Skip to main content
SpringerLink
Log in

Stochastic Models of Climate Extremes: Theory and Observations

  • Chapter
  • First Online:
Book cover Extremes in a Changing Climate

Part of the book series: Water Science and Technology Library ((WSTL,volume 65))

Abstract

One very important topic in climatology, meteorology, and related fields is the detailed understanding of extremes in a changing climate. There is broad consensus that the most hazardous effects of climate change are due to a potential increase (in frequency and/or intensity) of extreme weather and climate events. Extreme events are by definition rare, but they can have a significant impact on people and countries in the affected regions. Here an extreme event is defined in terms of the non-Gaussian tail (occasionally 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. Because weather and climate risk assessment depends on knowing the tails of PDFs, understanding the statistics and dynamics of extremes has become an important objective in climate research. Traditionally, stochastic models are extensively used to study climate variability because they link vastly different time and spatial scales (multi-scale interactions). However, in the past the focus of stochastic climate modeling hasn’t been on extremes. Only in recent years new tools that make use of advanced stochastic theory have evolved to evaluate the statistics and dynamics of extreme events. 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 chapter discusses the theoretical framework, observational evidence, and related developments in stochastic modeling of weather and climate extremes.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Albeverio S, Jentsch V, Kantz H (eds) (2006) Extreme events in nature and society. Springer, Berlin, 352 pp

    Google Scholar 

  • Alexander LV et al (2006) Global observed changes in daily extremes of temperature and precipitation. J Geophys Res 111. doi:10.1029/2005JD006 290

    Google Scholar 

  • Berner J (2005) Linking nonlinearity and non-gaussianity of planetary wave behavior by the Fokker-Planck equation. J Atmos Sci 62:2098–2117

    Article  Google Scholar 

  • Berner J, Branstator G (2007) Linear and nonlinear signatures in the planetary wave dynamics of an AGCM probability density function. J Atmos Sci 64:117–136

    Article  Google Scholar 

  • Brönnimann S, Luterbacher J, Ewen T, Diaz HF, Stolarski RS, Neu U (eds) (2008) Climate variability and extremes during the past 100 years. Springer, Dordrecht 364 pp

    Google Scholar 

  • Brooks CEP, Carruthers N (1953) Handbook of statistical methods in meteorology. Her Majesty’s Stationery Office, London, 412 pp

    Google Scholar 

  • Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661–703

    Article  Google Scholar 

  • Coles S (2001) An introduction to statistical modeling of extreme values. Springer, London, 208 pp

    Google Scholar 

  • Corti S, Molteni F, Palmer TN (1999) Signature of recent climate change in frequencies of natural atmospheric circulation regimes. Nature 29:799–802

    Google Scholar 

  • Crommelin DT, Vanden-Eijnden E (2006) Reconstruction of diffusions using spectral data from timeseries. Comm Math Sci 4:651–668

    Google Scholar 

  • DelSole T (2004) Stochastic models of quasigeostrophic turbulence. Surv Geophys 25:107–149

    Article  Google Scholar 

  • Easterling DR, Meehl GA, Parmesan C, Changnon SA, Karl TR, Mearns LO (2000) Climate extremes: observations, modeling, and impacts. Science 289:2068–2074

    Article  Google Scholar 

  • Farrell BF, Ioannou PJ (1995) Stochastic dynamics of the midlatitude atmospheric jet. J Atmos Sci 52:1642–1656

    Article  Google Scholar 

  • Farrell BF, Ioannou PJ (1996) Generalized stability theory. Part I: autonomous operators. J Atmos Sci 53:2025–2040

    Google Scholar 

  • Frankignoul C, Hasselmann K (1977) Stochastic climate models. Part II. application to sea-surface temperature anomalies and thermocline variability. Tellus 29:289–305

    Google Scholar 

  • Franzke C, Majda AJ, Vanden-Eijnden E (2005) Low-order stochastic mode reduction for a realistic barotropic model climate. J Atmos Sci 62:1722–1745

    Article  Google Scholar 

  • Friedrich R et al (2000) Extracting model equations from experimental data. Phys Lett A 271:217–222

    Article  Google Scholar 

  • Gardiner CW (2004) Handbook of stochastic methods for physics, chemistry and the natural science, 3rd edn. Springer, Berlin/New York, 415 pp

    Google Scholar 

  • Garrett C, Müller P (2008) Extreme events. Bull Am Meteor Soc 89:ES45–ES56

    Google Scholar 

  • Gumbel EJ (1942) On the frequency distribution of extreme values in meteorological data. Bull Am Meteor Soc 23:95–105

    Google Scholar 

  • Gumbel EJ (1958) Statistics of extremes. Columbia University Press, New York, 375 pp

    Google Scholar 

  • Hasselmann K (1976) Stochastic climate models. Part I. Theory. Tellus 28:473–484

    Google Scholar 

  • Holton JR (1992) An introduction to dynamic meteorology, 3rd edn. Academic, San Diego, 507 pp

    Google Scholar 

  • Holzer M (1996) Asymmetric geopotential height fluctuations from symmetric winds. J Atmos Sci 53:1361–1379

    Article  Google Scholar 

  • Horsthemke W, Léfèver R (1984) Noise-induced transitions: theory and applications in physics, chemistry, and biology. Springer, Berlin, 318 pp

    Google Scholar 

  • Hoskins BJ, McIntyre ME, Robertson AW (1985) On the use and significance of isentropic potential vorticity maps. Quart J Roy Meteor Soc 111:877–946

    Article  Google Scholar 

  • Houghton J (2009) Global warming – the complete briefing, 4th edn. Cambridge University Press, Cambridge, 438 pp

    Book  Google Scholar 

  • Ioannou PJ (1995) Nonnormality increases variance. J Atmos Sci 52:1155–1158

    Article  Google Scholar 

  • Isern-Fontanet J, Lapeyre G, Klein P, Chapron B, Hecht MW (2008) Three-dimensional reconstruction of oceanic mesoscale currents from surface information. J Geophys Res 113:C09005

    Article  Google Scholar 

  • Jaynes ET (1957a) Information theory and statistical mechanics. Phys Rev 106:620–630

    Article  Google Scholar 

  • Jaynes ET (1957b) Information theory and statistical mechanics. ii. Phys Rev 108:171–190

    Article  Google Scholar 

  • Jaynes ET (2003) Probability theory: the logic of science. Cambridge University Press, Cambridge, 758 pp

    Book  Google Scholar 

  • Katz RW, Naveau P (2010) Editorial: special issue on statistics of extremes in weather and climate. Extremes 13. doi:10.1007/s10 687–010–0111–9

    Google Scholar 

  • Kharin VV, Zwiers FW (2005) Estimating extremes in transient cimate change simulations. J Clim 18:1156–1173

    Article  Google Scholar 

  • Kharin VV, Zwiers FW, Zhang X, Hegerl GC (2007) Changes in temperature and precipitation extremes in the IPCC ensemble of global coupled model simulations. J Clim 20:1419–1444

    Article  Google Scholar 

  • Klaassen CAJ, Mokveld PJ, van Es B (2000) Squared skewness minus kurtosis bounded by 186/125 for unimodal distributions. Stat Prob Lett 50:131–135

    Article  Google Scholar 

  • Kloeden P, Platen E (1992) Numerical solution of stochastic differential equations. Springer, Berlin, 632 pp

    Google Scholar 

  • Kondrashov D, Kravtsov S, Ghil M (2006) Empirical mode reduction in a model of extratropical low-frequency variability. J Atmos Sci 63:1859–1877

    Article  Google Scholar 

  • Kravtsov S, Kondrashov D, Ghil M (2005) Multi-level regression modeling of nonlinear processes: derivation and applications to climate variability. J Clim 18:4404–4424

    Article  Google Scholar 

  • Kravtsov S, Kondrashov D, Ghil M (2010) Empirical model reduction and the modelling hierarchy in climate dynamics and the geosciences. In: Palmer T, Williams P (eds) Stochastic physics and climate modelling. Cambridge University Press, Cambridge, pp 35–72

    Google Scholar 

  • Krommes JA (2008) The remarkable similarity between the scaling of kurtosis with squared skewness for TORPEX density fluctuations and sea-surface temperature fluctuations. Phys Plasma 15:030703

    Article  Google Scholar 

  • Labit B, Furno I, Fasoli A, Diallo A, Müller SH, Plyushchev G, Podestà M, Foli FM (2007) Universal statistical properties of drift-interchange turbulence in TORPEX plasmas. Phys Rev Lett 98:255002

    Article  Google Scholar 

  • Lapeyre G, Klein P (2006) Dynamics of the upper oceanic layers in terms of surface quasigeostrophic theory. J Phys Oceanogr 36:165–176

    Article  Google Scholar 

  • Lind PG, Mora A, Gallas JAC, Haase M (2005) Reducing stochasticity in the north atlantic oscillation index with coupled langevin equations. Phys Rev E 72:056706

    Article  Google Scholar 

  • Majda AJ, Timofeyev I, Vanden-Eijnden E (1999) Models for stochastic climate prediction. Proc Natl Acad Sci 96:14687–14691

    Article  Google Scholar 

  • Majda AJ, Timofeyev I, Vanden-Eijnden E (2001) A mathematical framework for stochastic climate models. Commun Pure Appl Math 54:891–974

    Article  Google Scholar 

  • Majda AJ, Timofeyev I, Vanden-Eijnden E (2003) Systematic strategies for stochastic mode reduction in climate. J Atmos Sci 60:1705–1722

    Article  Google Scholar 

  • Majda AJ, Franzke C, Khouider B (2008) An applied mathematics perspective on stochastic modelling for climate. Phil Trans R Soc. 366:2429–2455

    Google Scholar 

  • Mo K, Ghil M (1988) Cluster analysis of multiple planetary flow regimes. J Geophys Res 93:10927–10952

    Article  Google Scholar 

  • Mo K, Ghil M (1993) Multiple flow regimes in the Northern Hemisphere winter: Part I: methodology and hemispheric regimes. J Atmos Sci 59:2625–2643

    Google Scholar 

  • Molteni F, Tibaldi S, Palmer TN (1990) Regimes in the wintertime circulation over northern extratropics. I. Observational evidence. Quart J R Meteor Soc 116:31–67

    Article  Google Scholar 

  • Monahan AH (2004) A simple model for the skewness of global sea-surface winds. J Atmos Sci 61:2037–2049

    Article  Google Scholar 

  • Monahan AH (2006a) The probability distributions of sea surface wind speeds Part I: theory and SSM/I observations. J Clim 19:497–520

    Article  Google Scholar 

  • Monahan AH (2006b) The probability distributions of sea surface wind speeds Part II: dataset intercomparison and seasonal variability. J Clim 19:521–534

    Article  Google Scholar 

  • Monahan AH, Pandolfo L, Fyfe JC (2001) The prefered structure of variability of the Northern Hemisphere atmospheric circulation. Geophys Res Lett 27:1139–1142

    Article  Google Scholar 

  • Monin AS, Yaglom AM (1971) Statistical fluid mechanics. mechanics of turbulence, vol I. MIT Press, Cambridge, 784 pp

    Google Scholar 

  • Monin AS, Yaglom AM (1975) Statistical fluid mechanics. mechanics of turbulence, vol II. MIT Press, Cambridge, 896 pp

    Google Scholar 

  • Müller D (1987) Bispectra of sea-surface temperature anomalies. J Phys Oceanogr 17:26–36

    Article  Google Scholar 

  • Nakamura H, Wallace JM (1991) Skewness of low-frequency fluctuations in the tropospheric circulation during the Northern Hemisphere winter. J Atmos Sci 48:1441–1448

    Article  Google Scholar 

  • Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law. Contempor Phys 46:323–351

    Article  Google Scholar 

  • Øksendal B (2007) Stochastic differential equations, 6th edn. Springer, Berlin/New York, 369 pp

    Google Scholar 

  • Paul W, Baschnagel J (1999) Stochastic processes: from physics to finance. Springer, Berlin/New York, 231 pp

    Google Scholar 

  • Peinke J, Böttcher F, Barth S (2004) Anomalous statistics in turbulence, financial markets and other complex systems. Ann Phys 13:450–460

    Article  Google Scholar 

  • Penland C (1989) Random forcing and forecasting using principal oscillation pattern analysis. Mon Weather Rev 117:2165–2185

    Article  Google Scholar 

  • Penland C, Ghil M (1993) Forecasting Northern Hemisphere 700-mb geopotential height anomalies using empirical normal modes. Mon Weather Rev 121:2355–2372

    Article  Google Scholar 

  • Penland C, Matrosova L (1994) A balance condition for stochastic numerical models with application to El Niño – the southern oscillation. J Clim 7:1352–1372

    Article  Google Scholar 

  • Penland C, Sardeshmukh PD (1995) The optimal growth of tropical sea surface temperature anomalies. J Clim 8:1999–2024

    Article  Google Scholar 

  • Petoukhov V, Eliseev A, Klein R, Oesterle H (2008) On statistics of the free-troposphere synoptic component: an evaluation of skewnesses and mixed third-order moments contribution to the synoptic-scale dynamics and fluxes of heat and humidity. Tellus A 60:11–31

    Google Scholar 

  • Pope SB (2000) Turbulent flows. Cambridge University Press, Cambridge, 771 pp

    Book  Google Scholar 

  • Reynolds RW, Smith TM, Liu C, Chelton DB, Casey KS, Schlax MG (2007) Daily high-resolution blended analyses for sea surface temperature. J Clim 20:5473–5496

    Article  Google Scholar 

  • Salmon R (1998) Lectures on geophysical fluid dynamics. Oxford University Press, New York, 378 pp

    Google Scholar 

  • Sandberg I, Benkadda S, Garbet X, Ropokis G, Hizanidis K, del Castillo-Negrete D (2009) Universal probability distribution function for bursty transport in plasma turbulence. Phys Rev Lett 103:165001

    Article  Google Scholar 

  • Sardeshmukh PD, Sura P (2009) Reconciling non-Gaussian climate statistics with linear dynamics. J Clim 22:1193–1207

    Article  Google Scholar 

  • Siegert S, Friedrich R, Peinke J (1998) Analysis of data sets of stochastic systems. Phys Lett A 243:275–280

    Article  Google Scholar 

  • Smyth P, Ide K, Ghil M (1999) Multiple regimes in norther hemisphere height fields via mixture model clustering. J Atmos Sci 56:3704–3732

    Article  Google Scholar 

  • Sornette D (2006) Critical phenomena in natural sciences. Springer, Berlin/New York, 528 pp

    Google Scholar 

  • Sura P (2003) Stochastic analysis of Southern and Pacific Ocean sea surface winds. J Atmos Sci 60:654–666

    Article  Google Scholar 

  • Sura P (2010) On non-Gaussian SST variability in the gulf stream and other strong currents. Ocean Dyn 60:155–170

    Article  Google Scholar 

  • Sura P, Barsugli JJ (2002) A note on estimating drift and diffusion parameters from timeseries. Phys Lett A 305:304–311

    Article  Google Scholar 

  • Sura P, Gille ST (2003) Interpreting wind-driven Southern Ocean variability in a stochastic framework. J Mar Res 61:313–334

    Article  Google Scholar 

  • Sura P, Gille ST (2010) Stochastic dynamics of sea surface height variability. J Phys Oceanogr 40:1582–1596

    Article  Google Scholar 

  • Sura P, Newman M (2008) The impact of rapid wind variability upon air-sea thermal coupling. J Clim 21:621–637

    Article  Google Scholar 

  • Sura P, Perron M (2010) Extreme events and the general circulation: observations and stochastic model dynamics. J Atmos Sci 67:2785–2804

    Article  Google Scholar 

  • Sura P, Sardeshmukh PD (2008) A global view of non-Gaussian SST variability. J Phys Oceanogr 38:639–647

    Article  Google Scholar 

  • Sura P, Sardeshmukh PD (2009) A global view of air-sea thermal coupling and related non-Gaussian SST variability. Atmos Res 94:140–149

    Article  Google Scholar 

  • Sura P, Newman M, Penland C, Sardeshmukh PD (2005) Multiplicative noise and non-Gaussianity: a paradigm for atmospheric regimes? J Atmos Sci 62:1391–1409

    Article  Google Scholar 

  • Sura P, Newman M, Alexander MA (2006) Daily to decadal sea surface temperature variability driven by state-dependent stochastic heat fluxes. J Phys Oceanogr 36:1940–1958

    Article  Google Scholar 

  • Taleb NN (2010) The black swan: the impact of the highly improbable, 2nd edn. Random House, New York, 480 pp

    Google Scholar 

  • Thompson KR, Demirov E (2006) Skewness of sea level variability of the world’s oceans. J Geophys Res 111:c05005. doi:10.1029/2004JC00283.

    Article  Google Scholar 

  • Trenberth KE, Mo KC (1985) Blocking in the southern hemisphere. Mon Weather Rev 113:3–21

    Article  Google Scholar 

  • van Kampen NG (2007) Stochastic processes in physics and chemistry, 3rd edn. Elsevier, North-Holland, 463 pp

    Google Scholar 

  • Whitaker JS, Sardeshmukh PD (1998) A linear theory of extratropical synoptic eddy statistics. J Atmos Sci 55:237–258

    Article  Google Scholar 

  • White GH (1980) Skewness and kurtosis and extreme values of Northern Hemisphere geopotential heights. Mon Weather Rev 108:1446–1455

    Article  Google Scholar 

  • Wilkins JE (1944) A note on skewness and kurtosis. Ann Math Stat 15:333–335

    Article  Google Scholar 

  • Wilks DS (2006) Statistical methods in the atmospheric sciences, 2nd edn. Academic, Burlington, 627 pp

    Google Scholar 

  • Winkler CR, Newman M, Sardeshmukh PD (2001) A linear model of wintertime low-frequency variability. Part I: formulation and forecast skill. J Clim 14:4474–4493

    Google Scholar 

Download references

Acknowledgements

The author thanks the anonymous reviewer whose comments greatly improved the chapter. 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”.

Author information

Authors and Affiliations

  1. Department of Earth, Ocean and Atmospheric Science, The Florida State University, 1017 Academic Way, Tallahassee, FL, 32306-4520, USA

    Philip Sura

  2. Center for Ocean-Atmospheric Prediction Studies, The Florida State University, 2035 E. Paul Dirac Drive, Tallahassee, FL, 32306-2840, USA

    Philip Sura

Authors
  1. Philip Sura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philip Sura .

Editor information

Editors and Affiliations

  1. , Dept. of Civil and Env. Engineering, University of California, Irvine, Engineering Gateway E/4130, Irvine, 92697, California, USA

    Amir AghaKouchak

  2. NOAA's National Climatic Data Center, Patton Avenue 151, Asheville, 28801-5001, North Carolina, USA

    David Easterling

  3. , Dept. of Civil and Env. Engineering, University of California, Irvine, Engineering Gateway E/4130, Irvine, 92697, California, USA

    Kuolin Hsu

  4. NASA/Goddard Space Flight Center, Greenbelt Road 8800, Greenbelt, 20771, USA

    Siegfried Schubert

  5. , Dept. Civil & Environmental Engineering, University of California, Irvine, Engineering Gateway E/4130, Irvine, 92697, California, USA

    Soroosh Sorooshian

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Sura, P. (2013). Stochastic Models of Climate Extremes: Theory and Observations. In: AghaKouchak, A., Easterling, D., Hsu, K., Schubert, S., Sorooshian, S. (eds) Extremes in a Changing Climate. Water Science and Technology Library, vol 65. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4479-0_7

Download citation

Publish with us

Policies and ethics

Search

Navigation