  We introduce a model of interacting lattices at different resolutions driven
by the two-dimensional Ising dynamics with a nearest-neighbor interaction. We
study this model both with tools borrowed from equilibrium statistical
mechanics as well as non-equilibrium thermodynamics. Our findings show that
this model keeps the signature of the equilibrium phase transition. Moreover
the critical temperature of the equilibrium models correspond to the state
maximizing the entropy and delimits two out-of-equilibrium regimes, one
satisfying the Onsager relations for systems close to equilibrium and one
resembling convective turbulent states. Since the model preserves the entropy
and energy fluxes in the scale space, it seems a good candidate for parametric
studies of out-of-equilibrium turbulent systems.
