Renato Vitolo

Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system

Tipping points associated with bifurcations (B-tipping) or induced by noise (N-tipping) are recognized mechanisms that may potentially lead to sudden climate change. We focus here on a novel class of tipping points, where a sufficiently rapid change to an input or parameter of a system may cause the system to ‘tip’ or move away from a branch of attractors. Such rate-dependent tipping, or R-tipping, need not be associated with either bifurcations or noise. We present an example of all three types of tipping in a simple global energy balance model of the climate system, illustrating the possibility of dangerous rates of change even in the absence of noise and of bifurcations in the underlying quasi-static system.

Serial clustering of intense European storms

This study has investigated how the clustering of wintertime extra-tropical cyclones depends on the vorticity intensity of the cyclones, and the sampling time period over which cyclone transits are counted. Clustering is characterized by the dispersion (ratio of the variance and the mean) of the counts of eastward transits of cyclone tracks obtained by objective tracking of 850 hPa vorticity features in NCEP-NCAR reanalyses. The counts are aggregated over non-overlapping time periods lasting from 4 days up to 6 month long OctoberMarch winters over the period 1950–2003. Clustering is found to be largest in the exit region of the North Atlantic storm track (i.e. over NE Atlantic and NW Europe). Furthermore, it increases considerably for the intense cyclones, for example, the dispersion of the 3-monthly counts near Berlin increases from 1.45 for all cyclones to 1.80 for the 25 % most intense cyclones. The dispersion also increases quasi-linearly with the logarithm of the length of the aggregation period, for example, near Berlin the dispersion is 1.08, 1.33, and 1.45 for weekly, monthly, and 3-monthly totals, respectively. The increases and the sampling uncertainties in dispersion can be reproduced using a simple Poisson regression model with a time-varying rate that depends on large-scale teleconnection indices such as the North Atlantic Oscillation, the East Atlantic Pattern, the Scandinavian pattern, and the East Atlantic/West Russia pattern. Increased dispersion for intense cyclones is found to be due to the rate becoming more dependent on the indices for such cyclones, whereas increased dispersion for longer aggregation periods is related to the small amounts of intraseasonal persistence in the indices. Increased clustering with cyclone intensity and aggregation period has important implications for the accurate modelling of aggregate insurance losses. Zusammenfassung

Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing

A low-dimensional model of general circulation of the atmosphere is investigated. The differential equations are subject to periodic forcing, where the period is one year. A three-dimensional Poincare mapping depends on three control parameters F, G, and , the latter being the relative amplitude of the oscillating part of the forcing. This paper provides a coherent inventory of the phenomenology of F,G,. For small, a Hopf-saddle-node bifurcation of fixed points and quasi-periodic Hopf bifurcations of invariant circles occur, persisting from the autonomous case = 0. For = 0.5, the above bifurcations have disappeared. Different types of strange attractors are found in four regions (chaotic ranges) in {F,G} and the related routes to chaos are discussed.

Temporal clustering of tropical cyclones and its ecosystem impacts

Tropical cyclones have massive economic, social, and ecological impacts, and models of their occurrence influence many planning activities from setting insurance premiums to conservation planning. Most impact models allow for geographically varying cyclone rates but assume that individual storm events occur randomly with constant rate in time. This study analyzes the statistical properties of Atlantic tropical cyclones and shows that local cyclone counts vary in time, with periods of elevated activity followed by relative quiescence. Such temporal clustering is particularly strong in the Caribbean Sea, along the coasts of Belize, Honduras, Costa Rica, Jamaica, the southwest of Haiti, and in the main hurricane development region in the North Atlantic between Africa and the Caribbean. Failing to recognize this natural nonstationarity in cyclone rates can give inaccurate impact predictions. We demonstrate this by exploring cyclone impacts on coral reefs. For a given cyclone rate, we find that clustered events have a less detrimental impact than independent random events. Predictions using a standard random hurricane model were overly pessimistic, predicting reef degradation more than a decade earlier than that expected under clustered disturbance. The presence of clustering allows coral reefs more time to recover to healthier states, but the impacts of clustering will vary from one ecosystem to another.

Quasi-periodic Bifurcations of Invariant Circles in Low-dimensional Dissipative Dynamical Systems

This paper first summarizes the theory of quasi-periodic bifurcations for dissipative dynamical systems. Then it presents algorithms for the computation and continuation of invariant circles and of their bifurcations. Finally several applications are given for quasiperiodic bifurcations of Hopf, saddle-node and period-doubling type.

Extreme Value Statistics of the Total Energy in an Intermediate-Complexity Model of the Midlatitude Atmospheric Jet. Part I: Stationary Case

A baroclinic model of intermediate complexity for the atmospheric jet at middle latitudes is used as a stochastic generator of atmosphere-like time series. In this case, time series of the total energy of the system are considered. Statistical inference of extreme values is applied to sequences of yearly maxima extracted from the time series in the rigorous setting provided by extreme value theory. The generalized extreme value (GEV) family of distributions is used here as a basic model, both for its qualities of simplicity and its generality. Several physically plausible values of the parameter T E , which represents the forced equator-to-pole temperature gradient and is responsible for setting the average baroclinicity in the atmospheric model, are used to generate stationary time series of the total energy. Estimates of the three GEV parameters-location, scale, and shape-are inferred by maximum likelihood methods. Standard statistical diagnostics, such as return level and quantile-quantile plots, are systematically applied to assess goodness-of-fit. The GEV parameters of location and scale are found to have a piecewise smooth, monotonically increasing dependence on T E . The shape parameter also increases with T E but is always negative, as is required a priori by the boundedness of the total energy. The sensitivity of the statistical inferences is studied with respect to the selection procedure of the maxima: the roles occupied by the length of the sequences of maxima and by the length of data blocks over which the maxima are computed are critically analyzed. Issues related to model sensitivity are also explored by varying the resolution of the system. The method used in this paper is put forward as a rigorous framework for the statistical analysis of extremes of observed data, to study the past and present climate and to characterize its variations.

Extreme value laws in dynamical systems under physical observables

Extreme value theory for chaotic deterministic dynamical systems is a rapidly expanding area of research. Given a system and a real function (observable) defined on its phase space, extreme value theory studies the limit probabilistic laws obeyed by large values attained by the observable along orbits of the system. Based on this theory, the so-called block maximum method is often used in applications for statistical prediction of large value occurrences. In this method, one performs statistical inference for the parameters of the Generalised Extreme Value (GEV) distribution, using maxima over blocks of regularly sampled observable values along an orbit of the system. The observables studied so far in the theory are expressed as functions of the distance with respect to a point, which is assumed to be a density point of the system’s invariant measure. However, at least with respect to the ambient (usually Euclidean) metric, this is not the structure of the observables typically encountered in physical applications, such as windspeed or vorticity in atmospheric models. In this paper we consider extreme value limit laws for observables which are not expressed as functions of the distance (in the ambient metric) from a density point of the dynamical system. In such cases, the limit laws are no longer determined by the functional form of the observable and the dimension of the invariant measure: they also depend on the specific geometry of the underlying attractor and of the observable’s level sets. We present a collection of analytical and numerical results, starting with a toral hyperbolic automorphism as a simple template to illustrate the main ideas. We then formulate our main results for a uniformly hyperbolic system, the solenoid map. We also discuss non-uniformly hyperbolic examples of maps (Henon and Lozi maps) and of flows (the Lorenz63 and Lorenz84 models). Our purpose is to outline the main ideas and to highlight several serious problems found in the numerical estimation of the limit laws.

Extreme value statistics of the total energy in an intermediate-complexity model of the midlatitude atmospheric jet. Part II Trend detection and assessment

Abstract A baroclinic model for the atmospheric jet at middle latitudes is used as a stochastic generator of nonstationary time series of the total energy of the system. A linear time trend is imposed on the parameter TE, descriptive of the forced equator-to-pole temperature gradient and responsible for setting the average baroclinicity in the model. The focus lies on establishing a theoretically sound framework for the detection and assessment of trend at extreme values of the generated time series. This problem is dealt with by fitting time-dependent generalized extreme value (GEV) models to sequences of yearly maxima of the total energy. A family of GEV models is used in which the location μ and scale parameters σ depend quadratically and linearly on time, respectively, while the shape parameter ξ is kept constant. From this family, a GEV model is selected with Akaike’s information criterion, complemented by the likelihood ratio test and by assessment through standard graphical diagnostics. The inferre...

Extreme value statistics of the total energy in an intermediate-complexity model of the midlatitude atmospheric jet. Part I Stationary case

A baroclinic model of intermediate complexity for the atmospheric jet at middle latitudes is used as a stochastic generator of atmosphere-like time series. In this case, time series of the total energy of the system are considered. Statistical inference of extreme values is applied to sequences of yearly maxima extracted from the time series in the rigorous setting provided by extreme value theory. The generalized extreme value (GEV) family of distributions is used here as a basic model, both for its qualities of simplicity and its generality. Several physically plausible values of the parameter T E , which represents the forced equator-to-pole temperature gradient and is responsible for setting the average baroclinicity in the atmospheric model, are used to generate stationary time series of the total energy. Estimates of the three GEV parameters-location, scale, and shape-are inferred by maximum likelihood methods. Standard statistical diagnostics, such as return level and quantile-quantile plots, are systematically applied to assess goodness-of-fit. The GEV parameters of location and scale are found to have a piecewise smooth, monotonically increasing dependence on T E . The shape parameter also increases with T E but is always negative, as is required a priori by the boundedness of the total energy. The sensitivity of the statistical inferences is studied with respect to the selection procedure of the maxima: the roles occupied by the length of the sequences of maxima and by the length of data blocks over which the maxima are computed are critically analyzed. Issues related to model sensitivity are also explored by varying the resolution of the system. The method used in this paper is put forward as a rigorous framework for the statistical analysis of extremes of observed data, to study the past and present climate and to characterize its variations.

Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms

Dynamical phenomena are studied near a Hopf-saddle-node bifurcation of fixed points of 3D-diffeomorphisms. The interest lies in the neighbourhood of weak resonances of the complex conjugate eigenvalues. The 1?:?5 case is chosen here because it has the lowest order among the weak resonances, and therefore it is likely to have a most visible influence on the bifurcation diagram. A model map is obtained by a natural construction, through perturbation of the flow of a Poincar??Takens normal form vector field. Global bifurcations arise in connection with a pair of saddle-focus fixed points: homoclinic tangencies occur near a sphere-like heteroclinic structure formed by the 2D stable and unstable manifolds of the saddle points. Strange attractors occur for nearby parameter values and three routes are described. One route involves a sequence of quasi-periodic period doublings of an invariant circle where loss of reducibility also takes place during the process. A second route involves intermittency due to a quasi-periodic saddle-node bifurcation of an invariant circle. Finally a route involving heteroclinic phenomena is discussed. Multistability occurs in several parameter subdomains: we analyse the structure of the basins for a case of coexistence of a strange and a quasi-periodic attractor and for coexistence of two strange attractors. By construction, the phenomenology of the model map is expected in generic families of 3D diffeomorphisms.