Examples of Natural Power Laws and Fractal Sets

River basins have a fractal nature.

Hack's law states that L=1.4 A^(D/2), where L is the length of the longest stream in miles, A the surface area of the drainage region in square miles and D is the fractal dimension, which in this case is D=1.2. Empirical power laws also exist that relate channel width Wc and channel depth H (both in meters) such that Wc=6.8 H^(1.54). The channel width is itself related to the meander width Wm by the expression Wm=7.44 Wc^(1.01) (Wm and Wc are in feet). The shape of a whole river system is also fractal. For instance, the fractal dimension for the Amazon river is 1.85 while the Nile's is estimated as 1.4. This result suggests that the fractal dimension of a river increases with the amount of rainfall.

The relationship between the size of an earthquake and its frequency of occurence obeys fractal statistics. It is well known that very large earthquakes are rare (events the size of the 1960 Chilean earthquake are estimated to occur once every 200 years or so) and very small earthquakes are very frequent (seismically active regions can register thousancs of small earthquakes per day). What is remarkable is that there is a power law relationship between how many large and how many small earthquakes th ere must be in a given region per unit time. For instance, every year and globally, there is on average just one earthquake of magnitude eight, ten magnitude seven, one hundred magnitude six, one thousand magnitude five, and so on.

The power law in this case is N(.m) = a A^-b, where N(.m) is the total number of earthquakes per unit time in a give region with magnitude m or greater, and A is the amplitude of ground motion, and m ~ logA. Taking logarithms the power law becomes log N (>m) = loga - bm, an empirical formula known as the Richter-Gutenberg law of earthquake statistics. Globally the value of b (usually called the "b-value") is observed to be around 1.0, the constant a is about 1x10^8 year^-1 (it represents the number of m agnitude 1 earthquakes in a year) and the fractal dimension can be shown to be related to the b value such that D=2b, so that D=2. This implies that if in the Earth one earthquake of magnitude 8 occurs in a year, then there will be about ten magnitude 7, one-hundred magnitude 6, one-thousand magnitude 6 etc., in the same year, all happening in a remarkably orderly fashion. Yet earthquakes are notoriously difficult to predict becuase the underlyhing mechanics that produces them is likely to be chaotic. Another interesting power law statistic about earthquakes is that the temporal frequency (e.g., number of events per unit time) of aftershocks following a major shock decays following a power law known as Omori's law. The law specifies that if N(t) denotes the number of aftershocks a time t after the main event, then N(t) ~ t^-a, where a varies between 1 and 1.5.