Emergence of tip singularities in dissolution patterns

Significance

Why are crests and spikes so often observed in nature when soluble rocks are dissolved by the action of water flow? In particular, concave depressions delimited by sharp crests called scallop patterns can be found on the walls of underground caves. Here, we show that the emergence of the spikes and crests is explained in a general way by a geometrical mechanism describing the evolution of a receding interface. From a dissolution pattern with a characteristic scale set by hydrodynamic mechanisms, the wall undulations will deform to transform the smooth humps into sharp crests while widening the troughs. This mechanism also applies to melting interfaces and more generally to any ablation process.

Abstract

Chemical erosion, one of the two major erosion processes along with mechanical erosion, occurs when a soluble rock-like salt, gypsum, or limestone is dissolved in contact with a water flow. The coupling between the geometry of the rocks, the mass transfer, and the flow leads to the formation of remarkable patterns, like scallop patterns in caves. We emphasize the common presence of very sharp shapes and spikes, despite the diversity of hydrodynamic conditions and the nature of the soluble materials. We explain the generic emergence of such spikes in dissolution processes by a geometrical approach. Singularities at the interface emerge as a consequence of the erosion directed in the normal direction, when the surface displays curvature variations, like those associated with a dissolution pattern. First, we demonstrate the presence of singular structures in natural interfaces shaped by dissolution. Then, we propose simple surface evolution models of increasing complexity demonstrating the emergence of spikes and allowing us to explain at long term by coarsening the formation of cellular structures. Finally, we perform a dissolution pattern experiment driven by solutal convection, and we report the emergence of a cellular pattern following well the model predictions. Although the precise prediction of dissolution shapes necessitates performing a complete hydrodynamic study, we show that the characteristic spikes which are reported ultimately for dissolution shapes are explained generically by geometrical arguments due to the surface evolution. These findings can be applied to other ablation patterns, reported for example in melting ice.