A Topological Phase Transition in the Scheidegger Model of River Networks

We investigate the canonical Scheidegger Model of river network morphology for the case of convergent and divergent underlying topography, by embedding it on a cone. We find two distinct phases corresponding to few, long basins and many, short basins, respectively, separated by a singularity in number of basins, indicating a phase transition. Quantifying basin shape through Hack's Law $l\sim a^h$ gives distinct values for the exponent $h$, providing a method of testing our hypotheses. The generality of our model suggests implications for vascular morphology, in particular differing number and shapes of arterial and venous trees.