JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,

Packing variations on a ripple of nearly monodisperse dry sandM. Y. Louge,1 A. Valance,2 A. Ould el-Moctar,3 and P. Dupont4

Abstract. Dielectric measurements reveal that bulk density on the surface of recently-mobilized dry ripples of a nearly monodisperse desert sand varies from the random jammedpacking at troughs to the minimum value for a stable packing at crests, suggesting thatactive ripples are relatively loose structures moving above a more consolidated substra-tum. Although the wind-driven evolution of static pressure on the surface induces airseepage through sand, we show that such internal flow is not strong enough to swell thebed and explain variations of bulk density on the surface.

1. Introduction

Wind mobilizes desert sands by overcoming static frictionand other surface forces of van der Waals, electrostatic, orcapillary origin. The resulting flow has low-energy grains“reptating” in a thin shear layer traveling on the sand bed,and fewer, faster-moving “saltating” particles acceleratedby the free stream before impacting the reptating layer ata grazing angle [Bagnold , 1942; Andreotti , 2004; Kok andRenno, 2008]. Such bombardment dislodges grains fromthe bed, brings them into reptation, and gives them ini-tial agitation. (Following Bagnold [1942], authors have alsodistinguished “creeping” grains rolling on the surface and“reptating” grains executing short hops after being ejectedby impact with saltating sand in a process called “splash”.Recently, this distinction among low-energy grains has beenblurred e.g., Manukyan and Prigozhin [2009]).

The formation, motion and rearrangement of ripples is arelatively slow process involving basal surface erosion, repta-tion, and gradual accumulation of sand, ultimately forminga pattern of troughs and crests perpendicular to wind blow-ing in the direction x. Sand accumulation on ripple crestsarises as streamwise variations of the solids mass flow ratem′ per unit width of the ripple crest line balance variationsin the local surface erosion-deposition rate

φ′′ =∂m′

∂x. (1)

In turn, φ′′ sets the relatively slow rate of change of bedelevation h that creates ripples,

h = − φ′′

ρsν. (2)

In this expression, ρs is the material density of sand, ν isthe solid volume fraction, the dot denotes the partial timederivative, and each prime indicates per unit length. Ignor-ing air mass, the solid volume fraction ν = ρb/ρs or “packingfraction” is the ratio of bulk density ρb and sand materialdensity ρs. Its complement (1 − ν) is the “voidage”. Vari-

1Sibley School of Mechanical and Aerospace Engineering,Cornell University, Ithaca, NY 14853, USA.

2Institut de Physique de Rennes, Universite de Rennes 1,France.

3Ecole Polytechnique de l’Universite de Nantes, France.4Institut National des Sciences Appliquees de Rennes,

France.

Copyright 2009 by the American Geophysical Union.0148-0227/09/$9.00

ations of m′ have been attributed to different rates of im-pact from saltating grains [Anderson, 1987] due to “shadow-ing” [Sharp, 1965], coupled with a BCRE model [Bouchaudet al., 1994; Csahok et al., 2000], to rolling and avalanch-ing [Hoyle and Woods, 1997; Prigozhin, 1999], and to vari-ations of the saltation flux along the ripple [Nishimori andOuchi , 1993]. Predictions of the ripple pattern and its evo-lution are derived from non-linear stability analyses thatmodel φ′′ [Csahok et al., 2000]. Manukyan and Prigozhin[2009] also proposed a numerical model capturing the roleof a wide particle size distribution on ripple formation.

Ripple crests are often noticeably looser than the under-lying sand surface. Meanwhile, the experiments of Rioualet al. [2003] suggested that the resuspension of bed particlesis more likely to occur when saltating grains impact a looseassembly. Thus, streamwise variations of bulk density onthe surface likely affect the bed erosion rate, a mechanismwhich Csahok et al. [2000] modeled in terms of local bedcurvature. In this context, we bring insight into this processby quantifying variations of bulk density on the surface of adry, recently mobilized sand ripple with a new capacitanceinstrument.

2. Instrument

We exploit the capacitance technique that Louge et al.[1997, 1998] developed to measure dielectric properties of thesnow pack. In this technique, sketched in Fig. 1, a genericprobe has three conductors, called the “sensor”, “ground”and “guard” electrodes. For our portable, self-containedelectronics (Capacitec model 4100 with PC-201 preampli-fier), an oscillator supplies a sinusoidal current of constantamplitude at a frequency fc = 15 kHz to the sensor usinga feedback control circuit that maintains a stable voltageacross a reference capacitance. A voltage follower of highinput impedance samples the resulting sensor signal withoutdisturbing it, driving the guard at precisely the same volt-age. Crucially, the guard envelops the sensor everywhereexcept at the probe surface. To that end, the amplifier issurrounded by a guarded cage, and the probe is connectedto the electronics using a high-quality coaxial cable withinner conductor and outer shield respectively at the sen-sor and guard voltages. The guard thus eliminates strayand cable capacitances, and permits the detection of ca-pacitances as low as a few femtoFarads [Louge et al., 1996](1 fF = 10−15 F).

Calibrations using a parallel-plate capacitance held infront of a grounded target plate by a precision microme-ter indicate that, if probes are exposed to a pure dielectric,the peak-to-peak guard voltage Vg of our own electronics isrelated to the capacitance C between sensors and groundusing Vg × C = Υ ' 945 V×fF. For robust desert opera-tions, electronics are powered by a car battery and carried

1

X - 2 LOUGE ET AL.: PACKING VARIATIONS ON SAND RIPPLES

in a backpack. They are warmed up 30 min before oper-ations, and their response is checked by holding a piece ofglass with known dielectric constant in front of the probes.

We capture the oscillating guard voltage on a portableFluke digital oscilloscope to find Vg and the phase ϕ of theguard with respect to the oscillator. Like snow, humid sandpossesses a complex dielectric constant

εe = ε′e − ıε′′e , (3)

where ı2 = −1; the real and imaginary parts ε′e(ν; Ω) andε′′e (ν; Ω) are functions of ν and the fraction Ω of the totalbed mass that is composed of water. (Ω is related to the“volumetric water content” θ ≈ ρsΩν/ρw, where ρw is thedensity of liquid water). Thus, the complex impedance be-tween sensors and ground is Z = [2πfcC0(ε′′e +ıε′e)]

−1, whereC0 is the capacitance of the probe held in air.

As Louge et al. [1997] showed, the Capacitec amplifierproduces a voltage related to the modulus

|εe| =pε′2e + ε′′2e = Υ/(VgC0), (4)

and a phase that is related to the “loss tangent”

ε′′e/ε′e = | tanϕ|. (5)

The probe of Figure 1 consists of two square sensors lo-cated symmetrically around the long axis of a rectangularguard. As Louge et al. [1998] calculated for a homoge-neous medium, their measurement volume is bounded bytwo half-toroidal surfaces penetrating the bed a distanceH = (a2 − b2)/2c ' 11 mm perpendicular to the probeface, where 2a ' 11.5 mm is the guard width in the di-rection joining the two sensors, 2b ' 8 mm is the distancebetween exterior sides of the two sensors, and 2c ' 1.6 mmis the width of the guarded gap separating them. The sen-sor thickness tw ' 3.1 mm on the long axis of the guarddetermines spatial resolution in the x-direction along theripple. (Figure 2 of Louge et al. [1998] marks a, b, andc on a sketch of the probe head, but uses d to denote thethickness tw). Nominally, the probe has a capacitance in airC0 = (2ς0tw/π) ln[(a + b)(a − c)/(a − b)/(a + c)] ' 25 fF,where ς0 = 8.854 10−12 F/m is the permittivity of free space.

Figure 1. Probe recording profiles of bulk density onthe surface of sand ripples. Because its guard has finiteextent in the direction perpendicular to the line joiningthe sensors, its actual capacitance in air is C0 ' 40 fF.

0.000

0.008

PSD

(µm

-1)

0 250 500d (µm)

psd.cg

psd (µm^-1) pt 11

Figure 2. Typical size distribution vs. diameter (µm) ofcollected particles, normalized with

R∞d=0

PSD(ξ)dξ ≡ 1.

We evaluated properties of dry sands in the field byclosely packing a sample to the volume fraction ν in an openrectangular container of 50 × 80 × 40 mm3, by placing thesurface probe on the dry sample, and by measuring ε′e atΩ ' 0. We calculated ρs and ν by later weighing the sampleand recording how much water volume it displaced. Follow-ing the suggestions of Louge et al. [1997] for non-sphericalparticles, we adopted the homogenization model of Bottcher[1952] to describe the dependence of the real part of theeffective dielectric constant on ν,

ε′e(ν; Ω)− 1

3ε′e(ν; Ω)= ν

ε′p(Ω)− 1

ε′p(Ω) + 2ε′e(ν; Ω). (6)

which we used to extract the dry material dielectric con-stant ε′p(0) from the measured ε′e(ν; 0) and ν of the sample.We then used ε′p(0) to calculate volume fractions on drybeds from surface measurements of |εe|. While measure-ments presented here concern dry sands at the dune surface(Ω = 0), our companion paper [Louge et al., 2009] will intro-duce another probe, and consider variations of humidity inthe depth after establishing how the real part of the materialdielectric constant ε′p varies with Ω.

3. Measurements

We recorded the evolution of bulk density along thesurface of ripples on a small barchan dune holdingnearly monodisperse sand (Figure 2) at the site studiedby Ould Ahmedou et al. [2007] near Akjoujt, Mauritania,at 1950.666′ N, 1408.842′ W . Measurements were carriedout on the convex “zone C” that Wiggs et al. [1996] identi-fied between the brink and summit of such dunes. To insurethat the surface was dry, and to allow optical profiling ofripple elevation, we operated shortly past sunset after thewind had stopped blowing. Figure 3 shows a photographtaken after completing measurements.

Profiling was achieved by intersecting the undulating sur-face with a thin triangular light sheet created by a cylindri-cal lens at the tip of a diode laser pointed ∼ 30 downward

LOUGE ET AL.: PACKING VARIATIONS ON SAND RIPPLES X - 3

Figure 3. Surface profiles of ν (top graph) and dimen-sionless elevation h† ≡ h/L (bottom graph) along the di-rection x† ≡ x/L of prevailing winds blowing from left toright. Lines are best fits ν = 0.614+0.049 cos(2πx†+ϕν)and h† = −(h0/L) cos(2πx†) with ripple amplitude h0 '2.8 mm and wavelength L ' 145 mm; from the phase lagϕν ' 195, minimum ν occurs shortly behind the crest.On the top graph, horizontal marks locate ν = νc ' 0.64and ν = νm ' 0.545, and the dotted line shows predic-tions of equation (A11) with R = Rc at y† = 0. On thebottom graph, lines mark h = ±3 mm and intervals of 100mm along x. The photograph shows the structure hold-ing camera and diode laser, the light sheet intersectingthe bed, and toothpicks planted where ν was recorded;the red triangle mimics spreading of the sheet.

along ripple crest lines. The trace was acquired by an over-head digital camera, and converted to ripple elevation afterrecording where a slender parallelepiped of known thicknessintersected the sheet. The resulting ratio h0/L ' 0.02 ofamplitude to ripple wavelength was consistent with similarmeasurements of Andreotti et al. [2006] for fully-developedripples.

Our procedure was to place a light metal structure hold-ing the laser and camera around the region of interest, andto take a photograph of the laser trace on undisturbed rip-ples. Then, we gently applied the capacitance probe ofFigure 1 to the bed surface at several locations along thewind direction x perpendicular to ripple crests. As Figure 3shows, the evolution of solid volume fraction ν is out-of-phase with the relative ripple elevation h† ≡ h/L, where thedagger † represents distances relative to the ripple wave-length. At ripple troughs, ν reaches a value just abovethe random jammed packing of spheres νc ' 0.64 [Torquatoet al., 2000], as expected from slightly polydisperse, nearly

spherical grains, and in good agreement with the measure-ments of Dickinson and Ward [1994]. At ripple crests, ν isapproximately the random loose packing [Onoda and Lin-iger , 1990] or the so-called “melting transition” νm ' 0.545of hard spheres [Zamponi , 2007].

The relatively large penetration H ' 11 mm of the ca-pacitance probe measurement volume in the vertical direc-tion further suggests that loose packings are not confined toa thin region near the crest surface, but instead permeatethe ripple. The qualitative observations in Figure 4 con-firm this view. Here, one of us rammed a foot through asoft dune region exhibiting little or no cohesive material un-derneath. The ensuing stresses propagated through forcechains in the bed [Radjai et al., 1998], and emerged exclu-sively from beneath ripple troughs, throwing particles ver-tically to a height of approximately 50 mm. This suggeststhat the material forming crests is made up of loose sandwith ν below the “glass transition,” which occurs at a vol-ume fraction νg ' 0.575 for hard spheres [van Megen et al.,1998]. Above νg, sheared granular assemblies must dilate,as ripple troughs were forced to do in this simple test.

Figure 4. Successive photographs of the forceful im-pact of a foot against a soft, dry dune at 1801.140′ N,1545.282′ W near Nouakchott, Mauritania, showing (a)the moment of impact; (b) as the foot sinks, sand isejected from ripple troughs, (c) eventually reaching analtitude approximately 50 mm above the bed; and finally(d) the foot and sand at rest. The bottom close-up pho-tograph shows aftermath of a similar event. The arrowshows where ejecta returned to ripple troughs whencethey jumped. Note how crests appear unaffected by thisprocess. The magnified region reveals a natural patternof crosses marking the planes of Mohr-Coulomb failure.From their apex angle 2α ' 57±2, we infer the internalfriction tanα ' 0.543 for this sand, see Appendix A. Ashort movie of this process is available on line.

X - 4 LOUGE ET AL.: PACKING VARIATIONS ON SAND RIPPLES

4. Discussion

We expect surface variations of bulk density on dry rip-ples to matter in two ways. First, at troughs, higher vol-ume fractions should raise the local threshold shear stressnecessary to lift particles. Conversely, at crests, the loosergrain assembly should facilitate the ejection of particles intoreptation. This section discusses possible origins of thesevariations. We first consider the seepage of air through theporous bed that is driven by variations of static pressure in-duced by the ripple’s presence. Although Louge et al. [2009]predict that such seepage plays an important role in trans-ferring dust and moisture through the surface, we dismiss itscapacity to swell the bed and cause the variations of bulkdensity that we observed.

4.1. Role of seepage

As Buckles et al. [1984] and Gong et al. [1996] found inwater and air, turbulent flows on wavy surfaces induce pres-sure oscillations due to streamline expansion and contrac-tion. Gunther and von Rohr [2003] reviewed experimentsand theory for such flows. In a porous sand bed, these os-cillations create a secondary gas “seepage”: air penetratesripple troughs where pressure is highest, and re-emerges atcrests, where it is lowest. A similar mechanism exists withunderwater ripples [Meysman et al., 2007; Huettel et al.,2007]. To establish whether seepage can cause variationsof bulk density at the surface by swelling the bed, we firstcalculate the air pressure and velocity fields within the sandbeneath a ripple.

We adopt a cartesian coordinate system with unit vectorsx along the wind direction and y downward in the directionof gravity g. We consider ripples as two-dimensional objectswith sinusoidal elevation h = −h0 cos(2πx/L), amplitudeh0 and wavelength L. Because h0/L 1, we treat thebed surface as flat to simplify calculations within. Simi-larly, because h0 is small compared to the seepage penetra-tion ∼ L/2π, we ignore surface variations of solid volumefraction reported in section 3 to calculate seepage pressureand velocity fields. Thus, we assume that the porous bedis made of identical cohesionless spheres of diameter d atthe uniform solid volume fraction ν = νc ' 0.64. We alsotake the seepage flow as steady. Its surface is subject to thestatic pressure profile

p(y = 0) ' pa + p0 cos(2πx†), (7)

where pa is the ambient pressure, and p0 is its ampli-tude. Gong et al. [1996] observed

p0 ' ρU2h0/(ηL), (8)

where ρ is gas density, U is wind speed, and η ' 0.14. Gonget al. [1996] measured U ≈ 10m/s at Y ≈ 1m above the wavysurface, or at a wall coordinate Y + ≡ ρY u∗/µ ≈ 26, 000.(Such evolution of static pressure would also arise in aone-dimensional, steady, incompressible, inviscid and irro-tational flow on a wavy surface h = −h0 cos(2πx†) at adistance ηL below a parallel flat wall).

In Appendix A, we verify that the seepage flow of airthrough the sand bed has a small Reynolds number. There-fore, it is subject to the viscous macroscopic Ergun’s equa-tion

u = −(K/µ)∇p, (9)

where u ≡ (1 − ν)v is the superficial gas velocity vector, vis the interstitial gas velocity,

K ' d2

150

(1− ν)3

ν2(10)

is the bed permeability, and µ is the gas viscosity. A conse-quence of equation (9) is that µ(1−ν)[(∇v)+(∇v)T ] ≡ 0, sothat gas shear stress vanishes identically within the porousbed. In principle, consistent with the Beavers and Joseph[1967] boundary condition, a thin boundary layer mightform within the sand bed just beneath the surface. Be-cause its thickness is on the order of

√K ≈ 0.03d, we ignore

its existence. Thus, turbulence above the dune transmitsno shear stress to the bed’s interstitial air, but rather tothe sand bed assembly, and seepage is entirely driven by thestatic pressure profile at the bed surface.

For steady flow, the mass conservation equation is

∇ · ρu = 0. (11)

Therefore, in a homogeneous bed with negligible gas com-pressibility, pressure satisfies the Laplace equation ∇2p = 0subject to equation (7) at the surface and p→ pa as y →∞.Its solution is

p = pa + p0 cos(2πx†) exp(−2πy†), (12)

yielding the local pressure gradient

∇p = −2πp0

Lexp(−2πy†)sin(2πx†)x + cos(2πx†)y. (13)

and dimensionless velocity field

u† ≡ uLµ/(2πp0K) = exp(−2πy†)[sin(2πx†)x

+ cos(2πx†)y]. (14)

Inspired by Eames and Gilbertson [2000] and Mourguesand Cobbold [2003], we now ask whether seepage can swellcrests by exerting an upward drag defeating gravity and fric-tional forces. Appendix A reveals that, if the wind is strong,the sub-surface sand swells with a pattern of bulk densitythat is consistent with our observations on the surface (Fig-ure 3, dotted line). However, it also shows that bed swellingis set by

R ≡ 2πp0/(ρsνcgL), (15)

which represents the fraction of gravity that seepage dragrelieves. (Note that, although swelling does not explic-itly depend on d, it is affected by grain size through L).For Akjoujt ripples, seepage only begins to defeat gravity(R > 1) for U > 46 m/s. Although it may release some graincontacts at typical wind speeds (R ' 5% at U = 10 m/s),seepage does not swell the bed directly. Therefore, otherexplanations must be sought for streamwise variations of νon ripples.

4.2. Ripple evolution

As this section suggests, ripple growth and evolution arelikely responsible for packing variations observed on dry, re-cently mobilized, wavy surfaces of a nearly monodispersesand. Using radioactive particle tracers in a wind tunnel,Barndorff-Nielsen et al. [1985] noted that grains are period-ically buried under a ripple, later re-emerging as the ripplemoves forward [Barndorff-Nielsen, 1986].

Grains advancing on the sand bed are moved by a surfaceshear stress τ0 that includes a contribution τ0g from air tur-bulence and another τ0s from the momentum that saltationimparts on them by collisions. Because for ripples of smallamplitude (h0 L) the total time-averaged shear stress isindependent of altitude in steady, fully-developed flow, itsmagnitude |τ0| at the surface is, to a good approximation,equal to its value above the particle cloud i.e., |τ0| ≈ ρu∗2,where u∗ is the turbulent shear velocity on a flat surfacewithout particles, and the bar denotes average over time.

LOUGE ET AL.: PACKING VARIATIONS ON SAND RIPPLES X - 5

In general, both τ0g and τ0s can vary along the ripple. Ona wavy surface, Gong et al. [1996] observed that the time-average air shear stress |τ0g| is high at crests and low attroughs, while boundary layer separation can occur on thedownward ripple surface behind crests. Motivated to finda simple theory for the dynamics of barchan dunes, Kroyet al. [2002] modeled the evolution of |τ0g| along a wavysurface by simplifying the treatment of Hunt et al. [1988].They expressed the stress in terms of local surface slope andan integral convolving slopes at distances effectively . o(L)away, |τ0g|(x) ∼ ρu∗21+B∂h/∂x+A

R +∞ξ=−∞(dξ/πξ)∂h(x−

ξ)/∂x, where A and B are adjustable constants [Hersen,2004]. Manukyan and Prigozhin [2009] adopted a similarframework to calculate the evolving shear stress on a rip-pled surface.

While τ0g is chiefly determined by nearby surface curva-ture, τ0s is affected by saltation, which borrows momentumfrom the wind farther away, and thus should be more uni-formly distributed along x at the ripple scale [Manukyan andPrigozhin, 2009]. However, where the incident inclination ofsaltating particles is less than the downward slant of the rip-ple surface, such as right behind steep crests, Sharp [1965]noted that impact “shadowing” protects surface grains, thuseffectively canceling the local contribution of saltation to τ0.More generally, the way by which saltation imparts stress onsurface grains should depend on local bed inclination andpacking. This concept is consistent with the standard ex-planation for growth of aeolian ripples on an initially flat,erodible bed, whereby saltating particles impact the ripple’swindward slope at an angle closer to the outward normalthan the downwind slope, thus ejecting more grains intoreptation, and producing a greater mass flow rate m′ thatpushes grains toward the ripple crest [Bagnold , 1935; An-derson, 1987]. In short, τ0g and τ0s both vary along x, thusaffecting the local rate at which reptating grains are mobi-lized.

In this context, for thin granular flows driven by a sur-face shear stress, the depth-averaged solid volume fractiondecreases as surface shear stress increases. Conversely, whenthis stress falls anywhere below an entrainment threshold,moving grains stop and deposit on the bed. (Appendix Billustrates these points by considering an idealized thin gran-ular flow with free surface subject to shear stress and grainagitation imposed by a turbulent gas). Meanwhile, lowervalues of |τ0| can arise behind ripples crests if τ0g collapsesby boundary layer separation and/or τ0s does so by salta-tion shadowing. Such mechanisms might explain how theradioactive grain tracers of Barndorff-Nielsen et al. [1985]could have been trapped by advancing ripples leeward oftheir crests.

In general, because free-surface granular flows are proneto jamming at solid volume fractions & 0.6, they dilatewhile being sheared, often maintaining a volume fraction0.545 < ν < 0.595 [Silbert et al., 2000]. We expect that mo-bile grains preserve such relatively loose fabric once trappedin a ripple. Because ripples are ephemeral objects with-out significant overburden, few mechanisms can pack themas they move, or for some time after they stop. As rip-ples travel slowly forward, compaction may arise by impactbombardment from saltating grains. An order of magni-tude analysis comparing this process to packing by hori-zontal vibration suggests that periodic bombardment andshear stress fluctuations cannot result in any significant com-paction deep within the ripple: in a unit length along thecrest, the mean horizontal fluctuating force is ∼ o(τ0L), act-ing on a ripple mass ∼ ρsνch0L; thus the relative accelera-tion is Γ ∼ ρu∗2/(ρsνch0g) [Ristow et al., 2003]; for Akjoujtripples, Γ ' 0.004, which is much smaller than the mini-mum value Γ ∼ 0.01 at which shaking begins to compress agranular assembly [D’Anna and Gremaud , 2001]. Nonethe-less, while saltation bombardment is unlikely to pack ripplesin depth, it is possible that surface grains might rearrangemore closely upon repeated impacts.

A longer-term compaction mechanism is due to ther-mal cycling through sand of thermal diffusivity αs overseveral days of J = 24 hrs. Although grains in a mo-bile, fully-developed ripple are everywhere within the depth∼pαsJ/π ' 0.07 m through which diurnal temperature

variations matter [Carslaw and Jaeger , 1959], a typical rip-ple turn-over time ∼ 105

pd/g ' 450 s [Andreotti et al.,

2006] is J , too short to pack recently mobile ripples.However, because the basal surface exposed by the rippletrough endures much longer, particularly near the dune apexwhere sand erosion and deposition rates balance, this sur-face may be subject to enough thermal cycles to explain itshigh packing ∼ νc, as the recent experiments of Divoux et al.[2008] suggest. Consistent with this, we report in the com-panion paper [Louge et al., 2009] that packings remain nearνc ' 0.64 within the first few cms below the dune surface,in sharp contrast with the loose packing recorded beneathcrests (except at the avalanche face and at point 3 just abovethe horn, where packing is loose). The observations of Fig. 4further support this view: ramming forces are chiefly trans-mitted to ripple troughs, suggesting that the latter form arelatively compact base connected by the “strong force net-work” of Radjai et al. [1998]; conversely, ripple grains looselydeposited beneath crests likely constitute a “weak force net-work.”

Because our observations were carried out in the field ona nearly monodisperse, recently mobilized sand, they mightnot represent ripples with wider size distribution. For ex-ample, “giant” ripples, which are armored by larger grainsrolling on the surface and accumulating at crests [Tsoar ,1990; Manukyan and Prigozhin, 2009], are less mobile, per-haps giving the ripple enough time to compact, and theirshape is no longer harmonic.

However, our observations and those of Barndorff-Nielsenet al. [1985] suggest that, in an active ripple of narrow sizedistribution, the region with elevation between trough andcrest is filled with loose, recently mobilized sand. Fromthis view, it is tempting to speculate that a ripple of nearlymonodisperse, round, dry sand is a loose, slowly advancingobject that “rolls” on a relatively passive denser substratumunder the action of turbulent air and saltation bombard-ment, repeatedly burying and dislodging the same popula-tion of reptating grains as it progresses.

5. Conclusions

Dielectric measurements on a barchan dune revealed thatbulk density at the surface of a dry ripple of nearly monodis-perse round sand varies from random jammed packing attroughs to random loose packing at crests. The kicking ofa soft sand bed also suggested that dry crests are relativelythick compliant regions of low density unable to transmitsignificant stress.

We introduced a model of the internal seepage flow drivenby variations of static pressure on the surface of a wind-swept ripple. An analysis of the frictional sand bed showedthat the resulting seepage drag contributes to relieving grav-ity, but is not able to swell the bed beneath crests to theobserved random loose packing at reasonable wind speed.Instead, the ripple picture that emerges is one of a hard,almost immobile flat base (the trough), perhaps compactedby diurnal thermal cycles, and a heap (with crest) slowly ad-vancing at the minimum bulk density for a stable solid. Fur-ther measurements are needed to confirm speculation thatripples of monodisperse dry sand are objects executing aslow rolling motion on a hard base.

X - 6 LOUGE ET AL.: PACKING VARIATIONS ON SAND RIPPLES

Appendix A: Seepage and bed expansion

We show that, for typical wind speed, the volumetricseepage drag working against gravity is insufficient to ex-pand the interconnected granular assembly to a lower solidvolume fraction. To that end, we ignore the possible role ofreptation in helping fluidize the bed, and we adopt pressurevariations from equation (12). Because seepage has negligi-ble inertia, its momentum conservation reduces to a balancebetween the drag force Fd exerted by the gas on solids in aunit volume and the local pressure gradient exerted on thefraction of the volume occupied by the gas, −(1−ν)∇p = Fd.Then, the total force F exerted by the gas on solids is thesum of Fd and the buoyancy force arising from the gas pres-sure gradient, F = Fd − ν∇p = −∇p = µu/K [Andersonand Jackson, 1967]. Because F ∝ K−1 and the superfi-cial gas velocity u is conserved, a small variation of ν awayfrom νc corresponds to F(ν) ' F(νc)[K(νc)/K(ν)], where∇p(νc) = −F(νc) is calculated at ν = νc. Thus,

F = −K(νc)

K(ν)∇p, (A1)

where ∇p is given by equation (13). Assuming that the bedis a homogeneous continuum with symmetric plane stresses,balancing forces on solids along x and y yields, respectively,

∂σx∂x

+∂τ

∂y+ F · x = 0 (A2)

and∂τ

∂x+∂σy∂y

+ F · y + ρsνg = 0, (A3)

where σ denotes a normal stress on surfaces perpendicular tothe index shown, and τ is the shear stress on these surfaces.In these equations, we adopt the convention that compres-sive normal stresses are negative, and that the shear stresson a surface of normal y directed along x is positive. Solvingequations (A2) and (A3) requires constitutive stress-strainrelations and boundary conditions on all sides.

However, bed expansion can be modeled without speci-fying granular constitutive laws and solving equations (A2)and (A3) formally in the periodic interval x ∈ [0, L] andy ∈ [0,∞[. To that end, without loss of generality, we fo-cus on the region beneath the free surface (y > 0) withL/4 < x < L/2. (A mirrored region of similar behavior islocated across the plane x = L/2). There, the y-body forceF · y + ρsνg may turn negative, thus raising the prospect oftensile stresses (σ > 0), which are prohibited if the granularmaterial is not cohesive.

Where equations (A2) and (A3) would predict such ten-sile stresses, we postulate that the granular assembly ex-pands to avoid them, while exhibiting incipient Mohr-Coulomb failure on slip planes with the same orientation

Figure 5. Mohr-Coulomb circle. Normal stresses σxand σy on planes perpendicular to x and y are on theabscissa; shear stress τ is on the ordinate along the posi-tive direction shown. Open circles represent the plane onwhich the granular assembly is postulated to fail with in-ternal friction µE = tanα. The lightly and heavily filledcircles are planes of normal x and y, respectively.

relative to the resultant body force than the surface of areposing heap relative to gravity. This implies that, in theexpanded bed, a slip plane has a normal directed an angle αcounterclockwise from the resultant body force F+ρsνgy ifthe shear stress τs on such plane is negative (Figure 5a), andclockwise if it is positive (Figure 5b). We identify α withthe angle of repose, and equate it to the internal frictioncoefficient µE ≡ tanα.

The signs of τs imply two possible states of stress for thegranular assembly in the expanded bed. Figure 5 illustratesthem both. For τs

><0, we find

σx = σyh1− sinα sin(3α∓ 2β)

1 + sinα sin(3α∓ 2β)

i(A4)

and

τ = ±σyh sinα cos(3α∓ 2β)

1 + sinα sin(3α∓ 2β)

i, (A5)

where β is the angle of the resultant body force with x,

tanβ =F · y + ρsνg

F · x . (A6)

An analysis of equation (A5) reveals that

(tanα− 1/√

3)× τ(β . +π/2)× τs < 0, ∀α. (A7)

For small winds, this condition implies τs < 0 when tanα <1/√

3 and > 0 otherwise.We substitute these expressions in equations (A2) and

(A3), assume that β varies slowly, combine them linearly to

Figure 6. Diagram of tanβ (equation A6) vs. inter-nal friction tanα. Dark shading indicates regions whereζ(tanβ, tanα) < 0. The diagram is plotted by upholdingcondition (A7) at low winds; therefore, τs must be nega-tive to the left of the vertical dash-dotted line, and pos-itive to the right. As wind strengthens, β moves down-ward in this diagram on a vertical line at fixed tanα = µEfrom β = +π/2 (no wind) until it reaches the boundary ofthe shaded region (solid line) where β = β0. For strongerwinds, β remains equal to β0 while the bed swells.

LOUGE ET AL.: PACKING VARIATIONS ON SAND RIPPLES X - 7

eliminate ∂σy/∂x, define σ† ≡ σ/(ρsνcgL), and find

∂σ†y∂y†

= −RK(νc)

K(ν)sin(2πx†) exp(−2πy†)

(1 + sinα sin(3α∓ 2β))h1 + 4 sin2 α

1 + tan2 β

iζ(tanβ, tanα), (A8)

whereR ≡ 2πp0/(ρsνcgL) (A9)

is the dimensionless group governing this system, having thestructure of a Shields parameter. We define the function

ζ(tanβ, tanα) ≡ (tanβ∓tanα)(tanβ−tanβ±1 )(tanβ−tanβ±2 ),(A10)

with

tanβ±j =∓2 tanα(1− tan2 α) + (−1)j

√ξ

1 + 5 tan2 α(A11)

for j = 1 or 2, and ξ ≡ (2 tanα − 1)(1 + 2 tanα +2 tan2 α + 4 tan3 α + tan4 α + 2 tan5 α). Because σ†y nearlyvanishes at the free surface, integration of equation (A7)from y† = 0 along y avoids positive tensile stresses in theregion x† ∈ [1/4, 1/2], iff ζ(tanβ, tanα) > 0.

Consider a region near the free surface. Without wind,the only body force is gravity, so β = +π/2. For weak winds,the surface experiences a small negative τ . Figure 6 mapsvalues of tanβ that avoid tensile stresses or, equivalently,where ζ(tanβ, tanα) > 0. To construct this diagram, weassume that the granular assembly retains the state of stressit had when the wind first built it i.e., τ < 0. Consistentwith condition (A7), this implies τs < 0 for tanα < 1/

√3.

Then, because ζ > 0 at β & π/2,∀ tanα, there are no ten-sile stresses near the surface at low wind, or deep in the bed.As wind strengthens, β decreases, and eventually reaches β0

such that ζ(tanβ0, tanα) = 0 (solid lines in Figure 6). De-creasing β further would let ζ < 0 and, near enough thesurface, this would produce tensile stresses. To avoid theseas wind strengthens further, the assembly must swell to can-cel ζ near the surface, thus maintaining σy = 0 anywhere ζwould otherwise be negative. In this swollen region, β re-mains fixed at β0, and ν satisfies F · y + ρsνg = tanβ0F · xor, equivalently, using equations (10), (13) and (A1)

ν

(1− ν)3=

νc(1− νc)3

×

cosβ0

R exp(−2πy†)[− cos(2πx† + β0)], (A12)

which defines lines of constant ν throughout the bed. Aninspection of this equation reveals that at a fixed ν, suchlines exist (y† > 0) in the interval x† ∈ [x†ν , 1/2], where

x†ν =1

2− 1

2πarccos

hcosβ0

R(1− ν)3

ν

νc(1− νc)3

i− β0

2π. (A13)

The smallest solid volume fractions arise on the free sur-face near the crest at x† = x†m ≡ (1/2) − (β0/2π) if β0 6 0or at x†m = 1/2 otherwise. If the wind speed is too low, Ris small, and drag within the porous bed is not sufficient toswell it, so that ν = νc everywhere. As the wind strength-ens, the first point to swell is at x† = x†m. This occurs forthe minimum value of R such that ν = νc at x†ν = x†m or,from equation (A13), for R = cosβ0. As R rises furtherwith increasing wind speed, the line ν = νc demarcating theswollen region from the underlying compacted bed movesdownward, while ν decreases near the crest.

However, ν cannot become too small, lest the bed be-come too dilute to have a solid, interconnected network ofcontacts. With simulations of hard spheres, Richard et al.

[1999] showed that packings below the “melting” transitionν < 0.545 [Zamponi , 2007] cannot form crystals. Thus agranular assembly at ν < 0.545, which lies just below thesmallest ν for minimum fluidization [Wen and Yu, 1966] orrandom loose packing of spheres [Onoda and Liniger , 1990],hardly transmits stress. In this view, we expect νm ' 0.545to be the smallest ν that crests can exhibit. For strongwinds, the point x† = x†m near the crest maintain ν = νmby losing particles to the saltating cloud, while R remainsinvariant. The latter is found by substituting ν = νm, y† = 0and x† = x†m in equation (A11),

R = Rc =“ νcνm

”“1− νm1− νc

”3

cosβ0. (A14)

Figure 7 illustrates the predictions of equation (A11)for ν for a wind at R = Rc. Although the correspond-ing magnitude of ν at the surface agrees well with themeasurements of Figure 3 (dashed line, top graph), thewinds required to reach this value of R are unrealisticallystrong. For Akjoujt sand with µE ' 0.5, tanβ0 ' 0.33,and ρs ' 2500 kg/m3 forming fully-developed ripples withh0/L ' 0.0195 and L ' 14.5 cm, the required wind speed tomaintain R = Rc ' 2.2 is much too large, Ucrit ' 70 m/s.(We also expect any sand cohesion to raise this velocity fur-ther). Therefore, seepage is not responsible alone for varia-tions of ν at the surface and in the depth.

Note that, even at the largest speed, the Reynolds numberRep ≡ ρud/µ within the porous bed remains small. Com-bining equations (9) and (10) at ν = νc with equation (13),we find that, for R < Rc, Rep < Ar(1 − νm)3/(150νm),where Ar ≡ ρsρgd3/µ2 is the Archimedes number. Then, atworst, Rep < 0.9 in the Sahara, which is smaller than theReynolds number ∼ 86ν ∼ 55 at the transition of a porousbed at ν = νc from viscous to inertial flow.

Although seepage fails to explain the variations of ν forrealistic wind speeds, its swelling model is instructive. First,

Figure 7. Top: seepage swelling predictions forR = Rc,and Akjoujt sand with tanα = 0.5 and β0 = 0.33. Al-though the model assumes a flat bed, we reproduce theelevation profile of Figure 3 for comparison. Arrows markseepage streamlines, which are invariant with p0, U or R.Dashes represent, from bottom to top, lines of ν = 0.64,0.60 and 0.552. The dark region is where ν = νc. The x-axis is along the wind; the y-axis is downward, as shown.Bottom: close-up sketch (not-to-scale) of geometry andnomenclature for the analysis in Appendix B.

X - 8 LOUGE ET AL.: PACKING VARIATIONS ON SAND RIPPLES

seepage counteracts a notable part of gravity even for typicalwinds. Second, because cosβ0 < 1 decreases with tanα, bedfriction facilitates such relief, and different sands may havedifferent swelling behavior. Third, the swelling model, whichpredicts vanishing solids stress where the bed expands, envi-sions wide regions sustaining little or no stress around andbeneath ripple crests (Figure 7). The model also predictsthat the depth of these regions is on the order of L/(2π),which is consistent with the intact crests in Figure 4: if crestsmerely featured a thin region of loose packing near the sur-face, then they would have erupted as troughs did. Fourth,seepage swelling may be responsible for the disappearance ofripples at very large winds [Bagnold , 1942]. Finally, becausethe critical swelling wind speed Ucrit grows with L, “giant”ripples should survive greater wind speeds.

Appendix B: Free-surface granular shearflow

We consider a thin shear layer of agitated grains sketchedin the bottom of Fig. 7. It travels on an erodible base, isdriven by a shear stress τ0 applied on its free surface, andborrows agitation at that surface from turbulent gas velocityfluctuations. Because typical rippled surfaces make a smallangle < arctan(2πh0/L) ' 7 to the horizontal, we neglectgravity along x, so that grains are moved by τ0 alone. Be-cause the thin shear layer has relatively dense particle load-ing, its grains entrain the surrounding air at their own veloc-ity. Thus, we ignore drag, making our analysis simpler thanthe more dilute, one-dimensional, steady sheet flow modelof Jenkins and Hanes [1998]. We also neglect electrostat-ics [Kok and Renno, 2008], buoyancy, and the force exertedby seepage emerging from the bed. We assume that the lo-cally fully-developed shear layer is sufficiently agitated thatgrains only interact by binary collisions. This differs frompassive, steady flows down inclines, where grains also expe-rience enduring, multiple contacts [Delannay et al., 2007].

Force balances in the x- and y-directions reduce to

∂τ/∂y = 0, (B1)

and∂σy/∂y + ρsgν = 0, (B2)

with the constitutive relation for shear stress

τ = f1(ν)ρsdT1/2∂vx/∂y, (B3)

in which f1(ν)ρsdT1/2 is a granular viscosity, and the “equa-

tion of state” for normal stress

σy = −f4(ν)ρsT. (B4)

For identical grains, particle agitation is measured with the“granular temperature” T ≡ (1/3) < v′2x +v′2y +v′2z >, wherev′i ≡ evi−vi, evi is the instantaneous grain velocity, vi is its av-erage over time and i = x, y, z. The local fluctuation energybalance (resembling the k-equation in turbulence) is

−∂q/∂y + τ∂vx/∂y − γ = 0, (B5)

where the flux of fluctuation energy is

q = −f2(ν)ρsdT1/2∂T/∂y, (B6)

in which f2(ν)ρsdT1/2 is a granular conductivity, and the

volume rate of dissipation of fluctuation energy (resemblingthe turbulent energy dissipation rate ε) is

γ = f3(ν, e)ρsT3/2/d. (B7)

We eliminate the velocity derivative in equation (B5) usingequation (B3) and write

∂q/∂y = τ2/[f1ρsdT1/2]− γ. (B8)

For nearly elastic, frictionless spheres of restitution coef-ficient e, Jenkins and Richman [1985] calculated

f1 =8

5√πν2g12

h1 +

π

12

“1 +

5

8νg12

”2i, (B9)

f2 =4√πν2g12

h1 +

9π

32

“1 +

5

12νg12

”2i, (B10)

f3 =12√πν2g12(1− e2), (B11)

andf4 = ν(1 + 4νg12), (B12)

where g12(ν) is the pair distribution of colliding grains “1”and “2” at contact, which, for two identical spheres, is wellrepresented by the Carnahan and Starling [1969] expression

g12 =2− ν

2(1− ν)3, (B13)

as long as ν stays below the “freezing” value νf ' 0.49. Forν > νf , we adopt the extension of Torquato [1995],

g12 =(2− νf )

2(1− νf )3× (νc − νf )

(νc − ν). (B14)

Equations (B1), (B2), (B6) and (B8) are non-linear ODEsin the dependent variables τ , σy, T and q. We deduce νfrom −σy/(ρsT ) in equation (B4) by inverting f4(ν) using alook-up table. On the free surface at y = 0, we prescribe theshear stress τ = τ0 < 0 and invoke the boundary conditionsof Jenkins and Hanes [1993] relating the normal stress

σy0 = −2ν0ρsT0 (B15)

and the temperature T0. There, the volume fraction satisfiesν0g12 = 1/4 or ν0 ' 0.16. For a horizontal free surface, Jenk-ins and Hanes [1993] also prescribed q = 0 at y = 0.

Parameters of this problem are τ0, T0 and e, in which e isan effective parameter characterizing energy lost in the im-pacts of two inelastic, frictional grains. Jenkins and Zhang[2002] calculated e ' en−(π/2)µf in terms of the kinematiccoefficient of normal restitution en and the Coulomb fric-tion coefficient µf . To represent grains of sand, we adopten ' 0.92 and µf ' 0.05 reported by Lorenz et al. [1997] forglass beads and find e ' 0.85.

We relate temperature and shear stress at the free sur-face by invoking Pourahmadi and Humphrey [1983], whowrote 3T0 = 2k/(1 + St), where St is the Stokes number,

k ≡< u′2x + u′2y + u′2z >= u∗2/C1/2µ is the turbulent fluctua-

tion kinetic energy per mass, Cµ ' 0.09 [Patel et al., 1985;Kussin and Sommerfeld , 2002] and u′i is the fluctuation gasvelocity. The Stokes number St = ωs/ωL divides the parti-cle relaxation time ωs = ρsd

2/[18µ(1 + 0.15Re0.687)] by theLagrangian turbulent integral time scale ωL in the particleframe. Clift et al. [1978] corrected the relaxation time for

high Re = ρd√

2k/µ = [2|τ0|/(ρsgd)]1/2Ar1/2/C1/4µ based on

rms fluid velocity, and Ar ≡ ρsρgd3/µ2 is the Archimedes

number. The integral time scale ωL = (CT /2)k/ε is roughlyhalf its counterpart in the fluid frame [Kulick et al., 1994],where CT ' 0.41 [Pourahmadi and Humphrey , 1983]. Theturbulent energy dissipation rate ε = εg + εs in a unit fluidmass has a contribution εg = Cερu

∗4/µ from the gas [Pa-tel et al., 1985] and εs = 2(ρs/ρ)[ν0/(1 − ν0)](k/ωs) from

LOUGE ET AL.: PACKING VARIATIONS ON SAND RIPPLES X - 9

solids [Kulick et al., 1994]. We adopt the peak value ofCε ' 0.25 at the wall [Patel et al., 1985].

Combining these expressions, we find

T0 '“ 2Rρ

3C1/2µ

” |τ0|/ρs(1 + St)

, (B16)

where Rρ ≡ ρs/ρ is the material density ratio and

St =Rρ

CT

hC1/2µ CεAr

18

|τ0|/(ρsgd)

(1 + 0.15Re0.687)+ 2“ ν0

1− ν0

”i.

(B17)For typical conditions in the Sahara, St > 5700 ≫ 1, justi-fying our implicit assumption in equation (B5) that granularagitation within the shear layer is unaffected by the gas.

We solve the four ODEs from the free surface at y = 0downward along the y-axis to the passive base using afourth-order Runge-Kutta procedure until the granular tem-perature, which decreases throughout the flow from its peakT0 at the free surface, eventually reaches T = 0. This bound-ary condition represents the passive, erodible sand base,which has little or no agitation (T ' 0). It is simpler thanthe prescription of Jenkins and Askari [1991], while nearlyidentical in result. Imposing this condition fixes the flowthickness y = h`, on the order of a few grain diameters, atwhich T = 0 is reached.

Agitated shear layers do not exist at arbitrarily small |τ0|.Because they occur on an erodible bed that only dissipatesfluctuation kinetic energy [Jenkins and Askari , 1991], theiragitation is sustained if and only if the volumetric produc-tion rate of fluctuation energy in the second term of equa-tion (B5) exceeds the volumetric dissipation rate γ. Thus,as |τ0| decreases, the shear layer loses its agitation and even-tually collapses at a minimum surface shear stress |τ0crit| be-low which equations (B1)-(B17) have no solution. By solv-ing the numerical model for a number of conditions within0.6 < e < 0.99 and 2 < Ar < 80000, we find that, locally,the minimum shear velocity u∗crit necessary to create an ag-itated granular shear layer obeys approximately

ρu∗2crit ≡ −τ0crit ' ρsgdn

0.09 + 0.135(1− e2)0.4 ×

exp[−(Ar/Ar0)1/2]o, (B18)

where Ar0 ' 6000. Although this result overestimates bya factor of 2.7 the threshold shear velocity that Shao andLu [2000] correlated from empirical data, it predicts, for ex-ample, how Martian winds must be faster than Earth’s toreach the particle entrainment threshold.

This model also yields the average volume fraction ofgrains in the shear layer

ν =1

h`

Z h`

y=0

νdy, (B19)

in which we exploit equation (B2) to evaluate the integral,

ν = −[σy(y = h`)− σy0]/(ρsgh`). (B20)

The result is conveniently represented by

ν = ν∞ + (νc − ν∞) exp−[(τ0 − τ0crit)/τ0s]1/2 (B21)

with ν∞ ' 0.27, −τ0s/(ρsgd) ' 0.17, and τ0crit from equa-tion (B17). This shows how the granular shear layer be-comes less dense as its mobility increases.

Because it ignores the role of saltation and “splash” insetting agitation T0 and flux q at the free surface, this “toymodel” of the shear layer can only describe qualitatively theflow of reptating grains. Because its excludes grain inter-actions other than binary collisions, it likely fails near the

threshold. However, the model indicates that, because |τ0|peaks at crests, reptation is sustained at lower wind speedson crests compared to troughs. Its predictions for the al-titude ∼ T0/g to which grains are ejected above the freesurface also yields a simple closed-form expression of thesaltation mass flow rate [Louge et al., 2009b] that agreeswell with correlations [Sørensen, 2004].

Finally, it is instructive to relate our calculations of nor-mal stress at the free surface to the predictions of Creysselset al. [2009], who recently reported how saltation interactswith the sand bed at steady state. Upon balancing the gran-ular influx hitting the ground and the subsequent outflux ofejecta, these authors showed that saltating grains have agranular temperature aloft that is invariant with u∗ and isempirically set by dimensionless impact parameters of thesplash, which do not depend on wind strength: because fast-hitting grains produce more ejecta, while slower grains re-main trapped, the velocity variance of grains in saltationevolves until there is no net imbalance between influx andoutflux. Creyssels et al. [2009] predicted the volume fractionνsalt of grains in saltation

νsalt '1

µspexp

“− gh

Tsalt

” |τ0 − τ0th|ρsTsalt

, (B22)

where h ≡ −y is altitude, Tsalt is the (invariant) tempera-ture of saltating grains, τ0th is a threshold shear stress, andµsp is an effective bed friction coefficient ∼ 1 that they cal-culated from horizontal and vertical momentum balances onthe bed surface in terms of parameters of the splash. Thesimulations of Creyssels et al. [2009] also revealed that thevolume fraction of saltating grains, which is ν0, decaysexponentially with altitude as predicted by the collisionlessBoltzmann equation, and increases with wind speed. Us-ing equation (B2), the normal stress that saltating grainsimpose on the free surface is

−σy0 =

Z ∞h=0

ρsνsaltgdh '1

µsp|τ0 − τ0th|, (B23)

which is independent of Tsalt. Returning to the granu-lar shear layer, because St ≫ 1, there are two limitsfor T0, depending whether solids or gas turbulent dissipa-tion dominate. If εg εs, then T0/(gd) ≈ 12CT (1 +0.15Re0.687)/(CεCµAr) varies weakly with τ0, but stronglywith particle size through Ar. This limit arises at low solidvolume fraction; it is reminiscent of the nearly invariant tem-perature that Creyssels et al. [2009] observed in saltation, al-though these authors invoked splash, – rather than particle-turbulence interactions –, to evaluate it. Instead if εs εg,

then T0 ≈ |τ0|CT (1 − ν0)/(3ν0ρsC1/2µ ) grows linearly with

τ0, but is independent of Ar. Because ν0 ' 0.16 is relativelyhigh, this limit typically dominates in the shear layer. Inthis case, −σy0 = 2ν0ρsT0 ≈ 2CT (1 − ν0)/(3C

1/2µ )|τ0| '

0.8|τ0|, which compares well with equation (B23) derivedfrom Creyssels et al. [2009] at wind speeds sufficiently abovethreshold.

Acknowledgments. Louge conceived instrumentation andexperiments, which he conducted in Mauritania with Ould el-Moctar, Valance, and Dupont. Louge developed the theory withinput from Valance. The authors gratefully acknowledge sugges-tions of anonymous referees, which we incorporated in the sectionon ripple dynamics, as well as fruitful conversations with BrunoAndreotti, Olivier Dauchot, Renaud Delannay, Daniel Goldman,and Jasper Kok. They are indebted to Dah Ould Ahmedou andAhmedou Ould Mahfoudh for their support in the field, and toPatrick Chasles, Stephane Bourles, Herve Orain, Mikael Fontaineand Alain Faisant for technical help with instruments. This work

X - 10 LOUGE ET AL.: PACKING VARIATIONS ON SAND RIPPLES

was funded by the “Ministere de la Recherche” and “Agence Na-tionale de la Recherche” under grants ACI-500270 and ANR-05-BLAN-0273. MYL’s travel was supported by NSF grant INT-0233212.

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M. Y. Louge, Sibley School of Mechanical and AerospaceEngineering, Cornell University, Ithaca, NY 14853, USA.(MYL3@cornell.edu)

A. Valance, Institut de Physique de Rennes, CNRS UMR 6251,Universite de Rennes 1, Campus Beaulieu, Bat 11A, 35042 RennesCedex, France. (alexandre.valance@univ-rennes1.fr)

A. Ould el-Moctar, Ecole Polytechnique de l’Universite deNantes, Laboratoire de Thermocinetique, UMR CNRS 6607,44306 Nantes Cedex 3, France. (ahmed.ouldelmoctar@univ-nantes.fr)

P. Dupont, Institut National des Sciences Appliquees deRennes, 20, Avenue des Buttes de Coesmes, 35043 Rennes Cedex,France. (Pascal.Dupont@insa-rennes.fr)