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.

Figure 1

(a) Intra-2D CS sand flow. (b) Inter-2D CS sand flow.

Figure 2

Analysis setup: (a) Two 2D CSs selected from the transverse dune obtained by simulation: first and second 2D CSs are represented by the solid and dashed lines, respectively. (b) System of two 2D CSs with lateral periodicity.

Figure 3

Nullclines, solution trajectories, and vector fields of Eqs. (4a, 4b) for (a) $H_0=25$, (b) $H_0=35$, (c) $D_u=0.06$, and (d) $D_u=0.10$. The thin and thick solid lines represent $y$ nullcline and $h_1$ nullcline, respectively. Both short and longer dashed lines represent the solution trajectories starting from the different initial values. The vectors are normalized in magnitude while maintaining the direction. The unvarying parameters are set as $q=0.5$ and $D_d=0.1$ in all figures. Other parameters are set as [(a), (b)] $D_u=0.1$ and [(c), (d)] $H_0=40$.

Figure 4

Bifurcation diagram of fixed points obtained by varying one control parameter: (a) $H_0$ and (b) $D_u$. The solid and dashed lines denote stable and unstable fixed points, respectively. B, W, and S denote the ranges of B phase, WT phase, and ST phase, corresponding to the dune shapes, respectively. The unvarying parameters ($q$, $D_d$) that are common between (a) and (b) are the same as the values in Fig. 3, whereas other parameters are set as (a) $D_u=0.1$ and (b) $H_0=40$.