Morphology and dynamics of active bed forms in the Terek R. Channel

ISSN 0097-8078, Water Resources, 2017, Vol. 44, No. 2, pp. 226–236. © Pleiades Publishing, Ltd., 2017. Original Russian Text © N.I. Alekseevskii, A.Yu. Sidorchuk, 2017, published in Vodnye Resursy, 2017, Vol. 44, No. 2, pp. 147–157. HYDROPHYSICAL PROCESSES Morphology and Dynamics of Active Bed Forms in the Terek R. Channel N. I. Alekseevskii† and A. Yu. Sidorchuk* Moscow State University, Moscow, 119991 Russia *Е-mail: aleksey.sidorchuk@geogr.msu.ru Received September 4, 2015 Abstract—Active bed forms of three major classes are formed in Terek lower reaches during summer floods. They include ripples, dunes of the first order, and dunes of the second order (from smaller to larger), which commonly form an incomplete hierarchy. The morphology of the bed forms is essentially stochastic and can be adequately described by probability distribution functions of bed form characteristics for some narrow ranges of hydraulic flow characteristics. At the same time, the mean values of bed form morphometric characteristics (length, height, and asymmetry) show stable relationships with flow velocity and depth. The celerity of active dunes can be adequately described by a modified Snishchenko–Kopaliani formula. The channelforming sediments that move as bed forms in Terek lower reaches account for 7% of sediment transport rate of all channel-forming sediments, a value near the lower limit for rivers with sand alluvium. Keywords: active bed forms, classification, rivers with flood regime, channel-forming sediment transport rate DOI: 10.1134/S0097807817010031 INTRODUCTION Bed forms in river channel and large laboratory flumes can be of two major types: with active motion (active bed forms) and passive motion (passive bed forms) [11]. Examples of studies of the morphology and dynamics of passive bed forms are numerous, as only passive bed forms can be measured in rivers with short spring floods (often with an ice drift) and long low-water periods. In large (and, the more so, in small) laboratory flumes, passive bed forms are commonly formed because of the small relative flow depths. The researchers believe that the bed forms on river bed are always asymmetrical and have a triangular shape with a steep downstream and gentle upstream slope, their motion being governed by bed load, resulting in the formation of cross bedding in alluvial strata (see, e.g., [9]). Studies in tropic rivers with typically long spring floods show active bed form generation in channels in high-water periods with large relative depths. These bed forms are asymmetrical symmetrical and have ellipsoidal shape with convex upstream and downstream slopes; their motion is governed by the dynamics of the wavelike flow structure that generates the bed form; the result of this motion is bedload sediment transport and the formation of a wavy, almost horizontal lamination in the channel alluvium [3, 11, 16, 17]. † Deceased. Active bed forms can also be studied in rivers with high summer floods. In Russia, these are, primarily, the rivers of the North-Caucasian and East-Siberian types according to B.D. Zaikov [6]. A typical example of rivers with the hydrological regime of North-Caucasian type is the Terek with its tributaries. The highest floods occur in the lower reaches of the Terek (downstream of the Sunzha mouth), where summer rain nourishment dominates (on the average, almost 33% of the annual runoff, according to [7]). The water and sediment discharges during such floods in some years can be higher than those during spring floods. Data on water and sediment discharges and channel transformations were collected during a summer flood in a 15-km reach of the Lower Terek (downstream of the Sunzha R. mouth). These materials were only partly published in [2, 10], though they contain very rare for Russian rivers data on the morphology and dynamics of active channel bed forms. STUDY AREA A series of seven bends with a mean step and amplitude of 1300 and 600 m, respectively, and the mean channel width of 260 m was studied in a 15-km reach of the Terek R. downstream of the Sunzha mouth (Fig. 1a). The dynamics and morphology of the bed forms were measured in a linear segment between two downstream bends in this series (Fig. 1b). During the construction of a new road (completed in 2012), a steep bend was cut by a canal, reducing the length of 226 MORPHOLOGY AND DYNAMICS OF ACTIVE BED FORMS 227 (a) (b) 1982 2014 (b) 12 10 13 11 9 8 6 2002 2 3 4 5 7 Longitudinal profile 1 Longitudinal profile 2 500 m Fig. 1. (a) Study area of 1982 in the Terek R. downstream of the Sunzha R. in SPOT 5 photograph of Sep. 2, 2006, and (b) positions of measurement profiles for studying the morphology and dynamics of channel bed forms. The SPOT image was presented by MSU Geoportal. this river segment to 11.5 km. The natural horizontal transformations of these bends are also considerable— the mean bank erosion rates over 50 years are ~3 m/year with local mean velocities reaching 10–15 m/year [2]. However, the banks of the linear channel segments where bed form measurements were carried out remained unchanged in 1982–2014 (Fig. 1b); therefore, data of 1982 are still quite representative. Channel shallows changed their positions from year to year, but the type of relief configuration remained unchanged, i.e., a linear channel with a middle bar (commonly submerged) in the upstream part of the segment; accordingly, two pool hollows are located here, merging to form one in the downstream part of the segment. The depths were measured on July 22–August 30, 1982, by a PEL-3 depth sounder along two longitudinal profiles with lengths of 970 and 500 m fixed by bank marks. The distances on the profiles were determined with the use of 13 transverse bank marks spaced 60–100 m apart. The measurements were made by downstream drifting, with the measurement of the mean surface water flow between the cross sections. Twice a day (in the morning and evening), bed relief was measured at the middle bar and in the right and WATER RESOURCES Vol. 44 No. 2 2017 joint pool hollows. Such measurement frequency was enough to reliably identify the same large bed forms in sound records made at different times and to evaluate the velocity of their displacement. Hydraulic (mean flow velocity and the depth over the bed form) and morphometric (the total length, the lengths of the upstream and downstream slopes, and the bed form height) characteristics were measured for all bed forms. The total of 57 measurements were processed (5927 bed forms) on a long longitudinal profile 1 and 44 measurements (3481 bed forms), on a short profile 2. HYDROLOGICAL–HYDRAULIC CONDITIONS OF THE LOWER TEREK The Lower Terek in the study region has a drainage area of 37400 km2, the mean water discharge of 294 m3/s, the mean maximal discharge of 970 m3/s; the absolute recorded maximum of water discharge was ~2000 m3/s during a flood in June 1914. The mean annual suspended sediment transport rate is estimated at 580 kg/s (18.3 million t/year), though it shows wide variations; for example, the sediment transport rate in 2002 was in excess of 63 million t [5]. 228 ALEKSEEVSKII, SIDORCHUK The measurements of 1982 were carried out at a peak of a flood (with water discharges of 570‒670 m3/s), which lasted from July 22 to August 3 and was followed by a slow decline with water discharge reaching 360 m3/s by August 30. The flow rate varied both along the measurement profiles and over time from 0.12 to 3.5 m/s; the depth varied from 0.4 to 10.3 m; and the Froude number, from 0.02 to 0.9. Two types of sediments can be identified in the lower Terek R.: transit sediments (with particle diameter <0.05 mm), which do not contribute to channel relief transformation (they are absent in the channel alluvium), and channel-forming sediments, of which channel relief forms are composed. The transit sediments occur in suspension, while the channel-forming sediments, both in the suspension and as bed forms. The transport rate of transit sediments at flood peak in cross-section number 12 reached 6000 kg/s at their concentration of 8.9 kg/m3 and dropped to 370– 440 kg/s at concentration of 1.0–1.2 kg/m3 by the end of the study. The temporal variations of transit sediment concentration were high: at flood peak in different measurement periods, at a narrow variation range of water discharges (620‒670 m3/s), it changed 4.5 times, i.e., from 2 to 9 kg/m3, at lesser water discharges, its variations dropped to 2 times. At the same time, the spatial variation of transit sediment concentration within the study reach was relatively low, averaging 20% (at a maximum of 60%). On the average, 50% of transit sediments are fractions with particle diameter of 0.05‒0.01 mm; 20% have the size of 0.01‒0.005 mm; and 30%, finer than 0.005 mm. Analysis of a vast body of data on the grain-size distribution of suspended sediments at different hydrological stations along the lower Terek shows that the content of transit sediments in their total concentration averages ~60%. Therefore, their annual yield in the study area is estimated at 11 million t. Transit sediments are generated on river basin; they are not deposited in the Lower Terek channel (only on its floodplain); and their relationship with the hydraulic characteristic of the channel flow is indirect. Conversely, the concentration of channel-forming sediments depends on the local hydraulic conditions in the channel. The bottom concentration (at a height of Δ above the bed) of suspended channel-forming sediments of the ith fraction Ci(Δ) is determined by the proportion of the sediments of the ith fraction of bed alluvium pi, the settling velocity of the particles ωi, and the flow velocity U. The bed alluvium in the studied river reach is sand with a mean diameter of 0.4 mm: on the average, 20% are coarse sands with diameter of 1‒0.5 mm; ~60% are medium-size sands with a diameter of 0.5‒0.25 mm; 20% are fine sands with a diameter of 0.25‒0.1 mm; and ~2% are fine sands with a diameter of 0.1‒0.05 mm. The distribution of sediments over fractions is close to lognormal. For the lower Terek, the following empirical relationship was derived [2]: () 3 (1) C i ( Δ ) = 1.2 × 10 −5 U pi , ω here, the value of Δ is determined by the height of the fish-like sink-stone, on which the nozzle of the vacuum bathometer was installed, and amounts to 12– 15 cm. The distribution of the channel-forming sediment concentration over flow depth can be well approximated by Rouse–Velikanov formula [4]. Both time and space variations of the concentration of channel-forming sediments and their proportion in the total suspension concentration are wide. Thus, at the flood peak of July 21, 1982, the proportion of channel-forming sediments (with a diameter of >0.05 mm), measured at 13 cross-sections, varied from 16 to 56% in the section-averaged suspended sediment concentration. Accordingly, the particlesize distribution of the suspended channel-forming sediments also varies widely, though commonly dominating are sediments of the fraction 0.25‒0.1 mm (up to 60% of channel-forming sediments). On the average, the proportion of channel-forming sediments in the total suspended sediment concentration is ~40%; and their annual runoff in the study area is estimated at 7.3 million t. SAND-WAVE RELIEF IN THE LOWER TEREK CHANNEL The lower reaches of the Terek, as well as many other rivers with sand alluvium, shows a complex hierarchic structure of channel sand-wave forms. Up to six levels of sand-wave hierarchy can be identified. Large sand waves—alternating bars and middle bars—with a mean length of 600 m commonly consist of mediumsize sand-waves, both larger, with a mean length of 230 m, and smaller with a mean depth of 90 m, modeling the larger ones. The large and medium-size sand waves are inundated during floods, while at low discharges, they rise over the surface and determine the configuration of the low-water channel. This complex is superimposed by small and smallest sand-waves, which can include up to three hierarchic levels: the largest second-order dunes (dunes_2), the smaller first-order dunes (dunes_1), and the smallest sandwaves—megaripples. The small and smallest sandwaves almost always are under water. The repeated measurements in the summer of 1982 were focused on this type of bottom bed forms. Classification of Bed Forms The common hierarchy of small and smallest bed forms in rivers with sand alluvium is as follows [1, 12, 14]: dunes_2 superimpose on the medium-size sandwaves; superimposed on dunes_2 are smaller dunes_1, which, in their turn, are modeled by megaripples. The WATER RESOURCES Vol. 44 No. 2 2017 MORPHOLOGY AND DYNAMICS OF ACTIVE BED FORMS 229 Flow direction July 21, 1982, 15.10 July 21, 1982, 18.50 July 22, 1982, 10.15 July 22, 1982, 19.20 500 cm July 23, 1982, 08.10 300 100 July 23, 1982, 16.10 Marks Distance from mark 5 9 10 400 11 500 13 12 600 700 m Fig. 2. Morphology and dynamics of active dunes_1 in the Lower Terek in the profile 1 (marks 9–13) during flood rise at conventional level of 800 cm. The distances are measured from mark 5. situation during floods in the lower Terek is different. Full three-member bed form hierarchy can be generated at some reaches of the river channel; the above classification is applicable to this case. In other reaches of the channel (or in the same reach, but in other time), no full hierarchy will form: it can be double-member or absent at all. Such incomplete structure of the bed form relief most often generates in the Lower Terek (Fig. 2); therefore, to classify bed forms in such structures requires the following additional analysis of channel relief. It is reasonable to assign the bed forms of the Lower Terek special indices. In the case of three-member hierarchy, we denote the largest bed forms as bed forms_003, on which bed forms_023 are superimposed, with bed forms_123 on the top level. In the case of double-member hierarchy, bed forms_020 will be at the lower level and bed forms_120 at the upper level. WATER RESOURCES Vol. 44 No. 2 2017 In the case of single-member channel relief structure, the index bed forms_100 will be used. Theoretical studies into the origin of the bed form relief of river channel, carried out by a method of small initial perturbations, showed that, for megaripples, the dimensionless wave number krD = 2πD/Lr is related to the Froude number Fr = U gD by an inverse proportion relationship [8, 12]: kr D = 1 , Fr (2) where Lr is the length of the megaripples, U is flow velocity, D is its depth, measured as the depth over the bed form crest plus half the bed form height, g is acceleration due to gravity. For the boundary between the small and medium-size sand-waves, a formula was obtained [10]: 230 ALEKSEEVSKII, SIDORCHUK N _003 _023 _23 Dunes_2 35 70 140 Dunes_1 Megaripples 30 60 120 25 50 100 20 40 80 15 30 60 10 20 40 5 10 20 0 –1.5 –1.3 –1.1 –0.9 –0.7 –0.5 –0.3 –0.1 0 0.1 0.2 0.3 0.4 0.5 log (kD × Fr) –1.4 –1.2 –1.0 –0.8 –0.6 –0.4 –0.2 Bed form_003 Bed form_023 Bed form_123 Fig. 3. Histograms and plots of functions approximating the values of log(2πDFr/L) for bed forms_003, _023, and _123. 0.3  2g  (3) k mD =  2  exp ( − 2.0Fr ) , C 0  here C0 is the coefficient in Chezy formula, km is the wave number for medium-size sand-waves. In the diagram 2πD/L ~ Fr, a domain of small sand-waves is located between these two lines, where theoretical calculations give no additional boundaries. The empirical data on the three-member hierarchical structure of dunes and megaripples in the river channel, collected in rivers with sand alluvium [12], confirm the theoretical relationships (2) and (3). The results of measurements for bed forms in the Lower Terek, which form a complete three-member hierarchy (bed forms _123, _023, and _003), imposed on the diagram 2πD/L ~ Fr, also confirm the earlier identified domains, despite the quite explainable stochastic character of their boundaries. The probability densities for the logarithms of coefficient a = 2πDFr/L for bed forms of these classes (Fig. 3) can be well approximated (Kolmogorov–Smirnov test 0.03–0.05) by the normal distribution (i.e., the distribution of coefficient a is lognormal). In the case of incomplete hierarchy, the bed forms with indices _100, _120, _103, and _023 can be referred to certain class only with some probability. Based on the size and hydraulic characteristics of the flow (by the value of coefficient a), the bed forms_100 and _120 can be referred to either megaripples or dunes_1, or dunes_2; the bed forms_103 are both megaripples and dunes_1; the bed forms_020 are mostly dunes_1. As a first approximation, the regional classification of such bed forms can be based on histograms of the logarithms of the coefficient a = 2πDFr/L for bed forms_123, _023, and _003 (Fig. 3). The number of megaripples nr in the group of N different bed forms, combined by the value of log a (in some specified range), can be evaluated as: N _123 (4) , N _123 + N _ 023 + N _ 003 and they can be separated from this group by random choice. To determine the number of dunes_1 in such group, N_123 is replaced by N_023 in the numerator of formula (4), and for dunes_2, by N_003. Here, N_123, N_023, and N_003 are the numbers of bed forms of appropriate types in measurements on the Lower Terek in this range of loga values, according to histograms in Fig. 3. Formula (4) was applied to data arrays with bed form characteristics of the Lower Terek, resulting in that the bed forms in the incomplete hierarchies were also classified as megaripples, dunes_1, and dunes_2. The probability densities for the logarithms of the coefficient a = 2πDFr/L for the entire complex of bed forms are also well approximated by a normal distribution with the means and standard deviations close to those obtained only for bed forms in three-member hierarchic complexes (Table 1). nr = N Morphology of Megaripples and Dunes The results of processing measurement data on the Lower Terek show (Table 1; Fig. 4), that the megaripple length in the Lower Terek for the mode of lognormal probability density function can be described by the formula k r D = 0.98 , Fr (5) Lr = 6.4DFr. (6) or WATER RESOURCES Vol. 44 No. 2 2017 MORPHOLOGY AND DYNAMICS OF ACTIVE BED FORMS 231 Table 1. Characteristics of the normal distribution function for logarithms of the coefficient a = 2πDFr/L for bed forms in the Lower Terek Bed forms Number of bed forms Mean value of log(a) Standard deviation Bed forms in three-member hierarchic complexes 825 –0.074 0.243 305 –0.538 0.211 158 –0.936 0.22 Combined bed form array after classification by formula (4) 5840 –0.009 0.196 1741 –0.451 0.195 575 –0.865 0.187 Megaripples Dunes_1 Dunes_2 Megaripples Dunes_1 Dunes_2 In the case of dunes_1, the coefficient in formula (5) is 0.35, and for dunes_2, it is 0.14 (Fig. 4). The appropriate coefficients in formula (6) are 17.7 and 46.0. A linear relationship appears at Fr < 0.6, and, in more kinetic flows, the relationship becomes exponential. For the probability density functions constructed for bed form lengths within narrow ranges of flow velocities (ΔU = 0.5 m/s) and depths (ΔD = 0.5 m), the distribution is still lognormal. As was the case with the dunes and megaripples in other rivers [12], the Lower Terek shows a close relationship between the standard deviation σ and the mean values Lav. Earlier, a linear relationship was found to exist between these variables, i.e., the coefficient of variation σ/Lav kept constant whatever the combination of flow velocity and depth. Data on the Terek, obtained for a wider range of hydraulic characteristics of the flow, show that the linear relationship between σ and Lav is still true here with reliability R2 = 0.94. The coefficient of variation is 0.43 for megaripple lengths and 0.28 for dunes_1; in the case of dunes _2, the available data are not enough to obtain a reliable relationship. During the flood of July–August 1982, most bed forms in the Lower Terek were isometric in the longitudinal section. For megaripples, the ratios of the upper and lower slopes A = Lu/Ll, in 75% of cases fall within 0.5‒2.0; and, in 30% of cases, within 0.8‒1.25. For dunes_1, 64% of A values lie within 0.5‒2.0; and 23%, within 0.8‒1.25; for dunes_2, these ranges contain 68 and 22% skewness values. The probability distribution for the logarithms of skewness A is described by Laplace function (Fig. 5) with a distinct mode at values logA ~ 0 (A ~ 1). Within narrow ranges of flow velocities and depths, a dependence of bed form asymmetry σ on the mean values Аav can be constructed. This dependence is nonlinear and it can be approximated by a power function with R 2 = 0.52‒0.58: q σ = rAav . (7) For megaripples, r = 0.8, q = 1.6; for dunes_1 – r = 1.0 and q = 2.0. Therefore, for bed forms with skewness WATER RESOURCES Vol. 44 No. 2 2017 Kolmogorov–Smirnov test 0.025 0.04 0.056 0.03 0.05 0.051 values less than unit (the upper slope is shorter than the lower slope), the coefficient of variation of skewness is less than that for bed forms with skewness greater than unit (the upper slope is longer than the lower). The isometry of bed forms is an important indicator to the active stage of their motion. Also typical of this stage is the convex shape of both upper and lower slopes, which is most distinct in large dunes (Fig. 2). The relative height of the megaripples h/D changes with flow velocity. At velocities of 0.1‒0.5 m/s, the bed of the channel is nearly flat; at the velocity of 1.3– 1.5 m/s, h/D reaches a maximum; with a further increase in the velocity, the relative height of the megaripples decreases. This regularity is not strong for the medium values of h/D (Fig. 6a), but it can be clearly seen in the upper envelope of the field of points. The relative height of dunes_1 increases with flow velocity increasing to 2.5‒2.7 m/s, while at a further increase in the velocities, the bed forms show a tendency toward erosion. Dunes_2 keep increasing their height at flow velocities of ~3 m/s (Fig. 6a). These regularities are clearly seen for both medium and maximal values of h/D. Thus, an increase in flow velocity causes flattening of bed forms; in this case, the megaripples require lesser flow velocities than the dunes_1 do; the dunes_2 show no such effect within the measured velocity range. Within narrow flow velocity ranges, the probability densities of the relative heights of all types of megaripples are well described by gamma distribution (Fig. 6b). The dependence of standard deviation σ of the relative bed form height on their mean values (h/D)av is linear, and the coefficient of variation is 0.52 for megaripples and 0.55 for dunes_1. In shallow areas, a drop in river water level may cause a short-time increase in flow velocity up to 2.5‒3.0 m/s in the conditions of low flow depth. In this case, the bed forms will be completely washed out to give place to a flat bed of the second stage. In some cases, such regime in shallow zones leads to the forma- 232 ALEKSEEVSKII, SIDORCHUK kD = 2πD/L 10 kd1D = 0.35/Fr Megaripples Dunes_1 krD = 0.98/Fr kd2D = 0.14/Fr 1 Dunes_2 0.1 0.01 0.1 1 Er Fig. 4. Correlation of the dimensionless wave number kD = 2πD/L for megaripples, dunes_1 and dunes_2 in the Lower Terek with the Froude number. tion of antidunes, which have concave slopes on both upstream and downstream sides (Fig. 7). Dune Dynamics The bed form celrity was determined by repeated measurements within the longitudinal profile 1 in a 160-m-long segment between marks 10 and 12 (Figs. 1, 2). This is a single pool hollow with uniform flow and steady depths under the bed forms; therefore, the mean hydraulic characteristics and the celerities were evaluated for individual bed forms over the time of their passage through this segment (Table 2). A histogram of the logarithms of coefficient a shows these bed forms to belong to the class of dunes_1. Commonly, for a part of their existence time, they are disturbed by megaripples (bed form_120), while in the other part of the time, they are simple dunes_1 (bed form_100). In some cases, dunes_1 disturb the surface of large dunes_2 (Fig. 2 gives the situation as of the morning of July 22, 1982). In all cases, these are active bed forms, and their shape does not show any significant unidirectional changes (skewness) over the movement time [11]. The celerity of dunes_1 (Cg, m/h) is determined by the hydraulic characteristics of the flow and can be WATER RESOURCES Vol. 44 No. 2 2017 MORPHOLOGY AND DYNAMICS OF ACTIVE BED FORMS N 2400 2200 2000 1800 1600 1400 1200 1000 800 600 400 200 0 Laplace (x; –0.0737; 0.2184) N = 4990, Mean = –0.074, StdDv = 0.31 233 (а) h/D 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 Dunes_2 Dunes_1 Megaripples –1.2 –0.8 –0.4 0 0.2 0.4 0.6 0.8 1.0 1.2 –1.0 –0.6 –0.2 log (Lu/Ld) 0 N 700 Fig. 5. Histogram and a plot of approximating function for the skewness of Lower Terek megaripples. 600 0.5 1.0 1.5 2.0 2.5 (b) 3.0 U, m/s N = 5770, Mean = 0.1, StdDv = 0.056 Gamma (x/0.0298; 3.3415)/0.0298 500 satisfactorily described by an expression similar to the Snishchenko–Kopaliany formula [13]: 3 (8) C g = bU . D The mean value of the empirical coefficient b in formula (8) is 1.91 s3/(m h). The coefficient tends to increase with the simplification of bed form hierarchy, simple dunes_1 commonly move faster than those disturbed by megaripples: b = b1 + b2 p. (9) Here, p is the proportion of time t/T, where t is the time within which the dunes_1, when passing a segment 160 m within time T, are simple bed forms; the coefficients b1 and b2 are 1.8 and 0.6 s3/(m h), respectively. Formula (9) indicates only a tendency, the scatter of points in this relationship is high; though, in some cases, this tendency is very clear (Fig. 3.13 in [12]). The comparison of data on the motion of active dunes_1 for the Lower Terek with the measured celerities of active dunes in the Niger R. (Table 1 in [11]) shows them to be in satisfactory agreement (Fig. 8). For the combined data, the mean value of coefficient b equals 1.9 (at the Cg expressed in m/h) with the reliability R 2 = 0.82. In addition, this data can be satisfactorily (with a reliability R2 = 0.71) approximated by Snishchenko–Kopaliani formula with coefficient 0.016 (with Cg expressed in m/s). 400 300 200 100 0 0.04 0.10 0.16 0.22 0.28 0.34 0.40 h/D Fig. 6. (a) The relative height of bed forms versus flow velocity and (b) probability distribution function for the relative height of megaripples. In such calculations, it is convenient to determine the mean values of the hydraulic characteristics of flow and, for bed form morphometry, the heights and celerities, not for individual bed forms, but for all bed forms in the area under consideration for the specified time interval (Table 3). It can be seen that, during the flood in July–August 1982, the unit sediment transport rate in the form of bed forms in the pool hollow, on the average, is 7.4% of the transport rate of susсm 750 Flow direction Edge 650 550 SEDIMENT TRANSPORT RATE The unit sediment transport rate qs3 in the form of bed forms is evaluated by the formula (10) q s3 = 0.5hgC g. WATER RESOURCES Vol. 44 No. 2 2017 0 20 40 60 80 100 m Fig. 7. Antidunes in shallows in the Lower Terek channel during flood recession. 234 ALEKSEEVSKII, SIDORCHUK Table 2. Morphology and dynamics of individual dunes_1 of the Lower Terek on profile 1 between marks 10–12 (N is the number of measurements with the particular dune_1, T is the duration of observations of the motion of this dune_1, U is the average flow velocity over the dune in this period, D is the mean depth of the flow over the dune + 1/2 of dune height, L is mean dune length, h is the mean dune height, C is the mean dune celerity, p is the part of the time when the dune was not disturbed by megaripples) Observation period Dune number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 beginning end date time date July 22 July 22 July 22 July 23 July 23 July 25 July 26 July 26 July 27 July 28 July 28 July 29 July 29 July 30 July 31 Aug. 1 Aug. 1 Aug. 3 Aug. 4 Aug. 4 Aug. 6 Aug. 6 Aug. 7 Aug. 9 Aug. 10 Aug. 11 Aug. 12 Aug. 14 10:30 10:30 17:10 8:10 16:10 9:00 13:40 13:40 9:00 9:45 9:45 10:40 10:40 13:20 19:20 12:15 12:15 9:25 13:40 13:40 9:45 19:20 9:45 18:35 8:00 16:10 10:00 13:00 July 22 July 23 July 23 July 24 July 24 July 25 July 26 July 26 July 27 July 28 July 29 July 29 July 30 Aug. 1 Aug. 3 Aug. 3 Aug. 3 Aug. 3 Aug. 6 Aug. 7 Aug. 7 Aug. 7 Aug. 9 Aug. 10 Aug. 12 Aug. 14 Aug. 13 Aug. 14 N T, h U, m/s D, m L, m H, m C, m/h p 2.25 2.51 2.62 2.29 2.53 2.53 2.36 2.48 2.73 2.48 2.29 2.24 2.02 2.07 2.08 2.14 2.19 2.2 2.02 2.04 2.05 2.08 1.96 2.31 2.05 1.93 1.72 1.77 3.9 4.6 4.57 5.22 5.2 4.75 7.48 7.65 6.22 5.17 5.3 5.35 5.84 6.44 7.33 7.01 6.36 5.93 6.72 6.19 5.71 5.92 6.32 6.44 5.99 6.12 6.5 6.35 60.84 71.03 81.92 46.94 84.89 113.4 45.64 114.43 104.28 18.32 46.92 37.57 115.34 68.31 32.72 40.8 33.09 67.33 40.05 44.89 60.47 54.83 43.81 53.99 68.61 68.44 48.04 43.69 2.77 3.1 3.53 2.47 2.77 2.45 2.47 2.4 2.3 1.4 1.52 1.3 2 2.13 2.48 2.35 1.62 3.05 2.31 2.52 2.52 1.23 1.7 1.96 3.39 2.51 1.67 3 6.49 6.22 5.23 5.41 6.1 6.16 4.7 3.59 5.26 7.5 3.37 5.14 3.06 2.14 2.74 2.69 2.74 3.52 2.23 2.36 2.53 2.75 2.5 3.73 2.54 1.66 1.67 3.91 0.33 0.25 0.25 0.75 0.33 0 0.33 0.5 0 0.67 0 0 0 0 0.5 1 0.5 0 0.38 0.1 0.6 0 0.33 0.6 0.13 0 0 0.5 time 19:20 8:10 16:10 13:20 15:50 12:30 18:40 17:10 19:10 17:50 18:50 18:50 17:10 15:25 9:25 15:25 15:25 15:25 19:20 16:07 16:07 16:07 18:35 16:15 10:00 18:55 19:05 18:55 3 4 4 4 3 2 3 2 3 3 5 2 4 4 6 6 6 2 8 10 5 3 3 5 8 8 3 2 8.8 21.7 23.0 29.2 23.7 3.5 5.0 3.5 10.2 8.1 33.1 8.2 30.5 50.1 62.1 51.2 51.2 6.0 53.7 74.4 30.4 20.8 56.8 21.7 50.0 74.8 33.1 5.9 pended channel-forming sediments qs2, and the total transport rates of channel-forming sediments account for >70% of transit sediment transport rate. These are the characteristics of the conditions in the pool for the given flood. On the average for the entire channel of the Lower Terek, the part of the channel-forming sediments amounts to 40% of all suspended sediments and, clearly, it is less than that in the pool hollow during spring flood. The proportions of the transport rates of the channel-forming sediments moving in different manner is more stable, as they are determined by the same hydrological characteristics of the flow and these transport rates satisfy the linear relationship (11) q s3 = 0.074q s2 with approximation reliability R 2 = 0.73. Now, with the annual yield of suspended channel-forming sediments estimated at 7.3 million t, the annual sediment yield in the form of bed forms will be 0.54 million t or ~0.3 million m3. As the alluvium in the Lower Terek is not coarse, and the turbulence and the mean flow velocity are high, a large amount of channel-forming sediments can be transported in suspension. ThereWATER RESOURCES Vol. 44 No. 2 2017 MORPHOLOGY AND DYNAMICS OF ACTIVE BED FORMS C1D, m2/h 80 70 60 C1D = 1.7U3.1 R2 = 0.77 50 fore, the sediment yield in the form of bed forms, estimated at 7.4% of the yield of channel-forming sediments in suspension, seems to be close to the lower boundary for such relationships in rivers with sand alluvium. For example, in the Lower Lena at Kyusyur station, this ratio is close to 100%, i.e., the amounts of channel-forming sediments transported as bed forms and as suspension are nearly equal [15]. 40 CONCLUSIONS Active dunes in the Terek 30 20 10 0 Active dunes in the Niger 0.5 1.0 1.5 235 2.0 2.5 3.0 U, m/s Fig. 8. Correlation of the celerity of active dune_1 in the Lower Terek and the Niger with the flow velocity and depth. Active bed forms of three main classes—megaripples, dunes_1, and dunes_2—form in the Lower Terek even during small summer floods with water discharges of 600–700 m3/s. The analysis of their morphology and dynamics generally confirms the major relationships obtained for bed forms in other rivers with sand alluvium. Data on the Lower Terek show these can be applied to a wider range of hydraulic flow characteristics, i.e., velocities, depths, and Froude numbers. The hierarchy of active bed forms in the Lower Terek channel during floods is rarely complete (threemember). Two-member hierarchy, if any, is common. This complicates the classification of bed forms. Table 3. Daily averages of flow hydraulic characteristics (flow velocity U and depth D) and morphodynamic characteristics of dunes_1 (height h and celerity Cg) in the 160-m-long segment of profile 1 between marks 10 and 12 during the flood of July 22–August 10, 1982 in the pool hollow on the Lower Terek; calculated by formula (10) specific (per unit channel width) transport rates of channel-forming sediments moving as bed forms qs3 are compared with the measured transport rates of transit sediments qs1 and the transport rates of suspended channel-forming sediments qs2 calculated by formula (1) and Rouse–Velikanov curve Date July 22 July 23 July 24 July 25 July 26 July 27 July 28 July 29 July 30 July 31 Aug 1 Aug. 2 Aug. 3 Aug. 4 Aug. 5 Aug. 6 Aug. 7 Aug. 9 Aug. 10 U, m/s D, m h, m Cg, 105 m/s qs3, m2/s 2.2 2.3 2.2 2.3 2.3 2.0 1.9 2.2 2.1 2.0 2.1 1.9 2.0 2.0 2.3 2.0 1.8 1.8 1.8 4.5 5.3 5.5 7.3 4.7 5.9 6.2 6.7 6.6 5.9 6.2 6.3 6.2 6.0 6.3 5.9 6.3 6.1 6.2 2.9 2.3 2.1 2.1 1.7 2.0 1.6 2.4 2.4 1.9 2.0 2.7 2.3 1.8 2.6 2.8 1.6 1.5 2.5 115.1 153.0 127.7 185.6 146.2 89.8 75.5 86.8 85.9 68.5 60.8 59.6 73.7 63.5 132.5 69.4 45.8 31.7 68.0 0.00162 0.00173 0.00134 0.00194 0.00120 0.00088 0.00060 0.00110 0.00105 0.00065 0.00061 0.00078 0.00084 0.00056 0.00167 0.00098 0.00040 0.00024 0.00087 WATER RESOURCES Vol. 44 No. 2 2017 qs1, m2/s qs2, m2/s (qs2 + qs3)/qs1, % qs3/qs2, % 0.012 0.028 0.027 0.076 0.035 0.026 0.030 0.030 0.018 0.013 0.012 0.035 0.012 0.016 0.016 0.011 0.011 0.010 0.018 0.0143 0.0212 0.0162 0.0242 0.0187 0.0099 0.0094 0.0217 0.0145 0.0107 0.0127 0.0090 0.0104 0.0102 0.0237 0.0110 0.0068 0.0055 0.0064 134.5 82.9 64.6 34.1 57.5 40.9 33.4 74.9 86.8 90.4 108.8 28.1 89.6 67.0 156.1 105.7 67.2 56.3 39.7 11.4 8.2 8.3 8.0 6.4 8.9 6.3 5.1 7.2 6.1 4.8 8.6 8.1 5.4 7.0 8.9 5.8 4.3 13.5 236 ALEKSEEVSKII, SIDORCHUK The morphology of active bed forms is essentially stochastic and can be adequately described by probability distribution functions of bed form characteristics for some ranges of hydraulic flow characteristics. The distribution functions are asymmetric and can be best described by lognormal or gamma distribution. The first two moments of these distributions are commonly interrelated, making the distributions oneparameter. The arithmetic or geometric means of the morphometric characteristics of bed forms show stable relationships with hydraulic flow characteristics, in some cases (as for megaripple lengths), such empirical relationships agree with theoretical calculations (formulas (5), (6)). The celerity of active dunes_1 can be acceptably described by Snishchenko–Kopaliani formula or, somewhat better, by its modification—formula (8). The dynamics of bed forms in the hierarchic system depends on the complexity of this system: the more complex the hierarchy, the slower the motion of the bed forms basic for this hierarchy. The part of the channel-forming sediments moving as bed forms in the Lower Terek is ~7% of the transport rate of all channel-forming sediments, i.e., it is close to the lower limit for such relationships in rivers with sand alluvium. ACKNOWLEDGMENTS This study was implemented under State funded program “The Evolution and Transformation of Erosion–Channel Systems under Changing Environment and Human impact.” REFERENCES 1. Alekseevskii, N.I., Bed load transport at a developed structure of channel relief, Meteorol. Gidrol., 1990, no. 9, pp. 100–105. 2. Alekseevskii, N.I. and Sidorchuk, A.Yu., Morphology and dynamics of channel relief in the Lower Terek, in Zemel’nye i vodnye resursy (Land and Water Resources), Moscow: Izd Mosk. Univ., 1990, pp. 87–95. 3. Berkovich, K.M. and Sidorchuk, A.Yu., Dynamics of channel relief of the Lower Niger, Izv. Akad. Nauk SSSR, Ser. Geogr., 1983, no. 4, pp. 66–72. 4. 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Sidorchuk, A.Yu., Morphology and dynamics of channel relief forms in Terek mouth area, in Kaspiiskoe more. Gidrologiya ust’ev rek Tereka i Sulaka (Caspian Sea. Hydrology of the Terek and Sulak River deltas), Moscow: Nauka, 1993, pp. 89–102. 11. Sidorchuk, A.Yu., Evaluating bedload transport rate in a river bed taking into account data on the active and passive bed-forms dynamics, Water Resour., 2015, vol. 42, no. 1, pp. 38–51. 12. Sidorchuk, A.Yu., Struktura rel’efa rechnogo rusla (Relief Structure of River Channels), St. Petersburg: Gidrometeoizdat, 1992. 13. Snishchenko, B.F. and Kopaliani, Z.D., On the celerity of bed forms in rivers and laboratory experiments, Tr. GGI (Gos. Gidrol. Inst., State Hydrological Institute), 1978, no. 252, pp. 30–37. 14. Alekseevskiy, N., Movement of bed forms and sediment yield of rivers, IAHS Publication, 2004, no. 288, pp. 395–403. 15. Alekseevskiy, N. and Sidorchuk, A., Total sediment yield of the Lena River, Eastern Siberia, Hydrological and Geochemical Processes in Large Scale River Basins. Manaus, 1999, p. 26. 16. Best, J., The fluid dynamics of river dunes: a review and some future research directions, J. Geophys. Res., 2005, vol. 110, no. 4, p. 1–21. 17. Vionnet, C., Marti, C., Amsler, M., and Rodriguez, L., The use of relative celerities of bedforms to compute sediment transport in the Paraná River, IAHS Publ., 1998, no. 249, pp. 399–406. Translated by G. Krichevets WATER RESOURCES Vol. 44 No. 2 2017