**Existence of Continuum Complexity in the Elastodynamics**
**of Repeated Fault Ruptures**

Bruce E. Shaw and James R. Rice

**Abstract:**

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

resolution.

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.