Drainage basins are essential to Geohydrology and Biodiversity. Defining those regions in a simple, robust and efficient way is a constant challenge in Earth Science. Here, we introduce a model to delineate multiple drainage basins through an extension of the Invasion Percolation-Based Algorithm (IPBA). In order to prove the potential of our approach, we apply it to real and artificial datasets. We observe that the perimeter and area distributions of basins and anti-basins display long tails extending over several orders of magnitude and following approximately power-law behaviors. Moreover, the exponents of these power laws depend on spatial correlations and are invariant under the landscape orientation, not only for terrestrial, but lunar and martian landscapes. The terrestrial and martian results are statistically identical, which suggests that a hypothetical martian river would present similarity to the terrestrial rivers. Finally, we propose a theoretical value for the Hack’s exponent based on the fractal dimension of watersheds, γ = D/2. We measure γ = 0.54 ± 0.01 for Earth, which is close to our estimation of γ ≈ 0.55. Our study suggests that Hack’s law can have its origin purely in the maximum and minimum lines of the landscapes.
We investigate the behavior of a two-state sandpile model subjected to a confining potential in one and two dimensions. From the microdynamical description of this simple model with its intrinsic exclusion mechanism, it is possible to derive a continuum nonlinear diffusion equation that displays singularities in both the diffusion and drift terms. The stationary-state solutions of this equation, which maximizes the Fermi-Dirac entropy, are in perfect agreement with the spatial profiles of time-averaged occupancy obtained from model numerical simulations in one as well as in two dimensions. Surprisingly, our results also show that, regardless of dimensionality, the presence of a confining potential can lead to the emergence of typical attributes of critical behavior in the two-state sandpile model, namely, a power-law tail in the distribution of avalanche sizes.Physical processes involving anomalous diffusion are typically associated with systems in which the mean square displacement of their elementary units follows a nonlinear power-law relationship with time, σ 2 ∝ t α , with an exponent α = 1, in contrast with linear standard diffusion (α = 1). Instead of being a rare phenomenon, as suggested by its own denomination, anomalous diffusion, however, appears rather ubiquitously in Nature, playing an important role in a variety of scientific and technological applications, such as fluid flow through disordered porous media [1], surface growth [2], diffusion in fractal-like substrates [3][4][5][6][7], turbulent diffusion in the atmosphere [8,9], spatial spreading of cells [10] and biological populations [11], cellular transport [12], and cytoplasmic crowding in cells [13]. Anomalous diffusion can also manifest its non-Gaussian behavior in terms of nonlinear Fokker-Plank equations [14][15][16][17][18], which is the case, for example, of the dynamics of interacting vortices in disordered superconductors [19][20][21][22], diffusion in dusty plasma [23,24], and pedestrian motion [24].The extreme case of nonlinear behavior in diffusive systems certainly corresponds to singular diffusion. For instance, in some physical conditions, the diffusion of adsorbates on a surface can be strongly nonlinear [25][26][27], with a surface diffusion coefficient that depends on the local coverage θ as, D ∝ |θ − θ c | −α . The study of surfacediffusion mechanisms is crucial for the understanding of technologically important processes related with physical adsorption [28] and catalytic surface reactions [29][30][31].In particular, a singularity in the coverage dependence of the diffusion coefficient is frequently associated to continuous phase transitions [27].A direct connection between singular diffusion and selforganized criticality [32] has been disclosed by Carlson et al. [33,34] in terms of a two-state one-dimensional sandpile model with a driving mechanism, where grains are added at one end of the pile and fall off at the other end. Besides exhibiting a self-organized state, the continuum limit of this simple model leads to a nonli...
Widespread valley networks (VNs) on Mars and other evidence point to an early warm and wet climate. However, ongoing debates still exist about VN’s formation processes and associated climatic conditions. The power law relationship between basin length and area (Hack’s Law) can be diagnostic of different fluvial processes related to climatic conditions. Past studies of Hack’s Law on Mars at local sites have produced inconclusive results. Here we used a parameter-free method to delineate watersheds globally on Mars based on mapped VNs and extracted their Hack’s Law exponent (h). The majority of h values on Mars are similar to those in arid areas on Earth, implying similar runoff processes and arid conditions for VN formation on early Mars. Statistical analyses show that the spatial distribution of h on Mars is not random, but with a few clustered high and low values, likely controlled by local conditions (e.g., slope or structures).
Widespread valley networks (VNs) on Mars and other evidence point to an early warm and wet climate. However, ongoing debates still exist about VN's formation processes and associated climatic conditions. The power law relationship between basin length and area (Hack's Law) can be diagnostic of different fluvial processes related to climatic conditions. Past studies of Hack's Law on Mars at local sites have produced inconclusive results. Here we used a parameter‐free method to delineate watersheds globally on Mars based on mapped VNs and extracted their Hack's Law exponent (h). The majority of h values on Mars are similar to those in arid areas on Earth, suggesting similar runoff processes and arid conditions for VN formation on early Mars. Statistical analyses show that the spatial distribution of h on Mars is not random, but with a few clustered high and low values, likely controlled by local conditions (e.g., regional topographic slope).
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