# (25) Morphodynamics of dunes under unidirectional wind

From Stratodynamics

## Contents |

## Author

Hirofumi Niiya, Akinori Awazu, and Hiraku Nishimori

Department of Mathematical and Life Sciences, Hiroshima University

## Abstract

### Introduction.

Erosion and deposition due to wind execute various types of sand dunes in deserts on the surface of Earth, Mars, and Titan [1]. As the dominant factors dictating several dune shapes, the steadiness of wind direction and the amount of available sand in each dune field are considered [2]. For example, a unidirectional wind generates barchans, crescent-shaped dune, or transverse dunes extending perpendicular to the wind direction. The former are formed in dune fields with small amounts of available sand, whereas the latter are formed in dune fields with larger amounts of available sand than barchan-rich fields.

Recent dune studies have made a significant progress in the quantitative analysis of dune morphodynamics. In particular, rescaled water tank experiments have successfully been conducted to form distinct dune shapes under controlled conditions for the wind direction [3]. Also, computer models have reproduced the qualitative and quantitative morphodynamics similar to aeolian and subaqueous dunes [4]. However, a theoretical methodology to explain the basic mechanism behind dune shape formation beyond a mere numerical reproduction is yet to be established. To analyze the formation process and dynamics of characteristic types of dunes, we propose the dune skeleton model consisting of coupled ordinary differential equations under unidirectional steady wind [5]. In this study, using the numerically and analytically approaches of this model, we investigate the shape transition of dunes depending on the amount of available sand and the stability of dune shapes.

### Model.

The dune skeleton model is a reduced model to describe the formation process and the dynamics of barchans and transverse dunes generated under a unidirectional steady wind, and this model is roughly based on three assumptions. First, the dunes consist of triangular slices with constant angle of upwind and downwind slope. Second, these slices are arrayed perpendicular to the wind direction at constant lateral interval between slices. Third, a combination of the intra-slice and inter-slice sand movements are considered to govern the macroscopic morphodynamics of dunes. Here, the variables in this model are given as the wind directional position and the height of slice’s crest because of constant angle of slopes. With consideration of the above intra- and inter-sand movements, the coupled ordinary differential equations for crests derived.

### Result.

Numerical simulations of this model are carried out using number of slices, N=1000. As the initial condition, the initial dune shape is set slightly fluctuated from a straight transverse dune. Moreover, the lateral boundary condition is set as periodic, whereas the wind directional boundary condition is set such that sand escaping into the leeward boundary is redistributed uniformly from the windward boundary. Thus, the total amount of sand constituting dune given as the initial condition is conserved throughout the simulation unless the annihilation of slice occurs. This model reproduces that three typical shapes of dunes, straight transverse dunes, wavy transverse dunes, and barchans, are formed depending on the amount of available sand and wind strength. Numerical simulations also show that the increase in the amount of available sand and inter-slice sand movement enhances the stability of transverse dunes, whereas the decrease in the amount of available sand and the increase in the intra-slice sand movement destabilize the shape of transverse dunes to enforce the deformation into barchans.

In order to elucidate the transition mechanism between different steady dune shapes obtained above simulation, we conduct a bifurcation analysis of the reduced dune skeleton model [6]. This model consists of two slices, each of which is assumed to represent dunes with a large number of slices. The analysis of the model shows the existence of stable two types of transverse dunes, straight and wavy. Additionally, the eventual dune shape is uniquely selected by a set of control parameters, that is, a coexistence of different dune shapes is not realized at the end-stage. These bifurcation structures reveal the transition mechanism between dunes generated under unidirectional wind, and these results qualitatively correspond to the morphodynamics of previous observations of real dunes, water tank experiments, and computer models.

### References

[1] M. C. Bourke, and A. S. Goudie, Aeolian Res. 1 (2009) 45.

[2] I. Livingstone, and A. Warren, Aeolian Geomorphology (1996).

[3] E. Reffet, S. Courrech du Pont, P. Hersen, and S. Douady, Geology 38 (2010) 491. [4] D. Zhang, C. Narteau, and O. Rozier, J. Geophys. Res. 115 (2010)

F03041.

[5] H. Niiya, A. Awazu, and H. Nishimori, J. Phys. Soc. Jpn. 79 (2010)

063002.

[6] H. Niiya, A. Awazu, and H. Nishimori, Phys. Rev. Lett. 108 (2012)

158001.