This article presents a bifurcation analysis of a simple Energy Balance Model (EBM) of the Earth’s climate, which suggests that topological change has occurred in the paleoclimate history of the Earth. In the theory of dynamical systems, two systems that are topologically equivalent have solutions with the same qualitative behavior. A change in the topological equivalence class, as parameters are varied, is called a bifurcation. Thus, a bifurcation demarcates a significant change in the behavior of the solutions of a dynamical system. If that system represents climate, then that topological change may represent an abrupt transformation of the climate, occurring even with a very small change in the forcing parameters. In this paper, the existence of a cusp bifurcation is proven in a climate EBM. The existence of this cusp bifurcation implies the co-existence of two distinct stable equilibrium climate states (bistability), as well as the existence of abrupt transitions between these two states (fold bifurcations) in the EBM. These transitions are dependent on the past history of the system (hysteresis). The two universal unfolding parameters for the cusp bifurcation have been determined as functions of the relevant physical parameters. These ideas lead to the proposal of a new explanation for the so-called warm equable climate problem of the mid-Cretaceous and early Eocene. The analysis presented here implies that the mid-Cretaceous and early Eocene climate systems are topologically equivalent to each other, but they are not topologically equivalent to the preindustrial modern climate. The transition from the warm, equable paleoclimate to today’s cooler nonequable climate occurs via fold (or saddle-node) bifurcations in the EBM, which correspond to the Eocene-Oligocene Transition (EOT) at the south pole and the Pliocene-Pleistocene Transition (PPT) at the north pole, in the paleoclimate record of Earth.

Abstract. This article presents the results of a bifurcation analysis of a simple energy balance model (EBM) for the future climate of the Earth. The main focus is on the following question: can the nonlinear processes intrinsic to atmospheric physics, including natural positive feedback mechanisms, cause a mathematical bifurcation of the climate state, as a consequence of continued anthropogenic forcing by rising greenhouse gas emissions? Our analysis shows that such a bifurcation could cause an abrupt change to a drastically different climate state in the EBM, which is warmer and more equable than any climate existing on Earth since the Pliocene epoch. In previous papers, with this EBM adapted to paleoclimate conditions, it was shown to exhibit saddle-node and cusp bifurcations, as well as hysteresis. The EBM was validated by the agreement of its predicted bifurcations with the abrupt climate changes that are known to have occurred in the paleoclimate record, in the Antarctic at the Eocene–Oligocene transition (EOT) and in the Arctic at the Pliocene–Paleocene transition (PPT). In this paper, the EBM is adapted to fit Anthropocene climate conditions, with emphasis on the Arctic and Antarctic climates. The four Representative Concentration Pathways (RCP) considered by the IPCC (Intergovernmental Panel on Climate Change) are used to model future CO2 concentrations, corresponding to different scenarios of anthropogenic activity. In addition, the EBM investigates four naturally occurring nonlinear feedback processes which magnify the warming that would be caused by anthropogenic CO2 emissions alone. These four feedback mechanisms are ice–albedo feedback, water vapour feedback, ocean heat transport feedback, and atmospheric heat transport feedback. The EBM predicts that a bifurcation resulting in a catastrophic climate change, to a pre-Pliocene-like climate state, will occur in coming centuries for an RCP with unabated anthropogenic forcing, amplified by these positive feedbacks. However, the EBM also predicts that appropriate reductions in carbon emissions may limit climate change to a more tolerable continuation of what is observed today. The globally averaged version of this EBM has an equilibrium climate sensitivity (ECS) of 4.34 K, near the high end of the likely range reported by the IPCC.

Climate models predict that the climate of the Earth is warming and will continue to warm in coming centuries, if there is no mitigation. A recent energy balance model [Kypke et al., Nonlin. Process. Geophys. 27 (2020) 391–409] forecasts that, if the current increase of carbon dioxide in the atmosphere continues unabated, then in the next century the climate of the Earth will not only get warmer, but will transition abruptly via a bifurcation, to a warm equable climate unlike any climate seen on Earth since the Pliocene. This transition to a new climate state is a topological change. That model includes the effects of water vapour feedback and ice albedo feedback, as well as ocean and atmospheric heat transport. This paper adds to that model further amplification by permafrost feedback. That is, as the Arctic warms, permafrost will thaw, releasing large amounts of the greenhouse gases carbon dioxide and methane, which cause further warming. Since knowledge of permafrost stores and release rates is limited, a range of permafrost carbon release sensitivities (Q 10) is considered. The model predicts that permafrost feedback accelerates the timing and increases the likelihood of a topological climate change in the Arctic, and reinforces the view that permafrost feedback should not be ignored in Anthropocene climate models.

Abstract. A column model of the Arctic atmosphere-ocean system is developed including the nonlinear responses of surface albedo and water vapor to temperature. The atmosphere is treated as a gray gas and the flux of longwave radiation is governed by the two-stream Schwarzschild equations. Representative carbon pathways (RCPs) are used to model carbon dioxide concentrations into the future. The resulting nine-dimensional two-point boundary value problem is solved under various RCPs and the solutions analyzed. The model predicts that under the highest carbon pathway, the Arctic climate will undergo an irreversible bifurcation to a warm steady state, which would correspond to an annually ice-free situation. Under the lowest carbon pathway, corresponding to very aggressive carbon emission reductions, the model exhibits only a mild increase in Arctic temperatures. Under the two moderate carbon pathways, temperatures increase more substantially, and the system enters a region of bistability where external perturbations could possibly cause an irreversible switch to a warm, ice-free state.

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