### 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.

Original language | English |
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Pages (from-to) | 163-204 |

Number of pages | 41 |

Journal | Studies in Applied Mathematics |

Volume | 109 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2002 |

### Keywords

- Horton-Strahler
- river networks
- cluster system
- time dynamics
- applied mathematics

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## Cite this

Da Costa, F. P., Wattis, J. A. D., & Grinfeld, M. (2002). A hierarchical cluster system based on Horton-Strahler rules for river networks.

*Studies in Applied Mathematics*,*109*(3), 163-204. https://doi.org/10.1111/1467-9590.00221