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corresponding lengths here are expressed in units of the

Moon Earth’s average radius. 700 150 Earth Now imagine flooding this global landscape in a way

600 100 that the continental land masses were criss-crossed by a series of narrow channels so that the resulting sea 500 50 level all over the Earth would coincide with a spherical h=-1080

400 0 − -8000 -4000 0 4000 8000 surface the geoid. All parts above the water level are

300 then colored differently as disjoint islands, and the rest is left white (Fig. 1). If the water level is high, there will 200 be small disconnected islands, and if it is low, there will h=-4280 100 be disconnected lakes. However, there may be a critical total length of the coastlines h=-760 h=320 value of the sea level h = hc at which a percolation 0 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 transition takes place [28, 29]. height The percolation problem [29, 31] is an example of the simplest pure geometrical phase transitions with nontriv- ial critical behavior, and it is closely related to the surface FIG. 2: (Color online) The total length of the coastlines as a function of the sea level (or height) for the Earth (main panel) topography [32, 33]. At the critical point in two dimen- and the Moon (inset). sions, the percolation clusters are some fractal objects whose outer perimeter is described by a fractal dimension of 4/3. By considering the dynamic sea level (height) as a percolation parameter, I examine a possible descrip- erties of oceans, continents and continental margins [7]. tion of Earth’s topography by means of the percolation Such a difference is also evident in the well-known bi- theory. modal distribution of the Earth’s topography [23] that re- The first quantity of interest is the total length of the flects the topographic dichotomy of continents and ocean coastlines at varying altitude (or sea level) which is shown basins, a consequence of plate tectonic processes. in Fig. 2. In all figures, the error bars are of the same Plate tectonic theory provides a framework that ex- order as the symbol size [37]. Having looked at Fig. 2, plains most of the major surface topographic features of this quantity closely resembles the height distribution the Earth. It also accounts for the connection between function of the Earth and the Moon [30]. The one for the processes that facilitate heat loss and the forces that the Earth is characterized by the presence of two lev- drive plate motion. The distribution of the plate areas covering the Earth has been shown to be a power law with exponent ∼ 0.25 for all plates [24, 25]. A remark- able relationship that provides one of the cornerstones of 1 plate tectonics is that, to a very good approximation, the Earth depth of the ocean floor beneath the ridge crest increases h=-3287 with square root of the age of the ocean floor, at least for 0.8 ocean lithosphere younger than about 80 Myr ago [26].

It plays an important role in topographical changes and 0.6 Islands fundamentally affects long-term variations in global sea The largest island level that would assume a surface equal to the geoid. Here The largest sea Seas I present the results of a statistical analysis based on per- 0.4 colation theory that provides new insight into the better understanding of various interrelationships between the relative surface area 0.2 above mentioned issues and their origins. h=-3640 I use the topographic data available for the 0 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 global relief model of the Earth’s surface that in- height tegrates land topography and ocean bathymetry [27] (the data information is presented in http://www.ngdc.noaa.gov/mgg/global/global.html). FIG. 3: (Color online) Relative surface area of the largest Current mean sea level is assumed as a vertical datum island (circles) and the largest sea (triangles) followed by the of the height relief which means that the data considers total surface area of the islands and the oceans (solid lines) to the imperfect ellipsoidal shape of the Earth. The height the total area 4π of the Earth, as a function of the sea level. relief h(r, θ, φ) is therefore assumed on a sphere of One critical level is distinguished by each order parameter. unit radius r = 1, that also coincides with the present The oceanic critical level is close to the level h = −3287 m at mean sea level (as zero height level) on the Earth. All which the total island and oceanic surface areas are equal. 3 els centered around the elevations 320 m and −4280 m in the continental platforms and oceanic floors, respec- 1 tively. The ratio of the total length of the coastlines at Earth 320 m to the zero height level is ∼ 2.71. Unlike Earth, 0.9 Correlation length 0.8 the Moon’s curve features only a single peak at around Mean island size −1080 m. 0.7

The usual order parameter is defined as the proba- 0.6 bility of any site to be part of the largest island. As 0.5 shown in Fig. 3, the order parameter for islands has a 0.4 sharp drop-off around the zero height level i.e, right at the 0.3 present mean sea level. According to the further evidence given in the following, it is an indicative of a geometri- 0.2 cal phase transition at this level. The same analysis for 0.1 the oceanic clusters (where disjoint oceans at each level 0 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 are differently colored, leaving islands white) gives rise height to a discontinuous jump in the oceanic order parameter at around −3640 m (Fig. 3). Figure 4 illustrates two other percolation observables FIG. 4: (Color online) Correlation length and mean island measured for the Earth, the mean island size (analo- (land) size vs the sea level, with a remarkable characteriza- gous to the susceptibility of the system), and the cor- tion of a geometrical phase transition at the present mean sea relation length. The mean island size χ is defined as level (zero hight level). The scale factors by which all the ′ 2 ′ χ = Ps s ns(h)/ Ps sns(h), where ns(h) denotes the numbers labeling the vertical axis should be multiplied to get average number of islands of size s at level h, and the the correct graph units are the radius and the square of the prime on the sums indicates the exclusion of the largest radius of the Earth for ξ and χ, respectively. island in each measurement. The correlation length ξ is also defined as average distance of sites belonging to the 2 ′ 2 2 ′ 2 − − same island, ξ = Ps 2Rss ns(h)/ Ps s ns(h), where Rs at 1320 m and 3360 m, respectively (Fig. 1). is the radius of gyration of a given s-cluster. As shown in In order to have a reference point for comparison, Fig. 4, both quantities χ and ξ become divergent at the as an example of the most heavily studied waterless present mean sea level. The divergence of the correlation body with a completely different surface properties and length is a signature of a continuous phase transition at interior mechanism, let me analyze the lunar topography. this level, implying that the critical fluctuations domi- I used the topogrd2 data set (accessible from http://pds- nate at each length scale and that the system becomes geosciences.wustl.edu/missions/clementine/gravtopo.html), scale invariant. These results provide a strong correlation which is measured relative to a spheroid of radius 1738 between the water and long-range topographic evolutions km at the equator−the zero height level. I rescale the on the Earth. Nevertheless, one may imagine a model in Moon’s average radius to 1. which water itself−through erosion, evaporation, precipi- tation and sedimentation, etc.−may have an active role in The percolation observables discussed above for the is- shaping topography i.e., the activity of water itself with lands are measured as a function of a hypothetical sea resulting plains of little height, shapes the landscape to level (Fig. 5). The order parameter shows two rather appear critical around the zero height. small jumps at altitude levels around −960 m and 1360 The measurement of χ and ξ for the oceanic clusters m. The correlation length and the mean island size have shows a dominant divergence that signals the oceanic also two dominant peaks at these levels (see the inset of critical level already observed in Fig. 3. The mean ocean Fig. 5). size reaches its absolute maximum at this critical level The Moon’s height distribution function features a single and the correlation length remains approximately con- global peak at level ∼−950 m which is quite close to the stant at its maximum for level interval −4280 . h . one of the critical levels located at ∼ −960 m. In addi- −3760 (see the supplementary material). tion, if we measure the correlation length and the mean Figure 1 gave an illustration of the percolation transi- cluster size for the oceanic clusters, they show only one tion at the present mean sea level. As can be seen from critical level very close to the one located at ∼−960 m. the figure, all major continental junctions occur at the These may imply that this critical level is more impor- level interval −80 . h . +80. At a sea level around tant for the description of the global topography of the −760 m, Greenland joins the Afro-Eurasia superconti- Moon. This is also quite close to the level h = −1049 m nent to the Americas and at the same level the total at which the total island and oceanic surface areas are length of the coastlines reaches its minimum (Fig. 2). equal, meaning that the island and oceanic percolation Australia and Antarctica continents join to the landmass thresholds coincide. 4

The illustrative figure 6 shows the connectivity of the is- lands on both sides of the critical level. At a height level h = −950 m, a little above the critical level, there exists a number of disjoint islands. At slightly below the criti- cal level at h = −1050 m, the islands merge together to form a giant percolating island which spans the Moon in h= -950 h= -1050 the longitudinal direction. Nevertheless, the other critical level at h = 1360 m, unravels a characteristic feature of the lunar farside. As it is known, one of the most striking geological features of the Moon is the elevation dichotomy [34] between the h= +1360 h= +1340 hemispheres: the nearside is low and flat, dominated by volcanic maria, whereas the farside is mountainous and FIG. 6: (Color online) Upper: disjoint islands with heights deeply cratered. The illustrations in Fig. 6 for elevation higher than the hypothetical sea level on the Moon for two levels at h = 1360 and 1340 m, at both sides of the critical different levels h = −950 m and −1050 m, slightly above level, indicate the aggregation of two main mountainous and below the critical level, respectively. Appearance of the islands that are separated by a very narrow passageway. spanning island (right figure) is an indicative of a percolation This may be a benchmark of a rather non-random origin transition. Lower: aggregation of the two main mountainous of the formation of the lunar farside highlands [35]. islands at the lunar farside around the second critical level at ∼ 1360 m. To summarize, the percolation description of the global Earth’s topography uncovers the important role that is played by the water on the Earth. The critical threshold ing within the Earth that are mainly responsible for the of the model coincides with the current mean sea level very long-wavelength topography of the Earth’s surface, on the Earth. This criticality is along with a sign of the rather than by the exogenic processes like erosion, weath- continental aggregation at this level which seems to be ering and precipitation. The criticality of the current sea more dominated by the endogenic processes (like volcanic level also justifies the appearance of the scale (and con- activity, Earthquakes and tectonic processes) originat- formal) invariant features on the Earth with an intriguing coincidence of the dominant 4/3 fractal dimension in the critical model and observation. The main critical level

0.5 for the Moon has the same amount of land and oceans h=-960 h=1360 at the threshold, indicating a purely geometrical phase Moon 0.4 transition. 1 Correlation length h=-1049 Mean island size 0.3 I would like to thank J. Cardy, M. Kardar, J. Krug, T.

0.2 Quella and M. Sahimi for their many useful comments, 0.8 and especially thank D. Stauffer for his many helpful 0.1 comments and suggestions. The author also thanks H. 0.6 0 Dashti-Naserabadi for his kind help with programming -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 height and M.J. Fallahi for the Earth’s data source. Financial

0.4 Islands support from the Deutsche Forschungsgemeinschaft via The largest island SFB/TR 12 and supports from Alexander von Humboldt Foundation, are gratefully acknowledged. I also acknowl- relative surface area 0.2 h=-960 edge partial financial supports by the research council of

h=1360 the University of Tehran. 0 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 height

∗ Electronic address: [email protected] FIG. 5: (Color online) Relative surface area of the largest [1] C. Lyell, The Principles of Geology; or, the Modern island followed by the total surface area of the islands to the Changes of the Earth and its Inhabitants Considered as total area 4π of the Moon, as a function of the hypothetical Illustrative of Geology 3 ∼ − , vol. , Murray, London (1830). sea level. At level 1049 m, the total island and oceanic [2] A. Cazenave, A. Souriau and K. Dominh, Nature 340, surface areas on the Moon are equal. The two jumps in the − 54-57 (1989). order parameter at levels around 960 m and 1360 m are also [3] J. Perrin, Les atomes, NRF-Gallimard, Paris (1913). decoded in the divergent behavior of the correlation length [4] B. B. Mandelbrot, Science 156, 636-638 (1967). and the mean island size (the inset). Such two jumps are [5] B. B. Mandelbrot, The Fractal Geometry of Nature,W. unusual for percolation. H. Freeman, New York (1983). 5

[6] J.-S. Gagnon, S. Lovejoy and D. Schertzer, Europhys. York, Dover, 246pp (1966). Lett. 62, 801-807 (2003). [24] P. Bird, Geochemistry Geosystems 4(3), 1027 [7] J.-S. Gagnon, S. Lovejoy and D. Schertzer, Nonlinear (2003). Processes Geophys. 13, 541-570 (2006). [25] D. Sornette and V. F. Pisarenko, Geophys. Res. Lett. 30, [8] A. Baldassarri, M. Montuori, O. Prieto-Ballesteros and 1105 (2003). S. C. Manrubia, J. Geophys. Res. 113, E09002 (2008). [26] D. L. Turcotte and G. Schubert, Geodynamics, Applica- [9] G. Boffetta, A. Celani, D. Dezzani and A. Seminara, Geo- tion of Continuum Physics to Geological Problems, John phys. Res. Lett. 35, L03615 (2008). Wiley, New York (1982). [10] C. P. Stark, Nature 352, 423-425 (1991). [27] C. Amante and B. W. Eakins, ETOPO1 1 Arc- [11] A. Maritan, F. Colaiori, A. Flammini, M. Cieplak and J. Minute Global Relief Model: Procedures, Data Sources R. Banavar, Science 272, 984-986 (1996). and Analysis. NOAA Technical Memorandum NESDIS [12] B. Sapoval, A. Baldassarri and A. Gabrielli, Phys. Rev. NGDC-24, 19 pp (March 2009). Lett. 93, 098501 (2004). [28] J. Cardy, Nature physics 2, 67-68 (2006). [13] F. A. Vening Meinesz, Proc. Koninkl. Ned. Akad. Weten- [29] M. Sahimi, Applications of Percolation Theory, Taylor & sch. ser. B 55, 212-228 (1951). Francis, London (1994). [14] B. Mandelbrot, Proc. Nat. Acad. Sci. U.S.A. 72, 3825- [30] P. R. Stoddard and D. M. Jurdy, Icarus 217, 524-533 3828 (1975). (2012). [15] R. S. Sayles and T. R. Thomas, Nature 271, 431-434 [31] D. Stauffer and A. Aharony, Introduction to Percolation (1978). Theory, 2nd ed. Taylor & Francis, London (1994). [16] J. D. Pelletier, J. Geophys. Res. 104, 7359-7375 (1999). [32] A. A. Saberi, M. A. Rajabpour and S. Rouhani, Phys. [17] H. Steinhaus, Colloquium Mathematicum, III, 1-13 Rev. Lett. 100, 044504 (2008). (1954). [33] A. A. Saberi, Appl. Phys. Lett. 97, 154102 (2010). [18] D. C. Harvey, H. Gaonach, S. Lovejoy, J. Stix and D. [34] M. T. Zuber, D. E. Smith, F. G. Lemoine and G. A. Schertzer, Fractals 10, 265-274 (2002). Neumann, Science 266, 1839-1843 (1994). [19] H. Gaonach, S. Lovejoy and D. Schertzer, Int. J. Rem. [35] M. Jutzi and E. Asphaug, Nature 476, 6972 (2011). Sens. 24(11), 2323-2344 (2003). [36] bathymetry is the underwater equivalent to topography. [20] M. Pilkington and J. Todoeschuck, Geophys. Res. Lett. [37] The ETOPO1 global relief model has an accuracy of 22, 779-782 (1995). about 10 meters at best, likely less accurate in the deep [21] S. Pecknold, S. Lovejoy and D. Schertzer, Geophys. J. ocean. Part of the reason for the large error bars are that Inter. 145, 127-144 (2001). each cell’s elevation value represents the average eleva- 2 [22] I. Rodriguez-Iturbe and A. Rinaldo, Fractal River tion over the entire roughly 2 × 2 km footprint of the Basins: Chance and Self-Organization, Cambridge Uni- cell [27]. All mentioned height levels here would indeed versity Press, Cambridge (1997). bear such uncertainties. [23] A. Wegener, The origin of continents and oceans. In: J. Biram (Ed.), Trans. from the 1929 4th German edn. New