The Platonic Ideal of Stalactite Growth

The chemical mechanisms underlying the growth of cave formations such as stalactites are well-known, yet no theory has yet been proposed which successfully accounts for the dynamic evolution of their shapes. Here we consider the interplay of thin-film fluid dynamics, calcium carbonate chemistry, and CO$_2$ transport in the cave to show that stalactites evolve according to a novel local geometric growth law which exhibits extreme amplification at the tip as a consequence of the locally-varying fluid layer thickness. Studies of this model show that a broad class of initial conditions is attracted to an ideal shape which is strikingly close to a statistical average of natural stalactites. A linear stability analysis shows is used to explain the instability of this state to the formation of centimeter-scale ripples, as commonly seen on a wide range of speleothem surfaces.