Climate Fluctuations and Non-equilibrium Statistical Mechanics: An Interdisciplinary Dialogue

The laws of thermodynamics can be extended to mesoscopic systems for which energy changes are on the order of the thermal energy are relevant. Therefore, thermodynamic observables associated with mesoscopic degrees of freedom are stochastic. A key example of such thermodynamic observable is the stochastic entropy production in nonequilibrium processes. Little is known beyond fluctuation theorems about universal or model-independent statistics of entropy-production fluctuations. Using Martingale theory [1] we have discovered novel universal statistics of stochastic entropy production in nonequilibrium steady states such as: (i) The distribution of the negative record (which we call infimum) of entropy production (ii) the passage probabilities of entropy production; (iii) the stopping-time fluctuations of entropy production. Our results have interesting implications for stochastic processes that can be discussed in colloidal systems, active molecular processes and nanoelectronic devices. For example, we make predictions for the distribution of the maximum backtrack depth of RNA polymerases during RNA transcription in eukaryotes. [1] I. Neri, É. Roldán, and F. Jülicher, Phys. Rev. X 7, 011019 (2017).

(Authors: Adrian Odenweller, Reik V. Donner) Over the last decade, complex network methods have been frequently used for characterizing spatio-temporal patterns of climate variability from a complex systems perspective, yielding new insights into time-dependent teleconnectivity patterns and couplings between different components of the Earth climate. Among the foremost results reported, network analyses of the synchronicity of extreme events as captured by the so-called event synchronization have been proposed to be powerful tools for disentangling the spatio-temporal organization of particularly extreme rainfall events and anticipating the timing of monsoon onsets or extreme floodings. Rooted in the analysis of spike train synchrony analysis in the neurosciences, event synchronization has the great advantage of automatically classifying pairs of events arising at two distinct spatial locations as temporally close (and, thus, possibly statistically -- or even dynamically -- interrelated) or not without the necessity of selecting an additional parameter in terms of a maximally tolerable delay between these events. This consideration is conceptually justified in case of the original application to spike trains in electroencephalogram (EEG) recordings, where the inter-spike intervals show relatively narrow distributions at high temporal sampling rates. However, in case of climate studies, precipitation extremes defined by daily precipitation sums exceeding a certain empirical percentile of their local distribution exhibit a distinctively different type of distribution of waiting times between subsequent events. This raises conceptual concerns if event synchronization is still appropriate for detecting interlinkages between spatially distributed precipitation extremes. In order to study this problem in more detail, we employ event synchronization together with an alternative similarity measure for event sequences, event coincidence rates, which requires a manual setting of the tolerable maximum delay between two events to be considered potentially related. Both measures are then used to generate climate networks from parts of the satellite-based TRMM precipitation data set at daily resolution covering the Indian and East Asian monsoon domains, respectively, thereby re-analysing previously published results. The obtained spatial patterns of degree densities and local clustering coefficients exhibit marked differences between both similarity measures. Specifically, we demonstrate that there exists a strong relationship between the fraction of extremes occurring at subsequent days and the degree density in the event synchronization based networks, suggesting that the spatial patterns obtained using this approach are strongly affected by the presence of serial dependencies between events. Given that a manual selection of the maximally tolerable delay between two events can be guided by a priori climatological knowledge and even used for systematic testing of different hypotheses on climatic processes underlying the emergence of spatio-temporal patterns of extreme precipitation, our results provide evidence that event coincidence rates are a more appropriate statistical characteristic for similarity assessment and network construction for climate extremes, while results based on event synchronization need to be interpreted with great caution.

The climate system may be understood as a chaotic dynamical system, which settles on an attractor with fractal geometry. The local properties of this set are believed to be closely related to the systems predictability and entropy. Recent advances in dynamical system theory allow us to estimate one of these properties, the local dimension $D_p$, from single trajectories with very little computational effort using EV-theory. In this talk, I would like to present an overview of my Msc.-thesis on this topic (currently work in progress). This would encompass an introduction to the theoretical background of fractal dimensions, as well as some results of numerical experiments on low-dimensional "toy"-models and real atmospheric forecast data. Much of my work is based on Faranda, Davide, Gabriele Messori, and Pascal Yiou. "Dynamical proxies of North Atlantic predictability and extremes." (2016). Lucarini, Valerio, Davide Faranda, and Jeroen Wouters. "Universal behaviour of extreme value statistics for selected observables of dynamical systems." Journal of statistical physics 147.1 (2012): 63-73.

Within the Gulf Stream jet-extension, the onset of turbulence leads to complex, transient behaviour. For example, meanders grow in amplitude downstream from Cape Hatteras and eddies are generated through a mix of barotropic and baroclinic instability. The Gulf Stream transport, mean position and meandering path exhibit year-to-year variability. We will explore the inter-annual variability of the statistics of dynamical properties in the Gulf Stream region using altimetry data. We will analyse the different statistical moments (mean, standard deviation, skewness and kurtosis) of jet latitude and along-jet speed. We will discuss their relationships to the underlying dynamics on interannual timescales and the role of instabilities in driving the flow. We will show that year-to-year statistics of jet latitude and along-jet speed are intimately related, specifically where the Gulf Stream destabilizes. The inter-annual variations of dynamics within the Gulf Stream can have significant impacts on the transport and mixing of heat and salinity, therefore affecting mid-latitude climate.

This would be a general and pedagogical introduction to large deviation theory, the theoretical tools it provides to physicist and dynamicist, and how they can be used in kinetic theory (with the example of large deviations for the Boltzmann equation), and dynamical systems (Freidlin-Wentzell theory, large deviation and averaging).

Consider equations of motion that generate dispersion of an ensemble of particles. For a given dynamical system an interesting problem is not only what type of diffu- sion is generated by its equations of motion but also whether the resulting diffusive dynamics can be reproduced by some known stochastic model. I will discuss three examples of dynamical systems generating different types of diffusive transport: The first model is fully deterministic but non-chaotic by displaying a whole range of nor- mal and anomalous diffusion under variation of a single control parameter [1]. The second model is a dissipative version of the paradigmatic standard map. Weakly per- turbing it by noise generates subdiffusion due to particles hopping between multiple attractors [2]. The third model randomly mixes in time chaotic dynamics generat- ing normal diffusive spreading with non-chaotic motion where all particles localize. Varying a control parameter the mixed system exhibits a transition characterised by subdiffusion. In all three cases I will show successes, failures and pitfalls if one tries to reproduce the resulting diffusive dynamics by using simple stochastic models. Joint work with all authors on the references cited below. [1] L. Salari, L. Rondoni, C. Giberti, R. Klages, Chaos 25, 073113 (2015) [2] C.S. Rodrigues, A.V. Chechkin, A.P.S. de Moura, C. Grebogi and R. Klages, Europhys. Lett. 108, 40002 (2014) [3] Y.Sato, R.Klages, to be published.

In this talk, I would discuss bistability situation for the dynamics of zonal jets in the barotropic quasigeostrophic model. I would present regimes with bistability (very rare transitions between attractors) and explain how rare event algorithm allow to study those very rare events. On the theoretical side, I would explain how to compute quasi potentials, transition rates, and reactive trajectories using large deviation theory. The talk could be very pedagogical, or more technical if intended to specialist of statistical physics.

I will present a brief introduction to non-equilibrium thermodynamics and statistical physics. This tutorial will review how the essential laws of thermodynamics apply to macroscopic systems, and how these laws can be extended to apply to microscopic systems where fluctuations dominate.

We study the statistics of infima, stopping times, and passage probabilities of entropy production in nonequilibrium steady states, and we show that they are universal. We consider two examples of stopping times: first-passage times of entropy production and waiting times of stochastic processes, which are the times when a system reaches a given state for the first time. Our main results are as follows: (i) The distribution of the global infimum of entropy production is exponential with mean equal to minus Boltzmann’s constant; (ii) we find exact expressions for the passage probabilities of entropy production; (iii) we derive a fluctuation theorem for stopping-time distributions of entropy production. These results have interesting implications for stochastic processes that can be discussed in simple colloidal systems and in active molecular processes. In particular, we show that the timing and statistics of discrete chemical transitions of molecular processes, such as the steps of molecular motors, are governed by the statistics of entropy production. We also show that the extreme-value statistics of active molecular processes are governed by entropy production; for example, we derive a relation between the maximal excursion of a molecular motor against the direction of an external force and the infimum of the corresponding entropy-production fluctuations. Using this relation, we make predictions for the distribution of the maximum backtrack depth of RNA polymerases, which follow from our universal results for entropy-production infima. [1]Statistics of Infima and Stopping Times of Entropy Production and Applications to Active Molecular Processes, Izaak Neri, Édgar Roldán, and Frank Jülicher, Phys. Rev. X 7, 011019

I will present an introduction to the climate system and its modeling, from the Kolmogorov scale to the global scale. I will introduce the concept of models for sub-grid (i.e., unresolved) processes, including stochastic models. I will discuss the relationship (or lack of) between climate variability, turbulent variability and forcing variability. Some of the key questions in climatology surround the societal and apparent physical importance of low-dimensional subspaces that describe large fractions of the variability. The ability of kitchen sink models to reproduce this variability (and how well the variability is known from observations) will be quantified. Whether or not these modes are predictable and toy models that share some of the climate system behaviors will be discussed.

Understanding the statistics of ocean geostrophic turbulence is of utmost importance in understanding its interactions with the global ocean circulation and the climate system as a whole. Here, a study of eddy-mixing entropy in a forced-dissipative barotropic ocean model is presented. Entropy is a concept of fundamental importance in statistical physics and information theory; motivated by equilibrium statistical mechanics theories of ideal geophysical fluids, we consider the effect of forcing and dissipation on eddy-mixing entropy, both analytically and numerically. By diagnosing the time evolution of eddy-mixing entropy it is shown that the entropy provides a descriptive tool for understanding three stages of the turbulence life cycle: growth of instability, formation of large scale structures and steady state fluctuations. Further, by determining the relationship between the time evolution of entropy and the maximum entropy principle, evidence is found for the action of this principle in a forced-dissipative flow. The maximum entropy potential vorticity statistics are calculated for the flow and are compared with numerical simulations. Deficiencies of the maximum entropy statistics are discussed in the context of the mean-field approximation for energy. This study highlights the importance entropy and statistical mechanics in the study of geostrophic turbulence.

Making predictions in systems with spatio-temporal chaos involves not only an analysis of error amplifications, coming from model uncertainties and assimilation defects, but also a study of the spatial propagation of perturbations. Suited perturbations are daily used in weather forecasting in the generation of initial ensembles (Kalnay, 2002). The so-called Ensemble Prediction System (EPS), which is the main operative tool used in today’s forecasting, estimates the error evolution by means of deterministic forecast integration given by an atmospheric/ocean model (Tracton and Kalnay, 1993; Molteni et al., 1996). The choice of a proper initial ensemble is crucial for the final result and some kind of control— in order to obtain a determined statistics or spread on the initial ensemble evolution— is necessary. Most of the existing studies limit their analysis to the behaviour of the perturbation amplitude (the norm), paying very little or no attention to the differences in the spatial structure of bred, Lyapunov or SVs (Smith, 2001). Here we present a diagnosis tool to differentiate the good from the bad perturbations to initialize an ensemble forecasting system. We study the evolution of finite perturbations in the Lorenz ‘96 model, a meteorological toy model of the atmosphere. The initial perturbations are chosen to be aligned along different dynamic vectors: bred, singular, and characteristic Lyapunov vectors. We discuss the corresponding advantages of using those different vectors for preparing initial perturbations to be used in ensemble prediction systems, focusing on key properties: dynamic adaptation to the flow, robustness, equivalence between members of the ensemble, etc. Among all the vectors considered here, the so-called characteristic Lyapunov vectors are possibly optimal, in the sense that they are both perfectly adapted to the flow and extremely robust. Kalnay, E. 2002. Atmospheric Modeling, Data Assimilation and Pre- dictability, E. Cambridge University Press, Cambridge. Tracton, M. S. and Kalnay, E. 1993. Operational ensemble prediction at the national meteorological center: practical aspects. Wea. Forecast. 8, 379–398. Molteni, F., Buizza, R., Palmer, T. N. and Petroliagis, T. 1996. The ECMWF ensemble prediction system: Methodology and validation. Quart. J. Roy. Meteor. Soc. 122, 73–119. Smith, L. A. 2001. Disentangling uncertainty and error: On the pre- dictability of nonlinear systems. In: Nonlinear Dynamics and Statis- tics (ed. A. I. Mees), Birkh ̈ user Verlag, Boston.

Authors: Freddy Bouchet, Francesco Ragone, and Jeroen Wouters. ENS de Lyon and CNRS For some aspects of climate dynamics, rare dynamical events may play a key role, for instance when they have a huge impact. We will study the paradigmatic example of extreme heat waves. In the recent past, new theoretical and numerical tools have been developed in the statistical mechanics community, in order to specifically study such rare events. Some of those approaches are based on large deviation theory for complex dynamical systems. We will study the probability of extreme heat waves in a comprehensive GCM. At a fixed numerical cost, several hundreds more heat waves are observed than in a control run. The thousands of sampled extreme heat waves open the door to their dynamical studies, precursor, and fluctuation paths, in a way that can not be foreseen using conventional tools based on direct numerical simulations. Moreover extreme events that can not be observed in a GCM at a reasonable cost can now be studied. This new tool open new perspectives for the study of climate extremes. As an example we discuss teleconnection patterns for extremes. The relation with statistical mechanics tools in general, and especially large deviation theory, will be emphasized.

It is an important open scientific question whether the global climate is a collection of coupled chaotic oscillators. The important El Niño Southern Oscillation display both chaotic deterministic and stochastic processes, but characterisation of the nature of this important individual climate oscillation is still far from complete. Explaining multidecadal climate variability is an important topic for applying the theory of chaotic coupled oscillators. In the instrumental temperature record, a 65-70-year fairly regular quasiperiodic cycle in global mean temperature can be seen. On decadal time scales, the long cycle is comparable in strength to global warming. The long cycle be seen in many climate model simulations even when excluding external forcings such as greenhouse gases and aerosol particles. For that and other reasons, it is almost certain that the cycle is mainly driven by internal dynamics of the ocean-atmosphere system. Following Tsonis et al. (GRL, 2007, 2009), a hypothesis to explain the long cycle is that higher-frequency (~3-6 y period) oscillations, including El Niño, North Atlantic Oscillation and North Pacific Oscillation turnwise synchronise and desynchronise to produce it. Coupled oscillations whose Lyapunov divergence times and coupling coefficients are close to balancing each other out are referred to as weak chaos. The solar system and nonlinear electronic and gene circuits with time-delayed coupling look like stable systems, but are chaotic on closer inspection. These systems can perhaps help climate science to get beyond the equilibrium plus noise linear paradigm. The ECHAM5-MPI-OM coupled ocean-atmosphere climate model developed at the Max Planck Institute for Meteorology is one of the best climate models in the world to reproduce the long cycle regarding period, amplitude and regional structure. The climate model provides more data than measurements allowing us to study parts of fluctuation mechanisms everywhere from the deep sea to the stratosphere. Climate model results agree well with properties of the observed long cycle. The mainstream North-Atlantic meridional overturning circulation driven theory is not validated strongly enough by data. While the ideas of Tsonis et al. seem to need some revision, oscillation coupling does seem to be involved in producing the cycle. Meanwhile also asymmetry, irregularity and sudden shifts in higher-frequency oscillations (3-6 y) of many types are observed at each turning point of the long cycle. Physical manifestations of the coupling between different modes of variability are looked into in detail. Most fluctuations are characterised by power laws instead of Gaussians and simple mathematical models are examined for insight into the underlying reasons.

Ocean mesoscale eddies are turbulent processes which strongly affect the strength and variability of large-scale ocean jets. Their horizontal scales, roughly 10 to 100 km, are too small to be adequately resolved in current ocean models. Representing eddy-mean flow processes in climate simulations remains a key challenge, especially quantifying the dependence of eddy effects on the underlying dynamics of the resolved flow and external forcing. I will present new ways to diagnose, analyse and parametrize turbulent eddy effects. The work is based on the analysis of idealised and state of the art ocean models, fluid mechanics and stochastic dynamics. I will discuss novel approaches to (1) improve current numerical mixing schemes using stochastic physics and (2) develop of a new turbulence closure based on non-Newtonian flow and wind-induced turbulence. The new schemes will be shown to improve drastically the mean and variability of the ocean in coarse resolution and eddy permitting simulations. In addition, the stochastic components represent a measure of uncertainty.

Symmetries and topology play central roles in our understanding of physics. Topology explains the precise quantization of the Hall effect and the protection of surface states in topological insulators against scattering from disorder or bumps. However discrete symmetries and topology have so far played little role in thinking about the fluid dynamics of oceans and atmospheres. I show that, as a consequence of the rotation of the Earth that breaks time reversal symmetry, equatorially trapped Kelvin and Yanai waves emerge as topologically protected edge modes. Thus the oceans and atmosphere of Earth naturally share basic physics with topological insulators. As equatorially trapped Kelvin waves in the Pacific ocean are an important component of El Niño Southern Oscillation, these new results demonstrate that topology plays a surprising role in Earth’s climate system.

Quasi-stationary planetary waves of large-amplitude have been linked to the occurrence of many of the most extreme weather events of the past decades in the Northern Hemisphere. This includes the European heat waves of 2003 and 2010 as well as the catastrophic Elbe flooding 2002. A resonance mechanism was proposed to explain the occurrence of large-amplitude planetary waves (Petoukhov et al. 2013) and a recent increase in the frequency of resonance events has been identified (Coumou et al. 2014). Updated analysis of more recent extreme events confirms a tendency to an increase in the frequency of occurrence of quasi-resonant conditions, favoring the emergence of persistent regional extremes in the NH mid-latitudes (Petoukhov et al. 2016). As a specific example, in May 2014 the Balkans were hit by a Vb-type cyclone that brought disastrous flooding and severe damage to Bosnia and Herzegovina, Serbia and Croatia. We have analysed this event in some detail and demonstrate a linkage to planetary wave resonance (Stadtherr et al. 2016). The talk will discuss this and further work on the role of planetary wave resonance in causing extreme weather. References Coumou D, Petoukhov V, Rahmstorf S, Petri S, and Schellnhuber HJ (2014) Quasi-resonant circulation regimes and hemispheric synchronization of extreme weather in boreal summer. Proc Natl Acad Sci USA 111: 12331-12336. Petoukhov V, Rahmstorf S, Petri S, Schellnhuber HJ (2013) Quasiresonant amplification of planetary waves and recent Northern Hemisphere weather extremes. Proc Natl Acad Sci USA 110(14): 5336-5341. Petoukhov V, Petri S, Rahmstorf S, Coumou D, Kornhuber K, Schellnhuber HJ (2016) The role of quasi-resonant planetary wave dynamics in recent boreal spring-to-autumn extreme events. Proc Natl Acad Sci USA 113: 6862-7. Stadtherr L, Coumou D, Petoukhov V, Petri S, Rahmstorf S (2016) Record Balkan floods of 2014 linked to planetary wave resonance. Science Advances, doi: 10.1126/sciadv.1501428

In 2015 many regions in the Northern Hemisphere (NH) registered record breaking annual temperatures (since 1880). At the same time, 2015 registered a widespread growing-season greening in terrestrial ecosystems over most of the NH, evaluated from satellite records. It is still under discussion whether the year-to-year variations in vegetation activity present some degree of predictability. We decomposed a vegetation index (NDVI) into the long-term and inter-annual variability components, and find that the Pacific Decadal Oscillation (PDO) and the Atlantic Multidecadal Oscillation (AMO) explain about half of inter-annual variability in NH vegetation activity. We show that the 2015 extreme greening can be explained as the vegetation response to temperature and water anomalies associated with the very strong state of the PDO in 2015. The link found between ocean oscillation patterns, atmospheric transport and vegetation activity indicates that there is potential predictability of changes to the carbon-cycle at inter-annual to decadal time-scales, which is relevant, for instance, for land-management activity.

We investigate the impact of strong climate change on a European energy system dominated by wind power. No robust trend can be observed regarding the change of the wind power yield for most countries. However, we find a higher stationarity of weather situations and a more pronounced seasonality for many European countries. This leads to higher backup needs. In contrast, for the Iberian Peninsula, Greece and Croatia we find a decreasing need for backup as the stationarity and intra-annual variability tend to decrease there. Our simulations are based on the results of five different Global Circulation Models (GCMs) using the Representative Concentration Pathway scenario 8.5 (RCP8.5). These results are dynamically downscaled with the regional atmospheric model RCA4 by the EURO-CORDEX initiative (Coordinated Downscaling Experiment - European Domain). A comparison was made between historical data (1970-2000) and mid century (2030-2060) and end of century (2070-2100) data, respectively. For all time frames we made the assumption that a certain amount of energy is provided by wind power plants. This implies that changes in wind power potentials are neglected and only temporal effects are considered. Wind speed time series are converted to power generation time series using an extrapolation to hub height and a standardized power curve. Assuming a scenario for the future distribution of wind turbines, we obtain a wind power generation time series aggregated on a national level. The operation of backup power plants and storage facilities is simulated on coarse scales assuming an optimal storage strategy. Backup is required whenever the storage facilities are empty.

I will give a gentle introduction to the effects of long-range temporal correlations in stochastic particle systems, focusing particularly on fluctuations about the typical behaviour. Specifically, in the first part of the talk, I will discuss how long-range memory dependence can modify the large deviation principle describing the probability of rare currents and lead, for example, to superdiffusive behaviour. In the second part, I will describe a more interdisciplinary project incorporating the psychological "peak-end" heuristic for human memory into a simple discrete choice model from economics. Throughout the talk, I will attempt to indicate connections between different approaches and, if time permits, I will finish by showing how new results on phase transitions in large deviations of reset processes can be obtained from a rather old model of DNA denaturation.

The probability distributions of large-scale atmospheric and oceanic variables are generally skewed and heavy-tailed. It is argued that these distinctive departures from Gaussianity arise fundamentally from the fact that in a quadratically nonlinear system with a quadratic invariant, the coupling coefficients between system components are not constant but depend linearly on the system state in a distinctive way. In particular, the skewness arises from a tendency of the system trajectory to linger near states of weak coupling. One result, consistent with observation, is that the fluctuation relations between positive and negative perturbations when perturbations are strong are opposite those relations when the perturbations are weak.

The climate system may be understood as a chaotic dynamical system, which settles on an attractor with fractal geometry. The local properties of this set are believed to be closely related to the systems predictability and entropy. Recent advances in dynamical system theory allow us to estimate one of these properties, the local dimension $D_p$, from single trajectories with very little computational effort using EV-theory. In this talk, I would like to present an overview of my Msc.-thesis on this topic (currently work in progress). This would encompass an introduction to the theoretical background of fractal dimensions, as well as some results of numerical experiments on low-dimensional "toy"-models and real atmospheric forecast data. Much of my work is based on Faranda, Davide, Gabriele Messori, and Pascal Yiou. "Dynamical proxies of North Atlantic predictability and extremes." (2016). Lucarini, Valerio, Davide Faranda, and Jeroen Wouters. "Universal behaviour of extreme value statistics for selected observables of dynamical systems." Journal of statistical physics 147.1 (2012): 63-73.

Records are rather common in everyday life: we are always talking of record rainfall, record temperature, records in sports and stock prices etc. When the random sequence consists of independent random variables, the record statistics is well known. In this talk, I'll discuss the record statistics in a strongly correlated random walk sequence and show that they are universal, i.e., independent of the noise (jump) distribution. Several applications and extensions will be discussed--such as the effect of a constant drift and the effect of measurement errors.

Multistability is a ubiquitous feature in systems of geophysical relevance and provides key challenges for our ability to predict a system's response to perturbations. Near critical transitions small causes can lead to large effects and - for all practical purposes - irreversible changes in the properties of the system. The Earth climate is multistable: present astronomical and astrophysical conditions support two stable regimes, the warm climate we live in, and a snowball climate, characterized by global glaciation. We first provide an overview of methods and ideas relevant for studying the climate response to forcings and focus on the properties of critical transitions. Following an idea developed by Eckhardt and co. for the investigation of multistable turbulent flows, we study the global instability giving rise to the snowball/warm multistability in the climate system by identifying the climatic edge state, a saddle embedded in the boundary between the two basins of attraction of the stable climates. The edge state attracts initial conditions belonging to such a boundary and is the gate facilitating noise-induced transitions between competing attractors. We use a simplified yet Earth-like climate model constructed by coupling a primitive equations model of the atmosphere with a simple diffusive ocean. We refer to the climatic edge states as Melancholia states. We study their dynamics, their symmetry properties, and we follow a complex set of bifurcations. We find situations where the Melancholia state has chaotic dynamics. In these cases, the basin boundary between the two basins of attraction is a strange geometric set with a nearly zero codimension, and relate this feature to the time scale separation between instabilities occurring on weather and climatic time scales. We also discover a new stable climatic state characterized by non-trivial symmetry properties.

coauthors: Nahal Sharafi and Marc Timme Critical transitions occur in a variety of dynamical systems. Here, we employ quantifiers of chaos to identify changes in the dynamical structure of complex systems preceding critical transitions. As suitable indicator variables for critical transitions, we consider changes in different features of covariant Lyapunov vectors. Studying changes in growth rates and directions of covariant Lyapunov vectors in several models of fast-slow systems, i.e., a network of coupled FitzHugh-Nagumo oscillators, models for Josephson junctions and the Hindmarsh-Rose model, we demonstrate that tangencies between covariant Lyapunov vectors are a generic feature during critical transitions. We further demonstrate that this deviation from hyperbolic dynamics is linked to the occurrence of critical transitions by using it as an indicator variable and evaluating the prediction success through receiver operator characteristic curves. In the presence of noise, we find the alignment of covariant Lyapunov vectors and changes in finite time Lyapunov exponents to be more successful in announcing critical transitions, than common indicator variables as, e.g., finite time estimates of the variance.

Three cyclically competing species are often described using the cyclic Lotka-Volterra model, also known as the zero-sum rock-paper-scissors game. It is well known that the mean field dynamics are characterised by a non-trivial constant of motion and neutrally stable orbits, while finite populations are driven to the eventual extinction of 2 species by demographic fluctuations (internal noise). In fact it has been shown that each species has a certain probability survive under two different scenarios: when the population size is very large, the only species to survive is the one with the smallest reaction rate (“law of the weakest”), whereas for small population sizes the most likely to survive is the species not involved in the reaction with the highest rate (“law of stay out”). So far these models have all assumed a constant environment, however since there is clear evidence that randomly changing external factors can greatly influence the evolution of populations, here we investigate the non-spatial dynamics of the cyclic Lotka-Volterra model in a fluctuating environment. For simplicity, environmental noise is modelled in terms of dichotomous noise by allowing one reaction rate to switch randomly between two values. By combining numerical methods and analytical arguments, we study the joint effect of environmental and demographic noise on the survival probabilities and analyse how they alter the “law of the weakest” and “law of stay out”. In particular we scrutinise circumstances where these previously established laws are violated and hold only in a static environment.

Neue globale Temperaturrekorde in den Jahren 2014, 2015 und 2016, schwächelndes Golfstromsystem, steigender Meeresspiegel und eine nicht abreißende Folge von Wetterextremen – wie sehen die neuesten Daten zur globalen Erwärmung aus? Wurden bereits Kipp-Punkte des Klimasystems überschritten? Gibt es einen Zusammenhang von Klimawandel und Fluchtursachen? Und was bedeutet das Pariser Klimaabkommen? Lässt sich die globale Erwärmung überhaupt noch deutlich unterhalb der 2-Grad-Grenze stoppen?

In this talk a methodology for data-driven stochastic modelling of unresolved scales and processes in weather and climate models is discussed. It has been shown that a stochastic subgrid model should be conditional on the state of the resolved variables. Here patterns in the space of resolved scales are identified and linked predictively to patterns in the space of subgrid forcing. On top of this deterministic subgrid scheme the subgrid patterns are forced stochastically. The method is explored in the Lorenz 1996 model and then applied to parametrisation of meso-scale eddies in ocean models. Ensemble prediction and long-term simulations are evaluated. Also an attempt on scale-invariant parametrisation is made.

Because weather is chaotic in the mathematical-physical sense, climate as the time-average of weather has chaotic components, too. These chaotic components -- termed internal climate variability -- severely complicate all attributions of observed climate events to a particular cause and also all evaluations of climate models against observations. A powerful new tool has recently arisen that allows us to diagnose unambiguously the degree of internal variability overlaying the climate response to external forcing: We simulate a particular period many times over, with a state-of-the-art climate model under identical forcing but ever so slightly different initial conditions in each different realisation. At the Max Planck Institute for Meteorology, we have constructed the largest currently existing of these ensembles -- our Grand Ensemble. We have constructed Grand Ensembles for the canonical time periods, the historical period since 1850 and the future out to 2100. I will give some examples of what insights can be gained through these ensembles, such as characterising the extent of irreducible uncertainty in climate predictions. I will also outline some of the conceptual and statistical challenges posed by the information treasure trove of our Grand Ensembles.

Over the past ten years, a key advance in our understanding of atmospheric variability is the discovery that between the weather and climate regimes lies an intermediate “macroweather” regime, spanning the range of scales from ≈10 days to ≈30 years. Macroweather statistics are characterized by two fundamental symmetries: scaling in space and in time and the factorization of the joint space-time statistics. In the time domain, the scaling has low intermittency with the additional property that successive fluctuations tend to cancel. In space, on the contrary the scaling has high (multifractal) intermittency corresponding to the existence of different climate zones. These properties have fundamental implications for macroweather forecasting: first, the temporal scaling implies that the system has a long range memory that can be exploited for forecasting; second, the low temporal intermittency implies that mathematically well-established (Gaussian) forecasting techniques can be used; and third, the statistical factorization property implies that although spatial correlations (including teleconnections) may be large, if long enough time series are available, they are not necessarily useful in making forecasts. In addition, they imply the existence of stochastic predictability limits which should also apply to GCM’s. In this presentation, we review theoretically and empirically the evidence for the symmetries showing that they accurately hold on both real world data and on GCM outputs, including (by hindcasting), we confirm the stochastic predictability limits on GCM control runs. This is the basis of a forecasting system: Stochastic Seasonal to Interannual Prediction System (StocSIPS) that has been operational since April 2016.

Better understanding of dynamics in complex systems, such as the Earth’s climate or human brain, is one of the key challenges for contemporary science and society. Large amount of experimental data requires new mathematical and computational approaches. Natural complex systems vary on many temporal and spatial scales, often exhibiting recurring patterns and quasi-oscillatory phenomena. The statistical inference of causal interactions and synchronization between dynamical phenomena evolving on different temporal scales is of vital importance for better understanding of underlying mechanisms and a key for modelling and prediction of such systems. This study introduces and applies information theory diagnostics to phase and amplitude time series of different wavelet components of the observed data that characterizes El Ni\~{n}o. A suite of significant interactions between processes operating on different time scales were detected. The mechanisms of these nonlinear interactions were further studied in conceptual low-order and state-of-the-art dynamical, as well as statistical climate models. Observed and simulated interactions exhibit substantial discrepancies, whose understanding may be the key to an improved prediction. Moreover, the statistical framework which we apply here is suitable for direct usage of inferring cross-scale interactions in e.g. human brain dynamics and other complex systems.

Extremes in weather and climate do not occur 'out-of-the-blue'. Their probability of occurrence is determined by the state of the dynamical system, which in turn is to some degree predictable. However, particularly on the atmospheric mesoscale, uncertainties are large and predictions are probabilistic in nature. Thus mesoscale weather ensemble prediction systems (EPS) not only issue one future state of the atmosphere but a sample of possible future states thereby accounting for several sources of uncertainty. We will present and discuss ensemble post-processing for spatial extremes. Statistical representation of spatial extremes naturally resorts to spatial max-stable processes. The multivariate analysis of extremes involves two tasks: first, estimating the marginals and second, characterizing the dependence structure of the spatial process. We present a Bayesian approach to estimate the spatial marginals conditional on the EPS forecasts, and discuss the Brown-Resnick process as a spatial model for observations.

(Authors: Reik V. Donner, Marc Wiedermann, Alexander Radebach, Jonathan Donges) As a by-product of the rising interest in the structural and dynamical characterization of networks in various scientific disciplines, complex network based methods have recently found wide use as an exploratory tool of data analysis across fields. In particular, functional networks encoding essential statistical interrelationships among multivariate time series have proven to be a versatile tool in climate sciences, neurophysiology and various other areas. In my presentation, I will present some recent applications of functional network analysis in the field of climatology to highlight the potentials as well as ongoing challenges of such “climate network” approaches. As one particular example, I will demonstrate how the El Nino Southern Oscillation, the most prominent mode of tropical climate variability, is reflected in the global correlation structure of surface air temperatures. I will show that several climate network characteristics are able to trace the emergence of El Nino and La Nina episodes. However, there is no one-by-one correspondence between anomalies of these features and the aforementioned climate phenomena, pointing towards (i) a distinctive difference in the spatial structure of global temperature teleconnections associated with two different flavors of El Nino and La Nina and (ii) the fact that other regional-scale climate anomalies like cooling trends following strong volcanic eruptions leading to qualitatively similar effects in the network structure. Finding (i) is further corroborated by the results of a statistical assessment of the simultaneous occurrence of different event types and seasonal precipitation anomalies across the globe using the new intuitive statistical approach of event coincidence analysis. Contributing to our understanding of coupling between different subsystems, I will present a straightforward methodological extension utilizing the idea of coupled networks and their appropriate structural quantification. Specifically, I will demonstrate the application of this idea to studying ocean-atmosphere interdependences in the Northern latitude extratropics during boreal winter, which reveals clear differences between the way correlations from the ocean (sea-surface temperatures) to the atmosphere (pressure levels) are spatially organized as compared with the opposite direction. Specifically, taking the two local measures cross-degree and local cross-clustering coefficient, the strongest correlations from the ocean towards the atmosphere exhibit a power-law between both characteristics pointing to some specific type of hierarchical structure. While recent work has largely focused on the presented climatological applications, the utilized frameworks of functional network analysis, event coincidence analysis and coupled network analysis are generic and widely applicable to other fields as well.

Paleoclimate records are extremely rich sources of information about the past history of the Earth system. We take an information-theoretic approach to analyzing data from the WAIS Divide ice core, the longest continuous and highest-resolution water isotope record yet recovered from Antarctica. We use weighted permutation entropy to calculate the Shannon entropy rate from these isotope measurements, which are proxies for a number of different climate variables, including the temperature at the time of deposition of the corresponding layer of the core. We find that the rate of information production in these measurements reveals issues with analysis instruments, even when those issues leave no visible traces in the raw data. These entropy calculations also allow us to identify a number of intervals in the data that may be of direct relevance to paleoclimate interpretation, and to form new conjectures about what is happening in those intervals—including periods of abrupt climate change.

When a control parameter of a system is suddenly changed, the accessible phase space changes too and the system needs its characteristic relaxation time to reach the final equilibrium distribution. An important and relevant question is whether it is possible to travel from an equilibrium state to another in an arbitrary time, much shorter than the natural relaxation time. Such strategies are reminiscent of those worked out in the recent field of Shortcut to Adiabaticity, that aim at developing protocols, both in quantum and in classical regimes, allowing the system to move as fast as possible from one equilibrium position to a new one, provided that there exist an adiabatic transformation relating the two. Proof of principle experiments have been carried out for isolated systems. Instead in open system the reduction of the relaxation time, which is frequently desired and necessary, is often obtained by complex feedback processes. In this talk, we present a protocol, named Engineered Swift Equilibration (ESE), that shortcuts time-consuming relaxations, We tested experimentally this protocol on a Brownian particle trapped in an optical potential first and then on an AFM cantilever. We show that applying a specific driving, one can reach equilibrium in an arbitrary short time. We also estimate the energetic cost to get such a time reduction. The ESE method paves the way for applications in micro and nano devices, in high speed AFM, or in monitoring mesoscopic chemical or biological process. References: (1) Engineered Swift Equilibration, I. A. Martinez; A. Petrosyan; D. Guéry-Odelin; E. Trizac; S. Ciliberto, Nature Physics, Vol 12, 843 (2016). (2) Arbitrary fast modulation of an atomic force microscope, A. Le Cunuder; I. A. Martinez; A. Petrosyan; D. Guéry-Odelin; E. Trizac; S. Ciliberto, . Applied Physics Letters, 109, 113502 (2016)

Numerical formulations of turbulent heat fluxes must lead to positive energy dissipation and positive internal entropy production. Current parameterization approaches only deliver positive dissipation rates for free convection, not for forced convection. This contribution explains how positive dissipation rates are achieved by a new formulation of subgrid-scale terms in the case of stable stratification. This is of importance for the numerical realization of the breakdown of gravity waves. A turbulent atmosphere tends to an isentropic stratification, because in addition to turbulent heat diffusion, pressure work leads to expansion when air is rising and contraction when air is sinking. This pressure work remains completely subgrid-scale when the atmosphere is unstably stratified. When the atmosphere is stably stratified, this pressure work has to be done by the outer environment. Therefore, a turbulent pressure gradient term is introduced in the vertical momentum equation and the equations account for the irreversible energy conversion from resolved kinetic energy into the model's internal energy. Numerical experiments for breaking gravity waves in the mesosphere highlight the different behavior of new and conventional approaches. The conventional approach leads to a deepening of the wave amplitudes, whereas the new approach supports wave overturning. The observed foliate structure of very sharp inversion layers is indeed simulated with the new approach for stable stratification. In contrast to the traditional setting the new scheme does not evolve into persistent non-physical wave structures on long timescales.

The atmosphere is a complex system involving many interacting scales. Therefore, subgrid-scale (SGS) parameterizations are essential for climate simulations and numerical weather prediction. Many of those parameterizations contain tuning parameters obtained by fitting model behavior to reference data statistics. Consequently, if the atmosphere is perturbed, and hence also the statistics, these parameters might become erroneous and the SGS parameterization may no longer be able to help simulating the dynamics of the perturbed atmosphere. Therefore, we propose a climate dependence of the tuning parameters using the Fluctuation-Dissipation Theorem (FDT). The FDT is able to predict the changes in the statistics of a system, caused by small external forcings. Those changes are then used to update the empirical components of the tuning parameters. This procedure is tested in a toy atmosphere provided by a three-layer quasi-geostrophic model (QG3LM). The corresponding climate model is given by a low-order model, based on a reduced number of QG3LM variance patterns, with an empirical linear closure as SGS parameterization. The external perturbation is given by some local anomalous heating in the extratropics. It is shown that the FDT is able to predict the required change in the closure parameters. The climate model with the FDT-corrected closure improves the agreement with the perturbed toy atmosphere, compared to the climate model without a corrected closure. In addition, we show that the climate model with FDT-corrected closure outperforms the direct FDT estimation of the response of the toy atmosphere, provided sufficiently many basis patterns are used.

The atmospheric circulation is observed to exhibit regime behaviour. Regimes are a set of preferred flow patterns that are qualitatively different from each other. When the flow is in one regime, it will persist for an extended period of time, before undergoing a rapid transition to another regime. An example of atmospheric regimes is sudden stratospheric warmings, which occur when the stratospheric polar vortex undergoes a sudden transition from an axisymmetric state to a split or displaced state. This talk will show that useful insights into atmospheric regime transitions may be gained by studying experiments on rotating fluids in the laboratory. In this simple setting, regimes may be studied without the often ad-hoc approximations of numerical and theoretical approaches. In laboratory experiments, we have found that transitions between large-scale flow regimes may be triggered by small-scale gravity waves. These transitions may be mimicked in a numerical model via the injection of stochastic noise, motivating stochastic parameterisations in atmospheric models. The laboratory experiments have also inspired a new interpretation of sudden stratospheric warmings, as being gravity wave noise-induced transitions.

In this talk I will summarise the progress made in determining the statistics of turbulent flows using Direct Statistical Simulation. I shall also describe recent advances in generalising the Quasilinear approximation and how this generalisation can improve the range of applicability of statistical methods.

The atmosphere is governed by continuum mechanics and thermodynamics yet simultaneously obeys statistical turbulence laws. Up until its deterministic predictability limit (τw ≈ 10 days), only General Circulation Models (GCM’s) have been used for prediction; the turbulent laws being still too difficult to exploit. However, beyond τw – in macroweather – the GCM’s effectively become stochastic with internal variability fluctuating about the model – not the real world – climate and their predictions are poor. In contrast, the turbulent macroweather laws become advantageous notably due to a) low macroweather intermittency that allows for a Gaussian approximation, and b) thanks to a statistical space-time factorization symmetry that (for predictions) allows much decoupling of the strongly correlated spatial degrees of freedom. The laws imply new stochastic predictability limits. We show that pure macroweather – such as in GCM’s without external forcings (control runs), can be forecast nearly to these limits by the ScaLIng Macroweather Model (SLIMM) that exploits huge system memory that forces the forecasts to converge to the real world climate. To apply SLIMM to the real world requires preprocessing to take into account anthropogenic and other low frequency external forcings. We compare the overall Stochastic Seasonal to Interannual Prediction System (StocSIPS, operational since April 2016) with a classical GCM (CanSIPS) showing that StocSIPS is superior for forecasts two months and further in the future, particularly over land. In addition, the relative advantage of StocSIPS increases with forecast lead time. In this presentation we give some details of StocSIPS and its implementation and we evaluate its skill both absolute and relative to CanSIPS.

Over the last decade, stochastic thermodynamics has emerged as a comprehensive framework for describing a large class of strongly driven systems [1]. I will introduce its basic concepts and discuss, as one major outcome, the fluctuation theorem for entropy production. The more recently discovered thermodynamic uncertainty relation gives a universal inequality relating the mean and dispersion of any current with the overall entropy production [2]. For molecular motors, it has been used as a tool of thermodynamic inference to reveal hidden properties of a system [3]. Since these results are mathematical identities, they hold for the stationary state of any Markovian stochastic dynamics. This fact offers the perspective of applying them beyond biophysics to other fields like climate science. [1] U Seifert, Rep. Prog. Phys. 75, 126001, 2012. [2] AC Barato and U Seifert, Phys. Rev. Lett. 114, 158101, 2015. [3] P Pietzonka, AC Barato, and U Seifert, J Stat Mech, 124004, 2016.

Entropy production (EP) $\sigma$ has played a central role in the fluctuation theorem, so-called the Jarzynski's integral fluctuation theorem (IFT) $\lagnle e^{-\sigma}\rangle=1$ in non-equilibrium systems. The EP is obtained during a forward and backward process of a particle or system. Even though the IFT has been verified in experiments in terms of the difference of external works applied and gained to and from the system during the forward and backward processes, there is little case in which the entropy production is directly measured. Here, we check and confirm the IFT in the data packet transport process on complex networks and obtain the distribution of entropy production distribution. This work could be extended to further problems on protein folding-unfolding problems.

Richard Blender, Frank Lunkeit and Denny Gohlke, Meteorological Institute and CEN, University of Hamburg In a large number of physical systems non-equilibrium fluctuations satisfy the Fluctuation Theorem, which relates the probabilities of negative and positive entropy production. Here we analyse the fluxes in the Lorenz energy cycle in simulations with the atmospheric model PUMA (Portable University Model of the Atmosphere, University of Hamburg). The Lorenz energy cycle includes the injected power, the total dissipated energy and captures the internal conversions of available potential and kinetic energy, and the energy flow from the zonal mean to the eddy scale. PUMA is a dynamical core based on the hydrostatic primitive equations with simplified forcing and friction parameterizations. The aim is to assess the possible validity of the Fluctuation Theorem and to determine the Large Deviation rate functions for the energy fluxes. In a sub-project of the Collaborative Research Center Transregio TRR181 “Energy transfers in Atmosphere and Ocean”, we intend to use periods with negative entropy production to constrain stochastic or counter-gradient parameterizations in models.

Accurate representation of atmospheric jets in general circulation models is vital for weather and climate, and for understanding how the atmospheric circulation responds to the applied external and internal forcings (e.g., surface warming, volcanic eruptions, solar variability, model parameter variation). If a model is unable to faithfully represent the present day climate or is acutely sensitive to model numerics, any predictions of circulation response to forcing with such a model is suspect. Several examples where different model numerics or variation of parameters in the boundary layer parametrisation scheme within observational constraints can lead to markedly different modelled jet magnitude and location are given, ranging from simple models of super-rotating planetary atmospheres to the complex climate models that are used to inform policy makers. The fluctuation dissipation theorem, which states that the response to forcing is proportional to the decorrelation time scale of the system, has become a standard theory in explaining the amplitude of the mid-latitude jet response to forcing in idealised dry general circulation models. However, it will be shown that when moisture is added to the general circulation model, the fluctuation dissipation theorem is unable to explain the jet response to forcing.

N. Watkins 1,2,3, S. Chapman 2, A. Chechkin 4, I. Ford 5, R. Klages 6, D. Stainforth 1 1 Centre for the Analysis of Time Series, London School of Economics and Political Science, UK 2 University of Warwick, UK 3 Open University, UK 4 Akhiezer Institute of Theoretical Physics, Kharkiv, Ukraine 5 UCL, UK 6 QMUL, UK Energy Balance Models (EBMs) have a long heritage in climate science for modelling global mean temperature anomalies. Many types of EBM have now been studied, including examination of spatiotemporal, and particularly latitudinal, dependence, as well as possible low dimensional effects. Particular interesting to statistical mechanics, and the interdisciplinary dialogue to be fostered by CANEID, are the stochastic EBMs. These may allow direct treatment of climate fluctuations and noise. Some recent stochastic EBMs (e.g. [1]) map on to the Langevin equation, with temperature anomaly replacing velocity, and other corresponding replacements being made. This raises the question as to how far the full technology of modern Langevin modelling could be applied to this aspect of the climate problem. This talk will discuss this question, and give some preliminary answers. We first note that the most familiar Langevin equation is a limiting case of the generalised Langevin equation (GLE) which so far has seen little or no application in climate science, raising the question of whether a GLE-type EBM could in fact be derived or motivated. This is particularly important because, as noted in [2], the EBM studied by Padilla et al [1] used a correlated (red) noise term but its analogous term to resistance is still constant. A fluctuation-dissipation theorem was thus precluded by its construction. We then note that long memory simplifies the GLE to a fractional Langevin equation (FLE). The still controversial experimental evidence for long range memory in global temperature has already motivatedstudy of a power law response model [3,4]. We go beyond this to ask whether an FLE-type EBM exists, and what its solutions would be. [l] Padilla et al, J. Climate 24, 5521 (2011). [2] Watkins, GRL, 40, 1–9, doi:10.1002/GRL.50103 (2013) [3] Rypdal, JGR, 117(D6), D06115 (2012) [4] Rypdal and Rypdal, J. Climate, 27(14), 5240 (2014)

Consider equations of motion that generate dispersion of an ensemble of particles. For a given dynamical system an interesting problem is not only what type of diffu- sion is generated by its equations of motion but also whether the resulting diffusive dynamics can be reproduced by some known stochastic model. I will discuss three examples of dynamical systems generating different types of diffusive transport: The first model is fully deterministic but non-chaotic by displaying a whole range of nor- mal and anomalous diffusion under variation of a single control parameter [1]. The second model is a dissipative version of the paradigmatic standard map. Weakly per- turbing it by noise generates subdiffusion due to particles hopping between multiple attractors [2]. The third model randomly mixes in time chaotic dynamics generat- ing normal diffusive spreading with non-chaotic motion where all particles localize. Varying a control parameter the mixed system exhibits a transition characterised by subdiffusion. In all three cases I will show successes, failures and pitfalls if one tries to reproduce the resulting diffusive dynamics by using simple stochastic models. Joint work with all authors on the references cited below. [1] L. Salari, L. Rondoni, C. Giberti, R. Klages, Chaos 25, 073113 (2015) [2] C.S. Rodrigues, A.V. Chechkin, A.P.S. de Moura, C. Grebogi and R. Klages, Europhys. Lett. 108, 40002 (2014) [3] Y.Sato, R.Klages, to be published.

Chapman, Stainforth, Watkins Global mean temperature (GMT) provides a simple means of benchmarking a broad ensemble of global climate models (GCMs) against past observed GMT which in turn provide headline assessments of the consequences of possible future forcing scenarios. The slow variations of past changes in GMT seen in different GCMs track each other [1] and the observed GMT reasonably closely. However, the different GCMs tend to generate GMT time-series which have absolute values that are offset with respect to each other [2]. Subtracting these offsets is an integral part of comparisons between ensembles of GCMs and observed past GMT. We will discuss how this constrains how the GCMs are related to each other. The GMT of a given GCM is a macroscopic reduced variable that tracks a subset of the full information contained in the time evolving solution of that GCM. If the GMT slow timescale dynamics of different GCMs is to a good approximation the same, subject to a linear translation, then the phenomenology captured by this dynamics is essentially linear; any feedback is to leading order linear in GMT. It then follows that a linear energy balance evolution equation for GMT is sufficient to reproduce the slow timescale GMT dynamics, provided that the appropriate effective heat capacity and feedback parameters are known. As a consequence, the GCM’s GMT timeseries may underestimate the impact of, and uncertainty in, the outcomes of future forcing scenarios. The offset subtraction procedure identifies a slow time-scale dynamics in model generated GMT. Fluctuations on much faster timescales do not typically track each other from one GCM to another, with the exception of major forcing events such as volcanic eruptions. This suggests that the GMT time-series can be decomposed into a slow and fast timescale which naturally leads to stochastic reduced energy balance models for GMT. [1] IPCC Chapter 9 P743 and fig 9.8, IPCC TS.1 [2] see e.g. [Mauritsen et al., Tuning the Climate of a Global Model, Journal of Advances in Modelling Earth Systems, 2012]4, IPCC SPM.6

Heavy particles suspended in an incompressible randomly mixing or turbulent flow form a ‘turbulent aerosol’. When the inertia of the particles is significant then the particles respond in intricate ways to the turbulent fluctuations of the carrying fluid: in- dependent particles may cluster together and form spatial patterns even though the fluid is incompressible, and the relative speeds of nearby particles may fluctuate strongly. Both phenomena depend sensitively on the inertia of the particles, and both phenomena affect collision rates and collision outcomes, and thus the long-term fate of the turbulent aerosol. In recent years it has become clear that important aspects of the dynamics of turbulent aerosols can be understood in terms of statistical models. In this talk I describe how statistical-model calculations have led to a detailed understanding of the mechanisms that determine inertial-particle dynamics in turbulent aerosols.

with D. A Stainforth, N. W. Watkins Estimates of how our climate is changing are needed both locally and globally. For local adaptation this requires quantifying the geographical patterns in changes at specific quantiles or thresholds in distributions of variables such as daily surface temperature and precipitation. This talk will first focus on these local changes and on a model independent method [1] to transform daily observations into patterns of local climate change. This method estimates how fast different quantiles across the distributions are changing. This determines which quantiles and geographical locations show the greatest change and also those at which any change is highly uncertain. For daily temperature changes in the distribution itself can yield robust results [2]. For fatter-tailed distributions such as precipitation we can focus on quantities that characterize the changes in time of the likelihood above a threshold and of the relative amount of precipitation in those days [3]. The fundamental timescales of anthropogenic climate change limit the identification of societally relevant aspects of changes but nevertheless it is possible to extract, solely from observations, some confident quantified assessments of change at certain thresholds and locations. We demonstrate this approach using E-OBS gridded data timeseries [4] from specific locations across Europe over the last 60 years. Global mean temperature (GMT) provides a simple means of benchmarking a broad ensemble of global climate models (GCMs) against past observed GMT which in turn provide headline assessments of the consequences of possible future forcing scenarios. The slow variations of past changes in GMT seen in different GCMs track each other [5] and the observed GMT reasonably closely. However, the different GCMs tend to generate GMT time-series which have absolute values that are offset with respect to each other [6]. Subtracting these offsets is an integral part of comparisons between ensembles of GCMs and observed past GMT. We will discuss how this constrains how the GCMs are related to each other. The GMT of a given GCM is a macroscopic reduced variable that tracks a subset of the full information contained in the time evolving solution of that GCM. If the GMT slow timescale dynamics of different GCMs is to a good approximation the same, subject to a linear translation, then the phenomenology captured by this dynamics is essentially linear; any feedback is to leading order linear in GMT. It then follows that a linear energy balance evolution equation for GMT is sufficient to reproduce the slow timescale GMT dynamics, provided that the appropriate effective heat capacity and feedback parameters are known. As a consequence, GMT may not be a sensitive indicator of non-linear dynamics and may underestimate the impact of, and uncertainty in, the outcomes of future forcing scenarios. The offset subtraction procedure identifies a slow time-scale dynamics in model generated GMT. Fluctuations on much faster timescales do not track each other from one GCM to another. This suggests that the GMT time-series can be decomposed into a slow and fast timescale which naturally leads to stochastic reduced energy balance models for GMT. 1] Chapman, S. C., D. A. Stainforth, N. W. Watkins, Phil. Trans. Royal Soc., A,371 20120287 (2013) [2] Stainforth, D. A. S. C. Chapman, N. W. Watkins, ERL 8, 034031 (2013) [3] Chapman, S. C., Stainforth, D. A., Watkins, N. W., ERL 10, 094018 (2015) [4] Haylock M. R. et al., J. Geophys. Res (Atmospheres), 113, D20119, (2008) [5] IPCC Chapter 9 P743 and fig 9.8, IPCC TS.1 [6] see e.g. [Mauritsen et al., Tuning the Climate of a Global Model, Journal of Advances in Modelling Earth Systems, 2012]4, IPCC SPM.6

I intend my talk to be informal and interactive, and to build some of the points arising from Friday’s discussions in CANEID. Responding to the questions in Holger’s talk about the present status of the idea of long range dependence, and based on my discussion in Watkins, EPJB submitted (see also https://arxiv.org/abs/1603.00738), I will describe three things: The Hurst effect, 1/f noise, and stationary long range dependence, which may all be present in a time series model, but which are not completely synonymous, as still seems often to be believed. I will describe how, for example, models such as fractional Gaussian noise and ARFIMA can show all these properties (see also Graves et al, Entropy, submitted, https://www.preprints.org/manuscript/201705.0194/v1 ) but how a different class of model, the fractional renewal process (known at least since the work of Mandelbrot in 1963-67, see Watkins op cit, and substantially developed in recent papers by Kantz, Barkai, Niemann and others) gives rise to quite a different origin for 1/f noise. This model is arguably not stationary LRD in the same sense as fGn. Consideration of the different ways in which such behaviours can arise, will illuminate the varieties of 1/f behaviour seen, giving a partial answer to Holger’s question. I also plan to summarise our recent papers on how, if one knows that the ARFIMA class is suitable, one can do Bayesian inference to obtain both the memory parameter d and the parameters specifying additional short-ranged ARMA(p,q) behaviour. This work is published in Graves et al https://www.nonlin-processes-geophys.net/22/679/2015/ and http://www.sciencedirect.com/science/article/pii/S0378437117300298 Time permitting, I hope also to talk a bit more about how the generalised and fractional Langevin equations could be motivated in a climate context, and how these complement the vector Langevin models used by Penland and others, and the fGn model used in Lovejoy and co-workers’ StocSIPS forecasting approach, as well as the extensive recent work by the Tromso group on long-ranged response in climate.

Climate history precedes the observational record. Indeed, important timescales of variability in the ocean and glaciers precede the invention of concepts such as temperature! The paleoclimatic record offers our only view into variability on these long timescales. I will present the basic ideas of climate reconstruction from paleo-proxies: sediments, ice cores, corals, isotopes, etc. The sparseness of the data, difficulties in interpretation, and some discussion as to what is revealed will be the foci.

Deeper understanding of complex dynamics of the Earth atmosphere and climate is inevitable for sustainable development, mitigation and adaptation strategies for global change and for prediction of and resilience against extreme events. Traditional (linear) approaches cannot explain or even detect nonlinear interactions of dynamical processes evolving on multiple spatial and temporal scales. Combination of nonlinear dynamics and information theory explains synchronization as a process of adjustment of information rates [1] and causal relations (\`{a} la Granger) as information transfer [2]. Information born in dynamical complexity or information transferred among systems on a way to synchronization might appear as an abstract quantity, however, information transfer is tied to a transfer of mass and energy, as demonstrated in a recent study using directed (causal) climate networks [2]. Recently, an information transfer across scales of atmospheric dynamics has been observed [3]. In particular, a climate oscillation with the period around 7--8 years has been identified as a factor influencing variability of surface air temperature (SAT) on shorter time scales. Its influence on the amplitude of the SAT annual cycle was estimated in the range 0.7--1.4~$^{\circ}$C and the effect on the overall variability of the SAT anomalies (SATA) leads to the changes 1.5--1.7~$^{\circ}$C in the annual SATA means. The strongest effect of the 7—8 year cycle was observed in the winter SATA means where it reaches 4--5~$^{\circ}$C in central European station and reanalysis data [4]. In the dynamics of El Niño-Southern Oscillation, three principal time scales have been identified: the annual cycle (AC), the quasibiennial (QB) mode(s) and the low-frequency (LF) variability. An intricate causal network of information flows among these modes helps to understand the occurrence of extreme El Niño events, characterized by synchronization of the QB modes and AC, and modulation of the QB amplitude by the LF mode. The latter also influences the phase of the AC and QB modes. These examples provide an inspiration for a discussion how novel data analysis methods, based on topics from nonlinear dynamical systems, their synchronization, (Granger) causality and information transfer, in combination with dynamical and statistical models of different complexity, can help in understanding and prediction of climate variability on different scales and in estimating probability of occurrence of extreme climate events. [1] M. Palus, V. Komarek, Z. Hrncir, K. Sterbova, Phys. Rev. E, 63(4), 046211 (2001) http://www.cs.cas.cz/mp/epr/sir1-a.html [2] J. Hlinka, N. Jajcay , D. Hartman, M. Palus, Smooth Information Flow in Temperature Climate Network Reflects Mass Transport, submitted to Chaos. http://www.cs.cas.cz/mp/epr/vetry-a.html [3] M. Palus, Phys. Rev. Lett. 112 078702 (2014) http://www.cs.cas.cz/mp/epr/xf1-a.html [4] N. Jajcay, J. Hlinka, S. Kravtsov, A. A. Tsonis, M. Palus,. Geophys. Res. Lett. 43(2), 902-909 (2016) http://www.cs.cas.cz/mp/epr/xfgrl1-a.html

The mathematical theory of records is well developed for sequences of i.i.d. random variables, but much less is known about records in time series with a trend. The talk will review results that have been obtained for the paradigmatic linear drift model originally introduced in the context of athletic records, and describe its application to time series of temperature and precipitation.

The atmosphere contains turbulence on all length scales from the planetary scale down to the Kolmogorov dissipation scale. An important mechanism for generating atmospheric turbulence is the Kelvin-Helmholtz shear instability. Climate change is not warming the atmosphere uniformly, and the temperature trend as a function of latitude and height has a well-defined structure. In order to remain in geostrophic balance, the mid-latitude winds must respond to this temperature pattern by becoming more sheared in the upper troposphere and lower stratosphere. This is a mechanism by which climate change could generate more turbulence, including on the length scales that affect aviation. Our calculations show that the amount of severe aircraft turbulence could double or even treble later this century as a consequence.

The Indian monsoon features strong variability between different years and decades. Patterns in the variability between normal, drought and flood years can be seen and are examined further in this study. El Niño anticorrelates with drought years while the multidecadal variability showed peaks around 1905 and 1975 and a minimum around 1935 and around now. The multidecadal variability correlates very strongly with the Indian Ocean Dipole, a weak mode of variability in sea surface temperatures of the Indian Ocean until the 1930s, then the correlation disappears. This study goes deeper into explaining the variability than previous studies. It'll be shown that standard indices such as El Niño and the Indian Ocean Dipole are not maximally effective in providing explanations for internal climate variability and monsoon rains. In particular, alternatives for the weak Indian Ocean index are suggested. Besides being a large-scale seasonal analogy of a sea breeze, the migration of the Inter-tropical convergence zone and jet streams are known to be important in producing the monsoon. When the ocean boundary conditions for atmospheric pressure, evaporation and therefore monsoon winds and rains are considered in more detail for each year, causality from sea surface temperature fields to monsoon rains is extracted on a more physical basis. Ocean areas of particular importance to create (un)favorable evaporation and monsoon wind fields are identified. Couplings between the Pacific and Indian Oceans are examined for added information as to why the patterns of internannual and multidecadal monsoon variability appear the way they do. Mechanisms are studied to some extent using those climate model simulations that reproduce the observed variability best.

Hybrid phase transition showing features of both second and first-order phase transitions at a transition point has been recently issued in complex systems, even though it was observed a long time ago in equilibrium physical systems such as a spin-glass model and liquid-helium systems. A hybrid percolation transition (HPT) induced by cascade dynamics with contact process is one of such examples found in $k$-core percolation and the cascading failure model on interdependent networks. A HPT of a restricted percolation model is another example. While such a transition type is identified by the behavior of the order parameter, microscopic mechanisms underlying those HPTs were not uncovered yet. In my talks, I will report our recent discoveries of the underlying mechanism and features of such transition behaviors. Moreover, the conventional scaling and hyperscaling relations between critical exponents are checked in hybrid percolation transitions.