A comprehensive picture for binary interactions of subaqueous barchans
aa r X i v : . [ phy s i c s . g e o - ph ] S e p manuscript submitted to Geophysical Research Letters
A comprehensive picture for binary interactions ofsubaqueous barchans
An edited version of this paper was published by AGU. Copyright 2020 AmericanGeophysical Union.Assis, W. R. and Franklin, E. M. (2020). A comprehensive picture for binary in-teractions of subaqueous barchans. Geophysical Research Letters, 47, e2020GL089464,DOI 10.1029/2020GL089464.To view the published open abstract, go to https://doi.org/10.1029/2020GL089464.
W. R. Assis , E. M. Franklin , School of Mechanical Engineering, UNICAMP - University of Campinas,Rua Mendeleyev, 200, Campinas, SP, Brazil
Key Points: • We identify five binary interactions of barchans and propose classification maps • We show indications that an ejected barchan has the same mass of the impact-ing one • We found that the asymmetry of the downstream dune is large in wake-dominatedprocesses
Corresponding author: Erick M. Franklin, [email protected] –1–anuscript submitted to
Geophysical Research Letters
Abstract
We investigate experimentally the short-range interactions occurring between two sub-aqueous barchans. The experiments were conducted in a water channel of transparentmaterial where controlled grains were poured inside, and a camera placed on the top ac-quired images of the bedforms. We varied the grain types (diameter, density and round-ness), pile masses, transverse distances, water flow rates and initial conditions. As a re-sult, five different patterns were identified for both aligned and off-centered configura-tions and we propose interaction maps that depend basically on the ratio between thenumber of grains of each dune, Shields number and alignment of barchans. In addition,we show experimental indications that an ejected barchan has roughly the same massof the impacting one in some cases, and that in wake-dominated processes the asymme-try of the downstream dune is large. The present results shed light on the size regula-tion of barchans found on Earth and other planets.
Plain Language Summary
Barchans are crescent-shaped dunes that are often organized in dune fields, wherebinary interactions and collisions play a significant role in regulating their dynamics andsizes. Barchan collisions are frequent in many environments, such as Earth’s deserts andon the surface of Mars, but their large time scales (the decade and the millennium foraeolian and Martian collisions, respectively) compared to the aquatic case (of the orderof the minute) make subaqueous barchans the ideal object of study. Taking advantageof that, we performed experiments in a water channel of transparent material, where pairsof barchans were transported by the water flow while a camera acquired images of them.We found five different types of barchan-barchan interaction, and propose maps that pro-vide a comprehensive classification for the short-range interactions of subaqueous barchans.In addition, we show that, in some cases, an ejected barchan has roughly the same massof the impacting one, and that the perturbation of the flow caused by the upstream barchangenerates large asymmetries in the downstream one. Our results represent a significantstep toward understanding the barcanoid forms and size regulation of barchans foundin water, air, and other planetary environments.
The interaction between a fluid flow and a granular bed gives rise to different kindsof bedforms. Of particular interest are the crescent-shaped dunes, called barchans, thatare formed under one-directional flow and limited amount of available grains, being en-countered in different environments such as rivers, water ducts, Earth’s deserts and otherplanetary environments (Bagnold, 1941; Herrmann & Sauermann, 2000; Hersen, 2004;Elbelrhiti et al., 2005; Claudin & Andreotti, 2006; Parteli & Herrmann, 2007). Althoughbarchans may grow as isolated bedforms (Alvarez & Franklin, 2017, 2018), they are of-ten organized in dune fields, where dune-dune interactions play a significant role in reg-ulating their dynamics and sizes (Hersen et al., 2004; Hersen & Douady, 2005; Kocureket al., 2010; G´enois, Hersen, et al., 2013; G´enois, du Pont, et al., 2013).Over the past decades, several studies investigated the collisions and short-distanceinteractions of aeolian barchans based on field measurements (Norris & Norris, 1961; Gay,1999; Vermeesch, 2011; Hugenholtz & Barchyn, 2012). Yet, because these measurementsare based on aerial images, the time series for barchan collisions are usually incompletegiven the long timescales of aeolian interactions (of the order of the decade), hinderinga comprehensive understanding of barchan collisions. Because of their much faster scales(of the order of the minute), some studies investigated the interactions of barchans inwater flumes and tanks (Endo et al., 2004; Hersen & Douady, 2005), from which differ-ent collision patterns were identified and their dynamics described. In addition, numer-ical simulations using continuum (Schw¨ammle & Herrmann, 2003; Dur´an et al., 2005; –2–anuscript submitted to
Geophysical Research Letters
Zhou et al., 2019) and simplified discrete models (Katsuki et al., 2011) could reproducesome of the collision patterns, shedding light on the essential mechanisms involved. How-ever, the simplifications present in those models precluded them from reproducing cor-rectly all barchan interactions, failing to predict the correct split of dunes in some casesand predicting soliton behavior in others.By observing that a solitary barchan within a dune field is marginally stable, tend-ing to grow or shrink once the stable size is disturbed, and the existence in nature of cor-ridors of barchans, Hersen et al. (2004) proposed a model for the formation of corridors,and Hersen and Douady (2005) showed that barchan collisions could be important forthe size regulation of barchans. In order to better understand the mechanisms behindthe formation of corridors with size-selected barchans, Dur´an et al. (2009) and G´enois,du Pont, et al. (2013) introduced simplified models based on sand flux balances and el-ementary rules for barchan collisions. With such models, Dur´an et al. (2009) showed thatcollisions are important for size regulation and inter-barchan spacing, while G´enois, duPont, et al. (2013), by adjusting sand fluxes, obtained corridors of sparse and large ordense and small barchans according to the balance between sand fluxes and collision types,showing that sand distribution due to collisions organizes barchans in narrow corridorsof size-selected dunes. Bo and Zheng (2013), based on numerical simulations using a scale-coupled model, found that the probability of barchan collisions varies with the flow strength,grain diameter, grain supply and height ratio of barchans. They quantified the proba-bilities for the occurrence of three different types of barchan collisions within a dune field(merging, exchange and fragmentation-exchange, described next), but not how the col-lision processes vary with the considered parameters.Although many previous studies were devoted to barchan collisions, the problemis still not completely understood and a general picture is lacking. The emerging pat-terns, though present in both aeolian and aquatic environments, have not yet had all theirimportant parameters identified, so that universal expressions or maps for predicting theresults of collisions do not exist. The identification of collision patterns from the approach-ing of subaqueous barchans until the end of the collisional process was performed by Endoet al. (2004) in the case of aligned dunes for different mass ratios, and by Hersen andDouady (2005) in the case of off-centered dunes for different transverse distances of cen-troids of colliding dunes (referred to as impact or offset parameter), while Bo and Zheng(2013) focused on the probabilities of barchan collisions in a dune field obtained fromnumerical computations. However, how the diameter, density and roundness of grains,flow strength and initial conditions affect the collision patterns remains to be investigated.In addition, mass transfers between barchans during collisions are not completely un-derstood.In this Letter we investigate extensively the binary interactions, including binarycollisions, of subaqueous barchans. We carried out exhaustive measurements of the short-range interactions between two barchan dunes, i.e., when their longitudinal separationis of the order of the size of the upstream bedform, by varying the mass of initial piles,their alignment (centered or off-centered), initial longitudinal separation, grain proper-ties (diameter, density and roundness), flow strength and initial conditions (downstreambarchan already formed or to be developed), most of them affecting the patterns emerg-ing from interactions. We identify five types of binary interactions for both aligned andoff-centered barchans, and show indications that an ejected barchan has roughly the samemass of the impacting one in cases involving collisions with exchange of grains and thatin wake-dominated processes the asymmetry of the downstream dune is large. We pro-pose a new classification for the binary short-range interactions of subaqueous barchansthat depends on the ratio between the number of grains of each dune, Shields numberand barchans alignment, shedding light on the size regulation of barchans in a dune field. –3–anuscript submitted to
Geophysical Research Letters
The experimental device consisted of a water reservoir, two centrifugal pumps, aflow straightener, a 5- m -long closed-conduit channel of transparent material and rect-angular cross section (width = 160 mm and height 2 δ = 50 mm), a settling tank, anda return line. A pressure-driven flow was imposed in the channel by means of the cen-trifugal pumps, and the flow followed the order just described. The channel test sectionwas 1 m long and started 40 hydraulic diameters, 40 × d h , downstream of the channelinlet, assuring a developed channel flow just upstream the bedforms, where d h = 3.05 δ is the cross-sectional area multiplied by four and divided by the cross-sectional perime-ter. With the channel previously filled with water, controlled grains were poured inside,forming a pair of bedforms in either aligned or off-centered configurations. By impos-ing a water flow, each bedform was deformed into a barchan shape and interacted witheach other. We used different initial conditions, in which we placed a first pile and letit deform into a barchan dune before placing an upstream pile, or we let it deform in half-way a barchan dune before placing the second pile, or we placed two conical piles andlet them deform together into barchan dunes, and the mass ratio of the piles, defined hereas the mass of the upstream pile (impacting) divided by that of the downstream one (tar-get), varied within 0.005 and 1. The initial longitudinal distance between bedforms wasof the order of the diameter of the upstream pile, D , being within 0.22 and 3.6 D betweendune borders (smaller distance between dunes in the longitudinal direction), and, becausethe dune velocity varies with the inverse of its size (Bagnold, 1941), the mass of the im-pacting dune was always equal or lesser than that of the target dune. A camera placedabove the channel acquired images of the bedforms and, therefore, we did not measuresystematically the barchan height. However, based on reported values of the aspect ra-tio of barchans (Andreotti et al., 2002a) and our experimental observations, we estimatethe crest heights as approximately 5 mm, i.e., 10 % of the channel height. The layoutof the experimental device, a photograph of the test section, and microscopy images ofthe used grains are shown in the supporting information.A total number of 123 tests were performed, for which we used tap water at tem-peratures within 22 and 30 ◦ C and different populations of grains (not mixed): roundglass beads ( ρ s = 2500 kg/m ) with 0 .
15 mm ≤ d ≤ .
25 mm and 0 .
40 mm ≤ d ≤ .
60 mm, angular glass beads with 0 .
21 mm ≤ d ≤ .
30 mm, and zirconium beads ( ρ s = 4100 kg/m ) with 0 .
40 mm ≤ d ≤ .
60, where ρ s and d are, respectively, the den-sity and diameter of grains. The cross-sectional mean velocities of water, U , varied be-tween 0.226 and 0.365 m/s, corresponding to Reynolds numbers based on the channelheight, Re = ρU δ/µ , within 1 . × and 1 . × , where µ is the dynamic vis-cosity and ρ the density of the fluid. The shear velocities on the channel walls (base state), u ∗ , were computed based on measurements with a two-dimensional particle image ve-locimetry (2D-PIV) device (Franklin et al., 2014; C´u˜nez et al., 2018; Alvarez & Franklin,2018) and found to follow the Blasius correlation (Schlichting, 2000), being within 0.0133and 0.0202 m/s. By considering the fluid velocities applied to each grain type, the Shieldsnumber, θ = ( ρu ∗ ) / (( ρ s − ρ ) gd ), varied within 0.019 and 0.106, where g is the acceler-ation of gravity (see supporting information for a description of the PIV tests, estimateddeviations in u ∗ and θ , and lists of all tested conditions). Five different patterns were observed as resulting from the short-range interaction,as can be seen in Figures 1 and 2, that show, respectively, snapshots of barchan inter-actions for the aligned and off-centered cases: 1) chasing (Figures 1a and 2a), when theupstream dune does not reach the downstream one. This pattern appears when the barchanshave almost the same size, and the wake of the upstream dune, by increasing turbulentlevels and creating channeling (Palmer et al., 2012; Bristow et al., 2018), promotes a largererosion on the downstream dune, which then shrinks and moves faster (even if it receives –4–anuscript submitted to
Geophysical Research Letters
Figure 1.
Snapshots of barchan interactions for aligned dunes. In the snapshots, the waterflow is from left to right, the upstream pile consisting of red (darker) glass beads and the down-stream pile of white (clearer) glass beads, and the corresponding times are shown in each frame.In Figure (a), 0 .
40 mm ≤ d ≤ .
60 mm and in the remaining figures 0 .
15 mm ≤ d ≤ .
25 mm.(a) Chasing; (b) merging; (c) exchange; (d) fragmentation-chasing; (e) fragmentation-exchange,and they correspond to test numbers 61, 65, 36, 5 and 22 in the table of Fig. S23 of the support-ing information, respectively. –5–anuscript submitted to
Geophysical Research Letters
Figure 2.
Snapshots of barchan interactions for off-centered dunes. In the snapshots, thewater flow is from left to right, the upstream pile consisting of red (darker) glass beads and thedownstream pile of white (clearer) glass beads of 0 .
15 mm ≤ d ≤ .
25 mm, and the correspond-ing times are shown in each frame. (a) Chasing; (b) merging; (c) exchange; (d) fragmentation-chasing; (e) fragmentation-exchange, and they correspond to test numbers 43, 38, 41, 31 and 5 inthe table of Fig. S24 of the supporting information, respectively.–6–anuscript submitted to
Geophysical Research Letters grains from the upstream barchan); 2) merging (Figures 1b and 2b), when the upstreamdune reaches the downstream one and they merge; 3) exchange (Figures 1c and 2c), when,once the upstream dune reaches the downstream one, a small barchan is ejected and, be-ing the smaller one, outruns the other and migrates downstream. The first impressionis that the impacting barchan traverses the target one, but the use of marked grains showsthat there is mass exchange between them, as can be seen in Figures 1c and 2c. In somecases, depending on the sum of sizes of the impacting and target barchans, the ejectedbarchan is so small that it is close to the minimum size (Franklin & Charru, 2011) andspreads out just after being ejected; 4) fragmentation-chasing (Figures 1d and 2d), when,due to the wake of the upstream dune (Palmer et al., 2012; Bristow et al., 2018), in par-ticular just downstream the reattachment point of the recirculation region, the down-stream dune splits before being reached. Because the divided dunes are smaller than theupstream one, they outrun the upstream dune; and 5) fragmentation-exchange (Figures1e and 2e), when fragmentation as in (4) initiates, but, the upstream barchan being stillfaster then the splitting dune, the former reaches the latter. Once they touch, an off-centerexchange occurs, and a small barchan is ejected. In the aligned configuration, the ejectedbarchan results from the interaction of an elongated horn with the remaining of the di-vided dunes, while in the off-centered configuration the ejected barchan is the smallerportion of the splitting dune. Finally, this redistribution of grains having finished, thesmaller dunes are downstream and, therefore, three resulting barchans migrate withoutreaching each other. Movies showing all the five dune-dune interactions for both con-figurations and snapshots for other grain types are available as supporting information.The presence of the five patterns in both aligned and off-centered configurationsshows that variations of the offset (or impact) parameter, although influencing the con-ditions where patterns can appear, are not crucial for their appearance. Also, the massratio alone cannot regulate the appearance of all collision patterns, Endo et al. (2004)and Dur´an et al. (2005) having not found the five patterns for aligned dunes by vary-ing only their mass ratio. Endo et al. (2004) identified only the merging, exchange andfragmentation-chasing patterns (which they named absorption, ejection and split), andDur´an et al. (2005), based on numerical simulations, the merging and exchange patterns(which they called coalescence and breeding), but the latter with a different behavior thanour experimental observations. In addition, they found a pattern called budding, whichcould be equivalent to the fragmentation-exchange, but, in fact, is different, the targetdune splitting only after the collision had happened, and also a solitary wave behavior,which is not observed experimentally. However, until now the different patterns emerg-ing from collisions have been described in terms of only the offset parameter and massratio (Katsuki et al., 2005; G´enois, du Pont, et al., 2013; G´enois, Hersen, et al., 2013).We observed in our experiments that, in addition to these parameters, the fluid shear-ing and mass of each grain are also of importance, the latter, combined with the pile masses,being equivalent to the number of grains forming the piles. If, in one hand, the differ-ence in the number of grains (or, also, the mass ratio) gives the time scale for collision,on the other hand the total number of grains (or the sum of pile volumes) gives the to-tal size of the system, indicating if the resulting barchan is too large, with tendency tosplit because of instabilities of hydrodynamic nature (Andreotti et al., 2002a, 2002b; Charru,2006; Franklin, 2015). In addition, the flow strength and the size and density of grainsare also related to hydrodynamic instabilities and to minimum sizes regulating the wave-length of bedforms and favoring the split of dunes or even their spread out (Andreottiet al., 2002b; Parteli et al., 2007; Franklin & Charru, 2011; Charru et al., 2013; Cour-rech du Pont, 2015), so that they also must be taken into account. For example, Khosronejadand Sotiropoulos (2017) showed that new barchans can be generated by a calving pro-cess on the horns of existing barchans, caused by the fluid shearing on the horn surface.Therefore, barchan collisions would be better described by the number of grains form-ing each pile, size and density of grains, flow strength and alignment of barchans, insteadof only the mass ratio of piles and the offset parameter. Another aspect not investigated –7–anuscript submitted to
Geophysical Research Letters in previous studies is the effect of initial conditions of bedforms on barchan collisions (tar-get barchan being initially a fully-developed barchan, a partially-developed barchan, ora conical pile). For the initial conditions, as well as the grain roundness, we did not ob-serve any significant difference in our experiments (see supporting information for snap-shots of barchan interactions with two conical piles as initial condition).We propose that the short-range interaction patterns can be described by the off-set parameter, the Shields number, and the number of grains forming each pile. For thelatter, the difference in the number of grains forming each pile, ∆ N , is proportional tothe relative velocity of dunes, while the sum of those numbers, Σ N , is proportional tothe total size of the bedform once dunes have collided. We then introduce the dimen-sionless particle number: ξ N = ∆ N Σ N (1)The Shields number is the ratio between mobile and resisting forces, linked to thefluid shearing and the grain weight, respectively, so that it takes into account theflow strength and the size and density of grains. Finally, the alignment of barchansis represented by the offset parameter σ (dimensionless), computed here as thetransverse distance between the centroid of approaching barchans, η , divided bytheir average width: σ = 2 η/ ( W U + W D ), where W U and W D are the widths of theupstream and downstream bedforms, respectively, and η is positive to the left ofthe target dune (with respect to the flow direction). Although we recognize threedimensionless groups, we decided to present all our data in two 2D maps in or-der to organize the patterns in the simplest and comprehensive way that we couldfind. Therefore, we plotted one interaction map for the aligned case (Figure 3a)and another one for the off-centered case (Figure 3b), where patterns are shown asfunctions of ξ N and θ . In addition, for the off-centered case the map is parametrizedby σ < σ ≥ ξ N and θ groups, independent of the initial longitudinal separation between bed-forms, with transition regions between them where patterns are sometimes difficult toclassify (their behavior in these regions is close to two patterns). Conscious of this dif-ficulty, we drew lines separating the different patterns, which we present in Figures 3cand 3d for the aligned and off-centered configurations, respectively. We drew such linesbased solely on the experimental observations, and they consist in a tentative way to clas-sify the different patterns in θ vs. ξ N maps. Although computation of those lines basedon stability analyses or other analytical method remains to be done, we believe that thepresent maps may be useful for predicting the output of short-range barchan-barchaninteractions under different conditions.Based on image processing, we tracked the bedforms along the acquired images foreach of the five patterns and identified some of their characteristic lengths. Because ofapproximately constant ratios between barchan dimensions (Hersen et al., 2002; Andreottiet al., 2002a), the projected area of a developed barchan multiplied by its height (around10% its width) is proportional to its volume, and, therefore, to its mass. However, in thepresent case barchans are being formed and deformed, interacting with each other, sothat those relations are not completely valid. Conscious of that, we decided to analyzethe projected areas of barchans as an indicator of the quantity of grains forming the dunes.Figure 3e presents the instantaneous values of the projected area of bedforms along timefor the exchange pattern, and Figure 3f the evolution of the ratio between the lengthsof the left and right horns (with respect to the flow direction), L hl and L hr , respectively. –8–anuscript submitted to Geophysical Research Letters
Figure 3.
Figures (a) and (b): Patterns of barchan-barchan interactions as functions of ξ N and θ for (a) aligned and (b) off-centered barchans. Stars, diamonds, circles, squares andtriangles correspond to chasing, merging, exchange, fragmentation-chasing and fragmentation-exchange, respectively. In Figure (b), open symbols correspond to σ < σ ≥ L hl /L hr , of the downstreamdune along time. Stars, squares and triangles correspond to chasing, fragmentation-chasingand fragmentation-exchange patterns, respectively (tests 61, 5 and 22 of Fig. S23, and 43, 31 and5 of Fig. S24 of the supporting information). In Figs (e) and (f), open symbols correspond to thealigned and solid symbols to off-centered cases. All individual images that were processed to plotFigures (e) and (f) are available on Mendeley Data (http://dx.doi.org/10.17632/jn3kt83hzh)–9–anuscript submitted to Geophysical Research Letters
We start by observing that the area of the upstream bedform increased in the be-ginning of all experiments because it was initially a conical pile, with a higher ratio be-tween its height and length, and, therefore, it spread out once the water flow was im-posed. While the upstream barchan was growing, the downstream one was already formedand lost grains by its horns without receiving much grains from the upstream bedform,so that its area decreased slightly in the beginning of each test. Figure 3e shows also thatthe area of the dune resulting from the collision increases along time due to its spread-ing, since just after collision the upstream dune (impact dune) climbs over the downstreamone (target dune), as can be seen on movies available as supporting information. Afterthat, a new born barchan is expelled with roughly the same area of the impact dune (seesupporting information for a table showing the areas of impacting and expelled dunesof 15 tests). This indicates that the mass of the generated barchan is approximately thatof the impacting one, though the constituent grains are not the same (Figures 1c and2c). Although this mass exchange of same value has been conjectured before (Vermeesch,2011), being even confounded with a solitary behavior in some cases (Schw¨ammle & Her-rmann, 2003), it had never been assessed from controlled experiments until now.Finally, from Figure 3f we observe experimental evidence that the asymmetry ofthe downstream dune is large in wake-dominated processes (i.e., when the growth of oneof the horns is due mainly to the fluid flow), the asymmetry being lower in the case ofcollision-generated asymmetries (not shown in Figure 3f, but presented in the support-ing information). This implies that the wake of upstream dunes (Palmer et al., 2012; Bris-tow et al., 2018), and not the collision itself, generates most of horns asymmetries. Al-though the origin of horns asymmetries has been studied previously (Parteli et al., 2014),it needs to be investigated further in the specific case of dune-dune interactions.Although our experiments were limited to the subaqueous case, the resulting anal-ysis may be useful for predicting barchan-barchan interactions in other environments,such as Earth’s deserts and on the surface of Mars. However, we expect differences re-lated to the larger quantities of grains involved in the aeolian and Martian dunes and,in particular, the trajectories followed by individual grains according to the state of thefluid. For the trajectories, grains move mainly by rolling and sliding and follow closelythe fluid flow in the subaqueous case, being susceptible to small vortices and other smallstructures of the flow. This has been shown to be especially important for the grains mi-grating to the barchan horns (Alvarez & Franklin, 2018, 2019). When the fluid is a gas,grains move by saltation and reptation, and those in saltation follow basically a straightline in the main flow direction (Bagnold, 1941), being undisturbed by the small struc-tures of the flow. Discrepancies between the present analysis and the behavior in otherenvironments are likely to occur, mainly for the wake-dominated processes, but whereand when they occur, and to what extent, remain to be investigated.
In conclusion, subaqueous barchan-barchan interactions result in five different pat-terns for both aligned and off-centered configurations, being well organized in two mapsas functions of ξ N and θ and parametrized by σ . These maps provide a comprehensiveand simple classification for the short-range interactions of subaqueous barchans and,although we have not analyzed the binary collisions in Earth’s deserts and other plan-etary environments, given their long timescales, they may be useful for predicting thecollisions of barchans in different environments. The present results represent a signif-icant step toward understanding the barcanoid forms, barchan asymmetries and size reg-ulation of barchans found in water, air, and other planetary environments. –10–anuscript submitted to Geophysical Research Letters
Acknowledgments
W. R. Assis is grateful to FAPESP (grant no. 2019/10239-7), and E. M.Franklin is grateful to FAPESP (grant no. 2018/14981-7), to CNPq (grant no.400284/2016-2) and to FAEPEX/UNICAMP (grant no. 2112/19) for the financialsupport provided. Data supporting this work are available in the supporting infor-mation and in Mendeley Data (http://dx.doi.org/10.17632/jn3kt83hzh).
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