A mathematical model for the morphological evolution of a volcano on an island

Abstract The present paper attempts a mathematical description, in two dimensions, of the morphological evolution of a volcano, as a result of erosional processes and volcanic activity. The whole morphological evolution is represented by a partial differential equation, in which erosion and volcanic activity are represented by the erosion coefficient K and a (mass) transfer function Trsf , respectively. The transfer function expresses the deposition rate of lava and pyroclastic material at the slopes of the volcano. The boundary conditions of the differential equation express a volcano which crowns an island area with length L and a time constant sea level. The solutions of the differential equation represent the morphological evolution of the volcano through time, under different initial states and geological conditions. It is concluded that the altitude of a dead volcano tends to zero with time. On the other hand, an active volcano with a time constant transfer function tends to a steady state of dynamic equilibrium. The dimensions of the profile at the steady state depend on the mass transfer rate and the erosion coefficient. The time at which the volcano comes to the steady state is proportional to the square length L 2 and inversely proportional to the erosion coefficient. The results and conclusions of this paper may be useful in understanding, in quantitative terms, how the relief of a volcano may evolve in time and which factors control the whole process.