Mathematical modeling of soil erosion by rainfall and shallow overland flow

New analytical and numerical solutions are developed to both the kinematic approximation to the St Venant equations and the Hairsine-Rose (HR) soil erosion model in order to gain a better physical understanding of soil erosion and sediment transport in shallow overland flow. The HR model is unique amongst physically based erosion models in that it is the only one that: considers the entire distribution of the soil s sediment size classes, considers the development of a layer of deposited non-cohesive sediment having different characteristics to the original underlying cohesive soil and considers separately the erosion processes of rainfall detachment, runoff entrainment and gravitational deposition. The method of characteristics and the method of lines were used to develop both the analytical and numerical solutions respectively. These solutions were obtained for boundary and initial conditions typical of those used in laboratory flume experiments along with physically realistic constant and time dependent excess rainfall rates. Depending on the boundary and initial conditions, interesting new solutions of the kinematic wave equation containing expansion waves, travelling shocks as well as solutions which split into an upslope and downslope drying profiles were found. Numerical solutions of the HR model were applied to the experimental flume data of Polyakov and Nearing (2003) obtained under flow conditions which periodically cycled between net erosion and net deposition conditions. While excellent agreement was found with suspended sediment data, the analysis suggested that an additional transport mechanisms, traditionally not included in soil erosion models, was occurring. While the inclusion of bed-load transport improved the ii overall model prediction, it was still not sufficient. Subsequent asymptotic analysis then showed that the interaction of the flow with an evolving bed morphology was in fact far more important than bed load transport. A very interesting finding from this work showed that the traditional criterion of validating sediment transport model based solely on suspended sediment data was not sufficient as reliable predictions could be obtained even when important transport mechanisms were neglected. Experimental plots of sediment discharge or suspended sediment concentration against water discharge in overland flow have been shown to contain significant hysteresis between the falling and rising limbs of the discharge hydrograph. In the final Chapter, the numerical solution developed for the complete system of soil erosion and kinematic flow was used to show that it was possible for the HR model to simulate three of the four hysteresis loops identified in the literature. Counter clock-wise loops, clock-wise loops and figure 8 loops could all be produced as a result of starting with different initial conditions, being mi(x; 0) = 0, mi(x; 0) = pimt and mi(x; 0) = 0:5pimt respectively. This is the first time that these types of hysteresis loops have been produced by any erosion model. The generation of these hysteresis loops are physically explainable in terms of sediment availability and is consistent with data obtained on the field scale.