$\displaystyle\displaystyle\lambda_{2,3}=\frac{-\alpha\epsilon^{2}\underline{h}-a\pm\sqrt{(a-\alpha\epsilon^{2}\underline{h})^{2}+4\alpha\epsilon^{2}\frac{am\underline{h}\underline{c}}{\bar{\rho}_{s}}}}{2}.$

Consequently the asymptotic development of the eigenvalues of the system when
$\eta^{2}\to+\infty$ and $\xi=0$, $K=0$ are

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\lambda_{1}(0,\eta)=0,\vspace{0.3cm}\\\
\displaystyle\lambda_{2}(0,\eta)=-\alpha\underline{h}\eta^{2}+o(\eta^{2}),\vspace{0.1cm}\\\
\displaystyle\lambda_{3}(0,\eta)=\frac{s\bar{\rho}_{s}}{e\underline{h}}\left(\frac{m\underline{c}}{\bar{\rho}_{s}}-1\right)+o(1).\end{array}\right.$

This achieves the proof of Proposition 4.2. ∎

##  Appendix D Numerical scheme

In this appendix, we give the numerical scheme used for the simulations of the
system.

#### Stationary regime for the water

The ratio between the erosion speed and the fluid velocity is small and we
will suppose that the time derivatives of $c$ and $h$ are small. In order to
justify this assumption, we rescale the equations by introducing the
characteristic variables $Z$ the eroded height, $L$ the characteristic length,
and $T=Z/e$ the characteristic time. We define the dimensionless variables:

$\displaystyle h^{\prime}:=\frac{h}{H},\quad z^{\prime}=\frac{z}{Z},\quad
v^{\prime}:=\frac{v}{V},\quad c^{\prime}:=\frac{c}{c_{sat}},$ $\displaystyle
x^{\prime}:=\frac{x}{L},\quad t^{\prime}:=\frac{Z}{e}t.$

Therefore, dropping the primes, System (8) is written as:

$\displaystyle\left\\{\begin{array}[]{llll}\displaystyle\partial_{t}h+\frac{1}{\epsilon}\operatorname{div}(hv)=0,\\\
\displaystyle\partial_{t}(ch)+\frac{1}{\epsilon}\operatorname{div}(chv)=\frac{\rho_{s}}{c_{sat}}\left(h^{m}|v|^{n}-\frac{s}{e}c\right),\\\
\displaystyle\partial_{t}z=K\Delta z-h^{m}|v|^{n}+\frac{s}{e}c,\\\
\displaystyle v=-\frac{\mu}{V}\nabla(h+z).\end{array}\right.$

Here, we defined $\epsilon:=\frac{Le}{HV}$. If $\epsilon\ll 1$, as generally
$\rho_{s}\gg c_{sat}$, we can neglect the terms $\partial_{t}h$ and
$\partial_{t}(ch)$ in the equation $h$ et $ch$. Thus, assuming enough
regularity on the solutions, we can write $\operatorname{div}(chv)=hv.\nabla
c+c\operatorname{div}(hv)=hv.\nabla c$. Moreover we fix $V=\mu$ for
simplicity. We obtain:

$\displaystyle\left\\{\begin{array}[]{llll}\displaystyle\operatorname{div}(hv)=0,\\\
hv.\nabla c=\frac{\rho_{s}}{c_{sat}}\left(h^{m}|v|^{n}-\frac{s}{e}c\right),\\\
\displaystyle\partial_{t}z=K\Delta z-h^{m}|v|^{n}+\frac{s}{e}c,\\\
\displaystyle v=-\nabla(h+z).\end{array}\right.$ (67)

#### Discretisation of the equations

The constraint on $h$ at each time $\operatorname{div}(h\nabla(h+z))=0$ leads
to a fully nonlinear problem that may be hard to solve numerically. Instead,
we discretize this equation as

$\displaystyle\operatorname{div}(h^{n-1}\nabla
h^{n})+\operatorname{div}(h^{n}\nabla z^{n})=0,$ (68)

where $h^{n}$ represents the fluid height at time $t_{n}=n\delta t$ whereas
the surface height at time $n$, $z^{n}$ has been calculated by an explicit
Euler method, using the solutions at time $n-1$, $h^{n-1}$ and $c^{n-1}$:
$\forall 1\leq i\leq N_{x}$, $1\leq j\leq N_{y}$,

$\displaystyle\displaystyle\frac{z_{i,j}^{n}-z_{i,j}^{n-1}}{\text{dt}}=K\frac{z_{i+1,j}^{n-1}-2z_{i-1,j}^{n-1}+z_{i,j}^{n-1}}{\text{dx}}+K\frac{z_{i,j+1}^{n-1}-2z_{i,j-1}^{n-1}+z_{i,j}^{n-1}}{\text{dy}}-(E-S)_{i,j}^{n}.$

We discretise the equation on $h^{n}$ by a centered finite volume scheme,
which writes:

$\displaystyle 0=$
$\displaystyle\frac{h_{i+1/2,j}^{n-1}(h_{i+1,j}^{n}-h_{i,j}^{n})-h_{i-1/2,j}^{n-1}(h_{i,j}^{n}-h_{i-1,j}^{n})}{\text{dx}^{2}}+\frac{h_{i,j+1/2}^{n-1}(h_{i,j+1}^{n}-h_{i,j}^{n})-h_{i,j-1/2}^{n-1}(h_{i,j}^{n}-h_{i,j-1}^{n})}{\text{dy}^{2}}$
$\displaystyle+\frac{h_{i+1/2,j}^{n}(z_{i+1,j}^{n}-z_{i,j}^{n})-h_{i-1/2,j}^{n}(z_{i,j}^{n}-z_{i-1,j}^{n})}{\text{dx}^{2}}+\frac{h_{i,j+1/2}^{n}(z_{i,j+1}^{n}-z_{i,j}^{n})-h_{i,j-1/2}^{n}(z_{i,j}^{n}-z_{i,j-1}^{n})}{\text{dy}^{2}},$

where $h_{i+1/2,j}^{n-1}=\frac{h_{i,j}^{n-1}+h_{i+1,j}^{n-1}}{2}$. Finally,
Equation (43) on $c$ is discretised by an upwind finite volume scheme,
considering the variable $x$ as a time variable. The time discretisation is an
explicit Euler scheme.

$\displaystyle\displaystyle\frac{c_{i+1,j}-c_{i,j}}{dt}+\frac{max(v_{x}[i,j],0)}{v_{y}[i,j]}\displaystyle\frac{c_{i,j}-c_{i,j-1}}{\text{dy}}+\frac{min(v_{x}[i,j],0)}{v_{y}[i,j]}\displaystyle\frac{c_{i,j}-c_{i,j+1}}{\text{dy}}=-\frac{\rho_{s}}{h_{i,j}v_{x}[i,j]}(E-S)_{i,j}^{n}.$

#### Boundary conditions

We choose periodic boundary conditions in the $y$ direction. There remains two
boundaries, the top and the bottom of the tilted plane. The boundary
conditions for the soil height are Neumann conditions: $\forall 1\leq j\leq
N_{y}$,

$\displaystyle z_{0,j}^{n}=z_{1,j}^{n},\quad
z_{N_{x},j}^{n}=z_{N_{x+1},j}^{n}.$

The boundary conditions for the water height and for the concentration of
sediments are a Dirichlet condition at the top because the incoming flow is
fixed, and a free flow Neumann condition at the bottom.

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle h_{0,j}^{n}=h_{0},\quad
h_{N_{x},j}^{n}=h_{N_{x+1},j}^{n},\\\ \displaystyle c_{0,j}^{n}=c_{0},\quad
c_{N_{x},j}^{n}=c_{N_{x+1},j}^{n}.\end{array}\right.$

#### Acknowledgments

PD holds a visiting professor association with the Department of Mathematics,
Imperial College London.

#### Availability of Data and Materials

The code of numerical simulations generated during the current study is
available at https://github.com/juliebinard/landscape_evol_finite_volum.

## References

  * [1] R. A. Adams and J. J. Fournier.  Sobolev spaces.  Elsevier, 2003.
  * [2] S. K. Anand, M. Hooshyar, and A. Porporato.  Linear layout of multiple flow-direction networks for landscape-evolution simulations.  Environmental Modelling & Software, 133:104804, 2020.
  * [3] A. L. Andrew, K.-W. E. Chu, and P. Lancaster.  Derivatives of eigenvalues and eigenvectors of matrix functions.  SIAM journal on matrix analysis and applications, 14(4):903–926, 1993.
  * [4] S. Bonetti, M. Hooshyar, C. Camporeale, and A. Porporato.  Channelization cascade in landscape evolution.  Proceedings of the National Academy of Sciences, 117(3):1375–1382, 2020.
  * [5] A. Chen, J. Darbon, G. Buttazzo, F. Santambrogio, and J.-M. Morel.  On the equations of landscape formation.  Interfaces and Free Boundaries, 16(1):105–136, 2014.
  * [6] A. Chen, J. Darbon, and J.-M. Morel.  Landscape evolution models: A review of their fundamental equations.  Geomorphology, 219:68–86, 2014.
  * [7] P. Cherrier and A. Milani.  Linear and quasi-linear evolution equations in Hilbert spaces.  American Mathematical Society Providence, 2012.
  * [8] W. E. H. Culling.  Analytical theory of erosion.  The Journal of Geology, 68(3):336–344, 1960.
  * [9] W. Davis.  The convex profile of bad-land divides.  Science, (508):245–245, 1892.
  * [10] L. C. Evans.  Partial differential equations, volume 19.  American Mathematical Soc., 2010.
  * [11] G. K. Gilbert.  Report on the Geology of the Henry Mountains.  US Government Printing Office, 1877.
  * [12] G. K. Gilbert.  The convexity of hilltops.  The Journal of Geology, 17(4):344–350, 1909.
  * [13] A. Guérin, J. Derr, S. C. Du Pont, and M. Berhanu.  Streamwise dissolution patterns created by a flowing water film.  Physical Review Letters, 125(19):194502, 2020.
  * [14] A. D. Howard and G. Kerby.  Channel changes in badlands.  Geological Society of America Bulletin, 94(6):739–752, 1983.
  * [15] M. Lebrun, M. Colom, J. Darbon, and J.-M. Morel.  Numerical simulation of landscape evolution models.  Image Processing On Line, 8:219–250, 2018.
  * [16] D. S. Loewenherz.  Stability and the initiation of channelized surface drainage: a reassessment of the short wavelength limit.  Journal of Geophysical Research: Solid Earth, 96(B5):8453–8464, 1991.
  * [17] G. Métivier.  Para-differential calculus and applications to the Cauchy problem for nonlinear systems.  2008\.
  * [18] J. Morris.  The routh and routh-hurwitz stability criteria: Their derivation by a novel method using comparatively elementary algebra.  Aircraft engineering and aerospace technology, 1962.
  * [19] T. R. Smith and F. P. Bretherton.  Stability and the conservation of mass in drainage basin evolution.  Water Resources Research, 8(6):1506–1529, 1972.