A semi-analytical model of barotropic and baroclinic flows for an open Panama Gateway

Dynamics of Atmospheres and Oceans 50 (2010) 55–77

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A semi-analytical model of barotropic and baroclinic flows for an open Panama Gateway Hiroshi Sumata a,∗, Shoshiro Minobe b, Tatsuo Motoi c, Wing-Le Chan d a

Division of Earth System Science, Graduate School of Environmental Earth Science, Hokkaido University, N10 W5 Kita-ward, Sapporo 060-0810, Japan b Division of Natural History Science, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan c Meteorological Research Institute, Tsukuba 305-0052, Japan d Research Institute for Global Change, JAMSTEC, Yokohama 236-0001, Japan

a r t i c l e

i n f o

Article history: Received 31 March 2009 Received in revised form 4 September 2009 Accepted 27 October 2009 Available online 13 November 2009 Keywords: Panama Gateway Panama Seaway Central American Seaway Island rule Vorticity balance Semi-analytical model

a b s t r a c t A semi-analytical model of the Panama throughflow is presented. The model expresses the throughflow transport as a function of deep water formation in the North Pacific and in the North Atlantic, and of the Panama Gateway depth. The model is derived from the integral of the momentum equation along a circumpolar path, and can be interpreted from the point of view of the vorticity balance. The important conditions are whether the deep water, whose location is considered to be above the bottom water formed around Antarctica, originates from the North Atlantic or from the North Pacific, and whether the Panama Gateway is shallower than the lower boundary of the deep water. The present model indicates that the barotropic transport through the Panama Gateway is eastward, except for the case where the deep water is formed in the North Pacific and the sill of the Panama Gateway is shallow. The baroclinic structure of the Panama throughflow depends on whether the deep water is formed in the North Pacific or in the North Atlantic. These qualitative implications of the model are consistent with recent numerical studies and proxy-based paleoceanographic studies. Numerical experiments performed in the present study reinforce confidence in the semi-analytical model. © 2009 Elsevier B.V. All rights reserved.

∗ Corresponding author. Tel.: +81 11 706 2288; fax: +81 11 706 4865. E-mail address: [email protected] (H. Sumata). 0377-0265/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.dynatmoce.2009.10.001

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1. Introduction The Earth experienced three major tectonic events associated with oceanic gateways during the Miocene to the Pliocene, i.e., separation of the Tethys Sea from the Indian Ocean, opening of the Drake Passage and emergence of the Panama Isthmus. Opening or closing of such oceanic gateways had large impacts on the oceanic circulation, climate system, and ecological system of the Earth (e.g. Woodruff and Savin, 1989; Wright et al., 1992; Mikolajewicz et al., 1993; Reynolds et al., 1999; Lear et al., 2003). Of particular importance is the closure of the Panama Gateway (or Central American Seaway) which was the most recent major tectonic event. The resultant oceanic and climatic environmental changes must be considered when discussing the extinction or evolution of flora and fauna leading up to the present day (Duque-Caro, 1990; Coates et al., 1992; Collins et al., 1996; Kameo and Sato, 2000; Haug et al., 2001; Schneider and Schmittner, 2006). To investigate the effects of an open Panama Gateway on global ocean circulation and climate, several numerical studies have been conducted in the last two decades (e.g. Maier-Reimer et al., 1990; Murdock et al., 1997; Nisancioglu et al., 2003; Motoi et al., 2005; Heydt and Dijkstra, 2005; Schneider and Schmittner, 2006). These studies showed a significantly different ocean circulation in comparison with the present day circulation. According to these numerical studies, an open Panama Gateway leads to a decrease in sea surface salinity (SSS) in the subtropical North Atlantic Ocean, and the resultant SSS distribution affects the formation of North Atlantic Deep Water (NADW). Some numerical studies showed weaker NADW formation than that today (Mikolajewicz and Crowley, 1997; Murdock et al., 1997; Nisancioglu et al., 2003), while others proposed a termination of NADW (Maier-Reimer et al., 1990; Motoi et al., 2005). Naturally, a weakening or termination of NADW formation would lead to a different global thermohaline circulation and climate system compared with those of the present day. These changes associated with the NADW formation suggested in the numerical studies imply that the direction and strength of the surface current of the Panama throughflow (hereafter PTF) are important. A westward surface current transports warm and saline surface water from the subtropical North Atlantic, resulting in substantial decrease in SSS in the Atlantic Ocean and termination of NADW (Motoi et al., 2005). On the other hand, an eastward surface current carries subtropical Pacific water into the Atlantic Ocean, resulting in a relatively small decrease in SSS, and maintains weakened but, nonetheless, active NADW formation (Nisancioglu et al., 2003; Steph et al., 2006). In addition to the influence of the surface current on the deep-water formation, the location and amount of deepwater formation should influence the direction and the strength of the surface current, which can be understood from the mechanisms for not only barotropic but also for baroclinic structures of the PTF. Consequently, it is important to clarify mechanisms relating the baroclinic structure of the PTF to deep-water formation in the Northern Hemisphere. The mechanisms of the barotropic PTF was proposed in two theoretical studies (Nof and Gorder, 2003; Omta and Dijkstra, 2003). Nof and Gorder (2003) applied the ‘island rule’ (Godfrey, 1989) to the area which encloses the South American continent and the South Atlantic Ocean and tried to explain the direction of the PTF. They also took into account effects of NADW formation, and proposed a westward PTF, which, however, disagrees with the eastward barotropic PTF as suggested in many numerical experiments. Omta and Dijkstra (2003), on the other hand, considered conservation of absolute vorticity over an area that lies between South America and Africa, and obtained an eastward barotropic PTF consistent with the numerical experiments. In their theoretical explanation, a meridional difference in the axes of the Antarctic Circumpolar Current (ACC) when entering and leaving this area causes a vorticity imbalance. An eastward PTF is then required to balance the vorticity budget in this area. Although Omta and Dijkstra (2003) succeeded in explaining the eastward barotropic transport through the Panama Gateway, issues regarding the mechanisms responsible for the baroclinic structure remain unanswered. As mentioned earlier, the baroclinic structure of PTF is important since it is not only related to the oceanic environment around the Panama Gateway, but also to deep water formation in the Northern Hemisphere. Therefore, the purpose of the present study is to clarify mechanisms responsible for the baroclinic structure of the PTF and its relationship to deep water formation in the Northern Hemisphere. In addition, the barotropic PTF is examined with respect to deep water formation.

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When one focuses on the effects of an open Panama Gateway on the global thermohaline circulation, the depth of the gateway is also an important factor to be considered. As will be shown later, the gateway depth has a substantial impact on the PTF strength and structure. Since the gradual closing of the Panama Gateway lasted from 15 Ma to the final closure at 2.7 Ma, the PTF and related deep water circulation in the Northern Hemisphere may have experienced a substantial transition during this period (e.g. Burton et al., 1997; Haug and Tiedemann, 1998; Frank, 2002; Frank et al., 2002). The deep water connection between the Pacific and the Atlantic Oceans closed by ∼15 Ma (Duque-Caro, 1990), and the intermediate water connection closed by ∼12 Ma (Duque-Caro, 1990; Frank, 2002). In the late Miocene, only surface water connection may have been possible (Keigwin, 1982; Haug and Tiedemann, 1998, Steph et al., 2006). Therefore, we have to take into account the effects of sill depth variation when formulating a relationship between the PTF and deep water formation. The rest of this paper is organized as follows. A semi-analytical model for the barotropic and baroclinic structure of the PTF is presented in Section 2, along with its relation to deep and bottom water formation. Comparisons of an analytically estimated PTF with previous numerical studies are also presented. In Section 3, we examine the validity of the semi-analytical model by comparing its estimations with numerical results for a wide parameter space. A summary and discussions are presented in Section 4. 2. A semi-analytical model for the Panama throughflow (PTF) 2.1. Model derivation We introduce a semi-analytical model that has three levels of constant thickness and prescribed basin-averaged vertical velocities across the level interfaces (Fig. 1). The interface between the first and second levels corresponds to the lower boundary of the surface flow in the Panama Gateway, whereas the depth of the interface between the second and third levels is taken to be the sill depth of the gateway. Thus, in the Panama Gateway, there are two levels, and the PTF transports in these levels are estimated by the present model. It should be noted that the level interfaces of the model do not necessarily coincide with the level of no-motion. Rather, the semi-analytical model implicitly assumes pressure gradient on the level interfaces. However, as we will see later, we do not need to specify the pressure gradient or resultant geostrophic flow on the level interfaces. Instead, we have to specify the basinaveraged vertical velocities related to deep and bottom water formation, which acts as a forcing in the present model. In this subsection, we first derive a formulation for the barotropic PTF corresponding to the sum of the first and second level transports, and then derive another for the baroclinic PTF.

Fig. 1. Schematic diagram of the semi-analytical model in the present study. The model is divided into three levels in order to represent the baroclinic structure in the Panama Gateway. The first two levels have constant thickness. −H1 and −H2 represent the interface depths between the levels. The interface depth between the first and second levels corresponds to the lower bound of the surface flow in the gateway, whereas that between the second and third levels is taken to be the sill depth of the gateway. There are two levels in the gateway, and the transports in each level are estimated by the present model. The vertical mass transports across the level interfaces in each basin are used for the analytical estimation of the Panama throughflow.

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We first express the barotropic PTF by integrating the momentum equation vertically, and next, we introduce a circumpolar integral of the momentum equation to derive a vorticity equation. The vorticity equation combined with mass conservation provides a relationship between PTF and deep or bottom water formation, as will be seen later. The horizontal momentum equation is written as   ∂ ∂u 1 ∂u  + f k × u   = − ∇p + + F, +A Av 0 ∂t ∂z ∂z

(1)

 = (u, v) is the horizontal velocity, f is the Coriolis parameter, 0 is the mean density of the where u ocean, k is the upward vertical unit vector, Av is the vertical eddy viscosity coefficient, F is the horizontal  is the nonlinear advection terms. The diffusion of momentum (the horizontal eddy viscosity) and A  may be written as u  ∇ v + w(∂v/∂z) +  ∇ u + w(∂u/∂z) + uv tan(/r) and u individual components of A (u2 tan /r 2 ) for the zonal and meridional components, respectively, where r is the radius of the earth. With the assumption of steady state, vertical integration of (1) in the first and second levels gives ¯ = − f k × U ¯ ≡ where U

0 −H1

 1 ¯ ∇ P¯ + + R, 0 0

 dz + u

 −H1 −H2

 dz = u

(2)

0 −H2

 dz is the vertically integrated horizontal velocity in the first u

and second levels with a rigid-lid approximation. −H1 and −H2 are the vertical coordinates of the interface between the first and second levels, and that between the second and third levels, respectively. 0 ¯ is the vertically integrated residuals Similarly, P¯ ≡ p dz is the vertically integrated pressure, R −H2

¯ ≡ −A (∂u given by the sum of the viscous and nonlinear terms R v  /∂z)|z=−H2 +

0

−H2

 dz, and  is (F − A)

the wind-stress applied to the ocean surface. Vertical integration of the momentum equation allows us to avoid direct consideration of the effects of bottom topography, whereas using the prescribed level depth, −H2 , entails consideration of the vertical velocities at the interface between the levels. Note that the interfaces between the levels do not necessarily coincide with the levels of no motion, and horizontal pressure gradients on the respective interfaces are allowed. The present formulation only requires horizontal continuity of in situ pressure over the level interfaces.

Fig. 2. A circumpolar integral path used in the present analytical formulation. The path C runs from the southern tip of Africa (F) to the western coast of South America (B) via the southern tip of Australia (A) along a circle of latitude, and runs northward along the coast until it reaches the northern end of South America (D). After passing through the Panama Gateway, it runs eastward along a circle of latitude until it reaches the western coast of Africa (E), and then runs southward along the western coast of Africa to close the path. The Coriolis parameters at the southern and northern bounds of the path are fP and fA, respectively. Ti represents the Panama throughflow, while TPi , TAi and TIi represent the meridional mass transport across the Pacific (AB), North Atlantic (DE) and Indian (FA) sectors on the path C, respectively, where the subscript i corresponds to the level number (i = 1 or 2). WPi , WAi and WIi are the amounts of vertical mass transport in the Pacific, Atlantic and Indian Oceans, respectively. The amounts are calculated in the basins north of the path C.

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Here we introduce a circumpolar integral path C (Fig. 2) and calculate the line integral of (2) along it. The circumpolar path runs from the southern tip of Africa to the western coast of South America along a circle of latitude, and then runs northward along the coast until it reaches the Panama Gateway. After passing through the gateway, the path runs eastward along another circle of latitude until it reaches the western coast of Africa, and then runs southward along the coast until it reaches the starting point. Such a path avoids locations of strong dissipation of vorticity associated with western boundary regions in the same manner as Godfrey’s ‘island rule’ (Godfrey, 1989). A similar integral path had been used to investigate the effects of meridional land barrier on the Antarctic Circumpolar Current (e.g., Ishida, 1994; Hughes and De Cuevas, 2002). We have the South American continent acting as a large meridional land barrier. It should be noted here that we assumed that major features of the bottom topography, such as seamounts, lie beneath the interface between the second and third level along the path C, so that the entire bottom topography is contained within the third level. The line integral along the closed path C eliminates the pressure term, assuming that the in situ pressure at z = H2 changes continuously along the path, and gives a relationship between the transport across the path, a line integral of the wind-stress and that of the residual terms:



−fP (T¯ P + T¯ I ) − fA T¯ A =

 · dl +  C 0



→ ¯ · d− R l ,

(3)

C

where fP and fA are the Coriolis parameters at the southern tip of Africa and at the Panama Gateway, respectively, and T¯ P , T¯ I and T¯ A are the meridional mass transports in the first and second levels across the Pacific, Indian and Atlantic sectors, respectively (see Fig. 2). Although we employed a line integral to obtain vorticity Eq. (3) for the sake of mathematical simplicity, one can, of course, obtain the same equation by calculating the curl of Eq. (2), in which the usual notation for the beta term is included (See Appendix A). We can rewrite the meridional mass transports, T¯ P , T¯ I and T¯ A , with the vertical mass transport in each basin and barotropic PTF transport. From mass conservation in the Pacific and Indian Oceans, T¯ P + T¯ I + WP2 + WI2 = T¯ ,



(4)



where WP2 ≡ w|z=−H2 dS and WI2 ≡ w|z=−H2 dS are the vertical mass transport across Pacific Indian the interface between the second and third levels in the Pacific and Indian Oceans, respectively, and T¯ is the barotropic PTF transport. For simplicity, we assume that the Bering Strait is closed or that the transport across the Strait is negligibly small. For the Atlantic Ocean, mass conservation gives T¯ A + WA2 + T¯ = 0,

(5)



where WA2 ≡ w|z=−H2 dS is the vertical mass transport across the interface between the secAtlantic ond and third levels in the Atlantic and Arctic Oceans. Substituting Eqs. (4) and (5) into (3) gives a relationship between the barotropic PTF transport, the vertical mass transport in each basin, the wind-stress and residual terms,



(fA − fP )T¯ + [fP (WP2 + WI2 ) + fA WA2 ] =

 · dl +  C 0



¯ · dl. R

(6)

C

Eq. (6) provides the vorticity balance of the area enclosed by the circumpolar path C. The left-hand side denotes planetary vorticity flux across C in the first and second levels, whereas the right-hand side denotes vorticity input through the sea surface due to wind-stress curl and the circumpolar integral of residual terms. The lateral planetary vorticity flux on the left-hand side consists of two parts. The first term represents the total planetary vorticity flux across C, since the total mass transports crossing FAB (See Fig. 2) and DE from the sea surface to the bottom (from the first to the third level) are equal to T¯ and −T¯ , respectively. The second term, in the square brackets, represents planetary vorticity flux across C in the third level, since the vertical mass transports, WP2 + WI2 and WA2 , are equivalent to the meridional mass transports crossing FAB and DE, respectively, in the third level. We now consider a physical constraint on the right-hand side of (6) in the present situation. If we consider a line integral of the momentum equation on a closed path along the coastline of a closed basin, incoming vorticity from the sea surface due to wind-stress mainly dissipates in the western

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boundary, and a steady circulation appears. On the other hand, since the circumpolar integral path C in the present study avoids western boundary regions, the vorticity dissipation may occur mainly through interactions between flow and bottom topographies. This situation is similar to circumpolar momentum balance of the Antarctic Circumpolar Current. In such situations, zonal momentum input due to wind-stress is transmitted downward and is balanced with dissipation due to the form drag of the bottom topography. In other words, vorticity input due to wind-stress curl is balanced with the bottom pressure torque (Olbers et al., 2004; Vallis, 2006). In the present formulation, however, Eq. (6) does not include such a dissipation process explicitly. Instead, the residual terms contain downward vorticity flux through the interface between the second and third levels due to viscosity and advection of relative vorticity. Therefore, if we assume a balance between the vorticity input due to wind-stress and the dissipation due to bottom topography, the two terms on the right-hand side of Eq. (6) cancel each other out:



 · dl +  C 0



¯ · dl∼0, R

(7)

C

where their sum is small enough to be ignored, in comparison to the other terms in (6). Eq. (7) is the key assumption employed in the present study to formulate a barotropic PTF, and its validity will be closely examined using results of General Circulation Model (GCM) experiments in Section 3. Substituting Eq. (7) into Eq. (6) gives a relationship between the barotropic PTF, T¯ , and vertical mass transports, WP2 , WI2 and WA2 , crossing the horizontal interface defined by the depth of the Panama Gateway: (fP − fA )T¯ = fP (WP2 + WI2 ) + fA WA2 .

(8)

The right-hand side refers to the forcing terms related to the global thermohaline circulation, acting as a vorticity input (or output) to the area enclosed by the path C and vertically sandwiched between the sea surface and the horizontal surface defined by the depth of the Panama Gateway. The barotropic PTF on the left-hand side is determined so as to balance the vorticity excess (or deficiency) induced by the thermohaline forcings on the right-hand side. From (8), the barotropic PTF is finally given in the following form: T¯ = −

fP (WP2 + WI2 ) + fA WA2 . fA − fP

(9)

We will examine the relationship between the thermohaline circulation and the barotropic PTF observed in the GCM in the next subsection. Next we consider the baroclinic PTF, beginning with a derivation of the PTF transport in the first level. The procedure is basically the same as the barotropic PTF except for the vertical limits of the integral of Eq. (1), which is now applied to the first level. The vorticity balance in the first level is given by



(fA − fP )T1 + fP (WP1 + WI1 ) + fA WA1 =





 · dl +  C 0



 1 · dl, R C

(10)



where WP1 ≡ w|z=−H1 dS, WI1 ≡ w|z=−H1 dS and WA1 ≡ w|z=−H1 dS denote Pacific Indian Atlantic vertical mass transport across the interface between the first and second levels in the Pacific, Indian,  1 is also defined and Atlantic Oceans, respectively, and T1 is the PTF in the first level. The residual term R 0  dz. Here, we assume that the right-hand side  1 ≡ −Av (∂u  /∂z)|z=−H + in the first level as R (F − A) 1

−H1

of Eq. (10) is small compared with each term on the left-hand side, and so the approximate vorticity balance in the first level can be estimated by the left-hand side of Eq. (10). In Section 3, the validity of this assumption will also be discussed with the results of a GCM experiment as well as the key assumption (7) used to derive the barotropic PTF. Neglecting the circumpolar integral of the wind-stress and residual terms, we write the PTF in the first level as T1 = −

fP (WP1 + WI1 ) + fA WA1 . fA − fP

(11)

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Fig. 3. Schematic diagram of the vertical mass transport in one basin associated with different vertical extensions of the MOC cell and depths where the mass transport measured. The thin solid and dotted lines denote mass stream function in a meridional section. The solid (dotted) line corresponds to clockwise (counter-clockwise) circulation. The thick dashed line denotes the depth where the vertical mass transports are measured. The solid large white arrow indicates the vertical mass transports across the depth defined by the thick dashed line. Deep water formation occurs in (b) and (c) but not in (a). Mass transport can be measured at large depths (b) or at shallow depths (c). If no deep water formation occurs in the basin, the vertical mass transport across an arbitrary depth is positive (a), while if formation does occur, then the vertical mass transport depends on the depth (b and c). The transport is positive at depths below the deep convective cell in the basin (b), but is negative within the cell (c).

Eq. (11) indicates that the PTF in the first level is driven by vertical mass transport between the first and second levels in the same manner as that of the barotropic PTF. Finally, we obtain the PTF in the second level by subtracting (11) from (9): T2 =

fP (WP1 − WP2 + WI1 − WI2 ) + fA (WA1 − WA2 ) . fA − fP

(12)

Eqs. (9), (11) and (12) indicate that the distribution of vertical mass transport associated with upwelling and downwelling determine both the barotropic and baroclinic PTF. 2.2. Flow directions of the modeled PTF In this subsection, we argue how the global distribution of the Meridional Overturning Circulation (MOC) and associated deep water formation are related to the direction of the barotropic and baroclinic PTF by using (9), (11) and (12). In the present situation, the Pacific and Atlantic Oceans are connected not only by the Panama Gateway, but also by the Southern Ocean. Therefore, if deep water formation does not occur in either the Atlantic or Pacific basin, then deep or bottom water will be transported to one basin from the other or from the Southern Ocean. Otherwise, the deeper part of the basin will be filled by water whose lower density is equivalent to that of the densest water formed in the basin, and vertical stratification in that basin will become extremely weak compared to that of the other basin. This causes a strong density gradient between this and the other basin, and leads to deep or bottom water inflow into the deeper part of the basin. The deep or bottom water inflow at the same time causes outflow of shallow water from the basin, and is accompanied by vertical mass transport in the basin. Therefore, if deep water formation does not occur in one basin, the basin-averaged vertical transport in the deeper part is expected to be upward (Fig. 3a). On the other hand, if deep water formation occurs in one basin but not in the other, then the basin-averaged vertical transport is expected to be downward in the former (Fig. 3c). The excess (net) downwelling transport is the source of upwelling in the other basins. However, if the downwelling reaches only a certain depth, upwelling prevails below that depth (Fig. 3b). This is equivalent to the case in which the MOC associated with deep water formation is entirely restricted to one basin and no deep water is supplied to the other basin. The vertical transport in (9), (11) and (12), which determines the PTF in the present model, is therefore influenced not only by the deep water formation in the respective basin, but also by the depths where vertical transport is measured. In the present model, the interface between the second and third levels corresponds to the sill depth at the Panama Gateway. Thus, whether the deep water circulation and

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the associated MOC reach the sill depth of the gateway or not is important when determining the PTF. During the period of an open Panama Gateway, it is believed that waters formed in both the Northern and Southern Hemisphere (Wright et al., 1992). Numerical studies suggest that bottom water formed in the Southern Hemisphere filled the abyss of the oceans and that deep water formed in the Northern Hemisphere occupied the shallower part above it (Motoi et al., 2005; Schneider and Schmittner, 2006; Heydt and Dijkstra, 2006). Thus, the global MOC related to deep water formation in the Northern Hemisphere and bottom water formation in the Southern Hemisphere may be common to both the present day and the era of an open Panama Gateway. Today, there are two dominant MOC cells. One is accompanied by NADW formation and its circulation (hereafter referred to as the NADW cell), and the other is accompanied by Antarctic Bottom Water (AABW) formation and its extent (hereafter referred to as the AABW cell). The NADW cell reaches depths of about 4000 m in the Atlantic Ocean, while the deepest part of the AABW cell lies beneath the NADW cell (e.g. Ganachaud and Wunsch, 2000; Tally, 2003). The depth of the Panama Gateway relative to the MOC cells may have experienced several transitions during the period of an open Panama Gateway. The Panama Gateway continued shoaling during the Miocene, while deep water formation in the Northern Hemisphere is also believed to have experienced variations in its strength and depth during that period (Wright et al., 1992; Frank, 2002; Frank et al., 2002). We therefore consider the directions of the barotropic and baroclinic PTFs in the model for two typical cases: one in which the MOC cell associated with deep water formation in the Northern Hemisphere extends beyond the depth of the Panama Gateway and one in which the MOC does not. The first scenario is referred to as the shallow Panama case, and the other as the deep Panama case. In the deep Panama case, the direction of the barotropic PTF is always eastward in this model. The vertical mass transports at the interface between the second and third levels, WP2 , WI2 and WA2 , are positive because deep water, which form in the Northern Hemisphere, does not reach depths where the vertical transports are defined (Fig. 4a). The terms in the numerators on the right-hand side of (9) have opposite signs. It may be reasonable to assume that the vertical velocities in each basin have the same order of magnitude. In this case, since the area on the Pacific side is larger than that on the Atlantic side, the vertical mass transport in the Pacific and Indian Oceans should also be much larger than that in the Atlantic and Arctic Oceans. Because the sign of –fP (WP1 + WI2 ) is positive due to negative fP , the sign of (9) is also positive. Consequently, the direction of the barotropic PTF is estimated to be eastward, regardless of the location of deep water formation.

Fig. 4. Schematic diagram showing the meridional overturning circulation cell in the global ocean and in the Panama Gateway. The meridional mass stream function is indicated by the solid and dotted lines, which correspond to a clockwise and counter-clockwise circulation, respectively. If the Panama Gateway is deeper than the deep water formation cell in the Northern Hemisphere (a), then the vertical mass transport across the dashed line is positive in both the Pacific and Atlantic Oceans. If it is shallower (b), then the vertical mass transport across the dashed line is negative in the basin where deep water formation occurs, and positive where it does not.

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Table 1 Properties of the semi-analytical model. Location of deep water formation in the Northern Hemisphere Atlantic side Barotropic transport

Pacific side Vertical structure

Barotropic transport

Deep Panama sill

Eastward

Eastward

Shallow Panama sill

Eastward

Westward

Vertical structure

The direction of the baroclinic PTF in the deep Panama case, however, may depend on the location of deep water formation. If deep water formation occurs on the North Atlantic side, we obtain negative WA1 and positive WP1 + WI1 . Eq. (11) then gives an eastward PTF in the first level. In the second level, the PTF direction cannot be simply estimated by the model. If deep water formation in the North Atlantic Ocean is sufficiently active to supply deep water to the other basins, the second term of the numerator in (12) becomes a large negative value. In this case, deep water upwells on the Pacific side and the amount, (WP1 + WI1 ), exceeds that of bottom water upwelling (WP2 + WI2 ). The first term of the numerator in (12) also becomes negative, indicating a westward PTF in the second level. On the other hand, if deep water formation in the Atlantic Ocean is not sufficient to supply deep water to the other basins, the vertical mass transport on the Pacific side will depend mainly on upwelling of bottom water formed in the Southern Hemisphere. In this case, both westward and eastward PTFs in the second level are possible. If deep water formation occurs on the Pacific side, then the baroclinic structure in the model exhibits properties opposite to that associated with deep water formation on the Atlantic side. The PTF direction in the first level is westward, since both terms in the numerator of (11) are positive. In the second level, the PTF direction is eastward since the barotropic PTF given by (9) is eastward. In the shallow Panama case (Fig. 4b), both the barotropic and baroclinic PTFs depend on the location of deep water formation. If deep water is formed on the Atlantic side, the barotropic PTF flows eastward since WA2 and WP2 + WI2 in (9) have a negative and positive sign, respectively. On the other hand, deep water formation on the Pacific side leads to a westward barotropic PTF, since WA2 and WP2 + WI2 are positive and negative, respectively. For the baroclinic structure, the PTF direction in the first level is estimated to be the same as that for the deep Panama case, while that in the second level cannot be estimated a priori from the location of deep water formation. The magnitudes of the deep and bottom water formation are required before we can use Eq. (12) to estimate the PTF in the second level. Properties of the barotropic and baroclinic PTFs discussed above are summarized in Table 1 with respect to the deep or shallow Panama cases and to Pacific or Atlantic deep water formation. The barotropic PTF generally flows eastward; an exception (westward flowing barotropic PTF) occurs only in the shallow Panama case with deep water formation on the Pacific side. Baroclinic PTF is controlled by the location and strength of deep water formation. Deep water formation on the opposite side of the Panama Gateway generally leads to an opposite baroclinic structure. The flow directions of the barotropic and baroclinic PTFs in the present model are consistent with the results of previous numerical studies. These studies reported that the barotropic PTF is eastward with an open Panama (or Central American Seaway), regardless of whether Northern Hemisphere deep water formation occurs in the North Atlantic Ocean (Maier-Reimer et al., 1990; Mikolajewicz and Crowley, 1997; Murdock et al., 1997; Nisancioglu et al., 2003; Schneider and Schmittner, 2006) or in the North Pacific (Motoi et al., 2005). It is noteworthy that the Pacific MOC in Motoi et al. (2005) is shallower than the sill of the Panama Gateway and is thus classified as a deep Panama case. In this case, where the Panama sill is deep and deep water formation occurs in either ocean, our model predicts

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an eastward barotropic PTF (Table 1), consistent with the previous studies. The baroclinic structure of the PTF was only reported in Nisancioglu et al. (2003) and Schneider and Schmittner (2006), which correspond to shallow and deep Panama cases, respectively. Both experiments showed an eastward PTF in the shallow depths and a westward PTF in the lower depths. This is, again, consistent with the implications of the present study. 3. Numerical experiments Although the present semi-analytical model qualitatively explains the direction of the barotropic and baroclinic PTFs in previous numerical studies, it is not clear whether the present model can estimate these PTFs quantitatively. Furthermore, since deep water formation on the Pacific side was only found in Motoi et al. (2005) which did not look at the baroclinic structure of PTFs, it would be of interest to closely examine whether the present model can also adequately explain the baroclinic PTFs when deep water is formed in the Pacific Ocean. We therefore conduct numerical experiments with an open Panama Gateway for further examination of the present model. Comparisons will be made between this semi-analytical model and numerical results over a wide parameter space, regarding the variation in vertical transport and Panama sill depth. Of course, the use of one numerical model cannot cover the whole parameter space because the location and strength of deep water formation is dependent on the type of model. However, we believe that our comparison between the semi-analytical and numerical models is useful, as it can serve as a framework for future comparisons with other numerical models. 3.1. AOGCM and experimental design The AOGCM used in the present study was developed at Geophysical Fluid Dynamics Laboratory (Manabe et al., 1991). The model simulates positive and negative feedbacks of salt and heat transports on the thermohaline circulation reasonably well (Manabe and Stouffer, 2000; Stouffer and Manabe, 2003), and was used to investigate the effects of an open Panama Gateway on the climate and ocean circulation (Motoi et al., 2005). Although the model employs surface flux corrections for heat and salt, it does not matter for our purpose. We intend not to carry out paleo-climate simulations but to examine the PTF for MOCs of various strengths and for a Panama Gateway of various depths. Since the AOGCM exhibits deep water formation on the Pacific side with an open Panama Gateway (Motoi et al., 2005), it is favorable for verifying the validity of the semi-analytical model with various conditions not mentioned in the previous section. The resolution of the atmospheric part is R15 with 9 levels, while that of the ocean part is 3.75◦ longitude and 4.4◦ latitude with 12 levels. The initial condition for our experiments is obtained from a 10,000-year integration with a closed Panama Gateway. After the integration, four ocean grid cells are removed from the surface to the 9th level (2560 m) in order to represent an open Panama Gateway in the same way as Motoi et al. (2005). The equilibrium solution is obtained after a 2000-year integration, and is termed “control run”. To examine the validity of the semi-analytical model quantitatively, we will compare the PTF calculated in the AOGCM with that obtained from the semi-analytical model for MOCs of various strengths. The vertical mass transports observed in the AOGCM will be used for forcing of the semianalytical model. To obtain the PTF for various MOC strengths, we carried out experiments in which the wind-stress applied to the ocean part is modified. Since the variation of wind-stress alters the MOC strength through variation of heat exchange between ocean and atmosphere (Rahmstorf and England, 1997; Tsujino and Suginohara, 1999; Hirabara et al., 2007), we can obtain MOCs and PTFs of various strengths by changing the wind-stress. In addition, this setting allows us to evaluate the validity of key assumption (7) over a wide range of wind-stress. In the AOGCM, wind-stress data is passed from the atmospheric part to the ocean part once per model-day. In our experiments, the wind-stress is modified by the following equation, ¯ = c  .

(13)

where  is the wind-stress calculated from the atmospheric part and ¯ is that passed to the ocean part. The constant c is a multiplication factor for the wind-stress (hereafter referred to as the wind-stress factor). We carried out 6 additional experiments with different values for c (c = 0, 0.25, 0.5, 0.75, 1.25

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Table 2 Depth of the Panama Gateway in each experiment. Experiment number

Numbers of vertical layers at the Panama Gateway

Panama sill depth (m)

P1 (closed Panama Gateway) P2 P3 P4 P5 P6 P7 P8 P9 (control run)

0 2 3 4 5 6 7 8 9

0 119 220 371 595 914 1347 1898 2560

and 1.5). The control run corresponds to the case of c = 1.0. The initial conditions for each experiment are given by the equilibrium solutions of the control run, and the integration time to obtain quasiequilibrium state is 500 years in each experiment. This spin-up time seems to be too short to obtain a steady state. However, the robust response to the wind-stress variation is apparent in the first 150 years, and variations after 400 years are sufficiently small to assume quasi-steady state for the present analysis. This short adjustment time is possibly attributable to the steadiness of the MOC pattern in the experiment. The temporal average of the last 100 years of the AOGCM output is used for analysis. In addition to the control experiment with the Panama Gateway depth of 2560 m, we carried out 8 experiments with different sill depths (Table 2). The initial condition for each experiment is obtained from a 10,000-year time integration with a closed Panama Gateway. After the integration, 4 grid cells are removed from the Isthmus of Panama in the same way as the control run, except for its depth. The equilibrium solution is obtained after a further 2000-year integration for each experiment, and the temporal average of the last 100 years of the AOGCM output is used for analysis. 3.2. Wind and accompanied thermohaline circulation changes Fig. 5 shows the response of the ocean part of the AOGCM to variations in the wind-stress factor c. Strictly speaking, the wind-stress applied to the ocean part in each experiment is not equivalent to c times the wind-stress in the control run, since the modified wind-stress changes the atmospheric

Fig. 5. AOGCM response to changes in the wind-stress factor. (a) A line integral of the wind-stress along the circumpolar path C. The integral of the wind-stress responds linearly to the wind-stress factor, suggesting that ocean–atmosphere feedback is relatively small and that the pattern of the wind-stress in each experiment remains virtually unchanged. (b) Transport of the MOC associated with the Antarctic Bottom Water (AABW) formation. The magnitude of the MOC is defined as the maximum value of the meridional stream function between depths of −5000 and −3000 m in the Southern Hemisphere. (c) MOC transport and vertical transport in the North Pacific Ocean. The solid and dotted lines indicate the negative maximum of the meridional stream function in the North Pacific Ocean and the vertical mass transport in the Pacific Ocean enclosed by the path C, respectively. The strong wind-stress causes an active meridional overturning circulation in both hemispheres.

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circulation throughout feedback from the ocean circulation. However, the effects of the feedback hardly change the pattern of the wind-stress (not shown), and the line integral of the wind-stress along the path C is almost linearly related to the wind-stress factor (Fig. 5a). A large wind-stress factor leads to an active MOC in both the Northern and Southern Hemispheres (Fig. 6). As the wind-stress factor increases, bottom water formation in the Southern Hemisphere is enhanced, and there appears

Fig. 6. Meridional overturning stream function (Sv) in response to changes in the wind-stress factor, c, ranging from 0.25 to 1.5. Positive (negative) value denotes clockwise (counter-clockwise) circulation.

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Fig. 7. (a) The barotropic PTF transport and (b) the PTF transport in each level in response to changes in the wind-stress factor. In each panel, the solid line indicates the transport observed in the AOGCM experiments, and the dashed line indicates that estimated from the analytical model.

to be a near-linear relationship with wind-stress factor above 0.5 (Fig. 5b). In addition, deep water formation in the northern part of the North Pacific Ocean increases almost linearly as the wind-stress factor is increased (Fig. 5c). It is worth noting that the variation in wind-stress applied to the Southern Ocean might influence the meridional transport crossing a circle of latitude in two ways. One way is through the direct influence of the wind-stress on the meridional circulation, whereby the northward Ekman transport and its compensating flow are expected to vary with the wind-stress strength (Nof, 2003). The other is through the influence of the wind-stress on the thermohaline circulation, which consists of deep and narrow downwelling in the Southern Ocean and extensive upwelling in both hemispheres. As discussed by Rahmstorf and England (1997) and Hirabara et al. (2007), the variation of wind-stress in the Southern Ocean influences not only the AABW formation, but also deep water formation in the Northern Hemisphere. As shown in Fig. 6, the meridional circulation driven directly by the wind-stress has a relatively small meridional extent (from 60◦ S to 30◦ S in control run), whereas the thermohaline circulation extends northward beyond the equator. The assumption of Eq. (7) was introduced to exclude any direct contributions from this wind-driven circulation, since this circulation is practically limited to the Southern Ocean and hardly affects the meridional circulation extending to the Northern Hemisphere. Fig. 7 shows, at each level, the barotropic and baroclinic PTF transports obtained in the AOGCM and estimated from the semi-analytical model. The semi-analytical estimations are calculated from (9), (11) and (12) by substituting the vertical mass transports in the Pacific, Indian and Atlantic Oceans in the AOGCM. The interface depth between the first and second levels is chosen to be 220 m, corresponding to the nodal point of the baroclinic structure of the PTF found in the AOGCM (Fig. 8). Since a larger wind-stress factor corresponds to stronger bottom and deep water formation in both hemispheres, the horizontal axis of Fig. 7 can be regarded as an index of the MOC. The figure shows that a strong meridional overturning circulation, expressed by a large wind-stress factor, leads to a larger barotropic and baroclinic transport of the PTF in both the AOGCM and the semi-analytical estimation. Furthermore, semi-analytical estimations reproduce the PTF transports in both levels and the barotropic PTF transport obtained by the AOGCM satisfactorily. Although there are systematic discrepancies between modeled and estimated transports, they are small compared with the transports when the wind-stress factor is larger or equal to 0.5. It should be noted here that the interface between the first and second levels, H1 , should be chosen so as to represent the nodal depth of the baroclinic structure of the PTF if one intends to capture the baroclinic structure in the semi-analytical model. Even a choice of H1 that is not in accordance with

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Fig. 8. Vertical profiles of the Panama throughflow in response to changes in the wind-stress, ranging from c = 0.25 to c = 1.5. Positive (negative) value denotes eastward (westward) velocity [cm/s].

the nodal depth gives a water mass transport in the respective levels comparable to that in the AOGCM solutions, since the semi-analytical solution is based on the vorticity and mass balance at each level, which is not affected by the choice of the interfacial depth. However, choosing an interfacial depth which is different from the nodal depth may lead to water mass transport that does not capture the baroclinic structure in the Panama Gateway. In the present AOGCM experiments, the nodal depth is not influenced by the MOC strength (Fig. 8). The reason for the invariable nodal depth is possibly due to

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Fig. 9. Amplitudes of each term in vorticity Eq. (6) in the first and second levels combined, evaluated over the area enclosed by the path C and obtained from the AOGCM results. In panel (a), the sum of the wind-stress and residual terms are compared with other terms, while in panel (b), the wind-stress and residual terms are plotted separately.

the steadiness of the MOC pattern occurring in both hemispheres. The strength of the MOC varies with the wind-stress factor, whereas the interfacial depth between the deep water formed in the Northern Hemisphere and the bottom water formed in the Southern Hemisphere appears to be invariant in the AOGCM (Fig. 6). These features found in the model MOC are probably due to the experimental design of the parameter study, in which only the wind-stress is modified to vary the MOC strength. If we modify the surface heat flux arbitrarily, not only the strength, but also the depth of the MOC will change, and the associated baroclinic structure of the PTF is also expected to be modified. In fact, different GCM experiments exhibit different nodal depths, as shown in Nisancioglu et al. (2003) and Schneider and Schmittner (2006). In those numerical experiments, the nodal depth is located at depths of about 1000 m, with a relatively deep MOC cell in the Northern Hemisphere compared to the present study. These deeper nodal depths are likely to be due, in part, to the deeper MOC cells in those experiments. Given the good agreement between the semi-analytical model and the AOGCM, it would be of interest to examine each term of the vorticity equation for the first two levels combined (6), which forms the basis of the semi-analytical model (9), along with the assumption that the right-hand side of (6) is small. The integrals of the respective terms in (6) along the path C are obtained from the AOGCM, except for the residual term, which was estimated from the differences of the other terms in (6). The two terms used in the semi-analytical model (9) are of the same magnitude, and are much larger than the sum of the wind-stress and residual terms (Fig. 9a), which nearly cancel each other (Fig. 9b). This confirms that the key assumption (7) is adequate for deriving the barotropic PTF transport. In addition, the respective terms of the vorticity equation for the first level (10) are examined (Fig. 10). Again, the two terms used for the semi-analytical model (11) are of the same magnitude. These terms are much larger than the sum of the wind-stress and residual terms, though the degree of cancellation between the wind-stress and residual terms is much smaller than that for the vorticity equation for the first two levels combined. This indicates that the assumption we used for deriving (11) is also appropriate.

3.3. Changes in the depth of the Panama Gateway Now we examine the AOGCM response due to changes in the sill depth of the Panama Gateway. As described in Section 2.2, the semi-analytical model suggests that the depth of the Panama Gateway influences the barotropic PTF when deep water is formed in the Pacific Ocean (Table 1).

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Fig. 10. Same as Fig. 9, but for the vorticity Eq. (10), applied to the first level.

Fig. 11 shows the maximum value of the meridional overturning stream function in the Northern Hemisphere, observed in the AOGCM, in response to changes in the sill depth of the Panama Gateway. The AOGCM reproduces a strong Atlantic MOC (16 Sv) and a much smaller Pacific MOC for the closed Panama Gateway, consistent with the present climate. With an open Panama Gateway, however, even if the depth is as shallow as 119 m, the Pacific MOC overwhelms the Atlantic MOC. The strength of the Pacific MOC is practically independent of the sill depth except for the case of complete closure of the Panama Gateway, while the Atlantic MOC is gradually enhanced as the sill shoals. The barotropic PTF estimated from the semi-analytical model captures the essential changes in the PTF with sill depth obtained in the AOGCM experiments (Fig. 12a). In particular, the westward

Fig. 11. The maximum value of the stream function associated with the North Pacific MOC (solid line) and with the North Atlantic MOC (dashed line), obtained in the AOGCM experiments for various Panama sill depths.

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Fig. 12. (a) Barotropic PTF transport and (b) PTF transports in the first and second levels for various Panama sill depth. The solid line denotes the PTF obtained in the AOGCM experiments, whereas the dashed line denotes that estimated from the semi-analytical model.

barotropic PTF in the experiments with the two shallowest sills, are consistent with the qualitative prediction of the semi-analytical model in the case of a shallow sill combined with deep water formation in the Pacific Ocean (Section 2.2, Table 1). As the sill depth increases, the barotropic PTF changes its direction to eastward for sill depths of about 250–400 m. Further increases in sill depth generally lead to a stronger eastward barotropic transport. These features are found in both the AOGCM and the semi-analytical model. The PTF transports in the first and second levels are also estimated from the semi-analytical model, and are compared to the PTF transport in the AOGCM (Fig. 12b). In this estimate, we used the same interface depth between the first and second levels as that in the previous subsection (220 m), since the nodal depth is not influenced by the sill depth except in the case where the sill depth is shallower than 220 m (not shown). On the other hand, the interface between the second and third levels varies with the sill depth. Thus, for the two shallowest sill depths, the sill becomes shallower or equal to the constant interface depth. In this situation, only the first level exists, and the transport in that level is the same as the barotropic transport. The westward transport in the first level of the AOGCM is nearly constant for deep to middle sill depths (2600–600 m), and decreases in the case of shallower sills (370–120 m). In contrast, the eastward transport in the second level decreases in response to shoaling of the sill. The analytical estimation also agrees well with the trends and signs of the PTF variation at each level. Although the analytical estimation tends to underestimate the magnitude of the transport at each level, the maximum error is less than 25%. 4. Summary and discussion In the present study, we proposed a semi-analytical model that estimates the barotropic and baroclinic transports of the PTFs from the vertical transports in the respective basins and from the depth of the Panama Gateway. In order to formulate the PTF, we introduced a circumpolar integral of the vertically integrated momentum equation, which is equivalent to a surface integral of the vorticity equation. This equation enables us to consider the vorticity budget over the area including the Pacific, Indian and Atlantic Oceans. The vorticity budget gives the PTF as a function of vertical mass transports in the basins enclosed by the circumpolar path. Since the vertical mass transports are closely related to the location and strength of deep and bottom water formation in both hemispheres, we can relate the PTF to water formation.

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Fig. 13. Integral path employed in Omta and Dijkstra (2003). The solid thick line, C , indicates the integral path employed in Omta and Dijkstra (2003). In their study, the barotropic PTF is given by TP ∼ (fSA − fD )/(fP − fSA )TD , where TP and TD are the barotropic PTF and ACC transport through the Drake Passage, respectively.

According to the present model, barotropic transport through the Panama Gateway (or the Central American Seaway) flows eastward, except when deep water formation occurs in the North Pacific Ocean with a shallow Panama Gateway. As for the baroclinic PTF, deep water formation in the North Atlantic Ocean is associated with an eastward flow in the upper depths, whereas formation in the North Pacific is associated with a westward flow. These qualitative findings from the model are consistent with recent numerical studies (see Nisancioglu et al., 2003; Prange and Schulz, 2004; Klocker et al., 2005; Motoi et al., 2005; Schneider and Schmittner, 2006). In addition, the numerical experiments in the present study showed quantitative agreement between the PTF obtained in the AOGCM and that estimated from the semi-analytical model for a wide range of MOC strengths and Panama Gateway sill depths. It is interesting to note the similarities and differences between the present formulation and the formulation by Omta and Dijkstra (2003). Omta and Dijkstra (2003) considered the vorticity budget over the South Atlantic Ocean and Atlantic sector of the Southern Ocean and employed an integral path enclosing this area for the barotropic PTF (see Fig. 13). Their semi-analytical model gives the barotropic PTF as a function of the transport through the Drake Passage and the meridional location of the ACC. Although the integral path employed in Omta and Dijkstra (2003) differs from that used in the present study, the foundation of their formulation is the same as that for the present study, i.e., the conservation of absolute vorticity in a specific region. In fact, the formulation used in Omta and Dijkstra (2003) provides a consistent barotropic PTF in response to wind-stress variations in the present experiment (Fig. 14a). However, their formulation is only applicable to the case in which the sill of the Panama Gateway is sufficiently deep. Fig. 14b shows the vorticity balance in their formulation in response to variations in the sill depth of the present study. When the sill is deeper than about 1300 m, their formulation provides a reasonable estimate of the barotropic PTF. On the other hand, a shallower sill depth leads to a less reliable estimate. The failure to estimate the PTF in shallow sills is possibly due to the range of the vertical integration used to derive the vorticity equation. In their formulation, the vertical integration is carried out over the entire water column, including the bottom topography. Significant effects of the sill topography on the vorticity balance may cause a large deviation between the analytical estimate and the numerical results. In contrast to Omta and Dijkstra (2003), the present formulation changes the range of the vertical integration as sill depth varies, and the vorticity budget within a water column shallower than the sill depth is considered. This formulation enables us to estimate the barotropic PTF, even in shallow sills, but does not allow us to use water mass transport crossing an arbitrary vertical section such as the Drake Passage. Instead of using the horizontal transport, we utilized the vertical transport across the level interface by introducing mass conservation. This formulation gives a relationship between the thermohaline circulation and the PTF transport. It is also worth noting that the present formulation

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Fig. 14. Vorticity balance of the formulation in Omta and Dijkstra (2003) in response to changes in (a) the wind-stress and (b) sill depth of the present study. The solid line indicates the vorticity flux through the Panama Gateway, while the dashed line indicates that through the Drake Passage. Both fluxes are given as the difference from the vorticity flux leaving the meridional section that connects the southern tip of Africa to Antarctica (see Fig. 13).

enables us to avoid direct consideration of pressure distribution on the level interfaces. In the cases of a shallow sill, the contribution from the bottom pressure torque is expected to be important (Wajsowics, 1993; Jackson et al., 2006). Since the entire bottom topography is contained within the third level in the present formulation, the contribution from the bottom pressure torque affects the transport in the second level through the pressure distribution on the level interface. Nevertheless, we do not need to specify these contributions explicitly as discussed above, because the geostrophic transport associated with the pressure gradient on the level interface is replaced by the vertical mass transport across the interface. In fact, the estimates from the semi-analytical model which does not consider the sill effect (Omta and Dijkstra, 2003) show a large deviation from the numerical results for a shallow Panama sill (Fig. 14b), whereas those in the present model agree well (Fig. 12a). Comparison of the two semi-analytical models demonstrates the use of vorticity conservation and points to constraints on its use when considering water mass exchange between basins. The implications of the semi-analytical model are consistent with proxy-based paleoceanographic studies. Reynolds et al. (1999) showed eastward flowing Pacific water through the Panama Gateway into the Caribbean Sea between 8 and 5 Ma from Nd- and Pb-isotope records. Since deep water formation in the North Atlantic Ocean has been present for at least the last 11 Ma (Woodruff and Savin, 1989), there appears to be an eastward PTF with deep water formation in the North Atlantic Ocean during the same period (See Frank, 2002 for a review). The relationship between the PTF and the location of deep water formation is consistent with the implications of the present semi-analytical model. Steph et al. (2006) showed an eastward PTF between 5 and 2.5 Ma from ı18 O records, also consistent with the present model. The implications of the PTF direction obtained from previous numerical and paleoceanographic studies and from the present model are summarized in Fig. 15. The present semi-analytical model has profound implications for our understanding of the paleoclimate. The model relates the PTF to the strength of the vertical transport in the respective basins, which should be related to deep water formation and MOCs. If we are able to evaluate the flow direction of the PTF from sediment data on both sides of the Panama Gateway, we can then estimate the location of deep water formation in the Northern Hemisphere during that period from the present model. Conversely, information on deep water formation in the Northern Hemisphere, combined with tectonic data on the Panama Gateway enables us to estimate the direction of flow for the PTF and its vertical structure. The model also enables us to consider the shoaling process of the Panama Gateway. If we can obtain a time-record of deep water formation and the direction of flow for the PTF from various proxy data, then we can relate this information to the shoaling process of the Panama Gate-

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Fig. 15. Schematic diagrams of the Panama throughflow with possible deep water formation in the Northern Hemisphere. Deep water formation in the North Atlantic Ocean (top) and in the North Pacific Ocean (bottom). The left and center panels show a rough sketch of the upper and deep ocean circulation, respectively. The shaded ellipses indicate the locations of deep water formation, while thick arrows indicate the direction of the current. The vertical profiles of the Panama throughflow are shown on the right panels. These schematic diagrams correspond to the deep Panama sill case. For the shallow sill case, refer to the text.

way. Thus, the semi-analytical model can relate our current understanding of the ocean circulation to tectonic events, and provide a more consistent picture of the paleo-climate during the period of an open Panama Gateway. The validity of the semi-analytical model given by (9), (11) and (12) should be further examined by using the output from OGCMs and AOGCMs. Although the present semi-analytical model is supported by the results of an AOGCM, its performance with respect to other AOGCMs and OGCMs may differ. However, large differences in the performance of the semi-analytical model, if there are any, can also impart useful information on the dynamics of the paleo-ocean. Even if the present semi-analytical model does not explain the results from an AOGCM or OGCM, the vorticity balances (6) and (10) should be satisfied, because we made no assumptions in their derivation. Therefore, differences in the performance of the semi-analytical model are likely to be related to the way different numerical models handle the vorticity balance. Analysis of the respective terms in the vorticity equations, as shown in Figs. 9 and 10, for different numerical models can thus reveal similarities and differences in the dynamics of the models. In particular, key assumption (7) should be examined in other models, since the residual terms in the present study is estimated from the differences of the other terms in (6), and its validity may depend on the accuracy of the integral balance in the GFDL R15 model. Such knowledge should be a basis for further understanding of how oceanic circulation in the past was formed under different configurations of land mass and ocean basins. Acknowledgements We would like to thank the two anonymous reviewers for their constructive comments and A. Ishida, A. Kubokawa and G. Mizuta for their fruitful discussions and encouragement. We also gratefully

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acknowledge funding from a 21st Century Center of Excellence (COE) for the “Neo-Science of Natural History” program (Leader: Hisatake Okada) at Hokkaido University. The GFD-DENNOU library and Grid Analysis and Display System (GrADS) were used to draw the figures. Appendix A. Here we provide an alternative derivation of Eq. (3) by calculating the curl of Eq. (2). We first calculate the curl of (2): ¯ = ∇ × ∇ × (f k × U)

 ¯ + ∇ × R. 0

(A1)

The left-hand side of this equation can be divided into advection of planetary vorticity (beta term) and vertical stretching of the water column. The vertical component of the left-hand side is ¯ · k = ˇV¯ + fw| (∇ × (f k × U)) z=−H2 ,

(A2)

¯ where the mass conservation equation, (∂U/∂x) + (∂V¯ /∂y) = w|z=−H2 , is applied. The vertically integrated vorticity equation from −H2 to the sea surface can then be represented in the following form with the usual notation for the beta term:



 0

∇×

ˇV¯ + fw|z=−H2 =



¯ , + (∇ × R) z

(A3)

z

 indicates the vertical component of vector A.  If we integrate Eq. (A3) over the area S enclosed where (A) z by the circumpolar path C, we obtain

 



 ˇV¯ dS + S

S

fw|z=−H2 dS =

S

 ∇× 0



 dS + z

S

¯ dS. (∇ × R) z

(A4)

The first term of the left-hand side of Eq. (A4) is,





ˇV¯ dS = S

yN



ˇ(

yS

xE



V¯ dx) dy =

xW

yN

ˇT¯ (y) dy,

(A5)

yS

where yN and yS denote the latitudes of the northern and southern bounds of the area S, and xE and xW denote the longitudes of the eastern and western boundaries of the basin. T¯ (y), the meridional mass transport across the zonal section at latitude y and from the surface to −H2 , is defined as T¯ (y) =

 xE  0 xW

−H2

v dz dx. By using partial differentiation, Eq. (A5) can be rewritten as:





y

S

N ˇV¯ dS = [f T (y)]yS −

yN

f

yS

∂T¯ (y) dy. ∂y

(A6)

In the Indo-Pacific sector, yS corresponds to the latitude of the southern tip of Africa, whereas in the Atlantic sector, yS corresponds to that of the Panama Gateway. Next, the first term of the left-hand side of Eq. (A4) becomes





ˇV¯ dS = −fP (T¯ P + T¯ I ) − fA T¯ A −

yN

yS

S

f

∂T¯ (y) dy, ∂y

(A7)

since in both sectors, the meridional mass transport at the northern bound of the area must be zero. On the other hand, the second term of the left-hand side of Eq. (A4) is





S

fw|z=−H2 dS =

yN

yS



f(

xE

xW

w|z=−H2 dx) dy.

(A8)

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From mass conservation in each zonal section, the zonal integration of the vertical velocity at z = −H2 is equivalent to the meridional gradient of the northward mass transport across the basin:



xE

xW

w|z=−H2 dx =

∂T¯ (y) . ∂y

(A9)

Therefore, Eq. (A8) becomes





S

fw|z=−H2 dS =

yN

f

yS

∂T¯ (y) dy. ∂y

(A10)

Substituting (A7) and (A10) into (A4) gives the vorticity equation which is represented by the horizontal mass transport crossing the perimeter of the area S:

 

∇×

−fP (T¯ P + T¯ I ) − fA T¯ A = S

 0





dS + z

S

¯ dS. (∇ × R) z

(A11)

Finally, by applying Stokes’ theorem, we obtain



−fP (T¯ P + T¯ I ) − fA T¯ A =

 · dl +  C 0



¯ · dl, R

(A12)

C

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