Raindrop erosion of tillage induced microrelief: possible use of the diffusion equation

The purpose of this paper is to evaluate the possibility of using the diffusion equation for raindrop erosion modelling. We wanted in particular to know if such a model could provide accurate interpolations of microrelief between two known dates. In a theoretical section, we show that the assumption that soil particles follow parabolic trajectories when splashed by raindrop impacts leads to a diffusion equation. This equation suggests a linear relation between Δz, the variation of height between two dates, and the Laplacian ∇2z (∇2z=∂2z/∂x2+∂2z/∂y2). This relation is confirmed by data from a simulated rainfall experiment carried out in the sandy soils of the Senegalese groundnut belt. Four square plots of side 4 m each were used. They were hoed with a traditional horse-drawn three-tined hoe. Three rains of 70 mm h−1 lasting 30 min each were applied. An automated relief meter designed and constructed by the authors was used to measure the distribution of heights for every 5 cm before the first rain, and after the first and the third rains. The mean correlation coefficient of the model was 62% for the first rain and 46% for the next two rains. Besides raindrop erosion, compaction occurred during the first rain. Adding a crude description of compaction enhanced the mean of the correlation coefficients of the model up to 70% for the first rain. Furthermore, the coefficient of variation of the four adjusted total diffusion lessens from 10 to 6%. The simulated surfaces were smoother than the real ones, which was an expected result, but the surface storage capacity was overestimated. The latter result illustrates the role of runoff in shaping the flow paths it follows and, consequently, in lessening the surface storage capacity. The main conclusion is that the diffusion equation provides a promising frame for further development of models simulating microrelief evolution during rainfall. Another conclusion is that these models should integrate existing routines for runoff erosion at small scale in order to simulate surfaces with realistic hydraulic properties.