Sandy coasts have a smooth profile while rocky coasts have a fractal nature. One characteristic feature of a rocky coast is that new details appear when we zoom in on it. And if we were to measure the length of the coast, the length would increase significantly when zooming in on the details. If we model this coast as a curve, then this curve would have an infinite length. One summarizes some characteristics of the coast through a number, the dimension, which describes the “complexity” of the curve. A smooth curve has a dimension of 1, while a surface has a dimension of 2. A dimension between 1 and 2 is typical of a self-similar object which is thicker than a curve but with empty interior. How does the dimension of a fractal coast depend on the coast? An article of Sapoval, Baldassarru and Gabrielli (Physical Review Letters 2004) presents a model suggesting that this dimension is independent of the coast and very close to 4/3.
The model of of Sapoval, Baldassarru and Gabrielli describes the evolution of the coast from a straight line to a fractal coast through two processes with two different time scales: a fast time and a slow time. The mechanical erosion occurs rapidly, while the chemical weakening of the rocks occurs slowly. The force of the waves acting on the coast depends on the length of the coast. Hence, the waves have a stronger destructive power when the coast is linear and a damping effect takes place when the coast is fractal. The erosion model is a kind of percolation model with the resisting Earth modeled by a square lattice. The lithology of each cell, i.e., the resistance of its rocks, is represented by a number between 0 and 1. The resistance to erosion of a site, also given by a number between 0 and 1, depends both on its lithology and on the number of sides exposed to the sea. Then an iterative process starts: each site with resistance number below a threshold disappears, and the resistances of the remaining sites are updated because new sides become exposed to the sea. This leads to the creation of islands and bays, thus increasing the perimeter of the coast. When the perimeter is sufficiently large, thus weakening the strength of the waves, the rapid dynamics stops, even if the power of the waves is nonzero! During this period the dimension of the coast is very close to 4/3.
This is when the slow dynamics takes over since chemical weakening of the remaining sites continues, thus reducing the resistance to erosion of sites. The slow dynamics is interrupted by short episodes of fast erosion when the resistance number of a site falls below the threshold. This dynamics of alternating short episodes of fast erosion and long episodes of chemical weakening is exactly what we observe now, since the initial fast dynamics occurred long ago.
The model presented here is a model of percolation gradient, with the sea percolating in the Earth, and such models of percolation gradient exhibit universality properties.