Linear stability analysis of transverse dunes

Sand-moving winds blowing from a constant direction in an area of high sand availability form transverse dunes, which have a fixed profile in the direction orthogonal to the wind. Here we show, by means of a linear stability analysis, that transverse dunes are intrinsically unstable. Any perturbation in the cross-wind profile of a transverse dune amplifies in the course of dune migration due to the combined effect of two main factors, namely: the lateral transport through avalanches along the dune’s slip-face, and the scaling of dune migration velocity with the inverse of the dune height. Our calculations provide a quantitative explanation for recent observations from experiments and numerical simulations, which showed that transverse dunes moving on the bedrock (or “transverse sand ridges”) cannot exist in a stable form and decay into a chain of crescent-shaped barchans.     

Transverse instability of dunes

The simplest type of dune is the transverse one, which propagates with invariant profile orthogonally to a fixed wind direction. Here we show, by means of numerical simulations, that transverse dunes are unstable with respect to along-axis perturbations in their profile and decay on the bedrock into barchan dunes. Any forcing modulation amplifies exponentially with growth rate determined by the dune turnover time. We estimate the distance covered by a transverse dune before fully decaying into barchans and identify the patterns produced by different types of perturbation.     

Bifurcation analysis of the transition of dune shapes under a unidirectional wind

A bifurcation analysis of dune shape transition is made. By use of a reduced model of dune morphodynamics, the Dune Skeleton model, we elucidate the transition mechanism between different shapes of dunes under unidirectional wind. It was found that the decrease in the total amount of sand in the system and/or the lateral sand flow shifts the stable state from a straight transverse dune to a wavy transverse dune through a pitchfork bifurcation. A further decrease causes wavy transverse dunes to shift into barchans through a Hopf bifurcation. These bifurcation structures reveal the transition mechanism of dune shapes under unidirectional wind.     

Modeling transverse dunes with vegetation

In the present work, we use dune modeling in order to investigate the evolution of transverse dunes in the presence of vegetation. The vegetation is allowed to grow up to a maximum height with a growth rate R that oscillates in time. We find that the presence of the vegetation establishes a maximum height for the transverse dunes. If the transverse dune is larger than this maximum size, then the vegetation traps a considerable amount of sand, leading to the formation of vegetation marks at the upwind side of the dune. We also investigate the formation of the transverse dune fields from a flat sand beach under saturated sand flux and vegetation growth. We find that the behavior of the field is determined by the maximum height, , of the vegetation cover.     

Selection of dune shapes and velocities Part 2: A two-dimensional modelling

We present in this paper a simplification of the dune model proposed by Sauermann et al. which keeps the basic mechanisms but allows analytical and parametric studies. Two kinds of purely propagative two dimensional solutions are exhibited: dunes and domes. The latter, by contrast to the former, do not present a slip face. Their shape and velocity can be predicted as a function of their size. We recover that dune profiles are not scale invariant (small dunes are flatter than the large ones), and that the inverse of the velocity grows almost linearly with the dune size. We furthermore get the existence of a critical mass below which no dune solution exists. It rises the problem of dune nucleation: how can dunes appear if any bump below this minimal mass gets eroded and disappears? The linear stability analysis of a flat sand bed shows that it is unstable at large wavelengths: dune can in fact nucleate from a small sand mass if the proto-dune is sufficiently long. Received 22 December 2001 / Received in final form 31 May 2002 Published online 31 July 2002     

Why do active and stabilized dunes coexist under the same climatic conditions?

Sand dunes can be active (mobile) or stable, mainly as a function of vegetation cover and wind power. However, there exists as yet unexplained evidence for the coexistence of bare mobile dunes and vegetated stabilized dunes under the same climatic conditions. We propose a model for dune vegetation cover driven by wind power that exhibits bistabilty and hysteresis with respect to the wind power. For intermediate wind power, mobile and stabilized dunes can coexist, whereas for low (or high) wind power they can be fixed (or mobile). Climatic change or human intervention can turn active dunes into stable ones and vice versa; our model predicts that prolonged droughts with stronger winds can result in dune reactivation.     

Spatiotemporal model for the progression of transgressive dunes

Transgressive dune fields, which are active sand areas surrounded by vegetation, exist on many coasts. In some regions like in Fraser Island in Australia, small dunes shrink while large ones grow, although both experience the same climatic conditions. We propose a general mathematical model for the spatiotemporal dynamics of vegetation cover on sand dunes and focus on the dynamics of transgressive dunes. Among other possibilities, the model predicts growth parallel to the wind with shrinkage perpendicular to the wind, where, depending on geometry and size, a transgressive dune can initially grow although eventually shrink. The larger is the initial area the slower its stabilization process. The model’s predictions are supported by field observations from Fraser Island in Australia.     

Quantitative visualization of flow fields associated with alluvial sand dunes: Results from the laboratory and field using ultrasonic and acoustic doppler anemometry

This paper presents results detailing the quantitative visualization of flow fields associated with natural sand dunes, Fraser River Estuary, Canada, using the complementary approaches of laboratory modelling and field instrumentation. Ultrasonic Doppler velocity profiling is used in the laboratory to elucidate the mean flow fields of low-angle dunes (leeside slope angle ≈14°) that are typical of many large natural rivers. These dunes do not possess a zone of permanent flow separation in the dune leeside and have a velocity structure that is dominated by the effects of flow acceleration and deceleration generated by topographic forcing of flow over the dune form. Turbulence associated with these dunes appears linked to both longer-period shear layer flapping and eddy generation along the shear layer. The field study uses acoustic Doppler profiling to reveal similar mean flow patterns and shows that flow is dominated by deceleration in the leeside without the presence of a region of permanent separated flow.     

Sand dunes mobility and stability in relation to climate

Sand dunes form an important and unique system that can be mobile or fixed by vegetation. The common mobility indices of sand dunes, which are related to the wind and the amount of precipitation and potential evaporation, do not work in many dune fields around the world. The reasons for that lie in the singular physical characteristics of the sandy soil. Sand has high hydraulic conductivity causing a high rate of infiltration of rain water to the groundwater. Sand particles lack cohesion and that makes wind erosion the main limiting factor for vegetation. Hence, wind power, manifested by the drift potential (DP), is a good index for the limiting factor of plants on sand. The physical–biological interaction is further developed by hysteresis, which shows that a dune can become vegetated when the wind power is sufficiently low. Once vegetated, a much higher wind stress is needed to destroy the vegetation and re-activate the dunes.     

The morphology of dunes

The motion of dunes and their morphology is a fascinating, largely unexplored subject. Already the barchan, the simplest moving dune, poses many questions. We will present some results of field measurements on desert and costal dunes. Then we will present a model which consists of three coupled equations of motion for the topography, the shear stress of the wind and the sand flux. These evolution equations are verified on the experimental data and new possibilities of simulations of dunes are put in perspective.     

Evolution and shapes of dunes

The motion of dunes and their morphology is a fascinating, largely unexplored subject. Already the barchan, the simplest moving dune, poses many questions. I will present some results of field-measurements on desert and coastal dunes. Then I will present a model which consists of three coupled equations of motion for the topography, the shear stress of the wind and the sand flux. These evolution equations are verified on the experimental data and new possibilities of simulations of dunes are put in perspective. To cite this article: H.J. Herrmann, C. R. Physique 3 (2002) 197–206.     

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.Received: 26 January 2004, Published online: 9 April 2004PACS: 45.70.-n Granular systems - 47.54. + r Pattern selection; pattern formation     

Saltation transport on Mars

We present the first calculation of saltation transport and dune formation on Mars and compare it to real dunes. We find that the rate at which grains are entrained into saltation on Mars is 1 order of magnitude higher than on Earth. With this fundamental novel ingredient, we reproduce the size and different shapes of Mars dunes, and give an estimate for the wind velocity on Mars.     

Vegetation against dune mobility

Vegetation is the most common and most reliable stabilizer of loose soil or sand. This ancient technique is for the first time cast into a set of equations of motion describing the competition between aeolian sand transport and vegetation growth. Our set of equations is then applied to study quantitatively the transition between barchans and parabolic dunes driven by the dimensionless fixation index theta which is the ratio between the dune characteristic erosion rate and vegetation growth velocity. We find a fixation index theta(c) below which the dunes are stabilized, characterized by scaling laws.     

Modelling a dune field

We present a model to describe the collective motion of barchan dunes in a field. Our model is able to reproduce the observation that a typical dune stays confined within a stripe. We also obtain some of the pattern structures which resemble those observed from aerial photos which we do analyse and compare with the specific field of Laâyounne.     

Song of the dunes as a self-synchronized instrument

Since Marco Polo it has been known that some sand dunes have the peculiar ability to emit a loud sound with a well-defined frequency, sometimes for several minutes. The origin of this sustained sound has remained mysterious, partly because of its rarity in nature. It has been recognized that the sound is not due to the air flow around the dunes but to the motion of an avalanche, and not to an acoustic excitation of the grains but to their relative motion. By comparing singing dunes around the world and two controlled experiments, in the laboratory and the field, we prove that the frequency of the sound is the frequency of the relative motion of the sand grains. Sound is produced because moving grains synchronize their motions. The laboratory experiment shows that the dune is not needed for sound emission. A velocity threshold for sound emission is found in both experiments, and an interpretation is proposed.     

A model of Barchan dunes including lateral shear stress

Barchan dunes are found where sand availability is low and wind direction quite constant. The two dimensional shear stress of the wind field and the sand movement by saltation and avalanches over a barchan dune are simulated. The model with one dimensional shear stress is extended including surface diffusion and lateral shear stress. The resulting final shape is compared to the results of the model with a one dimensional shear stress and confirmed by comparison to measurements. We found agreement and improvements with respect to the model with one dimensional shear stress. Additionally, a characteristic edge at the center of the windward side is discovered which is also observed for big barchans. Diffusion effects reduce this effect for small dunes.     

Surface waves in aeolian bedforms

Nature repeat the stripe pattern in aeolian bedform twice, one is sand ripples, the other is linear dunes. Being stripes, these two share many common features, e.g., the whole appearance, the processes of development from a homogeneous state, and the decorated defects. By computer simulations using coupled map lattices, the universal mechanism being at work behind the similarities is revealed. Furthermore, the pattern selection at distinct scales of ripples and linear dunes is discussed. So the results of ripple scale experiments may be used to infer dune scale property.