Existence of Continuum Complexity in the Elastodynamics
of Repeated Fault Ruptures
Bruce E. Shaw  and  James R. Rice


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.
We set for ourselves the most stringent conditions to address,
definitively, questions of nucleation and discreteness in obtaining
complexity: 1) that there be stability at the smallest scales, 2) that
it be done in at least two spatial dimensions, 3) that there be a finite
loading rate 4) with, during nucleation, stable sliding occurring at
length scales below a critical stiffness, 5) and dynamic break-out
occurring above that critical stiffness, 6) with grid resolution of the
critical stiffness scale and 7) independence of the results on grid
The model we use consists of a one dimensional fault boundary with friction,
a steady slowly moving one-dimensional boundary parallel to the fault, and
a two dimensional scalar elastic media connecting the two boundaries.
Features of complexity of interest are (I) a broad distribution of
event sizes with nonperiodic features and (II) a power-law frequency-size
distribution of Gutenberg-Richter type over some range of small events.
Using a friction which either weakens with slip, gradually restrengthening
with time, or weakens with velocity and
strengthens with gradients of velocity,
we meet all the criteria listed above.
The constitutive law used involves a pair of weakening processes, one
occurring over a small slip (or velocity) and accomplishing a small
fraction of the total strength drop, the other at larger slip (or velocity)
and providing the remaining strength drop.
Our main results are:
i)  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.
 ii) We find complexity of type (II), with numerous small events showing a
power law distribution,
only in a
restricted range of parameter space.
The restricted range occurs for parameter values where two conditions are met.
First, the large scale weakening process produces weakening at a rate which
is comparable to the stress drops associated with the sliding with the
stiffness of a seismogenic thickness.  For slip-weakening, this corresponds to a
large scale nucleation size comparable to the layer thickness, while for
velocity-weakening, this corresponds to a weakening at a large scale velocity comparable
to the radiation damping velocity.  A second needed condition
is the existence of a small initial drop in friction in the small
weakening process, going from
sticking to sliding.
The fact that we see numerous small events showing a power-law distribution
of sizes only over a restricted range of parameter space
suggests a basis for reconciling
different previously reported results.
iii) Bulk dispersion appears to be relatively unimportant to 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 off of the fault plane, both at long wavelengths and at short
wavelengths.  In contrast,
the fault properties are seen to be very important to the results.
iv) For events above the critical stiffness scale, the distribution of
sizes of events is the same for events nucleated
from finite loading as for events nucleated with a
time dependent drop; a number of other variations on the nucleation
process also give similar results using the class of constitutive
relations we examine, at least in the two dimensional geometries
studied here.  We caution, however, that not all approximations of
the nucleation process for all constitutive relations show the same
insensitivity; other work with other constitutive relations has
identified simplifications of the nucleation process, rendering the
system ``inherently discrete'' and grid size sensitive, which do
strongly affect the size distribution for large events.
v)  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 with this class of constitutive relations.
Continuum homogeneous dynamic complexity does
indeed exist.  Again, however, small event complexity exists, in
uniform continuum fault models, only under restricted circumstances.