Multi-fractal Formulation for General Hydraulic Geometry Relationships

The natural flow regime, usually determined by flow resistance and its inertial force interaction, and channel forming materials play important roles to the hydraulic behaviors and geomorphic patterns (in a long-term) of a river. Since power functional hydraulic geometry relationships for the spatio-temporal variation of the surface width, mean depth and mean velocity were proposed [Leopold and Maddock, 1953], the power-law relationships have been widely adopted. Behind the single power laws, the assumption of the homogeneity at a catchment scale resides. However, there exist rivers with heterogeneous geomorphological phases, where the mono-fractal represented by the single power law is not necessarily suitable. To this end, we generalize the formulation of hydraulic geometry relationships to cover both homogeneous and heterogeneous catchments. The proposed formulation is tested to the Colorado River.