Recent model results have suggested that there may be a scalar indicator Σ monitoring whether the Atlantic meridional overturning circulation (MOC) is in a multiple equilibrium regime. The quantity Σ is based on the net freshwater transport by the MOC into the Atlantic basin. It changes sign as soon as the steady Atlantic MOC enters the multiple equilibrium regime because of an increased freshwater input in the northern North Atlantic. This paper addresses the issue of why the sign of Σ is such a good indicator for the multiple equilibrium regime. Changes in the Atlantic freshwater budget over a complete bifurcation diagram and in finite amplitude perturbation experiments are analyzed in a global ocean circulation model. The authors show that the net anomalous freshwater transport into or out of the Atlantic, resulting from the interactions of the velocity perturbations and salinity background field, is coupled to the background (steady state) state freshwater budget and hence to Σ. The sign of Σ precisely shows whether this net anomalous freshwater transport is stabilizing or destabilizing the MOC. Therefore, it can indicate whether the MOC is in a single or multiple equilibrium regime.
Over the last few decades, serious concerns have been raised about possible anthropogenic changes in the ocean circulation and the resulting consequences for climate (Rahmstorf 2003). When the ocean velocity field is integrated in an east–west direction across an ocean basin, the resulting flow is referred to as the meridional overturning circulation (MOC). In the Atlantic Ocean, there is a net northward flow of surface and bottom waters, which is compensated for by a southward motion at intermediate depths. The Atlantic MOC transports a substantial amount (about 1.5 PW at 25°N) of heat northward. A reduction (collapse) of the MOC may therefore lead to serious climate change. Regions around the North Atlantic would experience significant cooling, and other parts of the world would also be affected (Vellinga et al. 2002). Paleoclimatic data suggest that such changes in the MOC may have occurred in the past (Clark et al. 2002; Rahmstorf 2002).
Although wind-generated motions influence the shallow part of the MOC, the deep overturning is dominantly driven by interior turbulent mixing of heat and salt, and it is therefore referred to as the thermohaline circulation (THC). The spatial structure of the THC is predominantly determined by the ocean’s density field (Wunsch 2002; Kuhlbrodt et al. 2007). Because the ocean flow itself influences the density field by advection, the THC is by implication a nonlinear phenomenon. Furthermore, the distribution and intensity of surface fluxes of heat and freshwater act to modulate the flow pattern and its temporal behavior.
The nonlinear nature of the THC may result in the existence of multiple equilibria (ME) for a given set of boundary conditions. Under present-day forcing conditions, two stable equilibria appear possible in the models: one usually referred to as the “on” state (or conveyor state) and the other as the “off” state (or collapsed state). Traditionally, large and abrupt changes in the Atlantic MOC have been related to the existence of multiple equilibria. These are a robust feature in models ranging from simple box ocean-only models (Stommel 1961) to the so-called earth system models of intermediate complexity (EMIC; Rahmstorf et al. 2005) and have also been found in early coupled climate models (Manabe and Stouffer 1988). It is, however, not clear whether this is a common characteristic of state-of-the-art coupled (ocean–atmosphere) general circulation models (CGCMs) simulating present-day climate. In analyzing nine different CGCM simulations of the response to the Intergovernmental Panel on Climate Change (IPCC) Special Report on Emissions Scenarios (SRES) A1B scenario of future CO2 increase, Schmittner et al. (2005), for example, found that none of these models predicted an abrupt change of the MOC.
As it is impossible at the moment to determine whether these CGCMs are in a multiple equilibrium regime, it is important to have (preferably scalar) indicators for the presence of such a regime. Using a simple box model, it was already pointed out by Rahmstorf (1996) that the multiple equilibrium regime may be related to the net freshwater budget over the Atlantic basin. This issue was revisited by De Vries and Weber (2005), who showed (using an EMIC) that the sign of the net freshwater export by the Atlantic MOC, indicated by Mov(θ), near the latitude θ = 35°S in the Atlantic may be controlling whether, in addition to the on state, a stable off state exists.
In Dijkstra (2007), it was shown that a measure of the divergence of the freshwater transport Σ of the Atlantic MOC over the Atlantic basin is a good indicator of the multiple equilibrium regime. When the freshwater transport at the northern boundary is neglected, Σ reduces to the indicator in De Vries and Weber (2005), because
where θn and θs are the northern and southern latitudes of the Atlantic domain. It appears that Σ changes sign (from positive to negative) when the Atlantic MOC on state enters the multiple equilibrium regime as the freshwater input in the northern North Atlantic is increased. There is a slight sensitivity to the choice of the southern boundary, but θs = 35°S seemingly is a “best” choice because the tip of Africa marks the southern boundary of the Atlantic basin. The sensitivity of Σ to the northern-latitude θn is relatively small as long as it is north of 60°N.
Although the indicator Σ seems able to serve as an indicator for the multiple equilibria regime and descriptive explanations were given in De Vries and Weber (2005) and Dijkstra (2007), there is a need for a better physical basis of this indicator that can explain why Σ crosses zero when entering the multiple equilibrium regime and why the southern boundary of 35°S is so important. The main purpose of this paper is to provide strong support that Σ is a correct indicator for distinguishing different MOC regimes.
The advantage of the approach and methodology followed in Dijkstra (2007) is that full bifurcation diagrams are available and that steady states on the on and off branches satisfy integrated (freshwater) balances with a relative error smaller than 0.1%. In particular, the availability of the unstable steady state—which was shown in Dijkstra et al. (2004) to separate the regions (the so-called attraction basins) of initial conditions going to either on or off states—enables targeted studies on the development of finite amplitude perturbations.
To be self-contained here, we start in section 2 with a very brief summary of the main results in Dijkstra (2007). In section 3, we analyze details of the Atlantic freshwater balance and provide a physical description of the processes that are involved in the changes of this balance along with a full bifurcation diagram. In section 4, the transient development of specific finite amplitude perturbations is studied with a focus on the processes determining MOC recovery and collapse. The analyses in sections 3 and 4 provide a physical explanation of why the sign of Σ is a good indicator for the multiple equilibrium regime. The results are summarized and discussed in section 5. Here, we also address the usefulness of Σ in CGCMs and observations and whether it is, in principle, possible to use the indicator to assess the stability regime of the present-day MOC.
2. The indicator Σ in a global ocean model
Bifurcation diagrams were computed in Dijkstra (2007) using a fully implicit global ocean model. Just to stress that this is no “toy” model, we mention that the governing equations of this ocean model are the hydrostatic, primitive equations in spherical coordinates on a global domain, which includes full continental geometry as well as bottom topography (Dijkstra and Weijer 2005). Horizontal and vertical mixing of momentum and heat and salt is represented by a Laplacian formulation, with prescribed constant “eddy” viscosities AH and AV and vertically dependent “eddy” diffusivities KH(z) and KV(z); see Dijkstra (2007). The ocean flow is forced by the annual-mean wind stress as given in Trenberth et al. (1989). The upper ocean is coupled to a simple energy balance atmospheric model (see the appendix in Dijkstra and Weijer 2005) in which only the heat transport is modeled (no moisture transport). The freshwater flux is prescribed, and the model has no sea ice component.
Starting from the steady-state solution determined under restoring conditions for the surface salinity field (Levitus 1994), steady states were calculated in Dijkstra (2007) versus a parameter γp under the freshwater flux FS = P − E with
where FSp is unity only in the domain (ϕ, θ) ∈ [60°W, 24°W] × [54°N, 66°N] and zero outside. Furthermore, FSe is freshwater flux diagnosed from the solution at γp = 0. The quantity Q is determined such that
where Soa is the total ocean surface and r0 is the radius of the earth, to ensure a net zero freshwater flux over the total ocean surface.
For the case where the vertical diffusivity KV increases from 1.2 × 10−4 m2 s−1 at the surface to 5.3 × 10−4 m2 s−1 near the bottom of the flow domain, the bifurcation diagram, where the maximum of the Atlantic MOC ψA is plotted versus γp, is shown in Fig. 1a. For clarity, the stable (drawn) branches are indicated with the on branch and with the off branch. The dashed branch represents unstable steady states. It is clear that the saddle-node bifurcations L− and L+ limit the multiple equilibria regime; we indicate the values of γp at these points with γL− and γL+, respectively.
Solutions of the Atlantic MOC along several labeled points of the bifurcation diagram in Fig. 1a are plotted in Figs. 2a–f. For small γp, the solution of the Atlantic MOC (the on state) is near to the unperturbed state, with strong northern sinking and no bottom water of southern origin (Fig. 2a). Along the bifurcation diagram the strength of the Atlantic MOC decreases (Fig. 2b), with γp increasing until the saddle-node bifurcation at γL+. In the pattern of the Atlantic MOC, the return flow shallows (Fig. 2c) and the deep flow from the south strengthens. Once on the unstable branch of steady states from L+ to L−, this southern sinking component increases, leading eventually to the stable off state (Figs. 2d–f) for values of γp > γL− on the lower (drawn) branch in Fig. 1. For values of γp between γL− and γL+, the MOC is in the ME regime. For γp < γL− and γp > γL+, only one stable steady state exists and the MOC is in the single equilibrium (SE) regime.
The indicator Σ in Dijkstra (2007) is based on the freshwater budget that arises when the stationary salinity equation is integrated over a volume V of the Atlantic Ocean bounded by the latitudes θs and θn. This integrated salinity budget can be written as
In (4), the left-hand side is the freshwater volume [in Sv (1 Sv ≡ 106 m3 s−1)] going through the ocean–atmosphere surface SAtl and the right-hand side is the net freshwater transport (in Sv) through the lateral boundaries at θn and θs. The salt flux Φ is defined as
where Sθ is the (zonal vertical) ocean section at latitude θ.
Here, η and 〈F〉 (for a function F) are given through
with υ′ = υ − 〈υ〉 and S′ = S − 〈S〉. The indicator Σ(θs, θn) is plotted (along the on branch of Fig. 1a) in Fig. 1b for θn = 60°N (in the sinking region) and θs = 35°S (at the southern tip of Africa). Here, Σ changes sign (from positive to negative) just as the multiple equilibrium regime is approached (near the saddle-node bifurcation L−). In the SE regime, the Atlantic MOC exports salt (Σ(θs, θn) > 0), whereas in the ME regime, it is exporting freshwater (Σ(θs, θn) < 0).
It is remarkable that the indicator Σ, which is evaluated on the on branch, is able to detect the presence of the saddle-node bifurcation L−, which is located on the off branch. Certainly, the on states for γp > γL− are linearly stable (i.e., very small perturbations on these states will decay), but they are susceptible to finite amplitude instabilities. As was shown in Dijkstra et al. (2004), the attraction domains of the on state and off state seem to be bounded by the unstable state, and therefore it requires a finite amplitude perturbation that crosses the unstable branch to make a transition from the on to the off state. But how would Σ provide any information on the presence of the multiple equilibrium regime and hence the behavior of finite amplitude perturbations? In the next sections, we will investigate this systematically by analyzing details in the freshwater balances along both the on and off branches (section 3) and by considering the transient development of finite amplitude disturbances on the on state (section 4).
3. Freshwater balance of the equilibria
When we represent the diffusive fluxes as
and use the notation
the total freshwater balance (4) can be written as
The terms of this equation are shown in Fig. 3a (southern boundary) and Fig. 3b (northern boundary). As was shown in Dijkstra (2007), the freshwater balance (10) is satisfied accurately in this model. We see that the terms at the northern boundary (Fig. 3b) are smaller than the corresponding terms at the southern boundary (Fig. 3a), also the diffusive terms are not small with respect to the other terms and that many parts of the curves are characterized by near-straight lines. In sections 3a and 3b, we study the latitudinal dependence of Mov and Maz on the off and on branches and the change of Mov(θs) along the on branch.
a. Behavior of Mov and Maz on the off branch
The function Maz(θ) is plotted in Fig. 4a. The azonal transport south of θ = 10°S is driven by the subtropical gyre in the South Atlantic, with southward (northward) flow in the western (eastern) part of the basin. The salinity field S′ (not shown) at the off solution in Fig. 2f for θs = 35°S is negative in the western part of the basin and positive in the eastern part, the value of Maz is negative and the gyres transport freshwater out of the basin. With decreasing γp, the salinity field S′ changes to become more positive at the western side of the basin and more negative at the eastern part. Hence, the value of Maz becomes less negative. The changes in S′ go down to about 1000 m while in deeper levels S′ is hardly affected by the addition of salt in the northern North Atlantic. It is interesting that Maz becomes zero near θ = 35°S at the saddle-node bifurcation L− (very close to the solution in Fig. 2d). This indicates that the saddle-node bifurcation, which is a signal of “nonexistence” of a steady solution, is connected to the gyre-driven freshwater transport over the southern boundary.
The function Mov(θ) is plotted in Fig. 4b and shows that the off state in the Atlantic MOC is exporting salt south of 10°S. For the solution in Fig. 2f, 〈υ〉 is strongly negative in the upper ocean and the mean salinity 〈S〉 − S0 > 0, which leads to a positive Mov(θ). When γp decreases (effectively adding salt in the northern North Atlantic), the mean salinity at 35°S increases over the whole depth and the mean surface velocity decreases slightly, leading to a slight increase in Mov(θ) at θ = 35°S (Fig. 4b).
b. Behavior of Mov and Maz on the on branch
The latitudinal dependence of Maz is plotted in Fig. 5a for the solutions in Figs. 2a–c. South of θ = 20°S, the gyre-driven freshwater transport is positive—that is, the gyres transport salt into the basin. Again the velocity distributions υ′ are fairly similar to those along the off branch (because the winds are the same); however, now the salinity distributions S′ are quite different, with positive values in the western part of the basin and negative ones at the eastern part of the basin. With increasing γp (adding freshwater in the northern North Atlantic), the fields S′ hardly change and hence Maz south of 20°S remains constant with γp. The reason is that the freshwater anomaly is transported southward at depth by the on state and therefore this does not affect the azonal transport.
In Fig. 5b the function Mov(θ) is positive at the solution in Fig. 2a, which is south of the equator as the ocean velocity at depth 〈υ〉 is negative where the salinity 〈S〉 is largest. Hence, the meridional overturning transports salt out of the basin. When γp is increased, one sees the decrease in mean salinity at depth, whereas the salinity at the surface increases. This implies that Mov must decrease with increasing γp as is seen in Fig. 2a.
Actually, it appears that there is linear behavior of Mov with γp on the on branch, resulting from the specific choice of the forcing. When we differentiate the forcing (2) with respect to γp, we find a positive constant (α), and hence
Using the notation Σaz = Maz(θs) − Maz(θn) and Σd = Md(θs) − Md(θn), we can write the Eq. (10) as
We argued above that Σaz does not change much along the on branch and it appears from Fig. 3 that the same holds for Σd. Using this result, (11) and the fact that for the reference solution, we find that
This indicates that Σ has a zero on the on branch with increasing γp (as α > 0) when Σ0/α < γL+. Physically, this simply means that the input of freshwater in the northern North Atlantic eventually leads to the export of freshwater by the Atlantic meridional overturning circulation.
Although from (13) it is clear that Σ decreases along the on branch, it is not at all guaranteed that it is changing sign near γL−. Therefore, why are the sign changes of the quantities Σ and the zonal salinity difference on the on branch associated with the existence (nonexistence) boundary of the off branch? This is a highly nonlinear problem because the issue is whether a finite amplitude perturbation to the on state is able to recover. In the unique regime, one can put any perturbation on the MOC but it will always recover. In the multiple equilibrium regime, there always exists a finite amplitude perturbation that is able (i.e., which is large enough) to induce a transition to the (stable) off state for the same value of γp. Hence, an analysis of the development of finite amplitude perturbations in both regimes is required and this is the focus of the next section.
4. Development of finite amplitude perturbations
In this section, we investigate why the on state of the MOC always recovers in the SE regime but that in the ME regime the off state can be reached. Thereto, we apply specific finite amplitude freshwater perturbations by considering the so-called thermohaline pulse response problem (similar to the approach in Dijkstra et al. 2004), where the freshwater perturbation is switched on instantly and after a certain time tm is suddenly reduced to zero. If we represent the time dependence as a block function B(t; tm), then the total freshwater flux can be written as
where H is the Heaviside function and Q(t) is determined from the condition that the surface integrated freshwater flux is zero. The value of γp refers to the value of γp at a steady state and we use Δγp as a perturbation of it. To facilitate the interpretation of the results, we choose to make the pattern of the freshwater perturbation associated with Δγp the same as that was computed for the bifurcation diagram (in γp).
a. Transient solutions
Now, let us assume we start from a steady-state solution determined for a certain γp on the on branch. When Δγp = 0, we will remain at that steady state. However, when Δγp is so large that
the solution will be attracted to the off state for γp + Δγp because this is the only steady state for this value of γp. After a time tm, the trajectory will reach a certain state and when the anomalous forcing is then suddenly released, the trajectory will be attracted to one of the stable steady states that are present for γp = γp. In the SE regime there is only one steady state at γp and all trajectories (for all values of tm) will be attracted to the on state, the MOC always recovers. In the ME regime, however, the MOC will recover for a small tm, whereas the off state will be reached for a large tm. In Dijkstra et al. (2004), it was shown that the critical time, that is t*m, is determined by the time when the unstable steady state is crossed.
We take a value γpS = 0.083 Sv in the SE regime and a value γpM = 0.166 Sv in the ME regime. Plots of the meridional velocity field and salinity field at 35°S are presented in Fig. 6 for both steady states. The meridional velocity fields look very similar for both cases, although the MOC is slightly shallower in Fig. 6c than in Fig. 6a. The salinity fields differ markedly, with a much larger salinity contrast between surface and deep ocean in the ME regime. This reflects the change in freshwater transport by the MOC over this section, which is positive (freshwater transport into the Atlantic basin) in the SE regime and negative (freshwater transport out of the basin) in the ME regime.
To analyze the development for a relatively small Δγp, we show results for both cases when the states in Fig. 6 undergo a change of ΔγpS = ΔγpM = 0.02 Sv for tm = 200 yr. The maximum of the Atlantic MOC (ψA) is plotted versus time in Fig. 7a and the drawn curves (black, SE regime; gray, ME regime) show the long time development under this change in freshwater flux. After about 200 years, the MOC strength has decreased less than 1 Sv for both cases. When the anomalous forcing is released at tm = 20 and tm = 100 yr (dashed curves), the original equilibria are obtained showing that the steady states are indeed linearly stable.
Subsequently, we take ΔγpS = 0.197 Sv and ΔγpM = 0.114 Sv, respectively, such that for tm → ∞ the same off state at γp = 0.280 Sv is reached (cf. Fig. 1). This is confirmed by the drawn curves in Fig. 7b. The dashed curves in Fig. 7b again show the development of the MOC strength, when the anomalous forcing is turned to zero (i.e., γp = γp) after a time tm. It is indeed seen that for every tm the trajectories in the SE regime (black curves) eventually end up on the on state. However, for the ME regime (gray curves) we see that for tm < t*m the Atlantic MOC recovers but that for tm > t*m the Atlantic MOC approaches the off state; here is the critical time t*m ≈ 400 yr.
We now turn to a physical explanation of the results in Fig. 7. Our task is to explain why 1) the MOC recovers in the ME regime for tm < t*m, and collapses for tm > t*m and 2) the MOC in the SE regime, when subjected to a similar perturbation, always recovers. Our ingredients are that Σ ≈ Mov(θs) < 0 for the steady on state in the ME regime and Σ ≈ Mov(θs) > 0 in the SE regime.
The existence of multiple equilibria in the MOC is caused by the salt–advection feedback, which is present in its simplest form in the Stommel (1961) box model. A freshwater perturbation on the on state of the MOC, for example, causes a weakening of the MOC and hence leads to a smaller meridional advective salt transport, which amplifies the original perturbation. As the advective meridional transport is crucial, this feedback has also been identified in two- and three-dimensional ocean models (Walin 1985; Dijkstra and Molemaker 1997; Kuhlbrodt et al. 2007).
If a perturbation (υ̃, S̃) is assumed on a mean state (v, S), the development of the salinity perturbation is determined [considering only meridional advective transport and hence neglecting all other (zonal/vertical, advection, and diffusive) transports] by the equation
When we integrate (16) over the Atlantic basin (longitude ϕ from coast to coast, z from bottom to surface, and latitude θ from θn to θs), then the development of the Atlantic basin averaged salinity anomaly is governed by the equation
where the fluxes through the northern boundary are neglected. This relation shows that the growth of the salinity anomaly in the Atlantic basin is related to the anomalous meridional salt transport terms integrated over the southern boundary.
In the following, we will investigate the time development of the different terms in the right-hand side of (17) along the trajectories in Fig. 7a. We will call the contribution of a certain term in the (17) “stabilizing” (destabilizing) when it increases (decreases) the salt content of the Atlantic basin and hence strengthens (weakens) the MOC. The terms are plotted in Fig. 8a for Δγp = 0.02 Sv. Note that because this value of Δγp is relatively small, both linear interaction terms are of larger magnitude than the nonlinear interaction term υ̃S̃. The S̃ term is positive for both SE and ME regimes and this transport is stabilizing the MOC. The nonlinear interaction term is negative (making the Atlantic fresher) and hence is destabilizing the MOC for both regimes. The central result is that the υ̃S term is stabilizing in the SE regime, whereas it is destabilizing for the ME regime.
To understand the sign of the terms in (17), plots of υ̃ and S̃ are shown in Fig. 9 for the solutions at year 200 (endpoints of drawn curves in Fig. 7a). For both ME and SE regimes, the S̃ field is positive at the surface and negative at depth, which is a typical response to the slowdown of the MOC. As the term is largest at the surface (cf. Fig. 6a), the anomalous salt transport associated with the term S̃ is into the basin and hence is stabilizing. As υ̃ is negative at the surface and positive at depth (Figs. 9a,c), the nonlinear interaction term υ̃S̃ always leads to salt export out of the Atlantic basin and hence this term is destabilizing.
The υ̃S field is plotted for the SE and ME regimes (again at year 200) in the Figs. 10a,b, respectively, and their difference is shown in Fig. 10c. The change in the steady-state salinity field with γp turns out to be crucial for the sign of the υ̃S term. The salinity S is smaller in the deep ocean (below ∼3 km) for the solution in the ME regime than for the SE regime. Hence, the transport of salt because of the υ̃S term is out of the Atlantic basin in the ME regime and into the Atlantic basin for the SE regime; hence, this term is stabilizing in the SE regime and destabilizing in the ME regime.
We can make the link of the section integral over υ̃S and Mov(θs) more explicit by looking at the relation between the profiles of 〈〉 (Fig. 11a) and 〈υ̃〉 (plotted for different times in Fig. 11b). By inspection, it appears that for both regimes it is a reasonable assumption that 〈υ̃〉 ≈ −ε(t)〈〉. Physically, this makes sense because the MOC decreases as a result of the imposed change in freshwater flux in the northern North Atlantic, and the overall spatial pattern of the MOC remains the same for small Δγp. In other words, there is a southward velocity perturbation in the upper layers and a northward velocity perturbation at depth.
When we now decompose υ̃ = 〈υ̃〉 + υ̃′, S = 〈S〉 + S′, then the section integral can be written as
Using 〈υ̃〉 ≈ −ε(t)〈〉, the first term in the right-hand side is proportional to Mov(θs). The second integral is dependent on the azonal components of υ̃ and S. For both SE and ME cases in Fig. 8a, the three integrals in (18) are plotted in Fig. 8b. For the ME regime (gray curves) the integral involving the υ̃′S′ term is much smaller than that involving the 〈υ̃〉〈S〉 term. Hence, the sign of the section integral of υ̃S is the same as that of Mov at 35°S. For the SE case, the integrals in the right-hand side of (18) are of the same order of magnitude, so here the relation between the sign of the υ̃S integral and Mov at 35°S is less obvious. However, from Fig. 8b, it is clear that the sign of the υ̃S term is most influenced by the changes in the sign of the 〈υ̃〉〈S〉 term.
Furthermore, from Fig. 8a it can be seen that the stabilizing term S̃ decreases and the destabilizing term υ̃S̃ becomes more negative (more destabilizing) with increasing γp. This demonstrates that when the on state of the MOC enters the ME regime, the salt transport near the southern boundary induced by changes in the northern North Atlantic freshwater flux tends to be more destabilizing. In the previous analysis, the salt transport at the northern boundary θn is assumed to be much smaller than that at the southern boundary θs but it can easily be taken into account in (17). Following (18), the integral of the term υ̃S over the northern and southern boundary can then be related to Σ. Hence, assuming a dominance of the meridional advective transport terms, the sign change in the υ̃S term, as directly linked to Σ, is the crucial effect determining whether the MOC is in the SE or ME regime. When the perturbations become very large, as in the results of Fig. 7b, the pattern of the MOC changes drastically with time; eventually, the magnitudes of all terms in (17) along the drawn trajectories in Fig. 7b will determine whether the MOC will collapse or recover when the forcing is released.
c. Further analysis in a box model
To substantiate the reasoning in the last section, we analyze the same connection between properties of the steady states and the transient development of perturbations in the Rahmstorf (1996) model. In this box model, only the meridional advective processes are taken into account. The model consists of two equatorial boxes connected to two polar boxes that are arranged as depicted in Fig. 12a. It is designed to mimic the pattern of the Atlantic circulation, with deep convection at subpolar latitudes (box 2), while low-latitude upwelling is limited (hence no connection between boxes 3 and 4) but instead takes place outside the domain of the Atlantic (box 1). The volume transport between the boxes has strength m and is proportional to the density difference ρ2 − ρ1. Assuming a linear equation of state, the flow strength is diagnosed from
where k, β, and α are constants and Ti and Si are the temperature and salinity in box i.
Mixed boundary conditions are imposed where temperatures will be simply prescribed and the surface freshwater forcing consists of two independent active fluxes, F1 and F2. The adjective “active” here means that not only atmospheric vapor fluxes are captured in the Fi, but also the salt transport by the wind-driven gyres as well as (subgrid scale) diffusion. The fluxes Fi are converted to equivalent salt transports by multiplication with −S0, a fixed reference salinity, and will be assumed constant. The salt conservation equation for the southern box is then given by
Here, V1 is the fixed volume of box 1. The equations for the other boxes follow in a similar fashion (Rahmstorf 1996).
For m ≥ 0, the steady-state salinity in box 2 is equal to that in box 4, S2 = S4. The combination of Eqs. (19) and (20) then yields the steady-state flow strength as function of the temperature contrast and the freshwater forcing:
For m ≤ 0, the solution is similar to (22) but with −F1 replaced by +F2.
We will only consider the case for which T1 > T2. For m ≥ 0, solutions exist for , where , the critical freshwater flux at the saddle-node bifurcation. We are primarily interested in the transition associated with the sign change of F1, which Rahmstorf (1996) identified with the transition from the single equilibrium to the multiple equilibrium regime. The bifurcation diagram for this case is plotted in Fig. 12b, showing all steady solutions for m ≥ 0. Note that for m ≤ 0 solutions exist for F2 ≥ 0, independent of F1. However, this inverse circulation implies that all upwelling takes place in the northern box, despite the absence of a physical mechanism to limit dominant upward motion to the northern high latitudes. Hence, the applicability of the box model ends when the flow reverses sign. For simplicity we define any situation for which m < 0 as the collapsed state, and we take F2 > 0 to ensure that the trajectories are attracted to this state when m becomes negative.
Suppose we perturb the equilibrium given by Eq. (21) by applying an anomalous freshwater flux of strength γp to box 2 for a time tm. Simultaneously, an equal amount of water is extracted from boxes 1 and 3, in a ratio defined by their respective surface areas A1 and A3, in order to preserve salinity. This will change the salinity contrast between boxes 1 and 2 such that the flow is weakened, but γp and tm are chosen such that m does not reverse sign during the application of the perturbation. In the first two experiments, F1 is taken slightly negative (F1 = −10−3 Sv), and in the latter two slightly positive (F1 = 10−3 Sv). For each of the two cases we apply a perturbation γp = 0.25 Sv for a period of tm = 20 yr and for a period of tm = 23 yr. The time scale can be easily increased by changing the parameters as summarized in Table 1 and is not essential here. Figure 12c shows that the system recovers after both perturbations when F1 < 0, which is in the SE regime. On the other hand, when F1 > 0, the system recovers for tm = 20 yr but collapses for tm = 23 yr.
The fluxes and salinities may be written as the sum of a mean state (m, Si) and a perturbation (m̃, S̃i). Since total salinity is conserved, the evolution of the Atlantic salinity is proportional to −dS̃1/dt, which for positive m is given by
which has a similar interpretation as (17) for the global ocean model. The time evolution of these three terms is shown in Fig. 12d for the case tm = 23 yr, both for the system residing in the single (black curves) and in the multiple equilibrium regime (gray curves). Similar to the results in the global ocean model, the term m(S̃4 − S̃1) is stabilizing (it leads to an increase in Atlantic salinity) and the term m̃(S̃4 − S̃1) is destabilizing. The two terms m(S̃4 − S̃1) and m̃(S̃4 − S̃1) are very similar in both ME and SE regimes.
The change from SE to ME regime is, just as in the global ocean model, also related to a sign change in the term m̃(S4 − S1). The nice element in the box model is that the steady-state salinity contrast is given by
and hence its sign is directly coupled to that of F1. Because m > 0 and m̃ < 0 in both regimes, the term m̃(S4 − S1) will switch sign when F1 (in the box model the equivalent to Mov in the global model) switches sign, which is exactly at the boundary between SE and ME regime (Fig. 12b).
5. Summary and discussion
Using a fully implicit global ocean model coupled to an energy balance atmosphere model, we revisited the problem of the characterization of the multiple equilibrium (ME) regime of the Atlantic MOC through an indicator Σ given by (1). Our ocean model certainly has many deficiencies (Dijkstra and Weijer 2005), such that western boundary currents are very broad, the wind-driven gyre flows are relatively weak, and eddy processes are completely ignored. The discussion below should be considered with these limitations in mind. The main advantages of the model approach here is that 1) full bifurcation diagrams can be computed versus the freshwater flux parameter γp and that 2) the freshwater balances over the Atlantic basin are accurately satisfied. A detailed connection can therefore be made between the position of the saddle-node bifurcations bounding the hysteresis regime of the Atlantic MOC and changes in the Atlantic freshwater budget.
Our aim was to provide a better physical picture of why the indicator Σ has a zero on the on branch for the value of γL−, where we find the saddle-node bifurcation L− on the off branch. This is a nonlinear problem because entering the ME regime from the SE regime when γp is increased is related to a change in the development of finite amplitude perturbations. The on state is stable to small perturbations in both SE and ME regime. Our approach was to add controlled perturbations (an anomalous freshwater flux) for a time tm and then analyze the differences in the evolution of the Atlantic freshwater budget between both ME and SE regimes.
Equation (17) clearly illustrates the importance of the freshwater fluxes at the southern boundary. Interactions between the perturbed flow and the background steady state, as well as nonlinear-driven freshwater transport, directly contribute to the tendency of the volume-averaged salinity perturbation in the Atlantic basin. At the value of γp where the steady-state value of Σ ≈ Mov(35°S) changes sign, the anomalous salinity transport associated with the υ̃S term also changes sign. As the steady-state meridional velocity does not change much with γp on the on branch, the sign of Σ is closely coupled to that of the steady-state salinity field at 35°S. With increasing γp, the salinity at depth decreases (there is freshwater export in the ME regime) and hence the υ̃S term becomes more destabilizing (again because υ̃ is very similar for the SE and ME regime).
This connection between properties of the steady states and the processes controlling the evolution of perturbations was most clearly illustrated with the box-model analysis in section 5c. When the MOC decreases because of the change in freshwater flux in the northern box, the interaction of the velocity perturbation and the steady-state salinity field transports salt into the Atlantic basin in the SE regime and freshwater in the ME regime. While in the box model the meridional advection terms in the salinity equation are the only relevant process, we have assumed that these terms are dominant in the global ocean model [as reflected in the Eq. (17)].
Climate models that were integrated as part of the SRES A1B scenario of the IPCC do not show any sign of abrupt change in the Atlantic MOC (Schmittner et al. 2005), although this does not prove that the models do not have a ME regime. Model intercomparison studies show no systematic differences in THC behavior and climate response between EMICs and atmosphere–ocean GCMs (CGCMs) (Gregory et al. 2005; Stouffer et al. 2006). The ME regime is present in simple coupled models (De Vries and Weber 2005; Rahmstorf et al. 2005; Weber et al. 2007) and in some more complex coupled models (Manabe and Stouffer 1988). Atmospheric feedbacks may indeed change the crucial role of the salt–advection feedback by affecting the east–west salinity difference at 35°S in response to a temporary decrease in the Atlantic MOC. However, when they would completely remove the ME regime, it becomes more difficult to explain the paleoclimatic record (Clark et al. 2002).
Before discussing the applicability of Σ as an indicator of multiple equilibria in GCMs and in observations, we mention explicitly that the model used here has a relatively large vertical diffusivity and hence the MOC is in the “mixing” regime. When KV is decreased, several cases can be distinguished: 1) the multiple equilibria disappear, 2) there are still multiple equilibria but because the MOC is more “wind driven” the quantity Σ would not be a good indicator, and 3) the multiple equilibria remain and Σ is still a good indicator. Support for 3) is provided by the results in Weber et al. (2007), but a detailed study of what happens with the bifurcation diagrams when decreasing KV would require much more work. It would also require a different model, as low vertical diffusivity regimes cannot be reached with the model used in this paper.
Calculating Σ to investigate multistability in coupled GCMs is rather straightforward and has the advantage of inferring the stability regime from the equilibrium solution, without having to perform hosing experiments. When the MOC changes in these models, the freshwater flux field changes in a complicated way and there are also changes in the wind field and the heat flux field. However, based on the increased knowledge of the physics behind the indicator Σ, we think that it is a relevant diagnostic for the stability properties of the MOC in coupled GCMs as it is related to (intrinsic) advective processes in the Atlantic freshwater budget.
Yin and Stouffer (2007) compared the development of the MOC to freshwater perturbations in two different versions of the Geophysical Fluid Dynamics Laboratory (GFDL) model. They attributed the different behavior of the MOC in these models to the different character of the off states and to different atmospheric feedbacks. However, from their analysis it can be deduced that Σ is negative in the model where the off state appears to be stable and is characterized by a reversed MOC. In the other model, Σ is positive and a weakened on state is found, which evolves back to the on-state MOC when the hosing stops, regardless of the atmospheric feedbacks. This behavior is consistent with the hypothesis that the sign of Σ determines the regime of the MOC.
It is interesting to see that the result on Σ strongly depends on the depth of the zero contour of the MOC. When the position of the zero contour is at a shallower depth, Σ tends to be more negative and hence there is a tendency towards the multiple equilibrium regime. This is actually seen in GCMs where the MOC shoals under increasing greenhouse gas concentrations (Stouffer et al. 2006). Also, in simulations of glacial climates, there are indications for a shoaling of the MOC (Weber et al. 2007), again pointing to a MOC that is more likely to be in a multiple equilibrium regime.
To address the important question of where the real ocean resides, estimates of Σ can be made. Using data from an inversion of the World Ocean Circulation Experiment (WOCE) data by Holfort (1994), Weijer et al. (1999) concluded that the MOC exports freshwater at 30°S. They determined a present-day value of Σ ≈ Mov(30°S) ≈ −0.3 Sv and no error estimate was given. Using a recent dataset (Gouretski and Koltermann 2004), we determined the zonally averaged salinity profile at 35°S (Fig. 13a) and calculated a zonal-mean velocity profile based on thermal wind balance with the same method as was used to obtain Fig. 8.2 in Van Aken (2007). An Ekman transport of 4 Sv was added to obtain a zero integral of the section-averaged volume transport (as required in steady state); the result is shown in Fig. 13b. From these profiles, we obtain Σ = Mov(35°S) ≈ −0.1 Sv; an Ekman transport change of 2 Sv gives a difference of 0.02 Sv in Σ. Similar to the results in Weijer et al. (1999), this would indicate that the present-day MOC is in the ME regime. Obviously, this result probably has a large error bar because Σ will be a highly fluctuating quantity affected by many processes not considered here (such as the effect of the Bering Strait transport). We hope, however, that this value for Σ and the results in this paper will stimulate analysis of combined datasets from observations and model simulations (such as in Garzoli and Baringer 2007) to provide better estimates of present-day values of Σ.
The main result of this paper is that we provide a physical justification that a negative sign of Σ is a good indicator for the multiple equilibrium regime of the MOC. The results in section 4b show that when a freshwater perturbation is imposed on the on state of the MOC, Σ is a measure for the anomalous freshwater transport into (out of) the Atlantic, induced by velocity perturbations and the background salinity field. This transport is stabilizing the MOC (making the Atlantic saltier) in the SE regime for which the background MOC is exporting salt. However, it is destabilizing (freshening the Atlantic) in case the MOC exports freshwater. The analysis fully supports and corroborates earlier descriptive explanations (Rahmstorf 1996; De Vries and Weber 2005; Dijkstra 2007) and provides the details of the processes involved.
The computations were done on the Huygens IBM p6 supercomputer at SARA Amsterdam. Use of these computing facilities was sponsored by the National Computing Facilities Foundation (NCF) under the project SH084-08 with financial support from the Netherlands Organization for Scientific Research (NWO). This work was supported by a NWO Toptalent Grant to one of the authors (MdT). We thank both Stefan Rahmstorf and another anonymous referee for very useful comments on the first version of this paper.
Corresponding author address: Henk A. Dijkstra, Institute for Marine and Atmospheric Research, Dept. of Physics and Astronomy, Utrecht University, Princetonplein 5, 3584 Utrecht, Netherlands. Email: firstname.lastname@example.org