dorsal/arxiv
View SchemaGeometry of River Networks I: Scaling, Fluctuations, and Deviations
| Authors | Peter Sheridan Dodds, Daniel H. Rothman |
|---|---|
| Categories | |
| ArXiv ID | physics/0005047 |
| URL | https://arxiv.org/abs/physics/0005047 |
| DOI | 10.1103/PhysRevE.63.016115 |
Abstract
This article is the first in a series of three papers investigating the detailed geometry of river networks. Large-scale river networks mark an important class of two-dimensional branching networks, being not only of intrinsic interest but also a pervasive natural phenomenon. In the description of river network structure, scaling laws are uniformly observed. Reported values of scaling exponents vary suggesting that no unique set of scaling exponents exists. To improve this current understanding of scaling in river networks and to provide a fuller description of branching network structure, we report here a theoretical and empirical study of fluctuations about and deviations from scaling. We examine data for continent-scale river networks such as the Mississippi and the Amazon and draw inspiration from a simple model of directed, random networks. We center our investigations on the scaling of the length of sub-basin's dominant stream with its area, a characterization of basin shape known as Hack's law. We generalize this relationship to a joint probability density and show that fluctuations about scaling are substantial. We find strong deviations from scaling at small scales which can be explained by the existence of linear network structure. At intermediate scales, we find slow drifts in exponent values indicating that scaling is only approximately obeyed and that universality remains indeterminate. At large scales, we observe a breakdown in scaling due to decreasing sample space and correlations with overall basin shape. The extent of approximate scaling is significantly restricted by these deviations and will not be improved by increases in network resolution.
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"abstract": "This article is the first in a series of three papers investigating the\ndetailed geometry of river networks. Large-scale river networks mark an\nimportant class of two-dimensional branching networks, being not only of\nintrinsic interest but also a pervasive natural phenomenon. In the description\nof river network structure, scaling laws are uniformly observed. Reported\nvalues of scaling exponents vary suggesting that no unique set of scaling\nexponents exists. To improve this current understanding of scaling in river\nnetworks and to provide a fuller description of branching network structure, we\nreport here a theoretical and empirical study of fluctuations about and\ndeviations from scaling. We examine data for continent-scale river networks\nsuch as the Mississippi and the Amazon and draw inspiration from a simple model\nof directed, random networks. We center our investigations on the scaling of\nthe length of sub-basin\u0027s dominant stream with its area, a characterization of\nbasin shape known as Hack\u0027s law. We generalize this relationship to a joint\nprobability density and show that fluctuations about scaling are substantial.\nWe find strong deviations from scaling at small scales which can be explained\nby the existence of linear network structure. At intermediate scales, we find\nslow drifts in exponent values indicating that scaling is only approximately\nobeyed and that universality remains indeterminate. At large scales, we observe\na breakdown in scaling due to decreasing sample space and correlations with\noverall basin shape. The extent of approximate scaling is significantly\nrestricted by these deviations and will not be improved by increases in network\nresolution.",
"arxiv_id": "physics/0005047",
"authors": [
"Peter Sheridan Dodds",
"Daniel H. Rothman"
],
"categories": [
"physics.geo-ph"
],
"doi": "10.1103/PhysRevE.63.016115",
"title": "Geometry of River Networks I: Scaling, Fluctuations, and Deviations",
"url": "https://arxiv.org/abs/physics/0005047"
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