Finite Amplitude of Free Alternate Bars With Suspended Load

Abstract

River bars are macroscale sediment patterns, whose main geometrical features (wavelength and amplitude) depend on the mutual interactions between hydrodynamics and sediment transport. River bars develop as an instability of the plane bed to an infinitesimal perturbation, which grows in time to eventually reach a finite amplitude. We here determine, with reference to both bed and suspended loads, a closed form for the finite amplitude, through the nonlinear Center Manifold Projection technique. Results show that suspension plays a destabilizing role in bar instability, affecting both the bar wavelength (linear analysis) and the bar amplitude (weakly nonlinear analysis). This proves the importance of considering suspended load for practical purposes. The outcomes of the model are satisfactorily compared with field observations.

Key Points

• The influence of suspended load on sandbar instability is investigated with linear and weakly nonlinear analyses
• An analytical relation for the amplitude of free alternate bars is obtained through Center Manifold Projection
• The matching of the analytical results with field observations is verified to be satisfactory

1 Introduction

The interactions between a fluid, such as water or air, and a deformable surface, such as rock, ice, or sand, create marvelous natural patterns. Desert dunes generated by the wind (Lancaster, 2013; Pye & Tsoar, 2008), ice and cave features shaped by thin water films (Camporeale, 2015, 2017; Chen & Morris, 2013), and sand ripples on the sea bottom (Blondeaux et al., 2000) are some of the most well known. Sediment patterns are formed by a flow that is sufficiently strong to trigger grain transport. This happens in rivers, which are one of the most powerful erosive forces on Earth. Interacting with sediments, rocks, and vegetation, rivers develop intriguing geometries such as meanders (Camporeale et al., 2005; Liverpool & Edwards, 1995), bifurcations (Federici & Paola, 2003; Zolezzi et al., 2006), and bedforms at different scales (Best, 2005; Seminara, 2010).

The present work concerns alternate sandy fluvial bars, which are macroscale sediment patterns, a graphical sketch of which is furnished in Figure 1. The two-dimensional sediment wave forming alternate bars has a longitudinal wavelength that is up to 10 times the river width, a transverse wavelength constrained by the river width and an amplitude that has similar scales to the flow depth. Owing to their considerable size and ubiquity in nature, bars constitute the most important fluvial patterns, strongly influencing the water flow and any anthropogenic or natural activity in a river. Bars are the key elements that trigger the meandering (Parker, 1976), and they cause localized erosion of river banks (Visconti et al., 2010). Moreover, bars limit the navigability of river channels and can play a fundamental role in river renaturalization, by increasing the bed channel area and favoring biodiversity (Gilvear & Willby, 2006). For these reasons, river bars have been studied extensively over the last decades through field observations (Bertoldi et al., 2009; Eekhout et al., 2013), experiments (Crosato et al., 2012; Lanzoni, 2000), numerical simulations (Defina, 2003; Siviglia et al., 2013), and analytical modeling.

Figure 1

Graphic framework. (a and b) Aerial sketch and channel section. The superscript hat refers to dimensional quantities, and is the amplitude of the bar, that is, the height of the bar crest with respect to the flat bottom condition.

From classical theoretical investigations of shallow water and Exner equations, it is known that a base plane bed becomes unstable when the aspect ratio β, which is the ratio between the half channel width and the uniform flow depth, exceeds a critical value (β > β_c; Blondeaux & Seminara, 1985). In such unstable conditions, any infinitesimal spatial perturbation is able to trigger the development of the bar pattern. Bars have been modeled, through this instability approach, by means of (i) linear analyses, which lead to the unstable domain in the parameter space and the bar wavelength (Blondeaux & Seminara, 1985; Callander, 1969; Parker, 1976) and (ii) nonlinear analyses, which furnish the bar height (Colombini et al., 1987) and a prevision of the wave packet formation (Schielen et al., 1993). Other noticeable works have included the effect of flow unsteadiness (Hall, 2004; Tubino, 1991) and the possibility for vegetation to spread on bar crests depending on flow variability (Bertagni et al., 2018). However, all the above-mentioned works only considered sediment transported as bedload. This approximation is satisfactory for the standard conditions in gravel-bed rivers, but it is not adequate for a correct morphodynamic modeling of those river in which the suspended load is the dominant transport mechanism. Much work still has to be done in this direction.

The problem of coupling bar formation with suspended sediment was first addressed by Tubino et al. (1999), through a three-dimensional approach. The authors showed that suspension has a strong quantitative effect on bar instability as it reduces β_c and leads to longer wavelengths at criticality. However, the linear stability analysis was solved numerically, thus limiting the possibility of additional analytical studies. A further advancement was made by Bolla Pittaluga and Seminara (2003), who revisited some previously formally incorrect asymptotic approaches and demonstrated the validity of a depth-averaged river modeling even in the presence of suspension. The main assumption is that advective and unsteady effects are smaller than gravitational settling and turbulent diffusion. Bars instability was linearly and analytically investigated through this approach (Federici & Seminara, 2006), and the semianalytical results of Tubino et al. (1999) were confirmed.

In this manuscript, we extend the linear stability analysis of Federici and Seminara (2006), which only provides the bar wavelength, to the weakly nonlinear order, which also provides the bar finite amplitude. The weakly nonlinear analysis is performed through the analytical technique of Center Manifold Projection (CMP; Guckenheimer & Holmes, 2013; Wiggins, 2003). This technique has already been verified as a tool to simplify and rigorize the bifurcation analysis of multiscale theories (Carr & Muncaster, 1983). The main assumption is that the stable modes have fast dynamics, which can be projected onto the slow dynamics of the quasi-neutral or weakly unstable mode (the alternate bar mode). Eventually, a differential equation for the dynamics of alternate bar amplitude is obtained, that is, the well-known Stuart-Landau equation. This equation quantifies the nonlinear terms that dampen the exponential growth expected from the linear theory, and its stationary solution is the finite amplitude of alternate bars. Thus, we provide a closed-form relation for the amplitude of alternate bars in presence of suspended load. Neglecting the role of suspended load, one obtains the Stuart-Landau equation for alternate gravel bars as in Bertagni et al. (2018). Another application of the CMP in a different geomorphological context can be found in Bertagni and Camporeale (2017).

From an experimental point of view, bedform formation characterized by a significant fraction of sediment transport in suspension is a very complicated task. For this reason, the experimental studies have only focused on the role of bedload (e.g., Crosato et al., 2012). In the laboratory experiments on bar patterns by Lanzoni (2000), some tests exhibited a certain amount of suspended load. However, bedload was still the main transport mechanism, and the theories that consider only bedload (Bertagni et al., 2018; Colombini et al., 1987) are sufficient to fit the experimental outcomes well. Due to the lack of experimental data, we validate the analytical results by addressing the field observations of alternate sandy bars available in literature. In particular, bathymetry data from the Mississippi river (Ramirez & Allison, 2013) and a straight artificial channel in the Netherlands (Eekhout et al., 2013) are used. In addition, some speculative considerations are made pertaining to partial data on the Yellow River in China (Ma et al., 2017).

The paper is structured as follows: The mathematical framework is presented in section 2, together with the linear and the nonlinear analyses that lead to the Stuart-Landau equation; the outcomes of the analytical theory are verified and discussed with field data in section 3; some conclusive considerations are given in section 4 . The material reported in the Appendices has already been shown in previous publications, and it is here reported for the sake of completeness and self-consistency.

2 Mathematical Model

2.1 Dimensionless Shallow Water Equations

Dimensionless Exner and shallow water equations under quasi-steady approximation (i.e., the flow adapts instantaneously to variations in the bed height) read as

(1)

(2)

(3)

(4)

with boundary conditions that impose a vanishing transversal flux of water and sediment at the river banks

(5)

In the system 1-5, s and n are the longitudinal and transversal coordinates, see Figure 1; U and V are the longitudinal and transversal depth-averaged velocities; D and η are the water depth and bottom height; ∂ refers to partial derivative; τ_s and τ_n are the two components of the bottom shear stress; and are the two components of the bedload solid discharge; and refer to the suspended solid discharge. Note that equations 1-5 have been made dimensionless through the following scaling:

(6a)

(6b)

(6c)

where the hat refers to dimensional quantities, is the channel half width, and are the uniform flow depth and velocity, is the sediment particle diameter (granulometry is assumed uniform), ρ the water density, g the gravitational acceleration, and Δ=(ρ_s−ρ)/ρ ∼ 1.65 with ρ_s the sediment density. The dimensionless parameters appearing in equations 1-5 are

(7)

where Fr is the Froude number; γ is the dimensionless solid discharge; λ_p is the porosity of the granular medium (around 0.3 for sand mixtures); and β is the aspect ratio, which is the main parameter controlling bar instability. The closure relationships for shear stress τ and bedload Q^b, which depend on the relative roughness and the Shield stress θ, are reported in Appendix A.

Suspended load is accounted for in equation 4 through the asymptotic approach derived by Bolla Pittaluga and Seminara (2003) and successively adopted by Federici and Seminara (2006) for the linear analysis of bar instability. Such an approach is based on an asymptotic expansion of the exact solution of the advection-diffusion equation for the sediment concentration, under the hypothesis that the flow is slowly varying. The main assumption is that advective and unsteady effects are smaller than gravitational settling and turbulent diffusion. Mathematically, this reduces to

(8)

where is the settling speed of a sediment particle (see Appendix B1), and is the scale of the longitudinal variation of the flow field, that is, the bar wavelength. In the following, to avoid to depend on the a priori unknown bar wavelength, we define

(9)

and formally assume that δ is a small parameter (as done in Federici & Seminara, 2006). In order to assure validity of the approach, the parameter has to be necessarily small (i.e., ). The suspended load, obtained as an asymptotic expansion of the advection-diffusion equation, reads

(10)

where the term O(δ²) contains the negligible second-order correction. At the leading order of the advection-diffusion equation, the classical Rouse profile for suspended concentration in uniform condition (ψ⁽⁰⁾) is obtained. The order δ correction (ψ⁽¹⁾) accounts for the nonequilibrium effect induced by spatial and temporal variations of the flow field. Bolla Pittaluga and Seminara (2003) found

(11)

where is the depth-averaged concentration of the Rouse profile, while K_0,1 are functions of the relevant physical parameters (see Appendix B2). The function K₁ had previously been obtained numerically (Bolla Pittaluga & Seminara, 2003; Federici & Seminara, 2006), while we have here adopted a completely analytical solution (see Appendix B3 for further details). Notice that, to address the closure relationships for suspended load, the Reynolds particle R_p needs to be introduced

(12)

where ν is the kinematic viscosity of water.

2.2 Stability Analysis

Free alternate bars develop as an instability of the uniform flow solution to an infinitesimal perturbation. Hence, the vector of the state variables U = (U,V,D,η) is recast as the sum of the base state U₀ and a small perturbation U₁ as follows:

(13)

where η₀=−Jβs is the bed height at the base state, J is the river slope, and Θ₁ is the bed height perturbation. By substituting 13 in the system 1-4 and truncating to quadratic nonlinearities, one obtains

(14)

where is a linear differential operator; L₀ is a matrix with null elements except for the lower-right entry, which is 1; and N(U₁) contains all the second-order nonlinearities. The CMP technique requires the unknown U₁ to be further expanded in terms of the u_m eigenfunctions through the following ansatz

(15)

where k is the longitudinal wavenumber, and A_[m,p](t) is the generic amplitude of the longitudinal mode p and lateral mode m. Such an expansion takes on slightly different meaning for linear and nonlinear analyses, as explained hereafter.

2.3 Linear Order

A linear analysis of the system 14 has already been investigated by Federici and Seminara (2006) with the closure relationship for suspended load provided by Bolla Pittaluga and Seminara (2003); thus, it is only briefly reported here. In general, linear analyses rely on the hypothesis that perturbations of different modes are infinitesimally small, decoupled, and grow/decay exponentially in time. Therefore, if reference is made to the expansion 15, the only longitudinal harmonic of interest is the fundamental one (p = 1). Each transversal harmonic m leads to a decoupled problem (m = 1 is an alternate bar mode; m = 2 is a central bar mode), and the amplitudes read , where Ω_m is a complex number, whose real and imaginary parts determine the growth rate and the angular phase. In order to satisfy the boundary conditions 5, the u_m eigenfunctions are written as

(16)

where c.c. means complex conjugate. If the closure relationships for suspended sediment 11 are expanded in the perturbations of the state variables, one obtains

(17)

(18)

with and K_n defined as in Appendix B2, while t_{1 − 4} are the same as in Federici and Seminara (2006). Moreover, τ_s, τ_n, , and are written as a first-order Taylor expansion of the state variable perturbation, as reported in Colombini et al. (1987). By substituting 16-18 in the system 14 and linearizing (i.e., neglecting N), one obtains

(19)

where the algebraic matrix is reported in Appendix C. After imposing , the dispersion relation and consequently the Ω_m eigenvalue are obtained. The analytical expression for Ω_m is reported in the supporting information S1. It should be noted that, unlike Federici and Seminara (2006), we have not expanded the perturbations 15 in the δ parameter, but we have instead solved the linearized problem directly.

The neutral condition for alternate bar formation is defined as the solution to the dispersion relation that satisfies Ω_1,r=0, where Ω_1,r is the growth rate for alternate bars (the subscript r refers to the real part). This condition manifests as a marginal curve in the (k,β) plane, with a minimum at the critical point (k_c,β_c). In Figures 2a and 2b, neutral stability curves for alternate bar formation (m = 1) and for higher transversal modes (m = 2, m = 3) are reported for both the plane-bed- and dune-covered configurations, which differ for the closure relationships reported in Appendixes A and B2.

Figure 2

Linear results. (a and b) The solid blue lines are the neutral stability curves for alternate bar formation in plane- and dune-covered configurations, with the unstable domains emphasized in gray (θ₀=0.5, d_s=10⁻³, and R_p=8). The blue-dashed lines stand for the neutral stability curves of higher transversal modes (m = 2, m = 3), and the red solid lines refer to the case where only the bedload is considered. (c) Growth rate for alternate bars Ω_1,r versus the wavenumber k for diffent values of β. k_s is the wavenumber of maximal instability (black-dashed lines in (a) and (b)).

The qualitative behavior of the alternate bar growth rate Ω_1,r versus k for different values of β is reported in Figure 2c. For β < β_c, all perturbations decay in time (stability of the flat riverbed). For β = β_c, the perturbation of wavenumber k_c does not decay nor grow in time (neutral critical condition). For β > β_c, the growth rate becomes positive for a band of wavenumbers (instability of the flat riverbed and bar formation). We here make the classical assumption of linear theories that the wavenumber of maximum instability k_s, that is, the fastest growing one, is the one that is most likely to be observed in nature. This assumption has already been verified on gravel bars (Blondeaux & Seminara, 1985). In Figures 2a and 2b, the black-dashed lines reveal the trend of k_s, which increases almost linearly with β.

The validity of the asymptotic approach 8-11 for suspended sediment is verified by the condition , which is obtained by manipulation of equation 8. Usually, β/k_s is one order of magnitude greater than the ratio between velocities.

In order to ensure continuity with previous works on sand bar instability, the figures were obtained by fixing d_s, R_p, and the Shield stress in the unperturbed condition θ₀ (however, a different combination of parameters may be chosen). Figure 2 also shows how suspension influences the domain of instability (blue solid lines) to a great extent, reducing both β_c and k_c, especially in the plane-bed case, compared to the case where only bedload is considered (red lines).

This aspect is further explored in Figure 3, where several neutral stability curves are plotted for different parameter combinations in the plane-bed configuration. Suspension is confirmed as a destabilizing mechanism that reduces β_c and leads to longer bars at the critical conditions (Federici & Seminara, 2006; Tubino et al., 1999). In the case where only bedload is considered, three dimensionless parameters, usually θ₀, d_s, and β, define the problem, while suspension needs a fourth parameter to characterize the suspended particle size, that is, R_p from equation 12. For this reason, the red curves in Figure 3a and Figure 3b are the same, while the blue curves show that the destabilizing effect of suspension is enhanced for finer sediments (  mm for Figure 3a and  mm for Figure 3b). Some comparisons of how sediment transport influences the linear results of bar instability are presented on the right of Figure 3. However, more precise considerations of general validity are jeopardized by the high number of control parameters, the two possible initial bed configurations (flat and dune covered), the instability of higher-order transversal modes (central bars), and the absence of closed-form relationships for k_s and the neutral stability curve. Overall, as suspension becomes the main sediment transport mechanism, neglecting it leads to incorrect predictions.

Figure 3

Influence of suspension on sandbar linear instability for the plane-bed configuration. (a and b) Neutral stability curves for different θ₀ and R_p (d_s=10⁻³). The addition of suspension to bedload enhances bar instability, reducing β_c and k_c, especially for finer sediments (for (a)  mm and for (b)  mm). The table on the right shows the influence of suspension on the numerical outcomes of sandbar instability (usually a ∼ 0.015).

2.4 Weakly Nonlinear Order

In this section, we present a weakly nonlinear analysis performed through CMP, with the aim of providing an analytical solution for the finite amplitude of the fundamental mode (m = 1). The CMP has been verified as a powerful analytical method to simplify and rigorize the bifurcation analysis of multiscale theories. In fact, the amplitude equations can be more easily derived, and they are not constrained around the critical point, as they are in multiscale theories (Carr & Muncaster, 1983). In addition, CMP can be applied to study subharmonic and superharmonic instabilities (Armbruster et al., 1988; Bertagni & Camporeale, 2017). A good theoretical introduction to CMP can be found in Wiggins (2003), while the mathematical procedure, which we adopted, is shown concisely in the work by Cheng and Chang (1992).

The first step requires finding the solution to the linear adjoint eigenvalue problem

(20)

where the star ^∗ refers to complex conjugate, while the symbol ^† refers to the adjoint operator. In this case, the internal product that defines the adjoint operator is

(21)

where x and y are generic vectors. Through 21 and the boundary conditions 5, one obtains that and is the complex conjugate of with the derivatives in n switched in sign. After proper normalization, the eigenfunctions of the linear problem u_m and the adjoint eigenfunctions become orthonormal with respect to the internal product 21 with , so that

(22)

Let us recall the expansion of the perturbation in terms of the linear eigenfunctions 15. Unlike the linear analysis, when nonlinearities are at play, the modes are coupled. Therefore, in order to evaluate the dynamics of the nearly neutral fundamental mode (p = m = 1), it is necessary to consider the dynamics of the stable modes (all other superharmonic and transversal modes). For the present analysis, the expansion 15 has been truncated to second-order harmonics (n_p=n_m=2), since higher lateral and longitudinal harmonics provide a negligible contribution (it should also be noted that ). By substituting 15 in the perturbation system 14, taking the internal product with the adjoint eigenfunctions and collecting the terms of the same Fourier modes, the equation for the nearly neutral amplitude (A_[1,1]) and the two equations for the stable amplitudes (A_[m,2]) are obtained

(23)

(24)

where we have posed A_1,1=A₁. Equations 23 and 24 represent a Galerkin-type projection of the full equations truncated at the second-order nonlinearities. The ellipses in the right-hand side of equation 24 refer to the omitted quadratic terms involving interactions of the stable modes, while the derivation of the coefficients P_i,S_j, which follows the procedure introduced by Cheng and Chang (1992), is reported in Appendix D. The fundamental mode, in the unstable domain close to the neutral stability curve (see Figures 2a and 2b), has a slow dynamics due to its weak instability. On the other hand, the stable modes have fast dynamics and can thus be projected onto the slow dynamics of the neutral or weakly unstable mode. Mathematically, this means that A_[m,2] can be recast as a nonlinear combination of the neutral mode A₁ and its complex conjugate as follows

(25)

where the projection coefficients a, b, and c still have to be determined, and the expansion has been truncated to as the higher-order corrections are negligible. The time derivative of 25 reads

(26)

By substituting equations 23-25 in equation 26, maintaining only the leading order terms and collecting like powers of A₁, , one obtains

(27)

From 27, it is straightforward to conclude that b = c = 0 and a =− S_m/(Ω_m(2k) − 2Ω₁); thus, the stable amplitudes can be written as

(28)

Finally, after substituting 28 into 23, the Stuart-Landau equation is obtained

(29)

where Ξ is the complex Landau coefficient, whose analytical expression is reported in the supporting information S1. Equation 29 describes the time dynamics of the fundamental amplitude. A graphical example of the amplitude dynamics, which refers to the real part of the equation, is reported in Figure 4a. Starting from an infinitesimal perturbation, the amplitude initially grows exponentially through the Ω₁ eigenvalue as expected from the linear analysis, then the rising effect of nonlinearities (Ξ|A₁|²) dampens the growth, until an equilibrium value is eventually reached.

Figure 4

Nonlinear results. (a) Amplitude dynamics in time as described by the Stuart-Landau equation 29. (b and c) Contour plots for the dimensionless amplitude A_s, with a value that increases from blue to green (the same cases as Figures 2a and 2b). The black-dashed lines refer to k_s.

This equilibrium stationary value can readily be obtained by setting , which gives, apart from the trivial solution, the finite amplitude . The contour plots of A_s in the instability domain are presented in Figures 4b and 4c for plane- and dune-covered bed configurations, respectively. In this case, the dune presence leads to higher bars. However, this is not a general rule, since the dune effect is negligible for other parameter combinations. The white areas in Figures 4b and 4c are due to positive values of D_1,r, which means there is no damping effect of nonlinearities, that is, no saturation (this is a sign that a quintic-order Stuart-Landau equation is formally needed). However, these areas correspond to modes that are not observed in nature, as they grow much slower than k_s, see Figure 2c, and they are easily destabilized by nonlinear interactions with neighborhood modes (Schielen et al., 1993). Dimensionally speaking, the amplitude reads , where the prefactor 2 is due to the complex conjugation. Neglecting the role of suspension, equation 29 reduces to the Stuart-Landau equation for bar amplitude with only bedload, as in Bertagni et al. (2018). Comparing the results of these two equations is not straightforward. In fact, we verified, in the previous section, that suspension drastically increases the unstable domain, and this leads to the presence of finite amplitude bars in a region of the parameter space where the bedload is not even sufficient to trigger the instability. For this reason, we have furnished the Mathematica® toolbox and the open-source Jupyter notebook®, so that specific numerical considerations can be made for the parameter combination of the case study of interest.

3 Validation and Discussion of the Results

In this section, the analytical results are validated by addressing the field observations of alternate sandy bars available in literature. In particular, we focus on a reach of bathymetry of the Mississippi river (Ramirez & Allison, 2013) and on a field experiment in a straight artificial channel in the Netherlands (Eekhout et al., 2013). Some explicative figures regarding these two field studies are reported in Figure 5.

Figure 5

Field studies used for the model validation. (a)–(c) have been edited from Ramirez and Allison (2013), (d)–(f) from Eekhout et al. (2013). (a) Location of the study reach on the Mississippi river in Louisiana, United States. (b and c) Submerged alternate bars revelead by the bathymetry and two cross sections. (d) Location of the field experiment in the Netherlands. (e) Aerial photo of the artificial channel. (f) Detrended bed topography highlighting the six alternate bars at different times from the beginning of the experiment.

Resuming the rationale of the analytical theory described in the previous section, a combination of four dimensionless parameters is necessary to define each case study. For continuity with previous theories (Federici & Seminara, 2006; Tubino et al., 1999), the 4-tuple (θ₀, d_s, R_p, and β) has been used. This 4-tuple is sufficient to determine the bar dimensionless wavenumber k_s, from the linear system 19, and finite amplitude A_s, from the Stuart-Landau equation 29. As in this section the focus is on the dimensional bar sizes; it is useful to recall that the bar wavelength ( ) and height ( ), measured from bar through to bar crest, read

(30)

In Table 1, the four dimensionless parameters necessary to define each case study are reported together with the field measurements and the theoretical predictions for the alternate bars. The closure relationships for sediment and suspended loads are the one used throughout the paper and reported in Appendixes A and B (Meyer-Peter Muller formula for the bedload; Meyer-Peter & Müller, 1948, and the asymptotic approach 8-11 for the suspended load; Bolla Pittaluga & Seminara, 2003). Finally, some speculative considerations are also made for partial data on the Yellow River in China (Ma et al., 2017).

Table 1. Data for the Model Validation

Case	θ0	ds(10^−4)	Rp	β	(m)	(m)		(m)	ΔL	(m)	ΔH
Miss.	0.66	0.113	13.6	20.6	6,000	20	0.1	7,500	25%	16	20%
								6,650	11%	7	65%
E1	0.94	12	13	20	55	0.27	0.08	70	27%	0.26	2%
								58	6%	0.14	48%
E2	0.9	6	13	10.4	75	0.3	0.09	125	66%	0.35	17%
								—	>100%	—	>100%

• Note. Miss. stands for the Mississippi River, and E1 and E2 for the artificial channel in the Netherlands after the formative events and . The subscripts f and t stand for field and theoretical, respectively. is the height of the bar measured from the bar trough to the bar crest, and is the bar length. The small values of the parameter prove the validity of the asymptotic approach 8-11. Δ_L (Δ_H) is the relative error between theoretical prediction and field measurement for the bar length (height). The model results for the case in which suspension is added to the bedload are presented in blue (considering only bedload leads to the results in red). Notice that, without considering suspended load, bar formation is not detected for E2.

3.1 The Mississippi River

The Mississippi River is the third longest river in the world (considering the Missouri-Jefferson), with a basin that covers one third of the United States, and it carries several kilotons of suspended sediment a day (Mossa, 1996). A field work conducted 100 km upstream of the delta in the Gulf of Mexico by Ramirez and Allison (2013) examined a 20-km reach of bathymetry and imaged five submerged alternate bars, which were usually obscured by murky water, see Figures 5a–5c. These bars were approximately 6 km long and had heights that varied from 16 to 24 m (with the height measured from the bar trough to the bar crest and neglecting the effect of the banks). Comparisons of the field measurements (subscript f) and theoretical predictions (subscript t) of the present model are reported in Table 1. As the bars are always submerged, they are constantly morphologically active. Thus, the hydrodynamic and sediment quantities for the model results have been fixed to averaged values:  m³/s is the mean flow rate recorded in the period 1961–2010; m/s is the averaged velocity for mean flow conditions;  m is the averaged depth of the channel centerline;  m and  mm (Ramirez & Allison, 2013). Moreover, the presence of 0.4-m-high and 10-m-long dunes, which were detected by the bathymetry, has been superimposed. The results show that the inclusion of suspension leads to longer and higher bars, which match the observations more satisfactorily for the bar heights and less precisely for the bar lengths. It should be pointed out that the constant values for the hydrodynamic and sediment quantities are approximations of reality that is unsteady and noisy. Moreover, other characteristics of this Mississippi reach differ from the model hypotheses, for example, the reach is not straight, and it is influenced to a great extent by anthropic activities that have not been considered in the present model. Nonetheless, the linear and nonlinear outcomes seem to adequately predict both the length and height of the bars.

3.2 The Artificial Channel in the Netherlands

The second field observations used to validate the model are the ones of Eekhout et al. (2013). The authors monitored the morphological evolution of a 600-m-straight channel under unsteady flow conditions in the Netherlands for almost 3 years, see Figures 5d–5f. The sediment was fine sand with  mm, and the channel was 7.5 m wide, with a slope that adjusted from 1.8‰ at the beginning of the experiment to 0.9‰ at the end. Six alternate bars were observed in the final part of the reach after the first survey, which was performed 250 days from the beginning of the experiment. The bars were emerged for ordinary flows, and they became morphologically active during the flood events. However, the duration of these events was not sufficient to trigger migration of the bars but only to adjust their amplitude and in second place their wavelength. For this reason, the bars were classified as nonmigrating (steady) by the authors. From the hydrograph reported in Eekhout et al. (2013), it is reasonable to suppose that the bar geometry was initially shaped by two formative events of intensity  m³/s (days 150 and 210). Subsequently, bars became higher and longer when two floods of  m³/s occurred (around day 550). After the events, the bars remained almost geometrically constant. This result is in agreement with the value of the unsteady coefficient U ≫ 1 found by Eekhout et al. (2013), which showed that bar development was much slower than the time evolution of the basic flow, which was thus unable to affect the bars (Tubino, 1991). The dimensionless parameters for the two formative events ( and ) and the measurements and model predictions for the associated bars are reported in Table 1 with the names E1 and E2. The theoretical outcomes show that, as in the Mississippi River, the inclusion of suspension in the E1 case leads to higher and longer bars, which match the observations more satisfactorily for the bar heights. While in the E2 case, the model detects bar instability only if suspended load is considered. In order to include the effect of the irregular channel bed in the field experiment, the closure relationships that account for dune presence have been used. Apart from the unsteady flow, other model hypotheses not fulfilled in the field experiment were the nonrectangular shape of the channel cross section and the finite longitudinal length of the reach.

3.3 The Yellow River

A further comment can be made regarding a field work conducted by Ma et al. (2017) on the Yellow River, which owes its name to the huge amount of suspended sediment. The main focus of the paper by Ma et al. (2017) concerned the correct evaluation of the suspended load, which was underestimated by one order of magnitude when classical formulas were used. Our interest has instead been on the 1.5-km-long longitudinal section of bathymetry surveyed by Ma et al. (2017) near one of the river banks. Because it is just a longitudinal section, it is not possible to make accurate deductions. However, the bathymetry data for the base flow revealed bedforms with a length/height ratio of 1,200. The authors attributed such bedforms to extremely long dunes, which were not in agreement with classical theories that predict a ratio of between 10 and 100 (e.g., Colombini, 2004). Our interpretation is that those bedforms could actually be bars, for which our model has predicted a length/height ratio of 1,100 (θ₀=1.05, d_s=610⁻⁵, β = 113, and R_p=3.5). However, this ratio has been calculated as the bars were alternate bars, while, in reality, the very high β value causes many transversal modes to be unstable (m > 1). Therefore, any further considerations on wavelength and amplitude would not be reliable without more bathymetry data.

4 Conclusions

In the last decades, bar patterns have been studied extensively because of their importance on river environments. However, the effect of suspended load on bar formation has been investigated using only a few analytical models and field studies. Federici and Seminara (2006) studied bar formation through a linear stability analysis, in which suspended load was included via the asymptotic approach of Bolla Pittaluga and Seminara (2003). In this work, we have extended the analysis of Federici and Seminara (2006) to the weakly nonlinear level using the analytical technique of CMP. In this way, the Stuart-Landau equation 29 for bar amplitude dynamics and an analytical relation for bar finite amplitude have been achieved. An advantage of CMP with respect to the classical multiple-scale method is that the results are valid for any condition close to neutrality and not only around the critical point.

Both the linear and nonlinear results show that suspension plays a quantitative role in bar formation: (i) suspended load enhances the instability, reducing the critical aspect ratio β_c (see Figures 2 and 3), and it may therefore lead to finite amplitude bars in conditions for which the bedload is not even sufficient to trigger the instability; (ii) the critical wavenumber k_c is smaller (longer bars at critical condition), and the selected wavenumber k_s has an almost-linear dependence on the aspect ratio β; and (iii) suspended load usually increases bar length and height (see Table 1).

In addition, both the linear and nonlinear outcomes have been satisfactorily verified with the few field observations of the sandy alternate bars present in literature. In particular, for one of the test cases, suspended load has proven to be crucial to detect bar instability, and for the other two cases, inclusion of suspension has increased the accuracy in the prediction of the bar height. Nonetheless, experiments would allow a further validation to be made.

Formally speaking, the Stuart-Landau equation obtained through the CMP is not rigorously valid for the entire unstable domain (the gray areas in Figures 2a and 2b). In fact, the application of CMP needs just one eigenvalue (Ω₁, the one related to alternate bars) to be unstable. If, for example, the central bar eigenvalue (Ω₂) is also unstable, the usage of equation 29 becomes a further approximation. For this reason, the analytical relationships for alternate bars should not be used for multiple bars in braided rivers, where many transversal modes are unstable and nonlinearly interact among them. The analytical expressions that have been used to obtain the bar length and height are reported in the supporting information S1. In particular, we furnish an open-source Jupyter notebook in Python® language and a Mathematica® ready-to-use toolbox, which more easily allows further analytical studies to be made. These results can be used in any engineering project that deals with river bedforms, such as bridges, groynes, or bed regulations, and to study organizational patterns of channel forms in natural rivers and support restoration projects.

Acknowledgments

The open-source Jupyter notebook in Python® language and the Mathematica® ready-to-use toolbox are furnished in the supporting information.

Appendix A: Closure Relationships for the Shallow Water and Exner Equations

The closure relationships necessary to solve the 1-4 system are furnished hereafter. The dimensionless shear stress is defined as

(A1)

where the friction factor C_f can be determined by means of the Einstein et al. (1950) formula for plane bed

(A2)

or from Engelund and Hansen (1967) for a dune-covered bed

(A3)

(A4)

In A3 and A4, θ and are the Shield stress and the relative roughness, respectively, and bedload transport is defined as follows:

(A5)

with α being the angle between the average particle path and the longitudinal direction, which is assumed to be small, so that

(A6)

with r = 0.56 (Talmon et al., 1995). Bedload intensity Φ in A5 is defined through the Meyer Peter and Muller formula

(A7)

where if there are no bedforms.

Appendix B: Asymptotic Expansion of ψ

B1 Settling Velocity

The settling velocity can be computed as

(B1)

where d is the dimensionless particle diameter, ν is the water kinematic viscosity, and , , q = 0.7 + 0.9S_p with shape parameter S_p=0.7 (Wu & Wang, 2006).

B2 Relationships of ψ⁽⁰⁾ and ψ⁽¹⁾

In order to avoid repetitions of previous works, only the most important analytical relationships—or the information that is lacking in Appendix B of Federici and Seminara (2006)—are reported here. Reference should be made to that work for the expressions of the Rouse number Z, as well as the integrals I₁, I₂, K₀, and K₂.

The expressions for suspended sediment, not yet expanded in the perturbations of the state variables, read

(B2)

where ψ is a depth-averaged concentration that also includes the self-similar vertical profile of the velocity field (Bolla Pittaluga & Seminara, 2003). In B2, is the depth-averaged concentration at the leading order, and it reads

(B3)

where ζ_r is the dimensionless value of the reference elevation at which C = C_e, and C_e is the reference concentration that is defined from Rijn (1984a). It should be noted that the effective Shield stress is defined as in A3, while the effective roughness that accounts for the effect of dunes reads

(B4)

where i = 0 (i = 1) for the plane (dune covered) bed. and are the dune wavelength and height, respectively (Rijn, 1984b)

(B5)

with θ_c=0.06. For the evaluation of the expanded ψ⁽⁰⁾ and ψ⁽¹⁾ in the state variables, equations 17 and 18, these last expressions are needed

(B6)

where K₁ is reported in the next section.

B3 Variation of Parameters for the Analytical Solution of K₁

An expression for is required to evaluate the order δ correction in B2, where C_f,0 is the friction coefficient for the uniform condition and κ is the von Karman constant. From the asymptotic theory developed in Bolla Pittaluga and Seminara (2003), we obtain

(B7)

where ζ₀ is the conventional dimensionless value of the reference elevation for no slip in uniform flows (ζ₀=ζ_r for a plane bed and for a dune-covered bed), and C₁₂ is obtained analytically by solving the differential equation

(B8)

with the following boundary conditions

(B9)

(B10)

Reference should be made to Bolla Pittaluga and Seminara (2003) for the expressions of ϕ₀(ζ), F₁(ζ), and F(ζ). By using the method of variation of parameters (Bender & Orszag, 2013), it is possible to write

(B11)

where y₁(ζ) =− (1 − ζ)^Z/(Zζ^−Z) and y₂=1 are two linearly independent solution of the homogeneous equation B8, and the particular solutions u₁(ζ) and u₂(ζ) are given as

(B12)

where W(ζ) =− (1 − ζ)^{−1 + Z}ζ^{−1 − Z} is the Wronskian of y₁(ζ) and y₂(ζ). The c₁ and c₂ constants are specified by imposing the boundary conditions B9 and B10 on B11. The final form of the analytical solution for C₁₂ is reported in the supporting information S1.

Appendix C: Linear Matrix L₁

The corresponding algebraic eigenvalue problem of a generic harmonic m reads , where is a matrix with null elements, except for the lower-right entry, which is 1, while reads

(C1)

(C2)

(C3)

(C4)

(C5)

(C6)

(C7)

where the subscript 0 refers to the undisturbed uniform flow solution and according to Colombini et al. (1987)

(C8)

(C9)

The dispersion relation is readily obtained by imposing  = 0. The analytical expression of Ω_m is reported in the supporting information S1.

Appendix D: CMP Coefficients

The nonlinear coefficients of the amplitude equations are

(D1)

(D2)

where N is the symmetric function that contains all the second-order nonlinearities of the system 14, for which the internal product 21 is needed; ^† and ^∗ refer to the complex conjugate and adjoint, respectively. The Landau coefficient reads

(D3)