Constraints on the long-term colluvial erosion law by analyzing slope-area relationships at various tectonic uplift rates in the Siwaliks Hills (Nepal)



[1] The large database of topographic form and uplift rates that exists for the Siwaliks Hills (central Nepal) makes possible a thorough analysis of the long-term erosion model. The study especially focuses on drainage areas larger than 5.10−3 km2, fixed by the database resolution, and smaller than 1 km2 above which a fluvial signature is recorded. This area range corresponds to colluvial valleys in which the dominant erosion process is likely debris flow. We evaluate a phenomenological model wherein erosion is considered to depend on drainage area and slope. We test this model by assuming that the uplift rate is in approximate equilibrium with erosion. The stream power law model, formulated by analogy to river incision and transport problems, is found to be consistent with data since an inverse power law relationship between slope and drainage area is systematically observed between 7.10−3 and ∼1 km2, with little variability on the exponent ∼−0.24. Thanks to the range of uplift rates, we obtain constraints on the slope dependency of erosion law, which appears linear and which predicts a significant erosion threshold. The linear dependence on slope in the debris-flow zone is consistent with findings by Kirby and Whipple [2001] in the fluvial downstream zone and with the linear relationship between local relief and uplift rate documented by Hurtrez et al. [1999]. The transition between this colluvial-channel regime and the fluvial regime appears quite sharp in contrast with recent studies, but the latter regime is not sufficiently documented to derive definite conclusions.

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 km2 and smaller than 1–10 km2. 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.

Figure 1.

View of the Bakeya valley (looking South) showing soil-mantled valley sides and forest cover. Note the incision of the Bakeya river into the sandstones of the Middle Siwaliks Formation, and the abandoned strath terrace. The vertical uplift rate in this location is close to 14 mm.yr−1 (courtesy of Jerôme Lavé).

[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.

2. Theoretical Interpretation of Slope-Area-Uplift Relationships

[9] In the following, we sum up briefly the interpretation in terms of erosion and transport laws that can be made from the observations of relationships between slope and drainage area in rivers. We voluntarily kept a heuristic approach for which the most general and simple erosion model that explains the slope-area power law is considered. In the case of rivers, this phenomenological model has a physical rationale that we do not develop here [Whipple et al., 2000; Willgoose et al., 1991b], whereas in the case of processes operating in these colluvial valleys (assumed to be debris flow), it remains purely empirical.

2.1. Stream Power Law Models and Slope-Area Relationships

[10] Numerous studies of river geometry have reported the existence of a power law relationship between local slope and drainage area in a wide range of tectonic, lithologic and climatic settings [Flint, 1974; Hack, 1957; Howard, 1980; Howard and Kerby, 1983; Ijjasz-Vasquez and Bras, 1995; Lague et al., 2000; Moglen and Bras, 1995; Montgomery, 2001; Montgomery and Foufoula-Georgiou, 1993; Sklar and Dietrich, 1998; Snyder et al., 2000; Tarboton et al., 1989, 1992; Tucker and Bras, 1998; Whipple and Tucker, 1999, 2002; Willgoose et al., 1991a]. This relationship is expressed as follow:

display math

where the coefficient k is called steepness index, and θ is a scaling exponent, also known as the concavity index, which ranges generally between 0.4 and 0.7, with some rare values close to 1 and lower than 0.2 (see Tucker and Whipple [2002] for a review). Several studies have demonstrated that this relationship can be theoretically explained if one assumes that (1) the channel incision rate is spatially uniform (as it is expected in a steady state case with uniform uplift along the river), or vary as power function of drainage area [Kirby and Whipple, 2001], that (2) climate and lithology are uniform or vary as a power function of drainage area, and that (3) the bedrock incision rate or the sediment transport flux can be expressed as a power function of slope and drainage area with a potentially non-negligible erosion threshold [Howard, 1980, 1994; Snyder et al., 2003a; Tucker and Bras, 1998; Whipple and Tucker, 2002; Willgoose et al., 1991a]. In the following theoretical derivation, we assume that (1) climate and lithology are spatially uniform, that (2) topography is in steady-state and that uplift rate is the only spatially variable forcing that caused a measurable change of topographic forms. These are plausible assumptions for the basins we have studied in the Siwaliks Hills and that we discuss in section 3.

[11] Depending on the sediment transport length Lt in the river [Davy and Crave, 2000], the long-term evolution of the river bed is driven either by the need to incise bedrock (i.e., detachment limited conditions when Lt is several times larger than the basin length), by the need to transport sediment (i.e., transport-limited conditions when Lt is several times smaller than the basin size) or by a mix of both conditions with a progressive transition from upstream detachment limited conditions to downstream transport limited conditions (i.e., when Lt is of the order of magnitude the basin length). Whipple and Tucker [2002] have recently examined the expected steady-state slope area relationship of rivers in function of the three different sediment transport conditions, and for different dependencies between incision rate and sediment flux. They found that the power law relationship between slope and drainage area is predicted at steady-state for almost all models and sediment transport conditions, so that there is no unique erosion-transport model that can be derived from equation (1) [see also Lague et al., 2003; Tucker and Whipple, 2002]. In this study, we face the same uncertainty regarding the pertinence of transport-limited, detachment-limited or mixed conditions for the long-term modeling of debris-flows dominated channels. For the sake of simplicity, we only retain the first two formulations that are the most used in river studies [Howard et al., 1994; Sklar and Dietrich, 1998; Smith and Bretherton, 1972; Tucker and Whipple, 2002; Whipple and Tucker, 1999; Willgoose et al., 1991a] and hillslope evolution [Braun et al., 2001; Carson and Kirkby, 1972; Fernandes and Dietrich, 1997; Kirkby, 1971; Roering et al., 1999]. In particular, transport-limited conditions are expected to be favored in comparison to detachment-limited ones by a high supply of colluvium from hillslopes or a high rate of bedrock weathering.

[12] The incision law and sediment transport models that we used are based on the stream power law family of models that is widely used for modeling river incision [Howard, 1980, 1994; Howard and Kerby, 1983; Snyder et al., 2000; Whipple and Tucker, 1999] and fluvial sediment transport [Howard et al., 1994; Smith and Bretherton, 1972; Willgoose et al., 1991b]. Several studies have pointed out that erosion thresholds such as the critical shear stress for incipient motion of bedload sediment, or initiation of bedrock plucking, might play a key role in the dynamics of topography and the scaling of topography with uplift rate [Howard, 1980; Lague et al., 2003; Snyder et al., 2003a; Tucker and Bras, 1998, 2000; Tucker and Slingerland, 1997]. We thus introduce explicitly a threshold–in a very general form–in our model.

[13] In the case of transport-limited processes, the local mass balance depends on the gradients of sediment fluxes along flow paths. A simple expression arises when integrating over the entire basin area:

display math

where 〈UA is upstream average uplift rate, A is drainage area, and Qs the total sediment flux over the entire flow width. We use a constitutional transport equation for Qs that is based on sediment transport laws for alluvial rivers [Howard, 1980, 1994]:

display math

where S is slope, kQ, m, n are constants and ξQ is a sediment transport threshold which might depend on A and S.

[14] In the case of detachment-limited processes, the mass balance is local, and the equations turn into:

display math
display math

where E is the local incision rate, U is the local uplift rate, kE, m′, n′ are constants and ξE is an incision threshold which might depend on A and S. Equation (5) is formulated by analogy with the bedrock incision law proposed by Howard [1980, 1994] and Howard and Kerby [1983].

[15] Equations (2) and (3), or (4) and (5), give the theoretical expression of the steady-state area-slope relationship in the case of transport limited processes and detachment limited processes respectively [Howard, 1994; Lague et al., 2003; Snyder et al., 2003a; Tucker and Bras, 1998]:

display math
display math

In the case of rivers, the model parameters in equations (3) and (5) can be related to the mechanics of incision [Whipple et al., 2000] or sediment transport [Howard et al., 1994; Willgoose et al., 1991b], and equations (6) and (7) thus give a physical meaning to the observed slope-area relationship in natural rivers (equation (1)). However, it is important to note that any power law relationship between slope and drainage area, whether it corresponds to rivers, colluvial valleys or hillslopes, can be interpreted in terms of a sediment transport or incision model of the stream power family model (equations (3) and (5)).

[16] As pointed out by several authors [Lague et al., 2000; Snyder et al., 2000; Tucker and Whipple, 2002; Whipple and Tucker, 1999, 2002; Willgoose, 1994; Willgoose et al., 1991a], an analysis of a single area-slope relationship gives only access to the relative value of m and n (resp. m′ and n′), whereas no information on the other model parameters can be derived. However, equations (6) and (7) show that if the dependency of the steepness and concavity indexes with uplift rate is known, then all the model parameters can be theoretically back-calculated [Lague et al., 2003; Snyder et al., 2000]. This point has motivated our study. The following section presents the expected steepness and concavity index relationship with uplift rate for erosion-transport processes that are relevant to this work.

2.2. Steepness and Concavity Indexes' Dependency on Uplift Rate

[17] A necessary condition for stream power law models of equations (3) and (5) to apply, is that the concavity index should be independent of uplift rate (equations (6) and (7)). If any significant correlation were found, that would suggest that either the uplift rate changes with drainage area [Kirby and Whipple, 2001], or that the slope and drainage area exponents m and n (resp. m′ and n′) depends on sediment flux.

[18] Some particular processes lead to specific concavity index and steepness index-uplift rate relationships. These are noteworthy, as they might be significant in the Siwaliks:

2.2.1. Linear Diffusion Processes With Negligible Sediment Transport Threshold (e.g., Soil Creep [Carson and Kirkby, 1972])

[19] Linear diffusion processes with a negligible sediment transport threshold belong to the transport-limited case with m = 0, n = 1, ξQ = 0 which gives k = 〈UA/kQ and θ = −1 (equation (6)), i.e. a positive scaling between slope and drainage area [Willgoose et al., 1991a]. This kind of scaling has been observed with high resolution DEMs on soil-mantled hillslopes over a distance of approximately 100 m [Montgomery, 2001; Montgomery and Foufoula-Georgiou, 1993]. In that case, the steepness index should be proportional to uplift rate.

2.2.2. Threshold Slope Landsliding

[20] In the case of threshold slope landsliding, all slopes are theoretically close to a threshold angle (S = Sc) which leads to approximately straight hillslopes (i.e., θ ∼ 0). In reality, Sc is expected to be variable in a drainage basin [Burbank et al., 1996; Montgomery, 2001; Schmidt and Montgomery, 1995], and in that case, numerical simulations show that the steady-state area slope relationship calculated for an entire basin might be, on average, a power law relationship with a slightly positive concavity exponent [Tucker and Bras, 1998]. Given that slopes are close to or at the threshold angle of landsliding, the steepness index should be independent of uplift rate.

2.2.3. Pore Pressure-Activated Landsliding

[21] In the case of pore pressure-activated landsliding, the theoretical slope-area relationship is not a power law, but rather a succession of two different relationships [Montgomery and Dietrich, 1994; Tucker and Bras, 1998]: for small drainage areas, the area-slope relationship is curved (convex-up) in log-log space (reflecting partially saturated landsliding), and then approaches horizontal for larger drainage area (saturated landsliding). This mechanism has a specific area-slope signature, but DEM errors and high variability in non-saturated and saturated failure angles may eventually blur it, and lead to an approximate power law relationship with a negative exponent given that non-saturated landsliding occurs for higher slopes than in the saturated case. However, if such a power law relationship is observed, the power law exponent that is measured should increase with uplift rate up to a constant value corresponding to the case where landsliding is completely non-saturated at small drainage area and saturated at large drainage area. In that last case, the steepness index is independent of uplift rate.

[22] Hence, except for erosion transport processes governed by threshold conditions, we expect that a spatial variation of uplift rate should induce a change in the steepness index value that is closely related to the parameters of the erosion-transport model. In particular, if the sediment transport (resp. incision) threshold is negligible, the steepness index-uplift rate relationship is a power law whose exponent is 1/n (resp. 1/n′) [Snyder et al., 2000; Whipple and Tucker, 1999]. Conversely, if the threshold is non-negligible, the steepness index-uplift rate relationship predicts a strictly positive steepness index for zero uplift rate [Lague et al., 2003; Snyder et al., 2003a, 2003b].

3. Data and Methods

[23] We have analyzed 10 basins located in the Siwalik Hills, on a large anticline located along the Main Frontal Thrust, 60 km to the south of Katmandu (Figure 2). This location corresponds to the southernmost active deformation in the Himalayas, and was shortened at a constant rate of ∼21 mm.yr−1 during the Holocene [Lavé and Avouac, 2000]. A recent study of folded fluvial terraces by Lavé and Avouac [2000] has shown that: (1) the fold growth obeys a fault-bend fold model of evolution that can be used to calculate uplift rate from bedding dip measurements and (2) the two main rivers crossing the anticline (the Bagmati and Bakeya rivers) have been at, or near, steady state between tectonic and erosion during the Holocene. Balanced geological cross-sections show that the present topography represents approximately 7% of the total volume of material that has been exhumed since the beginning of shortening (0.6 My ago, [Lavé, 1997]). This suggests that erosion rate over the period of exhumation is on average just less than or equal to the tectonic uplift, indicating that the hypothesis of steady-state for the whole topography (including hillslopes and colluvial channels) is likely plausible. This area is thus an ideal place for our study because: (1) the uplift rate is known everywhere and varies significantly between 6 and 15 mm.yr−1 (Figure 2); (2) the lithology of the anticline core is roughly uniform–it consists of fine-grained sandstones and mudstones (Middle Siwaliks formation)–; (3) the topography is likely close to a long term steady state; (4) the climate can be assumed to be spatially uniform.

Figure 2.

Topography (top) and uplift rates [Lavé, 1997] (bottom) of the Siwalik Hills, 60 km south of Kathmandu (central Nepal). Thin dash-dotted lines delineate the lithologic units: US, Upper Siwaliks; MS, Middle Siwaliks; LS Lower Siwaliks. MFT is Main Frontal Thrust. Position of the 10 studied basins is delineated by a dashed line.

[24] Given the very high uplift rates in the core of the anticline, the topographic relief of the Siwaliks is relatively small (maximum of 920 m with respect to the gangetic plain) (Figure 2). Annual precipitation is very high (>2000 mm.yr−1) and fall mainly during the monsoon period. The valley sides are soil-mantled and often covered by a dense forest (Figure 1), and erosion in this topographic domain is expected to occur mainly as a combination of soil creep, shallow landsliding and debris-flows that are triggered during the monsoon. Given that the two former processes have a specific signature in slope-area diagram (see previous section), we expect to be able to estimate their relative contribution in the shaping and erosion of the non-fluvial part of the topography. Large bedrock landsliding is rare except in the inner gorge of the Bagmati and Bakeya Valleys that have not been studied here.

[25] We use a DEM with a nominal resolution of 20 m derived from a pair of stereoscopic SPOT satellite images and resampled for the present study at 60 m by a median filter, in order to remove a white noise observed at wavelengths less than 50 m [Hurtrez et al., 1999]. The vertical precision of the DEM is about 15 m. As Hurtrez et al. [1999], we use the uplift rate map computed by Lavé [1997] from direct measurements of bedding dip assuming a fault-bent fold model that has been validated by Lavé and Avouac [2000]. After interpolation, the resolution of this map is approximately 100 m with an accuracy that depends on the density of dip measurements [Hurtrez et al., 1999; Lavé and Avouac, 2000]. The studied basins were chosen according to 4 criteria: (1) a high density of bedding dip measurements, in order to have the smallest possible error in uplift estimation, (2) a constant tectonic uplift through the basin, in order to have direct and unbiased estimates of the scaling exponent (we restrict our analysis to basins having a standard deviation in uplift rate equal or less than 1 mm.yr−1), (3) a sufficiently large basin size in order to have meaningful statistics when averaging raw data, (4) a location within the Middle Siwaliks, in order to limit the lithology variations. Applying these criteria results in selecting 10 basins spanning a range of uplift rates between 7.9 and 14.4 mm.yr−1 (Figure 2).

[26] For each basin, we compute the drainage area using an algorithm of steepest slope flow routing, and the local slope is calculated along the flow path over two contiguous pixels (i.e., 60 m). As observed by many authors [Lague et al., 2000; Montgomery and Foufoula-Georgiou, 1993; Willgoose et al., 1991a] a considerable scatter occurs even in a log-log plot of data points. A part of this scatter comes from DEM errors that affect the calculation of slope and drainage area. Incomplete steady-state, intra-basin variations of uplift rate that we have tried to reduce by our choice of basins, as well as possible variation of the lithology of the Middle Siwaliks formation, can also produce such scatter [Moglen and Bras, 1995]. To obtain a general tendency, the slopes are averaged in geometrically spaced bins of drainage area (that is equally spaced in logarithm). The slope-area parameters are then fitted over the average relationship using a least-square algorithm. This procedure gives the same weight to each drainage area, whereas a non-linear fit on raw data is biased by the very large number of small drainage areas which arises from the convergent structure of flow on the colluvial channels.

4. Results

4.1. Area-Slope Scaling of Colluvial Valleys

[27] Figure 3 shows a typical area-slope relationship data obtained for basin 4, in a log-log plot. A large scatter is observed on raw data, but the logarithmic-bin averaging exhibits a power law scaling between 7 10−3 and ∼1 km2, with an area-slope exponent θ of 0.24. Two cut-offs are observed: (1) the DEM resolution effect at 7.10−3 km2, and (2) a critical drainage area Ac of about ∼1 km2, above which slopes become significantly smaller than the power law relationship. The basin area is not sufficiently large to quantitatively interpret this large-scale relationship, but it appears consistent with the one observed for river profiles by Kirby and Whipple [2001] in the same area (θ ∼ 0.43). The cut-off area Ac of about 1 km2 is similar to the bend in the slope-area scaling that has been documented by various authors, and which corresponds to the transition from debris-flow dominated to fluvial channels [Montgomery, 2001; Montgomery and Foufoula-Georgiou, 1993; Snyder et al., 2000; Whipple and Tucker, 1999]. Strictly speaking, the occurrence of debris-flows channels is restricted to colluvial channels, whereas the portions of colluvial valleys where channelization scarcely occurs are also known as unchanneled valleys or valleys heads and are expected to occur for smaller drainage areas [Montgomery, 2001; Montgomery and Foufoula-Georgiou, 1993]. In the case of the Siwaliks we have no evidence of a distinct signature between these two domains, and we thus refer to the topographic domain characterized by the power law relationship that we document as “colluvial valleys”. The main channel profile for basin 4 shows that the colluvial/fluvial transition occurs at 1500 m from the ridge crest (Figure 3b), and that the colluvial relief represents ∼70% of the total basin relief. Note that the area-slope average does not exhibit an increase of slope with drainage area for small drainage area that would have revealed diffusive processes. Given the DEM resolution, if diffusive processes effectively shape the hilltops, it is over a distance smaller than 120 m. We have used the DEM at its nominal resolution (i.e., 20 m) and observed that even at this pixel size, although the analysis is arguably less reliable than at 60 m, the reversal from negative to positive correlation between slope and drainage area is not observed. We thus expect that the hillslope size is less than 40 m.

Figure 3.

(a) Typical area-slope relationship for the entire drainage basin and (b) mainstream profile in the Siwalik Hills (Basin 4, 8.7 km2, U = 13.2 mm.yr−1).

[28] Note that the power law relationship of colluvial valleys is consistent with the threshold slope landsliding case with random critical angles of failure described by Tucker and Bras [1998], but not with the pore pressure-activated landsliding case.

4.2. Colluvial Valley Form Dependency With Uplift Rate

[29] For the 10 basins studied, the critical area Ac of the colluvial/fluvial channel transition does not seem to depend on uplift rate, but this assertion has to be further checked on a larger dataset. Snyder et al. [2000] also observed in the King Range (California, United States) that Ac is roughly independent of uplift rate. Thus, if true in the Siwaliks, this result implies that the steady-state gradient of debris-flow channels and fluvial channels show the same sensitivity to uplift rate.

[30] The area-slope scaling of colluvial valleys have been calculated using a non-linear power law fit between 7 10−3 km2 (e.g., 2 pixels of drainage area) and 0.5 km2, this latter value being chosen smaller than the colluvial/fluvial channel transition to ensure that no points in the fluvial regime are used for calculation. The derived scaling exponent and steepness index are shown in Figure 4. The scaling exponent θ remains approximately constant around 0.24, with 8 basins falling into the 1σ error interval. Thus the stream power law model is likely consistent with data, and predictions of equation (6) or (7) can be used to derive erosion parameters (cf. section 2.2). In order to have a consistent comparison of the steepness index between drainage basins, we calculate it by fixing θ constant at 0.24 (this method was used by Snyder et al. [2000]). The steepness index increases significantly with mean basin uplift rate (Figure 4). This result confirms that threshold slope landsliding is not the dominant process of erosion in this area (cf. section 1.2). The best fit was obtained with a linear relationship that predicts a non-zero value of steepness index for zero uplift rate (black line in Figure 4r2 = 0.97, fit weighted by steepness index errors). According to equations (6) and (7), this fit predicts a significant sediment transport (resp. incision) threshold. The stream power law model without an erosion threshold would have predicted a power law dependency of the steepness index with uplift rate. When applied to data, the power law model predicts an exponent of ∼1/3 (dashed line in Figure 4) with a regression coefficient, r2 = 0.93 (fit weighted by steepness index errors).

Figure 4.

Colluvial valley scaling dependency with uplift rate. (top) concavity index (or power law exponent) θ. Bottom: (a) steepness index k (or amplitude factor) fitted for θ = 0.24; (b) Same data and fits showing the predicted steepness indexes for zero uplift rate. In both figures, the uplift rate error bars are equal to the standard deviation of the mean uplift rate of the corresponding drainage basin, whereas steepness and concavity indexes error bars corresponds to 1σ error of the non-linear fit used on the average slope-area relationship. The linear and non-linear fits for the steepness index-uplift rate relationship are weighted by the errors on steepness index.

[31] The two regression models are,

display math
display math

with k0 = 74.8 mm0.48, kl = 3.3 mm−0.52.yr and kn = 11.43 106 mm0.53 yr. Equations (8) and (9) differ essentially on the slope prediction at low uplift rates. Unfortunately, we cannot explore this domain, because such low uplift rates are encountered in the Upper Siwaliks area located immediately on the North of the fold (Figure 2), which has a very different lithology (loose conglomerate), and where both erosion features and slope-area characteristics are very different. We thus expect that the erosion model for colluvial channels in the Upper Siwaliks area is different from that in the Middle Siwaliks and cannot be used to complete the model presented in Figure 3 for uplift rates smaller than 5 mm.yr−1.

4.3. Interpretation in Terms of an Empirical Stream-Power Law Model

[32] Given the uncertainty on whether transport of sediment or incision of bedrock controls the long-term geometry of colluvial valleys, equations (8) and (9), can be interpreted either in terms of transport-limited or detachment-limited processes that obey a stream power law model using equations (6) and (7), respectively. Thus, from the comparison between the linear steepness index-uplift rate relationship (equation (8)) and equations (6) and (7), the model parameters for the transport-limited and detachment-limited conditions would be respectively: m = 1.24, n = 1, ξQ = k0A, and m′ = 0.24, n′ = 1, ξE = k0. This gives the following sediment transport and incision laws:

display math

[33] Likewise, the comparison between the power law relationship between steepness index and uplift rate (equation (9)) and equations (6) and (7) gives: m = 1.75, n = 3.13, ξQ = 0 and m′ = 0.75, n′ = 3.13, ξE = 0, which leads to the following sediment transport and incision laws:

display math

[34] Although no definite argument can be pushed forward, we argue in favor of the linear-threshold model (equation (10)) for 3 reasons:

  • The observation that the critical drainage area for the colluvial/fluvial transition does not seem to depend on uplift rate, suggests that the slope exponent is identical for debris-flows and bedrock channels. Kirby and Whipple [2001] found values of n′ between 0.7 and 1 for fluvial channels suggesting that the linear-threshold model is the most suitable for colluvial valleys.
  • Given that the topographic relief measured at length scales between 200 and 400 m is mainly set by the slope of colluvial valleys, the linear dependency between uplift rate and steepness index (equation (8)) that we document is consistent with the linear relationship between topographic relief and uplift rate found by Hurtrez et al. [1999].
  • The erosion threshold that appears in equation (10) is physically consistent with most mechanistic approaches, and with the observation that erosion of valley sides in mountainous areas often starts above a certain level of precipitation [Hovius et al., 2000]. Note that the threshold does not need to be large to produce significant steepness index at zero uplift [Lague et al., 2003; Snyder et al., 2003a, 2003b].

5. Discussion

5.1. A Stream Power Law Model for Debris-Flow Incision

[35] The previous results suggest that neither diffusive processes, nor threshold dominated processes such as threshold slope or pore pressure-activated landsliding are dominant in the range of drainage area that we study. We thus assume that debris-flows are the dominant erosion-transport processes in the Siwaliks Hills colluvial valleys, and that the incision (resp. transport) model that we derive (equation (10)) pertains to this process. It might be surprising that a model developed for the long-term incision and sediment transport in rivers, proves to be consistent–except for the exponent values–with the long-term erosion of debris-flow that have a significant different rheology and obey different hydraulics than fluvial rivers [Iverson, 1997; Iverson and Vallance, 2001; Takahashi, 1991]. Basically, we find that debris-flow erosion (1) increases linearly with slope, (2) increases non-linearly with drainage area and (3) needs a critical threshold to be overcome. The first point is not unrealistic, especially if incision of bedrock (or sediment transport) is somehow a function of the shear stress at the base of the debris flow. As the slope increases, the shear stress increases and so does the bedrock incision rate (resp. sediment transport capacity). An ad hoc shear-stress model for debris-flow erosion has even been used in a study of colluvial valley evolution [Lancaster et al., 2001]. The interpretation of the second point is less straightforward: drainage area (as a proxy for water discharge) might modulate the sediment concentration and consequently the flow rheology, but also the flow depth, depending on the variation of channel width with drainage area, for which we have no constraints. In particular, if flow depth increases with drainage area, so does the basal shear stress and the incision rate if it depends on this parameter. At last, the third point is consistent with the existence of a non-negligible critical shear stress for bedrock incision (if detachment limited conditions are considered) [Snyder et al., 2003a, 2003b].

[36] Stock and Dietrich [2003] have recently proposed that the slope-area relationship of channels dominated by debris-flows scour is curved (convex-up) in a log-log space, rather than a power law as we find. This finding would invalidate the use of a stream power like model as we did, or at least in its present version (for instance, an exponent on water discharge increasing with drainage area or sediment flux could fit their observations). As observed [Montgomery and Foufoula-Georgiou, 1993; Seidl and Dietrich, 1992; Snyder et al., 2000; Whipple and Tucker, 1999] or suggested [Sklar and Dietrich, 1998] by various authors, Stock and Dietrich found that the upstream extent of fluvial processes is limited by a critical slope Sc ranging between 0.03 to 0.12, slightly less than in the Siwaliks (according to equation (8) or (9) and using a colluvial/fluvial transition at 1 km2, Sc ranges from 0.13 to 0.16 for U varying between 8 to 15 mm.yr−1). Compared to our work, their analysis is based on a detailed topographic map analysis of single channel profiles where recent debris-flow scour and deposition has been documented in the field, rather than on a DEM-based study of the entire basin. Differences in methodology might explain the discrepancy in results, especially because our method averages several different colluvial channels. Some site specific properties and/or processes can also be invoked since the erosion rates we measured are at least 2 times higher than in their study for the same range of slopes, emphasizing the high erodability of the Middle Siwaliks sandstones.

[37] The fluvial regime is not sufficiently documented in this study (although the concavity exponent θ is consistent with those measured by many authors [Tucker and Whipple, 2002]) to definitively address the issue of the colluvial/fluvial channel transition. Given the wide range of debris-flow rheologies and the relative role of weathering and impact processes in colluvial channels [Howard, 1998] that is expected to depend on site-specific characteristics such as erosion rate, rock lithology, climate and vegetation, we do not argue that our model is general but rather that it should indicate that even if more complexity than fluvial processes can arguably be expected, the long-term erosion model for debris-flow might finally be quite simple, as it is for rivers.

5.2. Consequences for the Scaling of Relief With Uplift Rate

[38] Independent of the interpretation in terms of a stream power law model, we have documented a clear relationship between the steepness index and uplift in the Siwaliks Hills that is likely linear. Given that fluvial processes depend linearly on uplift rate in this region [Kirby and Whipple, 2001], these findings show that the entire topography (except maybe hillslopes, if they are at a critical angle of failure) is responding linearly to tectonic forcing. This gives an explanation in terms of erosion-transport law to the linear relationship between local relief (and also mean elevation) and uplift rate observed by Hurtrez et al. [1999]. As for fluvial channels [Kirby and Whipple, 2001], the geometry of colluvial channels can be used to infer uplift rate from the topography (provided that lithology and climate are identical to the Middle Siwaliks) [Lague et al., 2000; Snyder et al., 2000]. Our study also suggests that a threshold-slope or pore pressure-activated landsliding model is not adequate for modeling the long-term evolution of the non-fluvial part of the topography in the Siwaliks, even if the uplift rates are very high.

[39] Given the large proportion of basin relief represented by colluvial valleys in the specific case of the Siwaliks and in other settings [Stock and Dietrich, 2003], our study strongly advocates the use of a specific erosion-transport law–similar to equation (10)–for debris-flow processes between the range of drainage areas characterizing hillslopes and river processes. A binary description of the topography in terms of fluvial channels and landslide dominated hillslopes would probably not catch the relevant scaling relationship between relief and uplift rate in the Siwaliks area (and in others area where debris-flows are quantitatively important), and might possibly miscalculate the response time of the topography to tectonic perturbations.

[40] The existence of a non-negligible threshold of erosion has several important implications for relief form and dynamics. It predicts non-zero slopes in relaxation systems (that is without uplift). Equation (8), for instance, would predict slopes as large as 10° for a drainage area of about 0.1 km2 and no uplift if we assume that the erosion process remains similar–which is certainly not true. We also point out that including or excluding a threshold or not has a strong implication for the prediction that can be made on the slope exponent n (resp. n′) [Lague et al., 2003; Snyder et al., 2003a]. Indeed neglecting the erosion threshold in the erosion model would have led us to predict very large slope exponents n (resp. n′). This point addresses important issues with respect to the tectonic/erosion coupling since the exponent n (resp. n′) controls the response time of topographic dynamics with respect to uplift rate [Lague, 2001; Whipple and Tucker, 1999].

6. Conclusion

[41] We have shown that the stream-power law model, which is classical for fluvial sediment-transport processes, is also likely consistent with debris-flow scour in colluvial valleys. Theoretical predictions for slope-area-uplift relationship have been derived and have been compared to an analysis of the relationship between colluvial channel geometry and uplift rate in the Siwaliks Hills of central Nepal. We found that colluvial valley geometry is well described by a power law relationship between local slope and drainage area. The amplitude of this power law (steepness index) increases linearly with uplift rate, whereas the scaling exponent is roughly constant and equal to 0.24. This power law relationship is valid up to ∼1 km2; a different scaling is observed for larger areas that expresses fluvial erosion processes. The stream power law model of debris-flow erosion is consistent with these observations if the slope exponent = 1, the drainage area exponent = 0.24 if detachment limited are considered, or = 1.24 for transport limited conditions, and if a nonnegligible threshold for incision is taken into account. The linear dependency with local slope for colluvial channels is consistent with the linear dependency between channel slope and incision rate found by Kirby and Whipple [2001] in the downstream fluvial channels. These results reinforce the often-overlooked importance of geomorphic thresholds in landscape analysis discussed by others [Lague et al., 2003; Snyder et al., 2003a, 2003b], and point out that neglecting this aspect might result in incorrect parameter estimation for the slope exponent of the stream power law model.

[42] We cannot define whether the transport-limited, detachment-limited or even mixed models of evolution should be used to model the long-term evolution of colluvial valleys. This issue can only be addressed by an analysis of the transient dynamics [Lague et al., 2003; Tucker and Whipple, 2002; Whipple and Tucker, 2002]

[43] As underlined by Stock and Dietrich [2003], colluvial valleys can represent a large part of basin relief (between 50 and 70% in the Siwaliks), and thus are likely to play a key role in the long-term dynamics of mountainous drainage basins. Efforts should be directed towards a mechanistic description of debris-flow scour and the development of small-scale laboratory experiments that can provide insightful constraints on this process.


[44] We thank Jean-Philippe Avouac, Kelin Whipple and an anonymous reviewer for constructive criticism that helped improve this paper. Jerome Lavé and Jean-Emmanuel Hurtrez are greatly acknowledged for providing topographic and uplift rate data for the Siwaliks. We also thank Colin Stark for helpful discussions. This study was funded by the CNRS-INSU research program PNSE (Programme National Sol et Erosion).