# Computation of Extreme Values of Time Averaged Observables in Climate Models with Large Deviation Techniques

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## Abstract

One of the goals of climate science is to characterize the statistics of extreme and potentially dangerous events in the present and future climate. Extreme events like heat waves, droughts, or floods due to persisting rains are characterized by large anomalies of the time average of an observable over a long time. The framework of Donsker–Varadhan large deviation theory could therefore be useful for their analysis. In this paper we discuss how concepts and numerical algorithms developed in the context of with large deviation theory can be applied to study extreme, rare fluctuations of time averages of surface temperatures at regional scale with comprehensive numerical climate models. When performing this type of analysis, unless a rigorous study of the convergence to the large deviation limit is performed, it can be easy to be misled in thinking to have reached the asymptotic regime. In this paper we provide a systematic protocol to study the convergence of large deviation functions tailored for applications to climate problems. Referring to the existing literature on the subject, we provide explicit formulas to compute large deviation functions directly from time series of a deterministic dynamical system that can be applied to climate records, and we describe how to study the convergence. We show how using a rare event algorithm applied to a numerical model can improve the efficiency of the computation of the large deviation functions. As a case study we consider the time averaged European surface temperature obtained with the numerical climate model Plasim. We show how a precise analysis of the convergence leads to the conclusion that the large deviation limit is nor properly reached for time scales shorter than a few years, and is therefore of no practical interest to study midlatitude heat waves. Finally we show how, even in a case like this, rare event algorithms developed to study large deviation functions can be used to improve the statistics of events on time scales shorter than the one needed to reach the large deviation asymptotic regime.

## Keywords

Large deviation theory Rare event algorithms Climate extremes Heat waves## Notes

### Acknowledgements

The authors thank the editor and two anonymous reviewers for their constructive criticism and suggestions. The research leading to these results has received funding from the European Research Council under the European Union’s seventh Framework Programme (FP7/2007-2013 Grant Agreement No. 616811). The simulations have been performed on the machines of the Pôle Scientifique de Modélisation Numérique (PSMN) and of the Centre Informatique National de l’Enseignement Supérieur (CINES).

### Compliance with Ethical Standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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