Self-organization, the cascade model, and natural hazards

Abstract

We consider the frequency-size statistics of two natural hazards, forest fires and landslides. Both appear to satisfy power-law (fractal) distributions to a good approximation under a wide variety of conditions. Two simple cellular-automata models have been proposed as analogs for this observed behavior, the forest fire model for forest fires and the sand pile model for landslides. The behavior of these models can be understood in terms of a self-similar inverse cascade. For the forest fire model the cascade consists of the coalescence of clusters of trees; for the sand pile model the cascade consists of the coalescence of metastable regions.A number of complex natural phenomena exhibit power-law frequency-area statistics in a very robust manner. The classic example is earthquakes. For over 50 years, it has been accepted that earthquakes universally obey Gutenberg-Richter scaling; the number of earthquakes in a region with magnitudes greater than m, N CE , is related to m by (1)where b E and C are constants; b E is known as the b value and has a near-universal value b E ϭ 0.90 Ϯ 0.15. The constant C is a measure of the intensity of the regional seismicity. When Eq. 1 is expressed in terms of the earthquake rupture area A E instead of earthquake magnitude, this relation becomes a power law (2)Earthquakes satisfy fractal (power-law) scaling in a very robust manner, despite their complexity. Earthquakes occur in hot regions and cold regions, with compressional, shear, and tensional stresses, yet they universally satisfy the power-law scaling given in Eq. 2. In this article we will focus our attention on two other natural phenomena, forest fires and landslides. We will show that both also exhibit power-law frequency-area statistics in a robust manner.A fundamental question is: Why do earthquakes, forest fires, and landslides exhibit a power-law frequency-area dependence? In each case a simple cellular automata model has been proposed to explain the observed behavior. For earthquakes it is the slider-block model. The standard slider-block model (3, 4) considers an array of slider blocks each with a mass m. The blocks are attached to a constant velocity driver plate with puller springs, spring constant k p , and to each other with connector springs, spring constant k c . The blocks interact nonlinearly with the surface over which they are dragged through friction.If the sliding (dynamic) friction coefficient f d is less than the static friction coefficient f s , stick-slip behavior is observed. The area of a slip event is the number of blocks involved in the event. Under a wide variety (but not all) conditions, the frequency-area statistics of the slip events are well approximated by the powerlaw relationwhere N E is the number of events with area A E , the number of blocks slipping in an event. The exponent a E is generally found to be a E ϳ 1.0. For a noncumulative power-law distribution (Eq. 3) with a ϭ 1, the corresponding cumulative distribution obtained by summing or integrating will be logarithmic. For a noncum...

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