On the crescentic shape of barchan dunes

Aeolian sand dunes originate from wind flow and sand bed interactions. According to wind properties and sand availability, they can adopt different shapes, ranging from huge motion-less star dunes to small and mobile barchan dunes. The latter are crescentic and emerge under a unidirectional wind, with a low sand supply. Here, a 3d model for barchan based on existing 2d model is proposed. After describing the intrinsic issues of 3d modeling, we show that the deflection of particules in reptation due to the shape of the dune leads to a lateral sand flux deflection, which takes the mathematical form of a non-linear diffusive process. This simple and physically meaningful coupling method is used to understand the shape of barchan dunes. PACS. 45.70.-n Granular systems – 47.54.+r Pattern selection; pattern formation 1 Characteristics of barchan dunes R.A. Bagnold opened the way to the physics of dunes with his famous book: The physics of blown sand and desert dunes [1]. From then on, a great deal of investigations – laboratory experiments [1–3], field measurements [4–14] and numerical computations [15–23] – have been conducted by geologists and physicists. In particular, a large amount of work has been dedicated to the barchan, a dune shaped by the erosion of a unidirectional wind on a firm ground. A side view of a barchan – see Figure 1 – shows a rather flat aerodynamic structure. When viewed from above, a barchan presents a crescentic shape with two horns pointing downwind. A sharp edge – the brink line – divides the dune in two areas: the windward side and the slip face, where avalanches develop – see Figure 2. Because of a boundary layer separation along this sharp edge [1,4,5], a large eddy develops downwind and wind speed decreases dramatically. Therefore, the incoming blown sand is dropped close to the brink line. That is why the barchan is known to be a very good sand trapper. Sometimes, when the drift of sand is too large, an avalanche occurs and grains are moved down the slip face. In short, grains are dragged by the wind from the windward side of the dune to the bottom of the slipface and, grain after grain, the dune moves. Field observations show that barchans can move up to 70 m/year [14]. Their speed is dependent on wind power and on their size: for the same wind strength, the velocity a e-mail: [email protected] Fig. 1. Side view of a barchan dune. The main properties of barchan dunes are outlined: two horns pointing downwind, the slip-face and the flat main body. The barchan shown is approximately 20 meters long and wide, and 2 meters high. The slip-face angle is roughly 30◦ to the vertical, which corresponds to the angle of repose of a sand-pile. Fig. 2. Barchan dune properties. Grains follow the wind direction, and sand flux is not strongly deflected by the dune relief. As observed in the field, sand grains can escape from the horns, but not from the main dune body. Instead, they are trapped into the slip-face. This difference of behavior between the main body and the horns is the key to understand the barchans. 508 The European Physical Journal B of barchans is roughly inversely proportional to their heights [1,2,4,10,12]. Barchan dimensions range from 1 to 30 meters high, and from 10 to 300 meters long and wide [1,4,5]. However, large barchans are often unstable, leading to complex structures called mega-barchans [4,5]. More accurate analyses reveal that the height, width and length are related by linear relationships [2,13] and that no mature barchan dune smaller than one meter high can be found: there is a minimal barchan size. Finally, the last important characteristic of barchans, but much less well-documented, is the sand leak at the tip of the horns [1,4,5,24], where no recirculation bubble develops. This shows that barchan dunes are three dimensional structures, whose center part and horns have totally different trapping efficiency – see Figure 2. Because of the inherent difficulties of field-work (a typical mission duration is rather short compared to the lifetime of a dune and its shape movement), numerical modeling is, alongside laboratory experiment [3], an important method to explore barchan properties such as their morphogenesis, their stability and their interaction modes, which are still poorly understood. Obviously, these problems are related to the original structure of barchans, and, accordingly, they should be studied with a 3d approach. The aim of the present paper is to discuss how to extend existing 2d models to the 3d situation. Then, we will show that the crescentic shape can be explained by the existence of a sand-flux constraint and a lateral sand-flux. In the following part, we start by recalling briefly the main features of 2d modeling of dunes . Then a model for the 3d situation will be presented. 2 The Cc class of models Numerical modeling of dune requires a description of the effect of wind flow over a sand-bed. Obviously, computing exact turbulent numerical solution starting with the Navier-Stokes equations is possible, but this would take a very long time. An alternative is to use the Cc model [23], based on the following approach. 2.1 Numerical model for the 2d case Let us call h(x, t) the sand bed profile and q(x, t) the vertically integrated volumic sand-flux. They are linked by the mass conservation: