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The collision of a spherical grain with a granular bed is commonly parametrized by the splash function, which provides the velocity of the rebounding grain and the velocity distribution and number of ejected grains. Starting from elementary geometric considerations and physical principles, like momentum conservation and energy dissipation in inelastic pair collisions, we derive a rebound parametrization for the collision of a spherical grain with a granular bed. Combined with a recently proposed energy-splitting model [Ho et al., Phys. Rev. E 85, 052301 (2012)PLEEE81539-375510.1103/PhysRevE.85.052301] that predicts how the impact energy is distributed among the bed grains, this yields a coarse-grained but complete characterization of the splash as a function of the impact velocity and the impactor-bed grain-size ratio. The predicted mean values of the rebound angle, total and vertical restitution, ejection speed, and number of ejected grains are in excellent agreement with experimental literature data and with our own discrete-element computer simulations. We extract a set of analytical asymptotic relations for shallow impact geometries, which can readily be used in coarse-grained analytical modeling or computer simulations of geophysical particle-laden flows.

Starting from the physics on the grain scale, we develop a simple continuum description of aeolian sand transport. Beyond popular mean-field models, but without sacrificing their computational efficiency, it accounts for both dominant grain populations, hopping (or 'saltating') and creeping (or 'reptating') grains. The predicted stationary sand transport rate is in excellent agreement with wind tunnel experiments simulating wind conditions ranging from the onset of saltation to storms. Our closed set of equations thus provides an analytically tractable, numerically precise and computationally efficient starting point for applications addressing a wealth of phenomena from dune formation to dust emission.A drawback of the two-species description has been that it is still conceptually and computationally quite demanding. For reasons of simplicity and computational efficiency, many theoretical studies have therefore chosen to reduce the mathematical description even further, to mean-field-type 'single-trajectory' models [6]. This may be justified for certain purposes, e.g. for the mathematical modelling of aeolian sand dunes, which are orders of magnitude larger than the characteristic length scales involved in the saltation process and thus not expected to be very sensitive to the details on the grain scale. Their formation and migration is thought to predominantly depend on large-scale features of the wind field and of the saltation flux, chiefly the symmetry breaking of the turbulent flow over the dune and the delayed reaction of the sand transport to the wind [7]. Moreover, the reptating grains, although they are many, are generally thought to contribute less to the overall sand transport, because they have short trajectories and quickly get trapped in the bed again. Therefore, it seems admissible to concentrate on the saltating particles. On this basis, numerically efficient models for one effective grain species that can largely be identified with the saltating grain fraction have been constructed. The Sauermann model [8] is a popular and widely used example of such mean-field continuum models. Onespecies models have become a very successful means of gaining analytical insight into [9][10][11][12][13][14] and to conduct efficient large-scale numerical simulations of [15][16][17][18][19] the complex structure formation processes caused by aeolian transport. The reduction to a single representative trajectory makes the one-species models analytically tractable and computationally efficient. However, it is also responsible for some weaknesses concerning both the way in which the two species are subsumed into one [5] and how they feed back onto the wind [12]. These entail imperfections in the model predictions, most noticeably a systematic overestimation of the stationary flux at high wind speed (see figure 5). Therefore, the one-species models have been criticized for their lack of numerical accuracy and internal consistency [5,20]. There is also a number of interesting phenomena that cannot be quantitatively mode...

The mesoscale structure of aeolian sand transport determines a variety of natural phenomena studied in planetary and Earth science. We analyze it theoretically beyond the mean-field level, based on the grain-scale transport kinetics and splash statistics. A coarse-grained analytical model is proposed and verified by numerical simulations resolving individual grain trajectories. The predicted height-resolved sand flux and other important characteristics of the aeolian transport layer agree remarkably well with a comprehensive compilation of field and wind tunnel data, suggesting that the model robustly captures the essential mesoscale physics. By comparing the saturation length with field data for the minimum sand-dune size, we can reconcile conflicting previous models for this most enigmatic emergent aeolian scale and elucidate the importance of intermittent turbulent wind fluctuations for field measurements.

The emergence of order from disorder is a topic of vital interest. We here propose that long-range order can arise from a randomly arranged two-phase material under mechanical load. Using Small-Angle Neutron Scattering (SANS) experiments and Molecular Dynamics based finite element (FE) models we show evidence for stress-induced ordering in spider dragline silk. Both methods show striking quantitative agreement of the position, shift and intensity increase of the long period upon stretching. We demonstrate that mesoscopic ordering does not originate from silk-specific processes such as strain-induced crystallization on the atomistic scale or the alignment of tilted crystallites. It instead is a general phenomenon arising from a non-affine deformation that enhances density fluctuations of the stiff and soft phases along the direction of stress. Our results suggest long-range ordering, analogously to the coalescence of defects in materials, as a wide-spread phenomenon to be exploited for tuning the mechanical properties of many hybrid stiff and soft materials.