### Mathematical Representation of the Morphological Evolution of Rills.

Ruth Maria Bianchini de Quadros, Environment Ministry, SGAN 601 Lote 1 Ed. Codevasf 4º andar, Brasília, Brazil

A mathematical model described by the one-dimensional hydrodynamic and sediment continuity equations for simulating the morphological evolution of rills was developed. The equation to determine the soil particles flux was developed based on dimensionless parameters related with the balance between viscous and inertial forces and the balance between kinetic and potential energy, as well, with the soil physical properties and the friction coefficient. With this equation the variable critical shear stress can also be determined through algebraic manipulation. The differential equations were solved by MacCormack finite difference scheme and the boundary conditions by the method of characteristics. Numerical simulations were performed for both steady and unsteady flow. The experimental data of Elliot et al. (1989) in soils from areas of agricultural importance of the USA were utilized for the calibration and sensitivity analysis. The main variables of the calibration were related with both the soil and flow resistance, the aggregates diameter were found to be the most sensitive. Linear regressions were computed between soil detachment rate and soil particle flux to determine the main variable and coefficients responsible for the soil particle detachment. The results showed that the balance between kinetic and potential energy was the main factor responsible for the soil detachment rate (R=0.93), followed by critical shear stress (R=-0.83). Decreasing trends of soil detachment rate occurred with both the clay concentration (R=-0.63) and the aggregates diameter (R=-0.49). These trends confirm the arguments found in the literature that good soil aggregation ensures erosion resistance. The results showed also that the soil detachment process is very complex, but that the model developed can contribute to better understand it. The present model is very simple, the calibration and implementation are easy; however it still needs further effort to bring it to a completed state.