Green’s Function of Mass Transport and the Landform Equation

Transport of the earth surface material associated with diffusion brings a landform change, and the resulted sediment or debris thickness is approximated by a normal distribution. The essential feature of this geomorphic process is described in terms of the transport distance and the spatial spreading of the masses. A unit mass on a slope may move at a rate b downward and spread at a diffusivity a, simultaneously. Thus, we may introduce a kind of Green’s function which gives the diffusion over the area at and the transportation over the distance bt of the unit mass situated on a given point at t = 0 during the time t. The Green’s function is applied to any kind of mass movement, though the constants a and b may change by the processes. Topographic change due to the masses distributed continuously in the source area is derived by integration of the Green’s function of mass transport, and the result gives the general solution of the landform equation which means that the rate of erosion is proportional to the convexity and gradient of land surface. The solution derived from the landform equation shows the retreat and subduing of a vertical scarp at x = 0 with time in the simplest case. Application of the Green’s function to such a present day mass movement as the Ontake landslide in 1984 is also possible, though several problems are open to the future investigations including that from the statistical point of view.