A tempest is “a violent windstorm” or “a violent commotion, disturbance, or tumult.” So, tempestuous is an adjective meaning “characterized by or subject to tempests” or “resembling a tempest”.
The Online Etymology Dictionary says that tempest comes from Latin “tempestas (gen. tempestatis) ‘storm, weather, season,’ also ‘commotion, disturbance,’ related to tempus ‘time, season.’”
I did a search for tempestuous on Google News and found many usage examples like “tempestuous teenager”, “the tempestuous situation in the Middle East”, “the tempestuous struggle for power in early Rome”, “tempestuous girlfriend”, and of course “tempestuous weather”. Journalists sure love this word!