  Extreme events provide relevant insights into the dynamics of climate and
their understanding is key for mitigating the impact of climate variability and
climate change. By applying large deviation theory to a state-of-the-art Earth
system model, we define the climatology of persistent heatwaves and cold spells
in key target geographical regions by estimating the rate functions for the
surface temperature, and we assess the impact of increasing CO$_2$
concentration on such persistent anomalies. Hence, we can better quantify the
increasing hazard {\color{black}due} to heatwaves in a warmer climate. We show
that two 2010 high impact events - summer Russian heatwave and winter Dzud in
Mongolia - are associated with atmospheric patterns that are exceptional
compared to the typical ones, but typical compared to the climatology of
extremes. Their dynamics is encoded in the natural variability of the climate.
Finally, we propose and test an approximate formula for the return times of
large and persistent temperature fluctuations from easily accessible
statistical properties.

Atmospheric and Oceanic Physics;Statistical Mechanics;Fluid Dynamics;Geophysics