JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 94, NO. B 1l, PAGES 15,635-15,637, NOVEMBER 10, 1989 Earthquakes as a Self-Organized Critical Phenomenon PER BAK AND CHAO TANG Brookhaven National Laboratory, Upton, New York The Gutenberg-Richter power law distributionfor energy releasedat earthquakescan be understood as a consequence of the earth crust being in a self-organized critical state. A simple cellular automaton stick-slip type model yields D(E) • E -• with r = 1.0 and r = 1.35 in two and three dimensions, respectively. The size of earthquakes is unpredictable since the evolution of an earthquakedepends crucially on minor details of the crust. INTRODUCTION The distributionof energyreleased duringearthquakes has been found to obey the famous Gutenberg-Richter law [Gutenberg and Richter, 1956]. The law is based on the empirical observation that the number N of earthquakesof size greater than rn is given by the relation model must necessarily be grosslysimplified.The immediate goal is not to produce an accurate model but to point out a general mechanism leading to the power law distribution of earthquakes. In the following section an effort will be made to connect the concept of self-organizedcriticality to earth- quakes. log10 N = a -bm (1) The precise values of a and b depend on the location, but generally b is in the interval 0.8 < b < 1.5. The energy releasedduring the earthquakeis believed to increaseexpo- nentially with the size of the earthquake, loglo E = c -dm (2) so the Gutenberg-Richter law is essentially a power law connecting the frequency distribution function with the energy release E (or other physical quantities such as the "seismic moment") dN/dE o• m-• - b/d .__ m-" (3) with 1.25 < r < 1.5. Despite the universality of the Gutenberg-Richter relation, there is essentially no understanding of the underlying mech- anisms.It has been suggested that the power law is related to geometric features of the fault structure [Kagan and Knop- off, 1987], and indeed it has been pointed out by Mandelbrot [1982] that earthquakesoccur on "fractal" self-similar sets. But, of course, this just shiftsthe problem to identifying the dynamical mechanismproducingthese geometric structures which are ultimatelyresponsible for the earthquake dynam- ics. In fact, power laws (an,d the lack of understanding of those) are quite common in nature. Recently, we have shown that dynamical systems may self-organize into a critical state similar to that of systems undergoingcontinu- ous phase transitions,with power law spatial and temporal correlation functions. In the following section we show that this behavior can be related fairly directly to earthquakes. Thus the Gutenberg-Richter law can be interpreted as a manifestation of the self-organized critical behavior of the earth dynamics. The fractal geometric distribution and the earthquakedynamicsare the spatialand temporal signatures of the samephenomenon. Of course, any specificdynamical Copyright 1989 by the American GeophysicalUnion. Paper number 89JB01265. 0148-0227/89/89JB-01265 $05.00 8ELF-ORGANIZED CRITICALITY AND MODEL CALCULATIONS It is generally assumedthat the dynamics of earthquakes is due to a stick-slipmechanisminvolving slidingof the crust of the earth along faults [Stuart and Mavko, 1979; Sieh, 1978; Choi and Huberrnan, 1984]. When slip occurs at some location, the strain energy at that position is released, and the stress propagates to the near environment. While this picture is rather well established, no connection between stick-slip models and the actual spatial and temporal corre- lations has been demonstrated. It has been suggested that the stick-slip picture can be modeled as a branching process [Kagan and Knopoff, 1987]. The observedpower law behav- ior is then rather remarkable since one would naively expect some exponential distribution, e.g., D(E) • e-œ/œo, where E0 is roughly the energy released at a single slip. In simpledynamical systems with few degrees of freedom, and in extended equilibrium statistical systems,power laws are rare. One has to fine tune a parameter such as a dynamical coupling or temperature to arrive at a "critical point" in order to get power law correlations. But for dynamical systems in nature there is nobody to turn the knob, so where does the apparent criticality come from? We have found that certain interacting dynamical systems naturally evolve into a statistically stationary state, which is also critical, with power law spatial and temporal correla- tions [Bak et al., 1987, 1988; Tang and Bak, 1988a, b]. It is essentialthat the systemsare dissipative (energy is released) and that they are spatially extended with an "infinity" of degrees of freedom. Energy is fed into the system in a uniform way, either directly into the bulk or through the boundaries.The crust of the earth, subjectedto the pressure from tectonic plate motion, may be viewed as a system of this kind. At the stationary state there is a fragile balance between the local forces, adjusting the probability that a slip will propagateto a near neighborprecisely to unity. The proba- bility of branching of the activity is compensated by the probability of "death" of the activity. The stationary state can be thought of as a critical chain reaction. Visually, the critical state can be thought of as the state of a steep sand 15,635