Morphology of Dune-Like Relief in Rivers

ISSN 0097-8078, Water Resources, 2020, Vol. 47, No. 1, pp. 65–76. © Pleiades Publishing, Ltd., 2020. Russian Text © The Author(s), 2020, published in Vodnye Resursy, 2020, Vol. 47, No. 1, pp. 33–45. HYDROPHYSICAL PROCESSES Morphology of Dune-Like Relief in Rivers A. Yu. Sidorchuk* Moscow State University, Moscow, 119899 Russia *e-mail: fluvial05@gmail.com Received November 27, 2018; revised November 27, 2018; accepted April 11, 2019 Abstract—Linearized equations of two-dimensional hydraulics have been analyzed by the method of small perturbations within a wide range of alluvial features sizes at large values of Froude numbers and hydraulic resistance. The preservation of three-dimensional effects in depth-averaged equations of motion, continuity, and deformation enabled the authors to identify domains with combinations of flow hydraulic characteristics that correspond to alluvial features of different size and morphology. The study confirmed the results of the earlier analysis for subcritical flows with small hydraulic resistance and showed new types of relationships between alluvial features morphology and the hydraulic characteristics of flow for flows with large Froude numbers and high hydraulic resistance. In addition, a relationship between the length of two-dimensional ultramicroforms and hydraulic resistance has been established. A new class of macroforms—two-dimensional macroforms—have been identified. The verification of the results of theoretical analysis by measurements data on the morphology of channel formations and hydraulic characteristics of the flow has shown that the analysis of linearized equations of two-dimensional hydrodynamics by small-perturbation method can be used to determine the morphology and dimensions of channel forms in both subcritical and supercritical streams. The step–pool systems in torrential streams in mountain rivers are analogues of antidunes (in supercritical flows) and ripples (in subcritical flows), obtained in large flumes with sand alluvium. Three-dimensional macroforms are most common in rivers. If such macroforms are well developed in wide channels (at flow width greater than the half-width of the macroform), their length is determined by channel depth. Macroforms in narrower channels cannot develop completely, and their lengths are limited by channel width and can be calculated only through this width. Keywords: small-perturbation method, large Froude numbers, hydraulic resistance, ultramicroforms, step– pool systems, macroforms DOI: 10.1134/S0097807820010133 bed deformations. The main difficulty in applying this approach is that the modern mathematical methods cannot be applied for analytical solution of full nonlinear equations for perturbations of the flow and channel bed. These equations are to be simplified (linearized) to make them soluble without a loss of significant elements of solution. A simplification of the hydrodynamic equations has been found at which the results of theoretical calculations are in a good agreement with the data of measurements in flumes and natural rivers [9, 26]. Studying such equations by small-perturbation method has been carried out for a narrow range of the values of the governing variables. In the numerical calculations, the Froude number FR varied within the range of 0.1–0.5 and the hydraulic resistance was taken to be <0.02. Such values are typical of large lowland rivers. At the same time, the small-perturbation method was successfully applied to problems with large Froude numbers FR, though for solving specific problems, e.g., in studying the formation of antidunes [21]. Therefore, it is proposed to INTRODUCTION Attempts to give a theoretical description of the dune-like relief of river channel have been numerous. Thus, more than 50 hypotheses of the formation mechanisms of river bends have been proposed by as long ago as the mid-XX century [6]. In about the same time, the method of small perturbations came into practice [10, 21]. Many studies have shown this method to be effective for solving various problems. K.V. Grishanin [2] gave the following characteristic of the situation: “Taken together, the studies… allow to highly appreciate the contribution made by the small perturbation method to the solution of the problem of the origin of undulating relief of the movable bottom. Being applied after many years of searching and guesses, the method of small perturbations set, finally, solving of this problem on a scientific basis.” Of greatest importance are the studies of R. Callander [11], G. Parker [25], J. Fredsoe [16], and A.E. Mikhinov [7], which give special analysis of the solutions of twodimensional equations of flow dynamics and channel 65 66 SIDORCHUK consider the theory of dune-like relief of river beds within a wide range of bed waves size and at large values of the Froude number and hydraulic resistance and to compare the results of theoretical calculations with data of measurements in rivers and experimental flumes. HYDRODYNAMIC MECHANSISM OF DUNE-LIKE RELIEF FORMATION IN RIVER CHANNELS M.A. Velikanov [1] has formulated the main postulate of the theory of channel process—the principle of interaction between the flow and the channel, which underlies the hydrodynamic approach to solving the problem of river channel morphodynamics. N.I. Makkaveev [6] formulates this principle as follows: “…channel formation process can be determined as a process of “imprint” to the surface of a solid medium (i.e., deposits composing a bed) the features of the movement of water and sediment transported by it….” As proposed in [2, 21], a quantitative description of this “imprint” can be given with the use of the method of small perturbations. We will use the equations of flow motion in curvilinear orthogonal coordinates [3]. N.A. Kartvelishvili has integrated them over depth in the most general form [4]. The author studied these equations by the method of small perturbations taking into account the curvature of the flow and bed, the instability of the flow free surface, and, additionally, the instability of the deformable bed [9, 26]. Importantly, these two-dimensional equations account for the effects relating to the nonhydrostatic distribution of pressure over flow depth and flow circulation at a bend. The small perturbations of the flow and the channel within the framework of these equations are analyzed in the standard form [5]. In the integrands, velocity components, flow depth, its free surface elevation, and the curvature of orthogonal coordinates are represented as the sum of two components—for the main averaged flow and for perturbation. The simplest configuration of the main flow, i.e., linear channel with a rectangular cross-section and uniform depth and velocities is considered. The perturbed flow is assumed steady-state and nonuniform; the elevations of the free surface and the bed vary over space and time. The hydraulic resistance is evaluated with the use of Chezy formula. The equations are linearized, i.e., the terms containing products and powers of perturbation components are discarded. The result is the following system of equations for perturbations of the flow and channel bed in a dimensionless form: αuU ∂u + αU v ∂v + 1 2 ∂h ∂s1 ∂s2 Fr ∂s1 3 + β ∂ h3 + αuU λ u − αuU λ d = 0, 2 ∂s1 3 αU v ∂v − αU 2 γv + 1 2 ∂h + β ∂2 h + αvU λ v = 0, (1) ∂s1 2 Fr ∂s2 ∂s1 ∂s2 αu ∂u + αv ∂v + αU ∂d + ∂d = 0, ∂s1 ∂s2 ∂s1 ∂t ∂ (h − d ) M ∂u + M ∂v + (1 − ε ) = 0. ∂s1 4 ∂s2 ∂t Here, the dimensionless variables are 2g , C2 Fr = U , u = u* , v = v * , h = h* , U U D gD s* s* d = d * , s1 = 1 , s2 = 2 , t = U t* and M is the D D D D dimensionless sediment discharge 2 D 0.015αuU 2 U3 (U − U CR ) s ; ε is bed deposits porosD U CR ity; the dimensional variables—C is the coefficient in Chezy formula; U and D are the velocity and depth of the main flow; u*, v*, h*, and d* are the longitudinal and transverse velocity, free-surface elevation, and flow perturbation depth; s1* , s2* , and t* are the longitudinal and transverse coordinates, time, and the critical velocity of the start of sediment motion 13 16 U CR = 4.6Ds D , Ds is the diameter of particles of bed load. λ= The kinematic coefficients αi and β depend on the velocity distribution in the main and perturbed flow along the vertical coordinate. Based on some hypotheses regarding the shape of these distributions, the following values were taken αU 2 = 0.75; αvU = 0.86; αv = 1; αu = 1; αU = 0.86; β = 0.86; αuU2 = 0.75. The coeffi- 1 − 7.2 D C results from the replacement 7.4 − 13.0 g C of the curvature of the longitudinal lines of perturbed flow K1 by the transverse near bed velocity by I.L. Rozovskii formula [8]: cient γ = K1 = −γ vb* . UD (2) This coefficient commonly varies within the range of 0.08–0.1. The perturbations of flow velocities, free surface, and depth are recorded in the form of moving sinusoidal waves with amplitude varying over time. In a first approximation, the amplitude increases as an exponential function of time. Now, with the introduction WATER RESOURCES Vol. 47 No. 1 2020 MORPHOLOGY OF DUNE-LIKE RELIEF IN RIVERS of a dimensionless complex velocity c = perturbation waves can be written as u v h d cR + ici , the U = A ( s2 ) exp [ik1 ( s1 − ct )] , = B ( s2 ) exp [ik1 ( s1 − ct )] , = P ( s2 ) exp [ik1 ( s1 − ct )] , = T ( s2 ) exp [ik1 ( s1 − ct )] . 67 W = L1/2 W = LY (3) Here cR is the velocity of longitudinal motion of wave, ci is the rate of amplitude increase, k1 is the longitudinal wave number = 2πD/LX (LX is wave length). Only parts in expressions (3) have physical meaning. The complex functions A, B, P, and T describe the shape of perturbations in the transverse direction. Substituting (3) into (1) and eliminating functions A, B, and T from the obtained system of equations yields a second-order ordinary differential equation, similar to the well-studied differential equation, which describes free oscillations: d 2P = k 2P, 2 ds22 (4) here а1 k22 b ( −a3c1d4 − a4c3d1 + a1c3d4 + a4c1d3 ) = 2 , c2 ( a4b3d1 − a1b3d4 − a3b4d1 + a1b4d3 ) (5) 2 a1 = αuU (ik1 + λ ) , c1 = ik1  1 2 − β k1  − αuU λ ,  Fr  2 2 d1 = αuU λ , b2 = αU 2 −γ + λ , c2 =  1 2 − β ki  ,  Fr  2 2 a3 = αuik1, c3 = ik1 (1 − c ) , d3 = −ik1 (1 − c ) , a4 = Mik1, b4 = M , d4 = − (1 − ε ) ik1c. ( ) Equation (4) describes variations of perturbations of free-surface elevation P across the flow. The boundary conditions on P on flow boundaries can be nonzero. In this case, equation (4) has nontrivial solutions for all real eigenvalues. This leads to a continuous spectrum of perturbation waves in both longitudinal and transverse directions, which is an important distinction of the proposed solution from those obtained in other studies [7, 11, 16, 25], in which the spectrum of perturbation waves in the transverse direction was discrete. Thus, the perturbations of free-surface elevation (as well as the perturbations of all other hydraulic characteristics of flow, including dimensionless bed elevations Z 0) take the form of a bi-sinusoid with amplitude exponentially depending on time, moving along the flow:     µπs2 Z 0 = exp  2πD ci t  sin 2πD ( s1 − cRt ) sin D, (6) W  LX   LX  here W is channel width, µ is wave shape characteristic = 2W/LY, LY is wave length in the transverse direction WATER RESOURCES Vol. 47 No. 1 2020 а2 b1 b2 Fig. 1. Perturbations of channel bed elevations at wave shape characteristics µ = 2W/LY, equal to 1 (index a1, a2) and 2 (index b1, b2). If the wave cross section is expressed in terms of sine, the wave system is asymmetrical (index a1, b1); if it is expressed in terms of cosine, the system is symmetrical (index a2, b2). Here LY is wave length in the transverse direction (wave width), W is channel width. (wave width). If the transverse shape of the perturbation wave is expressed with the use of sine, the system of waves is asymmetric, while if it is expressed with the use of cosine, this system is symmetric (Fig. 1). THEORETICAL CHARACTERISTICS OF THE DUNE-LIKE RELIEF OF A RIVER CHANNEL Equation (5) is a dispersion relationship in the form of a second-order algebraic equation for the complex velocity c: (7) A1c 2 + A2c + A3 = 0. The coefficients in equation (7) are complex functions of the dimensionless wave numbers, hydraulic resistance, Froude number Fr, and the dimensionless bed load transport rate, as well as the coefficients αi, β, and γ. Their expressions are complicated (Appendix in [9]). Because of this, the sign and the magnitude of the rate of changes in the perturbation wave amplitude of the channel bed was evaluated numerically. The calculations were made in a wide range of dimensionless lengths LX/D and widths LY/D of waves with different hydraulic characteristics of the main flow. The pertur- 68 SIDORCHUK log(LY/D) N_2 2.0 28 .5 27.5 1.5 26.5 27 28 28 .5 B A 27 28 .5 28 .5 28 28 1.0 27 .5 28 .5 0.5 28 .5 –0.5 –0.5 0 28.5 28 .5 28 N_1 0 0.5 1.0 1.5 2.0 log(LX/D) Fig. 2. The field of amplitude growth rate of channel bed perturbations (i.e., channel dune-like features) in coordinates of logarithms of dimensionless length LX/D and width LY/D of alluvial features for Fr = 0.6 and λ = 0.1. The crest A corresponds to amplitude growth rate of three-dimensional alluvial features; the crest B is the same for two-dimensional ultramicroforms. Point N_1 corresponds to the local maximum of amplitude growth rate of three-dimensional ultramicroforms, N_2 is that for macroforms. bation waves with a negative velocity of changes in the amplitude are not considered below, while, for the waves with growing amplitude, which correspond to dune-like forms of channel relief, fields of amplitude growth velocity were constructed in the coordinates of dimensionless wave length and width (i.e., of channel dune-like features) for various combinations of the hydraulic characteristics of the main flow (Fig. 2). This field of the growth rate of perturbation wave amplitudes (spectrum) has two important features. First, it is continuous. Second, it has a complex relief. Peaks A and B of the growth rate maximums of perturbation wave amplitudes with characteristic inflections can be seen, as well as local maximums of amplitude growth rates. The combination of these lines and points can be used to identify perturbation wave groups with similar properties. Four such major groups were identified [9, 26]: ultramicroforms, microforms, mesoforms, and macroforms. For some characteristic points of the spectrum of perturbation waves in Fig. 2 (points N_1 and N_2, as well as points on line B), relationships between the dimensionless length and width of the waves (dune-like features) and the hydraulic characteristics of the main flow can be obtained. Most studies applying the small-perturbation method to solving the problem of the formation of dune-like features in river channels, both initial [11, 16, 25] and modern [12, 13] consider two-dimensional equations where the distribution of pressure over depth is described by a static law. In such case, the two-dimensional spectrum given in Fig. 2 shows neither ridge B nor local maximums on line A, which can be interpreted as alluvial features of different types. The ridge B in the spectrum, which reflects the smallest two-dimensional bedforms—ultramicroforms— was first obtained by A.E. Mikhinov [7] with the use of the equations of motion in Boussinesq approximation, taking into account the effect of the dynamic pressure distribution. This showed the necessity to preserve three-dimensional effects in two-dimensional equations. Ultramicroforms A.E. Mikhinov also proposed a formula for calculating the lengths of two-dimensional ultramicroforms [7]: LX _ 2D = 5.4DFr. WATER RESOURCES Vol. 47 (8) No. 1 2020 MORPHOLOGY OF DUNE-LIKE RELIEF IN RIVERS LX_2D/D 6 69 λ = 0.1 λ = 0.3 λ = 0.5 λ = 0.7 λ = 0.9 5 λ = 1.5 4 λ = 2.5 3 2 1 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Fr Fig. 3. Dependence of dimensionless lengths of two-dimensional ultramicroforms on number Fr at different hydraulic resistance. Detailed calculations for points on the ridge B at LY_2D/LX_2D > 10 showed that the dimensionless wave length of two-dimensional ultramicroforms LX_2D/D depends on both the Fr number and the hydraulic resistance values (Fig. 3). An approximation of this relationship (with a correlation coefficient >0.98) has the form: LX _ 2D (9) = a [1 − exp ( −bFr )] . D At Fr < 0.4 and hydraulic resistance values λ < 0.1, formula (9) yields a relationship similar to the linear formula (8) with a coefficient of 6.28. The coefficient b increases almost linearly with increasing hydraulic resistance λ: (10) b − bλ=0.2 = λ . 2 Here bλ = 0.2 = 1.333. The increase in coefficient b decreases the effect of Fr number on change in LX_2D/D; at large values of Fr, the wave length of twodimensional ultramicroforms is mostly determined by the value of coefficient a. The latter depends only on the values of hydraulic resistance and decreases with increasing λ: (11) a = 0.31λ2 − 1.84λ + 6.13. The point N_2 is a local maximum of the rate of increase in the amplitude of the smallest three-dimensional bedforms—three-dimensional ultramicroforms. This maximum was obtained in [9]. The dimensionless lengths of the three-dimensional ultramicroforms depend only on Fr, and this dependence can be described by a simple equation WATER RESOURCES Vol. 47 No. 1 2020 LX _3D (12) = 6.28Fr. D Two- and three-dimensional ultramicroforms occur in the undulating beds of river channels within a wide range of Fr numbers and hydraulic resistance values (numerical calculations were carried out for 0.01 < Fr < 4 and 0.02 < λ < 4). However, their amplitudes vary within the Fr–λ diagram. The rate of amplitude growth of the ultramicroforms 2πc/LX can be greater for either three-dimensional or two-dimensional ultramicroforms, depending on the combination of Fr and λ; the boundaries are given in Fig. 4. Two-dimensional ultramicroforms bar the channel over the entire width, since their width is significantly greater than their length and the width of the channel W. Three-dimensional ultramicroforms are isometric; their width LY_3D can be calculated by (12). If W ≤ 2 or W ≤ 12.56Fr, LY _ 3D D (13) then three-dimensional ultramicroforms also bar the channel over the entire width. In mountain rivers, ultramicroforms of both types are step–pool systems, where the dune-like features with crests composed of coarse alluvium (pebble and boulders) represent steps with a height of about half of the channel depth, separated by shallow pools (Fig. 5). If these features are two-dimensional, the flow forms a hydraulically single channel, and it separates into individual streams only by large alluvial particles and their clusters. They are also hydraulically linear—the alluvial features do not make the flow sinuous, and river bends follow the relief of the main banks. If the alluvial features are 70 SIDORCHUK λ 5 λ = 0.52Fr –0.76 Two-dimensional ultramicroforms 1 Fr = 0.3 Two-dimensional and three-dimensional 0.1 Three-dimensional ultramicroforms 0.01 0.003 0.1 1 14 15 22 23 18, ripples 24 27 5 Fr 18, antidunes 28 Fig. 4. Zones of combinations of the hydraulic characteristics of flow, which correspond to different types of ultramicroforms. The numbers correspond to the references from which data on ultramicroform formation conditions have been taken. three-dimensional, the channel can be sinuous and braided. In mountain rivers, it is often difficult to differentiate the ultramicroforms by types, as the presence of boulder–pebble alluvium causes the separation of flow even at two-dimensional step-pool systems. In lowland rivers, the ultramicroforms are commonly three-dimensional; these are the smallest bedforms in the hierarchy, ripples in subcritical flows or antidunes in supercritical flows. The morphological characteristics of such channels have been studied well [14, 15, 17–20, 22–24, 27, 28]. For measurements in flumes [15, 18, 20, 22, 27], the hydraulic characteristics are reliable enough, while for natural objects [17, 23, 24, 28], they are sometimes calculated rather than measured, as in [14]. Several points in Fig. 4 lie within the domain of existence of two-dimensional ultramicroforms. These are the upper reaches of the rivers of Kowai and Taramakau (Camp Creek), New Zealand [28]. Alluvium in their channels is represented by large boulders and blocks (Fig. 2 in [28]); therefore, the type of these Fig. 5. Step–pool systems in the Dzhergalan river channel, Kirgizia (the author’s photo). dune-like features is impossible to identify reliably. Anyway, the hydraulic conditions of two-dimensional ultramicroforms formation represent the physical limit for the formation of alluvial features. The major portion of the examined channels in mountain rivers and flumes with large slope show ultramicroforms of a mixed type, where two-dimensional and three-dimensional bedforms are combined (Fig. 4). The lengths of those ultramicroforms, calculated by (9) and (12), generally fit to linear relationship with measured lengths of alluvial features. The scatter in the measured and calculated lengths of the alluvial features is considerable. This is due not only to the inaccuracy of measurements, as the scatter is about the same for the relationships derived from flume experiments and field studies. The main cause here is the stochastic properties of the dune-like relief of the channel. Calculations by (9) considerably underestimate the lengths of two-dimensional ultramicroforms—by about one half compared with the measured values. The domain of existence of channels with threedimensional ultramicroforms contains some mountain rivers in New Zealand and channels in the experimental flumes where the hydraulic resistance λ < 0.075. The same is true for the results of measurements in large flumes with sand and gravel deposits, where ripples and antidunes are formed [18]. The lengths of three-dimensional ultramicroforms, calculated by (12) are in a good agreement with the linear relationships with the measured lengths (Fig. 6). Formula (12) underestimates the lengths of the ultramicroforms by 25%; therefore, the empirical coefficient is 8.4. Macroforms Point N_3 is a local maximum of the rate of increase in the amplitude of large three-dimensional alluvial features—macroforms. This maximum appears in the theoretical spectrum (Fig. 2) when WATER RESOURCES Vol. 47 No. 1 2020 MORPHOLOGY OF DUNE-LIKE RELIEF IN RIVERS 71 Lcalc, m 100 10 by formula (12) Lcalc = 0.74Lmeas 1 0.1 0.01 0.1 1 10 100 Lmeas, m Fig. 6. Correspondence between the ultramicroform lengths calculated by (12) and measured in step–pool systems in studies [14, 15, 17, 19, 20, 22–24, 27, 28], as well as ripples and antidunes in [18]. transverse flow circulation is taken into account in the two-dimensional equations [9]. The near-bed transverse velocity in this circulation is described by formula (2) based on the results of I.L. Rozovskii [8]. As the result, the structure of the coefficients of dispersion relationships (5) and (7) includes expression (–γ + λ/2). If the sign of this expression is plus, i.e., λ > 1.6–0.2, then the macroforms are not expressed in the theoretical spectrum in Fig. 2. At Fr > 1 and λ < 0.09ln (Fr ) + 0.02 , the macroforms in the theoretical spectrum merge with ultramicroforms (Fig. 7), and the hierarchic structure of the dune-like relief in the river channel becomes simpler. Earlier, it was supposed that the dependence of dimensionless lengths of macroforms on the major factors [9] can be expressed by the relatively simple equation: LX _ mak D LX _ mak D LX _ mak D LX _ mak D WATER RESOURCES LX _ mak D ) Fr ≤ 0.8, 2 0.55Fr −1.62Fr0.56 ) 2γ − λ ( 2.45 1.25 exp , 0.8 < Fr ≤ 1.3, = Fr λ λ ( )( ) 2γ − λ ( = 2.45 exp (1.25) ( Fr λ λ ) Vol. 47 ) No. 1 2020 2 0.04Fr − 0.2Fr0.43 0.04Fr 2 − 0.2Fr0.43) 2γ − λ ( , =λ λ ( (14) Numerical solutions of the dispersion relationship within a wide range of Fr numbers and resistance have shown that formula (14) gives a satisfactory approximation to this relationship at Fr < 0.6 and λ < 0.016. At larger Fr and λ, the dimensionless lengths of the macroforms LX_mak/D change in a complex manner with changes in the hydraulic characteristics of the main flow (Fig. 8). Overall, an inverse relationship with Fr holds, but the coefficients in the formula for this relationship has different approximations in different domains. Thus, the following approximations (they are not the only ones) can be proposed: −0.27Fr 2 + 0.13Fr −0.34 ) 2γ − λ ( 9.35 , = λ Fr λ ( = 6.28 . λ Fr ) , 1.3 < Fr ≤ 2.6, Fr > 2.6. (15) 72 SIDORCHUK log(2πс/LX) –3 –5 s rm fo ro ac M Ul tra mi cro for ms –4 Fr = 3.0 2.0 –6 1.0 0.5 –7 0.1 –8 1 10 100 1000 Lx/D Fig. 7. Amplitude growth rates of three-dimensional alluvial features along crest A as a function of Froude number Fr at λ = 0.1. The degree of elongation of macroforms changes in even more complex manner. It is mostly determined by the values of Fr; however, the form and the sign of the relationship depends on the hydraulic resistance: at λ < 0.07 the elongation of macroforms decreases with increasing Fr and otherwise it increases (Fig. 9). The ratio of macroform length to its width is approximated by the following formulas: LX _ mak 2 LX _ mak = 2W aFr + bFr + c . µ ( 2 = aFr + bFr + c, LY _ mak a = − 0.02 , b = 0.0073 , c = 0.27 , λ ≤ 0.07, λ λ λ (16) 0.8 2γ − λ 2γ − λ , b = −0.79 + 1.15, a = 0.24 λ λ 0.8 2γ − λ , λ > 0.07. c = 0.66 λ ( ) ( ( ) ) In this case, the macroforms can be two-dimensional: LX_mak/LY_mak < 1, and three-dimensional: LX/LY > 1 (Fig. 9). Two-dimensional macroforms lie in the domain 0.2 > λ ≥ At Fr > 0.9, the macroforms are three-dimensional. In the case of three-dimensional macroforms, the wave shape characteristic µ = 2W/LY is also a characteristic of channel shape: at µ ≤ 1, there is a single channel; at 1 < µ ≤ 2 the channel has two branches, etc. In this case, formula (16) can be written in terms of the channel width 0.0875 , Fr ≤ 0.9. 1.8 1.25 − Fr (17) ) (18) This relationship is an important supplement to formulas (14)–(15) for calculating the length of macroforms. The theory shows that macroforms appear in river channel relief in the presence of transverse flow circulation. The application of I.L. Rozovskii’s formula [8] to evaluating the bottom transverse velocity leads to the appearance in the dispersion relationship of expression (–γ + λ/2). If the sign of this expression is plus, i.e., λ > 0.16–0.2, then the macroforms are not expressed in the theoretical spectrum. Large values of hydraulic resistance are typical of mountain rivers. However, no values λ > 0.2 have been found in the literature for channels with macroforms. These empirical data confirm the results of theoretical calculations (Fig. 10). WATER RESOURCES Vol. 47 No. 1 2020 MORPHOLOGY OF DUNE-LIKE RELIEF IN RIVERS 73 λLX/D 1000 λ = 0.19 500 λ = 0.18 λ = 0.17 λ = 0.15 100 λ = 0.13 λ = 0.11 λ = 0.09 λ = 0.07 λ = 0.05 λ = 0.03 λ = 0.01 50 10 5 1 0.05 0.1 0.5 1 5 Fr Fig. 8. Dependence of dimensionless lengths of macroforms on number Fr at different hydraulic resistance. In lowland and piedmont rivers, macroforms determine the morphological type of the channel. In most cases, these are three-dimensional macroforms, which are developed either in single-channel (filled circles in Fig. 10) or braided (empty circles) channels. Three-dimensional macroforms are also typical for mountain rivers. Theoretical calculations have identified a domain of two-dimensional macroforms. This domain includes some rivers of Karelia and Kola Peninsula with small slopes and considerable hydraulic resistance. These rivers commonly flow in hard-rock beds with poor sediments and almost no alluvial relief of the bed. The type of macroforms for such rivers is difficult to determine. The few rivers with alluvial channel that lie in this domain show a low-sinuosity channel or very sharp bends. Both variants can testify in favor of the assumption regarding the two-dimensional character of the macroforms existing in these channels. However, this question requires further studies. The dependence of the dimensionless lengths of three-dimensional macroforms on the major factors is given by equations (15) and (19). WATER RESOURCES Vol. 47 No. 1 2020 Observations show that formula (15) gives a good description of the wave length of alternating bars and middle bars in a sufficiently wide channel (Fig. 11, triangles), where the condition 2W = µ @ 1 holds. Such LY channel presents conditions for the full development of macroforms. At µ ! 1 in a relatively narrow channel, the size of macroforms is commonly less than their theoretical size (Fig. 11, circles) calculated by formula (15). Formula (19) gives a good description of the size of such three-dimensional macroforms provided that the value of µ is chosen correctly. In narrow channels, shorter and less developed macroforms or even mesoforms appear. Such macroforms, and, even more so, mesoforms, are less mature than the most developed macroform are. CONCLUSIONS The presence of three-dimensional effects in depth-averaged equations of motion, continuity, and deformation considerably extend the potentialities of the linearized analysis of such equations. The complex relief that forms on the two-dimensional spectrum of the rate of amplitude growth of channel bed perturba- 74 SIDORCHUK LX/LY 10 Three-dimensional macroforms λ = 0.13–0.19 λ = 0.11 λ = 0.09 λ = 0.01 1 λ = 0.03 λ = 0.05 λ = 0.07 λ = 0.09 λ = 0.11 λ = 0.13 λ = 0.07 λ = 0.05 λ = 0.03 Two-dimensional macroforms λ = 0.15 λ = 0.17 λ = 0.18 λ = 0.19 0.1 0.01 0.1 1 10 Fr Fig. 9. Dependence of macroform elongation on number Fr at different hydraulic resistance. tions enables identifying individual domains of channel dune-like features of different size and morphology: ultramicroforms, microforms, mesoforms, and macroforms. In some cases (ultramicroforms and macroforms), theoretical relationships can be constructed between the morphological characteristics of channel alluvial features and the hydraulic characteristics of the flow. The extension of the analysis to the high-kinetic flows with large Fr numbers and high hydraulic resistance values revealed new features of such relationships. A new type of such relationships was obtained, which was found to be far more complex than that derived from the earlier analysis of subcritical flows with small hydraulic resistance values. Thus, for twodimensional ultramicroforms, a relationship was established between their length and the hydraulic resistance. A new class of macroforms—two-dimensional macroforms—was identified. The verification of the results of theoretical analysis based on data of measurements of channel dunelike features morphology and hydraulic characteristics of the flow has shown that the analysis of linearized equations of two-dimensional hydrodynamics by the method of small perturbations can be used to determine the morphology and size of channel features in either subcritical or supercritical flows. The step–pool systems in the supercritical flows of mountain rivers represent ultramicroforms (two-dimensional and three-dimensional), which are analogs of antidunes (in supercritical flows) and ripples (in subcritical flows), reproduced in large flumes with sand alluvium. The relationship between their size and the hydraulic characteristics of flow—channel depth, Fr number, and the values of hydraulic resistance—can be in general described by theoretical formulas with a quite acceptable accuracy. However, the coefficients in these formulas require calibration against field data—the lengths of theoretical three-dimensional ultramicroforms are 15–20% less than those obtained in field experiments with the same hydraulic characteristics of the flow; and the length of two-dimensional ultramicroforms with elements of three dimensions are less by half. The causes of such difference require special studies. Macroforms in natural flows do not form at a hydraulic resistance λ > 0.2, thus confirming the theoretical results. The domain of existence of twodimensional macroforms, obtained in the theory, contains a very small number (3–4) out of the 230 river reaches covered by the study. The morphology of such channel features requires additional studies. Threedimensional macroforms are most common in rivers. If such macroforms are well developed in wide channels (at flow width greater than half the width of the macroform), their length can be calculated by (15) using channel depth, Fr number, and the values of λ resistance, or by (19) using channel width, Fr number, and the values of hydraulic resistance. The macroforms in a narrower channel do not completely develop, and their lengths can be calculated only by (19), as they are limited by channel width. FUNDING This study was carried out under the project: “Evolution and Transformation of Erosion-Channel Systems under Changing Environment and Human Impact” (GO) (government financing, section 0110, no. I.13, TsITIS no. АААА-А16-116032810084-0). WATER RESOURCES Vol. 47 No. 1 2020 MORPHOLOGY OF DUNE-LIKE RELIEF IN RIVERS λ Two-dimensional macroforms 0.1 0.01 Three-dimensional macroforms 0.001 0.1 1 19 Lowland rivers 5 Fr Mountain rivers Fig. 10. Domains of combinations of flow hydraulic characteristics, which correspond to different macroform types. The number 19 shows data taken from [19]. Lmes 100 000 10 000 1000 100 1000 10 000 100 000 Lcalc Fig. 11. Correspondence between macroform lengths calculated by (15) and macroform lengths measured in piedmont braided rivers with a wide channel (triangles) and in meandering single-branch channels of lowland rivers (circles). WATER RESOURCES Vol. 47 No. 1 2020 75 76 SIDORCHUK REFERENCES 1. Velikanov, M.A., Gidrologiya sushi (Land Hydrology), Leningrad: Gidrometeoizdat, 1948. 2. Grishanin, K.V., Ustoichivost’ rusel rek i kanalov (Bed Stability of Rivers and Channels), Leningrad: Gidrometeoizdat, 1974. 3. Kartvelishvili, N.A., Potoki v nedeformiruemykh ruslakh (Flows in Nondeforming Channels), Leningrad: Gidrometeoizdat, 1973. 4. Kochin, N.E., Kibel’, I.A., and Roze, N.V., Teoreticheskaya gidromekhanika (Theoretical Hydromechanics), Moscow: Gostekhizdat, 1948, part 2. 5. 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