A Note on Fractal Channel Networks

Abstract

This paper studies the relation between the structure of river networks and the features of their geomorphologic hydrologic response. A mathematical formulation of connectivity of a drainage network is proposed to relate contributing areas and the network geometry. In view of the connectivity conjecture, Horton's bifurcation ratio R B tends, for high values of Strahler's order Ω of the basin, to the area ratio R A , and Horton's length ratio R L equals, in the limit, the single‐order contributing area ratio R a . The relevance of these arguments is examined by reference to data from real basins. Well‐known empirical results from the geomorphological literature (Melton's law, Hack's relation, Moon's conjecture) are viewed as a consequence of connectivity. It is found that in Hortonian networks the time evolution of contributing areas exhibits a multifractal behavior generated by a multiplicative process of parameter 1/ R B . The application of the method of the most probable distribution in view of connectivity contributes new inroads toward a general formulation of the geomorphologic unit hydrograph, in particular generalizing its width function formulation. A quantitative example of multifractal hydrologic response of idealized networks based on Peano's construct (for which R B = R A = 4, R L = 2) closes the paper.