We derive a phenomenological continuum saltation model for aeolian sand transport that can serve as an efficient tool for geomorphological applications. The coupled differential equations for the average density and velocity of sand in the saltation layer reproduce both the known equilibrium relations for the sand flux and the time evolution of the sand flux as predicted by microscopic saltation models. The three phenomenological parameters of the model are a reference height for the grain-air interaction, an effective restitution coefficient for the grain-bed interaction, and a multiplication factor characterizing the chain reaction caused by the impacts leading to a typical time or length scale of the saturation transients. We determine the values of these parameters by comparing our model with wind tunnel measurements. Our main interest are out of equilibrium situations where saturation transients are important, for instance at phase boundaries (ground/sand) or under unsteady wind conditions. We point out that saturation transients are indispensable for a proper description of sand flux over structured terrain, by applying the model to the windward side of an isolated dune, thereby resolving recently reported discrepancies between field measurements and theoretical predictions.

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We present a minimal model for the formation and migration of aeolian sand dunes in unidirectional winds. It combines a perturbative description of the turbulent wind velocity field above the dune with a continuum saltation model that allows for saturation transients in the sand flux. The latter are shown to provide a characteristic length scale, called saturation length, which is distinct from the saltation length of the grains. The model admits two different classes of solutions for the steady-state profile along the wind direction: smooth heaps and dunes with slip face. We clarify the origin of the characteristic properties of these solutions and analyze their scaling behavior. We also investigate in some detail the dynamic evolution of heaps and dunes, including the steady-state migration velocity and transient shape relaxation. Although the minimal model employs nonlocal expressions for the wind shear stress as well as for the sand flux, it is simple enough to serve as a very efficient tool for analytical and numerical investigations and opens up the way to simulations of large scale desert topographies.

We have investigated the viscosity and the plateau modulus of actin solutions with a magnetically driven rotating disk rheometer. For entangled solutions we observed a scaling of the plateau modulus versus concentration with a power of 7͞5. The measured terminal relaxation time increases with a power 3͞2 as a function of polymer length. We interpret the entanglement transition and the scaling of the plateau modulus in terms of the tube model for semiflexible polymers. [S0031-9007(98)07135-X] PACS numbers: 87.15. Da, 61.25.Hq, 83.50.Fc Networks of semiflexible macromolecules are major constituents of biological tissue. There is experimental evidence [1-4] that certain aspects of biologically important macromolecules, such as DNA and actin, are well described by the minimal theoretical model of a semiflexible macromolecule, also known as the wormlike chain model. This model represents the polymer as a smooth inextensible contour with an energy cost for bending and includes ideal flexible chains as a limiting case. The bending modulus of the single molecule can be expected to be constitutive also for the collective mechanical properties of gels and sufficiently concentrated solutions of semiflexible polymers. (Recently, possible contributions from twist have also been discussed [5].) However, very little is known about how semiflexible polymers build up statistical networks and how the macroscopic stresses and strains are mediated to the single molecules in such networks. This is also known as the entanglement problem. In this Letter, we report on experiments performed with a magnetically driven rotating disk rheometer, which elucidate some important aspects of the entanglement problem. The systems under scrutiny are in vitro polymerized actin solutions of various concentrations c and average polymer lengths L. Actin [6] forms large semiflexible polymers with a persistence length ᐉ p of about 17 mm [7,8] (comparable to typical filament lengths in our experiments) and is the most abundant cytoskeletal element in most eucariotic cells. We have analyzed the transition from the dilute to the semidilute phase (the entanglement transition) as a function of polymer length and concentration. The data can be interpreted in terms of a virial expansion for effective "tubes." For entangled solutions we observed a scaling of the plateau modulus G 0 versus actin concentration c. This is compared with various theoretical predictions [9][10][11][12][13][14]. Lastly, we analyzed the dependence of the zero shear rate viscosity on polymer length, which exhibits a much weaker length dependence than one would expect theoretically from work by Odijk [9] and Doi [15].Actin was prepared as previously described [16] and purified in a second step using gel column chromatog-raphy (Sephacryl S-300). Monomeric actin (called G-actin) was kept in G-buffer, consisting of 2 mM Imidazol (pH 7.4), 0.2 mM CaCl 2 , 0.2 mM DTT, 0.5 mM ATP, and 0.005 vol % NaN 3 . Polymerization was initiated by adding 1͞10 of the sample volume of 10-fold concentrated...

We propose a minimal model for aeolian sand dunes. It combines an analytical description of the turbulent wind velocity field above the dune with a continuum saltation model that allows for saturation transients in the sand flux. The model provides a qualitative understanding of important features of real dunes, such as their longitudinal shape and aspect ratio, the formation of a slip face, the breaking of scale invariance, and the existence of a minimum dune size.Sand dunes develop wherever sand is exposed to an agitating medium (air, water . . . ) that lifts grains from the ground and entrains them into a surface flow. The diverse conditions of wind and of sand supply in different regions on Earth give rise to a large variety of shapes of aeolian dunes [1][2][3]. Moreover, dunes have been found on the sea-bottom and even on Mars. Despite the long history of the subject, the underlying physical mechanisms of dune formation are still not very well understood. How are aerodynamics (hydrodynamics) and the particular properties of granular matter acting together to create dunes? How is the shape of a dune maintained when it moves? In the following we propose a "minimal model" for aeolian sand dunes to address such questions. Although it refers only to rather generic properties of the wind velocity field and the laws of aeolian sand transport, it can make interesting qualitative predictions that are not sensitive to the simplifying assumptions, e.g. about the surface profile, the development and position of the slip face, dune migration etc. Using results from turbulent boundary layer calculations [4-6], we will propose an approximate analytical description of the surface shear stress exerted by the wind onto a heap of sand. This will be combined with a saltation model [7] that allows for saturation transients of the sand flux, which are an essential element of a consistent description of dunes.We start by summarizing some basic knowledge about aeolian sand transport and saltation. The mean turbulent wind velocity above a plane surface increases logarithmically with height. Its magnitude is specified by a characteristic velocity called the "shear velocity" u * and defined by u 2 * ≡ τ 0 /ρ with τ 0 the average surface shear stress (far away from any obstacle) and ρ the density of air. On a surface covered with sand, the wind entrains some grains into a surface layer flow if the shear velocity exceeds a threshold value. The grains advance mainly by an irregular hopping process, thereby reducing the wind velocity in the surface layer. A unique relation between the shear stress τ and the sand flux q is thus established in the equilibrium state. If τ is not too close to the threshold, one has approximately [1, 2,8,9] q s ∝ τ 3/2 .(1)The index s emphasizes that this simple relation is restricted to situations where the flux is saturated. According to Eq.(1), the changing wind shear stress above a heap of sand h(x, y) is responsible for flux gradients, which cause erosion and deposition. Due to mass conservation, the flu...