Acta Geophysica vol. 60, no. 6, Dec. 2012, pp. 1607-1638

DOI: 10.2478/s11600-012-0075-z

________________________________________________ © 2012 Institute of Geophysics, Polish Academy of Sciences

Bedform Effect on the Reorganization of Surface and Subsurface Grain Size Distribution

in Gravel Bedded Channels

Arvind SINGH1, Michele GUALA1, Stefano LANZONI2, and Efi FOUFOULA-GEORGIOU1

1St. Anthony Falls Laboratory and National Center for Earth-surface Dynamics, Department of Civil Engineering, University of Minnesota, Minneapolis, USA

e-mail: sing0336@umn.edu 2University of Padova, Department IMAGE, Padova, Italy

A b s t r a c t

Quantification of river bedform variability and complexity is important for sediment transport modeling as well as for characterization of river morphology. Alluvial bedforms are shown to exhibit highly non-linear dynamics across a range of scales, affect local bed roughness, and vary with local hydraulic, hydrologic, and geomorphic properties. This paper examines sediment sorting on the crest and trough of gravel bed-forms and relates it to bed elevation statistics. The data analysed here are the spatial and temporal series of bed elevation, grain size distribution of surface and subsurface bed materials, and sediment transport rates from flume experiments. We describe surface topography through bedform variability in height and wavelength and multiscale analysis of bed eleva-tions as a function of discharge. We further relate bedform migration to preferential distribution of coarse and fine sediments on the troughs and crests, respectively, measuring directly surface and subsurface grain size distributions, and indirectly the small scale roughness variations as esti-mated from high resolution topographic scans.

Key words: bedforms, roughness, grain sorting, power spectral density, sediment transport.

Author e-mail: sing0336@umn.edu

Author e-mail: sing0336@umn.edu

University of Padova, Department IMAGE, Padova, Italy

Author University of Padova, Department IMAGE, Padova, Italy

A b s t r a c t

Author A b s t r a c t

Quantification of river bedform variability and complexity is

Author Quantification of river bedform variability and complexity is

important for sediment transport modeling as well as for characterization

Author important for sediment transport modeling as well as for characterization of river morphology. Alluvial bedforms are shown to exhibit highly non-

Author of river morphology. Alluvial bedforms are shown to exhibit highly non-linear dynamics across a range of scal

Author linear dynamics across a range of scalvary with local hydraulic, hydrologic, and geomorphic properties. This

Author vary with local hydraulic, hydrologic, and geomorphic properties. This

Author paper examines sediment sorting on the crest and trough of gravel bed-

Author paper examines sediment sorting on the crest and trough of gravel bed-forms and relates it to bed elevation statistics. The data analysed here are

Author forms and relates it to bed elevation statistics. The data analysed here aretheAuthor the spatial and temporal series of bed elevation, grain size distribution of Author

spatial and temporal series of bed elevation, grain size distribution of surface and subsurface bed materials, and sediment transport rates from Author

surface and subsurface bed materials, and sediment transport rates from flume experiments. We describe Author

flume experiments. We describe variability in height and wavelength and multiscale analysis of bed eleva-Author

variability in height and wavelength and multiscale analysis of bed eleva-tions as a function of discharge. We further relate bedform migration to Author

tions as a function of discharge. We further relate bedform migration to

copyof Surface and Subsurface Grain Size Distribution

copyof Surface and Subsurface Grain Size Distribution

, Stefano LANZONI

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, Stefano LANZONI2

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2,

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, and Efi FOUFOULA-GEORGIOU

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and Efi FOUFOULA-GEORGIOU1

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1

onal Center for Earth-surface Dynamics, copy

onal Center for Earth-surface Dynamics, Department of Civil Engineering, University of Minnesota, Minneapolis, USA co

pyDepartment of Civil Engineering, University of Minnesota, Minneapolis, USA

e-mail: sing0336@umn.edu copy

e-mail: sing0336@umn.edu University of Padova, Department IMAGE, Padova, Italy co

pyUniversity of Padova, Department IMAGE, Padova, Italy

A. SINGH et al.

1608

1. INTRODUCTION Bedforms present on the bed surface of a gravel-bed river are highly variable and strongly depend on the local bed shear stress and grain size distribution of bed material (Nordin 1971, Paola and Borgman 1991, Powell 1998, Buffington and Montgomery 1999, Lanzoni 2000, Kleinhans et al. 2002, Blom et al. 2003, Blom and Parker 2004, van der Mark et al. 2008, Singh et al. 2011). They evolve as a result of the complex interaction between turbu-lent flow field, sediment transport and underlying fluvial bed topography (Jerolmack and Mohrig 2005, Best 2005, Venditti et al. 2005, Venditti 2007, Singh et al. 2010, Singh and Foufoula-Georgiou 2012). Quantification of their formation and evolution is essential towards understanding their inter-action with flow turbulence and particle transport, interpretation of sedimen-tary structure, developing predictive models for sediment transport, river management as well as river habitat dynamics (Simons et al. 1965, Nelson et al. 1993, 1995, Yarnell et al. 2006, ASCE 2002, Best 2005, McElroy and Mohrig 2009, Wilcock 1998, Leclair 2002).

Several studies have focused on characterizing bedform variability using both numerical, theoretical and empirical approaches (Paola and Borgman 1991, Coleman and Melville 1994, Lanzoni and Tubino 1999, Blom et al. 2003, Jerolmack and Mohrig 2005, Coleman et al. 2006, Van der Mark et al. 2008, Singh et al. 2011). For example, Jerolmack and Mohrig (2005) devel-oped a nonlinear stochastic surface evolution model which reproduces labor-atory observations of evolution of bedforms and their long term dynamical behavior as observed in natural systems. Van der Mark et al. (2008) used controlled laboratory and field studies to characterize variability in bedform geometry and suggested that bedform variability can be represented by an exponential function for the coefficient of variation.

For a bed with a wide grain size distribution, as in the case of real rivers, bedform variability results in preferential movement and deposition of sedi-ment causing significant changes in the local bed roughness (see e.g., Lanzoni and Tubino 1999, Blom et al. 2003, Blom 2008). Traditionally, to characterize gravel bed roughness, percentiles of the grain size distribution of the surface patches are used (e.g., Nikora et al. 1998). However, it has been argued that the roughness characteristics of a bed cannot be only approximated with a single parameter, say d84, since other factors, e.g., parti-cle shape, orientation, structural arrangement of the particles and bedform geometry are also important. More recently, statistical approaches, such as variograms and higher order structure function analysis have been explored, using high resolution surface elevations, as an alternative to describe the bed roughness (see e.g., Butler et al. 2001, Nikora and Walsh 2004, Aberle and Nikora 2006).

Author Several studies have focused on char

Author Several studies have focused on char

both numerical, theoretical and empirical approaches (Paola and Borgman

Author both numerical, theoretical and empirical approaches (Paola and Borgman 1991, Coleman and Melville 1994,

Author 1991, Coleman and Melville 1994, Lanzoni and Tubino 1999, Blom

Author Lanzoni and Tubino 1999, Blom2003, Jerolmack and Mohrig 2005, Coleman

Author 2003, Jerolmack and Mohrig 2005, Coleman et al.

Author et al. 2011). For example, Jerolmack and Mohrig (2005) devel-

Author 2011). For example, Jerolmack and Mohrig (2005) devel-oped a nonlinear stochastic surface e

Author oped a nonlinear stochastic surface evolution model which reproduces labor-

Author volution model which reproduces labor-

atory observations of evolution of be

Author atory observations of evolution of bedforms and their long term dynamical

Author dforms and their long term dynamical

behavior as observed in natural systems. Van der Mark

Author behavior as observed in natural systems. Van der Markcontrolled laboratory and field studies to characterize variability in bedform

Author controlled laboratory and field studies to characterize variability in bedform geometry and suggested that bedform variability can be represented by an

Author geometry and suggested that bedform variability can be represented by an exponential function for the coefficient of variation.

Author exponential function for the coefficient of variation.

For a bed with a wide grain size distribution, as in the case of real rivers,

Author For a bed with a wide grain size distribution, as in the case of real rivers,

bedform variability results in preferential movement and deposition of sedi-

Author bedform variability results in preferential movement and deposition of sedi-ment causing significant changes in the local bed roughness (see Author

ment causing significant changes in the local bed roughness (see Lanzoni and Tubino 1999,Author

Lanzoni and Tubino 1999,Author

characterize gravel bed roughness, percentiles of the grain size distribution Author

characterize gravel bed roughness, percentiles of the grain size distribution of the surface patches are used (Author

of the surface patches are used (

copylent flow field, sediment transport and underlying fluvial bed topography

copylent flow field, sediment transport and underlying fluvial bed topography

2005, Venditti 2007,

copy 2005, Venditti 2007,

Singh and Foufoula-Georgiou 2012). Quantification of

copySingh and Foufoula-Georgiou 2012). Quantification of

their formation and evolution is essential towards understanding their inter-

copytheir formation and evolution is essential towards understanding their inter-

transport, interpretation of sedimen-

copy transport, interpretation of sedimen-

models for sediment transport, river

copy

models for sediment transport, river management as well as river habitat dynamics (Simons

copy

management as well as river habitat dynamics (Simons et al.

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et al. 1965, Nelson

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1965, Nelson . 2006, ASCE 2002, Best 2005, McElroy and co

py. 2006, ASCE 2002, Best 2005, McElroy and

acterizing bedform variability using copy

acterizing bedform variability using both numerical, theoretical and empirical approaches (Paola and Borgman co

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Lanzoni and Tubino 1999, Blomcopy

Lanzoni and Tubino 1999, Blom 2006, Van der Markco

py 2006, Van der Mark

GRAIN SORTING IN GRAVEL BEDFORMS

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In gravel bed rivers, to the best of our knowledge, statistical properties of bedform-dominated beds have not been related to the observed grain size distribution, except in the study of Blom et al. (2003). Blom et al. (2003) related the observed probability density function of the bed elevation, specif-ically bedform trough elevation, to the vertical sorting within the bedform. More recently, Blom et al. (2008) proposed a morphodynamic model which reproduces time evolution of the vertical sorting profile and grain size distri-bution of bed load transport. In contrast to the bedform-dominated beds, on plane-beds, only recently, a few studies have focused on multiscale statisti-cal characterization of bed elevation fluctuations and their relation to observed grain size distribution (Nikora et al. 1998, Butler et al. 2001, Aberle and Nikora 2006).

The goal of this study is to understand and quantify how an initial grain size distribution of bed material is redistributed preferentially within the macroscale structures (crest and trough of bedforms) of the bed, both on the surface and in the subsurface layers in bedform-dominated gravel bed rivers. The data analysed are the high resolution spatial and temporal bed elevation series, grain size distribution of surface and subsurface layers, and the sedi-ment transport rates collected in a large-scale experimental channel. Specifi-cally, we relate small-scale roughness due to grain sizes to the large-scale roughness due to bedforms. We also perform multi-scale statistical charac-terization of bed elevation fluctuations obtained under different flow condi-tions.

The paper is organized as follows. In Section 2 we briefly describe the experimental setup and the data collected in two laboratory experiments under low and high flow conditions. Section 3 focuses on physical character-istics of bed topography, while Section 4 discusses multiscale statistics of bed elevations. In Section 5, an analysis on grain size distribution is per-formed whereas the relation between observed grain size distribution and bed elevations of patches of crest and trough is explored in Section 6. Sec-tion 7 focusses on discussion of the results obtained whereas Section 8 pre-sents summary and the concluding remarks.

2. EXPERIMENTAL SETUP AND DATA COLLECTED The experiments reported here were conducted in a large-scale experimental channel, called Main Channel, at the St. Anthony Falls Laboratory, Univer-sity of Minnesota. These experiments were the follow-up of previous experiments conducted in the spring of 2006 known as StreamLab06 (Singh et al. 2012b). The Main Channel is 84 m long, 2.75 m wide, and has a maxi-mum depth of 1.8 m with a maximum discharge capacity of 8000 l/s. Only the first 55 m long upstream reach of the channel was used in these experi-

Author The data analysed are the high resolutio

Author The data analysed are the high resolutioseries, grain size distribution of surface and subsurface layers, and the sedi-

Author series, grain size distribution of surface and subsurface layers, and the sedi-ment transport rates collected in a large-scale experimental channel. Specifi-

Author ment transport rates collected in a large-scale experimental channel. Specifi-cally, we relate small-scale roughness

Author cally, we relate small-scale roughness due to grain sizes to the large-scale

Author due to grain sizes to the large-scale

Author roughness due to bedforms. We also

Author roughness due to bedforms. We also perform multi-scale statistical charac-

Author perform multi-scale statistical charac-terization of bed elevation fluctuations obtained under different flow condi-

Author terization of bed elevation fluctuations obtained under different flow condi-

The paper is organized as follows. In

Author The paper is organized as follows. In

experimental setup and the data collect

Author experimental setup and the data collectunder low and high flow conditions. Section 3 focuses on physical character-

Author under low and high flow conditions. Section 3 focuses on physical character-istics of bed topography, while Section 4 discusses multiscale statistics of

Author istics of bed topography, while Section 4 discusses multiscale statistics of bed elevations. In Section 5, an analysis on grain size distribution is per-

Author bed elevations. In Section 5, an analysis on grain size distribution is per-

Author formed whereas the relation between observed grain size distribution and

Author formed whereas the relation between observed grain size distribution and bed elevations of patches of crest and trough is explored in Section 6. Sec-Author

bed elevations of patches of crest and trough is explored in Section 6. Sec-tion 7 focusses on discussion of the results obtained whereas Section 8 pre-Author

tion 7 focusses on discussion of the results obtained whereas Section 8 pre-sents summary and the concluding remarks. Author

sents summary and the concluding remarks.

copyast to the bedform-dominated beds, on

copyast to the bedform-dominated beds, on

have focused on multiscale statisti-

copy have focused on multiscale statisti-

cal characterization of bed elevation fluctuations and their relation to

copycal characterization of bed elevation fluctuations and their relation to

1998, Butler

copy 1998, Butler et al.

copyet al. 2001,

copy 2001,

The goal of this study is to understand and quantify how an initial grain

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The goal of this study is to understand and quantify how an initial grain size distribution of bed material is redistributed preferentially within the

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size distribution of bed material is redistributed preferentially within the macroscale structures (crest and trough of bedforms) of the bed, both on the co

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bedform-dominated gravel bed rivers. copy

bedform-dominated gravel bed rivers. n spatial and temporal bed elevation co

pyn spatial and temporal bed elevation

series, grain size distribution of surface and subsurface layers, and the sedi-copy

series, grain size distribution of surface and subsurface layers, and the sedi-ment transport rates collected in a large-scale experimental channel. Specifi-co

pyment transport rates collected in a large-scale experimental channel. Specifi-

due to grain sizes to the large-scale co

pydue to grain sizes to the large-scale

A. SINGH et al.

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ments. The sediment was partially recirculated while the water, taken directly from the Mississippi River, was fed to the channel without recircula-tion.

The channel was filled with a 0.45 m thick layer of sediment, composed of a mixture of gravel and sand with density about 2650 kg/m3, with a medi-an particle size diameter d50 = 7.7 mm, and an overall grain size distribution (hereafter GSD) characterized by d16 = 2.2 mm and d84 = 21.2 mm.

To achieve dynamic equilibrium in transport and slope adjustment for both water surface and sediment bed, a constant water discharge Q was fed into the channel prior to the data collection. This dynamic equilibrium state was evaluated by checking the stability of the 60 min average total sediment flux at the downstream end of the test section. Continuous data collection occurred for about twenty hours after the channel had reached dynamic equi-librium.

The data presented here are the spatial and temporal series of bed eleva-tion, grain size distribution, and the sediment transport rates. The spatial series of bed elevations were collected by a three-axis positionable data acquisition (DAQ) carriage, capable of traversing the entire 55 × 2.74 m test reach and positioning probes with an accuracy of 1 mm along all the three

Fig. 1. Schematic of the Main Channel showing the location of sonars used to measure temporal bed ele-vations, of sediment pans used to monitor sediment transport rates and of the patches considered for sampling of sediment used to determine surface and subsurface grain size distri-butions. Spatial bed eleva-tions were sampled by laser scanner mounted on a po-sitionable data acquisition carriage that can move along the 55 m long test section. The dashed line represents the centerline of the channel while the direc-tion of the flow is from bot-tom to the top of the figure.

Author series of bed elevations were collected by a three-axis positionable data

Author series of bed elevations were collected by a three-axis positionable data acquisition (DAQ) carriage, capable of traversing the entire 55

Author acquisition (DAQ) carriage, capable of traversing the entire 55reach and positioning probes with an accuracy of 1 mm along all the three

Author reach and positioning probes with an accuracy of 1 mm along all the three

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Author copyTo achieve dynamic equilibrium in transport and slope adjustment for

copyTo achieve dynamic equilibrium in transport and slope adjustment for

both water surface and sediment bed, a constant water discharge

copyboth water surface and sediment bed, a constant water discharge Q

copyQ was fed

copy was fed

ection. This dynamic equilibrium state

copyection. This dynamic equilibrium state

of the 60 min average total sediment

copyof the 60 min average total sediment

flux at the downstream end of the test section. Continuous data collection

copyflux at the downstream end of the test section. Continuous data collection

occurred for about twenty hours after the channel had reached dynamic equi-

copy

occurred for about twenty hours after the channel had reached dynamic equi-

The data presented here are the spatial and temporal series of bed eleva-copy

The data presented here are the spatial and temporal series of bed eleva-tion, grain size distribution, and the sediment transport rates. The spatial co

pytion, grain size distribution, and the sediment transport rates. The spatial series of bed elevations were collected by a three-axis positionable data co

pyseries of bed elevations were collected by a three-axis positionable data acquisition (DAQ) carriage, capable of traversing the entire 55co

pyacquisition (DAQ) carriage, capable of traversing the entire 55reach and positioning probes with an accuracy of 1 mm along all the three co

pyreach and positioning probes with an accuracy of 1 mm along all the three

GRAIN SORTING IN GRAVEL BEDFORMS

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axes. The sampling resolution of the spatial bed elevation was 1 cm in both streamwise and spanwise direction. The temporal bed elevations were meas-ured through submersible sonar transducers of 2.5 cm diameter, deployed 0.3 m (on average) below the water surface, at the downstream end of the investigated reach (Fig. 1). The sampling interval of temporal bed elevation measurements was 5 s with a vertical precision of 1 mm.

Bedload traps located at the downstream end of the test reach, consisting of 5 weighing pans of equal size that spanned the width of the channel, were used for measuring sediment transport rates (Fig. 1). The weight of bedload sediment falling into the weigh pans was automatically recorded every 1.1 s. To remove the negative, unphysical values that sometimes appeared in the record of sediment transport rate sequence, a moving average window of 2 min was used (see Singh et al. 2009). A typical series of 2 min averaged sediment transport rates is shown in Fig. 2 for the discharges of 1500 l/s (Fig. 2a) and 2800 l/s (Fig. 2b). After the weigh pans filled with a maximum of 40 kg of sediment, their base tipped to release the sediment and reset the weigh pan. The released sediment was re-circulated and returned back into the channel at the upstream end of the 55 m test reach, to maintain equilibri-um conditions through a closed system.

In order to study the GSD of surface and subsurface composition of the bed material, the channel was drained at the end of each experiment and

Fig. 2. Two minutes averaged sediment transport rates for the discharges of: (a) 1500 and (b) 2800 l/s.

Author weigh pan. The released sediment was re

Author weigh pan. The released sediment was rethe channel at the upstream end of the 55 m test reach, to maintain equilibri-

Author the channel at the upstream end of the 55 m test reach, to maintain equilibri-um conditions through a closed system.

Author um conditions through a closed system. In order to study the GSD of surface and subsurface composition of the

Author In order to study the GSD of surface and subsurface composition of the bed material, the channel was drained at the end of each experiment and

Author bed material, the channel was drained at the end of each experiment and

Author

Author

Author copyof 5 weighing pans of equal size that spanned the width of the channel, were

copyof 5 weighing pans of equal size that spanned the width of the channel, were

tes (Fig. 1). The weight of bedload

copytes (Fig. 1). The weight of bedload

automatically recorded every 1.1 s.

copyautomatically recorded every 1.1 s.

lues that sometimes

copylues that sometimes appeared in the

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copyrecord of sediment transport rate sequence, a moving average window of

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2009). A typical series of 2 min averaged sediment transport rates is shown in Fig. 2 for the discharges of 1500 l/s

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of 40 kg of sediment, their base tipped to release the sediment and reset the copy

of 40 kg of sediment, their base tipped to release the sediment and reset the -circulated and returned back into co

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the channel at the upstream end of the 55 m test reach, to maintain equilibri-

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pyIn order to study the GSD of surface and subsurface composition of the

A. SINGH et al.

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Fig. 3. Photographs of the patches of bed surfaces obtained at the end of the experi-ments carried out with discharges of: (a) 1500 and (b) 2800 l/s. Note that marks 1 and 2 are located in the correspondence of bedform crests and troughs, respectively.

surface layers of 30 × 30 cm patch size were painted, after identifying the patch location on the bedform crest and trough (Fig. 1). The surface layers of these patches were carefully extracted for determining the surface GSD; a 10 cm deep layer of bed material was then excavated in order to determine the subsurface GSD. Figure 3 shows the marked patches on the crest and the trough of bedforms observed at the end of runs carried out for discharges of 1500 (Fig. 3a) and 2800 l/s (Fig. 3b).

Although measurements were taken over a range of discharges corre-sponding to different bed shear stresses, here, for the sake of brevity we con-sider only the data collected at discharges of 1500 l/s (low flow) and 2800 l/s (high flow), corresponding to Shields stress of 0.049 and 0.099, respectively (for details about the hydraulic conditions see Table 1). The critical Shields stress ( cτ

∗ ), determined on the basis of the median grain size of the mixture d50 = 7.7 mm, was assumed to be 0.03 as suggested by Buffington and Montgomery (1997) and references therein. The aspect ratio (channel width/ flow depth) corresponding to the investigated flows were 6.4 and 4.3 for the discharges of 1500 and 2800 l/s, respectively.

Table 1 Hydraulic conditions and statistics of spatial bed elevation

Q [l/s]

D [m]

V [m/s] Sw hR

[m] bτ∗ std(h(x))

[mm] 1500 0.43 1.27 0.0019 0.33 0.049 16.19 2800 0.64 1.59 0.0029 0.44 0.099 39.28

Explanations: Q – water discharge, D – average flow depth along test reach, V – average flow velocity, Sw – water surface slope, hR – hydraulic radius,

bτ∗ – dimensionless shields stress (computed using hydraulic radius),

std(h(x)) – standard deviation of spatial bed elevation.

Author the subsurface GSD. Figure 3 shows the marked patches on the crest and the

Author the subsurface GSD. Figure 3 shows the marked patches on the crest and the trough of bedforms observed at the end of runs carried out for discharges of

Author trough of bedforms observed at the end of runs carried out for discharges of 1500 (Fig. 3a) and 2800 l/s (Fig. 3b).

Author 1500 (Fig. 3a) and 2800 l/s (Fig. 3b). Although measurements were taken ov

Author Although measurements were taken over a range of discharges corre-

Author er a range of discharges corre-sponding to different bed shear stresses, here, for the sake of brevity we con-

Author sponding to different bed shear stresses, here, for the sake of brevity we con-sider only the data collected at disch

Author sider only the data collected at discharges of 1500 l/s (low flow) and 2800 l/s

Author arges of 1500 l/s (low flow) and 2800 l/s

Author (high flow), corresponding to Shields stress of 0.049 and 0.099, respectively

Author (high flow), corresponding to Shields stress of 0.049 and 0.099, respectively (for details about the hydraulic conditio

Author (for details about the hydraulic conditio

), determined on the basis of the median grain size of the mixture

Author ), determined on the basis of the median grain size of the mixture

= 7.7 mm, was assumed to be 0.03 as suggested by Buffington and

Author = 7.7 mm, was assumed to be 0.03 as suggested by Buffington and

Montgomery (1997) and references therei

Author Montgomery (1997) and references thereiflow depth) corresponding to the investigated flows were 6.4 and 4.3 for the

Author flow depth) corresponding to the investigated flows were 6.4 and 4.3 for the discharges of 1500 and 2800 l/s, respectively. Author

discharges of 1500 and 2800 l/s, respectively.

copysurfaces obtained at the end of the experi-

copysurfaces obtained at the end of the experi-

ments carried out with discharges of: (a) 1500 and (b) 2800 l/s. Note that marks 1

copyments carried out with discharges of: (a) 1500 and (b) 2800 l/s. Note that marks 1

and troughs, respectively.

copyand troughs, respectively.

30 cm patch size were pain

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30 cm patch size were painted, after identifying the

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pya 10 cm deep layer of bed material was then excavated in order to determine the subsurface GSD. Figure 3 shows the marked patches on the crest and the co

pythe subsurface GSD. Figure 3 shows the marked patches on the crest and the trough of bedforms observed at the end of runs carried out for discharges of co

pytrough of bedforms observed at the end of runs carried out for discharges of co

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GRAIN SORTING IN GRAVEL BEDFORMS

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3. BED TOPOGRAPHY 3.1 Physical characteristics The bedforms formed within the channel consisted mainly of bedload sheets at the low discharge (1500 l/s) and of three-dimensional dunes at the high discharge (2800 l/s). Figure 4 shows the bed elevation profile at the flume centerline for the two discharges here considered. It can be seen that the low flow run (Fig. 4a) produced a channel bed with only limited topographic variation, i.e., without obvious large-scale bed structures: the standard devia-tion of the detrended (linear trend removed) bed elevation being 16.19 mm as compared to a d50 grain size of 7.7 mm. Conversely, the high flow run (Fig. 4b) generated substantial bed variability (std. dev. = 39.28 mm) at large scale in the form of dunes, with intermediate to grain-scale fluctuations superimposed on them (Table 1). In particular, the standard deviation of the detrended bed elevation roughly doubled (from 16.19 to 39.28 mm) by increasing the discharge from 1500 to 2800 l/s, suggesting that bed fluctua-tions are more variable at higher discharge.

Fig. 4. Longitudinal transects of the spatial bed elevation sampled at the channel centerline at a resolution of 10 mm after the end of the experiment for the discharges of: (a) 1500 and (b) 2800 l/s. Note that the zero bed elevation (h(x) = 0) does not correspond to the base of the flume but corresponds to the lowest bed elevation (ref-erence point) below which no elevation fluctuations were observed for a given dis-charge. A linear trend was removed from the above shown transects to compute the standard deviation of detrended bed elevation (see the statistics of detrended bed elevation in Table 1).

Author

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nnel bed with only limited topographic

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nnel bed with only limited topographic structures: the standard devia-

copystructures: the standard devia-

moved) bed elevation being 16.19 mm

copymoved) bed elevation being 16.19 mm

grain size of 7.7 mm. Conversely, the high flow run

copy grain size of 7.7 mm. Conversely, the high flow run

tial bed variability (std. dev. = 39.28 mm) at large

copytial bed variability (std. dev. = 39.28 mm) at large

ediate to grain-scale fluctuations

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ediate to grain-scale fluctuations ticular, the standard deviation of the

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ticular, the standard deviation of the on roughly doubled (from 16.19 to 39.28 mm) by

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increasing the discharge from 1500 to 2800 l/s, suggesting that bed fluctua-copy

increasing the discharge from 1500 to 2800 l/s, suggesting that bed fluctua-copy

A. SINGH et al.

1614

3.2 Bedform extraction The geometric characteristics of bedforms were extracted from the longitu-dinal spatial transects of bed elevation, obtained at distances of 0.5 m apart across the width of the flume, using the methodology described in Singh et al. (2011). From these transects, first, the high-wavenumber fluctuations (wavenumber > 10–2 mm–1, corresponding to wavelengths smaller than 10 cm) induced by small scale bedforms or grain-scale variations were filtered out using the Fourier transform of the signal and then the signal was recon-structed with all wavenumbers < 10–2 mm–1 (note that the filtered wavenum-bers correspond to the high wavenumbers in the power spectral density of bed elevation, discussed later in Section 4, where the transition in the slope of the power spectral density occurs). Then, after determining the local maxima and minima in the filtered signal, the differences between consecu-tive minima and maxima were computed. Finally, the bedform heights above a certain threshold (threshold = 2 d50) were extracted. Figure 5 shows the probability density function (hereafter pdf) of the extracted bedform heights for the discharges of 1500 (Fig. 5a) and 2800 l/s (Fig. 5b), whereas Fig. 6 shows the corresponding pdfs of bedform lengths for the discharge of 1500 (Fig. 6a) and 2800 l/s (Fig. 6b). The statistics of the extracted bedform heights and lengths are summarized in Table 2 which indicates that the mean and the standard deviation of bedform heights increases with increasing dis-charge, while an opposite behaviour is exhibited by bedform lengths. A simi-lar trend is followed by the coefficient of variation, CV (standard deviation/ mean), which shows that with increasing discharge the CV for the bedform height increases, suggesting a wider range of bedform heights at higher

Fig. 5. Probability density function of the normalized bedform heights for the dis-charges of: (a) 1500, and (b) 2800 l/s. The dotted curve shows the fitted Gamma distribution with k as the shape parameter and θ as the scale parameter.

Author for the discharges of 1500 (Fig. 5a) and 2800 l/s (Fig. 5b), whereas Fig. 6

Author for the discharges of 1500 (Fig. 5a) and 2800 l/s (Fig. 5b), whereas Fig. 6 shows the corresponding pdfs of bedform lengths for the discharge of 1500

Author shows the corresponding pdfs of bedform lengths for the discharge of 1500 (Fig. 6a) and 2800 l/s (Fig. 6b). The

Author (Fig. 6a) and 2800 l/s (Fig. 6b). The statistics of the extracted bedform

Author statistics of the extracted bedform heights and lengths are summarized in

Author heights and lengths are summarized in Table 2 which indicates that the mean

Author Table 2 which indicates that the mean

Author and the standard deviation of bedform heights increases with increasing dis-

Author and the standard deviation of bedform heights increases with increasing dis-charge, while an opposite behaviour is

Author charge, while an opposite behaviour is exhibited by bedform lengths. A simi-

Author exhibited by bedform lengths. A simi-

lar trend is followed by the coefficient

Author lar trend is followed by the coefficient mean), which shows that with increasing discharge the CV for the bedform

Author mean), which shows that with increasing discharge the CV for the bedform height increases, suggesting a wider range of bedform heights at higher

Author height increases, suggesting a wider range of bedform heights at higher

Author

Author copy

induced by small scale bedforms or grain-scale variations were filtered out

copy

induced by small scale bedforms or grain-scale variations were filtered out using the Fourier transform of the signal and then the signal was recon-

copyusing the Fourier transform of the signal and then the signal was recon-

(note that the filtered wavenum-

copy (note that the filtered wavenum-

s in the power spectral density of

copys in the power spectral density of

bed elevation, discussed later in Section 4, where the transition in the slope

copybed elevation, discussed later in Section 4, where the transition in the slope

Then, after determining the local

copy

Then, after determining the local gnal, the differences between consecu-

copy

gnal, the differences between consecu-tive minima and maxima were computed. Finally, the bedform heights above

copy

tive minima and maxima were computed. Finally, the bedform heights above ) were extracted. Figure 5 shows the co

py) were extracted. Figure 5 shows the

f) of the extracted bedform heights copy

f) of the extracted bedform heights for the discharges of 1500 (Fig. 5a) and 2800 l/s (Fig. 5b), whereas Fig. 6 co

pyfor the discharges of 1500 (Fig. 5a) and 2800 l/s (Fig. 5b), whereas Fig. 6 shows the corresponding pdfs of bedform lengths for the discharge of 1500 co

pyshows the corresponding pdfs of bedform lengths for the discharge of 1500

statistics of the extracted bedform copy

statistics of the extracted bedform

GRAIN SORTING IN GRAVEL BEDFORMS

1615

Fig. 6. Probability density function of the normalized bedform lengths for the dis-charges of: (a) 1500 and (b) 2800 l/s. The dotted curve shows the fitted Gamma distribution with k as the shape parameter and θ as the scale parameter.

Table 2 Bedform statistics extracted from spatial and temporal bed elevations

Q [l/s]

Spatial bed elevation Temporal bed elevation

bfH< > [mm]

std( )bfH[mm]

CV(Hbf)

bfL< >

[m] std( )bfL

[m] CV(Lbf)

AR bfH< >

[mm] std( )bfH[mm]

CV (Hbf)

1500 32.6 12.2 0.37 3.92 2.43 0.62 120 33.8 9.8 0.29 2800 74.5 44.0 0.59 3.29 1.28 0.38 44 82.3 27.9 0.34

Explanations: bfH< > , std( )bfH – mean and standard deviation of bedform height obtained from the ensemble of bedform heights extracted from different transects of spatial bed elevations measured at the end of a run and from different probe loca-tions from temporal bed elevations; bfL< > , std( )bfL – mean and standard deviation of bedform length obtained from the ensemble of bedform lengths extracted from dif-ferent transects of spatial bed elevations measured at the end of a run; CV – coeffi-cient of variation of bedform height and bedform length, AR – aspect ratio ( bf bfL H= < > < > ).

discharge, whereas the CV for the bedform length decreases suggesting a narrower range of bedforms lengths at higher discharge. Similarly, the ratio of mean bedform length Lbf to mean bedform height Hbf (aspect ratio) decreases with increasing discharge (Table 2), i.e., with increasing bedform height bedform length decreases.

Previous literature has suggested that bedform heights and lengths can be represented with a Gamma distribution (e.g., van der Mark et al. 2008 and

Author Bedform statistics extracted from sp

Author Bedform statistics extracted from spatial and temporal bed elevations

Author atial and temporal bed elevations

Spatial bed elevation

Author Spatial bed elevation

Author CV

Author CV(

Author (H

Author Hbf

Author bfHbfH

Author HbfH )

Author )bf)bf

Author bf)bf

Author bf

Author bfL

Author LbfLbf

Author bfLbf<

Author < >

Author >

[m]

Author [m]

Author std( )

Author std( )bf

Author bfstd( )bfstd( )

Author std( )bfstd( )std( )Lstd( )

Author std( )Lstd( )[m]

Author [m]

CV

Author CV

Author

Author

Author

Author

Author 1500 32.6 12.2 0.37 3.92 2.43 0.62 120 33.8 9.8 0.29

Author 1500 32.6 12.2 0.37 3.92 2.43 0.62 120 33.8 9.8 0.29

Author

Author

Author

Author

Author 2800 74.5 44.0 0.59 3.29 1.28 0.38 44 82.3 27.9 0.34

Author 2800 74.5 44.0 0.59 3.29 1.28 0.38 44 82.3 27.9 0.34

Author

Author

Author

Author

Author

Author Explanations:

Author Explanations:

Author bf

Author bfH

Author H

Author < >

Author < >bf< >bf

Author bf< >bfH< >H

Author H< >HbfHbf< >bfHbf

Author bfHbf< >bfHbf ,

Author ,

Author std( )

Author std( )std( )Hstd( )

Author std( )Hstd( )

obtained from the ensemble of bedform heights extracted from different transects of

Author obtained from the ensemble of bedform heights extracted from different transects of spatial bed elevations measured at the end of a run and from different probe loca-

Author spatial bed elevations measured at the end of a run and from different probe loca-tions from temporal bed elevations; Author tions from temporal bed elevations; bedform length obtained from the ensemble of bedform lengths extracted from dif-Author

bedform length obtained from the ensemble of bedform lengths extracted from dif-ferent transects of spatial bed elevations measured at the end of a run; CV – coeffi-Author

ferent transects of spatial bed elevations measured at the end of a run; CV – coeffi-cient of variation of bedform height and bedform length, AR – aspect ratio Author

cient of variation of bedform height and bedform length, AR – aspect ratio Author

L HAuthor

L HL H= < > < >L HAuthor

L H= < > < >L H

copy

copy

copy

Fig. 6. Probability density function of the normalized bedform lengths for the dis-

copy

Fig. 6. Probability density function of the normalized bedform lengths for the dis-charges of: (a) 1500 and (b) 2800 l/s. The dotted curve shows the fitted Gamma

copy

charges of: (a) 1500 and (b) 2800 l/s. The dotted curve shows the fitted Gamma as the scale parameter. co

py as the scale parameter.

atial and temporal bed elevations copy

atial and temporal bed elevations copy

copy

A. SINGH et al.

1616

references therein). The pdf of a Gamma distributed random variable x can be defined as

11( , , ) ,( )

xk

kf x k x ek

θθθ

−−=Γ

(1)

where k is the shape parameter and θ is the scale parameter. Gamma distribu-tions were fitted here to the normalized bedform height and length and are shown in Figs. 5 and 6 for reference. These fitted distributions were tested for goodness of fit using the chi square test and the null hypothesis of data coming from Gamma distribution was accepted with a p-value ranging from 0.06 to 0.83. The estimated parameters of the fitted Gamma distribution (estimated using maximum likelihood estimation and shown in Figs. 5 and 6) suggest that both the shape parameter, k, and scale parameter, θ, change with increasing discharge (k decreases while θ increases) for the bed-form heights, whereas for the bedform lengths k increases while θ remains constant. Similar results for the bedform heights were obtained from the temporal series of bed elevations (Table 2), as already partially noted by Singh and Foufoula-Georgiou (2012) and Singh et al. (2012a).

4. MULTISCALE STATISTICS OF BED TOPOGRAPHY River bed topography and its evolution are found to exhibit variability across a range of scales. One common way to characterize this variability is via plotting the power spectral density (hereafter PSD). For a discrete signal F(x), the PSD can be defined as

2 *ˆ ˆ1 ( ) ( )( ) ( ) ,

2π2πi t F FS F x e ω ω ωω

∞−

−∞

= =∑ (2)

where ˆ ( )F ω is the discrete Fourier transform of F(x), *ˆ ( )F ω is its complex conjugate and ω is the wavenumber. Here we place special emphasis on identifying spectral scaling ranges, i.e., ranges of scales over which log-log linearity is observed in the power spectral density.

Figure 7 shows the PSD of the spatial bed elevation series for the dis-charges of 1500 (lower spectrum) and 2800 l/s (upper spectrum). It can be seen that the PSDs follow a power law-decay with a slope ~1.7 for the dis-charge of 1500 l/s and ~2.3 for the discharge of 2800 l/s, suggesting the presence of statistical scaling in the bed elevation series. Note that these slopes were estimated for the same range of scales (see Table 3). The largest length scale of bedform observed from these PSDs is of the order of 10 m for both discharges. Similar results of increasing slope of PSDs with increasing discharge are observed from the analysis of temporal bed elevation series. For example, the slope of the PSD for the discharge of 1500 l/s is ~1.9,

Author Singh and Foufoula-Georgiou (2012) and Singh

Author Singh and Foufoula-Georgiou (2012) and Singh

MULTISCALE STATISTICS OF BED TOPOGRAPHY

Author MULTISCALE STATISTICS OF BED TOPOGRAPHY River bed topography and its evolution

Author River bed topography and its evolution are found to exhibit variability across

Author are found to exhibit variability across a range of scales. One common way to characterize this variability is via

Author a range of scales. One common way to characterize this variability is via plotting the power spectral density (hereafter PSD). For a discrete signal

Author plotting the power spectral density (hereafter PSD). For a discrete signal

), the PSD can be defined as

Author ), the PSD can be defined as

Author

Author

Author 1 (

Author 1 (( ) ( )

Author ( ) ( )( ) ( )

Author ( ) ( )( ) ( )

Author ( ) ( )( ) ( )

Author ( ) ( )1 (( ) ( )1 (

Author 1 (( ) ( )1 (2

Author 2

( ) ( )2

( ) ( )

Author ( ) ( )

2( ) ( )S F

Author S F( ) ( )S F( ) ( )

Author ( ) ( )S F( ) ( )( ) ( )S F( ) ( )

Author ( ) ( )S F( ) ( )( ) ( )S F( ) ( )

Author ( ) ( )S F( ) ( )1 (( ) ( )1 (S F1 (( ) ( )1 (

Author 1 (( ) ( )1 (S F1 (( ) ( )1 (( ) ( )S F( ) ( )ω( ) ( )S F( ) ( )

Author ( ) ( )S F( ) ( )ω( ) ( )S F( ) ( )1 (∞1 (

Author 1 (∞1 (( ) ( )S F( ) ( )= =( ) ( )S F( ) ( )

Author ( ) ( )S F( ) ( )= =( ) ( )S F( ) ( )( ) ( )S F( ) ( )= =( ) ( )S F( ) ( )

Author ( ) ( )S F( ) ( )= =( ) ( )S F( ) ( )( ) ( )S F( ) ( )= =( ) ( )S F( ) ( )

Author ( ) ( )S F( ) ( )= =( ) ( )S F( ) ( )

where

Author where ˆ

Author ˆ ( )

Author ( )F

Author F ( )ω( )

Author ( )ω( ) is the discrete Fourier transform of

Author is the discrete Fourier transform of

conjugate and Author conjugate and Author

ωAuthor ω is the wavenumber. Here we place special emphasis on Author

is the wavenumber. Here we place special emphasis on identifying spectral scaling ranges, Author

identifying spectral scaling ranges, linearity is observed in the power spectral density. Author

linearity is observed in the power spectral density. Author

Figure 7 shows the PSD of the spatial bed elevation series for the dis-Author

Figure 7 shows the PSD of the spatial bed elevation series for the dis-charges of 1500 (lower spectrum) and 2800 l/s (upper spectrum). It can be Author

charges of 1500 (lower spectrum) and 2800 l/s (upper spectrum). It can be

copyThese fitted distributions were tested

copyThese fitted distributions were tested

test and the null hypothesis of data

copytest and the null hypothesis of data

-value ranging from

copy-value ranging from

of the fitted Gamma distribution

copy of the fitted Gamma distribution

estimation and shown in Figs. 5

copyestimation and shown in Figs. 5

, and scale parameter,

copy

, and scale parameter, decreases while

copy

decreases while θ

copy

θ increases) for the bed-

copy

increases) for the bed-θ increases) for the bed-θ

copy

θ increases) for the bed-θform heights, whereas for the bedform lengths

copy

form heights, whereas for the bedform lengths k

copy

k increases while

copy

increases while k increases while k

copy

k increases while kdform heights were obtained from the co

pydform heights were obtained from the

temporal series of bed elevations (Table 2), as already partially noted by copy

temporal series of bed elevations (Table 2), as already partially noted by Singh and Foufoula-Georgiou (2012) and Singhco

pySingh and Foufoula-Georgiou (2012) and Singh et al.co

py et al. (2012a). co

py (2012a).

MULTISCALE STATISTICS OF BED TOPOGRAPHY co

pyMULTISCALE STATISTICS OF BED TOPOGRAPHY

GRAIN SORTING IN GRAVEL BEDFORMS

1617

Fig. 7. Power spectral density of spatial bed elevation for the discharge of 1500 (lower spectrum: broken line) and 2800 l/s (upper spectrum: solid line). Note that the spectrum at higher discharge (2800 l/s) is displaced vertically by one order of mag-nitude.

Table 3 Multiscale statistics of bed elevations

Q [l/s]

Spatial bed elevation Temporal bed elevation

Spec-tral

slope

Spectral scaling range

Multi-fractal

parameters

Multi-fractal scaling range

Spec-tral

slope

Spectral scaling range

Multi-fractal

parameters

Multi-fractal scaling range c1 c2 c1 c2

1500 1.68 8 cm – 5 m 0.43 0.05 2 cm –

1.5 m 1.87 15 s – 55 min 0.48 0.09 0.5-8

min

2800 2.28 8 cm – 5 m 0.69 0.10 2 cm –

1.4 m 2.18 20 s – 25 min 0.55 0.13 0.5-7

min

whereas for 2800 l/s it is about 2.2. The increase of spectral slope (temporal PSD) with increasing discharge, along with the reduction in scaling regime (Table 3), suggests that the bedforms of comparable energy (height) move faster at higher discharge, as expected.

It is important to note that the PSD characterizes how the second order moment (variance) in the signal changes with scale/frequency and, as such, it

Author Fig. 7. Power spectral density of spatial bed elevation for the discharge of 1500

Author Fig. 7. Power spectral density of spatial bed elevation for the discharge of 1500en line) and 2800 l/s

Author en line) and 2800 l/s (upper spectrum: solid line). Note that the

Author (upper spectrum: solid line). Note that the spectrum at higher discharge (2800 l/s) is di

Author spectrum at higher discharge (2800 l/s) is displaced vertically by one order of mag-

Author splaced vertically by one order of mag-

Multiscale statistics of bed elevations

Author Multiscale statistics of bed elevations

[l/s]

Author [l/s]

Spatial bed elevation

Author Spatial bed elevation

Author Spec-

Author Spec-tral

Author tral

slope Author

slope

Spectral

Author Spectral scaling

Author scaling range Author

range

Multi-

Author Multi-

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

1500 1.68 Author

1500 1.68 Author

Author

Author

Author

Author

Author

Author copy

Fig. 7. Power spectral density of spatial bed elevation for the discharge of 1500copy

Fig. 7. Power spectral density of spatial bed elevation for the discharge of 1500copy

A. SINGH et al.

1618

Fig. 8. Quantile-quantile plots of bed elevation increments for the discharges of 1500 and 2800 l/s. The dash lines in the qq-plots represent the Gaussian pdfs. Note that tails of the pdf at higher discharge are thicker than tails at lower discharge.

fully characterizes only a Gaussian pdf over scales (Singh et al. 2011). Fig-ure 8 shows the quantile-quantile plots (qq-plot) of Δh(x), the bed elevation increments, (Δh(x) = h(x + Δx) – h(x), where Δx is the resolution of the measurement, i.e., 10 mm) for the investigated discharges. Negative incre-ments of the bed elevation series, Δh(x) < 0, correspond to depositional events (i.e., an increase of elevation at the point of measurement during an interval Δx) whereas positive values, Δh(x) > 0, to erosional events. The qq-plots of bed elevation increments reported in Fig. 8 show significant deviation from the Gaussian distribution (dashed lines).

As a consequence, it is important to test for scaling in higher order statis-tical moments. For this, a higher-order structure function analysis which quantifies the manner in which higher order statistical moments of the local fluctuations in the bed elevation series change with scale was performed. In particular, a statistical analysis was performed on the differences (or incre-ments) of the bed elevation time series h(x) at different scales a, denoted by Δh(x, a), and defined as ( , ) ( ) ( ) ,h x a h x a h xΔ = + − (3)

where x is the length and a is the scale. The qth order statistical moment estimates of the absolute values of the increments at scale a, M(q, a), are defined as

Author Fig. 8. Quantile-quantile plots of bed elevation increments for the discharges of

Author Fig. 8. Quantile-quantile plots of bed elevation increments for the discharges of and 2800 l/s. The dash lines in the qq-plots represent the Gaussian pdfs. Note

Author and 2800 l/s. The dash lines in the qq-plots represent the Gaussian pdfs. Note that tails of the pdf at higher discharge are thicker than tails at lower discharge.

Author that tails of the pdf at higher discharge are thicker than tails at lower discharge.

fully characterizes only a Gaussian pdf over scales (Singh

Author fully characterizes only a Gaussian pdf over scales (Singhure 8 shows the quantile-quantile plots (qq-plot) of

Author ure 8 shows the quantile-quantile plots (qq-plot) of

(

Author (x

Author x(x(

Author (x( ) =

Author ) = h

Author h(

Author (x

Author x(x(

Author (x( +

Author + Δ

Author Δx

Author xΔxΔ

Author ΔxΔ ) –

Author ) – h

Author h

measurement,

Author measurement, i.e

Author i.e., 10 mm) for the investigated discharges. Negative incre-

Author ., 10 mm) for the investigated discharges. Negative incre-

ments of the bed elevation series,

Author ments of the bed elevation series, events (

Author events (i.e

Author i.e., an increase of elevation at the point of measurement during an

Author ., an increase of elevation at the point of measurement during an

interval

Author interval Δ

Author Δx

Author xΔxΔ

Author ΔxΔ ) whereas positive values,

Author ) whereas positive values,

qq-plots of bed elevation increments reported in Fig. 8 show significant Author qq-plots of bed elevation increments reported in Fig. 8 show significant deviation from the Gaussian distribution (dashed lines). Author

deviation from the Gaussian distribution (dashed lines). As a consequence, it is important to tAuthor

As a consequence, it is important to tAuthor

tical moments. For this, a higher-order structure function analysis which Author

tical moments. For this, a higher-order structure function analysis which quantifies the manner in which higher order statistical moments of the local Author

quantifies the manner in which higher order statistical moments of the local Author copy

Fig. 8. Quantile-quantile plots of bed elevation increments for the discharges of copy

Fig. 8. Quantile-quantile plots of bed elevation increments for the discharges of and 2800 l/s. The dash lines in the qq-plots represent the Gaussian pdfs. Note co

pyand 2800 l/s. The dash lines in the qq-plots represent the Gaussian pdfs. Note

that tails of the pdf at higher discharge are thicker than tails at lower discharge. co

pythat tails of the pdf at higher discharge are thicker than tails at lower discharge.

copy

GRAIN SORTING IN GRAVEL BEDFORMS

1619

1

1( , ) ( , ) ,N

qM q a h x aN

= ∑ (4)

where N is the number of data points of the series (increments) at scale a. As an extension to second order (spectral) scaling, higher order statistical scal-ing, or scale-invariance, requires M(q, a) to be a power law function of the scale a, that is ( )( , ) ~ ,qM q a aτ (5)

where τ(q) is called the scaling exponent function. The most basic form of scaling, known as simple scaling or mono-scaling, occurs when the scaling exponents are a linear function of the moment order, i.e., when τ(q) = Hq. In this case, the single parameter H, known as the Hurst exponent, describes how the whole pdf changes over scales. If τ(q) is a nonlinear function of q, more than one parameter is required to describe the behavior of the pdf changes over scale and the investigated series is called multi-fractal (Cas-taing et al. 1990, Venugopal et al. 2006, Singh et al. 2009). The simplest, but not the unique, way to parameterize the nonlinear dependence of τ(q) on q is via a quadratic approximation defined as 2

1 2( ) 2q c q c qτ = − , where c1 and c2 are the coefficient of roughness and the coefficient of intermittency, respectively. The parameter c1 is a measure of the average “roughness” of the series whereas the parameter c2 gives a measure of the inhomogeneous arrangement of the local fluctuations in the series. The reader is referred to Singh et al. (2011) for more details about the structure function analysis.

Figure 9 shows the τ(q) curves computed from the slopes of the log-log plots of the moments M(q, a) (not shown here for brevity) within the scaling range for the bed elevations at the discharges of 1500 and 2800 l/s, respec-tively (see Table 3). It can be seen from Fig. 9 that the τ(q) has a nonlinear dependence on q, which is an indication of the presence of multi-fractality. A summary of the computed multifractal parameters c1 and c2 along with the scaling ranges for both the discharges of 1500 and 2800 l/s can be seen in Table 3, along with the multiscale statistics of temporal bed elevation. It is interesting to note that both the roughness coefficient c1 and the inter-mittency coefficient c2 increase with increasing discharge.

The increase of c2 with increasing discharge suggests a faster rate of change of the pdf’s shape across a range of scales (statistical interpretation) and a more inhomogeneous arrangement of abrupt bed elevation fluctuations over space (geometrical interpretation), whereas the increase of roughness parameter c1 with increasing discharge suggests that bed elevation fluctua-tions are smoother overall at higher discharge than at lower discharge. Note that the “smoothness” in the signal at higher discharge is associated with the higher Hurst exponent, whereas the abruptness of local, very infrequent fluc-

Author but not the unique, way to parameterize the nonlinear dependence of

Author but not the unique, way to parameterize the nonlinear dependence of

is via a quadratic approximation defined as

Author is via a quadratic approximation defined as are the coefficient of roughness and the coefficient of intermittency,

Author are the coefficient of roughness and the coefficient of intermittency, respectively. The parameter

Author respectively. The parameter c

Author c1

Author 1 is a measure of the average “roughness” of

Author is a measure of the average “roughness” of the series whereas the parameter

Author the series whereas the parameter c

Author c2

Author 2 gives a measure of the inhomogeneous

Author gives a measure of the inhomogeneous

Author arrangement of the local fluctuations in the series. The reader is referred to

Author arrangement of the local fluctuations in the series. The reader is referred to

(2011) for more details about the structure function analysis.

Author (2011) for more details about the structure function analysis.

Figure 9 shows the

Author Figure 9 shows the τ

Author τ(

Author (q

Author q) curves computed from the slopes of the log-log

Author ) curves computed from the slopes of the log-log

plots of the moments

Author plots of the moments M

Author M(

Author (M(M

Author M(M q

Author q,

Author , a

Author a) (not shown here for brevity) within the scaling

Author ) (not shown here for brevity) within the scaling

Author range for the bed elevations at the discharges of 1500 and 2800 l/s, respec-

Author range for the bed elevations at the discharges of 1500 and 2800 l/s, respec-tively (see Table 3). It can be seen from Fig. 9 that the

Author tively (see Table 3). It can be seen from Fig. 9 that the dependence on

Author dependence on q

Author q, which is an indication of the presence of multi-fractality.

Author , which is an indication of the presence of multi-fractality.

A summary of the computed multifractal parameters Author A summary of the computed multifractal parameters scaling ranges for both the discharges of 1500 and 2800 l/s can be seen Author

scaling ranges for both the discharges of 1500 and 2800 l/s can be seen in Table 3, along with the multiscale statistics of temporal bed elevation. Author

in Table 3, along with the multiscale statistics of temporal bed elevation. Author

It is interesting to note that both the roughness coefficient Author

It is interesting to note that both the roughness coefficient mittency coefficient Author

mittency coefficient

copy (5)

copy (5)

) is called the scaling exponent function. The most basic form of

copy) is called the scaling exponent function. The most basic form of

scaling, known as simple scaling or mono-scaling, occurs when the scaling

copyscaling, known as simple scaling or mono-scaling, occurs when the scaling

., when

copy., when τ

copyτ(

copy(q

copyq) =

copy) = Hq

copyHq. In

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pychanges over scale and the investigated series is called multi-fractal (Cas-

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et al. 2009). The simplest, copy

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pybut not the unique, way to parameterize the nonlinear dependence of

( ) 2copy

( ) 2q c q c qcopy

q c q c q( ) 2q c q c q( ) 2copy

( ) 2q c q c q( ) 2τcopy

τ ( ) 2q c q c q( ) 2= −( ) 2q c q c q( ) 2copy

( ) 2q c q c q( ) 2= −( ) 2q c q c q( ) 2 are the coefficient of roughness and the coefficient of intermittency,

copy

are the coefficient of roughness and the coefficient of intermittency,

A. SINGH et al.

1620

Fig. 9. Scaling exponents τ(q) estimated from the log-log linear regressions within the scaling regions of statistical moments of spatial bed elevation for discharges of 1500 l/s (bottom curve) and 2800 l/s (top curve). Notice the deviation of τ(q) from the linear line establishing the presence of multifractality.

tuations is not captured by H. This property is captured with the intermit-tency parameter c2 which is higher for higher discharge, indicating that sharp elevation increments due to the passing of steep bedforms or sub-bedforms facies are not homogeneously arranged in the signal (partly due to the fact that bedforms of a wide range of sizes are present at high discharge). Also note that in case of a mono-fractal (c2 = 0), the shape of pdf of the incre-ments does not change with scales and that the slope of the second moment of the structure functions is related to the slope of the PSD via the relation (β = 2H + 1), where β is the slope of power spectrum, and H is the Hurst exponent. Similar trends of increasing c1 and c2 with increasing discharge are observed from the multiscale analysis of temporal bed elevation (see Table 3).

5. GRAIN SORTING IN BEDFORMS At the end of each run, bedforms were visually identified on the channel bed surface and their crests and troughs were located (Figs. 1 and 3). Patches with dimension of 30 × 30 cm were marked on both crest and trough (Fig. 3) and their surface (top layer corresponding to one grain size) and sub-surface

Author ) estimated from the log-log linear regressions within

Author ) estimated from the log-log linear regressions within the scaling regions of statistical moments of spatial bed elevation for discharges of

Author the scaling regions of statistical moments of spatial bed elevation for discharges of (bottom curve) and 2800 l/s

Author (bottom curve) and 2800 l/s (top curve). Notice the deviation of

Author (top curve). Notice the deviation of the linear line establishing the presence of multifractality.

Author the linear line establishing the presence of multifractality.

tuations is not captured by

Author tuations is not captured by H

Author H. This property is captured with the intermit-

Author . This property is captured with the intermit-H. This property is captured with the intermit-H

Author H. This property is captured with the intermit-H

tency parameter

Author tency parameter c

Author c2

Author 2 which is higher for higher discharge, indicating that sharp

Author which is higher for higher discharge, indicating that sharp

elevation increments due to the passing of steep bedforms or sub-bedforms

Author elevation increments due to the passing of steep bedforms or sub-bedforms facies are not homogeneously arranged in

Author facies are not homogeneously arranged inthat bedforms of a wide range of sizes are present at high discharge). Also

Author that bedforms of a wide range of sizes are present at high discharge). Also note that in case of a mono-fractal (

Author note that in case of a mono-fractal (

Author

ments does not change with scales and that the slope of the second moment Author

ments does not change with scales and that the slope of the second moment of the structure functions is related toAuthor

of the structure functions is related to = Author

= 2 Author

2HAuthor

H + 1), where Author

+ 1), where H + 1), where HAuthor

H + 1), where Hexponent. Similar trends of increasing Author

exponent. Similar trends of increasing Author

Author copy

) estimated from the log-log linear regressions within copy

) estimated from the log-log linear regressions within the scaling regions of statistical moments of spatial bed elevation for discharges of

copy

the scaling regions of statistical moments of spatial bed elevation for discharges of co

pyco

py

GRAIN SORTING IN GRAVEL BEDFORMS

1621

Fig. 10. Typical sketch of investigated bedforms showing the location of patches of surface and subsurface samples, bedform height Hbf and bedform length Lbf. The subscript i represents patch number, whereas the subscript s and ss represent surface and subsurface, respectively. C and T denote bedform crest and trough, respectively.

Fig. 11. Grain size distribution obtained from the material sampled at the crest and trough of bedforms for the surface (top panel) and the subsurface (bottom panel) for the discharges of 1500 (left panel) and 2800 l/s (right panel). Ci and Ti (i = 1, 2) curves refers to crest and trough samples, respectively.

(10 cm deep material) GSDs were obtained. Figure 10 shows a typical sketch of the observed bedforms and of the sampling locations, whereas Fig. 11 shows the GSDs of both surface (upper panel) and subsurface (bottom

Author

Author

Author copy

copy

copyFig. 10. Typical sketch of investigated bedforms showing the location of patches of

copyFig. 10. Typical sketch of investigated bedforms showing the location of patches of

and bedform length

copyand bedform length L

copyLbf

copybf.

copy. bf. bf

copybf. bf The

copyThe

ss

copyss represent surface

copy represent surface

e bedform crest and trough, respectively.

copye bedform crest and trough, respectively.

copy

A. SINGH et al.

1622

Table 4 Statistics of grain size distribution

Q [l/s] Patch 

Surface Subsurface

d16 [mm]

d50 [mm]

d84 [mm]

Sand [%]

d16 [mm]

d50 [mm]

d84 [mm]

Sand [%]

1500

C1 4.38 8.86 16.54 0.99 4.19 9.25 15.79 5.21 C2 4.61 8.44 14.94 0.14 2.05 7.97 15.48 2.9 T1 6.39 15.19 22.69 1.37 1.19 8.15 19.35 14.32 T2 7.35 14.34 24.67 0.43 1.30 4.61 18.50 13.29

2800

C1 5.35 14.18 23.70 0.99 1.21 5.52 15.49 13.25 C2 4.41 7.35 14.37 0.11 4.49 7.23 12.85 0.01 T1 12.12 20.20 28.62 1.13 0.51 2.89 12.23 31.63 T2 7.67 19.56 29.95 1.50 0.87 4.41 9.22 18.44

panel) samples for the discharges of 1500 (left panel) and 2800 l/s (right panel), while a synthesis of their statistics is reported in Table 4. Below, we characterize the statistics of surface and subsurface GSD as a function of crest and trough for both low and high flow conditions.

5.1 Surface GSD Figure 11 (top panel) shows the GSDs of the surface patches for the crests and the troughs for the discharges of 1500 (Fig. 11a) and 2800 l/s (Fig. 11b), whereas Table 4 shows the statistics of these patches as a function of dis-charge. From Fig. 11a it can be seen that for the low discharge the GSDs of the two crests coincide with each other as also do the GSDs of the two troughs. However, for a high discharge the GSDs change their shape consid-erably (Fig. 11b). The median diameters d50, obtained from different bed-forms, for the crests as well as for the troughs are similar for different crests and troughs and increase with increasing discharge (Table 4). Similar trends are observed for d16 and d84.

To further compare the GSDs of crests and troughs as a function of dis-charge, we show in Fig. 12 the average of the crests GSDs (C = (C1 + C2)/2) versus the average of the troughs GSDs (T = (T1 + T2)/2) for the discharges of 1500 (Fig. 12a) and 2800 l/s (Fig. 12b). Clearly, sorting effects associated with bedforms are enhanced as the bedform amplitude grows and, conse-quently, the grain size distribution gets wider. For example, the differences between the surface composition at troughs and crests, summarized by the variations in d16, d50, and d84 percentiles, are equal to 2.34, 6.11, and 7.93 mm for the discharge of 1500 l/s, whereas they are 5.01, 9.11, and 10.25 mm for 2800 l/s discharge (see Table 5).

Author panel), while a synthesis of their statis

Author panel), while a synthesis of their statistics is reported in Table 4. Below, we

Author tics is reported in Table 4. Below, we

characterize the statistics of surface and subsurface GSD as a function of

Author characterize the statistics of surface and subsurface GSD as a function of crest and trough for both low and high flow conditions.

Author crest and trough for both low and high flow conditions.

Figure 11 (top panel) shows the GSDs

Author Figure 11 (top panel) shows the GSDs and the troughs for the discharges of 1500 (Fig. 11a) and 2800 l/s (Fig. 11b),

Author and the troughs for the discharges of 1500 (Fig. 11a) and 2800 l/s (Fig. 11b), whereas Table 4 shows the statistics of these patches as a function of dis-

Author whereas Table 4 shows the statistics of these patches as a function of dis-

Author charge. From Fig. 11a it can be seen that for the low discharge the GSDs of

Author charge. From Fig. 11a it can be seen that for the low discharge the GSDs of the two crests coincide with each ot

Author the two crests coincide with each ottroughs. However, for a high discharge th

Author troughs. However, for a high discharge therably (Fig. 11b). The median diameters

Author erably (Fig. 11b). The median diameters forms, for the crests as well as for thAuthor forms, for the crests as well as for thand troughs and increase with increasing diAuthor

and troughs and increase with increasing diare observed for Author

are observed for To further compare the GSDs of crests and troughs as a function of dis-Author

To further compare the GSDs of crests and troughs as a function of dis-charge, we show in Fig. 12 the average of the crests GSDs (C = (C1 + C2)/2) Author

charge, we show in Fig. 12 the average of the crests GSDs (C = (C1 + C2)/2)

copyC2 4.61 8.44 14.94 0.14 2.05 7.97 15.48 2.9

copyC2 4.61 8.44 14.94 0.14 2.05 7.97 15.48 2.9

copyT1 6.39 15.19 22.69 1.37 1.19 8.15 19.35 14.32

copyT1 6.39 15.19 22.69 1.37 1.19 8.15 19.35 14.32

copy

copyT2 7.35 14.34 24.67 0.43 1.30 4.61 18.50 13.29

copyT2 7.35 14.34 24.67 0.43 1.30 4.61 18.50 13.29

copy

copy

copyC1 5.35 14.18 23.70 0.99 1.21 5.52 15.49 13.25

copyC1 5.35 14.18 23.70 0.99 1.21 5.52 15.49 13.25

copy

copy

copy

copyC2 4.41 7.35 14.37 0.11 4.49 7.23 12.85 0.01

copyC2 4.41 7.35 14.37 0.11 4.49 7.23 12.85 0.01

copy

copy

copy

copy

T1 12.12 20.20 28.62 1.13 0.51 2.89 12.23 31.63

copy

T1 12.12 20.20 28.62 1.13 0.51 2.89 12.23 31.63

copy

copy

copy

copy

copy

T2 7.67 19.56 29.95 1.50 0.87 4.41 9.22 18.44

copy

T2 7.67 19.56 29.95 1.50 0.87 4.41 9.22 18.44

copy

copy

copy

copy

copy

copy

copy

copy

copy

copy

panel) samples for the discharges of 1500 (left panel) and 2800 l/s (right copy

panel) samples for the discharges of 1500 (left panel) and 2800 l/s (right tics is reported in Table 4. Below, we co

pytics is reported in Table 4. Below, we

characterize the statistics of surface and subsurface GSD as a function of copy

characterize the statistics of surface and subsurface GSD as a function of crest and trough for both low and high flow conditions. co

pycrest and trough for both low and high flow conditions.

GRAIN SORTING IN GRAVEL BEDFORMS

1623

Fig. 12. Comparison of the initial grain size distribution with averaged grain size distribution resulting from surface samples collected at bedform crest and trough for the discharges of (a) 1500 and (b) 2800 l/s.

Table 5 Averaged GSD over bedform crests and troughs

Q [l/s] Patch

Surface Subsurface

d16 [mm]

d50 [mm]

d84 [mm]

d16 [mm]

d50 [mm]

d84 [mm]

1500 C 4.5 8.65 15.74 3.32 7.69 15.54 T 6.87 14.76 23.68 1.24 6.37 18.92

2800 C 4.88 10.76 19.03 2.85 6.37 14.17 T 9.89 19.88 29.98 0.68 3.65 10.73

The averaged GSDs shown in Fig. 12 indicate a departure of the surface

grain size distribution with respect to the initial bed composition that, as a consequence of the preferential entrainment of finer particles, is particular-ly pronounced in bedform troughs. All the representative grain sizes d16, d50, and d84 (see Tables 4 and 5) are invariably larger than those characterizing the initial sediment distribution (approximately equal to 2, 8, and 21 mm, re-spectively) and increase as the discharge increases. An appreciable departure from the shape of the initial GSD is also observed on bedform crests, espe-cially for the lower discharge: the bed composition tends to get coarser ow-ing to a lack in finer fractions (both d16 and d50 are larger than those of the initial GSD) which compensate the concurrent reduction in coarser fractions (embodied by the decrease of d84 from 21 to 15.7 mm). For the higher dis-charge, this reduction is significantly attenuated (d84 = 19 mm) while the lack in finer fraction still persists. The recovering, with respect to initial sed-iment composition, of the upper GSD shape when increasing the discharge is

Author Averaged GSD over bedform crests and troughs

Author Averaged GSD over bedform crests and troughs

Surface Subsurface

Author Surface Subsurface

Author [mm]

Author [mm] d

Author d50

Author 50 d50 d

Author d50 d[mm]

Author [mm] d

Author d84

Author 84 d84 d

Author d84 d[mm]

Author [mm]

Author

Author

Author C 4.5 8.65 15.74 3.32 7.69 15.54

Author C 4.5 8.65 15.74 3.32 7.69 15.54

Author

Author

Author T 6.87 14.76 23.68 1.24 6.37 18.92

Author T 6.87 14.76 23.68 1.24 6.37 18.92

Author

Author

Author

Author C 4.88 10.76 19.03 2.85 6.37 14.17

Author C 4.88 10.76 19.03 2.85 6.37 14.17

Author

Author

Author T 9.89 19.88 29.98 0.68 3.65 10.73

Author T 9.89 19.88 29.98 0.68 3.65 10.73

Author

Author

Author

Author

Author

Author

Author

Author The averaged GSDs shown in Fig. 12

Author The averaged GSDs shown in Fig. 12

grain size distribution with respect to the initial bed composition that, as

Author grain size distribution with respect to the initial bed composition that, as a consequence of the preferential entrainment of finer particles, is particular-Author

a consequence of the preferential entrainment of finer particles, is particular-ly pronounced in bedform troughs. All the representative grain sizes Author

ly pronounced in bedform troughs. All the representative grain sizes and Author

and d Author

d Author

84Author

84d84d Author

d84d (see Tables 4 and 5) are invariably larger than those characterizing Author

(see Tables 4 and 5) are invariably larger than those characterizing the initial sediment distribution (approximately equal to 2, 8, and 21 mm, re-Author

the initial sediment distribution (approximately equal to 2, 8, and 21 mm, re-

copyFig. 12. Comparison of the initial grain size distribution with averaged grain size

copyFig. 12. Comparison of the initial grain size distribution with averaged grain size

rface samples collected at be

copy

rface samples collected at bedform crest and trough for

copy

dform crest and trough for

Averaged GSD over bedform crests and troughs copy

Averaged GSD over bedform crests and troughs

Surface Subsurface copy

Surface Subsurface copy

copy

dco

pydco

pyco

pyco

py

A. SINGH et al.

1624

likely associated with the establishment of a sediment transport condition much closer to equal mobility. On the other hand, the deficiency of finer fractions, that is observed independently of the discharge, is partly related to the winnowing of finer grains from the surface as well as the movement of finer grains into the pores of bed subsurface and partly to the recirculation feeding system, whereby the composition of transported sediment is signifi-cantly affected not only by dynamic armor, but also by the intense longitudi-nal and vertical sorting due to bedform dynamics (Lanzoni 2000).

5.2 Subsurface GSD The GSDs of the subsurface material sampled at crests and troughs are shown in the bottom panel of Fig. 11 for the discharges of 1500 (Fig. 11c) and 2800 l/s (Fig. 11d) along with their statistics in Table 4, whereas their average GSDs over the bedforms crests and troughs are shown in Fig. 13 for the discharges of 1500 (Fig. 13a) and 2800 l/s (Fig. 13b). The crucial role that bedform migration has on the reworking of the bed and, hence, on determining the composition of the active layer, is clearly attested by the lar-ger variability of GSD curves as the bedform amplitude increases (i.e., at the higher discharge). Although a significant fining of both crests and troughs material, with respect to the initial bed composition, generally characterizes the subsurface samples, the effects are more pronounced at the higher dis-charge (Fig. 13). Moreover, the material collected below bedform crests is coarser than that sampled below the troughs (Tables 4 and 5). This coarser material results from the inclusion in the bedform body, as a consequence of bedform migration, of the coarser grains previously entrapped in bedform troughs. The pattern of sorting is less clear at the lower discharge (i.e., for

Fig. 13. Comparison of the initial grain size distribution with averaged grain size distribution resulting from subsurface samples collected at bedform crest and trough for the discharges of (a) 1500 and (b) 2800 l/s.

Author determining the composition of the active la

Author determining the composition of the active lager variability of GSD curves as the bedform amplitude increases (

Author ger variability of GSD curves as the bedform amplitude increases (higher discharge). Although a significant fining of both crests and troughs

Author higher discharge). Although a significant fining of both crests and troughs material, with respect to the initial be

Author material, with respect to the initial bed composition, generally characterizes

Author d composition, generally characterizes the effects are more pronounced at the higher dis-

Author the effects are more pronounced at the higher dis-charge (Fig. 13). Moreover, the material collected below bedform crests is

Author charge (Fig. 13). Moreover, the material collected below bedform crests is coarser than that sampled below the tr

Author coarser than that sampled below the trmaterial results from the inclusion in the bedform body, as a consequence of

Author material results from the inclusion in the bedform body, as a consequence of

Author bedform migration, of the coarser grai

Author bedform migration, of the coarser graitroughs. The pattern of sorting is less clear at the lower discharge (

Author troughs. The pattern of sorting is less clear at the lower discharge (

Author

Author copyThe GSDs of the subsurface material sampled at crests and troughs are

copyThe GSDs of the subsurface material sampled at crests and troughs are

shown in the bottom panel of Fig. 11 for the discharges of 1500 (Fig. 11c)

copyshown in the bottom panel of Fig. 11 for the discharges of 1500 (Fig. 11c)

r statistics in Table 4, whereas their

copy

r statistics in Table 4, whereas their nd troughs are shown in Fig. 13 for

copy

nd troughs are shown in Fig. 13 for the discharges of 1500 (Fig. 13a) and 2800 l/s (Fig. 13b). The crucial role co

pythe discharges of 1500 (Fig. 13a) and 2800 l/s (Fig. 13b). The crucial role that bedform migration has on the reworking of the bed and, hence, on co

pythat bedform migration has on the reworking of the bed and, hence, on determining the composition of the active la co

pydetermining the composition of the active layer, is clearly attested by the lar-co

pyyer, is clearly attested by the lar-

ger variability of GSD curves as the bedform amplitude increases (copy

ger variability of GSD curves as the bedform amplitude increases (higher discharge). Although a significant fining of both crests and troughs co

pyhigher discharge). Although a significant fining of both crests and troughs

GRAIN SORTING IN GRAVEL BEDFORMS

1625

smaller bedforms). In this case, only the values attained by the d84 are invariably smaller than those characterizing the initial GSD, while the trends exhibited by the d16 and d50 are not univocal.

5.3 Surface, subsurface GSD: comparison of quantiles The upper 10 percentiles of the observed GSDs of the crests for the surface samples, in general, for both the discharges, show finer trend than the initial GSD, whereas troughs show coarser (Fig. 12). For the subsurface, both crests and troughs are finer than the initial GSD for both the discharges (see Fig. 13). A better visual comparison of quantiles of GSDs of crests and troughs as a function of discharge is shown in Fig. 14 for the surface (Fig. 14a) and the subsurface (Fig. 14b) material. Comparing the two crests,

Fig. 14. Comparison of percentiles of GSDs obtained from (a) surface and (b) sub-surface sampling at the crests and troughs of the bedform for discharges of 1500 and 2800 l/s.

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author

Author copy

arges, show finer trend than the initial

copy

arges, show finer trend than the initial ig. 12). For the subsurface, both

copyig. 12). For the subsurface, both

crests and troughs are finer than the initial GSD for both the discharges (see

copycrests and troughs are finer than the initial GSD for both the discharges (see

of quantiles of GSDs of crests and

copyof quantiles of GSDs of crests and

troughs as a function of discharge is shown in Fig. 14 for the surface

copytroughs as a function of discharge is shown in Fig. 14 for the surface

material. Comparing the two crests,

copymaterial. Comparing the two crests,

copy

copy

copy

copy

copy

copy

copy

copy

copy

copy

copy

copy

copy

copy

copy

copy

A. SINGH et al.

1626

C1 and C2, within the same discharge for the surface material, it can be seen from Fig. 14a that the crests are more variable at higher discharge. For example, the range of d16 to d84 is 5.35-23.74 mm for C1, while for C2 it is 4.41-14.73 mm. For the low discharge these ranges are 4.38-16.54 mm and 4.61-14.91 mm for C1 and C2, respectively. The higher variability in crest’s grain sorting at higher discharge is due to the fact that at higher discharge bedform heights are more variable (see Table 2). In fact, it is noticed that the bedform heights, extracted from spatial bed elevation series, corresponding to the C1 and C2 for the high discharge are 10.21 and 18.81 cm, while for low discharge they are 4.91 and 5.36 cm, suggesting a higher elevation dif-ference between sampled patches at higher discharge. This difference in C1 and C2 at higher discharge can also be noticed from the difference of mean elevation between C1 and T1, and C2 and T2, which are 3.59 and 9.46 cm, respectively. In contrast to the crests, the troughs do not show significant dif-ferences in the sampled patches for both the discharges of 1500 and 2800 l/s (Fig. 14a).

5.4 Integrated GSD Figure 15 shows the grain size distribution obtained after the integration of GSDs of crests and troughs extracted from surface and subsurface layers for the discharges of 1500 and 2800 l/s. It appears that the integrated GSD re-covers most of the grains present in the initial GSD. For example, the d50

Fig. 15. GSD obtained after the integration of GSDs of crests and troughs belonging to surface and subsurface for the discharges of 1500 and 2800 l/s. Dashed line repre-sents the initial GSD of the be material.

Author Figure 15 shows the grain size distribution obtained after the integration of

Author Figure 15 shows the grain size distribution obtained after the integration of GSDs of crests and troughs extracted from surface and subsurface layers for

Author GSDs of crests and troughs extracted from surface and subsurface layers for the discharges of 1500 and 2800 l/s. It a

Author the discharges of 1500 and 2800 l/s. It appears that the integrated GSD re-

Author ppears that the integrated GSD re-the discharges of 1500 and 2800 l/s. It appears that the integrated GSD re-the discharges of 1500 and 2800 l/s. It a

Author the discharges of 1500 and 2800 l/s. It appears that the integrated GSD re-the discharges of 1500 and 2800 l/s. It a

Author covers most of the grains present in the initial GSD. For example, the

Author covers most of the grains present in the initial GSD. For example, the

Author copybedform heights, extracted from spatial bed elevation series, corresponding

copybedform heights, extracted from spatial bed elevation series, corresponding

to the C1 and C2 for the high discharge are 10.21 and 18.81 cm, while for

copyto the C1 and C2 for the high discharge are 10.21 and 18.81 cm, while for

low discharge they are 4.91 and 5.36 cm, suggesting a higher elevation dif-

copylow discharge they are 4.91 and 5.36 cm, suggesting a higher elevation dif-

gher discharge. This difference in C1

copygher discharge. This difference in C1

be noticed from the difference of mean

copybe noticed from the difference of mean

and T2, which are 3.59 and 9.46 cm,

copy

and T2, which are 3.59 and 9.46 cm, the troughs do not show significant dif-

copy

the troughs do not show significant dif-ferences in the sampled patches for both the discharges of 1500 and 2800 l/s co

pyferences in the sampled patches for both the discharges of 1500 and 2800 l/s co

pyFigure 15 shows the grain size distribution obtained after the integration of co

pyFigure 15 shows the grain size distribution obtained after the integration of

GRAIN SORTING IN GRAVEL BEDFORMS

1627

for the initial GSD and for the integrated GSDs corresponding to the two investigated discharges are 7.7, 9.8, and 8.58 mm, respectively. The small discrepancies emerging from Fig. 15 can be attributed partly to the slope of the bedform stoss side and partly to the distance between the sampled patches. A smaller slope and a closer distance between the patches, associ-ated with smaller bedforms occurring at lower discharge, will produce less distinct differences between the initial GSD and that measured at the end of each experiment.

6. BED ELEVATION STATISTICS OF SAMPLED AND VIRTUAL PATCHES

The bed elevations corresponding to the sediment patches sampled at bed-form crests and troughs (Fig. 3) for both low and high discharges were extracted and their statistics were computed. Figure 16 shows the pdfs of the crests and troughs elevation for the discharge of 1500 (Fig. 16a) and 2800 l/s (Fig. 16b), respectively. It can be seen from these figures that while the pdfs of the bed elevation of both crests and troughs are qualitatively similar (i.e., partially overlapping) for the lower discharge, the pdfs observed at the higher discharge are quite different, especially for the crests as also seen in the case of grain size distribution curves (see Figs. 11a-b) (note that here similarity in pdfs is referred to the similarity in the statistics, for, e.g., mean and standard deviation, of the pdfs). The corresponding statistics (mean and standard deviation) reported in Table 6, suggest that bed roughness (meas-ured by the standard deviation of linearly detrended bed elevation) is higher for the troughs than the crests for both discharges. For example, the average standard deviation of the crest and the trough elevations is 3.36 and 4.3 mm, respectively, for the discharge of 1500 l/s, whereas it is 3.34 and 5.97 mm, respectively, for the discharge of 2800 l/s. Moreover, for the higher dis-

Fig. 16. Probability density functions of bed elevations corresponding to the sampled patches for GSD for the discharges of (a) 1500 and (b) 2800 l/s.

Author of the bed elevation of both crests and troughs are qualitatively similar (

Author of the bed elevation of both crests and troughs are qualitatively similar (partially overlapping) for the lower discharge, the pdfs observed at the

Author partially overlapping) for the lower discharge, the pdfs observed at the higher discharge are quite different, especially for the crests as also seen in

Author higher discharge are quite different, especially for the crests as also seen in the case of grain size distribution curves (see Figs. 11a-b) (note that here

Author the case of grain size distribution curves (see Figs. 11a-b) (note that here similarity in pdfs is referred to th

Author similarity in pdfs is referred to the similarity in the statistics, for,

Author e similarity in the statistics, for, and standard deviation, of the pdfs)

Author and standard deviation, of the pdfs). The corresponding statistics (mean and

Author . The corresponding statistics (mean and standard deviation) reported in Tabl

Author standard deviation) reported in Table 6, suggest that bed roughness (meas-

Author e 6, suggest that bed roughness (meas-

ured by the standard deviation of linear

Author ured by the standard deviation of linear

Author for the troughs than the crests for both

Author for the troughs than the crests for both standard deviation of the crest and the trough elevations is 3.36 and 4.3 mm,

Author standard deviation of the crest and the trough elevations is 3.36 and 4.3 mm, respectively, for the discharge of 1500 l/s, whereas it is 3.34 and 5.97 mm,

Author respectively, for the discharge of 1500 l/s, whereas it is 3.34 and 5.97 mm, respectively, for the discharge of 2800

Author respectively, for the discharge of 2800

Author

Author

Author copye sediment patches sampled at bed-

copye sediment patches sampled at bed-

form crests and troughs (Fig. 3) for both low and high discharges were

copy

form crests and troughs (Fig. 3) for both low and high discharges were extracted and their statistics were computed. Figure 16 shows the pdfs of the

copy

extracted and their statistics were computed. Figure 16 shows the pdfs of the crests and troughs elevation for the discharge of 1500 (Fig. 16a) and 2800 l/s co

pycrests and troughs elevation for the discharge of 1500 (Fig. 16a) and 2800 l/s

from these figures that while the pdfs copy

from these figures that while the pdfs of the bed elevation of both crests and troughs are qualitatively similar (co

pyof the bed elevation of both crests and troughs are qualitatively similar (partially overlapping) for the lower discharge, the pdfs observed at the co

pypartially overlapping) for the lower discharge, the pdfs observed at the higher discharge are quite different, especially for the crests as also seen in co

pyhigher discharge are quite different, especially for the crests as also seen in the case of grain size distribution curves (see Figs. 11a-b) (note that here

copy

the case of grain size distribution curves (see Figs. 11a-b) (note that here

A. SINGH et al.

1628

Table 6 Statistics of digital elevation model (DEM) of patches

Q [l/s]

Patch [mm] 

Observed patch h (x,y) Virtual patch h (x,y)

C1 C2 T1 T2 Crest Trough

1500 Mean 59.33 62.24 43.75 47.64 49.38 39.01 Std. dev. 3.36 3.36 4.07 4.52 3.44 3.80

2800 Mean 150.41 169.19 114.47 74.53 109.23 84.24 Std.dev. 4.09 2.58 5.08 6.87 3.49 4.22

charge the crest roughness decreases as the bedform height increases (see the differences in mean elevation for C2 and T2 in Table 6).

The higher discharge is also associated with larger variability in the bed topography, higher complexity and three-dimensionality, essentially related to the wider variety of bedforms that form at this discharge. While the visual selection of crest and trough was still possible, as assessed by the compara-tive analysis in grain size distribution, its statistical characterization turns out to be more difficult as the bedform topography becomes more complex.

In order to substantiate statistically the results obtained on the analysed patches and provide evidence of sorting effects at the full channel scale, we expand our analysis to virtual patches. As GSD was observed to depend on the specific location of the (sub)surface samples within the bedform (crest or trough), we identify on the digital elevation model (DEM) topography regions with the same size and characteristic location of the crest and trough studied above. The sediment distribution on these virtual patches is obvious-ly unknown but the signature of the coarser grains is expected to survive on the standard deviation of bed elevation computed on each linearly detrended virtual patch, using only DEM data. In other words, the bedform modulation of surface GSD resulting from grain size analysis performed on visually identified patches is expected to hold when comparing small scale roughness over virtual patches. To achieve this goal and further investigate the rough-ness of sediment patches over crests and troughs, the elevations correspond-ing to several virtual patches of dimensions 30 × 30 cm were extracted from the bed profile (Fig. 17). Figure 17 shows the stepwise procedure for extract-ing virtual patches from the bed elevation profiles for the discharge of 1500 (left column) and 2800 l/s (right column).

Panels (a) and (f) of Fig. 17 show the transects of the bed elevation obtained from the DEM of the bed topography passing through patch 1 for the discharges of 1500 (Fig. 17a) and 2800 l/s (Fig. 17f). Panels (b) and (g) of Fig. 17 show the location of local maxima and local minima superim-posed on the filtered (using Fourier transform) bed elevation series;

Author tive analysis in grain size distribution, its statistical characterization turns out

Author tive analysis in grain size distribution, its statistical characterization turns out to be more difficult as the bedform topography becomes more complex.

Author to be more difficult as the bedform topography becomes more complex. In order to substantiate statistically the results obtained on the analysed

Author In order to substantiate statistically the results obtained on the analysed patches and provide evidence of sorting effects at the full channel scale, we

Author patches and provide evidence of sorting effects at the full channel scale, we

Author expand our analysis to virtual patch

Author expand our analysis to virtual patches. As GSD was observed to depend on

Author es. As GSD was observed to depend on the specific location of the (sub)surface samples within the bedform (crest or

Author the specific location of the (sub)surface samples within the bedform (crest or trough), we identify on the digital elevation model (DEM) topography

Author trough), we identify on the digital elevation model (DEM) topography regions with the same size and characteristic location of the crest and trough

Author regions with the same size and characteristic location of the crest and trough studied above. The sediment distribution on these virtual patches is obvious-

Author studied above. The sediment distribution on these virtual patches is obvious-ly unknown but the signature of the coar

Author ly unknown but the signature of the coarthe standard deviation of bed elevati

Author the standard deviation of bed elevati

Author virtual patch, using only DEM data. In

Author virtual patch, using only DEM data. Inof surface GSD resulting from grain size analysis performed on visually Author of surface GSD resulting from grain size analysis performed on visually identified patches is expected to holdAuthor

identified patches is expected to holdover virtual patches. To achieve this goal and further investigate the rough-Author

over virtual patches. To achieve this goal and further investigate the rough-Author

ness of sediment patches over crests Author

ness of sediment patches over crests ing to several virtual patches of dimeAuthor

ing to several virtual patches of dime

copy150.41 169.19 114.47 74.53 109.23 84.24

copy150.41 169.19 114.47 74.53 109.23 84.24

copy4.09 2.58 5.08 6.87 3.49 4.22

copy4.09 2.58 5.08 6.87 3.49 4.22

copy

copy

copyecreases as the bedform height increases (see the

copyecreases as the bedform height increases (see the

differences in mean elevation for C2 and T2 in Table 6).

copy

differences in mean elevation for C2 and T2 in Table 6). with larger variability in the bed

copy

with larger variability in the bed topography, higher complexity and three-dimensionality, essentially related

copy

topography, higher complexity and three-dimensionality, essentially related to the wider variety of bedforms that form at this discharge. While the visual co

pyto the wider variety of bedforms that form at this discharge. While the visual selection of crest and trough was still possible, as assessed by the compara-co

pyselection of crest and trough was still possible, as assessed by the compara-tive analysis in grain size distribution, its statistical characterization turns out co

pytive analysis in grain size distribution, its statistical characterization turns out to be more difficult as the bedform topography becomes more complex. co

pyto be more difficult as the bedform topography becomes more complex.

In order to substantiate statistically the results obtained on the analysed copy

In order to substantiate statistically the results obtained on the analysed

GRAIN SORTING IN GRAVEL BEDFORMS

1629

Fig. 17: (a, f) Bed elevation profile, (b, g) location of local maxima (star) and local minima (dot) superimposed on filtered bed elevation using Fourier transform, (c, h) extracted bedform heights after threshold, (d, i) crest elevations, (e, j) trough eleva-tions in mm corresponding to the virtual patch of 30 × 30 cm. Left column repre-sents 1500 l/s discharge and right column represents 2800 l/s discharge.

Figs. 17c and h show the extracted bedform heights after thresh-holding (see Singh et al. 2011) for details about bedform extraction and thresh-holding), whereas panels (d), (i) and (e), (j) of Figs. 17 show the crest elevation and the trough elevation, respectively. The statistics (mean and standard devia-tion) of the extracted virtual patches can be seen in Table 6. As noticed from the sampled patches, the standard deviation (roughness) of the trough eleva-tion is higher than the crest elevation. Blom et al. (2003) observed a similar relation, i.e., the trough surface is coarser than the crest surface. They attrib-uted this to the mechanism of winnowing of fines from the trough surface.

7. DISCUSSION As discussed in the previous sections, the roughness of the gravel bed is usu-ally characterized by the percentiles of the GSD. However, in the presence of bedforms the roughness of the bed significantly changes. In fact, bedforms change the flow conditions, i.e., the flow becomes unsteady and non-uniform locally, especially in shallow channels, a characteristic of gravel channels. Moreover, in the presence of graded sediment, longitudinal and vertical pat-terns of sorting form as a consequence of the non-uniform bed shear stress distribution along the bedform profile and of the bedform migration. For steady uniform flow and plane bed condition, turbulence (for quantifying sediment transport rates) can be fully characterized by the local bed shear stress (Nelson et al. 1995, Schmeeckle and Nelson 2003). However, for

Author sents 1500 l/s discharge and right column represents 2800 l/s discharge.

Author sents 1500 l/s discharge and right column represents 2800 l/s discharge.

Figs. 17c and h show the extracted bedform heights after thresh-holding (see

Author Figs. 17c and h show the extracted bedform heights after thresh-holding (see 2011) for details about bedform extraction and thresh-holding),

Author 2011) for details about bedform extraction and thresh-holding), whereas panels (d), (i) and (e), (j) of Figs. 17 show the crest elevation and

Author whereas panels (d), (i) and (e), (j) of Figs. 17 show the crest elevation and the trough elevation, respectively. Th

Author the trough elevation, respectively. The statistics (mean and standard devia-

Author e statistics (mean and standard devia-

tion) of the extracted virtual patches ca

Author tion) of the extracted virtual patches cathe sampled patches, the standard de

Author the sampled patches, the standard detion is higher than the crest elevation. Blom

Author tion is higher than the crest elevation. Blom

i.e

Author i.e., the trough surface is coarser than

Author ., the trough surface is coarser than

Author uted this to the mechanism of winnowing of fines from the trough surface.

Author uted this to the mechanism of winnowing of fines from the trough surface.

7. Author

7. DISCUSSION Author

DISCUSSION As discussed in the previous sectionsAuthor

As discussed in the previous sectionsAuthor

ally characterized by the percentiles of Author

ally characterized by the percentiles of bedforms the roughness of the bed significantly changes. In fact, bedforms Author

bedforms the roughness of the bed significantly changes. In fact, bedforms

copy

Fig. 17: (a, f) Bed elevation profile, (b, g) location of local maxima (star) and local

copy

Fig. 17: (a, f) Bed elevation profile, (b, g) location of local maxima (star) and local minima (dot) superimposed on filtered bed elevation using Fourier transform, (c, h) co

pyminima (dot) superimposed on filtered bed elevation using Fourier transform, (c, h)

, i) crest elevations, (e, j) trough eleva-copy

, i) crest elevations, (e, j) trough eleva-tions in mm corresponding to the virtual patch of 30 × 30 cm. Left column repre-co

pytions in mm corresponding to the virtual patch of 30 × 30 cm. Left column repre-sents 1500 l/s discharge and right column represents 2800 l/s discharge. co

pysents 1500 l/s discharge and right column represents 2800 l/s discharge.

Figs. 17c and h show the extracted bedform heights after thresh-holding (see co

pyFigs. 17c and h show the extracted bedform heights after thresh-holding (see

copy

A. SINGH et al.

1630

locally non-uniform flow conditions typically occurring in the presence of bedforms, not only the skin friction varies along the bedform surface, but a form drag contribution to the total shear stress also arises (see Wiberg and Nelson (1992) and references therein), which is specifically modulated by the presence of bedforms. Hence in such flows sediment transport modeling generally requires more information than just the total boundary shear stress (Nelson et al. 1995, Sumer et al. 2003, Singh et al. 2012a), since one has to know a priori the characteristics of both small scale roughness elements, i.e., grain scale roughness, and large scale roughness due to bedforms.

From the statistics of grain sizes and elevation of sampled patches and from the extracted virtual patches, it was observed that the crests present a typically finer surface grain distribution as compared to troughs (Fig. 14a). Coarser particles then tend to accumulate at bedform troughs under the action of winnowing, forming a coarser matrix that, however, is progres-sively covered by new material as bedforms migrate downstream. This sort-ing process is governed by the spatial bed shear stress distribution associated with bedform shape and by the differential entrainment capacity of the flow regions above the troughs (characterized by a lower mean velocity) as com-pared to the flow regions above the crests. These observations are consistent with the observations of Singh and Foufoula-Georgiou (2012) where it was shown that with increasing discharge the bedform height increases, which creates more space for ejection events in the trough of the bedform. Due to higher ejection events and low velocity in trough of the bedforms, sediment in trough is preferentially entrained, i.e., smaller particles get entrained thus rendering trough surface coarser. As a consequence of the increased variabil-ity in bed topography, flow turbulence and grain entrainment with increasing discharge, also the sediment transport is more variable at higher discharge than at lower discharge. For example, in the present experiments, the stand-ard deviation of the 2 min averaged sediment transport rates changes from ~6.8 to ~25.5 kg/m/min as the discharge (and hence bedform height) increases from 1500 to 2800 l/s.

The comparative analysis of surface grain distribution over crests and troughs is not qualitatively affected by the increasing discharge and thus by the sediment transport process. The surface material over the trough is invar-iably coarser than that found over the crests, the variability of the grain dis-tribution curves tending to increase with bedform size (i.e., discharge). On the other hand, subsurface grain distributions show an opposite trend: the subsurface grain size statistics of the crests (specifically d16 and d50) point at a coarser composition with respect to the trough material and the difference increases with increasing discharge (Fig. 14a). A similar trend is observed for d84 at the higher discharge, whereas for the low discharge d84 of the crests is finer than that over the troughs (see Fig. 14b). While at the low discharge

Author with bedform shape and by the differential entrainment capacity of the flow

Author with bedform shape and by the differential entrainment capacity of the flow regions above the troughs (characterized

Author regions above the troughs (characterizedpared to the flow regions above the cr

Author pared to the flow regions above the crests. These observations are consistent

Author ests. These observations are consistent with the observations of Singh and Foufoula-Georgiou (2012) where it was

Author with the observations of Singh and Foufoula-Georgiou (2012) where it was shown that with increasing discharge the bedform height increases, which

Author shown that with increasing discharge the bedform height increases, which

Author creates more space for ejection events in the trough of the bedform. Due to

Author creates more space for ejection events in the trough of the bedform. Due to higher ejection events and low velocity

Author higher ejection events and low velocityin trough is preferentially entrained,

Author in trough is preferentially entrained, rendering trough surface coarser. As a con

Author rendering trough surface coarser. As a conity in bed topography, flow turbulence and grain entrainment with increasing

Author ity in bed topography, flow turbulence and grain entrainment with increasing discharge, also the sediment transport is more variable at higher discharge

Author discharge, also the sediment transport is more variable at higher discharge than at lower discharge. For example, in the present experiments, the stand-

Author than at lower discharge. For example, in the present experiments, the stand-ard deviation of the 2 min averaged sediment transport rates changes from

Author ard deviation of the 2 min averaged sediment transport rates changes from

Author

~6.8 to ~25.5 kg/m/min as the discharge (and hence bedform height) Author

~6.8 to ~25.5 kg/m/min as the discharge (and hence bedform height) increases from 1500 to 2800 l/s. Author

increases from 1500 to 2800 l/s. The comparative analysis of surface grain distribution over crests and Author

The comparative analysis of surface grain distribution over crests and troughs is not qualitatively affected by the increasing discharge and thus by Author

troughs is not qualitatively affected by the increasing discharge and thus by

copy the characteristics of both small scale roughness elements,

copy the characteristics of both small scale roughness elements,

grain scale roughness, and large scale roughness due to bedforms.

copygrain scale roughness, and large scale roughness due to bedforms.

From the statistics of grain sizes and elevation of sampled patches and

copyFrom the statistics of grain sizes and elevation of sampled patches and

was observed that the crests present

copywas observed that the crests present

a typically finer surface grain distribution as compared to troughs (Fig. 14a).

copya typically finer surface grain distribution as compared to troughs (Fig. 14a).

Coarser particles then tend to accumulate at bedform troughs under the

copy

Coarser particles then tend to accumulate at bedform troughs under the action of winnowing, forming a coarser matrix that, however, is progres-

copy

action of winnowing, forming a coarser matrix that, however, is progres-dforms migrate downstream. This sort-co

pydforms migrate downstream. This sort-

ing process is governed by the spatial bed shear stress distribution associated copy

ing process is governed by the spatial bed shear stress distribution associated with bedform shape and by the differential entrainment capacity of the flow co

pywith bedform shape and by the differential entrainment capacity of the flow

by a lower mean velocity) as com-copy

by a lower mean velocity) as com-ests. These observations are consistent co

pyests. These observations are consistent

with the observations of Singh and Foufoula-Georgiou (2012) where it was co

pywith the observations of Singh and Foufoula-Georgiou (2012) where it was

GRAIN SORTING IN GRAVEL BEDFORMS

1631

subsurface distributions differ only in the very fine sediment fraction, for the high discharge the grain size distributions between subsurface layers below crest and trough are observed to vary in the whole range of grain sizes (see Fig. 14b). Why do we observe this strong variability? This could be due to the following reason: the typical bedform height at the low discharge is about 4 cm with a standard deviation of roughly 1 cm. This implies that the subsurface layer (10 cm deep excavation) is statistically below the region intercepted by the moving bedform. It is thus reasonable to expect that sub-surface grain size distribution should not exhibit significant differences, regardless of the presence of a crest or trough above them. For larger dis-charge the mean bedform height is about 7 cm with a standard deviation of 4 cm. Therefore, the subsurface layers are still within the propagating bedforms for the crests. As coarse grain sizes tend to accumulate in the trough with little chance to get entrained or transported away, since in the trough the flow velocity is comparatively low, they get buried by the evolv-ing bedforms. Subsurface grain distributions within a still active layer in terms of transport and sorting mechanisms are in fact expected to show vari-ability in the GSD and thus present differences based on the above topogra-phy (crest or trough).

An interesting observation from our experiments is that the troughs at higher discharge for the subsurface material are significantly finer than the initial distribution (compare d50 of 2.89, 4.41 mm versus 7.7 mm for T1, T2 and initial distribution, respectively). What creates this re-organization? We suggest that this is due to the variability in bedform height (bedform height corresponding to C1 is about 10 cm whereas bedform height corresponding to C2 is approximately 18 cm). The higher bedform follows a low elevation trough (see the comparison of crests and troughs mean elevation in Table 6): due to this geometry the recirculation zone is bigger (taller) and the flow velocity is lower. As a result, finer material entrained from the following crest and the other upstream bedforms may get settled on the trough and work through the sediment pores to become a part of subsurface, causing the subsurface material to be finer. However, new experiments on flow veloci-ties above gravel bedforms are needed to verify this hypothesis.

It is interesting to note that variability in bed elevation series can be rep-resented as the sum of the small scale roughness (grain scale roughness) and the large scale roughness (bedform scale roughness). It is noticed that for the low discharge, grain scale features contribute about 25% whereas bedform contribute about 75% to the total roughness measured as the standard devia-tion of the bed elevation series. However, for the higher discharge, the roughness due to bedforms (standard deviation of the bedforms) is much higher than the standard deviation of bed elevation series (see Tables 1, 2, and 6).

Author terms of transport and sorting mechanisms are in fact expected to show vari-

Author terms of transport and sorting mechanisms are in fact expected to show vari-ability in the GSD and thus present differences based on the above topogra-

Author ability in the GSD and thus present differences based on the above topogra-

An interesting observation from our ex

Author An interesting observation from our experiments is that the troughs at

Author periments is that the troughs at higher discharge for the subsurface materi

Author higher discharge for the subsurface material are significantly finer than the

Author al are significantly finer than the initial distribution (compare

Author initial distribution (compare d

Author d50

Author 50d50d

Author d50d of 2.89, 4.41 mm

Author of 2.89, 4.41 mm and initial distribution, respectively). Wh

Author and initial distribution, respectively). Whsuggest that this is due to the variab

Author suggest that this is due to the variabcorresponding to C1 is about 10 cm

Author corresponding to C1 is about 10 cm to C2 is approximately 18 cm). The

Author to C2 is approximately 18 cm). The trough (see the comparison of crests and troughs mean elevation in Table 6):

Author trough (see the comparison of crests and troughs mean elevation in Table 6): due to this geometry the recirculation zone is bigger (taller) and the flow

Author due to this geometry the recirculation zone is bigger (taller) and the flow velocity is lower. As a result, fine

Author velocity is lower. As a result, finecrest and the other upstream bedforAuthor

crest and the other upstream bedforwork through the sediment pores to becoAuthor

work through the sediment pores to becosubsurface material to be finer. HoweAuthor

subsurface material to be finer. Howeties above gravel bedforms are needed to verify this hypothesis.Author

ties above gravel bedforms are needed to verify this hypothesis.

copyis thus reasonable to expect that sub-

copyis thus reasonable to expect that sub-

surface grain size distribution should not exhibit significant differences,

copysurface grain size distribution should not exhibit significant differences,

regardless of the presence of a crest or trough above them. For larger dis-

copyregardless of the presence of a crest or trough above them. For larger dis-

t 7 cm with a standard deviation of

copyt 7 cm with a standard deviation of

4 cm. Therefore, the subsurface layers are still within the propagating

copy4 cm. Therefore, the subsurface layers are still within the propagating

bedforms for the crests. As coarse grain sizes tend to accumulate in the

copy

bedforms for the crests. As coarse grain sizes tend to accumulate in the trough with little chance to get entrained or transported away, since in the

copy

trough with little chance to get entrained or transported away, since in the trough the flow velocity is comparatively low, they get buried by the evolv-co

pytrough the flow velocity is comparatively low, they get buried by the evolv-

tions within a still active layer in copy

tions within a still active layer in terms of transport and sorting mechanisms are in fact expected to show vari-co

pyterms of transport and sorting mechanisms are in fact expected to show vari-ability in the GSD and thus present differences based on the above topogra-co

pyability in the GSD and thus present differences based on the above topogra-

periments is that the troughs at co

pyperiments is that the troughs at

A. SINGH et al.

1632

Fig. 18. Predictive relationship between bed elevation standard deviation and d84 of grain size distribution of the sampled patches for both the discharges of 1500 and 2800 l/s. Note that the dotted line represents a linear fit.

For modeling accurately the flow field and the sediment transport rates over bedform-dominated river beds, it is required to have a priori knowledge of both grain scale roughness and the roughness due to bedforms. Currently available hydrodynamic models, for example, Large Eddy Simulation mod-els, require this information as boundary conditions for improved predictive modelling of flow and sediment transport. The results of this study may pro-vide a better understanding of grain size distribution and roughness parame-terization in the presence of bedforms. Figure 18 shows a predictive relation between the standard deviation of the bed elevation and the 84th percentile (d84) of GSD of the sampled patches, whereas Fig. 19 shows the percentage deviation (trough to crest) for the grain sizes (d84) of the sampled patches as a function of percentage deviation in the standard deviation of bed elevation for the same sampled patches, suggesting that local small scale roughness variations measured by the high resolution topography scan can be related to the actual variation in the d84, typically chosen as the parameter controlling frictional roughness effects. A similar behavior, i.e., a high correlation between standard deviation of bed elevation and d84 of grain size distribution was observed in Aberle and Nikora (2006), however in that study the bedforms were absent. Figure 20 shows the probability density function of percentage deviation between the crest and the trough for the virtual patches

Author Fig. 18. Predictive relationship between bed elevation standard deviation and

Author Fig. 18. Predictive relationship between bed elevation standard deviation and grain size distribution of the sampled patches for both the discharges of 1500 and

Author grain size distribution of the sampled patches for both the discharges of 1500 and 2800 l/s. Note that the dotted line represents a linear fit.

Author 2800 l/s. Note that the dotted line represents a linear fit.

For modeling accurately the flow fiel

Author For modeling accurately the flow fiel

over bedform-dominated river beds, it is required to have

Author over bedform-dominated river beds, it is required to have of both grain scale roughness and the roughness due to bedforms. Currently

Author of both grain scale roughness and the roughness due to bedforms. Currently available hydrodynamic models, for example, Large Eddy Simulation mod-

Author available hydrodynamic models, for example, Large Eddy Simulation mod-els, require this information as bounda

Author els, require this information as boundamodelling of flow and sediment transport.

Author modelling of flow and sediment transport.vide a better understanding of grain si

Author vide a better understanding of grain siterization in the presence of bedforms. Figure 18 shows a predictive relation Author terization in the presence of bedforms. Figure 18 shows a predictive relation between the standard deviation of the bed elevation and the 84th percentile Author

between the standard deviation of the bed elevation and the 84th percentile d Author

d84 Author

84 Author

) of GSD of the sampled patches, whereas Fig. 19 shows the percentage Author

) of GSD of the sampled patches, whereas Fig. 19 shows the percentage deviation (trough to crest) for the grain sizes (Author

deviation (trough to crest) for the grain sizes (a function of percentage deviation in the standard deviation of bed elevation Author

a function of percentage deviation in the standard deviation of bed elevation

copy

Fig. 18. Predictive relationship between bed elevation standard deviation and copy

Fig. 18. Predictive relationship between bed elevation standard deviation and grain size distribution of the sampled patches for both the discharges of 1500 and co

pygrain size distribution of the sampled patches for both the discharges of 1500 and co

pyco

py

GRAIN SORTING IN GRAVEL BEDFORMS

1633

Fig. 19. Percentage deviation (tough to crest) for the grain size (d84) of the sampled patch as a function of percentage deviation of the bed elevation standard deviation of the same sampled patch for the discharges of 1500 and 2800 l/s. Parameters d84,T and d84,C in the y axis represent 84th percentile of GSD of trough and crest patches, respectively, whereas σT and σC in the x axis represent the standard deviation in the elevation of the sampled patch at trough and crest, respectively.

Fig. 20. Probability density functions of percentage deviation in between crests and troughs of virtual patches obtained from the DEM of topography for the discharge of (a) 1500 and (b) 2800 l/s. Note that the dotted line in the pdfs corresponds to the value of percentage deviation (RPatch-DEM) obtained for the sampled patches (see Fig. 19) whereas the solid line represents mean of the distribution.

extracted from the bed elevation series for the discharge of 1500 (Fig. 20a) and 2800 l/s (Fig. 20b). The mean percentage deviation obtained for the vir-

Author

Author

Author patch as a function of percentage deviation of the bed elevation standard deviation of

Author patch as a function of percentage deviation of the bed elevation standard deviation of the same sampled patch for the discharges of 1500 and 2800 l/s. Parameters

Author the same sampled patch for the discharges of 1500 and 2800 l/s. Parameters axis represent 84th percentile of GSD of trough and crest patches,

Author axis represent 84th percentile of GSD of trough and crest patches, C

Author C in the

Author in the x

Author x axis represent the standard deviation in the

Author axis represent the standard deviation in the elevation of the sampled patch at

Author elevation of the sampled patch at trough and crest, respectively.

Author trough and crest, respectively.

copy

Fig. 19. Percentage deviation (tough to crest) for the grain size (copy

Fig. 19. Percentage deviation (tough to crest) for the grain size (dcopy

d84copy

84d84dcopy

d84d ) of the sampled copy) of the sampled

patch as a function of percentage deviation of the bed elevation standard deviation of copy

patch as a function of percentage deviation of the bed elevation standard deviation of the same sampled patch for the discharges of 1500 and 2800 l/s. Parameters co

pythe same sampled patch for the discharges of 1500 and 2800 l/s. Parameters

axis represent 84th percentile of GSD of trough and crest patches, copy

axis represent 84th percentile of GSD of trough and crest patches, axis represent the standard deviation in the

copy

axis represent the standard deviation in the co

pyco

py

A. SINGH et al.

1634

tual patches is about 15% for the low discharge, whereas it is approximately 30% for the higher discharge (Table 7). We acknowledge that the percent-age deviations for virtual patches are highly variable and also show some negative deviations (Fig. 20). This could be due to the fact that the sorting mechanisms may not always respond to a strong variability in bedform ge-ometry (e.g., in the case of merging or superimposed bedforms).

Table 7 Statistics of percentage deviation between crest and trough

for sampled GSD, sampled patch, and virtual patch

Q [l/s]

Rdg [%]

Rpatch-DEM (observed patch)

[%]

Rv-patch-DEM (virtual patch)

[%] (mean)

1500 31.34 19.11 14.77 49.13 29.44

2800 18.81 21.59 29.83 70.31 90.79

8. SUMMARY AND CONCLUDING REMARKS This paper investigates the effect of bedform geometry on the particle organization on the surface and subsurface of the crest and trough of the bed-form. The data used in this study are the high resolution spatial and temporal bed elevation and grain size distribution of surface and subsurface bed materials of crest and trough of bedforms for low flow and high flow condi-tions. The experiments were conducted in a large experimental channel at the St. An-thony Falls Laboratory, University of Minnesota. The main results of this study can be stated as follows:

Statistics of extracted bedform characteristics suggest that the mean and standard deviation of bedform heights increase with increasing dis-charge, whereas for bedform lengths, the mean and standard deviation decrease with increasing discharge.

With increasing discharge, bed elevation fluctuations become smoother as suggested by the increasing slope of the power spectral density and the c1 (multifractality parameter). However, as the variability in bedforms increases, the inhomogeneity in the bed elevation, measured with the intermittency parameter c2, increases resulting in the more variable grain size distribution on the crest of the bedform and a stronger sorting effect both in the surface and subsurface layers.

Bedform migration is observed to induce a preferential grain size distribution (GSD) along the bedform wavelength. In particular, coarser

Author 70.31 90.79

Author 70.31 90.79

Author SUMMARY AND CONCLUDING REMARKS

Author SUMMARY AND CONCLUDING REMARKS This paper investigates the effect of bedform geometry on the particle

Author This paper investigates the effect of bedform geometry on the particle organization on the surface and subsurface of the crest and trough of the bed-

Author organization on the surface and subsurface of the crest and trough of the bed-form. The data used in this study are

Author form. The data used in this study are bed elevation and grain size distribu

Author bed elevation and grain size distribumaterials of crest and trough of bedforms for low flow and high flow condi-

Author materials of crest and trough of bedforms for low flow and high flow condi-

Author tions. The experiments were conducted in

Author tions. The experiments were conducted in St. An-thony Falls Laboratory, University

Author St. An-thony Falls Laboratory, Universitythis study can be stated as follows:

Author this study can be stated as follows:

Author Statistics of extracted bedform characteristics suggest that the mean Author Statistics of extracted bedform characteristics suggest that the mean

and standard deviation Author

and standard deviation charge, whereas for bedform lengths, the mean and standard deviation Author

charge, whereas for bedform lengths, the mean and standard deviation decrease with increasing discharge. Author

decrease with increasing discharge. With increasing discharge, bed elevation fluctuations become Author

With increasing discharge, bed elevation fluctuations become

copyTable 7

copyTable 7

Statistics of percentage deviation between crest and trough

copyStatistics of percentage deviation between crest and trough

for sampled GSD, sampled patch, and virtual patch

copyfor sampled GSD, sampled patch, and virtual patch

R

copyRv-patch-DEM

copyv-patch-DEM

(virtual patch)

copy

(virtual patch) [%] (mean)

copy

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14.77 copy

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18.81 21.59 copy

18.81 21.59 29.83 copy

29.83 copy

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GRAIN SORTING IN GRAVEL BEDFORMS

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(finer) sediments were observed to predominantly accumulate in the trough (crest). Sorting effects due to bedform evolution can be interpreted as a scale interaction mechanism where small scale roughness (scaling at the grain scale) is modulated by large scale roughness (scaling at the bedform scale).

Sorting effects in the subsurface layers were observed to depend on bedform height distribution, suggesting that sorted surface layers are progressively buried creating layers of coarse and fine materials in the stra-tigraphy.

At high discharge, the statistics of GSD on the crests are more varia-ble than the corresponding ones at low discharge, whereas the GSD at the troughs do not show significant deviations at both discharges, suggesting that the dominant sorting mechanism is due to the entrainment potential rather than to preferential depositional processes.

Comparison between sampled and virtual patches shows that small scale roughness variations estimated by the local standard deviation of sur-face elevation at the patch scale are consistent with the variations in the grain size distribution. High-resolution surface topography measurements can be thus used to identify crest and trough regions, to estimate directly the varia-tion in small scale roughness and indirectly the key grain size parameters controlling transport and frictional drag (d50, d84).

In terms of total roughness, while bedform heights provide the dom-inant contribution, the variation in small scale roughness, e.g., d84, still con-tributes significantly to the total roughness. Local small scale roughness variation between crest and trough suggests that a realistic model of surface topography must resolve both large scale roughness features consistent with the expected bedforms and small scale roughness characteristics consistent with bedform modulated sorting (varying local d84 along the bedforms).

Acknowledgmen t s . This research was supported by the National Center for Earth-surface Dynamics (NCED), a Science and Technology Cen-ter funded by NSF under agreement EAR-0120914 as well as by NSF grants EAR-0824084 and EAR-0835789. The experiments performed for this study are the follow up of previous experiments (known as StreamLab06) con-ducted at the St. Anthony Falls Laboratory as part of an NCED program to examine physical-biological aspects of sediment transport (http://www.nced. umn.edu). The authors are thankful to Jeff Marr, Craig Hill, and Sara John-son for providing help in running the experiments. The authors are also thankful to the editors and reviewers whose suggestions and constructive comments substantially improved our presentation and refined our interpre-tations. Computer resources were provided by the Minnesota Supercompu-ting Institute, Digital Technology Center at the University of Minnesota.

Author size distribution. High-resolution surface topography measurements can be

Author size distribution. High-resolution surface topography measurements can be thus used to identify crest and trough regions, to estimate directly the varia-

Author thus used to identify crest and trough regions, to estimate directly the varia-tion in small scale roughness and indirectly the key grain size parameters

Author tion in small scale roughness and indirectly the key grain size parameters controlling transport and frictional drag (

Author controlling transport and frictional drag (d

Author d50

Author 50d50d

Author d50d ,

Author , d

Author dIn terms of total roughness, while bedform heights provide the dom-

Author In terms of total roughness, while bedform heights provide the dom-

Author inant contribution, the variation in small scale roughness,

Author inant contribution, the variation in small scale roughness, tributes significantly to the total r

Author tributes significantly to the total roughness. Local small scale roughness

Author oughness. Local small scale roughness

variation between crest and trough suggest

Author variation between crest and trough suggesttopography must resolve both large scale roughness features consistent with

Author topography must resolve both large scale roughness features consistent with the expected bedforms and small scale roughness characteristics consistent

Author the expected bedforms and small scale roughness characteristics consistent with bedform modulated sorting (varying local

Author with bedform modulated sorting (varying local

Acknowledgmen t s . This research was supported by the National Author Acknowledgmen t s . This research was supported by the National Author

Center for Earth-surface Dynamics (NCED), a Science and Technology Cen-Author

Center for Earth-surface Dynamics (NCED), a Science and Technology Cen-ter funded by NSF under agreement EAR-0120914 as well as by NSF grants Author

ter funded by NSF under agreement EAR-0120914 as well as by NSF grants EAR-0824084 and EAR-0835789. The experiAuthor

EAR-0824084 and EAR-0835789. The experiare the follow up of previous experiments (known as StreamLab06) con-Author

are the follow up of previous experiments (known as StreamLab06) con-

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pytion in small scale roughness and indirectly the key grain size parameters

A. SINGH et al.

1636

R e f e r e n c e s

Aberle, J., and V. Nikora (2006), Statistical properties of armored gravel bed sur-faces, Water Resour. Res. 42, W11414, DOI: 10.1029/2005WR004674.

ASCE Task Committee on Flow and Transport over Dunes (2002), Flow and trans-port over dunes, J. Hydraul. Eng. ASCE 128, 8, 726-728, DOI: 10.1061/ (ASCE)0733-9429(2002)128:8(726).

Best, J. (2005), The fluid dynamics of river dunes: A review and some future re-search directions, J. Geophys. Res. 110, F04S02, DOI: 10.1029/ 2004JF000218.

Blom, A. (2008), Different approaches to handling vertical and streamwise sorting in modeling river morphodynamics, Water Resour. Res. 44, W03415, DOI: 10.1029/2006WR005474.

Blom, A., and G. Parker (2004), Vertical sorting and the morphodynamics of bed form-dominated rivers: A modeling framework, J. Geophys. Res. 109, F02007, DOI: 10.1029/2003JF000069.

Blom, A., J.S. Ribberink, and H.J. de Vriend (2003), Vertical sorting in bed forms: Flume experiments with a natural and a trimodal sediment mixture, Water Resour. Res. 39, 2, 1025, DOI: 10.1029/2001WR001088.

Blom, A., J.S. Ribberink, and G. Parker (2008), Vertical sorting and the morphody-namics of bed form-dominated rivers: A sorting evolution model, J. Geo-phys. Res. 113, F01019, DOI: 10.1029/2006JF000618.

Buffington, J.M., and D.R. Montgomery (1997), A systematic analysis of eight dec-ades of incipient motion studies, with special reference to gravel-bedded rivers, Water Resour. Res. 33, 8, 1993-2029.

Buffington, J.M., and D.R. Montgomery (1999), Effects of hydraulic roughness on surface textures of gravel-bed rivers, Water Resour. Res. 35, 11, 3507-3521, DOI: 10.1029/1999WR900138.

Butler, J.B., S.N. Lane, and J.H. Chandler (2001), Characterization of the structure of river-bed gravels using two-dimensional fractal analysis, Math. Geol. 33, 3, 301-330, DOI: 10.1023/A:1007686206695.

Castaing, B., Y. Gagne, and E.J. Hopfinger (1990), Velocity probability density-functions of high Reynolds number turbulence, Physica D 46, 2, 177-200, DOI: 10.1016/0167-2789(90)90035-N.

Coleman, S.E., and B.W. Melville (1994), Bed-form development, J. Hydraul. Eng. ASCE 120, 5, 544-560, DOI: 10.1061/(ASCE)0733-9429(1994)120:5(544).

Coleman, S.E., V.I. Nikora, S.R. McLean, T.M. Clunie, T. Schlicke, and B.W. Melville (2006), Equilibrium hydrodynamics concept for developing dunes, Phys. Fluids 18, 10, 105104, DOI: 10.1063/1.2358332.

Jerolmack, D., and D. Mohrig (2005), Interactions between bed forms: Topography, turbulence, and transport, J. Geophys. Res. 110, F02014, DOI: 10.1029/ 2004JF000126.

Author Flume experiments with a natural and a trimodal sediment mixture,

Author Flume experiments with a natural and a trimodal sediment mixture,

Author , 2, 1025, DOI: 10.1029/2001WR001088.

Author , 2, 1025, DOI: 10.1029/2001WR001088.

Author Blom, A., J.S. Ribberink, and G. Parker (2008), Vertical sorting and the morphody-

Author Blom, A., J.S. Ribberink, and G. Parker (2008), Vertical sorting and the morphody-

Author namics of bed form-dominated rivers: A sorting evolution model,

Author namics of bed form-dominated rivers: A sorting evolution model,

Author , F01019, DOI: 10.1029/2006JF000618.

Author , F01019, DOI: 10.1029/2006JF000618.

Author Buffington, J.M., and D.R. Montgomery (1997), A systematic analysis of eight dec-

Author Buffington, J.M., and D.R. Montgomery (1997), A systematic analysis of eight dec-

Author ades of incipient motion studies, with

Author ades of incipient motion studies, with

Author Water Resour. Res.

Author Water Resour. Res. 33

Author 33, 8, 1993-2029.

Author , 8, 1993-2029.

Author Buffington, J.M., and D.R. Montgomery (1999), Effects of hydraulic roughness on

Author Buffington, J.M., and D.R. Montgomery (1999), Effects of hydraulic roughness on

Author

Author surface textures of gravel-bed rivers,

Author surface textures of gravel-bed rivers,

Author 3521, DOI: 10.1029/1999WR900138.

Author 3521, DOI: 10.1029/1999WR900138.

Author Butler, J.B., S.N. Lane, and J.H. Chandler

Author Butler, J.B., S.N. Lane, and J.H. Chandler

Author of river-bed gravels using two-dimensional fractal analysis,

Author of river-bed gravels using two-dimensional fractal analysis,

Author 3, 301-330, DOI: 10.1023/A:1007686206695. Author 3, 301-330, DOI: 10.1023/A:1007686206695. Author

Castaing, B., Y. Gagne, and E.J. Hopfinger (1990), Velocity probability density-Author

Castaing, B., Y. Gagne, and E.J. Hopfinger (1990), Velocity probability density-Author

functions of high Reynolds number turbulence, Author

functions of high Reynolds number turbulence, Author

DOI: 10.1016/0167-2789(90)90035-N. Author

DOI: 10.1016/0167-2789(90)90035-N. Author

Coleman, S.E., and B.W. Melville (1994), Bed-form development, Author

Coleman, S.E., and B.W. Melville (1994), Bed-form development,

copyBest, J. (2005), The fluid dynamics of river dunes: A review and some future re-

copyBest, J. (2005), The fluid dynamics of river dunes: A review and some future re-

copy, F04S02, DOI: 10.1029/

copy, F04S02, DOI: 10.1029/

copyBlom, A. (2008), Different approaches to handling vertical and streamwise sorting

copyBlom, A. (2008), Different approaches to handling vertical and streamwise sorting Blom, A. (2008), Different approaches to handling vertical and streamwise sorting

copyBlom, A. (2008), Different approaches to handling vertical and streamwise sorting

copyWater Resour. Res.

copyWater Resour. Res. 44

copy44, W03415, DOI:

copy, W03415, DOI:

copy

Blom, A., and G. Parker (2004), Vertical sorting and the morphodynamics of bed

copy

Blom, A., and G. Parker (2004), Vertical sorting and the morphodynamics of bed

copy

form-dominated rivers: A modeling framework,

copy

form-dominated rivers: A modeling framework, J. Geophys. Res.

copy

J. Geophys. Res.

copy

Blom, A., J.S. Ribberink, and H.J. de Vriend (2003), Vertical sorting in bed forms: copy

Blom, A., J.S. Ribberink, and H.J. de Vriend (2003), Vertical sorting in bed forms: copy

Flume experiments with a natural and a trimodal sediment mixture, copy

Flume experiments with a natural and a trimodal sediment mixture, copy

, 2, 1025, DOI: 10.1029/2001WR001088. copy

, 2, 1025, DOI: 10.1029/2001WR001088. copy

Blom, A., J.S. Ribberink, and G. Parker (2008), Vertical sorting and the morphody-copy

Blom, A., J.S. Ribberink, and G. Parker (2008), Vertical sorting and the morphody-copy

GRAIN SORTING IN GRAVEL BEDFORMS

1637

Kleinhans, M.G., A.W.E. Wilbers, A. De Swaaf, and J.H. van den Berg (2002), Sediment supply-limited bedforms in sand-gravel bed rivers, J. Sediment. Res. 72, 5, 629-640, DOI: 10.1306/030702720629.

Lanzoni, S. (2000), Experiments on bar formation in a straight flume: 2. Graded sediment, Water Resour. Res. 36, 11, 3351-3363, DOI: 10.1029/2000 WR900161.

Lanzoni, S., and M. Tubino (1999), Grain sorting and bar instability, J. Fluid Mech. 393, 149-174, DOI: 10.1017/S0022112099005583.

Leclair, S.F. (2002), Preservation of cross-strata due to the migration of subaqueous dunes: An experimental investigation, Sedimentology 49, 6, 1157-1180, DOI: 10.1046/j.1365-3091.2002.00482.x.

McElroy, B., and D. Mohrig (2009), Nature of deformation of sandy bed forms, J. Geophys. Res. 114, F00A04, DOI: 10.1029/2008JF001220.

Nelson, J.M., S.R. McLean, and S.R. Wolfe (1993), Mean flow and turbulence fields over two-dimensional bed forms, Water Resour. Res. 29, 12, 3935-3953, DOI: 10.1029/93WR01932.

Nelson, J.M., R.L. Shreve, S.R. McLean, and T.G. Drake (1995), Role of near-bed turbulence structure in bed load transport and bed form mechanics, Water Resour. Res. 31, 8, 2071-2086, DOI: 10.1029/95WR00976.

Nikora, V., and J. Walsh (2004), Water-worked gravel surfaces: High-order struc-ture functions at the particle scale, Water Resour. Res. 40, W12601, DOI: 10.1029/2004WR003346.

Nikora, V.I., D.G. Goring, and B.J.F. Biggs (1998), On gravel-bed roughness char-acterization, Water Resour. Res. 34, 3, 517-527, DOI: 10.1029/97 WR02886.

Nordin, C.F., Jr. (1971), Statistical properties of dune profiles, Geol. Surv. Prof. Pa-per 562, F, 1-41.

Paola, C., and L. Borgman (1991), Reconstructing random topography from pre-served stratification, Sedimentology 38, 4, 553-565, DOI: 10.1111/j.1365-3091.1991.tb01008.x.

Powell, D.M. (1998), Patterns and processes of sediment sorting in gravel-bed riv-ers, Progr. Phys. Geogr. 22, 1, 1-32, DOI: 10.1177/030913339802200101.

Schmeeckle, M.W., and J.M. Nelson (2003), Direct simulation of bedload transport using a local, dynamic boundary condition, Sedimentology 50, 2, 279-301, DOI: 10.1046/j.1365-3091.2003.00555.x.

Simons, D.B., E.V. Richardson, and C.F. Nordin Jr. (1965), Bedload equation for ripples and dunes, Geol. Surv. Prof. Pap. 462, H, 1-9.

Singh, A., and E. Foufoula-Georgiou (2012), On the interaction of bed topography, instantaneous shear stress and sediment flux in an experimental channel, In: J. Venditti, J. Best, M. Church, and R. Hardy (eds.) Coherent Flow Structures in Geophysical Flows at Earth’s Surface, Wiley-Blackwell (in print).

Singh, A., K. Fienberg, D.J. Jerolmack, J. Marr, and E. Foufoula-Georgiou (2009), Experimental evidence for statistical scaling and intermittency in sediment

Author

Author turbulence structure in bed load tr

Author turbulence structure in bed load tr

Author , 8, 2071-2086, DOI: 10.1029/95WR00976.

Author , 8, 2071-2086, DOI: 10.1029/95WR00976.

Author Nikora, V., and J. Walsh (2004), Water-wo

Author Nikora, V., and J. Walsh (2004), Water-worked gravel surfaces: High-order struc-

Author rked gravel surfaces: High-order struc-

Author

Author ture functions at the particle scale,

Author ture functions at the particle scale, Water Resour. Res.

Author Water Resour. Res.

Author 10.1029/2004WR003346.

Author 10.1029/2004WR003346.

Author Nikora, V.I., D.G. Goring, and B.J.F. Biggs (1998), On gravel-bed roughness char-

Author Nikora, V.I., D.G. Goring, and B.J.F. Biggs (1998), On gravel-bed roughness char-

Author Water Resour. Res

Author Water Resour. Res

Author Nordin, C.F., Jr. (1971), Statisti

Author Nordin, C.F., Jr. (1971), Statistical properties of dune profiles,

Author cal properties of dune profiles,

Author 562

Author 562, F, 1-41.

Author , F, 1-41.

Author

Author Paola, C., and L. Borgman (1991), Reconstructing random topography from pre-

Author Paola, C., and L. Borgman (1991), Reconstructing random topography from pre-

Author served stratification,

Author served stratification,

Author 3091.1991.tb01008.x.

Author 3091.1991.tb01008.x.

Author Powell, D.M. (1998), Patterns and processes of sediment sorting in gravel-bed riv-Author Powell, D.M. (1998), Patterns and processes of sediment sorting in gravel-bed riv-Author

ers, Author

ers, Progr. Phys. Geogr.Author

Progr. Phys. Geogr.Author

Author

Author

Schmeeckle, M.W., and J.M. Nelson (2003), Author

Schmeeckle, M.W., and J.M. Nelson (2003), Author

using a local, dynamic boundary condition, Author

using a local, dynamic boundary condition, Author

DOI: 10.1046/j.1365-3091.2003.00555.x. Author

DOI: 10.1046/j.1365-3091.2003.00555.x.

copy

copyLeclair, S.F. (2002), Preservation of cross-strata due to the migration of subaqueous

copyLeclair, S.F. (2002), Preservation of cross-strata due to the migration of subaqueous

copy, 6, 1157-1180,

copy, 6, 1157-1180,

copyMcElroy, B., and D. Mohrig (2009), Nature of deformation of sandy bed forms,

copyMcElroy, B., and D. Mohrig (2009), Nature of deformation of sandy bed forms,

copy

, F00A04, DOI: 10.1029/2008JF001220.

copy

, F00A04, DOI: 10.1029/2008JF001220.

copy

(1993), Mean flow and turbulence fields

copy

(1993), Mean flow and turbulence fields

copy

Water Resour. Res.

copy

Water Resour. Res. 29

copy

29, 12, 3935-3953,

copy

, 12, 3935-3953,

copy

Nelson, J.M., R.L. Shreve, S.R. McLean, and T.G. Drake (1995), Role of near-bed copy

Nelson, J.M., R.L. Shreve, S.R. McLean, and T.G. Drake (1995), Role of near-bed copy

ansport and bed form mechanics, copy

ansport and bed form mechanics, copy

, 8, 2071-2086, DOI: 10.1029/95WR00976. copy

, 8, 2071-2086, DOI: 10.1029/95WR00976. copy

rked gravel surfaces: High-order struc-copy

rked gravel surfaces: High-order struc-copy

Water Resour. Res.co

pyWater Resour. Res.

A. SINGH et al.

1638

transport rates, J. Geophys. Res. 114, F01025, DOI: 10.1029/2007 JF000963.

Singh, A., F. Porté-Agel, and E. Foufoula-Georgiou (2010), On the influence of gravel bed dynamics on velocity power spectra, Water Resour. Res. 46, W04509, DOI: 10.1029/2009WR008190.

Singh, A., S. Lanzoni, P.R. Wilcock, and E. Foufoula-Georgiou (2011), Multiscale statistical characterization of migrating bed forms in gravel and sand bed rivers, Water Resour. Res.47, W12526, DOI: 10.1029/2010WR010122.

Singh, A., E. Foufoula-Georgiou, F. Porte-Agel, and P.R. Wilcock (2012a), Coupled dynamics of the co-evolution of gravel bed topography, flow turbulence and sediment transport in an experimental channel, J. Geophys. Res., DOI: 10.1029/2011JF002323 (in print).

Singh, A., J.A. Czuba, J. Marr, C. Hill, S. Johnson, C. Ellis, J. Mullin, C.H. Orr, P.R. Wilcock, M. Hondzo, C. Paola, and E. Foufoula-Georgiou (2012b), StreamLab experiments and research synthesis, Water Resour. Res., DOI: 10:1029/2012WR012124 (in print).

Sumer, B.M., L.H.C. Chua, N.-S. Cheng, and J. Fredsøe (2003), Influence of turbu-lence on bed load sediment transport, J. Hydraul. Eng. ASCE 129, 8, 585-596, DOI: 10.1061/(ASCE)0733-9429(2003)129:8(585).

van der Mark, C.F., A. Blom, and S.J.M.H. Hulscher (2008), Quantification of vari-ability in bedform geometry, J. Geophys. Res. 113, F03020, DOI: 10.1029/ 2007JF000940.

Venditti, J.G. (2007), Turbulent flow and drag over fixed two- and three-dimensional dunes, Water Resour. Res. 112, F04008, DOI: 10.1029/2006 JF000650.

Venditti, J.G., M.A. Church, and S.J. Bennett (2005), Morphodynamics of small-scale superimposed sand waves over migrating dune bed forms, Water Resour. Res. 41, W10423, DOI: 10.1029/2004WR003461.

Venugopal, V., S.G. Roux, E. Foufoula-Georgiou, and A. Arneodo (2006), Revisit-ing multifractality of high-resolution temporal rainfall using a wavelet-based formalism, Water Resour. Res. 42, W06D14, DOI: 10.1029/2005 WR004489.

Wiberg, P.L., and J.M. Nelson (1992), Unidirectional flow over asymmetric and symmetric ripples, J. Geophys. Res. 97, C8, 12745-12761, DOI: 10.1029/ 92JC01228.

Wilcock, P.R. (1998), Two-fraction model of initial sediment motion in gravel-bed rivers, Science 280, 5362, 410-412, DOI: 10.1126/science.280.5362.410.

Yarnell, S.M., J.F. Mount, and E.W. Larsen (2006), The Influence of relative sedi-ment supply on riverine habitat heterogeneity, Geomorphology 80, 310-324, DOI: 10.1016/j.geomorph.2006.03.005.

Received 5 September 2012 Received in revised form 13 September 2012

Accepted 23 September 2012

Author lence on bed load sediment transport,

Author lence on bed load sediment transport,

Author 596, DOI: 10.1061/(ASCE)0733-9429(2003)129:8(585).

Author 596, DOI: 10.1061/(ASCE)0733-9429(2003)129:8(585).

Author van der Mark, C.F., A. Blom, and S.J.M.H. Hulscher (2008), Quantification of vari-

Author van der Mark, C.F., A. Blom, and S.J.M.H. Hulscher (2008), Quantification of vari-

Author ability in bedform geometry,

Author ability in bedform geometry, J. Geophys. Res.

Author J. Geophys. Res.

Author Venditti, J.G. (2007), Turbulent flow and drag over fixed two- and three-

Author Venditti, J.G. (2007), Turbulent flow and drag over fixed two- and three-

Author

Author dimensional dunes,

Author dimensional dunes, Water Resour. Res

Author Water Resour. Res

Author Venditti, J.G., M.A. Church,

Author Venditti, J.G., M.A. Church, and S.J. Bennett (2005), Morphodynamics of small-

Author and S.J. Bennett (2005), Morphodynamics of small-

Author scale superimposed sand waves

Author scale superimposed sand waves

Author Resour. Res.

Author Resour. Res. 41

Author 41, W10423, DOI: 10.1029/2004WR003461.

Author , W10423, DOI: 10.1029/2004WR003461.

Author Venugopal, V., S.G. Roux, E. Foufoula-Georgiou, and A. Arneodo (2006), Revisit-

Author Venugopal, V., S.G. Roux, E. Foufoula-Georgiou, and A. Arneodo (2006), Revisit-

Author

Author ing multifractality of high-resolution temporal rainfall using a wavelet-

Author ing multifractality of high-resolution temporal rainfall using a wavelet-

Author based formalism, Author based formalism, based formalism, Author based formalism, Author

WR004489. Author

WR004489. Author

Wiberg, P.L., and J.M. Nelson (1992), UnAuthor

Wiberg, P.L., and J.M. Nelson (1992), UnAuthor

symmetric ripples, Author

symmetric ripples, Author

92JC01228. Author

92JC01228.

copy

copy

, W12526, DOI: 10.1029/2010WR010122.

copy

, W12526, DOI: 10.1029/2010WR010122.

copySingh, A., E. Foufoula-Georgiou, F. Porte-Agel, and P.R. Wilcock (2012a), Coupled

copySingh, A., E. Foufoula-Georgiou, F. Porte-Agel, and P.R. Wilcock (2012a), Coupled

copydynamics of the co-evolution of gravel bed topography, flow turbulence

copydynamics of the co-evolution of gravel bed topography, flow turbulence

copyJ. Geophys. Res.

copyJ. Geophys. Res.,

copy, DOI:

copyDOI:

copy

C. Ellis, J. Mullin, C.H. Orr,

copy

C. Ellis, J. Mullin, C.H. Orr,

copy

P.R. Wilcock, M. Hondzo, C. Paola, and E. Foufoula-Georgiou (2012b),

copy

P.R. Wilcock, M. Hondzo, C. Paola, and E. Foufoula-Georgiou (2012b),

copy

StreamLab experiments and research synthesis,

copy

StreamLab experiments and research synthesis, Water Resour. Res.

copy

Water Resour. Res.

copy

Sumer, B.M., L.H.C. Chua, N.-S. Cheng, and J. Fredsøe (2003), Influence of turbu-copy

Sumer, B.M., L.H.C. Chua, N.-S. Cheng, and J. Fredsøe (2003), Influence of turbu-copy

J. Hydraul. Eng.copy

J. Hydraul. Eng. ASCEcopy

ASCEcopy

596, DOI: 10.1061/(ASCE)0733-9429(2003)129:8(585). copy

596, DOI: 10.1061/(ASCE)0733-9429(2003)129:8(585). copy

van der Mark, C.F., A. Blom, and S.J.M.H. Hulscher (2008), Quantification of vari-copy

van der Mark, C.F., A. Blom, and S.J.M.H. Hulscher (2008), Quantification of vari-copy

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