# An Idealized 3D Model of Interfacial Instability of Aeolian Bedform
## 4 Results  
The values of the parameters representative of the typical sand transport conditions on Earth are considered. The thicknesses of the creep layer is related to the grain size and the number of rolling or sliding grains per unit surface, \(R\sim d^{3}N_{c}\), where \(N_{c}\) is the number of rolling or sliding grains per unit surface. The erosion rate is \(E_{0}\sim nN_{0}d^{3}\), where \(N_{0}\) is the number of impacting grains per unit time and surface and \(n\) is a dimensionless ejection rate coefficient. Following [PERSON] et al. [28], the erosion rate can be expressed as \(E_{0}=nQ_{s}/l_{s}\), where \(Q_{s}\) is saltation flux and \(l_{s}\) is the mean saltation length. In our model, the reptation length distribution \(p(a)\) is extracted from the study of reference [19]. We first analyze the growth rate \(\mathrm{Re}(\omega_{\mathrm{c}})\), which depends on the creep and is negative and proportional to \(k_{x}^{2}\) and \(k_{y}^{2}\); therefore, the slope-dependence of the creep flux stabilizes short-wavelength perturbations, that is, the effect of the gravity dampens the largest wave numbers, which favors a smoothing of the bedform surface. The growth \(\mathrm{Re}(\omega_{\mathrm{r}})\) contains two contributions, \(\mathrm{Re}(\omega_{\mathrm{r,je}})\) and \(\mathrm{Re}(\omega_{\mathrm{r,se}})\); \(\mathrm{Re}(\omega_{\mathrm{r,je}})\) is positive, proportional to \(\cot\beta_{1}\) and \(\cot\beta_{2}\), and related to the probability distribution of the reptation length. The smaller the angle of impact, the larger \(\cot\beta_{i}(i=1,2)\) becomes; it plays a destabilizing role. Whereas \(\mathrm{Re}(\omega_{\mathrm{r,se}})\) is negative and stabilizes the sand bed, it is proportional to \(\mu_{i}(i=x,y)\); the parameter \(\mu_{i}\) represents the change in the reptation flux due to the slope effect. Physically, the saltating grain affects the sand bed and induces some grains to roll or slide down the slope. This mechanism is termed an \"impact-induced gravity flow\" and acts to reduce the flux upslope and to increase it downslope [15]. Hence, the pattern selection results from a compromise between the physical mechanisms of stabilizing and destabilizing. The competition between these two opposite mechanisms results in a region of amplified in the space \(\left(k_{x},k_{y}\right)\). By changing the relative strength of these two effects, we can control the instability of the system and approximate a marginal stable state. Figure 2 shows the growth rate \(\mathrm{Re}(\omega)\) of the bed perturbations as a function of the topographic wave numbers \(k_{x}\) and \(k_{y}\), in which (A) the total growth rate is the sum of the contributions of (B) reptation and (C) creep. In turn, the contributions to the growth rate due to reptation are given by the sum of (D) impact-induced reptating grains flow effect, which is the growing mechanism, and (E) impact-induced gravity flow effect, which is the decaying mechanism. The results show that the reptation is the growing mechanism and the slope effect is the decaying mechanism. Figure 3 shows marginal stability curve of dispersion relation for different parameter \(D\), the curve where \(\mathrm{Re}(\omega)=0\); the other parameters are the same as in Figure 2. The curve delimits the two regions for internal and external; the sand bed is unstable in the internal area, while it is stable in the external area. The stable region expands with the increase of parameter \(D\), where \(D\) is a dimensionless parameter that reflects the relative strength of reptation and creep.