The sea ice model used follows almost exactly, and its key features are depicted schematically in Fig. . The model contains four state variables: sea ice effective thickness (V, which is volume divided by the area of the model grid box), sea ice area (A), sea ice surface temperature (Ti), and mixed-layer temperature (Tml) for a single box representing the entire Arctic. Subsequent versions of this sea ice model have been used in , , and . Those versions are derived from the model used here, making a few further modest simplifications (using a hyperbolic tangent function for surface albedo, assuming the ice surface temperature is in a steady state, combining all prognostic variables into one, i.e., enthalpy) that do not affect the qualitative behavior of the model (i.e., the nature of summer and winter sea ice bifurcations). We choose to implement the earlier model because it explicitly represents the key physical variables of ice volume, area, ocean temperature, and ice temperature as prognostic variables – as opposed to combining them all into a single enthalpy – and thus provides more transparency and interpretability. We, therefore, do not expect our results to change if we use any of the later model versions.

In the model, the atmosphere is assumed to be in radiative equilibrium with the surface, and the model is forced with a seasonal cycle of insolation; of poleward atmospheric heat transport from the midlatitudes; and of local optical thickness of the atmosphere, which represents cloudiness. Sea ice growth and loss are primarily determined by the heat budget at the bottom of the ice and are therefore set by the balance between ocean–ice heat exchanges and heat loss through the ice to the atmosphere. When conditions for surface melting are met (when the ice surface temperature is zero and net fluxes on the ice are positive), all surface heating goes into melting ice and the surface albedo of the ice is set to the melt pond albedo. The ocean temperature is affected by shortwave and longwave fluxes in the fraction of the box that is ice-free and by ice–ocean heat exchanges. When the ocean temperature reaches zero, all additional cooling goes into ice production, while the ocean temperature remains constant. The full equations of the sea ice model can be found in the original paper and in the Supplement; here, we highlight a few minor ways in which our implementation differs. First, for simplicity, we do not model leads, which in the original model were represented by capping the ice fraction at 0.95 rather than 1. Second, we use an approximation to the seasonal cycle of insolation using a latitude of 75∘ N. The atmospheric albedo is set to 0.425 to produce the same magnitude of the seasonal cycle as in the original model of .

In our transient-forcing scenarios (described below), we vary CO2 in time which affects the prescribed near-surface atmospheric midlatitude temperature (Tmidlat) and the atmospheric optical depth (N; see the Supplement). Specifically, we increase the annual mean of Tmidlat by 3 ∘C per CO2 doubling and N by a ΔN that corresponds to 3.7 Wm-2 per doubling. All model parameters are as in except as mentioned below.

We configure the model in three different scenarios that yield a wide CO2 range of bistability in winter sea ice (Scenario 1), a small range of bistability in winter sea ice (Scenario 2), and no bistability in winter sea ice (Scenario 3). We do so by modifying the strength of the ice–albedo feedback by changing the albedos of bare ice (αi), melt ponds (αmp), and ocean (αo), as listed in Table S1 in the Supplement.

In each of the three scenarios, we tune the model (by adjusting the mean and amplitude of the atmospheric optical depth) to roughly match the observed seasonal cycle of ice thickness under pre-industrial CO2 ∼ 2.5–3.7 m,. We then run each scenario with multiple CO2 ramping rates (expressed in “years per doubling”) with an initial stabilization period (fixed pre-industrial CO2), a period of exponentially increasing CO2 concentration (which corresponds to linearly increasing radiative forcing), another period of stabilization at the maximum CO2, a period of decreasing CO2, and a final period of stabilization at the minimum CO2 value (see Fig. S2 in the Supplement). Scenarios 2 and 3 are ramped to higher final CO2 values than Scenario 1 so that they lose all their sea ice. We also directly calculate the steady-state behavior of the sea ice (as done in the original study) by running many simulations with fixed CO2 values until the seasonal cycle of all the variables stabilizes. Because we expect multiple equilibria (which could be ice-free, seasonal ice, or perennial ice) at some CO2 values in Scenarios 1 and 2, we run these steady-state simulations starting with both a cold (ice-covered) and a warm (ice-free) initial condition in order to find these different steady states. In the ice-free initial condition runs, the ice–albedo feedback will still play an important role if the temperature cools sufficiently for ice to develop. At CO2 values for which the sea ice is bistable, the ice-free initial condition evolves to a perennially ice-free steady state, and the ice-covered initial condition evolves to a seasonally ice-covered steady state (seen by the dotted and dashed lines, respectively, in Fig. a and c).

The main points we are trying to make about the transient versus equilibrium behavior of winter sea ice near a tipping point are not unique to the problem of winter sea ice, and in order to demonstrate this, we use the simplest mathematical model that can display tipping points, following other studies that have also used such simple dynamical systems . The cubic ODE used, while much simpler than the sea ice model above, has some of the key characteristics of the sea ice system (it is a non-autonomous system due to the time-dependent forcing and has saddle–node bifurcations), which allows for direct comparison between the two models. The ODE equation, 1dxdt=-x3+δx+β(t),β(t)=β0+μt, contains a time-changing forcing parameter, β(t) mimicking the effects of CO2 in the sea ice model. We consider this differential equation in three scenarios, paralleling those used with the sea ice model: in Scenario 1, δ=5 leading to a wide region of bistability; in Scenario 2, δ=1 leading to a narrow region of bistability; and finally, in Scenario 3, δ=0 leading to a mono-stable system. The different values of δ, therefore, produce the same three scenarios that were achieved in the sea ice model by modifying the strength of the ice–albedo feedback. We mimic the hysteresis experiments of the sea ice model with a sequence of ramping up and ramping down (using different ramping rates, μ) with values of β ranging from -10 to 10 to sweep the parameter space that contains the bifurcations. We calculate the steady states with fixed values of β (μ=0), starting with both a positive and a negative initial condition of x to yield two stable solutions when these exist.

We want to calculate the upper and lower CO2 values of the hysteresis region in runs with time-changing (i.e., transient) CO2 forcing. We do so by calculating the CO2 value at which the March sea ice area drops below a critical threshold (50 % ice coverage; results are insensitive to the specific value used) during increasing and decreasing CO2 integrations: we denote these CO2 values CO2i and CO2d, respectively (see Fig. S9). The difference between CO2i and CO2d is referred to below as the “hysteresis width” of the rate-dependent hysteresis whether an equilibrium hysteresis exists or not; this width approaches the width of bistability at very slow ramping rates.

One of our main goals (see the Introduction) is to efficiently estimate the equilibrium behavior of sea ice, including the location of tipping points, without running the model to a steady state for many CO2 values. This would show that such estimation could be calculated for GCMs where tipping points cannot be detected using steady-state runs due to their computational cost. In order to estimate the values of CO2i and CO2d that would have occurred for an infinitely slow ramping rate (in other words, the range of CO2 for which there is bistability) using only the transient runs, we fit a polynomial of the form f(x)=mxc+b to CO2i and CO2d as functions of the ramping rate x. Because c is negative, the fitted parameter b represents the prediction of CO2i and CO2d at infinitely slow ramping rates, i.e., in the steady state. We also calculate the uncertainty in the fitted parameter b by block bootstrapping to account for autocorrelation; see the Supplement. Other fits to CO2i and CO2d as a function of ramping rates, such as an exponential function f(x)=a+bexp(-cx), could in principle be used, although we found that fit to be less good in our case.

Our goal in this section is to understand the relationship of winter sea ice forced with time-changing CO2 to its equilibrium state, both in cases with and without a sea ice tipping point. In Fig. a, c, and e, we plot the results of running all three scenarios (wide range of bistability, Scenario 1; narrow range of bistability, Scenario 2; and no bistability, Scenario 3) under time-changing (transient) and fixed CO2 values. In all scenarios, the experiments run with time-changing CO2 exhibit rate-dependent hysteresis; the hysteresis width (lower horizontal gray bar in Fig. a) is larger for faster ramping rates (Fig. a, c, and e). For Scenarios 1 and 2, which have a region of bistability and equilibrium hysteresis (upper gray bar in Fig. a), this corresponds to a widening from the equilibrium hysteresis (that would exist even with infinitely slow ramping rates), while in Scenario 3, this hysteresis occurs only in transient simulations and is due to the inertia in the system (the sea ice cannot respond instantaneously to forcing changes). In Scenarios 1 and 2, whose equilibrium solutions (dashed and dotted black lines in Fig. ) have a tipping point and therefore an infinite gradient of sea ice thickness vs. CO2, the faster ramping rates also lead to more gradual (and finite) gradient of sea ice thickness vs. CO2.

The rate-dependent hysteresis loops across all scenarios at fast enough ramping rates (loops composed of the darkest blue and darkest red in Fig. 2a, c, and e) are qualitatively similar in shape, despite their different underlying steady-state structures. This similarity indicates that from a single hysteresis run with time-changing CO2, we cannot discern whether the underlying Arctic winter sea ice equilibrium behavior has a region of bistability or not nor how wide the region of true bistability is. In particular, a single hysteresis loop found from a time-changing forcing simulation would always overestimate the width of bistability if it was assumed to represent a quasi-steady state. This result demonstrates that the apparent sea ice hysteresis loop found by could be due to a system without an equilibrium hysteresis, as they suggest, or due to a system with a narrower equilibrium hysteresis than the one implied by their transient simulation.

We now discuss the behavior of the simple cubic ODE (Eq. ) under similarly time-changing forcing. Previous work in the dynamical systems literature e.g., has examined a variety of simple systems to understand the nature of bifurcations in the presence of a time-changing (“drifting” or “transient”) forcing parameter. In the climate literature, too e.g.,, idealized dynamical systems similar to our Eq. () have been used to understand the predictability of tipping points in the presence of noise, and the ability to recover from such tipping points (“overshoot” scenarios). These works, as well as the AMOC study of , found that a system with a bifurcation that is run with a time-changing forcing parameter can follow a given equilibrium value beyond the bifurcation value of the forcing parameter before undergoing the tipping point transition to the new equilibrium value. This is consistent with the out-of-equilibrium behaviors we find for sea ice in Scenarios 1 and 2. To our knowledge, the simple ODE used here has not yet been analyzed with our specific goal in mind: to compare the shape of rate-dependent hysteresis loops in generic dynamical systems both with and without bifurcations and to address the question of whether the equilibrium behavior can be inferred from the rate-dependent behavior of such systems.

To address these two goals, we configure Eq. () analogously to the sea ice model in three scenarios with wide bistability (Scenario 1), narrow bistability (Scenario 2), and no bistability (Scenario 3) and force it with a time-changing forcing parameter. In Fig. b, d, and f, we see that the three scenarios with similar dynamics (but different equilibrium structures) all display rate-dependent hysteresis, similar to the result from the sea ice model. Specifically, even when there is only one stable equilibrium solution in both models (Scenario 3, panels e and f of Fig. 2), there is still a narrow region of rate-dependent hysteresis. Thus, we find that the inability to tell if rate-dependent hysteresis in Arctic winter sea ice is accompanied by an underlying equilibrium hysteresis appears to be a generic feature of dynamical systems, which helps explain the challenges of interpreting the results of .

Mathematically, this 1D system is fundamentally different from the sea ice model because it is not periodically forced. We show in the Supplement that adding a sinusoidal forcing term to the ODE does not qualitatively change our results.

Our next objective is to demonstrate that it would require expensive runs in a GCM to approach the equilibrium behavior of sea ice using slower and slower-changing CO2 runs (hysteresis experiments). As we saw in Fig. , the rate of loss of sea ice with increasing CO2 is infinite (dashed and dotted black lines) in Scenarios 1 and 2 at the tipping points. On the other hand, the gradient of sea ice thickness with respect to CO2 is more gradual and finite under time-changing forcing (blue and red curves) but steepens as the ramping rate of CO2 decreases. We now quantify the rate of this steepening by examining the maximum gradient of sea ice loss during each transient simulation as a function of ramping rate (inverse of the years per doubling of CO2).

In Fig. a, we plot the maximum gradient of March sea ice thickness with respect to CO2 during each hysteresis experiment, as a function of the CO2 ramping rate. In Scenarios 1 and 2 (wide and narrow bistability, respectively), the maximum gradient gets greater as the ramping rate is slower (Fig. a, negative slopes of solid and dashed lines), consistent with Fig.  (e.g., steepening from dark-blue to light-blue curves in Fig. a and b). In particular, the gradient approximately follows a negative power law as a function of ramping rate on both warming and cooling time series. In Scenario 3, the maximum gradient is nearly insensitive to the ramping rate (relatively flat dash-dotted lines seen in Fig. 3a). In Fig. b, we see a similar result for the simple ODE, as seen by the shallowing of the power law from Scenarios 1 to 3 (though here the slope in Scenario 3 is clearly nonzero). Notably, in the cubic ODE the power law in the case with the largest region of bistability (Scenario 1) is approximately given by max(dx/dβ)∝μ-1, where μ again is the ramping rate. The Supplement further explains the above convergence rate of µ-1.

A dependence on the reciprocal of the ramping rate in the case of wide bistability suggests that running a climate model with twice as gradual CO2 ramping leads to a less than a factor of 2 increase in the gradient max(dV/dCO2). This is an important result because this implies that the distance between the CO2 at the simulated transient tipping point and the CO2 of the true (equilibrium) tipping point (which we want to estimate) also only reduces by a factor of 2 when the ramping rate is reduced by a factor of 2. A greater power law slope (e.g., a slope of -2) would imply a much faster convergence to the equilibrium location of the tipping point. Thus, using more and more gradual ramping experiments may be an inefficient way to approach the equilibrium behavior of this physical system, suggesting the need for a more efficient approach, discussed next.

Our main novel result, presented next, is a method for finding the CO2 concentration at which a bifurcation (if any) occurs in the equilibrium using computationally feasible transient model runs instead of fixed-forcing steady-state runs. We are interested in this CO2 concentration because it determines the threshold beyond which significant sea ice loss is practically irreversible . In our simple, inexpensive model, we can test the estimates of the bistability and associated tipping points derived from transient model runs against the known true tipping points and equilibrium structure that are found from fixed-forcing runs (see Methods). When used in a GCM, our method would provide a prediction for the existence and location of tipping points when the equilibrium value of sea ice is actually unknown. Thus, this section is a proof of concept that our new method can accurately determine whether observed rate-dependent hysteresis is caused by lag around a system with no bistability or tipping points or is caused by a rate-dependent widening of an equilibrium hysteresis loop in a system with tipping points.

In Fig. a, we plot a measure of the upper and lower CO2 values that correspond to the rightmost and leftmost edges of the rate-dependent hysteresis (by calculating the CO2 at which the March sea ice area crosses a critical threshold; see Methods and Fig. S9). We plot this measure for the warming (increasing greenhouse concentration) trajectories in blue (CO2i) and for the cooling (decreasing greenhouse) trajectories in red (CO2d), as a function of the ramping rate for all three scenarios. As expected, as the ramping rate gets slower CO2i and CO2d approach the CO2 values corresponding to the edges of the equilibrium hysteresis and the location of the true tipping points in the case of Scenarios 1 and 2 (denoted by the × symbols). In Scenario 3, CO2i and CO2d approach the same value (the rate-dependent hysteresis width approaches zero) because there is no bistability in the steady state.

Finally, we demonstrate that fitting a curve to the edges of the rate-dependent hysteresis (CO2i and CO2d) as a function of the ramping rate can be used to predict CO2i and CO2d at infinitely slow ramping rates (i.e., the edges of the equilibrium hysteresis). This would allow us to estimate the CO2 value corresponding to a bifurcation in the equilibrium behavior without running a model to a steady state. In Fig. a, we plot CO2i and CO2d and the curves that fit them (see Methods) as functions of the ramping rate and the predicted values of CO2i and CO2d at infinitely slow ramping rates with a 95 % confidence interval range shaded around them. We perform this fitting and estimation process using all the ramping experiments (18 different ramping rates total, as shown in Fig. a). We then repeat the fit using fewer and fewer experiments to explore how the uncertainty in predicted values of CO2i and CO2d increases as we move to only using a few fast ramping experiments that are more feasible when using full-complexity climate models. Figure b shows a summary of these analyses.

The predicted values of CO2i and CO2d are remarkably accurate for all scenarios (points approaching the red and blue x symbols in Fig. b), even when excluding several of the slower ramping experiments. This is an important test because when this method is applied to a GCM, one would only have a smaller number of faster-ramping experiments due to computational limitations. The uncertainties (indicated by the shaded blue and red bars around the points) in the predictions grow when excluding more experiments from the curve fitting process but still remain very low, especially for Scenarios 1 and 2. In predicting CO2d for Scenario 3, the uncertainties are a bit higher because the functional form of our fit does not represent this case as well as the others, leading to serial correlation in the residuals. The structure in the residuals can be used to guide the choice of the functional form used to fit such model output in future applications. This same method and functional form can also successfully predict the equilibrium structure of our simple ODE (Eq. 1), with even smaller uncertainties in the prediction when using very few ramping experiments (see Fig. S11). Finally, we can use the difference between the distributions CO2i and CO2d to calculate the probability that bistability – and thus a tipping point – exists (see the Supplement). Another very similar approach using only the difference between CO2i and CO2d (i.e., the hysteresis width) as a function of the ramping rate is also shown in Fig. S10.

Overall, these results demonstrate the potential for using several shorter runs with time-changing CO2 forcing to efficiently estimate the CO2 value of the tipping points and predict the existence of bistability in GCMs where equilibrium runs or long, slow-ramping hysteresis runs are computationally infeasible.