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'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 c...
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| Outros Autores: | , |
| Formato: | preprint |
| Idioma: | eng |
| Publicado em: |
2010
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| Assuntos: | |
| Texto completo: | http://hdl.handle.net/10400.2/1534 |
| País: | Portugal |
| Oai: | oai:repositorioaberto.uab.pt:10400.2/1534 |
| Resumo: | 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 |
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