35 research outputs found
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
Mathematical analysis of a model of river channel formation.
The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin
Testing fluvial erosion models using the transient response of bedrock rivers to tectonic forcing in the Apennines, Italy
The transient response of bedrock rivers to a drop in base level can be used to
discriminate between competing fluvial erosion models. However, some recent studies of
bedrock erosion conclude that transient river long profiles can be approximately
characterized by a transportâlimited erosion model, while other authors suggest that a
detachmentâlimited model best explains their field data. The difference is thought to be
due to the relative volume of sediment being fluxed through the fluvial system. Using a
pragmatic approach, we address this debate by testing the ability of endâmember fluvial
erosion models to reproduce the wellâdocumented evolution of three catchments in the
central Apennines (Italy) which have been perturbed to various extents by an
independently constrained increase in relative uplift rate. The transportâlimited model is
unable to account for the catchmentsâresponse to the increase in uplift rate, consistent with
the observed low rates of sediment supply to the channels. Instead, a detachmentâlimited
model with a threshold corresponding to the fieldâderived median grain size of the
sediment plus a slopeâdependent channel width satisfactorily reproduces the overall
convex long profiles along the studied rivers. Importantly, we find that the prefactor in the
hydraulic scaling relationship is uplift dependent, leading to landscapes responding faster
the higher the uplift rate, consistent with field observations. We conclude that a slopeâ
dependent channel width and an entrainment/erosion threshold are necessary ingredients
when modeling landscape evolution or mapping the distribution of fluvial erosion rates in
areas where the rate of sediment supply to channels is low