A general perspective of extreme events in weather and climate

https://doi.org/10.1016/j.atmosres.2011.01.012Get rights and content

Abstract

One of the most important problems in meteorology, physical oceanography, climatology, and related fields is the understanding and dynamical description of multi-scale interactions. Multi-scale interactions are closely related to extreme events in climate and, therefore, of great practical importance. Here an extreme event is defined in terms of the non-Gaussian tail (sometimes also called a weather or climate regime) of the data's probability density function (PDF), as opposed to the definition in extreme value theory, where the statistics of time series maxima (and minima) in a given time interval are studied. The non-Gaussian approach used here allows for a dynamical view of extreme events in weather and climate, going beyond the solely mathematical arguments of extreme value theory. Extreme events are by definition scarce, but they can have a significant impact on people and countries in the affected regions. Understanding extremes has become an important objective in weather/climate variability research because weather and climate risk assessment depends on knowing the tails of PDFs. In recent years, new tools that make use of advanced stochastic theory have evolved to evaluate extreme events and the physics that govern these events. Stochastic methods are ideal to study multi-scale interactions and extreme events because they link vastly different time and spatial scales. One theory attributes extreme anomalies to stochastically forced dynamics, where, to model nonlinear interactions, the strength of the stochastic forcing depends on the flow itself (multiplicative noise). This closure assumption follows naturally from the general form of the equations of motion. Because stochastic theory makes clear and testable predictions about non-Gaussian variability, the multiplicative noise hypothesis can be verified by analyzing the detailed non-Gaussian statistics of atmospheric and oceanic variability. This review paper discusses the theoretical framework, observational evidence, and related developments in stochastic modeling of extreme events in weather and climate.

Research highlights

► This paper discusses the theoretical framework, observational evidence, and related developments in stochastic modeling of non-Gaussian extreme events in weather and climate. ► We provide theoretical and observational evidence suggesting that a simple linear stochastic differential equation with correlated additive and multiplicative (CAM) noise is an excellent candidate (i.e., null hypothesis) to explain the non-Gaussian statistics of numerous weather and climate phenomena. ► The evidence is based on the fact that the CAM noise theory makes clear and testable predictions about non-Gaussian variability that are verified by analyzing the detailed statistics of atmospheric and oceanic variability.

Introduction

Extreme events in nature and society are by definition scarce, but they can have a significant physical and socioeconomic impact on people and countries in the affected regions (Albeverio et al., 2006). Here we deal with extreme events in weather and climate (i.e., nature), keeping in mind that there exists a strong connection to society, because an extreme event in nature often triggers an extreme socioeconomic event. For example, a natural disaster is often followed by a financial crisis. Alongside our intuitive knowledge that a hurricane, tornado, or earthquake might qualify, how can we define extreme events more precisely?

In non-technical terms an extreme event is a high-impact, hard-to-predict phenomenon that is beyond our normal (i.e., Gaussian bell curve) expectations. In popular culture an extreme event is sometimes called a “Black Swan”, because a Black Swan provides a metaphor for a highly improbable incident (e.g., Taleb, 2010); before the discovery of Australia, people in the Old World were convinced that all swans where white. Thus, in more technical terms an extreme event can be statistically defined as the non-normal (i.e., non-Gaussian) tail of the data's probability density function (PDF). That means, that in this paper an event is only considered extreme if its probability of occurrence is governed by non-Gaussian statistics. That implies, that a high amplitude event does not qualify as extreme if it is described by Gaussian statistics. It is important to recognize that in the following we, therefore, use the terms extreme event and non-Gaussian statistics (and related phrases) synonymously.

Understanding extremes has become an important objective in weather/climate variability research because weather and climate risk assessment depends on knowing and understanding the tails of PDFs. At this point we need to define what we mean by weather and climate a bit more. Here we typically think of representative atmospheric variables (for example, pressure, wind, vorticity, temperature, etc.) as varying on weather timescales of hours, days, to a few weeks, and the ocean (for example, sea surface temperatures, sea level heights, currents, etc.) varying on longer climate timescales of weeks, months, years, and decades. There is, of course, certain overlap and we use the terms weather and climate in a loose way, specifying the timescales as needed for particular applications. It should also be noted that nonlinear multi-scale interactions make a strict separation of timescales impossible.

There is broad consensus that the most hazardous effects of climate change are related to a potential increase (in frequency and/or intensity) of extreme weather and climate events (e.g., Houghton, 2009, Brönnimann et al., 2008, Alexander et al., 2006, Easterling et al., 2000). The overarching goal of studying extremes is, therefore, to understand and then manage the risks of extreme events and related disasters to advance strategies for efficient climate change adaptation. While numerous important studies have focused on changes in mean values under global warming, such as mean global temperature [one of the key variables in almost every discussion of climate change; see, for example, reports from the Intergovernmental Panel on Climate Change (IPCC)1], the interest in how extreme values are altered by a changing climate is a relatively recent topic in climate research. The reasons for that are primarily twofold. First, we need high-quality, high-resolution (in space and time) observational data sets for a comprehensive analysis of non-Gaussian climate variability. It is only recently that global high-quality daily observations became readily available to the international research community (Alexander et al., 2006). Second, we need extensive simulations of high-resolution climate models to (hopefully) simulate realistic non-Gaussian climate variability. Again, only recently long enough high-resolution numerical simulations of climate variability became feasible to study global non-Gaussian and higher-order statistics in some detail (e.g., Kharin and Zwiers, 2005, Kharin et al., 2007).

The general problem of understanding extremes is, of course, their scarcity: it is very hard to obtain reliable (if any) statistics of those events from a finite observational record. Therefore, we have to somehow extrapolate from the well sampled center of a PDF to the scarcely or unsampled tails. The extrapolation into the more or less uncharted tails of a distribution or, avoiding technical lingo, the approach to study extreme events in climate, can be roughly divided into three major, by no means mutually exclusive categories. In fact, the study of extreme events in weather and climate is most often done by combining the strategies of the following methods (Garrett and Müller, 2008).

The statistical approach, also called extreme value theory, is solely based on mathematical arguments (e.g., Coles, 2001, Garrett and Müller, 2008, Wilks, 2006, Gumbel, 1942, Gumbel, 1958). It provides methods to extrapolate from the well sampled center to the scarcely or unsampled tails of a PDF using mathematical tools. The key point of the statistical approach is that, in place of an empirical or physical basis, asymptotic arguments are used to justify the extreme value model. In particular, the generalized extreme value distribution (GEV) is a family of PDFs for the maximum (or minimum) of a large sample of independent random variables drawn from the same arbitrary distribution. While the statistical approach is based on sound mathematical arguments, it does not provide much insight into the physics of extreme events.

Extreme value theory is, however, widely used to explore climate extremes (Katz and Naveau, 2010). In fact, the foundation of extreme value theory is very closely related to the study of extreme values in meteorological data (Gumbel, 1942, Gumbel, 1958). Nowadays this is very often done in conjunction with the numerical modeling approach discussed below. That is, model output is analyzed using extreme value theory to see if statistics are altered in a changing climate.

The empirical–physical approach uses physical reasoning based on empirical knowledge to provide a basis for an extreme value model. The key point here is that, in contrast to the purely statistical method that primarily uses mathematical (asymptotic) arguments, physical reasoning is employed to perform the extrapolation into the scarcely sampled tails of the PDF. The empirical–physical method can itself be further split into either empirical or physical strategies, focusing on the empirical or physical aspects of the problem respectively. The empirical–physical approach lacks the mathematical rigor of the statistical method, but it provides valuable physical insight into relevant real world problems. An example for an empirical–physical application is the Gamma distribution which is often used to describe atmospheric variables that are markedly asymmetric and skewed to the right (Wilks, 2006). Well known examples are precipitation and wind speed, which are physically constrained to be positive or zero.

The numerical modeling approach aims to estimate the statistics of extreme events (the tails of the PDF) by integrating a general circulation model (GCM) for a very long period (e.g., Easterling et al., 2000, Kharin and Zwiers, 2005, Kharin et al., 2007). That is, this approach tries to effectively lengthen the limited observational record with proxy data from a GCM, filling the unsampled tails of the observed PDF with probabilities from model data. Numerical modeling allows for a detailed analysis of the physics (at least model-physics) of extreme events. In addition, the statistical and empirical–physical methods can also be applied to model data, validating (or invalidating) the quality of the model. It is obvious that the efforts by the IPCC to understand and forecast the statistics of extreme weather and climate events in a changing climate fall into this category.

Note that every approach effectively extrapolates from the known to the scarcely known (or unknown) using certain assumptions and, therefore, requires a leap of faith. For the statistical approach the assumptions are purely mathematical. For example, the assumption of classical extreme value theory that the extreme events are independent and drawn from the same distribution, and that we have enough data for convergence to a limiting distribution (the generalized extreme value distribution) may not be met (e.g., Coles, 2001, Wilks, 2006). The potential drawback of the empirical–physical approach is its lack of mathematical rigor; it primarily depends on empirical knowledge and physical arguments. The weakness of numerical modeling lies in the largely unknown ability of a model to reproduce the correct statistics of extreme events. At this point, GCMs are basically calibrated to reproduce the observed first and second moments (mean and variance) of the general circulation of the ocean and atmosphere. Very little is known about the credibility of GCMs to reproduce non-Gaussian statistics, that is, extreme events.

Overall it is fair to say that until recently the study of extreme meteorological events has been largely empirical. That is, most investigators used observations or model output to estimate the probabilities of, for example, extreme winds and temperatures, without actually addressing the detailed dynamical–physical reason for the shape of the probability density functions beyond the mathematical arguments of extreme value theory. In addition, many investigators typically study non-Gaussian statistics in a phase-space spanned by the two or three leading empirical orthogonal functions (EOFs) (e.g., Mo and Ghil, 1988, Mo and Ghil, 1993, Molteni et al., 1990, Corti et al., 1999, Smyth et al., 1999, Monahan et al., 2001, Berner, 2005, Berner and Branstator, 2007, Majda et al., 2003, Majda et al., 2008, Franzke et al., 2005). Significant exceptions are White, 1980, Trenberth and Mo, 1985, Nakamura and Wallace, 1991, Holzer, 1996 who present maps of observed skewness and (partly) kurtosis of Northern and Southern Hemisphere geopotential heights. More recently Petoukhov et al. (2008) calculated skewnesses and mixed third-order statistical moments for observed synoptic variation of horizontal winds, temperature, vertical velocity and the specific humidity. Again, none of those papers provide complete dynamical explanations for the observed non-Gaussian structures. One partial exception is Holzer (1996) who attributes negative midlatitude skewness bands to the rectification of velocity fluctuations by the advective nonlinearity. However, the dynamics of the general non-Gaussian structures and extreme events remain largely unexplained.

In recent years, new tools that make use of advanced stochastic theory have evolved to evaluate extreme events and the physics that govern these events (e.g., Berner, 2005, Berner and Branstator, 2007, Franzke et al., 2005, Majda et al., 2003, Majda et al., 2008, Monahan, 2004, Monahan, 2006a, Monahan, 2006b, Peinke et al., 2004, Sura, 2003, Sura et al., 2005, Sura et al., 2006, Sura and Newman, 2008, Sura and Sardeshmukh, 2008, Sardeshmukh and Sura, 2009). These tools take advantage of the non-Gaussian structure of the PDF by linking a stochastic model to the observed non-Gaussianity. The novel feature of those models is that the stochastic component is allowed to be state dependent (or multiplicative). The physical significance of multiplicative noise is that it has the potential to produce non-Gaussian statistics in linear systems. Because that phenomenon is at the heart of this paper, the basic physical principle is explained first (later we will explore the related mathematics in more detail). Fig. 1 illustrates the following discussion, using a non-Gaussian bimodal PDF as an accessible example (note that in reality we rarely observe bimodal PDFs).

Suppose climate dynamics are split into a slow (i.e., slowly decorrelating) contribution and a fast (i.e., rapidly decorrelating) contribution; this is an assumption well known in turbulence theory. The fast part is then approximated as noise. That is, we consider the dynamics of an n-dimensional system whose state vector x is governed by the stochastic differential equation (SDE)dxdt=A(x)+B(x)η(t),where the vector A(x) represents all slow, deterministic processes. The product of the matrix B(x) and the noise vector η, B(x)η, represents the stochastic approximation to the fast phenomena. The stochastic components ηi are assumed to be independent Gaussian white-noise processes: ηi(t)=0 and ηi(t)ηi(t)=δ(tt), where ... denotes the time average (or, assuming ergodicity, an ensemble average) and δ Dirac's delta function. In general, x in Eq. (1) will have non-Gaussian statistics and is, therefore, well suited to study extreme events. This is well known to be the case if A(x) is nonlinear, even if B(x) is constant (i.e., the noise is state-independent or additive; see left branch of Fig. 1). It is also true, yet less well known, if the deterministic dynamics are linear, represented for example as A0x with the matrix A0, as long as B(x) is not constant (i.e., the noise is state-dependent or multiplicative; see right branch of Fig. 1).

Let us look into the physics of each branch in more detail, starting with the left one. There it is obvious that transitions from one potential well to the other driven by additive noise will result in a bimodal PDF, as long as the additive noise is not too strong (in that case we get a monomodal PDF because the potential barrier does not inhibit the motion of the trajectory). This is not, however, the only dynamical system which can produce such a PDF. Consider instead a linear system (the right branch), represented by a unimodal deterministic potential, in which the trajectories are perturbed by multiplicative noise. Because of the larger noise amplitudes near the center of the monomodal potential, as compared to the strength of the noise to the right and left, the system's trajectory is more often found on either side of the central noise maximum, and this system will have a bimodal PDF as well. Thus the same non-Gaussian PDF can result from either a slow (deterministic) nonlinear dynamical system or a fast (stochastic) nonlinear dynamical system.

There are different ways to study and use Eq. (1) to understand non-Gaussian statistics. One is to derive an SDE of type (1), including a nonlinear deterministic and a multiplicative noise term, directly from the equations of motion. This method has been pioneered over the last decade by applied mathematicians (e.g., Majda et al., 2003, Majda et al., 2008, Franzke et al., 2005). It is sometime called the MTV method, after the names (Majda, Timofeyev, and Vanden-Eijnden) of its proponents (Majda et al., 1999, Majda et al., 2001, Majda et al., 2003). The drawback of this method is, while mathematically rigorous and scientifically useful, that it requires detailed knowledge of the underlying equations and processes. It is also unclear if it can be applied to very complex, state-of-the-art climate models. Most important, the MTV method cannot be used to analyze observational or model data directly.

That drawback can be overcome by trying to estimate the terms of Eq. (1) directly from data. Unfortunately, it is nontrivial to estimate both the nonlinear deterministic and multiplicative noise terms for higher (larger than 2) dimensional systems. However, for 1-d and 2-d systems both terms can be estimated empirically (e.g., Berner, 2005, Berner and Branstator, 2007, Crommelin and Vanden-Eijnden, 2006, Lind et al., 2005, Siegert et al., 1998, Friedrich et al., 2000, Sura and Barsugli, 2002, Sura, 2003, Sura and Gille, 2003, Sura et al., 2005, Sura et al., 2006, Sura and Newman, 2008, Sura and Sardeshmukh, 2009). For higher dimensions the multiplicative noise term poses the largest problem, whereas the deterministic contribution can still be empirically estimated using nonlinear regression techniques. Therefore, one approach is to simply use additive noise to close the system (the additive noise contribution is relatively easy to estimate). For example, an empirical multivariate nonlinear (with quadratic nonlinearities) stochastic model with additive (state-independent) noise captures many types (tropical SSTs and extratropical atmospheric variability) of non-Gaussian climate variability remarkably well (Kravtsov et al., 2005, Kravtsov et al., 2010, Kondrashov et al., 2006).

There is obviously a gap between purely mathematical approaches on the one hand, and predominantly empirical methods on the other hand. Here we describe and review a third approach, introduced by Sura and Sardeshmukh, 2008, Sardeshmukh and Sura, 2009, to study non-Gaussian statistics in climate. That approach combines mathematical and empirical methods to start filling the aforementioned gap. It attributes extreme anomalies to stochastically forced linear dynamics, where the strength of the stochastic forcing depends linearly on the flow itself (i.e., linear multiplicative noise). Most important, because the theory makes clear and testable predictions about non-Gaussian variability, it can be verified by analyzing the detailed non-Gaussian statistics of oceanic and atmospheric variability. In fact, Sura and Sardeshmukh, 2008, Sardeshmukh and Sura, 2009 did just that for sea surface temperature and atmospheric geopotential height and vorticity anomalies, thereby confirming the multiplicative noise hypothesis of extreme events for the respective variables. The theoretical framework, the observational evidence, and the implications on how to explore and interpret extreme events in weather and climate are discussed in the remainder of this paper.

The paper is structured as follows. In Section 2 the theoretical underpinnings of stochastically describing Gaussian and non-Gaussian climate variability are presented. The observational evidence for multiplicative noise dynamics in the ocean and atmosphere (and very briefly, in plasma turbulence) is discussed in Section 3. Finally, Section 4 provides a summary and discussion of the status quo, focusing on outstanding issues, challenges, and perspectives of future research on extreme events in weather and climate.

Section snippets

Some fundamentals of stochastic dynamics

This section reviews a few basic ideas of stochastic dynamics used in this paper. More comprehensive treatments may be found in many textbooks (e.g., Gardiner, 2004, Øksendal, 2007, Paul and Baschnagel, 1999, van Kampen, 2007, Horsthemke and Léfèver, 1984).

Observations and applications

In this section we present and discuss recent observational examples and applications of our non-Gaussian stochastic framework. We will see that several relevant weather and climate phenomena in the atmosphere (Sardeshmukh and Sura, 2009, Sura, 2010, Sura and Perron, 2010) and ocean (Sura and Sardeshmukh, 2008, Sura and Gille, 2010) conform to the non-Gaussian skewness–kurtosis and power-law statistics predicted by Eq. (18), allowing us to attribute the statistics of extreme events to a

Where do we stand?

Understanding extremes is an important goal in the atmospheric and ocean sciences because weather and climate risk assessment depends on knowing the tails of PDFs. In recent years new tools that make use of advanced stochastic theory have evolved to evaluate extreme events and the physics that govern these events. Stochastic methods are ideal to study extreme events because they link vastly different time and spatial scales. We have seen that non-Gaussian statistics of extreme anomalies can be

Acknowledgments

The reviewer's comments are gratefully acknowledged. This project was in part funded by the National Science Foundation through awards ATM-840035 “The Impact of Rapidly-Varying Heat Fluxes on Air–Sea Interaction and Climate Variability” and ATM-0903579 “Assessing Atmospheric Extreme Events in a Stochastic Framework”.

References (91)

  • A. Clauset et al.

    Power-law distributions in empirical data

    SIAM Rev.

    (2009)
  • S. Coles

    An Introduction to Statistical Modeling of Extreme Values

    (2001)
  • S. Corti et al.

    Signature of recent climate change in frequencies of natural atmospheric circulation regimes

    Nature

    (1999)
  • D.T. Crommelin et al.

    Reconstruction of diffusions using spectral data from timeseries

    Commun. Math. Sci

    (2006)
  • T. DelSole

    Stochastic models of quasigeostrophic turbulence

    Surv. Geophys.

    (2004)
  • D.R. Easterling et al.

    Climate extremes: observations, modeling, and impacts

    Science

    (2000)
  • B.F. Farrell et al.

    Stochastic dynamics of the midlatitude atmospheric jet

    J. Atmos. Sci.

    (1995)
  • B.F. Farrell et al.

    Generalized stability theory. Part I: autonomous operators

    J. Atmos. Sci.

    (1996)
  • C. Frankignoul et al.

    Stochastic climate models. Part II. Application to sea-surface temperature anomalies and thermocline variability

    Tellus

    (1977)
  • C. Franzke et al.

    Low-order stochastic mode reduction for a realistic barotropic model climate

    J. Atmos. Sci.

    (2005)
  • C.W. Gardiner

    Handbook of Stochastic Methods for Physics, Chemistry and the Natural Science

    (2004)
  • C. Garrett et al.

    Extreme events

    Bull. Am. Meteorol. Soc.

    (2008)
  • E.J. Gumbel

    On the frequency distribution of extreme values in meteorological data

    Bull. Am. Meteorol. Soc.

    (1942)
  • E.J. Gumbel

    Statistics of Extremes

    (1958)
  • K. Hasselmann

    Stochastic climate models. Part I. Theory

    Tellus

    (1976)
  • J.R. Holton

    An Introduction to Dynamic Meteorology

    (1992)
  • M. Holzer

    Asymmetric geopotential height fluctuations from symmetric winds

    J. Atmos. Sci.

    (1996)
  • W. Horsthemke et al.

    Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology

    (1984)
  • B.J. Hoskins et al.

    On the use and significance of isentropic potential vorticity maps

    Q. J. R. Meteorolog. Soc.

    (1985)
  • J. Houghton

    Global Warming — The Complete Briefing

    (2009)
  • P.J. Ioannou

    Nonnormality increases variance

    J. Atmos. Sci.

    (1995)
  • J. Isern-Fontanet et al.

    Three-dimensional reconstruction of oceanic mesoscale currents from surface information

    J. Geophys. Res.

    (2008)
  • R.W. Katz et al.

    Editorial: Special issue on statistics of extremes in weather and climate

    Extremes

    (2010)
  • V.V. Kharin et al.

    Estimating extremes in transient cimate change simulations

    J. Climate

    (2005)
  • V.V. Kharin et al.

    Changes in temperature and precipitation extremes in the IPCC ensemble of global coupled model simulations

    J. Climate

    (2007)
  • P. Kloeden et al.

    Numerical Solution of Stochastic Differential Equations

    (1992)
  • D. Kondrashov et al.

    Empirical mode reduction in a model of extratropical low-frequency variability

    J. Atmos. Sci.

    (2006)
  • S. Kravtsov et al.

    Multi-level regression modeling of nonlinear processes: derivation and applications to climate variability

    J. Climate

    (2005)
  • S. Kravtsov et al.

    Empirical model reduction and the modelling hierarchy in climate dynamics and the geosciences

  • J.A. Krommes

    The remarkable similarity between the scaling of kurtosis with squared skewness for TORPEX density fluctuations and sea-surface temperature fluctuations

    Phys. Plasmas

    (2008)
  • B. Labit et al.

    Universal statistical properties of drift-interchange turbulence in TORPEX plasmas

    Phys. Rev. Lett.

    (2007)
  • G. Lapeyre et al.

    Dynamics of the upper oceanic layers in terms of surface quasigeostrophic theory

    J. Phys. Oceanogr.

    (2006)
  • P.G. Lind et al.

    Reducing stochasticity in the North Atlantic Oscillation Index with coupled Langevin equations

    Phys. Rev. E

    (2005)
  • A.J. Majda et al.

    Models for stochastic climate prediction

    Proc. Natl. Acad. Sci.

    (1999)
  • A.J. Majda et al.

    A mathematical framework for stochastic climate models

    Commun. Pure Appl. Math.

    (2001)
  • Cited by (0)

    View full text