Stalactite Growth as a Free-Boundary Problem: A Geometric Law and Its Platonic Ideal

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 ${\mathrm{C}\mathrm{O}}_2$ 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.

Figure 1

Geometry of fluid flow along the surface of a stalactite (a) and model for determination of growth rate (b).

Figure 2

Aspects of the growth law. (a) Growth velocity $v_c$ versus $p\mathrm{H}$, using $\mathrm{C}{\mathrm{O}}_2$ partial pressure in the cave atmosphere of $3\times {10}^{-4}\mathrm{a}\mathrm{t}\mathrm{m}$, a temperature of $20°\mathrm{C}$, (i) $\overline{[\mathrm{C}{\mathrm{a}}^{2+}]}$ of 100 ppm and volumetric fluid flow $Q=2{\mathrm{c}\mathrm{m}}^3/\mathrm{h}$, and (ii) $\overline{[\mathrm{C}{\mathrm{a}}^{2+}]}=300\mathrm{p}\mathrm{p}\mathrm{m}$ and $Q=20{\mathrm{c}\mathrm{m}}^3/\mathrm{h}$. Formulas for constants taken from 13. (b) The function of tangent angle $\theta$ in Eq. (9). (c) A rounded initial condition evolves into a fingered shape. (d) Aligning the tips of the growing shapes in (c) shows at early times a rapid collapse to a common form.

Figure 3

Comparison between observed stalactite shapes and the Platonic ideal. Three examples (a)–(c) are shown, each next to an ideal shape of the appropriate aspect ratio and size (a′)–(c′). Scale bars in each are 10 cm. (d) Master plot of stalactite shapes, rescaled as described in text. The average of 20 stalactites is shown with green circles and red uncertainties, and compared with the ideal (black curve).