Percolation Description of the Global Topography of Earth and the Moon

Abbas Ali Saberi

Phys. Rev. Lett. 110, 178501 – Published 24 April 2013

Abstract

Remarkable global correlations exist between geometrical features of terrestrial surfaces on Earth, current mean sea level, and its geological internal processes whose origins have remained an essential goal in the earth sciences. Theoretical modeling of the ubiquitous self-similar fractal patterns observed on Earth and their underlying rules is indeed of great importance. Here I present a percolation description of the global topography of Earth in which the present mean sea level is automatically singled out as a critical level in the model. This finding elucidates the origins of the appearance of scale invariant patterns on Earth. The criticality is shown to be accompanied by a continental aggregation, unraveling an important correlation between the water and long-range topographic evolutions. To have a comparison point in hand, I apply such an analysis to the lunar topography which reveals various characteristic features of the Moon.

Received 13 August 2012

DOI:https://doi.org/10.1103/PhysRevLett.110.178501

Authors & Affiliations

Abbas Ali Saberi^*

Department of Physics, College of Science, University of Tehran, P.O. Box 14395-547, Tehran, Iran
Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany

^*ab.saberi@ut.ac.ir

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Issue

Vol. 110, Iss. 17 — 26 April 2013

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Images

Figure 1
Schematic illustration of the continental aggregation by decreasing the sea level from top to bottom. The first four snapshots are selected around the remarkable percolation transition at the present mean sea level around which the major parts of the landmass join together. This is followed by three other levels indicating the junction of Greenland, Australia, and Antarctica to the giant landmass. The lowest right figure shows the longitudinal percolation of the oceanic clusters (see the Supplemental Material 37 for enlarged figures and also for the ones at additional sea levels).Reuse & Permissions
Figure 2
The total length of the coastlines as a function of the sea level (or height) for Earth (main panel) and the Moon (inset).Reuse & Permissions
Figure 3
Relative surface area of the largest island (circles) and the largest sea (triangles) followed by the total surface area of the islands and the oceans (solid lines) to the total area $4 π$ of Earth, as a function of the sea level. One critical level is distinguished by each order parameter. The oceanic critical level is close to the level $h = - 3287 m$ at which the total island and oceanic surface areas are equal.Reuse & Permissions
Figure 4
Correlation length and mean island (land) size vs the sea level, with a remarkable characterization of a geometrical phase transition at the present mean sea level (zero hight level). The scale factors by which all the numbers labeling the vertical axis should be multiplied to get the correct graph units are the radius and the square of the radius of Earth for $ξ$ and $χ$ , respectively.Reuse & Permissions
Figure 5
Relative surface area of the largest island followed by the total surface area of the islands to the total area $4 π$ of the Moon, as a function of the hypothetical sea level. At level $\sim - 1049 m$ , the total island and oceanic surface areas on the Moon are equal. The two jumps in the order parameter at levels around $- 960$ and 1360 m are also decoded in the divergent behavior of the correlation length and the mean island size (the inset). Such two jumps are unusual for percolation.Reuse & Permissions
Figure 6
Upper: Disjoint islands with heights higher than the hypothetical sea level on the Moon for two different levels $h = - 950$ and $- 1050 m$ , slightly above and below the critical level, respectively. Appearance of the spanning island (right figure) is indicative of a percolation transition. Lower: Aggregation of the two main mountainous islands at the lunar farside around the second critical level at $\sim 1360 m$ .Reuse & Permissions

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