REE3: Desert landforms and coastal geomorphology

Project completed

Background

Several sand dune structures are found in nature, and each one depends on specific environmental conditions, namely wind direction, and the type, amount and moisture content of the sand. This research project focuses on deriving and solving a mathematical model to describe the evolution of a particular sand dune (the transverse dune), specifically taking into account the effect of airflow separation on the leeward side of the dune.

The transverse dune’s structure is a product of a strong wind, generally in a constant direction, and an abundance of sand. Its shape is roughly invariant in the transverse direction (the direction perpendicular to airflow). Thus, the structure of the dune is encompassed in its cross-sectional shape which simplifies the problem from three to two dimensions. This property makes dune transience mathematically tractable.

Techniques and Challenges

Akin to waves forming in an ocean, sand dunes form and move because turbulent airflow over the dune causes shear stress on the top layer of sediment, resulting in sand being moved along the surface of the dune. The first aim of the model is to find a formula for the shear stress due to turbulent airflow.

In this first model several assumptions are made. Firstly, the wind incident on the dune is assumed to be uni-directional. The resulting airflow over the dune is turbulent, but it is modelled using a constant eddy viscosity where small-scale effects of turbulence are ignored. The eddy viscosity characterises large-scale motion in the turbulent region and allows the effect caused by this airflow to be modelled as the effect of viscosity in a laminar flow. Then the airflow over the dune can be described by the familiar Navier-Stokes equations with a large turbulent Reynolds number. For the initial model, the airflow is assumed to remain attached to the dune, so separation at the crest is ignored.

Lastly, the dune’s movement is much slower than an individual particle of sand’s movement, and the dune can be considered stationary.

To derive the formula for shear stress, the effect of a perturbation to the dune’s surface on the fluid flow is considered. To arrive at a formula for sand transport, this formula for shear stress is coupled with an equation relating sand flux to shear stress. It is found that the force caused by the airflow over the dune results in an instability of the top sediment layer. This instability is what causes dunes to evolve and move.

A more realistic model is obtained when two assumptions from the initial model are removed. The dune is no longer assumed to be stationary since realistically sand dunes slowly move across the desert. Also, the effect of airflow separation in the lee of the dune is taken into account. In order for sand transport to occur, the shear stress in the top layer must be above a certain threshold. However, where flow separation occurs, the shear stress is below this threshold (i.e. it tends to zero). The effect of airflow separation is incorporated by modifying the sand flux-shear stress equation derived in the initial model so that in the separation zone there is a flux of sand through the region, but the shear stress remains zero.

Results

The results obtained by the research team at Oxford are shown in a simulation of the dune shape and the shear stress along the dune surface in Figure 3. The height of the dune is 34.8m, its length is 6km and its speed is 11.95m/yr. These results are comparable to those observed in nature, although the model led to the length of the dune being on the longer.

The Future

Future directions for this work will extend the model to three dimensions. This will allow for other types of dunes to be studied that do not have shape invariance. Since the shape of dunes formed in the presence of vegetation would require a three-dimensional model, this is a key step in the process of studying the effect of vegetation on dune formation and migration.

Related Publications

[10/26] Ellis A.S., Fowler A.C.: On an evolution equation for sand dunes

[1] Wiggs, G.F.S.: Desert dune processes and dynamics, Progress in Physical Geography, Vol. 25(1), pp. 53-79

[2] Schwammle, V., Herrmann, H.: Modelling transverse dunes, Earth Surface Processes and Landforms, Vol. 29, June, pp. 769-784, 2004

[3] Walker, I.J.: Secondary airflow and sediment transport in the lee of a reversing dune, Earth Surface Processes and Landforms, May 1999