the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Linear stability analysis of plane beds under flows with suspended load
Hajime Naruse
Norihiro Izumi
Abstract. Plane beds develop under flows in fluvial and marine environments; they are recorded as parallel lamination in sandstone beds, such as those found in turbidites. However, whereas turbidites typically exhibit parallel lamination, they rarely feature dune-scale cross lamination. Although the reason for the scarcity of dune-scale cross-lamination in turbidites is still debated, the formation of dunes may be dampened by suspended load. Here, we perform, for the first time, linear stability analysis to show that flows with suspended load facilitate the formation of plane beds. For a fine-grained bed, suspended load can promote the formation of plane beds and dampen the formation of dunes. These results of theoretical analysis were verified with observational data of plane beds under open-channel flows. Our theoretical analysis found that suspended load promotes the formation of plane beds, which suggests that the development of dunes under turbidity currents is suppressed by the presence of suspended load.
Koji Ohata et al.
Status: final response (author comments only)
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RC1: 'Comment on esurf-2022-33', Anonymous Referee #1, 29 Jul 2022
The manuscript deals with a linear stability analysis of flow over anerodible bed with suspended load. This is a resubmission of the manuscript esurf-2021-60, which I reviewed as well. At that time my suggestion was to reject the manuscript but I was open to a resubmission, which, however, "should involve a great deal of revision of the present manuscript, in particular in the way results are obtained, presented and discussed". Unfortunately, the present resubmission does not seem to have solved my concerns, so my suggestion remains that of a rejection. This time, I do not suggest a resubmission either.
Nonetheless, I remark in the following my general and specific comments, with some reference to the comments and rebuts of the original submission wherever the questions are relevant to the present resubmission too.
General comments:
The manuscript deals with a linear stability analysis of flow over an erodible bed with suspended load. The topic of dune-antidune formation has been deeply investigated in the past in terms of linear stability analysis, but the effect of suspended load has been neglected in recent theories, which assume bedload only. The inclusion of suspension represents therefore an interesting development.
My main concern remains the same. There is no such thing as the "formation of a plane bed". Plane bed is not a bedform with an extremely small wavenumber (or an infinite wavelength) as mentioned at line 61 of the revised manuscript. Plane bed is the result of the absence of bedforms, which corresponds to the stable "upper plane bed" region where neither dunes nor antidunes form and the growth rate is negative. Indeed, the problem under investigation is the stability of a uniform flow over an erodible plane bed with active sediment transport. The focus of the paper must be on the effect of suspension on the formation of dunes and antidunes. If the unstable regions expand, the stable "upper plane bed" region shrinks, and viceversa.
The choice of the governing parameters is unfortunate, with an unnecessary and awful mix of dimensional and non dimensional quantities which makes the analysis of the results quite cumbersome. Finally the discussion is too concise and leaves many points unaddressed. This is true for the conclusions as well.
Specific comments:
1) Please show stability plots in the Fr-k space. The Froude number is THE stability parameter for dune-antidune stability the sub-super critical character of these bedforms being well established. If you want to show the effect of suspension on the dune-antidune stability, you should start from the marginal curves (the boundaries of the unstable regions) in this space.
2) In your rebut to the original submission, you stated "Actually, the Shields stress does not necessarily increase with Fr because the Froude number is normalized by the square root of the product of the flow thickness and the gravity acceleration.". Â This is a nonsense. Stability plots in the Fr-k space are obtained for a constant value of the grain size to depth ratio D (or of the friction coefficient C_z), so that the Shields stress of the base uniform flow is strictly proportional to the square of the Froude number.
3) Related to the previous point, I confirm my concern: the role of suspension should become increasingly  important as Froude is increased, hence moving from dunes to the upper plane bed region to antidunes. I expect marginal curves in the Fr-k space to be deformed by the effect of suspension, the smaller the grain size to depth ratio D the lower the value of the Shields parameter at which the marginal curves for dunes and antidunes are affected. I would like to see clearly this effect before wandering in the Fr-D space, where any information on the
wavenumber is lost and the unstable regions for dunes and antidunes overlap (although they remain distinct in the Fr-k space) because the upper limit for dunes in terms of Froude number may be higher than the lower limit for antidunes, expecially for finer materials.4) Stop using dimensional parameters! Use Rep instead of the fixed dimensional grain size in figures 2 and 3. Use the grain size to depth ratio instead of the fixed dimensional flow depth in figures 4 and 5.
5) The wavenumber of maximum amplification is difficult to read in regions where dunes and antidunes overlap. Please use the growth rate of maximum amplification instead and show the curves Fr_ca(D) and Fr_cd(D) that bound the instability regions in Figures 2, 3, 4 and 5. Moreover, are the latter of any help for the reader in order to understand the results of your analysis? Â Five lines of text in the manuscript (280-284) do not justify two pages of figures.
6) Before attempting a comparison with experimental data, provide the reader with some plots of your stability analysis to explain your results and choose more wisely the values of the parameter you fix: in figure 2b dunes disappear, meaning that suspension completely inhibits dune formation. This does not help much to understand what happens in between. Moreover, some pictures are really obscure: what are those color leaks in figure 3b at D=0.007 and D=0.03? Â For such a coarse bed material suspension should be irrelevant in the dune region, whereas the plot is remarkably different from figure 3a in that region. Hard to explain.
Citation: https://doi.org/10.5194/esurf-2022-33-RC1 - AC1: 'Reply on RC1', Koji Ohata, 28 Oct 2022
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RC2: 'Comment on esurf-2022-33', Anonymous Referee #2, 23 Aug 2022
In this manuscript, the authors use linear stability analysis to show that suspended sediment load could promote the stability of plane beds for open-channel flows with fine bed-material sediment. They propose that this mechanism could explain the observations of parallel laminations in turbidites, which typically lack dune-scale cross stratification. The authors also use observational data to test their hypothesis. Overall, the manuscript is reasonably well-written; however, the writing and presentation still needs a lot of work to clarify the results and avoid repetition. Importantly, I found that this manuscript needs significant amount of work to clarify several aspects of the analysis before being ready for publication. I summarize my major comments below:
1. The definition of a plane bed in terms of dominant wave number seems rather confusing to me. By definition, a plane bed is not a bed form that has a large wavelength. So, defining the plane bed this way and then using linear stability analysis to find parameter space that correspond to a small dominant wave number seems odd to me. At least, there is no justification given for why this should correspond to a strict definition of a plane bed. This is a major point as this assumption is the foundation for the entire manuscript.Â
2. The limits on the parameter space explored here needs justification. For example, in lines 73-77, the authors describe the range of particle sizes and flow depths explored but also state that they set the grain size to 3 values and flow depths to 3 values. How is it that the data could not be recast into only dimensionless terms without the need for using a mix of dimensional and dimensionless variables?
3. The authors need to give more detail about the observational data that is used to support their hypothesis. How are data from a range of grain sizes and flow depths collated to plot on stability diagrams with a single value of grain size, for example? What is the sensitivity of these stability diagrams to the parameters?Â
4. What is the criterion for the success of the model? It appears from the results that a majority of the observations plotting in the stable region of the contour maps is enough to state that the model works. There is no discussion of how many points do not plot in the stable region and what it means for the model veracity. I think the authors need to lay out the metrics they will use to test the success of the model and then discuss how the field and flume data compare with this test. Right now, the entire model testing part of the manuscript is weak and arbitrary.Â
5. The figures need some more explanation. It is not clear to the reader where each of these data points should lie in terms of model expectations? For example, I would expect that if larger fraction of actual plane bed data lining up with the stable region in the contour plots would be a model success but I don’t see a lot of observational data matching up with stable regions on the contour plots. If I am mistaken about my interpretation here, then the authors need to do a better job of explaining the metrics for success of their model.Â
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Citation: https://doi.org/10.5194/esurf-2022-33-RC2 - AC2: 'Reply on RC2', Koji Ohata, 28 Oct 2022
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RC3: 'Comment on esurf-2022-33', Anonymous Referee #3, 30 Aug 2022
This paper conducts a stability analysis to explicitly include sediment suspension in order to determine the role of suspension on the suppression of bed topography. Their analysis shows that the presence of suspended load is a controlling factor on upper plane bed stability with implications for the understanding of hydrodynamics of deposits such as those formed by turbidites and other high suspended sediment concentration flows. The model framework was well described and understandable. However I do agree with most comments posed by Reviewers 1 and 2, particularly the contextualization and success criterion questions brought up by reviewer 2 and the appropriateness of using the dominant wavenumber to define the formation of plane bed.Â
There are a few additional contextualization issues to clarify and put the work into the broader picture should those prior issues be adequately addressed. First, in the abstract, there are a number of sentences that seem repetitive, and the mechanistic component of the role that suspension plays is not described. Again in the introduction, the hypothesized role of suspension is not mechanistically discussed. In the discussion, there is brief allusion to the fact that this analysis demonstrates that turbulent suppression for example is not required, but I think the exact mechanism by which the presence of suspended load is not fully described in the written work.Â
I should note that I don’t necessarily agree with the dimensional arguments made by Reviewer 1 - while it is certainly common practice and more relevant in predictive modeling to non-dimensionalize, the framing of the arguments in this paper does not necessarily require it in my opinion.
Citation: https://doi.org/10.5194/esurf-2022-33-RC3 - AC3: 'Reply on RC3', Koji Ohata, 28 Oct 2022
Koji Ohata et al.
Koji Ohata et al.
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