Robustness of multiple equilibria in the global ocean circulation



[1] In an idealized Atlantic-Pacific ocean model we study the steady state solutions versus freshwater input in the northern North Atlantic. We find that four different states, the Conveyor (C), the Southern Sinking (SS), the Northern Sinking (NS) and the Inverse Conveyor (IC), appear as two disconnected branches of solutions, where the C is connected with the SS and the NS with the IC. We argue that the latter has the intriguing consequence that the parameter volume for which multiple steady states exist is greatly increased.

1. Introduction

[2] The meridional overturning circulation (MOC) in the Atlantic is important for the temperate Western European climate, because of the northward heat transport (��1015W) in the Atlantic basin [Ganachaud and Wunsch, 2000]. A collapse of the Atlantic MOC can induce substantial cooling over the North Atlantic region [Vellinga et al., 2002]. It is therefore important to know the probability of such a collapse under present day forcing. Proxy data have indicated that relatively rapid climate changes have occurred in the past, for example the Dansgaard-Oeschger events during the last glacial period. Transitions associated with the existence of multiple equilibria of the Atlantic MOC form a plausible explanation for these events [Clark et al., 2002].

[3] In most previous modeling studies the salt-advection feedback in the Atlantic is the origin of the multiple equilibria of the MOC. This is a positive feedback where changes in the MOC affect the meridional salt transport which in turn affects the MOC [Stommel, 1961]. The MOC can collapse from an ‘on’ state (or ‘Conveyor’ state) with present-day North Atlantic Deep Water (NADW) formation to a so-called ‘off’ state (or ‘Southern Sinking’ state), where NADW formation has ceased. A typical feature is the hysteresis behavior in the strength of the MOC when the freshwater input in the northern North Atlantic is varied, giving rise to a range of freshwater flux amplitudes where both ‘on’ and ‘off’ state coexist. Such a hysteresis behavior is found in a hierarchy of models, from simple box models [Stommel, 1961], through zonally averaged ocean models [Stocker and Wright, 1991] and ocean general circulation models [Bryan, 1986] to intermediate complexity climate (ocean-atmosphere-sea ice) models [Rahmstorf et al., 2005].

[4] The climate models that were integrated under the SRES A1B-scenario of the IPCC do not show any sign of abrupt change in the Atlantic MOC up to 2100, challenging the results from simpler climate models [Schmittner et al., 2005]. It has been argued that coupling to a fully dynamical atmosphere model could potentially remove the multiple equilibrium regime of the Atlantic MOC [Schlesinger et al., 2006]. Furthermore, it has been proposed that the multiple MOC flows are caused by spuriously high vertical diffusivities in ocean models [Nof et al., 2007]. On the other hand, if atmospheric feedbacks or small vertical diffusivities do indeed remove the multiple equilibrium regime of the Atlantic MOC, it becomes more difficult to explain the paleoclimatic record.

[5] In idealized three-dimensional two-basin ocean-only models, four essentially different steady flow states were found [Marotzke and Willebrand, 1991; Hughes and Weaver, 1994; Von der Heydt and Dijkstra, 2008]. In addition to the ‘Conveyor’ (C) state and ‘Southern Sinking’ (SS) state, two states with North Pacific deep water formation exist, the so-called ‘Inverse Conveyor’ (IC) state and a ‘Northern Sinking’ (NS) state. The IC-state is characterized by sinking in the Pacific and upwelling in the Atlantic and the NS-state has northern sinking in both basins.

[6] The question arises of why there is a Conveyor in the present day climate. The often heard argument that EP is larger over the Atlantic than over the Pacific is probably incorrect. Recent estimates of the precipitation indicate P is quite similar over the North Atlantic and North Pacific [Emile-Geay et al., 2003] and the evaporation difference is due to the temperature difference between both basins. However, the latter is largely a consequence of the presence of the ocean circulation itself [Warren, 1983; Marotzke and Willebrand, 1991; Czaja, 2009]. The explanation of why there is a present-day conveyor likely involves a multiple equilibrium problem where asymmetries determine the preference for a certain state. For example, Von der Heydt and Dijkstra [2008] find a preference for the NS-state when the basin of the Pacific is extended northward.

[7] To analyse this problem within an idealized two-basin ocean model, similar to [Marotzke and Willebrand, 1991], it is crucial to know how the four (C, SS, NS, IC) states are linked to each other under a change of the freshwater flux in the northern North Atlantic, commonly used in studies of this type. In this paper, we demonstrate that there are two separate branches of steady solutions, where the C-state is connected to the SS and the IC to the NS. The fact that the two branches are disconnected has deep implications for the multiple equilibria regime of the global ocean circulation and will increase the parameter volume for which this regime exist.

2. Model and Experimental Design

[8] We employ version 3.1 of the GFDL Modular Ocean Model (MOM) [Pacanowski and Griffies, 2000]. The model domain consists of two equal, rectangular basins (‘Atlantic’ and ‘Pacific’) connected in the south via a zonal channel with a periodic boundary condition. The basins and channel have a constant depth of 4500 m. Each basin extends from 68°S to 68°N in latitude and is 60° wide in longitude. The basins are separated by continents of 15° wide that reach down to 52°S. The horizontal resolution is 3.75° × 4°, with 15 non-equidistant layers in the vertical (50 m deep at the surface to 550 m at the bottom). At the surface the rigid-lid approximation is used, at lateral boundaries there are no-slip conditions for the velocity and at the bottom boundary a free-slip condition is applied. At the bottom and the continental walls there is a zero-flux condition.

[9] The coefficients for horizontal (KH = 1.0 103m2s−1) and vertical (KV = 0.5 · 10−4 m2s−1) mixing of heat and salt and for horizontal (AH = 2.8 105 m2s−1) and vertical (AV = 1.0 · 10−3 m2s−1) mixing of momentum are constant. The model simulations were spun up with restoring boundary conditions (restoring timescale is 30 days) for temperature (T) and salinity (S), and under an idealized zonally uniform wind stress [Bryan, 1987]. The restoring profiles for T and S are a function of latitude only and follow a cosine law with an equator to pole difference of Δ T = 20° and Δ S = 2 psu, respectively, and reference values T0 = 15° and S0 = 34.5 psu. The spinup simulations are integrated to equilibrium, i.e., until the absolute heat flux averaged over the whole surface is smaller than 10−3 Wm−2. At the end of the spinup the (virtual) freshwater flux is diagnosed, symmetrized with respect to the equator and the simulation is continued under mixed boundary conditions.

3. Results

[10] As was done by Marotzke and Willebrand [1991], four equilibrium solutions are found under the symmetric freshwater flux; contour plots of the MOC streamfunction in the Atlantic (AMOC) and the Pacific (PMOC) are shown in Figure 1. The C-state (Figure 1a) has an AMOC strength of about 25 Sv, the SS-state (Figure 1b) has concentrated downwelling near the southern boundary, for the NS-state the PMOC and AMOC strengths are similar (Figure 1c) and the IC-state (Figure 1d) is like the C-state but with the Pacific and Atlantic interchanged. In all states the ACC transport is about 60 Sv and there is no formation of Antarctic deep waters.

Figure 1.

Meridional overturning streamfunction (Sv) of the Southern Ocean, Pacific and Atlantic for the four equilibrium solutions as found under equatorially symmetric freshwater forcing (γp = 0) for experiment SC (see Table 1). Data are averaged over the last 100 years of integration. (a) C-state, (b) SS-state, (c) NS-state and (d) IC-state. Contour interval is 2.5 Sv.

[11] To determine the sensitivity of the equilibria to the freshwater flux, we add a freshwater flux perturbation (FSp in Sv) with amplitude γp in the North Atlantic between 40°N and 64°N to the diagnosed equatorially symmetric freshwater flux FSd. Hence the total freshwater flux (FS) is given by

equation image

Here, Q is a uniform compensating flux in the rest of the domain, to ensure a net zero freshwater flux over the total ocean surface. Note that when γp is negative, salt is added in the northern North Atlantic.

[12] In Figure 2a the maximum value for the AMOC (ΨAtl) is plotted against γp. Starting from the C-state and increasing γp, the salinity in the North Atlantic decreases, making the water less dense. After a certain threshold is reached in γp (γI+ ∼ 0.061 Sv), the C-state ceases to exist and a collapse of the Atlantic MOC (due to the salt advection feedback) to the SS-state occurs. The reverse transition occurs when we remove freshwater from the North Atlantic (γp negative) starting from the SS-state. In this case, the northern North Atlantic becomes saltier and eventually (γI ∼ −0.035 Sv) a recovery to the C-state takes place. This hysteresis agrees with earlier results on the collapse of the Atlantic MOC [Rahmstorf et al., 2005]. The changes in the Pacific are much less dramatic (only a few Sv) when varying γp along the C–SS branch (Figure 2b), because the forcing is applied in the Atlantic.

Figure 2.

Maximum value of the steady meridional streamfunction Ψ (Sv) in (a) the Atlantic and (b) the Pacific versus the strength of the freshwater flux γp for experiment SC. The crosses and pluses are computed values (between 20°N and 68°N, 200 and 4500 m depth, averaged over the last 100 years of the simulation) and are connected by straight lines to obtain the branches. Positive (negative) values correspond to clockwise (anti-clockwise) MOC orientation (see Figure 1). The arrows in Figure 2a indicate the collapse and recovery of the Atlantic overturning cell at γ±, and those in Figure 2b indicate the changes in the Pacific overturning cell.

[13] The interesting new aspect in Figure 2 appears when we start from the NS-state and increase γp. At a certain threshold (γII+ ∼ 0.028 Sv) the positive overturning cell in the Atlantic again collapses (Figure 2a). The difference lies in the Pacific overturning cell (Figure 2b), which now has a positive sign. When increasing γp, the PMOC slightly intensifies, while the AMOC collapses, leading to the IC-state. When adding salt to the northern North Atlantic, the AMOC recovers (for γII ∼ −0.023 Sv), while the PMOC stays positive. Therefore, there is a second separate branch of steady solutions connecting the NS and IC states.

[14] As is already indicated in Figure 2, the NS-state and C-state remain in existence for large negative γp. For both states, the Atlantic becomes saltier with decreasing (negative) γp while the PMOC hardly changes. Similarly, the IC and SS-states remain existent for large positive γp, where the northern Atlantic becomes fresher, which further weakens any northern sinking. In the Pacific, there is only gradual change of the overturning with increasing γp.

[15] The fact that the C–SS and NS–IC branches are disconnected has profound consequences. The bifurcation diagram corresponding to Figure 2a is sketched in Figure 3a; the only qualitative difference is that the branches of unstable steady states (which we know to exist [Dijkstra and Weijer, 2005]) are drawn explicitly. Now consider experiments where other factors than γp are varied—for example using an asymmetric freshwater flux, applying different sizes of the Atlantic and Pacific or adding bottom topography to the model — then the bifurcation diagram would deform and when branches remain existent, three qualitatively different results may arise.

Figure 3.

Qualitative sketches of possible bifurcation diagrams where the C–SS and NS–IC branches exist and remain disconnected. Solid curves indicate stable steady state solutions and the dashed curves are the unstable parts of the branches. (a) Hysteresis behavior with γp is found on both branches as in Figure 2. Note that the branches do not interact: the intersection of the branches is just due to the chosen representation of the solutions. (b) Shifting branches with γp; hysteresis behavior is still present on both branches. (c) Hysteresis behavior disappears for the C–SS branch, but remains in the NS–IC branch; the reverse is also possible (not shown). (d) Hysteresis behavior disappears in both branches.

[16] First of all, the C–SS and NS–IC branches may shift with respect to each other as in Figure 3b. Then, the salt advection feedback is not affected and multiple equilibria exist (or hysteresis still occurs) for both C–SS states and NS–IC states. Secondly, multiple equilibria for one or both branches may disappear (Figures 3c3d) possibly mimicking the situation where the vertical mixing is very small [Nof et al., 2007] or where atmospheric feedbacks are able to counteract the salt advection feedback. But even in the case of Figure 3d, as long as the branches remain disconnected, there will always be two equilibrium states for every value of γp.

[17] Indeed, we performed four additional series of simulations (see Table 1) under modified boundary conditions. The cases considered ERA40 wind stress, a different value of the vertical diffusivity, the coupling of the ocean to a simple atmospheric energy balance model and a more realistic zonal salinity profile (with a saline Atlantic and a fresh Pacific), and a change in the meridional extent of ‘Africa’. For each case a new spinup run was performed and from the spinup state a point on each of the C–SS and NS–IC branches was found by temporarily adding freshwater perturbations in the North Atlantic and/or North Pacific as by Marotzke and Willebrand [1991]. For each part of the branch, at least six simulations were performed to equilibrium, with different values of γp to determine the sensitivity of the equilibrium state versus γp (see Table 1). Although the pattern and strength of the Atlantic and Pacific MOC in the five cases are very different (for example in case SA_EL, an Antarctic Bottom Water cell is present underneath the NADW cell) all four states are found in the simulations and the branches are organized in the way of Figures 3a3b. Of course, the bifurcation points are at different values of γp and the hysteresis widths are different for the different cases, but the shape is uniform.

Table 1. Overview of the Five Simulation Casesa
NameConfigurationKV (m2s)WindT, S spinup b.c.γI+γIγII+γII
  • a

    Case SC is our standard case with results in Figures 1 and 2. In case SC_LKV, the vertical diffusivity is increased. In the other three simulations, the geometry is slightly different: ‘Africa’ reaches only to 36°S, and a sill of 2100 m height is present in ‘Drake Passage’, reducing the water depth to 2400 m. Case SA is similar to SC in this new geometry, in case SA_ERA the wind stress curl is different and in case SA_EL, the ocean model is coupled to an atmospheric energy balance model (EBM) and a more realistic restoring salinity profile was prescribed in the spinup. The mean ACC transport is about 60 Sv for all cases except SA_ERA (90 Sv) and SA_EL (145 Sv). The transport is typically higher in the SS-state and lower in the NS-state. Values for γ are in Sv.

SCSymmetric0.5 · 10−4Bryancosine restoring0.061−0.0350.028−0.023
SC_LKVSymmetric2.3 · 10−4Bryancosine restoring0.11−0.120.061−0.062
SAShort Africa0.5 · 10−4Bryancosine restoring0.07−0.040.04−0.02
SA_ERAShort Africa0.5 · 10−4ERA40cosine restoring0.067−0.0660.038−0.039
SA_ELShort Africa0.5 · 10−4BryanEBM, Levitus-like S0.12+0.045∼0.06+0.04

[18] In four of the cases in Table 1, the C, SS, NS and IC states are present for γp = 0, but in the most realistic case, SA_EL, the values of γp± are all positive. The C-state is, for example, stabilized by the response of the atmosphere to SST anomalies (represented in an EBM). However, even in the SA_EL case at γp = 0, a transition from the C-state to the NS-state remains possible.

4. Discussion and Conclusions

[19] The new element in this study is the calculation of the sensitivity of steady solutions of the circulation in an idealized two-basin configuration to the parameter γp, the strength of an anomalous freshwater flux in the northern North Atlantic. The main result is that, under a range of boundary conditions, the steady solutions are connected in two separate branches, where the C-state is connected with the SS-state and the NS-state with the IC-state. This result has important implications for the existence of a multiple equilibrium regime. Even if the hysteresis behavior associated with the salt advection feedback disappears, and the branches do not disappear, there will always exist at least two solutions for every value for the freshwater forcing, i.e., the multiple equilibrium regime of the ocean circulation is robust.

[20] Many model studies have concentrated on multiple equilibria associated with the C–SS branch, starting with a C-state and hosing in the North Atlantic. There is no model study where a systematic search for the other branch of solutions has been performed. In the model study of Von der Heydt and Dijkstra [2008], the NS-state could still be found with one basin reaching to 76N and the other to 64N. In the study of Saenko et al. [2004], the IC-state was found by prescribing excess evaporation in the Pacific and excess precipitation in the Atlantic. Their model simulations indicate that equilibrium states with northern Pacific sinking may exist under realistic geometry.

[21] The robust multiple equilibrium regime as found here will always provide a possibility for rapid transitions, even if transitions between the C-state and SS-state are inhibited by atmospheric feedbacks or through small vertical mixing. With respect to the Dansgaard-Oeschger oscillations, the MOC transition would then not be from the C-state to the SS-state but from the C-state to the IC-state, the difference being the changes in the Pacific. This may explain evidence from proxy data that indicate warmer sea surface temperatures in the North Pacific during Dansgaard-Oeschger events (while North Atlantic sea surface temperatures were colder [Kiefer et al., 2001]). These transitions may also play a role in the future development of the global MOC and must be taken into account when trying to assess the risk of a change in ocean circulation due to global warming.


[22] The comments of two anonymous reviewers were greatly appreciated. A. von der Heydt acknowledges personal support through a VENI grant by the Netherlands Organization for Scientific Research (NWO). The authors would like to thank M. den Toom for his comments on the manuscript.