Percolation framework of the Earth's topography

Self-similarity and long-range correlations are the remarkable features of the Earth's surface topography. Here we develop an approach based on percolation theory to study the geometrical features of Earth. Our analysis is based on high-resolution, 1 arc min, ETOPO1 global relief records. We find some evidence for abrupt transitions that occurred during the evolution of the Earth's relief network, indicative of a continental/cluster aggregation. We apply finite-size-scaling analysis based on a coarse-graining procedure to show that the observed transition is most likely discontinuous. Furthermore, we study the percolation on two-dimensional fractional Brownian motion surfaces with Hurst exponent $H$ as a model of long-range correlated topography, which suggests that the long-range correlations may play a key role in the observed discontinuity on Earth. Our framework presented here provides a theoretical model to better understand the geometrical phase transition on Earth, and it also identifies the critical nodes that will be more exposed to global climate change in the Earth's relief network.

Figure 1

(a) The largest landmass cluster relative size $s$ vs the fraction total landmass $p$ for real (green square) and shuffled (red circle) Earth relief records. $g_1$–$g_5$ indicate the five largest gaps, defined in Eq. (2). (b) Snapshots of the landmass clusters of the Earth's surface topography network just before the percolation threshold (the largest jump at $p\approx 0.321$). Different colors represent different clusters; the grid resolution is 1 arc min; the star indicates the critical node. Only the clusters with relative size larger than 0.01 are shown.

Figure 2

Similar to Fig. 1 but for the oceanic clusters. (a) The largest oceanic cluster relative size $s$ vs $p$. (b) Snapshots of the oceanic clusters structure of the Earth's relief network just before the percolation threshold (the largest jump at $p\approx 0.379$). The star indicates the critical node.

Figure 3

Finite-size effects of the percolation in Earth's relief network. (a) Log-log plot of the largest gap $g_1$ vs the network system size $L$ for original real data (red square) and shuffled data (blue circle). (b) Log-log plot of the largest landmass cluster relative size $s_c$ at the percolation threshold vs $L$ for real data (red square) and shuffled data (blue circle). For the real data, the slope seems to approach zero, suggesting a discontinuous phase transition; for the shuffled data, the slope approaches $-0.10$, which suggests a continuous phase transition with a known critical exponent $\beta /\nu =5/48$ and $d-d_f=5/48$. The dashed lines are the best-fit lines with $R^2>0.98$. The shaded regions correspond to error bars, which are calculated by the standard deviation.

Figure 4

The largest landmass cluster relative size $s$ vs the fraction number of nodes or sites, $p$, for real, shuffled, and 2D fractional Brownian motion (fBm) surfaces with different Hurst exponents. The two dashed lines indicate the sea level ($h=0$ m) and the well-known site percolation threshold $p_c=0.5927$, respectively.

Figure 5

The percolation on 2D fractional Brownian motion surfaces. (a) The average of the largest jump $g_1(H,L)$ as a function of system size $L$; (b) the corresponding percolation threshold $p_c(H,L)$ as a function of $L$. The dashed line in (b) stands for the percolation threshold $p_c=0.321$ for real data. Here, $H=0.66$ is the Hurst exponent governing the correlations over the continental topography, following [2]. The shaded regions correspond to error bars, estimated by the standard deviation.

Figure 6

Snapshots of the landmass clusters structure of the Earth's relief network just before the percolation threshold at (a) the second largest jump with $p\approx 0.285$; (b) the third largest jump with $p\approx 0.578$; (c) the fourth largest jump with $p\approx 0.450$; and (d) the fifth largest jump with $p\approx 0.712$. The red represents the largest landmass cluster, and the stars indicate the critical nodes.