Water flow model on vegetated hillslopes with erosion

The water circulation in the Soil-Plant-Atmosphere continuum and particularly the soil erosion induced by water are problems of main concern in the new era of climate change. The present paper aims to provide a mathematical tool to investigate the water-soil and water-plant interactions involved in the complex process of water flow on plant-covered soil surfaces. Basically, the mathematical model consists of an extended Saint-Venant system of equations for water flow coupled with Hairsine-Rose equations for soil erosion. The classical Saint-Venant model is thus modified in order to take into account the presence of plants on the soil surface.
A numerical approximation of the solution of our model is built using a Finite Volume Method for the discretization in space and a fractional time-step scheme to discretize the time variable and resulting time derivatives. Several properties of the scheme with physical relevance are also discussed and investigated.
In order to validate both the model and the numerical method, and to see if essence of the reality is adequately reflected, a series of qualitative and quantitative tests are performed. Given that the mathematical model is flexible enough to reflect the variability of the environmental variables such as soil structure, soil surface roughness, or plant cover structure, each numerical experiment is constructed as an image of a target hydrological context. The dam break problem, flash floods, water-induced soil erosion in a catchment basin are all subjects of numerical analysis. It is shown that the presence of the plant cover drastically modifies the water dynamics and the distribution of the soil eroded particles and one can quantitatively evaluate such effects. The methods described in the paper can also help one to manage the environmental resources in order to avoid the water induced disasters.