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 oftenobserved on dissolving, melting or sublimating solid substrates. One of the most common isknown as scallops, and consists of a cellular pattern of cups-like concavities surrounded by verysharp crests. They can be found typically on the walls of limestone caves carved by undergroundrivers. Yet very similar patterns form on the immersed part of icebergs, on high-altitude glaciersor on the surface of meteorites during their entry into the atmosphere [1]. The similarity betweenthese patterns, despite the wide range of materials and hydrodynamic conditions, suggests acommon and general mechanism.By comparing field measurements, numerical models and experiments, we propose a geometricapproach to explain the generic emergence of scallops [2]. We first characterize the morphologyof scallops found on the walls of a limestone cave thanks to 3D reconstruction byphotogrammetry, and demonstrate the presence of crests which can be seen as singularstructures. Then, we discuss the results of numerical models of interface propagation. Theyallow us to interpret the appearance of crests and the formation of cellular structures as a directconsequence of the fact that the erosion velocity is always directed along the normal to theinterface. 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 ourmodel, 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 largelyindependent of the details of the flow but rather results from a geometric mechanism. Thisgeneral mechanism can also explain the common presence of crests or spikes on othergeological patterns created by dissolution.[1] P. Meakin, B. Jamtveit, Geological pattern formation by growth and dissolution inaqueous 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 oftip singularities in dissolution patterns. Proceedings of the National Academy of Sciences120(48), e2309379120 (2023).[3] C. Cohen, M. Berhanu, J. Derr, S. Courrech du Pont, Buoyancy driven dissolution ofinclined blocks: Erosion rate and pattern formation. Physical Review Fluids 5, 053802 (2020).