A rift storm was a phenomenon that occurred in the Cardiff Space-Time Rift.
The Sanctified farmed dinosaurs as livestock. Numerous dinosaur livestock of the Sanctified (including Brachiosaurids, Ankylosaurus and a Baryonyx) came through the Cardiff Space-Time Rift to Cardiff in 2008.
After the Sanctified sent dinosaurs through the Rift in 2008, the Torchwood Three team were able to capture the prehistoric creatures and send them back through a rift storm to where they came from. (COMIC: Rift War!)
Rift storms could be artificially created. A woman being pursued by Captain John Hart made one and sent canisters through it with instructions on how to find an Arcadian diamond. (TV: Kiss Kiss, Bang Bang) The Sanctified could also create rift storms. (COMIC: Rift War!)