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

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 ci,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→∞ are obtained; in particular, we prove that ci,j(t) → 0 and Ni(c(t)) → 0 as t→∞, 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.