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  • View in gallery

    The conveyor belt illustrated with Lagrangian trajectories. Color indicates the temperature and salinity composition, with red for warm water, green for cold water, yellow for saline water, and blue for less saline water. This figure is also shown as an animation in supplement 1.

  • View in gallery

    The global meridional overturning streamfunction as a function of (top) depth, (middle) temperature, and (bottom) salinity. Units are in Sv with a volume transport of 4 Sv in each of the layers (between the black streamlines).

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    Schematic illustration of the heat transport in the S direction as an integral of the streamfunction between two salinities lying on the same streamline.

  • View in gallery

    The thermohaline view of the global ocean as a function of temperature and salinity. (a) The thermohaline streamfunction. The conveyor belt cell is the blue clockwise circulation corresponding to water-mass conversion from warm water to more saline, then to cold, then to less saline, and back to warm water. The tropical red cell flows in opposite direction. Units are in Sv with a volume transport of 4 Sv in each of the layers. (b) The thermohaline streamfunction with isopycnals for σ0 (solid) and σ4 (dashed). (c) The stream layer between the 6- and 14-Sv streamlines. The colors change as a clock with its center in the middle of the cell so that each color indicates particular temperature and salinity intervals in the same way as in Fig. 1. (d) The turnover time for the different stream layers.

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    The water volume distribution for the different oceans in blue and the thermohaline streamlines for the global ocean superimposed in red for (a) the global ocean; (b) the Atlantic Ocean, which is defined as north of the latitude of Cape Agulhas, with the Arctic Ocean included; (c) the Indo-Pacific, which is also defined as north of this latitude; and (d) the Southern Ocean, which is defined as the ocean south of Cape Agulhas. Volume distribution units in 1011 m3 (°C PSU)−1. Streamline intervals are 8 Sv.

  • View in gallery

    The longitude–latitude projection of the clock phases of the conveyor belt cell between the streamlines 6 and 14 Sv defined by the colors in Fig. 4c. Arrows indicate the direction of the conveyor belt by following the clock clockwise. (a) The warm branch of the conveyor belt and (b) the cold branch. These figures are also shown as an animation in supplement 2.

  • View in gallery

    Schematic illustration of the tropical and conveyor belt cells, where the different segments in the cells are denoted by numbers in the tropical cell and letters in the conveyor belt. The numbers and letters are as in Fig. 8, which shows the geographical location of the cells.

  • View in gallery

    Schematic illustration of the geographical paths of tropical and conveyor belt cells. The different segments can be traced for the conveyor belt cell with (a) warm water flowing from the less saline Indo-Pacific into the more saline Atlantic, (b) the saline water of the North Atlantic cools and downwells into the deep Atlantic, (c) the cold water from the North Atlantic mixes with less saline water as it travels toward the Indo-Pacific, (d) the less saline water heats up in the Indo-Pacific before it returns to the Atlantic. The segments of the tropical cell are 1) poleward Ekman drift, 2) subducted equatorward flow along isotherms and isohalines, 3) upwelling in the eastern Pacific and then westward movement in the equatorial current with increasing temperature and salinity, and 4) westward propagation in the western Pacific in the equatorial current with freshening surface water.

  • View in gallery

    The longitude–latitude projection of the clock phases of the tropical cell between the streamlines 6 and 14 Sv defined by the colors in Fig. 4c. Arrows indicate the direction by following the clock counterclockwise. (a) The upper branch of the tropical cell and (b) the lower branch. These figures are also shown as an animation in supplement 3.

  • View in gallery

    The turnover time as a function of the conveyor belt clock, where 12 indicates an entire conveyor belt cycle. The different curves correspond to the different stream-layer intervals of Fig. 4a. The total turnover time for one entire cycle corresponds to the value at 12 o’clock, which is the same as shown in Fig. 4d for each layer.

  • View in gallery

    The heat and freshwater transports of the conveyor belt and tropical cells. (a) The heat transport as a function of salinity. Units are in PW (1015 W). (b) The freshwater transport as a function of temperature. Units are in 106 kg s−1.

  • View in gallery

    The model grid box i, j, k with its six adjacent T boxes (a) seen from above and (b) looking northward. The temperature and salinity are located at the center of the box (Ti,j,k). The zonal (U), meridional (V), and vertical (W) volume fluxes are located on the middle of the six walls of the i, j, k box. (c) The temperature and salinity discretization that the cross-isothermal and cross-isohaline fluxes are projected on. The blue arrows symbolize the cross-isothermal flux between two grid boxes and shows how the flux Ui,j,k crosses the three isotherms Tm−1, Tm, and Tm+1 on its way from Ti,j,k to Ti+1,j,k.

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The World Ocean Thermohaline Circulation

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  • 1 Department of Meteorology, Stockholm University, Stockholm, Sweden
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Abstract

A new global streamfunction is presented and denoted the thermohaline streamfunction. This is defined as the volume transport in terms of temperature and salinity (hence no spatial variables). The streamfunction is used to analyze and quantify the entire World Ocean conversion rate between cold/warm and fresh/saline waters. It captures two main cells of the global thermohaline circulation, one corresponding to the conveyor belt and one corresponding to the shallow tropical circulation. The definition of a thermohaline streamfunction also enables a new method of estimating the turnover time as well as the heat and freshwater transports of the conveyor belt. The overturning time of the conveyor belt is estimated to be between 1000 and 2000 yr, depending on the choice of stream layer. The heat and freshwater transports of these two large thermohaline cells have been calculated by integrating the thermohaline streamfunction over the salinity or temperature, yielding a maximum heat transport of the conveyor belt of 1.2 PW over the 34.2-PSU salinity surface and a freshwater transport of 0.8 Sv (1 Sv ≡ 106 m3 s−1) over the 9°C isotherm. This is a measure of the net interocean exchange of heat between the Atlantic and Indo-Pacific due to the thermohaline circulation.

Supplemental information related to this paper is available at the Journals Online website.

Corresponding author address: Kristofer Döös, Department of Meteorology, Stockholm University, SE-10691 Stockholm, Sweden. E-mail: doos@misu.su.se

Abstract

A new global streamfunction is presented and denoted the thermohaline streamfunction. This is defined as the volume transport in terms of temperature and salinity (hence no spatial variables). The streamfunction is used to analyze and quantify the entire World Ocean conversion rate between cold/warm and fresh/saline waters. It captures two main cells of the global thermohaline circulation, one corresponding to the conveyor belt and one corresponding to the shallow tropical circulation. The definition of a thermohaline streamfunction also enables a new method of estimating the turnover time as well as the heat and freshwater transports of the conveyor belt. The overturning time of the conveyor belt is estimated to be between 1000 and 2000 yr, depending on the choice of stream layer. The heat and freshwater transports of these two large thermohaline cells have been calculated by integrating the thermohaline streamfunction over the salinity or temperature, yielding a maximum heat transport of the conveyor belt of 1.2 PW over the 34.2-PSU salinity surface and a freshwater transport of 0.8 Sv (1 Sv ≡ 106 m3 s−1) over the 9°C isotherm. This is a measure of the net interocean exchange of heat between the Atlantic and Indo-Pacific due to the thermohaline circulation.

Supplemental information related to this paper is available at the Journals Online website.

Corresponding author address: Kristofer Döös, Department of Meteorology, Stockholm University, SE-10691 Stockholm, Sweden. E-mail: doos@misu.su.se

1. Introduction

The World Ocean thermohaline circulation is frequently idealized as a conveyor belt transporting heat and freshwater from the Indo–Pacific to the Atlantic (Broecker 1987). This interocean exchange of heat and freshwater, closely associated with the formation of North Atlantic Deep Water (NADW), is of key importance for the climatic and hydrographic differences between the North Atlantic and the North Pacific. It should be noted, however, that the freshwater transport into the Atlantic is to a large extent accomplished by the Southern Hemisphere subtropical gyres (Talley 2008). In fact, the interocean flow tied to the formation of North Atlantic Deep Water acts in many ocean circulation models to export freshwater southward out of the Atlantic (e.g., Rahmstorf 1996; Hawkins et al. 2011), thereby contributing to the salinization of the Atlantic basin. Nevertheless, the great ocean conveyor belt due to Broecker (1987), encapsulated in a well-known schematic of the ocean circulation, portrays some key element of the interocean circulation. The pathways of the conveyer belt–like circulation have been traced with different methods such as hydrographic sections (e.g., Gordon 1986), radiocarbon concentrations (e.g., Broecker 1991), inverse methods (e.g., Rintoul 1991), and ocean general circulation models (OGCMs).

The advantage of using model data is, of course, that there are nearly continuous spatial and temporal data of the physical variables. Nevertheless, it is still not clear how to best trace and understand the ocean circulation because it is necessary to simplify the time-dependent three-dimensional (3D) fields into a 2D image. When analyzing model integrations, the main challenge is therefore to extract the phenomenon under consideration, which in our case is the conveyor belt, associated with the interbasin exchange of water masses. This can be accomplished in several different ways, such as analyzing 1) model-simulated water-mass data in a similar way as any analysis based on observational data, 2) passive tracers (e.g., England 1995; Stevens and Stevens 1999), 3) Lagrangian trajectories (e.g., Döös 1995; Speich et al. 2002) and 4) overturning streamfunctions (e.g., England 1992; Döös 1996; Nycander et al. 2007; Döös et al. 2008).

Model-calculated Lagrangian trajectories provide a more detailed view than do the usual schematic illustrations of the ocean conveyor belt. Figure 1 and the animation (available as supplemental material at the Journals Online website: http://dx.doi.org/10.1175/JPO-D-11-0163.s1) show Lagrangian trajectories of the conveyor belt that have been calculated using Nucleus for European Modelling of the Ocean (NEMO) model-simulated velocity fields and the Tracing the Water Masses (TRACMASS) code (de Vries and Döös 2001).

Fig. 1.
Fig. 1.

The conveyor belt illustrated with Lagrangian trajectories. Color indicates the temperature and salinity composition, with red for warm water, green for cold water, yellow for saline water, and blue for less saline water. This figure is also shown as an animation in supplement 1.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0163.1

However, the limitation of most of these methods of tracing the conveyor belt is that they do not yield any quantification of the thermohaline circulation and hence no measure of its importance for the oceanic temperature and salinity distribution and the climate. A more quantitative measure is given by the meridional overturning streamfunctions in Fig. 2, which show the total volume transport as a function of latitude and depth/temperature/salinity. These streamfunctions are all based on zonal integration. Another possibility is to compute a streamfunction with depth and density as coordinates (Nycander et al. 2007). This streamfunction is well suited to analyze the transformation between kinetic and potential energy and also (e.g., the streamfunction in latitude–density coordinates) distinguishes between isopycnal and diapycnal flows.

Fig. 2.
Fig. 2.

The global meridional overturning streamfunction as a function of (top) depth, (middle) temperature, and (bottom) salinity. Units are in Sv with a volume transport of 4 Sv in each of the layers (between the black streamlines).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0163.1

In the present study, we present a new global streamfunction, denoted the thermohaline streamfunction, with temperature and salinity as coordinates. This makes it possible to display water-mass transformations in temperature and salinity in the same diagram, whether they are isopycnal or diapycnal. It is obtained by a global integration of the volume transport across the temperature and salinity surfaces and hence does not depend on any spatial variables.

A temperature–salinity streamfunction was first introduced by Blanke et al. (2006) but then by summing over selected Lagrangian trajectories, which made it possible to limit the study to the Atlantic Ocean. The thermohaline streamfunction in the present study is calculated directly from the Eulerian velocity fields from the OGCM and requires for mass conservation reasons that the velocity fields are integrated over the entire model domain with no open boundaries. In simultaneous and independent work, Zika et al. (2012) have made a similar analysis of the thermohaline streamfunction as the one in the present study. They find similar thermohaline cells as here, but with velocity fields from the ocean part of a climate model with coarse resolution.

The work is organized as follows: In section 2, we present the OGCM and introduce the thermohaline streamfunction. In section 3, we describe the structure of the thermohaline cells obtained with the streamfunction and how they are reprojected to horizontal large-scale circulation. In section 4, we calculate the World Ocean turnover time from the thermohaline cells. In section 5, we present the heat and freshwater transports that are associated with the thermohaline cells. In section 6, we summarize and discuss the results. The discretization of the thermohaline streamfunction is given in the appendix. In addition to this, there are three animations as supplementary material.

2. Methods

a. The ocean general circulation model NEMO

The ocean velocity field has been obtained from the NEMO ocean/sea ice general circulation model (Madec 2008). The model configuration employed, ORCA025, has a tripolar grid with a ¼° horizontal grid resolution (27.75 km) at the equator. The grid is finer with increasing latitudes, yielding a 13.8-km resolution at 60°S and 60°N. The spatial resolution is hence only eddy permitting, not truly eddy resolving. The water column is divided into 75 levels, with a grid spacing ranging from 1 m near the surface to 200 m at the bottom. The configuration was described and is developed by Barnier et al. (2006, 2007), who demonstrated its high capacity for representing strong currents and eddy variability, even compared with more highly resolved models. The Laplacian lateral isopycnal tracer diffusion coefficient is 300 m2 s−1 at the equator and decreases poleward in proportion to the grid size. The sea ice model was developed by Fichefet and Maqueda (1997). The atmospheric forcing is derived by Brodeau et al. (2010) and based on the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) and ERA-Interim reanalyses (Uppala et al. 2008). The model is forced using the “bulk” method as described in Large and Yeager (2004). The surface salinity is restored with a relatively strong damping coefficient of −26.9257 mm day−1. The velocity fields were archived in time means of 5 days. The model is fully prognostic (viz. it does not use assimilated data; neither satellite altimetry nor in situ temperature or salinity), and its results can therefore differ substantially from observations in some areas.

The initial conditions for the temperature and salinity fields are based on Levitus et al. (1998). The spinup takes place over the years 1957–99, and the analysis covers the period 2000–06.

b. The thermohaline streamfunction

The traditional meridional overturning streamfunction ψ(ϕ, z), where ϕ is latitude and z is depth, is defined as the northward volume transport through the vertical surface at the latitude θ below the depth z. Because of mass conservation, this is equal to the upwelling [or downwelling when ψ(ϕ, z) is negative] through the horizontal surface at depth z north of the latitude ϕ. This streamfunction is shown in Fig. 2 (top).

Figure 2 (middle) shows the streamfunction ψ(ϕ, T), which is defined as the northward volume transport across that part of the vertical surface at the latitude ϕ where the temperature is less than T. (Note that the temperature variable we use is the potential temperature, which is here simply denoted as T.) If the ocean is in a statistically steady state, this is equal to the volume transport through the isothermal surface T north of the latitude ϕ. If the temperature field is changing in time, on the other hand, the isothermal surfaces move in space and ψ(ϕ, T) is not equal the flow that penetrates the isothermal surfaces. The same applies for the streamfunction ψ(ϕ, S), which gives the northward volume transport across that part of the vertical surface at the latitude ϕ where the salinity is less than S.

We now introduce the thermohaline streamfunction ψ(S, T), which describes the volume flow in TS coordinates. Analogously to above, ψ(S, T) is defined as the volume transport through the isothermal surface T where the salinity is less than S. Thus, given the potential temperature, salinity, and velocity fields, we compute the thermohaline streamfunction, averaged over the time interval from t0 to t1, as
e1
where Ath(S, T) is the part of the isothermal surface (or family of surfaces) T where the salinity is less than S, V is the three-dimensional velocity, and is the surface element on Ath(S, T) pointing toward increasing temperature. Equivalently, the thermohaline streamfunction can be calculated from the volume transport that penetrates the isohaline surfaces,
e2
where Asa(S, T) is the part of the isohaline surface (or family of surfaces) S where the potential temperature is less than T and is the surface element on Asa(S, T) pointing toward increasing salinity. The numerical discretization of the thermohaline streamfunction is presented in the appendix.

In the computations, Ath(S, T) and Asa(S, T) are treated as “frozen” surfaces: that is, the fact that they might be moving is not accounted for. That is why Eqs. (1) and (2) are equivalent. However, this also means that the flow described by ψ(S, T) does not necessarily correspond to water-mass transformation (i.e., changing T and S in Lagrangian water parcels). For that to be true, the ocean must be in a statistically steady state. In this case, the time-mean volumetric TS distribution is constant. The global transformation of water masses due to air–sea fluxes and oceanic mixing therefore gives rise to a nondivergent flow in TS coordinates. Hence, in a statistically steady state, the water-mass transformation is described by the streamfunction ψ(S, T).

As defined here, ψ(S, T) represents volume transport through the boundaries of the volume in the ocean where the temperatures are less than T and the salinities are less than S. The volume flow exiting the bounding isothermal surface is −ψ(S, T), and volume flow exiting the bounding isohaline surface is ψ(S, T): that is, the net flow through the boundaries is zero. In analogy with a streamfunction in geometrical coordinates, a velocity is defined on the ST plane by (∂ψ/∂T, −∂ψ/∂S). The interpretation is as follows:
e3
represents the volume transport in the S direction in the potential temperature range from T to T + dT and
e4
represents the volume transport in the T direction in the salinity range from S to S + dS.

We note that Speer (1993) introduced a flow vector in TS coordinates that describes the transformations due to air–sea fluxes. However, when the transformations due to air–sea fluxes or oceanic mixing are considered separately, the resulting flow in TS coordinates is generally divergent, as illustrated by the observationally based computations for the North Atlantic presented by Speer. It can therefore not be represented by a streamfunction.

By using the fact that cpρT(∂ψ/∂T)dT is the heat transport in the S direction in the potential temperature range from T to T + dT, one can calculate the heat transport accomplished by a closed cell bounded by the contour ψ(S, T) = ψ0, where ψ0 is a constant. Integrating by parts, we find that the heat transport in the S direction, illustrated by Fig. 3, is given by
e5
where Tmin(S) and Tmax(S) are the potential temperatures on the boundary of the cell, ρ = 1035 kg m−3 is the average density of the seawater, and the specific heat for seawater is cp ≈ 4000 J (kg °C) −1. By similar arguments, we find that the salinity transport in the T direction accomplished by the same cell is given by
e6
where Smin(T) and Smax(T) are the salinities on the boundary of the cell. The oceanic freshwater transport in the in T direction is approximately given by −F(T)/Sr, where Sr = 35 is a constant reference salinity. If the integration limits in Eqs. (5) and (6) span over the whole oceanic potential temperature and salinity range, one obtains the global transports.
Fig. 3.
Fig. 3.

Schematic illustration of the heat transport in the S direction as an integral of the streamfunction between two salinities lying on the same streamline.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0163.1

The relations (5) and (6) show that a circulation cell for which ψ > 0 transports heat toward decreasing salinity and salt toward increasing temperature: that is, freshwater is transported toward decreasing temperature. The reverse holds if ψ < 0.

Note that the surface fluxes of freshwater generally cause the net volume transport across the isotherms [i.e., ψ(S = ∞, T)] or the isohaline surfaces [i.e., ψ(S, T = ∞)] to be nonzero. However, we ignore these generally small net transports and set ψ(S, T = ∞) = 0 and ψ(S = ∞, T) = 0, allowing the global heat and salt transport to be calculated from Eqs. (5) and (6). This is a reasonable approximation, because the net surface freshwater flux in regions with either positive or negative EP is less than 1 Sv (1 Sv ≡ 106 m3 s−1) in the present study, which is in good agreement with other studies (e.g., Wijffels et al. 1992).

The thermohaline streamfunction has been averaged between t0 = 1 January 2000 and t1 = 31 December 2006. Note that it is only in a statistically steady state that ψ(S, T) gives the transformation of water masses water. When the TS distribution changes in time, the volume transport described by ψ(S, T) may be associated partly with the motion of the isothermal and isohaline surfaces in space, rather than with water-mass transformation. This feature also applies to the latitude–density streamfunction (Döös and Webb 1994) as well as to the global density–depth overturning streamfunctions (Nycander et al. 2007). Walin (1982) and Marsh et al. (2000) have described how the velocity-based streamfunction and the time tendency of the volumetric distribution function can be combined to give the cross-isothermal or diapycnal transformation when the flow is unsteady. In principle, their methods could be generalized to the present a more complicated case where booth cross-isothermal and isohaline flows are considered. However, such an analysis is beyond the scope of the present paper and we have thus not quantified the effect of model drift on the computed ψ(S, T). We note that Zika et al. (2012), using a model with coarser-resolution model that is integrated to a steady state, obtained a qualitative similar thermohaline streamfunction. Thus, despite there being some model drift in the present simulation, we deem that it is small enough to allow the calculated thermohaline streamfunction to be interpreted as essentially the result of statistically steady water-mass transformations.

3. The thermohaline cells

The global thermohaline streamfunction, defined by Eq. (1), is shown in Fig. 4a and with superimposed isopycnals in Fig. 4b. The present model-based calculation yields a streamfunction comprised of two main cells: one clockwise cell, which we interpret to reflect the interocean conveyor belt circulation, and one anticlockwise cell at high temperatures, which we interpret to reflect flows and water-mass transformations primarily in the upper near-equatorial Pacific. There is also a minor anticlockwise cell at cold temperatures, which we interpret to be related to the Antarctic Bottom Water (AABW). We use the term “cell” in the present study because they appear as cells in TS space. They will to some extent also exist as cells in the horizontal–vertical space but far from everywhere.

Fig. 4.
Fig. 4.

The thermohaline view of the global ocean as a function of temperature and salinity. (a) The thermohaline streamfunction. The conveyor belt cell is the blue clockwise circulation corresponding to water-mass conversion from warm water to more saline, then to cold, then to less saline, and back to warm water. The tropical red cell flows in opposite direction. Units are in Sv with a volume transport of 4 Sv in each of the layers. (b) The thermohaline streamfunction with isopycnals for σ0 (solid) and σ4 (dashed). (c) The stream layer between the 6- and 14-Sv streamlines. The colors change as a clock with its center in the middle of the cell so that each color indicates particular temperature and salinity intervals in the same way as in Fig. 1. (d) The turnover time for the different stream layers.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0163.1

It is noteworthy that the amplitude of the conveyor belt cell is comparable to that of the meridional overturning in the World Ocean (Fig. 2). Essentially, this indicates that the flow directed chiefly poleward in the World Ocean, associated with transformation toward lower temperature, occurs at higher salinities than the flow along which the temperature increases. If the salinities of these two flow branches were nearly equal, the amplitude of the thermohaline streamfunction would be small compared to the meridional overturning streamfunction.

It is also important to stress that the streamlines of any streamfunction that is not based on a 2D stationary velocity field do not correspond to Lagrangian trajectories. A consequence of this is that velocities flowing in opposite directions will cancel each other when flowing on the same level. The three cells of the thermohaline streamfunction are no exception, and the cells will interfere with each other. The warm part of the conveyor belt cell may thus cancel a part of the tropical cell. A fair guess is that if one could isolate these two cells—for example, by tracing them with Lagrangian trajectories—the tropical cell would extend farther “down” into colder waters and the conveyor belt farther “up” into warmer waters. This might also explain why the AABW cell is so weak. It may in reality be both stronger and extend into warmer waters.

Although the thermohaline streamfunction is a novel way to portray the ocean circulation, it uses the same coordinates as the classical TS diagram. Figure 5 shows the global volumetric TS distribution for the present model. Because the model is initiated with climatological hydrography and uses restoring boundary surface conditions, this is close to the observed TS distribution (e.g., Worthington 1981). Traditionally, TS diagrams have been an important tool for analyzing how water masses are distributed, transported, and mixed. However, the classical TS analyses provide essentially qualitative information. With the construction of a thermohaline streamfunction, we can directly infer the rate of water-mass conversion in TS coordinates. In combination with the thermohaline streamfunction (Fig. 4a), the volumetric TS distribution (Fig. 5) is potentially useful for examining how the global water-mass distribution is maintained.

Fig. 5.
Fig. 5.

The water volume distribution for the different oceans in blue and the thermohaline streamlines for the global ocean superimposed in red for (a) the global ocean; (b) the Atlantic Ocean, which is defined as north of the latitude of Cape Agulhas, with the Arctic Ocean included; (c) the Indo-Pacific, which is also defined as north of this latitude; and (d) the Southern Ocean, which is defined as the ocean south of Cape Agulhas. Volume distribution units in 1011 m3 (°C PSU)−1. Streamline intervals are 8 Sv.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0163.1

The σ0 and σ4 surfaces are shown together with the thermohaline streamfunction in Fig. 4b in order to estimate whether the streamlines are isopycnal or diapycnal. The conveyor belt and AABW cells there are tilting in the same direction as the isopycnals. Most of the conveyor belt cell is however diapycnal, in contrast to the AABW cell.

Interior diffusive fluxes result from mixing water parcels with different TS properties and therefore make the TS distribution more compact. Hence, all extreme TS values must be created at the surface of the ocean, and the surface fluxes may play a role for the water-mass transformation everywhere in the TS plane. The relative importance of surface fluxes and interior diffusive fluxes remains to be investigated.

a. The conveyor belt cell

The thermohaline streamfunction does not provide any direct information about the geographical characteristics of the cells. It is, however, possible to indirectly project the thermohaline cells on geographical space by locating them where there are water masses in particular temperature and salinity intervals. For this reason, we have chosen the layer between the 6- and 14-Sv streamlines in Fig. 4a and graphed it in Fig. 4c, where the different colors indicate specific temperature and salinity intervals. These waters have hereafter been located in the longitude–latitude space of Fig. 6, respecting the same color conventions for the temperature and salinity intervals as in Fig. 4c and in the Lagrangian trajectories of the conveyor belt (Fig. 1). The colors follow the rainbow order of Goethe’s color wheel with red (green) representing warm (cold) waters and yellow (blue) representing saline (fresh) waters. The colors change gradually so that, for example, orange refers to both rather warm and rather saline waters. The colors are ordered as a clock (with a new color nuance for each half hour), making one 12-h revolution a complete conveyor belt cycle. It is now possible to geographically follow the cycle of the conveyor belt cell in Fig. 6, bearing in mind the order of the colors of the stream layer in Fig. 4c.

Fig. 6.
Fig. 6.

The longitude–latitude projection of the clock phases of the conveyor belt cell between the streamlines 6 and 14 Sv defined by the colors in Fig. 4c. Arrows indicate the direction of the conveyor belt by following the clock clockwise. (a) The warm branch of the conveyor belt and (b) the cold branch. These figures are also shown as an animation in supplement 2.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0163.1

The geographical projection of the conveyor belt cell is divided into a warm branch (Fig. 6a) and a cold branch (Fig. 6b), separated by the 9°C isotherm. The two branches are shown separately because they are often active at the same longitudes and latitudes but at different depths. The warm part of the conveyor belt, shown in Fig. 6a, is represented on the color wheel by water flowing in the order blue, purple, red, orange, and yellow from 9 to 3 o’clock. This warm part of the conveyor belt can be traced from the eastern midlatitude Pacific with blue waters, which become red at lower latitudes and then orange when passing through the tropical Indian up into the Atlantic, where they start to cool and eventually become yellow. The transition from the warm to the cold part of the conveyor belt takes place in the North Atlantic according to our color wheel, and the cycle then continues in Fig. 6b. The yellow water becomes green as the water cools and continues southward. Dark green, which is initially NADW, fills the major part of World Ocean. The waters eventually surface at some point in the Southern Ocean or North Pacific as less saline and less cold blue waters. As additional material, we provide an animation (available as supplemental material at the Journals Online website: http://dx.doi.org/10.1175/JPO-D-11-0163.s2) of the cycle as illustrated by Fig. 6, but with one color for each film frame and hence directly comparable to the Lagrangian trajectory animation (available as supplemental material at the Journals Online website: http://dx.doi.org/10.1175/JPO-D-11-0163.s1).

The blue clockwise conveyor belt cell, which has a maximum amplitude of about 32 Sv, can now be understood by following the water cycle illustrated in the blue cell in the schematic Fig. 7 and the schematic geographical representation in Fig. 8.

Fig. 7.
Fig. 7.

Schematic illustration of the tropical and conveyor belt cells, where the different segments in the cells are denoted by numbers in the tropical cell and letters in the conveyor belt. The numbers and letters are as in Fig. 8, which shows the geographical location of the cells.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0163.1

Fig. 8.
Fig. 8.

Schematic illustration of the geographical paths of tropical and conveyor belt cells. The different segments can be traced for the conveyor belt cell with (a) warm water flowing from the less saline Indo-Pacific into the more saline Atlantic, (b) the saline water of the North Atlantic cools and downwells into the deep Atlantic, (c) the cold water from the North Atlantic mixes with less saline water as it travels toward the Indo-Pacific, (d) the less saline water heats up in the Indo-Pacific before it returns to the Atlantic. The segments of the tropical cell are 1) poleward Ekman drift, 2) subducted equatorward flow along isotherms and isohalines, 3) upwelling in the eastern Pacific and then westward movement in the equatorial current with increasing temperature and salinity, and 4) westward propagation in the western Pacific in the equatorial current with freshening surface water.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0163.1

  1. The warm water flowing from the Indo-Pacific into the Atlantic is the top of the blue cell, where water increases its salinity as it enters the more saline Atlantic Ocean.
  2. The water then cools as it approaches the North Atlantic, which is the right-hand side of the blue cell.
  3. Once in the North Atlantic, the water downwells by convection into the deep ocean and travels southward into the less saline Southern Ocean and Indo-Pacific, which is the cold part of the conveyor belt cell. In the schematic view in Fig. 8, this is shown to happen in the deep ocean, which might be a result of NADW mixing with AABW. However, this freshening of the water might also occur near the surface of the ocean near Antarctica, as a result of sea ice melting and of the net precipitation–evaporation in the Southern Ocean. The freshening can be seen directly in the meridional overturning streamfunction as a function of salinity in Fig. 2 from the anticlockwise cell in the Southern Ocean with waters flowing from high to low salinities at around 60°S.
  4. The water eventually warms up as it travels toward the upper Indo-Pacific, which is the left side of the cell, and closes the cell by returning to its initial position.

This description of the conveyor belt cell is consistent with the water volume distribution for the different oceans shown in Fig. 5. In particular, Fig. 5b shows that the warming segment of the cell (i.e., segment d in Fig. 7) lies entirely outside of the Atlantic and therefore lies in the Indo-Pacific. The cooling segment (segment b in Fig. 7), on the other hand, may lie in either the Atlantic or the Indo-Pacific. We also see that the circulation in a central part of the cell is entirely confined to the Indo-Pacific and therefore does not correspond to the traditional conveyor belt circulation.

The method of successively mapping the water masses of a stream layer in order to follow the conveyor belt cycle is far from perfect because the water masses of the different cells overlap. Furthermore, we have here only followed the layer between the 6- and 14-Sv streamlines and have not taken account of the seasonal cycle. Still, it gives a new illustrative way of following the conveyor belt.

b. The tropical cell

The tropical cell (i.e., the anticlockwise cell in Fig. 4a) has an amplitude of about 25 Sv. Its strength is thus comparable to the conveyor belt cell, but it is confined to a smaller temperature interval. More than 50% of the transport in the cell occurs at temperatures above 24°C. This suggests that the tropical cell is primarily a thermohaline coordinate manifestation of the shallow wind-driven near-equatorial cells, which are commonly referred to as the subtropical cells (see, e.g., McCreary and Lu 1994; Pedlosky 1996; Gu and Philander 1997).

The tropical cell is projected on the longitude–latitude space in Fig. 9 and as an animation (available as supplemental material at the Journals Online website: http://dx.doi.org/10.1175/JPO-D-11-0163.s3) in the same way as the conveyor belt was projected in the previous subsection. Note, however, that the tropical cycle flows in the opposite direction so that the clock should be followed counterclockwise. In geometrical coordinates, the strength of the subtropical cells is related to the zonally integrated poleward Ekman transport in the surface layer (e.g., McCreary and Lu 1994). Accordingly, the tropical cell in TS coordinates is expected to be dominated by the circulation in the Pacific, whose zonal width is significantly greater than that of the Atlantic. Indeed, Fig. 5 shows that the tropical cell is centered around water masses encountered in the Indo–Pacific basin.

Fig. 9.
Fig. 9.

The longitude–latitude projection of the clock phases of the tropical cell between the streamlines 6 and 14 Sv defined by the colors in Fig. 4c. Arrows indicate the direction by following the clock counterclockwise. (a) The upper branch of the tropical cell and (b) the lower branch. These figures are also shown as an animation in supplement 3.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0163.1

Our geographical interpretation of the circulation gives rise to the triangularly shaped tropical cell in Fig. 4a, which in turn consists of four geographical segments. One of the geographical segments (number 2 in Fig. 7) does not involve any water-mass transformations and is hence a point in the triangle, whereas the three others do and correspond therefore to the three sides of the triangle. We begin at the top-left corner of the triangle, the warmest and least saline part of the cell. This TS relation corresponds to the warm pool in the near-equatorial western Pacific. Tracing the 10 Sv-contour from S ~ 34.5 and T ~ 29°C, four segments can be distinguished (with numbers as in Figs. 7, 8):

  1. From the warm pool, surface water flows essentially poleward and becomes cooler and more saline: that is, the density increases.
  2. At the coldest temperatures along the streamlines in the tropical cell (i.e., the bottom corner of the triangle), the surface water is subducted into the thermocline off the equator. From the subduction region, the water flows equatorward joining the eastward-flowing equatorial undercurrent (e.g., Gu and Philander 1997). Eventually, the water surfaces in the cold pool of the eastern Pacific. To lowest order, the flow in this thermocline segment is adiabatic and hence does not yield any flow in TS coordinates.1
  3. Following the 10-Sv contour (from T ~ 26°C, S ~ 35), the continued flow toward increasing temperatures and salinities in the tropical cell represents westward movement of near-equatorial surface water originating from the cold pool. There is net evaporation and strong heat flux into the ocean in the eastern equatorial Pacific (see, e.g., the ERA-40 atlas by Kållberg et al. 2005), resulting in a transformation with little density change.
  4. The final segment, which in geometrical coordinates starts near the dateline and close to the equator (the top-right corner of the triangle in Fig. 7, at T ~ 27°C, S ~ 35.5), involves a transformation toward lower salinities at nearly constant T: that is, the density decreases. This represents the freshening of the surface waters in the westward flow in the warm pool, where there is net precipitation and the surface heat flux is weak.

Admittedly, this is a highly idealized schematic of the upper-ocean circulation in the equatorial Pacific, which in reality exhibits pronounced north–south asymmetries (e.g., Wyrtki and Kilonsky 1984). Note that at colder temperatures cancelation occurs between transformation in the tropical and conveyor belt cells. Thus, the thermohaline tropical cell is not expected to capture the whole range of TS transformation that occurs in the oceanic subtropical cells. Our interpretation is furthermore based on the assumption that the TS transformations are dominated by air–sea fluxes, hence neglecting the turbulent mixing and transformation in the equatorial undercurrent (Gregg et al. 1985). The transformation in the first, poleward-flowing segment is consistent with the meridional profiles of the zonal-mean T and S in the surface water in the tropics, where the temperature tends to decrease and the salinity increase poleward. However, it is the near-equatorial zonal variations of the hydrography and surface fluxes that determine the shape and orientation of the tropical cell. Specifically, the strong oceanic heat gain and net evaporation in the Pacific cold pool is instrumental for the third segment. The fact that the TS relation of the surface water along the equator in the Pacific west of 260°E resembles the streamlines of ψ(S, T) in the third and fourth segments supports the notion that these segments reflect transformations of westward-flowing near-equatorial surface waters.

As noted, Fig. 5b shows that the Atlantic water masses lay almost entirely outside the main tropical cell. Hence, this cell is mainly due to circulation the Indo-Pacific, in agreement with the description above. Nevertheless, it is interesting to compare with the observationally based analysis of water-mass transformations in the Atlantic by Speer (1993). His analysis indicates that the air–sea fluxes tend to transform the North Atlantic surface waters in the temperature range between 26° and 30°C toward increasing salinity. Thus, the fourth segment of the tropical cell in Fig. 7, which involves freshening at high temperatures, appears to be absent in the Atlantic. The seemingly different transformation pattern in the tropical Atlantic that emerges in the analysis of Speer (1993) may be due to the stronger cross-equatorial transport of thermocline water associated with the Atlantic meridional overturning circulation (Kawase and Sarmiento 1985).

It can be noted that the transport of about 30 Sv in the tropical cell is roughly comparable to estimated transports of the equatorial undercurrent in the central Pacific, which range from 25 to 50 Sv (see, e.g., Wyrtki and Kilonsky 1984; McCreary and Lu 1994). As a comparison, the poleward surface Ekman transport in the entire ocean is about 50 Sv at 10° latitude in each hemisphere (e.g., Nilsson and Körnich 2008).

c. The Antarctic Bottom Water cell

There is also a third weak anticlockwise cell under the conveyor belt cell, which most likely corresponds to the mixing of the AABW. When formed, this water is fresh and cold but becomes, when mixed with the North Atlantic Deep Water, warmer and more saline, which explains the tilting of the cell. It is somewhat surprising that the AABW cell is so weak, because this is one of the major water masses in the World Ocean. One explanation is that the cells overlap and partly cancel each other, so that the AABW cell might in reality be stronger and extend into warmer waters. Another explanation could be that the model does not manage to simulate the abyssal water-mass conversions correctly. It could also be caused by the model drift, which should have the strongest effect on the AABW. The AABW cell is, however, present in all three the meridional streamfunctions shown in Fig. 2, though with latitude–depth coordinates this cell lies entirely below 2000 m and is disconnected from the surface.

4. The World Ocean turnover time

The World Ocean turnover time, defined as the time it takes for waters to make an entire circuit in the conveyor belt, can be calculated in a number of ways. It could, for example, be computed directly from the Lagrangian trajectories in Fig. 1, which were started in the Atlantic at the latitude of Cape Agulhas, 35°S, and were followed both northward and southward from this latitude. We hereafter selected the trajectories that reached at least 60°N in the Atlantic and the equator in the Pacific and returned back to 35°S in the Atlantic. We also required that the return to 35°S should occur within 200 yr. If this condition is eliminated, we could construct some sort of turnover time of the conveyor belt cycle. However, the choice of starting section and geographical extent is not obvious and would affect the result. The time it takes to return to the section is very different for different trajectories, and it is difficult to summarize this by a turnover time.

A more objective and quantitative way of determining the turnover time τ is to calculate it from the thermohaline streamfunction (Fig. 4a) and the corresponding volumetric TS distribution (Fig. 5). This can be done for each stream layer, so that if we choose, for example, the layer between the 6- and 10-Sv streamlines (Fig. 4c), we can calculate the turnover time by dividing the volume transport of the stream layer by the volume of the water corresponding to the layer between the streamlines,
e7
This is repeated for all stream layers, resulting in the turnover times of Fig. 4d. We could also compute a continuous distribution of the turnover time by using the limiting expression τ = ∂V/∂ψ. However, for illustrative purposes, we chose to use a rather coarse resolution in ψ in Fig. 4d.

The turnover time is also shown in Fig. 10 for the stream layers of the conveyor belt as it is integrated progressively around the conveyor belt cycle (clock). This shows clearly that the water remains for most of the time in the deep cold ocean, as seen from the big jump at 5 o’clock.

Fig. 10.
Fig. 10.

The turnover time as a function of the conveyor belt clock, where 12 indicates an entire conveyor belt cycle. The different curves correspond to the different stream-layer intervals of Fig. 4a. The total turnover time for one entire cycle corresponds to the value at 12 o’clock, which is the same as shown in Fig. 4d for each layer.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0163.1

As seen in Fig. 4d, the turnover time is small in the center of the conveyor belt cell. This is primarily because these circuits largely avoid the deep water that takes a long time to traverse, as seen in Fig. 10, but also because for ψ < −20 Sv the circuits do not enter the Atlantic (cf. Fig. 5b) and are therefore much shorter. The layer closest to the center only requires 23 yr to complete one turnover cycle but corresponds to very little water-mass conversions. This short cycle is not a part of any large-scale interocean circulation. The turnover time increases away from the center with a turnover time of 1000 yr or more for the three last layers and maximum turnover time of 1951 yr for the 2–6-Sv stream layer (black curve in Fig. 10).

5. The World Ocean heat and freshwater transports

The thermohaline streamfunction offers a novel perspective of the global heat and freshwater transports. The advective heat transport H(S) across global salinity surfaces as a function of salinity has been determined using Eq. (5) and is shown in Fig. 11a. It is also possible to obtain the heat transport of the cells individually by integrating Eq. (5) over the positive and negative values of the streamfunction separately. We should, however, not forget that it is not possible to isolate the cells completely since they overlap.

Fig. 11.
Fig. 11.

The heat and freshwater transports of the conveyor belt and tropical cells. (a) The heat transport as a function of salinity. Units are in PW (1015 W). (b) The freshwater transport as a function of temperature. Units are in 106 kg s−1.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0163.1

A positive heat transport implies that heat is transported across the isohalines toward increasing salinity and vice versa for negative transport values. Thus, the conveyor belt transports heat from low to high salinity (blue curve in Fig. 11a), whereas the tropical cell transports heat from high to low salinities (red curve in Fig. 11a). The conveyor belt cell has a maximum heat transport of about 1 PW over a salinity range from 34 to 35 PSU. At 35 PSU, for instance, Fig. 4a shows that the direction of the heat transport is a result of water warmer than about 15°C being transformed toward higher salinity, whereas the colder water is transformed toward decreasing salinity. An inspection of Fig. 5 reveals that the water warmer than 15° with a salinity of about 35 is primarily present in the Indo-Pacific, whereas the waters colder than 15°C are present in all basins. Obviously, the heat transport across the isohaline surfaces is global quantity with contributions from all basins. However, it is interesting to compare its magnitude with the northward heat transport into the Atlantic near 30°S, which is estimated to about 0.5 PW (Trenberth and Caron 2001). Note that the northward heat transport in the Atlantic is greater farther north, with a maximum of about 1 PW.

The heat transport of the tropical cell is less than half of that of the conveyor belt cell, despite the two cells having similar volume transport amplitudes. This is because the conveyor belt cell transports water over larger temperature intervals.

The freshwater transport as function of temperature, shown in Fig. 11b, is obtained from Eq. (6). It is divided between the cells in the same way as the heat transport. The positive sign, of the blue curve in Fig. 11b, for the conveyor belt indicates a freshwater transport from cold to warm waters. Between 7° and 15°C, the freshwater transport is about 0.7 Sv. This number can be compared with estimates of the poleward atmospheric freshwater, which is also about 0.7 Sv around 40° latitude in each hemisphere (Wijffels et al. 1992). Thus, the conveyor belt freshwater transport near 10°C is smaller but comparable to the net poleward atmospheric transport from the edges of subtropics. This might correspond to the fact that the conveyor belt imports freshwater into the North Atlantic. This is a result of southward-flowing NADW being replaced by less saline waters flowing in the opposite direction, at higher temperatures, in the warm branch of the conveyor belt. The volume transport of the less saline northward transport is greater than that of the southward NADW because it needs to compensate for the net evaporation in the North Atlantic. The direction of the freshwater transports follows from the fact that the warming of the water (occurring mainly in the Indo-Pacific; see segment d in Fig. 7) occurs at lower salinities than the cooling of the water (occurring mainly in the Atlantic; see segment b in Fig. 7).

The tropical cell transports both heat and freshwater in the opposite directions to the conveyor belt in the slightly more complicated cycle explained previously. The freshwater transport by the weak AABW cell is also visible in the red curve in Fig. 11b, with a freshwater transport of 0.1 Sv across the isotherm ~ 1°C. The AABW cell is, however, most likely overlapping too much with the conveyor belt cell for this sort of analysis of the freshwater and heat transports.

6. Discussion and general conclusions

We believe that the thermohaline streamfunction presented here opens up a new way of “observing” the model-simulated ocean circulation in temperature–salinity space. Nycander et al. (2007) observed three global ocean overturning cells in density–depth space. Here, three similar cells have been identified in TS space. However, the AABW cell is much weaker and also more uncertain because of model drift. The circulation is thus dominated by two cells, the conveyor belt cell and the tropical cell.

The reason for this two-cell structure can be understood if we assume the flow in TS space to be forced mainly at the sea surface and if we assume the surface temperature and the freshwater flux to be given by the boundary conditions. The surface temperature is largest at the equator and decreases toward the poles. The net freshwater flux into the ocean, on the other hand, is positive both near the equator and in the subpolar regions, whereas it is negative in the subtropics. Thus, both the warmest and the coldest waters get fresher, whereas the water at intermediate temperatures gets saltier. This means that the flow in TS space is leftward (decreasing S) at the top and the bottom of the plot, whereas it is rightward (increasing S) at the middle of the plot, which generates the observed two-cell structure. The salinity span of the cells should essentially be set by the strength of circulation in physical coordinates: a faster ocean circulation is expected to yield a smaller salinity span. Interestingly, the salinity range of the tropical cell may depend fairly weakly on the wind strength. The reason is that in the tropics both the oceanic flow and the atmospheric freshwater transport are roughly proportional to the near-surface Ekman transports, which are comparable in the ocean and the atmosphere but have the opposite directions (Nilsson and Körnich 2008).

The longitude–latitude reprojection of a chosen stream layer of the thermohaline streamfunction confirms that the main clockwise thermohaline cell to a large extent reflects a conveyor belt circulation, which transports water between the Indo–Pacific and the North Atlantic in connection with the North Atlantic Deep Water formation. This does not mean, however, that all the water contributing to this cell is as a part of the interocean conveyor belt. The water-mass mixing in TS space decreases as one approaches the center of the cells and a reprojection in geographical coordinates would be less interesting for these water masses.

To summarize, the thermohaline streamfunction introduced in the present study has made it possible to

  1. identify and quantify the volume transport of the thermohaline circulation cells;
  2. reproject a thermohaline stream layer to longitude–latitude space in order to clearly distinguish the conveyor belt;
  3. calculate the turnover time of the conveyor belt, which ranges between two centuries and two millennia depending on which stream layer is considered; and
  4. calculate the heat transport across the isohalines and the freshwater transport across the isotherms of the two dominant thermohaline cells.

We have here presented results from an eddy-permitting general circulation model with a rather short spinup from climatological temperature and salinity fields. The thermohaline streamfunction strictly speaking represents water-mass transformation only if the model is in statistical equilibrium. A model drift will always to some extent contaminate the calculations of any sort of streamfunction. The reason for this choice is that in longer integrations with spinups over several centuries the water-mass distribution tends to be displaced from the realistic distribution in TS space. It would therefore be desirable, in the future, to perform stable long spinups, which might be possible by using eddy-permitting model configurations, which presently demand too much computer resources.

Preliminary results from the calculation of the thermohaline streamfunction, for coupled climate integrations using the coarser-resolution non-eddy-permitting version of NEMO, show how thermohaline circulation is affected and changed during long centennial model integrations. By analyzing the combination of how the water volume is distributed in the TS space and the thermohaline streamfunction, it is possible to judge to what extent these changes are realistic and physical or numerical because of the shortcomings of the model. We thus believe that calculating of the thermohaline streamfunction is a powerful tool not only for analyzing the ocean circulation but also for comparing and “validating” different model integrations.

Acknowledgments

The authors wish to thank Peter Lundberg for constructive comments. This work has been financially supported by the Bert Bolin Centre for Climate Research and by the Swedish Research Council. The NEMO integrations were done on the computer cluster Ekman at the Centre for High Performance Computing in Stockholm.

APPENDIX

Discretization of the Thermohaline Streamfunction

In Eq. (1), the thermohaline streamfunction is defined from the cross-isotherm flux,
ea1
We will here explain how we computed ψ(S, T) by numerical discretization of Eq. (A1), using finite differences on an Arakawa C grid, which is used in the NEMO. (This grid is illustrated in Figs. A1a,b.) Essentially, we used the fact that (∂ψ/∂S)TΔS is the volume flux across an isothermal surface between two curves of constant salinities S and S + ΔS.
Fig. A1.
Fig. A1.

The model grid box i, j, k with its six adjacent T boxes (a) seen from above and (b) looking northward. The temperature and salinity are located at the center of the box (Ti,j,k). The zonal (U), meridional (V), and vertical (W) volume fluxes are located on the middle of the six walls of the i, j, k box. (c) The temperature and salinity discretization that the cross-isothermal and cross-isohaline fluxes are projected on. The blue arrows symbolize the cross-isothermal flux between two grid boxes and shows how the flux Ui,j,k crosses the three isotherms Tm−1, Tm, and Tm+1 on its way from Ti,j,k to Ti+1,j,k.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0163.1

Equivalently, the streamfunction may be computed from the cross-isohaline flux, as in Eq. (2). This is done similarly as described below, by switching temperature and salinity and changing the sign of the streamfunction.

We begin by defining discrete classes of salinity with the index l and temperature with the index m, so that SlSmin + ΔS(l − 1) and Tm = Tmin + ΔT(m − 1), as shown in Fig. A1c. The constants Smin and Tmin are here smaller than any values of S and T occurring anywhere in the water volume. The streamfunction is discretized as ψl,m = ψ(Sl, Tm). The volume flux across the isotherm Tm between the salinities Sl and Sl+1 is then Fl,mψl+1,mψl,m.

Now consider the two grid boxes in Figs. A1a,b with the temperatures Ti,j,k and Ti+1,j,k and the salinities Si,j,k and Si+1,j,k. The zonal transport between these boxes is Ui,j,k, which is computed numerically from the zonal velocity multiplied by the area of the surface between the boxes (Ui,j,k = ui,j,kΔyΔz) and is illustrated by the blue arrows in Fig. A1. In this particular case, there are three temperature classes between the temperatures Ti,j,k and Ti+1,j,k, and the flux Ui,j,k is therefore added to Fl,m−1, Fl,m, and Fl,m+1. The salinity class l is determined from the salinity of the upstream grid cell, Si,j,k in Fig. A1. This calculation is repeated for all pairs of adjacent grid cells, successively adding the corresponding cross-isothermal volume fluxes between the cells to the appropriate fluxes Fl,m.

The final values of the fluxes Fl,m correspond to (∂ψ/∂S)TΔS, from which the streamfunction ψ can be obtained by integration along S, setting ψ(Smin, T) = 0. Numerically, the streamfunction is hence given by
ea2
We here included an additional index n denoting time, and N is the total number of GCM datasets to be averaged over.

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1

The temperature and salinity range in the subsurface nearly adiabatic segment is roughly that of the core of the undercurrent (i.e., T ~ 15°–25°C and S ~ 34.5–35.5) (Wyrtki and Kilonsky 1984). Although T and S should be roughly conserved when the water moves equatorward, some subsurface water-mass transformation is expected in the equatorial undercurrent, where the vertical mixing is elevated (Gregg et al. 1985).

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