Existence of continuum complexity in the elastodynamics of repeated fault ruptures

What are the origins of earthquake complexity? The possibility that some aspects of the complexity displayed by earthquakes might be explained by stress heterogeneities developed through the self-organization of repeated ruptures has been suggested by some simple self-organizing models. The question of whether or not even these simple self-organizing models require at least some degree of material heterogeneity to maintain complex sequences of events has been the subject of some controversy. In one class of elastodynamic models, previous work has described complexity as arising on a model fault with completely uniform material properties. Questions'were raised, however, regarding the role of discreteness, the relevance of the nucleation mechanism, and special parameter choices, in generating the complexity that has been reported. In this paper, we examine the question of whether or not continuum complexity is achieved under the stringent conditions of continuous loading, and whether the results are similar to previously claimed findings of continuum complexity or its absence. The elastodynamic model that we use consists of a 1-D fault boundary with friction, a steady slowly moving 1-D boundary parallel to the fault, and a 2-D scalar elastic media connecting the two boundaries. The constitutive law used involves a pair of sequential weakening processes, one occurring over a small slip (or velocity) and accomplishing a small fraction of the total strength drop, and the other at larger slip (or velocity) and providing the remaining strength drop. The large-scale process is motivated by a heat weakening instability. Our main results are as follows. (1) We generally find complexity of type I, a broad distribution of large event sizes with nonperiodic recurrence, when the modeled region is very long, along strike, compared to the layer thickness. (2) We find that complexity of type II, with numerous small events showing a power law distribution, is not a generic result but does definitely exist in a restricted range of parameter space. For that, in the slip weakening version of our model, the strength drop and nucleation size in the small slip process must be much smaller than in the large slip process, and the nucleation length associated with the latter must be comparable to layer thickness. This suggests a basis for reconciling different previously reported results. (3) Bulk dispersion appears to be relatively unimportant o the results. In particular, motions on the fault plane are seen to be relatively insensitive to a wide range of changes in the dispersion in the bulk away from the fault, both at long wavelengths and at short wavelengths. In contrast, the fault properties are seen to be very important to the results. (4) Nucleation from slip weakening and time-dependent weakening showed similar large-scale behavior. However, not all constitutive laws are insensitive to all nucleation approximations; those making a model "inherently discrete" and hence grid-dependent, in particular, can affect large scales. (5) While inherent discreteness has been seen to be a source of power law small-event complexity in some fault models, it does not appear to be the cause of the complexity in the attractors examined here, and reported in earlier work, fortuitously in the special parameter ange, with the same class of continuum fault models and same or very similar constitutive relations. Continuum homogeneous dynamic complexity does indeed exist, although that includes type II small-event complexity only under restricted circumstances.