Well-posedness and stability analysis of a landscape evolution model

TL;DR

This paper analyzes a PDE system modeling landscape evolution, establishing well-posedness, studying pattern formation mechanisms like rills and gullies, and validating findings through numerical simulations.

Contribution

It provides the first well-posedness analysis for the landscape evolution PDE system and links spectral instability to pattern formation in landscapes.

Findings

System is well-posed for short time with non-vanishing initial fluid height.

Spectral analysis identifies conditions for pattern formation like rills and gullies.

Numerical simulations confirm the instability-driven pattern formation mechanism.

Abstract

In this paper, we consider a system of partial differential equations modeling the evolution of a landscape. A ground surface is eroded by the flow of water over it, either by sedimentation or dilution. The system is composed by three evolution equations on the elevation of the ground surface, the fluid height and the concentration of sediment in the fluid layer. We first consider the well-posedness of the system and show that it is well posed for short time and under the assumption that the initial fluid height does not vanish. Then, we focus on pattern formation in the case of a film flow over an inclined erodible plane. For that purpose, we carry out a spectral stability analysis of constant state solutions in order to determine instability conditions and identify a mechanism for pattern formations. These patterns, which are rills and gullies, are the starting point of the formation of rivers and valleys in landscapes. Finally, we make some numerical simulations of the full system in order to validate the spectral instability scenario, and determine the resulting patterns.