Hydraulic geometry, river sediment and the definition of bedrock channels

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Geomorphology 99 (2008) 26 – 38 www.elsevier.com/locate/geomorph

Hydraulic geometry, river sediment and the definition of bedrock channels Jens M. Turowski a,⁎, Niels Hovius a , Andrew Wilson a , Ming-Jame Horng b a

Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge, CB2 3EQ, United Kingdom b Water Resources Agency, Ministry of Economic Affairs, Hsin-Yi Road, Taipeh, Taiwan, ROC Received 2 July 2007; received in revised form 2 October 2007; accepted 8 October 2007 Available online 13 October 2007

Abstract A growing literature deals with bedrock channels, erosion processes within them, and their role in landscape evolution. Formal definitions currently in use classify channels as bedrock when alluvial cover is discontinuous or thin. This is equated with a physical property of the flow. Using up-to-date erosion laws and descriptions of channel dynamics, we show that existing definitions are difficult to apply and may lead to conflicting classifications. We propose the definition: A bedrock channel cannot substantially widen, lower or shift its bed without eroding bedrock. Applied to channels in Taiwan, this leads to a different classification than previous definitions. By analysing at-a-station hydraulic geometry of these channels, we show that the classification exposes different controls of sediment effects on dynamics of channels classified as alluvial and bedrock. © 2007 Elsevier B.V. All rights reserved. Keywords: Bedrock channels; Hydraulic geometry; Fluvial geomorphology; Sediment effects; Taiwan

1. Introduction Bedrock rivers play a key role in erosional landscape evolution. They set the baselevel for hillslopes, transport sediment to depositional basins, and communicate changes in tectonic and climatic boundary conditions throughout the landscape (Whipple, 2004). In recent years, considerable attention has been given to bedrock channel morphology and dynamics, and to fluvial erosion processes (e.g., Howard, 1994; Wohl et al., 1994; Tinkler and Wohl, 1998a; Stock and Montgomery, 1999; Whipple et al., 2000a,b; Lavé and Avouac, 2001; Wohl and Merritt, 2001; Hartshorn et al., 2002; van der Beek and Bishop, 2003; Snyder et al., 2003a,b; Duvall et al., 2004; Finnegan et al., 2005; Stock et al., 2005; Stark, 2006; Wobus et al., 2006a,b; Whittaker et al., 2007). It has been argued that bedrock channels adjust their shape by eroding bedrock until a steady state is achieved in which the vertical erosion equals tectonic advection of rock mass (e.g., Whipple and Tucker, 1999; Stark, 2006; Wobus et al., 2006b). In contrast, alluvial channels, set in unconsolidated sediment, ⁎ Corresponding author. Current address: WSL Birmensdorf, Zürcherstrasse 111, 8903 Birmensdorf, Switzerland. Tel.: +41 044 7392 424. E-mail address: [email protected] (J.M. Turowski). 0169-555X/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.geomorph.2007.10.001

adjust their shape by entraining and redepositing bed material, reaching a steady state configuration, or grade, when the flow can transport all sediment supplied from upstream without net deposition or erosion (e.g., Mackin, 1948; Parker, 1978a,b; Vigilar and Diplas, 1997, 1998). When the sediment supply exceeds the transport capacity of a river, its sediment load is said to be transport-limited and aggradation of the channel bed results. Bedrock channels are commonly deemed detachmentlimited as the capacity of the river to carry sediment exceeds the supply of material to the channel. Nevertheless, sediment transport and alluvial cover play a fundamental role in fluvial bedrock erosion (Sklar and Dietrich, 1998, 2001; Whipple and Tucker, 2002; Sklar and Dietrich, 2004, 2006) and channel evolution (Shepherd, 1972; Hancock and Anderson, 2002; Korup, 2006; Finnegan et al., 2007; Johnson and Whipple, 2007; Turowski et al., in press, 2007). Even in detachmentlimited conditions bedload can act as dynamic (mobile) cover, shielding parts of the channel bed (Sklar and Dietrich, 1998, 2001, 2004; Turowski et al., 2007), and the distribution of loose material in the channel sets the distribution of erosion in the channel cross section (Finnegan et al., 2007; Turowski et al., in press). Similarly, fluvial bedrock erosion can occur even when much sediment is present in the channel and the conditions are transport-limited. In any case, erosive events are rare (Molnar,

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2001; Lague et al., 2005), and the processes shaping bedrock channels may be inactive for a large fraction of the time. Existing definitions stress the lack of extensive bed cover as an essential attribute of bedrock channels. When this observational criterion was applied to 81 river channels in Taiwan, only two were classified as “bedrock channel”. However, many of these channels are located within a rapidly uplifting and eroding mountain belt and are bounded in some way by bedrock outcrops. Therefore they are commonly and informally described as bedrock channels, irrespective of the formal definition: a discrepancy exists between current definitions of bedrock channels and common use of this term in the field. Any functional definition should reflect channel properties and dynamics where they differ from alluvial channels on all scales. Furthermore, the definition should allow easy distinction of alluvial and bedrock channels, both in the field and by using maps, digital elevation models, and remote sensed imagery. This requires a focus on channel-shaping events or channel dynamics rather than on the bed configuration observed at a given time. Here we discuss two popular definitions of bedrock channels and argue that they have limited efficiency and discriminatory power. We propose a new definition based on processes and dynamics of bedrock channels. Applying this definition, we determine the bedrock or alluvial nature of 81 river channels in the Taiwan mountain belt and analyse the at-a-station hydraulic geometry of these channels. We show that the relationship between channel cross-sectional shape and mean suspended sediment concentration (used as a measure of the total river load) is different for alluvial and bedrock rivers in our sample. This difference can be explained by the different mechanics of the channel-shaping processes, and it confirms the resolving power of our definition for sets of channels. 2. Definitions of bedrock channels In accordance with earlier definitions of bedrock channels (Gilbert, 1877; Howard, 1980; Howard et al., 1994; Montgomery et al., 1996), Whipple (2004) wrote: “Bedrock channels lack a continuous cover of alluvial sediments, even at low flow, and exist only where transport capacity exceeds sediment flux over the long term.” A second definition by Tinkler and Wohl (1998b) identifies “bedrock channels as those reaches along which a substantial proportion of the boundary (≥ 50%) is exposed bedrock, or is covered by an alluvial veneer which is largely mobilised during high flows such that underlying bedrock geometry strongly influences patterns of flow hydraulics and sediment movement.” Both statements define bedrock channels according to the extent of alluvial cover on the bed and equate scarce cover with a physical condition in the river. Whipple (2004) assumed that large, uncovered stretches only occur when the sediment load of the river is detachment-limited in the long term. However, deposition of sediment may occur in bedrock channels, for example in the inner bend of an active meander, limiting bedrock exposure to thalweg and/or outer channel wall. More commonly, a riverbed may be covered by alluvium for most of the time, while excess sediment is removed episodically during high stage floods.

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Although the river in this scenario is detachment-limited in the long term, and thus complies with the second half of Whipple's (2004) definition, it is difficult to recognise in the field without detailed analysis of discharge and sediment transport data, or continuous observation over long periods of time. Tinkler and Wohl (1998b) postulated that the underlying bedrock influences hydraulics and sediment movement in bedrock channels. The cut-off value of 50% exposed or thinly covered bedrock is arbitrary. Again, to use the second half of the definition for classification, extensive field observations for a range of flow stages are necessary to enable classification. Data on flow dynamics and extent of sediment movement during high stage floods are rare, and methods for systematic observation have not yet been developed. Flood discharges are commonly assumed to strip cover and expose underlying bedrock channel floors. However, Turowski et al. (in press) have argued that abundant supply of sediment from upstream locations at high flow stage may enhance the probability of extended cover with increasing discharge and that bedrock exposure may in fact be more common during low flow stages in certain climatic and tectonic conditions. In the absence of much relevant data, the easy solution is to classify rivers with patches of exposed bedrock and gravel-bed reaches, i.e., most mountain rivers, as “mixed bedrock-alluvial” rivers. However, this classification lumps together two channel types with fundamentally different dynamics that are unlikely to alternate at reach scale. 3. Channel bed cover and a new definition of bedrock channels Sediment has a dual role in bedrock channels. Abrasion and quarrying, driven by the impact of moving sediment particles on the bedrock river bed, are the dominant mechanisms of fluvial bedrock erosion (Whipple et al., 2000a; Hartshorn et al., 2002; Sklar and Dietrich, 2004). Increasing sediment supply will increase the number of impacts per unit bed area and time and with it the erosion rate (the tools effect). However, a further increase of sediment supply may result in increased bed cover, shielding the bed from impacts, and decreasing the erosion rate (the cover effect). Therefore, as a first step toward a robust, workable definition of bedrock rivers considering bedload sediment transport and the development of an alluvial cover on the bedrock channel bed is important. Slingerland et al. (1997) and Sklar and Dietrich (1998) proposed a linear cover model, where the fraction of the bed covered by alluvium Ra is proportional to the ratio of sediment supply Qs and bedload transport capacity Qt: Ra ¼

Qs ; for Qs bQt Qt

ð1Þ

Ra ¼ 1; for Qs zQt : In accordance with the definition by Whipple (2004), Eq. (1) predicts the bed to be fully covered when Qs is equal to Qt, which occurs at the transition between detachment-limited and transport-limited conditions. If the fluvial bedrock erosion rate

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erosion by saltating bedload (Sklar and Dietrich, 1998, 2004), steady state erosion rate E in a channel can be written as (Turowski et al., 2007): E ¼ KQs e E ¼ 0;

Fig. 1. Illustration of the erosion model described by Eq. (3), using three different values for the cover factor φ. For comparison, the predicted form of the function using the linear cover model is shown as well. For Qs N Qt the channel is said to be transport-limited; for Qs b Qt the channel is said to be detachmentlimited. The erosional response is tools-dominated for the region to the left of the maximum and cover-dominated for the region to the right of the maximum.

is proportional to the fraction of exposed bed (Sklar and Dietrich, 1998), then it drops to zero when the system becomes transport-limited. Eq. (1) has no physical or observational basis. Turowski et al. (2007) have derived an alternative, exponential model of bed cover based on a probabilistic argument. It gives the fraction of bed cover in closed systems (e.g., recirculating flumes) and detachment-limited open systems (e.g., river channels where Qs b Qt) as: Qs

Ra ¼ 1  euQt

ð2Þ

In detachment-limited conditions (Qs b Qt), all transportable sediment is mobile and, in the absence of coarse, static grains, the cover is dynamic. In contrast, in transport-limited open systems (Qs N Qt) surplus sediment is deposited on the aggrading bed steadily increasing the extent of cover until all bedrock is protected from erosion (i.e., Ra = 1). The fraction of sediment that is transported accounts for dynamic cover, while the remaining sediment forms a static cover. The cover factor φ in Eq. (2) is dependent on bed topography and is equal to one for a flat bed in uniform flow conditions. Combining Eq. (2) with the model of

uQQs ; t

for Qs bQt

ð3Þ

for Qs zQt

Here, K is a parameter dependent on rock and substrate properties and local hydraulics. The function expressed by Eq. (3) has a maximum at Qs = Qt / φ (Fig. 1). On the rising limb of the function, the linear dependence of the erosion rate on Qs is due to the tools effect (Foley, 1980; Sklar and Dietrich, 2004). The falling limb is dominated by the cover effect. Clearly the detachment-limited region in Eq. (3) may extend beyond the tools-dominated domain if φ is larger than one (Fig. 1). Exposed bedrock and active erosion may occur in the transportlimited domain as long as φ is smaller than one. Therefore, both cover models discussed above (Eqs. (1) and (2)) predict active erosion even when bed cover is substantial but incomplete. This notion is carried into our definition of bedrock channels. We propose the following, qualitative definition: A bedrock channel cannot substantially widen, lower, or shift its bed without eroding bedrock. This definition is based on a consideration of channel dynamics and can easily be applied in the field or by using remotely sensed imagery and topographic data for classification. Field indicators are bedrock outcrops in the river channel, thin or discontinuous alluvial cover, and/or steep bedrock channel walls. Here, the term bedrock channel is used in contrast to alluvial channel, expressing a difference in dynamics rather than a channel state in time. The term rock-bed channel is proposed to describe channels with extensive stretches of exposed bedrock in the channel bed, in analogy to gravel-bed or sand-bed channels (compare to the use of the term “rockbed” in Tinkler, 1997a,b and with the comment by Tinkler and Wohl, 1998b, page 15, paragraph 1). End members of the range of channels classified as bedrock channel according to our definition are schematically depicted in Fig. 2. The field indicators listed above are similar to those used in previous classification schemes. In fact, channels classified as bedrock according to previous definitions form a subset of the range of bedrock channels according to our new definition. This range also includes so-called mixed bedrock-alluvial channels with alternating covered and uncovered stretches. However, importantly, our definition brings a shift from the observation of

Fig. 2. Schematic depictions of end members classified as bedrock according to the proposed definition. (A) Channel confined entirely in bedrock, showing both steep bedrock walls and exposed bedrock within the channel. (B) Channel with steep bedrock walls, but thick alluvial cover in the bed. (C) Channel with exposed bedrock bed, but set in an alluviated plain.

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the physical state of the river bed to the consideration of the channel-shaping processes and channel dynamics. Thus, there is a distinction between classifications that describe the channel at a specific time and location as a gravel-bed, sand-bed, or rock-bed channel and classifications that are based on channel-shaping processes and that distinguish between alluvial and bedrock channels. For example, there can be gravel-bed bedrock channels as well as gravel-bed alluvial channels. However, there cannot be a rock-bed alluvial channel. The proposed definition has a considerable interpretative component. For instance, what precisely is meant by “substantial”? We consciously decline to give limiting boundaries in

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some measurable physical parameters. Given our current state of knowledge any choice of parameters cannot be supported by data and would essentially be arbitrary. Furthermore, without specified parameter bounds the system as a whole can be judged by the observer, who can take more aspects into account than a few parameters. In the next section we apply the proposed definition to a set of channels in Taiwan and show that it exposes different statistical relationships for alluvial and bedrock channels, despite this interpretative component. Although these differences cannot be used to validate the classification of a single channel, they discriminate effectively between sets of channels.

Fig. 3. Location map of the gauging stations in Taiwan. Gauging stations in bedrock reaches are represented by triangles, stations in alluvial reaches by circles.

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4. Classification of Taiwanese rivers To test the efficiency and practicality of our definition of bedrock rivers, we selected 81 river channels in and around the active Taiwan orogen. The Taiwan mountains are uplifted and exhumed at rates of 5–10 mm/y (Dadson et al., 2003; Willett et al., 2003). Throughout, steep-sided valleys cut across the structural grain of the mountain belt, traversing metamorphic gradients and seismically active faults. Inside the mountain belt, valley floors are narrow with limited space for alluviation; but high rates of sediment supply from adjacent hillslopes cause extensive gravel mantling of channel beds. Toward the mountain front, valleys open up and broader floodplains have formed. The mountain belt is flanked by alluvial plains, most prominently in the west, where amalgamated alluvial fans grade into the Taiwan Strait. This setting presents a wide range of channel types with different degrees of alluviation; and given their tectonic context, on longer time scales many river channels inside the Taiwan orogen must cut into bedrock. Therefore, Taiwan is a demanding site for a comprehensive test of our definition. If the definition works well, then it should effectively discriminate between bedrock channels and alluvial channels with fundamentally different hydraulic geometries. We have determined the topographic context of 81 Taiwanese channel sites at hydrometric stations of the Water Resources Agency (WRA) using Landsat 7 images with 28.5-m pixels and a digital elevation model with 40-m grid cells (Fig. 3 shows a location map). Our routine consisted of a broad characterization of the valley cross section, a search for bedrock outcrop in the channel bed and/or bank (as far as recognisable at the available resolution), delineation of alluvial deposits on the valley floor, a comparison of the floodplain size to the size of the active channel, and assessment of the general topographic context. As an example, the available data for the Kaoping River near Maolin village is shown in Fig. 4. The Ta-ChinChiou gauging station (WRA number 1730H041) is located near the mountain front. On the satellite image (Fig. 4A) and in the topography (Fig. 4B) the high relief around the channel can be seen (compare to the low-relief region to the west). Shadow effects indicate steep walls (probably bedrock) adjacent to the channel (marked with I). The active channel takes up most of the valley floor upstream of the station. Several landslide scars can be seen in the surrounding terrain (II), indicating steep, active hillslopes and thin soil mantles. Field observations have confirmed these interpretations (Fig. 4C). Using similar observations, we classified 46 of 81 channels as bedrock channels. Most stations were inspected in the field in March 2006 to verify their classification and to adjudicate uncertain cases. At this time, only two channels classified as bedrock river had extensive stretches of exposed bedrock within the channel (WRA stations 1430H030 and 1430H042 on the Wu River). Thus, using the bed cover criterion in the definitions by Whipple (2004) and Tinkler and Wohl (1998b), nearly all channels in our test would have been classified as alluvial channels. Lack of time coverage made it impossible to assess whether streams are detachment-limited in the long term, and whether cover is mobilised and flow affected by underlying

Fig. 4. Illustration of the classification of channels: (A) Satellite image of the region around the Kaoping River at Maolin Village near Ta-Chin-Chiou Station (1730H041). The gauging station is marked with a box, and the location of the picture with a star. I denotes the location of shadows indicating steep walls near the channel and II. landslide scars. (B) Shaded DEM (200 m pixel size) of the region to show topographic context. Note the high relief in the surrounding area and the narrow valley. (C) Photograph of the channel, looking upstream. Note the steep bedrock valley walls, thick alluvial cover and channel braiding in the lower part of the picture.

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Fig. 5. Schematic cross sections of bedrock channels in Taiwan. (A) Channel set entirely in bedrock. Alluvial cover is thin and patchy. (B) Alluvial cover is frequent with patchy bedrock exposure, but thin. (C) The channel bed is wrapped in alluvium. Cover is of small to medium thickness. (D) As (C), the thickness of the alluvium is not known and may be substantial. (E) The channel is set primarily in alluvium, but one bank is delimited by a bedrock wall. The other bank is vegetated. The vegetated stretch is of comparable width as the active channel and end at a bedrock wall. (F) Planform view of a channel, including section of the types (D) and (E). The flow touches the bedrock valley walls in the outer bends. This geometry is very common in Taiwan. (G) Paotzuliao River (no gauging station). The water flows directly over bedrock, alluvial cover is patchy. H) Wu River near Chin-Fon Bridge gauging station (number 1430H042). Bedrock is exposed along the far bank, thin alluvial cover on the near bank. I) The active, gravel-bedded channel of the Lehe River is confined by steep bedrock walls. The flow touches the rock in bends. Occasional vegetated bars can be found in the valley. J) Photograph of the Peinan River near Hsin Wu Lu station (number 2200H020), looking downstream. The wide valley is flanked by bedrock walls. To the left of the picture, vegetated banks in the inner bend of a meander can be seen. A colour version of this figure can be found in the online material.

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bedrock at high stage floods. A classification according to these criteria was therefore not possible. Fig. 5 shows schematic representations of bedrock channels in Taiwan according the proposed definition, as well as field photographs of representative examples. The progression from (A) to (E) in Fig. 5 reflects increasing amounts of alluvium in the channel. While alluvium is largely absent in (A), it covers the channel floor in (B) and wraps the entire channel in (C). The thickness of cover in cases (D) and (E) is not known and may be substantial. However, the dynamics of the channel are limited by the bedrock valley walls. A very common geometry in Taiwan is depicted in (E) and (F), in which the valley floor is several times wider than the active channel and confined by steep bedrock walls on one or both sides. The channel floor is covered by gravel; and its planform is somewhat sinuous, touching bedrock in the outer channel bends, while inner bends may be alluviated and vegetated.

Fig. 6. Width–discharge relation at Hsin-Wu-Lu station (number 2200H020), which was classified as bedrock channel. For a picture see Fig. 5J. The trend line is the non-linear best fit to the data using a power law (Eq. (8)) with kw = 1.02 ± 0.02, ω = 0.46 ± 0.01, and R2 = 0.87.

5. Hydraulic geometry of Taiwanese rivers Bedrock and alluvial rivers have different channel-forming processes, and these processes should have an expression in the at-a-station hydraulic geometry of the channel, defined by the relations between channel width W, depth D, and average flow velocity V with discharge Q within a channel cross section. Leopold and Maddock (1953) found that these relations are well described by power laws of the form W ~Qx

ð4Þ

To evaluate the discriminating power of our definition, we have concentrated on the relationship between channel width and discharge to make use of available data. For each of the 81 river stations in this study, more than 1000 discharge and 100 channel width measurements are available. The at-a-station relationship between width and discharge is well described by a power law (Fig. 6):

D~Qd

ð5Þ

W ⁎ ¼ kw ðQ⁎ Þx

V ~Qm

ð6Þ

Here W⁎ is the flow width normalised by the average of all available width measurements for the station, and Q⁎ is the water discharge normalised by the average water discharge at the station. With this normalisation, we remove the mean discharge as a direct control on channel geometry, enabling direct comparison of measurements from stations with different upstream areas. The prefactor kw and exponent ω are dimensionless parameters, found by non-linear fit to the available data. Here, kw corresponds to the ratio between channel flow width at the mean discharge and the

Here ω, δ, and ν are dimensionless exponents. The relations denoted by Eqs. (4) and (5) build on the implicit assumption that cross-sectional geometry of the channel can be described by an equation of the form: ð7Þ

D~W d=x We have adopted this assumption in our work.

ð8Þ

Table 1 Average values from the literature and for Taiwan for the at-a-station exponent of the width–discharge relation (ω) and, if available, the depth–discharge relation (δ) with corresponding standard deviations (denoted by σω and σδ respectively) Stream / Region

ω

σω

δ

σδ

δ/ω

Number of stations

Bollin-Dean (Knighton, 1975) Midwest, USA (Leopold and Maddock, 1953) Brandywine Creek (Wolman, 1955) Ephemeral streams, USA (Leopold and Miller, 1956) White River (Fahnestock, 1963) Rio Manati (Lewis, 1969) New Zealand (Jowett, 1998) France (Lamouroux and Capra, 2002) France (Lamouroux and Souchon, 2002) Australia/England (Stewardson, 2005) Taiwan (all data) Taiwan (alluvial) Taiwan (bedrock)

0.12 0.26 0.04 0.25 0.38 0.17 0.18 0.13 0.13 0.11 0.34 0.35 0.34

0.12 0.16 0.03 0.10

0.40 0.40 0.41 0.41 0.33 0.33

0.10 0.17 0.05 0.14 0.10

3.33 1.54 10.25 1.65 0.87 1.94

0.28

0.09

2.55

12 20 7 10 112 channels 10 73 reaches 58 reaches 28 reaches 54 81 35 46

0.11 0.05 0.05 0.05 0.10 0.10 0.11

The ratio δ/ω is the exponent of the corresponding power law cross section (Eq. (7)).

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mean flow width. ω is a measure of the bank steepness. For the ensemble of rivers in Taiwan, ω = 0.34 ± 0.10 (error gives standard deviation) with maximum and minimum values of 0.64 and 0.14. For alluvial channels, ω = 0.35 ±0.10, and for bedrock channels ω = 0.34 ± 0.11. For comparison, values reported in the literature for rivers elsewhere are given in Table 1. The average prefactor value for all channels is kw = 1.10 ± 0.17, with maximum and minimum values of 0.65 and 1.70. For alluvial channels kw = 1.12 ± 0.10; and kw = 1.09 ± 0.11 for bedrock channels. Values for ω and kw for bedrock and alluvial channels are drawn from similar populations, and a comparison of averages is not sufficient to show differences. Instead, these values should be viewed in the context of other river attributes. The prefactor kw is independent of drainage area, mean discharge, mean suspended sediment concentration (calculated as a monthly weighted average to decrease errors because of biased sampling, as suggested by Dadson, 2004), the coefficients of variation of discharge and sediment concentration, and catchment-wide erosion rate derived by Dadson et al. (2003) for

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both alluvial and bedrock rivers (Fig. 7). To test for correlations, we calculated Kendall's rank correlation coefficient τ, which is a standard statistical method that tests whether a trend in the data is statistically significant or not (e.g., Snedecor and Cochran, 1967). All trends we report in this paper are statistically significant; most are highly significant. The exponent ω is independent of drainage area (Fig. 8A), catchment-wide erosion rate (Fig. 8C), and mean suspended sediment concentration (Fig. 8E) for alluvial channels, while the data exhibit positive trends for bedrock channels (Fig. 8B, D; F). Herein lies a fundamental difference between bedrock and alluvial channels. This difference is expressed most clearly in the relation between the exponent ω and the mean suspended sediment concentration (Fig. 8F). Similar trends for catchment-wide erosion rate and catchment size may derive (at least in part) from the fundamental relation between the mean suspended sediment concentration and the hydraulic geometry of the bedrock channel at a station, given that the catchment erosion rate is the total sediment load of a river divided by its drainage area.

Fig. 7. The prefactor kw of the at-a-station width–discharge relationship (Eq. (8)) is independent of (A) drainage area, (B) mean suspended sediment concentration, (C) the coefficient of variation of discharge (but note the increase in range of kw with increasing coefficient), and (D) the coefficient of variation of sediment concentration for rivers in Taiwan. Filled squares: alluvial channels; open squares: bedrock channels. Whiskers give 95% confidence limits of the best-fit values.

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Fig. 8. Relation of at-a-station width–discharge exponent ω (Eq. (8)) with drainage area catchment-wide erosion rate, and mean sediment concentration for alluvial (A, C, E) and bedrock channels (B, D, F). For alluvial rivers the exponent is independent of the parameters, while for bedrock channels the relations show a positive trend. On (F) data points are separated into basins draining to the west (solid squares), east (open squares), and north (open triangles). Whiskers give 95% confidence limits of the best-fit values.

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6. Interpretation and discussion To illustrate how the exponent ω reflects channel geometry, Fig. 9A shows the channel width–discharge relation at a station for the mean, minimum, and maximum value of the exponents found for Taiwanese bedrock channels. This plot shows that the increase in width with rising discharge is greater for higher values of the exponent. This can be understood as follows. Assuming a power law relation between depth and discharge with a constant exponent δ = 0.4, as found in many channel sets elsewhere (Table 1), width–discharge plots can be converted into channel cross sections (Eq. (7); Fig. 9B). We note that the ratio δ /ω is a more accurate descriptor of the cross-sectional shape of a channel than ω. This ratio shows a negative correlation with ω, well described by a power law, at least for average data reported in the

Fig. 10. Cross-sectional exponent δ/ω plotted against width exponent ω, using data given in Table 1. As expected, small values of ω correspond to steeper channel banks. Whiskers give the standard deviation. The line is a power law fit with δ/ω = (0.28 ± 0.06) ω(− 1.12 ± 0.07) and R2 = 0.99.

literature (Fig. 10). Hence, in general, steep channel walls lead to width–discharge relations with a small exponent ω, while gently rising channel walls lead to larger values for ω. To further validate the relationship between ω and bank steepness, we have measured bank gradients of 11 channel cross sections surveyed by the WRA (Table 2). As expected, ω decreases with increasing bank gradient (Fig. 11). Our results indicate that fluvial erosion shapes bedrock channels as a function of the river sediment load. When the mean suspended sediment concentration is low, bedrock channels in Taiwan are commonly narrow and deep; when the mean suspended sediment concentration is high, they are wide and shallow. This can be understood in terms of the balance of sediment supply to a river reach and the river transport capacity in that reach (Turowski et al., in press). A small sediment load relative to the river transport capacity equates to a scarcity of erosive tools in the channel (tools-dominated domain, cf. Eq. (3)). Mobile sediment is likely to concentrate in the central part of the channel where bed shear stress is greatest. As a result,

Table 2 Bank gradient measured of cross sections surveyed by the WRA for several gauging stations

Fig. 9. (A) Width–discharge relationship for observed minimum, mean and maximum at-a-station width–discharge exponent ω (Eq. (8)). The prefactor has been kept constant at kw = 1.1. (B) Cross-sectional shapes for minimum, mean and maximum exponents derived from the width–discharge relation, assuming a power law relation between depth and discharge with a constant exponent of δ = 0.4. Small exponents ω result from steeper channel banks than large exponents.

Gauging station number

Bank gradient

Exponent ω

1300H013 1300H014 1350H001 1510H024 1510H049 1580H001 2170H001 2180H002 2420H018 2420H036 2420H037

1.44 ± 0.57 0.77 ± 0.07 1.70 1.23 ± 1.60 0.22 ± 0.07 1.30 ± 0.50 0.33 ± 0.03 0.33 ± 0.09 0.05 ± 0.01 0.23 ± 0.04 0.27 ± 0.04

0.22 ± 0.20 0.26 ± 0.01 0.25 ± 0.01 0.23 ± 0.01 0.51 ± 0.02 0.32 ± 0.01 0.33 ± 0.01 0.40 ± 0.01 0.53 ± 0.01 0.51 ± 0.01 0.43 ± 0.01

The bank gradient is the slope of a line fitted to the appropriate stretch of the cross section. Errors give 95% confidence intervals for the best-fit values. The value for station 1350H001 has no error associated with the fit, since the bank was characterized by only two survey points. The data is plotted in Fig. 11.

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Fig. 11. At-a-station width–discharge exponent ω as a function of bank gradient, using the data given in Table 2. Smaller values of ω correspond to steeper channel banks. Whiskers give 95% confidence intervals for the best-fit values.

the river cuts downward. In the extreme, lateral erosion of the channel walls is rare and a slot gorge forms. In contrast, in channels with a large sediment load, episodes when sediment supply outstrips local transport capacity may occur. Then the channel bed will be alluviated and erosion will focus on the channel walls above the cover. Channels in which this condition dominates are wide and shallow with gently rising banks. The role of sediment load in alluvial channels is different. Alluvial channels are shaped by sediment transport processes and adjust their geometry such that the sediment transport capacity is equal to upstream sediment supply (e.g., Mackin, 1948; Parker, 1978a,b; Vigilar and Diplas, 1997, 1998). As such, the absolute amount of sediment transported by the river will affect the size of the channel (e.g., Schumm and Khan, 1972), but not the overall shape. Therefore, the lack of a relation between the exponent ω, reflecting the steepness of channel banks, and the mean suspended sediment concentration in alluvial channels in Taiwan matches with expectations. In this difference in the tie between sediment load and the hydraulic geometry of Taiwanese channels classified as bedrock and alluvial channels, we see a clear confirmation of the efficacy of our definition and recognition of bedrock rivers. It should be noted that our test is based on a consideration of mean suspended sediment. Importantly, bedload rather than suspended sediment is thought to be responsible for most erosion in bedrock river channels. In the absence of direct measurements of bedload sediment transport, the mean suspended sediment concentration in a river is a measure of the absolute amount of sediment supplied to the channel reach from upstream. Although the partitioning between suspended load and bedload at a given discharge can vary considerably in time (e.g., Gomez and Church, 1989; Lenzi and Marchi, 2000; Meunier et al., 2006), in general their concentrations are positively correlated (e.g., Mètivier and Meunier, 2003). Therefore, we are confident that our analysis of the link between channel shape and sediment load, and with it the test of our definition of bedrock channels, is robust.

Other factors are known to control the cross-sectional geometry of bedrock channels, including variability of sediment load (Turowski et al., in press), the variability of water discharge (Hartshorn et al., 2002; Snyder et al., 2003b; Stark, 2006), tectonics (Snyder et al., 2003a; Duvall et al., 2004; Stark, 2006; Turowski et al., 2006; Wobus et al., 2006b; Amos and Burbank, 2007), and substrate properties (Wohl and Ikeda, 1998; Wohl, 2000; Montgomery and Gran, 2001; Montgomery, 2004). The present lack of relevant data for Taiwanese rivers puts the role of variability of sediment load and local tectonic forcing beyond reach, but a first pass at the role of substrate is possible. In Taiwan, metamorphic grade of rocks decreases from east to west, tracked by a 30-fold decrease of measured compressive rock strengths (Dadson et al., 2003; Dadson, 2004). Of the 46 bedrock channel stations in Taiwan analysed in this paper, 27 are located in catchments draining to the west, 16 in catchments draining to the east, and the remaining 3 in catchments draining to the north. Bedrock river stations in the east of Taiwan have higher ω values than those in the west of Taiwan (Fig. 8F). The mean exponent and prefactor are respectively ω = 0.31 ± 0.09 and kw = 1.10 ±0.06 for west-draining channels and ω = 0.41 ± 0.09 and kw = 1.04 ± 0.13 for east-draining channels. The hypothesis that the values for the exponent are drawn from the same population can be rejected at the 5 and 1% level. This is quantitative confirmation of the substrate control on channel geometry. However, data from east- and west-draining channels plot on the same trend in the relationship between exponent ω and sediment concentration. Hence, substrate is a control of secondary importance when compared to sediment supply. An interesting question is whether rock type controls channel morphology directly, or whether it affects the sediment load and caliber of the river and through this channel geometry indirectly. Insufficient reliable rock strength data is available for the bedrock channel stations in Taiwan to take these issues further. 7. Conclusions We propose to define fluvial bedrock channels as channels that cannot substantially widen, lower, or shift their bed without eroding bedrock. Although this definition is not entirely devoid of an interpretative component, it classifies channels according to their dynamics rather than a state in time. It is easily applicable, both in the field and using remote sensed imagery and (digital) topographic data, and has in some cases returned different classifications for river channels than previous definitions. For rivers in Taiwan, a classification of mountain river channels based on the new definition exposes a difference in hydraulic geometry between bedrock and alluvial channels, which can be explained by the different roles of sediment in the processes that shape bedrock and alluvial channels. Specifically, we have found that the hydraulic geometry of Taiwanese bedrock channels is adjusted to the sediment load of the river, whereas the shape of alluvial mountain rivers does not depend on sediment load. In this finding, we see confirmation of the fact that our definition addresses channel processes rather than channel state. Nevertheless, the diagnostic tie between sediment load and channel hydraulic geometry has a shortcoming: this tie is evident

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only in populations of channels. Many individual channel segments have hydraulic geometries and sediment load characteristics within the normal range of bedrock and alluvial channels, and their classification relies fully on inspection of the channel boundaries. In many respects, it would be preferable to have one or a small set of observables that can be measured quickly in the field to distinguish reliably between bedrock and alluvial channels. We do not believe that such criteria exist. Labels such as “alluvial” and ”bedrock” help to order and interpret observations, but do not reflect a fundamental division in nature. There is a graded transition between bedrock and alluvial channels, as dominant controls on channel dynamics shift from one to the other. Our definition is less dependent on the time and location of observation than previous definitions. It reflects different dynamics of bedrock and alluvial rivers. As such, it fulfils the two requirements for a useful definition of bedrock channels we stated at the outset. Acknowledgements Thanks go to many people who commented on the concepts presented herein, and especially to J.R. Barbour for an excellent critique of an earlier version of this manuscript and to C.P. Stark for the stimulating discussions. N. Finnegan, N. Gasparini and three anonymous reviewers provided insightful comments, which helped to improve this work considerably. JMT was supported by a NERC Blue Skies Studentship, the Cambridge Trusts, and the Cambridge Philosophical Society. In addition, we acknowledge support form NSF EAR 06-17557 and CNRSINSU programme RELIEF. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.geomorph.2007.10.001. References Amos, C.B., Burbank, D.W., 2007. Channel width response to differential uplift. J. Geophys. Res. 112, F02010. doi:10.1029/2006JF000672. Dadson, S.J., 2004. Erosion of an active mountain belt. Ph.D. Thesis, Wolfson College, Cambridge, UK. Dadson, S.J., Hovius, N., Chen, H., Dade, W.B., Hsieh, M.-L., Willett, S.D., Hu, J.-C., Horng, M.-J., Chen, M.-C., Stark, C.P., Lague, D., Lin, J.C., 2003. Links between erosion, runoff variability and seismicity in the Taiwan orogen. Nature 426, 648–651. Duvall, A., Kirby, E., Burbank, D., 2004. Tectonic and lithologic controls on bedrock channel profiles and processes in coastal California. J. Geophys. Res. 109, F03002. doi:10.1029/2003JF000086. Fahnestock, R.K., 1963. Morphology and Hydrology of a Glacial Stream. US Geol. Surv. Prof. Pap., vol. 422-A. 70 pp. Finnegan, N.J., Roe, G., Montgomery, D.R., Hallet, B., 2005. Controls on the channel width of rivers: implications for modelling fluvial incision of bedrock. Geology 33 (3), 229–232. doi:10.1130/G21171.1. Finnegan, N.J., Sklar, L., Fuller, T.K., 2007. Interplay of sediment supply, river incision, and channel morphology revealed by the transient evolution of an experimental bedrock channel. J. Geophys. Res. 112, F03S11. doi:10.1029/ 2006JF000569. Foley, M.G., 1980. Bed-rock incision by streams. Geol. Soc. Am. Bull. 91, 2189–2213.

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