Bruno Andreotti

Selection of dune shapes and velocities. Part 1: Dynamics of sand, wind and barchans

Abstract:Almost fifty years of investigations of barchan dunes morphology and dynamics is reviewed, with emphasis on the physical understanding of these objects. The characteristic quantities measured on the field (shape, size, velocity) and the physical problems they rise are presented. Then, we review the dynamical mechanisms explaining the formation and the propagation of dunes. In particular a complete and original approach of the sand transport over a flat sand bed is proposed and discussed. We conclude on open problems by outlining future research directions.

A scaling law for aeolian dunes on Mars, Venus, Earth, and for subaqueous ripples

Abstract The linear stability analysis of the equations governing the evolution of a flat sand bed submitted to a turbulent shear flow predicts that the wavelength λ at which the bed destabilises to form dunes should scale with the drag length L drag = ρ s ρ f d . This scaling law is tested using existing and new measurements performed in water (subaqueous ripples), in air (aeolian dunes and fresh snow dunes), in a high pressure CO2 wind tunnel reproducing conditions close to Venus atmosphere and in the low pressure CO2 Martian atmosphere (Martian dunes). The difficulty is to determine the diameter of saltating grains on Mars. A first estimate comes from photographs of aeolian ripples taken by the rovers Opportunity and Spirit, showing grains whose diameters are smaller than on Earth dunes. In addition we calculate the effect of cohesion and viscosity on the dynamic and static transport thresholds. It confirms that the small grains visualised by the rovers should be grains experiencing saltation. Finally, we show that, within error bars, the scaling of λ with Ldrag holds over almost five decades. We conclude with a discussion on the time scales and velocities at which these bed instabilities develop and propagate on Mars.

Field evidence for surface-wave-induced instability of sand dunes

Field studies of barchans—crescent-shaped dunes that propagate over solid ground under conditions of unidirectional wind—have long focused on the investigation of an equilibrium between sand transport by wind and the control of air flow by dune topography, which are thought to control dune morphology and kinematics. Because of the long timescale involved, however, the underlying dynamic processes responsible for the evolution of dune fields remain poorly understood. Here we combine data from a three-year field study in the Moroccan Sahara with a model study to show that barchans are fundamentally unstable and do not necessarily behave like stable solitary waves, as suggested previously. We find that dune collisions and changes in wind direction destabilize the dunes and generate surface waves on the barchans. Because the resulting surface waves propagate at a higher speed than the dunes themselves, they can produce a series of new barchans of elementary size by breaking the horns of large dunes. The creation of these new dunes provides a mechanism for sand loss that prevents dune fields from merging into a single giant dune and therefore plays a fundamental role in the control of size selection and the development of dune patterns.

Selection of dune shapes and velocities Part 2: A two-dimensional modelling

Abstract:We present in this paper a simplification of the dune model proposed by Sauermann et al. which keeps the basic mechanisms but allows analytical and parametric studies. Two kinds of purely propagative two dimensional solutions are exhibited: dunes and domes. The latter, by contrast to the former, do not present a slip face. Their shape and velocity can be predicted as a function of their size. We recover that dune profiles are not scale invariant (small dunes are flatter than the large ones), and that the inverse of the velocity grows almost linearly with the dune size. We furthermore get the existence of a critical mass below which no dune solution exists. It rises the problem of dune nucleation: how can dunes appear if any bump below this minimal mass gets eroded and disappears? The linear stability analysis of a flat sand bed shows that it is unstable at large wavelengths: dune can in fact nucleate from a small sand mass if the proto-dune is sufficiently long.

Giant aeolian dune size determined by the average depth of the atmospheric boundary layer

Depending on the wind regime, sand dunes exhibit linear, crescent-shaped or star-like forms resulting from the interaction between dune morphology and sand transport. Small-scale dunes form by destabilization of the sand bed with a wavelength (a few tens of metres) determined by the sand transport saturation length. The mechanisms controlling the formation of giant dunes, and in particular accounting for their typical time and length scales, have remained unknown. Using a combination of field measurements and aerodynamic calculations, we show here that the growth of aeolian giant dunes, ascribed to the nonlinear interaction between small-scale superimposed dunes, is limited by the confinement of the flow within the atmospheric boundary layer. Aeolian giant dunes and river dunes form by similar processes, with the thermal inversion layer that caps the convective boundary layer in the atmosphere acting analogously to the water surface in rivers. In both cases, the bed topography excites surface waves on the interface that in turn modify the near-bed flow velocity. This mechanism is a stabilizing process that prevents the scale of the pattern from coarsening beyond the resonant condition. Our results can explain the mean spacing of aeolian giant dunes ranging from 300u2009m in coastal terrestrial deserts to 3.5u2009km. We propose that our findings could serve as a starting point for the modelling of long-term evolution of desert landscapes under specific wind regimes.

Elastic deformation due to tangential capillary forces

A sessile liquid drop can deform the substrate on which it rests if the solid is sufficiently “soft.” In this paper we compute the detailed spatial structure of the capillary forces exerted by the drop on the solid substrate using a model based on Density Functional Theory. We show that, in addition to the normal forces, the drop exerts a previously unaccounted tangential force. The resultant effect on the solid is a pulling force near the contact line directed towards the interior of the drop, i.e., not along the interface. The resulting elastic deformations of the solid are worked out and illustrate the importance of the tangential forces

Drops on soft solids: free energy and double transition of contact angles

The equilibrium shape of liquid drops on elastic substrates is determined by minimizing elastic and capillary free energies, focusing on thick incompressible substrates. The problem is governed by three length scales: the size of the drop R, the molecular size a and the ratio of surface tension to elastic modulus γ/E. We show that the contact angles undergo two transitions upon changing the substrate from rigid to soft. The microscopic wetting angles deviate from Young’s law when γ/(Ea)≫1, while the apparent macroscopic angle only changes in the very soft limit γ/(ER)≫1. The elastic deformations are worked out for the simplifying case where the solid surface energy is assumed to be constant. The total free energy turns out to be lower on softer substrates, consistent with recent experiments.

Droplets move over viscoelastic substrates by surfing a ridge

Liquid drops on soft solids generate strong deformations below the contact line, resulting from a balance of capillary and elastic forces. The movement of these drops may cause strong, potentially singular dissipation in the soft solid. Here we show that a drop on a soft substrate moves by surfing a ridge: the initially flat solid surface is deformed into a sharp ridge whose orientation angle depends on the contact line velocity. We measure this angle for water on a silicone gel and develop a theory based on the substrate rheology. We quantitatively recover the dynamic contact angle and provide a mechanism for stick–slip motion when a drop is forced strongly: the contact line depins and slides down the wetting ridge, forming a new one after a transient. We anticipate that our theory will have implications in problems such as self-organization of cell tissues or the design of capillarity-based microrheometers.

Elasto-capillarity at the nanoscale: on the coupling between elasticity and surface energy in soft solids

The capillary forces exerted by liquid drops and bubbles on a soft solid are directly measured using molecular dynamics simulations. The force on the solid by the liquid near the contact line is neither oriented along the liquid vapor interface nor perpendicular to the solid surface, as usually assumed, but points towards the liquid. It is shown that the elastic deformations induced by this force can only be explained if, in contrast to an incompressible liquid, the surface stress is different from the surface energy. Using thermodynamic variations we show that the surface stress and the surface energy can both be determined accurately by measuring the deformation of a slender body plunged in a liquid. The results obtained from molecular dynamics fully confirm those recently obtained experimentally [Marchand et al., Phys. Rev. Lett., (2012), 108, 094301] for an elastomeric wire.

Evidence of rayleigh-hertz surface waves and shear stiffness anomaly in granular media.

Using the nonlinear dependence of sound propagation speed with pressure, we evidence the anomalous elastic softness of a granular packing in the vicinity of the jamming transition. Under gravity and close to a free surface, the acoustic propagation is only possible through surface modes guided by the stiffness gradient. These Rayleigh-Hertz modes are evidenced in a controlled laboratory experiment. The shape and the dispersion relation of both transverse and sagittal modes are compared to the prediction of nonlinear elasticity including finite size effects. These results allow one to access the elastic properties of the packing under vanishing confining pressure.