This does not seem like a formal term, but according to the quoted site, your example is called a VERBLESS CLAUSE.
"A verbless clause . . . is considered a clause because it is dealing with a separate piece of information in relation to the main clause. For example, in the sentence, In the interests of the local children, the council should reconsider its decision, there are two separate pieces of information: the main clause--the council should reconsider its decision; and a dependent clause that deals with issues that interest local children. In this clause, however, the verb has been nominalized resulting in a verbless clause. Verbless clauses are different from adverbial phrases. The latter provide some information to do with the time, place, or manner in which something happens within an existing clause. Verbless clauses, on the other hand, provide a separate piece of information outside of an existing clause."
(Peter Knapp and Megan Watkins, Genre, Text, Grammar: Technologies for Teaching and Assessing Writing. UNSW Press, 2005)