the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 06 Sep 2024
Investigating the celerity of propagation for small perturbations and dispersive sediment aggradation under a supercritical flow
Abstract. The manuscript presents an investigation of the scales of propagation for sediment aggradation in an overloaded channel. The process has relevant implications for land protection, since bed aggradation reduces channel conveyance and thus increases inundation hazard; knowing the time needed for the aggradation to take place is important for undertaking suitable actions. Attention is here focused on supercritical flow, under which the process is dispersive and a depositional front cannot be clearly recognized; in these conditions, one needs to define propagation scales locally and instantaneously. Based on spatial and temporal rates of variation of the bed elevation we quantify a celerity of propagation for the sediment aggradation wave. Furthermore, considering that morphological processes are modeled by differential equations, the eigenvalues of the latter’s system are the celerities of the so-called small perturbations. With reference to a laboratory experiment with temporally and spatially detailed measurements, and after a review of existing approaches to determine the celerity of small perturbations considering or discarding the concentration of transported sediment, the manuscript shows how the celerities of propagation correlate with one another, while their values differ by orders of magnitude. It is argued that accounting or not for the solid concentration in the governing equations does not significantly impact the correlation trends, even one of the eigenvalues changes significantly. Finally, a bulk value of a dimensionless aggradation celerity is provided, that can serve as a rule-of-thumb estimation, useful for engineering purposes.
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RC1: 'Comment on egusphere-2024-414', Anonymous Referee #1, 05 Nov 2024
This manuscript presents an interesting and valuable contribution to our understanding of sediment transport processes, specifically focusing on the propagation of aggradation waves under supercritical flow conditions. The work addresses an important topic in geomorphology with potential implications for flood risk assessment and channel management. The experimental approach, featuring detailed spatial and temporal measurements of bed elevation changes, provides a strong foundation for investigating the relationship between theoretical predictions of perturbation celerity and observed aggradation patterns.
The study's primary strength lies in its methodical comparison between experimentally observed celerities and theoretical eigenvalues derived from governing equations, offering insights into how different mathematical formulations correlate with physical observations. The laboratory dataset appears robust and well-suited to the investigation's objectives, although only one case was considered.
However, the manuscript would benefit from several improvements to enhance its impact and accessibility. The presentation of the material requires restructuring to improve flow and clarity, particularly in separating results from their interpretation. The discussion section could be expanded to better explore the broader implications of the findings and their potential applications. Additionally, some technical aspects need attention, including more precise definitions of key concepts and a more rigorous approach to statistical analysis.
The discussion section needs to be improved significantly as it appears to repeat the results section without substantially expanding on implications and broader applications, or considering cases beyond the single experiment conducted.
With appropriate revisions, this work has the potential to make a significant contribution to our understanding of sediment transport processes in supercritical flows and provide valuable guidance for practical applications.
- AC1: 'Reply on RC1', Alessio Radice, 12 Nov 2024
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RC2: 'Comment on egusphere-2024-414', Anonymous Referee #2, 14 Nov 2024
In the manuscript, Eslami et al. (2024) have investigated the celerity of aggradations in a controlled flume experiment and explored the relationship between the celerity of aggradation waves with the celerity of propagation of small perturbations. This is a nice work and detailed analysis of the experiments; however, I have a number of questions, partly regarding the theoretical conceptualization and clarifications on several points.
It is very nice that the authors found the ratio C/u<0.04 based on the experiment. If I interpret it correctly, doesn't this suggest that during high Froude number flows, sediment transport is more dispersive, making aggradation much harder to occur? I wonder what the threshold for the ratio C/u would be under field conditions. If the results presented here are valid, I am curious whether this has implications for understanding aggradation at field conditions. For example, how long would it take for sediment transport under high Froude number flow to alter riverbed and channel morphology?
More specific comments are outlined below:
The authors seem to accept the celerity parameters are eigenvalues of the coefficient matrix of a system of partial differential equations as proposed previously in the literature. I am skeptical about whether these eigenvalues are indeed representing the celerity of a disturbance (of bed or surface water). Thus, I would recommend adding a theoretical background section to introduce how the eigenvalues become celerity from a mathematical perspective. For example, after linearizing the non-linear partial differential equation and establishing a system of differential equations, dU/dt + AdU/dx = 0, explain how this concept translates into the celerity of a disturbance happening at a water surface or streambed. Essentially, you can mention that this problem can be treated as Reimann invariant (e.g., Lyn and Altinakar, 2002) under a characteristic path of dx/dt = lambda, where lambda is characteristic speed or wave speed, which I think is how the term celerity arises. In this path, the disturbance waves move at constant speed over spatial and temporal scales.
In the introduction, the concept of celerity of propagation of small perturbations and aggradation wave is not clearly defined. I do not understand what are referred to as small perturbations, and in what criteria the authors define a “small” perturbation here. As I read the introduction of the manuscript, I assumed that the celerity of propagation of small perturbations is in this context, water wave celerity and bed wave celerity. It would be nice to introduce/explain this concept in a clearer way. As mentioned in the manuscript, the celerity of the aggradation is easily quantified in low Froude number flow since the translational migration of bedload can be captured visually. The authors noted that in the high Froude number flow, sediment and water are moving more dispersive; thus, the celerity of aggradation needs to be more precisely defined before diving into the analysis of the partial differential equations in chapter 2, and later quantified by equation 20.
The goal of the present manuscript is not clear according to the results presented there. The authors aim to answer 3 questions. (1) How can one quantify the scales of propagation in an aggradation process? (2) What is the relationship between the aggradation celerity and the celerity of small perturbations? And finally, (3) Which is the impact of considering or not the sediment concentration on the previous point? However, results and discussions were not focused on answering them; instead, the authors presented some graphical results between the aggradation celerity and the celerity of small perturbations, and no specific formula or relationships between have been clearly determined. What are some findings for questions 1 and 2, I did not see it? How one can predict the spatial and temporal scales of an aggradation wave, given the information to solve the system of differential equations here? Having read the title and introduction, I hope the present study can show when and where the aggradations are likely to happen based on the celerity of propagation of aggradations.
In the result section, I assumed the author performed many experiments and listed the average results or best results, but I only see a table summary of the experiment. Is this experiment (and results) reproducible? Updated: I see now in the results section, the authors mentioned that “This result, shown here for a single experiment, was confirmed by the others run in the current experimental campaign”. If the authors run other experiments to confirm the findings here, they should be presented in the manuscript, or at least should be in supplemental information.
In the discussion section, there is little interpretation of values of lamda1, 2, and 3, as well as c, in the context of the performed experiment. What exactly do these lambdas represent? The authors just presented results with minimal intuition and without comparison with previous work (i.e., Zanchi and Radice, 2021) to distinguish between subcritical and supercritical cases. I expected to see more on the comparison of how changing lambdas can impact the bed aggradation and bed elevations.
Eigenvalues are the solutions of characteristic equations that encompass information of matrix coefficients of a system of differential equations. Hence, it is difficult to say, for example, that lambda 1 is celerity of water flow, as this manuscript and previous authors suggested, or lambda 2 and lambda 3 are celerity of bed perturbations. Rather, it may be a combined effect of both water flow and streambed on these eigenvalues. What are the viewpoints of the authors for these parameters, according to the experiment performed here? Additionally, does this characteristic equation det(A-I) = 0 always have 3 real roots? Is there any scenario such that the equation yields two negative roots instead of one, as shown in this paper?
Line 67-68: Why is the celerity of propagation of small perturbations not the celerity of propagation of the aggradation wave? For example, if the aggradation wave is moving downstream and generating disturbances, can these disturbances generate perturbations of water and bed?
Line 69: What is “something different” here?
Line 73: Missing parenthesis.
Line 129, equation 10: Lack of definition for Froude number.
Line 165, equation 19: I don’t understand why setting dX/dt = 0 in Equation 19. Does that imply that C = -(dX/dt)/(dX/dx) = 0? Later, the authors used bed elevation for their calculations, but dzb/dt is not 0, which is a contradiction. Also, in lines 163 and 164, if I understand correctly, celerity is a scalar quantity, while velocity is a vector quantity; they are not the same. On line 67, you mentioned “the celerity of propagation of small perturbations is not the celerity of propagation of the aggradation wave." Then, in equation 19, you defined the local and instantaneous Cx = dx/dt, which is the same as the celerity of a disturbance. This is again another contradiction.
Line 215, table 1: It seems that only one experiment was performed (e.g., Table 1). I’m worried about the reproducibility of the results presented in this manuscript.
Line 221: The authors mentioned that they did not set the camera to capture the profile before 140 cm, which piques my curiosity. Was there any aggradation or erosion in this section during the experiment?
Line 255, equation 20: Is equation 20 derived from equation 19? It seems to me that celerity can be simply defined as C = dx/dt (as in de Vries, 1993; if x is difficult to quantify, you can then use dx/dt = (dX/dt)/(dX/dx) where X is easily to quantify (e.g., bed elevation in the paper). I am not sure about the minus sign in this equation, as it contradicts the standard definition from de Vries or from Morris and Williams.
Line 320, 335: Were results shown in Figures 8 and 10 for the entire profile of the riverbed in this experiment (e.g., 140 to 520 cm) or only for some selected locations along this section?
Line 345: I don’t quite understand how the correlation trends are obviously consistent. Which one is consistent with the other?
Line 347: The authors stated: “The second correlation (𝐹𝑟 - 𝐶/𝑢) has returned the dimensionless celerity as a decreasing function of the Froude number”. However, there are some clusters (e.g., 3 lines on top) where C/u increases with respect to Froude number (Figure 8).
Line 360: If lambda2 and lambda3 are attributed to the bed perturbations, can you explain more on the negative values of lambda 2?
Line 378: The authors claimed that it is always possible to find a trend linking celerity of small perturbations and celerity of aggradation wave. This is a bold statement that needs to be tested not only in experimental studies but also at field scale, where things are much more complicated. And I am highly skeptical about this.
Line 383: What are other processes in this context?
Citation: https://doi.org/10.5194/egusphere-2024-414-RC2 - AC2: 'Reply on RC2', Alessio Radice, 22 Nov 2024
Data sets
Raw data for the present experiment Hasan Eslami et al. https://doi.org/10.5281/zenodo.10641001
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