Barchan dunes in two dimensions: Experimental tests for minimal models

A well-defined two-dimensional single barchan dune under the force of a shearing water flow is investigated experimentally. From an initially prepared triangular heap a rapid relaxation to a steady-state solution is observed with constant mass, shape, and velocity. This attractor exhibits all characteristic features of barchan dunes found in nature, namely a gently inclined windward side, crest, brink, and steep lee face. The relaxation time towards the steady state increases with mass. For small dunes we find significant deviations from a fixed height-length aspect ratio. As predicted by recent theoretical models, the migration velocity scales reciprocal to the length of the dune.

Figure 1

Temporal evolution of a developing two-dimensional barchan dune subject to unidirectional water flow ($\mathrm{Re}=27000$, $m=13\mathrm{g}$).

Figure 2

Scheme of the experimental setup (top view).

Figure 3

Horizontal component $u$ of the flow velocity as a function of the vertical position $h$ for three different Reynolds numbers: $\mathrm{Re}=13000$ (solid line), $\mathrm{Re}=28500$ (dashed line), and $\mathrm{Re}=44000$ (dotted line).

Figure 4

A zoom to the picture at $t=10\mathrm{s}$ in Fig. 1: The found border of the barchan is marked white, the five striking points of a barchan are shown as dots, and the length $L$ and the height $H$ are plotted as double arrows.

Figure 5

The time evolution of the mass $m$, the velocity $v$ of the center of mass and the temporal evolution of the aspect ratio $H∕L$ for the barchan dune in Fig. 1: The solid line is an exponential fit according to Eq. (1).

Figure 6

Final aspect ratio $H_{\infty }∕L_{\infty }$ and relaxation time $\tau$ for different barchan sizes and three different flow velocities: $\mathrm{Re}=25500$ (triangles), $\mathrm{Re}=27000$ (squares), and $\mathrm{Re}=28500$ (circles). The dotted, dashed, and solid lines are guides to the eyes.

Figure 7

Relation between mean barchan velocity $\overline{v}$ and the mass of the barchan dune $m$ for three different flow velocities: $\mathrm{Re}=25500$ (triangles), $\mathrm{Re}=27000$ (squares), and $\mathrm{Re}=28500$ (circles). The dotted, dashed, and solid lines are fits to $\overline{v}\sim m^{-\alpha }$. The inset shows the same data on double-logarithmic scales.

Figure 8

Relation between mean barchan velocity $\overline{v}$ and height $H_{\infty }$, respectively, length $L_{\infty }$ for three different flow velocities: $\mathrm{Re}=25500$ (triangles), $\mathrm{Re}=27000$ (squares), and $\mathrm{Re}=28500$ (circles). The dotted, dashed, and solid lines are fits to $\overline{v}\sim H_{\infty }^{-\beta }$, respectively, $\overline{v}\sim L_{\infty }^{-\gamma }$. The insets show the same data within double-logarithmic plots.