Figure b also shows the AMOC response to two cases of transient forcing that overshoot the tipping point γ¯A. The freshwater forcing grows linearly in time as shown in Fig. a where the threshold is denoted by a horizontal line: in one case (blue curve) the forcing goes beyond the tipping point after 900 model years. In the other case (red curve), the tipping point is passed after 7400 model years. In Fig. b, the AMOC trajectories of both forcing scenarios initially follow the stable steady-state branch. Since the forcing keeps increasing beyond the tipping point, the AMOC undergoes a change of state. The slower forcing (red) causes the AMOC to stay closer to the steady-state branch, whereas the faster forcing leads to a larger overshoot, making the AMOC reach the off state for higher values of the parameter γA.

The AMOC is considered a slow-onset tipping system, which means that crossing a tipping point threshold does not always result in an immediate transition as seen in Fig. b. This leaves the possibility that when the freshwater forcing is reversed, the AMOC may undergo a safe overshoot; i.e., it does not collapse . To investigate safe vs. unsafe overshoots of the AMOC, we consider freshwater forcing scenarios γA(t) as piecewise linear functions: 4γA(t)=m1t0≤t<t1,-m2t+ht1≤t<t2,-m2t2+ht≥t2, where m1 and m2>0 and h=(m1+m2)t1 is such that γA is continuous. Moreover, we chose t2 such that the forcing after t2 reaches a constant value, set to be half of the value of the tipping point γ¯A, i.e., 5t2=-1m2γ¯A2-h. For t<t1, the forcing represents a linear growth of the freshwater anomaly in the North Atlantic. To have an overshoot beyond the tipping point, the parameters m1 and t1 will be chosen such that the maximum value of γA (reached at t=t1) is above the tipping point γ¯A; hence 6γA(t1)=m1t1=-m2t1+h>γ¯A. The subsequent linear decay tries to capture in the simplest way possible the decrease in the freshwater perturbation, which could be faster or slower depending on the effort of society to lower the global emission rates of greenhouse gases.

We investigated three different properties of the freshwater perturbation applied in the region P that can influence the safe or unsafe overshoot of the AMOC: the rate of decline (case A), the rate of increase (case B) and the height of the peak (case C). To understand the physics of the overshoot behavior, only case A, shown in Fig. , is needed. The results for the other cases will only be shortly mentioned at the end of Sect. 3.3.

Case A represents two scenarios differing only in the rate at which the forcing decreases. The rate m1 corresponds to 6.0×10-4 Svyr-1, which is slightly smaller than the rate of meltwater release from Greenland (about 2.0×10-3 Svyr-1 over the period 2002–2021). The two freshwater forcing trajectories of case A (Fig. a) reach the same maximum value of 0.2384 Sv but have different decrease rates: -1.0×10-3 Svyr-1 for the blue scenario vs. -5.7×10-4 Svyr-1 for the (dashed) green scenario. Since both trajectories exceed the threshold value γ¯A, one would expect the AMOC to tip to the off state. However, as can be seen in Fig. b, the blue scenario results in a recovery and hence a safe overshoot, while the green one does not. In the safe overshoot scenario, the AMOC spends a shorter time beyond the tipping point (131 years), enabling it to recover. Conversely, in the unsafe overshoot, the AMOC remains beyond the tipping point for a longer time (167 years), and the slower decreasing forcing causes a collapse. In summary, a decrease that is too slow in the forcing prevents a recovery.

In order to understand the physics underlying the results presented in Sect. , we consider the integral balance of salinity over the Atlantic basin, with meridional boundaries Sθ located at 35°S in the south and 60°N in the north. The overall salinity balance is given by : Φb=Φs-Φa(θs)+Φa(θn)-Φd(θs)+Φd(θn)7+ddt∫AtlSdV. Here Φs, given by 8Φs=-∫SoaS0FSr02cos⁡θdλdθ, represents the (equivalent) salt flux through the ocean–atmosphere surface of the basin (Soa), positive (negative) when evaporation is larger (smaller) than precipitation; S0=35 psu indicates the reference salinity. The quantities Φa and Φd are the advective and diffusive salt fluxes through the boundary Sθ and are given by Φa(θ)=∫SθvSr0cos⁡θdλdz,9Φd(θ)=-∫SθKH∂S∂θcos⁡θdλdz, where KH is the horizontal diffusivity. The last term on the right-hand side of Eq. () measures the changes in time of the salt content stored in the Atlantic basin. This term will be zero for steady states, while it will play an important role in transient solutions. Finally, the residual Φb is used to monitor how well the salt balance is closed as in that case it should be zero. It was shown in that the salt balance is closed in the steady case, i.e., along the bifurcation diagram in Fig. b.

In the transient situation, we focus first on case A (Fig. a and b), where a higher rate of forcing decline results in a recovering AMOC, while a slower decline rate weakens the AMOC until it collapses. Since both scenarios have forcing trajectories with identical peaks at the same time (355 model years), they behave in the same way during the first 355 years. Due to the different rates at which the forcing decreases, we will focus on this phase to characterize the safe and unsafe overshoots. In Fig. a, three main contributions of the salt balance are considered: the surface flux Φs, the tendency of the integrated salt content, and the net salt flux Φlat through the lateral boundaries Sθs and Sθn, where θs=35°S and θn=60°N. Here, 10Φlat=Φ(θs)-Φ(θn), where Φ(θ)=Φa(θ)-Φd(θ); Φlat is positive when salt is transported into the basin. Figure b shows the decomposition of the lateral fluxes into advective and diffusive fluxes at the northern and southern boundary of the Atlantic Ocean.

While γA increases, the surface salt flux Φs decreases linearly (purple curve in Fig. a) due to the input of freshwater in the Atlantic. However, this flux is still positive, which means that there is a net salt buildup in the Atlantic basin that needs to be compensated through salt transport out of the basin. This transport occurs through the lateral boundaries, where (note that Φlat is negative) salt is transported out of the basin. Its absolute value is decreasing with increasing γA, and hence less salt is transported out through the lateral boundaries. However, since the integrated salt content is decreasing (ddt∫AtlSdV<0), the lateral salt outflow is larger than that needed to compensate the surface salt input. This implies that the lateral salt transport does not adjust quickly enough to the changing forcing, remaining stronger than needed for a steady balance. An important contribution to the lateral fluxes is due to Φa(θs) (red curve in Fig. b), which indicates the salt was transported northwards at the southern boundary. As Φa(θs) is negative, the salt is being transported out of the basin. Note that here the flux at the southern boundary dominantly determines the behavior of Φlat.

At the time when γA starts to decrease, Φs increases in both forcing scenarios, reflecting that less freshwater enters the ocean through its surface. At the same time, the dashed curves and the solid ones in Fig. a begin to diverge. When γA decreases more rapidly (solid curves), the increase in Φs is faster (solid purple curve). At the same time, Φlat (solid blue curve) reaches a minimum and then starts to increase again, indicating that the amount of salt transported out of the basin is reduced. These two factors make the Atlantic basin saltier as can be seen by the positive ddt∫AtlSdV (solid green curve): less freshwater is put in and less salt is transported out. In the case of collapse (dashed curves), the combination of freshwater input and salt transport outwards does not make the term ddt∫AtlSdV positive: the salt flux through the lateral boundaries remains too strong to balance the slow increase in Φs and hence the salt storage in the Atlantic keeps decreasing.

To explain the different behavior of the two scenarios in more detail, two boxes are defined: one in the North Atlantic and one in the South Atlantic. The northern box spans latitudes from 40 to 60°N and the southern one extends from 15 to 35°S. Both are bounded zonally by the land bordering the Atlantic. We compute the box-averaged densities ρi, salinities Si and temperatures Ti (i=s,n) by integrating over the total volume of each box. The location of the southern box has been motivated by earlier work , showing a linear relation between the AMOC strength and the meridional density difference. The meridional density difference Δρ:=ρn-ρs is shown in Fig. a together with the AMOC strength. The density difference behaves in the same way in the two scenarios up to the point where the freshwater forcing reaches its peak. Afterwards, the different decrease rates lead to changes in Δρ. In the collapse case, Δρ keeps decreasing, while it has a minimum when the AMOC recovers after the overshoot. As shown in Fig. b, Δρ is well correlated with the AMOC strength, consistent with other low-resolution ocean model studies and a consequence of the thermal wind balance. In the case of collapse, the density is plotted vs. the AMOC strength for the whole time of the simulation, while in the recovery case, the relation is linear only during the increase in the forcing; in the period of the forcing decrease the AMOC has a more complex, time-dependent response (not shown).

We further decompose the density difference contribution by salinity (ΔS) and temperature (ΔT) meridional differences in Fig. d and c. Note that ΔT has a smaller impact on Δρ compared to ΔS because the thermal expansion coefficient αT=10-4C°-1 is smaller than the haline contraction coefficient αS=7.6×10-4psu-1 used in the linear equation of state. An increasing amount of freshwater is being introduced into the northern region of the Atlantic Ocean, which decreases the salinity difference (ΔS) between the northern and southern boxes until it reaches 0 psu at around 300 model years. After that point until t=t1 (when the forcing stops increasing) ΔS keeps decreasing, becoming more negative. The salinity in the north changes more than in the south (not shown), not only in range (in the north, it spans a range between 0.65–0.85 psu depending on the scenario while only a range of around 0.1 psu in the south) but especially in shape. The main contribution to ΔS results from changes in the northern box. This is not surprising given the northern location of the prescribed freshwater input anomaly γA. The freshwater anomaly is weakening the AMOC, which results in a reduced meridional heat transport northward. This cooling effect in the northern region makes ΔT smaller. The northern temperature spans a range almost 4 times as big as the one in the south in the safe overshoot scenario and 9 times as big as in the unsafe overshoot one; hence the main contribution to ΔT comes from the northern box. In the safe overshoot scenario (solid lines), ΔT and ΔS hit their lowest value before rising again, eventually stabilizing at a positive value. Conversely, in the unsafe overshoot scenario, both ΔS and ΔT continue to decline. In the collapsed state, the northern regions become less saline and cooler than the southern regions.

As most density changes occur in the northern box, we apply the salt balance (Eq. ) to the northern box that extends over the Atlantic basin from 40 to 60°N. The values of the fluxes in the northern region (Fig. a) show that, unlike in the whole Atlantic basin case, there is a net freshwater input (Φs<0) through the surface and a net saline water input through lateral boundaries. When γA starts to decrease, the response of the northern box reveals the precise mechanism of the AMOC safe/unsafe overshoot. In the safe overshoot scenario, a faster decline in freshwater forcing allows the lateral salt transport to surpass the effect of the freshwater input. This results in the change in sign of the tendency of the integrated salt content from negative to positive. Consequently, the northern box experiences an increase in salinity, reinforcing the AMOC and promoting its recovery. In contrast, in the unsafe overshoot scenario, the freshwater forcing remains dominant over the lateral salt transport, leading to a persistent net loss in salt storage (ddt∫northboxSdV<0). This results in a further freshening of the northern box and in a consequent decrease in ΔS and Δρ, which keeps weakening the AMOC and drives it toward a collapse. A detailed examination of the lateral fluxes within the northern box, shown in Fig. b, reveals that the advective salt transport through its southern boundary of this box is the dominant flux. This advective flux, which transports saline waters northwards, plays a crucial role in compensating the surface freshwater perturbation and thereby influences the AMOC behavior under different forcing scenarios.

Salt balance computations were similarly performed for the southern box but the corresponding plots have been omitted, as the salt fluxes in the South Atlantic have a negligible impact on the AMOC recovery dynamics. In the South Atlantic, the range of the freshwater anomaly is approximately an order of magnitude lower – of magnitude about 20 times less – than that computed in the North Atlantic, making its influence on the recovery or collapse minimal. Moreover, the lateral salt fluxes exhibit a smaller range as well, being roughly 11 times smaller than in the AMOC collapse scenario and 4 times smaller than in the safe overshoot scenario in comparison to their North Atlantic counterparts. Therefore, the essence of the mechanism can be found in the interplay between lateral salt transport and freshwater input in the northern box. It is the relatively early overcoming of freshwater surface flux by lateral salt transport that characterizes the recovery phase of the AMOC. This is quantitatively reflected in the shift from a negative to a positive time derivative of the integrated salt content, a change that signals the key transition in the dynamics. The Atlantic basin, particularly its northern regions, begins to retain more salt, thereby restoring the meridional salinity (and density) gradients crucial for maintaining the AMOC.

The same analysis done for case A was applied to two other cases (B and C). For case B, two freshwater forcing trajectories reach the same maximum value of 0.235 Sv but at different rates. The forcing then decreases linearly at the same rate in both scenarios. The slower increasing forcing causes a collapse, while the faster increasing forcing allows a recovery. As in case A, the AMOC is unable to recover when it has a more prolonged exposure to forcing levels beyond the tipping point. This makes the time spent beyond the tipping point an important factor that influences whether the AMOC collapses or recovers, as was also shown in using a conceptual box model. In case C, both forcing trajectories have the same rate of increase, but now they differ in terms of maximum values of γA(t). One peaks after 350 years, while the other reaches a higher peak above the threshold after 370 years. They both get to the same fixed level after 600 years, which means that the rate at which their decrease is different. The lower peak forcing leads to a weaker AMOC decrease and a subsequent recovery, while the higher forcing makes the AMOC tip to the off state. In both cases, the same mechanism as in case A is responsible for causing the safe/unsafe overshoot behavior.

To be able to compare with the ocean model later in Sect. 4.4, we need to use the general form of the saddle-node bifurcation 13dXdt=aX2+bX+c-γ(t), where γ(t) is defined as in Eq. (). In order to get the equation in the form of Eq. (), a rescaling is needed. We thereby apply the following change of variables: 14x=-attipX+b2a,τ=tttip,α=-am1ttip3,β=-am2ttip3, with ttip=cm1-b24am1. Rewriting Eq. () as dXdt=aX+b2a2+c-b24a-γ(t), we can substitute the new variables and get for t∈[t0,t1] dxdτ=dxdtdtdτ=ttipdxdt=ttip-a⋅ttipdXdt=-attip2dXdt=-x2+α(1-τ). For t>t1, similar calculations lead to dxdτ=-x2-β(1-τ)-k. In this way, the problem is in the form of Eq. ().

Analytical solutions to Eq. () can be found in terms of Airy functions . We first compute the bifurcation diagram of the corresponding autonomous system assuming τ as the bifurcation parameter (instead of time). The steady states are given by 15x±(τ)=±-f(τ), or more explicitly x1+(τ)=α(1-τ), x1-(τ)=-α(1-τ), x2+(τ)=-β(1-τ)-k and x2-(τ)=--β(1-τ)-k. The functions xi±(τ),i=1,2 form two parabolas which represent the bifurcation diagram. When a full solution of Eq. () crosses one of these curves, its derivative becomes zero, which means that xi±(τ), i=1,2 separate regions in the phase plane (t,x) where the derivative has different signs. Following the same methodology as in , the solution x(τ) is given in the Appendix.

Figure a shows two examples of forcing that differ in the value of β, and Fig. b shows the associated solutions. Continuous solutions as the one in blue reproduce the recovery of the AMOC, while the solutions with a vertical asymptote (in red) are the collapses. The bifurcation diagram for both cases (treating τ as parameter) is also shown as the thin curves. As the analytical approximation is tailored for a neighborhood of a single saddle-node bifurcation, it excludes the off state of the AMOC. Consequently, the post-collapse behavior cannot be captured within this analytical framework.

Given that the main scenarios analyzed in Sect.  are the ones of case A, we focus here on studying conditions for a safe overshoot in the case of a fixed value of α. We pick α=84 for reasons that will become clear in Sect.  below. First of all, the solution in the time interval with an increasing forcing (τ∈[τ0,τ1]) has a collapse time at τ=τ*, meaning that the solution reaches -∞. The value of τ* depends on the parameter α, and it is given by the asymptote of the pullback attractor, which can be explicitly computed: τ*(α)=1+2.338α-1/3 . This scenario corresponds to a prolonged increasing forcing that makes the AMOC collapse before the forcing starts decreasing. To avoid this kind of unsafe overshoot, we need τ*∉[τ0,τ1], i.e., 16τ1<τ*(α). From now on, we make sure that the condition in Eq. () is satisfied and the solution is continuous on the interval [τ0,τ1]. For α=84, this means that the forcing must have its peak before τ=1.5453.

Solutions for different values of τ1 and β are computed, and a fraction of those is shown in Fig. a and b. For the values of (τ1,β) for which a collapse happens, the collapse time τ* is computed (where τ=τ* is the equation of the asymptote). For each value of τ1, Fig. c displays the collapse times τ* as a function of the rate of decline β of the forcing. As anticipated, a smaller value of β corresponds to a smaller τ*. This indicates that as the forcing decreases more rapidly, the collapse is postponed, and if the decline is sufficiently fast, the collapse can be entirely avoided.

Finally, for each τ1 we can take the largest value of β, say β̃, for which the solution exhibits a collapse. For all β<β̃, the forcing is decreasing in a slower way and hence all the associated solutions collapse. On the other hand, for all β>β̃, the solutions recover. Thus, the curve (τ1,β̃(τ1)), plotted in Fig. d for α=84, divides the parameter space into safe and unsafe overshoot regions. The value of β̃ decreases as τ1 decreases. In fact, given that α is fixed, a smaller τ1 corresponds to a lower peak of the forcing, which in turn leads to a smaller decline in the AMOC. This allows for a longer AMOC overshoot (hence a slower decrease in the forcing) while still having a recovery. The analytic solutions provide a clear picture of the response of the AMOC to the freshwater forcing taking into account the rate of the forcing decrease β and time τ1 when the forcing begins to decrease (in the same way one could take the peak value of the forcing, as was done in the analysis of case C in Sect. ). Hence, within this simple analytical framework, a critical curve has been found defining stability regions similar to that in the conceptual model used in .

We now compare the results obtained from the simple ODE in Eq. () with the numerical simulations obtained with the global ocean model for which results are presented in Sect. . A quadratic fit is made near the tipping point in the bifurcation diagram (Fig. b) where ΨA is computed as a function of the bifurcation parameter γA. Hence, the AMOC strength ΨA=X is used as the primary variable in Eq. () and the fit to determine the constants a, b and c yields a=-0.0060,b=0.0658 and c=0.0056. In order to have recovery and collapse simulations while staying as close as possible to the saddle-node bifurcation, results for four additional global ocean model simulations (labeled a–d) are shown in Fig. a and b. They all have m1=6.1972×10-4 Svyr-1 and γ(t1)=0.22 Sv (with t1=355 years), while they differ for the value of m2={8.7,3.7,2.3,1.1}×10-4 Svyr-1.

To be able to compare the simulations to the analytical solutions, we transform the simulation data ΨA(t) and the simulation parameters using Eq. (). The parameters m1,t1 and m2 translate respectively to α=84.061, τ1=1.18 (max⁡(f)=f(τ1)=15.13) and β={118.71,49.89,31.58,15.03}.

The transformed AMOC trajectories are plotted vs. time in Fig. c–f (dashed curves) together with the analytical solution of the ODE in Eq. () computed with the same forcing parameters. The green part (sol A) corresponds to the solution in the time interval of forcing growth and the red part (sol B) to the solution in the time interval of forcing decline. Figure c shows that the simulation with the largest β(118.71) and hence the fastest decrease is the best-approximated simulation. The smaller β gets, the worse the approximation becomes. In Fig. d, for instance, the minimum of the analytical solution is larger (in absolute value) and is reached with a certain delay compared to the numerical simulation from the global ocean model. Figure e shows an AMOC recovery in the ocean model, whereas the corresponding analytical solution of the ODE instead exhibits a collapse. In Fig. f, both the numerical simulation and the analytical solution show an AMOC collapse. However, the simulation spends a longer time between the on and off state (it reaches the collapsed state at τ=2.25), while the analytical solution goes to -∞ much quicker (for τ*=1.65).

One of the possible reasons why the approximation may not always work well is that the ODE approximates the behavior of the AMOC only locally around the saddle-node bifurcation and it is possible that the simulations performed were not close enough to the tipping point. Another reason could be the choice of ΨA as the main variable X. The state variable in the global ocean model is multi-dimensional, while the bifurcation diagram used for the quadratic fitting is just a one-dimensional projection. Although the strength of the AMOC does exhibit a saddle-node bifurcation, it is also part of the projection of the state variable onto the eigenvector associated with this bifurcation. Incorporating the Lyapunov–Schmidt reduction method or center manifold reduction method could help in addressing this issue. These methods reduce the complexity of the multi-dimensional problem by decomposing it into a simpler, lower-dimensional form while capturing the essence of the bifurcation behavior. They would allow us to identify the behavior of the global ocean model near the bifurcation point, but its application to this large-dimensional model is outside the scope of this paper.