling of 5—6 Sv is achieved is not entirely obvious. Recent measurements of diapycnal diffusivities (Ledwell et al., 1993; Toole et al., 1994; Kunze and Sendord, 1996) suggest an interior mixing coefficient of no more than 0.1 cm s\. For an area of 15,000;10,000 km and a mean thermocline depth of 500 m, such a diffusivity coefficient would lead to an upwelling of merely three Sv. However, this slow mixing coefficient is not in agreement with budget calculations that suggest much higher coefficients in regions where there are intense currents. For example, Qiao and Weisberg (1998) suggest mixing coefficients that would allow 20 Sv or more to be easily upwelled into the thermocline over the equatorial Atlantic. Similarly, more than a decade ago, Roemmich (1983) suggested that, in the Atlantic, 6 Sv are converted to thermocline water somewhere between 8°S and 8°N. Both of these studies suggest that most of the mixing probably occurs in special locations and not in the ocean interior. This is supported by Lueck and Mudge (1997), who, on the basis of measurements of energy dissipation in the Pacific, suggest that shallow seamounts induce mixing that can be 100 to 10,000 times larger than the mixing away from the seamount. Bottom topography also produces very strong mixing (Polzin et al., 1996, 1997). Our model requires a vertical diffusivity coefficient that, on average, is twice as large as that observed in the interior of the ocean. Although this value (0.2 cm s\) is large in some sense, it is five times smaller than that required by most global models (1 cm s\). It is quite possible that, as others have suggested, intense mixing and entrainment take place along the boundaries of the ocean, along the equator, and in the vicinity of seamounts. An alternative to the upwelling scenario is that the model is flawed in some other aspect. This would be possible if the neglected lower layer motions were somehow directly responsible for the upper water transfer, i.e. that the neglected pressure gradients in the lower layer are responsible for forcing 5—7 Sv of upper thermocline water into the Indonesian seas. Using scaling arguments, it is easy to demonstrate that, in order for this to happen, the speeds in the lower layer must be of the same order as the upper water velocity. Since, on average, the lower layer is 8—10 times thicker than the upper layer, this implies a highly unlikely transport of 50—70 Sv in the lower layer. In view of this, this possibility is rejected. It should be pointed out here that other effects, such as JEBAR and bottom torque (Myers et al., 1996), corners and local geography (Dengg, 1993; Ou and DeRuijter, 1986), and lateral inflows (Ezer and Mellow, 1992; Thompson and Schmitz, 1989), are also excluded from our model. Although these processes affect the position of the separation, their effect is limited to several degrees. As already mentioned, our formula is very insensitive to the position of the separation (because it involves the wind stress rather than the curl of the wind stress), so that these issues have no impact on our results.

6. Relation to the baroclinic island rule calculations In this section we shall attempt to reconcile our no-upwelling adiabatic findings of virtually no wind-driven flux through the Indonesian Passages with the contradicting

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baroclinic island rule calculations (see, e.g. Godfrey, 1989) showing a transport of 15 Sv (without upwelling). The theoretical island rule calculations for a baroclinic ocean are essentially based on the same set of basic equations that we have used, but there are two important differences between the two methods. The first is that the baroclinic island rule calculations assume that Australia is an island far away from any continents. This point was already touched upon in the Introduction, where it was mentioned that no such assumption is made in our present study. The second difference is that by considering the fluid underneath the separating isopycnal (26.20p ) to F be at rest, our model does not include any waters below this level. Godfrey’s calculation, on the other hand, includes all the waters down to a depth of no motion of 1500 m. The first difference is important because, in reality, Australia is situated a distance of the order of the equatorial Munk layer thickness d [roughly 20 km based on a horizontal eddy viscosity l of 10 cm s\ and the relationship d& O(l/b)] away from the Indonesian Islands. Furthermore, the width of the various Indonesian Passages is also of the order of the Munk layer, and their length scale is comparable to the basin length mode. This means that friction in the passages is probably as important as the friction underneath most western boundary currents implying that it cannot be neglected. Recent calculations by Pedlosky et al. (1997) suggest that an island situated a Munk layer thickness away from a western boundary corresponds to a circulation that is only 15—20% of the transport calculated by the island rule. The second difference implies that Godfrey’s calculations may involve more water than that considered in our model, but it is hard to say how much more water is included in his computations. Wyrtki’s (1987) computations suggest that the bulk of the throughflow occurs within the upper 200 m, which corresponds to 26.20p in the F western equatorial Pacific. Since we include all the water from the free surface down to 26.20p , this would suggest that the excess water considered in the island rule is very F small. However, recent observations (Wijffels et al., 1996) suggest that there might be a considerable transport below 26.20p , suggesting that, even though our model capF tures a good fraction of the transport, it might neglect significant amounts captured by the baroclinic island rule. Although the above arguments are sensible with regard to the baroclinic island rule calculations, one can perhaps claim that, since the island rule calculation does not distinguish between the barotropic and the baroclinic transports, application of the barotropic island rule would also give a throughflow of 15 Sv. Since diabatic processes do not enter the barotropic calculations, one can then say that this barotropic transport of 15 Sv contains the mass flux due to diabatic processes (neglected in our formula). That is, one can say that the differences between the transport predicted by the separation formula ((15 Sv) and the barotropic island rule calculation (15 Sv) results from the neglect of adiabatic processes in the separation formula calculations. The difficulty with the above scenario is that the barotropic island rule cannot be applied to the throughflow because of the neglect of bottom topography. Godfrey (1989) recognized this and, consequently, applied the baroclinic island rule which, just like the separation formula, does not contain diabatic processes.

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An additional complication, which was already touched upon, is that, in reality, there are motions in the lower layer beyond the separation lines (e.g. the Oyashio). To see the issue associated with this aspect recall that, in our model, the speeds (though not necessarily the transports) beyond the separation lines are neglected because in these regions the lower layer is exposed to the atmosphere so that the wind is acting on an infinitely deep fluid. The same potential difficulty is associated with Parson’s (1969) original model (and with many other related models). This weakness of the model probably is not serious, because when the Oyashio transport is distributed over the entire lower layer thickness, the resulting speeds are small as assumed in the model. An additional effect that the Oyashio may have is to force the Kuroshio separation latitude away from the position that the Kurosio would separate on its own. This is certainly possible, but studies of a similar situation in the Atlantic (Thompson and Schmitz, 1989) show that the resulting meridional displacements are no more than 50—100 km, an amount that will have virtually no effect on our calculations.

7. Discussion and summary The foregoing calculations focused on the adiabatic upper-layer circulation in the Pacific. By taking advantage of the fact that western boundary currents separate at mid-latitude, we were able to compute the adiabatic transport forced by the wind into the Indonesian seas (Figs. 3—6). We showed that this transport can be determined from the wind stress alone [relations (2.12) and (3.10)], because the pressures along the boundaries are all related to each other. We have presented three examples that illustrate the importance of asymmetries in the hemisphere (necessary for a significant throughflow to exist). Using COAD’s winds we showed that the wind alone cannot force any significant amount of water through the passages. Observations suggest, on the other hand, that about 5 Sv flows from the Pacific to the Indian Ocean. These two aspects can be reconciled by noting that diabatic processes are absent from our model. Solving the entire wind-heat exchange problem is very difficult (primarily due to the three-dimensionality of the system). However, the effects of the cooling by the lower layer can be estimated by the addition of specified upwelling (a` la Goldsbrough) to our model. It shows that any upwelling within the Pacific ends up in the Indonesian seas. This suggests that the observed mass flux of 5 Sv in the Indonesian seas does not enter the Pacific (as surface water) between Australia and South America but rather upwells (through the base of the thermocline) somewhere between 40°S and 40°N. In this aspect, our model supports the original version of the global conveyor (Broecker, 1991). Our adiabatic purely wind-driven (i.e. no upwelling) findings are not in agreement with the calculations based on the baroclinic island rule, which suggest that 15 Sv or so are forced directly by the wind (i.e. without upwelling) into the Indonesian Passages. In principle, the presence of separation does not violate the rule (i.e. the rule is valid even if currents detach from the island), and the difference between the two

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methods is attributed to two possible causes. The first is friction within the Indonesian seas, which can dramatically reduce the computed ‘‘island’’ mass flux (see, e.g. Pedlosky et al., in press). The second is the chosen level of no motion in the two methods and the absence of separation in the island rule calculations. Our present model employs a level of no motion corresponding to 26.20p (the separating isopycF nal) whereas Godfrey’s baroclinic island rule calculation uses a deeper level of no motion (1500 m). It appears that our choice of 26.20p is reasonable as a bounding F density, but it is possible that a significant amount of water flows underneath this level. The different choices of level of no motion and the absence of separation from Godfrey’s model means that speeds beyond the separation lines (i.e. the intersection of the 26.20p with the free surface) are included in the island rule but excluded from our F model. [Note, however, that, due to the large thickness, our transports beyond the separation lines are not required to be small because even small speeds (with negligible contribution to the upper layer pressure) can correspond to large transports.] It is difficult to say how important these neglected motions are, but it is planned to examine this aspect in the future through the use of process-oriented numerical models with a moving lower layer. Knowing the observed pressure difference is probably not sufficient for an examination of the issue at hand, as one does not clearly know how to distinguish between the pressure differences due to wind and those due to diabatic processes. In conclusion, we used an idealized case of adiabatic motions in a one-and-a-half layer ocean confined between two separation latitudes as a model for the throughflow. As such, the model fails to give reasonable values for the throughflow. It is argued that diabatic motions are essential, and it is the neglect of such motions that causes the discrepancy between the model and the observations, i.e. it is suggested that upwelling of lower layer water into the thermocline is of utmost importance. Even though the upwelled water is specified in our scenario, the model (with upwelling) does provide useful information. It shows that water upwelled in the Pacific cannot exit between Australia and South America; instead, it is always forced into the Indonesian seas. It is believed that the separation model discussed above provides a new point of view— though not necessarily a better view—of what may drive the throughflow.

Acknowledgements S. Godfrey as well as an additional anonymous reviewer provided very useful comments. Conversations and discussions with L. Pratt and G. Weatherly were very useful. Also, communications with S. Wijffels, N. Bray and A. Field were very helpful. This study was supported by the National Science Foundation (NSF) under grants OCE 9633655 and OCE 9503816, National Aeronautics and Space Administration grants NAGW 4883 and NAGS-4613, and Office of Naval Research grant N0001496-1-0541. Computations were done by Steve Van Gorder.

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