  We report on an analytical and numerical study of the dynamics of a
three-dimensional array of identical magnetic flux tubes in the
reduced-magnetohydrodynamic description of the plasma. We propose that the
long-time evolution of this system is dictated by flux-tube mergers and that
such mergers are dynamically constrained by the conservation of the pertinent
(ideal) invariants, {\it viz.} the magnetic potential and axial fluxes of each
tube. We also propose that in the direction perpendicular to the merging plane,
flux tubes evolve in critically-balanced fashion. These notions allow us to
construct an analytical model for how quantities such as the magnetic energy
and the energy-containing scale evolve as functions of time. Of particular
importance is the conclusion that, like its two-dimensional counterpart, this
system exhibits an inverse transfer of magnetic energy that terminates only at
the system scale. We perform direct numerical simulations that confirm these
predictions and reveal other interesting aspects of the evolution of the
system. We find, for example, that the early time evolution is characterized by
a sharp decay of the initial magnetic energy, which we attribute to the
ubiquitous formation of current sheets. We also show that a quantitatively
similar inverse transfer of magnetic energy is observed when the initial
condition is a random, small-scale magnetic seed field.
