Self‐similarity and multifractality of fluvial erosion topography: 2. Scaling properties

Abstract

Abstract. In a companion paper [Veneziano and Niemann, this issue] the authors have proposed self-similarity and multifractality conditions for fluvial erosion topography within basins and have shown that topographic surfaces with this property can evolve from a broad class of dynamic models. Here we use the same self-similarity and multifractality conditions to derive geomorphological scaling laws of hydrologic interest. We find that several existing relations should be modified, as they were obtained using definitions of the quantities involved or measurement techniques that are inappropriate under selfsimilarity. These relations include Hack's law, the power law decay of the distributions of contributing area and main channel length, the scaling of channel slope with contributing area, and the self-similarity condition for river courses. Most results are further generalized by replacing main stream flow length and drainage area with generic measures of basin size. The relations we obtain among properly measured topographic variables have simple universal exponents. For example, the exponent of Hack's law is 0.5, the exponent of the distribution of contributing area is -0.5, and the exponent of the distribution of main stream length is -1.0. We also suggest a stochastic condition of drainage network self-similarity that incorporates topological as well as geometric and hydrologic features and a reformulation of Horton's laws using drained area rather than stream order.

A quantitative evaluation of Playfair's law and its use in testing long‐term stream erosion models

“…The bias from the profile concavity has been discussed previously within the context of basin self-similarity by Veneziano and Niemann (2000b). The conclusion of that study is that for an equilibrium topography governed by Equation 4, the length R over which slopes are measured should vary as:…”

Normal fault control on bedrock channel incision and sediment supply: Insights from numerical modeling

“…In recent years much research has been undertaken into the processes controlling stream network development and, more generally, landscape evolution, leading to the development and testing of a variety of stream network and landscape evolution models (LEMs) [e.g., Stark , 1991; Rigon et al , 1994; Howard , 1994; Tucker and Slingerland , 1996, 1997; Rinaldo et al , 1995; Caldarelli et al , 1997; Rodriguez‐Iturbe and Rinaldo , 1997; Densmore et al , 1998; Van der Beek and Braun , 1999; Pelletier , 1999; Stock and Montgomery , 1999; Whipple and Tucker , 1999; Snyder et al , 2000; Veneziano and Niemann , 2000a, 2000b]. However, while these studies have provided a better understanding of the major controls on drainage basin morphology and evolution, the relationship between stream network/landscape evolution and sediment supply to adjacent sedimentary basins has received only limited study [e.g., Howard , 1994; Tucker and Slingerland , 1996, 1997; Densmore et al , 1998].…”

An evaluation of the geomorphically effective event for fluvial processes over long periods

“…To calculate the discharge at any point i along the profile, it is assumed that where L i is the distance of the location from the divide. This expression assumes the profile occurs in a self‐similar watershed [ Mandelbrot , 1983; Veneziano and Niemann , 2000; Hack , 1957], and it neglects the effects of channel sinuosity and the discrete occurrences of channel junctions [ Tucker and Slingerland , 1994; Veneziano and Niemann , 2000]. Therefore the water discharge Q i during a storm can be written as where P is a rainfall rate generated according to the exponential distribution.…”