  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 relation 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. We
study the convergence of large deviation functions for the time averaged
European surface temperature obtained with direct numerical simulation of the
climate model Plasim, and discuss their climate implications. We show how using
a rare event algorithm can improve the efficiency of the computation of the
large deviation rate functions. We discuss the relevance of the large deviation
asymptotics for applications, and we show how rare event algorithms can be used
also to improve the statistics of events on time scales shorter than the one
needed for reaching the large deviation asymptotics.
