  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 work
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 work formulates a
mathematical model for the formation of ablation morphologies as a function of
measurable parameters. The dependence of ablation morphologies on weather
conditions and initial dirt thickness are 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 travelling instability for a range of parameters.
