Equilibrium distance from long-range dune interactions

Vérité et al. (2024) preprint

General comments

The authors present an application of a well-established and widely-used numerical model (ReSCAL) to uncover the influence of long-range interactions on two-body (and briefly several-body) dynamics of simulated bedforms exposed to different flow regimes.  The manuscript is well-structured and well-written, and the figures are of an excellent standard and the authors should be congratulated.  The work will be of interest to those who study aeolian systems and compliments some of the existing literature, particularly the properties of systems of linear dunes and quasi-two-dimensional flume experiments.

However, there are some aspects of the work that this reviewer believes require additional investigation of clarification before the work is ready for publication.  Chief among these concerns is that the authors have highlighted in this manuscript the importance of the “dune” shape in governing the interactions that they observe.  The shape of the dune, in particular the aspect ratio, is found to govern much of the dynamical behaviour which is observed.  However, to the knowledge of this reviewer, the effect of changing the elementary aspect ratio (i.e. height to length or width) of the sand slabs (cells) from which the bedforms are formed has never been investigated (other than a brief mention in Appendix A of Narteau et al. (2009)).  Thus, one cannot be certain whether the results presented here are universal outputs of the model or hold only in the specific case of a 1:1 cell height:width ratio. 

Changing the elementary aspect ratio may affect coupling between the bedform and lattice gas since the bedforms are likely to be comprised of a greater number of sand slabs if the cell height is less than  and so the removal of a single slab due to erosion/transport will have less of an impact on the shape/size of the bedform and thus result in a smaller perturbation of the flow.  Changing the height of the slabs would also affect the avalanche procedure which could in turn influence the shape (and migration etc) of the bedforms.

 Although this reviewer concedes that there is a precedent for publication of studies applying ReSCAL without an investigation of the affect of changing the cell height:width ratio, this manuscript places a significant emphasis on the role of shape and bedform aspect ratio on governing the interactions.  As such, this reviewer feels that it is important to ascertain how the findings of this study would change if the elementary aspect ratio is varied before publication.

Below follow some more specific comments regarding the manuscript as presented.

Specific comments

1. Introduction

Lines 34-38 “Instead, another set…” – Génois et al., 2013 presents an agent-based model of barchan systems in which there is no consideration of the flow and so it is not accurate to say that the work “indirectly highlights the long-range flow-induced interactions”.  Long-range interactions in that work occur only through the sand-flux exchange/shadowing i.e. an upwind dune prevents a downwind one from absorbing incoming flux.  This has nothing to do with the fluid flow – it is a sediment-induced interaction.

Lines 39-42 “Under unidirectional flows…” – It is not yet established whether this shearing at a long-distance takes place in aeolian dunes or solely in subaqueous experiments and highly turbulent numerical simulations.  Some authors have claimed that in aeolian settings it is the case that it results from increased erosion induced by an upwind dune (e.g. Bourke Icarus (2010)) but others (e.g Elbelrhiti Geomorphology (2012)) have instead suggested that is it due to sand flux absorption and not due to erosion.  It is worth mentioning that this debate for aeolian dunes has not been resolved.

1. Methods

Line 80 “The physical environment…” – This sentence makes it clear that this work considers only the case that the height and length of the cells are equal (i.e. a square lattice) and changing this aspect ratio (e.g. a rectangular lattice) has not been considered.

Line 85 “To take into account…” – How is this angle of repose achieved, surely at an angle of 35 degrees two adjacent cells that differ by a single slab in height would avalanche and so the only stable state would be a flat bed.  If two adjacent cells are able to have a difference of a single slab (which is what this reviewer believes was applied in e.g. Zhang et al. JGR (2010)) then the angle of repose is in fact 45 degrees, which is higher than stated here and what it should be for aeolian bedforms.

Line 114 “The elementary length…” – in Narteau et al. (2009) the height:length ratio of the cells in all simulations shown was set to unity and it is not clear in that work how changing the ratio would impact the results due to both flow-coupling changes and avalanche changes as previously mentioned.

Line 121 “Theoretically, this rescaling…” – same comment as above. 

Line 126 “To prevent collisions…” – presumably this means that below this initial separation collisions do occur.  Since collisions are observed in nature, does this mean that dunes which collide must have formed closer to each other than this distance?

Line 127 “Except for the simulations…” – many linear dunes e.g. those shown in figure 1 form with an angular separation of less than 180 degrees.  If the 2D bedforms shown in this work are taken to be cross-sections of these linear dunes then that would mean that different parts of the dunes would be interacting in the primary and secondary modes (since there is a also a longitudinal component to the wind).  How is this likely to affect the results presented here?

Line 136 “We save the…” – again, these morphological characteristics might be influenced by choice of the cell shape.

1. Results

Line 154 “Regardless of the…” – provided that the initial spacing is above a predetermined length to avoid collisions.

Line 158 “Under these conditions…” – again, if the angular separation of the wind modes is less than 180 degrees different cross-sections of the dunes will be interacting under each mode.

Line 164 “Since this equilibrium…” – in Appendix A of Narteau et al. (2009) it is conceded that changing the aspect ratio of the cells in the model would “allow for the modeling of smaller structures of the flow”.  Thus the results here are certain to be influenced by the choice of a square lattice.

Line 172 “In the reattachment…” – the bedforms shown in this work are around 30 cells high, how would this “a few dune heights” change if the cells in the model were e.g. 1/10 as high as they are long and so dunes were 300 cells high?

Section 3.3 (whole thing) – This section goes into detail about the reshaping timescale of the dunes and their shape.  All of these processes are likely to be affected by the shape of the cells.  If, it turns out that they are unaffected then that is an interesting result worth including.

Line 185 “In the limit…” – As before, how is avalanching achieved at 35 degrees in a square lattice?

Line 219 “The drop 220 observed…” – again, how do the cells affect these aspect ratios and dune size?

Line 226 “Wake flow perturbations…” – what are the characteristic wavelengths in barchan dune field which have non-erodible interdune areas?  If it is the flow then surely there would be a coupling between the size of the dunes and their spacing but this has been shown not to be the case in Durán et al. Granular Matter (2009)

Figure 6 – This is an excellent figure.

1. Discussion

Line 238 “The mechanism of…” – Given this, it is even more important to understand how the cell shapes are influencing the bedform shape.

Line 262 “This could be…” – as mentioned, how is the distance of the reattachment changed by changing the cell height in this model?

Line 280 “These changes in…” – same comment as previously

Line 290 “For example, before…” – why then do we see collisions between barchans?  Is it simply that the fact that you have a three-dimensional dune rather than a two-dimensional cross-section changes the whole behaviour, in which case why should the results of two-dimensional simulations matter for studying barchans?

Line 309 “However, the results…” – this is a really important point and could be emphasised further.

Overall comments

This is a really nice study on dune interactions. The problem is well posed and the simulations address it properly. The discussion feels complete and balanced. The scope is appropriate, but there is plenty of further questions the manuscript produces that leave the reader hoping for more! I have some comments below which should be addressed, but none undermine the quality and novelty of the work. Looking forward to the final publication – I hope my review is useful.

General comments

Can you verify with the cross-sectional area of the dunes that there is no between-dune transport influencing the result? It’s not clear how an initial spacing of 120l_0 is a sufficient guarantee.

How do your results depend on the height of the domain? 100l_0 is about 1/3 of the dune height for the S=2e+3l_0^2 case (as per Figure 2) – there is almost certainly ‘pinching’ of the flow over the dunes in this case. For lower S values, maybe the pinching is less important, but the varying levels of flow pinching between the dune crest and the domain top (akin to fluvial dunes), and it’s role in influencing the stress across the dune stosses may make these results deviate from what could be observed in real aeolian dune systems. I think it’s especially important to test this and discuss it since most of the results in the paper depend on it.

I think it would be worth mentioning if the simulated dune aspect and shape ratios are comparable to observed dunes in nature, to give more weight to the results shown here.

Line comments

Line 20: Flow perturbation by aeolian dunes over long distances (i.e., beyond pair-wise interaction, at the dune-field scale) has been observed if the authors want a reference: https://doi.org/10.1029/2020GL088773

Line 28: This sentence is true for dunes with a sufficiently steep lee side, it might be worth qualifying that a separation bubble does not form for all dunes.

Line 45: “In other” should be “Another”.

Line 109: Controlling flow strength indirectly by \tau_1 is ok, but can you confirm that when you vary it, if you also vary \tau_2 commensurately? A little more explanation of how this isn’t quite the same as explicitly changing flow strength could be useful, e.g., non-linearity of flux-speed relationship, implicit Re-invariance of separation bubble size and speed-up effect, etc.

Line 146: Is \langle\tau_{flat}\rangle measured on the flat topography of an identical domain with no dunes, or on flat topography in a domain with dunes in it? If it’s the latter, then it should be clear what the condition for “away from any topography” is.

Figure 2: “Unidirectionnal” typo in top left plot title. X-axis label on panel b says “x-ct”, while panel f this is omitted.

Figure 3: It would be useful to know what timestep in which experiment panels a and b correspond to, and if it is currently repulsing, attracting, or in equilibrium. In panel b, where does the dispersion band come from? Is it from the 10 random seeds or from multiple timesteps? Panels d and e aren’t labelled.

Line 172: This sentence is critical, yet I don’t think it’s reflected that well in Figure 3b. Here it says that downwind of a dune, once the flow has reattached, the stress is higher than if the dune wasn’t there. That is only shown for the downwind dune, not the upwind dune – it’s probably worth mentioning that when referencing the figure here.

Line 174: Since \Delta\langle\tau_S\rangle is normalised by \langle\tau_{flat}\rangle I think maybe it’s useful to put a \hat{} over it so it’s clear it’s normalised. I know they are all already non-dimensional but this one doubly so. Readers may take the quantities of ~1e-1 on the y-axis of Fig 3e literally.

Section 3.3: This whole section and Figures 4 & 5 are fantastic. Very well explained.

Figure 5: caption for panel a should say “shape ratio (bottom)”. The panel b is really nice, I wonder if it’s possible to draw an analogy to pair-wise interaction potentials (e.g., Lennard-Jones) – especially since at even shorter ranges (i.e., in the recirculation bubble) the curve will be attractive again, right? But for panel b, I still do not think that it’s demonstrated anywhere how the long-range attractive regime can be seen in the surface stress profile downwind of a dune. Shouldn’t you see effectively a vertically flipped version of this curve in the \tau profile downwind of an isolated single dune? It would be great to see that – since in Figure 3b that attractive regime is missing.

Figure 6: It would be very useful to see if there’s any way of collapsing these curves onto a master curve given the control parameters – it may not be simple though. Also, why is the control parameter set sparse? E.g., reversing flow \tau_1/\tau_0=30 data, and (\tau_1/\tau_0<30)&(S<2e+3l_0^2) data, are not shown. Now also, it should be stated when exactly the aspect ratio shown on this graph is being measured – is this the time-averaged aspect ratio, or at the flow reversal, or at t/\DeltaT=1/4, etc.? Figure 5a demonstrates well that this specificity is required.

Line 246: isn’t this perturbation decay the simplest explanation for the slope change in Figure 3e?

Line 253: The sentence “Our model…” should probably come with an explanation of why you think your model is not accurate enough – i.e., provide its limitations.

Line 262: For the sentence “This could be…”, I feel like this could’ve been tested using the simulations. You did not quantify the lengthscales of the recirculation bubble or of the upstream shear-stress drop, or show how they depend on dune shape or size.

Section 4.3: This is an interesting discussion, but I think it’s definitely worth noting earlier on (rather than at the end) that we expect from various data (ReSCAL simulations included) that dune spacing during dune field formation is predominantly controlled by a pattern coarsening process, even when resultant fluxes are small. A ‘steady state’ dune field—or one where coarsening rate is low enough that pair-wise potential between similarly-sized dunes is the dominant control on dune spacing—may not be very common on Earth since climate changes faster than large dunes can keep up with (e.g., https://doi.org/10.2110/jsr.2018.55 and https://doi.org/10.1130/G50837.1). Overall though I believe it’s a provocative and balanced section.