GEOPHYSICAL RESEARCH LETTERS, VOL. 9, NO. 7, PAGES 735-738, JULY 1982 CRACK FUSION DYNAMICS: A MODEL FOR LARGE EARTHQUAKES William I. Newman Department of Earth and Space Sciences, and Leon Knopoff Department of Physics and Institute of Geophysics and Planetary Physics University of California, Los Angeles, California 90024 Abstract. The physical processes of the renormalize the constitutive equations describing fusion of small cracks into larger ones are crack fusion events. A relevant example of how nonlinear in character. A study of the nonlinear scale invariance has led to a dramatic properties of fusion may lead to an understanding simplification of the description of a complex of the instabilities that give rise to clustering physical phenomenon is the application to of large earthquakes. We have investigated the hydrodynamic turbulence. There, in particular, properties of simple versions of fusion processes it was possible to obtain a description of the to see if instabilities culminating in repetitive onset of nonlinear fluid behavior and the cascade massive earthquakes are possible. We have taken of energy through a sequence of events wherein into account such diverse phenomena as the eddies of smaller dimensions were formed. We production of aftershocks, the rapid extension of contend that the application of similar large cracks to overwhelm and absorb smaller techniques to the dynamics of crack fusion can cracks, the influence of anelastic creep-induced provide a physical description (in contrast with time delays, healing, the genesis of "juvenile" a statistical one) of fusion events and the cracks due to plate motions, and others. A preliminary conclusion is that the time delays introduced by anelastic creep may be responsible for producing catastrophic instabilities characteristic of large earthquakes as well as aftershock sequences. However, it seems that nonlocal influences, i.e. the spatial diffusion of cracks, may play a dominant role in producing episodes of seismicity and clustering. Introduction Observations of seismic gaps and clustering phenomena are indications that more traditional models of earthquake occurrence, such as that of a simple Poisson random process, are inappropriate. There is an indication from laboratory experiments on fracturing that large- scale fractures are the end result of a sequential process of fusion of smaller cracks into larger ones. We study how the fusion of small cracks into larger ones can produce a catastrophic cascade of events which culminates in a massive earthquake as its endpoint. We include in our model such diverse phenomena as instabilities that give rise to such phenomena as aftershock sequences, clustering, episodes of seismicity, and seismic gaps. If we assume that we know the rates of crack fusion, healing, and other processes that may be involved, it is possible to describe the evolution of cracks by an integro-differential equation for the probability of finding a crack with a given length and orientation that is centered at a particular location. The spatial dependence of this formulation of crack fusion dynamics assumes a form commonly encountered in problems of nonlinear diffusion. This is especially significant since nonlinear diffusion processes combined with nonlinear sources and sinks can give rise to wave-like behavior that is spatially confined. [See, for example, Newman's (1981) treatment of certain nonlinear diffusion problems encountered in population genetics and combustion.] In combination with spatial inhomogeneities, nonlinear diffusion may have an important role in producing the properties of seismicity listed above that we think are evidence for instability, Nonlinear diffusion imparts a multidimensional character to the the production of aftershocks, the rapid problem. extension of large cracks to overwhelm and absorb smaller cracks, the influence of anelastic creep- induced time delays, and others. The processes wherein cracks of different sizes can interact are essentially independent of size, so long as the cracks considered are larger than the dimensions characteristic of individual grains or crystals (i.e. a few millimeters) and smaller than the distances between the triple junctions of the major plates. This property of cracks is a profound feature of rock physics and in this paper, we consider a preliminary version of the problem in order to expose the instabilities that result from the introduction of nonlinearities into a system describing cracks. To do this in the simplest way, we eliminate the spatial dependence of the crack distribution by reconstituting our dynamical equations in terms of the populations of cracks of different sizes, at a fixed, isolated region on a fault, for example. The integral equation character of these expressions can be simplified is manifest in the scale invariance of cracksize by ca•egorizing crack lengths into different distributions in rocks and in the spectrum or "bins, partitioned according to the logarithm of time signature of crack fusion events ranging their length. In that way, our "renormalization" from microcrack formation to earthquake exploits the scale invariance of crack size aftershock sequences (Kagan and Knopoff, 1978 and distributions. It is the formation of these 1980). What is particularly significant about cracks or their abrupt lengthening due to fusion the observed scale invariance of earth materials that we envision as being descriptive of an is that it may be possible, by employing an earthquake event. A particularly simple approximate description of the rock physics, to illustration of this approach, characterized by only two crack size categories is described Copyright 1982 by the American Geophysical Union. below. In this context, we will see that aperiodic earthquake sequences are a (possible) Paper number2L0801. direct consequence of time delays introduced by 0094-8276/82/002L-080153.00 anelastic creep and stress weakening. 735