Spacing of faults at the scale of the lithosphere and localization instability: 1. Theory

Large‐scale tectonic structures such as orogens and rifts commonly display regularly spaced faults and/or localized shear zones. To understand how fault sets organize with a characteristic spacing, we present a semianalytical instability analysis of an idealized lithosphere composed of a brittle layer over a ductile half‐space undergoing horizontal shortening or extension. The rheology of the layer is characterized by an effective stress exponent, n e . The layer is pseudoplastic if 1/ n e = 0 and forms localized structures if 1/ n e < 0. Two instabilities grow simultaneously in this model: the “buckling/necking instability” that produces broad undulations of the brittle layer as a whole, and the “localization instability” that produces a spatially periodic pattern of faulting. The latter appears only if the material in the brittle layer weakens in response to a local perturbation of strain rate, as indicated by 1/ n e < 0. Fault spacing scales with the thickness of the brittle layer and depends on the efficiency of localization. Localization is more efficient for more negative 1/ n e , leading to more widely spaced faults. The fault spacing is related to the wavelength at which different deformation modes within the layer enter a resonance that exists only if 1/ n e < 0. Depth‐dependent viscosity and the model density offset the instability wavelengths by an amount a L that we determine empirically. The wave number of the localization instability, is k j L = π( j + a L )(−1/ n e ) −1/2 / H , with H the thickness of the brittle layer, j an integer, and 1/4 < a L < 1/2 if the strength of the layer increases with depth and the strength of the substrate decreases with depth.