A hierarchical cluster system based on Horton-Strahler rules for river networks

摘要

We consider a cluster system in which each cluster is characterized by two parameters: an "order" i, following Horton-Strahler rules, and a "mass" j following the usual additive rule. Denoting by c(i,j)(t) the concentration of clusters of order i and mass j at time t, we derive a coagulation-like ordinary differential system for the time dynamics of these clusters. Results about the existence and the behavior of solutions as t --> infinity are obtained; in particular, we prove that c(i,j)(t) --> 0 and N-i(c(t)) --> 0 as t --> infinity, where the functional Ni(.) measures the total amount of clusters of a given fixed order i. Exact and approximate equations for the time evolution of these functionals are derived. We also present numerical results that suggest the existence of self-similar solutions to these approximate equations and discuss their possible relevance for an interpretation of Horton's law of river numbers. [References: 23]