# An Idealized 3D Model of Interfacial Instability of Aeolian Bedform
## 2 Mathematical Model
### Sand Transport  
Sand transport is the sum of three contributions: saltation, reptation, and creep. The saltating grains only provide the energy for the system and do not directly contribute to the formation of sand ripples [6; 9]. We therefore assume that the saltation flux is spatially homogeneous (i.e.,\(\
abla\cdot Q_{S}=0\)).  
The saltation impact causes the ejection of particles from the sand bed; the ejected particles, which have much lower energy than the saltating particles, make single small hop along the surface of the sand bed. The saltating particles are highly energetic and continue in saltation after rebounding from the sand bed [26].  
We assume that the saltating particles impact the sand bed surface at an angle \(\beta_{1}\) to the x-axis and \(\beta_{2}\) to the \(y\)-axis (see Figure 1). The impacts cause erosion of sand bed, while the saltating grains continue to rebound from the sand bed. The erosion rate \(E_{ej}(x,y,t)\) of sand grains ejected at \((x,y)\) per unit time is proportional to the impact intensity, which depends on the surface orientation with respect to the direction of saltation, based on the geometrical consideration [19; 27], giving  
\[E_{ej}(x,y,t)=E_{1}\cos\theta_{1}\bigg{(}1+\frac{\tan\theta_{1}}{\tan\beta_{1 }}\bigg{)}+E_{2}\cos\theta_{2}\bigg{(}1+\frac{\tan\theta_{2}}{\tan\beta_{2}} \bigg{)} \tag{2}\]  
where \(\theta_{1}=\arctan(\partial h(x,y,t)/\partial x)\), \(\theta_{2}=\arctan(\partial h(x,y,t)/\partial y)\), and \(E_{i}(i=1,2)\) are the erosion rate of the grains discharged from the horizontal surface into reptation by the impacts of saltation, which are determined by the intensity of saltation. However, if the point \((x,y,t)\) is in shadow zone and unreachable by saltating grains, \(E_{i}(i=1,2)\) are set to zero.  
To evaluate the reptation flux per unit width over a horizontal bed, we use the approach of [PERSON] et al. [9]; thus,  
\[\begin{split} Q_{r}=Q_{rx}+Q_{\gamma y}=E_{1}(1-\mu_{x}\partial_{ x}h)\int_{0}^{\infty}p(a)\int_{x-a}^{x}\cos\theta_{1}\Big{(}1+\frac{\tan\theta_{1}}{ \tan\theta_{1}}\Big{)}dxda\\ +E_{2}\big{(}1-\mu_{y}\partial_{y}h\big{)}\int_{0}^{\infty}p(a) \int_{y-a}^{y}\cos\theta_{2}\Big{(}1+\frac{\tan\theta_{2}}{\tan\theta_{2}} \Big{)}dyda\end{split} \tag{3}\]  
where \(p(a)\) is the probability distribution of the reptation length and is chosen from a normal distribution [19]; the parameter \(\mu_{i}(i=x,y)\) represents the change in the reptation flux due to the slope [9; 15].  
As for the creep flux of grains caused by gravity-driven, the sand grains tend to roll in the direction of steepest descent under the influence of gravity; for the three-dimensional sand ripples, the gravity-induced velocity of the sand grains can be written as [12]  
\[v_{r}=-\frac{\sqrt{gd}}{r}\sin\theta \tag{4}\]  
Figure 1: Sketch of the model’s geometry.  
where \(g\) is the acceleration of gravity and \(r\) is a function of the grain packing; in the present work, for simplicity, we set \(r\) to be constant. The horizontal creep flux of sand grains is proportional to the horizontal speed of rolling, \(v_{r}\cos\theta\); hence, the horizontal creep flux of grains is  
\[Q_{c}=Q_{cx}+Q_{cy}=-R\sqrt{gd}\frac{\partial_{x}h+\partial_{y}h}{1+\left( \partial_{x}h\right)^{2}+\left(\partial_{y}h\right)^{2}} \tag{5}\]  
where \(R\) is the equivalent partial thicknesses of the creep layer.