Theory of structure formation in snowfields motivated by penitentes, suncups, and dirt cones

Penitentes and suncups are structures formed as snow melts, typically high in the mountains. When the snow is dirty, dirt cones and other structures can form instead. Building on previous field observations and experiments, this paper presents a theory of ablation morphologies, and the role of surface dirt in determining the structures formed. The glaciological literature indicates that sunlight, heating from air, and dirt all play a role in the formation of structure on an ablating snow surface. The present paper formulates a minimal model for the formation of ablation morphologies as a function of measurable parameters and considers the linear stability of this model. The dependence of ablation morphologies on weather conditions and initial dirt thickness is studied, focusing on the initial growth of perturbations away from a flat surface. We derive a single-parameter expression for the melting rate as a function of dirt thickness, which agrees well with a set of measurements by Driedger. An interesting result is the prediction of a dirt-induced traveling instability for a range of parameters.