Erosion on a line

An erosion model is proposed to calculate erosion rates for plane-strain models in which the Earth's surface is represented on a line. The fundamentals of river erosion networks are captured by two principles, Hack's Law, which describes the drainage area structure of river network and a stream-power erosion law, which describes the rate of incision of a river. For a simple morphology of parallel transverse rivers with rectangular drainage basins, this allows the earth's surface to be parameterized by two heights: the trunk stream channel height and the interfluvial ridge height. The resulting expressions are solved for the simple cases of constant uplift rate and a constant mean slope as occurs in critical wedge problems. In the latter case, the uplift rate is variable and changes in space so that the trunk channel elevation and the interfluvial ridge elevation average to maintain a constant mean slope. A general, numerical solution is presented for application to any numerical model with arbitrary surface velocity, variable rock erodibility and precipitation. This algorithm is coupled to a plane-strain, plastic-deformation model to demonstrate the utility of the model.