Analytical derivation of at-a-station hydraulic–geometry relations

This paper uses generalized expressions for both cross-section geometry and hydraulics (generalization of the Chezy and Manning relations) to derive explicit equations for the exponents and coefficients in the power-law at-a-station hydraulic–geometry relations. The exponents are shown to depend only on the depth exponent in the hydraulic relation (p) and the exponent that reflects the form of the cross-section (r). The coefficients depend on p and r, but also on the slope exponent in the generalized hydraulic relation and on the physical characteristics of the section: bankfull width, bankfull maximum depth, hydraulic conductance, and slope. The theoretical ranges of coefficient and exponent values derived herein are generally consistent with the averages and individual observed values reported in previous studies. However, observed values of the exponents at particular cross-sections commonly fall outside the theoretical ranges. In particular, the observed value of the velocity exponent m is commonly greater than the theoretical value, suggesting that hydraulic conductance often increases more strongly with discharge than predicted by the assumed hydraulic relations. The developments presented here provide new theoretical insight into the ways in which hydraulic and geometric factors determine hydraulic geometry. This insight should help to explain the variation of at-a-station hydraulic geometry and may facilitate prediction of hydraulic geometry at river reaches where detailed measurements are unavailable.