# An Idealized 3D Model of Interfacial Instability of Aeolian Bedform
## 3 Solution Method
### Linear Stability Analysis  
The stability of the planar sand bed is investigated by a normal mode analysis. We introduce small perturbations of the sand bed (strictly infinitesimal), hence the problem can be linearized and the small-amplitude sinusoidal topographic perturbations of the form can be considered:  
\[h(x,y,t)=h_{0}+\epsilon h_{1}(t)\cos(k\cdot(x+y)) \tag{8}\]  
where \(\epsilon h_{1}(t)\) is the perturbation amplitude, \(\epsilon\) is a small quantity, and \(k=k_{x}e_{x}+k_{y}e_{y}\) is the topographic wave vector. Substituting Equation (8) into Equation (6), using the linear approximation and neglecting all nonlinear terms, keeping only the linear terms we obtain the perturbed bed evolution equation  
\[ \begin{array}{l}\frac{\partial h_{1}(t)}{\partial t}=\left(-D\left(k_{x}^{ 2}+k_{y}^{2}\right)-\overline{a}\left(\mu_{x}k_{x}^{2}+\epsilon\mu_{y}k_{y}^{ 2}\right)\right.\\ \left.+\,i\left(\cot\beta_{1}(\hat{p}(k_{x})-1)k_{x}+\epsilon\cot\beta_{2} \left(\hat{p}\left(\,k_{y}\right)-1\right)k_{y}\right)\right)h_{1}(t)\end{array}  \tag{9}\]  
where \(\hat{p}(k_{x})\) and \(\hat{p}(\,k_{y})\) are the Fourier transform of \(p(a)\), \(\overline{a}=\int_{0}^{\infty}ap(a)da\). The solution to (9) is obtained by writing  
\[h_{1}(t)=\tilde{h}e^{\omega t} \tag{10}\]Then, we obtain the complex growth rate of the bed perturbation as a function of the topographic wave vector; its real and imaginary parts can be written as:  
\[ \begin{array}{c}\mathrm{Re}(\omega)=\cot\beta_{1}\exp\bigl{(}-k_{x}^{2}\sigma^{ 2}/2\bigr{)}\sin(\overline{a}k_{x})k_{x}+\epsilon\cot\beta_{2}\exp\Bigl{(}-k_{ y}^{2}\sigma^{2}/2\Bigr{)}\sin\bigl{(}\overline{a}k_{y}\bigr{)}k_{y}\\ \qquad\qquad\qquad-D\Bigl{(}k_{x}^{2}+k_{y}^{2}\Bigr{)}-\overline{a}\Bigl{(} \mu_{x}k_{x}^{2}+\epsilon\mu_{y}k_{y}^{2}\Bigr{)}\end{array}  \tag{11}\]  
\[ \begin{array}{c}\mathrm{Im}(\omega)=\cot\beta_{1}\bigl{(}\exp\bigl{(}-k_{x}^ {2}\sigma^{2}/2\bigr{)}\cos(\overline{a}k_{x})-1\bigr{)}k_{x}\\ \qquad\qquad+e\cot\beta_{2}\Bigl{(}\exp\bigl{(}-k_{y}^{2}\sigma^{2}/2\bigr{)} \cos(\overline{a}k_{y})-1\bigr{)}k_{y}\end{array}  \tag{12}\]  
The complex growth rate \(\omega\) contains two pieces of information: the real part gives the growth or decay rate of the amplitude of perturbation (i.e., by \(\mathrm{R}(\omega)\)), and the imaginary part describes the propagation of perturbation in the space (i.e., by \(\mathrm{Im}(\omega)\)). The bed is linearly stable, if the amplitudes decay for all bed perturbation modes (\(\mathrm{R}(\omega)<0\)), whereas if at least one mode bed of the amplitude grows (\(\mathrm{R}(\omega)>0\)), the bed is unstable. In order to investigate the mechanisms responsible for the sand ripples dynamics, it is necessary to identify the various contributions to the complex growth rate. The growth rate Equation (11) consists of two contributions involving the creep flux and reptation flux in the \(x\) direction and the \(y\) direction, which depend on the slope of the bed and on the impact of saltation, respectively. It is the sum of the growth rates due to creep and reptation, according to  
\[\mathrm{Re}(\omega)=\mathrm{Re}(\omega_{\mathrm{c}})+\mathrm{Re}(\omega_{ \mathrm{r}}) \tag{13}\]  
These contributions are given by  
\[\mathrm{Re}(\omega_{\mathrm{c}})=-D\Bigl{(}k_{x}^{2}+k_{y}^{2}\Bigr{)} \tag{14}\]  
\[ \begin{array}{c}\mathrm{Re}(\omega_{\mathrm{r}})=\cot\beta_{1}\exp\bigl{(}- k_{x}^{2}\sigma^{2}/2\bigr{)}\sin(\overline{a}k_{x})k_{x}\\ \qquad\qquad+e\cot\beta_{2}\exp\Bigl{(}-k_{y}^{2}\sigma^{2}/2\Bigr{)}\sin \bigl{(}\overline{a}k_{y}\bigr{)}k_{y}\end{array} -\overline{a}\Bigl{(}\mu_{x }k_{x}^{2}+\epsilon\mu_{y}k_{y}^{2}\Bigr{)} \tag{15}\]  
In turn, the reptation growth rate \(\mathrm{Re}(\omega_{\mathrm{r}})\) can be written as  
\[\mathrm{Re}(\omega_{\mathrm{r}})=\mathrm{Re}(\omega_{\mathrm{r,je}})+\mathrm{ Re}(\omega_{\mathrm{r,se}}) \tag{16}\]  
This consists of the contributions due to the impact-induced reptation flow \(\mathrm{Re}(\omega_{\mathrm{r,je}})\), and impact-induced gravity flow \(\mathrm{Re}(\omega_{\mathrm{r,se}})\).