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Description
Stalagmites are a classic example of a natural reactive transport system, where the evolution of the solid domain is coupled to the hydrodynamics of a thin fluid film and the precipitation kinetics of calcium carbonate. Nearly sixty years ago, Franke [1] formulated a mathematical model for this process, effectively casting it as a thin-film transport and reaction problem on a moving boundary. He argued that under steady cave conditions the stalagmite approaches an “ideal shape” that translates upward at constant speed without changing form. While this scenario has been reproduced numerically [2], the analytic structure of the invariant shapes has remained unresolved.
We show that Franke’s model admits an exact analytical solution and yields a family of invariant growth forms, organized by a Damköhler-type control parameter that captures the competition between along-surface transport in the film and precipitation-driven depletion [3]. Besides the previously reported columnar solution, the theory predicts flat-top stalagmites with a finite, selected apex radius and conical solutions with sharp tips. These forms correspond to distinct reactive transport regimes, controlled by drip flux, effective CO2 loss to cave air, and precipitation kinetics. We discuss how these ideal shapes can serve as benchmarks for interpreting more complex speleothem geometries observed in nature. Finally, we show how the discrete, finite-volume nature of drip feeding breaks the scale invariance of continuous-film models and selects a non-zero minimal stalagmite radius through the coupling of viscous drop spreading (a reactive gravity current) and precipitation kinetics.
[1] H. Franke, The theory behind stalagmite shapes. Stud. Speleol. 1, 89–95 (1965).
[2] D. Romanov, G. Kaufmann, and W. Dreybrodt, Modeling stalagmite growth by first principles of chemistry and physics of calcite precipitation. Geochim. Cosmochim. Acta 72, 423–437 (2008).
[3] P. Szymczak, A.J.C. Ladd, M. Lipar, and D. Pekarovic, Shapes of ideal stalagmites. Proc. Natl. Acad. Sci. U.S.A. 122, e2513263122 (2025).
| Country | Poland |
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