the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 28 Jul 2021
Linear stability analysis of plane beds under flows with suspended load
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 suggest that the development of dunes under turbidity currents is suppressed by the presence of suspended load.
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Interactive discussion
Status: closed
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RC1: 'Comment on esurf-2021-60', Anonymous Referee #1, 12 Aug 2021
Dear all,Â
please find attached my comments on the manuscript Linear stability analysis of plane beds under flows with suspended load by Ohata et al.
Note that I have ticked 'reconsidered after major revisions', as my concerns on the sediment transport model could lead to substantial modifications, and therefore need time.
-
AC1: 'Reply on RC1', Koji Ohata, 17 Oct 2021
Thank you very much. We are pleased to hear that you find that our manuscript may deserve publication. Every comment from you is very helpful to improve our manuscript, which we greatly appreciate. Our responses to your comments are shown on the supplement pdf.
-
AC1: 'Reply on RC1', Koji Ohata, 17 Oct 2021
-
RC2: 'Comment on esurf-2021-60', Anonymous Referee #2, 13 Sep 2021
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. Â I have no major concerns on the way the problem is formulated and the linear analysis performed, except for the way suspended and bed loads are arbitrarily turned on and off using the binary alpha coefficients, which I will discuss in the following section. 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.
My main concern on this manuscript is about the choice of focussing on the upper plane bed regime rather than discussing in depth the effect of suspension on the formation of dunes and antidunes, the former being a byproduct of the latter. Moreover, results are poorly presented and it is difficult to draw any conclusion except for some very general statements about the broadening of the stable region. My suggestion is therefore to reject the manuscript. I would favor a resubmission, though, which however should involve a great deal of revision of the present manuscript, in particular in the way results are obtained, presented and discussed.
Specific comments
1) The analysis is based on the perturbation of a uniform flow on an erodible bed with active sediment transport. Stability plots like the one presented in Figure 1 (those in figure 2 and 3 follow as a direct consequence) are obtained by varying the wavenumber of the mode AND the Froude number of the basic state. The Shields stress increases with Fr, and so I would expect the suspended load to be negligible at low Fr (where there is bedload only) and to become increasingly important as Fr increases (and dominant over bedload for very large Fr). It is not clear if this behaviour is correctly represented in the model, where the alpha_s coefficient is abruptly set to 1 as the the threshold for suspension (69) is exceeded. The latter condition, which implies a specific value Shields stress (and Fr) for the basic flow, should also correspond to a negligible amount of suspension with respect to bedload. Is that so?
2) In the search for the growth rate, the use of a spectral collocation method does not seem to be able to provide the required resolution: isolines in Figures 2 and 3 are quite wiggly, in particular close to the thresholds for transport and suspension. Perhaps more collocation points are needed?
3) The choice of the parameters is really unfortunate and provides an awkward mix of dimensional and nondimensional quantities. Indeed, (101) provides the parameters of the problem, so figures 2 and 3 can be more meaningfully represented in the Fr-Cz space for different values of Rp. Moreover, plots of the wavenumber of maximum amplification are confusing: is the wavenumber plotted that of a dune or of an antidune? Plot the growth rate of maximum amplification instead. Section 2.2 should be rewritten accordingly.
4) Since the work of Engelund it is well known that the role of suspension should be to inhibit the formation of dunes and to enhance that of antidunes. This is the picture I would like to see emerge from the linear analysis. Apart from a generic broadening of the upper plane region, which is compatible with the above framework, this picture does not surface clearly from the analysis of Figures 2 and 3.
5) In the stability plots with suspension some inconsistencies or, at least, some strange behaviours, appear. More precisely, in 2b and for relatively deep flows, the bed becomes plane as soon as the threshold for suspension is reached, even though the amount of suspension should be negligible there. The same happens in 2c, but not for the deepest flows. In 3b the modifications with respect to the bedload only case (3a) are inexplicable and unexplained.
6) The appearance of a sheet flow regime has been invoked in the past as the mechanism controlling the transition between dunes, plane bed and antidunes. Although this may be more a consequence than a cause, the bedload model adopted does not handle sheet flow transport mode. I would rather drop the starred experimental observations from Figure 3 than attempt to fit them in the picture.
Technical corrections
23) Therefore ... must be considered - Therefore, the influence of suspended load on the formation of dunes and antidunes is worth investigating.
33) Therefore ... we performed linear stability analyses - we performed a linear stability analysis
62) The section starts with the same sentence than 41. Please rephrase.
224) amplification -> amplitude
333) wider -> larger, broader
368) reasonably?
371) Ultimately, our linear analysis provides a possible explanation ...
376) We -> we
380) I would drop the whole section: the discussion about sheet flows is totally speculative, since the model adopted does not consider bed
load moving as sheet flow.Â
Citation: https://doi.org/10.5194/esurf-2021-60-RC2 - AC2: 'Reply on RC2', Koji Ohata, 17 Oct 2021
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EC1: 'Comment on esurf-2021-60', Eric Lajeunesse, 16 Sep 2021
Dear Authors,
We have now received two anonymous reviews of the manuscript « Linear stability analysis of plane beds under flows with suspended load ». Both of them identify major issues.  The topic of your manuscript is important : the inclusion of the influence of the suspended load in the formation of dunes and anti-dunes represents an interesting development. However, I share the opinion of the referees that considerably more work is  required for the present manuscript to be published in E-Surf.  I would therefore advise you to fully revise your manuscript in line with the reviewers recommendations. If you feel that you cannot handle the great deal of revision necessary to clarify the content of the paper within the next few weeks, we may grant you an extension of the deadline. Another possibility is to withdraw your paper, for the time being, and to resubmit an entirely new version, when it is ready.
Sincerely yours,
Eric Lajeunesse
Citation: https://doi.org/10.5194/esurf-2021-60-EC1 -
AC3: 'Reply on EC1', Koji Ohata, 17 Oct 2021
Thank you very much. We are pleased to hear that you find that our manuscript is important and appreciate every comment from anonymous reviewers. We already replied to reviewers' comments for your consideration on the publication of this manuscript. We will revise our manuscript and reconduct linear analyses to follow reviewers' comments. Â
Citation: https://doi.org/10.5194/esurf-2021-60-AC3
-
AC3: 'Reply on EC1', Koji Ohata, 17 Oct 2021
Interactive discussion
Status: closed
-
RC1: 'Comment on esurf-2021-60', Anonymous Referee #1, 12 Aug 2021
Dear all,Â
please find attached my comments on the manuscript Linear stability analysis of plane beds under flows with suspended load by Ohata et al.
Note that I have ticked 'reconsidered after major revisions', as my concerns on the sediment transport model could lead to substantial modifications, and therefore need time.
-
AC1: 'Reply on RC1', Koji Ohata, 17 Oct 2021
Thank you very much. We are pleased to hear that you find that our manuscript may deserve publication. Every comment from you is very helpful to improve our manuscript, which we greatly appreciate. Our responses to your comments are shown on the supplement pdf.
-
AC1: 'Reply on RC1', Koji Ohata, 17 Oct 2021
-
RC2: 'Comment on esurf-2021-60', Anonymous Referee #2, 13 Sep 2021
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. Â I have no major concerns on the way the problem is formulated and the linear analysis performed, except for the way suspended and bed loads are arbitrarily turned on and off using the binary alpha coefficients, which I will discuss in the following section. 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.
My main concern on this manuscript is about the choice of focussing on the upper plane bed regime rather than discussing in depth the effect of suspension on the formation of dunes and antidunes, the former being a byproduct of the latter. Moreover, results are poorly presented and it is difficult to draw any conclusion except for some very general statements about the broadening of the stable region. My suggestion is therefore to reject the manuscript. I would favor a resubmission, though, which however should involve a great deal of revision of the present manuscript, in particular in the way results are obtained, presented and discussed.
Specific comments
1) The analysis is based on the perturbation of a uniform flow on an erodible bed with active sediment transport. Stability plots like the one presented in Figure 1 (those in figure 2 and 3 follow as a direct consequence) are obtained by varying the wavenumber of the mode AND the Froude number of the basic state. The Shields stress increases with Fr, and so I would expect the suspended load to be negligible at low Fr (where there is bedload only) and to become increasingly important as Fr increases (and dominant over bedload for very large Fr). It is not clear if this behaviour is correctly represented in the model, where the alpha_s coefficient is abruptly set to 1 as the the threshold for suspension (69) is exceeded. The latter condition, which implies a specific value Shields stress (and Fr) for the basic flow, should also correspond to a negligible amount of suspension with respect to bedload. Is that so?
2) In the search for the growth rate, the use of a spectral collocation method does not seem to be able to provide the required resolution: isolines in Figures 2 and 3 are quite wiggly, in particular close to the thresholds for transport and suspension. Perhaps more collocation points are needed?
3) The choice of the parameters is really unfortunate and provides an awkward mix of dimensional and nondimensional quantities. Indeed, (101) provides the parameters of the problem, so figures 2 and 3 can be more meaningfully represented in the Fr-Cz space for different values of Rp. Moreover, plots of the wavenumber of maximum amplification are confusing: is the wavenumber plotted that of a dune or of an antidune? Plot the growth rate of maximum amplification instead. Section 2.2 should be rewritten accordingly.
4) Since the work of Engelund it is well known that the role of suspension should be to inhibit the formation of dunes and to enhance that of antidunes. This is the picture I would like to see emerge from the linear analysis. Apart from a generic broadening of the upper plane region, which is compatible with the above framework, this picture does not surface clearly from the analysis of Figures 2 and 3.
5) In the stability plots with suspension some inconsistencies or, at least, some strange behaviours, appear. More precisely, in 2b and for relatively deep flows, the bed becomes plane as soon as the threshold for suspension is reached, even though the amount of suspension should be negligible there. The same happens in 2c, but not for the deepest flows. In 3b the modifications with respect to the bedload only case (3a) are inexplicable and unexplained.
6) The appearance of a sheet flow regime has been invoked in the past as the mechanism controlling the transition between dunes, plane bed and antidunes. Although this may be more a consequence than a cause, the bedload model adopted does not handle sheet flow transport mode. I would rather drop the starred experimental observations from Figure 3 than attempt to fit them in the picture.
Technical corrections
23) Therefore ... must be considered - Therefore, the influence of suspended load on the formation of dunes and antidunes is worth investigating.
33) Therefore ... we performed linear stability analyses - we performed a linear stability analysis
62) The section starts with the same sentence than 41. Please rephrase.
224) amplification -> amplitude
333) wider -> larger, broader
368) reasonably?
371) Ultimately, our linear analysis provides a possible explanation ...
376) We -> we
380) I would drop the whole section: the discussion about sheet flows is totally speculative, since the model adopted does not consider bed
load moving as sheet flow.Â
Citation: https://doi.org/10.5194/esurf-2021-60-RC2 - AC2: 'Reply on RC2', Koji Ohata, 17 Oct 2021
-
EC1: 'Comment on esurf-2021-60', Eric Lajeunesse, 16 Sep 2021
Dear Authors,
We have now received two anonymous reviews of the manuscript « Linear stability analysis of plane beds under flows with suspended load ». Both of them identify major issues.  The topic of your manuscript is important : the inclusion of the influence of the suspended load in the formation of dunes and anti-dunes represents an interesting development. However, I share the opinion of the referees that considerably more work is  required for the present manuscript to be published in E-Surf.  I would therefore advise you to fully revise your manuscript in line with the reviewers recommendations. If you feel that you cannot handle the great deal of revision necessary to clarify the content of the paper within the next few weeks, we may grant you an extension of the deadline. Another possibility is to withdraw your paper, for the time being, and to resubmit an entirely new version, when it is ready.
Sincerely yours,
Eric Lajeunesse
Citation: https://doi.org/10.5194/esurf-2021-60-EC1 -
AC3: 'Reply on EC1', Koji Ohata, 17 Oct 2021
Thank you very much. We are pleased to hear that you find that our manuscript is important and appreciate every comment from anonymous reviewers. We already replied to reviewers' comments for your consideration on the publication of this manuscript. We will revise our manuscript and reconduct linear analyses to follow reviewers' comments. Â
Citation: https://doi.org/10.5194/esurf-2021-60-AC3
-
AC3: 'Reply on EC1', Koji Ohata, 17 Oct 2021
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