Statistical properties of armored gravel bed surfaces



[1] Roughness of successively developed armor layers in a large laboratory flume is analyzed considering gravel surfaces as two-dimensional random fields of bed elevations. Statistically derived roughness parameters are evaluated from precise digital elevation models (DEMs) of bed surfaces and are discussed with respect to successive development of stable armor layers. The key results relate to evolution of the probability distribution of bed elevations and its moments and structure functions of various orders with respect to armoring discharge. Our results suggest that statistical analysis of bed elevations is a helpful tool for defining roughness parameters, quantifying armoring effects, and identifying particle orientation, flow direction, and surface-forming processes.

1. Introduction

[2] The accurate description of armor layer roughness in rivers is of fundamental importance for fluvial geomorphology and river hydraulics since it determines such flow properties as mean flow velocity, turbulence, and sediment transport. Energy losses caused by multiscale bed roughness can be broadly subdivided into losses due to particle and form roughness [e.g., Griffiths, 1989]. Energy losses due to form roughness are mainly caused by large-scale roughness elements (or bed forms) and associated form drag. Energy losses due to particle roughness, on the other hand, result from viscous drag on the bed surface and form drag due to small-scale roughness elements such as grains of the bed material. These losses depend on particle size, shape, and arrangement [Schlichting and Gersten, 1997].

[3] Traditionally, particle roughness in gravel bed rivers is associated with a characteristic grain size of the bed material (e.g., d50, d84, or d90). Such percentiles of particle size distributions can be evaluated from laboratory sieve analysis or using Wolman's [1954] sampling method or Fehr's [1987] line-by-number procedure in field studies. A recent comparison of various methods is given in the work of Bunte and Abt [2001]. The basic idea of using a single characteristic grain size as a roughness parameter goes back to the fundamental work of Nikuradse [1933]. The equivalent sand roughness ks for gravel bed rivers can be roughly estimated as d50,d84, or multiples of these characteristic grain sizes [e.g., Hey, 1979; Bathurst, 1985; Graf, 1991; Wiberg and Smith, 1991]. However, gravel bed roughness in natural streams can be adequately described by a single characteristic percentile d only if the shape of the particle size distribution, bed arrangement, particle shape and orientation, packing, spacing and sorting, particle clusters, etc. are universal [Nikora et al., 1998]. In other words, the use of d as a roughness parameter requires that independently formed gravel bed surfaces with the same d show exactly the same resistance to flow for identical hydraulic conditions. The abundance of flow resistance equations for gravel bed rivers, on the other hand, indicates that this is not the case, as many studies showed that bed structure may significantly affect the overall hydraulic resistance [Bathurst, 1978; Bray, 1982; Furbish, 1987; Robert, 1990; Clifford et al., 1992; Baiamonte and Ferro, 1997; Nikora et al., 1998; Ferro, 1999; Aberle and Smart, 2003; Smart et al., 2004].

[4] An alternative way to take into account bed surface structure and to describe bed roughness is to consider the bed as a random field of surface elevations Z(x,y,t), where x and y are longitudinal (main flow direction) and transverse coordinates, and t is time. Until the early nineties, only a few studies have considered gravel bed roughness following this approach by analyzing bed profiles [e.g., Furbish, 1987; Robert, 1988, 1990, 1991; Clifford et al., 1992]. In recent years, this approach has become more popular due to technological advances in bed surface measurements such as photographic techniques and laser scanning methods [Nikora et al., 1998; Goring et al., 1999; Butler et al., 2001; Aberle and Smart, 2003; Marion et al., 2003; Nikora and Walsh, 2004; Smart et al., 2004]. These advances also stimulated the development of methods for representing complex surface topography in high-resolution three-dimensional computational fluid dynamic models [e.g., Lane et al., 2004; Hardy et al., 2005;Carney et al., 2006].

[5] In this paper we explore the applicability of the random field approach to describe the structure of successive water-worked, armored gravel bed surfaces using laboratory data. In addition to conventional statistical parameters such as standard deviation, skewness, and kurtosis coefficients, we apply second-order and higher-order structure functions to reveal statistical properties and potential multiscaling behavior of bed elevation fields. These parameters and methods are used to investigate the evolution of successively developed armor layers. Our target is the grain-scale range, from a few millimeters to the maximum bed particle size on the bed.

2. Data and Methods

[6] The gravel bed topography data analyzed in this paper were obtained from laboratory experiments in a 20 m long, 0.90 m wide and 0.60 m high tilting flume in the laboratory of the Leichtweiss-Institute for Hydraulic Engineering, Technical University of Braunschweig, Germany. These experiments were specifically designed to study the interaction between flow and roughness for flows with low and medium relative submergence [see Aberle and Koll, 2004; V. Nikora et al., Double averaging concept for rough-bed open-channel and overland flows: Theoretical background, submitted to Journal of Hydraulic Engineering, 2006, hereinafter referred to as Nikora et al., submitted manuscript, 2006a; also Double averaging concept for rough-bed open-channel and overland flows: Applications, submitted to Journal of Hydraulic Engineering, 2006, hereinafter referred to as Nikora et al., submitted manuscript, 2006b]. Two different coarse sediment mixtures with 0.63 mm < d < 64 mm (see Figure 1 for grain size distributions) were used as movable bed material for the development of water-worked roughness structures.

Figure 1.

(left) View of the flume during laser scan and (right) grain size distributions of initial bed material.

[7] At the beginning of an experiment, the well-mixed sediment was placed in the flume over a length of 13.40 m and flattened to a thickness of 0.20 m to ensure that the bed surface was parallel to the flume plastic bottom. The distance from the flow straightener at the flume inlet to the sediment bed was 1.00 m and the gap between was bridged with bricks to avoid extensive scouring at the inlet. At the downstream end the sediment body was stabilized by a 0.20 m high sill. The sill was constructed of perforated metal to allow flow in the subsurface layer.

[8] The surface was allowed to armor for a given constant discharge Qa1 which was adjusted by a valve and measured by an inductive flowmeter. When the sediment transport rate became negligible, the discharge was reduced to zero and photographs of the armor layer were taken and analyzed by applying the line-by-number method of Fehr [1987] to obtain the grain size distribution of the armor layer. The method was calibrated by manually sampling the surface layer for a selected experiment and adjusting parameters so that differences between a sieve-based grain size distribution and a distribution from photographs were minimized. A detailed description of the applied method is beyond the scope of this paper, and further information on gravel size analysis from photographs and on statistical approaches to determine the grain size distribution from bed material surface samples is given, e.g., by Adams [1979], Fehr [1987], Fraccarollo and Marion [1995], and Bunte and Abt [2001].

[9] The bed topography was measured along the 2.40 m long test section with an Optimess laser displacement meter (Figure 1) over a width of 0.716 m. The test section was located 8 m from the transition bricks sediment so that neither the scour at the inlet section nor the downstream sill influenced the development of bed topography. Longitudinal bed profiles were recorded with a sampling interval of Δx = 1.0 mm with a lateral spacing between profiles of Δy = 4 mm, and with a vertical precision of ±0.1 mm. The different sampling intervals in x and y directions reflected the footprint of the sampling volume of the Optimess laser system (0.2 mm in flow direction and 4 mm in transverse direction). In total, 180 longitudinal profiles were recorded per surface scan.

[10] The bed survey was followed by velocity measurements made for the bed forming water discharge and for lower discharges with a Dantec three-dimensional (3-D) laser Doppler anemometer (LDA) to study the turbulent flow field over water-worked gravel surfaces. After the velocity measurements had been completed, the bed forming discharge Qa1 was increased to a value Qa2 (> Qa1), so that the existing armor layer was destroyed and a new one developed. The above measurements were repeated and the procedure continued as long as the bed could stabilize itself without a considerable loss of slope or before the sediment was eroded to the plastic bottom of the flume. Table 1 summarizes available data for three experiments named A, B, and C and provides information on applied discharges and associated bed shear stresses. We note that for some of the experiments the armor layer was scanned a second time after the hydraulic measurements were finished. The analysis of these scans revealed no significant differences in armor layer geometry before and after the hydraulic measurements, and therefore these results are not presented here.

Table 1. Summary of the Experimental Conditionsa
ExperimentMatSFQa, L s−1τ0, N m−2d50, mmd84, mmσz, mmSkKuHx
  • a

    Mat, initial bed material; SF, flume slope; Qa, armoring discharge; τ0, bed shear stress; d50, d84, characteristic grain sizes; σz, standard deviation of bed elevations; Sk, skewness; Ku, kurtosis; Hx, scaling exponent. All values correspond to conditions at completion of armoring at the indicated water discharge Q.


[11] All LDA measurements were carried out within the 2.40 m long and 0.364 m wide test section along the centerline of the flume to reduce sidewall effects. Therefore, in this paper we focus on this bed area in order to obtain bed surface characteristics which will be linked with turbulence characteristics in a follow-up study. Ninety-two longitudinal profiles in total were analyzed for each armor layer covering a strip approximately 6dmax wide. The detailed description of the hydraulic measurements is also beyond the scope of this paper as it focuses on the analysis of the topography data. Preliminary results of the hydraulic measurements are given by Aberle and Koll [2004] and Nikora et al. (submitted manuscript, 2006b).

[12] Before the DEMs were analyzed, the planar trend (which was chosen since bed forms were not present) was removed from the data using a least squares fit procedure. Figure 2 shows shaded relief plots of detrended surfaces for experiment A and corresponding armoring discharges Qa of 120, 180, 220, 250a, and 250b L s−1. The lines in each subplot indicate the width of the DEM sections used in the analysis below. It should be noted that the bed was armored twice at a discharge of 250 L s−1 because the bed was accidentally destroyed in a local upstream section during LDA measurements after the first armoring process (250a L s−1). Thus the bed was armored again and the corresponding armoring discharge was assigned index 250b L s−1.

Figure 2.

Digital elevation models shown as shaded relief maps after armoring with Qa = 120, 180, 220, 250a, and 250b L s−1 for experiment A (from left to right). Flow direction is from the bottom to the top. Axis units are in mm.

[13] As a first step in our analysis, the detrended DEMs were used to estimate probability distribution functions (PDF) by removing the mean bed elevations first and then by building smoothed histograms using 100 bins, which cover the overall measuring range of the laser displacement meter of 100 mm (from –50 to +50 mm). The second step of our investigation of surface structure was the evaluation of moment functions. For this purpose we applied the generalized two-dimensional structure function as from Nikora et al. [1998] and Nikora and Walsh [2004]:

equation image

where lx = nδx, ly = mδy, δx and δy are the sampling intervals (1 and 4 mm, respectively), N and M are the total numbers of measuring points of bed elevations in x and y direction, and p is the order of the structure function.

[14] So far, most attention has been devoted to second-order structure functions (p = 2), which are identical to semivariograms [Robert, 1988, 1991; Clifford et al., 1992; Bergeron, 1996; Nikora et al., 1998; Butler et al., 2001; Marion et al., 2003]. The second-order structure function of a globally homogeneous random field has the following relationship with the correlation function R(lx, ly):

equation image

where σz is standard deviation of bed elevations.

[15] Thus, at large lags lx and ly when R(lx, ly) → 0 and DG2 → 2σz2 the data are spatially uncorrelated and the corresponding lags lx and ly at which DG2 → 2σz2 may be used to derive characteristic longitudinal and transverse length scales [Nikora et al., 1998].

3. Results and Discussion

[16] In the following sections we focus on the analysis of bed elevations Z (x, y) with regard to the history of surface forming flow. All DEMs were obtained directly after each armoring step. Thus we focus on stable armor layers rather than on description of gravel bed geometry during the armoring process itself, as was done by Marion et al. [2003].

[17] We first consider the PDFs of bed elevations and their moments (e.g., variance, skewness, and kurtosis), and then we evaluate moment functions such as structure functions for the armored gravel bed surfaces.

3.1. Probability Distribution Functions and Their Moments

[18] Figure 3 shows the PDFs of bed elevations for the surfaces from Figure 2 for the different armoring discharges. Note that we omitted the PDF of the initially screeded bed since it is narrow and steep, with a maximum probability density of 0.31 for zero mean. From Figure 3 it follows that with rising armoring discharge the range of bed elevations increases and the number of observations around zero mean decreases. Thus the PDFs visualize the increase of geometrical roughness due to the accumulation of coarser grains on the surface during the successive armor layer development. The increased range of bed elevations can be quantified by the standard deviation of bed elevations σz (see Table 1), which can be interpreted as a characteristic vertical roughness length of gravel beds [Nikora et al., 1998; Aberle and Smart, 2003]. Furthermore, the increased range of bed elevations becomes also apparent from the characteristic grain sizes d50 and d84 (Table 1).

Figure 3.

Probability distribution functions for armor layers at experiment A for Qa = 120, 180, 220, 250a, and 250b L s−1.

[19] In Figure 4 we examine the relationship between σz, d50 and d84. To compare our laboratory experiments with field measurements, Figure 4 also includes additional field data from New Zealand gravel bed rivers published by Smart et al. [2004]. Figure 4 reveals close relationships between σz and d50 and d84. Both, the σzd50 and σzd84 plots show surprisingly little scatter of data points, while the field data follow the same trend as the laboratory data.

Figure 4.

Standard deviation versus characteristic grain diameters d50 and d84. The circles are for New Zealand field data from Smart et al. [2004]. The solid symbols characterize d50, and the open symbols relate to d84.

[20] In our earlier works [Nikora et al., 1998; Aberle and Smart, 2003] we argued that the standard deviation σz of bed elevations is a better roughness scale compared to conventional d50 or d84. Indeed, Aberle and Smart [2003] demonstrated that roughness of gravel beds with step pool morphology can be better described by σz compared to a characteristic grain size, which did not correlate with σz. Furthermore, Aberle [2000] and Aberle and Smart [2003] were able to show the superiority of σz over characteristic grain sizes in their resistance equation for steep gravel bed flows. Vogel [2003] confirmed the applicability of this flow resistance equation based on σz. The present study of armored gravel surfaces, however, reveals that σz and d50 (or d84) are highly correlated, and therefore are equally suitable as measures of the roughness height. Figure 4 shows that this correlation is also applicable for field data of Smart et al. [2004]. The most plausible reason for this discrepancy is that in the present study and that of Smart et al. [2004] bed forms were not present and, as a consequence, the data from the flume and from the field closely collapsed. This result demonstrates that there are situations when the characteristic grain sizes may be as good roughness descriptors as σz. As our data show, one such situation corresponds to armored beds formed under conditions of a constant water discharge.

[21] A closer investigation of the PDFs shown in Figure 3 reveals that the shapes of the PDFs are not symmetrical. The data spread to the right of zero mean for the water worked beds and this spread is reflected by positive skewness Sk given in Table 1, which we directly relate to armoring effects. Only the initially screeded man-made beds (Qa = 0 L s−1) are characterized by negative skewness. In contrast to an initially screeded bed, the surface of an armored bed is composed of larger stones with finer particles filling depressions between them, thus reducing the magnitude of surface elevations below mean bed level. Therefore the PDFs are positively skewed [Nikora et al., 1998; Smart et al., 2004]. Table 1 shows that Sk values increase with armoring discharge, reflecting the accumulation of larger particles on the surface during successive armorings. One can also note that all measured PDFs are characterized by Ku > 3 (Table 1) indicating an increased (compared to the normal distribution for which Ku,N = 3) intermittency of bed elevations that can be attributed to the effect of rare large particles. Similar values for Sk and Ku for PDFs of longitudinal and transverse profiles of water-worked beds in gravel bed rivers have been reported by Nikora et al. [1998].

[22] All higher statistical moments (σz, Sk, Ku) show a high degree of reproducibility for experiments A and B, which have been carried out with the same bed material and armoring discharges. However, a closer examination of σz, Sk, Ku values given in Table 1 indicates that for similar discharges, all three moments are larger for experiment B compared to experiment A. Two possible explanations can be given for this behavior. First, the initial beds were prepared by different persons and thus the characteristics of the initial sediment bodies (e.g., compactness) may have been slightly different. Another possible explanation is that after the initial bed of experiment A was prepared and armored with Qa = 120 L s−1, more than half a year passed before the next armoring discharge Qa = 180 L s−1 was applied. During this period, the 3-D LDA and a new traverse system were installed to the flume and hydraulic measurements have been carried out with an Acoustic Doppler Velocimeter (ADV) at various discharges equal and smaller than the armoring discharge [Aberle and Koll, 2004]. The DEM for Qa = 120 L s−1 for experiment A used in this study was obtained directly before the LDA measurements were carried out. Thus the bed could settle for more than half a year before the surface was scanned while for all other armoring steps the surface was scanned directly after armoring. Therefore the differences in statistical parameters for experiments A and B for identical discharges may be attributed to additional compaction of the bed.

[23] Table 1 also shows that σz, Sk, and Ku do not change from armoring step of 180 to 240 L s−1 for experiment C, although the bed stabilized itself again during the armoring process. We associate this behavior with bed degradation, which became apparent from the planar fit parameters of the surface and the water surface slope. The slope reduction indicates that the critical shear stress for the mixture, which corresponds to the armor layer, has been reached between Qa = 180 and 240 L s−1. Once this critical shear stress is exceeded, the bed can only stabilize again by degradation of the bed slope [Günter, 1971], which is associated with a reduction of bed shear stress as indicated by τ0 values given in Table 1.

[24] Applying the Kolmogorov-Smirnov test to the data showed that the hypothesis of a normal distribution must be rejected at significance levels of 0.05 and 0.01 for all PDFs under investigation, due to the increased skewness and kurtosis coefficients. Smart et al. [2004] concluded that skewed distributions of natural gravel bed surfaces may well be fitted by a lognormal frequency distribution. However, as for the normal distribution, our analysis showed that the hypothesis of a lognormal distribution must be also rejected. Hence, although all PDFs are bell shaped and may be considered as near-Gaussian, the positive skewness coefficients and high kurtosis coefficients are the main reason for the rejection of the hypothesis of a normal or lognormal distribution.

3.2. Second-Order Structure Functions

[25] In Figure 5 we present second-order structure functions according to (1) for the surfaces shown in Figure 2. The line graphs show DG2(lx, ly = 0), the line and symbol graphs show DG2(lx = 0, ly), and the straight lines define 2σz2 for the corresponding surface.

Figure 5.

Second-order structure functions for ly = 0 (straight lines) and lx = 0 (line and symbol) for the surfaces shown in Figure 2 and for the initial bed (Qa = 0 L s−1). The horizontal lines indicate the corresponding values of 2σz2.

[26] The second-order structure functions follow the pattern that has been observed in various investigations focusing on the analysis of longitudinal and transverse bed profiles [Robert, 1988, 1991; Nikora et al., 1998] as well as surfaces [Butler et al., 2001; Nikora and Walsh, 2004]. The region at small spatial lags, where the structure functions can be approximated by power functions DG2l2H, corresponds to the scaling region, while at large lags the structure functions approach the saturation region with DG2(∞) = 2σz2. The scaling exponent H is known as the Hurst exponent and estimates of Hx are presented in Table 1. Between these two regions, the structure functions are curved in the so-called transition zone [Nikora et al., 1998; Nikora and Walsh, 2004].

[27] Bearing this separation zone in mind, Figure 5 provides important information on surface structure and scaling intervals. The extent of the scaling region increases with rising armoring discharge and for our data, the upper limit for the extent of the scaling region may be defined within lx/d50 < 0.5. Furthermore, the data in Table 1 show a dependency of the scaling exponents Hx on armoring discharge and subsurface material. Beds armored at higher discharges are characterized by larger scaling exponents Hx and largest Hx values are observed for experiment C, which has been carried out with the rougher material II. The scaling exponent Hx can be interpreted as a measure of complexity of topography, where larger values of Hx indicate a less complex topography [Bergeron, 1996]. Increasing Hx values with armoring discharge can therefore be explained by the fact that the smooth surface of large individual particles creates a less complex topography than a large number of small particles. These results support the observations of Robert [1988] and Bergeron [1996], that gravel bed rivers seem to be smoother at the scaling range when the surface is composed of large particles. Our estimated Hx values for water-worked beds range from 0.59 to 0.67, and these values are in the same range as those reported by Nikora et al. [1998], Butler et al. [2001], and Nikora and Walsh [2004] for field and flume studies. Thus the scaling exponent for gravel bed surfaces seems to be in a narrow range although our data also suggest its dependence on grain geometry.

[28] While the data density is large enough in the flow direction (δx = 1 mm) to evaluate the scaling exponents Hx using DG2image Figure 5 shows that the data density in the transverse direction (δy = 4 mm) is not large enough for a reliable estimation of Hy. For all structure functions the second data point of DG2(lx = 0, ly), which corresponds to a lag of 8 mm, is already located in the transition zone. A direct comparison of DG2(lx, ly = 0) and DG2(lx = 0, ly) for the same surfaces shows that the shapes of the curves are similar, although the curves are separated by a small offset [DG2(lx = 0, ly) > DG2(lx, ly = 0)]. The studies of Nikora et al. [1998], Butler et al. [2001] and Nikora and Walsh [2004] indicate that scaling is isotropic at small spatial lags, i.e., Hx = Hy. Such an assumption may be deduced from the parallel lines of DG2(lx = 0, ly) and DG2(lx, ly = 0) in the log-log coordinates. The observed vertical offset between DG2(lx = 0, ly) and DG2(lx, ly = 0) is consistent with shape of the two-dimensional second-order structure functions DG2(lx, ly).

[29] The plots of DG2(lx, ly) are useful to identify isotropy/anisotropy of the random field [Butler et al., 2001; Nikora and Walsh, 2004] and are given in Figure 6 for surfaces shown in Figure 2. All contour lines in Figure 6 are normalized with 2σz2 to enable a direct comparison of the plots during successive armor layer development. The maximum spatial lag shown in both x and y direction has been chosen as ±100 mm, being larger than the maximum particle size of 64 mm. The contour lines for all water-worked beds (Qa > 0 L s−1) are characterized, at relatively small spatial lags, by an elliptical shape. Geometrically, the elliptical shape of the contour lines reflects the elliptical form of the particles. The contour minor axis is aligned with the prevailing orientation of the b (intermediate) axis of the particles and the major axis with the orientation of the a (largest) axis of the particles. Figure 6 shows that the longest axes of the elliptical contours of DG2(lx, ly) at small spatial lags are aligned in flow direction indicating that the majority of particles tend to rest on the bed with their longest axis along the flow. In our study, the constant water discharge was applied until the transport rate became negligible. At this later stage of armoring, smaller particles were not rolling anymore as the applied bed shear stress became close to the critical bed shear stress. Instead, the particles responded to the flow by aligning their longest axis in the flow direction before they finally rested, at armoring completion. This hypothesis of particle alignment finds support in qualitative observations of Allen [1982]. On the other hand, large particles that did not move during armoring were arbitrarily orientated. This explains misalignment of larger elliptical contours of DG2(lx, ly) from the flow direction (Figure 6).

Figure 6.

Contour plots of the second-order structure functions for surfaces at experiment A armored with Qa = 0, 120, 180 L s−1 (top, from left to right), and 220, 250a, 250b (bottom, from left to right).

[30] If the applied bed shear stress would suddenly reduce during active bed load with rolling particles then they would rest on the bed with longest axes across the flow, i.e., differently to our case study. The elliptical contour lines of the structure functions corresponding to this case would also have their longest axis across the flow. Such a scenario has been discussed by Komar and Li [1986], Robert [1991], Nikora et al. [1998], Butler et al. [2001], and Nikora and Walsh [2004]. There may be even more complex particle arrangements such as, for instance, diamond-shaped clusters. In this later case the 2-D structure functions should also exhibit diamond-shaped topography oriented along the flow direction. It should be also noted that there are always particles on the bed deviating from the overall rule, depending on grain shape and position within particle clusters [de Jong, 1995]. Thus our results confirm the findings of Butler et al. [2001] and Nikora and Walsh [2004] that the structure functions of bed elevation provide useful information on the history of surface-forming mechanisms for the water-worked gravel beds.

[31] Figure 6 further reveals that the elliptical contour lines for DG2(lx, ly)/(2σz2) < 0.5 are growing in size for the water-worked beds with increasing armoring discharge. Bearing in mind that the shape of the contour lines reflects particle geometry, the expansion of the contour lines shows that during successive armor layer development an increasing number of coarse particles forms the surface, while finer material is washed out (see Figure 2). In Figure 6 the main axis of the ellipse formed by the contour line DG2(lx, ly)/(2σz2) = 0.5 increases from lx = 18 mm at Qa = 120 L s−1 to lx = 33 at Qa = 250b L s−1. The minor axis increases correspondingly from ly = 15 mm to ly = 28, still being smaller than maximum particle size of 64 mm. Similar results were observed for experiments B and C.

[32] For the initial bed of experiment A, the contour lines of DG2(lx, ly) at very small spatial lags are circular, indicating at isotropy of the surface structure. However, the data for experiment B demonstrate the elliptical contours. We associate the differences in contour shape with bed preparation, since different persons prepared the initial beds. Note that for experiment C no initial scan is available due to a corrupted data file.

3.3. Higher-Order Structure Functions

[33] Nikora and Walsh [2004] recently investigated higher-order structure functions (p = 1 – 6) of gravel bed surfaces of New Zealand rivers studied in the work of Smart et al. [2004]. The higher-order structure functions provide additional information on scaling behavior of water-worked gravel surfaces. In Figure 7 we show structure functions for orders p = 1 – 4 for experiment A with Qa = 250a. Similar to the second-order structure functions, the higher-order structure functions can also be subdivided into a scaling region, a transition region, and a saturation region. The nearly constant value of the structure functions in the saturation region indicates that the data are statistically homogeneous.

Figure 7.

Higher-order structure functions (p = 1 – 4) for ly = 0 (lines) and lx = 0 (symbols) for the gravel surface armored with Qa = 250a L s−1.

[34] There are two types of scaling that can be identified using high-order structure functions: simple scaling corresponds to ξp = Hp while for multiscaling ξpHp, where ξp is the scaling exponent for the scaling region (DGplxξp). Thus the scaling behavior may be identified by plotting the exponents ξp versus order p of the structure functions [Nikora and Walsh, 2004]. Such a plot, shown in Figure 8, indicates that the initial beds are characterized by simple-scaling behavior while the water-worked beds more likely relate to multiscaling. Indeed, with increase in p the deviation of ξp from a linear relation increases indicating multiscaling of water-worked gravel beds. Similar results were obtained by Nikora and Walsh [2004]. The origin of multiscaling behavior in gravel surfaces may relate to surface forming processes [e.g., Nikora, 2005]. However, we also have to keep in mind potential effects of data acquisition, resolution, and detrending, which smooth the real topography and may bias scaling exponents ξp. Proper evaluation of these effects requires highly accurate and extensive data sets, which have not been available yet to the writers.

Figure 8.

Scaling exponents versus structure function order.

3.4. Determination of Flow Direction

[35] As we pointed out above, the elliptical contours of the surface plots of the second-order structure functions may indicate a flow orientation. However, they do not reveal flow direction, i.e., from left to right or from right to left. A possible approach for identifying flow direction is to use local bed inclinations at small spatial lags [Monzavi, 1972; Smart et al., 2004] by defining an average positive bed slope:

equation image

and average negative bed slope

equation image

where np and nn are total numbers of positive and negative slopes, respectively. Note that relationships (3) and (4) are similar to the sign-specific structure functions based on increments of the same sign [Nikora et al., 1994; Frisch, 1995].

[36] Our data show that ɛp < |ɛn| for small lags lx except for experiment A and Qa = 120 L s−1 where ɛp is slightly larger than ɛn for lx = 1 mm and ɛp < |ɛn| for lags 2 mm < lx < 8 mm. This result may be associated with the aforementioned bed aggregation, although the second scan directly after the LDA measurements revealed that ɛp < |ɛn| throughout. We note that standard deviations σz of the two scans could be interpreted as identical when tested for equality at a significance interval of 0.01 using the procedure outlined by Kreyszig [1991] for identical variances of two non-Gaussian distributions.

[37] At small lags ɛp < |ɛn| is expected [Monzavi, 1972; Smart et al., 2004; Nikora and Walsh, 2004], because negative increments of surface elevations in flow direction have higher magnitudes than positive slopes due to particle imbrication and arrangement. We found that at small lags, negative slopes are less frequent than positive slopes as was described by Smart et al. [2004] and Nikora and Walsh [2004].

4. Summary and Conclusions

[38] Previous studies of armor layer development have been restricted to interpretation of grain size distributions at various stages and the roughness of armor layers has been described by characteristic grain diameters. This paper investigates, for the first time, armor layer geometry of successively developed stable armor layers by means of statistical parameters, which were obtained from digital elevation models. The key findings of the study include: (1) the characteristic vertical roughness length of gravel beds may be unambiguously defined by the standard deviation of bed elevations σz, (2) the PDFs of water-worked gravel beds are sensitive to bed armoring suggesting that skewness and kurtosis coefficients can be used for quantifying armoring effects; and (3) the coarsening of the surface during armor layer development as well as particle orientation can be effectively quantified using generalized structure functions of surface elevations.

[39] Relating σz to the applied armoring discharge showed that σz increases during successive armor layer development due to accumulation of coarser grains on the gravel bed surface. Furthermore, close relationships between σz and characteristic grain diameters d50 and d84 have been found for beds without bed forms and prominent clusters. Additional field data from New Zealand rivers, which follow the same relationship as the laboratory data, showed that these relationships are not flume related.

[40] All PDFs of all water-worked armor layers were found to be positively skewed which was related to the armoring process. During the armoring process, finer particles are settling in holes in the bed surface and thus reduce the magnitude of surface elevations below mean bed level, resulting in positive skewness coefficients. Furthermore, skewness coefficients were found to increase with armoring discharge which was associated with accumulation of coarser grains on the bed surface. Besides, kurtosis was always larger than for a normal distribution and the hypothesis of a Gaussian distribution of the PDFs could not be accepted, although the PDFs were bell-shaped.

[41] The coarsening of the bed surface was also observed in the second-order generalized structure functions of bed elevations. The scaling region at small spatial lags was found to be related to approximately 0.5d50 and the structure functions showed that the scaling region depends on armoring discharge. The scaling exponents Hx of DG2(lx, ly = 0), which can be interpreted as a measure of complexity of the surface, were as well found to increase with armoring discharge and to be dependent on subsurface bed material. The increase of scaling exponents with armoring discharge was explained by the fact that the smooth surface of large individual particles creates a less complex topography than a large number of small particles. All evaluated scaling exponents were found to be in a similar range as previously reported ones, indicating at similar scaling-forming mechanism for gravel bed surfaces at small lags. The 2-D generalized structure functions provided information and insight on the coarsening of the surface as well as particle arrangement. Smaller particles were found to be orientated with their longest axis in flow direction and this was reflected by ellipticity of contours of DG2(lx, ly). A method to identify flow direction by means of negative and positive local slopes was also proposed.

[42] The results of this study show that the random field approach for the description of bed roughness provides deeper insight into surface forming processes of gravel beds than is possible by conventional description of bed roughness and texture using characteristic grain sizes. The next step to enhance our understanding of relevant physical processes of the interaction of complex roughness and flow is to link statistical parameters with the turbulent flow field. The double averaging methodology, which is based on temporal and spatial averaging of the Navier-Stokes equation provides a theoretical background for such investigations. This methodology, which was recently revisited by Nikora et al. (submitted manuscript, 2006a; submitted manuscript, 2006b), requires detailed knowledge of surface geometry. This paper attempted to provide such knowledge.


grain diameter.

d50, d84

characteristic grain sizes.


generalized structure function of order p.


scaling exponent.


equivalent sand roughness.



lx, ly

lag in mm in flow and transverse direction, respectively.

m, n


M, N

number of measuring points of bed elevations in x and y direction.


order of the structure function.


armoring discharge.


correlation function.





x, y

spatial coordinates in flow and transverse direction, respectively.

δx, δy

sampling interval in flow and transverse direction, respectively.


bed shear stress.


standard deviation of bed elevations.


scaling exponent


[43] This study was funded by the Deutsche Forschungsgemeinschaft (DFG, grant DI 651/4-2). K. Koll, A. Dittrich, and R. Spigel provided useful comments and suggestions. U. Ecklebe provided technical support. The writers are grateful to three reviewers and the Associate Editor for helpful comments and suggestions.