## 1. Introduction

[2] Erosion and transport in tectonically active settings occur by a wide range of processes ranging from mass flow processes on hillslopes and colluvial valleys to bedrock incision and sediment transport in fluvial channels. Understanding the macroscopic response of the topography to tectonic or climatic perturbations both in terms of characteristic timescales and shapes, requires a detailed understanding and modeling of each of these processes, and their potential couplings [*Whipple*, 2001].

[3] In recent years, the processes and models of bedrock incision have received considerable attention [*Howard et al.*, 1994; *Seidl and Dietrich*, 1992; *Tinkler and Wohl*, 1998; *Whipple et al.*, 2000]. In settings where the variations of uplift rate were a-priori known, several studies have demonstrated that the geometry of river channels is sensitive to variations in tectonic uplift in a manner consistent with the stream power law family of models that relates the river incision rate (or sediment transport capacity) to a power of the slope and drainage area with a possible threshold of erosion [*Kirby and Whipple*, 2001; *Lague et al.*, 2000; *Lavé and Avouac*, 2001; *Snyder et al.*, 2000, 2003a, 2003b]. Conversely, in high-relief settings, hillslopes undergoing bedrock landsliding are supposed to reach threshold angles whose values are primarily set by rock strength [*Schmidt and Montgomery*, 1995], and which should thus be statistically independent of uplift rate [*Burbank et al.*, 1996; *Montgomery*, 2001]. If the hillslope length remains constant, the local relief in these settings should be independent of uplift rate [*Schmidt and Montgomery*, 1995], especially for very high uplift rates [*Montgomery*, 2001].

[4] In this study we aim at continuing this effort to characterize the long-term erosion processes that shape topography. Basically, this requires a large database where erosion fluxes can be measured and correlated to flow and topographic variables. A direct measurement of the net erosion flux is possible in tectonically active areas providing that the uplift rate is known, and that the topography is in approximate steady-state. Thanks to the work of [*Hurtrez et al.*, 1999; *Lavé*, 1997; *Lavé and Avouac*, 2000], the Siwaliks Hills of central Nepal constitute a remarkable case-study to explore this issue. The uplift rate was inferred from field measurements [*Lavé and Avouac*, 2000] and extrapolated all over the area by using a fold deformation model [*Hurtrez et al.*, 1999; *Lavé*, 1997; *Lavé and Avouac*, 2000]. We have also reasonable arguments to assume a steady state between erosion and tectonics. Moreover the lithology and climate is rather uniform all over the area so that we can concentrate on the dependency of erosion flux on drainage area and topographic slopes, two parameters that can be derived from the available 60 m-DEM.

[5] Sampling and spatial resolution of this database fix for a part the range of parameters that can be explored and thus the type of geomorphological structures and erosion processes that are characterized. In this case, the correlations are statistically sound for areas larger than ∼5.10^{−3} km^{2} and smaller than 1–10 km^{2}. This scale range (200 m–2 km) is typical of the forms observed in Figure 1. The resolution is likely not sufficiently high to observe convex hillslope sensu stricto as defined for instance by *Montgomery* [2001]. On the other hand, the sampling is not large enough to have good statistics on fluvial channels [*Kirby and Whipple*, 2001]. Actually this range of area corresponds to colluvial valleys [*Montgomery*, 2001; *Montgomery and Foufoula-Georgiou*, 1993] that make the connection between hillslopes and rivers, and where erosion and transport is dominated by shallow landsliding, episodic debris flows, infilling by colluvium and weathering of bedrock [*Benda*, 1990; *Benda and Dunne*, 1997; *Dietrich and Dunne*, 1978; *Howard*, 1998; *Montgomery and Foufoula-Georgiou*, 1993; *Stock and Dietrich*, 2003]. Most importantly, these colluvial valleys may represent up to 75% of total basin relief [*Stock and Dietrich*, 2003], and should thus play an important role on the response time of topography to tectonic or climatic perturbations. We still lack a long-term erosion model for this topographic domain [*Stock and Dietrich*, 2003], which makes the analysis of the Siwaliks Hills database an exceptional opportunity to address this problem.

[6] A recent study in this area have demonstrated that relief for length scales ranging between 200 and 400 m (calculated as the amplitude factor of the 2D topographic variogram) is proportional to uplift rates varying between 6 and 15 mm.yr-1 [*Hurtrez et al.*, 1999]. Given that the relief at the scales studied by Hurtrez and coworkers is mainly set by hillslopes and colluvial valleys, their observation suggests that the geometry of colluvial valleys is very sensitive to tectonic uplift, as are fluvial channels in the same area [*Kirby and Whipple*, 2001; *Lavé and Avouac*, 2001].

[7] We have thus studied the relationship between colluvial valley geometry and uplift rate, and derived the parameters of an erosion model that is relevant to the colluvial erosion-transport processes that are dominant between hillslopes and rivers. In the studied area, these processes are likely dominated by debris flows, although further field studies are required to clearly assess this point. By analogy with studies on fluvial systems, we assume that erosion depends on both local slope and drainage area, which is a proxy for water discharge, so that we can use the slope-area relationship to derive the erosion model [*Howard*, 1980, 1994; *Lague et al.*, 2003, 2000; *Seidl and Dietrich*, 1992; *Snyder et al.*, 2000, 2003a; *Tucker and Whipple*, 2002; *Whipple and Tucker*, 1999; *Willgoose et al.*, 1991a]. The limit of this formalism is that any effect due to lithology, vegetation or sediment concentration [*Sklar and Dietrich*, 1998; *Whipple and Tucker*, 2002] is averaged without knowing precisely which parameter is concerned by this averaging.

[8] As a starting point, we present the relationships between slope-drainage area and uplift rate that are predicted by the stream power law family of models for steady-state topography. This model that is used for river incision and sediment transport problems [*Davy and Crave*, 2000; *Howard*, 1980, 1994; *Howard and Kerby*, 1983; *Seidl and Dietrich*, 1992; *Tucker and Whipple*, 2002; *Whipple and Tucker*, 2002; *Willgoose et al.*, 1991b] was indeed found to be consistent with data. It permits us to derive analytically the expected relationship between measurements (slope, drainage area, uplift rate) and erosion parameters. We first give this theoretical framework and then analyze data to derive the consequences on erosion model.