64 research outputs found
An Analytical and Numerical Study of Optimal Channel Networks
We analyze the Optimal Channel Network model for river networks using both
analytical and numerical approaches. This is a lattice model in which a
functional describing the dissipated energy is introduced and minimized in
order to find the optimal configurations. The fractal character of river
networks is reflected in the power law behaviour of various quantities
characterising the morphology of the basin. In the context of a finite size
scaling Ansatz, the exponents describing the power law behaviour are calculated
exactly and show mean field behaviour, except for two limiting values of a
parameter characterizing the dissipated energy, for which the system belongs to
different universality classes. Two modified versions of the model,
incorporating quenched disorder are considered: the first simulates
heterogeneities in the local properties of the soil, the second considers the
effects of a non-uniform rainfall. In the region of mean field behaviour, the
model is shown to be robust to both kinds of perturbations. In the two limiting
cases the random rainfall is still irrelevant, whereas the heterogeneity in the
soil properties leads to new universality classes. Results of a numerical
analysis of the model are reported that confirm and complement the theoretical
analysis of the global minimum. The statistics of the local minima are found to
more strongly resemble observational data on real rivers.Comment: 27 pages, ps-file, 11 Postscript figure
Unified View of Scaling Laws for River Networks
Scaling laws that describe the structure of river networks are shown to
follow from three simple assumptions. These assumptions are: (1) river networks
are structurally self-similar, (2) single channels are self-affine, and (3)
overland flow into channels occurs over a characteristic distance (drainage
density is uniform). We obtain a complete set of scaling relations connecting
the exponents of these scaling laws and find that only two of these exponents
are independent. We further demonstrate that the two predominant descriptions
of network structure (Tokunaga's law and Horton's laws) are equivalent in the
case of landscapes with uniform drainage density. The results are tested with
data from both real landscapes and a special class of random networks.Comment: 14 pages, 9 figures, 4 tables (converted to Revtex4, PRE ref added
Basins of attraction on random topography
We investigate the consequences of fluid flowing on a continuous surface upon
the geometric and statistical distribution of the flow. We find that the
ability of a surface to collect water by its mere geometrical shape is
proportional to the curvature of the contour line divided by the local slope.
Consequently, rivers tend to lie in locations of high curvature and flat
slopes. Gaussian surfaces are introduced as a model of random topography. For
Gaussian surfaces the relation between convergence and slope is obtained
analytically. The convergence of flow lines correlates positively with drainage
area, so that lower slopes are associated with larger basins. As a consequence,
we explain the observed relation between the local slope of a landscape and the
area of the drainage basin geometrically. To some extent, the slope-area
relation comes about not because of fluvial erosion of the landscape, but
because of the way rivers choose their path. Our results are supported by
numerically generated surfaces as well as by real landscapes
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