the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
The probabilistic nature of dune collisions in 2D
Clement Narteau
Olivier Rozier
Nathalie M. Vriend
Abstract. Dunes are bedforms of different size and shape, appearing throughout aeolian, subaqueous and extra-terrestrial environments. Collisions between dunes drive dune field evolution, and are a direct result of interacting dunes of different heights, travelling at different speeds. We perform 2D cellular automaton simulations of collisions between dune pairs migrating in a steady flow. Modelled collisions can result in either downstream- or upstream-dominant coalescence (merging of dunes) or ejection, where dunes exchange mass before separating. For each of these three elementary types of interaction, we identify the mass exchange mechanism and the distinctive intermediate morphologies. Surprisingly, we show that the collision outcome depends probabilistically on the initial dune area ratio r and can be described by a narrow sigmoidal function centred on r = 1/2. Finally, we compare our simulations with laboratory experiments of dune collisions, finding good agreement concerning the intermediate morphology and the collision outcome. Our results can motivate further observational or experimental studies that validate our probabilistic collision predictions and fully determine the controls on the coalescence-ejection transition.
Paul A. Jarvis et al.
Status: final response (author comments only)
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RC1: 'Comment on esurf-2022-55', Dominic Robson, 04 Dec 2022
General comments:
Overall, this work represents an interesting development in the field of Earth Surface dynamics and, in particular, the study of bedform interactions. The finding that dune interactions are probabilistic represents a novel result and may come as something of a surprise to many researchers in the community. I welcome this finding and believe that this work represents a sufficient advancement to warrant publication. The authors should also be commended for the style of the report which is generally well written, organised, and presented.
I do, however, have some general problems with the study. These issues centre primarily around the limitations of 2-dimensional regimes and the question of how well these results scale to realistic 3-dimensional systems. More specifically, I find the attention paid to exactly determining the form of the observed stochasticity rather irrelevant since any real-world system will behave differently because of the increased dimensionality. Additionally, the comparison with the experiments presented in Jarvis et al. (2022) is perhaps an odd choice since, as the authors point out, those experiments involved a “train” of interacting dunes rather than the binary system considered in the simulations presented in this work. Although the authors mitigate some of the issues this might present by considering only interactions where there was no physical overlap with the additional bedforms, there may be wake effects such as those identified in Bacik et al. (2020) which are not accounted for in this work. I believe that the discussion of the limitations of this work should be made more detailed as, in my opinion, these problems are more substantial than they are made to seem in the current manuscript. Nevertheless, the essential finding that collisions are stochastic rather than deterministic is an important one and the authors should be congratulated for their work.
I will now provide some more specific comments.
Specific Comments:
Line 25 - “pattern coarsening, whereby a larger number of smaller dunes transition to become a smaller number of larger dunes…”
Coarsening has been observed in many experimental and numerical studies of dune dynamics. However, many natural dune systems have been shown to be homogeneous rather than coarsening. This point should be explicitly made here.
Line 41-42 - “Turbulence, however, is an inherently 3D phenomenon…”
Interesting claim given that the water tank experiments of Bacik et al. (2020) reported that induced turbulence led to repulsion even in their quasi-2D setup. Furthermore, the claim that only the size ratio controls the collision outcome is likely only true for sufficiently large bedforms where the assumption of scale invariance applies.
Line 59 - “However this means we can exhaustively…”
The merit of such an exhaustive study is severely limited however by the fact that 3D and 2D systems are inherently different. Although you may be able to fully understand the problem in 2D, one must recognise that this is still a toy model and that a study in 3D is going to have more real-world impact even if it cannot be quite as exhaustive.
Line 60 - “By performing large numbers…”
This makes it sound as though thousands of simulations have been performed but we later find out that it was only ~50 and that, in fact, the number of simulations that could be performed was a limiting factor on the uncertainty of the findings.
Line 126 - “Two distinct types of coalescence…”
This phrasing makes it sound like they are very different processes, however the authors themselves describe that the intermediate stages are only “slightly different” and later (line 151) state explicitly that “...close to the transition it is very difficult to distinguish the two types of behaviour”. This suggests that the types of coalescence are not really “distinct” as claimed here but two regimes of a single coalescence process between which there exists a continuous transition.
Line 149 - “...so many simulations would be required to gain meaningful outcomes.”
But the authors claimed in the introduction that they had performed “large numbers” of simulations and were able to “exhaustively study” the phenomena. This is a direct contradiction and makes it seem as though the claims made in the introduction were unwarranted.
Line 160 - “...creating multiple small bedforms”
Would it not make sense to define the cases where different numbers of bedforms were generated as different types of collisions, particularly as in 3D it may be possible for these new bedforms to escape from between the dunes? This would also be more consistent with the distinction between the types of coalescence identified by the authors for which the intermediary stages were key.
Line 188 - “Performing further simulations…”
Again, this ought to be mentioned earlier as the introduction makes it seem that the study was not restricted in this manner.
Line 194 - “finding a = 14±2 and b = 0.509±0.005.”
Given that the results of 2D experiments are not likely to be fully scalable to 3D I do not believe that exact determinations of these constants are particularly relevant to real-world systems. As such, I think these values could easily be removed to an appendix.
Figure 3 caption - “...is due to only 50 simulations…”
Same point made previously, this is a major shift in tone from the introduction.
Line 204 - “...only simulated interactions between two discrete dunes… train of interacting dunes”
I think this is a more important caveat than the authors make it seem. Other similar studies (e.g. Bacik et al. (2020) have found that in these systems wake induced turbulence plays a critical role. The turbulence generated by multiple interacting bedforms in these experiments is likely to be greatly affecting the outcomes. Whereas, this is not the case in the simulations where only two dunes were present.
Line 205 - “...the number of simulations… is relatively small”
Same point made previously!
Line 209 - “However, additional simulations…”
If these simulations have been performed already and the authors wish to compare with the experimental results of Jarvis et al. (2022) then why not simply present the results from the experiments where θ = 18⦠rather than those where θ = 35⦠?
Line 225 - “There is reasonable agreement…”
I would like to see a table containing these data and ideally some statistical tests about whether the experimental observations are statistically similar to the stochastic rules defined in this work.
Line 253 - “...agree well with numerical determined…”
Again I would like to see these data.
Citation: https://doi.org/10.5194/esurf-2022-55-RC1 - AC1: 'Reply on RC1', Paul Jarvis, 23 Feb 2023
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RC2: 'Comment on esurf-2022-55', Anonymous Referee #2, 12 Dec 2022
The manuscript presents a numerical study of dune-dune collisions in two-dimensions. For that, the authors carried out cellular automaton simulations and compared the results with quasi-2D experiments. The subject is interesting, and the manuscript is well written and should be considered for publication. However, I have some concerns that I list below.
General comments
- You affirm that the collisional processes are not deterministic. In my opinion, you must better justify this affirmation, or reformulate some of your sentences. For me, the physics here is deterministic, since the motion of each sand grain is deterministic. Of course, one can analyze or model the problem as probabilistic, but, in principle, it is (I believe) deterministic.
- After briefly discussing the turbulent wake shed by the upstream dune (line 39), you state that details of turbulence are negligible in the 2D simulations because turbulence is inherently 3D. However, the presence of a recirculation bubble in the wake of the upstream dune (independent of turbulence, since it can simply be a recirculation region) can affect significantly the dune-dune collision (even avoiding it, as shown in the experiments of Bacik et al., PRL, 2020). In addition, 2D dunes in nature (or in labs) have a finite thickness, and, therefore, the flow can be turbulent. Please consider reformulating your sentences.
- You compare your numerical results against those of Jarvis et al. J. Geophys. Res: ES, 2022, in which a train of dunes was present. Please consider comparing your results also with the experiments of Bacik et al., PRL, 2020. For example: can your simulations reproduce the dune-dune repulsion observed by Bacik et al.? If not, why?
- In my opinion, in their current form the comparisons with experiments are most qualitative. In order to be more quantitative, you should present the dune profiles (perhaps superposed), celerity of crests, values of mass exchanges, etc. This would strengthen your conclusions.
Specific comments
- The manuscript is well written and agreeable to read, congratulations for that.
- Line 24, "Such collisions have been frequently observed in subaqueous experiments and numerical simulations…". They have also been observed, although with incomplete time series, for aeolian dunes.
- Lines 27-32: It should be stated clearly here that these collisional processes are specific for 2D dunes. For 3D dunes, processes are more complex, with the existence of splitting mechanisms and other types of ejection (of new dunes), as you explain briefly in the next paragraph. In addition, it is not only the dune-dune collision that redistributes sand, but also dune-dune interactions through disturbances of the fluid flow (mainly the wake of the upstream dune), which can cause, for instance, surface waves (on the downstream dune) and, consequently, calving.
- Lines 41-42: the presence of a recirculation bubble in the wake of the upstream dune (independent of turbulence, since it can be simply a recirculation region) can affect significantly the dune-dune collision (even avoiding it, as shown in the experiments of Bacik et al., PRL, 2020). In addition, 2D dunes in nature (or in labs) have a finite thickness, and, therefore, the flow can be turbulent. Please consider reformulating your sentence.
- Line 51: Please consider stressing here that those results are, in principle, valid only for subaqueous barchans, and have not been tested against aeolian barchans.
- Line 58: Ok. But please note that 2D dunes in laboratories have finite thicknesses and are not strictly 2D.
- Line 64, about comparison with experiments: Please consider comparing your results also with the experiments of Bacik et al., PRL, 2020.
- Lines 75-76: On the one hand, experiments and DNS show that the slope angle of the leeside is important, and, on the other hand, your results are based on a probabilistic approach/analysis: should not the model consider the slope angle as a (stochastic) variable?
- Lines 76-77: I find this sentence strange: for me, the physics here is deterministic. One can analyze or model the problem as probabilistic, but, in principle, the motion of each sand grain is deterministic.
- Lines 95-96: I do not totally agree. I would expect variations in quasi-2D experiments (or simulations), since the flow disturbances (wake) generated by the upstream dune vary with the flow strength.
- Line 100, about comparison with experiments: Again, please consider comparing your results also with the experiments of Bacik et al., PRL, 2020.
- Lines 135-136: Please note that you compare your data with dunes that are not strictly 2D (then, turbulence may be important)...
- Lines 163-179. This part deserves a deeper discussion: you could present some statistics of mass exchange in order to strengthen your points. Another thing is that you should better justify your analysis, since your simulations do not compute the trajectory of each grain based on Newton's second law: as a reader I would expect a discussion on how accurate the simulations are and if they are physically consistent.
- Lines 182-183: This affirmation is rather strong. In my opinion, the processes are deterministic (since each grain follows Newton's second law). What happens is that one can analyze the problem (from experiments, for example) from a probabilistic point of view, or use a probabilistic model to compute/simulate the processes. The fact that we can use a probabilistic model that works does not mean that the problem is not deterministic in essence.
- Line 199: I do not agree. In my opinion, the comparisons are most qualitative. In order to be more quantitative, you should present the dune profiles (perhaps superposed), celerity of crests, values of mass exchanges, etc.
- Lines 210-211: You should better justify this assertion.
- Lines 214-215: So, you could not compare the results for this case. Can you find published data of experiments for this case?
- Lines 236-237. Again: you need to better justify this affirmation. In my opinion, the process is not probabilistic in essence (it is deterministic), but can, perhaps, be analyzed (or computed) in a probabilistic way.
Citation: https://doi.org/10.5194/esurf-2022-55-RC2 - AC2: 'Reply on RC2', Paul Jarvis, 23 Feb 2023
Paul A. Jarvis et al.
Paul A. Jarvis et al.
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