Self-Stabilized Fractality of Seacoasts through Damped Erosion

Erosion of rocky coasts spontaneously creates irregular seashores. But the geometrical irregularity, in turn, damps the sea waves, decreasing the average wave amplitude. There may then exist a mutual self-stabilization of the wave amplitude together with the irregular morphology of the coast. A simple model of such stabilization is studied. It leads, through a complex dynamics of the earth-sea interface, to the appearance of a stationary fractal seacoast with a dimension close to $4/3$. Fractal geometry here plays the role of a morphological attractor directly related to percolation geometry.

Figure 1

Time evolution of the coastline morphology. (Left) beginning of the erosion process: the coast is irregular but not yet fractal. (Right) coastline at the end of the rapid erosion process.

Figure 2

Illustration of the erosion process. The thick numbers in the square centers represent the lithology $\{x_i\}$. The numbers in the corners are the corresponding resistances $r_i$ which depend on the local environment as explained in the text. The sites marked with 1 are earth sites with no contact with the sea. Left and right pictures represent, respectively, the situations before and after an erosion step with $f(t)=0.5$. After this step resistances are updated due to the new sea environment.

Figure 3

Time dependence of the erosion force $f(t)$. The upper plot shows the evolution of the sea-erosion force acting on the coastline during a rapid sea-erosion process [different values of the scale gradients $g$ and of $f(0)$]. This dynamics spontaneously stops at a value weakly dependent on $g$ (systems with $L_0=1000$, averaged over ten different realizations). The lower plot is the erosion force $f(t)$ during the complete dynamics (slow weathering process triggering rapid erosion).

Figure 4

(Left) box-counting determination of the coast fractal dimension. The straight line is a power law with a slope $-4/3$. The best fit gives $D_f=1.332(3)$. The data refer to a large system with $L_0={10}^4$, and a small gradient $g={10}^{-4}$. (Right) scaling behavior of the coast width $\sigma$. The straight line is a power law fit with the GP exponent $-4/7$ (each point is an average of over 400 samples with $L_0=5000$).

Figure 5

Snapshots taken during the long term erosion dynamics for a small system ($L_0=3000$) with a moderate gradient $g=0.002$. Note that the measured fractal dimension fluctuates around the universal value $4/3$ corresponding to a very small gradient [see Fig. 4 (left panel)]. The color codes for successive arrest times.