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Computation of extreme heat waves in climate models using a large deviation algorithm
Francesco Ragone, Jeroen Wouters, Freddy Bouchet
TL;DR
Extremely rare climate events are difficult to study because direct simulations cannot run long enough to observe them quantitatively. The paper adapts a statistical-physics rare-event algorithm for finite-time climate observables and return times, producing large sampling gains and revealing teleconnection patterns associated with extreme European heat waves.
Problem
Rare climate extremes are too infrequent for direct simulations to provide sufficient quantitative evidence about their probabilities and dynamics.
Method
The paper applies the GKLT rare-event algorithm to finite-time, time-averaged temperature observables and adapts it to compute return times.
Results
The algorithm computes return times up to 10^6−10^7 years at a computational cost of about 10^3 years, a gain of more than three orders of magnitude in sampling efficiency.
Takeaways & Limitations
The method enables statistical and dynamical studies of extreme heat waves that direct numerical simulation could not observe, including their global teleconnection pattern.
Abstract
from arXiv · showhide
Studying extreme events and how they evolve in a changing climate is one of the most important current scientific challenges. Starting from complex climate models, a key difficulty is to be able to run long enough simulations in order to observe those extremely rare events. In physics, chemistry, and biology, rare event algorithms have recently been developed to compute probabilities of events that cannot be observed in direct numerical simulations. Here we propose such an algorithm, specifically designed for extreme heat or cold waves, based on statistical physics approaches. This gives an improvement of more than two orders of magnitude in the sampling efficiency. We describe the dynamics of events that would not be observed otherwise. We show that European extreme heat waves are related to a global teleconnection pattern involving North America and Asia. This tool opens a full range of studies, so far impossible, aimed at assessing quantitatively climate change impacts.
Rare event algorithms
Rare-event algorithms use specialized numerical strategies to estimate extremely unlikely events with much lower computational effort than direct simulation.
- Rare-event algorithms have been applied to magnetic transitions, chemical reactions, biomolecular conformational changes, and turbulent flows.
Significance Statement
The paper applies statistical-physics rare-event methods to climate-model heat waves, adapting them to finite-time observables and return-time estimation. This enables efficient sampling of otherwise inaccessible extremes and analysis of their teleconnection patterns.
- The study proposes a climate-model rare-event algorithm costing hundred to thousand times less than direct sampling.
- The resulting ensemble describes extreme heat waves and their teleconnection pattern involving Europe, North America, and Asia.
- The method targets extreme heat waves using the GKLT algorithm for large deviations of time-averaged quantities.
- The algorithm is adapted from infinite-time rate-function estimation to finite-time observables and return-time computation.
The jet stream dynamics and extreme heat waves
Midlatitude weather is governed by jet-stream dynamics, whose nonlinear meandering is linked to anticyclonic and cyclonic anomalies associated with heat waves.
- Jet streams are strong, narrow eastward air currents near 45° latitude, with maximum velocities of about 40 m.s−1.
- The Northern Hemisphere jet stream is represented in the model by time-averaged horizontal kinetic energy at 500 hPa.
- Nonlinear Rossby-wave meandering produces alternating anticyclonic and cyclonic anomalies in midlatitude weather.
- Figure 1 contrasts a North American upper-tropospheric jet-stream snapshot with the model’s averaged Northern Hemisphere jet stream.
Heat waves in the Plasim model
The study defines heat waves as rare, persistent regional surface-temperature anomalies and analyzes their time-averaged distribution in the Plasim climate model.
- The Plasim model represents atmospheric, land-surface, ocean mixed-layer, radiative-transfer, and cloud dynamics with about 10^5 degrees of freedom.
- Heat waves are defined as rare, long-lasting surface-temperature anomalies over an extended area.
- The heat-wave observable averages surface-temperature anomaly over Western Europe and a duration T varied from weeks to several months, with results discussed for T = 90 days.
- Figure 2 identifies the European averaging region and contrasts 6-hour temperature anomalies with a 90-day running mean, including a 2 K, 90-day example.
- The instantaneous surface-temperature anomaly has standard deviation σ ≈ 1.6 K and autocorrelation time τc ≈ 7.5 days.
- Positive k exponentially tilts the sampling measure toward larger time-averaged European temperatures, targeting different extreme heat-wave classes.
Importance sampling and large deviations of time averaged temperature
The method uses trajectory-level importance sampling and the GKLT large deviation algorithm to make rare temperature states common while recovering model statistics. A bias parameter selects time-averaged European temperatures through large-deviation sampling.
- Importance sampling: Importance sampling estimates rare-event probabilities by sampling from a distribution where the event is common and correcting with the likelihood ratio.The relative error scales as 1/√(NγB) for direct sampling, while importance sampling can reduce the required sample size substantially.
- Importance sampling: 100-fold sampling gain is obtained for a probability γB of order 10^-2 when estimating it with 1% relative error using importance sampling.The required sample size is of order 10^4, and the gain grows like the inverse of γB.
- Trajectory sampling: Trajectory-level importance sampling is required because the climate is a nonequilibrium dynamical system with trajectories distributed according to the model PDF P0.The GKLT algorithm selects trajectories from an importance-sampling PDF Pk.
- Trajectory sampling: The algorithm resamples trajectories using a score based on recent dynamics, cloning trajectories moving toward target extremes and killing poorly scoring ones.Successful copies are slightly perturbed so they can evolve differently; τ is the resampling time.
- Large deviations: The algorithm performs importance sampling by making +2° K, 90-day heat waves common in its statistics while they remain rare in the control model.The model statistics can be recovered from the biased distribution using L = ρ/˜ρ.
- Large deviations: The scaled cumulant generating function and rate function are related by Legendre–Fenchel transforms, allowing the bias k to select common time-averaged temperature states.Here, the dynamical energy a is the time-averaged European temperature.
Return times for 90 day heat waves
The large deviation algorithm estimates return times for 90-day European heat waves using biased simulations calibrated against a 1,000-year control run. It reproduces control-run behavior over 10–300 years and reaches far rarer events at much lower sampling cost.
- Experimental setup: Return times are computed for 90-day European surface-temperature averages using a 1,000-year control run and six biased experiments.The bias parameter k ranges from 10 to 40, and each biased simulation costs about 182 years.
- Rare-event reach: 10^6–10^7-year return times are computed with a total computational cost of order 10^3 years, yielding more than three orders of magnitude greater sampling efficiency.These events could not be observed in direct numerical simulations with current or foreseeable computational possibilities.
- Statistical quality: Several hundred 90-day heat waves exceeding 2 K occur in the k = 50 experiment, compared with one in the control run.This improves return-time estimation either by reducing numerical cost or by reducing relative error at fixed cost.
- Statistical quality: Improved statistics enable dynamical analysis involving temperature and pressure fields.The paper identifies this improvement as crucial for analyzing the dynamics of extreme heat waves.
- Validation: 10–300 years shows good overlap between return times from the control run and the large deviation algorithm at equal computational cost.The control-run estimates are shown in black and the algorithm estimates in red.
Teleconnection patterns for extreme heat waves
Conditional statistics for 90-day, 2 K European heat waves reveal a global temperature and circulation pattern. The pattern includes linked anomalies across Europe, Asia, North America, Russia, and Greenland, with regional jet-stream changes.
- 90-day, 2 K heat waves produce warming over Europe accompanied by a strong anticyclonic geopotential-height anomaly.The anomaly is located above the area experiencing maximum warming.
- Surface temperatures warm over South Eastern Asia and North America while cooling occurs over Russia and Greenland.The reported anomalies are approximately 1–3 K for warming and −1 to −2 K for cooling.
- A strongly nonlinear stationary jet-stream pattern, dominated by wavenumber 3, underlies the temperature teleconnection.The geopotential-height anomaly reveals this circulation structure.
- The kinetic-energy anomalies indicate a northward jet-stream shift over Europe, while the jet remains positioned over Greenland and North America but becomes more intense.Over Asia, increased kinetic energy reflects more intense cyclonic activity rather than a jet-position change.
- During long heat waves, weekly synoptic fluctuations of roughly 5–10 degrees persist around a higher-than-usual temperature.This behavior does not appear consistent with a blocking phenomenology as hypothesized in other publications.
Conclusions
The study demonstrates that statistical-physics rare-event algorithms improve computation of extreme heat-wave return times and dynamics. It anticipates applications to more complex climate models, quantitative model comparison, and new dynamical studies.
- Rare-event algorithms improve computation of return times and dynamical aspects of extreme heat waves.
- Selecting suitable algorithms and score functions for each type of climate extreme remains a future challenge.
- The method could enable rare-event studies with state-of-the-art climate models without unaffordable simulation times.
- Several orders of magnitude in sampling-efficiency gain could support quantitative comparisons of models’ ability to predict extreme events.
- In the Plasim model, numerous heat waves indicated that European events mainly affecting Scandinavia relate more to a northward jet-stream shift than to Rossby wave breaking.
SI Appendix
The Supporting Information Appendix documents the algorithm, return-time calculations, Plasim implementation, statistical post-processing, and dynamical quantities represented in the article.
- The appendix provides a complete description of the GKLT algorithm and the method for computing return times with rare-event algorithms.
- It also describes the Plasim implementation, statistical post-processing, and dynamical quantities represented in the article.
SI Data and Methods
The SI methods describe an ensemble-based large-deviation algorithm that selectively clones and kills trajectories to favor rare values of a time-averaged observable. Backward reconstruction then supports estimation of rare-event statistics, with asymptotic assumptions and sampling-error considerations.
- The GKLT algorithm evolves trajectories in parallel, then kills or clones them according to weights based on their past evolution.The weighting tilts the trajectory measure toward extremes of a chosen observable.
- For large N, the algorithm’s estimator is asymptotically valid, with typical averaging error of order 1/√N under the mean-field approximation.Formula (4) applies at times Ta that are integer multiples of τ.
- Backward reconstruction attaches ancestors to final trajectories, producing an effective ensemble distributed according to the tilted measure Pk.Killed trajectories are discarded from the statistics.
- The algorithm uses N trajectories with distinct initial conditions, a total integration time Ta, observable A(X(t)), and resampling interval τ.The study’s observable is European temperature.
- The selection strength k controls targeting, with larger k favoring trajectories having larger time-averaged observables.
- Large-deviation theory relates the scaled cumulant generating function λ(k) and rate function I(a) through Legendre–Fenchel transforms.
- Backward-reconstructed trajectories enable estimators of rare-event statistics with dramatically lower statistical error because relevant rare events are more numerous in the effective ensemble.
Algorithm implementation.
The algorithm maintains a fixed ensemble through repeated propagation, weighting, selection, cloning, and reinitialization, with perturbations separating deterministic copies. Tests in climate and Ornstein–Uhlenbeck models show substantially more efficient rare-event sampling than direct control runs.
- Ensemble evolution: The algorithm propagates each trajectory for a fixed resampling interval, assigns weights, and computes the number of copies produced.The ensemble is then adjusted through killing and cloning operations.
- Ensemble evolution: Killing or cloning restores the ensemble to a constant size, after which the system is reinitialized and iterated.After an initial transient, the ensemble reaches a statistically stationary state and loses memory of its initial conditions.
- Plasim model and setup: Independent initial conditions are sampled from a 1000-year control run, while the model uses T42 horizontal resolution, 10 vertical levels, and perpetual summer conditions.The control run also serves as a benchmark for algorithm performance.
- Plasim model and setup: The climate experiment uses k = 2, N = 128 trajectories, 800-day trajectories, and an approximately 284.4-year computational cost.The resampling time is τ = 8 days, close to the Europe-averaged temperature autocorrelation time of 7.5 days.
- Validation: The k = 2 algorithm estimate of λ at 248 years is at least as good as the 1000-year control estimate and much better than the 248-year control estimate.This comparison is presented as a consistency test supporting the robustness of the procedure, including trajectory perturbations.
- Validation: For the Ornstein–Uhlenbeck benchmark, algorithm return-time curves agree with a long control run at costs 160 and 1600 times smaller.The benchmark also reports about 10% error for return times between 5·10^7 and 5·10^8.