Analysis of small-scale gravel bed topography during armoring



[1] In evaluating the resistance of sediment particles to entrainment by the action of the flow in a river, the grain geometry is usually characterized using representative sizes. This approach has been dictated, initially by lack of physical insight, but more recently by the lack of analytical tools able to describe the 3-D nature of surface grain organization on water-worked sediment beds. Laboratory experiments are presented where mixed grain size beds were mobilized under a range of hydraulic and sediment input conditions. Detailed bed topography was measured at various stages. Statistical tools have been adopted which describe the degree of surface organization on water-worked sediment bed surfaces. The degree of particle organization and the bed stability can be evaluated in relative terms using the properties of the probability density distribution of the bed surface elevations and in absolute terms using a properly defined 2-D structure function. The methods described can be applied directly to natural water-worked surfaces given the availability of appropriate bed surface elevation data sets.

1. Introduction

[2] Early field and laboratory observations indicated that water worked mixed grain size beds were characterized by a surface layer whose composition was generally coarser than the underlying deposit sediment [Wolman, 1954; Little and Mayer, 1976; Parker et al., 1982; Day and Egginton, 1983; Drake et al., 1988; Leopold, 1992; Marion and Fraccarollo, 1997]. Consequently, a number of scientists have developed transport rate prediction methods which account for changes in the bed surface particle size distribution [Parker, 1990; Wilcock and MacArdell, 1993; Wilcock, 2001].

[3] However, field and laboratory experiments by Church et al. [1998] indicated that significant stability could be achieved by the formation of cellular type structures on water worked beds provided the shear stresses generated by the flow did not exceed twice Shields' threshold. Their data indicated that in situations of partial transport particularly when there was a low upstream sediment supply, the formation of bed scale structures could significantly increase the shear stress required to entrain grains of a particular size fraction. The development of resilient structures occurred over a longer timescale than any coarsening of the bed and occurred in periods of low transport. These observations are supported by those of Kirchner et al. [1990] who measured the distribution of friction angles within grain size fractions for water-worked and non-water-worked beds. They discovered that not only does the average friction angle increase within a size fraction in water worked beds but also its spread. This is particularly the case for fine grain sizes. Although a number of other studies have suggested that the self-organization of the bed grains is important, little work has been carried out to quantify objectively the degree of self-organization in a water worked sediment bed and its impact on stability.

[4] A new approach has recently been proposed, based on detailed survey of the bed elevations at the grain scale. This tries to account for the spatial arrangement of grains. We will use the term “texture” to describe the patterns of surface grain arrangement. Texture not only depends on the size of the grains but also on their spatial distribution on the surface. Some work has been done on sampling and mathematically describing bed texture and then relating texture measurements to flow resistance [Furbish, 1987; Robert, 1988; de Jong, 1992]. Gomez [1993] tried to provide better tools than the well-known Nikuradse's equivalent roughness approach, by examining linear streamwise step sequences of the bed profiles. These studies have attempted to describe the surface topography using 2-D data, even though the surface arrangement of stable grains on a water worked bed is an inherently 3-D system.

[5] The measurement of bed elevations and the full 3-D quantification of texture may in the future provide a more refined approach to the description of bed material mobility. This is possible as bed elevation data contains some information on the surface size distribution [McEwan et al., 2000] and of the distribution of grain areas exposed to the flow. Although the examination of bed elevation data has produced some promising observations [Marion et al., 1997; Tait et al., 1997] and its application to natural rivers appears feasible [Lane, 2001], appropriate tools for objective data interpretation have not been developed yet.

[6] The main aim of this paper is to illustrate the development of objective mathematical tools capable of defining the degree of surface particle organization in water-worked sediment beds. The paper will also report on the application of these techniques to a laboratory data set which comprises detailed surface measurement of gravel bed armor layers.

2. Experimental Conditions

[7] This paper describes the analysis of data collected as part of a larger research program, led by the Universities of Aberdeen and Glasgow at HR Wallingford in the UK. This program of laboratory tests aimed to investigate the physical mechanisms responsible for the development of armored layers in mixed grain size sediment deposits under different flow and upstream sediment feed conditions in a compound trapezoidal channel. Further details of the experimental program are given by Willetts et al. [1998]. This paper reports on the collection and subsequent analysis of detailed measurements of the bed surface topography during 7 experiments. The experiments that are examined will compare two different flow conditions (bed slope and discharge), three different sediment feed rates and both inbank and overbank flow. Experiment conditions are reported in Table 1.

Table 1. Summary of Tests Conditions
ExperimentDischarge, L/sUpstream Boundary Feed Rate, g/sBed SlopeNumber of Texture SurveysFlow Condition

2.1. Initial Bed, Discharge, and Sediment Feeding

[8] In all the experiments a unimodal sediment mixture was produced by manually mixing three different river gravels to provide a deposit with the following properties: d84 = 8 mm, d50 = 4 mm and d16 = 1 mm. This mixture was placed in the flume and then exposed to a steady flow rate. In five of the seven experiments sediment was fed into the flume for a set time period, then the feed was stopped so that in the final stages of these experiments the bed was subjected to zero sediment feed at the upstream boundary.

[9] In the first two experiments (f and g) with zero upstream sediment feed, the flow discharges were applied to the original mixed grain size bed. The bed degraded until a static armor layer formed and the bed load transport rate diminished to a low level. During the development of the static armor layer bed load transport rates and compositions were measured using a slot type trap spanning the whole width of the channel. In the feed tests (h, i, l, m, n) a sediment input rate was imposed on the upstream boundary using a conveyor belt system. A sediment mixture, containing the two finer river gravels was mixed and was input into the flume at a fixed controlled rate. This feed rate was maintained until a state of equilibrium was believed to have been achieved, as estimated by examining the weight of the collected bed load samples. A dynamic armor had now formed, in equilibrium with the imposed flow and sediment feed rate and composition. Once a state of dynamic equilibrium had been achieved, the upstream sediment feed was stopped and the bed degraded further with a static armor layer forming which then propagated down the flume.

2.2. Erosion/Deposition Phases

[10] Experiments f and g were purely degradational, as no upstream sediment feed was applied. Sediment transport at the downstream end of the flume rapidly decreased, as the bed developed toward a static armor (Figure 1). In these tests the applied hydraulic conditions were close to the critical conditions for entrainment. The flow therefore removed only a small amount of the bed volume and the bed slope did not change significantly. The difference between the two experiments is the discharge level, and the resulting overbank flow in test g that was expected to produce a somewhat higher value of the shear stress on the sediment bed.

Figure 1.

Sediment transport as measured for all tests at the slot trap. Solid line plots show sediment input at the upstream end of the flume.

[11] Experiment h had similar hydraulic conditions to Experiment f but was subjected to upstream feed at a constant rate of 2.5 g/s that lasted 67 hours. The aim of this experiment was to force the system to an equilibrium condition of a mobile armor, then to stop sediment feeding so that the adjustment to a new static armoring condition could be observed. It is apparent from transport data (Figure 1) that a dynamic equilibrium was reached after about 30 hours, while the degradational phase reached the sediment trap shortly before the end of the test.

[12] Experiments i, l, m, and n were all tests with a higher upstream sediment input of 5 g/s. This produced in all experiments an equilibrium condition with mobile armoring. The feed was then stopped, producing a degradational phase tending toward a clear-water static armoring equilibrium. Experiment i (inbank, low slope) was fed for 50 hours, reached a dynamic equilibrium after about 22 hours, and the purely degradational phase reached the position of the bed load trap after about 60 hours, i.e., 10 hours after the feed stop (Figure 1). Experiment l (overbank, low slope) was fed for 42 hours, the bed surface just upstream of the trap reached a dynamic equilibrium after about 13 hours, and the degradational phase reached this point after about 52 hours, i.e., 10 hours after the feed stopped (Figure 1). Experiment m (inbank, high slope) was fed for 32 hours, reached a dynamic equilibrium after about 20 hours, and had a degradational phase starting after about 42 hours, i.e., 10 hours after the feed stop (Figure 1). Experiment n (overbank, high slope) was fed for 32 hours, reached a dynamic equilibrium after about 18 hours, and had a degradational phase starting after about 48 hours, i.e., 16 hours after the feeding ceased (Figure 1).

3. Observation of the Condition of the Bed Surface

3.1. Measurements of Bed Topography

[13] Traditional types of hydraulic and sediment mobility measurements, e.g., bed load transport rate, bed load composition, and bed surface grain size distribution, were measured and are reported by Willetts et al. [1998]. Observations were made of the bed surface as it developed during each test. This was done to discover if the changing surface grain arrangement patterns, indicated by photography and by detailed bed elevation measurements, were characteristic of different types of armor layer. These data were also used to find out if there was the potential to quantify the influence of bed surface topography on sediment mobility.

[14] The shape of the bed surface was estimated from an orthogonal grid of bed surface point elevation measurements. The elevation of each point was measured using a Keyence LAS8010 laser displacement sensor. It was moved over the bed surface in a pre-determined grid of measurement points by an automated positioning system. The laser displacement sensor could measure to an accuracy of ±0.1 mm in the vertical, with this measurement being integrated over an area of approximately 50 μm in diameter. This sensor measured point levels of the bed surface, at 0.5 mm spacings, over a 192 × 192 mm area of the bed which was located on the flume centerline, 1.5 m upstream of the bed load trap. The bed load trap was located 18.7 m downstream of the sediment feeder. The resolution of the sensor and the measurement grid was believed to produce bed surface topography data capable of describing the features of the bed surface arrangement at a grain scale.

[15] The physical collection of these data took several hours to complete, due to the slow speed of the automated positioning system. In order to collect the data the sediment mobility in the experiments had to be temporarily stopped. Each experiment lasted several thousand minutes. At the end of each day the flow rate was reduced to near zero, but the flow depth maintained by raising a downstream weir. The bed topography measurements were made overnight. Each bed surface measurement therefore represents the condition of the bed at an ‘instant’ in time.

3.2. Data Analysis, Matrix Handling, Detrending, and Interpolation

[16] The data generated by the topographical measurement system are 384 × 384 matrices of point elevations. The vertical elevations were measured relative to an arbitrary datum position that was dependent on the vertical position of the laser displacement sensor. The laser displacement sensor has a valid measurement range of ±8mm relative to this zero position; given the grain size distribution used in the original sediment mixture, the vast majority of the point level measurements were within the sensor's range. However, due to the occasional large variations in bed elevation and the fact that the sensor needed to be lowered slightly as the general bed level degraded a small number of out of range values were recorded. In the 42 topography data sets reported in this paper the worst data set contained 0.21% out of range readings while the typical “out of range” rate was around 0.06%. In order to facilitate matrix handling, each out of scale reading was replaced by the average elevation of the eight adjacent points of the grid. Because of the extremely small number of out of scale readings, it may be fairly assumed that this correction does not significantly alter the results or the analysis presented in the following sections.

[17] One further source of data collection error was associated with the physical positioning of the automated positioning system relative to the degrading sediment bed surface. The frame on which the positioning system was located was parallel to the original bed slope. As the bed degraded the slope of the bed changed slightly while the position of the measurement system was fixed. It was decided to de-trend the data in both the lateral and streamwise directions using a simple linear interpolation method to remove any spatial bias in the data sets arising from the change in the sediment bed slope underneath the measurement frame. All data reported have been de-trended in this manner.

[18] The data were originally examined visually. In the later stages of the analysis of the data two different techniques are developed: statistical analysis of the probability distributions of the bed elevations and the use of a structure function to describe surface organization. The first technique can utilize data in an unstructured grid while the second technique requires data to be available in a structured grid.

4. Grain-Scale Properties of the Bed

4.1. Visual Observations

[19] Changes of the bed topography throughout all the runs are apparent from three-dimensional elevation plots of the bed surface. In all experiments the bed topography of the initial screeded bed appeared to be irregular with random grain-scale protrusions around the mean bed elevation. As each experiment progressed, the surface developed more distinct peaks and troughs which had larger deviations from the mean bed level. It is worth noting that visual observation of the surfaces show variations that appear to reflect the sequences of pure degradation and static armor formation as well as feed and dynamic armor formation that were created in the various experiments, as described above.

[20] Virtual images of plane view of the bed surface were obtained from the topography data sets. In Figure 2a the bed has a fairly flat uniform surface at the start of the experiment. Later a more organized pattern develops where depressed zones, many grain diameters in extent, are evident (Figure 2b). These appear to be bounded by chains of prominent coarse particles forming structures which appear to be similar to those described by Church et al. [1998].

Figure 2.

Virtual images of the bed surface obtained from texture data sets of experiment g, (a) for the initial bed and (b) for the final bed (after 72 h 25 m). Elevation is plotted as a gray level so that higher points appear lighter (scales in mm).

4.2. Probability Density Function of the Bed Elevations

[21] The bed condition during the experiments was examined using the probability density function (pdf) of the bed elevations from the mean bed level (Figure 3). Some preliminary applications of this tool were originally presented by Marion et al. [1997]. It can be seen that in each experiment the initial physically screeded bed produces a pdf which is relatively narrow and symmetrical. This is representative of a large number of elevation data points being very close to the mean value. No skewness is expected as a result of the screeding process. As water-working proceeds particles are re-arranged by the action of the flow and the bed can be expected to take a new form which will reflect the nature of the processes that have worked it. Therefore it is reasonable to suppose that changes in the pdfs will in some way follows the different phases of transport within the experiments.

Figure 3.

Probability density of the bed elevations for all experiments.

[22] The pdfs for pure degradation Experiments f and g follow similar trends. As expected, the initial bed has a fairly symmetrical distribution of bed elevations with a relatively small standard deviation. This is consistent with the appearance of the initial bed in Figure 2a. In both Experiments f and g, the subsequent pdfs, at 12 and 13 hours respectively, indicate that the bed rapidly adjusts to the imposition of transport. Further adjustment toward the zero transport equilibrium continues throughout these experiments but the pdfs indicate that the most dramatic adjustment has occurred in the first few hours of each experiment. These observations enable us to recognize the characteristic pdf signature of degradation, namely the pdf becomes flatter and wider.

[23] The feed experiments all contain a more complex sequence of adjustments to the pdfs. Broadly three distinct phases can be identified from the transport rate measurements. First, a period of low transport, possibly degradational, occurs in the interval between the start of the experiment and the arrival of the fed material of the downstream trap. Second a period of quasi-equilibrium develops in which the transport rates roughly matches the feed rate. Third, a period of degradation follows the cessation of sediment feed although there is a lag between feed cessation and the commencement of this phase due to the time it takes for changes in upstream sediment supply to influence events downstream.

[24] The pdfs of the bed elevations are consistent with this conceptual description of the system. In all the feed experiments the pdf for the initial bed is fairly symmetrical and peaked. In Experiments h and i the pdfs, 12 and 13 hours respectively, manifest the typical degradation signature that is they have become flatter and wider in comparison with the initial bed. This is also the case in Experiments l, m and n but the change in the pdfs was smaller. However in the second phase of these Experiments, during the passage of the fed sediment, this trend is reversed and the bed becomes smoother again as indicated by the increased peakiness of the pdfs. In the third phase, after the sediment feed had been turned off and the system had entered a purely degradational phase the pdfs became flatter and more skewed as in the first phase.

[25] Although all the experiments follow this general trend, there are a few exceptions. In experiment i, the pdf at 49 hours becomes very flat, even during the feed stage, and at 69 hours it is quite peaked even though the system has been in a purely degradation phase for a number of hours. In test l the pdf continues to become flatter even though feed material has reached the trap, similar in pattern observed in the pure degradation stage. In experiment m the general pattern is followed except for the final pdf which becomes more peaked even though a degradation phase is dominant.

[26] It appears that the pdfs indicate that the bed surface undergoes some adjustments and that they may be reversible. They also suggest that changes in the pdf pattern reflect change in the sediment mobility, although there are a number of exceptions to this trend particularly in experiments which experience phases of dynamic armoring.

4.3. Structure Function of the Bed Elevations

[27] Nikora et al. [1997, 1998] developed a one dimensional second order Kolmogorov structure function to investigate the structure of stream-wise transects. Goring et al. [1999] extended the concept to two-dimensions using the second order structure function defined in (1). The aim of the work was to investigate statistical properties of the bottom elevations z (x, y) using this function. In this paper we apply this method to a larger data set and relate it explicitly to the prevailing sediment transport conditions. In order to discuss our laboratory results, we are only considering the descriptive effects of the structure function, commonly labeled as D (Δx, Δy) where Δx and Δy are spatial distances between bed elevation readings in the streamwise and lateral flow directions respectively. The structure function D (Δx, Δy) of bed elevation z (x, y) is defined as an average squared increment. The two-dimensional structure function can be estimated from:

equation image

where Δx = nδx and Δy = mδy, δx and δy are the sampling intervals, N and M are the total numbers of measuring points of bed elevations in directions x (streamwise) and y (lateral), respectively. In the case of a full data set for the experiments reported here, Δx = Δy = 0.5 mm and N = M = 384.

[28] The structure function has the following property [Monin and Yaglom, 1975]:

equation image

where Rx, Δy) is the correlation function of the bed elevations and σz is the standard deviation of the bed elevations. The D values are therefore normalized with 2σz2 and then plotted as a surface to show the variation in the structure function with regard to the sampling lags in the streamwise and lateral directions. Examples of these plots, useful in the forthcoming discussion are presented in Figures 49.

Figure 4.

Plot of the structure function D(Δx, Δy) for experiment f indicating typical initial screeded bed.

Figure 5.

Structure function plots for experiment i: (a) at 13 h 56 m and (b) at 56 h 16 m.

Figure 6.

Structure function plots for experiment m: (a) at 1 h 51 m and (b) at 48 h.

Figure 7.

Structure function plots for experiment n: (a) at 7 h 12 m, (b) at 23 h 51 m, and (c) at 64 h 38 m.

Figure 8.

Structure function plots for experiment g: (a) at 10 h 37 m and (b) at 72 h 25 m.

Figure 9.

Structure function plots for experiment l: (a) at 42 h 55 m and (b) at 59 h 34 m.

4.4. Interpretation of the Structure Plots

[29] In this section of the paper the variation in the value of D(Δx, Δy) for different streamwise and lateral lags (Δx and Δy) will be examined. The patterns exhibited in the plots will be linked with different sediment behavior present during the experiments. Two fundamental types of sediment behavior were present at the measurement section. Pure degradation occurs when there is little or no sediment being supplied from upstream. In this type of sediment behavior the bed load transport rate drops quickly to near zero. The second type of behavior occurs when appreciable amounts of mobile sediment enter the measurement section from upstream. While some experiments (f and g) involve a single long duration pure degradation phase the other tests involve phases of degradation, followed by highly mobile sediment input from upstream and then followed by a final phase tending toward pure degradation. (e.g., experiments i, l, m and n).

[30] Initial inspection of the structure plots clearly show two general types of surface topography. Some plots clearly have a single well defined slope in the direction of increasing lateral lag (e.g., Figures 5a and 5b) while other plots indicate a bi-directional slope in both the lateral and streamwise directions (e.g., Figure 8b). All structure function plots tended toward a flat plateau at higher values of streamwise and lateral lags. An explanation of the physical significance of the 2-D structure function must be attempted before meaningful interpretation is possible. Figure 4 shows a typical structure function plot for an initial screeded bed.

[31] Low values of D(Δx, Δy) indicate high levels of correlation between bed levels with a streamwise and lateral lag of Δx and Δy. As the value of D(Δx, Δy) increases the degree of correlation decreases. When the nondimensional value of D(Δx, Δy) reaches unity then the bed elevations have no correlation at such lags and the bed topography can be sensibly taken to be random in organization. D values larger than 1 are associated with negative correlation. One interesting aspect of the structure function is that an isoline of D (Δx, Δy) indicates that the bed topography has a uniform level of bed elevation correlation in the direction tan−1 (Δy/Δx) to the streamwise direction of flow. This property of the structure function may therefore be used to identify the dominant correlation direction of grain scale structures.

[32] Consider the 2-D structure plots in experiment i. These plots, presented in Figure 5, indicate the type of bed organization after approximately 14 and 56 hours of flow. The plot shows spatial lags characterized by values of m and up to 80, corresponding to a maximum lag distance of 40 mm in both the streamwise and lateral directions. It was decided to use only lags up to these maximum values, as the summation of (3) using larger lags has a reduced number of data used and is therefore statistically less representative. At both times the bed material was mobile with measured bed load transport rates of between 2 and 4 g/s. The 2-D structure function plots show very similar patterns of correlation. There is a very strong streamwise coherence, particularly at the smaller lateral lags. This indicates the presence of distinct longitudinal grain-scale structures. This correlation appears to extend well beyond the lateral size of the plot, indicating long streamwise bed structures. These structures do not also seem to have a single dominant lag in the lateral direction. In both Figures 5a and 5b the lateral correlation becomes negligible at lateral lags of around 20 mm indicating that the longitudinal structures are relatively narrow in the lateral direction. Examination of Figure 6a indicates a similar type of bed elevation correlation. This also corresponds with the bed being highly mobile with a transport rate of around 2 g/s. This bed had been subjected to this bed load transport rate for less than 2 hours suggesting that this type of structure develops quickly.

[33] The behavior of the structure function in experiment n demonstrates the transient nature of surface grain organization in this test. The plots in Figure 7 represent three phases, an initial phase in which transport rates were low, followed by a phase of high sediment mobility caused by the upstream sediment feed followed by a final degradation phase. In the two phases in which there is degradation, the 2-D structure functions indicate a degree of coherence in both the streamwise and lateral directions. Examining the isolines of D(Δx, Δy) indicates the grain scale structures that have developed are now aligned at an angle of approximately 30° to the streamwise direction of flow. This pattern of behavior is confirmed by examining the plots from test g in which there is only a single phase of pure degradation (Figure 8b). It is clear that these oblique grain structures appear to be associated with stabilizing “static armoring” sediment beds. The rate at which these grain structures develop is also slower than the longitudinal grain scale structures found in the highly mobile beds (Figure 5). Examination of the structure functions from experiment g (Figure 8) suggests the attainment of an equilibrium correlation in the lateral direction is slower than in the lateral direction. These plots are taken from data collected after 10 and 72 hours of flow and although the longitudinal coherence is similar, at around 10 mm (m = 20), the range of lateral lags over which there is observable coherence has increased from 10 mm (n = 20) to 20 mm (n = 40) in the 62 hours between the measurements.

[34] It is clear that in the absence of large amounts of mobile sediments, that the bed surface organization develops coherence that has components in the streamwise and lateral directions. This development occurs over longer time periods than the longitudinal structures identified in test i (Figure 5). It is also probable, given the small growth rate of the oblique stabilizing grain structures that in the later stages of tests h, i, l, m and n a state of equilibrium was not obtained.

[35] In summary, examination of the 2-D structure function plots has clearly indicated two distinct types of grain-scale bed structure which correspond with different levels of observed sediment mobility (i.e., highly mobile and stabilizing beds). As well as having an impact on sediment mobility it is clear that the formation of these different structures occurs over different timescales. No significant differences were noted in these patterns in experiments with overbank flow.

5. Discussion

[36] A series of experiments were conducted where the grain-scale topography of a mobile sediment bed was measured periodically under variable conditions of discharge, slope, sediment feed conditions and inbank/overbank flow regimes. Visual observation and examination of “virtual” photographs of the bed surface clearly indicated that different grain surface arrangements were visible during different tests and at different times during the same test. The different bed surfaces appeared to be linked to different levels of sediment mobility. This occurred even during tests in which traditional surface sampling methods e.g., wax sampling, indicated no significant change in surface grain size composition [Marion, 1996]. Given the lack of significant grain size coarsening, detected by the wax sampling, it was deemed important that the bed surface topography was measured at a grain scale. This was achieved at a limited number of points during tests. The significance of the measurements lie in the insight provided into the influence of grain arrangement on sediment mobility. While the physical measurement of the bed surface can be easily achieved, standard analysis procedures are not available. This paper reports on attempts to develop appropriate analytical tools.

[37] Initially a purely statistical approach was adopted. Probability density functions of the bed surface elevations were plotted, initially these curves were narrow and as time progressed, in the pure degradation cases (f and g) the curves became progressively flatter and the skewness increased. This is indicative of an increasing proportion of the bed being less than the average elevation. The same pattern is observed in experiments h and i. The results show that after the initial change, the widening of the curves is not constant over time, but reverses. This is evident, for example, in experiment i, where the peak of the curve at the final time is higher than the one before. These changes were associated with the sequence of depositional and erosional phases that affected the bed.

[38] Although the probability density curves show some evidence of changes occurring on the bed topography, they failed to provide a reliable tool to represent particle stability and the degree of organization of mobile beds. In principle they do not carry enough information on the bed structure. The probability density does not carry any ‘directional’ information, i.e., is not capable of representing the geometrical properties of the bed associated with different orientations. It cannot differentiate between beds organized in transverse, rather than longitudinal or oblique particle clusters or grain-scale forms. In Figure 10 a pdf based analysis can distinguish between the bed arrangement in Figures 10a and 10b but is unable to recognize the differences between grain arrangements in Figures 10b and 10c. It is clear that a greater degree of grain interlock and sheltering, resulting in different distributions of the angles of particle alignment and inter particle contact, is likely to produce a more stable bed. This difference would not be identified purely by this statistical approach.

Figure 10.

Schematic of the bed (a) with a small number of coarse surface grains, (b) with a larger number of coarse surface grains, and (c) with the same set of coarse surface grains as Figure 10b but arranged differently.

[39] Therefore a 2-D structure function was employed, because it would be able to differentiate between the different bed structures. The 2-D structure function proved able to reliably show that in degradational phases, grain scale structures developed with coherence in both lateral and streamwise directions, similar to the pattern in Figure 10c. The structure functions also indicated that these patterns took a considerable time to develop. This suggests that in order for a mixed grain bed to achieve a true equilibrium it will require to be exposed to flows for a considerable time, even though the transport rate is low. Strong coherence in both lateral and streamwise directions is always associated with highly stable beds. This suggests that this type of bed texture produces distributions in particle alignment and contact angles which restrict grain mobility. In the dynamic armoring (sediment feed dominated) phases, the features formed had strong coherence only in the streamwise direction. They formed quickly and were associated with high mobility phases. These distinguishing patterns were apparent in all tests. The static armoring pattern at the end of test n still showed streamwise and lateral coherence, although the pattern was slightly more confused than in the pure degradation tests, e.g., test f. This may be due to either the lack of time to achieve bed surface equilibrium or the large depth scale flow features caused by overbank flow.

[40] The experiments presented were all stopped before the purely degradational phase reached a “true” equilibrium. Nevertheless, the sediment transport data appear good enough to identify the sequences of transport phases occurring in the system.

[41] The size of the window used for the measurement of the bed topography (192 × 192 mm) was sufficient to represent the symmetrical structure function associated with the direction-independent static armor development, but it appears too small to determine accurately the correlation of the bed elevations over the larger spatial scale associated with mobile armoring processes.

[42] The bed topography was sampled overnight. It appears that the data sets were too infrequent to describe the transport phenomena in detail. Some attempts made to develop more quantitative discussion of the significance of the structure function, using lump statistical parameters, have been hampered by the large time intervals between data sets. In our experiments, the sampling procedure was highly time-consuming and it was not feasible to perform more frequent measurements within the schedule set for the tests. Future observations of the link between the stability of the particles on a mobile bed and the statistical properties of the bed topography must incorporate more frequent observations of the bed surface. It may be also more appropriate to begin experiments with more “natural” water worked beds rather than with the randomly mixed, screeded sediment beds used in this study.

6. Conclusions

[43] A series of experiments were conducted for the first time in which the grain-scale bed topography of a mobile bed was measured under variable conditions of discharge, slope, sediment feed conditions and inbank/overbank flow regime.

[44] Statistical analysis of the bed elevation data has indicated that different bed surface topography was associated with different sediment mobility conditions. However, probability density functions of the bed elevations did not uniquely identify particular grain scale features.

[45] A 2-D structure function, developed by Goring et al. [1999], consistently indicated the development of two classes of grain scale bed feature. One, slow forming feature, with strong lateral and streamwise coherence was associated with stable beds formed under static armoring conditions. The other, quick forming, with very strong streamwise coherence was associated with phases of dynamic armoring.

[46] The detailing of the characteristics and behavior of these grain scale features, now needs to be explicitly linked with sediment mobility. This will involve more frequent topographic data sets than were available in this study. The analysis procedures reported here provide a promising route to substantial progress given some refinement of the data collection method.


particle dimension bigger than i% of the population.


structure function.

m, n

longitudinal and lateral ratio between space lag and measurement step.


correlation function.

x, y

Cartesian coordinates.

δx, δy

longitudinal and lateral measurement steps.

Δx, Δy

longitudinal and lateral space lags.


bed elevation.


standard deviation of bed elevations.


[47] The collection of the experimental data was part of a large EPSRC funded program (McEwan GR/L22065). Participation of Marion and Tait was through the EC Human Capital and Mobility Programme at HR Wallingford. Subsequent data analysis and collaboration between the Universities of Padova (Italy), Sheffield and Aberdeen (UK) was supported by the EPSRC grants GR/N24803/01 and GR/N24810/01, by the Italian National Research Council, GNDCI, U.O. Padova, by the Italian MURST 40% Project on “Morphodynamics of Fluvial Networks” and by the Marsden Fund administered by the Royal Society of New Zealand (contract NIW001). The authors acknowledge the assistance of John McNeill, Laura Dal Monte and Giacomo Fasolato at various stages of the data processing. Helpful comments on earlier manuscripts were also provided by Professor Brian Willetts.