Variability in Bedform Morphology and Kinematics with Transport Stage

Sedimentology (2016) doi: 10.1111/sed.12247

Variability in bedform morphology and kinematics with transport stage

JEREMY G. VENDITTI*, C.-Y. MARTIN LIN* and MOSLEM KAZEMI† *Department of Geography, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 (E-mail: [email protected]) †Brain Corporation, San Diego, CA 92121, USA

Associate Editor – David Mohrig

ABSTRACT Bedform geometry is widely recognized to be a function of transport stage. Bedform aspect ratio (height/length) increases with transport stage, reaches a maximum, then decreases as bedforms washout to a plane bed. Bedform migration rates are also linked to bedform geometry, in so far as smaller bedforms in coarser sediment tend to migrate faster than larger bedforms in finer sediment. However, how bedform morphology (height, length and shape) and kinematics (translation and deformation) change with transport stage and suspension have not been examined. A series of experiments is presented where initial flow depth and grain size were held constant and the transport stage was varied to produce bedload dominated, mixed-load dominated and suspended-load dominated conditions. The results show that the commonly observed pattern in bedform aspect ratio occurs because bedform height increases then decreases with transport stage, against a continuously increasing bedform length. Bedform size variability increased with transport stage, leading to less uniform bedform fields at higher transport stage. Total translation-related and deformation-related sediment fluxes all increased with transport stage. However, the relative contribution to the total flux changed. At the bedload dominated stage, translation-related and deformation-related flux contributed equally to the total flux. As the transport stage increased, the fraction of the total load contributed by translation increased and the fraction contributed by deformation declined because the bedforms got bigger and moved faster. At the suspended-load dominated transport stage, the deformation flux increased and the translation flux decreased as a fraction of the total load, approaching one and zero, respectively, as bedforms washed out to a plane bed. Keywords Bedform deformation, bedform translation, bedforms, dunes, sand-bedded rivers, sediment flux.

INTRODUCTION between two successive erosional surfaces are one of the most common structures in sedimen- Bedforms in sand-bedded rivers are the domi- tary strata (Allen, 1970; Rubin & Hunter, 1982). nant source of flow resistance, and their forma- These cross-sets have characteristic signatures tion, growth and kinematics control sediment that record bedform size, shape, migration rate flux; they also leave characteristic signatures of and direction (Allen, 1973). Cross-set thickness their presence in the sedimentary record that can be used to estimate bedform height (Paola & allow for reconstruction of palaeoflows. Cross- Borgman, 1991; Leclair, 2002; Jerolmack & Moh- stratified units made up of cross-sets preserved rig, 2005; Ganti et al., 2013) which is frequently

© 2015 The Authors. Sedimentology © 2015 International Association of Sedimentologists 1 2 Jeremy G. Venditti et al. used to estimate formative flow depths in rivers when u*/ws ≤ 1. It is possible to recast this sus- (e.g. Bridge & Tye, 2000). The controls bedforms pension threshold into a transport stage for a exert on flow resistance and sediment flux in given sand size because Shields stress and the rivers, and their use in palaeoflow estimation, is suspension criteria have the same basic form (a critically dependent on current understanding of shear stress metric divided by a grain size met- 1 2 how bedform dimensions and kinematics ric) and s* (u*/ws) . This approach has the respond to flow strength. advantage of placing the threshold for motion It is widely recognized that bedform dimen- into the same rubric as the threshold for suspen- sions are a function of transport stage (Yalin, sion. 1972; Yalin & Karahan, 1979a, Allen, 1982; van Yalin (1972) showed that bedform geometry Rijn, 1984; Venditti et al., 2005; Venditti, 2013; changes with transport stage. Bedform aspect Lin & Venditti, 2013) and bedform dimensions ratios (bedform height divided by length, H/L) are thought to influence migration rates (Allen, are low when s*/s*c is just above 1, increases 1973; Raudkivi & Witte, 1990; Coleman & Mel- with transport stage under BLD conditions, are ville, 1994; Venditti et al., 2005; Lin & Venditti, largest under MXD conditions and decline with 2013). As such, there is good reason to suspect a increasing transport stage under SSD conditions, relation between transport stage and migration until they washout to a flat bed. Ripples and rate. dunes follow the same pattern, but ripple H/L A common measure of transport stage is the peaks at lower values of s*/s*c. Data compiled non-dimensional Shields stress divided by its by van Rijn (1984, 1993) and perturbation theory value at the threshold for sediment entrainment proposed by Fredsoe (1982) both demonstrate or s*/s*c. The Shields stress is calculated as: similar patterns, but the functional forms of the relations are somewhat different. However, ¼ s ð Þ experiments designed to examine how bedform s ð Þ 1 qs qw gD dimensions adjust to changes in transport stage, holding other factors that influence bedform where s is the shear stress at the bed, qs and qw dimensions constant, have not been undertaken. are the sediment and water densities, respec- This is particularly concerning because there are tively, g is gravitational acceleration and D is relatively few observations of bedform dimen- the grain size of the sediment, which is usually sions under SSD conditions. taken to be the median size D50 (Raudkivi, Recognizing the relations between transport 1967). Values of the critical value of s* for sedi- stage, bedform dimensions and migration rates, ment entrainment (s*c) vary with grain size (cf. Lin & Venditti (2013) examined the relation Miller et al., 1977; Yalin & Karahan, 1979b; between transport stage and migration rates Brownlie, 1981). using data from a wide variety of laboratory Church (2006) established ranges of s*/s*c for flume experiments and field observations. For bedload dominated (BLD) conditions when dune fields that are in equilibrium with an < < 1 s*/s*c 3 3, mixed-load dominated (MXD) imposed flow, these authors demonstrated that conditions when 33 < s*/s*c < 33 and sus- migration rates increase with transport stage for pended-load dominated (SSD) conditions when a given grain size. The derived model between s*/s*c > 33. The BLD and SSD conditions were transport stage and translation rates holds across defined using the thresholds for motion and orders of magnitude of bedform size. Lin & Ven- suspension, established empirically for reach- ditti (2013) also found that dunes move faster in scales. The ranges were not intended to repre- coarser sediment than in finer sediment at a sent local particle dynamics. More useful given transport stage, probably because finer boundaries for BLD, MXD and SSD conditions, sediment is more likely to bypass the crestline that describe particle dynamics in terms of and not contribute to downstream migration local flow conditions, can be derived from a (Mohrig & Smith, 1996). This implies that the common suspension threshold as the ratio of suspension potential of the sediment is impor- the shear velocity to the settling velocity, u*/ws tant for controlling translation rates. whereby suspension occurs when u*/ws > 1 The results of Lin & Venditti (2013) apply (e.g. Bagnold, 1966; Nino & Garcia, 1998; Lopez strictly to bedform translation rate, which is the & Garcia, 2001). BLD conditions occur when s*/ downstream migration of bedforms without > < s*c 1 and u*/ws 1, MXD conditions occur changes in the shape, size or spacing. McElroy & when u*/ws ≥ 1 and SSD conditions occur Mohrig (2009) show that the elevation change at

© 2015 The Authors. Sedimentology © 2015 International Association of Sedimentologists, Sedimentology Bedform morphology and kinematics 3 a point with time in a bedform field can be wide and 06 m deep, it has an adjustable slope decomposed as: (05to20%) and can accommodate flows up to 02m3 s 1. The flume re-circulates water and og og þ þ d ¼ ð Þ sediment. Sediment used in the experiment was o Vb o b 0 2 = t x well-sorted quartz sand with D50 550 lm (Fig. 1). Mineralogical analysis showed that the where g is the bed elevation, t is time, x is dis- sand was 999% quartz, with trace amounts tance alongstream, Vb is the bedform migration (<01%) of sillimanite, garnet, sphalerite, mus- rate and db is the deformation rate of the bed. covite and gold attached to quartz. The first term on the left-hand side is the change in bed elevation at a point through time. The second term represents downstream translation Experimental design and procedure of the waveform. Deformation is the sum of all The experiment consisted of three runs under changes to the topographic profile of the bed BLD, MXD and SSD conditions (Table 1), that are not associated with the downstream designed in consideration of the Church (2006) translation and may include changes in bedform reach-scale ranges of s*/s*c for each transport shape, size and spacing. As such, deformation stage and the values of u*/ws that would pro- can represent an important component of bed- duce a given transport stage. Ultimately, the form kinematics and bedform-related sediment flow conditions were set based on visual obser- flux. Surprisingly little is known about bedform vation of the modes of transport. The BLD run translation and deformation in sand-bedded had no suspension; the SSD run had a substan- river channels and how they vary with transport tial suspended component and was just at the stage, which may be problematic when examin- threshold for the dunes washing out; the MXD ing bedform migration rates across a wide range run was above the threshold suspension. To of transport stages because McElroy & Mohrig ensure that a flow condition would produce (2009) reasoned that downstream bedform trans- dunes, the 10°C-equivalent mean velocity and lation contributes to bedload transport in the grain size were calculated and compared to the channel while deformation contributes to the bedform phase diagrams of Southard & Boguch- suspended load because, by definition, deforma- wal (1990). The BLD, MXD and SSD conditions tion occurs by vertical exchange of sediment plot in the lower, middle and upper portions of between the bedform and the flow. the dune phase space (Lin, 2011). Here, bedform dimensions, morphodynamics The transport stage and u*/ws were calculated and migration rates are examined experimen- to ensure that they were reasonably close to the tally under BLD, MXD and SSD conditions. Ini- design values, s* was calculated by assuming qw tial flow depth and bed material grain size were 3 3 and qs were 1000 kg m and 2650 kg m , held constant because they are thought to be respectively. The boundary shear stress was cal- important controls on dune size and migration culated from the depth-slope product: rates. Rather than a less detailed examination of ¼ ð Þ many variants of the experimental conditions, so qw ghS 3 which has been done before, three experimental runs are examined in detail. Answers to the following questions are sought: (i) how do bedform dimensions change with transport stage; (ii) how do bedform translation and deformation change with transport stage; and (iii) what is the relation between bedform translation, deformation, and the dominant transport modes?

METHODS

Laboratory experiments were conducted in the River Dynamics Laboratory (RDL) at Simon Fra- ser University, Canada. The RDL ﬂume is 15 m Fig. 1. Grain-size distribution of sediment used in long with a 12 m long working section, 1 m experiments.

Table 1. Summary of flow parameters. BLD, bedload and 30 h prior to the SSD run. After reaching dominated conditions; MXD, mixed-load dominated equilibrium conditions, each run was approxi- conditions; SSD, suspended-load dominated condi- mately 18 h, over two days. The flume was tions. shut down at the end of each day, but the bed Parameter BLD MXD SSD was left fully submerged to ensure negligible effect of the interruption. The data showed no 1 Mean velocity, U (m s )0433 0587 0867 evidence of the interruption on the bed mor- Mean flow depth, h (mm) 151 152 134 phology. Froude number 036 048 076 Reynolds number 64 418 87 907 114 463 Bed and water surface measurements Slope, S 9 10 3 132 337 613 Observations were made using a Swath Mapping System (SMS; Fig. 2) that measured alongstream † Shear stress , s (Pa) 1 90 4 89 7 86 profiles of bed topography and water surface ele- 1 Shear velocity u* (m s )0044 0070 0089 vation. The SMS consists of: (i) a 32 transducer ‡ Design s /s 423 204351 Seatek echo-sounding system (Seatek Inc., Gai- * *c nesville, FL, USA); (ii) three MassaSonic ultra- Design u*/ws 0 252 1 21 2 09 sonic sensors (Massa Products Corporation, ‡ Observed s*/s*c 713 183297 Hingham, MA, USA); (iii) a mechanical stepper- motor driven arm that moves sensors vertically; Observed u /w 0657 104 133 * s (iv) a mechanical, stepper-motor driven system † Sidewall corrected using Williams (1970). that moves the SMS in the streamwise direction; ‡ = Calculated for 0 55 mm sand using s*c 0 03 (van and (v) an onboard computer that records all Rijn, 1993). sensor signals and positions. The SMS traverses the flume on rails attached to the top of the tank (Fig. 2). where h is flow depth and S is the water surface The Seatek transducers are mounted in a Plexi- slope. This calculation represents the sum of the glasâ beam (Fig. 2) that is submerged a few mil- grain-related and bedform-related shear stress. limetres below the water surface to measure bed Sidewall corrections were implemented to cal- elevations. When the SMS is moving down- culate the shear stress applied to the bed using stream, the rate of movement and sampling fre- the empirical equation of Williams (1970): quency determines measurement point density, and the sampling frequency is dependent on the s s ¼ o ð4Þ number of transducers sampled. Sixteen odd- ðÞ1 þ 018h=w2 numbered (1, 3, 5 ... 31) transducers were sampled during each pass over the bed producing where w is the width of flume.pffiffiffiffiffiffiffiffiffiffi The shear velo- parallel profiles of bed topography with measure- city was calculated as u so=q and ws was ments at 0 5 mm intervals in the alongstream x- calculated using Dietrich (1982). Flow depth direction. The alongstream profiles were spaced was held constant at ca 015 m, by filling the at 61 mm in the cross-stream y-direction. The recirculating flume to the desired depth. The profile from transducer 17 is closest to the centre slope was then adjusted to the design value and of the channel at 486 mm from the right-hand the flow depth adjusted independently to the flume wall, looking downstream. The ultrasonic imposed slope when the pumps were started sensors (Fig. 2) measure the water surface eleva- (Table 1). Dune H and L are influenced by flow tion at y-positions of 206 mm, 499 mm and depth, so by holding the depth approximately 805mmfromtherightflumewall. constant, the observed variation in morphology Measurements of bed and water surface topo- is due to changes in the transport modes. graphy along at least 6 m of the flume (stream- After the desired h and S were set for a wise direction) were made with the SMS every given run, the flume was allowed to run until 10 min during the BLD and MXD runs. The per- an equilibrium was reached where S and h iod between SMS measurements was decreased became constant (slope of the time series of h to every 5 min during the SSD run because the and S approached zero) and the bedforms were bed evolved so quickly. The speed of the SMS fully adjusted to the flow condition. This pro- was also increased during the SSD run to cess took 72 h prior to the BLD and MXD runs decrease the time when the Plexiglas bar dis-

Fig. 2. The Swath Mapping System (SMS).

turbed the water surface, decreasing the along- ws stream distance between raw measurements to C ¼ h z a bju ð Þ 5 ca 45 mm. Ca z h a

where C is the concentration of suspended-sedi- Sediment transport measurements ment at height z above the sediment bed, Ca is a Suspended-sediment samples were collected reference concentration measured at elevation with a syphon system composed of an L- a = 40 mm above the bed, b is a coefficient that shaped copper tube, a nylon tube and a describes the difference in diffusion between a variable speed peristaltic pump. The syphon sediment particle and a fluid particle (assumed sampler was mounted on a point gauge to posi- to be 1), j is the von Karman constant (041) and * 05 tion it in the flow. The point gauge was incre- u is the shear velocity calculated as (s/qw) . mented at 03 mm. Samples were taken at Suspended-sediment flux per unit width is cal- 40 mm above the bed in the centre of the chan- culated as: nel over dune crests. While suspended-sedi- q ¼ U Ch ð6Þ ment flux may vary along the dune profile due ss to deposition in the bedform trough and resus- where U is the mean streamwise velocity and pension over the dune stoss slope, erosion and the overbar indicates a depth-average. Calcula- deposition should be balanced at the dune tion of q as the product of mean velocity and crest, making this the least biased position from ss mean concentration can induce a bias in the flux which to take suspended-sediment flux mea- when compared to the mean of the velocity-con- surements. In order to make the system isoki- centration product (i.e. UC) but the resultant netic, the pump rate was calibrated to the bias is negligible for the present conditions. calculated flow velocity at the intake by assum- A small amount of fine silt present in the bed ing a logarithmic velocity profile over the dune sediment made up 1% of the bed material crests. Samples were filtered using glass which, once entrained in the flow, remained in microfibre filters with a pore size of 16 lmto suspension. After a series of bedforms had obtain sediment concentrations. passed through the flume, the concentration of The suspended-sediment flux was calculated silt suspended in the flow was approximately using the Rouse equation (Rouse, 1939): constant. Values of qss were corrected by mea-

© 2015 The Authors. Sedimentology © 2015 International Association of Sedimentologists, Sedimentology 6 Jeremy G. Venditti et al. suring this washload concentration at the BLD pension near the bed because the bedload trap transport stage, where the only sediment in sus- measured all sediment travelling between the pension was the fine silt, and subtracting it from bed and z = 20 mm. Therefore, the Helley-Smith measured concentrations at all transport stages. samples are biased at the MXD and SSD trans- Bedload flux was measured with miniaturized port stages where there was substantial bed Helley-Smith samplers (Helley & Smith, 1971) material suspension. Measured bedload flux was that were scaled down to a 20 mm square mouth corrected by calculating the height of the salta- (Fig. 3A; Dietrich & Smith, 1984; Mohrig & tion layer using a model from van van Rijn Smith, 1996) with a 75 lm mesh bag. The sam- (1984). The Bagnold (1973) and Sklar & Dietrich plers were mounted such that one sampler col- (2004) saltation height models produce nearly lected sediment moving within the bottom identical values to the van Rijn (1984) model 20 mm of the flow and the other collected sedi- over the range of conditions in the experiment. ment between 20 mm and 40 mm above the bed A linear vertical concentration profile was (Fig. 3A). Bedload samples were collected in assumed, from the bed to z = 20 mm, and the sets of three over bedform crests (Fig. 3B) at the Helley-Smith measurements were split into bed- downstream end of the bed topography measure- load and suspended load, the latter of which ment section. The bedload flux over the dune was added to qss calculated using Eq. 6 to get crest is delivered to the lee side, which causes the total suspended bed material flux. More downstream migration, making this the best complicated, nonlinear profile shapes (for exam- place to estimate bed material bedload transport ple, Eq. 5) were examined, but the different pro- associated with the dune. Some sample sets did file shapes produce very similar corrections. not contain samples from all three samplers either because the operator could not identify a Data analysis sampling spot or because placement of the sampler substantially altered the dune crest. Bed- Bed surface data were filtered to remove acous- load flux qbl was calculated as the average of tic noise using a three-step process: sample weights across the channel, divided by 1 A filter removed elevations that were obvi- the sample times of a set. Observations from the ously noise (values below the Plexiglas flume top samplers are not presented, because isoki- floor and 018 m above the flume floor (near the netic suspended samples are a more reliable water surface). method for measuring suspended-sediment flux. 2 A 20 point running average of elevation was The Helley-Smith samples included bedload calculated and elevation readings that exceeded (material transported in traction and saltation), the average value by 3 mm or were less than the as well as material carried in intermittent sus-

Fig. 3. Miniaturized Helley-Smith samplers constructed by the authors.

"# average value by 1 mm were removed. These Xn 0 5 1 values were not replaced. WðlÞ¼ ðÞg g 2 ð7Þ n i 3 Step 2 was repeated, but all removed values i¼1 were replaced by the local mean, effectively smoothing the profile. for a continuous range of bed profile lengths l to produce a function that increases asymptoti- The reason for the difference in the filtering cally (n is the number of bed elevations in an thresholds for Steps 2 and 3 has to do with the alongstream bed profile, g is the bed elevation bottom detection algorithm used by the Seatek and the overbar represents an average over l). The manufacturer. It was found that the echo-soun- value of l when the W(l) curve approaches the ders were more prone to falsely identifying the asymptote defines a ‘saturation length’ L .In bottom as being above the bed than below, so sat order to estimate L unambiguously, McElroy the filtering was adjusted accordingly. Compar- sat (2009) calculates a new curve that is the logarith- ison of the filtered and unfiltered data showed mic derivative of W(l). The saturation length is that the process was effective at removing noise, taken as the value of l when the logarithmic but did not affect the dune shape. The same pro- derivative of W(l) is half the value of the first peak cess was used to filter the water surface eleva- in the curve. A ‘characteristic bedform length’ L tion profiles. c for an alongstream profile can be calculated from Water surface slope was calculated as the sum L as: of the water surface slope relative to the flume sat tank, measured using alongstream profiles, and p ¼ Lsat ffi ð Þ the slope of the flume tank. The latter was mea- Lc 1 57Lsat 8 2 sured with an electronic height gauge mounted to the flume and the laboratory floor that has a McElroy (2009). The corresponding value of W(l) resolution of 0 01 mm, but a practical resolution at L defines a ‘saturation height’ H that can sat sat of 0 1 mm. be used to calculate a ‘characteristic bedform Bedform dimensions were calculated from the height’ as: filtered bed profiles using both a manual and an automated method, which are compared here. Hc ffi 2 8Hsat ð9Þ The manual method used detrended, alongstream bed profiles obtained along the channel centreline. Heights were measured from the McElroy (2009). In order to apply the method, peak to the trough in the lee side of each bed- bed profiles linearly interpolated to a 0 5mm form. Reported height H is the mean for an spacing were used. alongstream transect. Bedform lengths L were Bedform translation rates were also obtained measured for a profile by measuring the total with a manual method and an automated distance of a train of bedforms and dividing by method. Both methods used consecutive along- the number of bedforms. As such, it also is a stream bed elevation surveys separated by some mean value for an alongstream transect. Bed ele- time interval. In the manual method, bedforms vation changes <10 mm and lengths <03m were identified in both surveys and the transla- were omitted from the calculations of H and L tion distance was measured using alongstream to keep the work tractable. This omission con- profiles from the centre Seatek sensor. When a strained the analysis to larger scale bedforms, bedform in the earlier survey split into two excluding smaller scale, superimposed bedforms smaller bedforms in the later survey, the nearest and perturbations in the bedform field, both of downstream crest in the later survey was which were observed. selected to calculate the translation distance. This manual method produced statistics for However, when two bedforms in the earlier sur- the channel centreline, but was too labour inten- vey merged into a single bedform in the later sive to pursue for all 16 alongstream profiles, so survey, the crest further upstream in the earlier an automated method, developed by McElroy survey was used. Some bedforms changed their (2009) was also used. This permitted the calcu- geometry instead of changing in streamwise lation of spatial averages of bedform dimen- position and their translation distance was set at sions. The standard deviation of detrended zero. This method was not performed on profiles alongstream profiles of bed elevation was calcu- from the SSD run because bedform shape chan- lated as: ged so much between two consecutive surveys,

© 2015 The Authors. Sedimentology © 2015 International Association of Sedimentologists, Sedimentology 8 Jeremy G. Venditti et al. it was not possible to reliably match them by stages (Fig. 4). This pattern of variability is mir- eye. rored in the bedform dimensions and translation The automated method calculated translation rates. The sampled sediment transport rates are distance using autocorrelation of two consecu- summarized in Table 2 and the calculated qbl, tive alongstream profiles that were linearly interpolated at a spacing of 1 mm. The profiles were lagged relative to one another and coefficient of determination (r2) was calculated for each lag distance. The profiles were lagged 1 to 1000 times (up to 1 m) for BLD, 1 to 1500 times (up to 15 m) for MXD, and 1 to 3000 times (up to 3 m) for SSD conditions. Translation distance between two surveys was taken as the lag distance that corresponded to the maximum r2 value. As a quality control measure for the SSD condition, a lag of 5000 steps (5 m) was also used. If the lag distance was different between 3000 and 5000 steps, the profiles were deemed too different to reliably calculate a translation distance. This automated method produced a translation distance for all 16 alongstream profiles in a survey, which were then averaged and divided by the survey interval to get a spatially- averaged translation rate for the survey.

RESULTS

The flume slope, initial water depth and channel discharge were imposed in the experiment, but the water surface slope, depth and transport stage adjusted to these initial conditions. Flow depth remained at ca 015 m for the BLD and MXD transport stages but dropped to 0134 m at the SSD transport stage (Table 1). The observed transport stages deviated from the design values slightly. The water surface slope relative to the flume was assumed to be negligible during the BLD run, but subsequent analysis revealed that this was not true and that s*/s*c was actually 1 7 times greater than the design value (Table 1). The observed s*/s*c for the SSD transport stage is 14% less than the design value because the depth declined in response to the flow (Table 1). However, visual observations of sediment movement at each transport stage were consistent with BLD, MXD and SSD conditions. Further- more, the BLD condition is below the threshold for suspension (u*/ws =066); the MXD condition is just above the threshold of suspension Fig. 4. Time series of water surface slope, flow depth, transport stage (s*/s*c) and the ratio of the shear velo- (u*/ws =104); the SSD condition well exceeds city to the settling velocity (u*/ws) for the BLD, MXD the threshold (u*/ws =133). and SSD conditions. The dashed line indicates the Variability in S, h and s /s increased slightly * *c suspension threshold u*/ws = 1. BLD, bedload domi- from BLD to MXD transport stages, but increased nated conditions; MXD, mixed-load dominated condi- dramatically from the MXD to SSD transport tions; SSD, suspended-load dominated conditions.

Table 2. Sampled sediment transport rates; n, number of sampling sets.

Helley-Smith Syphon Depth-integrated suspended Mean Mean bed material load n (g sec 1 m 1) n (g sec 1 m 1) (g sec 1 m 1)

Bedload dominated conditions (BLD) 24 258150759 0 Mixed-load dominated conditions (MXD) 21 111 17 810 706 Suspended-load dominated conditions (SSD) 23 459 18 767754

qss and total bed material flux qs = qbl + qss are larger bedforms. Bedform troughs were more given in Table 3. Both bedload and suspended deeply scoured at the flume walls, presumably bed material flux increased with transport due to increased wall drag. stage. At the BLD stage, qs is equivalent to qbl. The change in topography between the sur- At the MXD and SSD transport stages, qbl is ca veys is not great (Fig. 5D and E). The bedform 30% of qs. shapes did evolve, but they are readily identifi- Although there was clearly more material in able from one survey to the next (Fig. 5A to C). suspension during the SSD run than the MXD The irregular patterns in Dg (Fig. 5D and E) run, it is difficult to distinguish the runs based highlight the effect of bedform three-dimension- on the ratio of suspended-sediment flux to the ality on migration rates. Lobe-shaped portions of total flux. In fact, qss/qs was slightly higher in crestlines moved faster than saddle-shaped por- the MXD run than in the SSD run. With a nar- tions of the crest. Lobes were observed to catch rowly graded sand, it is not possible to generate up with the slower moving saddle shapes, form- a condition where suspended bed material dom- ing a new lobe at the sides of the pre-existing inates over bedload flux and retains bedforms, lobe or saddle, contributing to the change in because as the suspended bed material load bedform shape. Ridges sometimes shifted from increases, so does the bedload. side to side, also contributing to alongstream bedform profile change, but not contributing substantially to downstream bedform transla- Bedform morphodynamics tion. Bedload dominated transport stage At the BLD transport stage, sediment moved in Mixed-load dominated transport stage contact or within a few grain diameters of the Sediment was readily entrained into suspen- bed and transport was generally patchy and sion during the MXD transport stage (Table 3), intermittent. Although the occasional grain was bypassing the dune crestline, and the morpho- suspended from bedform crests, there was no dynamics of the bedforms reflect this beha- measureable bed material suspension (Table 3). viour. Figure 6A to C shows a series of bed Figure 5A to C shows the bed topography for elevation surveys and Dg calculated between three consecutive bed surveys and the change in surveys (Fig. 6D and E). The bedforms had the bed elevation Dg between the surveys (Fig. 5D same morphological characteristics as at the and E). The bedforms were angle of repose, BLD stage: angle of repose, asymmetrical, asymmetrical bedforms, commonly observed in three-dimensional features composed of lobes, small channels and flumes, as opposed to the saddles and spurs, but the bedforms are higher larger scale low lee angle dunes typically (ca 65 mm) and longer (ca 13 m). The observed in large river channels (cf. Kostaschuk increase in height appears to be due to deeper & Villard, 1996; Bradley et al., 2013). The three- scour in the trough. Sandsheets migrating over dimensional bedforms had mean heights of ca the backs of dunes are readily identifiable over 45 mm and mean lengths of ca 09 m. Saddle- the larger bedforms (Fig. 6A to C). Bedform shaped portions of crestlines formed down- crests became rounder and the lee slopes were stream of lobe-shaped crestlines along with more gentle, but the slipface slopes remained spurs (ridges parallel to the flow; Fig. 5A to C). at approximately the angle of repose. Compar- There were some sandsheet features (cf. Venditti ing Fig. 5A to C with Fig. 6A to C, it appears et al., 2005) superimposed on the backs of the that the bedforms were higher because of

ss deeper troughs, presumably caused by more q / vigorous mixing in the bedform lee during the D ∞ q MXD run. The change in topography between the sur- bl q 94 0.615

veys was greater than at the BLD stage (Fig. 6D / 481 876 1.06 T

q and E). Once the bedforms had migrated about one wavelength, it became difﬁcult to identify

D the same features between surveys (Fig. 6A to C) q 32 1 / 925 0 412 0 T because of greater bedform merging and split- q ting. There are obvious bands of larger Dg in

) Fig. 6E and D that represent bedform translation, 1 but there are also a wider range of bed elevation m

1 changes than during the BLD run, in part because the bed was evolving more quickly 41 40

D between surveys. q (g sec

) Suspended-load dominated transport stage 1 2) 51 3) 13 8) 379 0 At the SSD transport stage, the bedload layer m

1 became deeper and the ﬂux rates increased sub- (38 ‡ 6 (23 4 (17 , a stantially compared to the BLD and MXD runs † T

q (Table 3). Figure 7A to C shows that the bed- hi (g sec form morphology became unstable and it was

) difﬁcult to track individual bedforms, but 1

scoured troughs did persist. Through time, the 4) 67 7) 12 8) 156 m

1 bed appeared to alternate through three phases: (51 (i) plane bed with washed-out dunes (Fig. 7C); ‡ a Based on the ﬁnal 8 h of the experiment due to unsteadiness in the height 1 (42 5 (29 ‡

T (ii) a train of large dunes (Fig. 7B); and (iii) a q (g sec train of small dunes (Fig. 7A). In Phase 1, the

) bed was mostly ﬂat and bedforms had lengths 1

ranging from 15to25 m and heights ranging 6) 60 4) 13 1) 166 m

1 from 20 to 50 mm. Some bedforms were as high (54 as 80 mm. In Phase 2, the bed was populated by ‡ m 5 (50 4 (31

T large bedforms up to 100 mm in height and 3 m q (g sec long. In Phase 3, the bed was populated by bedforms less than 1 m long and 20 to 40 mm ) 1

high. m Visual observations by the present authors 1 suggest that the transition between these phases 814 was continuous rather than punctuated. In s q (g sec Phase 1, ﬂow separation and localized, intense erosion occurred on extended ﬂat areas of the ) 1

bed, forming bedforms observed in either Phase

m 2 or 3. Transition occurred by formation of a 1 localized incision, often followed by evenly 6 119 53 spaced incision events upstream and down- ss q (g sec stream of the initial incision. In Phase 2, large dunes emerged that often split into smaller ) 1

dunes forming Phase 3 and the smaller dunes m values that were different between the 3 m and 5 m windows. characteristic of Phase 3 often combined to form b 1 V the large dunes of Phase 2. Dune fields in Phase 983 8025 2 or 3 would sometimes wash out to the nearly bl Corrected sediment transport rates. Values in brackets are the coefficient of variation. BLD, bedload dominated conditions; MXD, mixed-load do- (g sec q flat bed of Phase 1, which would then be dis- sected again. The time a phase existed varied from a few minutes to more than half an hour. There is no manual bedform velocity, so the automated bedform velocity from the channel centreline was substituted, filtered to remove any negative SSD 178 357 535 159 MXD 34 Table 3. minated conditions; SSD, suspended-load dominated conditions. BLD 25 † velocities and time series. Transformation between phases could occur

Fig. 5. Bed topography and elevation change at the bedload dominated transport stage.

Fig. 6. Bed topography and elevation change at the mixed-load dominated transport stage. over a few seconds or minutes. Washed-out, bedform evolution resulted in longer and lower low-amplitude bedforms existed for longer than amplitude bedforms (Fig. 7A to C). Sediment by- the small and large dune phases. This pattern of passed the crest moving into suspension and

Fig. 7. Bed topography and elevation change at the suspended-load dominated transport stage. bedform crests were round with lee slope angles they occupied short sections of the lee face. lower than in the MXD run. Slipface slopes Angle of repose asymmetrical bedforms existed were approximately at the angle of repose, but but were typically washed out quickly. Maps of

Dg (Fig. 7D and E) are difficult to interpret by Eq. 7 did not reach an asymptote that corre- because each phase represents a complete rear- sponds to the bedform lengths due to residual rangement of the bedform field. trends in the bed profiles after linear detrending. Excluding these values, Lc ¼ 105Lm for the SSD run. Bedform dimensions The overestimation of Hm and Lm by the auto- Method comparison mated method along the channel centreline is a Figure 8 shows comparisons between the man- curious result, but is consistent with the results ual method for estimating bedform height and of McElroy (2009). The manual method is inhe- length (Hm and Lm) with the automated methods rently biased towards larger bedforms because (Hc and Lc) along the channel centreline. The bed elevation changes smaller than 10 mm in automated method yielded larger H and L values height and 300 mm in length were ignored. The than the manual method at all transport stages. automated method is inherently biased towards The manual measurements are regarded as more smaller heights and lengths, assuming that there reliable measures of H and L because they are is a continuous distribution of bed elevations direct measurements of observed waveforms. below the largest dunes. Nevertheless, the con- The reliability of the automated measurements sistency of the overestimation of Hm and Lm by is not known, so deviations from Hm and Lm are the automated method suggests that it can be treated as a bias inherent to the method. The corrected for by dividing Hc by 126 and Lc by bias in Hc relative to Hm is remarkably consis- 107, the mean overestimation ratio across all tent, but declines slightly with transport stage: three runs, before calculating the spatial aver- Hc=Hm is 129, 126 and 123 for the BLD, MXD ages. and SSD runs, respectively. The present sample size of three is too small to determine whether Spatially-averaged bedform dimensions this pattern is real. The bias in Lc is less, but Figure 9 shows spatial averages of bedform also consistent for the BLD and MXD runs height and length , calculated using where Lc=Lm is 107 and 109, respectively. The the McElroy (2009) method, corrected for bias. bias is somewhat larger for the SSD run because There is some minor non-stationarity in ca 10% of Lc are >3 m, the maximum manually and length , but the change in H or L is gen- measured Lm. In these cases, the curve described erally on the order of the instrument resolution

Fig. 8. Comparison between (A) bedform heights and (B) bedform lengths obtained along the channel centreline using the manual method (Hm and Lm) and the automated method (Hc and Lc) of McElroy (2009). Open triangles are Lc >3 m. SSD, suspended-load dominated conditions; MXD, mixed-load dominated conditions; BLD, bedload dominated conditions.

< > < > < > Fig. 9. Time series of spatially averaged bedform (A) height Hc and (B) length Lc and histograms of (C) Hc and (D) . The solid and dashed lines in panels (C) and (D) are for the first and last 8 h of the SSD run, respectively. SSD, suspended-load dominated conditions; MXD, mixed-load dominated conditions; BLD, bedload dominated conditions. at the BLD and MXD transport stages and so is so the first 8 h of the experiment are removed < > ignored. The non-stationarity in Hc is more from all further analysis. There is also some substantial at the SSD transport stage (Fig. 9A). unsteadiness in the L time series at the SSD Bedforms grew by ca 10% of the mean in the transport stage (Fig. 9B), but the distributions, first half of the experiment, which had some mean and median are nearly identical between impact on the distribution of sizes (Fig. 9C). The the first and second half of the experiment change in H was on the order of the instrument (Fig. 9D). Nevertheless, those L data are also resolution during the second half of the experi- removed from further consideration to be consis- ment. The mean and median are ca 5mm tent with the H data. greater in the second half of the experiment The results show that H increased then compared to the first half. It is difficult to deter- decreased with transport stage (Fig. 9A; see mine whether the increase in H was an adjust- Table 4 for mean values). The level of variability, ment towards equilibrium or simply related to quantified by the coefficient of variation (stan- the morphodynamics of a rapidly evolving bed, dard deviation divided by the mean), increases

Table 4. Mean bedform characteristics at different transport stages. Values in brackets are the coefﬁcient of variation (%). BLD, bedload dominated conditions; MXD, mixed-load dominated conditions; SSD, suspended-load dominated conditions.

4 1 hiHc (mm) hiHc =h hiLc (m) hiLc =h hiHc =hiLc hiVbc (10 ;ms )

BLD 448(72) 1/340906 (99) 6000498 (10) 309 (15) MXD 657(91) 1/23126 (130) 8300529 (11) 117 (18) SSD* 388 (17) 1/35206 (220) 15 00196 (27) 455 (36)

*Based on the final 8 h of the experiment due to unsteadiness in the height time series. with transport stage (Table 4). The distribution of sured, rather than select bedforms. There is no = H is more peaked and narrow for the BLD run obvious bias in Fig. 10, but Vbc 0 92 Vbm for = than for the MXD and SSD runs, which have simi- the BLD run and for Vbc 1 13 Vbm the MXD lar shapes (Fig. 9C). Bedform lengths increased run. No comparison between Vbc and Vbm for with transport stage from the BLD to the SSD the SSD run is presented because there was too transport stages (Fig. 9B; Table 4) as did the level much uncertainty in manually tracking bedform of variation (Fig. 9D). Bedform aspect ratios were crests. Given the similarity between Vbc and Vbm typical of dunes (L 20 H) at the BLD and MXD and that the calculated bias is not consistent, no transport stages, but much lower at the SSD trans- corrections are made to Vbc. port stage (L 57 H). Spatially averaged bedform translation rates Figure 11 shows spatially averaged bedform Bedform translation rates migration rates for each run. The time ser- Method comparison ies are stationary, all increasing by <1% of the Figure 10 compares the automated method used mean over the experiment. Migration rate to estimate bedform translation rates (Vbc) with increases with transport stage, as does the level the manual method Vbm. The automated method of variability (Fig. 11A; Table 4). The distribu- < > should be a more reliable estimator of Vbm tions of Vbc go from being narrow and peaked because the whole profile translation is mea- at the BLD stage to very broad for the SSD transport stage (Fig. 11B). This reflects the morphodynamics of the bedforms. Although the time series are stationary, values from the first half of the SSD experiment are removed from further consideration to be consistent with the bedform dimension data.

Bedform-related sediment flux rates Bedform translation flux rates Sediment flux associated with translation is calculated using the Simons et al. (1965) approach: ¼ ð Þ ð Þ qT bb 1 p VbH 10

where p is the porosity of the bed (04) and bb is a bedform shape factor equivalent to: ¼ = ð Þ bb HL Ab 11

where A is the cross-sectional area of the bed- Fig. 10. (A) Bedform translation rates obtained using b form (Venditti et al., 2005). If the bedform is tri- the manual method (Vbm) and the automated method (Vbc) along the channel centreline. BLD, bedload angular, as assumed in the Simons et al. (1965) dominated conditions; MXD, mixed-load dominated derivation, bb = 05. However, values of 056 are conditions. typical of asymmetrical bedforms (Van den Berg,

Fig. 11. Time series of spatially averaged bedform (A) migration rates and (B) histograms for each transport stage. The solid and dashed line in panels (C) and (D) are for the ﬁrst and last 8 h of the SSD run, respectively. SSD, suspended-load dominated conditions; MXD, mixed-load dominated conditions; BLD, bedload dominated conditions.

1987; Ten Brinke et al., 1999; Venditti et al., graphic profiles of bedform topography obtained 2005). Calculation of bb for a subset of the data at separate times. The two topographic profiles indicated that the 056 value is a suitable are lagged by the translation distance and the approximation. average of elevation changes that are indepen- Table 3 summarizes the mean translation dent of translation are summed over the bed fluxes calculated using: (i) Hm and Vbm (qTm); profile as: X Dxg (ii) Hc and Vbc along the channel centreline D ¼ jjPðÞx ð12Þ b 2n (qTa); and (iii) and (). x Because there are no Vbm values for the SSD ðÞ run, Vbc measured along the channel centreline where P x is the at-a-point elevation difference was substituted for that run. Regardless of how between the translated topography divided by the values are calculated, the translation-related the time between the profiles, Dxn is the length flux increased with transport stage. The values of the translated topographic profiles and n is of qTm and qTa, which should be the most clo- the number of points in a profile. As such, Db is sely related sediment fluxes because they are simply mean volume change in topography both measured along the channel centreline, are along the translated topographic profiles. McEl- within ca 10% of one another suggesting that roy & Mohrig (2009) further propose that the the automated method provides estimates of qT deformation flux can be calculated by scaling Db that are similar to the dune tracking method of to a horizontal flux using: Simons et al. (1965). The value of qTa is larger hi us than qTa for all runs which may be reason- qD ¼ ðÞ1 p Db ð13Þ ably expected since the latter is a spatial average ws that includes bedforms that may be slowed due where u is the horizontal sediment velocity and to interactions with the sidewalls. The coeffi- s w is the sediment fall velocity. The ratio u /w cient of variation for the different q values mir- s s s T is the ratio of the distances that a grain travels rors that for V and H, increasing with transport b in each direction during its trajectory. stage, and are smaller for the spatial averages Application of the McElroy & Mohrig method than along the channel centreline. is difficult for the current data set. Equation 13 uses streamwise transects through the bedform Bedform deformation flux rates field and assumes that the sediment involved McElroy & Mohrig (2009) calculate the deforma- in deformation has a trajectory through the tion rate from two detrended alongstream topo-

© 2015 The Authors. Sedimentology © 2015 International Association of Sedimentologists, Sedimentology 18 Jeremy G. Venditti et al. water column. The bedforms in the present forms and aggradation or degradation of the bed. experiment were highly 3D and some apparent Sediment was recirculated in the experiments, deformation occurred due to lateral fluxes and water surface slopes were parallel to the bed changes in bedform spacing, in many cases surface and the observations were made under without suspension. The sediment involved in equilibrium conditions. Therefore, aggradation deformation of the bedforms had much smaller or degradation of the mean bed elevation should trajectories than would be predicted with be negligible and Eq. 14 should provide a rea- Eq. 13, which can lead to overprediction of sonable approximation of deformation flux in deformation flux. Furthermore, deformation flux the bedform field. Calculated in this way, qD should be calculated over the same time scale increases with transport stage (Table 3); but rela- as the translation flux (Ganti et al., 2013), tive to translation, it decreases from the BLD which is integrated over a dune length (Simons transport stage to the MXD stage, then increases et al., 1965). The most appropriate time scale from the MXD to the SSD stage. is the time required for a bedform to migrate one wavelength (ca 48 min for BLD, ca 17 min for MXD and ca 75 min for SSD). The 3D bed- DISCUSSION forms in the present experiments rapidly evolved as they migrated so that, by the time a How do bedform dimensions change with bedform migrated one wavelength, they bore transport stage? no resemblance to the original bedform. As a result, it is difficult to know which two bed- It is conventional in sedimentology to assume forms to match to calculate deformation flux that equilibrium bedform dimensions scale with from transects through the bedform field at the flow depth (H = h/6 and L = 5 h) (cf. Bridge, translation time scale. Calculations of deforma- 2003), yet there is little empirical evidence to tion are further complicated by the level of support a specific scaling ratio between H, L and small-scale variability in the high resolution h (Allen, 1982; Venditti, 2013). Bedforms in the dune profiles caused by superimposed bed- present experiments were larger than the forms and spurs, laterally migrating lobes and expected scaling ratios for all runs (Table 4). saddles, and other sources of small-scale vari- There is more evidence in the literature that ability. These sources of deformation all have bedform geometry scales with transport stage. different time scales. How to remove these Yalin and collaborators (Yalin, 1972; Yalin & smaller scale sources of variability and retain Karahan, 1979a) showed that bedform aspect the deformation signal of the major bedforms is ratios change with transport stage, increasing not clear. Simply filtering this information out from BLD conditions to MXD conditions, then or removing the central part of a probability decreasing for SSD conditions. Data compiled distribution to eliminate small-scale variability by van Rijn (1984, 1993) and Lin & Venditti is arbitrary. One way to ameliorate the latter (2013) support this trend, as do the present two difficulties (rapid 3D evolution and small- observations (Fig. 12A). scale variability) is to calculate deformation The data show that the changes in bedform rate of individual bedforms following Ganti aspect ratio with increasing transport stage, doc- et al. (2013). The problem with this approach umented in previous work (Yalin, 1972; Yalin & is that it is not clear how to turn this indivi- Karahan, 1979a; Lin & Venditti, 2013), are a dual bedform deformation rate into a sediment reflection of the changes in H (Fig. 12B) against flux and the method is impractical for produc- continuously increasing L (Fig. 12C). This sug- ing a time series of data. gests that bedforms should grow with transport As an alternative to Eq. 13, the present stage until they reach their maximum H,at authors approximate the deformation flux as: which point H declines with transport stage. As H approaches zero under suspended dominated qD ¼ qs qT ð14Þ conditions, effectively washing out, L increases until they disappear. However, ‘low-amplitude This limits the authors to examination of the bedwaves’ on upper stage plane beds (Bennett mean fluxes for each experiment. By calculating et al., 1998) suggest that some residual topogra- qD using Eq. 14, it is assumed that all bed mate- phy may remain. rial sediment flux not associated with bedform The observed increase in the variability of H, translation is the result of changes in the bed- L and H/L with transport stage implies that a

Fig. 12. Bedform (A) aspect ratio /, height , (C) length and (D) dimensionless translation rate < > Vbc /ws versus transport stage s*/s*c. The dashed line in (A) is from Yalin (1972). The long dashed line in (D) is Eq. 12 from Lin & Venditti (2013). The short dashed line and dotted lines in (D) are least-squares regression through the scatter and mean values, respectively. The solid vertical line indicates the suspension threshold. BLD, bedload dominated conditions; MXD, mixed-load dominated conditions; SSD, suspended-load dominated conditions. wider distribution of bedform sizes and less uni- needed to ensure that measurements of bedform bedform ﬁelds at higher transport stage forms are indeed representative samples of the would be expected. The bed topography maps full distribution of features on the bed. (Figs 5, 7 and 8) appear to support this idea, as do the distributions of H and L (Fig. 9C and D). How does bedform translation change with Furthermore, there is an intimate linkage transport stage? between the variability in the ﬂow conditions and bedform dimensions. This link suggests that Lin & Venditti (2013) proposed an empirical as transport stage increases, greater care is model of bedform migration in rivers that

© 2015 The Authors. Sedimentology © 2015 International Association of Sedimentologists, Sedimentology 20 Jeremy G. Venditti et al. showed translation rate increasing with trans- may explain the wide scatter in the data from port stage. These authors showed that grain size the literature (Lin & Venditti, 2013). The results exerted a strong control on dune migration rates. suggest that observations of bedform migration Bedforms formed in coarser sediment move fas- rates based on tracking a small population of ter than in finer sediment at the same transport bedforms or a single bedform in a larger bedform stage. Lin & Venditti (2013) reasoned that this is field should be regarded with caution, which is because it is more difficult to move coarser par- especially the case at SSD transport stages ticles into suspension and normalized the bed- where observed migration rates vary by two form translation rate by the particle settling orders of magnitude at constant flow in the velocity (ws) yielding the following relation: experiments. 0:989 Vb ¼ : s ð Þ 0 00127 BCF 15 What is the relation between bedform ws sc translation, deformation and the dominant where BCF = 1316 and is a bias correction fac- transport modes? tor required when a log transformed variable is McElroy & Mohrig (2009) reasoned that transla- retransformed back into its original units. tion-related sediment flux should be equivalent The results support the general form of the to the bedload flux and that the deformation- power law relation between V /w and transport b s related sediment flux should be equivalent to stage. A line fit through all bed surveys matches the suspended-sediment flux in a channel the Lin & Venditti (2013) relation with a slope because deformation is defined as a vertical flux. of 119; however, a line fit to the mean values is The total sediment flux rate, q and q all much steeper (18; Fig. 12D). Nevertheless, the T D increased with transport stage in the experi- observed mean bedform translation rates fall ments, exhibiting power law behaviour within the scatter of the data used to derive the (Fig. 13A), although the specific functional form relation reported in Lin & Venditti (2013), of these relations is difficult to ascertain with although the BLD and MXD runs lie towards the only three data points. lower end of the scatter for a given transport The data do not show q = q or q = q stage and grain size. This result might be reason- bl T ss D (Table 3), but it might be overly ambitious to ably expected because the translation rates expect such a relation, given the inherent diffi- reported here are spatially averaged across the culties in accurately sampling bedload transport channel and most of the data used to derive the and the ad hoc method for calculating deforma- Lin & Venditti (2013) relation were obtained by tion flux. Nevertheless, the pattern in the ratios tracking individual dunes. of bedload to translation flux and deformation to As with the bedform geometry, the level of suspended-load flux are sensible. In the BLD variability in the migration rate increased with run q 05 q , q increased to ca 2 q in the transport stage. Under BLD conditions, most of T bl T bl MXD run, then declined to ca 09 q at the SSD the bedform change between surveys occurred bl transport stage (Table 3). The increase at the in the lee of the bedforms with some cross-chan- MXD transport stage is noteworthy, because it nel change, suggesting orderly downstream bed- suggests that suspension plays an important role form translation with a minor cross-stream in the downstream translation of bedforms in translation. At the MXD stage, orderly down- rivers, as has been previously suggested (Mohrig stream bedform translation still occurs, but there & Smith, 1996; Kostaschuk et al., 2009). The is a wider range of bed elevation change ratio of deformation-related flux to q at the between surveys, there is greater cross-stream ss BLD transport stage was infinity (there is no sus- translation and greater bedform merging and pended-sediment flux). q was 061 q in the splitting. At the SSD transport stage, down- D ss MXD run and 115 q in the SSD run (Table 3). stream translation still occurs, but the bed ss While the results do not directly support the evolves rapidly, cycling through different types contention of McElroy & Mohrig (2009) that of bedform morphology (small dunes, flat bed q = q or q = q , the results are consistent and large dunes). The level of variability in the bl T ss D with the idea that deformation and suspended data cloud for the SSD run is so large that defin- load are intimately linked. ing a single migration rate is only possible by The suggestion by McElroy & Mohrig (2009) examining the central tendencies of data. The that deformation should be equivalent to the high level of variability at high transport stage

cross-stream transport of sediment due to the 3D nature of the bedforms. This lateral translation is obvious in Fig. 7, where elevation changes are not uniform across the channel. It is likely that cross-stream transport is an important component of non-translational flux that needs to be accounted for in assessing deformation flux. As a fraction of the total sediment flux, qT and qD contributed equally to the qs at the BLD transport stage (Fig. 13B). However, as the transport stage increased, the fraction of qs contributed by translation increased from BLD to MXD and the fraction of qs contributed by deformation went down (Fig. 13B). This result is sur- prising, because it would be expected that deformation rate should increase as sediment is suspended. However, the translation rates also increased as the bedforms got bigger and the bedforms moved faster in the MXD run (Fig. 12). At the SSD transport stage, the opposite occurred: deformation flux increased and the translation flux decreased as a fraction of qs.In the limit, it would be expected that qD/qs would approach one and that qT/qs would approach zero as the bedforms washout to a plane bed, which is consistent with the patterns in Fig. 13B.

CONCLUSIONS

1 The typical relation between bedform aspect ratio and transport stage (increasing then decreasing as more sediment is moved into suspension) is a reflection of changes in the bedform height against a background of continuously increasing bedform length. In the limit of washed-out dunes, heights approach zero and lengths approach infinity. Fig. 13. (A) Change in the total (qS), translation- 2 As transport stage increases, the distribution related (qT) and deformation (qD) flux rates with transport stage. (B) The fraction of the total load con- of bedform heights and lengths broadens and the features become less uniform. This suggests tributed by qT and qD. The solid vertical line indicates the suspension threshold. that, as transport stage increases, greater care is needed to ensure that measurements of bedforms are indeed representative samples of the suspended-sediment flux is predicated on the full distribution of features on the bed. idea that there is a vertical exchange of sediment 3 Bedform translation rates generally conform between the bedform and the flow. However, to the power law relations proposed by Lin & these authors also noted that changes in shape, Venditti (2013). size and spacing of the bedforms contributed to 4 As with bedform dimensions, the distribu- deformation flux; these need not be associated tion of bedform translation rates became broader with sediment suspension. In the present experi- with transport stage, to the point where translation ments, there are substantial changes in the bed- rates can only be partly characterized by a mea- form shape, size and spacing that are caused by sure of central tendency.

5 As a fraction of the total sediment flux, trans- Lc, Characteristic bedform length and lation-related and deformation-related fluxes are its spatial average approximately equivalent under bedload domi- Lm Manually measured bedform height nated conditions. As transport stage increases, Lsat Saturation length the fraction of the flux attributed to translation n Number of bed profile elevations q Suspended-sediment flux increases and the fraction attributed to deforma- ss qbl Bedload sediment flux tion declines. However, as the bedforms begin to qs Sum of qbl and qss washout, the opposite happens. In the limit, the qT, qTm, qTa Translation-related sediment flux, fraction of the total load associated with transla- computed with Hm and Vbm, < > < > tion approaches zero and the fraction associated computed with Hc and Vbc with deformation approaches one, confirming qD Deformation-related sediment flux the close association between deformation and S Water surface slope t Time suspended-sediment flux suggested by McElroy u* Shear velocity & Mohrig (2009). U10 °C 10°C-equivalent mean velocity U Mean velocity us Horizontal sediment velocity ACKNOWLEDGEMENTS Vb Bedform translation rate Vbm Bedform translation rate from This work was supported by an NSERC Discov- manual method V , Bedform translation rate from ery Grant to JV. Robert Humphries and Natalia bc bc automated method and its Domarad helped undertake the experiments. We spatial average would like to thank Brandon McElroy for pro- â w Channel width viding MATLAB codes for automated calcula- ws Settling velocity of sediment tion of bedform heights and lengths. W(l) Elevation variance of bed profile Constructive reviews by David Mohrig, Vamsi length l Ganti and an anonymous reviewer greatly x Distance downstream z Elevation above the bed improved the manuscript. ML and JV designed bb Bedform shape factor the experiments; ML conducted the experiments b Coefficient in Rouse equation that and processed the data with assistance from describes the difference in sediment MK. JV wrote the paper and did the data analy- and fluid diffusivity sis presented herein. db Rate of vertical bed profile elevation change g Bed elevation j Von Karman constant NOTATION PðÞx Rate of ‘at a point’ elevation a Reference elevation above the bed difference between the translated in Rouse equation topography profiles C Concentration of qs and qw Sediment and water densities suspended-sediment at height s0 Shear stress from depth-slope product z above the sediment bed s Wall corrected shear stress Ca A reference concentration measured s*, s*c Shields number and its critical at bed elevation a in Rouse equation value for sediment entrainment C Depth averaged concentration s*/s*c Transport stage D Grain size Dxn Length of translated topographic D10 °C 10°C-equivalent grain size profiles D50 Median grain size Db Mean volume change in topography along translated topographic profiles g Gravitational acceleration h Flow depth REFERENCES H Bedform height Hm Manually measured bedform length < > Allen, J.R.L. (1970) A quantitative model of climbing ripples Hc, Hc Characteristic bedform height and and their cross-laminated deposits. Sedimentology, 14, its spatial average 5–26. Hsat Saturation height Allen, J.R.L. (1973) Features of cross-stratified units due to l Bed profile length random and other changes in bed forms. Sedimentology, L Bedform length 20, 189–202.

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