Violent storm encountered on the way up – bother! (7)
I believe the answer is:
'violent storm' is the definition.
(tempest is a kind of violent storm)
'encountered on the way up bother' is the wordplay.
'encountered' becomes 'met' ('meet' can be a synonym of 'encounter').
'on the way up' says the letters should be written in reverse.
'bother' becomes 'pest'.
'met' backwards is 'tem'.
(Other definitions for tempest that I've seen before include "A violent storm like Shakespeare's" , "Outburst" , "Shakespeare's 5" , "Violent Shakespearian storm" , "Pets met in violent storm" .)