  Motivated by the refreezing of melt water in firn we revisit the
one-dimensional percolation of liquid water and non-reactive gas in porous ice.
We analyze the dynamics of infiltration in the absence of capillary forces and
heat conduction to understand the coupling between advective heat and mass
transport in firn. In this limit, we formulate a kinematic wave theory that
results in a 2X2-system of hyperbolic partial differential equations (PDEs)
corresponding to the conservation of composition and enthalpy. For simple
initial conditions (Riemann problems) this system admits self-similar solutions
that illuminate the structure of melting/refreezing fronts and analytical
solutions are provided for 12 basic cases of physical relevance encountered in
the literature. Further we develop an extended kinematic theory that
encompasses the cases when the firn saturates completely to form a perched
water table governed by elliptic PDE so that the model is no longer fully
hyperbolic (local). These solutions provide benchmarks for numerical models of
melt infiltration into firn. They also provide insight into important physical
processes such as the formation of frozen fringes, the perching of meltwater on
pre-existing low porosity layers and the conditions required for impermeable
ice lens formation. Lastly, these analytic solutions can be utilized to improve
and compare the performance of the firn hydrology, ice-sheet and Earth system
models. Our analysis provides a theoretical framework to understand these
important processes in firn which affect the partitioning between meltwater
infiltration and surface runoff and therefore determine the surface mass loss
from ice sheets and its contribution to sea level rise.
