Mathematical model of ammonia nitrogen transport from soil to runoff on irregular slopes

The loss of nitrogen on sloping land causes a decline in land productivity and non-point source pollution. Research on nitrogen loss typically occurs on land with a uniform slope. However, irregular slopes are the most widely distributed types of slope in nature. Ignoring the effects of irregular slopes may cause misestimations of nitrogen loss on sloped land. In this study, a mathematical model was established to describe the process of ammonia nitrogen transfer from the surface soil to runoff under different slope shapes and rainfall intensities. The slopes were generalized as parts of circles; a slope curvature parameter was proposed to describe topographic and hydraulic differences. The model was evaluated using nine slope degrees (concave slopes of middle-down: 10, 20, 30, and 40 cm; uniform slope; and convex slopes of middle-up: 10, 20, 30, and 40 cm) and three rainfall intensities (0.125, 0.083, and 0.042 cm min−1). Runoff rates were fitted using the established runoff model based on Horton's infiltration equation. The ammonia nitrogen transfer from the soil layer to runoff was built based on the diffusion model. The governing equations of the models were solved numerically. The study found that the mean runoff rates of concave slopes were significantly lower than those of convex slopes under rainfall intensities of 0.083 and 0.042 cm min−1. At a rainfall intensity of 0.042 cm min−1, the mean ammonia nitrogen loss rates of concave slopes decreased by 21.70% to 34.17% compared to the uniform slope, while the loss rates of convex slopes increased by 18.49% to 58.97% (p < 0.05). Similar trends were observed at rainfall intensities of 0.083 and 0.125 cm min−1. Overall, the runoff rates, ammonia nitrogen concentration and ammonia nitrogen loss rate tended to decrease with an increasing slope curvature parameter. The simulated results for the runoff rate and ammonia nitrogen transport process were consistent with the observed data, with R2 values ranging from 0.80 to 0.98 for the runoff rate and 0.64 to 0.95 for the ammonia nitrogen concentration. Further correlation analyses were performed to determine the relationships among the slope curvature, rainfall intensity, hydraulic, and nutrient transport parameters. The results showed that these parameters can be described as a combined linear function of the slope curvature parameter and rainfall intensity. Moreover, the sensitivity of the model to various parameters showed that the slope shape and curvature had an important impact on nitrogen transport. This model can be used to improve the theoretical basis for developing nitrogen-loss control methods.