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A Hierarchical Cluster System Based on Horton–Strahler Rules for River Networks
Author(s) -
Da Costa F. P.,
Grinfeld M.,
Wattis J. A. D.
Publication year - 2002
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00221
Subject(s) - cluster (spacecraft) , mathematics , ordinary differential equation , order (exchange) , differential equation , statistical physics , mathematical physics , physics , combinatorics , mathematical analysis , computer science , programming language , finance , economics
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 c i,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 c i,j ( t ) → 0 and N i ( c ( t )) → 0 as t →∞, where the functional N i (·) 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.