The relation between particle path length distributions and channel morphology in gravel-bed streams: a synthesis

Geomorphology 56 (2003) 167 – 187 www.elsevier.com/locate/geomorph

The relation between particle path length distributions and channel morphology in gravel-bed streams: a synthesis Richard S. Pyrce1, Peter E. Ashmore * Department of Geography, The University of Western Ontario, London, Ontario, Canada N6A 5C2 Received 11 February 2002; received in revised form 6 January 2003; accepted 13 January 2003

Abstract The path length (downstream displacement over a given time period) of individual bed particles in gravel-bed rivers is central to morphological methods for measuring bed load transport rate and is also fundamental to understanding the bed load transport process and the development of channel morphology. Previous studies of particle movement using tracers report predominantly strongly positively skewed frequency distributions of path length with modes close to the point of entrainment. However, gravelbed rivers often have regularly spaced erosion (scour pools) and deposition (channel bars) sites that are several channel widths apart and it is reasonable to expect that particle path length would reflect this morphological scale, at least during flows large enough to create and modify the morphology. Here, we synthesize and re-analyze results from published bed load tracing experiments in gravel-bed rivers to identify the variety of possible path length distributions for differing channel morphology, channel dimensions, bed particle size, and particle mobility (i.e. flow magnitude) and to look for occurrences of path length coinciding with the length scale of the morphology. The results show that path length distributions may be positively skewed, symmetrical, and uni-, bi-, or multi-modal and may include modes that coincide with known or expected pool – bar spacing. Primary path length modes equivalent to possible pool – bar spacing are more probable at higher non-dimensional bed shear stress, from which it is inferred that both particle mobility and channel morphology exert an influence on particle path lengths and that particle movement is unlikely to be stochastic except at relatively low particle mobility. Existing data are inadequate for more than a preliminary analysis of this problem consequently there is a need for new data collected explicitly and systematically to confirm these preliminary results, isolate the effect of the several variables that influence the characteristics of path length frequency distributions and identify the conditions under which path length coincides with the length scale of the dominant morphology. D 2003 Elsevier Science B.V. All rights reserved. Keywords: Bed load transport; Channel morphology; Path length; Field experiment; Sediment tracers

1. Introduction

* Corresponding author. Fax: +1-519-661-3750. E-mail addresses: [email protected] (R.S. Pyrce), [email protected] (P.E. Ashmore). 1 Current address: Watershed Science Centre, Trent University, Symons Campus, 1600 West Bank Drive, Peterborough, Ontario, Canada K9J-7B8.

The displacement of bed sediment particles in gravel-bed rivers during transport can be represented by frequency distributions of path lengths (distance of travel) of individual grains. ‘‘Path length’’ (Einstein, 1937) denotes the total distance moved by particles from initial entrainment to final deposition and is

0169-555X/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0169-555X(03)00077-1

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distinct from ‘‘step length’’, which denotes intermittent bursts of movement between ‘‘rest periods’’ as the particle moves along the path. The path length is important for two reasons: (i) it is a fundamental characteristic of the bed load transport process because, when combined with an estimate of the quantity of material mobilized and the elapsed time, path length provides a formal definition of the bed load transport rate (Einstein, 1937), and therefore a basis for measurement and prediction of bed load transport rate; and (ii) particle displacement relates directly to the development of river bed topography, which is the net outcome of individual particle displacements from discrete erosion loci. Previous analyses of particle path length, primarily based on deployment and recovery of tracers in the field, have tended to focus on hydraulic control of path length and neglect the possible relation between path length and channel morphology. In this paper, we review and re-analyze published bed load tracing data from gravel-bed streams in order to describe the range of observed path length frequency distributions, determine whether path length frequency distributions are related to the length scale of pool – bar morphology and, if so, identify the conditions under which this occurs.

2. Background The governing equation for spatially averaged bed load transport rate, modified from the definition of Einstein (1937), can be written as: g ¼ rL=t

ð1Þ

where g is the weight rate of bed load transport per unit channel width (kg m 1 s 1); r is the weight of the eroded material per area of bed (kg m 2); L is the mean path length (m); and t is elapsed time (s). However, there remains considerable uncertainty about the controls on mean path length and the form of the characteristic distribution(s). In general, L increases with elapsed time (e.g., see results of Einstein’s, 1937 experiments), particle size, and particle mobility (e.g., unit stream power (x) in excess of critical; Hassan et al., 1991; Hassan and Church, 1992; Haschenburger and Church, 1998). In addition, there is evidence that larger particles have shorter path lengths (Hassan and Church, 1992). In some previous

work, the effect of elapsed time was accounted for by using virtual velocity (path length/elapsed time) as the dependent variable. Existing field data for L are widely scattered in relation to stream power, particle size, and event duration (Hassan et al., 1992) because ‘‘there are mechanisms which tend to equalize travel distances for different size fractions’’ (Hassan et al., 1991, p. 625). These include differences in reach characteristics, recent history of movement, differences in bed structure, and the relative size of grains in the bed material. Observed frequency distributions of path length are typically strongly positively skewed (e.g., gamma, exponential) (Hassan et al., 1991; Hassan and Church, 1992; Schmidt and Ergenzinger, 1992; Gintz et al., 1996). The peaks of these distributions are found near the point of tracer input; thus most particles either do not move or move only a short distance, and fewer and fewer grains are transported farther downstream. Although the effect of particle mobility on mean path length has been analyzed, there appears to have been little consideration of whether the shape of path length frequency distributions changes according to event duration or excess stream power. An additional factor potentially affecting L is the relation between bedload transport and stream bed morphology, especially the effect of regularly spaced bars in constraining downstream particle movement. It is expected that under some conditions particles move directly from erosion site to the nearest depositional site, which could produce, for example, modes in path length distributions corresponding to pool – bar spacing (e.g., Hassan and Church, 1992; Ferguson and Ashworth, 1992; Sear et al., 2000; Ferguson et al., 2002). The significance of this relation between particle movement and the occurrence and development of bar or riffle topography in gravel-bed rivers has been identified previously (Hassan and Church, 1992; Sear et al., 2000). Hassan and Church (1992, p. 175) are explicit on this point: ‘‘bar spacing is apt to emerge as a significant transport length scale, especially in large events’’. They recognize that some particle displacements are comparable with the ‘‘morphological scale’’ of the channel and this is significant because these features systematically influence particle movement and also because these features are formed by the transport process (Hassan and Church, 1992, p. 168). The implication is that, if pool –bar spacing significantly affects particle displacement, particle movement

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would cease to be the purely stochastic process described by Einstein (1937) and control of path length by particle mobility and event duration would become secondary to the effect of the morphology. The possible dominance of morphological control of path length is explicit in some morphological estimates of transport rate in rivers based on Eq. (1) (Ashmore and Church, 1998). For example, in the absence of direct measurements of path length it has often been assumed that the path length is equivalent to the spacing of the principle erosion and deposition sites, i.e. the spacing of sequential pools and bars (Neill, 1971; Ashmore and Church, 1998) and that particle transfer is directly (and continuously) from pool to bar. In the case of river meanders, Neill (1971) is explicit about the hypothetical connection between morphology and path length: Qs ¼ Lhye=yt

ð2Þ

where Qs is volumetric bed load transport rate (m3 day 1), L is path length (m), h is bank height (m), and

169

ye/yt is the rate of lateral erosion (m2 day 1). Neill (1971) assumed that path length was equal to the distance from the scour pool adjacent to an eroding bank (erosion site) to the first downstream point bar (deposition site). Thus, path length equals k/2 (k is meander wavelength) and is inherently linked to channel morphology. Further, the lateral erosion rate can be replaced by the product of bend amplitude and rate of down-valley migration of bends (Neill, 1987) so that transport is explicitly linked to morphodynamics and assumes that mean path length is constrained by, and contributes to the development of, the meandering morphology. The general assumption that path length can be inferred from morphology has often been made (Table 1). Path length has been variously assumed or inferred to be (i) equal to the spacing between major accumulation zones (Church et al., 1986; McLean, 1990); (ii) equal to k/2 (Neill, 1987), mean pool –bar/riffle spacing or a multiple of channel width (often 5) (e.g., Carson and Griffiths, 1989; McLean, 1990; Martin and

Table 1 Spatial and temporal scales of bed load estimation using the morphological method Study Large scale Church et al. (1986, Mackenzie, Canada) Neill (1987, Tanana, USA)a McLean and Church (1999, Fraser, Canada)b Medium scale Carson and Griffiths (1989, Waimakariri, New Zealand) Martin and Church (1995, Vedder, Canada)b Ham and Church (2000, Chilliwack, Canada) Small-scale Ferguson et al. (1992, Sunwapta, Canada) Ferguson and Ashworth (1992, White, USA) Goff and Ashmore (1994, Sunwapta, Canada) a b

Basin size (km2)

Transport distances (km)

Bed load estimate (tons year 1)

Time of study (years)

>105

1 – 10

105 – 106

30 – 45

f 103

0.4 – 1.0

102 – 104

1 – 40

pro-glacial

0.025 – 0.10

100 – 105

daily data collection

kg m 1 day 1

for f one month

< 500

Some background information provided by Burrows et al. (1981) and Collins (1990). Although transport distance information is given, method of bed load estimation is by sediment budget.

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Church, 1995; Ham and Church, 2000); or (iii) equal to the distance between surveyed sites of erosion and deposition (Ferguson and Ashworth, 1992; Ferguson et al., 1992; Goff and Ashmore, 1994; McLean and Church, 1999; Eaton and Lapointe, 2001). In many of these field studies, bedload transport rates calculated using Eq. (1) agree well with more traditional methods of bedload estimation (trap sampling or bed load functions) indicating validity in the assumption that path length scales with channel morphology under some conditions. Thus, there are good reasons to expect a relation between channel morphology and particle path length. However, there is only passing, if any, mention of this effect in most tracer studies. There has been no systematic analysis of whether it occurs and under what conditions. Hassan and Church (1992, p. 174) conclude that their analysis of the variation and hydraulic controls on path lengths raises many questions about particle transfer, one of which is whether there is a relation between path length and morphology. The question appears to have gone unanswered in more than a cursory way. More generally, it has been suggested that positively skewed (exponential and gamma) path length distributions may only represent low-intensity (or short) transport events or single flood events (Hassan and Church, 1992; Schmidt and Ergenzinger, 1992). At large discharges (higher particle mobility), particles will be influenced by, and will contribute to, the development of the morphological character of the channel and that path length frequency distributions for these events are expected to differ from those for short, low intensity events. Consequently, gamma and exponential models likely describe only part of the transport process under a limited range of conditions, or are applicable to only certain types of stream morphology. There has been no systematic experimentation or analysis intended to reveal the full range of observed path length distributions, the existence of path length distributions that show a relation to the scale of channel morphology and the conditions under which the morphological effect is apparent. The necessary starting point, before embarking on new measurement programs, is the synthesis and re-analysis of published path length data.

3. Methods Published field and flume data were re-analyzed and synthesized to identify the range of path length distributions of gravel sediment from a variety of channel types and flow conditions. Bed load tracers were used in all studies to determine path length distributions. In some studies, the raw data or distributions were published; but in other cases the distributions had to be extracted from diagrams or maps of particle movement. Each distribution is represented as the number of tracer particles against distance downstream. Data groupings vary from study to study, and these were not standardized here. Distributions include all tracer particles, not just those that moved. In cases where only mobile particles were used in the original analysis this produces distributions different from those originally published. Available information on channel dimensions, morphology, and sediment size, along with magnitude of flow events was compiled in order to relate differences in the distributions to these variables. Reporting of the hydraulic conditions under which tracers moved is erratic in the published studies. Some tracers were recovered after a single movement or flow event, others were recovered only after several (number unknown in some cases) events of differing magnitude and duration. The most consistently reported flow variable is peak flow for the period of time between deployment and recovery of tracers. In order to put all of the data onto a common hydraulic basis we estimated the non-dimensional bed shear stress for the reported peak flow, based on the available information. In most cases channel width, grain size, slope and discharge were given and we calculated iteratively the flow depth and velocity using a resistance function for gravel-bed rivers (Parker and Peterson, 1980). The depth then provided an estimate of bed shear stress from which the nondimensional shear stress is derived. Raw tracer data and shear stress estimation is available upon request from the authors.

4. Results The first systematic investigation of bed load transport distance was the work of Einstein (1937). In his

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experimental work, Einstein (1937) introduced instantaneously into the flow a large number of painted natural gravel clasts at a fixed cross-section in a straight-sided flume with gravel bed material (D50 = 24 – 34 mm). Particles moved intermittently, but continuously, until flow was turned off (in a few cases grains left the downstream end of the flume before experiments were terminated). The position of the tracers was recorded for different flow strength and flow duration. The appendices (Einstein, 1937, pp. C85 –C-94) contain plots of downstream tracer data for six (of about 60) experimental runs, which were presented to characterize distributions from a wide range of running time (Einstein, 1937, p. C-50). The number of recovered tracers average 480 grains per run. Table 2 summarises the experimental conditions, and Fig. 1a – f displays the data from the six plots presented in the appendix. The distributions range from positively skewed (Fig. 1a – d) to symmetrical (Fig. 1e,f) in a sequence of increasing running time (running time appears to range from a few minutes up to 1 h). Thus, on a plane bed, the distributions evolve with longer running time as the tracers disperse downstream as a traveling wave. The path length distribution depends primarily on the particle mobility and running time, giving rise to a variety of shapes and mean values for the path length distribution under constant flow conditions (hence the use of virtual velocity in place of path length in some subsequent studies). More important from the point of view of the relation of path length to morphology is that Einstein disregarded some of the

171

runs in which he found that the ‘‘condition of Eq. (32) is not fulfilled’’ (Einstein, 1937, p. C-51). By this he apparently means that in these individual cases the displacement process is not stochastic (as he theorized) but that grains were constrained by the influence of ‘‘sand-bars’’ that arrested their downstream movement even though flow conditions were steady and transport was continuous in other parts of the flume. Thus, the first tests of the controls on path length and virtual velocity for gravel particles are also the first to hint at the existence of an important relation between particle path length and channel morphology. The role of pool to bar transport is much more clearly apparent in subsequent qualitative experiments by Friedkin (1945) using physical models of meanders. Model channels ranged from 0.30 to 1.50 m wide by 15.2– 45.7 m long, with flow depths of 0.015 – 0.091 m. A preset meandering pattern with one concave bank composed of green sand and the next composed of red sand was used to show (after 108 h running time) that almost all of the sediment eroded from the channel banks in a bend was transported to the first bar downstream (a distance of k/2). These results strongly suggest a path length that is directly related to channel morphology. In Einstein’s (1937) and Friedkin’s (1945) experiments, channel bars trap bed load material even though the morphology of the bars is different—in Einstein’s experiments, the bars were formed from coarse bed sediment in a straight flume channel under conditions of high flow depth whereas in Friedkin’s experiments, bank erosion trans-

Table 2 Summary of Einstein’s (1937) and Friedkin’s (1945) flume experiments Channel Tracer Q (l s 1) Notes slope size (mm)

Study

Location

Tracers and input

Channel pattern and character

Einstein (1937) (Tests 39, 16, 29r, 51z, 28w, 18; pp. C85 – C90)

Hydraulic Experiment Station (Zurich, Germany)

Painted river gravel, placed instantaneously in the flow

Friedkin (1945) (Plate 6)

U.S. Waterways Experiment Station (Vicksburg, MS)

Two adjacent river banks composed of coloured sand, eroded in situ

Straight channel, ? with occasional development of sand bars (2 – 3 m in length) Low-sinuosity 0.0038 meandering point bar pattern

24 – 34

?

sand/silt

0.7 – 14.1

Evidence of positively skewed distributions and symmetrical distributions, as the influence of sand bars increase. Transport of tracer grains to the first point bar downstream.

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Fig. 1. Path length distributions from Einstein’s (1937) experiments.

ported tracer grains to developing point bars in a meandering channel. Despite these early results pointing to a relation between path length and morphology, subsequent research in the 1960s and 1970s focused on the idea that the particle transport process was purely stochastic. The research often employed sand-bed flumes with ripples and dunes, lacking point bar and scour pool development (Sayre and Hubbell, 1965; Crickmore, 1967; Grigg, 1970; Yang and Sayre, 1971; Lee and Jobson, 1977; Nakagawa and Tsujimoto, 1977;

Rathbun and Kennedy, 1978; Stelczer, 1981). Revitalized interest in the morphology and dynamics of gravel-bed rivers in the late 1970s, stimulated a number of field studies of particle displacement that are summarized in Table 3 and discussed in sequence below. These studies cover a wide variety of channel patterns, channel slope, basin area, tracer type, input conditions, and hydrologic regimes. The data from these studies are displayed in Fig. 2. A list of gravelbed tracing experiments is also presented in Sear et al. (2000).

Table 3 Summary of gravel-bed tracing experiments (1966 – 1996) Study

Location

Tracers and input

Channel pattern and character

Channel Channel Tracer slope width (m) size (mm)

Peak Q (m3 s – 1)

Thorne and Lewin (1982, Fig. 6)

River Severn, UK

Actively meandering with bars, sinuosity = 1.83

0.0018

f 30 – 40 D50bed = 40 80 (bankfull = 70, >25 competent flow events)

Laronne and Carson (1976, Fig. 10)

Seale’s Brook, QE, Canada

Painted clasts, introduction on five cross-channel transects Painted clasts placed on the channel bed

Curved with high bed packing and clustering

0.021 – 0.086

f 4–8

4 – 256

Mosley (1978, Fig. 3)

West Tamaki 3 m3 of gold limeRiver, New Zealand stone aggragate placed in the river

Meandering, across 0.022 a wide gravel bed between steep erodible banks Longitudinal channel 0.0048 bars, pool – riffle sequences

10 – 15

8 – 300

f 55

28 – 82

Negative exponential type distribution. 68% recovered move < 20 m from input. < 5 (peak flood) Positively skewed distribution with secondary mode 150 m downstream. 4.5 (R.I. = 0.5 yrs) Positively skewed distribution, tracer deposition with mean spacing of 62 m. 217 (114 mm of Positively skewed rain in 14 h) distribution; minor mode 10 – 13 m from input, perhaps related to riffle position. 1.64 – 86.6 Secondary mode apparent; extensive movement of gravel over sand. 34 (R.I. = Bi-modal distribution, 15 – 20 years) with a short movement mode at 10 m and a transport mode at 150 m.

Synthetic painted clasts, emplaced on a dry river bed in a pool

Leopold et al. (1966, Fig. 155, 3 sites)

Painted tracers arranged in squares on dry bed

Dominantly sand bed with scattered fine gravels and cobbles

0.021

5 – 35

65 – 230

Metallic tracers, placement along transects, replacing in situ clasts on bed Magnetic tracers, emplaced at the upstream riffle

Gravel-bed perennial stream

0.0037

10 – 20

32 – 128

31

2 – 20

33.1 (regulated to 1.3 – 16.0)

Butler (1977, Fig. 3)

Main Project Reach, Arroyo de los Frijoles, NM, USA Horse Creek, WY, USA

Sear (1996, Fig. 4)

River North Tyne, UK

Cobble-bed, with well- 0.0018 defined pool – riffle sequences

Kondolf and Matthews (1986, Fig. 3, 1983 – 1984)

Carmal River, CA, USA

Distinct lithology Heterogeneous mix of as rip-rap, erosion sand and gravel bars in situ at high flows

0.003

f 50

D50 = 130

273 (R.I. = 15 years)

Arkell et al. (1983, Fig. 7)

River Wye (Cefn Brwyn), Wales

Magnetic tracers, emplaced in a riffle

?

?

?

?

Minor bed form development, bare rock at or near surface

173

Bed load transport to mid – pool and pool – tail regions, 160 and 240 m from input, respectively. Tracer deposition at 350, 750 and 950 m, correlating with gravel bar locations. Transport is toward storage sites, at high flows, associated with channel margins and zones of reduced velocity.

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Hattingh and Illenberger Swartkops River, (1995, Fig. 3) South Africa

Comments

174 R.S. Pyrce, P.E. Ashmore / Geomorphology 56 (2003) 167–187 Fig. 2. Path length distributions from nine individual gravel-bed studies (1966 – 1996). The data display the range of downstream transport including positively skewed, and bi- and multi-modal distributions indicative of bed load trapping by bar-scale features.

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Fig. 3. Path length distribution from Hassan et al. (1991, 1992, 1999). 175

176 R.S. Pyrce, P.E. Ashmore / Geomorphology 56 (2003) 167–187

Fig. 4. Path length distributions from the Lainbach River, Germany (1992 – 1996).

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4.1. River Severn, UK (Thorne and Lewin, 1982) (Fig. 2a)

of path length distribution influenced by the occurrence of regularly spaced bars.

Despite a well-developed pool–bar morphology and >25 transporting flows over the duration of the study, a positively skewed path length distribution dominates the data. Five tracer sets of 100 stones each were spaced evenly along five cross-sections around a single bend. Tracer recovery was 49%. Twothirds of the recovered grains did not travel farther than 20 m from their input position, although some grains were introduced on the point bar so potential for erosion would be small compared to grains introduced in the scour pool. Although maximum discharge was just greater than bankfull, little relation between path length and channel morphology was apparent, as bed load transport was generally along the thalweg with minimal evidence of pool to bar transfer.

4.4. Swartkops River, South Africa (Hattingh and Illenberger, 1995) (Fig. 2d)

4.2. Seale’s Brook, Canada (Laronne and Carson, 1976) (Fig. 2b)

The authors examined gravel movement over a mixed gravel and sand-bed arroyo. Combining tracer data for the investigation, most of the tracers were deposited close to the input point although some grains moved a substantial distance (103 m). Preferential tracer deposition occurred nearly 1000 m from input.

This study combined data from four different input locations, resulting in a skewed-peak distribution with a minor secondary mode located 150 m from input. The authors indicated that deposition was influenced by a decrease in channel slope and by bed clustering of grains rather than bar position. 4.3. West Tamaki River, New Zealand (Mosley, 1978) (Fig. 2c) This study provides some of the clearest indication of path length related to channel morphology. Tracers were dumped in a single mass into the channel and the positions recorded after a single moderate flow event (return period approximately 0.5 years). The overall distribution is strongly skewed but has multiple, symmetrical peaks, the first of which is about 50 m downstream of the input site and the others are regularly spaced averaging about 60 m apart. The channel lacks a clear riffle/bar and pool morphology but the peaks in the distribution are five to six channel widths apart, which is consistent with typical observations of pool – bar spacing. There is also passing mention, but no details, of a second test, in the Mangapuaka Stream, that yield even clearer evidence

These are results from a single flood event, involving tracing of both natural and synthetic particles. The majority of tracers (f 52% of 400 grains) did not move but the mobile particles show a minor secondary mode 10 –13 m downstream, which was likely related to deposition on a riffle downstream of the input pool. Channel width was 55 m, so this peak seems to be strongly influenced by the proximity of a riffle to the tracer input location rather than the overall pool –riffle spacing. 4.5. Arroyo de los Frijoles, NM, USA (Leopold et al., 1966) (Fig. 2e)

4.6. Horse Creek, USA (Butler, 1977) (Fig. 2f) Tracers were deployed at several cross-sections around a bend and recovered at the end of the annual flood season during which the peak flow has a recurrence interval of 15 – 20 years. The path length distribution from this study was bimodal, containing a primary short movement mode 10 m from the input point and a transport mode 150 m from input in a stream with average width of about 20 m. Some individual particles were observed to move to the next point bar downstream. Horse Creek contained longitudinal bars at the beginning of the study; however, the bars disappeared after the spring flood. 4.7. River North Tyne, England (Sear, 1996) (Fig. 2g) Magnetic and painted tracers were introduced at a riffle in a reach in which flow is regulated by an upstream dam. Cumulative movement of several size

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fractions was monitored through the downstream pool during summer low flows and also during a large release from the reservoir (45 m3 s 1). The resulting path length distribution combining medium (6 – 20 mm) and fine (2– 6 mm) gravel tracers resulted in a large non movement mode and two smaller downstream modes related to particle deposition at the midpool (f 160 m) and pool – tail (f 240 m) regions, respectively. 4.8. Carmel River, USA (Kondolf and Matthews, 1986) (Fig. 2h) Tracers were angular dolomite stones placed to protect bridge abutments. Movement was traced over 3 years (1981 – 1983) with successively higher flood peaks. After the first year, in which the peak flow had a return period of about 2 years, the bulk of tracers remained near the ‘input’ site, giving an exponential distribution, although a small proportion moved to a bar 400 m (about 8  channel width) downstream. In subsequent years, several larger floods produced a path length distribution with peaks corresponding to locations of major gravel bars located f 350, 750, and 950 m from the point of tracer input (Fig. 2h). The authors indicated that tracer deposition at 750 m represented a stable bar from previous years, but tracers at 350 and 950 m reflected new bar locations, although it is not possible to assess the re-distribution accomplished by the individual events. The authors emphasize that the path length distributions strongly reflect the influence of channel morphology and are not simple exponential distributions. During sequential events the modes ‘jump’ from bar to bar, rather than progressing gradually down the channel as was the case in Einstein’s (1937) experiments.

4.10. Nahal Hebron, Nahal Og, Israel (Hassan et al., 1991) (Table 4) (Fig. 3a– d) Tracer distributions were measured for a series of flood events in each of these two desert streams. Particles were recovered after each event. Path length distributions for these two gravel-bed rivers were fitted with positively skewed models, although the authors noted that the influence of bed morphology led to poor model fits in several cases. The E – H – S (Einstein – Hubbell –Sayre) distribution is similar to the gamma distribution and stems from Einstein’s (1937) function of travel distances, which was further developed by Hubbell and Sayre (1964) using a probabilistic approach (Hassan et al., 1991). Four path length distributions from tracer studies in the Nahal Hebron and Nahal Og display the range of downstream transport conditions in these rivers. Fig. 3a,b is from the Nahal Hebron; and here, the E – H – S distribution successfully modelled the field data. Fig. 3c is from the Nahal Og, where the E –H – S distribution did not describe the path length distribution. The majority of bed load tracers travelled ~125 m downstream of the input position, beyond which < 25% of the recovered tracers were found. The authors indicated that this distribution was influenced by the presence of channel bars that trapped sediment. In Fig. 3d, nearly complete tracer erosion from the input point was found for tracers in January 1983 in the Nahal Hebron, resulting in a symmetrical distribution with a peak located at about 60 m, which also was poorly described by the E – H – S distribution. The peaked distributions are common in events when most of the particles move, and the authors note that when path length approaches the spacing of large bars the bars exert a large influence on the particle movement. This tends to produce more complex path length distributions at higher flows.

4.9. River Wye, Wales (Arkell et al., 1983) (Fig. 2i) Under a single high flow event, this study found that tracer transport was toward morphologically dependent storage sites, resulting in a primary short movement mode and a significant secondary path length mode. Unfortunately, further interpretation is impossible because of minimal information on channel morphology and hydraulics of the relevant flows.

4.11. Harris and Carnation Creeks, Canada (Hassan and Church, 1992) (Table 4) (Fig. 3e– f) Tracers were recovered after individual flow events, but these were complex, multi-peak rainfall or snowmelt events. Bed-sediment movement in Harris Creek displayed minor path length modes located 112.5 and 237.5 m downstream of input, along with a significant short movement peak at 19 m. Sediment movement in

Table 4 Summary of Hassan et al.’s gravel-bed tracing experiments (1991 – 1999) Study

Location

Hassan et al. (1991) Nahal Hebron, (Fig. 4A, 19.01.83) Israel (Fig. 4C, 08.11.86)

Tracers and input

Channel pattern and character

Channel Channel Tracer size Peak Q slope width (m) (mm) (m3 s1)

Magnetic tracers, less constrained surrounding particles

Alternation of gravel than bars and sandy pools

0.008 – 0.016

3–5

30 – 180 D50=80

(Fig. 4E, 08.01.86) Nahal Og, Israel Magnetic tracers, less constrained than surrounding particles

Point and alternate bars, intrabar channels, two small rocky waterfalls

0.014

5 – 12

Hassan and Church Harris Creek, (1992, Fig. 8.2a)a B.C., Canada

Cobble-sized gravels, stable banks, equilibrium reach

0.013b

f15 – 30 6 – 512

Pool – riffle units, possible migration of bed forms, straight reaches with sharp bends forced by bedrock and stable banks

0.0056 – f15 0.012

Magnetic tracers and lithology

Hassan and Church Carnation Creek, Magnetic tracers (1992, Fig. 8.3a)c BC, Canada

Sand bed with flat areas, 0.00215 20 – 40 Hassan et al. (1999) Metsemotlhaba Magnetic tracers, weakly developed point bars, (Fig. 5a, 87 – 88) River, Botswana positioned along three cross-channel stable island (Fig. 5c, 88 – 89) lines, evenly spaced (Fig. 5d, 90 – 91)

30 – 180 D50=80

16 – 180

4A:Qmax=9.20 E – H – S fit is good. 4C:Qmax=49.8 E – H – S fit is good, transport mode evident, study reach is mostly depositional. 4D:Qmax=33.0 E – H – S significantly different from measured data; symmetrical distribution. 4E:Qmax=36.7 E – H – S fit fails to model data, due to bar trapping of bed load tracer grains.

One complex melt event

E – H – S fit fails to model data. Tracer transport to 112.5 and 237.5 m, related to morphology, and a complex snowmelt event.

One major, multi-peak rainstorm event

E – H – S fit fails to model data. Secondary mode related to bar trapping of tracer grains and complex rainstorm induced flows.

synthetic, f3 – 5 discrete D50=18.5 events per year limestone, D50=47 – 90

E – H – S fit fails all three distributions. 5a: Symmetrical distribution, island likely traps tracer grains. 5c: Non – movement of tracers. 5d: Large single event, tracers moved f800 m from initial position.

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(Fig. 4D, 23.01.83)

Notes

a

Some background information provided by Church et al. (1991). An estimate of channel slope is from Tribe and Church (1999). c Some background information provided by Haschenburger and Church (1998). b

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180

Table 5 Summary of the Lainbach River tracing experiments (1992 – 1996) Location

Schmidt and Gintz (1995; Fig. 3.8a, July 14/89; Fig. 3.8b, July 25/91)

Lainbach River, Artificial magnetic S. Germany tracers exchanged with natural stones of similar shape

Schmidt and Ergenzinger Lainbach River, S. Germany (1992; Fig. 3., 4.7.86; Fig. 3., 22.8.86; Fig. 5a, HW2, 1988; Fig. 5b, HW3, 1988; Fig. 5c, HW4-8, 1988)

Gintz et al. (1996; Fig. 4g, June 27/91)

Tracers and input

Passive (iron, magnetic) and active (radio) tracers eroded from a pool

Lainbach River, Artifical magnetic S. Germany tracers exchanged with bar, step, and pool clasts

Busskamp (1994; Fig. 5b) Lainbach River, Active radio tracers S. Germany assess step lengths a

Channel pattern and character

Channel slope

Channel Tracer Peak Q width (m) size (mm) (m3 s 1)

Step pools with side bars. 1989 = 0.020 f 10 A June 1990 rain event 1991 = 0.035 (165 m3 s 1) completely changed the existing morphology. The original step-pool pattern was re-established by 1992. Step pools with side bars. 1989 = 0.020 f 10

Notes

30 – 170

3.8a Qmax = 12.2, (dur. = 9.25 h) 3.8b Qmax = 7.4, (dur. = 15.75 h)

Deviations from the fitted gamma curve demonstrate the morphologic control of the step – pool sequence.

50 – 170

1986=?

Step pools with side bars. 1991 = 0.035 f 10

D50 = 30 – 170

HW2 = 3.9 HW3 = 8.9 HW4 = 18.5 Qmax = 8.7, (dur. = 10.5 h)

Step pools with side bars. 1991 = 0.035 f 10

D50 = 60 – 130a

Initially a positively skewed function, eventually with modes at 50 and 75 m related to morphologic change from pool to step. The HW floods display tracer movement over time. Sediment bars halt the downstream movement of tracers, despite high amounts of bed load that do not move from their initial position. Step lengths modelled using a gamma distribution.

Some background information from Schmidt and Ergenzinger (1992).

Qmax = 18.5a

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Study

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Carnation Creek displayed a strong bimodal distribution. The authors suggested that a channel bar in Carnation Creek trapped transported bed material. The E – H –S model did not describe either the Harris Creek or Carnation Creek data because of the morphological influence of the river-bed and the complex flow events. 4.12. Metsemotlhaba River, Botswana (Hassan et al., 1999) (Table 4) (Fig. 3g – i) This is an unusual case because it involves gravel movement in a predominantly sand-bed river. Tagged particles were placed at the upstream end of a low sinuosity straight reach with weakly developed point bars. Seasonal gravel tracer recovery (after several flow events) on the sand-bed Metsemotlhaba River displayed three different bed load distributions. In 1988 –1989, >80% of the tracers did not move (Fig. 3g). In 1990 –1991, a bimodal downstream distribution was evident, related to a short movement mode, and a minor mode located f 800 m downstream (Fig. 3h). In 1987 – 1988, bed load transport distances were symmetrical in character (Fig. 3i). The E – H –S model did not describe the path length distributions from any of the Metsemotlhaba River studies. Channel morphology was again important because large proportions of tracer particles were found in the vicinity of a midchannel island and on point bars. 4.13. Lainbach River, Southern Germany (Schmidt and Ergenzinger, 1992; Busskamp, 1994; Schmidt, 1994; Schmidt and Gintz, 1995; Gintz et al., 1996) (Table 5, Fig. 4)

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positions were recorded, without recovery, during successive flood events with all tracers beginning from the same initial location, during floods in 1986 and then from a series of floods in 1988 and 1989 (Fig. 4d – e,g – i; Schmidt and Ergenzinger, 1992). The initial flood resulted in positively skewed distributions (Fig. 4d,g) that became less skewed as tracer movement continued downstream during subsequent transport events (Fig. 4e,h,i). This sequence of distributions from successive events is reminiscent of the evolution of distributions with elapsed time observed in the flume by Einstein (see Fig. 1). The minor peaks in these distributions, for example at 50 and 75 m in Fig. 4e, are related to tracer deposition at downstream pool – step and step – pool transitions, respectively (Schmidt and Ergenzinger, 1992). Schmidt and Gintz (1995) report preferential entrainment from, and deposition in, pools and very little displacement from bars. In the summer of 1990, the catchment experienced a flood with a recurrence interval >100 years, that modified the channel bed and transformed the entire river system (Schmidt, 1994). The 1990 flood on the Lainbach transferred half of the population of recovered bed load tracers to a gravel bar located 300– 350 m downstream of input. These tracers were distributed over the entire bar surface, whereas the remaining tracers were recovered over the inspected reach with no major concentrations in specific locations (Schmidt, 1994). Tracer tests subsequent to the 1990 flood (Fig. 4c) showed average path lengths up to 10 times higher than those of similar pre-1990 events (Fig. 4b).

5. Analysis and discussion The Lainbach is a steep, gravel and boulder-bed, step – pool, mountain river that was the subject of a series of tracer tests, under a variety of conditions from 1986 to 1991. Fig. 4a – c,f display the path length distributions of tracers (mostly artificial) based on the deployment and recovery of tracers before and after individual events from 1988 to 1991. The distributions were approximated by gamma functions (Busskamp, 1994; Schmidt and Gintz, 1995; Gintz et al., 1996) but in many of the results there is evidence of irregularity in the distributions that coincides with morphological features, especially the step – pool features of the channel. There are also data from tests in which tracer

Path length distributions from these studies might be categorized into three groups: (i) positively skewed distributions with short modal path length (e.g., Leopold et al., 1966; Laronne and Carson, 1976; Mosley, 1978; Thorne and Lewin, 1982; Schmidt and Ergenzinger, 1992; Hassan et al., 1999). (ii) bi-modal or multi-modal distributions in which the shorter modes are due to non-movement or short path lengths and the longer mode(s) may be related to channel morphology (e.g., Butler, 1977;

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Arkell et al., 1983; Kondolf and Matthews, 1986; Hassan and Church, 1992; Hattingh and Illenberger, 1995; Schmidt and Gintz, 1995; Sear, 1996). (iii) uni-modal symmetrical distributions with modal path length some distance downstream of the input point and associated with bed sediment trapping on bar surfaces (e.g., Hassan et al., 1991, 1999). Distributions vary widely in form and a substantial proportion are not positively skewed with a mode close to zero, but rather have modes at longer path lengths and there is at least descriptive information in several cases indicating that these modes are related to the length scale of the dominant morphology. Thus a variety of path length distributions occur depending on both channel morphology and flow conditions (and therefore, mobility of particles). Reviewing the various distributions and the conditions under which they occur, symmetrical distributions or path length modes related to bar development seem to occur in the cases that are known to have been large flow events (e.g., Butler, 1977; Kondolf and Matthews, 1986) or in rivers in which a regular pool – riffle/bar morphology exists or might be anticipated (e.g., Arkell et al., 1983; Hassan and Church, 1992; Sear, 1996). In contrast, positively skewed distributions are often associated with small, steep channels with poorly developed pool – bar morphology, but even these cases often show secondary modes that might coincide with the scale of the morphology, such as step –pool length. In addition, the positively skewed distributions may occur in larger rivers with known pool – bar morphology, but in which the tracer tests occurred in only moderate flood events that were below the flow necessary to develop the pool – bar morphology (e.g., Thorne and Lewin, 1982). It is significant that the bulk of the existing data are derived from relatively small, steep channels often with only rudimentary channelscale topography and with flashy hydrographs in which event duration may limit path lengths. These are the cases in which the influence of morphology on path length distributions would be expected to be minimal. All of the cases from lower gradient pool – riffle channels (although there are only a few) show clear morphological influence on path lengths except in cases in which only the effects of relatively modest flows are present.

The path length distributions alone are not sufficient to identify whether the shape of the path length distributions is related to channel morphology, because the data are reported only as absolute distance (or path length relative to mean path length). Clearly information on channel morphology is also required to relate the path lengths to morphology. However, the published results seldom report information on channel morphology (e.g., bar spacing) sufficient to make this direct comparison. In the absence of this information, channel width may be used as a surrogate with the expectation that pool – bar spacing is (if present) typically, five to seven times the channel width (e.g., McLean, 1990; Ham and Church, 2000) in pool – bar type channels. In small, steep step – pool channels (e.g., Lainbach) the characteristic length of step – pools appears to be about three channel widths. Fig. 5 shows the relationship between channel width and path length for the field data presented in Figs. 2 –4. Note that the analysis is done using the modal values of the distribution to identify the location of significant accumulations of tracers rather than the mean path length that is typically used in studies of the hydraulic control of path length that, for strongly skewed distributions, may be different from the mode. All primary and secondary path length modes are plotted so that some rivers or tracer tests are represented by more than one point. Note that the graph combines data from single events and multiple events, and from a variety of types of stream. The graph also includes zero transport modes (for tests in which the bulk of tracers were immobile). Although many cases for which path length is substantially < 5 times width are apparent (from which it could be inferred that there is no morphological signal in the path length distance), a number of cases exist in which the modal path length is in the range consistent with pool –bar transfer (Kondolf and Matthews, 1986; Hassan et al., 1991; Hassan and Church, 1992; Sear, 1996). In addition, there are a number of cases (especially for the Lainbach data) in which modes coincide with possible step –pool lengths. Fig. 5 also shows a substantial range of path length relative to channel width, even at a single field site. For example, multiple bed load tracer studies on the Nahal Hebron show path length distances from 2.0 to 57.5 m, with an extreme value of 301 m; and from 0 to 170 m on the Lainbach River, with an extreme of 325 m associated with the 100-year flood event. Therefore, in a given stream,

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Fig. 5. Relationship between channel width and bed load transport distance.

modal path lengths related to morphology occur in some events and not in others. The extreme cases translate into path lengths much greater than 15 w so that in some cases particles are apparently moving beyond the first bar downstream. The reasons for this cannot be discerned from published information. A variety of circumstances are possible and require systematic investigation: bed load movement may be unconstrained because channel bars are absent; flow magnitude may be so great that there is simultaneous destruction, or ‘‘rescaling’’ of the bar morphology; larger events may be sufficiently long for particles to be deposited and re-entrained more than once and so moved through several pool – bar sequences in a single event; events may be sufficiently energetic and long

enough for entire bars to be removed and/or re-deposited during the event. Path length modes differ substantially between streams and between tests, and in some cases modes that are apparently at distances commensurate with the length scale of the channel morphology appear for some tests and not for others. Are different hydraulic conditions associated with these different distributions? The important data here are those that relate to tracer deployment and recovery for a single mobilizing event. These are the cases for which the hydraulic conditions can be derived (for multiple events it will be unclear which events accomplished what proportion of the total displacement). Also, movement over distances similar to the morphological scale during a single

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Fig. 6. Relationship between non-dimensional bed shear stress and relative bed load path length.

event will demonstrate that this movement occurs as a single long path from erosion to deposition site, rather than as a series of short paths in successive events leading eventually to deposition on a bar. Fig. 6 shows the modal path lengths (as a proportion of channel width) plotted against non-dimensional bed shear stress. Modal path lengths of less than two channel widths (and therefore unrelated to channel morphology) occur at non-dimensional shear stress less than 0.06 with three exceptions. Two of these are tracer tests in Lainbach in which the path length mode for the mobile particles is several channel widths but the primary mode relates to non-movement of a significant proportion of the tracers (these non-movement data are sometimes difficult to interpret because some relate to tracers on bars that are unlikely to be entrained during moderate flows). The third case is the

Swartkops River in which there is a riffle only a few metres downstream of the input site that acted as a local sediment trap. Primary and secondary modes with path lengths equivalent to several channel widths generally occur at non-dimensional shear stress greater than 0.06. The pool –bar channel types (Harris Cr., Carnation Cr., Tamaki R. and Carmel R.) show primary or secondary modes at distances commensurate with pool – bar spacing. Carmel River shows this at unusually low non-dimensional shear stress. The explanation may be that the tracers in this case were armour stones that were entrained from the bank and, once moving, may have had low rolling resistance. Several of the Lainbach tests have modes at three to four channel widths that may relate to the length of step – pool sequences in this stream.

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There is uncertainty in these hydraulic re-constructions from numerous sources including the resistance function chosen, but the inference from Fig. 6 is that at low non-dimensional shear stress, positively skewed distributions with short modal path length dominate and that increasing non-dimensional shear stress favours the appearance of path length modes equivalent to several channel widths or multiples of this distance.

6. Discussion Previous analyses of some of these data have demonstrated that mean path length is controlled by hydraulic conditions (Hassan and Church, 1992; Sear et al., 2000) and make clear mention of the apparent influence of channel morphology on path length distributions. This analysis suggests that those longer path lengths are often constrained by, or coincide with, the characteristic length scale of the channel morphology. Bimodal distributions are common, with one mode related to non-movement and the other related to the channel morphology. In many cases, the bars are not an absolute constraint on path length—some particles move further downstream—but the key point is that substantial evidence exists showing that the particle transfer process is not purely stochastic, especially for events in which flows are sufficient to mobilize most of the particle sizes on the stream-bed. Similar statements have been made on the basis of some of these individual studies (Kondolf and Matthews, 1986; Church and Hassan, 1992) and this synthesis adds weight to those statements. If particle path lengths are related, at least under some conditions, to channel morphology this provides some support for the assumption used in some bed load transport estimates that mean path length during large events coincides with pool –bar spacing. More significantly, it implies that the functional control of path length involves both flow hydraulics (particle mobility) and channel morphology and that the relative influence of these two controls is a function of event magnitude. This proposition needs further assessment, but this will require new data. Existing data are inconclusive for several reasons: the data often involve multiple flow events between deployment and recovery; data are cumulative positions over several events, so that the conditions associated with particular path lengths can-

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not be derived; there is simply insufficient information on hydraulic conditions or of the location of tracer emplacement and recovery relative to the channel morphology. Beyond these operational issues, nature confounds results by delivering complex hydrographs of variable duration and magnitude and very few events sufficiently large to mobilize large numbers of tracers and to actually develop the morphology during tracer tests. Furthermore, very few previous studies are from pool –bar channel types in which the morphological effect is expected to be most significant. The bulk of existing tests are from small steep channels with large relative roughness and only rudimentary channel-scale morphology in which the particle transfer process is more likely to be stochastic and governed by grainscale interactions. The resolution to a number of these difficulties may be to conduct experiments under controlled flow regimes or in laboratory conditions using physical models in which a variety of channel types and hydraulic conditions can be reproduced.

7. Conclusions A re-analysis of existing particle tracer data reveals a much greater variety of path length distributions than the gamma or exponential distributions emphasized in previous work. Some of the modal path lengths are at least of the correct approximate magnitude to be commensurate with the likely length scale of the morphology (pool – bar or step – pool spacing) and there is descriptive information supporting this result. The path length modes related to channel morphology are more probable at higher flows and therefore higher values of non-dimensional bed shear stress. Unfortunately, the overall outcome is inconclusive because data on channel morphology, flow conditions and tracer deployment are not well reported in all of the studies and in many of the studies, it is impossible to separate out results for single movement events from those involving multiple events. It is clear that there is a need for deliberate, systematic investigation of the relation between path length distributions and channel morphology to establish the types of channels and particle mobility conditions in which particle transfer is, or is not, primarily stochastic, and the conditions under which particular path length frequency distributions occur. In particular, this requires

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more field tracer tests in pool – bar channel types, under conditions in which the morphology develops and is measured, and under controlled conditions in the laboratory where controlling factors can be isolated and the relevant flow conditions responsible for particle displacement can be varied and precisely known. Acknowledgements This research was supported by a Natural Sciences and Engineering Research Council of Canada grant. An Ontario Graduate Scholarship provided further funding to R.S.P. We are grateful to Dr. Joe Desloges and Dr. Chris Smart for comments on an earlier version of this manuscript. Comments provided by Dr. Bernard O. Bauer, Dr. Richard Marston, and an anonymous reviewer greatly improved the final version of this paper. References Arkell, B., Leeks, G., Newson, G.M., Oldfield, F., 1983. Trapping and tracing: some recent observations of supply and transport of coarse sediment from upland Wales. In: Collinson, J.D., Lewin, J. (Eds.), Modern and Ancient Fluvial Systems. Special Publication of the International Association of Sedimentologists, vol. 6. Blackwell Scientific, Oxford, UK, pp. 107 – 119. Ashmore, P., Church, M., 1998. Sediment transport and river morphology: a paradigm for study. In: Klingeman, P.C., Beschta, R.L., Komar, P.D., Bradley, J.B. (Eds.), Gravel-Bed Rivers in the Environment. Water Resources Publications, Highlands Ranch, CO, pp. 115 – 148. Burrows, R.L., Emmett, W.W., Parks, B., 1981. Sediment Transport in the Tanana River near Fairbanks, Alaska, 1977 – 1979. United States Geological Survey, Water Resources Investigation, 81-20, 62 pp. Busskamp, R., 1994. The influence of channel steps on coarse bed load transport in mountain torrents: case study using the radio tracer technique ‘‘Petsy’’. In: Ergenzinger, P., Schmidt, K.-H. (Eds.), Dynamics and Geomorphology of Mountain Rivers. Lecture Notes in Earth Sciences, vol. 52. Springer-Verlag, NY, pp. 129 – 139. Butler, P.R., 1977. Movement of cobbles in a gravel-bed stream during a flood season. Geological Society of America Bulletin 88, 1072 – 1074. Carson, M.A., Griffiths, G.A., 1989. Gravel transport in the braided Waimakariri River: mechanisms, measurements and predictions. Journal of Hydrology 109, 201 – 220. Church, M., Hassan, M.A., 1992. Size and distance of travel of unconstrained clasts on a streambed. Water Resources Research 28, 299 – 303.

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