Preservation of planar laminae due to migration of low-relief bed waves over aggrading upper-stage...

Sedimentology (1997) 44, 253-262

Preservation of planar laminae due to migration of low-relief bed waves over aggrading upper-stage plane beds: comparison of experimental data with theory JOHN BRIDGE* a n d JIM BESTt *Department of Geological Sciences, Binghamton University, PO Box 6000, Binghamton,

tDepartment of Earth Sciences, University of Leeds, Leeds LS2 9JT, West Yorkshire, UK NY 13902-6000, USA

ABSTRACT

Experimental studies of the formation of planar laminae by migration of low-relief bed waves over aggrading upper-stage plane beds show that the average thickness of laminae at a point increases with both aggradation rate and the variance of the heights of bed waves passing that point. In general, the preserved laminae represent only a small proportion (generally less than 50%) of the height of the largest bed waves in the population. The theory developed by Paola & Borgman (1991), relating the probability density function of stratal thickness to that of bed wave height for the case of no net aggradation, was adapted for aggrading conditions and shown to agree well with data. These results suggest that the theory can be used to estimate, from the distribution of thickness of planar laminae, either the distribution of bed wave heights, mean aggradation rate, mean bed wave length or mean bed wave celerity, provided the other parameters can be estimated.

INTRODUCTION

The downstream migration of low-relief (mm high) bed waves on aggrading upper-stage plane beds of sand beneath subcritical, turbulent water flows gives rise to characteristic sets of planar laminae (Bridge & Best, 1988,1990; Best & Bridge, 1992; Paola et al., 1989). Best & Bridge (1992) demonstrated experimentally that the preservation of laminae of a given thickness beneath any point on the bed depends on both the mean aggradation rate and the sequence of different height bed waves passing that point. An attempt to explain lamina preservation using a quanti- tative theoretical model is now possible in view of the development by Paola & Borgman (1991) of an exact theory for the probability density function of stratal thickness from knowledge of the probability density function of the heights of the bed waves responsible. Although their theory is only strictly applicable to the case of zero aggradation rate, it may be possible to adapt this theory to the more realistic aggradational case. In this paper, we use new experimental data on the geometry and migration rate of low-relief bed

8 1997 International Association of Sedimentologists

waves and lamina thickness to test and extend Paola and Borgman’s theory under aggrading conditions.

THEORY

In the case of bed waves of constant geometry and downstream migration rate under conditions of steady aggradation, the thickness of sets of cross strata, s (or planar laminae in the case of low-relief bed waves), is constant and given by

s=l r/c=l tan6 (1)

where 1 is the length (spacing) of bed waves, r is aggradation rate, c is bed wave celerity (downstream migration rate), and 6 is the angle of climb relative to mean bed level. However, the geometry and migration rate of bed waves cannot in reality be assumed to be constant, and Eq. (1) is therefore not generally applicable. In order to account for the effect on stratal thickness of the variability in bed wave heights, Paola & Borgman (1991) derived the following equation for the probability density function of stratal thickness, p(s):

253

254 J. Bridge and J. Best

Table 1. Flow and sediment transport conditions.

Run number 1 2 3

Mean flow depth, d (m) Vertically-averaged flow velocity, U (m/s) Froude number Water surface slope, S Bed shear stress, To=pgdS (Pa) Shear velocity, U+=(7,1p)0'5 (m/s) Dimensionless bed shear stress,@=.r,l(o - p)gD,, Friction coefficient, f =8(Ue/U)' Water temperature, "C Mean size of bed sediment, D,, (mm) Median size of bed sediment, D,, (mm) Standard deviation of bed sediment size (mm) Aggradation period (s) Aggradation in test section (mm) Aggradation rate, r (mm/s)

0.11 0.91 0.92 0.93 0.88 0.885 0.89

4.85 & 0.5 0.07 z t 0.003

0.7 & 0.07

0.0045 & 0.0005

0.047 0.046 0.045 17-19 12-1 5 18

0.43 0.43 1.42

29 995 19 080 11 100 20 44 42

0.0038 0.00065 0.0023

p(s)=a e p n s ( e P a S + a s - z)/(z - e p a S ) * ( 2 )

where Z/a is the mean value of the exponential tail of the probability density function of topographic height relative to a datum, and a is inversely related to the breadth of the tail of the height distribution. The mean value of s is 1-64493/a and the standard deviation is 144997Ia.

In order to test the applicability of Eq. ( 2 ) , it is necessary to measure distributions of stratal thickness and bed wave height at a specific location on the bed under conditions of no net aggradation. As explained by Paola & Borgman (1991), when a is derived from bed wave height data, its value must be doubled before it is used in Eq. ( 2 ) to predict the distribution of stratal thickness. Thus the mean stratal thickness, s,, will be 0.8225Ia and the standard deviation, ssd, is given by 0.725Ia. Paola and Borgman pointed out that it is useful to fit a two-parameter gamma density function to the measured distribution of bed wave heights, h:

p(h)= h" - 'e h'P//3T(a) (3)

where a and p are parameters and r is the gamma function. This particular density function is useful because p= l / a =hSd2/h,, a=hm2/hSd2, and therefore the expected stratal thickness and standard deviation are 0.8225 hSd2/h, and 0.725 hSd21 h,, respectively. Here, h, and h,, are the mean and standard deviation of the bed wave height distribution, respectively.

In this paper, we calculate a by fitting a two- parameter gamma distribution to our data, and then test the hypothesis that, under aggrading

conditions, the expected mean stratal thickness is given by:

s,=l r/c+0*8225/a=l r/c+0.8225 hSd2/h,. (4)

Equation ( 4 ) is reasonable in that mean stratal thickness depends only on the distribution of bed wave heights if aggradation rate is zero, and only on mean aggradation rate, bed wave length and celerity if there is no variation in bed wave height. Equation ( 4 ) does not describe any linkage between the parameters in the two terms. The absence of a linkage between aggradation rate and the density function of bed wave height appears to be the case for the data presented here and in Best & Bridge (1992), but further examination of the possibility of such linkages is desirable.

EXPERIMENTAL APPARATUS AND PROCEDURE

Experiments were conducted at the Department of Geological Sciences, Binghamton University, using a hydraulic flume 7.6 m long, 0.6 m wide and 0-4 m deep that recirculates water and sediment (see Best & Bridge, 1992). A conveyer belt at the flume entrance allowed sediment to be added to the flume over a wide range of accurately controlled feed rates. Such steady sediment feed resulted in steady aggradation throughout the length of the flume. Three experiments were conducted with different aggradation rates but similar flow conditions (Table 1). All the sediment used was natural silica sand with median grain size of 0.43 mm and standard deviation of 1.42 mm.

g> 1997 International Association of Sedimentologists, Sedirnentology, 44, 253-262

Preservation of planar laminae 255

150 I

loo 90 1 Run 1 80 1

1 5 0 , 0 5000 10000 15000 20000 25000 30000

0 5000 10000 15000 20000 100

90

80

70

60

40 Run 3 30

5000 10000 15000 0

t ime (s) Fig. 1. Records of bed height vs. time obtained using the ultrasonic bed profiler for runs 1 through 3. Fluctuations in bed height are due to the passage of low-relief bed waves. The overall linear sloping trends indicate constant aggradation rate. The horizontal trends at the ends of Runs 2 and 3 are due to cessation of aggradation at the end of these experiments.

At the start of each experiment, the flume slope, water depth and discharge were set to produce upper-stage plane beds (see Best & Bridge, 1992). The flume was allowed to run for a few hours without sediment feed to establish uniform, equi- librium flow. A vertical profile of flow velocity was then measured at the centre-line of the test section using a Marsh-McBirney electromagnetic current meter with a 12 mm-diameter sensor (details in Best & Bridge, 1992). The vertically averaged flow velocity was calculated by numeri- cal integration. Water surface elevation was then measured with a point gauge at nine centre-line points along the flume, allowing calculation of water surface slope. The water surface slope, S, and depth, d, in the test section were used to calculate centre-line bed shear stress

z,=pgdS (Table 1). Water temperature was also recorded.

Aggradation at a fixed rate was initiated by starting the conveyor belt. The desired aggradation rate was obtained by use of a constant thickness of dry sediment on the conveyor belt and a fixed belt speed. Bed height was measured in the centre of the test section for 10-15 min before the conveyor belt was started, and then throughout the rest of the run. Bed height was measured at 5 s intervals to an accuracy of 0.1 mm using an ultrasonic profiler interfaced with a PC (details of profiler in Bridge & Best, 1988; Best & Bridge, 1992; Best & Ashworth, 19941. These bed height records allowed measurement of the height and period of all bed waves passing under the profiler in the test section, plus the rate of

0 1997 International Association of Sedinientologists, Sedimentology, 44, 253-262

256 J . Bridge and J. Best

015 1 1 1 1 1 1 1 1 1 1 1

Run 3 Mean =7.9mm S D = 5 4 m m

~

= 127

~

e

n

0.05

0 00 0

0 0 0 -

Run 1 Mean =6.3mm S D =3.6 mm

= 281

-

I I I I 1

0 5 10 15 20 25 30

5 10 15 20 Bed-wave height (mm)

0.15

Mean =5.6mm S D =3.6 mm

p O.’O ; e

0.05

0.00 5 10 15 20 25

Bed-wave height (mm)

h n 1 Bed-wave height (mm)

B 30

25

20

8 15

10

5

0

Run 1: mean=99.9

30

25

20

$ 15

10

5

0

~86.7 9

rT_

0 50 100 150

30 Run 3: mean=97.9

SD=41.2 n=127

0 50 100 1

bed-wave period, seconds io

Fig. 2. Histograms of (A) bed wave height and (B) bed wave period for each run, obtained from the bed profiler records (Fig. 1). Best-fit two-parameter gamma density functions of bed wave height are also shown.

aggradation. In addition, observations were made, and 35 mm photographs were taken, through the side walls of the flume in order to determine bed wave lengths and migration rates (unobtainable from bed-profiler records), and the nature of sediment sorting over the bed waves. In order to ensure that the hydraulic conditions remained constant during an experiment, it was necessary to add water to the flume periodically to replace that ‘lost’ in the pore spaces of deposited sediment. Constant flow depth and vertically

averaged velocity were achieved throughout all runs.

After 20-50mm of sediment had aggraded in the test section, the flow was stopped carefully and the flume was drained slowly. Box cores were taken from the test section and epoxy-resin peels were made from the cores (details in Best & Bridge, 1992). Thin sections were then made from the parts of the peels that had been immediately beneath the ultrasonic profiler. The thin sections were photographed on an SEM and large-format

0 1997 International Association of Sedimentologists, Sedimentology, 44, 253-262


Table 2. Bed wave characteristics.

Run number 1 2

Aggradation rate, r (mm/s) Number of bed waves in bed profiler records Mean height, h,, (mm), from bed profiler records Standard deviation of height, h,, (mm), from bed profiler records u=hnl / hsd2 (mm ~ ') Mean of best-fit gamma probability density function of height (mm) Standard deviation of best-fit gamma probability density function of height (mm) a value for gamma probability density function (mm - ') Mean period from bed profiler records (s) Standard deviation of period (s) Mean height from side-wall observations (mm) Standard deviation of height from side-wall observations (mm) Mean bed-wave length, 1 (m), from side-wall observations Standard deviation of bed-wave length (m) Mean celerity, c (mm/s), from side-wall observations Standard deviation of celerity (mm/s)

0-00065 281 6.3 3.6 0.47 6.1 4.2 0.34 100

38 10.2

5.5 1.32 0-24

11.8 2.2

0.0023 221 5.6 3.6 0.43 5.4 3-6 0.41 87 38

8.2 3.5 1.2 0.22

11.2 3.1

3

0.0038 127 7.9 5.3 0.27 7.3 5.3 0.26 98 41 - - - - - -

2

A E

5 1.5

s 3 9 1 3 u a 0.5

4 a 6 -

. r . m

0 5 10 15 20 26

bed-wave height, mm 7 20 , I

ul

E E 15 s 5 l o 8 5 5 ? d o

c .-

m

0

0 5 I 0 15 20 25

7 20 ul

15 s c .- - kI l o 8 $ 5 3 6 d o

m

bed-wave height, mm

C

i . 0.5 I 1.5 2

bed-wave wavelength, rn

Fig. 3. Relationships among bed wave migration rate (celerity), height and length for runs 1 and 2 , using data obtained from flume side-wall observations. These data are biased towards the larger bed waves.

photographs were produced. A series of horizontal transect lines was drawn on each photograph with a constant vertical spacing equivalent to 0.5 mm on the thin section. The long axis of every grain intersecting each transect line was then measured. As the length of each transect line was ~ 1 2 mm, up to 25 grains were measured per transect. This methodology is consistent with that of Cheel & Middleton (1986), but we used a greater vertical sampling increment due to the larger grain sizes in this study. The purpose of this methodology was to determine as accurately as possible the vertical sequence of lamina thickness (as defined by grain-size trends) and then to compare this sequence with that predicted from the bed-height records. Each lamina in the sequence could then be matched with a specific formative bed wave.

RESULTS

Bed wave geometry and celerity The overall linear trends in the plots of bed height vs. time indicate that aggradation rate was constant in each experiment (Fig. 1; Table 1). Figure 1 also shows that bed wave height and period did not change systematically during the course of each experiment. These bed height records were used to define distributions of bed wave height and period (Fig. 2; Table 2). Most bed-height fluctuations with heights of less than 1-5 mm and periods of less than 40s were excluded from consideration because of the difficulty in accurate

0 1997 International Association of Sedimentologists, Sedirnentology, 44, 253-262


I

130 , , I , , , , , ( ( , , I ,

16500 17000 17500 181

time, seconds

Fig. 4. Detail of bed profiler record of Run 2 illustrating the migration of bed waves of different height and period under the profiler, and the use of the records to predict the thickness of preserved laminae.

Table 3. Lamina characteristics.

Run number ~~

1 2 3

Aggradation rate, r (mmis) Number of laminae predicted from bed profiler records Mean lamina thickness, s, (mm), from bed profiler records Standard deviation of lamina thickness, sSd (mm) from bed profiler records Mean lamina thickness from box-core peels and thin sections (mm) Standard deviation of lamina thickness from box-core peels and thin sections Mean height of bed waves forming laminae (mm) Ratio of mean lamina thickness: mean height of formative bed waves Ratio of mean lamina thickness: mean height of all bed waves Mean lamina thickness from Eq. (2) (mm) Standard deviation of lamina thickness from Eq. (2) (mm) Mean lamina thickness due to aggradation = I r / c (mm) Mean lamina thickness from Eq. (4) (mm)

0-00065 0.0023 12 23 1.4 2.1 0.83 2.67 2.4 2.6

12-4 10.0 (mm) 1.0 1.45

0.11 0.21 0.22 0.38 2.4 2.0 2.13 1.77 0.073 0.25 2.47 2.25

0.0038 13 3.5 2.69 3.34 2.3

13-9 0.25 0.44 3.16 2.79 0.41t 3.57

tusing 1 =1.2 m and c=11.2 mm/s (see Table 2).

measurement. There is no systematic relationship between the height and period of individual bed waves, nor between mean bed wave height and aggradation rate (Table 2). Side-wall observations of the length, height and celerity of bed waves are biased towards the larger forms (Table 2). There is no clear relationship between celerity and either height or length (Fig. 3).

Theoretical distribution of bed wave heights The two-parameter gamma density function Eq. (3) was fitted to the histograms of bed wave height

using least squares minimization techniques (Powell, Marquadt-Levenberg, Simplex; each giving results similar to the others). Figure 2 and Table 2 show that the bed wave height data are described well by the gamma density function. Correlation coefficients associated with the fitted curves are between 0-93 and 0.97. However, values of the parameters of the fitted gamma density functions (particularly the variance) are critically dependent on the shape of the exponential tail of the distribution, that in turn depends on the number of bins used to define the histogram.



Bridge, 1992). Of those bed waves partially preserved as laminae, lamina thickness is generally substantially less than 50% of the formative bed wave height (Fig. 5). Thus, in general, preserved laminae represent less than half the height of only the largest bed waves in the population.

14

Run2

E

a 5 10 15 20 25 30 formative bed-wave height, mm

Fig. 5. Lamina thickness plotted against formative bed wave height for each run. Lines denote the percentage of bed wave height preserved as a lamina. Note that less than 50% of the height of a bed wave is preserved in these experiments.

Bed wave preservation and lamina thickness: comparison with theory

Figure 4 illustrates how trough elevations of successive bed waves are used to predict the thickness of laminae preserved at the bed profiler position, and how each lamina can be related to a specific formative bed wave. The mean and standard deviation of lamina thickness increase as aggradation rate and variance in height of bed waves increase (a decreases), in remarkable agree- ment with theory (Tables 2 and 3 ) . This means that, by measuring the thickness distributions of planar laminae, it is possible to calculate using Eq. (4) either the mean and standard deviation of bed wave height, the mean bed wave celerity, the mean bed wave length, or the mean aggradation rate, given estimates of the other parameters. Furthermore, if the contribution to lamina thickness of the aggradation rate term can be con- sidered to be minor (see Table 3), an estimate of the density function of bed wave heights can be obtained directly.

The ratio of mean lamina thicknesdmean bed wave height also increases as aggradation rate and variance in height of bed waves increase (Table 3) , and this ratio is always greater than the ratio of mean lamina thickness/mean formative bed wave height. The formative bed waves (i.e. those that are actually partially preserved as laminae) are larger than the 71st, 51st and 32nd percentiles of bed wave height as aggradation rate increases from experiment 1 through 3. Larger aggradation rates favour partial preservation of more of the smaller bed waves (see also Best &

Lateral variation in lamina thickness

Lamina thicknesses predicted from the bed height records can be matched closely with those observed in the parts of the box-core peels corresponding to the position of the bed profiler (Fig. 6). Laminae in the box cores can be traced downstream for on the order of a metre, and they typically vary in thickness downstream due to varying elevations of the upper and/or lower bounding surfaces (Fig. 7). Downstream variation in the elevation of a lower bounding surface results from variation of the trough elevation (and height) of the formative bed wave as it migrates downstream, whereas variation in elevation of an upper bounding surface results from changes in the trough elevation of a succeeding bed wave as it migrates. Some of the thickest laminae show internal, low-angle inclined laminae (up to 10 degrees; Fig. 7) and vary in thickness markedly in the across stream direction (see also Best & Bridge, 1992).

Texture of laminae

Our measurements of the vertical variation of grain size in the thin sections of peels reveal that the thickest laminae are commonly the coarsest and may contain mm-scale vertical variations in grain size of tens of microns that reflect the presence of internal low-angle laminae (Fig. 6). Many laminae show an upward coarsening fol- lowed by an upward fining. Observations of the sorting of sediment over low-relief bed waves confirm our previous suggestion that the markedly fine-grained tops of laminae are generated by deposition from suspension on the distal lee sides of bed waves (Fig. 8). This fine sediment infil- trates the upper parts of the underlying laminae, similar to the glazing process described by Paola et al. (1989). Most bed waves do not have avalanche faces, and the grain size of bed-load sediment tends to decrease from the crest to the toe of the lee side, resulting in the coarsening upwards parts of laminae. The largest bed waves may be associated with periodic lee-side flow separation and a spatially fluctuating re-attachment zone. The associated turbulence



Fig. 6. Close-ups of box-core peels from the position of the bed profiler for each run, and corresponding grain-size profiles obtained from thin sections. Laminae are correlated between the peels and grain-size profiles. Laminae predicted from the analysis of bed profiler records (e.g. Fig. 4) are also shown to the right of the grain-size profiles.

reworks sediment on the lee side and may produce laminae with little vertical sorting. How- ever, the low-angle laminae within the thickest laminae are related to turbulent variations in the transport of sediment to and/or on the lee side of bed waves.

CONCLUSIONS

1 The average thickness of planar laminae formed by the migration of low-relief bed waves over an aggrading upper-stage plane bed increases with

aggradation rate and the variance in height of the bed waves. 2 In general, only a small proportion (less than 50% in our experiments) of the height of the largest bed waves in a population becomes preserved as planar laminae. 3 Paola & Borgman’s (1991) theoretical model for the probability density function of stratal thickness in terms of the probability density function of bed wave heights has been adapted for aggrading conditions, and agrees with experimental data remarkably well.



Fig. 7. Lateral variation of planar laminae as shown in selected box-core peels oriented in the down-current direction. Note erosional bases of some laminae, and the low-angle inclined laminae within some of the thicker planar laminae.

bed wave celerity, provided the other parameters can be estimated. 5 Further experiments are needed to understand the lateral variations in lamina thickness (or cross- set thickness) that arise from variations in the height and celerity of bed waves as they migrate.

ACKNOWLEDGMENTS

This work was supported by grants GR9/564 and GR3/8235 from the UK Natural Environment Research Council, for which we are grateful. Many thanks to Neil Woodhouse for grain-size analysis of the SEM photographs, and to Steve Dickman for assistance with the software used to fit the two-parameter gamma density function. JLB is grateful for a Nuffield Foundation Research Fellowship during the conduct of this research.

REFERENCES

Fig. 8. Photographs taken sequentially at 6-second intervals showing migration of a low-height bed wave through the test section. The deposition of relatively fine sediment in the lee of the bed wave (arrowed) results in grain-size fining at the top of the underlying lamina.

4 The theory can be used to estimate, from the distribution of thicknesses of planar laminae, either the distribution of bed wave heights, mean aggradation rate, mean bed wave length, or mean

Best, J.L. and Ashworth, P.J. (1994) A high-resolution ultrasonic bed profiler for use in laboratory flumes. 1. Sediment. Res., A64, 674-675.

Best, J.L. and Bridge, J.S. (1992) The morphology and dynamics of low amplitude bed waves upon upper stage plane beds and the preservation of planar laminae. Sedimentology, 39, 737-752.

Bridge, J.S. and Best, J.L. (1988) Flow, sediment transport and bedform dynamics over the transition from dunes to upper-stage plane beds: implications for the formation of planar laminae. Sedimentology, 35,

Bridge, J.S. and Best, J.L. (1990) Flow, sediment transport and bedform dynamics over the transition from dunes to upper-stage plane beds: implications for the formation of planar laminae: reply. Sedimentology,

Cheel, R.J. and Middleton, G.V. (1986) Measurement of small-scale laminae in sand-sized sediments. J. Sedi- ment. Petrol., 56, 547-549.

Paola, C. and Borgman, L. (1991) Reconstructing random topography from preserved stratification. Sedimentology, 38, 553-565.

753-764.

37, 551-553.


262

Paola, C., Wiele, S.M. and Reinhart, M.A. (1989) Upper- regime parallel lamination as the result of turbulent sediment transport and low-amplitude bedforms. Sedimentology, 36, 47-60.

J. Bridge and J. Best

Manuscript received 5 December 2 995; revision accepted 25 June 1996

LIST OF SYMBOLS

Inverse of mean value of exponential tail of probability density function of topographic height Bed wave celerity [downstream migration rate) Mean flow depth Mean size of bed sediment Median size of bed sediment Darcy-Weisbach friction coefficient Gravitational acceleration Bed wave height Mean bed wave height

Standard deviation of bed wave height Length [spacing) of bed waves Probability density function of stratal thickness Probability density function of bed wave height Aggradation rate Thickness of stratum (lamina or bed) or cross-stratal set Mean thickness of strata Standard deviation of stratal thickness Water surface slope Vertically averaged flow velocity Shear velocity Parameters in gamma density function Gamma function Angle of climb of bed waves relative to mean bed level Dimensionless bed shear stress Fluid density Sediment density Bed shear stress


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