Journal of Conference Abstracts

A Unification of Scaling Laws for River Networks

Peter S. Dodds (dodds@mit.edu) & Daniel H. Rothman

¹Rm 54-627 MIT, 77 Massachusetts Avenue, Cambridge MA, 2139, U.S.A.

All presently known scaling laws ascribed to river networks are shown to be derivable from a few simple assumptions. We require only that Horton's laws of stream number and stream length be obeyed, that drainage density be uniform and that single streams possess a given fractal dimension. Horton's laws provide a description of the all-pervading self-similarity observed in real networks and drainage density quantifies the degree of dissection in a landscape. Many relationships are shown to follow. These include Hack's law (which states that basin area scales with the main stream length, typically in non-Euclidean fashion), a redundancy in Horton's laws themselves, the self-affinity of drainage basins, probability distributions of drainage area and main stream length, and a complete set of laws connecting the various exponents involved. Moreover, several corrections and extensions of results found in the literature are obtained. All results are tested with data from real landscapes and with the exactly known properties of Scheidegger's random networks. The ramifications for the utility of the laws so derived are discussed with reference to the robustness of the stated assumptions. The major question raised by this work is how much do these scaling laws depend on the physics of erosion as opposed to the geometry of branching networks?