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.
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
2.2 Stability Analysis
2.3 Linear Order
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.
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 θ_{0} (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 θ_{0}, 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.
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).
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.
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).
Case | θ_{0} | d_{s}(10^{−4}) | R_{p} | β | (m) | (m) | (m) | Δ_{L} | (m) | Δ_{H} | |
---|---|---|---|---|---|---|---|---|---|---|---|
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: m^{3}/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 m^{3}/s (days 150 and 210). Subsequently, bars became higher and longer when two floods of m^{3}/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 (θ_{0}=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 (Ω_{1}, the one related to alternate bars) to be unstable. If, for example, the central bar eigenvalue (Ω_{2}) 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
Appendix B: Asymptotic Expansion of ψ
B1 Settling Velocity
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 I_{1}, I_{2}, K_{0}, and K_{2}.
B3 Variation of Parameters for the Analytical Solution of K_{1}
Appendix C: Linear Matrix L_{1}
The dispersion relation is readily obtained by imposing = 0. The analytical expression of Ω_{m} is reported in the supporting information S1.