# An Idealized 3D Model of Interfacial Instability of Aeolian Bedform
## 4 Results
For \(D=0.1\), the perturbation component is characterized by the maximum growth rate, i.e., the component that dominates the subsequent development of sand bed instability is characterized by \(\left(k_{x}^{\text{max}},k_{y}^{\text{max}}\right)=(2.9,\pm 2.3)\) (see Figure 5B). These wavenumbers correspond to the wavelength of about 2 cm. For \(D=1\), compared with the last case, it is clear that the instability region (i.e., regions with positive growth rates) decreases (see Figure 5C); it reflects the fact that the most unstable growth rate reaches 0.64 at \(\left(k_{x}^{\text{max}},k_{y}^{\text{max}}\right)=(1.5,\ 0)\). and the initial wavelength is about 4 cm. The initial wavelength scale is similar to the observed wavelengths [4] and simulates wavelengths that are about 25 times the grain diameter [10]. It is also found that the most unsble mode is longitudinal (i.e., \(k_{y}^{\text{max}}\approx 0\)) and therefore corresponds to aeolian ripples having crests perpendicular to the wind. When \(D=1.5\), the instability region shrinks to a small contour, almost becoming a single point (see Figure 5D). Hence, with the increase of \(D\), the marginal instability curve asymptotically approaches a point; if \(D\) continues to increase, the sand bed becomes stable and sand ripples will not be formed. We investigate the effect of the impact angle for the sand ripples formation. The saltating grains typically impact the bed at angles between \(9^{\circ}\) and \(16^{\circ}\)[29; 30; 31]. In doing so, we fix the topographic wave vector in \(y\) direction as 0, that is, \(k_{y}=0\). Figure 6 shows the growth rate \(\text{Re}(\omega)\) of the sand bed perturbations versus \(k_{x}\) for different values of versus of \(\beta_{1}\). When the impact angle is increased, the growth rate \(\text{Re}(\omega)\) shows that the flat sand bed is more stable. Moreover, the value of \(k_{x}\), which gives rise to the maximum value of \(\text{Re}(\omega)\), increases, thus showing that bigger impact angle generates longer sand ripples; the wavelength \(\lambda_{\text{max}}\) of the preferred mode increases slightly with the increase of impact angle at the initial stage (see Figure 6 (insets)). This possible reason is that the sand ripples are very small at the initial stage of sand ripple development; the greater the impact angle, the more effective the saltation impact, which will produce greater the reptation flux so that the sand ripples will grow. However, for the subsequent nonlinear growth stage, the shadow effect can be one of major mechanisms contributing to the growth of sand ripples; the smaller the impact angle, the larger the shadow zone and the longer the wavelength [11; 14]. The influence of impact angle on the dynamics of sand ripples is different in the initial linear stage and the later nonlinear stage. It remains for future further studies to determine the influence of impact angle in different stages of ripples formation. Figure 4: Marginal stability conditions in the plane \((k_{x},D)\) (blue line). Red line represents the location of the relative maximum of the growth rate \(\text{Re}_{\text{max}}(\omega)\). It is interesting to discuss the unstable transverse mode, although the mode does not prevail in the linear regime; since its growth rate is smaller than that of the preferred mode, we may wonder whether they play an important role in the transverse development of sand ripples. The influence of topography-induced gravity transport on the transverse instability is illustrated in Figure 7A; it shows that for small values of \(D\) (for example \(D=0.01\)), a wavy crestline with finite wavelength is formed, which corresponds to the most unstable mode \(k_{y}^{\rm max}\). This is the early stage of sand ripple formation, and the ripples are small (i.e., \(D\) is small), so the lateral transport of gravity flow is small, which reduces the lateral coupling and produces sand ripples with more irregular and lateral instability.