Has there been a study of circuits operating on arrays?
Not really an answer, but would've been too long a comment. The result of Ben-Or and Cleve that arithmetic formulas can be simulated by branching programs of width 3, works over arbitrary non-commutative rings (in particular over the matrix ring $M_n(F)$). A nice consequence of this is that $\# NC^1$ circuits over $M_n(\mathbb{Z})$ captures ${\text{GapL}}$ over $\mathbb{Z}$ (for which the Determinant is a hard function). See for example, the paper on catalytic space by Buhrman et al. for an interesting generalization of the work of Ben-Or and Cleve. There is a result of Capelli, Durand and Mengel which gives a characterization of the arithmetic complexity class VP as equivalent to formulas whose nodes compute tensor contractions (note that here you even lose associativity).