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.Comment: 16 pages, 13 figures, Revtex4, submitted to PR
