Scaling Laws of Drainage Networks for Global River Basins

River networks self-organize into tree-like network patterns that exhibit certain scaling laws. These networks have been reported to follow fairly constant scaling exponents despite differences in underlined geography and climatic controls. We investigated these scaling properties for global river networks. We considered the three important scaling laws, including Hack's law, exceeding probability distribution for the contributing area, and upstream length. We used the HydroSHEDS database to calculate the scaling exponents for rivers of global river basins divided into nine different regions: Africa, Arctic, Asia, Australia, Europe, Greenland, North America, South America, and Siberia. We performed analysis for area-wise top one percent largest basins in each region. We found that the exponents of scaling laws did fall within the reported range for all the continental landscapes except for Greenland. This suggests that scaling laws might be a peculiar property of fluvial drainage network evolution because the landscape evolution in Greenland is mostly dominated by glacial erosion.

We also looked at the relationship between these scaling exponents. Earlier studies by Dodds and Rothman (1999) had analytically derived expressions for these scaling laws and their interrelation. As per those expressions, the exponents for distribution of contributing area and upstream length should be inversely correlated with Hack's exponent. However, we did not find any relationship between Hack's exponent and the other two scaling exponents. Whereas the contributing area exponent and Upstream length exponent were found to be directly correlated. The values of these scaling exponents need to be investigated further for their applications in characterizing a landscape.