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

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. Received 22 December 2001 / Received in final form 31 May 2002 Published online 31 July 2002     

Non-linear dynamics of spinodal decomposition

We develop a new technique describing the non linear growth of interfaces. We apply this analytical approach to the one dimensional Cahn-Hilliard equation. The dynamics is captured through a solvability condition performed over a particular family of quasi-static solutions. The main result is that the dynamics along this particular class of solutions can be expressed in terms of a simple ordinary differential equation. The density profile of the stationary regime found at the end of the non-linear growth is also well characterized. Numerical simulations are compared in a satisfactory way with the analytical results through three different fitting methods and asymptotic dynamics are well recovered, even far from the region where the approximations hold. Received 16 October 2001 / Received in final form 15 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: josseran@lmm.jussieu.fr RID="b" ID="b"UMR CNRS 7607     

Spike autosolitons and pattern formation scenarios in the two-dimensional Gray-Scott model

We performed an extensive numerical study of pattern formation scenarios in the two-dimensional Gray-Scott reaction-diffusion model. We concentrated on the parameter region in which there exists a strong separation of length and/or time scales. We found that the static one-dimensional autosolitons (stripes) break up into two-dimensional radially-symmetric autosolitons (spots). The traveling one-dimensional autosolitons (wave fronts) can be stable or undergo breakup. The static two-dimensional radially-symmetric autosolitons may break up and self-replicate leading to the formation of space-filling patterns of spots, wave fronts, or spatio-temporal chaos due to the competition of self-replication and annihilation of spots upon collision. Received 6 November 2000 and Received in final form 27 February 2001     

Pulses of tunable size near a subcritical bifurcation

We show that a nonlinear gradient term can be used to tune the width of pulse-like solutions to a generalized quintic Ginzburg-Landau equation. We investigate the dynamics of these solutions and show that weakly turbulent patches can persist for long times. Analogies with turbulent spots in plane Couette flows are discussed. Received: 29 May 1997 / Revised: 16 March 1998 / Accepted: 30 April 1998     

Self-organized branching process for a one-dimensional rice-pile model

A self-organized branching process is introduced to describe one-dimensional rice-pile model with stochastic topplings. Although the branching processes are generally expected to describe well high-dimensional systems, our modification highlights some of the peculiarities present in one dimension. We find analytically that the crossover behavior from the trivial one-dimensional BTW behaviour to self-organized criticality is characterised by a power-law distribution of avalanches. The finite-size effects, which are crucial to the crossover, are calculated. Received 21 June 2001 and Received in final form 14 November 2001     

Fragmentation dynamics within shear bands--a model for aging tectonic faults?

A numerical model for packing of fragmenting blocks in a shear band is introduced, and its dynamics is compared with that of a tectonic fault. The shear band undergoes a slow aging process in which the blocks are being grinded by the shear motion and the compression. The dynamics of the model have the same statistical characteristics as the seismic activity in faults. The characteristic magnitude distribution of earthquakes appears to result from frictional slips at small and medium magnitudes, and from fragmentation of blocks at the largest magnitudes. Aftershocks to large-magnitude earthquakes are local recombinations of the fragments before they reach a new quasi-static equilibrium. The aftershocks satisfy Omori's law. Local precursor activity at a few times the normal background level appears at a short time before a major earthquake. Seismic gaps appear as a natural consequence of the aging process of a fault. Explanation of the heat flux and principal stress direction anomalies at the faults both involve the value of fracture stress of the blocks in the gouge. The final form of a tectonic fault is predicted to involve a gouge dominated by fine-grained and rather rounded blocks so that it cannot withstand large shear stresses. Received 26 July 2000     

Granular flow from a silo: Discrete-particle simulations in three dimensions

Molecular (or granular) dynamics methods are used to study the gravity-driven flow of granular material through a horizontal aperture in three dimensions. The grains are spherical and modeled using a short-range repulsive interaction, together with normal and tangential frictional damping forces. The material is contained in a rough-walled cylindrical container with a circular hole in its base, and to permit flow measurements under steady-state conditions a continuous feed approach is employed in which exiting grains are replaced at the upper surface of the material. The dependence of flow velocity and discharge rate on aperture diameter is found to agree with experiment; other quantities such as the kinetic energy and pressure distributions are also examined. Received 5 June 2000 and Received in final form 21 September 2000     

Electric Nusselt number characterization of electroconvection in nematic liquid crystals

We develop a characterization method of electroconvection structures in a planar nematic liquid crystal layer by a study of the electric current transport. Because the applied potential difference has a sinusoidal time dependence, we define two electric Nusselt numbers corresponding to the in-phase and out-of-phase components of the current. These Nusselt numbers are predicted theoretically using a weakly nonlinear analysis of the standard model. Our measurements of the electric current confirm that both numbers vary linearly with the distance from onset until the occurence of secondary transitions. A systematic comparison between our theoretical and experimental results, using no adjusted parameters, demonstrates moderate agreement, but discrepancies remain. Electric transport measurements during electroconvection represent a quantitative test of the standard model completely independent from optical probes. Thus, the technique described here can be a useful complement to traditional structural measurements. Received 21 January 2001 and Received in final form 1st February 2002     

Initiation and evolution of current ripples on a flat sand bed under turbulent water flow

We investigate the formation and dynamics of sand ripples under a turbulent water flow. Our experiments were conducted in an open flume with spherical glass beads between 100 and 500μm in diameter. The flow Reynolds number is of the order of 10 000 and the particle Reynolds number of the order of 1 to 10. We study the development of ripples by measuring their wavelength and amplitude in course of time and investigate the influence of the grain size and the flow properties. In particular, we demonstrate two different regimes according to the grain size. For fine grains, a slow coarsening process (i.e., a logarithmic increase of the wavelength and amplitude) takes place, while for coarser grains, this process occurs at a much faster rate (i.e., with a linear growth) and stops after a finite time. In the later case, a stable pattern is eventually observed. Besides, we carefully analyze the wavelength of ripples in the first stages of the instability as a function of the grain size and the shear velocity of the flow, and compare our results with other available experimental data and with theoretical predictions based on linear stability analyses.     

Random inelasticity and velocity fluctuations in a driven granular gas

We analyze the deviations from Maxwell-Boltzmann statistics found in recent experiments studying velocity distributions in two-dimensional granular gases driven into a non-equilibrium stationary state by a strong vertical vibration. We show that in its simplest version, the “stochastic thermostat” model of heated inelastic hard spheres, contrary to what has been hitherto stated, is incompatible with the experimental data, although predicting a reminiscent high-velocity stretched-exponential behavior with an exponent 3/2. The experimental observations lead to refine a recently proposed random restitution coefficient model. Very good agreement is then found with experimental velocity distributions within this framework, which appears self-consistent and further provides relevant probes to investigate the universality of the velocity statistics. Received: 27 May 2002 / Accepted: 6 May 2003 / Published online: 27 May 2003 RID="a" ID="a"e-mail: Alain.Barrat@th.u-psud.fr     

Heated granular fluids: The random restitution coefficient approach

We introduce the model of inelastic hard spheres with random restitution coefficient α, in order to account for the fact that, in a vertically shaken granular system interacting elastically with the vibrating boundary, the energy injected vertically is transferred to the horizontal degrees of freedom through collisions only, which leads to heating through collisions, i.e. to inelastic horizontal collisions with an effective restitution coefficient that can be larger than 1. This allows the system to reach a non-equilibrium steady state, where we focus, in particular, on the single-particle velocity distribution f (v) in the horizontal plane, and on its deviation from a Maxwellian. Molecular Dynamics simulations and Direct Simulation Monte Carlo (DSMC) show that, depending on the distribution of α, different shapes of f (v) can be obtained, with very different high-energy tails. Moreover, the fourth cumulant of the velocity distribution quantifying the deviations from Gaussian statistics is obtained analytically from the Boltzmann equation and successfully tested against the simulations. Received 24 November 2000 and Received in final form 8 February 2001     

Spatio-temporal patterns in the thermoconvection of a planar nematic layer: II. Experiments

We study experimentally the evolution of thermoconvection in a laterally extended planar nematic layer, at zero or weak stabilizing magnetic field. As the applied thermal gradient is increased, a cascade of symmetry breakings occurs, towards structures of increasing spatial complexity, and ultimately towards oscillating states. The patterns are characterized optically, and simple models for the distortion of the vertical (out of plane) component of the director field are proposed. Received: 1st December 1997 / Revised: 25 May 1998 / Accepted: 2 June 1998     

Dynamics of spatio-temporal defects in the Taylor-Dean system

The dynamics of defects in a pattern of traveling inclined rolls has been investigated. Two regimes were identified in the neighborhood of defects: a diffusive regime, with a negative phase diffusion coefficient, and a coalescence regime in which the phase gradient diverges in time following a power law behavior. The observed periodic nucleation of defects is related to the frequency inhomogeneity induced by the disymmetry of the wave amplitude. Amplitude holes have been observed in the secondary modulated pattern. Received 23 February 1999 and Received in final form 26 May 1999     

Super lattice formation of an array of volatile wetting droplets

For an ordered array of critical volatile wetting droplets the formation of a super lattice by an Ostwald-ripening-like competition process is considered. The underlying diffusion problem is treated within a quasistatic approximation and to first order in the inverse droplets distance. The approach is rather general but a square lattice and a triangular lattice are studied explicitly. Dispersion relations for the super lattice growth of these arrays are calculated. Received 29 November 1999 and Received in final form 15 February 2000     

Nonlinear competition between asters and stripes in filament-motor systems

A model for polar filaments interacting via molecular motor complexes is investigated which exhibits bifurcations to spatial patterns. It is shown that the homogeneous distribution of filaments, such as actin or microtubules, may become either unstable with respect to an orientational instability of a finite wave number or with respect to modulations of the filament density, where long-wavelength modes are amplified as well. Above threshold nonlinear interactions select either stripe patterns or periodic asters. The existence and stability ranges of each pattern close to threshold are predicted in terms of a weakly nonlinear perturbation analysis, which is confirmed by numerical simulations of the basic model equations. The two relevant parameters determining the bifurcation scenario of the model can be related to the concentrations of the active molecular motors and of the filaments, respectively, which both could be easily regulated by the cell.     

Strength and failure of cemented granular matter

Cemented granular materials (CGMs) consist of densely packed solid particles and a pore-filling solid matrix sticking to the particles. We use a sub-particle lattice discretization method to investigate the particle-scale origins of strength and failure properties of CGMs. We show that jamming of the particles leads to highly inhomogeneous stress fields. The stress probability density functions are increasingly wider for a decreasing matrix volume fraction, the stresses being more and more concentrated in the interparticle contact zones with an exponential distribution as in cohesionless granular media. Under uniaxial loading, pronounced asymmetry can occur between tension and compression both in strength and in the initial stiffness as a result of the presence of bare contacts (with no matrix interposed) between the particles. Damage growth is analyzed by considering the evolution of stiffness degradation and the number of broken bonds in the particle phase. A brutal degradation appears in tension as a consequence of brittle fracture in contrast to the more progressive nature of damage growth in compression. We also carry out a detailed parametric study in order to assess the combined influence of the matrix volume fraction and particle-matrix adherence. Three regimes of crack propagation can be distinguished corresponding to no particle damage, particle abrasion and particle fragmentation, respectively. We find that particle damage scales well with the relative toughness of the particle-matrix interface with respect to the particle toughness. This relative toughness is a function of both matrix volume fraction and particle-matrix adherence and it appears therefore to be the unique control parameter governing transition from soft to hard behavior.     

Stress fluctuations and macroscopic stick-slip in granular materials

This paper deals with the quasi-static regime of deformation of granular matter. It investigates the size of the Representative Elementary Volume (REV), which is the minimum packing size above which the macroscopic mechanical behaviour of granular materials can be defined from averaging. The first part uses typical results from recent literature and finds that the minimum REV contains in general 10 grains; this result holds true either for most experiments or for Discrete Element Method (DEM) simulation. This appears to be quite small. However, the second part gives a counterexample, which has been found when investigating uniaxial compression of glass spheres which exhibit stick-slip; we show in this case that the minimum REV becomes 10⁷ grains. This makes the system not computable by DEM. Moreover, similarity between the Richter law of seism and the exponential statistics of stick-slip is stressed. Received 8 March 2002 and Received in final form 12 July 2002