Introduction to the Special Issue on the Statistical Mechanics of Climate
aa r X i v : . [ phy s i c s . a o - ph ] J un Introduction to the Special Issue on the StatisticalMechanics of Climate
Valerio Lucarini
Department of Mathematics and Statistics,University of Reading,Reading UKCentre for the Mathematics of Planet Earth,University of Reading, Reading,UK andCEN, University of Hamburg, Hamburg,Germany ∗ (Dated: June 25, 2020) Abstract
We introduce the special issue on the
Statistical Mechanics of Climate pub-lished on the Journal of Statistical Physics by presenting an informal dis-cussion of some theoretical aspects of climate dynamics that make it a topicof great interest for mathematicians and theoretical physicists. In partic-ular, we briefly discuss its nonequilibrium and multiscale properties, therelationship between natural climate variability and climate change, thedifferent regimes of climate response to perturbations, and critical transi-tions. ∗ Email: [email protected] http://mpe.dimacs.rutgers.edu ), which has paved the way for many scientific initiativesand funding opportunities, and, more generally, for bringing to the spotlight a vast range ofinterdisciplinary research activities of great relevance in terms of science per se as well as ofsocio-environmental challenges they can contribute to. Obviously, there is a long history oftwo-way interactions between Earth system sciences, on the one hand, and mathematics andtheoretical physics, on the other hand see the examples of chaos theory, (geophysical) fluiddynamics, fractals, extreme value theory, stochastic dynamical systems, data assimilation,just to name a few. Additionally, since the very start of the computer age, modelling exer-cises dedicated to the simulation of the weather, and later of the ocean and of the climateas a whole have consistently been some of the heaviest users of high-performance comput-ing. Finally, the current revolution in data science is finding very important applications inEarth system science as well as receiving many challenging inputs from it (Buchanan, 2019;Faranda et al. , 2019; Hosni and Vulpiani, 2018).The goal of this special issue is to contribute to such interdisciplinary challenges by hostingscientific contributions that, on the one hand, try to move forward through theory, numericalsimulations, and analysis of data the understanding of the climate system through the lensof statistical mechanics and, on the other hand, develop ideas of relevance for mathematicsand physics taking inspiration from problems emerging in climate science.Using mostly a plain language, we introduce here some theoretical aspects of climatedynamics that make it a topic of great interest for mathematicians and theoretical physicists,and try to convince the reader of the existence of a great potential for important results bothat fundamental level and in terms of usable tools for studying specific problems associatedwith the understanding of the dynamics of the climate system. A more detailed expositionof the topics presented here (and of much more) can be found in a recent review paper(Ghil and Lucarini, 2020); see also Lucarini et al. (2014) and Ghil (2015, 2019).2 . THE CLIMATE AS A NONEQUILIBRIUM SYSTEM
Nonequilibrium statistical mechanics has made substantial progresses in recent decades(Gallavotti, 2006, 2014) and, as we will see below, the investigation of the climate systemseems to be a perfect setting for applying its tools and for finding intellectual challengesfor developing new general ideas. The nonequilibrium conditions of the climate system areprimarily set by the inhomogeneous absorption of solar radiation (Peixoto and Oort, 1992).In addition, the climate system also receives a direct mechanical forcing from solar andlunar tides. Such a forcing, while indeed important for some specific phenomena, is muchless relevant than the radiative forcing coming from the Sun and can be altogether neglectedin the discussion below.The absorption of solar radiation preferentially takes place a) near the surface rather thanin the deeper levels of the atmosphere and of the ocean; and b) at low rather than at highlatitudes. An approximate steady state is reached in the climate system through a complexset of dynamical and thermodynamical processes that reduce the temperature gradients thatwould be established were only radiative processes involved (Lucarini et al. , 2014). By andlarge, convective motions are key to reducing the inhomogeneity resulting from a), whilelarge scale atmospheric and ocean heat transport is responsible for reducing the equator-to-pole temperature difference associated with b) above, thus providing a mechanism of globalstability to the system. The hydrological cycle plays a fundamental role in terms of energeticsof the climate system, mainly because of the large latent heat associated with water phasechanges and of the possibly very large spatial scale associated with water transport processes(Lucarini et al. , 2010; Lucarini and Ragone, 2011; Pauluis, 2007; Trenberth et al. , 2009).It is possible to provide a succinct view of climate dynamics based on thermodynamicalarguments: the climate can be seen as an imperfect engine able to transform available po-tential energy associated with temperature gradients into kinetic energy in the form of windsand oceanic currents through a variety of dynamical processes. The dominating processesare convective instability near the tropics, and baroclinic (and, to a much lesser extent,barotropic) instability in the mid-latitudes (Peixoto and Oort, 1992; Vallis, 2006). The ki-netic energy is continuously dissipated through various mechanism of friction, while availablepotential energy is dissipated through diffusive processes. This is the so-called Lorenz energycycle (Lorenz, 1967, 1955), which can be augmented by defining the efficiency and the en-3ropy production of the climate (Ambaum, 2010; Kleidon and Lorenz, 2005; Lalibert´e et al. ,2015; Lucarini, 2009; Pauluis and Held, 2002a,b). The Lorenz energy cycle angle on thedynamics of climate leads to clearly pointing out the separate roles played by the two maingeophysical fluids. To a first approximation, the atmosphere is heated from below, and so isthermodynamically active (Lorenz, 1967; Schneider, 2006), while the ocean is heated fromabove, which leads to imposing - by and large - a stable stratification. As a result, the oceancurrents are mainly mechanically driven by surface winds, even if a non-trivial role is playedalso by localised density perturbations at the surface (Cessi, 2019; Dijkstra and Ghil, 2005;Kuhlbrodt et al. , 2007). As of today, at least in the author’s opinion, despite the many mer-its of the description of the climate system as a thermal engine, a comprehensive and closedtheory of climate dynamics able to explain coherently instabilities and stabilization mech-anisms on the basis of the fundamental astronomical, physical, chemical, and geometricalparameters of the Earth system has not yet been formulated.
II. THE CLIMATE AS A MULTISCALE SYSTEM
The nature of the external forcings, the inhomogeneity of the physical and chemical prop-erties of the components of the climate system, as well as the great variety of dynamicalprocesses occurring within each climatic component and of the coupling mechanisms betweendifferent components lead to the presence of non-trivial variability on a vast range of scales,covering over ten orders of magnitude in space - from Kolmogorov’s dissipation scale to theEarth’s radius - and even more than that in time - from microseconds to hundreds of millionsof years (Ghil, 2002). Our knowledge of the system is extremely limited in terms of observa-tional data: direct measurements of the climate system obtained with different and evolvingtechnology, and thus having a moderate amount of synchronic and diacronic coherence, areavailable, to a first approximation, only in the industrial era (Ghil and Malanotte-Rizzoli,1991), whereas for the more distant past one can only resort to indirect measurements inthe form of proxy data (Cronin, 2010). The partial and inaccurate observational data aremerged dynamically with a numerical model describing the evolution of the geophysical flu-ids through the process of data assimilation, whose aim is to provide the best time-dependentestimate of the state of the system given a set of available observations (Carrassi et al. , 2018;Ghil and Malanotte-Rizzoli, 1991; Kalnay, 2003).4dditionally, it is unthinkable, given our current scientific understanding at large and ouravailable or foreseen technological capabilities, to create a numerical model able to directlysimulate the climate system in all details for a time frame covering all the relevant timescales. Furthermore, following the Poincar´e parsimony principle, even if we had such amodel, it would not serve the scope of advancing scientific knowledge, but would rather bea virtual reality emulator that would overwhelm a user by details to the point of obscuringthe overall understanding of the problem; see a related discussion in, e.g., Held (2005). Asa result, each numerical model used for studying the climate system is formulated in such away that only a (very limited) range of scales and processes are directly simulated, whereasthe rest are either approximately parametrized and/or used to define suitable boundaryconditions. Correspondingly, in order to study particular classes of phenomena, approximateevolution equations - which provide the basis for the numerical modelling - are derivedfrom the fundamental equations describing the dynamics of climate (basically, Navier-Stokesequations for multicomponent and multiphase thermodynamical fluids in a rotating frame ofreference with a vast array of time-dependent forcings and non-trivial boundary conditions)in order to filter out certain physical processes that are heuristically assumed to play onlya minor role at the temporal and spatial scales of interest (Holton and Hakim, 2013; Klein,2010; Vallis, 2006).Therefore, the problem of constructing accurate and efficient reduced-order models (or,equivalently, of defining the coarse-grained dynamics) is an essential and fundamental aspectof studying the dynamics of climate, both theoretically and through simulations. Tradition-ally, parametrization schemes are formulated in such a way that one expresses the impacton the scales of interest of processes occurring within the unresolved scales via deterministicfunctions of the resolved variables. It has more recently become apparent, in the spiritof what is implied by the Mori-Zwanzig projection operator (Mori, 1965; Zwanzig, 1961),that, instead, parametrizations should involve stochastic and non-markovian components(Berner et al. , 2017; Franzke et al. , 2015; Palmer and Williams, 2009). Many strategiesfor constructing theoretically rigorous parametrizations have been devised, which can bebroadly divided into top-down - see e.g., Grooms and Majda (2013); Majda et al. (2001);Vissio and Lucarini (2018); Wouters and Lucarini (2012, 2013); Wouters and Gottwald(2019) - and data-driven approaches - see, e.g., Kondrashov et al. (2015, 2006); Kravtsov et al. (2005); Wilks (2005)). 5
II. CLIMATE VARIABILITY AND CLIMATE RESPONSE
One needs to remark that, additionally, the climate system is only in an approximatesteady state, because the incoming radiation is subject to quasi-random (e.g. sunspots, solarflares) as well as slow quasi-periodic modulations (e.g. Milankovitch cycles), and, on verylong time scales, to changes in the intensity of the solar irradiance, resulting from the Sun’sevolution. Additionally, the boundary conditions of the system change at a very slow pacein correspondence to geological processes, while the atmospheric composition is affected byvolcanic eruptions, and, in more recent times, by Humanity itself, which acts as very rapidgeological agent (Saltzman, 2001). Understanding climate change and its relationship tounperturbed natural climate variability is a grand challenge for contemporary science, withclear implications on: • better understanding and predicting how the ongoing anthropogenic climate changewill manifest itself at different spatial and temporal scales, and how it will impactdifferent subdomains of the climate system; • gaining a more detailed knowledge on the co-evolution of the Earths climate and oflife on Earth; • better defining planetary habitability, i.e. the potential to develop and maintain envi-ronments hospitable to life (at least in the form we know or can envision) in a planetor in a satellite.When trying to relate forced and free variability of a system, the default option is to try touse (one of the variants of) the fluctuation-dissipation theorem (Kubo, 1966; Marconi et al. ,2008). Indeed, the fluctuation-dissipation theorem, which was originally formulated forsystems that are near thermodynamic equilibrium, can be seen as a dictionary able totranslate the statistics of free (thermal) fluctuations into a prediction of the response of thesystem to external perturbations. In the case of the climate system, this amounts to saying, grosso modo , that the statistical properties of a different climate can be reconstructed bychanging the statistical weight of the natural modes of variability of the reference climate.This viewpoint seems unable to account for the possibility of climatic surprises, i.e.the occurrence in the perturbed climate of events that were absent in the reference case,as in the case of erratic variations in extreme events. This aspect hints at the need of6dopting a slightly different approach when relating climate variability and climate change(Gritsun and Lucarini, 2017).Indeed, Ruelle’s response theory (Ruelle, 1998, 2009) - see Liverani and Gou¨ezel (2006)for a rigorous functional analysis viewpoint - indicates that for nonequilibrium systemsobeying deterministic dynamics the fluctuation-dissipation theorem does not hold in itsusual formulation. A nontrivial relationship between unforced fluctuations and response ofthe system to perturbations can be recovered by using the formalism of the transfer operatorand studying the properties of the so-called Ruelle-Pollicott poles (Pollicott, 1985; Ruelle,1986) of the unpertubed system (Chekroun et al. , 2014; Lucarini, 2018) .An extremely fascinating aspect of climate variability is associated with the occurrenceof extreme events, such as heat waves, cold spells, droughts, floods, wind storms, and manyothers. Extreme meteo-climatic events can be wildly different in terms of spatial and tem-poral scales of interest (e.g. droughts are typically associated with much longer time scalesand much large spatial scales than floods) because of the variety of physical processes re-sponsible for them. The special importance given to the study of extremes in climate comesessentially from their relevance in terms of impacts - while, by definition, rare in terms ofoccurrence, they are disproportionally responsible for damage inflicted to society and ecosys-tems (IPCC, 2012). Extreme value theory (Coles, 2001) allows for a detailed description ofextreme meteo-climatic events (Ghil et al. , 2011; Katz et al. , 2005). Recently, it has beenshown that the investigation of extremes allows for understanding the dynamical propertiesof the system generating them (Lucarini et al. , 2016, 2014b). These results are finding ap-plications for providing a new viewpoint for the investigation of atmospheric predictability(Faranda et al. , 2017). The application of large deviation theory in the study of the climateis rather recent, and it has shown great potential in describing the properties of persistentextreme events like heat waves (G´alfi et al. , 2019). At a more abstract level, large deviationtheory-based tools have been instrumental in nudging climate model simulations towardsvery rarely explored regions of the phase space, thus enhancing tremendously the possibilityof studying mechanisms behind extreme events (Ragone et al. , 2018).Finally, we have to keep in mind that we should abandon the hypothesis of considering A rigorous formulation of response theory in the context of stochastic dynamics has been proposed byHairer and Majda (2010). Recently, Wormell and Gottwald (2019) have clarified the link between thedeterministic and the stochastic viewpoints. Note that in the case of stochastic systems (a general formof) the fluctuation-dissipation theorem is valid (Marconi et al. , 2008). See Gottwald (2020) for a recentspecial issue on linear response theory and its applications. et al. , 2013; Chekroun et al. , 2011; Ghil et al. , 2008), whichare the support of a time-dependent measure. While it is possible to provide precise mathe-matical definitions for the pullback attractor, the construction of the corresponding measurein a given numerical model and its use for computing the time-dependent values of observ-ables of interest is very challenging at practical level, because one needs many ensemblemembers for approximating the actual measure with the empirical one. Under suitable hy-potheses of structural stability - namely the chaotic hypothesis (Gallavotti and Cohen, 1995)- Ruelle’s response theory appears, despite the difficulties in constructing the response op-erators (Abramov and Majda, 2007), as an efficient and flexible tool for calculating climateresponse to weak and moderate forcings, greatly generalising classical concepts like equilib-rium climate sensitivity (long term change in the globally averaged surface temperature asa result of doubling in the CO concentration (von der Heydt et al. , 2016; IPCC, 2014a)),for explaining (Nijsse and Dijkstra, 2018) the theory of emergent constraints (Collins et al. ,2012) and, more generally, for reconstructing the properties of the pullback attractor from asuitably defined reference background state (Lembo et al. , 2020; Lucarini et al. , 2017); seealso an interesting application in (Aengenheyster et al. , 2018). Changes in the statistics ofextreme events can also be predicted using Ruelle’s response theory (Lucarini et al. , 2014b). IV. THE CLIMATE CRISIS: NON-SMOOTH CLIMATE RESPONSE AND CRIT-ICAL TRANSITIONS
The previous discussion, as well the usual narrative about climate change, often givesthe impression that climate change manifests itself essentially through gradual modulations.Time goes by, the CO concentration increases, and the temperature goes up. In this regard,the expression climate crisis has recently become prominent in the public discourse and someprominent media outlets do not use anymore the expression climate change (see, e.g., thestatement made by editorial board of The Guardian at https://tinyurl.com/yyu54ae5 ).On the one hand, climate crisis evokes a strong emotional response and conveys a sense ofurgency; on the other hand, it is, indeed, a very appropriate technical term, because one8f the most pressing challenges in climate science is achieving a much deeper understand-ing of its critical transitions (Kuehn, 2011; Scheffer, 2009), which are usually referred to as tipping points in the Earth system science jargon (Ashwin et al. , 2012; Boers et al. , 2017;Lenton et al. , 2008). These can be seen, by and large, as nonequilibrium phase transitionsleading to drastic and possibly catastrophic changes in the climate. When a system nearsa critical transition, its properties have a rough dependence on its parameters, because re-sponse to perturbations is greatly enhanced (Chekroun et al. , 2014). Conversely, the radiusof expansion of response theory becomes very small (Lucarini, 2016) and the decay of corre-lation for physically meaningful observables slows down, as a result of a vanishing spectralgap associated to the subdominant Ruelle-Pollicott pole(s) (Tantet et al. , 2018).Critical transitions in the climate system are especially relevant because they often ac-company the property of multistability. If one considers the case of deterministic dynamics,in a certain range of values of the parameters of the system, there are two or more competingsteady states that can be reached by the system. Important climatic subsystems such asthe Atlantic Meridional Overturning Circulation (Rahmstorf et al. , 2005) and the Amazonecosystem (Wuyts et al. , 2017) are considered to be bistable. And the climate as a wholeis indeed multistable, as the current astrophysical and astronomical conditions support atleast two possible climates - the one we live in, and the ice-covered one, often referred toas snowball state (Boschi et al. , 2013; Budyko, 1969; Ghil, 1976; Hoffman and Schrag, 2002;Sellers, 1969).The competing steady states are associated with attractors that are the asymptotic setsof orbits starting inside their corresponding basins. On the boundary of such basins wehave invariant sets, the Melancholia states, that attract initial conditions on the basinboundary. These states can be constructed using the edge tracking algorithm originallydevised for studying turbulent fluids (Skufca et al. , 2006). Similarly to the case of simplegradient systems, such unstable saddles define the global stability properties of the system(Grebogi et al. , 1983; Lai and T´el, 2011; Lucarini and B´odai, 2017). The Melancholia statesare the gateways of noise-induced transitions, no matter the noise law one selects. Largedeviation theory (Touchette, 2009) provides us with tools to compute the invariant mea-sure of the system and the statistics of transition times between different metastable states(Lucarini and B´odai, 2020, 2019). 9 . THIS SPECIAL ISSUE We present below a brief narrative of this special issue, and propose a rough thematicgrouping of the included contributions, which cover most of the topics mentioned in theprevious sections as well as exploring further exciting research directions. • Weiss et al. (2019) provide a conceptually elegant characterisation of the nonequilib-rium properties of the climate system by computing persistent probability currents,which cannot be found in equilibrium systems. The corresponding current loopsare used to characterise key climatic features like the El-Ni˜no Southern Oscillation(ENSO) and the Madden–Julian Oscillation and define a new indicator in the formof a probability angular momentum. Instead of focusing on steady state properties,Gottwald and Gugole (2019) propose a data-driven method based on the Koopmanoperator to detect eventual regime changes and fast transient dynamics in time se-ries, and provide convincing applications in a chaotic partial differential equation andin atmospheric data of the Southern and of the Northern Hemisphere. Equilibriumstatistical mechanics is instead the framework of the contribution by Conti and Badin(2020), who investigate the so-called generalized Euler equations, including those de-scribing surface quasi-geostrophic dynamics, which is especially relevant for small scaleoceanic features associated with horizontal gradients of buoyancy. They propose a gen-eralised selective decay principle able to explain the equilibrium state of the flow. Sucha principle imposes that the solutions of these equations approach the states that min-imise the generalized potential enstrophy compatibly with the value of the generalizedenergy. • The investigations of the multiscale nature of climate is a fil rouge of this special issue.Chekroun et al. (2019) proposes an innovative framework for constructing nonlinearparameterizations of unresolved scales of motions using a variational approach andpresents applications in the context of the primitive equations describing the dynam-ics of the atmosphere and the Rayleigh–B´enard convection. Tondeur et al. (2020),conversely, explores new aspects of coupled data assimilation schemes able to mergeobservational and model generated data pertaining to from both the atmosphere andthe ocean. A model reduction technique based on a stochastic variational approach for10eophysical fluid dynamics is used for developing a new ensemble-based data assimila-tion methodology for high-dimensional fluid dynamics models in Cotter et al. (2020).Eichinger et al. (2020), instead, address the impact of adding noise in the form of ad-ditive fractional Brownian motion on fast-slow systems and investigates the problemof estimating how likely is for the trajectories to stay nearby the slow manifold of thedeterministic system. • A new paradigm for climate science is suggested in three closely linked contributions.Alonso-Or´an et al. (2020) propose in the case of two-dimensional Euler-Boussinesqequations a closed theory of weather and climate - intended as statistics of fluctuationsand expectation value of the quantities of interest, respectively. This is achievedby taking into account the corresponding
Lagrangian averaged stochastic advectionby Lie transport equations, which are nonlinear and non-local, in both physical andprobability space (Drivas et al. , 2020). This formalism is further used to explore theproperties of a stochastically perturbed low-dimensional chaotic dynamical system.Geurts et al. (2019) show how to construct an effective dynamics for the expectationvalues of the solutions, which can be mapped to the original deterministic system byconsidering a renormalised dissipation. • A three-part contribution tries to address the overall problem of inferring informationon the natural variability of a system by studying the Ruelle-Pollicott resonances ex-tracted from time series of partial observations. The overall theoretical framework ispresented in Chekroun et al. (2020), while in Tantet et al. (2020) it is shown how toinvestigate Hopf bifurcations for a stochastic system and extract from data a clearfingerprint of nonlinear oscillations taking place within a stochastic background. Fur-thermore, the theoretical findings are exploited for studying the oscillations in theCane-Zebiak stochastic ENSO model (Tantet et al. , 2019). • The concept of natural variability requires using a different framework when consider-ing nonautonomous systems. As discussed above, the characterisation of the propertiesof the pullback attractor requires considering ensemble simulations, and it is in generalnot clear how large the ensemble should be in order to be able to make meaningfulstatements in the statistical properties of the system. This issue is addressed by Pierini(2019) using a simplified model of the ocean. Time-dependent correlations between11he parallel climate realizations defining the pullback attractor are used in T´el et al. (2019) to provide an alternative definition for teleconnections, i.e. correlation betweenanomalies of climatic fields in regions that are geographically very far away. • The problem of studying how the climate system responds to forcings is investigatedin different directions. Ashwin and von der Heydt (2019) contrast the usual definitionof equilibrium climate sensitivity, which relies, as discussed above, on taking a linearapproximation to the climate response, with a fully nonlinear version, and aim atstudying the behaviour of such quantities in the presence of tipping points in simpleclimate models. Climate response is investigated on paleoclimatic data by Ahn et al. (2019) with the goal of understanding whether it is possible to define causal linksbetween different proxy records as a result of the presence of cross-correlations, anddiscuss the need for conjecturing the presence of a separate forcing responsible for theobserved signals. The problem of predicting climate response to forcings motivatesthe study by Santos Guti´errez and Lucarini (2020), who provide general formulas forcomputing linear and nonlinear response to fairly general forcings in the context offinite Markov chains and use them, after performing a discretisation of the phasespace, to study the sensitivity of a deterministic and a stochastic dynamical system.Furthermore, Marangio et al. (2019) study the linear response of a stochastically forcedArnold circle map with respect to the frequency of the driving frequency. This map istaken as a climate toy model, and the goal of the study is, specifically, to gain insightsinto the ENSO phenomenon. Finally, the problem of exploring the multistabilityproperties of the climate system acts as primary motivation for the study by B´odai(2020), who proposes a new algorithm for evaluating efficiently in a stochasticallyperturbed system the quasi-potential barrier that confines a given metastable stateand tests the approach in a classical energy balance climate model. • The study of extreme events is a key aspect of several contributions to this special issue.Ragone and Bouchet (2019) discuss critically how concepts and algorithms informedby large deviation theory can be applied to study persistent extreme events in climateand present an application focusing on persistent and spatially extended heat wavesperformed using a comprehensive climate models. A different angle on persistentextremes is given in the study by Caby et al. (2019), who provide a very informative12eview of the extremal index. The extremal index is a quantity one defines in thecontext of extreme value theory that can be used to quantify the presence of clustersof extremes events, and can, in the context of dynamical systems, provide informationon local and global properties of the attractors. Caby et al. (2019) study both the caseof deterministic and of stochastic dynamical systems, and present also an example ofanalysis of actual atmospheric data. Extreme value theory is used by Pons et al. (2020)to propose a way to address well-known problem of the curse of dimensionality inestimating the Hausdorff dimension of the attractor from time series. The methodologyis applied to study recurrences in synthetically generated as well as real-life financialand climate data and define the degree of non-randomness of the system. Finally,motivated by the interest in extreme events near coastal continental shelves for waterwaves, Majda and Qi (2019) investigate the change in the statistical features foundwhen looking at water surface waves going across an abrupt depth change. The flow ismodelled using a truncated version of the Kortewegde Vries equation and the transitionbetween the statistical regimes of the incoming and of outgoing field is studied byconstructing mixed Gibbs measures with microcanonical and canonical components.
ACKNOWLEDGMENTS
VL wishes to thank all the authors of the special issue for their fascinating contributions;Elena Baroncini, Alessandro Giuliani, and Aldo Rampioni for their support and patience inthe preparation of this special issue; Giovanni Gallavotti, Michael Ghil, Joel Lebowitz, andDavid Ruelle for amazing intellectual inspiration and personal exchanges. VL acknowledgesthe support received from the EU Horizon2020 project TiPES (grant No. 820970). VLtakes the liberty of dedicating this special issue to Vito Volterra (1860-1940), who hadthe rare gift of matching scientific genius with moral strength and steadfastness againstfascism. As a trailblazing interdisciplinary scientist who pioneered mathematical modellingfor environmental sciences, he would have probably enjoyed the current developments of themathematics and physics of climate. 13
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