Modeling stalagmite growth by first principles of chemistry and physics of calcite precipitation

Growth rates and morphology of stalagmites are determined by the precipitation kinetics of calcite and the supply rates of water to their apex. Current modeling attempts are based on the assumption that precipitation rates decrease exponentially with distance along the surface. This, however, is an arbitrary assumption, because other functions for decrease could be used as well. Here we give a process-oriented model based on the hydrodynamics of a water sheet in laminar radial flow spreading outwards from the apex, and the well known precipitation rates F = α(c − c_eq); c is the actual calcium concentration at distance R from the growth axis, c_eq the equilibrium concentration of calcium with respect to calcite, and α is a kinetic constant. This enables us to calculate the concentration profile c(R) for any point of an actual surface of a stalagmite and consequently the deposition rates of calcite there. The numerical results show that under conditions constant in time the stalagmite grows into an equilibrium shape, which is established, when all points of its surface are shifting vertically by the same distance during a time interval. We also show this by strict mathematical proof. This new model is based entirely on first principles of physics and chemistry. The results show that the modeled precipitation rates can be approximated by a Gaussian decrease along the equilibrium surface. In general from the mathematical proof one finds a relation between the equilibrium radius of the stalagmite, Q the supply rate of water, and α the kinetic constant. This is also verified by numerical calculations. An interesting scaling law is found. Scaling all stalagmites by 1/R_eq and presenting them with the origin at their apex yields identical shapes of all. The shapes of the modeled stalagmites are compared to natural ones and show satisfactory agreement. Finally we explore the effect of varying water supply Q and kinetic constant α on the shape of a growing stalagmite, and estimate the minimum period of change that can be imprinted into the morphology of the stalagmite.