### Abstract

Language | English |
---|---|

Pages | 163-204 |

Number of pages | 41 |

Journal | Studies in Applied Mathematics |

Volume | 109 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2002 |

### Fingerprint

### Keywords

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

### Cite this

*Studies in Applied Mathematics*,

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

}

*Studies in Applied Mathematics*, vol. 109, no. 3, pp. 163-204. https://doi.org/10.1111/1467-9590.00221

**A hierarchical cluster system based on Horton-Strahler rules for river networks.** / Da Costa, F.P.; Wattis, J.A.D.; Grinfeld, M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A hierarchical cluster system based on Horton-Strahler rules for river networks

AU - Da Costa, F.P.

AU - Wattis, J.A.D.

AU - Grinfeld, M.

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - Horton-Strahler

KW - river networks

KW - cluster system

KW - time dynamics

KW - applied mathematics

UR - http://dx.doi.org/10.1111/1467-9590.00221

U2 - 10.1111/1467-9590.00221

DO - 10.1111/1467-9590.00221

M3 - Article

VL - 109

SP - 163

EP - 204

JO - Studies in Applied Mathematics

T2 - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

SN - 0022-2526

IS - 3

ER -