Rationalizing the Differences Among Hydraulic Relationships Using a Process-Based Model

The use of power law forms to describe hydraulic geometry is a classic subject with a history of over 70 years. Two distinct forms of power laws have been proposed: at-a-station hydraulic geometry (AHG) and downstream hydraulic geometry (DHG). Although the utility of these semiempirical expressions is widely recognized, they remain poorly understood in terms of the mechanisms underlying the differences between AHG and DHG, as well as the variability among different systems. In this study, we attempt to address these basic issues. Two hypotheses are proposed: (a) the different geomorphic relationships represented by AHG and DHG result from the control of lateral adjustment of the bank and flow turbulence over short and long timescales, respectively; and (b) the systematic variability of the AHG and DHG exponents is related to the description of the frictional resistance. These two hypotheses are embedded in our theoretical models and lead to explicit functional forms for AHG and DHG. The verification of our hypotheses is based on a large data set consisting of over 550 b-f-m exponents and 120 power law hydraulic relations. The analysis highlights the role of uncertainties in data acquisition and theoretical/statistical explanations. In addition, the theoretical expressions of AHG also provide an explanation of at-many-stations hydraulic geometry (AMHG) in a physical sense. Overall, our work provides new insights into the fundamental theory of power laws and hydraulic geometry.