Both of us sternly dismiss north wind (8)
I believe the answer is:
westerly
'wind' is the definition.
(I know that westerly is a type of west wind)
'both of us sternly dismiss north' is the wordplay.
'both of us' becomes 'we'.
'dismiss' is a deletion indicator.
'north' becomes 'N' (abbreviation).
'sternly' with 'n' removed is 'sterly'.
'we'+'sterly'='WESTERLY'
(Other definitions for westerly that I've seen before include "Wet lyres (anag.)" , "Type of wind blowing east" , "Our prevailing wind" , "Wind from Atlantic" , "It blows extremely" .)