Exact solutions of the Hairsine–Rose precipitation-driven erosion model for a uniform grain-sized soil

SummaryHairsine and Rose developed a mechanistic, one-dimensional, precipitation-driven erosion model that, since its appearance, has been validated by several sets of experimental results. The model allows any sediment particle to be present in one of three zones, viz., the flow zone, the deposited layer, or the original soil. The model has the general form of a two-region model, in which advection is the only transport process. For the special case of a soil composed of a single particle size and for overland flow that occurs at a steady rate and with a uniform depth, it is possible to derive fully explicit analytical solutions to the model. Details of the solutions for a slightly generalized mathematical form of the model are provided. The Goldstein J function, which appears commonly in two-region model solutions, was modified to accommodate some of the solutions presented. The form of the model analyzed indicated that, based only on sediment concentrations in runoff water, it is not possible to distinguish one mechanistic feature of the Hairsine–Rose model, i.e., that raindrop-induced detachment of the undisturbed soil moves directly into the flowing water. From the point of view of the model, it is equally plausible for raindrop impact to move sediment directly into the deposited layer.