  Severe geomagnetic storms appear to be ordered by the solar cycle in a number
of ways. They occur more frequently close to solar maximum and declining phase,
are more common in larger solar cycles and show different patterns of
occurrence in odd- and even-numbered solar cycles. Our knowledge of the most
extreme space weather events, however, comes from the spikes in
cosmogenic-isotope ($^{14}$C, $^{10}$Be and $^{36}$Cl) records that are
attributed to significantly larger solar energetic particle (SEP) events than
have been observed during the space age. Despite both storms and SEPs being
driven by solar eruptive phenomena, the event-by-event correspondence between
extreme storms and extreme SEPs is low. Thus it should not be assumed a priori
that the solar cycle patterns found for storms also hold for SEPs and the
cosmogenic-isotope events. In this study we investigate the solar cycle trends
in the timing and magnitude of the 67 SEP ground-level enhancements (GLEs)
recorded by neutron monitors since the mid 1950s. Using a number of models of
GLE occurrence probability, we show that GLEs are around a factor four more
likely around solar maximum than around solar minimum, and that they
preferentially occur earlier in even-numbered solar cycles than in odd-numbered
cycles. There are insufficient data to conclusively determine whether larger
solar cycles produce more GLEs. Implications for putative space-weather events
in the cosmogenic-isotope records are discussed. We find that GLEs tend to
cluster within a few tens of days, likely due to particularly productive
individual active regions, and with approximately 11-year separations, owing to
the solar cycle ordering. But these timescales do not explain
cosmogenic-isotope spikes which require multiple extreme SEP events over
consecutive years.
