The Olami-Feder-Christensen Model of Earthquakes

The sandpile model of Bak, Tang, and Wiesenfeld displays self-organised criticality due to an inherent conservation law of the dynamical variable. However, there exists also slowly driven models which violate the conservation law but nevertheless display the characteristic features of SOC: after a transient, the size distribution of events is a power law with a cutoff that diverges when the system size is increased. The Olami-Feder-Christensen (OFC) model of a earthquake fault serve as an example of such a nonconservative model.

The convective flow in the asthenosphere causes tectonic plates to move relatively to one another causing strain to build up at the boundary of the plates. The lithosphere is so rigid that the strain is released very suddenly through earthquakes. Thus the worldwide occurrence of earthquakes outline the plate boundaries.


Figure 1: A simple model of the San Andreas fault. The single line is a symbol for a transform fault. The (half) arrows show the direction of the relative movement. The tectonic plates move with a relative velocity of about 4 cm/year. Strain builds up along the boundary of the plates. The strain is released through an earthquake when the friction cannot sustain the strain any more.


Figure 2: The number of earthquakes N(E) with energy release larger than E per year (open circles). The red dashed line is the Gutenberg-Richter law Log10N(E) = -B Log10(E), B = 0.95. The deficit at small energy, is related to the problems with detecting small earthquakes. The data approximates a power law over 5.5 orders of magnitude in energy E and N(E). The data are from California during the period 1984 - 2000.

In the model, the fault is represented by a two-dimensional network of blocks interconnected by springs. Additionally, each block is connected to a single rigid driving plate by another set of springs as well as connected frictionally to a fixed rigid plate.


Figure 3: The geometry of the spring-block model. The force on the blocks increases uniformly as a response to the relative movement of the tectonic plates. Strain builts up and is released through an earthquake when the friction cannot sustain the strain any more.

The blocks are driven by the relative movement of the two rigid plates. When the force on one of the blocks is larger than some threshold value Fth (the maximal static friction), the block slips. We assume that the moving block slips to the zero force position. Slip of one block will redefine the forces on its nearest neighbours. This may lead to instabilities of the neighbouring blocks and thus, as a result, in further slips and a chain reaction (earthquake) can evolve. The total number of slips following a single initial slip event is a measure of the size (seismic moment) of the earthquake.


Figure 4: The probability P(E) of energy release E during an earthquakes exhibits a power-law behaviour with a cutoff that increases with system size L. Hence the model is critical despite the inherent nonconservative nature and reproduces the observed Gutenberg Richter law for the frequency of earthqukes versus size.

The origin of SOC in nonconservative models is conceptually different from the origin of SOC in conservative models. The main objective in future research is to understand the necessary conditions under which critical scale invariant behaviour occur spontaneously in weakly driven many-body systems.

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From Yukon Ho!, a Calvin and Hobbes Collection by Bill Watterson, 1989.
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