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

In a companion paper [Veneziano and Niemann, this issue] the authors have p roposed 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 geomorphologi cal scaling laws of hydrologic interest. We find that several existing rela tions should be modified, as they were obtained using definitions of the qu antities involved or measurement techniques that are inappropriate under se lfsimilarity. These relations include Hack's law, the power law decay of th e distributions of contributing area and main channel length, the scaling o f channel slope with contributing area, and the self-similarity condition f or river courses. Most results are further generalized by replacing main st ream flow length and drainage area with generic measures of basin size. The relations we obtain among properly measured topographic variables have sim ple universal exponents. For example, the exponent of Hack's law is 0.5, th e exponent of the distribution of contributing area is -0.5, and the expone nt of the distribution of main stream length is -1.0. We also suggest a sto chastic condition of drainage network self-similarity that incorporates top ological as well as geometric and hydrologic features and a reformulation o f Horton's laws using drained area rather than stream order.