On the crescentic shape of barchan dune

Aeolian desert sand dunes birth from wind flow and sand bed interactions: depending on wind properties and sand availability, they can adopt several different shapes from huge motionless star dunes to small and fast mobile barchan dunes. These latter have a characteristic crescentic shape and form on hard ground with low sand supply under a monodirectional wind . Here, a 3d barchan model based on existing 2d model is proposed. After describing the intrinsic difficulties of 2d model due to the barchan original 3d structure, we show that taking into account the reptating grains leads naturally to a lateral slice coupling, which takes the mathematical form of a diffusive process. This simple and physically meaningful coupling allow us to understand the formation of the crescentic shape of barchan dunes. PACS. 45.70.-n Granular systems – 47.54.+r Pattern selection; pattern formation 1 Properties of Barchan dunes Studies about sand dunes have began fifty years ago with R. A. Bagnold and his so-comprehensive book : the physics of wind blown sand and desert dunes [1]. From then on, many studies, both experimental, on the field, [2,3,4,5,6,7,8,9,10,11,12] and numerical [13,14,15,16,17,18,19,20,21,22,23,24,25], have been conducted with combined efforts of geologists and physicists. On that account, dunes understanding has been greatly improved. In particular, a great amount of work has been dedicated to the barchan dune : a crescenticshaped dune which forms with small sand supply under monodirectionnal wind. Fig. 1. Barchan dune side view. Main properties of barchan dunes are outlined : two horns pointing downwind, presence of a slipface and flatness of the main body. This barchan is approximately 30 meters long and width, while it is 3 meters high. The slipface angle is roughly 30 to the vertical, which is the angle of equilibrium of a sand-pile. Therefore, even though field measurements are difficult, the main properties of barchan dunes can be outlined. First of all, its main part is almost flat with an aspect ratio of roughly 1/10. It has a crescentic shape, its two horns pointing downwind a typical barchan is depicted on the Fig. 1. The inner sandless area is the trace of a recirculation bubble, due to boundary layer separation near the dune crest[1,2,3]. This recirculation bubble force the sand to be deposited near the crest, and consequently lead to the formation of a slip face on the downwind side. About barchans morphology, their dimensions range from 1 to 30 meters high, and 10 to 300 meters long and wide [1,2,3,26]. Larger barchan dunes exist but instabilities often arise on their windward side, leading to more complicated shapes called mega-barchan[2,3]. Field observations reveal that height, width and length of barchan dunes are linked by linear relationships [11,26,20], and that no mature barchan dune smaller than one meter high can be found in deserts. This last point reveals the existence of a minimal size for barchan dune. Secondly, Barchans may be very mobile and their speed is strongly dependent on wind power and on their size: geologist field observations show that they can move up to 70 meters a year[12]. Their motion is intrinsically linked with sand grains motion : when wind blows, sand grains are dragged by airflow, and are eventually deposited near the crest. If the deposit is too large, an avalanche occurs, and grains are moved down the slip face. Therefore, one can say that the dune is a very good sand trapper. Hence, grains from the windward side of the dune pass to the bottom of the slip face, and the dune moves. More accurately, for the same average wind strength, barchans dunes velocities are roughly inversely proportional to their heights[1,2,8,10,26,28]. 2 P. Hersen: On the crescentic shape of barchan dune Finally, the last important characteristic of barchan dunes, but the less documented, is that some sand can escape from the horns [1,2,3,29], where no recirculation eddy develops. Thereby, barchan dunes is a true three dimensional structure, where the center part and the border have totally different properties. Despite this great amount of work, there are still many unsolved fundamental issues on the physics of barchan dunes. As a matter of fact, the genesis of barchan dunes on sea-shore, their stability and their solitons-like behavior are poorly understood, as their dynamical shape-response to wind variations. Because, of the inherent difficulties of field experiments, (just think about the timescales and lengthscales involved), numerical modeling is, aside from laboratory experiment[27], a proper way to search for barchan dunes properties. Fig. 2. Barchan dune properties. Grains follow wind direction, and sand flux is not too much deviated 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 slipface. This difference of behavior between the main body and the horns is the key to understand the 3d structure of barchan. As a matter of fact, recent 2d models have already given excellent results for transversal dunes. However, it is unsuitable to keep studying barchan from a 2d model : barchans are 3d structures, where the differences between the main body property sand trap property and the horns are crucial. The aim of this paper is to discuss how to extend conveniently 2d Cc dune-maker model to a 3d situation, and then to show that the crescentic shape can be simply explained by looking at sand flux constraint and lateral sand flux redistribution. Hence, it is useful to start by recalling briefly main features of the Cc model. 2 The Cc class of model 2d models are based on the following idea : Taking an initial 2d profile for the sand bed, the induced perturbation of the fluid velocity field is computed. Then the rate of erosion is deduced and finally the sand bed is updated, and so forth. The hard part of this approach is to determine the flow velocity perturbations due to the topography of the 2d dune. This will be all too expansive in time to compute exact turbulent numerical solution starting from the Navier-Stokes equations. We prefer to use what we called C c c model [25], which is low-cost in computation time and which has already given excellent results for 2d dune shape study [21,22,24,25]. 2.1 numerical model for the 2d case We consider a sand bed profile h(x, t). where x is along the wind direction. First, the mass conservation can be written in function of the local height of the sand bed and of the sand flux q(x, t) as :