Woman cold in rising river (5)
I believe the answer is:
erica
'woman' is the definition.
'cold in rising river' is the wordplay.
'cold' becomes 'C' (eg on taps).
'in' indicates putting letters inside.
'rising' is a reversal indicator (in a down clue, letters go up).
'river' becomes 'aire' (River Aire in Yorkshire).
'aire' written backwards gives 'eria'.
'c' put into 'eria' is 'ERICA'.
(Other definitions for erica that I've seen before include "Heathland shrub; girl's name" , "Moorland plant, girl's name" , "Woman" , "Heather, for instance" , "Girl's name - heather" .)