A geometric mechanism explains the shape of scallops and other sharp patterns in dissolution or melting

In glaciology, karstology, speleology or planetology, regular and periodic patterns are often
observed on dissolving, melting or sublimating solid substrates. One of the most common is
known as scallops, and consists of a cellular pattern of cups-like concavities surrounded by very
sharp crests. They can be found typically on the walls of limestone caves carved by underground
rivers. Yet very similar patterns form on the immersed part of icebergs, on high-altitude glaciers
or on the surface of meteorites during their entry into the atmosphere [1]. The similarity between
these patterns, despite the wide range of materials and hydrodynamic conditions, suggests a
common and general mechanism.
By comparing field measurements, numerical models and experiments, we propose a geometric
approach to explain the generic emergence of scallops [2]. We first characterize the morphology
of scallops found on the walls of a limestone cave thanks to 3D reconstruction by
photogrammetry, and demonstrate the presence of crests which can be seen as singular
structures. Then, we discuss the results of numerical models of interface propagation. They
allow us to interpret the appearance of crests and the formation of cellular structures as a direct
consequence of the fact that the erosion velocity is always directed along the normal to the
interface. Finally, we carry out a simple experiment in which patterns are created by dissolution,
on the surface of a block of salt, by a solutal convection instability [3]. In accordance with our
model, we report the emergence of a cellular pattern of concavities surrounded by sharp crests,
very reminiscent of natural scallops. It confirms that the formation of scallops is largely
independent of the details of the flow but rather results from a geometric mechanism. This
general mechanism can also explain the common presence of crests or spikes on other
geological patterns created by dissolution.

[1] P. Meakin, B. Jamtveit, Geological pattern formation by growth and dissolution in
aqueous systems. Proceedings of The Royal Society A 466, 659 (2010).
[2] M. Chaigne, S. Carpy, M. Massé, J. Derr, S. Courrech du Pont, M. Berhanu, Emergence of
tip singularities in dissolution patterns. Proceedings of the National Academy of Sciences
120(48), e2309379120 (2023).
[3] C. Cohen, M. Berhanu, J. Derr, S. Courrech du Pont, Buoyancy driven dissolution of
inclined blocks: Erosion rate and pattern formation. Physical Review Fluids 5, 053802 (2020).