Direct numerical simulations of aeolian sand ripplesOrencio Durána,b,1, Philippe Claudina, and Bruno Andreottia

aLaboratoire de Physique et Mécanique des Milieux Hetérogènes, UMR 7636, CNRS, Ecole Supérieure de Physique et de Chimie Industrielles, Université ParisDiderot, Université Pierre et Marie Curie, 75005 Paris, France; and bMARUM–Center for Marine Environmental Sciences, University of Bremen, D-28359Bremen, Germany

Edited by Harry L. Swinney, The University of Texas at Austin, Austin, TX, and approved September 17, 2014 (received for review July 10, 2014)

Aeolian sand beds exhibit regular patterns of ripples resultingfrom the interaction between topography and sediment trans-port. Their characteristics have been so far related to reptationtransport caused by the impacts on the ground of grains entrainedby the wind into saltation. By means of direct numerical simu-lations of grains interacting with a wind flow, we show that theinstability turns out to be driven by resonant grain trajectories,whose length is close to a ripple wavelength and whose splashleads to a mass displacement toward the ripple crests. The patternselection results from a compromise between this destabilizingmechanism and a diffusive downslope transport which stabilizessmall wavelengths. The initial wavelength is set by the ratio ofthe sediment flux and the erosion/deposition rate, a ratio whichincreases linearly with the wind velocity. We show that this scalinglaw, in agreement with experiments, originates from an interfaciallayer separating the saltation zone from the static sand bed,where momentum transfers are dominated by midair collisions.Finally, we provide quantitative support for the use of thepropagation of these ripples as a proxy for remote measurementsof sediment transport.

aeolian transport | wind ripples | resonance | self-organization

Observers have long recognized that wind ripples (1, 2) do notform via the same dynamical mechanism as dunes (3). Cur-

rent explanations ascribe their emergence to a geometrical effectof solid angle acting on sediment transport. The motion of grainstransported in saltation is composed of a series of asymmetrictrajectories (4–7) during which they are accelerated by the wind.These grains, in turn, decelerate the airflow inside the transportlayer (1, 7–12). On hitting the sand bed, they release a splash-likeshower of ejected grains that make small hops from the point ofimpact (1, 13, 14). This process is called reptation. Previous windripple models assume that saltation is insensitive to the sand bedtopography and forms a homogeneous rain of grains approach-ing the bed at a constant oblique angle (15–20). Upwind-slopingportions of the bed would then be submitted to a higher im-pacting flux than downslopes (1). With a number of ejecta pro-portional to the number of impacting grains, this effect wouldproduce a screening instability with an emergent wavelength λdetermined by the typical distance over which ejected grains aretransported (15–17), a few grain diameters d. However, observedsand ripple wavelengths are about 1,000 times larger than d,on Earth. The discrepancy is even more pronounced on Mars,where regular ripples are 20–40 times larger than those ona typical Earth sand dune (21, 22). Moreover, the screeningscenario predicts a wavelength independent of the wind shearvelocity up, in contradiction with field and wind tunnel mea-surements that exhibit a linear dependence of λ with up (23–25).

ModelTo unravel the dynamical mechanisms resolving these discrep-ancies, we perform direct numerical simulations of a granularbed submitted to a turbulent shear flow. This flow is driven bya turbulent shear stress ρf u2p imposed far from the bed, where ρfdenotes air density or more generally that of the fluid consti-tuting the atmosphere. The grains, of density ρp, are subject togravity g and interact through contact forces. Based on the work

presented in ref. 26, we explicitly implement a two-way couplingbetween a discrete element method for the particles and acontinuum Reynolds averaged description of hydrodynamics,coarse-grained at a scale larger than the grain size. This couplingoccurs by means of drag and Archimedes forces in the equations ofmotion of the grains and via a body force term in the Reynoldsaveraged Navier–Stokes equations (SI Text). This method enablesus to perform runs over long periods of time using a large 2Dspatial domain, while keeping the whole complexity of the granularphase (Movie S1). In particular, we do not have to introducea splash function to describe reptation: the ejecta generated bya grain that collides with the sediment bed are directly obtainedfrom the interaction of the particles with their neighbors in contact.

ResultsSand Ripple Instability. Starting from a flat sediment bed, dis-turbed only by the randomness in the granular arrangement, oneobserves in the simulations the emergence and the propagationof ripples (Fig. 1, Fig. S1, and Movie S2). Tracking the graintrajectories, one can see that the saltation rain above the rippledbed is strongly modulated (Fig. 1B). As observed in experiments(Fig. S2), grains in saltation preferentially hit the bed upwind ofthe ripple crests. As ejected grains make small hops, the repta-tion flux tends to be enhanced on the windward side and reducedon the lee side. This results in a net transport from the troughstoward the crests that amplifies topographic disturbances, hencethe instability. The spontaneous ripple pattern has a wavelengthλ and a propagation speed c, both varying linearly with the im-posed wind shear velocity (Fig. 2), in agreement with experi-mental observations (23–25). Both λ and c are found to vanishwhen up tends to the threshold value uth, above which sedimenttransport takes place. The key issue addressed in this article isthe origin of these scaling laws, which results from the interplaybetween saltation and reptation transport modes.

Significance

Wind ripples decorate the flanks of dunes in amazingly regularpatterns, on both Earth and Mars. Their emergence at a wave-length much larger than the grain size is currently unexplained.We report direct numerical simulations of grains interactingwith a wind flow that are, for the first time to our knowledge,able to reproduce the spontaneous growth of ripples with aninitial wavelength and a propagation velocity linearly increasingwith the wind speed. We propose a new formation mechanism,involving resonant grain trajectories tuned with the ripplewavelength. We also show that the product of the ripple wave-length and velocity is a proxy for the sediment flux, opening apromising perspective from which to perform remote measure-ments of sand mass transfers, on Mars in particular.

Author contributions: O.D., P.C., and B.A. designed research, performed research, ana-lyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: oduran@marum.de.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1413058111/-/DCSupplemental.

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To investigate quantitatively the dynamical mechanisms lead-ing to the ripple instability, we have also performed simulationsstarting from a modulated bed whose topography follows a si-nusoidal profile of given wavenumber k and of small initialamplitude

��Z��ð0Þ. The phase and the modulation amplitude��Z��ðtÞ are measured as a function of time, by a simple Fouriertransform of the elevation profile at the wavenumber k. Asexpected for a linear instability, the growth or the decay of thedisturbance can be fitted to an exponential of the form��Z��ðtÞ= ��Z��ð0Þeσt (Fig. S1), which gives the growth rate σ. Theresulting dispersion relation σðkÞ, obtained for each wind ve-locity, is typical of a long-wave instability (Fig. 3A): smallwavenumbers (large wavelengths) are unstable (σ > 0), whereaslarge wavenumbers (small wavelengths) are stable (σ < 0). Themost unstable mode, determined by the wavenumber that max-imizes σ, coincides with the wavenumber 2π=λ of the pattern thatspontaneously emerges from a flat sediment bed.

Destabilizing Effect of Reptation. We have determined the contri-bution of the grains in reptation to the growth rate by selectingthe grains with a hop height smaller than 3d and measuring thedifference between deposition and erosion rates. As shown inFig. 3B, we find that reptation has a destabilizing effect (σ > 0)and contributes to σðkÞ linearly in k. The ratio σ=k is homoge-neous to an erosion/deposition rate and has therefore the di-mension of a velocity. This linear scaling in k must then originatefrom a characteristic value of such a rate associated with repta-tion. To determine it, we have measured the vertical flux densityprofile φðzÞ, defined as the volume of the grains crossing a unithorizontal surface at altitude z per unit time (SI Text). φðzÞsystematically presents a maximum at the surface of the staticbed (Fig. S3), which defines the basal erosion/deposition rate φb.The scaling properties of sand ripples directly originate from anunexpected dependence of φb on the wind speed, which musttherefore be discussed in detail.The vertical flux density profile φðzÞ reveals the existence of

a yet unnoticed interfacial layer separating the saltation zonefrom the static bed (Fig. S3). In this layer, which is a few grainsizes thick, the grain volume fraction is close to that of the staticbed, and midair collisions are frequent (27–30). As evidenced bythe large collision probability in this layer (Fig. S3A), only a smallfraction of the grains arriving from the upper transport layertruly impact the static bed: most of them actually bounce backbefore. Because of these collisions, the shear stress carried by thegrains is transferred from a kinetic form, i.e., a flux of momen-tum associated with a particle flux, to a contact stress. As thetypical grain velocity in this layer is set by φb, the associatedcollisional stress scales as ∼ ρpφ

2b. In a situation of steady and

homogeneous transport, this stress must balance the grain-borneshear stress in the upper transport layer, which scales with ρf and

with the square of the excess shear velocity δu= up − uth. Thebasal erosion/deposition rate therefore varies as

φb ∼ffiffiffiffiffiffiffiffiffiffiffiffiρf�ρp

qðup − uthÞ; [1]

in agreement with the numerical data computed with differ-ent values of the ratios up=uth and ρp=ρf (SI Text) (Fig. 2B). Thisscaling law is found to hold even close to the threshold (Fig. S3B),

A

B C

Fig. 1. Ripples emerging from a flat bed in a simulation ðu*=uth = 3Þ. (A) Large-scale view of the system composed of 45,000 grains in a quasi 2D xyz box ofrespective dimensions 3,400  d × 1  d × 1,000  d. Periodic boundary conditions are used in the x (wind) direction. The results presented here are obtained fora density ratio ρp=ρf = 500, a grain Reynolds number R=d=ν

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðρp=ρf − 1Þgd

q= 22 (ν is the air kinematic viscosity), and shear velocities in the range

u*=uth =1‒5. The colored background codes for the wind velocity; see wind profile (on the left). (B) Close-up view at the scale of the ripple wavelength,featuring saltation trajectories, with hop height between 15 and 30d. The average resonant trajectory is shown in red. (C) Zoom at the level of the interfacial.A collision between a grain in saltation (orange) and a grain in reptation (green) is sketched.

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Fig. 2. Selection of ripple wavelength and propagation speed. Blue circles,data consistently deduced from the most unstable mode of the dispersionrelation and measured starting from a flat sand bed; red squares, experimentaldata (8). (A) Ripple wavelength λ as a function of the wind shear velocity u*(left axis). Yellow triangles, ratio of the saltation flux qs to the erosion/deposition rate φb (right axis). (B) Propagation velocity c as a function of u* (leftaxis). Yellow triangles, erosion/deposition rate φb, measured for many values ofdensity ratios ρp=ρf and grain Reynolds numbers Re (right axis). Dashed lines,linear fits to the data. Statistical error bars are of the size of the symbols.

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which means that the ejection process and the redistribution ofmomentum in the interfacial layer must involve a large numberof grains, even when splashing grains are well separated in time.This observation challenges the common view on aeolian sedi-ment transport and calls for the seeking of collective effects inthe splash process.The grain hop length distribution ψðℓÞ (SI Text) reflects the

dynamical mechanisms dominating each of these transport lay-ers. Although ψ decreases with ℓ as a stretched exponential in theupper layer, it behaves as a scale-free power law ψbðℓÞ∼φbℓ

−1 inthe interfacial layer (Figs. S4 and S5). This, surprisingly, meansthat there is no characteristic hop length (a given number ofgrain sizes) for grains departing from the static bed—for repta-tion in particular—whereas there is one for grains flying in theupper transport layer (several tens of d). Grains ejected from thestatic layer inherit their scaling laws from the impacting grains.The scale-free behavior observed in the interfacial layer is an-other indication of the collective processes at work. As an im-portant consequence, the erosion/deposition rate φb is the singlecharacteristic quantity of this layer. Accordingly, we have foundthat varying the wind velocity and the density ratio ρp=ρf in thesimulations, the contribution of reptation to the growth rate σalways takes the form aφbk, with multiplicative factor a of order1 (Fig. 3B).

Stabilizing Effect of Saltation. The contribution of grains in salta-tion (i.e., grains with a hop height larger than 3d) to the growthrate σ is found to be negative and quadratic in k (Fig. 3B).Varying up=uth and ρp=ρf , we can further establish that the sal-tation-induced growth rate has the form −bqsk2, where qs is thetotal sediment flux over a flat bed and the multiplicative factorb is of order 1 (Fig. 3B). Saltation therefore has a direct stabi-lizing effect consistent with a topographic diffusion, i.e., adownslope component of the saltation flux. As evidenced in ex-periments (31), the overall sand transport—not only reptation—is sensitive to the bed slope, with a diffusion coefficient pro-portional to the total sediment flux qs. This slope effect can beunderstood from a momentum balance. Most of the sedimentflux takes place in the upper part of the transport layer and iscontrolled by the transfer of momentum from the wind to thegrains. At steady state, this momentum is balanced by resistiveforces due to collisions with the bed. Because these forces areeffectively smaller downslope than upslope, the surplus of mo-mentum from the wind can set more grains into motion and thusincreases the flux downslope.The measurement of qs in the simulations (SI Text) shows

a quadratic dependence on up (Fig. S6), in agreement withcontrolled experiments (31):

qs ∼ρf uthρpg

ðup − uthÞ2: [2]

This scaling behavior is fairly well established and has beeninterpreted as a consequence of the feedback of the grains onthe wind flow. This feedback keeps the characteristic saltationvelocity constant and proportional to the shear velocity thresholduth (1, 8, 9, 11, 26), a necessary condition to reach steady statesediment transport when one impacting grain is replaced, onaverage, by a single ejected grain (5).

Dispersion Relation. Summing up the contributions of reptationand saltation, the dispersion relation σðkÞ is well fitted, for allshear velocities up, by the parabolic function

σ = aφbk− bqsk2; [3]

with multiplicative constants a and b of order 1. This constitutesa major difference with existing models, which predict that boththe destabilizing and the stabilizing terms grow like k2 at smallwavenumber. The most unstable wavelength computed from thedispersion relation is proportional to the sediment transportlength, defined as the ratio of the horizontal flux to the ero-sion/deposition rate qs=φb (Fig. 2A). Because qs is quadratic,whereas φb is linear with respect to the shear velocity, we get(see Table S1 for a list of symbols)

λ∼qsφb

∼

ffiffiffiffiffiρfρp

suthgðup − uthÞ: [4]

We have checked that this scaling law is robust to the value ofthe grain to fluid density ratio ρp=ρf and of the grain Reynoldsnumber. Although our system is 2D and composed of rather softparticles, the emergent wavelength in the simulations is onlya factor 1.8 away from that obtained in wind tunnel experiments(Fig. 2A).In the field, initial ripples develop toward a statistically steady

pattern. Their wavelength eventually results from fluctuatingwind conditions and from ripple nonlinear interactions.However, measurements show that the linear dependence of thewavelength on the wind velocity still holds for developed ripples

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Fig. 3. Dispersion relation measured from the numerical simulations. (A)Ripple growth rate σ as a function of the wavenumber k=2π=λ for differentwind speeds (red, u*=uth = 3; green, u*=uth = 4; blue, u*=uth = 5). The solidline is the fit of Eq. 3. (B) Contributions of grains in saltation (right axis) andgrains in reptation (left axis) to σ. Same color code as in A.

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Fig. 4. Focusing of saltating trajectories. (A) Rescaled vertical flux densityprofiles of grains in reptation (green symbols) and grains in saltation (orangesymbols) for three different wind velocities (squares, u*=uth = 3; circles,u*=uth = 4; triangles, u*=uth = 5). Fluxes are evaluated on horizontal lines tosubtract the screening effect (SI Text). The solid red line corresponds toa uniform saltation rain. (B) Normalized ripple elevation profiles. Black solidlines, sinusoidal fits.

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(25). Furthermore, Martian ripple wavelength (in the range2–6 m; Table S2 and Fig. S7) is ’ 30 times larger than on a typicalEarth sand dune. Under the assumption that the most probablewind velocity for ripple formation is the static transport thresh-old, our simulations effectively predict that for these conditionsthe ratio qs=φb is 20 times larger on Mars than on Earth, due todifferent values of atmospheric characteristics (density and vis-cosity) and gravity.Experimental data and numerical simulations also agree

on the scaling law obeyed by the propagation speed of ripples(Fig. 2B):

c∼φb ∼ffiffiffiffiffiffiffiffiffiffiffiffiρf�ρp

qðup − uthÞ: [5]

The product of the wavelength and the speed is thereforeproportional to the total sediment flux qs ∼ cλ. This relationhas been successfully tested in the field, where the three quan-tities λ, c, and qs have been measured independently (Fig. S8).This opens a promising perspective from which to track sandmass transfers in real time, by performing remote measurementsof sediment fluxes from aerial views of sandy area, on Mars inparticular (Fig. S7).

DiscussionThe origin of the ripple instability can be understood as follows.As previously described, grains are eroded from the troughs andaccumulate on the crests because of the modulation of the

reptation flux, which is maximal on the windward side andminimal on the lee side (Fig. 4). In contrast with the geometricscreening scenario, simulations and experiments show that thesaltation rain is also strongly modulated (Fig. 4 and Fig. S2). Theimpacting flux is nonetheless larger on upwind slopes due tothe solid angle effect but also because of a stochastic focusing ofthe trajectories. Indeed, the rate at which grains leave the bed ismodulated similarly to the impacting rate: if there are moregrains arriving at the upwind slope inflection point, there are alsomore grains departing from that position. The key point is thatthese grains must statistically be the same: the modulation ofsediment transport is essentially due to grains departing from theinflection point and arriving at the following inflection pointlocated downwind. The consequence for the ripple emergence isfundamental: the modulation of the reptation flux is driven byresonant trajectories, those whose hop length is comparable tothe wavelength λ. Because processes at work in the interfaciallayer are scale-free, the reptating grains that contribute the mostto the instability have hop lengths proportional to λ and not tod: they inherit their characteristic scale from that of saltatinggrains. The new picture proposed here therefore substantiallychanges the paradigm of aeolian transport.

ACKNOWLEDGMENTS. We thank O. Pouliquen and H. Elbelrhiti for dis-cussions and assistance with the field work. We are grateful to J. Tavacoli,A. B. Murray, and G. Wiggs for a careful reading of the manuscript anduseful comments. This study was supported by an Agence Nationale de laRecherche Zephyr Grant.

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