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Timetable (MFEW01)

Non-equilibrium Statistical Mechanics and the Theory of Extreme Events in Earth Science

Tuesday 29th October 2013 to Friday 1st November 2013

Tuesday 29th October 2013
08:30 to 08:50 Registration
08:50 to 09:00 Welcome by John Toland, Director of the Institute
09:00 to 09:35 M Pollicott (University of Warwick)
Geodesic flows: Mixing, zeta functions and resonances
Historically important examples of ``chaotic'' dynamical systems are Anosov flows, in particular, and geodesic flows on negatively curved manifolds. In particular, they provide a concrete setting to explore a wealth of interesting topics: (i) mixing rates (which can be studied using zeta function and resonances); (ii) large deviations and fluctuation theorems (Gallavotti-Cohen theorem in non-equilibrium statistical mechanics); and (iii) escape rates (the rate at which mass escapes from an open system) and extremes.
INI 1
09:35 to 10:10 S Galatolo (Università di Pisa)
Rigorous computation of invariant measures and Lyapunov exponents
Co-author: Isaia Nisoli (Universitade Federal Rio de Janeiro)

We will consider the problem of computation of invariant measures and other aspects related to the statistical behavior of the dynamics up to certified errors.

In this way the output of a computation represent some rigorous quantitative estimation on the behavior of the dynamics under study, going towards more reliable tools for the simulation of dynamical models.

We will show some general approach which can be applied in several cases of systems having some hyperbolic behavior, including maps with indifferent fixed points.

Time permitting we will also consider a class piecewise hyperbolic maps related to the Lorenz attractor.

INI 1
10:10 to 10:45 P Imkeller (Humboldt-Universität zu Berlin)
Paleo-climatic time series: statistics and dynamics
Co-authors: Arnaud Debussche (ENS Cachan), Jan Gairing (HU Berlin), Claudia Hein (HU Berlin), Michael Högele (U Potsdam), Ilya Pavlyukevich (U Jena)

Dynamical systems of the reaction-diffusion type with small noise have been instrumental to explain basic features of the dynamics of paleo-climate data. For instance, a spectral analysis of Greenland ice time series performed at the end of the 1990s representing average temperatures during the last ice age suggest an $\alpha-$stable noise component with an $\alpha\sim 1.75.$ We model the time series as a dynamical system perturbed by $\alpha$-stable noise, and develop an efficient testing method for the best fitting $\alpha$. The method is based on the observed $p$-variation of the residuals of the time series, and their asymptotic $\frac{\alpha}{p}$-stability established in local limit theorems.\par\smallskip

Generalizing the solution of this model selection problem, we are led to a class of reaction-diffusion equations with additive $\alpha$-stable L\'evy noise, a stochastic perturbation of the Chafee-Infante equation. We study exit and transition between meta-stable states of their solutions. Due to the heavy-tail nature of an $\alpha$-stable noise component, the results differ strongly from the well known case of purely Gaussian perturbations.

INI 1
10:45 to 11:10 Morning Coffee
11:10 to 11:45 T Kuna (University of Reading)
Typical behaviour of extremes of chaotic dynamical systems for general observables
In this talk we discuss the distribution of extreme events for a chaotic dynamical system for a general class of observables. More precisely, we link directly the distribution of events over threshold to the local geometrical structure on the surface of the attractor. It is shown how this can provide us with information about the local stable and unstable dimensions. Using Ruelle's response theory, we discuss the sensitivity of the parameters of the distribution under perturbations. This is a joint work with Vlaerio Lucarini, Davide Faranda and Jeroen Wouters.
INI 1
11:45 to 12:20 S Wang (Indiana University)
Interplay between Mathematics and Physics
Co-author: Tian Ma (Sichuan University)

In this talk, we shall present three first principles and a few examples, demonstrating the symbiotic interplay between theoretical physics and advanced mathematics.

We start with a general principle that dynamic transitions of all dissipative systems can be classified into three categories: continuous, catastrophic and random. We shall illustrate this principle with a few examples in both equilibrium and non-equilibrium phase transitions, including the metastable oscillation mechanism of the El Nino Southern Oscillation (ENSO) and the existence of 3rd-order transitions beyond the Andrews critical point.

Then we present two basic principles: the principle of interaction dynamics (PID) and the principle of representation invariance (PRI), to study the nature's fundamental interactions/forces. Intuitively, PID takes the variation of the action functional under energy-momentum conservation constraint. PRI requires that physical laws be independent of representations of the gauge groups. These two principles give rise to a unified field model for four interactions, which can be naturally decoupled to study individual interactions. With PID, for example, we derive new gravitational field equations with a vector field $\Phi_\mu$, which can be considered as a spin-1 massless bosonic particle field. The field equations induce a natural duality between the graviton (spin-2 massless bosonic particle) and this spin-1 massless bosonic particle, leading to a unified theory for dark matter and dark energy. In addition, the PID offers a completely different and much simpler way of introducing Higgs fields.

INI 1
12:30 to 13:30 Lunch at Wolfson Court
13:40 to 14:15 M Branicki (New York University)
Quantifying uncertainty and improving reduced-order predictions of partially observed turbulent dynamical systems
Co-author: A. J. Majda (Courant Institute, NYU)

The issue of mitigating model error in reduced-order prediction of high-dimensional dynamics is particularly important when dealing with turbulent geophysical systems with rough energy spectra and intermittency near the resolution cut-off of the corresponding numerical models. I will discuss a new framework which allows for information-theoretic quantification of uncertainty and mitigation of model error in imperfect stochastic/statistical predictions of non-Gaussian, multi-scale dynamics. In particular, I will outline the utility of this framework in derivation of a sufficient condition for improving imperfect predictions via a popular but heuristic Multi Model Ensemble approach. Time permitting, the role and validity of 'fluctuation-dissipation' arguments for improving imperfect predictions of externally perturbed non-autonomous turbulent systems will also be addressed.
INI 1
14:15 to 14:50 S Vaienti (Centre de Physique Théorique, Marseille)
Extreme value theory for randomly perturbed systems: getting the local dimensions
We present some new results for extreme values distributions in dynamical systems perturbed "via" random transformations and with observational noise. In both cases the linear scaling parameters of the Gumbel law will allow to get informations on the local behavior respectively of the stationary measure (random transformations), and of the invariant measure (observational noise). This collects work done with Aytac, Faranda, Freitas, Lucarini and Turchetti.
INI 1
14:50 to 15:25 D Faranda (CEA/Saclay)
A new recurrences based technique for detecting robust extrema in long temperature records
Co-author: Sandro Vaienti (University of Marseille)

By using new techniques originally developed for the analysis of extreme values of dynamical systems, several long records of temperatures at different locations are analysed by showing that they have the same recurrence time statistics of a chaotic dynamical system perturbed with dynamical noise and by instrument errors. The technique provides a criterion to discriminate whether the recurrence of a certain temperature belongs to the natural climate variability or can be considered as a real extreme event with respect to a specific time scale fixed as parameter. The method gives a self-consistent estimation of the convergence.

INI 1
15:25 to 15:50 Afternoon Tea
15:50 to 16:25 I Didenkulova (Tallinn Technical University)
Extreme sea waves in the coastal zone
INI 1
16:25 to 17:00 J Vollmer (Max-Planck-Institut fur Dynamics and Self-Organisation)
Dew droplets and cloud droplets: droplet growth, size distributions, and corrections to scaling
I present the results of comprehensive laboratory experiments and numerical studies addressing droplet growth and droplet size distributions in systems where droplets grow due to sustained supersaturation of their environment.

Both, for droplets condensing on a substrate (like dew) and droplets entrained in an external flow (like in clouds), we observe remarkable shortcomings of classical scaling theories addressing these growth processes. The origins of the discrepancies are identified, and appropriate extensions of the theories are discussed.
INI 1
17:00 to 18:00 Welcome Wine Reception
Wednesday 30th October 2013
09:00 to 09:35 G Jona-Lasinio (Università degli Studi di Roma La Sapienza)
On thermodynamics of stationary states of diffusive systems Coauthors L. Bertini, A. De Sole, D. Gabrielli, C. Landim
Thermodynamic transformations connecting nonequilibrium stationary states have the peculiarity of dissipating, to keep the system out of equilibrium, an amount of energy which diverges for a quasi static transformation. By subtracting the divergent part one can define a renormalized work that satisfies a Clausius type inequality and with respect to which quasi static transformations are optimal. A different way of analyzing the energy balance and optimality criteria is to consider transformations over a long but finite time T developing the total work and the dissipated energy in powers of 1/T. The diverging terms cancel and one obtains relations among finite quantities.
INI 1
09:35 to 10:10 C Beck (Queen Mary, University of London)
Environmental superstatistics
Complex systems in driven nonequilibrium situations often consist of a superposition of several dynamics on well-separated time scales. Sometimes the parameters of the system fluctuate as well, on a much larger time scale than the local dynamics. The resulting marginal distributions typically have fat tails, which can be understood by superstatistical techniques. After a short review of the field I will concentrate on some examples relevant for planet earth: The dynamics of tracer particles in turbulent flows, the surface temperature statistics at various locations on planet earth, and the dynamics of sea levels.
INI 1
10:10 to 10:45 D Kondrashov (University of California, Los Angeles)
Data-driven model reduction and climate prediction: nonlinear stochastic, energy-conserving models with memory effects
Co-authors: Mickael D. Chekroun (University of California, Los Angeles), Michael Ghil (University of California, Los Angeles)

This talk will focus on theoretical understanding and climate applications of a data-driven reduction strategy that leads to low-order stochastic-dynamical models with energy-conserving nonlinearities and conveying memory effects. New opportunities for climate prediction will be illustrated in the framework of "Past Noise Forecasting", by utilizing on the one hand estimated history of the driving noise by the low-order model, and on the other hand the phase of low-frequency variability estimated by advanced time series analysis.

INI 1
10:45 to 11:10 Morning Coffee
11:10 to 11:45 DTC Crommelin (Centrum voor Wiskunde en Informatica (CWI))
Efficient sampling of rare events by splitting
Standard (or crude) Monte Carlo (MC) simulation is known to be inefficient for simulating rare events. For events with low probability, the squared relative error on estimates obtained from straightforward MC simulation is inversely proportional to the number of samples, so that an excessively large number of samples may be required to reach a desired accuracy for the estimation of rare event probabilities.

To improve the efficiency of MC sampling for rare events, various techniques have been developed in the past, for applications in e.g. communication networks and reliability analysis. Such techniques can be of interest for studying extremes in geophysical models. I will discuss a technique called multilevel splitting, in which model sample paths are split into multiple copies each time they cross thresholds (or levels) that lead closer to the rare event set.

INI 1
11:45 to 12:20 F Kwasniok (University of Exeter)
Regime-dependent modelling of extremes in the extra-tropical atmospheric circulation
The talk discusses data-based statistical-dynamical modelling of vorticity and wind speed extremes in the extra-tropical atmospheric circulation. The extreme model is conditional on the large-scale flow, consisting of a collection of local generalised Pareto distributions, each associated with a cluster or regime in the space of large-scale flow variables. The clusters and the parameters of the extreme models are estimated from data, either separately or simultaneously. The large-scale flow is represented by the leading empirical orthogonal functions (EOFs). Also temporal clustering of extremes in the different large-scale regimes is investigated using an inhomogeneous Poisson process model whose rate parameter is conditional on the large-scale flow. The study is performed in the framework of an intermediate complexity atmospheric model with realistic mean state, variability and teleconnection patterns. The methodology can also be applied to data from GCM scenario simulations, predicting future extremes.
INI 1
12:30 to 13:30 Lunch at Wolfson Court
13:40 to 14:15 C Maes (Katholieke Universiteit Leuven)
The modified second fluctuation-dissipation theorem
Baths produce friction and random forcing on particles suspended in them. The relation between noise and friction in (generalized) Langevin equations is usually referred to as the second fluctuation-dissipation theorem. We show, beyond formalities, what is the proper nonequilibrium extension, to be applied when the environment is itself active and driven.
INI 1
14:15 to 14:50 M Colangeli (Politecnico di Torino)
On the use of Ruelle's formalism in response theory
We use Ruelle’s formalism to express the response of a generic observable to a certain perturbation in terms of correlation functions computed with respect to the unperturbed invariant measure, for deterministic as well as stochastic dynamics. We discuss the onset of two relevant terms for the entropy production, comment on the Hamiltonian version of the resulting formulae and also propose a connection with similar results, reported in the literature, allowing to extend the Fluctuation-Dissipation formalism to nonequilibrium steady states.
INI 1
14:50 to 15:25 R Klages (Queen Mary, University of London)
Anomalous fluctuation relations
Co-authors: Aleksei V. Chechkin (Institute for Theoretical Physics NSC KIPT, Kharkov, Ukraine), Peter Dieterich (Institut fuer Physiologie, Medizinische Fakultaet Carl Gustav Carus, Dresden, Germany), Friedrich Lenz (Queen Mary University of London, School of Mathematical Sciences, London, UK)

We study Fluctuation Relations (FRs) for Gaussian stochastic systems that are anomalous, in the sense that the diffusive properties strongly deviate from the ones of Brownian motion. For this purpose we use a Langevin approach: We first briefly review the concept of transient work FRs for simple Langevin dynamics generating normal diffusion [1]. We then consider two different types of additive, power law correlated Gaussian noise [2,3]: (1) internal noise with a fluctuation-dissipation relation of the second type (FDR2), and (2) external noise without FDR2. For internal noise we find that FDR2 leads to conventional (normal) forms of transient work FRs. For external noise we obtain various forms of violations of normal FRs, which we call anomalous FRs. We show that our theory is important for understanding experimental results on fluctuations in systems with long-time correlations, such as glassy dynamics [1].

[1] R.Klages, A.V.Chechkin, P.Dieterich, Anomalous fluctuation relations, book chapter in: R.Klages, W.Just, C.Jarzynski (Eds.), Nonequilibrium Statistical Physics of Small Systems, Wiley-VCH, Weinheim (2013) [2] A.V.Chechkin, F.Lenz, R.Klages, J.Stat.Mech. L11001 (2012) [3] A.V.Chechkin, R.Klages, J.Stat.Mech. L03002 (2009)
INI 1
15:25 to 15:55 Afternoon Tea
15:55 to 16:30 J Vanneste (University of Edinburgh)
A large-deviation approach to passive scalar advection, diffusion and reaction
Co-authors: Peter H. Haynes (University of Cambridge), Alexandra Tzella (University of Birmingham)

The dispersion of a passive scalar in a fluid through the combined action of advection and molecular diffusion can often be described as a diffusive process, with an effective diffusivity that is enhanced compared to the molecular value. This description fails to capture the tails of the scalar concentration in initial-value problems, however. This talk addresses this issue and shows how the theory of large deviation can be applied to capture the concentration tails by solving a family of eigenvalue problems. Two types of flows are considered: shear flows and cellular flows. In both cases, large deviation is shown to generalise classical results (Taylor dispersion for shear flows, homogenisation results for cellular flows). Explicit asymptotic results are obtained in the limit of large Péclet number corresponding to small molecular diffusivity. The implications of the results for the problem of front propagation in reacting flows are also discussed.

INI 1
19:30 to 22:00 Conference Dinner at Emmanuel College
Thursday 31st October 2013
09:00 to 09:35 M Ghil ([Ecole Normale Supérieure, Paris & UCLA])
Extreme Events and Coupled Climate-Economics Modeling
In this talk, I will review some recent work on extreme events, their causes and consequences. The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deterministic modeling of extreme events, via continuous and discrete dynamic models. The applications include climatic, seismic and socio-economic events, along with their prediction. Two important results refer to (i) the complementarity of spectral analysis of a time series in terms of the continuous and the discrete part of its power spectrum; and (ii) the need for coupled modeling of natural and socio-economic systems. Both these results have implications for the study and prediction of natural hazards and their human impacts. A substantial part of the talk will deal with an endogenous business cycle (EnBC) model and with the way that EnBCs affect the impact of natural hazards on a dynamic economy. An out-of-equilibrium fluctuation-dissipation result for macroeconomics is inferred from the model and confirmed by the analysis of US economic data.
INI 1
09:35 to 10:10 RC Dewar (Australian National University)
Kinetic energy dissipation and the stability of stationary turbulent flows
Variational principles of fluid turbulence offer an attractive alternative to numerical solution of the Navier-Stokes equation, especially for global climate studies. I discuss the principle (Max-D) that certain stationary turbulent flows maximize the rate of kinetic energy dissipation of the mean flow. Following its conjecture as an organizational principle for atmospheric circulation [1], Max-D has gained numerical support from global climate model simulations [2]. Max-D has also been derived for turbulent shear flow in a channel from considerations of dynamic stability, and yields realistic predictions for the mean velocity profile at all Reynolds numbers [3]. Further theoretical support for Max-D in channel flow has emerged from the statistical principle of maximum entropy [4]. Tying these threads together may lead to a clearer understanding of the theoretical basis and range of validity of Max-D for global climate studies. I outline possible approaches to doing this.

[1] Lorenz EN (1955) Generation of available potential energy and the intensity of the general circulation. Scientific Report No. 1, UCLA Large Scale Synoptic Processes Project.

[2] Pascale S, Gregory JM, Ambaum MHP, Tailleux R (2012) A parametric sensitivity study of entropy production and kinetic energy dissipation using the FAMOUS AOGCM. Clim. Dyn. 38, 1211-1227 and references therein.

[3] Malkus WVR (2003) Borders of disorders: in turbulent channel flow. J. Fluid Mech. 489, 185-198.

[4] Dewar RC, Maritan A (2013) A theoretical basis for maximum entropy production. In Beyond the Second Law: Entropy Production and Non-equilibrium Systems (eds. RC Dewar, CH Lineweaver, RK Niven, K Regenauer-Lieb), Springer, in press.
INI 1
10:10 to 10:45 N Glatt-Holtz (Virginia Polytechnic Institute and State University)
Inviscid Limits for the Stochastic Navier Stokes Equations and Related Systems
One of the original motivations for the development of stochastic partial differential equations traces it's origins to the study of turbulence. In particular, invariant measures provide a canonical mathematical object connecting the basic equations of fluid dynamics to the statistical properties of turbulent flows. In this talk we discuss some recent results concerning inviscid limits in this class of measures for the stochastic Navier-Stokes equations and other related systems arising in geophysical and numerical settings. This is joint work with Peter Constantin, Vladimir Sverak and Vlad Vicol.
INI 1
10:45 to 11:10 Morning Coffee INI 1
11:10 to 11:45 J Wouters (Universität Hamburg)
A statistical mechanics approach to stochastic parametrizations
Co-author: Valerio Lucarini (University of Hamburg)

Current computer simulations of climate and weather prediction models can only take into account a limited number of the relevant degrees of freedom of the climate system. Therefore the physical dynamical equations need to be reduced to a smaller subset of variables.

The reduction of the number of degrees of freedom (also known as parametrization in the modeling community) is a central task in statistical mechanics and it is therefore not surprising that many techniques used in this field are also used in the derivation of stochastic parametrizations.

In this talk I will discuss some of these techniques and how they have been applied to climate modeling. I will then discuss how we have used the Mori-Zwanzig formalism and response theory to derive parametrizations for weakly coupled dynamical systems.

INI 1
11:45 to 12:20 PM Cox (University of Exeter)
Emergent Constraints on Earth System Sensitivities
Co-author: Chris Huntingford (Centre for Ecology and Hydrology)

Climate and Earth System Models are designed to project changes in the climate-carbon cycle system over the coming centuries. These models have ever higher spatial resolution and are based on an improving understanding of key processes. However, climate modelling still suffers from a significant timescale problem – we need to find constraints on the huge range of projected changes in the climate-carbon system over the next century, but the observational data that we have relates to much shorter timescales. This talk will summarise one promising way around the timescale problem - the use of “Emergent Constraints”. An Emergent Constraint is a relationship between some climate system sensitivity to anthropogenic forcing and an observable (or already observed) feature of the climate system. We call it emergent because it emerges from the ensemble of models, and it is described as a constraint because it enables an observation to constrain the estimate of the cli mate system sensitivity in the real world. As an example, I will describe an emergent constraint on the projected loss of tropical land carbon under climate change, which has a huge range amongst climate-carbon cycle projections for the 21st century. We have recently identified an emergent linear relationship across the ensemble of models between the sensitivity of tropical land-carbon storage to warming and the sensitivity of the annual growth-rate in atmospheric CO2 to tropical temperature anomalies. When combined with contemporary observations of the atmospheric CO2 concentration and the tropical temperature, this relationship provides a tight constraint on the sensitivity of tropical land carbon to warming in the real climate system (Cox et al., Nature, 2013). The talk will conclude by hypothesising how such emergent constraints may relate to (a) the Fluctuation-Dissipation Theorem and (b) Time-series Precursors of Tipping Points.

INI 1
12:30 to 13:30 Lunch at Wolfson Court
13:40 to 14:15 P Embrechts (ETH Zürich)
The modeling of rare events: from methodology to practice and back
In this talk I give a brief overview of the historical development of Extreme Value Theory (EVT), discuss some applications, highlighting EVT's strengths and weaknesses, and indicate relevant research themes going forward.
INI 1
14:15 to 14:50 P Friederichs (Universität Bonn)
Bayesian approaches for wind gust and quantitative precipitation forecasting
Co-author: Sabrina Bentzien (Meteorological Institute, University of Bonn, Germany)

Due to large uncertainties, predictions of high-impact weather on the atmospheric mesoscale are probabilistic in nature. Mesoscale weather ensemble prediction systems (EPS) are developed to obtain probabilistic guidance for high impact weather. An EPS not only issues a deterministic future state of the atmosphere but a sample of possible future states. Ensemble postprocessing then translates such a sample of forecasts into probabilistic measures.

We discuss Bayesian approaches for wind gust and quantitative precipitation forecasting. The Bayesian hierarchical model (BHM) for wind gusts uses extreme value theory, namely a generalized extreme value distribution (GEV), in the data level. A process level for the parameters is introduced which, on the one hand, models the spatial response surfaces of the GEV parameters as Gaussian random fields, and, on the other hand, incorporate the information of the COSMO-DE forecasts. The spatial BHM provides area wide forecasts of wind gusts in terms of a conditional GEV. It models the marginal distribution of the spatial gust process and provides not only forecasts of the conditional GEV at locations without observations, but also uncertainty information about the estimates. At this stage, the BHM ignores the conditional dependence between gusts at neighboring locations. However, an outline is given how this will be incorporated in a subsequent study using max-stable random fields.

For quantitative precipitation forecasting we use Bayesian quantile regression and its spatially adaptive extension together with a variable selection based on a Bayesian LASSO. All this is illustrated for the German-focused mesoscale weather prediction ensemble COSMO-DE-EPS, which runs operationally since December 2010 at the German Meteorological Service (DWD). We further discuss the issue of objective out-of-sample verification, where performance is measured using proper scoring rules and their decomposition.

INI 1
14:50 to 15:25 V Chavez-Demoulin (ETH Zürich)
Generalized additive modelling of hydrological sample extremes
Co-authors: Anthony Davison (EPFL, Lausanne), Marius Hofert (ETHZ, Zurich)

Estimation of flood frequencies and severities is important for many water management issues. We present a smoothing extreme value method fitted by penalized loglikelihood. Spline smoothing is used to estimate the parameters of the frequency and size distributions of extremes, depending on covariates in a non- or semiparametric way. The frequency process of high level extremes is modelled by a Poisson process, either homogeneous or non-homogeneous. The extreme sizes are considered to follow a generalized Pareto distribution. Being given by two parameters, the method of spline smoothing is not straightforward to apply. An efficient fitting algorithm based on orthogonal reparametrisation is developed to achieve this task. The method is applied to the daily maximum flows of an hydrological station in Switzerland and is used to estimate 20-year return levels.
INI 1
15:25 to 15:50 Afternoon Tea
15:50 to 16:25 S Lovejoy (McGill University)
Extreme events and the multifractal butterfly effect
Scaling processes abound in geophysics and this has important consequences for the probability distributions of the corresponding intensive and extensive geophysical variables. Classical scaling processes – such as in classical turbulence – are self-similar, they are characterized by exponents which are invariant under isotropic scale changes. However, the atmosphere and lithosphere are strongly stratified so that we must generalize the notion of scale allowing for invariance under anisotropic zooms. When this is done, it is often found that scaling can apply over huge ranges, up to planetary in extent. It is now clear that the generic scaling process is the multifractal cascade in which a scale invariant dynamical mechanism repeats (multiplicatively) from scale to scale; anisotropic scaling – and multifractal universality classes - imply that multifractals are widely relevant in the earth sciences. General (canonical) multifractal processes developed over finite ranges of scale and analyzed at their smallest scale (the “bare” process), have “long-tailed” distributions (e.g. the lognormal). However the small scale cascade limit is singular so that the integration/averaging of cascades developed down to their small scale limits leads to “dressed” properties characterized notably by “fat-tailed” power law probability distributions Pr(x>s)=s**-qD where x is a random value, s a threshold and qD the critical exponent implying that the moments for q>qD diverge. For cascades averaged over scales larger than the inne r cascade scale, the moments q>qD are no longer determined by the large scale finite by the small scale details: the “multifractal butterfly effect”. The sampling properties of such processes can be understood with “multifractal phase transitions”; we review this as well as evidence for the divergence of moments in laboratory, atmospheric and climatological series, and in data from the solid earth and discuss implications (abrupt changes, etc.).
INI 1
16:25 to 17:00 J Kurths (Potsdam Institute for Climate Impact Research (PIK))
Complex networks identify spatial patterns of extreme rainfall events of the Indian and the South American monsoon system
Co-authors: Niklas Boers (Potsdam Institute for Climate Impact Research ), Veronika Stolbova (Potsdam Institute for Climate Impact Research ), Bodo Bookhagen (UC Santa Barbara, Geography Department)

We investigate the spatial characteristics of extreme rainfall synchronicity of the Indian and South American Summer Monsoon System by means of Complex Networks (CN). By introducing a new combination of CN measures and interpreting it in a climatic context, we investigate climatic linkages and classify the spatial characteristics of extreme rainfall synchronicity. Although our approach is based on only one variable (rainfall), it reveals the most important features of the Monsoon Systems, such as the main moisture pathways, areas with frequent development of Mesoscale Convective Systems, and the major convergence zones. In addition, our results reveal substantial differences between the spatial structures of rainfall synchronicity above the 90th and above the 95th percentiles.

References Arenas, A., A. Diaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Phys. Reports 2008, 469, 93. Boers, N., B. Bookhagen, N. Marwan, J. Kurths, and J. Marengo, Geophys. Res. Lett. 2013, DOI: 10.1002/grl.50681 (online). Donges, J., Y. Zou, N. Marwan, and J. Kurths, Europhys. Lett. 2009, 87, 48007. Malik, N., B. Bookhagen, N. Marwan, and J. Kurths, Climate Dynamics, 2012, 39, 971.
INI 1
17:00 to 17:35 J Bertram (Australian National University)
Statistical stability arguments for maximum kinetic energy dissipation
The hypothesis that stationary turbulent flows have maximal mean-flow kinetic energy dissipation (Max-D) is intriguing because mean-flow properties can be predicted without modelling the turbulent component of the flow. Our knowledge of Max-D is largely restricted to relatively simple laboratory flows. Measured Poiseuille flow profiles match Max-D predictions closely and, under these simplified conditions, Malkus's statistical stability argument provides some theoretical justification for Max-D [1]. However, it is not clear whether Max-D is applicable to more complicated fluid systems, like Earth's atmosphere [2]. Recent global climate model simulations have found that the calibrated values of important tunable parameters are indeed consistent with Max-D [3]. Furthermore, the maximum entropy framework [4], which naturally gives a Max-D principle in the case of simple laboratory flows, can be readily applied to more complicated systems. I will discuss attempts to gener alise the Malkus statistical stability argument and how this connects with maximum entropy arguments. In doing so I hope to compare the physical insights of statistical stability, which emphasises dynamical resilience to perturbations, with maximum entropy considerations, which ignore system dynamics.

[1] W. V. R. Malkus. Borders of disorder: In turbulent channel ow. Journal of Fluid Mechanics, 489:185{198, 2003. [2] Richard Goody. Maximum entropy production in climate theory. Journal of the atmospheric sciences, 64(7):2735-2739, 2007. [3] Salvatore Pascale, Jonathan M. Gregory, Maarten H.P. Ambaum, and Remi Tailleux. A parametric sensitivity study of entropy production and kinetic energy dissipation using the FAMOUS AOGCM. Climate Dynamics, 38(5-6):1211-1227, 2012. [4] Dewar R and Maritan A. A theoretical basis for maximum entropy production. 2013. In Beyond the Second Law: Entropy Production and Non-equilibrium Systems (eds. R Dewar, C Lineweaver, R Niven, K Regenauer-Lieb), Springer, In Press
INI 1
17:35 to 18:10 M Vlad ([National Institute of Laser, Plasma and Radiation Physics])
Statistics of eddy transport
Co-author: Florin Spineanu (National Institute of Laser, Plasma and Radiation Physics)

A semi-analytical method for the study of eddy transport in presented. It determines the statistics of particle trajectories using the decorrelation trajectories, which are determined from the Eulerian correlation of the velocity. The fraction of free trajectories that effectivy determines the transport decreases with the increase of the Kubo number. The statistical method is able to describe the transport in these conditions where it is produced by a minority of the events. The effect of particle collisions is analysed. We show that eddy diffusion is strongly amplified by weak collisions and that the effective diffusion coefficient can be much larger than both the collisional diffusion coefficient and the eddy diffusion coefficients.

INI 1
Friday 1st November 2013
09:00 to 09:35 DP Ruelle (IHES)
Hydrodynamic turbulence as a problem in non-equilibrium statistical mechanics
The problem of hydrodynamic turbulence is reformulated as a heat flow problem along a chain of mechanical systems which describe units of fluid of smaller and smaller spatial extent. These units are macroscopic but have few degrees of freedom, and can be studied by the methods of (microscopic) non-equilibrium statistical mechanics. The fluctuations predicted by statistical mechanics correspond to the intermittency observed in turbulent flows. Specifically, we obtain the formula

$$ \zeta_p={p\over3}-{1\over\ln\kappa}\ln\Gamma({p\over3}+1) $$

for the exponents of the structure functions ($\langle|\Delta_rv|^p\rangle\sim r^{\zeta_p}$). The meaning of the adjustable parameter $\kappa$ is that when an eddy of size $r$ has decayed to eddies of size $r/\kappa$ their energies have a thermal distribution. The above formula, with $(\ln\kappa)^{-1}=.32\pm.01$ is in good agreement with experimental data. This lends support to our physical picture of turbulence, a picture which can thus also be used in related problems.
INI 1
09:35 to 10:10 GM Buttazzo (Università di Pisa)
Optimal location problems with routing cost
Co-authors: Serena Guarino (University of Pisa (Italy)), Fabrizio Oliviero (University of Pisa (Italy))

A model problem for the location of a given number $N$ of points in a given region $\Omega$ and with a given resources density $\rho(x)$ is considered. The main difference between the usual location problems and the present one is that in addition to the location cost an extra {\it routing cost} is considered, that takes into account the fact that the resources have to travel between the locations on a point-to-point basis. The limit problem as $N\to\infty$ is characterized and some applications to airfreight systems are shown.
INI 1
10:10 to 10:45 F Bouchet (ENS - Lyon)
Phase transitions and large deviations in geophysical fluid dynamics
Geophysical turbulent flows (atmosphere and climate dynamics, the Earth core dynamics) often undergo very rapid transitions. Those abrupt transitions change drastically the nature of the flow and are of paramount importance, for instance in climate. By contrast with most theoretical models of phase transitions, for turbulent flows it is difficult to characterize clearly the attractors (they are not simple fixed points of a deterministic dynamics or statistical equilibrium states) and the trajectories that lead to transitions from one attractor to the others.

I will review recent researches in this subject, including experimental and numerical studies of turbulent flows. Most of the talk will focus on theoretical works in the framework of the 2D stochastic quasi-geostrophic Navier-Stokes equations, the quasi-geostrophic equations, and the stochastic Vlasov equations. We will discuss predictions of phase transitions, validity of large deviation results of the Freidlin-Wentzell type, or more involved approaches when the Freidlin-Wentzell approach is not valid.

The results involve several works that have been done in collaborations with J. Laurie, M. Mathur, C. Nardini, E. Simonnet, J. Sommeria, T. Tangarife, H. Touchette, and O. Zaboronski.
INI 1
10:45 to 11:10 Morning Coffee
11:10 to 11:45 B Eckhardt (Philipps-Universität Marburg)
Turbulence transition in shear flows: coherent structures, edge states and all that
Pipe flow, plane Couette flow and several other shear flows show a transition to turbulence for flow rates where the linear profile is still stable. The turbulent dynamics is transient, so that the transition is related to the formation of a chaotic saddle in the state space of the system. The saddle is supported by exact coherent states and their heteroclinic connections. I will summarize the common features that appear across all these shear flows, sketch the numerical techniques used to identify and track the relevant structures in the state space of the system and point out possible applications beyond fluid mechanics.
INI 1
11:45 to 12:20 V Lucarini ([Universität Hamburg/University of Reading])
Noise, Fluctuation, and Response in Geophysical Fluid Dynamics
Response theory provides formidable methods for addressing many problems in statistical mechanics. Recently, it has been proposed as a gateway for various challenges in geophysical fluid dynamics, such as the provision of a rigorous conceptual framework for computing climate response to a variety of forcings and for deriving effective equations for coarse-grained variables, thus paving the way for constructing accurate parametrization of unresolved processes in numerical models. In this contribution, we first would like to present some new results showing how one can use response theory to compute the impact of adding stochastic forcing to deterministic chaotic systems. Then, we will discuss the applicability of the fluctuation-dissipation theorem in the context of non-equilibrium systems, focusing on the role played by the choice of observable. Finally, we will present some applications of response theory in geophysical fluid dynamical systems, ranging from low-order models such as the Lorenz 63 and Lorenz 96 models to General Circulation Models of the atmosphere.
INI 1
12:20 to 12:55 E Vanden-Eijnden (Courant Institute of Mathematical Sciences)
Nonequilibrium statistical mechanics of climate variability: modelling issues and applications to data assimilation techniques
Stochastic models and computational tools for the study of transitions between different metastable states (or regimes) in climate system are discussed using the barotropic quasi-geostrophic (QG) equation as a test case. Specifically, a stochastic partial differential equation (SPDE) is obtained by adding appropriate forcing and damping terms to the QG equation to make this equation dynamically consistent with the predictions of equilibrium statistical mechanics, while allowing to study nonequilibrium phenomena such as transitions between different regimes. In the small noise regime, the most likely states of the invariant measure for this SPDE coincide with the selective decay states and we establish conditions under which these states are not unique, implying the existence of different climate regimes. We also analyze the mechanism and rate of the dynamical transitions between these regimes by computing the most likely paths connecting them. Finally we will discuss how the se results can be used in the context of data assimilation procedure based on Kalman or ensemble filters to improve the efficiency of these methods in the presence of regime shifts.
INI 1
12:55 to 13:30 Lunch at Wolfson Court
University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons