Combining faulting and ductile deformation in long-term models of continental deformation

The spatial variation of strain rate in broad regions of continental collision, extension, or shear can often be well represented by the deformation of a thin viscous shell representing the lithosphere. The simplest explanation of this observation is that the deformation of the lithosphere is to first order a ductile process, even though shallow focus earthquakes imply slip on faults and release of elastic strain. In the thin-viscous-shell concept the strain of the upper brittle layer is assumed to simply follow the ductile strain of the stronger layers beneath, at least in the inter-seismic period. If the faults extend only to depths of 10 or 20 km, the brittle upper layer is not sufficiently thick or strong to do otherwise, and the concept of the brittle upper layer controlled by the ductile substrate is consistent with ductile models of the displacement-rate field constrained by GNSS observations. However, some large-scale faults do not comply with this concept and, rather than following the deformation of the ductile layer beneath, the strain localization on these structures appears to constrain the ductile deformation field of the adjoining regions. There are multiple lines of evidence from seismology and geodesy that great continental strike-slip faults, such as the San Andreas fault of California, the Alpine Fault of New Zealand, or the Altyn Tagh fault of China, extend through the crust and at least the upper part of the mantle lithosphere, even though earthquakes on these structures occur only in the upper 20 km. Taken together, the strain-localization and the lack of deep earthquakes suggest that these fault systems might be represented for the purpose of long-term continental deformation models as narrow ductile shear zones. A simple mechanical representation of localized strain on a ductile shear zone is defined by assuming traction is proportional to slip rate, with the proportionality constant described as a fault resistance coefficient. At the cost of ignoring the complexity of the earthquake cycle in such a model, we obtain a simple mechanical representation which we suggest is valid in the representation of long-term (and inter-seismic) continental deformation. In conceptual terms the fault resistance coefficient would be proportional to the effective viscosity of a ductile shear zone and inversely proportional to its width. However, an effective numerical implementation in a two-dimensional finite-element model is obtained by collapsing the narrow ductile shear zone to a one-dimensional structure characterised locally by the fault resistance coefficient. We illustrate the application of this conceptual model to the deformation field around the Alpine Fault in New Zealand, as constrained by an extensive array of GNSS displacement rates. The region as a whole is represented by a thin viscous shell that obeys a non-Newtonian viscous constitutive law, but we enable slip on model faults where there are steep local gradients in the geodetic displacement rates. The magnitude of the fault resistance coefficient is constrained by the requirement to fit the displacement rates and balance the stress.