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

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.