The European Physical Journal B manuscript No. (will be inserted by the editor) 4 0 On the crescentic shape of barchan dunes 0 2 Pascal Hersen n a Laboratoire de PhysiqueStatistiquede l’ENS, 24 rue Lhomond,75005 Paris, France. J 6 2 February 2, 2008 ] t f Abstract. Aeolian sand dunes originate from wind flow and sand bed interactions. According to wind o propertiesandsandavailability,theycanadoptdifferentshapes,rangingfromhugemotion-lessstardunes s tosmallandmobilebarchandunes.Thelatterarecrescenticandemergeunderaunidirectionalwind,with . t a low sand supply.Here,a 3d model forbarchan based on existing2d model is proposed. Afterdescribing a the intrinsic issues of 3d modeling, we show that the deflection of reptating particules due to the shape m of the dune leads to a lateral sand flux deflection, which takes the mathematical form of a non-linear - diffusiveprocess. Thissimpleandphysicallymeaningfulcouplingmethodisusedtounderstandtheshape d of barchan dunes. n o c PACS. 45.70.-n Granular systems – 47.54.+r Pattern selection; pattern formation [ 2 1 Characteristics of Barchan Dunes whereavalanchesdevelop-seeFig.2.Becauseofabound- v ary layer separation along this sharp edge [1,4,5], a large 0 0 R.A.Bagnoldopenedthewaytothephysicsofduneswith eddy develops downwind and wind speed decreases dra- 1 his famous book: The physics of blown sand and desert matically.Therefore,the incoming blownsandis dropped 8 dunes [1]. From then on, a great deal of investigations - close to the brink line. That is why the barchan is known 0 laboratory experiments [1,2,3], field measurements [4,5, tobeaverygoodsandtrapper.Sometimes,whenthedrift 3 6,7,8,9,10,11,12,13,14] and numerical computations [15, of sand is too large, an avalanche occurs and grains are 0 16,17,18,19,20,21,22,23]-havebeenconductedbygeolo- moveddown the slip face. In short,grains are draggedby / t gistsandphysicists.Inparticular,alargeamountofwork the wind from the windward side of the dune to the bot- a has been dedicated to the barchan, a dune shaped by the tomoftheslip-faceand,grainaftergrain,thedunemoves. m erosion of a unidirectional wind on a firm ground. Field observations show that barchans can move up to - 70m/year [14]. Their speed is dependent on wind power d n and on their size: for the same wind strength, the veloc- o ity of barchans is roughly inversely proportional to their c heights [1,2,4,10,12]. Barchan dimensions range from 1 v: to 30 meters high, and from 10 to 300 meters long and i wide [1,4,5]. However, large barchans are often unstable, X leading to complex structures called mega-barchans [4,5]. r More accurate analyses reveal that the height, width and a length are related by linear relationships [2,13] and that no mature barchandune smaller thanone meter highcan Fig. 1. Side view of a barchan dune . The main properties be found: there is a minimal barchan size. Finally, the of barchan dunes are outlined: two horns pointing downwind, last important characteristic of barchans, but much less the slip-face and the flat main body. The barchan shown is well-documented, is the sand leak at the tip of the horns approximately20meterslongandwide,and2metershigh.The [1,4,5,24], where no recirculation bubble develops. This slip-faceangleisroughly30o tothevertical,whichcorresponds shows that barchan dunes are three dimensional struc- to theangle of repose of a sand-pile. tures, whose center part and horns have totally different trapping efficiency - see Fig. 2. A side view of a barchan - see Fig. 1 - shows a rather flat aerodynamic structure. When viewed from above, a Becauseoftheinherentdifficultiesoffield-work,(atyp- barchanpresentsacrescenticshapewithtwohornspoint- icalmission duration is rather short comparedto the life- ing downwind. A sharp edge - the brink line - divides the time of a dune and its shape movement), numerical mod- dune in two areas: the windward side and the slip face, eling is, alongside laboratory experiment [3], an impor- 2 P.Hersen: On the crescentic shape of barchan dune inertia:if the wind speed increases,it takessome time for thegrains,initiallyatrest,toreachthewindvelocity.Dif- ferentapproachesarepossibletodescribethe evolutionof thesandfluxtowardsitssaturatedvalueq .Howeverfor sat thesakeofsimplicity,thissaturationprocessisheretaken into account with the equation, q −q sat ∂ q = . (2) x l sat This equation keeps the most important effect: the exis- tence of a characteristic length-scale l . Moreover, q , sat sat depends onthe wind shearvelocityu∗.Evenifthe nature of this dependency is still debated, the saturated flux is a Fig.2.Barchanduneproperties.Grainsfollowthewinddirec- growingfunctionofu∗ [25,26,27].Thenalinearexpansion tion,andsandfluxisnotstronglydeflectedbythedunerelief. of q , as proposed in the innovative work of Sauermann Asobservedinthefield,sandgrainscanescapefromthehorns, sat et al. [21,22] from the model of Jackson& Hunt [28]leads but not from the main dune body. Instead, they are trapped to: intotheslip-face.Thisdifferenceofbehaviorbetweenthemain body and the hornsis thekey tounderstand thebarchans. q (x) dχ sat =1+A ∂ h (x−χ)+B∂ h (x), (3) Q Z χ x e x e tant method to explore barchan properties such as their where Q is the saturated flux value on a flat ground and morphogenesis,theirstabilityandtheirinteractionmodes, h (x) is the envelope of the dune. This envelope encages which are still poorly understood. Obviously, these prob- e the dune and its recirculation bubble [12], and is used to lemsarerelatedtotheoriginalstructureofbarchans,and, includetheboundarylayerseparationheuristically.Notice accordingly, they should be studied with a 3d approach. that in the latter equation, h (x,t) appears only through The aim of the present paper is to discuss how to extend e itsfirstspatialderivative.Itisconsistentwiththeassump- existing2dmodelstothe3dsituation.Then,wewillshow tion that atmospheric turbulent boundary layer is fully that the crescentic shape can be explained by the exis- tence of a sand-flux constraint and a lateral sand-flux. In developed, so changes in u∗, and accordingly in qsat are scale-invariant.Fromanaerodynamicalpointofview,this the following part, we start by recalling briefly the main relation takes into account two effects: a pressure effect, features of 2d modeling of dunes . Then a model for the controlled by A, where the whole shape acts on the wind 3d situation will be presented. flow;and a destabilizing effect, controledby B, which en- suresthatthemaximumspeedoftheflowisreachedbefore c thedunesummit.Althoughinprinciplewecouldcompute 2 The Cc class of models the parameters A and B, we prefer to consider them as tunable parameters. Finally, avalanches are simply taken Numerical modeling of dune requires a description of the intoaccount:ifthelocalslopeexceedsacriticalvalue,the effect of wind flow over a sand-bed. Obviously, comput- sand flux is increased strongly along the steepest slope. ing exact turbulent numerical solution starting with the Wewillcomebacktothe descriptionofavalanchesinsec- Navier-Stokes equations is possible,but this would take a c tion4.TheonlyscalingquantitiesareQandl andthey verylongtime.AnalternativeistousetheCc model[23], sat are used to adimensionalize the problem. based on the following approach. c In fact, this model belongs to what we called the Cc class. It does not depend strongly on the model use for the shear stress: other models [29] of shear stress pertur- 2.1 Numerical model for the 2d case bationscouldbeusedprovidingthattheyincludetherole of the whole shape (A) and the asymmetry of the flow Let us callh(x,t) the sandbed profile andq(x,t) the ver- (B). The same remarks apply to the charge equation[21, tically integrated volumic sand-flux. They are linked by the mass conservation: 22] and to the equation linking qsat and u∗. This shows therobustnessofthephysicsingredientsusedinthisclass ∂ h+∂ q =0. (1) of model. Even though this method does not provide the t x most detailed results, this kind of model has been used Thesandfluxq(x,t)isthemainphysicalparameterneeded successfully to model 2d barchan profile [21,22,23]. tounderstanddunephysics.Itcannotlocallyexceedasat- urated value q (x,t), which is the maximum number of sat grains that the wind is able to drag per unit of time at x. 2.2 Speed dispersion and trapping efficiency Previousworkhavealreadyfocusedonthe importantrole playedbythesaturationofthesandfluxandtheso-called As a matter of fact, simulations in 2d show the existence saturation length, l [3,22,23]. The nature of this satu- of two kinds of solutions: dune and dome [23]. The dune sat ration process can be understood in terms of sand grain solutionhasaslip-facethatcatchesalltheincomingsand: P.Hersen: On the crescentic shape of barchan dune 3 the dune can only grow - except if the input sand flux perturbationsinthelateraldirectionandseewhatitgives. is null. On the contrary, for a dome, a large amount of This has been done recently [30], but there is hardly any sand can escape, and if the loss of sand is not balanced evidence that it is the main physical process responsible by an influx, the dome can only shrink. Hence, these two for a lateral sand flux on the dune. On the other hand, it solutionscannotco-exist.Tocomputethesteadyshapeof appears to be an important effect for the development of a 3d barchan,it is possible to cut it in ”slices” parallelto lateral instabilities along transversaldunes [31]. the winddirectionand,foreachslice,to usethe 2dmodel tocomputeitsevolution.However,inarealbarchan,a2d slicefromthehornsseemstobehavelike2ddomesolution, 3.3 Saltation and reptation coupling whilea2dslicefromthemainbodycorespondsto2ddune solution. However, in order to make slices from the main bodyandslicesfromthehornscoexist,wemustintroduce asandfluxwhichwillredistribuatelaterallythesandfrom the center towards the horns. This sand flux coupling is also needed to overcome the speed dispersion effect. As a matter of fact, if all the slices have initially the same shape - but at a different scale ratio - the saturated flux at the crest is the same for all the slices, as imposed by turbulencescaleinvarianceandthespeedofasliceisgiven by: q −q c out c= (4) h c where q and q are respectively the flux at the crest c out and the output flux, and h is the height at the crest c of the slice. Hence, the smaller the slice, the faster its motion. This dispersion explains why the barchan takes a crescentic shape. But to reach a steady state, all the slicesmustmoveatthesamespeed.Therefore,thelateral coupling must also induce a speed homogenization. Fig. 3. Influence of gravity. While trajectories of saltons are 3 Different lateral coupling mechanisms deflected randomly at each collision, the reptons are always pusheddown along thesteepest slope. What physical mechanisms can lead to a redistribution of the sand flux on the dune surface? We can think of A better candidate is the grain motion. Sand trans- three differentpossibilities:avalanches,lateralwindshear portation can be described in terms of two species [27]: stress perturbations and grain motions (saltation and/or grains in saltation - the saltons - and grains in reptation reptation). - the reptons - see Fig. 3. Saltons are dragged by wind, collide with the dune surface, rebound, are accelerated againbytheairflow,andsoforth.Ateachcollision,saltons 3.1 Avalanches dislodge many reptons, which travel on a short distance, rollingdownthesteepestslope,andthen,waitforanother First, avalanches develop in three dimensions along the salton impact. steepestslope.Thiscreateslateralsandfluxintheslipface Asthewinddeflectionisweak,weassumethatsaltons area.However,forrealbarchandunes,thatsand-transport follow quasi2dtrajectoryin a verticalplane (see lastsec- isdirectedfromtheedgestowardsthecenter.Accordingly, tion), except when they collide with the dune. At each the center part grows and slows down while the border collision,theycanreboundinmanydirections,depending slices shrink and accelerates. This does not constitute a on the local surface properties, and this induces a lateral stabilizing mechanism. sandflux.Giventhatthedeflectionbycollisionisstrongly dependent on the surface roughness, we assume that, on average, the deflection of saltons is smaller than for rep- 3.2 Wind deflection tons, which are always driven towards the steepest slope. In the following derivation, we will neglect the sand flux Thelateralwinddeflectionisanotherpossibility.Fieldob- deflection due to saltation collisions. servations[1]andnumericalsimulations[21]tendtoshow Despite the major role played by saltons in dune dy- that,due toits flatness,the dunedoes notmakethe wind namics, the presence of reptons should not be dismissed. flowdeviate too muchlaterally.Consequentlytrajectories Accordingtofieldobservations[32,33],reptonsarestrongly of grains are not dragged into the lateral direction. Nev- dependentonthelocalslope:thiscanbeobservedbylook- ertheless,it is always possible to compute the wind speed ingatthe relativeorientationofthe wind andripples.On 4 P.Hersen: On the crescentic shape of barchan dune hard ground, ripples are perpendicular to wind direction, Then, the mass conservation equation becomes: but on a dune their relative orientations change with the localslope[33].Evenifoneisreluctanttousethesaltation ∂th+∂xqsal+∇qrep =0. (8) coupling because of its inherent difficulties, the reptons CallingD =αβ/(1+α)andq˜=(1+α)q ,thetwolatter would still be rolling on the dune surface: gravity nat- sal equations can be rewritten as : urally tends to move the grains along the steepest slope, leadingtolateralcouplingoftheduneslices.Inthefollow- q˜ −q˜ sat ingpart,wewillfocusonthiscouplingbyreptation,which ∂xq˜= (9) l has never been used for the study of barchan dunes. The sat saltation process remains important in this model, since ∂ h+∂ q˜=D(∂ (q˜∂ h)+∂ (q˜∂ h)) (10) t x x x y y it induces reptation coupling. Finally, we obtain the same set of equations as in the 2d c Cc model, but with one more phenomenological parame- 4 A 3d model with reptation coupling ter,D,whichcanbe understoodas the importance ofthe lateral coupling, because of lateral deflection of reptons. Thisformulationappearstobe anonlineardifusionequa- 4.1 Formulation of the model tion driven by the non dimensionnal coefficient D. For a homogeneous flux solution, QD appears to be a diffusion coefficient, showing the diffusion-like role of the coupling coefficientD.Noticethatq˜is nolongerthe saltationflux, but the part of the flux that does not depend on the bed slope. Avalanches are computed in three dimensions us- ing a simple trick. If the local slope exceeds the threshold µ , the sandflux is stronglyincreasedby adding an extra d avalanche flux: q =E(δµ)∇h, (11) a whereδµisnullwhentheslopeislowerthanµ andequal d to (δµ = |∇h|2 −µ2) otherwise. For a sufficiently large d coefficient E, the slope is relaxed independently of E. Fi- nally, note that quasi-periodic boundary conditions (the total output flux is reinjected homogeneously in the nu- merical box) are used to perform the numerical simula- tions. These boundary conditions are used, first to work Fig. 4. Typical initial and finalshapeof barchan dunesgiven c with a constant mass, and second to force the system to by the 3d Cc model. Parameters used are A = 9.0, B = 5.0, convergetowardsitssteadystate.Obviouslythesebound- D=0.5 with initially: W0 =30lsat, H0 =3lsat. ary conditions constrain the final shape. As reptons are created by saltons impacts, the part of 4.2 The origins of the crescentic shape the flux due to reptons is assumed to be proportional to the saltation flux, qsal [1]. Hence, the flux of reptons is Our 3d Ccc model depends on three phenomenologicalpa- simply written as [34]: rameters : A and B take into account aerodynamics ef- fects,andD describestheefficiencyoftheslicescoupling. q =αq (e −β∇h) (5) rep sal x The final shape, if one exists, depends on these parame- ters. For example, Fig. 4 shows a typical 3d final steady The α coefficient represents the fraction of the total sand stateofacomputedbarchandune,whichlookslikesareal flux due to reptation on a flat bed. In case the bed is not aeolian one. The arrows on Fig. 5 indicate the direction flat,the fluxis correctedtothe firstorderofthehderiva- of the total sand flux on the whole dune shape. The de- tives, by a coefficient β and directed along the steepest viation towards the horns is clearly visible as the sand slopeto takeintoaccountthe deflectionofreptonstrajec- capturedbytheslip-face.LookingatFig6helpsustoun- tories by gravity.Assuming that saltationtrajectoriesare derstandthe formationof the crescentic shape ofbarchan 2d, the total sand flux q is given by: dune. Starting from a sand-pile, some horns will appear q =q (1+α)e −βαq ∇h (6) andsincetheyarefasterthanthecentersliceofthedune, sal x sal a crescentic shape appears. Now, let us consider a slice Moreover,reptationfluxisassumedtoinstantaneouslyfol- and let us call qx, the excess flux due to reptons coming low the saltation flux, so that there is no other charge fromthelateralsandflux,qin theincomingsaltationsand equation than the saltation flux one: flux brought by the wind and qe the flux due to erosion. The existence of a saturated flux imposes : q −q sat sal ∂xqsal = . (7) q +q +q <qmax. (12) lsat e x in sat P.Hersen: On the crescentic shape of barchan dune 5 Fig. 6. Barchan formation. a) evolution of an initial cosine bump sand pile : h(x,y) = cos(2πx/W0)cos(2πy/W0). In the beginning,thehornsmovesfasterthanthecentralpartofthedune,leadingtotheformationofthecrescenticshape.Theshape reaches an equilibrium thanks to the lateral sand flux, which feeds the horns. Parameters used are the same than for figures 4 and 5: A=9.0,B =5.0,D =0.5 with initially: W0 =30lsat, H0 =3lsat. b) Evolution of a bi-cone sand-pile (same parameters than in (a)). This shows that emergence of a crescentic shape is independant on initial shape: the hole in the middle is filled up,and a crescentic shape forms. In both boxes theheight between two level-lines is 0.2lsat. is governed by the erosion, the slice slows down. After a while, the horns receive enoughsand from the main body todecreasetheirspeedandtocompensatetheoutputflux. Finally the speed ofallslicesis the same andthe barchan moves without changing its shape. Moreover, the center part,whichwouldgrowwithoutlateralcoupling,cannow haveanequilibriumshape,sinceallthe extraflux is devi- atedtowardsthehorns.Furthermorethiscouplingprocess is stabilizing with respect to local deformation. If a slice increases in height, the excess sand flux leaving the slice will increase, and the deformation will shrink. Similarly, startingfromatwomaximashape,thepartofthefluxsen- sitivetothelocalslopetendstofillupthegapbetweenthe two maxima: the whole mass is redistributed and a single barchanshape is finally obtained - see Fig. 6. This agrees with the apparent robustness of the crescentic shape ob- servedon the field. Whatever the externalconditions are, the same morphology is roughly found everywhere where Fig. 5. Lateral diffusion of sand flux. The angle of the flux barchansdevelop. Hence, this lateralcoupling helps us to vectors are magnified 3 times to be clearly visible. Both the understandsuchbarchanstructures.However,wehaveno deflection towards the horns and the presence of avalanches clue about the possible value of this coupling coefficient canbeobserved.Parametersusedarethesamethanforfigure D anditisthereforeusefulto study the influence ofD on 4 and 6: A=9.0,B =5.0,D =0.5 with initially: W0 =30lsat, the barchanic shape. H0=3lsat. where qmax is the maximum value of the saturated flux sat on the given slice. Thus, if the excess flux q increases, x the erosion flux, q , decreases. As the speed of the slice e 6 P.Hersen: On the crescentic shape of barchan dune the width of the horns in the desert may be a convenient way to estimate the value of D from field observations. Finally the aspect ratio of the dune, h/w, varies with D, Fig. 7. Different 3d shapes obtained for different values of D Fig. 8. Dependance of barchan dune properties on the cou- andforthesameinitialsand-pile.Parametersused(exceptfor pling coefficient D. The case with D = 0 is taken has a D coefficient) are the same than for figures 4, 5 and 6. Com- reference. The properties are: (a) width of a horn, (b) to- putations are performed with quasi-periodic boundary condi- tal width, (c) equilibrium input flux, (d) speed of the dune tions. The barchan central slice is represented on the right. , (e) total length (including horns), (f) length of the center Notice that for D = 0, a steady state is reached because of slice, (g) maximum height. A typical output flux cross sec- lateral sand redistribution duetoavalancheson thewindward tion is shown in the upper left part of the graph: the width side. However, there is a sharp edge which separates the dune of the horns is measured by measuring the width of the flux in two part, leading to an arrow shape rather than to a cres- peaks. The initial sand pile is always a cosine bump defined centic shape. by h(x,y)=H0cos(2πx/W0)cos(2πy/W0) with W0 =20 and H0=2.0. 5 Significance of the coefficients becausethecouplingmechanismtendstodiminishslopes. 5.1 The influence of the coupling coefficient D In other words, the dune tends to spread out laterally when D increases, sometimes leading to structures with- As can be seen on Fig. 7, D has a crucial influence on out a slip-face: 3d dome solution. In this case, sand can the global morphology. First of all, for a small D, (i.e. a escapefromthe wholetransversalsectionofthedune and small lateral sand flux), the flux escaping from the tip of the output flux is then higher than for the dune solution. the horns is mainly due to qin and to the erosion flux, Basically the transition from 3d dune to 3d dome situa- qe. Therefore, erosion may be important, and the horns tionsoccurswhenthehornwidthisequaltohalfthedune maymovequickly.Thus,duringthetransient,thebarchan width. This corresponds approximately to D = 2.0 with elongates.Onthe contrary,fora largerD,the lateralflux the parameters A=9.0 and B =5.0. For too large a dif- receivedby the horns,qx,is largerandthenthe erosionis fusion, the dune becomes very wide and flat, without a lessimportant.Thehornsmoveslowerandtheelongation slip-face, and remains unsteady. is shorter. Secondly, the width of the horns is also dependent on 5.2 The influence of coefficients A and B D. For a large D, the deflected flux is large. However, as the output sand flux cannot exceed a maximum value, However, looking only at the D influence is not accurate hornshavetobewideinordertotransportawayallthein- enoughtodescribequalitativelythe influenceofdiffusion. comingsand,andindeed,thewidthofthehornsincreases In fact, the shape is the result of three physical effects with D. This simple explanation is confirmed by numeri- governed by the parameters A, B and D. Fig. 9 presents calsimulation,asdepictedinFig.7andFig.8.Measuring a phase diagram with D kept constant at a value of 0.5. P.Hersen: On the crescentic shape of barchan dune 7 Manydifferentmorphologiescanbeobserved,fromalarge andthin unsteady crescentto a flat unsteady sandpatch, andwithsteadybarchansinbetween.First,Agovernsthe stabilityofthesandbed,andthenalargeAtends toflat- ten the sand-pile. Increasing B, on the other hand, forces the slope to increase and to nucleate a slip-face. Thus, whentheratioA/B ishigh,thereishardlyaslipface.On the contrary, when it is low, the slip face develops on the whole dune. Thus the A/B parameter controls the way the slip-face appears. Secondly, one can observe that the dunes obtained with a constant ratio A/B have different morphologies.ThisshowsthatforsmallAandB thecou- pling coefficient D plays an importantrole in the shaping of the dune. Field measurements of the horns’ width and the minimal size ofbarchansshouldhelp us to give anes- timationoftheseparameters.Thefollowingtableprovides with a summaryof the differentqualitative effects of A,B and D. Fig. 9. Evolution of the same initial cosine bump sand pile physical mechanism Main effect when increasing (h(x,y)=cos(2πx/W0)cos(2πy/W0) with W0 =30lsat, H0 = A curvature effect L increases, 3lsat)changingthevalueoftheparametersAandB.Diskept constant and equal to 0.5. The dashed lines separate qualita- W decreases. tivelythedifferentstabilityandshapedomains.Thedomedo- B slope effect Slip-face appears easily, main is divided into two parts depending on the steadiness of ∂ h increases, W increases. x thedome solution. D Lateral coupling H decreases, W and horns width increase, otherwords,barchanicshapeappearstobetheverybasic output flux increases. shape taken by any sand-pile blown by a unidirectional wind. Now that the steady shape of barchan dunes seems Finally, the initial sand-pile does not always reach a con- to be well described, it would be interesting to study the stant shape. For large A/B no steady state appears.This dynamics of such a structure placed far from equilibrium is due to the fact that the erosion is so strong that it conditions, because, in the field, input flux has no reason cannot be balanced by the influx, and the sand-pile just to be related to the output flux of a dune. becomes flatter and flatter. On the contrary, for a large As a matter of fact, in this paper, barchanshave been B/A,athinunsteadycrescentappears.Finally,foralarge numerically obtained with quasi-periodic boundary con- D (comparedto A and B) steady and unsteady dome so- ditions. If standardboundary conditions are used, a typi- lutions are produced by the numerical model. This shows cal barchandune takes the crescentic shape, for the same the importance of A, B and D on the shape and on the reasons as the ones exposed in this article, but it keeps minimal size of computed barchan dunes. growingorshrinkingdepending onthe incidentsandflux, eitherhigherorlowerthanthesandfluxescapingfromthe horns[24]. This point is crucial because it suggests that 6 Conclusion single real barchan dunes are not steady structures and, moreover,itshowsthatacrescenticshapecanformevenif We have seen that reptation can be used to describe the thedune isnotinanequilibriumstate.Therefore,further 3d crescentic shape of barchan dunes. As reptons are sen- investigations are still required to understand the long- sitive on the local slope, they induce a lateral sand flux. term existence and shape of barchan dune in the field. This allows a given sand-pile blown by the wind to reach In our model we have neglected saltation coupling; but an equilibrium shape, despite varying ”slices” speed and we believe that it is not the dominant effect. Moreover,it intrinsicdifferencesbetweenthemainbodyandthehorns. could easily be taken into account by adding a new cou- The crescentic shape comes from two mechanisms, which pling coefficient dedicated to lateral deflection of saltons. compete with each other. The first one is the speed dis- Finally, it appears that depending on the coupling co- persion within the dune: the small height slices are faster efficient, D, the solution can either be a 3d dome or a 3d than higher ones, leading to the crescentic shape, and barchan.Similarly,changingAandB canleadtodifferent eventually to the destruction of the barchan dune. The shapes, and different behavior. it might be interesting to second one, is the lateral flux deflection that reduces the understandthe transformationofa 3dsandpatch,steady horns’ speed by decreasing the erosion flux on them. It or unsteady, into a barchan dune because of fluctuations leads to the homogenization of the speed of the different of these parameters, and particularly of D variations. As slices,andeventuallyto the propagationofa steadystate a matter of fact, these parameters may depend on exter- dune or a steady sand dome. This mechanism is efficient nal conditions such as grains size distribution, humidity since the overallexcessflux canescape fromthe horns.In or wind fluctuation. 8 P.Hersen: On the crescentic shape of barchan dune Acknowledgement : The author is grateful to T. 21. G.Sauermann.Modeling of WindBlown Sand and Desert Bohr for his constructive remarks. The author whishes Dunes. PhD thesis, Universitat Stuttgart, (2001). to thank K.H. Andersen, B. 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