Revisiting the relationships between channel geometry and sediment grain size in mountain rivers

Empirical equations relating the river geometry with discharge and sediment grain size have been proposed since the 1950s, by considering various spatial scales (i.e., reach and catchment scales). For instance, at the cross-sectional scale, Parker et al. (2007) and Wilkerson & Parker (2011) found dimensionless relationships between flow discharge, water depth, the river width, and channel slope in bankfull conditions for gravel-bed and sand-bed rivers, respectively. At the catchment scale, many authors found empirical correlations between drainage area and local morphological properties, in terms of bed slope and grain size distribution (Gasparini et al., 2004; Golden & Springer, 2006; Eaton, 2013, among others). One of the pioneer works proposing such empirical equations was Hack (1957), who found a power law relationship between the local bed slope, s, and the catchment-area-to-mean-grain-size ratio (A / D50) for various rivers in the US (Figure 1a). To our knowledge, a theory explaining the empirical formulation has not been proposed yet in literature. In this work, we derive a physically-based relationship relating local bed slope to the mean grain size and the upstream drainage area, which helps clarifying the origin of the early proposed empirical relationships and the reasons for further extensions.