Particle dynamics of a cartoon dune

The spatio-temporal evolution of a downsized model for a desert dune is observed experimentally in a narrow water flow channel. A particle tracking method reveals that the migration speed of the model dune is one order of magnitude smaller than that of individual grains. In particular, the erosion rate consists of comparable contributions from creeping (low-energy) and saltating (high-energy) particles. The saltation flow rate is slightly larger, whereas the number of saltating particles is one order of magnitude lower than that of the creeping ones. The velocity field of the saltating particles is comparable to the velocity field of the driving fluid. It can be observed that the spatial profile of the shear stress reaches its maximum value upstream of the crest, while its minimum lies at the downstream foot of the dune. The particle tracking method reveals that the deposition of entrained particles occurs primarily in the region between these two extrema of the shear stress. Moreover, it is demonstrated that the initial triangular heap evolves to a steady state with constant mass, shape, velocity and packing fraction after one turnover time has elapsed. Within that time the mean distance between particles initially in contact reaches a value of approximately one quarter of the dune basis length. S Online supplementary data available from stacks.iop.org/NJP/12/063025/ mmedia ‘We are right in the open desert,’ said the doctor. ‘Look at that vast reach of sand! What a strange spectacle! What a singular arrangement of nature!’ [1]. In this fictive story of Jules Verne from the 19th century, Dr Samuel Ferguson and his two companions discover the beauty of sand dunes in the deserts of Africa by traveling five weeks in a balloon across the continent. Today, scientists are still overwhelmed by these self-organized granular structures, but they use satellites for observations instead of balloons [2, 3]. 3 Author to whom any correspondence should be addressed. New Journal of Physics 12 (2010) 063025 1367-2630/10/063025+20$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 2 The more descriptive science of the past century [4–6] has changed to a detailed approach in understanding dune dynamics [7]. The crescent-shaped barchan dune was chosen as a suitable object because of its relatively fast dynamics. So-called ‘minimal models’ were established to describe the basic dynamics of barchan dunes [8–10]. These two-dimensional (2D) models deal with dune slices along the direction of the driving wind. Briefly, they combine an analytical description of the turbulent shear flow over low elevations [11, 12] with a continuum description, which models the saltation on the surface of the dune [13]. In the next step of complexity, these 2D slices are coupled in the cross-wind direction to model 3D barchan dunes [14]. Laboratory experiments [7, 15–18] and field measurements on Earth [19–21] or satellite observations of Mars [22, 23] reveal the quality of such models. Some aspects of the minimal models have been checked quantitatively in a narrow water flow channel [24–26], and the existence of a shape attractor for barchan dunes [27, 28] has been demonstrated experimentally [29]. Minimal models are continuum models: they deal with the overall shape of the dunes, neglecting the particulate nature of these granular systems. However, the dynamics of dunes is determined by the transport mechanism of the individual grains of sand [4]. Aeolian sand transport consists of two modes of transport: reptation in the low-energy regime and saltation in the high-energy regime [30]. The saltating grains are carried with the wind and have flight lengths of thousands of grain diameters. The paths of the reptating grains are much shorter. Their motion is initiated by impacts of the saltating grains onto the granular bed [31]. Experiments in wind tunnels with sand traps [32–34] and with particle tracking methods [35–37] give insight into the aeolian grain transport on the level of particle dynamics. In order to investigate desert dunes under laboratory conditions, it is convenient to use water instead of air [7, 18, 27], since this creates small replicas of aeolian dunes on a much shorter time scale. However, the subaqueous sand transport differs slightly from the aeolian one. Neglecting the possibility of suspension at high shear velocities, the bed-load transport in water can also be separated into two energy regimes: saltation and surface creep [38–41]. The saltating particles have much shorter flight lengths than in air and, in contrast to the reptating particles, the creeping particles are directly dragged by the fluid and always stay in contact with the bed surface. Particle tracking experiments in flow channels with denser fluids than air, like water or silicon oil, have been performed to investigate the erosion, transport and deposition of grains of sand on a granular bed [42–46]. The particle dynamics at the surface transfers itself into the bed. The packing density of the granular bed decreases from the inner layers towards the outer fluidized layer, which has a thickness of a few grain diameters in a laminar flow [45]. The observation of segregation effects indicates that the sediment in the sand bed is mixed during the process of sand transport and associated ripple formation [47]. Besides global granular transport, the transition between creeping and saltating particles is also a matter of interest [32, 41–43, 48]. The research presented here addresses three questions: (i) How does the fluid transport the particles? (ii) How is particle transport related to overall dune motion? (iii) How is fluid flow influenced by the presence of the dune? For this purpose we created an experimental realization of a minimal dune model—a cartoon dune. It is a migrating isolated heap on a plane surface reminiscent of a barchan dune in the desert. Our experiment reveals the details of particle transport above, on and inside this solitary dune, together with the driving water flow field. For direct access to the particles, our dune is designed with a thickness of less than two particle diameters and is, therefore, quasi-2D. New Journal of Physics 12 (2010) 063025 (http://www.njp.org/)