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# Direct numerical simulations of aeolian sand ripples

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

## Significance

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

## Abstract

Aeolian sand beds exhibit regular patterns of ripples resulting from the interaction between topography and sediment transport. Their characteristics have been so far related to reptation transport caused by the impacts on the ground of grains entrained by the wind into saltation. By means of direct numerical simulations of grains interacting with a wind flow, we show that the instability turns out to be driven by resonant grain trajectories, whose length is close to a ripple wavelength and whose splash leads to a mass displacement toward the ripple crests. The pattern selection results from a compromise between this destabilizing mechanism and a diffusive downslope transport which stabilizes small wavelengths. The initial wavelength is set by the ratio of the sediment flux and the erosion/deposition rate, a ratio which increases linearly with the wind velocity. We show that this scaling law, in agreement with experiments, originates from an interfacial layer 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 the propagation of these ripples as a proxy for remote measurements of sediment transport.

Observers have long recognized that wind ripples (1, 2) do not form via the same dynamical mechanism as dunes (3). Current explanations ascribe their emergence to a geometrical effect of solid angle acting on sediment transport. The motion of grains transported in saltation is composed of a series of asymmetric trajectories (4⇓⇓–7) during which they are accelerated by the wind. These grains, in turn, decelerate the airflow inside the transport layer (1, 7⇓⇓⇓⇓–12). On hitting the sand bed, they release a splash-like shower of ejected grains that make small hops from the point of impact (1, 13, 14). This process is called reptation. Previous wind ripple models assume that saltation is insensitive to the sand bed topography and forms a homogeneous rain of grains approaching the bed at a constant oblique angle (15⇓⇓⇓⇓–20). Upwind-sloping portions of the bed would then be submitted to a higher impacting flux than downslopes (1). With a number of ejecta proportional to the number of impacting grains, this effect would produce a screening instability with an emergent wavelength *λ* determined by the typical distance over which ejected grains are transported (15⇓–17), a few grain diameters *d*. However, observed sand 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 on a typical Earth sand dune (21, 22). Moreover, the screening scenario predicts a wavelength independent of the wind shear velocity *λ* with

## Model

To unravel the dynamical mechanisms resolving these discrepancies, we perform direct numerical simulations of a granular bed submitted to a turbulent shear flow. This flow is driven by a turbulent shear stress *g* and interact through contact forces. Based on the work presented in ref. 26, we explicitly implement a two-way coupling between a discrete element method for the particles and a continuum Reynolds averaged description of hydrodynamics, coarse-grained at a scale larger than the grain size. This coupling occurs by means of drag and Archimedes forces in the equations of motion of the grains and via a body force term in the Reynolds averaged Navier–Stokes equations (*SI Text*). This method enables us to perform runs over long periods of time using a large 2D spatial domain, while keeping the whole complexity of the granular phase (Movie S1). In particular, we do not have to introduce a splash function to describe reptation: the ejecta generated by a grain that collides with the sediment bed are directly obtained from the interaction of the particles with their neighbors in contact.

## Results

### Sand Ripple Instability.

Starting from a flat sediment bed, disturbed only by the randomness in the granular arrangement, one observes in the simulations the emergence and the propagation of ripples (Fig. 1, Fig. S1, and Movie S2). Tracking the grain trajectories, one can see that the saltation rain above the rippled bed is strongly modulated (Fig. 1*B*). As observed in experiments (Fig. S2), grains in saltation preferentially hit the bed upwind of the ripple crests. As ejected grains make small hops, the reptation flux tends to be enhanced on the windward side and reduced on the lee side. This results in a net transport from the troughs toward the crests that amplifies topographic disturbances, hence the instability. The spontaneous ripple pattern has a wavelength *λ* and a propagation speed *c*, both varying linearly with the imposed wind shear velocity (Fig. 2), in agreement with experimental observations (23⇓–25). Both *λ* and *c* are found to vanish when

To investigate quantitatively the dynamical mechanisms leading to the ripple instability, we have also performed simulations starting from a modulated bed whose topography follows a sinusoidal profile of given wavenumber *k* and of small initial amplitude *k*. As expected for a linear instability, the growth or the decay of the disturbance can be fitted to an exponential of the form *σ*. The resulting dispersion relation *A*): small wavenumbers (large wavelengths) are unstable (*σ*, coincides with the wavenumber

### Destabilizing Effect of Reptation.

We have determined the contribution of the grains in reptation to the growth rate by selecting the grains with a hop height smaller than *B*, we find that reptation has a destabilizing effect (*k*. The ratio *k* must then originate from a characteristic value of such a rate associated with reptation. To determine it, we have measured the vertical flux density profile *z* per unit time (*SI Text*).

The vertical flux density profile *A*), only a small fraction of the grains arriving from the upper transport layer truly impact the static bed: most of them actually bounce back before. Because of these collisions, the shear stress carried by the grains is transferred from a kinetic form, i.e., a flux of momentum associated with a particle flux, to a contact stress. As the typical grain velocity in this layer is set by *SI Text*) (Fig. 2*B*). This scaling law is found to hold even close to the threshold (Fig. S3*B*), which means that the ejection process and the redistribution of momentum in the interfacial layer must involve a large number of grains, even when splashing grains are well separated in time. This observation challenges the common view on aeolian sediment transport and calls for the seeking of collective effects in the splash process.

The grain hop length distribution *SI Text*) reflects the dynamical mechanisms dominating each of these transport layers. Although *ψ* decreases with ℓ as a stretched exponential in the upper layer, it behaves as a scale-free power law *d*). Grains ejected from the static layer inherit their scaling laws from the impacting grains. The scale-free behavior observed in the interfacial layer is another indication of the collective processes at work. As an important consequence, the erosion/deposition rate *σ* always takes the form *a* of order 1 (Fig. 3*B*).

### Stabilizing Effect of Saltation.

The contribution of grains in saltation (i.e., grains with a hop height larger than *σ* is found to be negative and quadratic in *k* (Fig. 3*B*). Varying *b* is of order 1 (Fig. 3*B*). Saltation therefore has a direct stabilizing effect consistent with a topographic diffusion, i.e., a downslope component of the saltation flux. As evidenced in experiments (31), the overall sand transport—not only reptation—is sensitive to the bed slope, with a diffusion coefficient proportional to the total sediment flux

The measurement of *SI Text*) shows a quadratic dependence on

### Dispersion Relation.

Summing up the contributions of reptation and saltation, the dispersion relation *a* and *b* of order 1. This constitutes a major difference with existing models, which predict that both the destabilizing and the stabilizing terms grow like *A*). Because *A*).

In the field, initial ripples develop toward a statistically steady pattern. Their wavelength eventually results from fluctuating wind conditions and from ripple nonlinear interactions. However, measurements show that the linear dependence of the wavelength on the wind velocity still holds for developed ripples (25). Furthermore, Martian ripple wavelength (in the range 2–6 m; Table S2 and Fig. S7) is

Experimental data and numerical simulations also agree on the scaling law obeyed by the propagation speed of ripples (Fig. 2*B*):*c*, and

## Discussion

The origin of the ripple instability can be understood as follows. As previously described, grains are eroded from the troughs and accumulate on the crests because of the modulation of the reptation flux, which is maximal on the windward side and minimal on the lee side (Fig. 4). In contrast with the geometric screening scenario, simulations and experiments show that the saltation rain is also strongly modulated (Fig. 4 and Fig. S2). The impacting flux is nonetheless larger on upwind slopes due to the solid angle effect but also because of a stochastic focusing of the trajectories. Indeed, the rate at which grains leave the bed is modulated similarly to the impacting rate: if there are more grains arriving at the upwind slope inflection point, there are also more grains departing from that position. The key point is that these grains must statistically be the same: the modulation of sediment transport is essentially due to grains departing from the inflection point and arriving at the following inflection point located downwind. The consequence for the ripple emergence is fundamental: the modulation of the reptation flux is driven by resonant trajectories, those whose hop length is comparable to the wavelength *λ*. Because processes at work in the interfacial layer are scale-free, the reptating grains that contribute the most to the instability have hop lengths proportional to *λ* and not to *d*: they inherit their characteristic scale from that of saltating grains. The new picture proposed here therefore substantially changes the paradigm of aeolian transport.

## Acknowledgments

We thank O. Pouliquen and H. Elbelrhiti for discussions 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 and useful comments. This study was supported by an Agence Nationale de la Recherche Zephyr Grant.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: oduran{at}marum.de.

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

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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

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