# An Idealized 3D Model of Interfacial Instability of Aeolian Bedform
## 4 Results
This is consistent with the characteristics of the early formation stage of sand ripple observed in wind tunnel and field. With the increase of \(D\), most unstable transverse mode is \(k_{y}^{\rm max}\to 0\). For example, when \(D=0.5\), we get \(k_{y}^{\rm max}\approx 0\). It reflects that the most unstable mode is a longitudinal mode in both modes; thus, the corresponding sand ripples has more straight  
Figure 5: The contour plot of growth rate in the parameter space \((k_{x},k_{y})\) for different values of D. (**A**) \(D=0\). (**B**) \(D=0.1\). (**C**) \(D=1\). (**D**) \(D=1.5\). Figure 6: The growth rate plotted versus the wavenumber \(k_{x}\), fixing \(k_{y}=0\) for different impact angle. The insets show the wavelength of the preferred mode as function of the impact angle. and continuous crests perpendicular to the wind direction. In fact, as the sand ripples grow, the gravity flow increases (i.e., \(D\) increases). The lateral transport of gravity flow has the capability of reducing along-crest topographic irregularities and producing sand ripples with straighter and more continuous crests perpendicular to the wind. This is also consistent with the features observed in wind tunnels and field. Figure 7B plots the growth rate versus \(k_{y}\) for \(k_{x}=k_{x}^{\text{max}}\) for different values of \(\mu_{y}\); it shows similar results, where \(\mu_{y}\) measures the strength of the sand flux by impact-driven along the \(y\) direction, which is responsible for the lateral coupling [15]. A larger lateral (along-crest) sand transport tends to stabilize the transverse mode, and the wavelength of the transverse mode becomes larger, which has the capability of reducing along-crest topographic instability and irregularities and producing bedform-like structures with straighter and more continuous crests. The numerical simulations of [PERSON] et al. [15] support our analysis results that owing to the large lateral flux, the along-crest height of sand ripples is almost the same; the transverse tends to be stable, reducing along-crest ripples sinuosity, creating a straighter crestline. In fact, the aeolian sand ripples are generally straight and continuous, which requires the physical coupling process along the crest of the sand ripples, and that lateral (along-crest) sand transport provides effective physical mechanisms to couple the topography at different along-crest locations and to overcome local instability. In the proposed model, the lateral transport is mainly contributed to by the along-crest diffusion of topography driven by impact and gravity; with the increase of \(\mu_{y}\) and \(D\), the lateral transport increases, which provides a mechanism for extending sand ripples in the along-crest direction. The normal ripples composed of monodisperse fine sand particles have larger transverse transport, which can fill downwind concavities and flatten downwind convexities; the sand ripple is straight and continuous [32]. However, for meageripples, coarse particles accumulate at the crest, which reduces the lateral sand transport along the ripple crest and increases the irregularities and transverse instability along-crest [15]. The transverse aeolian dunes are not straight, while the origin of transverse instability is due to different drift velocities caused by the different heights along the cross-wind direction [33] and due to the fact that the lateral transport on the dune slip face prevails over that on the upwind face [34]. This difference may arise from the strong scale separation between the grains' average hop length and the typical size of the dune, while there is no such scale separation for smaller-scale sand ripples. The transverse modes do not grow as fast as the longitudinal modes in the linear regime, but they could couple each other in the nonlinear regime. A further nonlinear analysis should be made to investigate the interaction between different modes and predict the defect behavior and bedform straightening in the non-transverse cases.