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

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, which, by contrast to the former, do not show avalanche slip face. Their shape and velocity can be predicted as a function of their size. We recover in particular 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 stable dune exists. However, the linear stability analysis of a flat sand sheet shows that it is unstable at large wavelengths and suggests a mechanism of dune initiation.Comment: submitted to Eur. Phys. J. B, 13 pages, 17 figure

Comment on "Minimal size of a barchan dune"

It is now an accepted fact that the size at which dunes form from a flat sand bed as well as their `minimal size' scales on the flux saturation length. This length is by definition the relaxation length of the slowest mode toward equilibrium transport. The model presented by Parteli, Duran and Herrmann [Phys. Rev. E 75, 011301 (2007)] predicts that the saturation length decreases to zero as the inverse of the wind shear stress far from the threshold. We first show that their model is not self-consistent: even under large wind, the relaxation rate is limited by grain inertia and thus can not decrease to zero. A key argument presented by these authors comes from the discussion of the typical dune wavelength on Mars (650 m) on the basis of which they refute the scaling of the dune size with the drag length evidenced by Claudin and Andreotti [Earth Pla. Sci. Lett. 252, 30 (2006)]. They instead propose that Martian dunes, composed of large grains (500 micrometers), were formed in the past under very strong winds. We show that this saltating grain size, estimated from thermal diffusion measurements, is not reliable. Moreover, the microscopic photographs taken by the rovers on Martian aeolian bedforms show a grain size of 87 plus or minus 25 micrometers together with hematite spherules at millimetre scale. As those so-called ``blueberries'' can not be entrained by reasonable winds, we conclude that the saltating grains on Mars are the small ones, which gives a second strong argument against the model of Parteli et al.Comment: A six page comment on ``Minimal size of a barchan dune'' by Parteli, Duran and Herrmann [Phys. Rev. E 75, 011301 (2007) arXiv:0705.1778

Stress Propagation and Arching in Static Sandpiles

We present a new approach to the modelling of stress propagation in static granular media, focussing on the conical sandpile constructed from a point source. We view the medium as consisting of cohesionless hard particles held up by static frictional forces; these are subject to microscopic indeterminacy which corresponds macroscopically to the fact that the equations of stress continuity are incomplete -- no strain variable can be defined. We propose that in general the continuity equations should be closed by means of a constitutive relation (or relations) between different components of the (mesoscopically averaged) stress tensor. The primary constitutive relation relates radial and vertical shear and normal stresses (in two dimensions, this is all one needs). We argue that the constitutive relation(s) should be local, and should encode the construction history of the pile: this history determines the organization of the grains at a mesoscopic scale, and thereby the local relationship between stresses. To the accuracy of published experiments, the pattern of stresses beneath a pile shows a scaling between piles of different heights (RSF scaling) which severely limits the form the constitutive relation can take ...Comment: 38 pages, 24 Postscript figures, LATEX, minor misspellings corrected, Journal de Physique I, Ref. Nr. 6.1125, accepte

Force chain splitting in granular materials: a mechanism for large scale pseudo-elastic behaviour

We investigate both numerically and analytically the effect of strong disorder on the large scale properties of the hyperbolic equations for stresses proposed in \protect\cite{bcc,wcc}. The physical mechanism that we model is the local splitting of the force chains (the characteristics of the hyperbolic equation) by packing defects. In analogy with the theory of light diffusion in a turbid medium, we propose a Boltzmann-like equation to describe these processes. We show that, for isotropic packings, the resulting large scale effective equations for the stresses have exactly the same structure as those of an elastic body, despite the fact that no displacement field needs to be introduced at all. Correspondingly, the response function evolves from a two peak structure at short scales to a broad hump at large scales. We find, however, that the Poisson ratio is anomalously large and incompatible with classical elasticity theory that requires the reference state to be thermodynamically stable.Comment: 7 pages, 6 figures, An incorrect definition of the Poisson ratio in dimensions not equal to 3 was amended. The conclusions are unchange