Jonathan Flowerdew, Ensemble forecasts of hazardous weather

Providing advance warning of hazardous and extreme weather is one of the primary roles of the UK Met Office. Ensemble forecasts support a more risk-based approach to civil protection by estimating the range of possible outcomes and their associated probabilities. Whereas many previous ensemble systems have focussed on the 3-15 day timescale, the Met Office ensemble has been specifically designed for the 0-2 day time range which is the focus for civil protection actions.

This talk will describe the Met Office ensemble systems and the perturbation strategies they employ. Short-range performance depends critically on the quality of the initial analysis, which in turn depends on the error covariance which the data assimilation system ascribes to the previous forecast. The Met Office has recently introduced 'hybrid' data assimilation, where the case-specific ensemble predictions of forecast uncertainty are used to augment the data assimilation's error covariance model.

Like all Numerical Weather Prediction, the grid resolution of ensemble forecasts is limited by the available computing power. The Met Office regional ensemble currently runs with a grid spacing of around 18km, much larger than the individual convective systems which are responsible for many cases of extreme precipitation. The new supercomputer will allow us to test an ensemble which attempts to represent convection using the underlying model dynamics rather than a convection parameterisation, using a 2.2km grid covering the UK alone.

Another key aspect of forecasting extreme events is the translation of atmospheric forecasts into the relevant impact and establishing when action should be taken. Under contract to the Environment Agency for England and Wales, the Met Office uses a barotropic storm surge model to turn its ensemble forecasts of surface wind and pressure into an ensemble of water level predictions. This permits direct assessment of the risk of coastal flooding. More recent work has extended these forecasts from 2 to 7 days, and examined the possibility of producing a similar ensemble prediction of ocean wave characteristics. The recent revisions to the National Severe Weather Warning Service, making greater use of ensemble data, will also be described.

Stephane Vannitsem, Statistical properties of extreme events in an intermediate order atmospheric model.

Valerio Lucarini, Extremes as dynamical indicators.

The Extreme Value Theory (EVT), originally introduced for series of independent variables, has been extended to dynamical systems in a nice theoretical framework which allows to infer stability and geometrical properties of the orbits. The main results of EVT developed for the investigation of the observables of dynamical systems rely, up to now, on the Gnedenko approach. In this framework, extremes are basically identified with the block maxima of the time series of the chosen observable, in the limit of infinitely long blocks. It has been proven that, assuming suitable mixing conditions for the underlying dynamical systems, the extremes of a specific class of observables are distributed according to the so called Generalized Extreme Value (GEV) distribution. Direct calculations show that in the case of quasi-periodic dynamics the block maxima are not distributed according to the GEV distribution. Reversing the argument, this opens up the possibility of understanding whether the underlying dynamics of a dynamical system is chaotic or regular by observing the statistical properties of the extremes of given observables. Moreover, we show that, in order to obtain a universal behaviour of the extremes, the requirement of a mixing dynamics can be relaxed if the Pareto approach is used, based upon considering the exceedances over a given threshold. We prove that the limiting distribution for the exceedances of the observables previously studied with the Gnedenko approach is a Generalized Pareto distribution where the parameters depend only on the Hausdorff dimension of the attractor and the value of the threshold. This result allows for extending the extreme value theory for dynamical systems to the case of regular motions. In order to provide further support to our findings, we present the results of numerical experiments carried out considering classical maps.

Davide Faranda, Breakdown of turbulence in a plane Couette flow. Can extreme fluctuations be used to understand critical transitions?

Critical transitions are observed in many natural phenomena and it is a scientific challenge to find out whether there are suitable observables to get early warnings of them. Among all the relevant physical problems that exhibit critical transitions, the breakdown of the turbulence in a plane Couette Flow is of great interest as varying the Reynolds number (Re) we observe three different dynamic regimes: if for higher Reynolds number the flow is completely turbulent, when 325< Re<410 plane Couette forms alternately turbulent and laminar oblique bands out of featureless turbulence. Eventually, when Re<325 turbulence is suppressed and a laminar behaviour prevails. We focus on the transition between the intermediate bands regime and the laminar behaviour trying to analyse the fluctuations of the so called perturbation energy. In particular we find that studying extreme fluctuations of the perturbation energy transient through the classical Extreme Value Theory (EVT) helps in understanding the mechanism of the suppression of turbulence: when the Reynolds number is decreased below Re=300, minima fluctuations amplitude increases considerably whereas maxima fluctuations remain about the same. This is compatible with the idea that the system is eventually going to suppress turbulence increasing the probability to observe very low values of turbulent perturbation energy. Although EVT was originally derived in the setting of stochastic variables, the application to fluid dynamics has been made possible by recent progresses on EVT in more general dynamical systems. We believe that testing EVT in an intermediate complexity fluid model could help in understanding what are the real possibilities in applying it to geophysical systems that represent complex real phenomena. Moreover, in the last years a lot of research effort has been directed towards understanding the role of early indicators of critical transitions both as diagnostic or prognostic tool: linking the behaviour of a system near the tipping points to modifications on its extreme fluctuations may improve our understanding of the dynamics when critical transitions occur.

Mike Todd, Clustering of rare events in dynamical systems

Rare events in dynamical systems can be approached through: 1) hitting time statistics (HTS), hits of iterates of points to a sequence of asymptotically small balls around a point $\zeta$; or 2) extreme value theory (EVT), thinking of the random variables as observations along orbits of points, where the observation takes its maximum at $\zeta$. In this talk, I'll explain the equivalence of these two approaches and discuss the tests in each context which guarantee the existence of a limiting distribution. In the context of EVT, clustering of random variables leads to a limiting law with an extremal index $\theta\in [0,1)$ related to the cluster size. For the HTS of a dynamical system, I'll show that the equivalent phenomenon is caused by underlying periodic behaviour. This perspective leads to new tests for limiting laws in the presence of clustering for general random variables. This is joint work with A.C. Freitas and J.M. Freitas.

Sebastian Wieczorek, Critical values and critical rates: Novel types of tipping points.

Consider a system in the vicinity of a stable state. There is a general belief that sudden changes in the state of a system, known as "tipping points" or "critical transitions", are associated with a slow passage of a system parameter through a "critical value" at which the stable state disappears or destabilises in a bifurcation, causing the system to move to another state. However, in addition to critical values, there also exist "critical rates" of parameter change [1]. Above a critical rate, the system is unable to keep pace with continusly changing stable state and tips to another state. This happens even though there are no critical parameter values! Such rate-induced tipping is conceptually different from previously studied mechanisms [2] and may be relevant to the current and future human-dominated climate epoch that is not so much about the ultimate magnitude of warmth but its rate of change. This talk will classify different tipping mechanism and use the theory of rate-dependent tipping to explain the curious "compost-bomb instability" observed in climate-carbon cycle models [1].

This is joint work with Peter Ashwin, Peter Cox and Renato Vitolo.

[1] S. Wieczorek, P. Ashwin, C.M. Luke, and P. Cox,

    "Excitability in ramped systems: the compost-bomb instability"
     Proc. Roy. Soc. A (May 8 2011) 467, 1243-1269.

[2] P. Ashwin, S. Wieczorek, R. Vitolo, and P. Cox,

    "Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system"
     Phil. Trans. Roy. Soc. A (2012) 370, 1166-1184

Alef Sterk, Predictability of extreme values in dynamical systems

Extreme value theory in deterministic systems is concerned with unlikely large (or small) values of an observable evaluated along evolutions of the system. In this talk I discuss the predictability of extremes in geophysical toy models by studying whether initial conditions that lead to extremes have larger error growth rates. General statements on the predictability of extreme values seem to be impossible: the predictability of extreme values depends on (1) the observable, (2) the attractor of the system, and (3) the prediction lead time.

Frank Kwasniok, 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 extreme value or 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 simultaneously from data. The large-scale flow is represented by the leading empirical orthogonal functions (EOFs). Also 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 dynamical framework of a three-level quasigeostrophic atmospheric model with realistic mean state, variability and teleconnection patterns. The methodology can also be applied to data from GCM simulations, predicting future extremes.

Pau Rabassa, Extreme value laws in dynamical system under physical observables

Classical extreme values theory concerns with the maximum over a collection of random variables. This theory can be applied to a process generated by (chaotic) deterministic system composed with an observable (a cost function). This is the basic idea behind the extreme value theory for chaotic deterministic dynamical systems, which is a rapidly expanding area of research. The observables which are typically studied in the literature are expressed as functions of the distance with respect a point within the attractor. This is at odd with the structure of the observable functions typically encountered in physical applications, such as windspeed or vorticity in atmospheric models. We consider extreme value limit laws for observables which are not necessarily functions of the distance from a density point of the dynamical system. In this case, the limit law is no longer determined by the functional form of the observable alone, but it also depends on the local dimension of the invariant measure and on the geometry of the underlying attractor.

Daan Crommelin, Regimes and metastability from a stochastic perspective

Andrew Ferguson, Extreme value theory in dynamical systems - the perturbative approach

I will discuss a recent development in the theory of extreme value theory in the context of dynamical systems. The approach is based on the perturbation theory of Keller and Liverani which guarantees, provided the correct functional setup exists, that one may obtain a first order expansion of the spectral radius of a perturbed bounded linear operator. This result has applications to distributional results for hitting, return times and extreme value theory as well as the computation of the dimension of survivor sets associated with small holes.