Da Costa, F.P. and Wattis, J.A.D. and Grinfeld, M.
(2002)
*A hierarchical cluster system based on Horton-Strahler rules for river networks.*
Studies in Applied Mathematics, 109 (3).
pp. 163-204.
ISSN 0022-2526

## Abstract

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.

Item type: | Article |
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ID code: | 2061 |

Keywords: | Horton-Strahler, river networks, cluster system, time dynamics, applied mathematics, Mathematics, Applied Mathematics |

Subjects: | Science > Mathematics |

Department: | Faculty of Science > Mathematics and Statistics |

Depositing user: | Strathprints Administrator |

Date Deposited: | 14 Jan 2007 |

Last modified: | 10 Dec 2015 16:13 |

URI: | http://strathprints.strath.ac.uk/id/eprint/2061 |

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