1. Introduction
Though the earth’s global thermohaline circulation is dominated by temperature (the deepest water is generally the coldest), salinity variations play a crucial role in determining the location of deep-water formation. The surface temperature distribution is roughly symmetric about the equator, but surface salinity has notable north–south asymmetries [e.g., cf. Figs. 8.8 and 8.10 in Peixoto and Oort (1992)], with winter northern North Atlantic water roughly 0.5 psu saltier than austral winter Weddell Sea water (Levitus and Boyer 1994). As a result, deep-water formation is equatorially asymmetrical within the World Ocean, with substantial transports of North Atlantic Deep Water crossing the equator and flowing into the Southern Hemisphere [Speer and McCartney (1991); Warren (1981) for a review of earlier work]. This asymmetry is interesting for purely oceanographic reasons as well as for its influence on climate. For example, the heat transport in the South Atlantic is equatorward rather than poleward (e.g., Macdonald 1993).
As Rooth (1982) and Bryan (1986) have shown, using a box model and a general circulation model, respectively, such an asymmetry in salinity and meridional overturning can occur even when the boundary conditions are equatorially symmetric. Surface boundary conditions form a relatively tight constraint on surface temperature but can be thought of as equivalent to specifying a flux of salinity rather than the salinity itself. Because of these “mixed boundary conditions,” more than one circulation state can exist given such a set of boundary conditions (Stommel 1961). In a single ocean basin spanning the equator, it is possible to have either equatorially symmetric temperature-dominated (T-dom) sinking near the poles, equatorially symmetric salinity-dominated (S-dom) sinking near the equator, or equatorially asymmetric deep-water formation at one pole only (Welander 1986).
How overturning strength depends on external parameters has received relatively thorough scrutiny in single-basin systems driven by temperature-only boundary conditions (Bryan 1986; Colin de Verdiere 1988; Winton 1996; Marotzke 1997) or freshwater-only boundary conditions (Huang and Chou 1994). The two-hemisphere, mixed-boundary-condition case has been explored with two-dimensional models by Marotzke et al. (1988), Thual and McWilliams (1992), Quon and Ghil (1992, 1995), Cessi and Young (1992), Schmidt and Mysak (1996), Vellinga (1996), and Dijkstra and Molemaker (1997). When salinity forcing is sufficiently stronger than temperature forcing, only the S-dom state exists, and when temperature forcing is sufficiently stronger than salinity forcing, only the T-dom state exists. It is only when both temperature and salinity forcing are in some sense at intermediate strengths that the equatorially symmetric states and the asymmetric state can all exist [see Thual and McWilliams (1992) Fig. 3]. These two-dimensional studies did not relate the flow strength and regime boundaries to parameters that could be readily applied to three-dimensional basins in which basin width, planetary rotation, and planetary radius play a role. There has been no three-dimensional study of this system in which forcing parameters were systematically varied, though Weaver and Sarachik (1991) used such a model to study the time evolution of transitions from state to state.
The thermohaline circulation is a global system in which the flow in each basin is influenced by that in the other basins (Warren 1983; Gordon 1986; Rintoul 1991; Marotzke and Willebrand 1991; Stocker and Wright 1991; England 1993; Macdonald and Wunsch 1996). This system has an extremely complicated dependence on a multiplicity of parameters, including basin geometry (Hughes and Weaver 1994), atmospheric freshwater transports between basins (Marotzke and Willebrand 1991; Stocker et al. 1992), and the coupling to the atmosphere (e.g., Mikolajewicz and Maier-Reimer 1994; Rahmstorf and Willebrand 1995; Weber 1998). Therefore, it is useful to understand the less complicated, single-basin cell as a stepping stone to understanding the more complete system.
The basic question of this paper is: How do the external parameters in a two-hemisphere, mixed-boundary-condition system determine the temperature, salinity, and overturning in the equatorially asymmetric state? We subdivide this into two smaller problems. In the asymmetric state, one hemisphere will possess the densest surface water in the basin. Peterson (1979) and Cox (1989) have shown that this water must dominate the bottom of the entire basin (when the equation of state is linear), but the resulting state has not been carefully studied. Thus, given such an asymmetric surface density distribution, what is the resulting overturning strength and pycnocline structure? This will be examined in section 3, which discusses experiments in which the surface density is restored to a reference profile. Then the full question can be answered if, for a given mixed boundary condition, we can predict the resulting surface density. We consider this question via mixed-boundary-condition experiments in section 4.
Hughes and Weaver (1994) showed in an idealized global GCM under mixed boundary conditions that Atlantic overturning strength is linearly related to the basinwide meridional gradient in zonally and vertically averaged steric height (i.e., a double vertical integral of zonally averaged density). Rahmstorf (1996) found in his global GCM coupled to a diffusive atmospheric energy balance model that Atlantic overturning is proportional to the zonally averaged middepth density difference between northern and southern Atlantic boundaries. In contrast to these studies, we want to relate the strength of the thermohaline circulation directly to external parameters; to clarify the relation we use simpler geometry and forcing. Wang et al. (1999a,b) used an idealized, global, hybrid coupled GCM and Scott et al. (1999) uncoupled and coupled box models to study interhemispheric thermohaline flow and its interaction with the atmosphere, in particular the effects of equatorially asymmetric atmospheric water vapor transports. Here, we leave out asymmetries in forcing (except in restoring-boundary-condition experiments) and detailed considerations of ocean–atmosphere interactions, to focus on the rotating fluid dynamics of the interhemispheric thermohaline circulation. The impact of wind, nonlinearities in the equation of state, and north–south asymmetries in the forcing are all important factors that deserve careful treatment, and we defer these to a later paper. For simplicity we use a relatively idealized system here.
2. Numerical model
Experiments are started with initial conditions of either a resting, isothermal, isohaline (35 psu) ocean, or a previous experiment. Each run is integrated until a nearly steady state is reached, unless noted otherwise. The usual criteria for steady state being reached is that the drift in meridional volume transport exponentially decreases over the last several centuries of the integration and is no more than 0.0025 Sv per century (Sv ≡ 106 m3 s−1) at the end. Spot checks on selected runs show that, when these criteria are met, other measures such as temperature and salinity are also nearly steady. In order to satisfy the Courant–Friedrichs–Levy criterion at the lowest computational cost, a shorter time step is used for the momentum equations than the tracer equations (Bryan 1984). Each experiment generally takes from 1000 to 5000 tracer years to reach steady state, which takes on the order of 1 day of cpu time on a DEC AlphaStation 250 4/266 workstation.
Most of the experiments are conducted using horizontal diffusion of properties to parameterize mixing induced by geostrophic eddies. Some runs with restoring boundary conditions are also repeated with the “Gent–McWilliams” parameterization, which supplements isopycnal diffusion of T and S with additional advection by a tracer velocity representing the untilting of isopycnals by baroclinic instability (Gent and McWilliams 1990; Gent et al. 1995). This parameterization has a stronger dynamical justification than horizontal diffusion, and allows numerical models to better represent the relatively thin thermocline and small deep-water formation regions of the real ocean and to eliminate spurious diapycnal diffusion in regions of strong horizontal gradients such as western boundary currents (Veronis 1975; Böning et al. 1995; Danabasoglu et al. 1994). The Gent–McWilliams runs are conducted with a flux-corrected transport scheme added to MOM-2 by Weaver and Eby (1997). Flux-corrected transport is a finite difference scheme that attempts to retain the accuracy of centered-difference schemes while also suppressing nonphysical overshoots of temperature and salinity (a property shared by the less accurate upstream differencing scheme).
3. Restoring boundary conditions
a. Experiments
We conduct two-hemisphere experiments, which are forced only by restoring to a temperature profile that is asymmetric about the equator. We wish to understand how the maximum meridional overturning volume transport and the pycnocline structure of a double-hemisphere overturning cell differ from the case of a single-hemisphere system. As section 4 will show, the surface density given by (2), with different values of ΔT in the northern and southern hemispheres, is a reasonable approximation to the shape of the surface density profiles produced by mixed-boundary-condition experiments.
In all the experiments, the southern hemisphere is made the “dominant” hemisphere by giving it the densest surface water. Thus the dominant deep water, analogous to North Atlantic Deep Water in the Atlantic Ocean, is formed in the southern hemisphere in our experiments. We chose the southern hemisphere because it happened to be the dominant hemisphere in the mixed-condition experiments (section 4); since our experiments have completely equatorially symmetric geometry (unlike the real ocean), the choice of dominant hemisphere is arbitrary. The weaker and deeper flow of Antarctic Bottom Water does not appear in these experiments because the equation of state is linear and there is no circumpolar channel corresponding to the Southern Ocean.
The relationship between the behavior in one-hemisphere (“1H”) and two-hemisphere (“2H”) basins is demonstrated by runs with ΔTS set to 30°, 6°, and 1°C. In each of these experiments, we set ΔTN = 0, thus exploring the case of most extreme equatorial asymmetry first. For each 2H experiment, a 1H experiment is run with forcing identical to the dominant hemisphere of the corresponding 2H run. Outside of section 3b, “1H experiment” refers to an experiment with equatorial symmetry rather than an actual wall at the equator. These two variations are nearly identical (11.86 Sv overturning with the wall and 11.84 Sv with symmetry).
In another series of experiments, ΔTS is fixed while ΔTN is varied. We are especially interested in the situation in which ΔTN is almost as large as ΔTS. In this case, the degree of asymmetry between the hemispheres is small, yet the circulation must be qualitatively different from a symmetric experiment because deep water is required to spread from the dominant hemisphere to fill the deepest region of the other “subordinate” hemisphere. In this series of experiments, ΔTS = 30°C, while ΔTP ≡ ΔTN − ΔTS is set to 15°, 6°, 3°, 1.5°, 0.6°, and 0°C.
b. Dependence on dominant hemisphere temperature gradient
The zonally integrated thermohaline circulation is characterized by small, relatively intense downwelling regions associated with deep convection and large areas of weak, diffusively driven upwelling (Fig. 1). In runs with ΔTN = 0, the presence of the nonconvecting hemisphere means that a strong thermocline covers about two times the area that it does in a single-hemisphere run.
The maximum buoyancy difference between the eastern and western boundaries is approximately bE − bW = 0.25Δb for all 1H and 2H runs, confirming the theoretical result of Marotzke (1997) and the hypothesis that the zonal buoyancy difference scales like Δb. The zonal buoyancy difference has a weak dependence on L (2H runs have about a 20% smaller proportionality constant). All these minor factors can be ignored in (9).
The factors of L1/3 in (9) imply that, since 21/3 = 1.3, each 2H run should have a 30% broader thermocline and 30% weaker upwelling than the corresponding 1H run. We measure the thermocline depth at the equator as before, and average the maximum zonal average w from 30°S to 62°N. The numerical experiments display the relationship between 2H and 1H D and W predicted by the scaling.
The combination of a somewhat weaker W and a larger area of upwelling cause the total overturning volume transport of the 2H experiments to be somewhat less than double that of the 1H experiments: the scaling predicts a factor of 22/3 = 1.6, whereas actual values are slightly higher (Table 2). Roughly equal amounts of upwelling occur in both hemispheres. For ΔTS of 30°C and 6°C, the convecting hemisphere has somewhat greater upwelling than the nonconvecting hemisphere. The western boundary current is stronger in this hemisphere (see Fig. 1) so that horizontal mixing there brings additional diapycnal mixing and enhances upwelling (Veronis 1975; Böning et al. 1995). In the ΔTS = 1°C run, the smaller upwelling area in the convecting hemisphere makes this hemisphere’s fraction of total upwelling smaller. Ignoring these details, however, the striking feature of these experiments is the simplicity of the relationship between flow magnitudes in the 2H and 1H cases. As the experiments show, the same relationship between 2H and 1H cases persist for ΔTS varied by a factor of 30 and corresponding thermocline depths ranging from a narrow surface region to nearly the entire water column.
c. Dependence on subordinate hemisphere temperature gradient
For 0 < ΔTP < ΔTS, the southern-sinking cell still dominates the circulation, but a weaker subordinate cell now forms in the near-surface water of the northern basin (Fig. 2). We assume that the ΔTP = ΔTS case can provide a crude estimate of how deep the northern cell extends. The minimum temperature of water sinking in the northern hemisphere is ΔTP (actually a little less, due to the restoring boundary condition). Thus, for ΔTP = 3°C, for example, the northern sinking should not penetrate the 3°C isotherm. Since the vertical temperature gradients are small below the top kilometer, the southern cell will dominate almost the entire northern hemisphere water column unless ΔTP is nearly zero. However, as Fig. 2 and Table 3 show, the maximum depth of the northern overturning cell actually reaches much deeper than either the ΔTP isotherm or the northern mixed layer. In sum, the penetration depth of the subordinate cell is clearly related to the mixed layer depth but, since the meridional overturning is proportional to a double vertical integral of the density difference between eastern and western walls, one cannot expect a simple correspondence [see Marotzke (1997) for a detailed discussion].
Asymmetrical overturning carries heat across the equator from the subordinate hemisphere to the dominant hemisphere. Although the cross-equatorial volume transport is not strongly dependent on the degree of asymmetry between northern and southern hemisphere surface density, the cross-equatorial heat transport is roughly proportional to
d. Gent–McWilliams runs
The experiments with ΔTS = 30°C and ΔTP = 0°, 0.6°, 3°, and 30°C are repeated with the Gent–McWilliams parameterization. These runs display the known features of isopycnal mixing: thinner thermocline and upper limb of overturning circulation, more compact downwelling region, and reduction of midlatitude upwelling. However, the key features described in the subsections above remain in these runs. The experiment with ΔTP = ΔTS has a broader thermocline and weaker w than the ΔTP = 0 run (because of the imposed symmetry, the latter is essentially a 1H run). In runs with a subordinate cell, the cell reaches deeper than would be predicted by looking at the northern mixed layer or the ΔTP = 0 stratification. Thus neither of these features can be attributed to the deficiencies of horizontal mixing. The volume transports in each cell and across the equator are somewhat less sensitive to ΔTP for weak asymmetries and more sensitive for strong asymmetries, compared to the horizontal mixing runs [equivalent to a larger exponent in (12)], but the magnitudes are still quite similar (Fig. 3). Why we find an apparently much reduced sensitivity to the eddy stirring parameterization than Weaver and Eby (1997) is not clear. Their Gent–McWilliams experiment had an overturning strength consistent with our 1H Gent–McWilliams run, but their horizontal-mixing experiment had much stronger overturning than ours.
4. Mixed boundary conditions
a. Temperature-dominated, salinity-dominated, and asymmetric states
The previous section took the surface density distribution as an almost prescribed external parameter. In reality, however, it is a function of salinity whose distribution is influenced by the ocean circulation more strongly than is temperature. Hence, we now turn to a model driven by mixed boundary conditions, where we still restore surface temperature as before but apply a fixed surface freshwater flux. Apart from showing the model’s response to large variations in the latter, we will, through scaling arguments, link the strength of the salt forcing to pole-to-equator and pole-to-pole salinity and density difference, at which point the results from the previous section apply.
We now consider the two-hemisphere system with surface temperature restored to the reference profile given by (2) and with surface salt flux given by (3). According to the work of Thual and McWilliams (1992) and others, one would expect that given the imposed temperature difference ΔT, if the salt flux QS is sufficiently small, the only possible circulation is a symmetric T-dom overturning. Similarly, for sufficiently high QS, the only solution should be a symmetric S-dom overturning. It is only at intermediate values of QS that the asymmetric mixed state occurs. Numerical experiments are conducted for ΔT = 30°C and 3°C using many values of QS and for ΔT = 10°C using a few values of QS. The unrealistically small temperature differences (3°C and 10°C) are used in order to better understand how circulation depends on ΔT. To isolate the importance of the diffusion term in the salt transport equation below, additional ΔT = 30°C experiments are performed with κH five times the default value. Large κH is not realistic for the ocean, but salt transport box models can use horizontal diffusion to mimic the effect of wind-driven gyres (Thual and McWilliams 1992), which is crucial for the stability of the thermohaline circulation under mixed boundary conditions (Marotzke 1990). Hence, it is interesting to establish diffusive behavior for comparison with future wind-driven experiments. Other ΔT = 30°C experiments are conducted with weaker restoring to the reference temperature (a timescale of τ = 300 d rather than the default 30 d timescale). This complicates the relationship between surface density and surface salinity, but may be more realistic for the large spatial scales considered here (e.g., Rahmstorf and Willebrand 1995; Marotzke and Pierce 1997).
Figure 5 shows a broad brush picture of where in the (ΔT, QS) parameter space asymmetric states exist (see also Table 4). For comparison, typical large-scale surface freshwater flux in the ocean is around 50 cm yr−1 (Schmitt et al. 1989; Wijffels et al. 1992). For a surface salinity of 35 psu, a freshwater flux of 1 cm yr−1 is equivalent to salt flux of about 10−6 psu cm s−1. For convenient comparisons, all QS values will be stated in units of 10−6 psu cm s−1 unless noted otherwise.
For most of the runs, the solution eventually approaches steady state based on the criteria described in section 2. These states generally have overturning cells and pycnocline structures qualitatively similar to the restoring boundary condition experiments of section 3. Typical surface salinity has a broad equatorial maximum, a weak southern polar minimum, and a strong northern polar minimum (Fig. 6a). Southern polar salinities are nearly identical between the experiments because most of the model volume is filled with this water. This means that southern polar salinity is near the globally averaged salinity, which is the same for all experiments. The freshness of the northern surface water makes the surface density profile asymmetric (Fig. 6b), as assumed in section 3. The surface density range in the dominant hemisphere is only modestly affected by the strength of the salinity forcing, whereas QS is a dominant influence on the density range in subordinate hemisphere. The surface temperature range is greater in the subordinate hemisphere than in the dominant hemisphere because the residence time of high-latitude surface water is somewhat longer in the subordinate hemisphere, giving the water more time to cool to the restoring temperature. In experiments with strong surface restoring, the surface temperature range in the two hemispheres is the same to within a few percent. However, in the weak restoring experiments, the temperature range is about 20% less in the dominant hemisphere, where the coldest surface water is around 5°C.
For large QS, no steady states are found. One such run (default κH, ΔT = 30°C, QS = 110) has an asymmetric overturning in which the dominant cell oscillates from a relatively weak, shallow overturning to a deeper, stronger overturning; the period is roughly 300 yr. A counterrotating high-latitude cell also appears at irregular periods. In the S-dom runs, the meridional overturning is usually confined to the top kilometer or so of the water column, with “flushes” of strong, deep equatorially asymmetric overturning occuring now and then. The shift from steady to oscillating states as salinity forcing is increased has also been seen in single-hemisphere systems (Weaver et al. 1991, 1993; Winton and Sarachik 1993; Huang 1994); thermohaline oscillations in two-hemisphere basins have been reported by Weaver and Sarachik (1991).
Dijkstra and Molemaker (1997) showed that for a two-dimensional model, the temperature restoring timescale and the exact form of qS(ϕ) can change qualitative aspects of the equilibrium curves of Φ+(QS). If qS(ϕ) has maxima at the polar boundaries [as in our experiments and in Thual and McWilliams (1992)] the system can have a subcritical pitchfork bifurcation, which allows stable equilibria to coexist in some range of QS [Fig. 7a; see also Dijkstra and Molemaker (1997) Figs. 15 and 16]. However, if the qS(ϕ) maxima are equatorward of the polar boundaries (as in Marotzke et al. 1988; Vellinga 1996), the system has a supercritical pitchfork bifurcation, in which the symmetric state is unstable when the asymmetric state is stable [Fig. 7b; also Dijkstra and Molemaker (1997, Fig. 4)].
We explored the multiple equilibria of the system by varying the initial condition for given parameters (Fig. 8). The behavior of the asymmetric state in our experiments is consistent with a subcritical bifurcation. All three series of experiments shown in Fig. 8 have both symmetric (Φ+ = Φ−) and asymmetric (Φ+ ≠ Φ−) circulations for overlapping ranges of QS. In a supercritical bifurcation, as QS is lowered, the dominant and subordinate transports converge until they become equal at a critical QS. In our experiments, no such transition region is evident; a small decrease in QS pushes a strongly asymmetric system into a T-dom state (see, e.g., Fig. 8b, 8 < QS < 9). In several experiments, we vary QS continuously over the course of a few thousand years. For the ΔT = 30°C, default κH series, these confirm the abrupt transition to the T-dom state and shows that the T-dom state is stable for QS < 26, above which the equatorial asymmetry begins to grow. We further test the stability of the T-dom state by imposing a pole-to-pole surface salinity difference of 0.2 psu for ten years on one high-QS symmetric state (ΔT = 30, default κH, QS = 10). This produces a small asymmetry in the overturning, which virtually disappears within 1000 yr.
The series of experiments with ΔT = 30°C (Figs. 8a,b) exhibit the states displayed in two-dimensional experiments: a T-dom, S-dom (not shown), and asymmetric state. The ΔT = 3°C case is more complicated (Fig. 8c). For overlapping ranges of QS, there are at least three asymmetric states, a “very” asymmetric state in which the dominant state cell is much stronger than the subordinate cell (Fig. 9a), an “intermediate” asymmetric state (Fig. 9b), and a “slightly” asymmetric state in which the dominant and subordinate cells are nearly equal in strength (Fig. 9c).
The streamfunction disturbance in the equatorial deep ocean in Fig. 9c may be due to the numerical instability identified by Weaver and Sarachik (1990). We attempted to eliminate this disturbance both by increasing the vertical viscosity from 1 cm2 s−1 to 10 cm2 s−1 and by roughly doubling the vertical resolution. Neither method worked; however, the higher resolution runs helped us discover the intermediate asymmetric state, which we were then able to maintain with the default vertical resolution.
Which state a given experiment falls into depends on the initial condition for the experiment. Starting at the very asymmetric state with QS = 3, when QS is decreased in small increments, the solution remains in a very asymmetric state for QS ≥ 1.2. Starting with the same QS = 3 initial condition, but immediately lowering QS to 1, the system moves into the slightly asymmetric state, which it maintains as QS is increased to 1.6. When the slightly asymmetric run with QS = 1 is used as an initial condition for a high vertical resolution run with the same QS, the system falls into the intermediate asymmetric state. This state persists when QS is raised to 1.2 and when this QS = 1.2 run is continued at the default (low) vertical resolution. These pathways between various states indicate that relatively small perturbations in the initial state can change the final state of the system. It is not clear how many more asymmetric states exist in this parameter range. Various ΔT = 30°C experiments with different initial conditions do not reveal any multiple asymmetric states.
b. Properties of asymmetric states
1) Salt transport and surface salinity range
The meridional salt transport is imposed by the surface salt flux QS, but the manner in which this constraint is satisfied is determined by the flow field. Given some knowledge about the behavior of the flow field, we can predict the meridional surface salinity gradient in the subordinate hemisphere as a function of QS.
The volume transport Φ is not an external parameter. According to the experiments of the previous section, it should be about twice the size of the overturning in a symmetric QS = 0 system. This overturning can be related to external parameters by Eq. (9d). We assume that about one-quarter of the upwelling occurs in the northern quarter of the basin, so a = ¼. We also assume that Δz is approximately the pycnocline depth [obtainable from Eq. (9a)].
The actual ranges of s for the asymmetric state are shown in Table 4. Whereas asymmetric states exist for quite different ranges in QS for ΔT = 3°C and ΔT = 30°C, the ranges in s are quite similar. Similarly, curves of QS(ΔT) for s constant are nearly parallel to the regime boundaries of the asymmetric state (see Fig. 5). It is also interesting that the two series of experiments with a strong diffusive term in (14) have nearly identical ranges of s.
2) Surface density and overturning strength
The northern hemisphere salinity range, ΔSN, is estimated in the preceding section. There is a relatively large gyre contribution to the meridional salt transport in the southern hemisphere (figure not shown), which makes it difficult to estimate ΔSS, the southern hemisphere range. However, because ΔSS is small, a very rough estimate may be good enough, so we ignore all but the overturning component in (13). The meridional volume transport at 32°S is fed by three times the upwelling area that feeds the transport at 32°N, so (13) implies that ΔSS ≈ (⅓)ΔSN. This prediction is approximately true for all the experiments except for the slightly asymmetric state runs. Thus ΔbP can be predicted from first principles.
Section 3 showed that for a density profile similar to the mixed boundary condition experiments, Φ±/Φ0 is given by (12), with Φ0 ∝ Δ
The heat transport in the mixed-boundary-condition experiments is quite similar to that in the temperature-only experiments (Fig. 4). The temperature-only experiments impose a strong asymmetry between northern and southern hemisphere temperatures, whereas the temperature in mixed-boundary-condition experiments is approximately symmetrical about the equator. However, the thermocline in the dominant hemisphere and in much of the subordinate hemisphere does look similar in temperature-only and mixed-boundary-condition experiments. Thus the more idealized experiments capture the behavior of the more complicated experiments. The mixed-boundary-condition runs have somewhat smaller heat transports at the equator and in the dominant hemisphere because they have a larger gyre component opposing the overturning component.
3) Low salinity tongue
In the asymmetric state, a subsurface tongue of low salinity water extends southward of its surface source (Fig. 12); in some cases the subsurface minimum extends for thousand of kilometers. This fresh tongue is reminiscent of the salinity signal associated with Antarctic Intermediate Water in the South Atlantic.
The low salinity tongue appears to be governed by a balance between southward advection of deep, low salinity from the northern boundary and downward diffusion of high salinity from the surface. The low salinity signal is injected below the surface by convection in the downward limb of the subordinate cell. This can be seen by comparing the maximum depth of the northern mixed layer and the mean depth of the fresh tongue in Fig. 12;the salinity signal reaches somewhat deeper than the mixed layer depth due to diffusion. When the subordinate-cell maximum density is nearly as great as the dominant-cell maximum density, the low salinity anomaly is relatively deep. In this case, it can travel for a relatively long distance before it is eroded by high salinity diffusing from above (Fig. 12a). Progressively shallower salinity injection results in progressively less prominant fresh tongues (Figs. 12b,c). These ideas are demonstrated by Fig. 13: the fresh tongue reaches farther as the density difference between the two polar boundaries decreases. The length of the tongue is defined here as the latitude range over which there is a subsurface minimum in the zonal average salinity. The buoyancy ranges are defined in terms of zonal average surface values; ΔbP is the difference between the southern hemisphere and northern hemisphere maxima, whereas ΔbS is the difference between the southern hemisphere maximum and the basin-wide minimum.
It is interesting that for the same ΔbP/ΔbS, the fresh tongue extends about equally far for experiments with ΔT = 30°C, ΔT = 3°C, and high κH. For the same ΔbP/ΔbS, the salinity minimum penetrates more deeply in the ΔT = 3°C experiments than in the ΔT = 30° experiments. However, the vertical advective-diffusive balance is also different in the two series, so that high surface salinity also diffuses to a greater depth in the ΔT = 3°C experiments; apparently the two effects are equally strong. The result is insensitive to κH because the horizontal salinity transport is dominated by advection.
5. Conclusions
In a system driven by mixed boundary conditions, a hallmark of equatorially asymmetric circulation is the different roles played by temperature and salinity. One hemisphere, which we call the “dominant” hemisphere, has the strongest meridional circulation, minimizing the exposure of surface water to the surface fluxes of salinity (or, more realistically, of freshwater) at any particular latitude. This hemisphere is thus dominated by temperature forcing, and contains the densest surface water, which then forms the deep water for the entire system. The other, subordinate hemisphere is marked by slower circulation and hence greater sensitivity to the salinity forcing. Here, no surface water is dense enough to penetrate to the bottom, but the local density maximum produces intermediate water that drives a relatively shallow overturning cell. To summarize, the buoyancy range of the entire system is largely controlled by the imposed temperature range ΔT, while the degree of asymmetry is controlled by both ΔT and the salinity flux scale QS.
Experiments with surface buoyancy directly controlled through a restoring boundary condition show how meridional overturning cell transports of mass and heat depend on the surface buoyancy range in each hemisphere. Even a small asymmetry in surface buoyancy makes one overturning cell much stronger than the other, with a strong volume transport forced to cross the equator. A much greater buoyancy asymmetry does not produce a much greater circulation asymmetry. In the mixed boundary system, therefore, deep-water formation rate and cross-equatorial flow are primarily governed by ΔT. As long as QS is in the range that permits asymmetric flow, QS will have a smaller influence on the dominant cell strength. The subordinate cell strength Φ− is more sensitive to ΔbP, the difference between the surface buoyancy maxima in the two hemispheres, so Φ− does depend on QS. The subordinate cell is likely to be strongly influenced by wind stress, which is not included in this model, but the deepening of the subordinate cell with decreasing ΔbP, and hence with decreasing QS, should be a robust result. This deepening also deepens (and hence cools) the warm limb of the cross-equatorial circulation so that QS has a significant influence on cross-equatorial heat transport.
A two-hemisphere basin with strongly asymmetric circulation has a somewhat broader pycnocline, correspondingly weaker upwelling, and nearly double the meridional volume transport of a one-hemisphere basin with equivalent forcing. The differences can be accounted for by scalings based on the classical assumptions of large-scale geostrophy and vertical advective–diffusive balance. However, care must be taken to close the system of scale relations with the continuity equation rather than with the commonly applied linear vorticity relation. The latter equation does not include important contributions to vorticity in the western boundary current.
It is more difficult to explain quantitatively how overturning strength depends on ΔbP, but it is clear that the general pattern depends on the existence of a pycnocline. The total upwelling is controlled by diffusion in the pycnocline; the strengths of the two cells are largely determined by how they divide this fixed amount of upwelling. Even a modest ΔbP forces the relatively small subpycnocline range of buoyancy to be filled by the dominant cell. This gives the dominant cell access to at least the base of the pycnocline over most of basin so that it can grow strong at the expense of the subordinate cell.
The power law for overturning Φ found here appears to be more complicated than the linear relationships between flow strength and meridional density differences found in global GCMs (Hughes and Weaver 1994;Rahmstorf 1996). On closer inspection, however, we have shown that the Hughes and Weaver result is consistent with the classical scaling. This has implications in formulating box models, which for simplicity typically assume that Φ is proportional to Δb between boxes (e.g., Rooth 1982; Scott et al. 1999). The classical ⅓ power law is appropriate to surface buoyancy and the linear law is appropriate to a double vertical integral of density, but a box model should be based on a single vertical integral. In that case, the reasoning in section 3b yields Φ ∝ Δb1/2.
In contrast to the Hughes and Weaver (1994) model, Rahmstorf (1996) pointed out that much of the North Atlantic Deep Water in his model did not upwell in the midlatitude Atlantic; hence, the Atlantic thermocline depth was unrelated to overturning strength. This significantly changed the scaling as expressed in (9), leading to linear dependence of overturning on meridional density differences. However, how global upwelling is distributed is still a somewhat open question. For instance, the horizontal (rather than isopycnal) eddy diffusivity commonly used in general circulation models may overstate the amount of upwelling in the Antarctic Circumpolar Current (Danabasoglu and McWilliams 1995).
The approximate surface distribution of salinity, and hence of density, can be related to QS and ΔT via approximations for the meridional salt transport. Our approximation is fairly crude since it does not take into account the effects of QS on the circulation, such as the existence of the subordinate cell. It can be thought of as essentially a box model, with the attributes more closely attuned to the dynamics of three-dimensional circulations than in previous work, such as Rooth (1982) or Thual and McWilliams (1992). In the more sensitive subordinate hemisphere, the salt transport is dominated by zonal-average (hence two-dimensional) properties such as meridional overturning. However, the strength of these processes is different for two- and three-dimensional systems. The degree of asymmetry of the system can be described by the relative salinity range s = βΔS/αΔT, where ΔS is the salinity contrast in the subordinate hemisphere and α and β are the thermal and haline expansion coefficients, respectively. Here, s measures the relative contribution of salinity to density variations; for the three-dimensional system, s is constant along curves of QS ∝ ΔT4/3.
The system appears to be governed by a “subcritical pitchfork bifurcation,” in which both symmetric and asymmetric solutions can coexist for the same parameters. This is consistent with the two-dimensional results of Dijkstra and Molemaker (1997), in which the nature of the bifurcation depends on details of the surface forcing. Horizontal diffusivity raises the value of the minimum QS for which the asymmetric state exists and is stable. This is also consistent with two-dimensional results (Thual and McWilliams 1992) and suggests that the minimum QS rises when wind forcing is added to the thermohaline circulation. The location of the regime boundaries must be viewed with some caution because, in reality, the ocean is coupled to an atmosphere in which meridional transports of heat and moisture can affect the stability of a state (Nakamura et al. 1994; Saravanan and McWilliams 1995; Marotzke 1996). We show that capturing the atmospheric temperature feedback by weakening the surface temperature constraint raises the minimum QS for which a stable asymmetric state exists. However, a two-dimensional model shows that such a configuration is more unstable than a coupled model, in which moisture feedbacks are also included (Capotondi and Saravanan 1996).
It is not clear how many asymmetric steady states are possible for a given set of parameters. We stumbled upon at least three stable equilibria for ΔT = 3°C, with significant basinwide changes in overturning strength and T–S–ρ characteristics. However, we could not find multiple asymmetric states for the more realistic ΔT = 30°C. Dijkstra and Molemaker (1997) also show a hint of multiple asymmetric states (their Figs. 15 and 16). We attempted to construct a box model that included some of the subtleties of the relationships among salinity, overturning strength, and meridional salt transport. None of our attempts produced the multiple states seen here, nor did they display the vanishing of the asymmetric state at low QS described above, so we do not describe the model details in this paper.
The differences in circulation in our multiple states are larger than those between multiple convection states found by Lenderink and Haarsma (1994) in a three-level three-dimensional model, so purely local processes are unlikely to be responsible. If the real ocean displays such a multiplicity of states, it would indicate that significant changes in the global circulation could be forced by temporary climate events. The existence of such states would imply a much richer range of responses for the thermohaline circulation, which is usually portrayed as choosing between convection or no-convection states for various oceans (Marotzke and Willebrand 1991) or ocean regions (Hughes and Weaver 1994).
The system studied here is rather idealized, though it does capture such Atlantic features as the north–south asymmetry in polar salinity, the concomitant dominance of one hemisphere over the other in deep-water production, and the tongue of low-salinity intermediate water (AAIW in the Atlantic) produced by the fresher hemisphere. The usefulness of such an idealization lies partly in the simplicity of the salt transport, which allows for a rough prediction of circulation properties from external parameters. The system also shows that the qualitative behavior illustrated by two-dimensional models holds for three-dimensional dynamics. An obvious next step would be to include more realistic features from the real world, building on the more idealized results to derive the rules governing the more complicated system. Prominent “complications” include the wind-driven gyres, nonlinearity in the equation of state, and the Antarctic Circumpolar Current.
Acknowledgments
Work on this paper was done under NSF Grant OCE-9521138. Financial support for JM was granted by the Tokyo Electric Power Company through the TEPCO/MIT Environmental Research Program. Kevin Kohler provided programming assistance. Thanks go to Andrew Weaver and Michael Eby for donating their flux-corrected transport code and for their help in using it, and to Ron Pacanowski and Stephen Griffies for assistance with MOM 2. We acknowledge Claes Rooth and Steve Meacham for many interesting and useful discussions about thermohaline circulation and the mathematics of bifurcations, Jeff Scott, Thomas Stocker, and two anonymous reviewers for extensive suggestions, and Bob Hallberg for his comments.
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Meridional overturning streamfunction (contours) and zonal average temperature (shading) for restoring BC experiments, (a) 1H and (b) 2H, ΔTN = 0. Overturning contours are 2 Sv apart; isotherms are at 0.05, 0.1, 0.2, 0.4, and 0.8 of ΔTS = 30°C. In this and all subsequent plots of overturning streamfunction, dashed, solid, and dotted contours represent negative (southern sinking), positive (northern sinking), and zero values, respectively.
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Meridional overturning streamfunctions for runs with ΔTS = 30°C and various ΔTN < ΔTS. Contour intervals are 2 Sv for dashed contours and 1 Sv for solid. Shading represents the temperature range ventilated in both hemispheres (all water warmer than subordinate hemisphere minimum sea surface temperature).
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Volume transport as a function of ΔTP for the dominant cell, subordinate cell, sum of dominant and subordinate cells, and cross-equatorial flow. (a) Experiments with horizontal diffusion. (b) Experiments with Gent–McWilliams mixing (advective plus eddy components).
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Meridional heat transport as a function of ΔbP/ΔbS for temperature-only (dashed) and mixed-boundary-condition (solid) experiments, including values at the equator (×), the southward (dominant) maximum (+), and the northward (subordinate) maximum (○).
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Salt flux QS and imposed temperature difference ΔT for mixed-boundary-condition experiments with κH = 1000 m2 s−1. Locations in parameter space are denoted by a “+” if only T-dom states were found, by a “×” if only S-dom states were found, and by an“○” if at least one asymmetric state was found. Solid lines represent estimates of regime boundaries. Dotted lines are estimates of QS(ΔT) for constant s values of 0.1,
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Zonal average surface (a) salinity and (b) density for several ΔT = 30 experiments (QS = 8.6, 10, 26, 52, 80). Salinity range and temperature asymmetry both increase with QS.
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Schematic of overturning transport maximum vs QS showing (a) subcritical and (b) supercritical pitchfork bifurcation (Drazin 1992). Solid lines are stable equilibria, while dashed lines are unstable equilibria. Upper curve is Φ+, lower curve is Φ−, and central curve is symmetric state.
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Total volume transport of the dominant (“+”) and subordinate (“○”) meridional overturning cells as a function of QS (in units of 10−6 psu cm s−1, which roughly corresponds to cm yr−1 of surface freshwater flux) for (a) κH = 5000 m2 s−1, (b) ΔT = 30°C, and (c) ΔT = 3°C experiments. Each dashed line connects an experiment to the experiment that is used as an initial condition (initial condition runs were generally at higher QS than succeeding runs). Solid lines show values for runs in which QS is changed continuously during the run. The S-dom experiments are not shown. In (b), the highest QS experiment data is based on a time-average of the oscillating series.
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Meridional streamfunction and zonal average density for (a) very asymmetric, (b) intermediate asymmetric, and (c) slightly asymmetric states with ΔT = 3°C and QS = 1.2. Streamfunction contour interval is 0.5 Sv; isopycnals are at (28 kg/m3 − ρ), where ρ is 0.025, 0.05, 0.1, 0.2, and 0.4 kg m−3.
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Dimensionless salinity gradient s as a function of dimensionless salinity flux QS/Q0 for asymmetric-state runs. ΔT = 3 runs are represented by circles; ΔT = 30 runs are represented by squares (default κH), diamonds (high κH), and triangles (weak temperature restoring). Black lines are estimates based on (14), with dashed line for high κH runs. For reference, gray lines mark slopes of 0.5, 1, 2, 4, 8, and 16.
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Total volume transport of dominant and subordinate cells as a function of polar buoyancy difference ΔbP for ΔT = 3°C (circles) and ΔT = 30°C (squares for default κH, diamonds for high κH, and triangles for weak restoring). Volume transport and buoyancy difference in each series of experiments are normalized by values from corresponding QS = 0 (T-dom state) experiments. Filled symbols represent experiments, open symbols represent prediction based on ΔbP and ΔbS (i.e., changes in dominant hemisphere buoyancy differences are taken into account), and dashed lines represent predictions based on ΔbP alone (dominant hemisphere buoyancy differences are assumed constant).
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Zonal average salinity (shaded), overturning streamfunction (thin contours), and densest isopycnal to outcrop at the surface in subordinate hemisphere (thick contours) for ΔT = 30°C experiments with (a) QS = 8.6, (b) QS = 26, and (c) QS = 52. For each experiment, salinity contour interval is 1/20 of salinity range of the experiment. Darker shading represents fresher water in tongue protruding from northern boundary, but saltier water in tropical surface patch. Streamfunction contour interval is 2 Sv, with negative values dashed and zero contour dotted.
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Length of low salinity tongue (in degrees latitude) as a function of ratio of polar buoyancy difference to dominant hemisphere buoyancy range, for ΔT = 3°C runs (circles), ΔT = 30°C, default and high κH runs (squares and diamonds, respectively).
Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0382:BODHTF>2.0.CO;2
Summary of numerical experiments.
Restoring runs, one and two hemisphere (hem) overturning. All transports given in Sv. For 2-hem runs, ΔTN = 0.
Subordinate overturning cell depth. Dcell is the maximum depth of the subordinate cell,
Series of mixed BC experiments, asymmetric state. External parameters and some solution characteristics of the mixed-boundary-condition runs. Min and max refer to minimum and maximum values for which an asymmetric state was found (not including slightly asymmetric states in ΔT = 3° runs). Here, ΔT is in °C, κH is in m2 s−1, τ is in days, and QS is in 10−6 psu cm s−1
