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Geometry of river networks. I. Scaling, fluctuations, and deviations
1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. dodds@segovia.mit.edu
Summary
River networks exhibit scaling laws, but this study reveals significant deviations and fluctuations. These variations, influenced by network structure and scale, restrict the universality of scaling in river systems.
Area of Science:
- Geomorphology
- Hydrology
- Network Science
Background:
- River networks are universal branching structures crucial for material distribution.
- Scaling laws are observed in river networks, but exponent values vary, indicating a lack of universality.
- Understanding deviations from scaling is key to describing branching network structure.
Purpose of the Study:
- To investigate fluctuations and deviations from scaling laws in river networks.
- To provide a fuller description of branching network structure.
- To examine the scaling of dominant stream length with subbasin area (Hack's Law) and its generalizations.
Main Methods:
- Theoretical and empirical study of river network geometry.
- Analysis of continent-scale river networks (e.g., Mississippi, Amazon).
- Inspiration from a simple model of directed, random networks.
- Generalization of Hack's Law to a joint probability density.
Main Results:
- Substantial fluctuations about scaling grow with system size.
- Small-scale deviations explained by linear network structures.
- Intermediate-scale slow drifts in exponent values indicate approximate scaling.
- Large-scale scaling breakdown due to decreasing sample space and basin shape correlations.
Conclusions:
- Approximate scaling in river networks is significantly restricted by observed deviations.
- Network resolution does not improve the extent of approximate scaling.
- Universality of scaling in river networks remains indeterminate due to scale-dependent deviations.
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