## 1. Introduction

[2] Hydraulic geometry (HG) refers to the power laws relating stream width *W*, average depth *D*, and mean velocity *V* to discharge *Q*: *W* = *aQ*^{b}, *D* = *cQ*^{f}, *V* = *kQ*^{m} [*Leopold and Maddock*, 1953]. These relationships have been observed to hold either for different discharges at an individual cross section (referred to as at-station HG), or for different downstream locations related through some characteristic discharge, e.g., mean annual discharge (referred to as downstream HG). This paper is concerned with the at-station HG.

[3] In a recent paper, *Dodov and Foufoula-Georgiou* [2004] presented empirical evidence that the exponents of at-station HG systematically depend on scale, i.e., drainage area upstream (*A*), and showed that this empirical trend can be captured by a multiscaling formalism of hydraulic geometry factors. Specifically, they postulated and confirmed via analysis of observations that the probability distributions of discharge *Q* and cross-sectional area *C*_{A} remain statistically invariant under proper rescaling with a random function which depends on scale only (notion of multiscaling). As a result, lognormal multiscaling models were fitted to *Q* and *C*_{A} and revised at-station HG relationships (i.e., relationships between *C*_{A} and *Q* and *V* and *Q*) whose coefficients were explicit functions of scale, were derived. These relationships were called generalized HG and were tested on 85 stations in Oklahoma and Kansas with good agreement to observations.

[4] In this paper, we attempt to provide a physical explanation of the empirically observed and statistically described scale dependence of at-station HG in terms of downstream variations in fluvial instability. First, we briefly review the multiscaling formalism of HG that gives rise to generalized at-station HG relationships. Then, we present an analysis of fluvial instability [*Parker*, 1976] as a function of contributing area to show that channel planform geometry (e.g., sinuosity, curvature and wavelength) and, particularly, the transition between straight and meandering channels, are scale-dependent. To relate channel planform geometry and channel shape, we use the model of *Johannesson and Parker* [1987, 1989] to calculate the bed topography of representative meander bends of a given Strahler order, and then, the HG of these bends. This model is based on a small perturbation approach which linearizes the governing equations maintaining full coupling between the flow field, bed load transport and bed topography. We show that the at-station HG that emerges from this physical model is scale-dependent and agrees with the empirical trends and the proposed statistical model. We also show by direct analysis of observations that the velocity HG exponent depends inversely on channel sinuosity and that sinuosity increases as a function of scale. These results together with findings from the physical model are interpreted as evidence that the physical origin of the scale-dependent HG is the systematic increase of channel asymmetry downstream induced by scale-dependent fluvial instability.