Geometry and kinematics of dunes during steady and unsteady flows in the Calamus River, Nebraska,...

Sedimentology (1993) 40,237-269

Geometry and kinematics of dunes during steady and unsteady flows in the Calamus River, Nebraska, USA

S H A R O N L. G A B E L *

Department of Geological Sciences, State University of New York Binghamton, N Y 13902-6000, USA

ABSTRACT

The geometry and kinematics of river dunes were studied in a reach of the Calamus River, Nebraska. During day-long surveys, dune height, length, steepness, migration rate, creation and destruction were measured concurrently with bedload transport rate, flow depth, flow velocity and bed shear stress. Within a survey, individual dune heights, lengths and migration rates were highly variable, associated with their three-dimensional geometry and changes in their shape through time. Notwithstanding this variability, there were discernible changes in mean dune height, length and migration rate in response to changing discharge over several days. Changes in mean dune height and length lagged only slightly behind changes in discharge. Therefore, during periods of both steady and unsteady flow, mean dune lengths were quite close to equilibrium values predicted by theoretical models. Mean dune steepnesses were also similar to predicted equilibrium values, except during high, falling flows when the steepness was above that predicted.

Variations in mean dune height and length with discharge are similar to those predicted by theory under conditions of low mean dune excursion and discharge variation with a short high water period and long low water period. However, the calculated rates of change of height of individual dunes vary considerably from those measured. Rates of dune creation and destruction were unrelated to discharge variations, contrary to previous results. Instead, creations and destructions were apparently the result of local variations in bed shear stress and sediment transport rate.

Observed changes in dune height during unsteady flows agree with theory fairly well at low bed shear stresses, but not at higher bed shear stresses when suspended sediment transport is significant.

INTRODUCTION

Dunes are the most common bed configuration in alluvial channels. Understanding the interaction between water flow, sediment transport, dune geometry and migration is fundamental to predicting flow resistance and sediment transport rates in alluvial channels and in linking the characteristics of cross- stratification produced by dune migration with the flows that created them. Much remains unknown about these interactions, particularly in natural flows, which are unsteady and non-uniform. Thus, concur-

*Present address : Department of Earth Sciences, State University of New York, College at Oswego, Oswego, NY 13126, USA.

rent measurements of dune geometry and migration, flow characteristics and sediment transport rates were made during steady and unsteady flows in a reach of the Calamus River, Nebraska, USA. Measurements weremade from bridges spanning the channel, thereby enhancing measurement accuracy and minimizing interference with the flow. A primary objective was to test models for dune geometry and migration in steady and unsteady, non-uniform flows. In this paper, observations of dune geometry and kinematics are presented as well as measurements of the flow parameters needed to test models for dune geometry in steady and unsteady flows. Further details of flow and sediment transport characteristics in the study

237

238 S. L. Gabel

area are presented elsewhere (Gabel, 1991 ; Bridge & Gabel, 1992).

B A C K G R O U N D

Dunes in steady, uniform flows

The main approaches to the study of dune stability and geometry in steady, uniform flows are linear stability analyses, theoretical models and empirical or semi-empirical models. Linear stability analyses have been used to determine hydraulic stability criteria for dune existence (reviews in Raudkivi, 1972; Engelund & Fredsere, 1982a). Flume studies have been used as well to determine the hydraulic stability limits of dunes (reviews in Allen, 1984; Middleton & Southard, 1984; Southard & Boguchwal, 1990). Empirical and theoretical studies have been used to determine dune dimensions as a function of flow parameters. In particular, dune length has been related to flow depth (e.g. Yalin, 1977) and dune height has been linked to bed shear stress (Fredsere, 1982; Allen, 1984). How- ever, such correlations from field data are commonly dominated by scatter (e.g. Jackson, 1976; Bridge & Jarvis, 1982; Terwindt & Brouwer, 1986). This lack of correlation is commonly attributed to flow unsteadiness and the influence of channel bends on the structure of macroturbulence (e.g. Bridge & Jarvis, 1982).

Dunes in unsteady, non-uniform flows

Flow in natural rivers is unsteady and non-uniform. Field studies have shown that during unsteady flows, dune geometry commonly changes more slowly than, or lags behind, river discharge (Znamenskaya, 1966; Nasner, 1978; Van Urk, 1982). Heights and lengths of dunes are commonly smaller during rising discharges and larger during falling flows, giving rise to loop shaped curves when dune height or length is plotted against discharge (Allen, 1973). Associated changes in hydraulic roughness may result in loop shaped stage discharge curves, which have important implications for predicting river stage during floods.

Allen (1976a, c, d, 1978b) modelled changes in mean height and length for a dune population under periodically varying discharge. Population characteristics were modified by changes in the geometry of individual dunes, and by the creation and destruction of dunes. The length of each dune was assumed constant through time, so that changes in mean dune

length for the population were controlled solely by dune creation and destruction. Heights of individual dunes were allowed to change at a rate proportional to the difference between the dune’s actual height and a theoretical value for the height of a dune in equilibrium with instantaneous flow conditions. The proportionality was assumed to be a constant called the ‘coefficient of change’. Dune creation and destruction were modelled by assigning an ‘excursion’, the distance a dune migrates during its lifetime (divided by its length), to each dune at the time of its creation. Additional simplifying assumptions appeared in Al- len’s model, such as a constant friction coefficient, even though dune geometry and discharge were changing. Allen presented results from his model as plots of population mean dune height or length against discharge, which yield characteristically loop shaped curves. The size and orientation of loops were used to quantify the degree of lag between dune geometry and discharge. Allen suggested that rates of dune creation and destruction were related to discharge and rate of change of discharge, and that dune creation rates influenced the variance in dune geometry within a population.

Fredsere’s (1979) theoretical model predicts the initial change in height of an individual dune in response to an abrupt change in discharge. The change in height depends on the change in bed shear stress and bedload transport rate at the dune crest.

Few experimental data are available for evaluating either Fredsere’s or Allen’s models. In flume experiments, Gee (1975) measured changes in flow depth due to changing dune dimensions in response to changing discharge. Fredsere (1979) compared Gee’s results to his model, but doing so required making assumptions about the effect of dune geometry on hydraulic roughness. Wijbenga & Klaassen (1983) also conducted flume studies in which water discharge was changed abruptly. They found that Fredsere’s model predicted changes in dune height for small changes in discharge more accurately than for larger discharge changes, and that results from the model were greatly affected by the sediment transport function used.

Wijbenga & Klaassen also showed that the rate of change of height of an individual dune is not a linear function of the difference between the individual’s height and the equilibrium height, in contrast to Allen’s model. Allen (1978a) used field data collected by Nasner (1974, 1978) in qualitative comparisons with phase diagrams generated from his model. Other than Wijbenga & Klaassen’s (1983) indirect measure-

Dunes in steady and unsteady flows, Nebraska 239

ments of dune lifespans in their experiments, no experimental studies of dune lifespan, excursion, or rates of creation and destruction have been made. However, this study yielded data on dune creation and destruction rates that are needed for evaluating significant aspects of Allen’s model.

STUDY LOCATION AND METHODS

Study location

The study reach is on the Calamus River, which flows within the Sandhills Region of Nebraska, USA (Fig. 1). Measurements were made at three locations around a mid-stream island, each of which has different degrees of flow non-uniformity (Fig. 2). These are referred to as the right channel (as viewed in the downstream direction), left channel and downstream study areas.

Bedform geometry

A depth sounder capable of resolving changes in bed elevation as small as 0.001 m in height (Dingler et al., 1977) was used to measure bedform geometry. The depth sounder was moved in the streamwise direction along railways suspended beneath the bridges (Fig. 3). The railways could move laterally across the channel.

The length and lateral spacings of sounding lines in the three locations are shown on Fig. 2. Bed profiles were measured along each line at least eight times per day at roughly 1-h intervals in order to document migration and changes in shapesof dunes. In addition, visual observations and sketches of dunes were made at least twice daily during low flow surveys, although increased suspended sediment transport during high flows made visual observation impossible.

Bed profiles were drawn on a chart recorder in the field and were later digitized and replotted at a fixed horizontal scale (Fig. 4). Dunes were identified as bedforms roughly triangular in cross-section and greater than 0.60 m in wavelength (see Allen, 1984). The height of each dune was measured as the elevation difference between the lowest point in the trough downstream of the avalanche face and the highest point on the dune. Dune steepness is the ratio of height to length.

Individual dunes could almost always be recognized on successive profiles taken throughout a day, thereby allowing calculation of their migration rates, changes in shape through time and minimum values of excursion and lifespan. Two-dimensional bed profiles have limitations for describing the geometry of dunes because dunes are three-dimensional features. Sinuous crested dunes with the same spatially averaged lengths and heights will show lateral variations in length and height when viewed in streamwise cross-sections (see

240 S. L. Gabel

.- - 2

I I I I

/

0 20

METRES -

Fig. 2. Map of study reach, made at the end of the April-May 1985 field season. Contours represent bed elevation with respect to survey datum; contour interval is 0.25 m. Bankfull level is approximately - 1.5 m. MN indicates magnetic north. Thick lines crossing the channel represent measurement platforms. Study areas are outlined, with depth sounding lines marked, and velocity profiling and bedload transport sampling stations on measurement platforms are indicated by dots. Study areas are: (A) right channel area; (B) left channel area; (C) downstream area. Thick arrow indicates flow direction. Cut banks are indicated by heavy, solid lines, and sand-vegetation boundaries are dashed lines.

Fig. 5). Three-dimensional mapping of dune crestlines from depth soundings was attempted, but uncertainty about lateral continuity of crestlines made identifying individual dunes on adjacent sounding lines subjective and unreliable. Visual observations of dunes a t low flow showed that continuity of crestlines across the channel was variable and ranged between approximately 1 m to more than 7 m at any given time. Crestlines did not always maintain their shape as a dune migrated. Sediment transport was more vigorous on certain parts of the dune crest, and the crest advanced faster in those areas. Therefore, variances in dune shape and migration rate will reflect the cross- stream and temporal variability of individual dunes as well as variations between different dunes.

Dune creations and destructions were also identified by comparing successive bed profiles from each sounding line. Four types of creation or destruction were observed (Fig. 6). Dune splitting, defined as the appearance of two or more dunes in the space occupied by one dune in previous records from a sounding line, resulted in the destruction of one dune and two (or more) dune creations (Fig. 6c, d). Dunes often combined, with a single dune appearing where two or more dunes had been (Fig. 6a, b). Combination commonly occurred by a faster moving dune progressively overtaking a slower moving dune. There were many instances when one dune migrated on to the back of another dune, but the dunes were not

considered ‘combined’ until only one discrete avalanche face could be recognized. This type of dune superposition is different from that described in other field studies (e.g. Coleman, 1969; Allen & Collinson, 1974; Jackson, 1976), where distinctly smaller dunes persist on the backs of larger dunes. When dunes combined, a single new dune was created, and the dunes which combined were counted as destroyed. Dunes which disappeared from records not as the result of splitting or combination, but simply became smaller and smaller on successive profiles, were considered to have been destroyed due to dying out (Fig. 6e). Finally, ‘spontaneous’ dune creations occurred when a small dune appeared on a profile in spaces between existing dunes (Fig. 6f).

The three-dimensional geometry of dunes presented difficulties in unequivocally recognizing dune creations and destructions from the bed profiles. What appeared to be a dune creation may have been the result of the sounding line temporarily intersecting the edge of a dune laterally adjacent to the line. Con- versely, disappearance of dunes may have resulted from the edge of a dune no longer intersecting the sounding line. In order to minimize these errors it was required that new dunes must migrate a t least one wavelength, and that destroyed dunes must not reappear on the next successive sounding record.

Rates of dune creation and destruction were calculated for each survey as the number of dunes

Dunes in steady and unsteadypows, Nebraska 241

Fig. 3. Photographs of depth sounding railways: (a) 23 m long railway used in the right channel; (b) 14.25 m long railway used in the left channel and downstream area.

created or destroyed divided by the time elapsed during each survey. Because the length of sounding lines differed between areas, and because fewer dunes were surveyed during high flows than in low flows (during high flows dunes were longer, so fewer fit on each sounding line), more creations and destructions would be observed on the longer sounding lines a t low flows. Therefore, creation and destruction rates were normalized by the total number of dunes on all sounding lines in the survey. For example, on 15 April

1985, the average dune length was 2.18m and the length of the sounding line was 23 m. Therefore about 10.5 dunes were visible on each of five sounding lines, so creation and destruction rates were divided by 52.75 (23 x 5/2.18).

Flow velocity measurements

Flow velocity measurements were made from the bridges midway along each depth-sounding line in the

242 S . L. Gabet

II 25

I253

14 33 - 15 44

< 17 09

H 0.2 m I

2 m

Fig. 4. Bed profiles taken along one sounding line during one survey. Also shown is an example of dune height and length measurements, and times when the profiles were made. Flow is from left to right.

B’

4 4 B A

Fig. 5. Sketch of variation in cross-sectional geometry of an individual dune due to three-dimensional plan geometry. Plan view of dunes is modified from Rubin (1987). Cross-sections A-A’ and B-B’ show how cross-sectional geometry of individual dunes varies laterally. Note difference in dune length (especially of middle dune) due to sinuous crestlines being out of phase.

Dunes in steady and unsteadyflows, Nebraska 243

Fig. 6. Examples of dune creation and destruction mechanisms. In all diagrams, flow is from left to right, horizontal scales represent 2 m, and vertical scales represent 0.2 m. (a) Dunes 3 and 4 combine (are destroyed) to form N 1. (b) Dunes 2 and 3 combitle to create N1. (c) Dune 4 splits to create N1 and N2; N3 is a new (spontaneously created) dune. (d) Dune 4 splits, creating N1 and N2; in the final profile, humpback dune 5 splits, creating dunes 3 and 4. (e ) Dune 4 becomes progressively smaller and dies out. (f) Spontaneous creation of dune NI, which first appears on the third profile.

left-hand channel and downstream of the island, and on the downstream bridge in the right-hand channel (Fig. 2). Measurements were repeated at least five times each day at each station in order to obtain many samples from different locations on dunes. A Marsh- McBirney electromagnetic current meter, Model 532M, with a sensing probe 12.7 mm in diameter, gave the velocity in two orthogonal directions in the horizontal plane. The sampling rate was 0.2 s- * for each direction. At each station, minute-long velocity measurements were made at 0~02,0~03,0~05,0~07 and 0.10 m above the bed, plus readings near the mean flow depth and the water surface. Resultant velocity at each height above the bed, v, is given by v = ,/ui + ut. in which u, and u, are the time averages of readings from the downstream and cross-stream probe axes, respectively. The mean downstream direction for each profile was taken as the direction of the resultant at 0.37 times the flow depth, d (see also Bathurst et al., 1977, 1979; Bridge & Gabel, 1992). Whenever possible, locations of velocity measurements on individual dunes were observed directly.

Depth sounding records were also used to estimate the position of the current meter relative to individual dunes (i.e. trough, crest, upper, middle or lower dune back).

Plots of the component of velocity in the mean downstream direction, u, against the logarithm of height above the bed were used to calculate local values of bed shear stress, qoc=pu2*, by applying the Law of the Wall,

(1) -- U l Y - -ln-, u* K Yo

to measurements in the lowest 15% of the flow depth. In Eq. (I) , u is the velocity at height y above the bed, yo is the height where u=O, K is von Karman’s ‘constant’ and u* is the shear velocity. The Law of the Wall holds only for profiles that are semi-logarithmic in the lowest 10-20% of the flow depth. Profiles measured in dune troughs, or with errant data (caused by the current meter sinking into the bed or becoming fouled by vegetation) were not semi-logarithmic, and therefore were not used for calculating bed shear

244 S. L. Gabel

stress. The clear-water value of K is 0.4. In this study, effects of suspended sediment on velocity profiles were accounted for by calculating an apparent K (denoted K'), following the methods of Itakura & Kishi (1980) and Adams & Weatherly (1981a,b). Data from this study indicate that K' values were roughly 0.36 during low flows and 0.34 during high flows (Gabel, 1991).

Water surface topography and spatially averaged bed shear stress

Water surface elevations were measured throughout the study reach during each survey. A point gauge with attached level was used to measure the distance to the water surface from fixed points on bridges and banks. Elevations of the fixed points relative to a survey datum were measured using a n automatic level (details in Bridge & Gabel, 1992). This technique yielded an accuracy of f 3 mm (level can be read to 1 mm; error in the water surface measurements was about 2 mm) for each water surface elevation measurement. Because water surface elevations were measured mainly from the bridges, the best estimates of water surface slopes are those in the right channel where the sounding lines extend between two bridges. These are the only water surface slopes used in subsequent calculations of bed shear stress.

Spatially averaged values of bed shear stress (tJ

were calculated from

z,, = pgdX (2) where g is the acceleration due to gravity, S is water surface slope and d is mean flow depth. Equation (2) is a simplified version of the conservation of momen- tum equation for steady flows and ignores convective accelerations due to channel curvature, flow diver- gence and convergence. Because the right channel is nearly straight and downstream changes in 0 (local vertically averaged velocity) and d are very small (Bridge & Gabel, 1992), and because flow velocity changes slowly through time, this simplified version is considered adequate.

Sediment transport

Bedload transport rate was measured using a standard, cable suspended Helley-Smith sampler with 76.2 x 76.2 mm opening and 0.125 m m mesh bag. Samples were taken for 60 s at stations 0.25 m to the side of velocity measurement stations (to minimise disruption of the bed). During each survey 35-40 samples were taken, roughly seven or eight from each station. All

samples were dried, weighed, and sieved a t whole phi intervals, and the mean grain size was calculated.

Survey averaged values for bedload transport rate were also estimated using dune migration rates and heights. If dunes are triangular in cross-section, bedload transport in terms of dry mass per unit width and time is

Q b = 0.5 (1 - L) D H U ~ , (3)

where ,I is sediment porosity, D is sediment density, H is mean dune height and Ub is mean dune migration rate. This equation has been used in many studies of bedload transport dunes (Simons ef a/., 1965; Boku- niewicz er a/., 1977; Dietrich & Smith, 1984; Rubin, 1987). Factors ranging from about 0.33 to 1 have been used instead of 0.5 in Eq. (3), depending on consider- ations regarding dune shape, the influence of suspended sediment settling on dune avalanche faces and the size of the wake region in the lee of dunes (Engel & Lau, 1980; Yang, 1986; Van den Berg, 1987). The shape factor in Eq. (3) was determined to be suitable for this study based on comparisons between calculated rates and those measured using the Helley- Smith sampler (Gabel, 1991).

RESULTS

Discharge variations

The hydrographs in Fig. 7 show discharge variations for the undivided channel during the study periods. Measurements for April and May 1984 were made in the downstream study area during falling flows (Fig. 7a). Measurements were made in the right channel during low, steady and high, falling flows in April and May 1985 (Fig. 7b). Also in April and May 1985, four surveys were made in the left channel. Three low flow surveys were made in August 1985 in the right channel (Fig. 7c). April 1986 measurements, also made in the right channel, cover high and falling stages of a discharge peak (Fig. 7d). Dates of measurement surveys and flow conditions are listed in Table 1. Discharges listed in Table 1 were computed from mean flow depth, velocity and study section width, in contrast to those in Fig. 7.

Dune morphology and kinematics

Migrating dunes covered the bed in all study areas during all measurement surveys. During low flows, when dunes were visible, their crestlines were sinuous

Dunes in steady and unsteadyjows, Nebraska 245

1

I MAR APR MAY

I JUL AUG MAR APR

Fig. 7. Hydrographs for field seasons. Data are from US Geological Survey gauging station at Harrop, NE, and have been adjusted to reflect discharge at the study site using data from Bridge & Gabel (1991). (a) Data from 1984 study period with measurement surveys (all made in the downstream area) denoted by + . (b) Spring 1985 study period showing right channel surveys (0) and left channel surveys (*). (c) Summer 1985 surveys from right channel (a). (d) Spring 1986 surveys from right channel (A). These symbols will be used throughout this paper to distinguish between different study periods and areas.

and oriented perpendicular to flow, and linear spurs jutted downstream from avalanche faces. The water surface was relatively tranquil, with small, weak boils occasionally breaking the surface. Boils were larger and erupted more vigorously during high flow stages, indicating that dune crestlines were more strongly curved than during low flows. Ripple backed dunes were frequently observed during the lowest flow stages, but were less common during slightly higher flow stages (ripple backed dunes could be recognized on some depth-sounding records).

The mean values for dune height, length, steepness and migration rate, as well as variances and coefficients of variation, for each survey are listed in Table 2. Figure 8 shows how the values for dune height, length, steepness and migration rate varied with discharge during the 1985 and 1986 surveys. Mean dune height and length increase with discharge from about 0.1 m and just over 2 m, respectively, during low flows, to roughly twice those values during high flow stages. In April and May 1985, mean height and length were observed from low, steady discharges,

over a small discharge peak (23-24 April 1985), and during a larger discharge peak (26 April through to 8 May 1985; see Fig. 7). For the large discharge peak, an anticlockwise loop can be traced for the variation in both height and length with discharge. This pattern suggests that mean dune height and length were slightly larger during falling flows compared with rising flows.

Mean dune steepness varied over a fairly narrow range, from 0.050 to 0.065 (Table 2). Scatter in these values is considerable, particularly during low, falling flows. Data obtained during spring 1985 and 1986 show that maximum dune steepness occurred during high, falling flows 1-2 days following the peak discharge.

Mean dune migration rates ranged between 0.8 and 1.56 m h- '. These survey averaged values generally increase with discharge, but data are widely scattered.

Table 1. Measurement surveys and flow conditions.

Date Discharges Flow conditions (m3 s-')

Right-hand channel 15 April 1985 20 April 1985 21 April 1985 23 April 1985 24 April 1985 29 April 1985 I May 1985 7 May 1985 8 May 1985

12 August 1985 13 August 1985 16 August 1985

5 April 1986 6 April 1986 7 April 1986 9 April 1986 10 April 1986 11 April 1986

1.14 1.09 1.02 1.33 1.18 1.73 1.53 1.08 1.10

0.95 0.82 0.85

1.62 1.66 1.59 1.30 1.17 1.01

Left-hand channel 14 April 1985 1.29 27 April 1985 1.99 28 April 1985 1.91 2 May 1985 1.64

Downstream from island 19 April 1984 2.03 12 May 1984 1.74 13 May 1984 1.68 15 May 1984 1.63 18 May 1984 1.50

Low, steady Low, steady Low, steady

High, unsteady Low, falling High, steady High, falling Low, falling Low, steady

Low, falling Low, steady Low, steady

High, rising High, steady High, steady High, falling Low, falling Low, steady

Low, steady High, rising High, steady High, falling

High, falling Low, falling Low, steady Low, steady Low, steady

246 S . L. Gabel

Table 2. Dune geometry and migration.

Dune migration rate Dtine height (m) Dune length (m) Dune steepness (m h-I)

Survey date R u2 cv 8 u2 cv K u2 cv K u2 cv Right-hand channel

15 April 1985 0.114 20 April 1985 0.097 21 April 1985 0.108 23 April 1985 0.130 24 April 1985 0.109 29 April 1985 0.184 1 May 1985 0.188 7 May 1985 0.115 8 May 1985 0.121

12 August 1985 13 August 1985 16 August 1985


0.127 0.114 0.1 17

0.191 0.195 0.194 0.165 0.109 0.119

Left-hand channel 14April1985 0.100 27 April 1985 0.174

2 May 1985 0.134 28 April 1985 0.178

19 April 1984 0.163 12 May 1984 0.179 13 May 1984 0.187 15 May 1984 0.159 18 May 1984 0.127

0.223 0.185 0.202 0.330 0.254 0.392 0.580 0.248 0.241

0.206 0.253 0.288

0.717 1.059 1.072 0.45 1 0.237 0.272

0.41 2.18 0.48 0.44 2.02 1.87 0.42 2.08 0.57 0.44 2.74 1.05 0.46 2.09 0.64 0.34 3.19 0.94 0.40 3.24 1.64 0.43 2.34 0.81 0.40 2.18 0.56

0.36 2.57 0.85 0.44 2,27 0.59 0.46 2.28 0.95

0.44 3.74 1.56 0.53 3.64 2.21 0.53 4.05 2.92 0.41 3.00 1.48 0.45 2.51 1.16 0.44 2.38 0.82

0.32 0.30 0.36 0.37 0.38 0.30 0.39 0.39 0.34

0.36 0.34 0.42

0.33 0.41 0.42 0.40 0.43 0.38

0.185 0.43 2.00 0.55 0.37 0.498 0.41 3.34 2.11 0.44 0.539 0.41 3.05 0.91 0.31 0.368 0.45 2.45 0.81 0.37

0.701 0.51 3.02 2.77 0.55 0.688 0.46 3.26 2.91 0.52 0.851 0.49 3.06 1.60 0.41 0.515 0.45 2.82 1.17 0.38 0.403 0.50 2.37 0.86 0.39

0.056 0.054 0.057 0.054 0.058 0.062 0.065 0.055 0.059

0.00064 0.00072 0.00071 0.00098 0.00098 0.00056 0~00110 0.00068 0.00070

0.44 0.50 0.47 0.57 0.53 0.39 0.51 0.47 0.44

0.056 0.00060 0.44 0.056 0.00070 0.48 0.057 0.00057 0.43

0.055 0.00064 0.45 0.059 0.00084 0.50 0.059 0.00087 0.49 0.062 0.00085 0.47 0.050 0.00075 0.54 0.056 0.00076 0.49

0.052 0.00051 0.41 0.056 0.00078 0.51 0.061 0.00055 0.38 0.057 0.00081 0.49

0.058 0.00075 0.47 0.062 0.00093 0.48 0,064 0~00110 0.52 0.060 0.00064 0.42 0.057 0.00079 0.49

1.08 0.41 0.59 0.98 0.57 0.76 1.19 0.33 0.52 1.21 0.77 0.73 1.22 1.06 0.84 1.31 0.62 0.60 1.16 0.37 0.53 0.93 0.33 0.57 1.01 0.34 0.57

0.80 0.22 0.55 0.99 0.27 0.51 1.27 0.57 0.60

1.39 1.13 0.46 1.56 0.95 0.62 1.33 0.97 0.72 1.18 0.62 0.66 0.98 0.48 0.70 1.27 0.80 0.70

1.04 0.40 0.61 1.10 0.97 0.89 1.38 0.79 0.64 1.35 0.69 0.61

1.06 0.90 0.89 0.99 0.81 0.91 1.06 0.78 0.83 0.81 0.59 0.73 1.05 0.55 0.70

The April-May 1985 surveys from the right channel study area show a clockwise variation in mean migration rates with discharges (Fig. S), indicating that migration rates were higher during rising flows than during falling flows following the discharge peak. This variation is expected if bedload transport rates are similar during rising and falling flows, because U, E Qb/H (Eq. 3). If dunes are smaller during rising flows, migration rates will be large, whereas during falling flows, transporting sediment at the same rate in larger dunes will result in slower migration rates.

Between study areas, only minor differences were observed in the mean values for dune geometry and migration. Any differences in dune geometry and migration between areas are smaller than the variability in mean values measured within the right channel area over the three field seasons during similar flow conditions.

The variance for dune height and length within a survey is very large and generally increases with discharge (Table 2). Coefficients of variation for height, length and steepness ranged between 30 and 60%, comparable with other field studies (e.g. Kapi- tonov, 1979). Dune migration rates had coefficients of variation ranging between about 50 and 90%, which also increased with discharge. It is difficult to determine how much of the variability in heights, lengths and migration rates of individual dunes was related to spatial and temporal variations in local flow conditions as opposed to temporal changes in overall flow conditions. The increase in variability with discharge may be related to the increasing sinuosity of dune crests, or to some aspect of the rates of change or destruction-creation of individuals.

Heights of individual dunes are plotted against time in Fig 9 for each survey in the 1986 field season. For


2 0 1 2

f

a

.... E - c 0.16 r m

0.08 -t

In In W

a W In c

W

3 0

c -f F "75 1.50 1 + 4 -1

1 1 - - I I I I

D~scharqe, 0 (m3 5-'1

Fig. 8. Plots of mean dune height versus discharge (top row), mean dune length versus discharge (second row), mean dune steepness versus discharge (third row) and mean dune migration rate (U,) versus discharge (bottom row). In each row, the left- hand diagram shows data from spring 1986. The middle diagram shows data from spring and summer 1985 from the right channel. Data from both the left channel and the downstream area are plotted on the right-hand diagram in each row. On all diagrams, lines connect consecutive surveys. Error bars are shown at lower right on each graph and represent the largest standard deviation of the mean measured from the data plotted on that graph.

clarity, only heights of dunes from one or two of the five sounding lines were graphed. Individual dune heights vary greatly through time, up to 0.30 m over periods of a few hours, with some increasing, some decreasing and others fluctuating about some average value. The average height did not change much during a survey (with the exception of an increase in average height on 6 April), although average height varied

between surveys (Fig. 8). Similarly, there is no general trend toward increasing or decreasing length within a survey (Fig. 10). The average length of dunes from all sounding lines varies little throughout each day, except on 9 April when average length decreased during a falling flow period. Temporal fluctuations in migration rates of individual dunes (Fig. 11) are much larger than for height or length changes. Also,

248 S . L. Gabel

5 APRIL 86 0.5

0.4 1 0.3 7

8 10 12 14 16 18 6 APRIL 86

0.5 7

0.4

0.3

0.2

0.1

0.0

0.5

0.4

0.3

0.2

- 0.1

2 0.0 E - 5. 0) .-

0.3 c

32 34 36 38 40 42 7 APRIL 86

7

0.0 104 106 108 110 112 114

10 APRIL 86 0.3 7

0.1

0.0 128 130 132 134 136 136

1 1 APRIL 86

0.1 o ' 2 ] 3 0.0 -,

152 154 156 158 160 162

5 APRIL 86

--1. 8 10 12 14 8 APRIL 18 18 86

k 32 34 36 38 7 APRIL 40 42 86

56 58 €4 8%

104 k 106 108 110 10 APRIL 112 114 86

128 L 130 132 134 1 1 APRIL 136 138 86

I

152 c 154 156 158 160 162

Time ( h )

fluctuations in mean migration rate with time obscure any trend in mean migration rate within a survey.

Although mean dune height, length and migration rate commonly did not show clear increasing or decreasing trends within a survey, there is clearly a gradual decrease of mean height and length corre- sponding to falling discharge (Fig. 12). Figure 12 shows that there was a small delay (1-2 days) between the time when discharge began to fall and times when mean dune height and mean dune length began to decline for both the 1985 and 1986 study periods. Systematic changes in mean migration rate in response to changes in discharge are less clear, and the variability in average migration rate within a single day is considerably greater than the overall change in mean migration rate.

In addition to changes in the geometry of individual dunes, a dune population can change by the creation and destruction of individuals. Rates of creation and destruction are somewhat variable for a given discharge, and creation and destruction rates do not appear to vary with discharge (Fig. 13) or rate of change of discharge (Fig. 14). Likewise, the rates of occurrence for each mechanism of creation and destruction did not depend on discharge or rate of change of discharge. For 75% of the surveys, creation rates were approximately equal to destruction rates.

Table 3 shows that, of the four mechanisms by which dunes were created and destroyed (Fig. 6), splitting and combining were the most common. Owing to temporal gaps between bed profiles and uncertainties in identifying creations and destructions, it is difficult to generalize about how splitting, combining, dying out and spontaneous creation occurred. For example, dunes commonly combined by one dune overtaking another. Some of the dunes that split were humpback dunes (Fig. 6c, dune 5 in 6d). The split may have resulted from flow separation and localized, intense erosion on the dune back downstream of the hump, as described by Saunderson & Lockett (1983). Some dunes appeared to have died out because they became trapped in the flow separation zone of a larger dune, and were starved of sediment.

Fig. 9. Diagrams on the left show the variation in heights of individual dunes through time for each survey of the 1986 study period. For clarity only dunes from one or two of the five sounding lines are shown. On the right-hand side, the average dune heights for dunes from all sounding lines are plotted against time. Zero hours corresponds to midnight, 5 April 1986.


5 APRIL 86

8'o 4 6.0

2.0

0.0 8 10 12 14 16 18

6 APRIL 86

6.0

4 . 0 ] 2.0 9 0.0 -,

32 34 36 38 40 42 7 APRIL 86

6.0 1

104 106 108 110 112 114 10 APRIL 86

6.0 7

2.0

4'0 I 0.0 +, 6.0 1

128 130 132 134 136 138 1 1 APRIL 86

1

0.0 -, 152 154 156 158 160 162

5 APRIL 86

k 8 10 12 14 6 APRIL 16 18 86

k 32 34 36 38 7 APRIL 40 42 86

56 58 60 62 64 66 9 APRIL 86

k 104 106 108 110 10 APR86 112 114

1:-::: 128 130 132 134 11 APRIL 136 138 86

152 i_: 154 156 158 1 6 0 162

Time (h)

These occurrences suggest that dune creations and destructions are related to localized, unsteady sediment transport conditions.

Lifespans and excursions of individual dunes proved difficult to measure. For all but newly created dunes, the time of creation of the dune was unknown. Time limitations and finite rail length made it possible to estimate the lifespans of only short lived dunes. Many dunes observed on the first sounding run of a day were still migrating downstream a t the end of the survey, 7-10h later. The surveys with the best potential for measuring dune lifespan and excursion were conducted on 7 and 8 May 1985, on the 23 m long rail. Even though these surveys were more than 10 h in duration, information on dune lifespan and excursions was still limited. Rates of dune creation and destruction were high for both surveys, with 35- 50% of all dunes destroyed during the 10-1 1 h period of observation. The rest of the dunes continued to migrate downstream. Lifespans greater than 10 h were common, during which time dunes migrated 10-15 m, giving excursions between 4.5 and 7 dune lengths. These values represent lower limits and indicate that long sounding lines and round the clock observations are needed to measure dune lifespans.

Flow depth and velocity

Mean flow depths ranged from 0.34 to 0.61 m during this study, and mean velocity (Dave) for each survey ranged between 0.51 and 0.77 m s- ' (Table 4). Depths were about the same for the three study areas, but the area downstream from the island had slightly lower mean velocities than the channels on either side of the island. Depths during rising and peak flows in the 1986 surveys were slightly lower than during falling flows, and velocities were concurrently a bit higher. However, well-defined loops due to dune lag are not evident in the depth versus discharge and velocity versus discharge curves (Fig. 15).

Individual flow depth and velocity measurements differ depending on where the measurement was taken on a dune (i.e. trough, crest or back). The variation in

Fig. 10. Diagrams on the left show the variation in lengths of individual dunes through time for each survey of the 1986 study period. For clarity only dunes from one or two of the five sounding lines are shown. On the right-hand side, the average dune lengths for dunes from all sounding lines are plotted against time. Zero hours corresponds to midnight, 5 April 1986.

250 S . L. Gabel

*1 5 APRIL 86

1

32 34 36 38 40 42 7 APRIL 86

128 130 132 134 136 138 11 APRIL 86

5 APRIL 86

m 6 6 APRIL 86

I

32 34 38 38 40 42 7 APRIL 86

1

56 M 80 62 64 66 0 m L 66

104 106 108 110 112 114 10 APRIL 88

126 130 132 134 136 138 11 APRIL 86

0- 152 154 156 158 160 162

i - m T T T n 152 154 156 158 160 162

Time (h l

velocity with position on dunes for three low-flow surveys is shown in Fig. 16. Local mean (vertically averagqd) velocity (ti> is lowest in dune troughs and increases slightly from lower to upper dune backs. The difference in local mean flow velocity between troughs and upper dune backs can be as large as 0.10- 0.15 m s - ’ , which is 10-25% of the survey averaged velocity, and half the difference between low flow and high flow survey averages. This underscores the need for many measurements of local mean velocity when determining reach or cross-section averages of velocity, and especially when comparing these averages between surveys.

Water surface topography and bed shear stress

Contour maps of water surface elevation showed the same general patterns as described in Bridge & Gabel (1992). Downstream water surface slopes were calculated for the right channel only and ranged from 0.0007 to 0-0011 (Table 4). The highest slopes were recorded during falling flows immediately following discharge peaks.

In alluvial channels with dune beds, bed shear stress, 7, is a function of bedload and bed grain roughness (7’) and form drag over bedforms (7”). Bed shear stress values calculated by applying the Law of the Wall to individual velocity profiles should primarily reflect r’, whereas the spatially-averaged value calculated from Eq. (2), 7ws, reflects the sum o f t ’ and 7”. Spatially averaged bed shear stress, measured in the right channel, ranged from 2.7 to 5.4 N m-’. These values are 1.5-3 times larger than the average of local bed shear stress measurements from velocity profiles (zlOc; Table 4).

The relationship between dimensionless bed shear stress due to bedload/bed grain roughness on dune backs, W , and dimensionless spatially averaged bed shear stress, 0, was modelled by Engelund & Hansen (1972). As a modification of Engelund & Hansen’s original equation, 8’ = 0.06+ 0.402, Engelund & Fredsere (1982a) found

8’ = 0.06 + 0.303’*. (4)

Fig. 11. Diagrams on the left show the variation in migration rates of individual dunes through time for each survey of the 1986 study period. For clarity only dunes from one or two of the five sounding lines are shown. On the right-hand side, the average dune migration rates for dunes from all sounding lines are plotted against time. Zero hours corresponds to midnight, 5 April 1986.

Dunes in steady and unsteadyjows, Nebraska

1 0 7 -i

: c . P ," 1 0 - " "l 0 -

0 0 -I---

25 1

\ , . , . , . , . , . , , , . , , ,

These relationships were derived empirically from plots of 0 measured by Guy et al. (1966) against 8' values calculated from the theoretical model developed by Engelund & Hansen (1972). Equation (4) does not specify bedform drag as a function of dune geometry, and therefore implies that for a given bed shear stress, bedform drag accounts for the same proportion of total drag in any alluvial channel. Other models account for drag over bedforms explicitly by including dune steepness and/or ratios of dune height to water depth (e.g. Fredsrae, 1982; Wiberg & Smith, 1989). For Calamus data, 0 was calculated from Eq.

(2), 8' was calculated as the average of individual 8' measurements from velocity profiles obeying the Law of the Wall, and the relationship between 6 and 8' is shown in Fig. 17. The Calamus data are widely scattered, which is not surprising given the difficulty in measuring 0' over dunes. None the less, the Engelund-Fredsse model (Eq. 4) falls within the error bars for most field data. Some 8' measurements are greater than the model predicts. It is possible that velocity gradients may have been affected by roughness due to ripples on dune backs, which would result in greater 8' values.

252 S . L. Gabel

0.15 7 0.15

0.05

-- 0.04 I c v

p) 0.03

0.02

4-

L 0

ol C .- z 0.01 a ._ -

v, 0.00

_1. 8 , 0 A + A +

A +

t

* -rlcrrrrl

0.05

- 0.04

h - !c

0,

? 0.03 c

c .? 0.02

'E 0.01 E 0 0.00

c 0 C

0

1 O 0

3

0.05 q -- 0.05 3

1 , ( ( , , , , , , I , , , , , ( ( , , I 0

: 0.00 0.5 -

1.0 1.5 2.0 2.5

(u

$ 0.03 L

5 0.02 0

~ 0 . 0 1

0.5

0

* + ago*;

po ++

n o v 0 A

I 1 1-

1.0 1.5 2.0 2.5

Discharge (m3 s-')

Fig. 13. Rates of creation, destruction, splitting, combination, new (spontaneously created) dunes and dunes dying out versus study section discharge. Symbols correspond to study periods (see caption to Fig. 6).

TESTS OF MODELS FOR DUNE GEOMETRY

Steady, equilibrium models

Dune length

Survey-averaged dune length shows a positive correlation with mean flow depth (Fig. 18). For most surveys, mean dune length ranges between five and seven times mean flow depth, which compares well with other studies (Allen, 1984). Least squares regression of dune length on flow depth gives L=6.42d- 0.27, with correlation coefficient, r = 0.77. Yalin's (1977) theoretical model for dune length as a function

of flow depth, L=2nd, is plotted on Fig. 18 for comparison. Mean dune lengths measured in most surveys fall somewhat below Yalin's predicted values. Most of these lower values occurred during low, steady flows, suggesting that equilibrium dune lengths for the Calamus are slightly lower than Yalin's model predicts. There is considerable scatter in data, and as noted earlier, such scatter in field data is commonly attributed to flow unsteadiness. However, there seems to be as much scatter in data from low, steady flows as from unsteady flow periods. Dune length also depends on sediment grain size according to Allen (1984) and Yalin (1977), but mean grain size in the Calamus varies little between surveys and does not correlate with mean dune length.


0.15

5 0.05 ._ -Id 0 f 0

0.00

0.05

-- 0.04 I .r. v

Q) 0.03

0.02

Y

Y 0, C .- t: 0.01 ._ - a v, 0.00

0.05 - I L v

u) 0.04 C 0 .- % 0.03 ?? 0

u) 0.02 3 0

6 0.01

g 0.00 Y c

m -

0.1 5 h - I c v

0) 0.10 Y z C

g 0.05

0" 0.00

0 2 Y v)

0.05

- 0.04 - - I .c

9) Y

2 0.03 * C

2 0.02 Y

o b K

0 5 0.01 E u 0.00

A

0

1

1 0 0

0

; , , , , , , , , , , , , 0

0 0, 0.03

L

Rate of change of discharge (m3 6' h-'1

Fig. 14. Rates of creation, destruction, splitting, combination, new (spontaneously created) dunes and dunes dying out versus rate of change of discharge. Symbols correspond to study periods (see caption to Fig. 6).

Dune height

Mean dune height shows a positive correlation with mean flow depth (Fig. 19). Least-squares linear regression gives H=0.36d-0.026, with r=0-75. Flume data examined by Yalin (1964) and Allen (1984) show that, for a given sediment grain size, the ratio of height to depth increases with bed shear stress up to some maximum value, then decreases a t higher bed shear

stress. Fredsere (1 982) developed a theoretical model for H/d as a function of local bed shear stress, which predicted a maximum H / d of roughly 0.28 if the effects of suspended sediment (which acts to reduce dune height) are ignored. Yalin's (1964) model gave a maximum H/d of 0.167. Calamus data for H/d range between 0.21 and 0.38, far above Yalin's predicted maximum, and commonly exceeding predictions from Fredsoe's model as well (Fig. 20). Dune height versus

254 S. L. Gabel

Table 3. Dune creation and destruction data

Rate Rate of Rate of new dunes Rate dunes

art Rc R d splitting combining formed died out Survey date N+ (h) (h-I) (h-') (h-') (h -7 (h-? (h- ')

Right-hand channel 15 April 1985 20 April 1985 21 April 1985 23 April 1985 24 April 1985 29 April 1985 1 May 1985 7 May 1985 8 May 1985 12 August 1985 13 August 1985 16 August 1985 5 April 1986 6 April 1986 7 April 1986 9 April 1986 10 April 1986 11 April 1986

55 60 55 45 55 35 35 50 50 45 50 50 30 30 30 35 45 50

Left-hand channel 14 April 1985 42 27 April 1985 24 28 April 1985 30 2 May 1985 36

Downstream from island 19 April 1984 25

13 May 1984 25 15 May 1984 25 18 May 1984 30

12 May 1984 20

6.43 7.94 6.21 7.00 7.51 6.36 6.46 9.60

11.10 10.33 10.10 4.64 5.71 7.33 7.1

6.87 6.14 4.65

6.37 5.32 5.67 6.57

6.75 4.90 4.70 6.18 4.75

0.062 0.132 0.091 0.05 1 0.082 0.009 0.124 0.085 0.077 0.054 0.071 0.082 0.053 0.086 0.052 0.067 0.112 0.077

0.049 0.078 0.035 0.080

0.036 0-082 0.068 0.065 0.084

0.057 0.132 0.091 0.092 0.080 0.036 0.124 0.063 0.072 0.058 0.063 0.065 0.076 0.086 0.052 0.050 0.069 0.073

0.07 1 0.070 0.041 0.072

0.065 0.071 0.068 0.084 0.077

0.0141 0.0113 0.0399 0.0420 0.0205 0.0264 0.0095 0.0317 0.0242 0.0218 - 0.0045

0.0310 0.0354 0.0229 0.0167 0.0252 0.0180 0.0151 0.0172 0.0218 0.0139 0.0302 0.0129 0.01 17 0.0234 0.0273 0.0273 0.0188 0.0094 0.0166 0.0166 0.0362 0.0145 0.01 72 0.0215

0.0075 0.0224 0.0235 0.0235 0.0059 0.0176 0.0254 0.0127

- 0.0237 0.0204 0.0204 0-0255 0.0170 0.0065 0.0388 0.0210 0.0281

0.0085 0.0084 0.0146

0.0121 0.0045 0.0088 0.0 188 0.0072 0.0065 0.01 39 0.0086 0.0058 0.0045

0.0166 0.0181 0.0129

-

-

0.0112 0.0078 0.0059 0.0127

0.0059 0.0204

0.0129 0.0070

-

0.0013 0.0147 0.0176 0.0032 0.0097 0.0270 0.0133 0,0063 0.0108 0.0043 0.0139 0.0043 0.0175 0.0045 0.0141

0.0036 0.01 72

-

0.0150 - -

0.021 1

0.01 19 0.0102 0.0085 - -

* Approximate number of dunes in area. t Time elapsed in survey. R, = dune creation rate; Rd = dune destruction rate.

flow depth ratios measured in flume experiments under steady, equilibrium flow conditions commonly exceed 0.3 (see Allen, 1984, his fig. 8-12). Therefore, high H/d values for Calamus data are not an indication that dune heights were not in equilibrium with flow depth. More recently Fredsere (1989) suggested that high H/d values result when dune crestlines are sinuous, as they are in the Calamus.

Dune steepness

For equilibrium dunes, dune steepness, HIL, like H / d , rises initially and then falls as bed shear stress increases. Yalin & Karahan's (1979) empirical model

z gives HIL versus -, where 7c is the value of bed

shear stress at the initiation of sediment transport. Fredsere's (1975) empirical model gives H / L as a function of 0. Fredsere (1982) later developed a theoretical model that relates dune steepness to 8', where 8' represents the local value of dimensionless bed shear stress measured at the top of the dune. In Fig. 21, mean dune steepnesses are plotted with the three models as a function of 8'. In the diagram, 8' was calculated from T~~ and the Engelund-Fredsrae model (Eq. 4). The three models bracket the majority of field measurements, although dune steepness tends to be lower than predicted by Fredsse's (1982) model

7,

Dunes in steady and unsteadyjfows, Nebraska 255

Table 4. Hydraulic and sediment transport parameters.

Survey date Q d 0 Slope T~~ f ioc o w s Q ~ - H s Q b - ~ u ~ e r D (m3s-l) (m) (ms- ') (N m-') (N m-') (kg ms- ') (kg ms-') (mm)

Right-hand channel 15 April 1985 20 April 1985 21 April 1985 23 April 1985 24 April 1985 29 April 1985 1 May 1985 7 May 1985 8 May 1985

12August 1985 13 August 1985 16August 1985


Left-hand channel 14 April 1985 27 April 1985 28 April 1985 2 May 1985

1.14 0.44 0.65 1.09 0.42 0.65 1.02 0.42 0.61 1.33 0.49 0.68 1.18 0.44 0.67 1.73 0.61 0.71 1.53 0.54 0.71 1.08 0.41 0.66 1.10 0.43 0.64

0.95 0.39 0.61 0.82 0.34 0.61 0.85 0.34 0.63

0.00084 0.00078 0.00089 0.00088 0.00086 0.00076 0~00102 0.0 0 0 6 8 0.00076

0.00079 0.00098 0.00088

1.62 0.55 0.69 0.00089 1.66 0.53 0.77 0.00083 1.59 0.53 0.73 0.00080 1.30 0.47 0.69 0.00110 1.17 0.43 0.68 0.00086 1.01 0.40 0.63 0.00075

1.29 0.43 0.60 1.99 0.54 0.74 1.91 0.58 0.66 1.64 0.49 0.67

Downstream from island 19 April 1984 2.03 0.53 0.66 12 May 1984 1.74 0.51 0.53 13 May 1984 1.68 0.49 0.51 15 May 1984 1.63 0.47 0.49 18 Mav 1984 1.50 0.43 0.47

3.63 3.20 3.66 4.23 3.12 4.57 5.40 2.72 3.19

3.04 2.97 2.93

4.26 4.33 4.18 5.07 3.63 2.94

1.70 0.64 0.30 2.44 0.64 0.49 1.60 0.73 0.32 1.38 0.79 0.26 1.75 0.67 0.32 2.05 0.72 0.32 1.63 0.88 0.25 1.49 0.52 0.29 1.34 0.60 0.25

1.90 0.46 0.29 1.21 0.47 0.19 1.94 0.50 0.33

1.82 0.71 0.30 1.29 0.67 0.20 1.41 0.64 0.22 0.93 0.87 0.16 0.66 0.66 0.12 1.75 0.53 0.33

0.044 0.057 0.055 0.042 0.048 0.055 0.065 0.051 0.045 0.044 0.057 0.05 1

0.047 0.039 0.043 0.039 0.052 0.042

0.042 0.061 0,042 0.046

0.087 0.054 0.046 0.040 0.05 1

0.027 0.021 0.028 0.035 0.029 0.053 0.048 0.026 0.027 0.022 0.026 0.033

0.059 0.065 0.053 0.046 0.023 0.033

0.023 0.039 0.049 0.040

0.038 0.039 0.044 0.028 0.030

0.35 0.3 1 0.31 0.33 0.34 0.39 0.38 0.32 0.33 0.41 0.39 0.36

0.37 0.40 0.40 0.36 0.34 0.33

0.29 0.34 0.32 0.31

0.25 0.39 0.35 0.34 0.33

(Fig. 21). In that model, Fredsee assumed that sediment moves only as bedload for lower 8' values. As 8' increases, sediment can be transported in suspension. Suspended sediment attenuates bed waves, and therefore dune steepness decreases with increasing suspended sediment concentrations (Fredsse, 1982; Bridge & Best, 1988; Nelson &Smith, 1989). In the Calamus, minor amounts of suspended sediment transport occurred at all flow stages, which may account for lower dune steepness in the field data compared with the model.

For two high, falling flow surveys (1 May 1985 and 9 April 1986) dune steepness exceeds the predictions of the models (Fig. 21). Large dune steepness during falling flows increases hydraulic roughness and spatially averaged bed shear stress. The models predict that steepness should decrease at the 8' values observed during these flows. This discrepancy may indicate

that steepness is not in equilibrium with flow conditions. Alternatively, Eq. (4) may overestimate 8' when dune steepness is large. The average value of 8', calculated from 11 velocity profiles taken on 1 May 1985 was 0.27, fairly close to the value (0.31) derived from Eq. (4). Therefore, the large values of dune steepness during these two high, falling flow surveys are more likely due to disequilibrium between dune steepness and bed shear stress.

Unsteady flow models

Allen's model

Allen (1976b,c,d; 1978b)modelled the change in mean dune height and length for a population of dunes under periodically varying discharge. This model included the effects of dune creation and destruction

256 S. L. Gabel

c O

f

0 .

m .3 1

.3 I I I , I ,

.3 1 10

Discharge (m3 s-I)

Fig. 15. (a) Mean flow depth versus discharge. (b) Mean flow velocity versus discharge. Symbol’s are the same as in Fig. 6 Error bars (standard deviations for mean flow velocity and depth) are smaller than symbols.

as well a$ changes in heights of individual dunes. In this study, dunes occupying the bed within each study area constitute a population. However, Calamus flow conditions and those used in Allen’s model are quite different. Allen modelled flows with discharges roughly 1000 times greater than Calamus discharge, and discharge ratios (ratio of maximum discharge during a flow period to the minimum discharge) input to the model ranged from about 5 to 10, compared with about 2 for the Calamus. Dune excursion, an input parameter in Allen’s model, could rarely be measured in the field. Nevertheless, judging from the large number of dune destructions observed during most surveys, dune lifespans were probably of the order of a day or two, giving excursions of approximately 6-24 dune lengths. The hydrograph for Allen’s (1976d) ‘Run D’ experiments are closest to Calamus conditions, with a short flood period and a longer low flow period. Phase diagrams for dune height and

length generated In these runs are similar to those from the Calamus in terms of the orientation of hysteresis loops (Fig. 22). Loops for field data more closely resemble the model results for a dune excursion value of 6 rather than 13. Field data for falling flows are lacking for the data set shown, but falling flow data for 1986 (Fig. 8) do not show the large heights and lengths predicted for the larger value of dune excursion.

Allen (1 976c,d) plotted histograms showing distributions of dune heights and lengths during rising, falling and low, steady flows. Characteristics of these histograms for a particular flow stage depended mainly on dune excursion, shape of the hydrograph and discharge ratio, all specified as input. Frequency distributions of dune heights and lengths generated by Allen’s model were polymodal during certain parts of a flow cycle, when dunes with a geometry in equilibrium with an earlier set of flow conditions persist into periods when flow conditions are distinctly different. The older, lagged dunes then coexist with dunes whose geometries are in equilibrium with new flow conditions. The ability for dunes to persist when their geometry is in disequilibrium with flow conditions depends on dune excursion, migration rates and the magnitude and range of discharge changes. Based on behaviour of the model, Allen (1978a) concluded that polymodal dune assemblages in nature are the result of dune lag, and that characteristics of the population (e.g. variance in height and length) are controlled by rates of creation and destruction.

Polymodal dune assemblages are common in rivers, but often occur as sets of smaller dunes superimposed on the backs of larger forms (e.g. Coleman, 1969; Allen & Collinson, 1974; Jackson, 1976; Smith & McLean, 1977). In contrast to Allen, some authors interpreted dune superposition as an equilibrium bed configuration, hydrodynamically distinct from solitary dunes. It remains unclear whether superimposed dunes and polymodal assemblages of solitary dunes are primarily the result of flow unsteadiness (Bridge, 1982).

In general, histograms of dune length and height for 1985 field data (Figs 23 & 24, respectively) show features similar to Allen’s. Both show approximately unimodal or bimodal, positively skewed to symmetrical distributions during falling flows, and unimodal, symmetrical distributions with low variance during low flows. Distributions of height and length shift to greater values during high, rising flows, and exhibit decreasing variability within distributions as flow recedes. As flow continues to fall the distributions

Dunes in sreudj) and unsieadj' fiottlr, Nebraska 257

0.40

0.20

Q) > 0

0.00 13

1-

1 3-

0 0 0

-0.20

-0.40

X

+ 8 8 b v * P

+

+ X

* t

X +

1 1 I I 2 3 4 -5

Position on dune Fig. 16. Variation in individual measurements of vertically averaged flow velocity with position on dunes: y-axis represents the difference between the average velocity at a vertical (u,wa,) from the survey mean velocity (ua,J. Data from three surveys are shown: + = I ? August 1985 (DAY, =0.61 m s - I ) ; x = 13 August 1985 (ua,,=0.61 m s - ' ) : 0 = 16 August 198.5 (uA,e= 0.63 m s - I ) . Inset: numbering scheme for position on dunes.

become unimodal. Field data for 1986 differ from Allen's results in that variance decreases more quickly, and height distributions tend to be polymodal for peak discharges (6 and 7 April) rather than for low, falling flows.

In this study, bi- or polymodality occurred without superposition of dunes of different sizes. Therefore these results d o not help solve the controversy regarding whether superimposed dunes are the result of dune lag or are an equilibrium dune configuration. The only clear superposition of dunes in this study occurred when a faster moving dune migrated on to the back of a slower moving dune. Because repeated bed profiles were made, it was clear that these were short lived occurrences. This indicates that caution must be exercised when interpreting bed configura- tions based on single depth sounding records.

Data from this study also provide a means for examining the two components of Allen's model that are effective in changing mean dune geometry: changes in the heights of individual dunes, and dune creation and destruction. Allen calculated the rate of change of height for an individual dune as

where H,.(t) is the theoretical height of a dune in equilibrium with flow conditionsat time, r , I N t ) is the actual height of the individual dune and A, is the coefficient of change. The factor Ub/H(r) represents Allen's (1976~) assumptions that larger dunes have a lower capacity to change in height, and that increasing the sediment transport rate (Allen used U b as a

258 S. L. Gabel

i

0

I 1 I , I 1

0.l 1 e

Fig. 17. Plot of local dimensionless bed shear stress from velocity profile against spatially averaged values. Open circles denote survey averages derived from fewer than seven measurements of local bed shear stress. Solid circles are considered to be more reliable estimates of 0’. Error bars represent the standard deviation of the mean value. Solid line represents Engelund & Freds~re’s (1982) model, short dashed line represents Freds~re’s (1982) model and long dashed line represents Wiberg & Smith’s (1989) model.

surrogate for sediment transport rate) will result in more rapid changes in height.

In order to apply Eq. ( 5 ) , H, must be known, Allen (1976d) used the equation H=0.167d to calculate H, for a given depth. Calamus data show an approximately linear relationship between mean dune height and flow depth for each survey, but mean heights are closer to 0.4d. Therefore, the regression equation for Has a function of d(given above) was used to calculate H, from mean flow depth. Because measurements of Hand Ub for individual dunes vary greatly with time, mean values of dH/dt, u b and H were calculated for each dune that was measured at least four times during a survey. These ‘dune average’ values were used in comparisons with Allen’s model (Table 5). Even when using averages for each dune, plots of observed dH/dt against UbIH (H,-H) are dominated by scatter (Fig. 25a-c). In formulating Eq. (3, Allen assumed that faster moving dunes can change height more quickly than slower moving dunes. No such trend is apparent in Calamus data (Fig. 25d,e). Similarly, the rate of change of height of an individual dune did not appear to be related to its average height (Fig. 25f-i), contrary to Allen’s (1976~) assertion that larger dunes would have less ability to change in height. Allen’s

E v

- 4.0 t+ c3 Z W -I

w 5.0 Z 3 0

Z 2 2.0

1.0 1 1 1 I I I 1 0.2 0.3 0.4 0.5 0.8 0.7 0.8

MEAN FLOW DEPTH (m)

Fig. 18. Plot of mean dune length against mean flow depth for each survey. Error bars are standard deviations of the mean values for that survey.

0.08 0.2 0.3 0.4 0.5 0.8 0.7 0.8

MEAN FLOW DEPTH (m)

Fig. 19. Plot of mean dune height against mean flow depth for each survey. Error bars are standard deviations of the mean values for that survey.

assumptions followed from the equation of sediment continuity, but that equation applies only when dunes migrate without changing shape.

Wijbenga & Klaassen (1983) also tested Eq. (5) in flume experiments in which discharge was changed

Dunes in steady and unsteady$ows, Nebraska 259

J 0.40 -

\

$ 0.20 I

- 0.0.3 mm -- -- -_ . '.

0.00 0.20 0.40 0.60 0.80 1 .oo e'

Fig. 20. Dune height versus flow depth ratio plotted against 8' (from Eq. 4). Both solid and dashed curves are from Freds0e.s (1982) model. Error bars (standard deviations of mean dune height versus flow depth ratios) are shown where they are larger than symbols.

4 > 0.04 - .. . . . -.., .. .. ...,

. I

- . - . _ ._ -,-.

-.-.

I - - _ - - - _ _ - - - - - 0.02 - , -. 0.00 r I I 1 I I I I O I " " I I I I t I 1 I I 1 I I I I I I I I I I 1 1 I I I I I I I I I I I I ' 1

0.00 0.20 0.40 0.60 0.80 1 .oo 8'

Fig. 21. Plot of mean dune steepness against 8' (from Eq. 4). Solid curve is Fredsere's (1982) model for O', dashed curve is Fredsne's (1975) equation. Dash-dot curve is Yalin & Karahan's (1979) model. Error bars indicate standard deviation of mean dune steepness.

Table 5. Comparison with Allen's model

20-21 April 1985 21-23 April 1985 29 April-1 May 1985 7-8 May1985

5-6 April 1986 6-7 April 1986 7-9 April 1986 9-10 April 1986 10-1 I April 1986

27-28 April 1985

12-13 May1984 13-15May1984

12-13 August 1985

0.000458 0.00048 1 0.000083 0.000250

0.000208 - 0.000542

- 0.000083 - 0.000604 - 0.002330

0.0004 17

0.000 167

0.000333 -0.000583

0.125 0.154 0.174 0.129 0.092 0.170 0.170 0.146 0.129 0.1 17

0.191

0.154 0.146

0.172 0.502

0.124 -0,071

-0.215 -0.233 - 0-200 - 0.304 - 0.273

0.072

0,122

-0.138 - 0.232

A ,

0,00266 0.00096

- 0.001 17 0.00202 0.00252

- 0.00089 0.0004 1 0,00199 0,00853 0.00579

0.00 I37

- 0,0024 1 0.0025 1

260 S, L. Gabel

120- 3.5 - 100

(C)

c 3.0

f s 80- 2.5 - m

0

0 2.0 - 20 -

0 I I I I I I 1.5 0.5 1.0 I 1.5 I 2.0 1 2.5 1 0 2 4 6 8 1 0 1 2

Discharge (m3 s-’ X d) Discharge (m3 s-’)

Fig. 22. (a) Phase diagrams showing mean dune height against discharge for Allen’s (1976d) Run D. Dashed line represents dune excursion value of 6; solid line is run using dune excursion value of 13. (b) Calamus data for mean dune height against discharge for the spring 1985 season, right channel surveys. (c) Phase diagrams showing mean dune length against discharge for Allen’s (1976d) Run D experiments with a dune excursion value of 6 (dashed line) and 13 (solid line). (d) Calamus data for mean dune length against discharge from spring 1985, right channel surveys.

abruptly. In evaluating A, they used values of H and U, averaged for each run (analogous to survey averaged run here) rather than individual values, and the overall rate of change of height for each flume test. They found that A, was not constant, but increased with the magnitude of discharge change. Mean values from the Calamus surveys also show a non-linear relationship between dHldt and U,/H (Hm-H), in agreement with Wijbenga & Klaassen’s results (Fig. 26).

Dune creation and destruction are the other important factors influencing the response of dune geometry to unsteady flows, according to Allen’s model. Due to the lack of any systematic studies of dune creation and destruction, no information was available to Allen on rates of creation and destruction as a function of flow characteristics, modes of creation and destruction, or whether these processes have any relationship to the geometrical properties of dune populations.

Allen assigned a value of dune excursion to each dune at the time of its creation. Excursion is the distance the dune travels during its lifespan (divided by dune length), and that distance depends on the dune migration rate. Upon completion of its excursion, the dune is replaced by one or a number of dunes. The height and length of each new dune is in equilibrium with flow conditions at the instant of creation.

By examining changes in the number of dunes with discharge, Allen (1976d) determined that dune destruction rates were greater during rising and high flows, and creation rates were greater during falling flows and low flows. Calamus data showed no relationship between creation or destruction rates and discharge or rate of change of discharge, as noted above. However, the ratio of creation rate to destruction rate, or the mechanism by which dunes are created or destroyed may be the important factors. Allen did not speculate about the mechanisms respons-

Dunes in steady and unsteadyjows, Nebraska 26 1 :h :I,. , _ k, . I k;L,;[ , , , ,

! 0 8 p (Lb, ~ ~ ~ k ~ ~ l , , , Z I , , ’

;l2kz - E

, ~ Ir, kiki[, , ,-;I, , . , 1

- D E 10 - 4

‘YI

E

P

*I

a! 2 4 6 8 1 0 2 4 0 a 1 c 2 4 6 6 ~ ) z 4 8 8 t o 2 . 6 8 1 0 -

6 A 30

g 2 0 20 10 2 0 . 10 I 0 10 U

10

2 4 6 8 1 0 1 4 6 8 1 0 2 * 8 8 1 0 2 4 6 8 1 0 2 . 6 8 1 0

0 10 2 0

$ 4

5 C D

ii %r “rr *r r 40 80 110 40 8 120 40 80 120 40 80 120 4 0 80 120

Time Dune length (m)

Fig. 23. Top row: graph at extreme left shows discharge hydrograph for part of the spring 1985 field season. Graphs on right show histograms of frequency,f, against dune length. Surveys shown are: (A) 24 April 1985; (B) 29 April 1985; (C) 1 May 1985; (D) 7 May 1985; (E) 8 May 1985. Middle row: hydrograph for spring 1986 field season is shown at left. At right are histograms showing dune length for: (A) 5 April 1986; (B) 6 April 1986; (C) 7 April 1985; (D) 9 April 1986; (E) 10 April 1986. Bottom row: results for Allen’s (1976d) Run D experiments with input hydrograph at left. Hydrograph plots discharge against time in fractions of a flow cycle, T. Distribution of dune lengths at each time marked on the hydrograph are shown at right.

ible for dune creation and destruction in his model. It seems reasonable to suggest that higher destruction rates during rising flows may reflect a predominance of dunes combining since, by the definition used here, combining. results in the destruction of two or more dunes but the creation of only one larger dune. Similarly, high creation rates could be produced by dunes splitting, whereby one dune is destroyed but two or more smaller dunes are created. Assuming constant lengths for extant dunes, a predominance of splitting over combining, and thus creation over destruction, would result in a decrease in mean dune length. Using the same reasoning, dune combining would increase mean dune length.

There is a weak tendency for creation rates to

exceed destruction rates during low, falling flows, and destruction rates can exceed creation rate during some high flows and some rising flows (Fig. 27a,b). The rate of dunes splitting and rate of combining are poorly correlated with changing mean dune length, however (Fig. 27c,d). Thus it appears that dune splitting and combining do not exert a major control on mean dune length. The changes in mean dune length with discharge are strongly dependent on changes in lengths of individuals.

Allen’s results from Run D experiments also showed that the rate of dune creation was related to the rate of change of coefficient of variation of dune height and length. Allen (1976d) found a positive correlation between creation rate and increasing variability. Data

262 S . L. Gabel

7 ‘

1

al 0.1 0.2 03 0.4 0.1 0 1 0.3 0.4 0.1 0 1 0.3 0 1

1” I‘

I ” I‘

laY 01 0.2 03 0. 0 1 0 2 03 0 .

Dune height (m) Time

Fig. 24. Top row: graph at extreme left shows hydrograph for part of the spring 1985 field season. Graphs on right show histograms of frequency,f; against dune height. Surveys shown are: (A) 24 April 1985; (B) 29 April 1985; (C) 1 May 1985; (D) 7 May 1985; (E) 8 May 1985. Middle row: hydrograph for spring 1986 field season is shown at left. At right are histograms showing dune height for: (A) 5 April 1986; (B) 6 April 1986; (C) 7 April 1985; (D) 9 April 1986; (E) 10 April 1986. Bottom row: results for Allen’s (1976d) Run D experiments with input hydrograph at left. Hydrograph plots discharge against time in fractions of a flow cycle, T. Distribution of dune heights at each time marked on the hydrograph are shown at right:

from this study show that rates of creation and destruction have no relationship with coefficients of variation of height or length (Fig. 28).

Fredsc~e’s model

Fredsrae’s (1979, 1981) model predicts the initial rate of change in height of an individual dune in response to a stepwise change in discharge. The change in discharge causes a change in bed shear stress at the dune crest, which in turn causes bedload transport rate to increase (resulting in a decrease in height) or decrease (causing an increase in height). The initial rate of change in height is given by

where F = (lid) (d+/d6), A = 82/6,, and subscripts 1

and 2 denote conditions before and after the discharge change, respective!y.

In comparing field data with predictions from the model, several important points must be considered : (1) changes in discharge in the Calamus were gradual, not abrupt; (2) field measurements of dH/dt for individual dunes are not necessarily the initial rates of change, and are highly variable with time and among individual dunes; (3) dH/dt for field data were calculated from 8, and it is uncertain whether these are equivalent to 8’ at dune crests as Fredsere (1981) assumed. As an approximation, conditions were compared from one survey to the next, provided that the surveys were not spaced more than 2 days apart. Mean hydraulic conditions during the first survey were used as initial conditions, and mean values on the subsequent survey represented conditions after the discharge change. The rate of change of height


0 (b)

0 0

0 0 i p, 1.0

o O $ 0 0 '4 -0.0

\ X D

-1.0

-2.0 -0.10 -0.05 -0.00 0.05 0.10

dH/dt observed

1 (d)

-0.1 0.0 0.1 0.2 0.3 0.4

/f (m)

-0.10 -0.05 -0.00 0.05 0.10 d t t / d t ObS8Wed

0 1 - 0 1 2

0

0

0.0 0.1 0.2 0.3 0.4

H (m)

Fig. 25. (a-c) Examples of data for rate of change of height for individual dunes compared with Allen's model (Eq. 5). (d-f) Plots of observed rates of change of heights of individual dunes against the average migration rate of the dune. (g-i) Plots of observed rates of change of heights for individual dunes against the average height of the dune. For each row, the left-hand graphs are plots of data obtained on I May 1985; middle graphs are from 7 May 1985; right-hand graphs are from 9 April 1986. In all graphs, dH/dt, height and migration rate are averages for individual dunes measured at least four times.

[ (~~~)e]-"4 ( 3 - 0 . 7 ,/&). (7) 4(') = -F -

This equation and the bedload transport equations of Bridge & Dominic (1984), Meyer-Peter & Muller (1948) and Engelund & Fredsse (1982b) were used to calculate dH/dt from Eq. (6). In order to apply these models to transport over dunes, 8' (here calculated

was taken to be the difference in mean dune height between the two surveys divided by the time that had elapsed between them. Mean dune length and grain size from the earlier survey were used in Eq. 6. The values used in comparisons with Fredsere's model are listed in Table 6.

The bedload transport model used by Fredsere (1979) is Engelund & Fredsse's (1976) equation,

264 S . L. Gabel

0.0006

0.0003

-0.0003

-0.0006 -

a a

a

a

a

a

a - a

I 1 1 - I I I I I I I I I I I I I l ~ - - - r l

50 -0.25 0.00 0.25 0.50

Fig. 26. Graph of observed rate of change of dune height against Allen's model, (Ub/H) (H, - H), for the transitions listed in Table 5. Hand U, are survey averaged values for the initial surveys in each transition. H , was calculated using the regression of H on d , taking d as the mean flow depth from the initial survey of each transition.

Table 6. Comparison with Fredscae's model.

D L dH/dtobserved dHAjdt dHBjdt d f l j d t dHDjdt Transition (mm) (m) 8'' 8'* (mh- ' ) (mh- ' ) (mh- ' ) (mh- l ) (mh-I )

21-23 April 1985 0.31 2.08 0.25 0.27 0.000458 23-24 April 1985 0.33 2.73 0.27 0.22 -0.000917

5-6 April 1986 0.37 3.74 0.24 0.22 0.000208 6-7 April 1986 0.40 3-63 0.22 0.21 -0.000042 7-9 April 1986 0.40 4-04 0-21 0.30 -0.000604 9-10 April 1986 0.36 3.00 0.30 0.22 -0.002333 10-1 1 April 1986 0.34 2.51 0.22 0.17 0.000417

29 April-I May 1985 0.39 3.19 0.24 0.31 0.000083

A Fredsee(l979): I $ = = 5[1 + ( 0 . 2 6 7 / @ - 0 , ) 4 1 - ' ' 4 ( ~ - 0 . 7 ~ ) . Meyer-Peter & Muller (1948): I$ = 8(@ - OJ3'', Bridge & Dominic (1984): I$ = lO(0' - O , ) ( f i -JK). Engelund & Fredsee (1982b): 4 = lo(@ - O,)(,/O' - 0.7&).

0.01 188 - 0.02235

0.03926 - 0.007 10 - 0.00359

0.03772 - 0.04384 - 0.01509

0.00087 -0.00148

0.003 15 - 0.00052 -0.00032

0.00334 - 0.00 199 - 0-00 174

0.00090 ~ 0.001 46

0.00340 ~ 0.00053 ~ 0,00032

0.00369 ~ 0-00209 - 0.00 I53

0.00084 -0.001 39

0.003 13 - 0.00050 - 0-00030

0.00336

-0,001 53 -0.00195

Dunes in steady and unsteadyflows, Nebraska

1.5

0.5 :

0.0

265

+ A

+ 0

6

, , l ~ l l l ~ l l l ~

1.5 *-O 1

0.05 - I -

i 0.04 - r al v

0.03 I c

0.5

(d)

0

0.02 - .- I= n E

0.01 - T

*

- A A

+

0.0 1, 0.5 1.0 1.5 2.0 2.5

o (m3 S-9

- 0.04 c v

a 0.03 4-

E .c 0.02

u, 0.01

m

+ 4- .-

0

t@

A

* +

A

I

0

0.00 -0.04 -0.02 0.00 0.02 0.04

0

6 +

C O A

4 O O

0

0

0

A

0.00 -I, -0.04 -0.02 0.00 0.02 0.04

d L / d f (m h-I) dL/df (rn h-'1

Fig. 27. Ratio of dune creation rate to dune destruction rate (RJR,) plotted against (a) discharge and (b) rate of change of discharge. (c) Rate of dune splitting plotted against rate of change of dune length. (d) Rate of dune combining plotted against rate of change of dune length. In all graphs, each point represents values for an individual survey.

from Eq. (4)) must be used in place of 8. Values for dH/dt are nearly the same when the equations of Bridge & Dominic, Meyer-Peter & Muller and Engelund & Fredsere are used (Fig. 29). Use of Eq. (7) yields dH/dt values that are a n order of magnitude greater than observed values, because 4 and d4jd8 are overpredicted by the equation (Table 6).

Fredsne's (1979) model predicts the rate of change of dune height fairly well for lower flows when changes in bed shear stress are rapid (e.g. transitions of 23-24 April 1985 and 9-10 April 1986; Fig. 29). During high flow periods, bed shear stress sometimes varied substantially between surveys, but only small changes in mean dune height resulted. At other times during high flows, dune height decreased in response to increasing bed shear stress (e.g. 7-9 April 1986). Fredsoe's (1979) model assumes that sediment is

transported as bedload only. Therefore the rate of change in height predicted is directly proportional to the change in bed shear stress, in contrast to these examples. Modelling the variation in dune height with bed shear stress a t large bed shear stresses requires including the effects of suspended sediment transport (Fredscae, 198 1). Due to lack of detailed measurements of suspended sediment transport, the effects of suspended sediment cannot be examined in this study.

CONCLUSIONS

(1) The geometry and movement of dunes showed tremendous variability within each day long survey due to lateral and temporal variations in individual dunes and differences between dunes.

266 S . L. Gabel

0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 Creation rate (h-') Destruction rate (h-')

Fig. 28. Coefficient of variation of dune height for each survey plotted against (a) dune creation rate and (b) dune destruction rate. Coefficient of variation of dune length for each survey plotted against (c) dune creation rate and (d) dune destruction rate.

Local values of flow velocity, bed shear stress, and bedload transport rate were also highly variable. Changes in shapes and migration rates of individual dunes over periods of hours reflect local changes in bed shear stress and sediment transport rates.

(2) Mean height and length of dunes reflect changes in discharge when observed over several days. Changes in mean dune height and length for individual surveys lagged very little behind dis-

.charge changes. The largest mean heights and lengths were recorded only 1 or 2 days after the discharge maximum.

(3) Changes in mean dune length with flow depth during unsteady flows were similar to the predictions of Yalin's equilibrium model, despite the fact that length-discharge relationships were loop shaped. Height versus depth ratios for a given bed shear stress generally exceed predictions of Fredsere's (1982) model for equilibrium dunes, and greatly exceed Yalin's (1 964) theoretical maximum

of 0.167. Dune steepnesses generally agreed with equilibrium models of Fredsere (1975, 1982) and Yalin & Karahan (1979) for all but rapidly falling flows, when dune heights may have been slightly larger than equilibrium values.

(4) There are no theoretical or empirical models that give equilibrium dune geometry as a function of river discharge. Therefore, monitoring dune geometry as a function of discharge actually gives little information on whether dune geometry is in equilibrium with flow conditions. Dune lag may be better defined in terms of how much the dune geometry deviates from equilibrium. This requires detailed measurements of bed shear stress.

( 5 ) Histograms of height and length distributions for surveys made during high and falling flow periods show qualitative similarities with histograms produced by Allen's model. Both results from this study and Wijbenga & Klaassen's (1983) work show that the 'coefficient of change' in Allen's model is not a constant, but depends on flow


1 0.004 A B A

81

-0.004 1 I I I I -0.003 -0.002 -0.001 0.000 0.001 0.002

d H / d t observed (m h-I)

Fig. 29. Comparisons of predictions from Fredsee’s (1979) model for rate of change of dune height with observed rates of change. Symbols correspond to calculations using different bedload transport equations: (x) Meyer-Peter & Muller (1948); (A) Bridge & Dominic (1984); (D) Engelund & Fredsee (1982b). The solid line represents perfect agreement between observed values and the model.

conditions. Disagreement between observed and modelled changes in heights of individual dunes may be due partly to difficulty in specifying the equilibrium height of dunes as a function of flow conditions. In contrast to Allen’s model, no correlation was found between rates of creation or destruction and discharge or rate of change of discharge. Likewise, the mechanisms by which creation and destruction occurred did not vary with discharge or rate of change of discharge. Field observations suggest that creation and destruction of dunes are related to localised variations in bed shear stress and bedload transport rates.

(6) Fredsse’s (1979) model correctly predicts changes in mean dune height for surveys in which bed shear stress is low (little suspended sediment transport).

ACKNOWLEDGMENTS

This research was funded by NSF grants EAR-920580 and EAR-8419256. The study formed part of a PhD thesis supervized by John S. Bridge. The invaluable

assistance with field-work provided by Peter Farwell, Paul Bernstein, Kendra Marks, Brian Willis and Kathryn Pollard is greatly appreciated. Peter Farwell, Daniel Bartlett and Paul Richards assisted in labora- tory analyses. Landowners Tom Liddy and Alfred Meeks allowed free access to the field site, and the Wubbenhorst family provided assistance at the site on numerous occasions. Jon Palmatier designed and machined several important pieces of field equipment. John Bridge, Brian Willis, Robert Demicco, Steve Dickman and Rudy Slingerland provided helpful comments on this study. The manuscript benefited from the comments of John Bridge, Lawrence Boguch- wahl, Jsrgen Fredsse and an anonymous reviewer.

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(Manuscript received 25 February 1992; revision received I I November 1992)

Geometry and kinematics of dunes during steady and unsteady flows in the Calamus River, Nebraska,...

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