Abstract In this study, based on the number of modes, the original three-dimensional Lorenz model (3DLM) is generalized with two additional modes [five-dimensional Lorenz model (5DLM)] to examine their role in the predictability of the numerical solutions and to understand the underlying processes that increase the solution stability. As a result of the simplicity of the 5DLM with respect to existing generalized Lorenz models (LMs), the author is able to obtain the analytical solutions of its critical points and identify the role of the major nonlinear term in the solution’s stability, which have previously not been documented in the literature. The nonlinear Jacobian terms of the governing equations are analyzed to highlight the importance of selecting new modes for extending the nonlinear feedback loop of the 3DLM and thus effectively increasing the degree of nonlinearity (i.e., the nonlinear mode–mode interactions) in the 5DLM. It is then shown that numerical solutions in the 5DLM require a larger normalized Rayleigh number r for the onset of chaos and are more predictable than those in the 3DLM when r is between 25 and 40 and the Prandtl number σ is 10. The improved predictability is attributable to the negative nonlinear feedback enabled by the new modes. The role of the (negative) nonlinear feedback is further verified using a revised 3DLM with a parameterized nonlinear eddy dissipative term. The finding of the increased stability in the 5DLM and revised 3DLM with respect to the 3DLM is confirmed with the linear stability analysis and the analysis of the Lyapunov exponents using different values of r and σ. To further understand the impact of an additional heating term, results from the 5DLM and a higher-dimensional LM [e.g., the six-dimensional LM (6DLM)] are analyzed and compared.

The ability of machine learning (ML) models to "extrapolate" to situations outside of the range spanned by their training data is crucial for predicting the long-term behavior of non-stationary dynamical systems (e.g., prediction of terrestrial climate change), since the future trajectories of such systems may (perhaps after crossing a tipping point) explore regions of state space which were not explored in past time-series measurements used as training data. We investigate the extent to which ML methods can yield useful results by extrapolation of such training data in the task of forecasting non-stationary dynamics, as well as conditions under which such methods fail. In general, we find that ML can be surprisingly effective even in situations that might appear to be extremely challenging, but do (as one would expect) fail when "too much" extrapolation is required. For the latter case, we show that good results can potentially be obtained by combining the ML approach with an available inaccurate conventional model based on scientific knowledge.

Ice sheets appeared in the northern hemisphere around 3 Ma (million years) ago and glacial-interglacial cycles have paced Earth's climate since then. Superimposed on these long glacial cycles comes an intricate pattern of millennial and sub-millennial variability, including Dansgaard-Oeschger and Heinrich events. There are numerous theories about these oscillations. Here, we review a number of them in order to draw a parallel between climatic concepts and dynamical system concepts, including, in particular, the relaxation oscillator, excitability, slow-fast dynamics and homoclinic orbits. Namely, almost all theories of ice ages reviewed here feature a phenomenon of synchronization between internal climate dynamics and astronomical forcing. However, these theories differ in their bifurcation structure and this has an effect on the way the ice age phenomenon could grow 3 Ma ago. All theories on rapid events reviewed here rely on the concept of a limit cycle excited by changes in the surface freshwater balance of the ocean. The article also reviews basic effects of stochastic fluctuations on these models, including the phenomenon of phase dispersion, shortening of the limit cycle and stochastic resonance. It concludes with a more personal statement about the potential for inference with simple stochastic dynamical systems in palaeoclimate science.

The fact that both the Lorenz 1963 and 1969 models suggest finite predictability is well known. However, less well known is the fact that the mechanisms (i.e., sensitivities) within both models, which lead to finite predictability, are different. Additionally, the mathematical and physical relationship between these two models has not been fully documented. New analyses, along with a literature review, are performed here to provide insights regarding similarities and differences for these two models. The models represent different physical systems, one for convection and the other for barotropic vorticity. From the perspective of mathematical complexities, the Lorenz 1963 (L63) model is limited-scale and nonlinear; and the Lorenz 1969 (L69) model is closure-based, physically multiscale, mathematically linear, and numerically ill-conditioned. The former possesses a sensitive dependence of solutions on initial conditions, known as the butterfly effect, and the latter contains numerical sensitivities due to an ill-conditioned matrix with a large condition number (i.e., a large variance of growth rates). Here, we illustrate that the existence of a saddle point at the origin is a common feature that produces instability in both systems. Within the chaotic regime of the L63 nonlinear model, unstable growth is constrained by nonlinearity, as well as dissipation, yielding time varying growth rates along an orbit, and, thus, a dependence of (finite) predictability on initial conditions. Within the L69 linear model, multiple unstable modes at various growth rates appear, and the growth of a specific unstable mode (i.e., the most unstable mode during a finite time interval) is constrained by imposing a saturation assumption, thereby yielding a time varying system growth rate. Both models were interchangeably applied for qualitatively revealing the nature of finite predictability in weather and climate. However, only single type solutions were examined (i.e., chaotic and linearly unstable solutions for the L63 and L69 models, respectively), and the L69 system is ill-conditioned and easily captures numerical instability. Thus, an estimate of the predictability limit using either of the above models, with or without additional assumptions (e.g., saturation), should be interpreted with caution and should not be generalized as an upper limit for atmospheric predictability.

Abstract ‘Past Tipping Points’ shows how the climate and human societies have undergone tipping point changes in the past and therefore they could do so again. It starts with the discovery of over twenty abrupt climate changes during the last ice age, recorded in ice cores from Greenland. Each time a tipping point was crossed, the regional climate jumped by nearly 10°C in a matter of decades. The last abrupt warming around 12,000 years ago tipped the climate out of the last ice age. Then a social tipping point from hunting and gathering to farming began – the ‘Neolithic Revolution’. It recurred independently in different parts of the world, as the climate settled into the relative stability of the ‘Holocene’. At first, populations flip-flopped between foraging and farming before reinforcing feedbacks made the transition to farming irreversible. Around 7,000 years ago, the first complex societies started to emerge, tipped by reinforcing feedbacks of escalating internal complexity, escalating conflict with others, and sometimes the challenges of a deteriorating climate. Civilizations spread, but as they aged, they tended to lose resilience. This made them vulnerable to smaller shocks which occasionally tipped them to collapse back to simpler levels of social organization. Finally, capitalism reached a tipping point into sustained economic growth in the Industrial Revolution and came to dominate the world. With the recent ‘Great Acceleration’, its self-propelling feedbacks are now destabilizing the climate and destroying nature, tying our future to that of the Earth system.

We study the behaviour at tipping points close to (smoothed) non-smooth fold bifurcations in one-dimensional oscillatory forced systems. The focus is the Stommel-Box, and related climate models, which are piecewise-smooth continuous dynamical systems, modelling thermohaline circulation. These exhibit non-smooth fold bifurcations which arise when a saddle-point and a focus meet at a border collision bifurcation. By using techniques from the theory of non-smooth dynamical systems we are able to provide precise estimates for the general tipping behaviour at the non-smooth fold as parameters vary. These are significantly different from the usual tipping point estimates, showing a much more rapid rate of tipping. We also see very rapid, and non-monotone, changes in the tipping points due to the effect of non-smoothness in the system. All of this has important implications for the prediction of tipping in climate systems.

Abstract. Simple climate models that are computationally inexpensive, transparent, and easy to modify are useful for assessing climate policies in the presence of uncertainties. This motivated the creation of SURFER v2.0, a model designed to estimate the impact of CO2 emissions and solar radiation modification on global mean temperatures, sea level rise, and ocean pH. However, SURFER v2.0 is unsuitable for simulations beyond a few thousand years because it lacks some carbon cycle processes. This is problematic for assessing the long-term evolution of the Earth system, particularly the dynamics of ice sheets and the resulting sea level rise. Here, we present SURFER v3.0, an extension to SURFER v2.0 that allows for accurate simulation of the climate, carbon cycle, and sea level rise on timescales ranging from decades to millions of years. We incorporated a dynamic cycling of alkalinity in the ocean, a carbonate sediments reservoir, and weathering fluxes into the model. With these additions, we show that SURFER v3.0 reproduces results from a large class of models, ranging from centennial Coupled Model Intercomparison Project Phase 6 (CMIP6) projections to 1 Myr runs performed with the cGENIE model of intermediate complexity. We show that compared to SURFER v2.0, including long-term carbon cycle processes in SURFER v3.0 leads to a stabilisation of the Greenland ice sheet for the middle of the road emission scenarios and to a significant reduction in the sea level rise contribution from Antarctica for high-emission scenarios.

Abstract. We investigate the probabilities of triggering climate tipping points under various shared socioeconomic pathways (SSPs), and how they are altered by including the additional carbon emissions that could arise from tipping points within the Earth's carbon cycle. Crossing of a climate tipping point at a threshold level of global mean surface temperature (threshold temperature), would commit the affected subsystem of the Earth to abrupt and largely irreversible changes with negative impacts on human well-being. However, it remains unclear which tipping points would be triggered under the different SSPs, due to uncertainties in the climate sensitivity to anthropogenic greenhouse gas emissions, the threshold temperatures of climate tipping points, and the response of tipping points within the Earth's carbon cycle to global warming. We include those uncertainties in our analysis to derive probabilities of triggering for 16 previously-identified climate tipping points within the Earth system. To conduct our analysis, we use the intermediate complexity climate model FaIR which is coupled to a conceptual model of the tipping processes within the Amazon rainforest and permafrost, which are the two major tipping elements within the Earth's carbon cycle. Uncertainties are propagated by employing a Monte Carlo approach for the construction of large model ensembles. We find that intermediate emission scenarios like SSP2-4.5 are highly unsafe with regard to triggering climate tipping points, with an average probability of triggering until the year 2500 of 65 %. Furthermore, the highest long-term temperature increase among all SSPs caused by carbon emissions from the Amazon and permafrost becomes possible under this scenario with 0.16 °C (0.03–0.91 °C) in 2500, which increases the average probability of triggering tipping points by 3.3 percent points (pp). This is due to the fact that maximum carbon emissions from tipping of the Amazon and permafrost become possible under this scenario, and they cause most warming when cumulative anthropogenic emissions are lower due to the saturating response of radiative forcing to increasing greenhouse gas concentrations. The risk of triggering climate tipping points is reduced significantly under SSP1-2.6 and even more so under SSP1-1.9, with average probabilities of triggering of 38 % and 28 % respectively, which are increased by 2.3 pp and 1.1 pp due to carbon emissions from the Amazon and permafrost.

We formulate and analyze a simple dynamical systems model for climate–vegetation interaction. The planet we consider consists of a large ocean and a land surface on which vegetation can grow. The temperature affects vegetation growth on land and the amount of sea ice on the ocean. Conversely, vegetation and sea ice change the albedo of the planet, which in turn changes its energy balance and hence the temperature evolution. Our highly idealized, conceptual model is governed by two nonlinear, coupled ordinary differential equations, one for global temperature, the other for vegetation cover. The model exhibits either bistability between a vegetated and a desert state or oscillatory behavior. The oscillations arise through a Hopf bifurcation off the vegetated state, when the death rate of vegetation is low enough. These oscillations are anharmonic and exhibit a sawtooth shape that is characteristic of relaxation oscillations, as well as suggestive of the sharp deglaciations of the Quaternary. Our model's behavior can be compared, on the one hand, with the bistability of even simpler, Daisyworld-style climate–vegetation models. On the other hand, it can be integrated into the hierarchy of models trying to simulate and explain oscillatory behavior in the climate system. Rigorous mathematical results are obtained that link the nature of the feedbacks with the nature and the stability of the solutions. The relevance of model results to climate variability on various timescales is d

Abstract. We formulate and analyze a simple dynamical systems model for climate–vegetation interaction. The planet we consider consists of a large ocean and a land surface on which vegetation can grow. The temperature affects vegetation growth on land and the amount of sea ice on the ocean. Conversely, vegetation and sea ice change the albedo of the planet, which in turn changes its energy balance and hence the temperature evolution. Our highly idealized, conceptual model is governed by two nonlinear, coupled ordinary differential equations, one for global temperature, the other for vegetation cover. The model exhibits either bistability between a vegetated and a desert state or oscillatory behavior. The oscillations arise through a Hopf bifurcation off the vegetated state, when the death rate of vegetation is low enough. These oscillations are anharmonic and exhibit a sawtooth shape that is characteristic of relaxation oscillations, as well as suggestive of the sharp deglaciations of the Quaternary. Our model's behavior can be compared, on the one hand, with the bistability of even simpler, Daisyworld-style climate–vegetation models. On the other hand, it can be integrated into the hierarchy of models trying to simulate and explain oscillatory behavior in the climate system. Rigorous mathematical results are obtained that link the nature of the feedbacks with the nature and the stability of the solutions. The relevance of model results to climate variability on various time

Extreme Events an Geophysical Mon © 2012. American 10.1029/2011GM The Hurst phenomenon, which reflects long-term fluctuations in geophysical time series, has attracted the attention of the research community for nearly 60 years due to its practical and theoretical importance. Yet a geophysical understanding of the Hurst phenomenon has remained elusive despite mutually conflicting hypotheses of long-term memory and nonstationarity proposed in the hydrology literature. The hypothesis of long-termmemory in the climate system has recently gathered support from climate scientists through the publication of spectra that are consistent with the Hurst phenomenon for decadal to centennial time scales. A possible geophysical basis for this statistical feature has been explored elsewhere using complex global climate models. Here we explore a complementary pathway by means of the dynamics of nonuniformly mixing systems. As these systems pass bifurcations, they can exhibit the characteristics of tipping points and critical transitions that are observed in many complex dynamical systems, ranging from ecosystems to financial markets and the climate. The specific case of a Hopf bifurcation, which marks the transition from a stable system to a cyclic system, is investigated using simple dynamical systems that have been used as climate models involving energy balance and negative feedback with and without the hydrological cycle. Analyses of the model simulations reveal complex nonstationary be

of models called “Earth models of intermediate complexity,” and thought about the relationship between those models and the concept of dynamical complexity

such as the logistic map, can exhibit strange attractors whatever their dimensionality. In contrast, for continuous dynamical systems, the Poincaré–Bendixson Chaos theory is a branch of mathematics and an interdisciplinary area of scientific study. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities. The theory states that within the apparent randomness of chaotic complex systems, there are u Chaos: When the present determines the future but the approximate present does not approximately determine the future. Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and… Topol…