Effective interface formalism: Application to global instability in viscous fingering

We develop an effective interface formalism design to provide an analytical understanding of the global stability properties of array structures in interfacial pattern-forming systems. This formalism is applied to perform the linear stability analysis of long-wavelength perturbations of a periodic array of Saffman-Taylor fingers. The breaking of the translational symmetry in the direction of propagation leads to an unstable mode analogous to the one which develops from the planar interface. The breaking of the translational symmetry in the transverse direction leads to a mode that, unlike the Eckhaus mode characteristic of small amplitude periodic structures, is always stable.