# An Idealized 3D Model of Interfacial Instability of Aeolian Bedform
## 4 Results
Taking into account that \(R\sim d^{3}N_{c}\) and \(E_{0}\sim nN_{0}d^{3}\), \(D\) can be written as  
\[D\sim\frac{N_{c}\sqrt{gd}}{\pi nN_{0}} \tag{19}\]  
The numerator represents the intensity of creep transport on the inclined plane, and the denominator is the horizontal flux of reptation grains on the flat bed, so the ratio between the creep flux and the reptation flux is \(\eta\propto D\), in which \(D\) measures the ratio of the creep flux to the reptation flux, which responsible for the relative transport intensity of the two transport modes. The next important issue is to determine the range of possible values for the model parameter \(D\) in real physical situations. Note that the experimental  
Figure 2: The growth rate as function of the topographic wave numbers \(k_{x}\) and \(k_{y}\) for \(n=1\), \(\beta_{1}=10^{\circ}\), \(\beta_{2}=12^{\circ}\), \(\mu_{i}(i=x,y)=0.4\), \(D=0.5\), and \(\epsilon=0.25\). The total growth rate (**A**) is the sum of the growth rates due to reptation (**B**) and creep (**C**). The reptation growth rate is the sum of two contributions: (**D**) the impact-induced reptation flow and (**E**) the impact-induced gravity flow. determination of reptation and creep is rather inadequate, especially the creep transport driven by gravity. It is therefore instructive to study the influence of these two transport modes on the marginal stability curve. In Figure 3, we find that the unstable region is very sensitive to a change of the constitutive parameter. The results show that the formation and evolution of the sand ripples occur under the combined action of saltation, reptation, and creep; the saltating grains energize the reptating population; and the gravity causes sand grain creep. So, it is proved that the instability of the sand bed is determined by the competition of the reptation and creep. For small values of \(D\), such as \(D<0.01\), the effect of the impact-induced gravity flow is much stronger than the topography-gravity flow (i.e., \(\mu_{i}(i=x,y)\gg D\)); the topography-induced gravity flow is less than 1% of the flux of reptating grains, and it has almost no effect on the stability of sand ripples. Therefore, the short-wavelength stabilization originates from impact-induced gravity flow. In contrast, for large values of \(D\), the topography-induced gravity flow plays a leading stabilizing role; a band of unstable modes appears only if reptation flow prevails over topography-induced gravity flow (or more precisely, if \(D\) is greater than a critical value given by \(D=D_{cr}\), see Figure 4). In order to efficiently investigate the effect of the parameter \(D\) on the aeolian ripples formation, a one-dimensional investigation has been made by fixing \(k_{y}=0\). The marginal stability conditions shown in Figure 4 were plotted in the space \((k_{x},D)\). The critical value \(D_{\sigma}\) of \(D\), above which sand ripples do not appear, and sand bed stable can be easily estimated from the graph, which also shows that the sand bed becomes unstable in a finite range of wave numbers \(k_{x}\) once \(D\) is less than its critical value. The maximum growth rate of sand bed perturbation versus its wavenumber \(k_{x}\) for different values of \(D\) is represented by the red line shown in Figure 4. The maximum wavenumber \(k_{x}^{\rm max}\) tends to an infinitesimal value (\(k_{x}^{\rm max}\to 0\)) as \(D\) tends to \(D_{\rm cr}\), and the wavelength of preferred mode tends to infinity (\(\lambda_{\rm max}\to\infty\)), so the sand bed is stable, and the sand ripples do not appear. For \(D=0\), the creep flux disappears; the model is similar to the model of [PERSON] et al. [9]. Figure 3: Marginal stability curves for different values of \(D\). Figure 5 plots the contour plot of growth rate \(\text{Re}(\omega)\) in the parameter space \((k_{x},k_{y})\) for different values of \(D\). We find that the characteristics of these contour plots highly depend on the value of \(D\). The zero growth contours on the graph locate the marginal instability curve.