Examples of Natural Power Laws and Fractal Sets
Rivers, channels, and drainage basins
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.
Size-frequency distribution of earthquakes
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.