Stability analysis of crenulations on speleothem surfaces: the role of hydrodynamics

Speleothem like stalactites ad stalagmites develop a formidable variety of surface patterns, the most evident and recognized by speleologists being the crenulations, i.e., some regular ripples of centimeter-scale. Despite the main chemical mechanisms of calcite precipitation/dissolution processes are well understood, and some recent works have provided a rational and physically-based justification to the overall shape of stalactites, to date, the mathematical problem related to the pattern instability of crenulations remained quantitatively unsolved. The reason of this gap is due to the role of the hydrodynamics of the thin drip water film which interacts with the surface pattern and which transport the solutes by convection and diffusion and is the site of the chemical reactions. A recent attempt to develop a stability analysis with a simplified shallow-water model in fact fails the prediction of instability. In the present work, we develop a novel linear stability for the water/calcite interface that is able to foresee the pattern instability associated to crenulation correctly. The adoption of lubrification theory for small Reynolds numbers (Re<1) and Orr-Sommerfeld equation otherwise has provided the correct solution for the 2D flow-field perturbation and the corresponding surface perturbation, the phase lag thereof being fundamental to trigger the instability. The calcite flux on the interface has been modeled by the PWP (Plummer-Buhmann-Dreybodt) equation and advective-diffusive equations for the perturbation of calcium and carbon dioxide perturbations have been resolved in term of Frobenious series. The results appear encouraging: the order of magnitude of the selected wavenumber, λ, are in fact successfully detected. In a reasonable range of the Reynolds number (Re=0.01-10) the outcomes of the model show a little dependence on [Ca+], temperature and atmospheric concentration of CO2, hence a rough linear approximation of the relationship λ= λ(Re) can be derived. Theoretical results are compared with data from Bossea Cave (Itay). Such results can provided a simple tool to reconstruct the ancient flows with applications in paleoclimatology. More info: Carlo Camporeale (see also our PRL pubblication)