Generalized hydraulic geometry: Derivation based on a multiscaling formalism

Abstract

Relationships between channel characteristics (e.g., mean depth, water surface width, mean velocity) and discharge, known as hydraulic geometry (HG), have been extensively used by hydrologists and geomorphologists since the seminal work of Leopold and Maddock [1953]. On the basis of recent empirical evidence that the parameters of at‐site HG depend systematically on the contributing area (scale) and that the parameters of downstream HG depend on the frequency of discharge, we propose a multiscaling formalism within which to model and interpret both at‐site and downstream HG in a homogeneous region. In particular, we postulate and test multiscaling models for cross‐sectional area and discharge and derive generalized HG relationships that explicitly account for scale‐frequency dependence. The multiscaling formalism is tested in several basins in Oklahoma and Kansas for drainage areas ranging from 2 to 20,000 km2 and shows good agreement with the data. To quantify the effects that scale dependence in HG has on the hydrologic response of a basin, a geomorphologic nonlinear cascade of reservoirs model has been used to compute attributes of a representative hydrologic response function for various levels of catchment‐averaged effective rainfall and different basin orders. The numerical experiment shows substantial differences in hydrologic response when using classical versus generalized HG. Finally, a preliminary effort is reported to generalize even further the HG relationships such that they can account for deviations from a single power law (e.g., consideration of two different power laws in low‐ and high‐flow regimes) through the introduction of a bivariate mixed multiscaling framework.