Fractality of eroded coastlines of correlated landscapes

Using numerical simulations of a simple sea-coast mechanical erosion model, we investigate the effect of spatial long-range correlations in the lithology of coastal landscapes on the fractal behavior of the corresponding coastlines. In the model, the resistance of a coast section to erosion depends on the local lithology configuration as well as on the number of neighboring sea sides. For weak sea forces, the sea is trapped by the coastline and the eroding process stops after some time. For strong sea forces erosion is perpetual. The transition between these two regimes takes place at a critical sea force, characterized by a fractal coastline front. For uncorrelated landscapes, we obtain, at the critical value, a fractal dimension $D=1.33$, which is consistent with the dimension of the accessible external perimeter of the spanning cluster in two-dimensional percolation. For sea forces above the critical value, our results indicate that the coastline is self-affine and belongs to the Kardar-Parisi-Zhang universality class. In the case of landscapes generated with power-law spatial long-range correlations, the coastline fractal dimension changes continuously with the Hurst exponent $H$, decreasing from $D=1.34$ to $1.04$, for $H=0$ and $1$, respectively. This nonuniversal behavior is compatible with the multitude of fractal dimensions found for real coastlines.

Figure 1

Snapshots of typical configurations obtained with the erosion model for uncorrelated lithology on a lattice with ${512}^2$ sites. The sea sites are in blue (dark gray) and all the other sites are land sites (light gray). Three different regimes are obtained: (a) a subcritical, weak sea force regime ($f<f_c$): the coastline is rough but not fractal; (b) a critical regime ($f=f_c$): the coastline is a self-similar fractal: and (c) a supercritical, strong sea force regime ($f>f_c$): the coastline is no longer self-similar, but rather self-affine.

Figure 2

Dependence of the sea-trapping probability $P_s$ on the erosion force $f$, for landscapes generated with a uniform distribution of the lithology parameter $\ell$. A transition is observed from a trapped, subcritical regime—for weak sea force—to a perpetual, supercritical invasion regime—for strong sea force. The transition occurs for $f_c=0.523\pm 0.001$, estimated from the crossing of the lines. Each curve corresponds to a different system size $L^2$, with $L=\{128,256,512,1024\}$, and results have been averaged over $\{800,400,200,100\}$ samples, respectively.

Figure 3

Size dependence of the mass of the coastline (number of sites) at the critical sea force, $f=f_c$. A fractal dimension $D=1.34\pm 0.01$ is obtained from the best fit of the data points. Square lattices, with uniform distribution of the lithology parameter, of size $L^2$ have been considered with $L=\{32,64,128,256,512,1024\}$. Results have been averaged over $\{3200,1600,800,400,200,100\}$ independent realizations. The inset shows, for the yardstick method, the number of sticks needed to cover the line as a function of the inverse of the stick size, for a system with $L=1024$, averaged over ${10}^2$ samples. From the fit, a fractal dimension $D=1.33\pm 0.01$ is obtained. From both methods the fractal dimension of the coastline in the critical regime is estimated to be $D=1.33\pm 0.01$.

Figure 4

Finite-size scaling of the time dependence of the coastline roughness in the supercritical regime ($f=0.9$). Square lattices with $L_x\times L_y$ sites have been considered, with $L_x=L$ and $L_y=128L$. The time $t$ has been rescaled by $L^z$, where $z$ is the dynamic exponent, and the roughness has been rescaled in units of $L^{\alpha }$, where $\alpha$ is the roughness exponent. The best data collapse is obtained for $\alpha =0.48\pm 0.04$ and $z=1.55\pm 0.11$, values that are consistent with the Kardar-Parisi-Zhang universality class [32]. The straight line has slope $\beta$, corresponding to the growth exponent. The inset shows the time dependence of the coastline roughness. Systems with $L=\{32,64,128,256\}$ have been sampled over $\{3200,1600,800,400\}$ independent runs, respectively.

Figure 5

Size dependence of the growth exponent, $\beta$. These results indicate that, in the thermodynamic limit, $\beta$ converges to $\beta =1/3$ (dashed line) as in the Kardar-Parisi-Zhang universality class. Systems with $L=\{32,64,128,256\}$ have been sampled over $\{3200,1600,800,400\}$ independent runs, respectively.

Figure 6

Snapshots of the initial distribution of the lithology parameter (a, b, and c) and the corresponding eroded system (d, e, and f) at the critical force ($f=f_c$), for three landscapes generated with long-range spatial correlations, for different Hurst exponent: (a) and (d) $H=0.0$; (b) and (e) $H=0.2$; and (c) and (f) $H=0.8$. The color scheme of coast sites represents the value of the lithology parameter $l$ and sea sites were assigned with $l=0$. Pictures have been obtained for square lattices with ${512}^2$ sites.

Figure 7

Critical coastline fractal dimension $D$ as a function of the Hurst exponent $H$. A continuous decrease from $D=1.34$ at $H=0$ to $D=1.04$ at $H=1.0$ is observed. The dashed line stands for the fractal dimension of the uncorrelated case ($D=1.33$). Each point is obtained from the size dependence of the number of coastline sites for system sizes $L=\{32,64,128,256,512,1024\}$ averaged over $\{3200,1600,800,400,200,100\}$ samples.