The South Atlantic Dipole via multichannel singular spectrum analysis.

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.

Improved subseasonal prediction of South Asian monsoon rainfall using data-driven forecasts of oscillatory modes.

Predicting the temporal and spatial patterns of South Asian monsoon rainfall within a season is of critical importance due to its impact on agriculture, water availability, and flooding. The monsoon intraseasonal oscillation (MISO) is a robust northward-propagating mode that determines the active and break phases of the monsoon and much of the regional distribution of rainfall. However, dynamical atmospheric forecast models predict this mode poorly. Data-driven methods for MISO prediction have shown more skill, but only predict the portion of the rainfall corresponding to MISO rather than the full rainfall signal. Here, we combine state-of-the-art ensemble precipitation forecasts from a high-resolution atmospheric model with data-driven forecasts of MISO. The ensemble members of the detailed atmospheric model are projected onto a lower-dimensional subspace corresponding to the MISO dynamics and are then weighted according to their distance from the data-driven MISO forecast in this subspace. We thereby achieve improvements in rainfall forecasts over India, as well as the broader monsoon region, at 10- to 30-d lead times, an interval that is generally considered to be a predictability gap. The temporal correlation of rainfall forecasts is improved by up to 0.28 in this time range. Our results demonstrate the potential of leveraging the predictability of intraseasonal oscillations to improve extended-range forecasts; more generally, they point toward a future of combining dynamical and data-driven forecasts for Earth system prediction.

Random templex encodes topological tipping points in noise-driven chaotic dynamics.

Random attractors are the time-evolving pullback attractors of deterministically chaotic and stochastically perturbed dynamical systems. These attractors have a structure that changes in time and that has been characterized recently using Branched Manifold Analysis through Homologies cell complexes and their homology groups. This description has been further improved for their deterministic counterparts by endowing the cell complex with a directed graph (digraph), which encodes the order in which the cells in the complex are visited by the flow in phase space. A templex is a mathematical object formed by a cell complex and a digraph; it provides a finer description of deterministically chaotic attractors and permits their accurate classification. In a deterministic framework, the digraph of the templex connects cells within a single complex for all time. Here, we introduce the stochastic version of a templex. In such a random templex, there is one complex per snapshot of the random attractor and the digraph connects the generators or "holes" of successive cell complexes. Tipping points appear in a random templex as drastic changes of its holes in time, through their birth, splitting, merging, or death. This paper introduces random templexes and computes them for the noise-driven Lorenz system's random attractor.

Asymptotic behavior of the forecast-assimilation process with unstable dynamics.

Extensive numerical evidence shows that the assimilation of observations has a stabilizing effect on unstable dynamics, in numerical weather prediction, and elsewhere. In this paper, we apply mathematically rigorous methods to show why this is so. Our stabilization results do not assume a full set of observations and we provide examples where it suffices to observe the model's unstable degrees of freedom.

The PaleoJump database for abrupt transitions in past climates.

Tipping points (TPs) in Earth's climate system have been the subject of increasing interest and concern in recent years, given the risk that anthropogenic forcing could cause abrupt, potentially irreversible, climate transitions. Paleoclimate records are essential for identifying past TPs and for gaining a thorough understanding of the underlying nonlinearities and bifurcation mechanisms. However, the quality, resolution, and reliability of these records can vary, making it important to carefully select the ones that provide the most accurate representation of past climates. Moreover, as paleoclimate time series vary in their origin, time spans, and periodicities, an objective, automated methodology is crucial for identifying and comparing TPs. To address these challenges, we introduce the open-source PaleoJump database, which contains a collection of carefully selected, high-resolution records originating in ice cores, marine sediments, speleothems, terrestrial records, and lake sediments. These records describe climate variability on centennial, millennial and longer time scales and cover all the continents and ocean basins. We provide an overview of their spatial distribution and discuss the gaps in coverage. Our statistical methodology includes an augmented Kolmogorov-Smirnov test and Recurrence Quantification Analysis; it is applied here, for illustration purposes, to selected records in which abrupt transitions are automatically detected and the presence of potential tipping elements is investigated. These transitions are shown in the PaleoJump database along with other essential information about the records, including location, temporal scale and resolution, as well as temporal plots. This open-source database represents, therefore, a valuable resource for researchers investigating TPs in past climates.

Noise-driven topological changes in chaotic dynamics.

Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways. To study these modifications, the present work compares the topological structure of the deterministic Lorenz (1963) attractor with its stochastically perturbed version. The deterministic attractor is well known to be "strange" but it is frozen in time. When driven by multiplicative noise, the Lorenz model's random attractor (LORA) evolves in time. Algebraic topology sheds light on the most striking effects involved in such an evolution. In order to examine the topological structure of the snapshots that approximate LORA, we use branched manifold analysis through homologies-a technique originally introduced to characterize the topological structure of deterministically chaotic flows-which is being extended herein to nonlinear noise-driven systems. The analysis is performed for a fixed realization of the driving noise at different time instants in time. The results suggest that LORA's evolution includes sharp transitions that appear as topological tipping points.

Reduced-order models for coupled dynamical systems: Data-driven methods and the Koopman operator.

Providing efficient and accurate parameterizations for model reduction is a key goal in many areas of science and technology. Here, we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parameterizations of weakly coupled dynamical systems. Such parameterizations yield a set of stochastic integrodifferential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integrodifferential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equation-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings, on the one hand, support the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parameterizations.

Tipping points induced by parameter drift in an excitable ocean model.

Numerous systems in the climate sciences and elsewhere are excitable, exhibiting coexistence of and transitions between a basic and an excited state. We examine the role of tipping between two such states in an excitable low-order ocean model. Ensemble simulations are used to obtain the model's pullback attractor (PBA) and its properties, as a function of a forcing parameter [Formula: see text] and of the steepness [Formula: see text] of a climatological drift in the forcing. The tipping time [Formula: see text] is defined as the time at which the transition to relaxation oscillations (ROs) arises: at constant forcing this occurs at [Formula: see text]. As the steepness [Formula: see text] decreases, [Formula: see text] is delayed and the corresponding forcing amplitude decreases, while remaining always above [Formula: see text]. With periodic perturbations, that amplitude depends solely on [Formula: see text] over a significant range of parameters: this provides an example of rate-induced tipping in an excitable system. Nonlinear resonance occurs for periods comparable to the RO time scale. Coexisting PBAs and total independence from initial states are found for subsets of parameter space. In the broader context of climate dynamics, the parameter drift herein stands for the role of anthropogenic forcing.

Ocean circulation, ice shelf, and sea ice interactions explain Dansgaard-Oeschger cycles.

The last glacial interval experienced abrupt climatic changes called Dansgaard-Oeschger (DO) events. These events manifest themselves as rapid increases followed by slow decreases of oxygen isotope ratios in Greenland ice core records. Despite promising advances, a comprehensive theory of the DO cycles, with their repeated ups and downs of isotope ratios, is still lacking. Here, based on earlier hypotheses, we introduce a dynamical model that explains the DO variability by rapid retreat and slow regrowth of thick ice shelves and thin sea ice in conjunction with changing subsurface water temperatures due to insulation by the ice cover. Our model successfully reproduces observed features of the records, such as the sawtooth shape of the DO cycles, waiting times between DO events across the last glacial, and the shifted antiphase relationship between Greenland and Antarctic ice cores. Our results show that these features can be obtained via internal feedbacks alone. Warming subsurface waters could have also contributed to the triggering of Heinrich events. Our model thus offers a unified framework for explaining major features of multimillennial climate variability during glacial intervals.

Economic networks: Heterogeneity-induced vulnerability and loss of synchronization.

Interconnected systems are prone to propagation of disturbances, which can undermine their resilience to external perturbations. Propagation dynamics can clearly be affected by potential time delays in the underlying processes. We investigate how such delays influence the resilience of production networks facing disruption of supply. Interdependencies between economic agents are modeled using systems of Boolean delay equations (BDEs); doing so allows us to introduce heterogeneity in production delays and in inventories. Complex network topologies are considered that reproduce realistic economic features, including a network of networks. Perturbations that would otherwise vanish can, because of delay heterogeneity, amplify and lead to permanent disruptions. This phenomenon is enabled by the interactions between short cyclic structures. Difference in delays between two interacting, and otherwise resilient, structures can in turn lead to loss of synchronization in damage propagation and thus prevent recovery. Finally, this study also shows that BDEs on complex networks can lead to metastable relaxation oscillations, which are damped out in one part of a network while moving on to another part.

Synchronization of world economic activity.

Common dynamical properties of business cycle fluctuations are studied in a sample of more than 100 countries that represent economic regions from all around the world. We apply the methodology of multivariate singular spectrum analysis (M-SSA) to identify oscillatory modes and to detect whether these modes are shared by clusters of phase- and frequency-locked oscillators. An extension of the M-SSA approach is introduced to help analyze structural changes in the cluster configuration of synchronization. With this novel technique, we are able to identify a common mode of business cycle activity across our sample, and thus point to the existence of a world business cycle. Superimposed on this mode, we further identify several major events that have markedly influenced the landscape of world economic activity in the postwar era.

Introduction to focus issue: Synchronization in large networks and continuous media-data, models, and supermodels.

The synchronization of loosely coupled chaotic systems has increasingly found applications to large networks of differential equations and to models of continuous media. These applications are at the core of the present Focus Issue. Synchronization between a system and its model, based on limited observations, gives a new perspective on data assimilation. Synchronization among different models of the same system defines a supermodel that can achieve partial consensus among models that otherwise disagree in several respects. Finally, novel methods of time series analysis permit a better description of synchronization in a system that is only observed partially and for a relatively short time. This Focus Issue discusses synchronization in extended systems or in components thereof, with particular attention to data assimilation, supermodeling, and their applications to various areas, from climate modeling to macroeconomics.

Publisher's Note: Comment on "Nonparametric forecasting of low-dimensional dynamical systems" [Phys. Rev. E 93, 036201 (2016)].

This corrects the article DOI: 10.1103/PhysRevE.93.036201.

Comment on "Nonparametric forecasting of low-dimensional dynamical systems ".

The comparison performed in Berry et al. [Phys. Rev. E 91, 032915 (2015)] between the skill in predicting the El Niño-Southern Oscillation climate phenomenon by the prediction method of Berry et al. and the "past-noise" forecasting method of Chekroun et al. [Proc. Natl. Acad. Sci. USA 108, 11766 (2011)] is flawed. Three specific misunderstandings in Berry et al. are pointed out and corrected.

Pathogens trigger top-down climate forcing on ecosystem dynamics.

Evaluating the effects of climate variation on ecosystems is of paramount importance for our ability to forecast and mitigate the consequences of global change. However, the ways in which complex food webs respond to climate variations remain poorly understood. Here, we use long-term time series to investigate the effects of temperature variation on the intraguild-predation (IGP) system of Windermere (UK), a lake where pike (Esox lucius, top predator) feed on small-sized perch (Perca fluviatilis) but compete with large-sized perch for the same food sources. Spectral analyses of time series reveal that pike recruitment dynamics are temperature controlled. In 1976, expansion of a size-truncating perch pathogen into the lake severely impacted large perch and favoured pike as the IGP-dominant species. This pathogen-induced regime shift to a pike-dominated IGP apparently triggered a temperature-controlled trophic cascade passing through pike down to dissolved nutrients. In simple food chains, warming is predicted to strengthen top-down control by accelerating metabolic rates in ectothermic consumers, while pathogens of top consumers are predicted to dampen this top-down control. In contrast, the local IGP structure in Windermere made warming and pathogens synergistic in their top-down effects on ecosystem functioning. More generally, our results point to top predators as major mediators of community response to global change, and show that size-selective agents (e.g. pathogens, fishers or hunters) may change the topological architecture of food webs and alter whole ecosystem sensitivity to climate variation.

Lessons on Climate Sensitivity From Past Climate Changes.

Over the last decade, our understanding of climate sensitivity has improved considerably. The climate system shows variability on many timescales, is subject to non-stationary forcing and it is most likely out of equilibrium with the changes in the radiative forcing. Slow and fast feedbacks complicate the interpretation of geological records as feedback strengths vary over time. In the geological past, the forcing timescales were different than at present, suggesting that the response may have behaved differently. Do these insights constrain the climate sensitivity relevant for the present day? In this paper, we review the progress made in theoretical understanding of climate sensitivity and on the estimation of climate sensitivity from proxy records. Particular focus lies on the background state dependence of feedback processes and on the impact of tipping points on the climate system. We suggest how to further use palaeo data to advance our understanding of the currently ongoing climate change.

Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances.

Despite the importance of uncertainties encountered in climate model simulations, the fundamental mechanisms at the origin of sensitive behavior of long-term model statistics remain unclear. Variability of turbulent flows in the atmosphere and oceans exhibits recurrent large-scale patterns. These patterns, while evolving irregularly in time, manifest characteristic frequencies across a large range of time scales, from intraseasonal through interdecadal. Based on modern spectral theory of chaotic and dissipative dynamical systems, the associated low-frequency variability may be formulated in terms of Ruelle-Pollicott (RP) resonances. RP resonances encode information on the nonlinear dynamics of the system, and an approach for estimating them--as filtered through an observable of the system--is proposed. This approach relies on an appropriate Markov representation of the dynamics associated with a given observable. It is shown that, within this representation, the spectral gap--defined as the distance between the subdominant RP resonance and the unit circle--plays a major role in the roughness of parameter dependences. The model statistics are the most sensitive for the smallest spectral gaps; such small gaps turn out to correspond to regimes where the low-frequency variability is more pronounced, whereas autocorrelations decay more slowly. The present approach is applied to analyze the rough parameter dependence encountered in key statistics of an El-Niño-Southern Oscillation model of intermediate complexity. Theoretical arguments, however, strongly suggest that such links between model sensitivity and the decay of correlation properties are not limited to this particular model and could hold much more generally.

Multivariate singular spectrum analysis and the road to phase synchronization.

We show that multivariate singular spectrum analysis (M-SSA) greatly helps study phase synchronization in a large system of coupled oscillators and in the presence of high observational noise levels. With no need for detailed knowledge of individual subsystems nor any a priori phase definition for each of them, we demonstrate that M-SSA can automatically identify multiple oscillatory modes and detect whether these modes are shared by clusters of phase- and frequency-locked oscillators. As an essential modification of M-SSA, here we introduce variance-maximization (varimax) rotation of the M-SSA eigenvectors to optimally identify synchronized-oscillator clustering.