We analyze statistically extreme time-integrated Ap events in 1958-2007, which occurred during both strong and weak geomagnetic storms. The tail of the distribution of such events can be accurately fitted by a power-law with a sharp upper cutoff, in close agreement with a second fit inferred from Extreme Value Theory. Such a behavior is suggestive of a self-organization of the solar wind-magnetosphere-ionosphere system appearing during strong and sustained solar wind driving. The 1 in 10 years to 1 in 100 years return levels of such extreme events are calculated, taking into account possible solar cycle modulations. The huge October 2003 event turns out to be a 1 in 100 (+/- 40) years event. Comparisons with the distribution of extreme time-integrated aa events collected in 1870-2010 support the reliability of our results over the long run. Using data from Time History of Events and Macroscale Interactions during Substorms (THEMIS) satellites and the Van Allen Probes, we show that extreme time-integrated $ap$ events produce hard fluxes of energetic electrons and ions in the magnetotail and high fluxes (>1000 000 e/cm2/sr/s/MeV) of 1.8 MeV electrons in the heart of the outer radiation belt.
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