Channel stability in bed load dominated streams with nonerodible banks: Inferences from experiments in a sinuous flume

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2007jf000902, 2009 Channel stability in bed load dominated streams with nonerodible banks: Inferences from experiments in a sinuous flume Brett C. Eaton 1 and Michael Church 1 Received 26 August 2007; revised 29 September 2008; accepted 3 December 2008; published 28 February [1] Experimental and computational evidence shows that bed state adjustments can accommodate a twofold to fourfold change in sediment supply without otherwise affecting the morphology of a sinuous gravel bed stream channel with nonerodible banks. In contrast, previously published experiments of the same kind but with erodible channel banks suggest that changes in sediment supply are taken up almost exclusively by changes in channel sinuosity and hence the reach-average water surface gradient. It appears that stream channel response to changes in sediment supply is strongly conditioned by the nature of the stream boundary, and that determines which of several potential adjustments are possible. Citation: Eaton, B. C., and M. Church (2009), Channel stability in bed load dominated streams with nonerodible banks: Inferences from experiments in a sinuous flume, J. Geophys. Res., 114,, doi: /2007jf Introduction [2] Channel morphology is generally accepted to be determined by the sediment supply, the flow available to transport it, and the topographic gradient down which fluid and sediment must be passed, and constrained by the properties of the boundary materials [Mackin, 1948; Lane, 1957; Leopold et al., 1964; Carson, 1984; Carson and Griffiths, 1987; Church, 1992; Montgomery and Buffington, 1997; Millar, 2000]. At the reach scale, channel pattern in single thread channels is often dominated by repeated pools, riffles and bars on alternate sides of the stream channel, which become meanders in regularly sinuous channels [Tinkler, 1970; Keller, 1972]. These features constitute important flow resistance elements and their ubiquity suggests that their formation is intimately linked to stabilization of the stream channel [Eaton et al., 2006]. [3] Existing observations and models of channel response demonstrate that multiple adjustments are possible to an imposed change in the governing conditions [Schumm, 1969, 1971; Kellerhals and Church, 1989; Montgomery and Buffington, 1998; Eaton et al., 2004]: stability can be produced by various system responses. The geomorphic history and the boundary conditions determine which of the possible responses actually does occur [Gardner, 1977; Desloges and Church, 1992; Schumm, 1993; Church, 1995; Eaton and Lapointe, 2001; Talbot and Lapointe, 2002]. [4] Recent work on reach-scale channel stability in the context of rational regime theory indicates that, to adopt a stable configuration, flow resistance for the system [Eaton et al., 2004] is maximized or (nearly equivalently) the rate of energy expenditure within the system is minimized 1 Department of Geography, University of British Columbia, Vancouver, British Columbia, Canada. Copyright 2009 by the American Geophysical Union /09/2007JF [Huang et al., 2004]. System-scale flow resistance (f sys = 8gRS v = u 2), where g is the acceleration of gravity, R is the hydraulic radius, S v is the valley slope, and u is the mean flow velocity, incorporates the effects of reach-scale flow resistance, f 000, due to the channel planform (e.g., the development of a meandering habit [Eaton et al., 2004]), the within-channel form resistance, f 00, due to bars, dunes and other in-channel features [after Parker and Peterson, 1980], and the grain resistance, f 0 [after Millar, 1999]. [5] Stream table experiments with erodible channel banks designed to test the idea that channel stability derives from maximizing f sys demonstrate that, in fact, channel sinuosity (f 000 ) and, thereby, channel gradient is the primary adjustable quantity as sediment concentration is varied when all components of flow resistance are adjustable [Eaton et al., 2004]. This is a reach-scale adjustment. Flow resistance adjustments at other scales appeared to have no discernable differential effect. The trajectory by which equilibrium was approached was consistent with maximizing system-scale flow resistance. [6] The generality of those experimental results is limited by the fact that, in all the experiments, the channel banks were composed of the same material as the channel bed and, consequently, were equally erodible. Obviously, in nature, additional strength is imparted to channel banks due both to the effect of vegetation (see data presented by Andrews [1984] and Hey and Thorne [1986]) and the deposition of cohesive sediment not typically found in the bed. These bank strengthening factors arguably are critical to the longterm maintenance of a single-thread, meandering channel pattern [Paola, 2002]. The question then arises what role the other dimensions of possible channel adjustment may play. [7] Accordingly, the experiments reported herein were designed to challenge the apparent implication of the experiments presented by Eaton and Church [2004] that channel gradient is the only significant adjustable quantity. In particular, we examine the role of bed sediment texture, 1of17

2 Figure 1. Annotated photograph after 4 hours (experiment 4 4), looking downstream near the middle of the study reach. Note the thalweg location, the riffle crest in the foreground, the bar deposited along the left bank, and the pool along the right bank. The bench marks for several cross sections are also visible in Figure 1. that is, the sources of the grain resistance component (f 0 )of channel adjustment. 2. Experimental Strategy [8] We proceed, as did Eaton and Church [2004], to run prescribed flows and sediment feed rates through a prepared channel and allow the channel to adjust to approach an equilibrium state, when we measure the morphology of the channel and sediment texture of the bed. In the present experiments, however, the morphological adjustment was deliberately constrained. To understand this strategy, it is important to recognize that the expression of equilibrium fundamentally depends on the spatial and temporal scales of the phenomena being observed. In a river reach that is tens of channel widths long, the time required to establish an equilibrium channel pattern is probably on the order of a decade. It involves adjustment of the number and configuration of bars, pools and riffles in the reach and, more generally, an adjustment of channel gradient via changing channel sinuosity. Such changes typically require the shifting of substantial volumes of bed material, hence the relatively long timescale for the emergence of an equilibrium form. [9] On the synoptic timescale of an individual flood event, the channel configuration remains nearly constant for all but exceptionally large events. At this timescale, it is appropriate to apply the concept of equilibrium response to the bed state. We use the term bed state to refer to the adjustment of surface sediment texture by armoring [Parker and Klingeman, 1982; Parker et al., 1982], the occurrence of sediment patches [Paola and Seal, 1995; Lisle and Hilton, 1999], and the development of sediment structures [Brayshaw, 1985; Reid et al., 1992; Church et al., 1998], all of which can affect the entrainment threshold and the intensity of bed material transport in gravel bed rivers, and all of which are implicated in the grain component of resistance to flow. In sand bed channels, the development of primary bed forms constitutes a similar response. [10] Over the longer timescales, it is appropriate to apply the related concepts of channel grade [Mackin, 1948; Lane, 1957] and river regime [Kirkby, 1977; Chang, 1979; White et al., 1982]. The experiments of Eaton and Church [2004] demonstrated that, in situations where the banks are relatively erodible, an equilibrium channel slope is established relatively quickly while the bed state appears to remain the same for a wide range of choices of formative discharge and imposed sediment load. In systems where the channel banks are more resistant, it may take much longer for channel slope to respond to changes in long-term average sediment supply or formative discharge. In these situations, the bed state is likely to change in response to short-term variations in sediment supply or discharge. Bed state is also likely to vary following a prolonged step change in order to maintain a conditional equilibrium as the channel slowly adjusts. Brooks [2003] has documented an example of this kind of behavior on the Red River of Manitoba, where slow but persistent lateral migration has produced a consistent decline in stream gradient and unit stream power over the past 8000 years. At the scale of an individual flood event, however, channel slope is effectively constant and equilibrium is presumably approached via other degrees of freedom, such as the bed state. [11] In order to isolate and study these other degrees of freedom, it is necessary to fix the channel pattern to remove degrees of freedom associated with longer-term changes in channel sinuosity and channel slope. This implies that the timescale of enquiry is shortened to something like the timescale for a synoptic flow event. In gravel bed rivers, short-term adjustments in bed surface characteristics appear to occur primarily by selective transport of the bed material [Hassan and Reid, 1990; Hassan and Church, 2000]. Therefore, in order to establish an equilibrium morphology by changing the bed state, it is necessary for a system to exhibit at least temporary disequilibrium in the bed material sediment transport (i.e., not all grains on the bed can be equally mobile). The only way to isolate this behavior is to hold both the rate and grain size distribution of the sediment supply constant: use of a recirculating flume would add an additional degree of freedom (i.e., the grain size distribution of sediment input at the upstream end of the flume) that could potentially obscure the connection between the bed state and the sediment supply. 3. Experimental Methods [12] Tests were carried out by deliberately constraining both the lateral adjustment of the thalweg and the wetted width of the experimental channels by fixing the channel banks. The experiments were conducted at the University of British Columbia on a 10 m long stream table that is 15 cm deep and 1.5 m wide. At the inlet of the UBC stream table, a broad-crested weir with a cross-sectional width of 50 cm was oriented at 45 to the stream table centerline. The fixed banks were constructed of strips of foam rubber with a triangular cross section having 1:1 side slopes, the surface of which was covered with a layer of particles, about 1 to 2 mm in diameter, in order to generate a surface roughness similar to that of alluvial banks (Figure 1). The area between 2of17

3 Figure 2. Bed material grain size distributions for the fixed bank experiments reported herein and for the mobile bank experiments reported previously. Below a grain size of about 350 mm (shaded in gray) fluid viscosity effects become important, introducing a scale distortion with respect to relative ease of particle entrainment: therefore, both distributions were truncated near this limit. the fixed banks was filled to a depth of 6 cm with sediment having the grain size distribution shown in Figure 2. [13] The grain size distribution and channel dimensions for these experiments are Froude-scaled to the experimental apparatus used by Eaton and Church [2004], such that both sets of experiments represent a moderately steep (1%) gravel bed field prototype with a median surface grain size of between 32 mm and 64 mm. An important reason to proceed in this way is to take advantage of the close relation between sediment concentration and channel gradient established under the conditions of the 2004 experiments. [14] The design criteria are based on typical morphological dimensions of the experiments reported by Eaton and Church [2004] (channel width, W = 60 cm; wavelength = 365 cm), but at 3/4 scale. The alluvial bed between the foam banks was approximately 44 cm wide, while the channel centerline was a sinusoid with wavelength 275 cm and amplitude 40 cm, producing a channel centerline sinuosity of about 1.2 (Figure 3). The meander amplitude was made as large as possible given the width of the stream tray. [15] The sediment feed rates, flume slope, and discharges used in the experiments were scaled to the ranges imposed in the 2004 experiments. The imposed discharge and sediment supply were varied (see Table 1) on the basis of a nominal design discharge of about 3.4 L s 1 for the Eaton and Church [2004] experiments which, under the 3/4 scale Froude-scaling, becomes about 1.7 L s 1 (since the discharge ratio, Q r, and length scale ratio, L r, are related by Q r / L r 5/2 ). The experiments were run for a total of 4 or 5 hours, typically involving a bed survey after 1 hour and then again at the end of the experiment. The typical experimental run time corresponds to a peak flow lasting for one to two days in the field prototype. The range of conditions over which the bed state can establish and maintain an equilibrium channel morphology represents the elasticity of the system to absorb event-scale variations in bed material sediment supply or the discharge available to transport the sediment supply. [16] Sediment was introduced to the system at a constant rate over the weir crest at the upstream end of the stream table, using material with a grain size distribution identical to that of the material used to construct the channel bed, with D 50 = 720 mm (Figure 2). All sediment leaving the stream table at the downstream end was captured in a sediment trap. The accumulated sediment transport was recorded every 15 min, at which time a sample was taken to determine the grain size distribution. Relative to the erodible bank equivalents upon which the design channel geometry was based, all of the experiments except for experiment 4 1 represent systems with an excess supply of bed material. That is, had the banks been erodible, one would expect the channels to stabilize at a higher channel gradient with a lower sinuosity [Eaton and Church, 2004]. [17] Descriptions of the alluvial system presented herein are based on measurements from the middle half of the stream table (the study reach) extending from x 2.25 m at the upstream end to x 6.75 m at the downstream end (refer to scale in Figure 3), away from any potential inlet and outlet effects, for a total study length along the channel of about 5.4 m. Measurements of the water surface elevation were made using a point gauge, and cross sections were surveyed using a laser displacement meter. The horizontal Figure 3. Experimental apparatus for the fixed bank experiments. Flow is from left to right. The average thalweg alignment is indicated with a dashed line, as are the typical riffle configurations. The bars are shown (schematically) in light gray, as is the sediment feeder at the upstream end of the channel; the pools are shown in dark gray. The channel bank crests and the thalweg location are drawn to scale; the dimensions are in centimeters. 3of17

4 Table 1. Governing Conditions and Equilibrium Response Experiment Q (L s 1 ) Governing Conditions Q b (g min 1 ) Q b /Q (g L 1 ) S (%) Equilibrium Response Sinuosity (m m 1 ) D 50 (Surface) D 50 (Bed Material) a n/a a b 0.83 b a Experiment did not reach equilibrium with respect to sediment supply (Q b ). b While never reaching equilibrium with the sediment supply, Q b, the system attained this stable channel gradient and sediment output of about 1.32 g L 1 during the last 2 hours of the experiment, which is used to determine the plotting position in Figures 8, 11, and 12. Reported sinuosity values refer to the self-formed thalwegs, which are shown in Figure 4. location of each point (x and y) was recorded with a resolution of ±1 mm, while the vertical resolution for both the point gauge and laser was approximately ±0.2 mm. [18] The slope of the stream table was held constant at 1.1%. However, at the beginning of each experiment, the bank alignment immediately produced strong secondary currents that quickly eroded the bed at the outside of each bend, creating a series of bars and pools along the stream channel and a transient high rate of sediment transport, before settling down to much lower transport rates associated with a stable bar-pool morphology. This reconstruction of the within-channel geometry was completed within about 5 to 10 min. Since the sediment feed at the inlet did not replicate this short-lived burst of rapid sediment transport, the upstreammost bar and pool unit was constructed primarily via erosion of the pool, while the pool-and-bar units downstream were formed because of a combination of deposition on the bar tops and erosion of the pools. The result was that the mean bed elevation, relative to the floodplain surface, increased by about 8 mm from the top of the study reach to the bottom. Therefore, the effective initial valley slope for these experiments was about 0.9%, not 1.1%. [19] The average channel gradient (S) was estimated by fitting linear regressions to the water surface elevation, plotted against distance along the thalweg. Cross sections were situated at each apex and crossover, as well as halfway between successive apices (see Figure 3). The same bench marks were reoccupied for all but the first experiment, and the cross sections are therefore directly compared from experiment to experiment. The mean bed elevation between the fixed banks was calculated for each cross section surveyed and used to derive an estimate of the bed surface gradient (S b ) in the down-valley direction, rather than along the channel thalweg. Since the average water depth along the channel was essentially constant, the bed gradient represents the down-valley water surface gradient and thereby the effective valley slope (S v ) for the experiments. [20] The bed state in these experiments was characterized by sampling the surface texture at the riffle crests: observation of surface structure and the development of surface patches was not feasible given the experimental apparatus. Samples of the bed surface texture were collected following each experiment using a flexible rubber plate covered with a layer of wet clay. A 4 cm by 8 cm plate was used, covered with clay about 1 to 2 mm thick. The plate was pressed firmly against the bed surface, then dipped into water to release any sediment adhering by soil moisture tension: these grains were excluded from the grain size analysis. The remaining grains were then separated from the clay by wet sieving, and analyzed at 1/2 phi increments. One sample was taken from each of the riffles in the study reach (i.e., just downstream of cross sections 3, 6 and 9) at the thalweg where it crossed each riffle: all three samples were combined for analysis, giving an average riffle surface texture for the reach. The sample locations were chosen to represent the zones over which sediment transport occurred which were observed (visually) to undergo significant textural adjustment during a run as equilibrium was achieved. The objective was to sample the surface layer, one grain thick, but there was invariably some plastic deformation of the clay paste around the surface grains. To avoid the errors associated with transforming these surface-based samples to bulk equivalents, reference samples of the undisturbed bed surface were also taken using the same method prior to the beginning of the experiment. These samples represent the subsurface material in the channel, against which the surface samples were compared. 4. Results 4.1. Trajectories Toward Equilibrium [21] Regardless of the combination of discharge and sediment supply (which in our experiments refers only to the supply of bed material, since wash load fractions were absent from the sediment feed), the general morphologic structure developed during every experiment was essentially the same (as depicted in Figure 3). A sequence of alternating bars and pools developed within the first 5 to 10 min of each experiment wherein the bars were deposited downstream of the bank apices, and the pools developed across the channel from the bars, producing a thalweg that was slightly out of phase with the sinusoidal bank alignment. Riffles extended across the channel near the upstream side of the bank apices, disappearing under the bar on the downstream side of the apex. Figure 1 shows the typical arrangement of riffle, bar and pool that developed at each meander bend. The thalweg alignments for the various experiments are shown in Figure 4, and the thalweg sinuosity for each experiment is reported in Table 1. The 4of17

5 Figure 4. The surveyed thalweg locations for all experiments are shown. The bank alignments are shown for reference. The scale is in centimeters. Since the thalweg alignment and sinuosity ratio are nearly constant, the various experiments have not been individually identified in Figure 4. thalweg locations shown in Figure 4 were generally established after about 10 min, and then remained stable during the remainder of the experiment. [22] Since sinuous gravel bed channels are characterized by alternating bar-pool-riffle sequences, the initially imposed flat bed was in no way related to the flow patterns created by the sinuous banks in our flume. The initial, rapid period of adjustment described above is associated with the coevolution of the flow field and the bed morphology, and represents a sort of spin up period. The observed rates of channel change are not likely to occur in natural streams where the flow structure and channel morphology have coevolved. We consider the channel condition after this period of rapid adjustment to represent the true beginning of the experiment. [23] There are two ways in which to assess the approach of an experiment toward equilibrium. First, indices of the channel morphology, including cross sections, channel slope and thalweg sinuosity, can be repeatedly measured. This is a basis for assessing the morphologic stability of the system. This is appropriate for evaluating equilibrium at the timescale of an individual flow event. Second, the sediment output rate and grain size distribution can be measured and compared against the sediment input rate and distribution, which is a basis for assessing the equilibrium of the sediment transport field. This kind of equilibrium is appropriate at river regime timescale (i.e., decades), but strict sediment transport equilibrium must be violated at timescales at which the bed state responds, since the modification of the bed state by definition involves some degree of size-selective transport. Both approaches were employed in the analysis of these experiments. [24] The primary method by which equilibrium conditions were identified was by comparing repeated cross section surveys at a number of fixed locations along the channel. After 1 or 2 hours, when it appeared that morphologic stability had approximately been reached, the flow was stopped and the bed was surveyed. The same cross sections were surveyed again after 4 hours of runtime. Typically, almost no difference between the surveys was detectable. [25] The cross-sectional surveys for experiment 4 2 (an experiment with a moderate sediment concentration of 0.98 g L 1 ) (which are typical of all the experiments that reached morphologic stability) are shown in the left-hand column of Figure 5. There appears to have been very slight degradation at the first apex (XS 4), and some shifting of the thalweg location at the second crossover (XS 6) but, otherwise, there is little detectable change. A comparison of the bed survey at 2 hours and at 4 hours for Experiment 4 4 (which had a high sediment concentration) also reveals nearly identical cross-sectional configurations (Figure 5). In contrast, comparison of the surveys for experiment 4 3 (high sediment concentration), which did not reach morphologic equilibrium, shows substantial net aggradation in the upper part of the study reach between 1 hour and 4 hours (Figure 5). All of the sections in the upstream half of the study reach aggraded, while the sections in the lower half remained relatively stable. [26] The secondary means by which equilibrium conditions were identified involved examining the sediment transport rates and grain size distributions. Use of this approach was hampered by the occurrence of a slight backwater at the outlet of the flume which, in some experiments, resulted in significant net deposition downstream of the study reach of the flume but upstream of the outlet. This effect was most pronounced for experiments with the lowest discharges (i.e., experiments 4 3 and 4 4) because the largest grains were near their entrainment threshold. Since our observations were made in the middle part of the stream table, some distance from the outlet, equilibrium is reached first in the study reach of the stream table, and only later at the outlet, where the transported material is collected. [27] The average transport rate, calculated for 15-min intervals, is plotted in Figure 6 against time for each experiment, along with the imposed sediment supply for reference. Some experiments reached equilibrium almost immediately (4 2, 4 6, and 4 7 in Table 1). These experiments had sediment concentrations ranging from low (0.83 g L 1 ) to high (1.14 g L 1 ), but all had relatively high discharges and were thus not affected by the backwater at the outlet. Experiment 4 1, which had the lowest sediment concentration (0.68 g L 1 ) exhibited a gradual decline in the transport rate as the system approached equilibrium. It attained a transport rate equal to the sediment supply after about 3 hours, after which equilibrium was maintained. [28] The trajectory by which sediment transport equilibrium was attained during experiment 4 4 requires more careful consideration, and can best be understood with reference to the backwater effect produced by the outlet. (It should be remembered that, on the basis of the morphologic criteria, this experiment clearly reached equilibrium; see Figure 5). Initially, the sediment transport leaving the channel was noticeably lower than transport throughout the rest of the channel (on the basis of qualitative, visual 5of17

6 Figure 5. Cross sections for two experiments that reached equilibrium (4 2 and 4 4) and one that did not (4 3). The initial survey is illustrated with a dashed line, and the survey at the end of the experiment is indicated with a solid line. The dimensions are reported in centimeters, and the vertical scale is exaggerated by a factor of 2. The cross section numbers increase in the downstream direction. Sections 3, 6 and 9 represent the crossovers, just upstream of the bar-pool unit. Sections 4, 7 and 10 represent the thalweg apex and cut through the bar and adjacent pool. Sections 5, 8 and 11 represent reference sections halfway between subsequent bars. appraisal) and, importantly, it was lower than in the study reach. The recorded sediment output increased progressively as a fan-like deposit developed near the outlet. After about 2 hours, the fan had steepened sufficiently to bring the sediment output in line with the transport rates characteristic of the study reach, and to establish equilibrium, on the basis of the recorded transport rates. This equilibrium condition persisted for about 30 min, after which the sediment output declined precipitously. During the interval of equilibrium behavior, the surface of the fan seemed to have stopped aggrading, but was modified texturally, as a layer of coarse sediment accumulated over much of the surface. The surface coarsening is thought to have produced the observed drop in the sediment transport rate after 2 1/2 hours by inducing aggradation just upstream. However, after about 3 1/2 hours, the armor near the left bank was breached, and a scour pool developed through which sediment was efficiently routed to the outlet; this again brought the system back into equilibrium at the outlet. However, during the entire experiment, the sediment transport rate in the study reach appeared to be unaffected by the dynamics near the outlet. [29] In the remaining experiments, neither of which reached morphologic equilibrium, (4 3, and 4 5, both with sediment concentrations >1.5 g L 1 ), transport rates remained consistently below sediment supply. Experiment 4 5 did appear to reach a quasi-equilibrium configuration during the last 2 hours, with nearly the same channel gradient and sediment transport rate as experiment 4 4, but the cross sections for that experiment showed persistent net aggradation within the study reach, and so did not reach equilibrium in any sense. [30] In Figure 7a, samples of the transported load collected at the outlet of the flume are compared against the sediment feed for the longest running experiment, 4 1. The distributions represent the average load collected during the last 15 min of each hour. All of the distributions except for the one taken at 5 hours are similar to the sediment feed distribution, and the average D 50 for the samples of transported load (725 mm) is quite close to the D 50 of the sediment feed (710 mm). In order to highlight any systematic differences in the distributions, the proportion of the transported load in each size class was normalized by the 6of17

7 Figure 6. Recorded sediment output for all experiments is plotted against time. The data represent 15-min average transport rates and are reported as a sediment concentration (g L 1 ) to permit direct comparison. For each experiment, the experiment number is shown in the top right corner. The sediment feed rate (horizontal dashed line) is shown for comparison, as is the quasi-equilibrium sediment output during the last 2 hours of experiment 4 5 (dotted horizontal line). sediment feed: the results are shown in Figure 7b. Most of the samples exhibit an under-representation of the sediment feed in both the coarse and the fine tails of the distributions. The trap at the end of the flume allowed a portion of the finest sediment to escape while retaining 100% of the coarser part of the sediment load. The median size of the sediment collected from the bottom of the tank in which the sediment trap was suspended was about 0.34 mm, corresponding to the size range that is most underrepresented in the fine tail of the distribution. Thus, the differences between the sediment feed and the sediment load at the outlet in the finer fractions is likely an artifact of the sediment trapping efficiency. [31] The deviations for the coarse end of the distribution are not artifacts. Almost all samples indicate that the coarsest two or three size fractions are under-represented in the sediment load collected at the outlet. Some of this under-representation is attributable to the trapping of coarse particles immediately upstream of the backwatered outlet, but some of it is presumably attributable to the development of an armor layer within the study reach as a result of selective transport of the initially unarmored bed material. This sort of disequilibrium in the sediment transport grain size distribution is necessary for an adjustment of the bed state, and thus is to be anticipated. The distribution at 5 hours is quite different from the others, and exemplifies the short-term variations in sediment transport characteristics that occurred during these experiments, despite the fact that sediment feed rate, grain size distribution, and formative discharge were all held constant. [32] The range of distributions between experiments, exemplified by samples taken during experiments 4 2, 4 4, 4 5 and 4 6 are shown in Figures 7c and 7d. The between-experiment range is similar to the temporal range shown in Figures 7a and 7b. There is a general trend for under-representation of the finest sizes (because of trapping efficiency) and of the coarsest sizes (as a result of armor development), while the occasional sample indicates an 7of17

8 Figure 7. Comparison of the grain size distribution of the transported load and the sediment feed. (a) The distributions of the sediment load for experiment 4 1 at the end of each hour are presented, along with the distribution of the sediment feed in order to illustrate the temporal variation in transport for a single experiment. (b) The fraction in each size class for the hourly sediment load samples from experiment 4 1 divided by the corresponding fraction for the bed material is also presented. (c) Randomly selected transport samples for experiments 4 2, 4 4, 4 5, and 4 6 are presented to illustrate the between-experiment range. (d) The fractional analysis for the transport samples from experiments 4 2, 4 4, 4 5, and 4 6 is presented as well. While the center of the distribution is fairly well represented, the tails are not. The under-representation of the finest part of the distribution is due to a systematic decline in the efficiency of the sediment trap at the end of the flume with decreasing particle size: sediment collected from the bottom of the tank in which the sediment trap was suspended had a median grain size of about 0.34 mm, which corresponds with the zone of maximum deviation from the sediment feed distribution. overrepresentation of the coarsest particles in the load. The D 50 for these transport samples range from 750 mm to 870 mm, which is still close to the D 50 of the sediment supply (720 mm) Equilibrium Results [33] The dominant response reported by Eaton and Church [2004] to changes in the governing conditions (indexed by the bed load sediment concentration, Q b /Q) under the condition of erodible banks was an incremental, linear change in channel gradient, achieved primarily by changing the thalweg sinuosity through bank erosion. The average channel gradient during the period of equilibrium for those experiments is plotted against Q b /Q in Figure 8, along with the data of the present experiments. The quasiequilibrium values for experiment 4 5 are plotted, and the channel gradient measured after 4 hours in experiment 4 3 is also plotted against the imposed sediment concentration. 8of17

9 Figure 8. Equilibrium channel gradient plotted against sediment concentration. The data for the fixed bank experiments presented herein are represented by black circles: a linear trend was fit to these data. Experiment 4 5 is indicated with a black triangle: the plotting position is based on the quasi-equilibrium conditions during last 2 hours. Experiment 4 3 is indicated by the gray triangle, and the gradient recorded at 4 hours is plotted on the basis of the imposed sediment supply. The trend for the mobile bank experiments reported by Eaton and Church [2004] is shown in gray for comparison. The mean bed elevations for 4 3 and 4 5 are consistently higher than for the other experiments. The estimated bed gradients derived from these data are plotted against Q b /Q in Figure 9b: all of the bed gradients are statistically similar to each other, except for experiment 4 3 (at a = 0.05). The slight trend between channel gradient and sediment concentration evident in Figure 8 is not evident in Figure 9b. [36] The adjustments in bed state, as reflected in the surface grain size distributions for the various experiments, are presented in Figure 10. The difference between the proportion of the surface distribution in a particular size class and the proportion of the bed material in the same size class is plotted against grain size for all experiments. Values greater than zero indicate that the surface distribution is overrepresented for the given size class, relative to the bed material, and values less than zero indicate that it is underrepresented. The equilibrium experiments exhibit coarsening of the uppermost part of the distribution (especially in the +2 mm (8 < 1) fractions) relative to the bed subsurface, while the nonequilibrium experiments do not exhibit this characteristic coarsening. The equilibrium experiment for which this coarsening is most weakly developed is 4 4, which is nearest the threshold for the onset of aggradation. If the maximum sediment transport capacity is associated with a bed having no armor layer, then we would expect no systematic differences in the surface-subsurface grain sizes: experiment 4 4 very nearly approaches this condition. [34] For our experiments, thalweg sinuosity did not vary with Q b /Q (refer to Figure 4 and Table 1), and the channel gradients were nearly constant (see Figure 8), except for the nonequilibrium experiments 4 3 and 4 5. The linear regression fit to the equilibrium data (i.e., excluding 4 3 and 4 5) does have a positive slope, but the regression slope is not statistically significant (a > 0.14). In contrast, the regression in Eaton and Church s [2004] experiments is much steeper, and the trend in the data is more consistent. The equilibrium slopes for the present experiments are substantially lower than for Eaton and Church s [2004] experiments with the same bed load sediment concentration. This is likely due to the fact that fixed banks permit a much greater degree of channel cross-section asymmetry to develop, producing deeper, narrower pools, and thereby increasing the net transport capacity for a given slope [Ferguson, 2003; Eaton et al., 2006]. As a result, the same bed load sediment concentration can be carried at a lower channel gradient when the banks are not erodible. [35] Slope-related changes in the channel morphology can also be detected on the basis of the surveyed bed elevations. Since the thalweg alignment changed very little between experiments, increases in the channel gradient should appear in our estimates of the bed gradient, S b.in Figure 9a mean bed elevation at each surveyed cross section is plotted against the down valley distance for each cross section for each experiment. Experiment 4 1 has the lowest mean bed elevation, while the mean bed elevations for experiments 4 2, 4 3, 4 6 and 4 7 are all quite similar. Figure 9. (a) The mean bed elevation evaluated at each surveyed cross section is plotted against down-valley distance. (b) The mean bed gradient, S b, (±2 standard deviation range) for all of the experiments is plotted against sediment concentration (g L 1 ) on the basis of the sediment supply rate. The bed gradient represents the effective valley slope for the experimental runs. Note that the steep upper part of the profile in 4 3 is evident in Figure 9a. 9of17

10 Figure 10. Differences between the surface grain size distribution for fixed bank experiments and the distribution for the bed material. The differences in the proportion of the surface in each size class are shown for all experiments, except 4 2 (data missing). The experiments that did not reach equilibrium are shown in lighter gray. Experiment 4 5 exhibits slight surface coarsening in the largest size class range, but fining for the remaining size classes, while experiment 4 3 exhibits limited coarsening in the very coarse sand range. The bed state response of these two experiments is difficult to interpret because they did not reach equilibrium nor do they exhibit the systematic variation of the equilibrium experiments. [37] Generally, Figure 10 indicates that the surface fines with increasing sediment concentration up to the onset of aggradation. Figure 11, in which selected percentiles of the surface grain size distribution are plotted against sediment concentration, shows that the change in surface texture occurs quite rapidly as Q b /Q falls below the limit transport for equilibrium and changes only slightly as sediment concentration is further reduced, principally by continuing increase of D 90. The results in Figures 10 and 11 imply that, in order to facilitate higher sediment transport rates, the surface texture fines [cf. Parker et al., 1982; Parker and Klingeman, 1982], thereby increasing the exposure of the grains passing over it and reducing the potential for grain hiding within the armor layer. After aggradation begins, there does not appear to be any further adjustment of the bed state. [38] In order to quantitatively assess changes in the bed state, linear regressions were fit relating the proportion of the surface in each size fraction (P i ) to the bed load sediment transport concentration (Q b /Q). The regression results are presented in Table 2. Two systematic trends are evident in the analysis. First, the proportion of the surface in the largest size fraction (2.8 mm to 4.0 mm) is negatively correlated with Q b /Q with an R 2 of This relation is evident for the next largest size class (2.0 mm to 2.8 mm), though it is weaker (R 2 = 0.57). Second, the proportion of the surface in the middle of the distribution (0.70 mm to 10 of 17

11 Figure 11. Analysis of surface grain size distributions, showing the relation between three indices of the cumulative grain size distribution and the governing condition, Q b / Q. Dashed lines representing the general trend of the data have been fit by eye to the data from equilibrium experiments. The reference indices for the bed material are indicated using dotted horizontal lines. 1.0 mm) is positively correlated with Q b /Q (R 2 = 0.91) as is the next largest size fraction (1.0 mm to 1.4 mm, R 2 = 0.61). Generally, both results point toward a decline in proportion of the largest grains on the bed surface as the bed load sediment concentration increases, with the largest compensatory increases occurring in the size fractions that bracket the median surface grain size. The net result of these changes is that the bed surface becomes smoother with fewer opportunities for grains to hide in the vicinity of large grains. We conclude from our experiments that, in channels that have nonerodible banks, significant variations in sediment supply can be accommodated by changing the bed surface texture Modeling the Effects of Surface Coarsening on Sediment Transport Rate [39] In this section, we attempt to quantify the relation between the degree of armoring and the sediment transport capacity of a stream in order to assess the range of sediment supply rates that can be accommodated by changing the bed surface texture alone. It is well known from various flume experiments [Parker et al., 1982; Kuhnle and Southard, 1988; Dietrich et al., 1989; Lisle et al., 1991, 1993] that a reduction in sediment supply to gravel bed streams (or, more accurately, their scaled equivalents) produces a coarsening of the bed surface and a concomitant decrease in the sediment transport rate for the stream. A recent analysis of field data [Pitlick et al., 2008] also suggests that surface armoring has a large effect on bed material transport rate. We have modeled the effect of armoring on the sediment transport rate using Parker s [1990] fractional sediment transport equations to estimate the transport rate for beds with a range of surface textures: Parker [1990] uses a hiding function to calculate the entrainment shear stress for all the grains found on the bed surface, and then calculates transport rates for each size range, which are then summed to give a total sediment transport rate. [40] In order to use these equations, we have to specify grain size distributions for the bed surface: we have assumed that surface grain size distributions are lognormally distributed. Furthermore, we assume that, when the supply of bed material to a stream is varied, the surface is modified by selective transport such that it either becomes finer or coarser, but remains lognormally distributed with the same maximum particle size. These assumptions are based on the understanding that all possible armor surfaces are derived from the subsurface bed material, and are constrained by the range of grain sizes present in the bed subsurface. This approach is fundamentally different from those that use the surface D 50 as an index of the textural adjustment because those approaches cannot distinguish between a narrowing of the surface distribution, as described above, and the wholesale shift of the distribution. A comparison between the transport rates modeled by applying Parker s [1990] equations and transport rates observed in our experiments allows us to evaluate the hypothesis that surface armoring is a viable means by which short-term, morphologic equilibrium is achieved. [41] Beginning with the finest possible surface, in which the surface and subsurface had the same grain size distribution, we progressively narrowed and coarsened the distribution and modeled the effect on the predicted transport rate using Parker s [1990] fractional sediment transport formula. The initial distribution to which the Parker [1990] equation is applied is similar to the bed material grain size distribution for our experiments (D mm, D mm); we generated additional, coarser distributions by holding constant the coarse end of the distribution (defined to be the grain size that is 3 standard deviations above D 50 ) and reducing the standard deviation of the distribution, producing lognormally distributed surfaces Table 2. Regression Results Relating the Proportion of the Surface Grain Size Distribution in Each Size (P i ) to Sediment Concentration (Q b /Q) Regression Statistics a (mm) C SD (0.013) (0.065) (0.044) (0.010) (0.010) (0.037) (0.045) (0.030) (0.010) C o SD (0.014) (0.071) (0.047) (0.010) (0.011) (0.040) (0.049) (0.032) (0.011) R (R 2 ) (0.85) b (0.57) (0.10) (0.61) (0.91) b (0.34) (0.33) (0.33) (0.13) a Regression model: P i = C 1 (Q b /Q) +C o (the standard error for the estimates of C 1 and C o are provided in brackets). b Regressions with an R 2 greater than of 17

12 Figure 12. Prediction of dimensionless transport ratio (q*) using Parker s [1990] fractional sediment transport equation, as described in section 4.3. (a) The lognormal grain size distributions, numbered 1 to 6, that were used to simulate the effect of surface coarsening on q* are shown. (b) The predicted value of q* based on Parker s [1990] equation for each distribution shown in Figure 12a is plotted against the armor ratio (Ar), and a curve has been fit to those data (dashed line): the equation of the fitted curve is shown in Figure 12b (R 2 = 1.0). Experimental data from this study and from previously published flume experiments summarized by Buffington and Montgomery [1999] are also shown in Figure 12b. The plotting position for experiment 4 7 based on a D 50 of 1.4 mm, consistent with the trend in Figure 11, is shown using an open circle. with increasingly coarse distributions. The resulting distributions are shown in Figure 12a. We calculated armor ratios by dividing the D 50 for the coarsened surfaces by the D 50 for the initial distribution: the distributions span a range of armor ratios from 1.0 to 4.2, which covers most of the armor ratios typically observed in field and laboratory studies (see summaries by Buffington and Montgomery [1999] and Hassan et al. [2006]). [42] By specifying a shear stress of 1.5 Pa (which is the estimated average shear stress at the riffles in our experiments), we applied Parker s [1990] equations to calculate the sediment transport rate per unit width of channel (q b )for the finest grain size distribution (i.e., the bed subsurface grain size distribution) and designated that the reference sediment transport rate for the system. We then calculated q b for all of the coarser lognormal distributions (which represent varying degrees of armoring) using the same shear stress and normalized these estimates by the reference transport rate. This ratio is equivalent to the dimensionless sediment transport ratio, q*, originally defined by Dietrich et al. [1989] as the ratio of the observed transport rate in the presence of an armored bed surface to the transport rate for an unarmored surface. The q* values predicted using Parker s [1990] equations are plotted against the armor ratio (Ar) for our hypothetical lognormally distributed bed surfaces in Figure 12b, along with a curve describing the trend. The fitted function (q*=1/ar) has an R 2 value of 1.0. Because of the use of dimensionless ratios to define the curve in Figure 12b, the value of shear stress used to calculate q b has no effect on the form of the curve. Constructing the model using a much coarser grain size distribution that has a subsurface D 50 of 16 mm generates almost identical relations between Ar and q*. [43] Two sets of experimental data also appear in Figure 12b: our experimental results, presented above, and similar experiments using traditional straight flumes [Parker et al., 1982; Kuhnle and Southard, 1988; Dietrich et al., 1989; Lisle et al., 1991, 1993]. In order to calculate q* for our experiments we assumed that the bed load transport concentration (based on the constant sediment feed rate) for experiment 4 4 is a good estimate of the reference sediment transport rate for our system: the armor ratio for this experiment was close to unity and it had the highest equilibrium sediment transport concentration, which is consistent with Dietrich et al. s [1989] original analysis. The equilibrium bed load concentration (again based on sediment feed rate) for the other experiments that reached equilibrium was normalized by the reference transport rate, and is plotted against the measured armor ratio in Figure 12b. For experiment 4 5, we calculated the bed load concentration using the average sediment output measured at the end of the flume (149 g min 1 )ratherthan using the sediment feed rate. [44] We used Buffington and Montgomery s [1999] analysis of the traditional straight flume experiments, wherein they reported the observed volumetric transport rate (q b ), surface and subsurface D 50, water or bed surface slope (S), and estimated the shear stress acting on the bed (summarized in their Table 1). In many of those experiments, changing sediment supply provoked both modification of the bed state and a change in the average energy gradient. In 12 of 17