Climate change is causing rapid temperature increases in the polar regions. A fundamental question is whether these temperature increases are reversible. If humanity fails to prevent a substantial warming of the planet in the next few decades, which is appearing to be more and more likely, will it be possible in the future to reverse our effects on climate enough to restore lower temperatures, or will we have passed a tipping point beyond which return to the present state is impossible? We address this question in particular for the Arctic, where the observed climate change is the most dramatic.

The Earth's climate is an extremely complex system. Modelling efforts range from simple models attempting to isolate the most pertinent features to very complicated numerical models trying to capture as many details as possible. The model presented here is close to the simple end of this spectrum, although not as simple as some, in that it is a 9D nonlinear two-point boundary value problem. The advantage of relatively simple models is that they allow more direct analysis of cause and effect, which is often obscured in highly complicated models.

The term “tipping point” is used by different researchers in various ways; see and for some definitions and discussion of the term. In all cases, however, tipping points are associated with large qualitative changes in a system due to relatively small changes in the parameters or “forcings” that drive the system. In the present paper, tipping points arise as a result of saddle-node and cusp bifurcations in the mathematical model. The mathematical theory of bifurcations is well-developed and employed here. Figure  illustrates the typical behaviour associated with these bifurcations. In Fig. a there are two saddle-node bifurcations resulting in a parameter interval of bistability; that is, two stable states coexist for an interval of μ values. If the system is on the lower branch of stable states, then, as μ increases through a critical value μcrit, there is an abrupt jump to the upper branch of stable states. In contrast, as μ decreases, the jump back to the lower state does not occur until a much smaller critical value of μ. This phenomenon is called hysteresis. If present in the Earth's climate system, it implies that once the upward jump occurs, it may be very difficult to achieve the reverse jump back to the original climate state. Figure b illustrates the situation of a cusp bifurcation, where the two saddle-node bifurcations in Fig. a have coalesced, for example, as another parameter of the system is varied. In this situation a small change in μ will also cause a large change in the system F, although it will be smooth and reversible. In Fig. c, even though the saddle-node bifurcations may be outside the parameter interval of interest, abrupt large transitions in the system can result from a small noise- or perturbation-induced change to the system even when the parameter value remains constant. It is the presence of saddle-node bifurcations in a mathematical model, even if not occurring precisely at the system's current parameter value, that is the root cause of all of the behaviours shown in Fig. .

For a tipping point to be present, the underlying mathematical model will be characterized by nonlinearity, generally in the form of a positive feedback that accelerates change once change has begun. For the Arctic, one of the primary positive feedbacks is the surface albedo. When the Arctic Ocean is frozen, the surface reflects a significant portion of the insolation back into space, but open water absorbs much more heat from the Sun. Timing of the melt in the spring has significant impact . An earlier melt means considerably more heat is absorbed by open water, raising the water temperature and delaying freeze-up in the autumn. The freeze-up date for the Beaufort, Chukchi, Laptev, and Kara seas, for example, has been getting later by 6–11 d per decade since 1979 . September sea ice extent has been decreasing at an accelerating rate. The linear trend from 1979 to 2001 is -7 % per decade, but including data up to 2013, the linear trend is -14 % per decade . Thus the observational evidence indicates that the processes behind this phenomenon are not linear at all but nonlinear.

Past studies on general circulation models (GCMs) have given mixed results regarding the presence of multiple stable states for ice conditions in the Arctic. Some indicate that there appears to be a continuous transition from perennial ice cover to annually ice-free that is reversible . Other studies have shown evidence for nonlinear behaviour in sea ice loss, especially in the transition from seasonally ice-free to annually ice-free . On the other hand, smaller conceptual models generally show bistability and abrupt transitions in sea ice cover . The most common result from all these models seems to be that sea ice will likely transition from perennial to seasonally ice-free in a continuous, reversible manner, but significant warming beyond that point will likely cause an abrupt change to annually ice-free . See the introduction in . The model we present here is an annually averaged model with no seasonal component. It is not a model of sea ice in particular but rather a column model of the atmosphere that incorporates a nonlinear albedo response to surface temperature. Bistability in our model with both warm and cold solutions corresponds to annually averaged ice-covered or ice-free situations.

The Arctic climate model presented here is motivated by three observations. First is the observation that the climate changes taking place on the Earth today are most dramatic in the high Arctic. Therefore, it is prudent to put a special focus on understanding Arctic climate change. Second, irreversible change is inevitably the result of nonlinear geophysical processes. So, while this model is kept very simple, it does include significant nonlinear phenomena that can lead to tipping points. Third, the 3D spherical shell of the atmosphere of the Earth is rotationally symmetric about the polar axis if annually and zonally averaged. Due to the rotation of the Earth, Hadley, Ferrel and polar cells form in the global circulation. If perfect rotational symmetry is assumed, the polar axis becomes flow-invariant, and this remains approximately true for the real Earth. Thus, a 1D model restricted to the polar axis can be expected to give useful information about climate in a neighbourhood of the pole. The study of a rotationally symmetric spherical shell model by  gives support to this hypothesis. A vertical column of atmosphere at other points on Earth would have a horizontal component of velocity, invalidating the type of analysis used here. Globally averaged climate models do reduce to one (vertical) dimension, but they give little information specific to the Arctic.

The present model builds on the simple energy balance slab model of Dortmans et al. (2019), which was applied to paleoclimate transitions, and the model of Kypke et al. (2020), which was applied to anthropogenic climate change. The primary improvement of the present model is a more physically accurate description of the atmosphere. Instead of using a slab to represent a uniform atmosphere with absorption properties similar to the real atmosphere, here we use the Schwarzschild two-stream equations to model absorption in the atmosphere explicitly as a function of altitude p. 191.

A bifurcation analysis is performed on the model, tracking the steady-state solutions as carbon dioxide levels increase. The question of reversibility is a question of whether the current cold state simply warms but persists. The disappearance of this cold state through a saddle-node bifurcation would result in an abrupt change in climate that may be practically irreversible. The simpler model of showed this behaviour under certain CO2 representative concentration pathway scenarios. We seek here to determine whether the present, more accurate model also displays this behaviour.

Section  and Appendix  provide a detailed derivation of the model. The model parameter values and calibration of some of them to empirical data are presented in Appendix . Although much of the detail is relegated to the Appendices, the authors feel this detail constitutes an essential part of the contribution of the paper, providing clarity, justification of choices, and the information necessary for replication. Hence we consider them essential reading. Section  presents the results, and the conclusions are in Sect. .

The model is developed from first principles and has the following features.

The atmosphere is a 1D column at the North Pole with physical properties that vary with altitude, from the surface to the tropopause.

The annually and zonally averaged Earth atmosphere is rotationally symmetric around the polar axis, which is invariant under the flow. Therefore, if one considers a column of the atmosphere near the North Pole, it is reasonably approximated by a 1D model with altitude-varying quantities. This approximation becomes exact in the limit as the diameter of the column shrinks to zero. Alternatively, one can view the model as a meridional and zonal average over a cylinder centred at the North Pole. Further, although the Arctic Ocean is not zonally symmetric, in the above view, the contribution of ocean heat transport can be reasonably captured as a scalar quantity. Thus our model is more precisely a model of the North Pole rather than the Arctic. Nonetheless, we do use some empirical data for the region north of 70∘ to calibrate the model for two reasons: (1) data further north are not readily available, and (2) the data we use are not likely to alter too much if they were measured closer to the pole. The values for atmospheric heat transport and ocean heat transport are two that may change significantly as one moves north from 70∘, and we therefore analyse the behaviour of the model over a wide range for these parameter values.

The model domain is a vertical cylinder of cross-sectional area A (m2) and circumference Cb (m). The atmosphere is assumed to be uniform in the cross-sectional direction, so that the model's dependent variables can be interpreted as cross-sectional averages that vary with the 1D vertical coordinate z∈[0,zT], where zT (m) is the height of the tropopause. This domain is divided into a surface boundary layer with height zB (m), zB≪zT, and the troposphere proper, z∈[zB,zT]. The model consists of a set of initial value problems (IVPs), with an independent spatial variable z on [0,zB], which can be solved analytically, and a two-point boundary value problem (BVP) on [zB,zT] that depends on the solutions to the IVPs. The model has equations governing the vertical wind speed, w (ms-1), the air density, ρ (kgm-3), the upward and downward longwave radiation, I+ and I- (Wm-2), the downward shortwave radiation, IS (Wm-2), the latent and sensible heat transport, FC (Wm-2), and the temperature T (K). Any reflection of shortwave radiation from either the surface or the atmosphere is ignored and is simply considered to have left the system. The model is depicted in Fig.  and is derived and explained in detail in the following subsections. Many of the details of the model, including its nondimensionalization, the vanishing conduction limit, and modelling choices used for various functional forms, are in Appendix . Calibration of the model parameters to empirical data is described in detail in Appendix , and the reader is referred to Tables  and  of that appendix for the values of the parameters.

For the Arctic parameter values given in Appendix  and for a given CO2 concentration, μ, the model can be solved numerically. We used MATLAB's built-in BVP solver “bvp5c” to solve individual instances of the (nondimensionalized) model and AUTO for continuation calculations. The results of the model for μ=390 ppm are shown in Fig. . The altitude dependence of the mass flux ϕ and the atmospheric meridional heat transport, FA, were calibrated to an empirical Arctic temperature profile from as detailed in Appendix . From this figure we see that the model fits the temperature profile very well, with some discrepancy near the surface. The overall atmospheric heat transport, Fig. c, indicates that, for our model, the upper half of the troposphere receives the most input of heat, while the bottom fifth actually has a net outward heat transport. This removal of heat near the bottom is likely the cause of the discrepancy between our model values and the Cronin and Jansen data. The negative values of FA near the surface are due to our modelling choice for FA. It is possible that alternative modelling formulations for FA could yield a better fit to the Cronin and Jansen data; however, the forms that we tried (including the ones reported in Appendix  and a few others) did not improve upon the one used here. Nonetheless, given the relative simplicity of our model, we feel the fact that it can fit the data as well as it does is remarkable.

For μ=390, the surface temperature from the solved model is -19.7 ∘C. Starting from this solution, we numerically continued the solution to other values of μ, resulting in the S-shaped curve in Fig. a. Between approximately μ=464 and μ=859, there are three solutions. The lower and upper solutions are stable cold and warm solutions, while the middle branch is unstable. Currently the Arctic is on the cold branch of this curve. The model predicts that, as CO2 levels rise, the equilibrium surface temperature in the Arctic will increase gradually at first. However, when μ exceeds 859 ppm, the model displays a saddle-node bifurcation where the stable cold solution is annihilated together with the unstable solution. At this point the climate would rapidly approach the warm stable solution where surface temperatures are significantly higher. The temperatures on the warm branch may seem unreasonably high. Although our model includes specific details of some of the natural phenomena governing the Arctic climate behaviour and is calibrated with real data, it is primarily a qualitative rather than quantitative model. It is the fact that the model predicts the qualitative feature of a saddle-node bifurcation that is the important result, not the precise temperature of the model's warm solution. Part of the reason the model has relatively high temperatures on the warm branch is because it has constant values for FO and FAtot. It is likely that, should the Arctic's annually averaged temperature rise dramatically, both FO and FAtot would be affected in a downward direction, which would reduce the warm equilibrium temperatures to some degree. Another important emergent feature of the model is that there is bistability for μ in the range [464,859]. Thus, even though the Arctic may in the future be on the cold branch in this range, it is possible that a strong enough climate disturbance could push the climate out of the basin of attraction of the cold solution and into the basin of attraction of the warm solution. The necessary strength of such a disturbance decreases as one moves closer to the upper end of this range. Our model does not incorporate the seasonal variation of solar input to the Arctic but rather uses an annually averaged value. Thus the equilibria in our model are annual averages. The seasonal variation of insolation would effectively result in oscillations around the annual average. These oscillations themselves may make a significant contribution to the “disturbance” needed to push the system out of the basin of attraction of the cold branch.

The International Panel on Climate Change (IPCC) has published various CO2 emission scenarios for the future based on possible levels of global action to suppress such emissions Box TS.6; . These scenarios are called Representative Concentration Pathways (RCPs) and are numbered based on the radiative forcing in the year 2100 due to anthropogenic emissions compared to the year 1750. The original four published RCPs are RCP2.6, RCP4.5, RCP6.0, and RCP8.5, representing strong mitigation (2.6) through “ignore the problem” (8.5) responses by world governments. Each RCP indicates likely CO2 concentration levels in the atmosphere out to the year 2100. From 2100 to 2200, the scenarios assume a “constant composition commitment”, which essentially freezes emission levels and eventually leads to a constant CO2 level in the atmosphere for all but RCP8.5. These RCPs are plotted in Fig. b. The CO2 levels for the four different pathways at the year 2200 are continued as dashed lines into Fig. a. It is clear from the figure that RCP8.5 leads to CO2 concentrations that far surpass the saddle-node bifurcation, whereas the other three RCPs do not. This result is in agreement with our simpler model . Both RCP6.0 and RCP4.5 end at levels within the bistable range, and indeed all the RCPs except RCP2.6 are within that range from about the year 2050 onward. The black dashed-dotted line extending from the bifurcation in the upper panel to RCP8.5 in the lower panel illustrates that RCP8.5 reaches the bifurcation near the year 2092.

The curve of equilibria in Fig. a displays hysteresis: CO2 levels rising past 859 ppm will cause a jump from the cold equilibrium state to the warm state, but a return to the cold state will not happen until CO2 levels are brought below 464 ppm, where the saddle-node bifurcation of the warm equilibrium is located (left bend of the S curve). If CO2 levels follow a trajectory similar to RCP8.5, Arctic climate may change drastically in less than 100 years, but a return to the current cold state may be essentially impossible for thousands of years afterward, assuming humankind can develop and implement the required technology to reduce atmospheric carbon levels sufficiently.

Total atmospheric heat transport, FAtot, and ocean heat transport, FO, are inputs to the model that are empirically based and that are not altered by the state of the model itself. We investigated the persistence of the saddle-node bifurcation and hysteresis in the presence of the uncertainty of these two values. Figure  shows two 2D bifurcation diagrams plotting the locations of the saddle-node bifurcations in the model (the left and right bends of the S curve in Fig. ). Figure a shows the bifurcations as both FAtot and μ vary. Figure b shows a similar diagram where FO replaces FAtot. The right curve in each of the panels is a curve of saddle-node bifurcations corresponding to the right bend in the S curve in Fig. ; they are transitions from a cold state to a warm state. Similarly, the left curve in each of the panels is a curve of saddle-node bifurcations that correspond to the left bend in the S curve and are the transitions from a warm state to a cold state. The shaded area between the two curves is the region of bistability. The curves meet in a cusp bifurcation, but it is important to note that this only occurs when CO2 levels are mathematically negative. Thus the two saddle-node bifurcations and the hysteresis phenomenon will be present for all physically possible CO2 levels and for all reasonable values of FAtot and FO. These bifurcation curves also make it evident that a transition from a cold state to a warm state occurs if either FAtot or FO are increased, even if CO2 levels are constant. If FAtot and FO increase along with carbon dioxide levels in the near future, which seems reasonably likely, then the saddle-node bifurcation will occur at lower levels of CO2. For example, a 10 % increase in FAtot to 110 Wm-2 from 100 would cause the saddle-node bifurcation to occur at about μ=754 ppm, which would mean that both RCP8.5 and RCP6.0 would pass through the saddle-node bifurcation.

Although the model presented here is clearly a simplification of the climate, made possible by the near invariance of the vertical flow on the polar axis, we believe it captures some of the most important aspects relevant for Arctic climate change. The model predicts that if humanity keeps carbon emission levels close to RCP6.0 or lower, then the Arctic will not likely undergo a sudden dramatic rise in annual average temperature. However, if carbon emissions are much worse than RCP6.0, such a change is likely, and the cause is a saddle-node bifurcation of the stable cold equilibrium. Such a change would clearly have catastrophic effects on the Arctic environment, leading to massive global effects. These results are in agreement with those of , who, from a study of various CMIP6 models of Arctic climate, predict that under a low emissions scenario sea ice loss will be seasonal but that for a high emissions scenario it will be year-round for all areas of the Arctic. Further, the hysteresis displayed by the model indicates that a change of this nature may be practically irreversible. Although some comfort may be taken that only the worst of the four carbon pathway scenarios ends in such a catastrophic change, the model shows that both RCP6.0 and RCP4.5 lie in the region of bistability from the year 2070 onward. In a bistable situation, external disturbances could cause the system to jump into the basin of attraction of the warm equilibrium, effectively bringing about the catastrophic change prior to the bifurcation. The model is too simple to allow for any reasonable measure of the likelihood of such an occurrence, but the important thing is that the model exhibits bistability in the parameter region where the system is likely to reside within 50 years' time. This bistability has been shown to persist regardless of the values of the two biggest uncertainties in the model, the atmospheric and ocean heat transport to the Arctic from the mid-latitudes.

Our model addresses the equilibrium state only and represents the Arctic temperature as an annual average. The real Arctic climate undergoes massive seasonal changes, which effectively means that the system is actually oscillating around the equilibrium temperatures of our model. Such temperature oscillations may be sufficient to effectively push a system located on a cold solution in the bistable regime to “above” the unstable solution and so into the basin of attraction of the warm solution.

Seasonal variations in Arctic temperatures and sea ice are studied in many works, including , who argue that a tipping point to a year-round ice-free Arctic is not likely to occur while the Arctic is ice-covered for a significant portion of the year, but once the Arctic is seasonally ice-free for a sufficient number of months in a year, such a tipping point becomes more likely. A possible enhancement to our model would be to include seasonal solar variation and an ice model as in .