Existence of continuum complexity in the elastodynamics of repeated fault ruptures

What are the origins of earthquake complexity? The possibility that some as pects of the complexity displayed by earthquakes might be explained by stre ss heterogeneities developed through the self-organization of repeated rupt ures has been suggested by some simple self-organizing models. The question of whether or not even these: simple self-organizing models require at lea st some degree of material heterogeneity to maintain complex sequences of e vents has been the subject of some controversy. In one class of elastodynam ic models, previous work has described complexity as arising on a model fau lt with completely uniform material properties. Questions were raised, howe ver, regarding the role of discreteness; the relevance of the nucleation me chanism, and special parameter choices, in generating the complexity that h as been reported. In this paper, we examine the question of whether or not continuum complexity is achieved under the stringent conditions of continuo us loading, and whether the results are similar to previously claimed findi ngs 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 connect ing the two boundaries. The constitutive law used involves a pair of sequen tial weakening processes, one occurring over a small slip (or velocity) and accomplishing a small fraction of the total strength drop, and the other a t 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 mo deled 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 v ersion of our model, the strength drop and nucleation size in the small sli p process must be much smaller than in the large slip process, and the nucl eation length associated with the latter must be comparable to layer thickn ess. This suggests a basis for reconciling different previously reported re sults. (3) Bulk dispersion appears to be relatively unimportant to the resu lts. In particular, motions an the fault plane are seen to be relatively in sensitive to a wide range of changes in the dispersion in the bulk away fro m 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) Nu cleation from slip weakening and time-dependent weakening showed similar la rge-scale behavior. However, not all constitutive laws are insensitive to a ll nucleation approximations; those making a model "inherently discrete" an d hence grid-dependent, in particular, can affect large scales. (5) While i nherent 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, f ortuitously in the special parameter range, with the same class of continuu m fault models and same or very similar constitutive relations. Continuum h omogeneous dynamic complexity does indeed exist, although that includes typ e II small-event complexity only under restricted circumstances.