What is the appropriate term for a modifying clause that generalizes, rather than limits, the preceding statement? For example: "It is crucial to bring a canteen on the trip, even if the sky is cloudy."
Carla Lowe, in 'Daily Writing Tips', gives the most appropriate definition of modifier that I've come across:
But she should probably add extends the boundary conditions of, and/or emphasises after 'limits', as John implies.
Here, 'even if the sky is cloudy' modifies the whole main clause.
It is important to bear in mind the linguistics usage of modify in such analyses; the main 'everyday' sense is 'to adjust in a relatively minor way' - although to the heroes at Top Gear, that might involve converting a saloon car to a boat.
This an adverbial prepositional phrase of concession. Adverbs tell time, place, degree, manner, cause, purpose, condition, or concession.