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We study the hemispheric to continental scale regimes that lead to summertime heatwaves in the Northern Hemisphere. By using a powerful data mining methodology - archetype analysis - we identify characteristic spatial patterns consisting of a blocking high pressure systems embedded within a meandering upper atmosphere circulation that is longitudinally modulated by coherent Rossby Wave Packets. Periods when these atmospheric regimes are strongly expressed correspond to large increases in the likelihood of extreme surface temperature. Most strikingly, these regimes are shown to be typical of surface extremes and frequently reoccur. Three well publicised heatwaves are studied in detail - the June-July 2003 western European heatwave, the August 2010 "Russian" heatwave, and the June 2021 "Heatdome" event across western North America, and are shown to be driven by blocking high pressure systems linked to stalled Rossby Wave Packets. We discuss the implications of our work for long-range prediction or early warning, climate model assessment and post-event diagnosis., Comment: 12 figure

Tipping points (TP) are often described as low-dimensional bifurcations, and are associated with early-warning signals (EWS) due to critical slowing down (CSD). CSD is an increase in amplitude and correlation of noise-induced fluctuations away from a reference attractor as the TP is approached. But for high-dimensional systems it is not obvious which variables or observables would display the critical dynamics and carry CSD. Many variables may display no CSD, or show changes in variability not related to a TP. It is thus helpful to identify beforehand which observables are relevant for a given TP. Here we propose this may be achieved by knowledge of an unstable edge state that separates the reference from an alternative attractor that remains after the TP. This is because stochastic fluctuations away from the reference attractor are preferentially directed towards the edge state along a most likely path (the instanton). As the TP is approached the edge state and reference attractor typically become closer, and the fluctuations can evolve further along the instanton. This can be exploited to find observables with substantial CSD, which we demonstrate using conceptual dynamical systems models and climate model simulations of a collapse of the Atlantic Meridional Overturning Circulation (AMOC).

The fluctuation-dissipation theorem is a cornerstone result in statistical mechanics that can be used to translate the statistics of the free natural variability of a system into information on its forced response to perturbations. By combining this viewpoint on response theory with the key ingredients of Koopmanism, it is possible to deconstruct virtually any response operator into a sum of terms, each associated with a specific mode of natural variability of the system. This dramatically improves the interpretability of the resulting response formulas. We show here on three simple yet mathematically meaningful examples how to use the Extended Dynamical Mode Decomposition (EDMD) algorithm on an individual trajectory of the system to compute with high accuracy correlation functions as well as Green functions associated with acting forcings. This demonstrates the great potential of using Koopman analysis for the key problem of evaluating and testing the sensitivity of a complex system.

Nonlinear dynamical systems are ubiquitous in nature and they are hard to forecast. Not only they may be sensitive to small perturbations in their initial conditions, but they are often composed of processes acting at multiple scales. Classical approaches based on the Lyapunov spectrum rely on the knowledge of the dynamic forward operator, or of a data-derived approximation of it. This operator is typically unknown, or the data are too noisy to derive a faithful representation. Here, we propose a new data-driven approach to analyze the local predictability of dynamical systems. This method, based on the concept of recurrence, is closely linked to the well-established framework of local dynamical indices. Applied to both idealized systems and real-world datasets, this new index shows results consistent with existing knowledge, proving its effectiveness in estimating local predictability. Additionally, we discuss its relationship with local dynamical indices, illustrating how it complements the previous framework as a more direct measure of predictability. Furthermore, we explore its reflection of the scale-dependent nature of predictability, its extension that includes a weighting strategy, and its real-time application. We believe these aspects collectively demonstrate its potential as a powerful diagnostic tool for complex systems.

The Atlantic Meridional Overturning Circulation (AMOC) is a much studied component of the climate system, because its suspected multistability is associated with tipping behaviour yielding potentially large regional and global climatic impacts. In this paper we investigate the global stability properties of the system using an ocean general circulation model. We construct an unstable AMOC state, i.e., an unstable solution of the flow that resides between the stable regimes of a vigorous and collapsed AMOC. Such a solution, also known as a Melancholia or edge state, is a dynamical saddle embedded in the boundary separating the competing basins of attraction. It is physically relevant since it lies on the most probable path of a noise-induced transition between the two stable regimes, and because tipping occurs when one of the attractors and the Melancholia state collide. Its properties may thus give hints towards physical mechanisms and predictability of the critical transition. We find that while the AMOC Melancholia state as viewed from its upper ocean properties lies between the vigorous and collapsed regimes, it is characterized by an Atlantic deep ocean that is fresher and colder compared to both stable regimes. The Melancholia state has higher dynamic enthalpy than either stable state, representing a state of higher potential energy that is in good agreement with the dynamical landscape view on metastability.

We present a wrapper that allows Abaqus user material subroutines (UMATs) to be used as an External Material library in the software COMSOL Multiphysics. The wrapper, written in C language, transforms COMSOL's external material subroutine inputs and outputs into Fortran-coded Abaqus UMAT inputs and outputs, by means of a consistent variable transformation. This significantly facilitates conducting coupled, multi-physics studies employing the advanced material models that the solid mechanics community has developed over the past decades. We exemplify the potential of our new framework, UMAT4COMSOL, by conducting numerical experiments in the areas of elastoplasticity, hyperelasticity and crystal plasticity. The source code, detailed documentation and example tutorials are made freely available to download at www.empaneda.com/codes.

We consider the following variants of the classical minimum dominating set problem in graphs: locating-dominating set, locating total-dominating set and open locating-dominating set. All these problems are known to be hard for general graphs. A typical line of attack, therefore, is to either determine the minimum cardinalities of such sets in general or to establish bounds on these minimum cardinalities in special graph classes. In this paper, we study the minimum cardinalities of these variants of the dominating set under a graph operation defined by Mycielski in~\cite{Mycielski1955} and is called the Mycielski construction. We provide some general lower and upper bounds on the minimum sizes of the studied sets under the Mycielski construction. We apply the Mycielski construction to stars, paths and cycles in particular, and provide lower and upper bounds on the minimum cardinalities of such sets in these graph classes. Our results either improve or attain the general known upper bounds.

In multistable dynamical systems driven by weak Gaussian noise, transitions between competing states are often assumed to pass via a saddle on the separating basin boundary. By contrast, we show that timescale separation can cause saddle avoidance in non-gradient systems. Using toy models from neuroscience and ecology, we study cases where sample transitions deviate strongly from the instanton predicted by Freidlin-Wentzell theory, even for weak finite noise. We attribute this to a flat quasipotential and present an approach based on the Onsager-Machlup action to aptly predict transition paths., Comment: Resubmitted version 2

The Indian monsoon brings around 80% of the annual rainfall over the summer months June--September to the Indian subcontinent. The timing of the monsoon onset and the associated rainfall has a large impact on agriculture, thus impacting the livelihoods of over one billion people. To improve forecasting the monsoon on sub-seasonal timescales, global climate models are in continual development. One of the key issues is the representation of convection, which is typically parametrised. Different convection schemes offer varying degrees of performance, depending on the model and scenario. Here, we propose a method to compute a convective timescale, which could be used as a metric for comparison across different models and convection schemes. The method involves the determination of a vertical convective flux between the lower and upper troposphere through moisture budget analysis, and then relating this to the total column moisture content. The method is applied to a WRF model simulation of the 2016 Indian monsoon, giving convective timescales that are reduced by a factor of 2 when the onset of the monsoon occurs. The convective timescale can also be used as an indicator of monsoon transitions from pre-onset to full phase of the monsoon, and to assess changes in monsoon phases under future climate scenarios., Comment: 21 pages, 9 Figures plus 10 Supplementary Figures

While one can unequivocally identify past climate transitions, we lack comprehensive knowledge about their underlying mechanisms and timescales. Our study employs a dimensional analysis of benthic stable isotope records to uncover, across different timescales, how the climatic fluctuation of the Cenozoic are associated with changes in the number of effective degrees of freedom. Precession timescales dominate the Hothouse and Warmhouse states, while the Icehouse climate is primarily influenced by obliquity and eccentricity timescales. Notably, the Coolhouse state lacks dominant timescales. Our analysis proves effective in objectively identifying abrupt climate shifts and extremes. This is also demonstrated using high-resolution data from the last glacial cycle, revealing abrupt climate shifts within a single climate state. These findings significantly impact our understanding of the inherent stability of each climate state and the evaluation of (paleo-)climate models' ability to replicate key features of past/future climate states and transitions.

Detecting recurrent weather patterns and understanding the transitions between such regimes are key to advancing our knowledge on the low-frequency variability of the atmosphere and have important implications in terms of weather and climate-related risks. We adapt an analysis pipeline inspired by Markov State Modelling and detect in an unsupervised manner the dominant winter mid-latitude Northern Hemisphere weather patterns in the Atlantic and Pacific sectors. The daily 500 hPa geopotential height fields are first classified in $\sim 200$ microstates. The weather dynamics is then represented in the basis of these microstates and the slowest decaying modes are identified from the spectral properties of the transition probability matrix. These modes are defined on the basis of the nonlinear dynamical processes of the system and not as tentative metastable states as often done in Markov state analysis. In the Atlantic and Pacific sectors slow relaxation processes are mainly related to transitions between blocked regimes and zonal flow. We also find strong evidence of a dynamical regime associated with the simultaneous Atlantic-Pacific blocking. When the analysis is performed in a broader geographical region of the Atlantic sector, we discover that the slowest relaxation modes of the system are associated with transitions between dynamical regimes that resemble teleconnection patterns like the North Atlantic Oscillation and weather regimes like the Scandinavian and Greenland blocking, yet have a much stronger dynamical foundation than classical methods based e.g. on EOF analysis. Our method clarifies that, as a result of the lack of a time-scale separation in the atmospheric variability of the mid-latitudes, there is no clear-cut way to represent the atmospheric dynamics in terms of few, well-defined modes of variability., Comment: 32 pages, 4 figures, plus supplementary material. Final accepted version

Anticipating critical transitions in the Earth system is of great societal relevance, yet there may be intrinsic limitations to their predictability. For instance, from the theory of dynamical systems possessing multiple chaotic attractors, it is known that the asymptotic state depends sensitively on the initial condition in the proximity of a fractal basin boundary. Here, we approach the problem of final-state sensitivity of the Atlantic Meridional Overturning Circulation (AMOC) using a conceptual climate model, composed of a slow bistable ocean coupled to a fast chaotic atmosphere. First, we explore the occurrence of long chaotic transients in the monostable regime, which can mask a loss of stability near bifurcations. In the bistable regime, we explicitly construct the chaotic saddle using the edge tracking technique. Quantifying the final-state sensitivity through the maximum Lyapunov exponent and the lifetime of the saddle, we find that the system exhibits a fractal basin boundary with almost full phase space dimension, implying vanishing predictability of the second kind near the basin boundary. Our results demonstrate the usefulness of studying non-attracting chaotic sets in the context of predicting climatic tipping points, and provide guidance for the interpretation of higher-dimensional models such as general circulation models.

This paper introduces a novel approach to quantifying ecological resilience in biological systems, particularly focusing on noisy systems responding to episodic disturbances with sudden adaptations. Incorporating concepts from non-equilibrium statistical mechanics, we propose a measure termed `ecological resilience through adaptation,' specifically tailored to noisy, forced systems that undergo physiological adaptation in the face of stressful environmental changes. Randomness plays a key role, accounting for model uncertainty and the inherent variability in the dynamical response among components of biological systems. Our measure of resilience is rooted in the probabilistic description of states within these systems, and is defined in terms of the dynamics of the ensemble average of a model-specific observable quantifying success or well-being. Our approach utilizes stochastic linear response theory to compute how the expected success of a system, originally in statistical equilibrium, dynamically changes in response to a environmental perturbation and a subsequent adaptation. The resulting mathematical derivations allow for the estimation of resilience in terms of ensemble averages of simulated or experimental data. Finally, through a simple but clear conceptual example, we illustrate how our resilience measure can be interpreted and compared to other existing frameworks in the literature. The methodology is general but inspired by applications in plant systems, with the potential for broader application to complex biological processes.

The predictability of weather and climate is strongly state-dependent: special and extremely relevant atmospheric states like blockings are associated with anomalous instability. Indeed, typically, the instability of a chaotic dynamical system can vary considerably across its attractor. Such an attractor is in general densely populated by unstable periodic orbits that can be used to approximate any forward trajectory through the so-called shadowing. Dynamical heterogeneity can lead to the presence of unstable periodic orbits with different number of unstable dimensions. This phenomenon - unstable dimensions variability - implies a serious breakdown of hyperbolicity and has considerable implications in terms of the structural stability of the system and of the possibility to describe accurately its behaviour through numerical models. As a step in the direction of better understanding the properties of high-dimensional chaotic systems, we provide here an extensive numerical study of the dynamical heterogeneity of the Lorenz '96 model in a parametric configuration leading to chaotic dynamics. We show that the detected variability in the number of unstable dimensions is associated with the presence of many finite-time Lyapunov exponents that fluctuate about zero also when very long averaging times are considered. The transition between regions of the attractor with different degrees of instability comes with a significant drop of the quality of the shadowing. By performing a coarse graining based on the shadowing unstable periodic orbits, we can characterize the slow fluctuations of the system between regions featuring, on the average, anomalously high and anomalously low instability. In turn, such regions are associated, respectively, with states of anomalously high and low energy, thus providing a clear link between the microscopic and thermodynamical properties of the system., Comment: 28 pages, 11 figures, final accepted version

The eddy-driven jet stream and storm tracks in the mid-latitude atmosphere are known to shift in latitude on various timescales, but the physical processes that cause these shifts are still unclear. In this study, we introduce a minimal dynamical system derived from the classical Phillips two-level model with the goal of elucidating the essential mechanisms responsible for the interaction between eddies and mean flow. Specifically, we aim to understand the link between the structure of the eddies and the shift of the latitudinal maximum of the zonal flow. By varying the horizontal shape of the eddies, we find three distinct dynamical regimes whose occurrence depends on the intensity of the external baroclinic forcing: a purely zonal flow, a barotropic eddy regime with net poleward momentum flux, and a baroclinic eddy regime with both net poleward momentum and temperature flux. For weak baroclinic forcing, the classical zonal flow solution with latitudinal maximum at the centre of the beta-channel is found. For strong forcing, if eddies are southwest-northeast tilted and zonally elongated, the system is in the baroclinic eddy regime, resulting in a poleward shift of the jet. The intermediate barotropic eddy regime also features a poleward shifted jet, yet with eddies structurally distinct from the baroclinic regime. Changing the parameters yields transitions between the regimes that can be either continuous or discontinuous in terms of the properties of the atmosphere. The findings of this study also provide insights into the properties of the storm track change between the jet entrance and jet exit regions of the North Atlantic., Comment: 15 pages, 4 figures; final accepted version

Tipping points (TP) in climate sub-systems are usually thought to occur at a well-defined, critical forcing parameter threshold, via destabilization of the system state by a single, dominant positive feedback. However, coupling to other sub-systems, additional feedbacks, and spatial heterogeneity may promote further small-amplitude, abrupt reorganizations of geophysical flows at forcing levels lower than the critical threshold. Using a primitive-equation ocean model we simulate a collapse of the Atlantic Meridional Overturning Circulation (AMOC) due to increasing glacial melt. Considerably prior to the collapse, various abrupt, qualitative changes in AMOC variability occur. These intermediate tipping points (ITP) are transitions between multiple stable circulation states. Using 2.75 million years of model simulations, we uncover a very rugged stability landscape featuring parameter regions of up to nine coexisting stable states. The path to an AMOC collapse via a sequence of ITPs depends on the rate of change of the meltwater input. This challenges our ability to predict and define safe limits for TPs.

A novel FFT-based phase-field fracture framework for modelling fatigue crack initiation and propagation at the microscale is presented. A damage driving force is defined based on the stored energy and dislocation density, relating phase-field fracture with microstructural fatigue damage. The formulation is numerically implemented using FFT methods to enable modelling of sufficiently large, representative 3D microstructural regions. The early stages of fatigue cracking are simulated, predicting crack paths, growth rates and sensitivity to relevant microstructural features. Crack propagation through crystallographic planes is shown in single crystals, while the analysis of polycrystalline solids reveals transgranular crack initiation and crystallographic crack growth.

Klaus Hasselmann's revolutionary intuition in climate science was to take advantage of the stochasticity associated with fast weather processes to probe the slow dynamics of the climate system. This has led to fundamentally new ways to study the response of climate models to perturbations, and to perform detection and attribution for climate change signals. Hasselmann's program has been extremely influential in climate science and beyond. We first summarise the main aspects of such a program using modern concepts and tools of statistical physics and applied mathematics. We then provide an overview of some promising scientific perspectives that might better clarify the science behind the climate crisis and that stem from Hasselmann's ideas. We show how to perform rigorous model reduction by constructing parametrizations in systems that do not necessarily feature a time-scale separation between unresolved and resolved processes. We propose a general framework for explaining the relationship between climate variability and climate change, and for performing climate change projections. This leads us seamlessly to explain some key general aspects of climatic tipping points. Finally, we show that response theory provides a solid framework supporting optimal fingerprinting methods for detection and attribution., Comment: 18 pages, 5 figures; final accepted version

We consider a class of nonequilibrium systems of interacting agents with pairwise interactions and quenched disorder in the dynamics featuring, in the thermodynamic limit, phase transitions. We provide conditions on the microscopic structure of interactions among the agents that lead to a dimension reduction of the system in terms of a finite number of reaction coordinates. Such reaction coordinates prove to be proper nonequilibrium thermodynamic variables as they carry information on correlation, memory and resilience properties of the system. Phase transitions can be identified and quantitatively characterised as singularities of the complex valued susceptibility functions associated to the reaction coordinates. We provide analytical and numerical evidence of how the singularities affect the physical properties of finite size systems.

The Earth's climate has experienced numerous critical transitions during its history, which have often been accompanied by massive and rapid changes in the biosphere. Such transitions are evidenced in various proxy records covering different timescales. The goal is then to identify, date, and rank past critical transitions in terms of importance, thus possibly yielding a more thorough perspective on climatic history. To illustrate such an angle, which inspired the punctuated equilibrium angle on the theory of evolution, we have analyzed 2 key high-resolution datasets: the CENOGRID marine compilation (past 66 Myr), and North Atlantic U1308 record (past 3.3 Myr). By combining recurrence analysis of the individual time series with a multivariate representation of the system based on the theory of the quasi-potential, we identify the key abrupt transitions associated with major regime changes that differentiate various clusters of climate variability. This allows interpreting the time-evolution of the system as a trajectory taking place in a dynamical landscape, whose multiscale features are associated with a hierarchy of tipping points., Comment: 21 pages, 5 figures plus appendix plus supplementary material; final published version

Detection and attribution (D&A) studies are cornerstones of climate science, providing crucial evidence for policy decisions. Their goal is to link observed climate change patterns to anthropogenic and natural drivers via the optimal fingerprinting method (OFM). We show that response theory for nonequilibrium systems offers the physical and dynamical basis for OFM, including the concept of causality used for attribution. Our framework clarifies the method's assumptions, advantages, and potential weaknesses. We use our theory to perform D&A for prototypical climate change experiments performed on an energy balance model and on a low-resolution coupled climate model. We also explain the underpinnings of degenerate fingerprinting, which offers early warning indicators for tipping points. Finally, we extend the OFM to the nonlinear response regime. Our analysis shows that OFM has broad applicability across diverse stochastic systems influenced by time-dependent forcings, with potential relevance to ecosystems, quantitative social sciences, and finance, among others., Comment: 11 pages, 3 figures, Final accepted version by PRL

We study a two-layer energy balance model, that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being captured by the other layer, and the effect of all non radiative vertical exchanges of energy. Therefore, an essential parameter is the absorptivity of the atmosphere, denoted $\epsilon_a$. The value of $\epsilon_a$ depends critically on greenhouse gases: increasing concentrations of $CO_2$ and $CH_4$ lead to a more opaque atmosphere with higher values of $\epsilon_a$. First we prove that global existence of solutions of the system holds if and only if $\epsilon_a \in (0, 2)$, and blow up in finite time occurs if $\epsilon_a > 2$. (Note that the physical range of values for $\epsilon_a$ is $(0, 1]$.) Next, we explain the long time dynamics for $\epsilon_a \in (0, 2)$, and we prove that all solutions converge to some equilibrium point. Finally, motivated by the physical context, we study the dependence of the equilibrium points with respect to the involved parameters, and we prove in particular that the surface temperature increases with respect to $\epsilon_a$. This is the key mathematical manifestation of the greenhouse effect., Comment: 56 pages, 17 figues

The South and East Asian summer monsoons are globally significant meteorological features, creating a strongly seasonal pattern of precipitation. The stability of the monsoon is of extreme importance for a vast range of ecosystems and for the livelihoods of a large share of the world's population. Simulations are performed with an intermediate complexity climate model, PLASIM, to assess the future response of the monsoons to changing concentrations of aerosols and greenhouse gases. The aerosol loading consists of a mid-tropospheric warming and a surface cooling, which is applied to India, Southeast Asia and East China, both concurrently and independently. The primary effect of increased aerosol loading is a decrease in summer precipitation in the vicinity of the applied forcing, although the regional response varies significantly. The decrease in precipitation is only partially ascribable to a decrease in the precipitable water, and instead derives from a reduction of the precipitation efficiency, due to changes in the stratification of the atmosphere. When the aerosol loading is added in all regions simultaneously, precipitation in East China is most strongly affected, with a quite distinct transition to a low precipitation regime as the radiative forcing increases beyond 60 W/m^2. The response is less abrupt as we move westward, with precipitation in South India being least affected. This lower sensitivity in South India is attributed to aerosol forcing over East China. Additionally, the effect on precipitation is approximately linear with the forcing. The impact of doubling carbon dioxide levels is to increase precipitation over the regions and weaken the circulation. When the carbon dioxide and aerosol forcings are applied at the same time, the carbon dioxide forcing partially offsets the surface cooling and reduction in precipitation associated with the aerosol response., Comment: 37 pages, 14 figures plus 12 Supplementary figures

The frequency and magnitude of weather extreme events have increased significantly during the past few years in response to anthropogenic climate change. However, global statistical characteristics and underlying physical mechanisms are still not fully understood. Here, we adopt a statistical physics and probability theory based method to investigate the nature of extreme weather events, particularly the statistics of the day-to-day air temperature differences. These statistical measurements reveal that the distributions of the magnitudes of the extreme events satisfy a universal \textit{Gumbel} distribution, while the waiting time of those extreme events is governed by a universal \textit{Gamma} function. Further finite-size effects analysis indicates robust scaling behaviours. We additionally unveil that the cumulative distribution of logarithmic waiting times between the record events follows an \textit{Exponential} distribution and that the evolution of this climate system is directional where the underlying dynamics are related to a decelerating release of tension. The universal scaling laws are remarkably stable and unaffected by global warming. Counterintuitively, unlike as expected for record dynamics, we find that the number of quakes of the extreme temperature variability does not decay as one over time but with deviations relevant to large-scale climate extreme events. Our theoretical framework provides a fresh perspective on the linkage of universality, scaling, and climate systems. The findings throw light on the nature of the weather variabilities and could guide us to better forecast extreme events.

We consider a class of models describing an ensemble of identical interacting agents subject to multiplicative noise. In the thermodynamic limit, these systems exhibit continuous and discontinuous phase transitions in a, generally, nonequilibrium setting. We provide a systematic dimension reduction methodology for constructing low dimensional, reduced-order dynamics based on the cumulants of the probability distribution of the infinite system. We show that the low dimensional dynamics returns the correct diagnostic properties since it produces a quantitatively accurate representation of the stationary phase diagram of the system that we compare with exact analytical results and numerical simulations. Moreover, we prove that the reduced order dynamics yields also the prognostic, i.e., time dependent properties, as it provides the correct response of the system to external perturbations. On the one hand, this validates the use of our complexity reduction methodology since it retains information not only of the invariant measure of the system but also of the transition probabilities and time dependent correlation properties of the stochastic dynamics. On the other hand, the breakdown of linear response properties is a key signature of the occurrence of a phase transition. We show that the reduced response operators capture the correct diverging resonant behaviour by quantitatively assessing the singular nature of the susceptibility of the system and the appearance of a pole for real value of frequencies. Hence, this methodology can be interpreted as a low dimensional, reduced order approach to the investigation and detection of critical phenomena in high dimensional interacting systems in settings where order parameters are not known., Comment: Accepted Version of the manuscript