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

• Author(s): Costa, Fernando Pestana da ; Grinfeld, Michael ; Wattis, Jonathan A. D.
• Date: 2002
• Persistent ID: http://hdl.handle.net/10400.2/1534
• Origin: Repositório Aberto da Universidade Aberta
• Subject(s): Coagulation equations; Cluster dynamics; Horton-Strahler rules
• Document Type: Preprint
• Language: English

Description

We consider a cluster system in which each cluster is characterized by two parameters: an \order" i; following Horton-Strahler's rules, and a \mass" j following the usual additive rule. Denoting by ci;j (t) the concen- tration of clusters of order i and mass j at time t; we derive a coagulation- like ordinary di erential system for the time dynamics of these clusters. Results about existence and the behaviour of solutions as t ! 1 are ob- tained, in particular we prove that ci;j (t) ! 0 and Ni(c(t)) ! 0 as t ! 1; where the functional Ni( ) measures the total amount of clusters of a given xed order i: Exact and approximate equations for the time evolution of these functionals are derived. We also present numerical results that sug- gest the existence of self-similar solutions to these approximate equations and discuss its possible relevance for an interpretation of Horton's law of river numbers