Meltwater is produced on the surface of glaciers and ice sheets when the seasonal energy forcing warms the snow to its melting temperature. This meltwater percolates into the snow and subsequently runs off laterally in streams, is stored as liquid water, or refreezes, thus warming the subsurface through the release of latent heat. We present a continuum model for the percolation process that includes heat conduction, meltwater percolation and refreezing, as well as mechanical compaction. The model is forced by surface mass and energy balances, and the percolation process is described using Darcy's law, allowing for both partially and fully saturated pore space. Water is allowed to run off from the surface if the snow is fully saturated. The model outputs include the temperature, density, and water-content profiles and the surface runoff and water storage. We compare the propagation of freezing fronts that occur in the model to observations from the Greenland Ice Sheet. We show that the model applies to both accumulation and ablation areas and allows for a transition between the two as the surface energy forcing varies. The largest average firn temperatures occur at intermediate values of the surface forcing when perennial water storage is predicted.

Meltwater percolation into surface snow and firn plays an
important role in determining the impact of climate forcing on glacier and
ice-sheet mass balance. Percolated meltwater may refreeze, run off, or be
stored as liquid water. Since meltwater that runs off from the surface
ultimately contributes to sea-level rise, and can influence ice dynamics if
it is routed to the ocean via the ice-sheet bed, understanding the proportion
of meltwater that runs off is important in assessing the health of glaciers
and ice sheets under atmospheric warming

Liquid water that is produced at the surface holds a substantial quantity of
latent heat. If the meltwater percolates into the snow and refreezes, it
releases the latent heat to warm the snow.

Our approach in this paper is to construct a continuum model for meltwater
percolating through porous snow, along similar lines to

We now summarize an outline of the paper. In Sect. 2, we construct our continuum model for the firn layer and describe its conversion to an enthalpy formulation that facilitates the numerical solution method. In Sect. 3, we analyze two test problems that involve the propagation of a refreezing front moving into cold snow and a saturation front filling pore space. These act as benchmarks for the numerics and elucidate some of the generic dynamics that occur within the model. In Sect. 4, we impose a more realistic surface energy forcing, corresponding to a periodic seasonal cycle, to examine the effect of climate variables on the fate of the meltwater and the resulting thermal structure of the snow.

Here we describe our model for the flow of meltwater through porous,
compacting snow. We keep track of the flow of water, mechanical compaction,
and the melt–refreezing of water into the snow. A volume fraction

The three components of meltwater-infiltrated snow: air, water, and
ice. Panel

This term is always negative, i.e., refreezing, and in fact is zero except on
refreezing interfaces. We assume that the air density is negligible and
henceforth neglect Eq. (

The flow of water is governed by Darcy's law, i.e.,

We must now distinguish between partially saturated (

If the snow is fully saturated, water pressure

One of the difficult aspects of modeling firn in a percolation zone is that
both mechanical compaction and refreezing combine to control the changes in
snow density. There are various empirical parametrizations of dry compaction
that can be used; these typically relate the rate of change of density, or
equivalently porosity, to quantities such as depth, accumulation rate,
temperature, and grain size. In our context, these can be expressed using the
material derivative

Combining with Eq. (

We assume that ice and water are at the same temperature and therefore any
region containing meltwater (

Here we write boundary conditions on the surface

Our complete model is given by ice and water conservation (

In this section, we rewrite the equations in a form that we use for our
numerical solutions. There are two steps: first, we combine the equations as
conservation equations for total water (ice and liquid water) and enthalpy
(sensible and latent heat). Using this approach, commonly referred to as the
enthalpy method, we can avoid tracking the phase change interfaces and can
solve for their location using inequalities

We define the total water as

We combine the conservation Eqs. (

The surface boundary conditions (at

Table of physical values, derived scales, and nondimensional
parameter values (defined in Appendix

We discretize the conserved fluxes in space using a finite volume method
implemented in MATLAB (see the Supplement for code). In this construction,
the value of each variable is constant in each cell center and the velocities
and fluxes are evaluated at cell edges, thereby transferring fluxes of each
variable from one cell to another. We evolve
Eqs. (

In this section we consider two test problems that demonstrate the model
behavior and validate the numerical method. The two problems that we consider
here are designed to explore the boundaries between frozen and unfrozen snow
(refreezing interfaces) as well as the boundaries between partially and fully
saturated snow (saturated interfaces). Both problems ignore mechanical
compaction. We start by describing the propagation of rainwater into dry
snow. This is similar to the problem studied by

Schematic of the test problems considered in

We consider the infiltration of rain into cold, dry snow as a test problem.
We start with a patch of dry snow (

Evolution of a refreezing front at three instances of time,
partitioned between the three components of the enthalpy. The green, red, and
yellow colors show the porosity, saturation, and temperature profiles,
respectively. The dashed lines show the approximate analytical solutions
described in Appendix

Note that if

The refreezing and release of latent heat as a front of meltwater moves
through a firn layer allows the percolation of meltwater to be observed in
englacial temperature data.

We now compare these data to the approximate solution for the temperature
field ahead of a refreezing front, as given in Eq. (

Data from

We now consider the propagation of rainwater into isothermal, temperate snow
of decreasing porosity such that fully saturated fronts develop. The porosity
decreases exponentially with depth as

Evolution of fully saturated fronts at three instances in time,
showing saturation (red), water flux (cyan), and water pressure (magenta).
The porosity (green) decreases exponentially with depth over length scale

In Appendix

We now examine the solutions to the full model with prescribed seasonal
energy forcing, which we parametrize as a sinusoid, using the annual mean as
a control parameter. In principle, we could also incorporate diurnal
periodicity, but we choose to ignore it because we expect diurnal variability
to affect only a small surface layer (

We apply an oscillating surface forcing in Eq. (

We run a suite of numerical simulations varying the accumulation rate and
annual mean surface forcing, each time allowing the dynamics to reach an
annual periodic state (typically this takes around 10 years). Four
representative space–time diagrams of these simulations are shown in
Fig.

Space–time diagrams showing the evolution of porosity

Each case shows a different value of

The four simulations in Fig.

Increasing

Figure

Above a critical

Average meltwater partition (right) and annual mean temperature at
the ice surface

In Fig.

For

The thermal structure and water content of the lower firn are strongly tied
to the amount of meltwater produced, which in this model is tied directly to
the annual mean surface forcing. In a warming world, one can imagine a
particular location transitioning from an accumulation to ablation region.
Our results in Fig.

We have described a continuum model for the evolution of firn hydrology, compaction, and thermodynamics. The model is capable of determining the evolution of the firn including the temperature, porosity, and water content. The model differs from other models of firn hydrology in its treatment of the percolation of water, for which we use Darcy's law and a parametrization of capillary pressure. Our treatment for runoff also differs in that we assume that water runs off when the surface layer of snow is fully saturated rather than assuming runoff at depth when the percolating water first reaches an impermeable ice layer.

The model applies to both accumulation and ablation areas. Given the forcing
(energy flux and accumulation rate), the model selects which of these applies
to any particular region. One of the useful outputs of the model is an
indication of how the firn may change as function of climate warming, as
revealed by moving from left to right in
Fig.

In the future, we hope to extend this work beyond the one-dimensional
solutions presented here. In principle the model applies to fully
three-dimensional geometries, when the slope of the saturated surface (the
“water table” in the firn) will allow meltwater to flow laterally as well
as vertically. The data from

The use of Darcy's law requires an estimate for the permeability and the
relative permeability. The comparison of our model behavior with the data
from

Although we have focused on idealized, periodic simulations, the model can be
forced by real climatological data or coupled to a regional atmospheric
model. The model could also be coupled to an ice-sheet model, using the deep
firn temperature

Typical numerical values for the surface energy balance

The data associated with this paper are contained in Humphrey et al. (2012) or can be produced from the code attached in the Supplement.

The surface energy balance is given by

Linearizing this equation around the melting temperature

We nondimensionalize the lengths by

Using the change of variables

Here we detail the approximate solution for the refreezing front considered
in Sect.

We next solve for the temperature evolution in the lower region. Assuming
that the freezing front moves quickly, i.e.,

To capture the smoothing of the front due to capillary pressure, we can
examine the narrow boundary layer behind the front. The relevant scale for
this region is of order

Here we calculate the motion of the fully saturated fronts for isothermal
conditions with fixed porosity

We can therefore calculate the position of the front, and the time, at which
full saturation occurs by setting

Now in the fully saturated region, between the upper and lower saturation
fronts

The authors declare that they have no conflict of interest.

We wish to thank the 2016 Geophysical Fluid Dynamics summer program at the Woods Hole Oceanographic Institution, which is supported by the National Science Foundation and the Office of Naval Research. We also acknowledge financial support by NSF grants DGE1144152 and PP1341499 (CRM) as well as Marie Curie FP7 Career Integration Grant within the 7th European Union Framework Programme (IJH). Edited by: Valentina Radic Reviewed by: two anonymous referees