Modeling of Wind Blown Sand and Desert Dunes

G. Sauermann, K. Kroy$^*$, J.S. Andrade Jr.$^{**}$,
H.J. Herrmann
$^*$ Klaus Kroy, University of Edinburgh, Edinburgh, UK
$^{**}$ Department of Physics, Universidade Federal do Ceará,
Fortaleza, Brazil

Diploma Thesis

Sand dunes can be found in deserts, on the sea-bottom, and even on Mars. Geoscientists have found innumerable classes of Dune shapes. The shape of a Dune depends mainly on the amount of available sand and on the change in the direction of the wind over the year. If the wind blows steadily from the same direction throughout the year and if there is not enough sand to cover the entire area, the dune shape develops into a crescent, so-called barchan form. Typical measurements of the relationship between height and width or length led to the assumption, that barchan dunes are shape invariant objects. Our field measurements of entire dune shapes in the desert of Morocco have disproved this shape invariance.

We have figured out that the ratio of the horn length to the length of the dune's windward side increases with height, which leads to varying positions of slip face and brink (for the definitions of the technical terms see Figure 6). Further, the necessary cubic scaling of the volume was not confirmed. Nevertheless, we showed that the windward side could be described by a paraboloid. A result of our analysis is that the difference between small and large dunes can be expressed by the position and the shape of the slip face or brink as shown in Figure 7, and leads implicitly to the separation of brink and crest as confirmed by our measurements.

The final goal of the project was the prediction of desert dune motion by numerical simulations. To reach this aim one had to understand the physics involved in dune formation and to derive a minimal model that consistently describes dune formation and that can predict the time evolution of a free surface exposed to wind. We have shown that the coupled problems of sand transport and dune formation can be treated separately due to the fact that the characteristic time scales differ by more than seven orders of magnitude. Hence, in order to get a complete model that can describe the formation of dunes, we had to treat three different physical processes: The global perturbation of the wind field onto a dune, a continuum saltation model that incorporates the local interaction between the sand grains and the wind near the ground, and a model for avalanches that maintains the sand transport due to gravity on the slip face.

Fig 6:
Sketch of a barchan dune. For each horn we define the lengths $L_a$, $L_b$ and the widths $W_a$, $W_b$ independently, as did Finkel [``The barchans of southern Peru''. Journal of Geologie, 67, 614-647 (1957)]. The orientation of the measuring axis is chosen according to the wind direction, which coincides with the symmetry line for a totally symmetric dune. Furthermore we introduce the length of the slip face $L_s$ and the length $L_0$ from the dune's toe on the windward side to the brink. Finally, the height of the slip face $H$ is defined at the highest point of the brink, which is the intersection of the brink and the longitudinal centerline of the dune.

To calculate the sand flux based on the idea of saltation we developed a quantitative non-equilibrium model. The dynamics of the sand flux lead to saturation transients and thus to a characteristic length scale. This length scale is important for dune formation and can explain the different shapes found for small and large dunes. Furthermore, we model the flow on top of a dune by solving well known boundary layer equations that we extended by introducing a separation bubble in order to model the recirculation on the lee-side of a dune. The wind field calculations agree with results calculated by FLUENT 5.0. Our theory reproduces the parabolic or cosine shaped symmetry profile derived from the field data.

a)

b)

Fig 7:
Shape of two barchan dunes of different height: a) 3 m and b) 9 m, according to the proposed model using parameters obtained from the data of the barchans of Morocco.

In the course of the project several field measurement expeditions were performed in Morocco and in Brazil. We measured the three dimensional shape of barchan dunes in a desert dune-field at Laâyoune in southern Morocco (Spring 1999), and we made correlated measurements of the wind field and the sand flux at the coast of Ceará in north-eastern Brazil, on a dune-field near Jericoacoara, about 300 km north-west of Fortaleza (Winter 2000)

Our minimal sand dune model comprises a non-local equation for the air shear stress at the ground, a continuum model for aeolian sand transport, and a model for avalanches. From the analytical model several predictions concerning barchan dunes could be drawn, e.g. about the shape, the scaling of the velocity, and the stability. Finally, we presented numerical calculations of the time evolution of an initial Gaussian shaped heap in two and three dimensions. We demonstrated that the solution converges to an invariantly moving profile in the steady state. These steady state forms displayed good agreement with the data obtained by our field measurements.

This project was granted by the Deutsche Forschungsgemeinschaft.