Snow-lines can be thermally unstable
  Volatile species in protoplanetary discs can undergo a phase change from
vapour to solid. These "snow-lines" can play vital roles in planet formation at
all scales, from dust coagulation to planetary migration. In the outer regions
of protoplanetary discs, the temperature profile is set by the absorption of
reprocessed stellar light by the solids. Further, the temperature profile sets
the distribution of solids through sublimation and condensation at various
snow-lines. Hence the snow-line position depends on the temperature profile and
vice-versa. We show that this coupling can be thermally unstable, such that a
patch of the disc at a snow-line will produce either run-away sublimation or
condensation. This thermal instability arises at moderate optical depths, where
heating by absorption of reprocessed stellar light from the disc's atmosphere
is optically thick, yet cooling is optically thin. Since volatiles in the solid
phase drift much faster than volatiles in the vapour phase, this thermal
instability results in a limit-cycle. The snow-line progressively moves in,
condensing volatiles, before receding, as the volatiles sublimate. Using
numerical simulations, we study the evolution of the CO snow-line. We find the
CO snow-line is thermally unstable under typical disc conditions and evolves
inwards from $\sim50$ to $\sim30$~AU on timescales from 1,000-10,000 years. The
CO snow-line spends between $\sim 10\%-50\%$ of its time at smaller
separations, where the exact value is sensitive to the total opacity and
turbulent viscosity. The evolving snow-line also creates ring-like structures
in the solid distribution interior to the snow-line. Multiple ring-like
structures created by moving snow-lines could potentially explain the
sub-structures seen in many {\it ALMA} images.
