Abrupt transitions in the North Atlantic Subpolar Gyre’s (SPG’s) behavior are a major source of uncertainty in decadal-scale climate predictability, as well as having potentially strong impacts on the Atlantic Meridional Overturning Circulation (AMOC), European climate, and marine ecosystems. Climate model simulations suggest that the SPG can undergo irreversible transitions from a regime of deep convection and strong circulation to one characterized by weak convection and reduced transport. Such a collapse would substantially cool the North Atlantic and could interact with a weakening AMOC in complex and nonlinear ways.SPG transitions emerge from the interplay between high-dimensional ocean dynamics and unresolved stochastic processes, making it difficult to represent them faithfully in current Earth system models and challenging for deterministic prediction frameworks. Here, we use CESM2 pre-industrial control simulations to perform a data-driven analysis of SPG dynamics. We construct a machine-learning-based stochastic neural emulator designed to learn, forecast, and quantify uncertainty in SPG evolution. The model simultaneously learns the conditional mean dynamics and state-dependent ensemble spread, enabling fully probabilistic predictions of key prognostic variables.This approach provides a tractable framework for investigating the mechanisms and precursors of SPG weakening and deep-convection collapse, and for assessing associated climate risks. When generalized across models, our approach also offers a pathway for systematically evaluating long-timescale North Atlantic dynamics in Earth system simulations.

This Focus Issue forms one part (NDS-G) of the Double Focus Issue on nonautonomous dynamical systems in the sciences that was published together with a second Focus Issue on nonautonomous dynamical systems in climate (NDS-C). Nonlinear dynamics has achieved a pre-eminent status in the mathematical, physical, and life sciences over the last couple of decades. A very significant recent direction is dealing with the role of time dependence in the forcing and parameters of the models under study. Paramount examples concern the role of anthropogenic effects on climate and biodiversity as well as machine learning and biomedical problems. The mathematical theory of such effects is developing rapidly via the study of nonautonomous and random dynamical systems. Applications are blossoming and have in return been driving further developments of the theory, as we discuss in this Focus Issue.

This Focus Issue is, along with the one on Nonautonomous dynamical systems: Theory, methods, and applications, part of the Double Focus Issue on Nonautonomous dynamical systems in the sciences. We refer to the two twin Focus Issues as NDS-G and NDS-C, for short, where "G" stands for General and "C" for Climate. A key area of inquiry for understanding climate behavior in this century is the impact of anthropogenic and natural forcing on a system that is highly nonlinear and exhibits both chaotic and random aspects. The theory of NDSs is the perfect framework for exploring this impact. The 16 papers in this issue address several questions within this broad area: (i) various types of tipping that have arisen in the system's history or may occur in its future; (ii) the effects that one component of the climate system might have on another one; and (iii) how we may better learn from observations and model simulations about all that is going on within the system.

This Double Focus Issue consists of two linked Focus Issues-(a) Nonautonomous dynamical systems: Theory, methods, and applications and (b) Nonautonomous dynamics in the climate sciences, which we refer to as NDS-G and NDS-C, for short, where "G" stands for general and "C" for climate. Nonlinear dynamics has achieved a leading status in the mathematical, physical, and life sciences over the last couple of decades. A very significant recent direction is dealing with the role of time dependence in the external forcing and the parameters of the models under study. Paramount examples concern the role of anthropogenic effects on climate and biodiversity or of ageing on biomedical problems. The mathematical theory of such effects is developing rapidly via the study of NDSs and random dynamical systems. Applications are blossoming and have in return been driving further developments of the theory. These theoretical advances and their application to an increasing number of areas of the physical, life, and socioeconomic sciences were on view in two recent meetings: a three-day minisymposium within the framework of Dynamics Days Europe 2023 in Naples, Italy, and an International Seminar and Workshop in Dresden, Germany, in October 2023. Discussions at these two events led to this Double Focus Issue, taking contributions from a wide community of researchers in the relevant areas.

Data assimilation, a vital process in areas such as numerical weather prediction, integrates observational data into computational models to provide accurate forecasts. In this study, we conceptualize the forecast-assimilation (FA) process as a dynamic-stochastic system driven by time-dependent observational data. The core objective is to investigate the stability of this process with respect to variations in its initial conditions, particularly when the underlying system dynamics, referred to here as the signal, exhibit instability. We provide a rigorous analysis for both linear and nonlinear dynamics to determine conditions under which the FA process remains stable. In the nonlinear case, we identify an exponential semi-group whose stability is used to prove a uniform in time bound on the expected Wasserstein distance between the true FA process and one that is incorrectly initialized. For linear dynamics, we prove that the FA process converges both weakly and in the Wasserstein topology to a "nominal" one. For this, we use a representation of the FA process by means of the classical Kallianpur-Striebel formula. We show that the Wasserstein distance between the FA process correctly initialized and one which is incorrectly initialized converges to 0 exponentially fast provided the wrong initial condition is absolutely continuous with respect to the correct initial condition.

Templexes are topological objects that encode the branching organization of a flow in phase space. We build on these objects to introduce the concept of topological modes of variability (TMVs). TMVs are defined as dynamical manifestations of algebraically defined cycles, called generatexes, in the templex; they provide a concrete link between abstract topological invariants and time-dependent behavior in a model or in observations. We apply this approach to a low-order model of the wind-driven ocean circulation, subject to both periodic and aperiodic forcing, and show how TMVs emerge or vanish over time in nonautonomous settings. The analysis reveals that TMVs allow for a qualitatively new understanding of variability in complex systems where linear modes fail to describe the nonlinear dynamics.

There is a history of simple forecast error growth models designed to capture the key properties of error growth in operational numerical weather prediction (NWP) models. We propose here such a scalar model that relies on the previous ones and incorporates multiplicative noise in a nonlinear stochastic differential equation (SDE). We analyze the properties of this SDE, including the shape of the error growth curve for small times and its stationary distribution, and prove well-posedness and positivity of solutions. Next, we fit this model to operational NWP error growth curves, and show good agreement with both the mean and probabilistic features of the error growth. These results suggest that the dynamic-stochastic error growth model proposed herein and similar ones could play a role in many other areas of the sciences that involve prediction.

Heatwaves represent a major health hazard, as was the case for the 2003 summer heat wave, responsible for more than 70 000 deaths in western Europe. The interplay of mean flow, quasi-periodic and random fluctuations — associated with the westerly jet, Rossby and gravity waves, and eddies — of the large-scale temperature field results in complex heat wave–related amospheric conditions. Understanding how the different physical processes interact is thus crucial for prediction of heat wave events. In this study, we use the triple decomposition of turbulent flow (Hussain and Reynolds, JFM, 1972) to compute mean, quasi-periodic and random energy fluctuations of the temperature field, that is the 1-point energy budget of temperatures. This decomposition takes into account all interactions between the zonal jet, Rossby waves, gravity waves, and eddies. Both spectral and dynamical systems analyses are applied to the computed terms. More specifically, the concept of extremal length (Ahlfors, Vol. 371, AMS, 2010) is integrated into the equations to quantify how each term of the energy budget equations contributes to the "trapping" of temperature anomalies over Europe. Results show that, amid positive sea surface temperatures and negative soil moisture anomalies, during the first half of August, i.e., the hottest days of the heat wave, quasi-periodic oscillations of polar air increased, resulting in meridional migration of cold air over Canada, and subsequent mixing with warmer air coming from North America. This mixing triggered baroclinic instabilities that led to production of turbulent eddies, which by August 5th suddenly stopped their eastward progression, creating a cyclonically stalled regime over the Mid-Atlantic; this stationary cyclone interacted positively with North African warm air propagating northward over Europe, thus sustaining the dry conditions over France and much of Western Europe. The cyclonic block finally disappeared, stopping the warm air advection from North Africa, with temperatures falling just after that. The study reveals that interactions between quasi-periodic and random processes of production, diffusion and dissipation of a scalar field’s energy play an important role in the evolution of a major heatwave. Hence, the 2003 event was not just the result of a superposition of Rossby waves and eddy anomalies. Extremal length analysis thus reveals that the zonal advection of temperature anomalies was blocked by interactions between quasi-periodic and random production and diffusion processes. This study highlights the complex turbulent interactions that lead to major heat wave events, and the fact that each such event is thus unique.

The wind-driven ocean circulation comprises the oceanic currents that are visible at the surface. In this paper, we use algebraic topology concepts and methods to study a highly simplified model of the evolution of this circulation subject to periodic winds. The low-order spectral model corresponds to a midlatitude ocean basin. For steady forcing, the model's intrinsic oscillations undergo a bifurcation from small-amplitude harmonic ones to relaxation oscillations (ROs) of high amplitude as the forcing increases. The ROs, in turn, give rise to chaotic behavior under periodic forcing. Topological invariants help identify distinct flow regimes that ensemble simulations visit under the action of the underlying deterministic rule in such a nonautonomous framework. We introduce topological variability modes of this idealized ocean circulation, based on the previously defined invariants.

Low-order climate models can play an important role in understanding low-frequency variability in the atmospheric circulation and how forcing consistent with anthropogenic climate change may affect this variability. Here, we study a conceptual model of the mid-latitudes' atmospheric circulation from the perspective of nonautonomous dynamical systems. First, a bifurcation analysis is carried out under time-independent forcing in order to identify different types of behavior in the autonomous model's parameter space. Next, we focus on the study of the nonautonomous system in which the cross-latitudinal heat flux varies seasonally, according to insolation changes. The forward attractor of the seasonally forced model is compared with the attractor of the autonomous one. The seasonal forcing results in a clear change of the attractor's shape. The summer attractor loses its periodicity, and, hence, predictability, when the forcing is seasonal, while the winter attractor favors energy transport through one of the model's two wave components. Climate change forcing produces several remarkable effects. Thus, the analysis of the model's forward attractor under climate trends suggests that the jet speed does not always follow the sign of the change in equator-to-pole thermal contrast, while the change in the energy transported by the eddies does. Chaotic behavior can be completely suppressed in favor of a regular periodic one and vice versa. Circulation patterns can change, suddenly disappear, and rebuild. The model's forward attractor in the presence of time-dependent forcing proves to be a robust tool to study model changes in internal variability due to climate trends, both positive and negative.

Causal inference is at the heart of the scientific method as usually practiced. Still, Karl Popper (The Logic of Scientific Discovery, 1935/1959) tells us that a theory in the empirical sciences can never be proven: it can only be falsified, meaning that it can, and should, be scrutinized with decisive experiments. Even so, nobody that I know writes or publishes papers to disprove one’s own theory, only an opposing theory. And the debate rages on.At the heart of this session lies the question of whether, and how, one can prove, rather than just disprove, a causal link between phenomena in the empirical sciences. The session deals specifically with statistical, as opposed to dynamical methods. These methods have the advantage that they are essentially indifferent to any laws of, or other accumulated heuristic ideas on, the field to which they are being applied: whether the time series one considers are from the environmental sciences, biology or medicine does not matter, only their length and accuracy does.Judea Pearl (e.g., Stat. Surveys, 2009) made an important observation on how to transcend the saying that “Correlation is not causation” by pointing out that standard methods of statistical analysis rely on the stationarity hypothesis of the phenomena being examined. Crucial questions, however, like the causal role of anthropogenic forcing in climate change, deal precisely with the causes of nonstationarity. In particular, Pearl suggested counterfactual analysis as an essential approach in establishing criteria for the necessary and sufficient character of a given cause for a given phenomenon. Thus, the common approach of detection and attribution in the climate sciences only covers the sufficiency aspect of anthropogenic forcing, and more can be done (Hannart et al., BAMS, 2016; Clim. Change, 2016).The present talk will cover four specific aspects of these broad issues: (i) the distinction between information transfer, including both linear correlations and nonlinear extensions thereof, and true causation; (ii) the divergent results of some widely, and not so widely, used methods of studying information transfer (Krakovska et al., PRE, 2018; Kossakowski et al., Psychol. Methods, 2021; Delforge et al., HESS, 2022); (iii) shared variability of climatic time series (De Viron, GRL, 2013; ); and (iv) the uses of data assimilation in applying counterfactual theory to nonstationary phenomena (Carrassi, QJRMS, 2017; Metref et al., QJRMS, 2019).Conclusions will include the obvious one that statistical studies of causal inference have to be complemented by dynamical ones.

Theoretical and numerical studies have shown that transient atmospheric motions leading to weather extremes can be classified through the instantaneous dimension and stability of a state of a dynamical system [Faranda et al., Sci. Rep., 2017]. The asymptotic values of these quantities can be computed theoretically only for specific systems, while their numerical counterpart for climate observables provides information on the rarity, predictability, and persistence of specific states. In this work, we present a first attempt to relate the presence of extreme events with the elements that make up a templex of the system under study, both in the deterministic [Charó et al., Chaos, 2022] and stochastic frameworks [Charó et al., Chaos, 2023]. The templex provides the key characteristics of the topological structure underlying a dynamical system. This work will present results for the classical, deterministic Lorenz [JAS, 1963] attractor and for the Lorenz Random Attractor, dubbed LORA [Ghil & Sciamarella, NPG, 2023].

The climate system is nonlinear and affected by both natural variability and several types of forcing(1). The impact of anthropogenic forcing and environmental change on several of the system’s nonlinear processes has led to considerable concern about the crossing of planetary boundaries(2) and the tipping of regional subsystems(3–5), due to their potentially irreversible consequences. On the global level, these nonlinear effects have been shown to have given rise to bistability(6) and chaotic behavior(7) in the system’s past(8–10) and could do so in its future(11). However, specific mechanisms for a sudden tipping to an alternate stable “hothouse,”(12) several degrees warmer than the present climate, have not been explored so far to a satisfactory extent(13,14). Here we show that a highly simplified energy balance model of globally averaged temperature T representing the radiative budget, coupled with global carbon dioxide dynamics, does exhibit such an alternate stable hothouse climate with T higher by roughly 10 C than the present. The model captures two biogeophysical mechanisms, which lead to the crossing of planetary boundaries that in turn trigger a global tipping to such a hothouse. The two regional mechanisms are (i) the decrease of terrestrial albedo due to the darkening of ice sheets by pervasive algal blooming(15–19); and (ii) the limits of vegetation adapting to increased environmental stress and, hence, the reduction of its carbon absorbtion(20–22).

This study delves into the predictability of atmospheric blocking, zonal, and transition patterns utilizing a simplified coupled model. This model, implemented in Python, emulates midlatitude atmospheric dynamics with a two-layer quasi-geostrophic channel atmosphere on a beta-plane, encompassing simplified land effects. Initially, we comprehensively scrutinize the model's responses to environmental parameters like solar radiation, surface friction, and atmosphere-ground heat exchange. Our findings confirm that the model faithfully replicates real-world Earth-like flow regimes, establishing a robust foundation for further analysis. Subsequently, employing Gaussian mixture clustering, we successfully delineate distinct blocking, zonal, and transition flow regimes, unveiling their dependencies on surface friction. To gauge predictability and persistence, we compute the averaged local Lyapunov exponents for each regime. Our investigation uncovers the presence of zonal, blocking, and transition regimes, particularly under conditions of reduced surface friction. As surface friction increases further, the system transitions to a state characterized by two blocking regimes and a transition regime. Intriguingly, periodic behavior emerges under specific surface friction values, returning to patterns observed under low friction coefficients. Model resolution increase impacts the system in a way that only two regimes are then obtained with the clustering: the transition phase disappears and the predictability drops to roughly 2 days for both of the remaining regimes. In accordance with previous research findings, our study underscores that when all three regimes coexist, zonal patterns exhibit a more extended predictability horizon compared to blocking patterns. Remarkably, transition patterns exhibit reduced predictability when coexisting with the other regimes. In addition, within a specified range of surface friction values where two blocking regimes are found, it is observed that blocked atmospheric situations in the west of the applied topography are marked by instabilities and reduced predictability in contrast to the blockings appearing on the eastern side of the topography.

Boolean delay equations (BDEs) are equations with discrete variables evolving in continuous time. They serve as exploratory tools in the study of nonlinear and complex systems. In this review paper, we outline their formulation and illustrate their properties by application to a solid-earth problem and a climate one. The first problem is the seismotectonic description and prediction of earthquakes and of their clustering. The second one is that of the coupled atmosphere-ocean phenomenon of the ElNi & ntilde;o-Southern Oscillation (ENSO). Both involve irregular behavior that is hard to predict, although some form of cyclicity is present in both, especially in ENSO. The paper concludes with broad perspectives on the further use of BDEs in the geosciences and elsewhere.

The goal of this thesis is to build a reduced-complexity model of coupled climate─economy─biosphere interactions, which uses the minimum number of variables and equations needed to capture the fundamental mechanisms involved and can thus help clarify the role of the different variables and parameters. The Coupled Climate─Economy─Biosphere (CoCEB) model described herein takes an integrated assessment approach to simulating global change. By using an endogenous growth module with physical and human capital accumulation, this thesis considers the sustainability of economic growth, as economic activity intensifies greenhouse gas emissions that in turn cause economic damage due to climate change. Various climate change mitigation policy measures are considered. While many integrated assessment models treat abatement costs merely as an unproductive loss of income, this thesis considers abatement activities also as an investment in increase of overall energy efficiency of the economy and decrease of overall carbon intensity of the energy system. One of the major drawbacks of integrated assessment models is that they mainly focus on mitigation in the energy sector and consider emissions from land-use as exogenous. Since greenhouse gas emissions from deforestation and current terrestrial uptake are significant, it is important to include mitigation of these emissions in the biota sinks within integrated assessment models. Several studies suggest that forest carbon sequestration can help reduce atmospheric carbon concentration significantly and is a cost efficient way to curb the prevailing climate change. This thesis also looks at relevant economic aspects of deforestation control and carbon sequestration in forests as well as the efficiency of carbon capture and storage (CCS) technologies as policy measures for climate change mitigation. Because full realistic coupled climate models are so complex, analyses of the various potential feedbacks between climate, economy, and biosphere have been rather limited. Potentially important mechanisms are better initially described in low or intermediate complexity models. vi The CoCEB is a formal framework in which it is possible to represent in a simple way different elements of the coupled system and their interactions. The model developed, being an exercise in simplicity and not a predictive tool for climate change impacts, brings together and summarizes information from diverse literature on climate change mitigation measures and their associated costs, and allows comparing them in a coherent way. The model is, of course sensitive, to the choice of key parameters and in particular the parameters setting the costs of the different means of climate change mitigation: the parameter values tested span the range of cost values found in literature. The thesis shows that: i) investment in low-carbon technologies helps to reduce the volume of industrial carbon emissions, lower temperature deviations, and lead to positive effects in the long term economic growth; ii) low investment in CCS contributes to reducing industrial carbon emissions and to increasing gross domestic product (GDP), but further investment leads to a smaller reduction in emissions, as well as in the incremental GDP growth; iii) enhanced deforestation control contributes to a reduction in both deforestation emissions and atmospheric carbon dioxide concentration, thus reducing the impacts of climate change and contributing to a slight appreciation of GDP growth, an effect that is very small, though, compared to that of lowcarbon technologies or CCS; and iv) the results in i) and ii) remain very sensitive to the formulation of technological improvements costs. To the contrary, the results for deforestation control are less sensitive to the formulation of its cost. A large range of hypotheses on these costs appear in the literature, and our modeling framework permitted to span this range and check the sensitivity of results The sensitivity study is not intended to make precise calibrations; rather, it is meant to provide a tool for studying qualitatively how various climate policies affect the economy.

This study analyzes coupled atmosphere-ocean variability in the South Atlantic Ocean. To do so, we characterize the spatio-temporal variability of annual mean sea-surface temperature (SST) and sea-level pressure (SLP) using Multichannel Singular Spectrum Analysis (M-SSA). We applied M-SSA to ERA5 reanalysis data (1959-2022) of South Atlantic SST and SLP, both individually and jointly, and identified a nonlinear trend, as well as two climate oscillations. The leading oscillation, with a period of 13 years, consists of a basin-wide southwest-northeast dipole and is observed both in the individual variables and in the coupled analysis. This mode is reminiscent of the already known South Atlantic Dipole, and it is probably related to the Pacific Decadal Oscillation and to El Niño-Southern Oscillation in the Pacific Ocean. The second oscillation has a 5-year period and also displays a dipolar structure. The main difference between the spatial structure of the decadal, 13-year, and the interannual, 5-year mode is that, in the first one, the SST cold tongue region in the southeast Atlantic's Cape Basin is included in the pole closer to the equator. Together, these two oscillatory modes, along with the trend, capture almost 40% of the total interannual variability of the SST and SLP fields, and of their co-variability. These results provide further insights into the spatio-temporal evolution of SST and SLP variability in the South Atlantic, in particular as it relates to the South Atlantic Dipole and its predictability.