## 1. Introduction

[2] Steep streams (here, streams with gradients greater than about 2%) occupy the majority of the total stream length in mountainous areas. Typically they differ from low-gradient streams by their greater roughness, characterized by wide grain size distributions, variable channel widths, large bedforms, and shallow flows. These streams are important agents of erosion and sediment transfer from headwaters to lower basins. Sediment mobility controls natural channel dynamics, and high-intensity sediment transport episodes pose hazards for buildings, water intakes, and other infrastructure in or near streams. To accurately predict bed load transport rates it is often necessary to estimate the reach-average shear stress, i.e., flow resistance. In steep streams, however, reliable flow resistance estimates require a better understanding of the effects of channel roughness.

[3] Flow velocity is mainly a function of the flow depth *h* (or the hydraulic radius *R _{h}*), the gravitational acceleration

*g*in the direction of the channel slope

*S*, and the channel roughness. The relationship between these parameters is most commonly described with the Darcy-Weisbach equation:

where *v* is the mean flow velocity and *f*_{tot}is the dimensionless Darcy-Weisbach friction factor, which scales with a roughness length. The friction factor is an empirical value that is highly variable over time and between individual stream reaches and is difficult to measure in the field. However, it remains crucial for the calculation of flow velocity.

[4] For rough mountain streams it was found that the performance of depth-based flow resistance equations is poor [*Marcus et al.*, 1992]. Alternatively, several authors have proposed nondimensional hydraulic geometry equations that link the mean flow velocity to total water discharge *Q* [*Rickenmann*, 1994; *Rickenmann*, 1996] or unit discharge *q* [*Aberle and Smart*, 2003; *Comiti et al.*, 2007; *Ferguson*, 2007; *Rickenmann*, 1994; *Rickenmann*, 1996; *Zimmermann*, 2010], because discharge is much easier to determine in rough streams than flow depth. These equations are given in dimensionless form:

where *v** = *v*/(*gD*_{84})^{0.5}, *q** = *q*/(*gD*_{84}^{3})^{0.5}, *q* is the discharge per unit channel width, *D*_{84} is the 84th percentile of the grain size distribution, and *c* and *m* are an empirically determined prefactor and exponent, respectively. *Ferguson* [2007]showed that this type of equation better describes flow velocity measurements in natural streams than other equations. The dimensionless variables were particularly successful in describing at-a-site variations of flow resistance. To better account for the variations between different sites, it was suggested to include the water surface or channel slope as a further factor [*Aberle and Smart*, 2003; *David et al.*, 2010; *Ferguson*, 2007; *Rickenmann and Recking*, 2011; *Zimmermann*, 2010]. As a consequence, *Rickenmann and Recking* [2011] introduced two new dimensionless variables to describe a large data set using an alternative form of equation (2):

where *v*** = *v*/(*gSD*_{84})^{0.5} and *q*** = *q*/(*gSD*_{84}^{3})^{0.5}. These new variables resulted in a similarity collapse of a large data set in the study by *Rickenmann and Recking* [2011].

[5] Both *Ferguson* [2007] and *Rickenmann and Recking* [2011] used the characteristic grain size *D*_{84}as the single explicit roughness measure. A characteristic grain size is also used in the standard logarithmic (Keulegan type) or power law equations (Manning-Strickler type). However, in order to allow for form drag on protruding clasts in shallow flows, the characteristic grain size is usually increased by multiplying it by an empirical factor [*Bathurst*, 1985; *Bray*, 1979; *Hey*, 1979; *Thompson and Campbell*, 1979].

[6] *Aberle and Smart* [2003] and *Lee and Ferguson* [2002], among others, have argued that grain size might not be an appropriate roughness measure in steep streams. *Aberle and Smart* [2003] found that hydraulic roughness varied among different sites or different flows even though the characteristic grain size (for example *D*_{84}) remained the same. Instead, they identified the standard deviation of bed elevation *s* as a roughness parameter that additionally accounts for the arrangement of grains [*Aberle and Smart*, 2003; *Smart et al.*, 2002].

[7] Steep streams typically feature large grains that can be randomly distributed in the channel, organized in patches or clusters [e.g., *Lamarre and Roy*, 2008; *Nelson et al.*, 2009], or in channel-spanning steps [e.g.,*Chin and Wohl*, 2005; *Church and Zimmermann*, 2007; *Whittaker and Jaeggi*, 1982; *Zimmermann et al.*, 2008]. These macroroughness features lead to additional flow resistance that is absent in lower gradient channels. In low-gradient streams the main source of resistance is skin friction, i.e., from drag on individual particles and viscous friction on their surfaces [*Ferguson*, 2007]. In steep streams, by contrast, flow resistance mainly results from macroroughness, including form drag around large boulders due to acceleration, deceleration, and turbulent wakes, as well as spill loss, particularly behind steps or larger particles if flow is locally supercritical [*Chin*, 2003; *Ferguson*, 2007]. *Zimmermann* [2010]concluded that a major part of the flow energy in steep streams is dissipated by form and spill drag around roughness elements like step-pools, as it was also discussed by*Comiti et al.* [2009], *MacFarlane and Wohl* [2003], and *Wilcox et al.* [2006]. The contribution of these structures to total flow resistance increases with increasing relative protrusion (or equivalently, with decreasing relative submergence of the bed).

[8] Macroroughness features are rarely taken into account explicitly in flow resistance equations. The effects of boulder diameter and areal boulder concentration on flow resistance have been studied in laboratory flumes, resulting in empirical or semitheoretical resistance equations [*Pagliara and Chiavaccini*, 2006; *Whittaker et al.*, 1988; *Yager*, 2006; *Yager et al.*, 2007]. Other equations include the effects of steps and pools on flow resistance, using step height and step length as the relevant measures of macroroughness [*Canovaro and Solari*, 2007; *Egashira and Ashida*, 1991; *Whittaker*, 1986]. There are few systematic tests of these approaches using field observations [e.g., *Nitsche et al.*, 2011].

[9] For some mountain streams with very pronounced step-pool structures,*Comiti et al.* [2007] and *David et al.* [2010] found that variations in flow resistance were mostly explained by unit discharge and slope, whereas *R _{h}*/

*D*

_{84}was not an appropriate explanatory variable.

*David et al.*[2010] also found that the relations between flow resistance and these variables were distinct for different channel types.

*MacFarlane and Wohl*[2003]found a significant positive correlation between flow resistance and step height-to-length ratio in some step-pool reaches, demonstrating the increasing effect of spill resistance with increasing step height. Detrended standard deviation of bed elevations and relative bed form submergence explained a large portion of the variance in measured flow resistance coefficients and dimensionless velocity in a study of

*Yochum et al.*[2012]. Their field data indicate an empirical relation between flow resistance and the relative submergence of bedforms, which supports previously published laboratory findings. Moreover, using detrended standard deviation of elevations instead of

*D*

_{84}provided relatively accurate flow velocity predictions for their data set. For some cascade and plane bed channels

*Reid and Hickin*[2008] found that among various roughness measures the sorting coefficient

*D*

_{84}/

*D*

_{50}correlated well with form roughness.

[10] Currently there is no agreement on how best to relate flow resistance to bed properties in steep or shallow streams. This is partly due to (1) a disagreement over how to quantify roughness, and (2) a scarcity of combined flow and roughness measurements in the field. Furthermore, roughness is often described using a single parameter, and none of the roughness measures proposed so far can completely explain the observed variability of flow velocity among different sites.

[11] In the present study we measured flow velocity over a wide range of discharges, in six stream reaches with widely varying channel bed slopes and grain size distributions. In addition, we quantified macroroughness for each of these stream reaches by measuring characteristic grain sizes, boulder concentrations, and the roughness of the longitudinal channel profiles. These data were used (1) to test how macroroughness can be measured, (2) to evaluate the relations among various measures of macroroughness and channel slope, (3) to compare the flow parameters of different rough streams, and (4) to explain the observed between-site variations in flow velocity and macroroughness, using nondimensional variables and regression analysis.