Computation of extreme heat waves in climate models using a large deviation algorithm

Francesco Ragonea,b, Jeroen Woutersa,c,d, and Freddy Boucheta,1

aLaboratoire de Physique, Ens de Lyon, Universite´ Claude Bernard, Universite´ Lyon, CNRS, F-69342 Lyon, France; bDepartment of Earth and Environmental Sciences, University of Milano–Bicocca, 20126 Milan, Italy; cSchool of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia; and dMeteorological Institute, University of Hamburg, 20146, Hamburg, Germany

Edited by Jonathan Weare, University of Chicago, Chicago, IL and accepted by Editorial Board Member Peter J. Bickel November 21, 2017 (received for review July 15, 2017) Studying extreme events and how they evolve in a changing cli- of past studies to either use models which are of a lesser quality mate is one of the most important current scientific challenges. than the up-to-date IPCC top class models or focus on a sin- Starting from complex climate models, a key difficulty is to be able gle event or a few events, which does not allow for a quanti- to run long enough simulations to observe those extremely rare tative statistical assessment. Making progress for this sampling events. In physics, chemistry, and biology, rare event algorithms issue would also allow a better understanding of those rare event have recently been developed to compute probabilities of events dynamics and strengthen future assessment of which class of that cannot be observed in direct numerical simulations. Here we models is suited for making quantitative predictions. propose such an algorithm, specifically designed for extreme heat or cold waves, based on statistical physics. This approach gives an Rare Event Algorithms improvement of more than two orders of magnitude in the sam- In physics, chemistry, and biology rare events may matter: Even pling efficiency. We describe the dynamics of events that would if they occur on timescales much longer than the typical dynam- not be observed otherwise. We show that European extreme heat ics timescales, they may have a huge impact. During recent waves are related to a global teleconnection pattern involving decades, new numerical tools, specifically dedicated to the com- North America and Asia. This tool opens up a wide range of possi- putation of rare events from the dynamics but requiring a con- ble studies to quantitatively assess the impact of climate change. siderably smaller computational effort, have been developed. They have been applied for instance to changes of configurations climate extremes | statistical physics | large deviation theory | in magnetic systems in situations of first-order transitions (10– rare event algorithms | heat waves 12), chemical reactions (13), conformal changes of polymer and biomolecules (14–17), and rare events in turbulent flows (18–22). are events, for instance extreme droughts, heat waves, rain- Since their appearance (23), the analysis of these rare event algo- Rfall, and storms, can have a severe impact on ecosystems rithms also became an active mathematical field (24–28). Sev- and socioeconomic systems (1–3). The Intergovernmental Panel eral strategies prevail, for instance genetic algorithms where an on Climate Change (IPCC) has concluded that strong evidence ensemble of trajectories is evolved and submitted to selections, exists indicating that hot days and heavy precipitation events minimum action methods, or importance sampling approaches. have become more frequent since 1950 (4, 5). However, the mag- Here we apply a rare event algorithm for sampling extreme nitude of possible future changes is still uncertain for classes events in a climate model. Given the complexity of the models of events involving more dynamical aspects, for instance hur- and phenomena, this has long been thought to be impracticable ricanes or heat waves (4, 6, 7). Estimates of the average time between two events of the same class, called return time (or Significance return period), are key for assessing the expected changes in extreme events and their impact. This is crucial on a national level when considering adaptation measures and on the inter- We propose an algorithm to sample rare events in climate national level when designing policy to implement the Paris models with a computational cost from 100 to 1,000 times less Agreement, in particular its Article 8 (https://en.wikisource.org/ than direct sampling of the model. Applied to the study of wiki/Paris Agreement). Public or private risk managers need to extreme heat waves, we estimate the probability of events know amplitudes of events with a return time ranging from a few that cannot be studied otherwise because they are too rare, years to hundreds of thousands of years when the impact might and we get a huge ensemble of realizations of an extreme be extremely large. event. Using these results, we describe the teleconnection The 2003 Western European heat wave led to a death toll of pattern for the extreme European heat waves. This method more than 70, 000 people (8). Similarly, the estimated impact should change the paradigm for the study of extreme events of the 2010 Russian heat wave was a death toll of 55,000 peo- in climate models: It will allow us to study extremes with ple, an annual crop failure of ∼25%, more than 1 million ha of higher-complexity models, to make intermodel comparison easier, and to study the dynamics of extreme events with burned areas, and ∼US$15 billion (∼1% of gross domestic prod- uct) of total economic loss (9). During that period, the tempera- unprecedented statistics. ture averages over 31 d at some locations were up to 5.5 SDs away Author contributions: F.B. proposed the project and initiated and directed the work; F.R., from the 1970–1999 climate (9). As no event similar to those heat J.W., and F.B. performed research; F.R. made the core of the numerical work that led to waves has been observed during the last few centuries, no past the rare event computation of heat waves and the resulting analysis; and F.R., J.W., and observations exist that would allow us to quantify their return F.B. wrote the paper. times. If return times cannot be estimated from observations, we The authors declare no conflict of interest. must rely on models. This article is a PNAS Direct Submission. J. Weare is a guest editor invited by the Editorial Several scientific barriers need to be overcome, however, Board. before we can obtain quantitative estimates of rare event return Published under the PNAS license. times from a model. One of them is that extreme events are 1To whom correspondence should be addressed. Email: [email protected]. observed so rarely that collecting sufficient data to study quanti- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. tatively their dynamics is prohibitively costly. This has led authors 1073/pnas.1712645115/-/DCSupplemental.

24–29 | PNAS | January 2, 2018 | vol. 115 | no. 1 www.pnas.org/cgi/doi/10.1073/pnas.1712645115 Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 h esblt fti prahfrtpcasIC oesi the in models IPCC advocate top-class to for future. and approach near models this potential of of class huge feasibility this the (for the for demonstrate report algorithms to event IPCC is rare last models aim of the of Our in 33). class waves ref. the heat instance, extreme of in discuss projection nevertheless to the is used assessing It for increase. used temperature models IPCC the about top-class features the with Plasim and While radia- dynamics. surface 10 of cloud parameterizations land and includes transfers it the tive ocean; with the of dynamics interactions layer mixed atmospheric their gives of of Plasim representation (32). and realistic model (Plasim) reasonably Simulator a Planet the use Model We Simulator Planet the in Waves Heat surface a pressure. on fixed value a height by geopotential geopotential defined the the at look of it to Equivalently, value customary coordinate). is some vertical convenient at most maps (the height pressure at key One look fields. 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APPLIED MATHEMATICS √ relative error of this estimate is of the order of 1/ N γB . For A −2 instance, if γB is of the order of 10 , estimating γB with a rel- ative error of 1% requires a huge sample size, of the order of 80 6 10 . However, if we rather sample N random numbers X˜n from the distribution ρ˜ (see Fig. 3(a)), where ρ(x) = L(x)˜ρ(x), with L 70 conveniently chosen, then the event may become common: this is importance sampling. From the formula ρ = Lρ˜, we have the     60 1 PN ˜ ˜ estimate γ˜B = N n=1 L Xn 1B Xn , where 1B is the indi-

lat cator function of the set B. If the rare event is actually√ common 50 for ρ˜, this estimate gives a relative error of order 1/ N (see (24) for a precise formula). Then, in order to estimate γB with a rel- 40 ative error of 1%, we need a sample size of order of 104; this is

30 A 0.5 −40 −20 0 20 40 lon B 8 0.4 6 hours 90 days 6 local max 6 hours 0.3 local max 90 days 4 PDF 0.2 2

0 0.1 a (K) B -2 0 8 -6 -4 -2 0 2 4 6 4 -4 0 X a (K) -4 B -6 -8 1.6 0 250 500 750 1000 CTRL_LONG time (years) 1.4 k50 -8 -180 -120 -60 0 60 120 180 1.2 time (days)

Fig. 2. (A) The red color marks the area of Europe over which the tempera- 1 ture is averaged. (B) Time series of European surface temperature anomaly, 6 h (light blue) and 90 d running mean (dark blue), during 360 d and 1,000 0.8 y (Inset). The red triangles and circles feature one local maximum of the PDF temperature anomalies, as an example of a 2 K heat wave lasting 90 d. 0.6

tail of the probability distribution function (PDF) of a, denoted 0.4 P(a, T ). The instantaneous TS has SD σ ≈ 1.6 K and is slightly skewed 0.2 toward positive values. Its autocorrelation time is τc ≈ 7.5 d, which is the typical time for synoptic fluctuations (at a scale of about 1,000 km). An example of a time series of the 90-d aver- 0 aged Europe temperature anomaly is shown in Fig. 2B. -3-2-10123 a (K) Importance Sampling and Large Deviations of Time-Averaged Temperature Fig. 3. (A) We want to estimate the probability to be in the set B, for We first explain importance sampling, a crucial probabilistic con- the model PDF ρ(x). We are able to sample instead from the PDF ρ˜(x) for N which the rare event becomes common. We know the relation L = ρ/ρ˜ and cept for the following discussion. We sample independent and can recover the model statistics ρ, from the importance sampling ρ˜. (B) PDF identically distributed random numbers from a PDF ρ and want of the time-averaged temperature a (T = 90 d) for the model control run to estimate γB , the probability to be in a small set B (Fig. 3A). (black) and for the algorithm statistics with k = 50, illustrating that the We will obtain about N γB occurrences in the set B, from which algorithm performs importance sampling and that +2 K heat waves become we can estimate γB . An easy calculation (24) shows that the common for the algorithm while they are rare for the model.

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APPLIED MATHEMATICS atmosphere during extreme heat wave events. Fig. 5A shows conditioned on the occurrence of a 90-d 2 K heat wave (com- the temperature and the 500-hPa geopotential height anomalies, posite statistics). Those conditional statistics are reminiscent of the teleconnection pattern maps sometimes shown in the climate community. However, while usual teleconnection patterns are A 0 computed from empirical orthogonal function (EOF) analysis, 0 0 and thus describe typical fluctuations, our extreme event condi- 10 tional statistics describe very rare flows that characterize extreme 20 heat waves. Those global maps are a unique way to consider rare event and atmosphere fluctuation statistics, which is extremely

0 10 interesting from a dynamical point of view. 0 By definition, as we plot statistics conditioned on a = 10 -10 10 1 R T -20 A(xn (t))dt > 2K, with T = 90 d, Fig. 5A shows a warming 10 T 0 0 pattern over Europe. The geopotential height map also shows a strong anticyclonic anomaly right above the area experiencing 20 -30 -20 the maximum warming, as expected through the known positive 10 0 correlation between surface temperature and anticyclonic condi- 40 -10 -20 30 -30 20 30 -10 0 tions (34). A less expected and striking result is that the strong 30 -20 40 20 10 warming over Europe is correlated with a warming over south- -100 50 10 0 eastern Asia and a warming over North America, both with sub- 1020 -30 30 40 10 stantial surface temperature anomalies of order of 1 K to 3 K, 70 50 60 and anticorrelated with strong cooling over Russia and Green- 40 30 0 land, of the order of -1 K to -2 K. This teleconnection pattern is 20 due to a strongly nonlinear stationary pattern for the jet stream, 10 with a wavenumber 3 dominating the pattern, as is clearly seen

0 from the geopotential height anomaly. In Fig. 5B, the anomaly 0 of the kinetic energy gives a complementary view: Over Europe, the succession of a southern blue band (negative anomaly) and a northern red band (positive anomaly) should be interpreted as -4-3-2-101234a northward shift of the jet stream there. Strikingly, over Green- Temperature anomaly (K) land and North America, the jet stream is at the same position (but it is more intense) for the large deviation algorithm statis- tics as for the control run, while it is shifted northward over B Europe and very slightly southward over Asia. This is related to the strong southwest–northeast tilt of the geopotential height anomalies over the Northern Atlantic. The extended red area (positive anomaly of kinetic energy) over Asia is rather due to a more intense cyclonic activity there, than to a change of jet stream position. Inspection of the time series of the daily temperature shows that along the long duration of heat waves, the synoptic fluctua- tions on timescales of weeks are still present (Fig. 2B). The tem- perature is thus fluctuating with fluctuations of order of 5–10 ◦C, as usual, but they fluctuate around a larger temperature value than usual. This is also consistent with the northward shift of the jet stream over Europe, but does not seem to be consistent with a blocking phenomenology as hypothesized in many other pub- lications. This calls for using similar large deviation algorithms with other models and other setups to test the robustness of the present observation.

Conclusions We have demonstrated that rare event algorithms, developed using statistical physics ideas, can improve the computation of the return times and the dynamical aspects of extreme heat waves. One of the future challenges in the use of rare event algo- -60 -40 -20 0 20 40 60 rithms for studying climate extremes will be to identify which 2 -2 algorithms and which score functions will be suitable for each Kinetic energy anomaly (m s ) type of rare event. We anticipate that this tool will make avail- able a range of studies that have been out of reach to date. Fig. 5. (A) Northern Hemisphere surface temperature anomaly (colors) First, it will pave the way to the use of state of the art climate and 500-hPa geopotential height anomaly (contours), conditional on the 1 R T models to study rare extreme events, without having to run the occurrence of heat wave conditions T 0 A(xn(t))dt > a, with T = 90 d and a = 2 K, estimated from the large deviation algorithm. (B) North- model for unaffordable times. The demonstrated gain of sev- ern Hemisphere anomaly of the averaged kinetic energy for the zonal eral orders of magnitude in the sampling efficiency will also help velocity at 500 hPa conditional on the occurrence of heat wave condi- to make quantitative model comparisons, to assess on a more h 1 R T i tions E KE500 | T 0 A(xn(t))dt > a , with T = 90 d and a = 2 K, estimated quantitative basis the skill to predict extreme events, for the from the large deviation algorithm, with respect to the long-time average existing models. It will also make available a range of dynami- E [KE500] computed from the control run. cal studies. As an example, having a high number of heat waves

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Research and this AXA discussions the of useful aspects for various Lestang on Thibault and Lunkeit, Frank ACKNOWLEDGMENTS. represented quantities dynamical article. the this postprocess- in of statistical description the the implemen- of times and the aspects ing, return of model, Plasim description compute the the of to algorithm, tation method event rare the the of with and algorithm Methods GKLT and Data SI Methods and Data u tn plMt Ser Math Appl Stand Bur instantons. of calculation the for equations ton’s equation. equations. geostrophic enls tessi oa e yais arXiv:1706.08810. dynamics. jet zonal in stresses Reynolds’ equations. Euler two-dimensional and Phys model Stat quasi-geostrophic the for instantons 478:1–69. heatwaves. European wave. heat region. Mediterranean Russian the 2010 the of attribution 39:L04702. to approaches model. friendly time. dynamics. Lett Rev Phys Appl Anal arXiv:1405.1352. case. idealized an in rithms Applications with tems UK). Chichester, ruF uae 20)Aatv utlvlsltigfrrr vn analysis. event rare for splitting multilevel Adaptive (2007) A Guyader F, erou ´ he E eiveT ose 21)Aayi faatv utlvlsltigalgo- splitting multilevel adaptive of Analysis (2014) M Rousset T, Lelievre CE, ehier ´ ttMc hoyExp Theory Mech Stat J 156:1066–1092. hsAMt Theor Math A Phys J a Phys Nat 25:417–443. 96:120603. rC(00 ossetgorpia atrso hne nhigh-impact in changes of patterns geographical Consistent (2010) C ar ¨ eerlZ Meteorol nItouto oRr vn Simulation Event Rare to Introduction An ena-a omleGnaoia n neatn atceSys- Particle Interacting and Genealogical Formulae Feynman-kac 3:203–207. a Geosci Nat frT adnEjdnE(04 rlnt aaerzdHamil- parametrized Arclength (2014) E Vanden-Eijnden T, afer ¨ Srne,NwYork). New (Springer, Phys J New frT(03 ntno leigfrtesohsi burgers stochastic the for filtering Instanton (2013) T afer h uhr hn ulir ai,Eibr Kirk, Edilbert Badin, Gualtiero thank authors The ¨ 12:27–30. PNAS aeEetSmlto sn ot al Methods Carlo Monte Using Simulation Event Rare nio e Lett Res Environ 14:299–304. otisacmlt ecito fthe of description complete a contains 2007:P03004. 46:062002. 3:398–403. | 17:015009. aur ,2018 2, January 7:014023. utsaeMdlSimul Model Multiscale | o.115 vol. Srne,NwYork). New (Springer, epy e Lett Res Geophys | o 1 no. 12:566–580. hsRep Phys (Wiley, Stoch | Natl 29 J

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