A Continuum Model for Meltwater Flow Through Compacting Snow

The Cryosphere, 11, 2799–2813, 2017 https://doi.org/10.5194/tc-11-2799-2017 © Author(s) 2017. This work is distributed under the Creative Commons Attribution 4.0 License.

A continuum model for meltwater ﬂow through compacting snow

Colin R. Meyer1 and Ian J. Hewitt2 1John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA 2Mathematical Institute, Woodstock Road, Oxford, OX2 6GG, UK

Correspondence to: Colin R. Meyer ([email protected])

Received: 2 July 2017 – Discussion started: 19 July 2017 Revised: 13 October 2017 – Accepted: 23 October 2017 – Published: 11 December 2017

Abstract. Meltwater is produced on the surface of glaciers health of glaciers and ice sheets under atmospheric warming and ice sheets when the seasonal energy forcing warms the (Harper et al., 2012; Enderlin et al., 2014; Forster et al., 2014; snow to its melting temperature. This meltwater percolates Koenig et al., 2014; Machguth et al., 2016). The balance be- into the snow and subsequently runs off laterally in streams, tween runoff, refreezing, and storage is controlled by the me- is stored as liquid water, or refreezes, thus warming the sub- chanics and thermodynamics of the porous snow. These pro- surface through the release of latent heat. We present a con- cesses also underlie the rate of compaction of firn into ice and tinuum model for the percolation process that includes heat therefore control the average temperature and accumulation conduction, meltwater percolation and refreezing, as well as rate that provide surface boundary conditions to numerical mechanical compaction. The model is forced by surface mass ice-sheet models (which typically do not include the com- and energy balances, and the percolation process is described pacting firn layer explicitly). using Darcy’s law, allowing for both partially and fully sat- Liquid water that is produced at the surface holds a sub- urated pore space. Water is allowed to run off from the sur- stantial quantity of latent heat. If the meltwater percolates face if the snow is fully saturated. The model outputs include into the snow and refreezes, it releases the latent heat to warm the temperature, density, and water-content profiles and the the snow. Humphrey et al.(2012) observe that the snow at surface runoff and water storage. We compare the propaga- 10 m depth in Greenland is often more than 10 ◦C warmer tion of freezing fronts that occur in the model to observa- than the mean annual air temperature because of the refreez- tions from the Greenland Ice Sheet. We show that the model ing of meltwater. If, however, this water runs off through applies to both accumulation and ablation areas and allows supraglacial streams or drains to the bed through moulins, the for a transition between the two as the surface energy forcing latent heat is carried away and subsequent cooling of the sur- varies. The largest average firn temperatures occur at interface in the winter means that the remaining snow is relatively mediate values of the surface forcing when perennial water cold. Since the capacity to store and/or refreeze meltwater is storage is predicted. tied to the porosity of the snow, which is in turn linked to the amount of storage and refreezing that have occurred in previous years, it is of interest to know how the partitioning of meltwater between runoff, refreezing, and storage, as well as 1 Introduction the firn temperature and density profiles, depends on climatic forcing (air temperature and radiative forcing as well as accu- Meltwater percolation into surface snow and firn plays an mulation). This question is of interest even under steady cli- important role in determining the impact of climate forcing mate conditions (i.e., seasonally periodic, without any year- on glacier and ice-sheet mass balance. Percolated meltwater on-year trend), and this forms the focus of our study. A fur- may refreeze, run off, or be stored as liquid water. Since melt- ther question of current interest is how the firn responds tran- water that runs off from the surface ultimately contributes to siently to year-on-year increases in melting (Harper et al., sea-level rise, and can influence ice dynamics if it is routed 2012; Koenig et al., 2014), but we consider the steady prob- to the ocean via the ice-sheet bed, understanding the propor- lem a prerequisite to understanding such transient response. tion of meltwater that runs off is important in assessing the

Published by Copernicus Publications on behalf of the European Geosciences Union. 2800 C. R. Meyer and I. J. Hewitt: Meltwater ﬂow through snow

Our approach in this paper is to construct a continuum (a) Accumulation area (b) Ablation area model for meltwater percolating through porous snow, along similar lines to Gray(1996). This contrasts cell-based numer- Runoff ical models that are often applied to the Greenland Ice Sheet, such as the Firn Densification Model (IMAU-FDM) that is incorporated in the regional climate model RACMO and is described by Ligtenberg et al.(2011) and Kuipers Munneke et al.(2014, 2015). That model includes mechanical compaction and a “tipping-bucket” hydrology scheme, where the firn is divided into distinct layers and water fills each layer up to the irreducible water content and then trickles instantaneously into the lower layers. Runoff occurs when the water reaches an impermeable layer and the water is removed (representing the lateral flow that occurs in real- ity). Steger et al.(2017a,b) used a similar tipping-bucket method in the SNOWPACK model (Bartelt and Lehning, 2002) and compared the results to the IMAU-FDM. Wever et al.(2014) compared the tipping-bucket and Richards equation formulations within SNOWPACK to field observations and found that using Richards’ equation provided a better fit. The Richards equation approach, in which water flow is Velocity < 0 Velocity > 0 driven by gravity and capillary pressure, is similar to the The three components of meltwater-infiltrated snow: air, model we adopt in this study. This has also been used in Figure 1. water, and ice. Panel (a) shows water infiltrating an accumulation a number of more theoretical models for the percolation of area where the snow density increases with depth and snow ad- meltwater through snow (Colbeck, 1972, 1974, 1976). Gray vects down. The water partially saturates the snow near the surface and Morland(1994, 1995) and Gray(1996) provide detailed (S < 1) whereas, at depth, all of the air is replaced by water and the descriptions of this approach in the context of mixture theory. snow is fully saturated (S = 1). Panel (b) shows an ablation area We now summarize an outline of the paper. In Sect. 2, we where the there is fully saturated porous snow in a thin layer near construct our continuum model for the firn layer and describe the surface and the underlying ice is solid, advecting into the do- its conversion to an enthalpy formulation that facilitates the main from upstream. Ice grains make contact in the third dimension numerical solution method. In Sect. 3, we analyze two test (into the page) and similarly many of the air and water pockets are problems that involve the propagation of a refreezing front connected in the third dimension. moving into cold snow and a saturation front filling pore space. These act as benchmarks for the numerics and elu- ∂(Sφρw) + ∇ · (Sφρ u ) = m, (2) cidate some of the generic dynamics that occur within the ∂t w w model. In Sect. 4, we impose a more realistic surface energy ∂ [(1 − S)φρ ] + ∇ · [(1 − S)φρ u ] = 0, (3) forcing, corresponding to a periodic seasonal cycle, to exam- ∂t a a a ine the effect of climate variables on the fate of the meltwater and the resulting thermal structure of the snow. where the subscripts i, w, and a indicate ice, water, and air, respectively. The densities ρi, ρw, and ρa are constants. The velocities of the ice, water, and air are given by ui, uw, and 2 Model ua. The variable density of the snow is (1 − φ)ρi + φSρw + φ(1 − S)ρa. The rate at which ice melts and turns into melt- 2.1 Percolation through porous ice water internally is given by m and is therefore a source in Eq. (2) and a sink in Eq. (1). Here we describe our model for the flow of meltwater This term is always negative, i.e., refreezing, and in fact is through porous, compacting snow. We keep track of the flow zero except on refreezing interfaces. We assume that the air of water, mechanical compaction, and the melt–refreezing of density is negligible and henceforth neglect Eq. (3). water into the snow. A volume fraction 1−φ is solid ice while The flow of water is governed by Darcy’s law, i.e., the void space φ is composed of water and air. We define the saturation S as the fraction of the void space that is filled by k(φ) φS (uw − ui) = − kr(S) ∇pw + ρwgzˆ , (4) water (see the schematic in Fig.1). Conservation of mass for µ ice, water, and air is expressed as where pw is the water pressure, k(φ) is the permeability, ∂ kr(S) is the relatively permeability, and µ is the viscosity of [(1 − φ)ρ ] + ∇ · [(1 − φ)ρ u ] = −m, (1) ∂t i i i the water. For the permeability we use a simplified Carman–

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Kozeny relationship, given by rate of refreezing that is calculated from the thermodynamics. Therefore, we have d2 p 3 3 k(φ) = φ = k0φ , (5) ∂φ m 180 + ui · ∇φ = − cφ, (9) ∂t ρi where dp is a typical grain size (Gray, 1996). Table1 provides the parameter values we use later. where the specific compaction rate we choose is the Herron We must now distinguish between partially saturated (S < and Langway(1980) model, which is written as = cφ. The − C 1) and fully saturated (S = 1) flow. When the snow is coefficient c in units of yr 1 is given by partially saturated, capillary forces drive flow along liquid  n o − 1222 bridges connecting ice crystals (Bear, 1972). Thus, we re- 11a exp T if φ > 0.4 = c = √ n o (10) late the water pressure to the capillary pressure pc by pw − 2574 ≤ 575 a exp T if φ 0.4, pa − pc, where pa is the air pressure (taken as zero). Both the capillary pressure and the relative permeability are pre- where a is the accumulation rate in meters of water equiv- scribed functions of the saturation S. We take alent per year and T is the absolute temperature. This is an β γ −α kr(S) = S and pc(S) = S , (6) empirical parametrization, and the two forms reflect a change dp in dominant compaction processes at a certain snow density. where γ is surface tension, and we choose the exponents Other parametrizations for compaction that could easily be α and β such that β = α + 1, which avoids a singularity in incorporated in this framework are discussed by Zwally and 0 Li(2002), Reeh(2008), and Morris and Wingham(2014). kr(S)p (S) at S = 0 (Gray, 1996). c We have chosen to use the Herron and Langway model here If the snow is fully saturated, water pressure pw is con- strained by mass conservation. Combining Eqs. (1), (2), and for simplicity; from the experiments we have conducted, dif- (4) gives ferent formulations do not appear to qualitatively change our results. k(φ) 1 1 Combining with Eq. (1), we note that Eq. (9) is equivalent ∇ · ui − ρgzˆ + ∇pw = m − , (7) µ ρw ρi to p ∂wi which is an elliptic problem for w. Boundary conditions for (1 − φ) = −cφ, (11) this equation are provided by the constraint that pw must be ∂z continuous across the interfaces between partially and fully where w is the vertical component of the ice velocity. saturated regions and the constraint of no flow across imper- i meable boundaries (e.g., ice lenses). 2.3 Temperature

2.2 Compaction We assume that ice and water are at the same temperature and therefore any region containing meltwater (S > 0) is at One of the difficult aspects of modeling firn in a percolation the melting point T . In regions without water, we solve the zone is that both mechanical compaction and refreezing com- m temperature evolution equation, bine to control the changes in snow density. There are vari- ous empirical parametrizations of dry compaction that can be ∂T ρ cp(1 − φ) + ρ cp(1 − φ)u · ∇T = ∇ · K∇T − m, (12) used; these typically relate the rate of change of density, or i ∂t i i L equivalently porosity, to quantities such as depth, accumula- where the heat capacity is c and the thermal conductivity is tion rate, temperature, and grain size. In our context, these p K = (1 − φ)K. The latent heat term − m operates on inter- can be expressed using the material derivative L faces of refreezing, where it is singular and causes disconti- ∂φ nuities to occur in the temperate gradient. + u · ∇φ = − , (8) ∂t i C 2.4 Surface boundary conditions where is a parametrization of the rate of compaction (Arth- C ern et al., 2010). The appropriateness of such models for Here we write boundary conditions on the surface zs(t), snow containing meltwater is uncertain. The parametriza- which we assume is locally flat, and we write wi and ww tions represent the rearrangement and growth of snow crys- as the vertical velocities. The kinematic conditions are tals and the accompanying closure of air voids as functions of temperature and accumulation rate, and these processes ρi(1 − φ)(wi −z ˙s) = ρw(M − a), (13) may be modified by the presence of liquid between crys- ρwφS(ww −z ˙s) = ρw(M − R + r), (14) tals. In the absence of a more developed theory for wet compaction, we take the approach of using these dry parametriza- where z˙s is the velocity of the surface, M is the rate of melt- tions but modify the material density derivative to include the ing, a is the accumulation rate, R is the rainfall rate, and r is www.the-cryosphere.net/11/2799/2017/ The Cryosphere, 11, 2799–2813, 2017 2802 C. R. Meyer and I. J. Hewitt: Meltwater flow through snow runoff, all expressed in units of water equivalent per year. The and we define the enthalpy as the sum of sensible and latent compaction Eq. (9) requires a boundary condition, φ = φ0, heat as when the accumulation rate is greater than the rate of melt- − ˙ − = ρcp (T − Tm) + ρ Sφ. (17) ing (i.e., wi zs < 0), where (1 φ0)ρi is the bulk density H W L of freshly deposited snow. The energy balance on the surface The inverse relationships that relate the enthalpy and total provides a boundary condition for the temperature equation H water to the temperature T , saturation S, and porosity φ when the surface temperature is less than T , and determines W m are the rate of melting M when T = T . These conditions are m combined as T = T + min 0, H , φ = 1 − + max 0, H , m W ρ ∂T W L ρicp (1 − φ)(wi −z ˙s)(T − Tm) − K and S = max 0, H . (18) ∂z ρ φ L = −Q + h(T − Tm) + ρw M, (15) L We define the depth below the ice surface Z, and the relative downward ice velocity wei, as along with the conditions M = 0 when T < Tm, and M ≥ 0 T = T Q(t) = − = ˙ − when m. The forcing energy flux includes the Z zs(t) z, wei zs wi. (19) combined effects of radiative, turbulent, and sensible heat fluxes. We assume that this is prescribed in order to provide We combine the conservation Eqs. (1) and (2) with Darcy’s a simple parametrization of the climate forcing. However, it law (4) and temperature evolution (12) as can be related to more specific components of the energy bal- ∂ ∂ ance as described in Appendix A. The heat transfer coeffi- W + (w + q) = 0, (20) ∂t ∂Z eiW cient h represents a combination of radiative and turbulent ∂ ∂ ∂T + + − + − heat transfer. We expect Q to have a typical magnitude on H wei q ρcp(T Tm) ρ K = 0, (21) −2 ∂t ∂Z H L ∂Z the order of Q0 = 200 W m with a comparable seasonal amplitude and take h = 14.8 W m−2 K−1 as a representative where the downward water flux is constant (Cuffey and Paterson, 2010; van den Broeke et al., k(φ)k (S) ∂p 2011). q = r ρg − w . (22) µ ∂Z 2.5 Numerical method Combining ice conservation (1) with compaction (9) gives ∂w Our complete model is given by ice and water conservation (1 − φ) ei = −cφ, (23) (1) and (2), Darcy’s law (4), compaction (9), and temperature ∂Z evolution (12), subject to the boundary conditions (13)–(15). and water pressure is given by The model is forced by a prescribed energy flux Q, accumu- γ − γ lation a, and precipitation R, and it predicts the temperature, p = − S α (S ≤ 1) or p ≥ − (S = 1). (24) w d w d porosity, and saturation profiles as well as the surface melt p p rate, runoff, refreezing, and storage of liquid water. The surface boundary conditions (at Z = 0), re-expressed In this section, we rewrite the equations in a form that we in terms of and , are given as use for our numerical solutions. There are two steps: first, W H we combine the equations as conservation equations for total w + q = a + R − r, (25) W ei water (ice and liquid water) and enthalpy (sensible and la- ∂T w + q ρcp(T − T ) + ρ − K = Q − h(T − T ) tent heat). Using this approach, commonly referred to as the eiH m L ∂Z m enthalpy method, we can avoid tracking the phase change + ρ R − ρ r, (26) interfaces and can solve for their location using inequalities L L a − M a − M (Hutter, 1982; Aschwanden et al., 2012; Hewitt and Schoof, = = ≤ wei (wei > 0) or wei (wei 0). (27) 2017). The second step is to change variables into a frame 1 − φ0 1 − φ that moves with the ice surface. At this stage we also sim- In these conditions, the runoff r is assumed to be zero un- plify the model to write it in one vertical dimension, and we less the snow at the surface reaches full saturation, in which make the Boussinesq approximation to ignore density differ- case Eqs. (25) and (26) determine r. At the bottom of our do- = = ences so that ρi ρw ρ. main, we assume that the conductive heat flux and pressure We define the total water as , which is the sum of liquid W gradients vanish to replicate effective matching conditions to and solid fractions, i.e., the deep interior of the ice sheet. On internal interfaces between fully saturated and partially saturated regions we apply = 1 − φ + Sφ, (16) p = −γ /dp to ensure pressure continuity. W w

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Table 1. Table of physical values, derived scales, and nondimensional parameter values (deﬁned in AppendixB). (a) (b)

−3 −2 ρ 917 kg m Q0 200 W m 100 2 −2 −1 −2 −1 U cp 2050 m s K h 14.8 W m K 12 S 334 000 m2 s−2 1T 13.5 K 260 L B K 2.1 kg m s−3 K−1 6 × 10−4 kg m−2 s−1 Pe 11 M g 9.806 m s−2 ` 20.6 m α 1 −1 −11 2 γ 0.07 N m k0 5.6 × 10 m β 2 −4 7 dp 10 m t0 3.15 × 10 s µ 10−3 Pa s

We discretize the conserved fluxes in space using a fi- nite volume method implemented in MATLAB (see the Sup- plement 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 Figure 2. Schematic of the test problems considered in (a) Sect. 3.1 Eqs. (20)–(21) in time using explicit forward Euler time step- and (b) Sect. 3.3. In both panels, rain falls at a rate R on the surface = ping, which involves evaluating the fluxes on the cell edges of the snow. White shading indicates dry snow (S 0), grey indi- < S < using the quantities from the previous timestep. For advec- cates partially saturated snow (0 1), and dark shading indicates fully saturated snow (S = 1). In panel , the snow is initially tion, we use an upwind scheme where the value of the vari- (a) cold with T = T∞ and dry, with uniform porosity φ . The rainwa- able advected depends on the velocity direction. For edges 0 ter percolates through the snow, refreezes at the interface Zf(t), and between partially saturated cells, we evaluate the water fluxes releases latent heat that warms the snow. The refreezing decreases p + using the capillary pressure for w on the adjacent cells. For the porosity in the upper region so that φ < φ0. In panel (b), the edges between fully saturated cells, we solve Eq. (20) with snow is temperate, T = Tm, with a porosity profile that decays ex- S = 1 as an elliptic equation for pw on the saturated cells, ponentially with depth. After the snow fully saturates two saturation which we then use to evaluate the water fluxes. In order to al- fronts emerge with Zl propagating downward and Zu upward. low cells to switch from fully to partially saturated, we com- pute the fluxes using both of these methods on edges between fully and partially saturated cells and choose that which gives the largest flux away from the saturated region. gates down. At a certain point the snow fully saturates and a saturated front propagates up toward the snow surface.

3 Test problems 3.1 Rainfall into cold snow

In this section we consider two test problems that demon- We consider the infiltration of rain into cold, dry snow as strate the model behavior and validate the numerical method. a test problem. We start with a patch of dry snow (S = 0) The two problems that we consider here are designed to ex- with constant porosity (φ = φ0) and temperature (T = T∞ < plore the boundaries between frozen and unfrozen snow (re- Tm). We assume no accumulation and ignore compaction so freezing interfaces) as well as the boundaries between par- that the ice is stationary. Furthermore, the porosity is large tially and fully saturated snow (saturated interfaces). Both enough that the snow never fully saturates. At time t = 0 a problems ignore mechanical compaction. We start by de- fixed flux of rain R with a temperature T = Tm is applied at scribing the propagation of rainwater into dry snow. This the surface Z = 0 and a wetting front at Z = Zf moves down ˙ is similar to the problem studied by Colbeck(1972), Gray at velocity Zf (we show a schematic in Fig.2 and the numeri- (1996), and Durey(2014) and has an approximate analytical solutions in Fig.3). Since the capillary pressure gradients cal solution that provides a useful test case for the enthalpy are small and the flow is largely driven by gravity, the wet- method. We also compare the results of the analytical solu- ting front behaves as a smoothed shock front. Some of the tion for the propagation of the meltwater front to temperature water at the shock front refreezes, warming the snow ahead. data from Humphrey et al.(2012). Secondly, to test the prop- As shown in more detail in AppendixC, the behavior of this agation of saturated fronts, we consider an isothermal prob- shock can be understood by ignoring the diffusive capillary lem in which the porosity profile is prescribed to decrease pressure term. This approximation relegates Eqs. (1) and (2) with depth. We again investigate how rainwater propagates to hyperbolic partial differential equations for the porosity into the snow, with saturation increasing as the front propa- and saturation as well as simplifying the temperature Eq. (12) www.the-cryosphere.net/11/2799/2017/ The Cryosphere, 11, 2799–2813, 2017 2804 C. R. Meyer and I. J. Hewitt: Meltwater flow through snow so that 3.2 Data comparison 0 ∂S ρgk(φ)kr(S) ∂S The refreezing and release of latent heat as a front of meltwa- + = 0, (0 < Z < Zf), (28) ∂t φµ ∂Z ter moves through a firn layer allows the percolation of melt- ∂φ water to be observed in englacial temperature data. Harper = 0, (0 < Z < Z ) and (Z > Z ), (29) ∂t f f et al.(2012) and Humphrey et al.(2012) collected temper- ∂T K ∂2T ature data in the accumulation zone on the western flank of = , (Z > Zf), (30) the Greenland Ice Sheet and inferred the movement of melt- ∂t ρcp ∂Z2 water by warming of the snow due to the release of latent 0 = where kr(S) dkr/dS, and with initial and boundary condi- heat. They set up a vertical string of thermistors to determine tions the temperature profile in the upper 10 m of the ice sheet. Data from one vertical string between the dates of 5 and 25 S = 0, φ = φ0,T = T∞ at t = 0, (31) July 2007 (days 185–203) are shown in Fig.4. From these T = T∞ as Z → ∞, (32) data it is clear that the ice at depth progressively warmed, likely due to the refreezing of liquid meltwater. Over the 12 T = Tm on Z = Zf, (33) days between day 185 and day 197, the warming front prop- ρgk(φ)k (S) r = R on Z = 0. (34) agated about a meter, while over the course of the next 6 µ days from day 197 to day 203 the meltwater penetrated two Equations (28)–(30) have corresponding jump conditions additional meters, showing a 4-fold increase in front veloc- across the shock which incorporate the refreezing rate −mi ity. Humphrey et al.(2012) infer that the warming spike on at that front. These are day 199 is due to an influx of meltwater from lateral sources. A minimum temperature is observed at around 5 m depth and ρg ˙ the temperature recorded on the lower thermistors is warmer, k(φ)kr(S) − φSZf = −mi, (35) µ + which could be due to prior warming by meltwater pulses or ˙ + Zf[φ]− = mi, (36) a manifestation of the seasonal thermal wave. We now compare these data to the approximate solution ∂T (1 − φ)K = ρ mi, (37) for the temperature field ahead of a refreezing front, as given ∂Z − L ˙ in Eq. (39). We fit front speed Zf for the days 185–197 and a larger front speed for days 197–203. The increase in the where + refers to the region above the front (Z < Zf). Us- ing these jump conditions and the solutions to Eqs. (28)–(30) front speed is likely due to an increase in surface melt. We = ◦ subject to the boundary conditions (31)–(34), we find an ap- set the melting temperature Tm 0 C, fit a constant far- proximate expression for the front velocity: field temperature T∞, and use the heat diffusivity for ice −6 2 −1 K/(ρcp) = 1.1 × 10 m s (Table1). In light of the sim- R ˙ = plified analysis, the fit between Eq. (39) and the Humphrey Zf + + L . (38) φ S + (1 − φ )(T − T∞) et al.(2012) data is quite good. L 0 m

Note that if T∞ = Tm, i.e., isothermal snow, the front 3.3 Isothermal saturation fronts simply propagates at the speed of the draining rainwater + + We now consider the propagation of rainwater into isother- R/(φ S ) (Bear, 1972). The effect of refreezing due to mal, temperate snow of decreasing porosity such that fully T∞ < Tm is to slow the front and to cause a decrease in + saturated fronts develop. The porosity decreases exponen- the porosity as the front passes by an amount φ − φ = 0 tially with depth as (1 − φ0)cp(Tm − T∞)/L. This is a mechanism by which ice lenses can form: if the pre-existing porosity is small enough, −Z/Z0 φ(Z) = φ0e , (40) the porosity above the front can decrease to zero and the pores freeze shut. In this case the front stops propagating, where Z0 is a constant. We continue to ignore compaction and a saturated region forms above the lens in a similar way and accumulation, and since the snow is isothermal the to that described in Sect. 3.3. We also determine approximate porosity is therefore constant in time. Initially, the rain par- analytical solutions for temperature and saturation, which are tially saturates the snow and a wetting front moves down- compared to the numerical solutions in Fig.3. The agree- ward, as shown in Fig.5a. Then, at a certain depth, the ment between the numerical and approximate solutions is maximum saturation reaches unity and two saturation fronts very good. The approximate temperature proﬁle ahead of the emerge, one that propagates up and the other down, as shown refreezing front is given by in Fig.5b and c. " # In AppendixD, we derive the locations of the upper Zu ρc Z˙ (Z − Z ) p f f and lower Zl fronts by neglecting ﬂow driven by gradients T = T∞ + (Tm − T∞)exp − . (39) K in capillary pressure. This analysis results in two differential

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0 0 0

0.5 0.5 0.5

1 1 1

1.5 1.5 1.5

t =0.08t0 t =0.16t0 t =0.24t0 2 2 2 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Nondimensional quantities Nondimensional quantities Nondimensional quantities

FigureFigure 3. Evolution 3. Evolution of a of refreezing a refreezing front front at at three three instances instances of of time, time, partitioned between between the the three three components components of the of enthalpy.the enthalpy. The green, The green, red, andred, yellow and yellow colors colors show show the the porosity, porosity, saturation, saturation, and and temperature temperature proﬁles, respectively. respectively. The The dashed dashed lines lines show show the approximate the approximate analytical analytical ˆ solutions described in AppendixC. The temperature is made nondimensional by T = Tm + (T∞ − Tm)T and the parameters are φ0 = 0.4 solutions described in Appendix C. The temperature is made nondimensional by T = Tm +(T Tm)Tˆ and the parameters are 0 =0.4, and R = 0.54, along with other values in Table1. 1 and R =0.54, along with other values in table 1.

where Z and t are the location and time at which full sat- Equations0 (28)-(30) have corresponding jump conditions across the shock1 which incorporate1 the refreezing rate mI at that uration initiates. We solve these coupled, nonlinear ordinary front. These are differential equations (ODEs) using a numerical integrator in ⇢g2 MATLAB and compare these semi-analytical solutions to the k()k (S) SZ˙ = m , (35) µ r f I full numerical solutions in Fig.5 (the dashed black lines). + 4 + The slight differences are due to neglecting the gradient in Z˙f [] = mI , (36) capillary pressure in our approximate solutions. @T 5 (1 )K = ⇢L mI , (37) 6 @Z 4 Results where + refers to the region above the front (Z

0 0 0

0.5 0.5 0.5

1 1 1

1.5 1.5 1.5

t =0.4t0 t =0.7t0 t = t0 2 2 2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

(a) Nondimensional quantities (b) Nondimensional quantities (c) Nondimensional quantities Figure 5. Evolution of fully saturated fronts at three instances in time, showing saturation (red), water flux (cyan), and water pressure Figure 5. 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 Z0 = `/2, where ` is the characteristic length scale (magenta). The porosity (green) decreases exponentially with depth over length scale Z0 = `/2, where ` is the characteristic length scale defined in Appendix B and given in table 1. Panel (a) shows the position of the front before the firn fully saturates. Panels (b) and (c) show defined in AppendixB and given in Table1. Panel (a) shows the position of the front before the firn fully saturates. Panels (b, c) show the bidirectionalthe bidirectional motion of motion the fully of thesaturated fully saturated fronts. fronts. Dashed Dashed black black lines lines show show semi-analytical semi-analytical solutions solutions from solving solving equation Eq. (41 (41).).

3.3 Isothermal saturation fronts ically this takes around 10 years). Four representative space– Figure6III shows a region which is an accumulation area time diagramsWe now of consider these thesimulations propagation are of shown rain water in Fig.into6 isothermal,. temperatebut with snow more of decreasingmelting than porosity in Fig. such6II. that Interestingly, fully saturated this sit- Each case shows a different value of Q with the same ac- uation no longer has a perennial aquifer and all of the melt- cumulationfronts rate develop. (1.7 m The water porosity equivalent decreases per exponentially year) and poros- with depth aswater that is produced refreezes. Although still a percola- = ity of fresh snow φ0Z/Z 0.64. While the ice surface moves up tion zone it is different in character than the region shown (Z)= e 0 , (40) and down during0 the simulation, we plot the quantities as a in Fig.6II. The porosity decreases more rapidly with depth function of depth below the surface Z = zs(t)−z and plot ice in this case so that despite more water being produced on the 5 where Z is a constant. We continue to ignore compaction and accumulation and since the snow is isothermal, the porosity is streamlines to show0 the relative motion of the ice. In Fig.7 surface during the summer, this larger quantity of water is we showtherefore how the constant mean in temperature time. Initially, at the the rain bottom partially of thesaturates do- the snownot able and a to wetting percolate front movesas far downward,into the snow. as shown As a in consequence, figure main T 5(a).∞ and Then, the at mean a certain surface depth, temperature the maximumT saturations change reaches as unityit is and not two so saturation well insulated fronts emerge, from the one coldthat propagates surface during up the the meanand surface the other forcing down, asvaries, shown for in threefigures different 5(b) and 5(c). values of winter and all of the water refreezes. This greater quantity of accumulation rate. Each point in this figure corresponds to refreezing is in turn responsible for the more rapid decrease In Appendix D, we derive the locations of the upper Z and lower Z fronts by neglecting flow driven by gradients in an annual average of a periodic simulation such as those inu in porosityl with depth that prevents the liquid water percolat- Fig.106 (andcapillary which pressure. are labeled This analysis in Fig.7 resultsb). in two differential equationsing for as the deep evolution as it does of upper in Fig. front6IIZu (moreand lower refreezing front Zl means, the The four simulations in Fig.6 represent the spectrum of pore space3 is filled in more effectively with ice). In contrast, qs qs R 3k00⇢g(Zu Zl) Z˙l = and Z˙u = with qs = , (41) possible surface types on glaciers and ice sheets,1/ encompass- the reason3Zu/Z0 a perennial3Zl/Z0 aquifer is sustained in Fig.6II is be- l µRe3Zu/Z0 µZ0 e e u 1 ⇢gk 3 ing both accumulation and ablation regions.0 0 If we interpret cause the water penetrates sufficiently far that it is insulated increasing Q as a parametrization of⇣ slow climate⌘ warming, from⇥ the cold surface⇤ (Kuipers Munneke et al., 2014). we mightsubject expect to the a location initial conditions that is initially an accumulation Above a critical Q there is too much melting for the firn to area to transition through each of these states. Figure6I is accommodate and runoff begins (this occurs at a value of Q an accumulationZu = Zl = areaZ1 whereat time theret = t is1, no melting at any point intermediate between Fig.6III and IV and is clearest(42) to see during the year. The ice streamlines show that the ice ad- in Fig.7b). The transition from an accumulation area to an vects downwardwhere Z1 and as moret1 are snowthe location accumulates and time on at which the surface. full saturationablation initiates. area We solve occurs these when coupled, runoff nonlinear exceeds ODEs the accumulation. using a The15 snownumerical compaction integrator isvisible in MATLAB, from and a convergence compare these of semi-analytical the Figure solutions6IV shows to the anfull ablation numerical area solutions where in figurethe surface 5 (the melt- streamlinesdashed with black time. lines). The The temperature slight differences variation are due with to neglecting depth thewater gradient isonly in capillary able to pressure enter in a our few approximate meters into solutions. the snow and in this case is just the thermal wave and the variations in reaches the impermeable barrier of the glacial ice surface. p surface temperature are only felt around K/(ρcpω) ∼ 6 m During the course of the summer all of the snow is melted as into the snow. 12 well as some of the glacial ice. The streamlines show net up- Increasing Q above −Q0 leads to melting during summer. ward motion in this case indicating that there is net ablation Figure6II shows an accumulation area where the temper- over the course of the year. ature and porosity profiles are significantly affected by the In Fig.7 we also calculate the total quantity of surface meltwater that drains into the snow during the summer. Here melt and the partitioning of the melt between runoff, liquid there is water below 10 m throughout the year fed by perco- storage in the ice, and refreezing in the firn. Runoff and melt lation each summer. This is a perennial aquifer, as found in are calculated from the model output, liquid storage is taken a number of field observations (Forster et al., 2014; Koenig to be the total water flux passing out of the bottom of the et al., 2014). domain (the domain represents only the surface firn layer, so this represents water that is stored within the upper part of the

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I II (a) (a)

(b) (b)

III IV (a) (a)

(b) (b)

Figure 6. Space–timeFigure 6. diagramsSpace-time showing diagrams showing the evolution the evolution of porosity of porosityφ (a)(top),, saturation saturation S S(middle),(b), and and temperature temperature T (bottom)T (c) as as a a function of time for −1 1 the accumulationof rate timea for= the1. accumulation7 m w.e. yr rateanda =1 four.7 mwe values yr ofand forcing: four values (I) of cold forcing: accumulation (I) cold accumulation zone where zone wherethe mean the mean forcing forcing is isQ = −Q0 and (II) accumulationQ = areaQ0 with. (II) accumulationmean forcing areaQ with= mean −0.707 forcingQ Q. In= this0.707 case,Q0. In a this clear case, perennial a clear perennial aquifer aquifer develops. develops. (III) (III) Accumulation accumulation area with 0 = − = − larger forcing Qarea with0.575 largerQ forcing0. (IV)Q Ablation= 0.575Q zone0. (IV) with ablation mean zone forcing with meanQ forcing0Q.146= Q0.0146. InQ0 all. In simulations all simulations the the porosity porosity of the of falling the falling snow is = φ0 0.64 and thesnow black is 0 lines=0.64 showand the ice black streamlines. lines show ice streamlines.

Figure 7. Average meltwater partition (right) and annual mean temperature at the ice surface T s and bottom of the domain T ∞ (left) as a func- − − tion of the annual mean surface forcing, with accumulation increasing from left to right: (a) a = 0.68 m w.e. yr 1, (b) a = 1.7 m w.e. yr 1, −1 and (c) a = 3.4 m w.e. yr . For Q > −Q0 melting occurs at the surface and meltwater percolation warms the bottom of the domain. Dashed lines in panels (a, b) mark the transition from an accumulation to ablation zone and the roman numerals in panel (b) correspond to the solutions in Fig.6.

ice sheet), and the amount of refreezing is computed as the For Q < −Q0 no melting occurs and the domain top and residual. As shown in Fig.7b and c, the maximum storage bottom temperatures are identical. However, as soon as the −1 is 0.56 and 1.5 m w.e. yr for accumulation rates of 1.7 and annual mean surface forcing increases above −Q0, the do- 3.4 m w.e. yr−1, respectively. main top and bottom temperatures diverge due to the re- www.the-cryosphere.net/11/2799/2017/ The Cryosphere, 11, 2799–2813, 2017 2808 C. R. Meyer and I. J. Hewitt: Meltwater flow through snow lease of latent heat which warms the snow. Depending on Table 2. Typical numerical values for the surface energy balance the accumulation rate, the average bottom firn temperature (Cuffey and Paterson, 2010; van den Broeke et al., 2011). can reach the melting point, corresponding to a perennial −2 firn aquifer. This does not occur for smaller accumulation Sw 292 W m Net shortwave radiation rates, i.e., Fig.7a, but does for larger accumulation rates, α 0.6 Ice albedo i.e., Fig.7b and c. Additionally, all three panels show that 0.97 Emissivity σ 5.7 × 10−8 W m−2 K−4 Stefan–Boltzmann constant when Q increases further the bottom firn temperature de- −2 Lw 279 W m Longwave radiation creases again. This corresponds to the second type accumu- − − χ 10.3 W m 2 K 1 Turbulent transfer coefficient lation area shown in Fig.6III, in which water only penetrates a 9.5 × 10−9 m s−1 Accumulation Q 0 part of the way into the domain before refreezing. When Ta 267 K Average air temperature is large enough such that the region has become an ablation zone, the bottom temperature (now the temperature of incoming glacial ice) is almost the same as the surface temperature. is sufficiently high accumulation and sufficient melting oc- The largest bottom temperatures occur at intermediate values curs. of surface forcing, considerably lower than the value required In the future, we hope to extend this work beyond the to transition to an ablation region. one-dimensional solutions presented here. In principle the The thermal structure and water content of the lower firn model applies to fully three-dimensional geometries, when are strongly tied to the amount of meltwater produced, which the slope of the saturated surface (the “water table” in the in this model is tied directly to the annual mean surface forc- firn) will allow meltwater to flow laterally as well as ver- ing. In a warming world, one can imagine a particular loca- tically. The data from Humphrey et al.(2012) suggest the tion transitioning from an accumulation to ablation region. occurrence of “piping events”, where meltwater forms a ver- Our results in Fig.7 show that storage and refreezing can ac- tical channel and breaks through to depths where the snow commodate much of the melt that occurs when the warming is much colder. These events could be captured in a two- is not too large. Once the forcing is sufficient for runoff to dimensional framework, and it is possible that a theory al- start, the amount of refreezing decreases slightly so that an lowing the solid ice and liquid water to have different tem- increasingly large fraction of the melt runs off. Most of this peratures may help explain these features. On a larger scale, runoff is presumably routed to the glacier bed and then the the horizontal scales of the ice sheet are much larger than the ocean. As well as a form of mass loss, the timing and quantity depth of the firn, so a reduced, vertically integrated version of meltwater delivery to the bed will determine the style of of this theory may also be useful. subglacial drainage system that develops and the subsequent The use of Darcy’s law requires an estimate for the perme- ice dynamics (Zwally et al., 2002; Schoof, 2010; Tedstone ability and the relative permeability. The comparison of our et al., 2015). model behavior with the data from Humphrey et al.(2012) in Fig.4 is encouraging and suggests that these parameters could be determined with detailed measurements of surface 5 Conclusions melt and snow temperatures. Here we have interpreted the porosity and the permeability as grain-scale properties. An We have described a continuum model for the evolution of alternative interpretation that might be appropriate on larger firn hydrology, compaction, and thermodynamics. The model scales would treat these as averages over fractures, pipes, and is capable of determining the evolution of the firn includ- ice lenses to give a macroscopic effective porosity and per- ing the temperature, porosity, and water content. The model meability. differs from other models of firn hydrology in its treatment Although we have focused on idealized, periodic simula- of the percolation of water, for which we use Darcy’s law tions, the model can be forced by real climatological data or and a parametrization of capillary pressure. Our treatment for coupled to a regional atmospheric model. The model could runoff also differs in that we assume that water runs off when also be coupled to an ice-sheet model, using the deep firn the surface layer of snow is fully saturated rather than assum- temperature T ∞ as the surface boundary condition for the ing runoff at depth when the percolating water first reaches ice sheet. an impermeable ice layer. The model applies to both accumulation and ablation areas. Given the forcing (energy flux and accumulation rate), Data availability. The data associated with this paper are contained the model selects which of these applies to any particular re- in Humphrey et al. (2012) or can be produced from the code at- gion. One of the useful outputs of the model is an indication tached in the Supplement. of how the firn may change as function of climate warming, as revealed by moving from left to right in Fig.7. In agree- ment with Kuipers Munneke et al.(2014) and Steger et al. (2017a) we find that perennial firn aquifers occur when there

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Appendix A: Surface energy balance = θ + φS, (B3) H W S ∂ ∂ The surface energy balance is given by W + (w + q) = 0, (B4) ∂t ∂Z eiW ∂T ∂ ∂ θ − K = −(1 − α)S − L + σ T 4 − χ(T − T) H + w + q (θ + ) − W = 0, (B5) ∂z w w a ∂t ∂Z eiH S Pe Z − ρ c a(T − T) − ρ c R(T − T) + ρ M, (A1) dw w i a w w a wL (1 − φ) e = −cφ, (B6) dZ where the terms represent, in order, conduction into the ice, ∂pw incoming shortwave radiation S (α is the albedo), incom- q = k(φ)k (S) 1 − , (B7) w U r ∂Z ing longwave radiation Lw, outgoing longwave radiation ( is the emissivity and σ is the Stefan–Boltzmann constant), 1 −α 1 pw = − S (S < 1) or pw ≥ − (S = 1), (B8) turbulent heat transfer with coefficient χ, sensible heat fluxes B B associated with solid and liquid precipitation, which is as- subject to the boundary conditions sumed to fall with the air temperature Ta, and latent heat flux associated with melting. 1 ∂θ wi + q (θ + ) − Linearizing this equation around the melting temperature e H S PeW ∂Z T gives Eq. (15) in the text, where the components of Q are = [Q − θ + R − r] on Z = 0, (B9) m S given by w + q (θ + ) = a + R − r on Z = 0, (B10) eiW S 4 a − M Q(t) = (1 − α)Sw + Lw − σ T + χ(Ta − Tm) = m wei (wei > 0) or 1 − φ0 + ρwcia(Ta − Tm) + ρwcwR(Ta − Tm), (A2) a − M wi = (wi ≤ 0) on Z = 0, (B11) and the effective heat transfer coefficient h includes contri- e 1 − φ e butions from turbulent heat transfer and outgoing longwave ∂p − k(φ)k (S) w (θ + ) → 0 as Z → ∞, (B12) radiation: U r ∂Z S = + 3 1 ∂θ h χ 4σ Tm. (A3) − → 0 as Z → ∞. (B13) PeW ∂Z Using the values shown in Tables1 and2, we determine that a reasonable scale for Q is Q = 200 W m−2 and h = 0 Appendix C: Refreezing front 14.8 W m−2 K−1. Here we detail the approximate solution for the refreezing front considered in Sect. 3.1. The schematic is shown in Appendix B: Nondimensional model Fig.2a. We use dimensionless variables and the equations We nondimensionalize the lengths by ` = Q t /(ρ ) and we solve are 0 0 L time by the annual period t0. We write T = Tm + 1T θ and ∂S ∂q = φ + = 0,(0 < Z < Zf) (C1) choose the temperature scale as 1T Q0/h. Enthalpy is ∂t ∂Z scaled with ρ c 1T , ice velocity with `/t , water velocity i i 0 1 0 ∂S with (ρ gk )/µ, and water pressure with ρ g`. We define q = k(φ)kr(S) 1 + p (S) ,(0 < Z < Zf) (C2) w 0 w U B c ∂Z the parameters ∂φ 2 = 0,(0 < Z < Zf) and (Z > Zf) (C3) ρgk t ρcp` ρgdp` ∂t = 0 0 , = L , Pe = ,B = , (B1) 2 U `µ S cp1T Kt0 γ ∂θ 1 ∂ θ = ,(Z > Z ). (C4) Pe 2 f where is the scale for the water percolation relative to ice ∂t ∂Z U motion, is the Stefan number, Pe is the Péclet number, and The boundary conditions for Eqs. (C1)–(C4) are S B is the Bond number. Typical parameter values are shown in Table1. Both and are large; this indicates that the θ = θ∞ as Z → ∞, (C5) U B water percolates relatively quickly and that the percolation is θ = 0,S = 0 on Z = Zf. (C6) mainly driven by gravity rather than capillary pressure gradi- q = R on Z = 0, (C7) ents. Both of these could be seen as justification for tipping- bucket-type models. where θ∞ < 0 is the cold far-field temperature, and R is the = − = ˙ − Using the change of variables Z zs(t) z, with wei zs prescribed constant rainfall rate. Integrating across the front wi, we write the full nondimensional equations as at Zf(t) gives the nondimensional jump conditions + = 1 − φ + φS, (B2) q + φS w − Z˙ = −m , (C8) W ei f − i www.the-cryosphere.net/11/2799/2017/ The Cryosphere, 11, 2799–2813, 2017 2810 C. R. Meyer and I. J. Hewitt: Meltwater flow through snow

− − ˙ + = (1 φ) wei Zf − mi, (C9) front. The relevant scale for this region is of order 1/B, so we + write Z − Z = Z/Bˆ and determine the leading-order quasi- 1 ∂θ f (1 − φ) = − mi, (C10) static approximation Pe ∂Z − S ∂S ∂ ∂S − ˙ + 0 + = − φZf k(φ)kr(S) pc(S) 1 0, (C18) which states that the mass mi that freezes from the liquid ∂Zˆ U ∂Zˆ ∂Zˆ phase enters the solid phase and that the latent heat from refreezing warms the dry ice below. We can simplify these with the boundary conditions θ = + φ = φ + equations since 0 in the upper portion ( ), 0 and S → S as Zˆ → −∞ and S = 0 on η = 0. (C19) = − S 0 in the lower portion ( ), and the ice velocity wei is zero, so We can integrate this once and ﬁnd

+ + q − φ S Z˙ = −m , (C11) ˙ 0 ∂S f i k(φ)kr(S) − φSZf + k(φ)kr(S)pc(S) + U U ˆ (φ − φ )Z˙ = m , (C12) ∂Z 0 f i + + ˙ = k(φ)kr(S+) − φ S Zf, (C20) 1 ∂θ U (1 − φ0) = mi. (C13) Pe ∂Z − S where the constant comes from the matching condition. If we −α β now make use of pc = S , kr = S and take β = 2, α = 1, After a short initial transient, the solution approximates a then Eq. (C20) becomes traveling wave in which the upper region 0 < Z < Zf has + θ = 0, φ = φ (to be determined shortly), and q = R. Since ∂S 2 2 = S − S+ − ψ(S − S+), (C21) B 1, this means k(φ+)k (S+) ≈ R, which determines ∂Zˆ U r the constant S+ in the upper region (there is a narrow bound- ˙ = φZf ary layer behind the front, in which S+ changes rapidly but where ψ Uk(φ) , which can be integrated to give q − φ+S+Z˙ is constant; see below). f ψ 2S+ − ψ ψ 2S+ − ψ We next solve for the temperature evolution in the lower S = + tanh arctanh − Zˆ , (C22) 2 2 ψ − 2S+ 2 region. Assuming that the freezing front moves quickly, i.e., |Z˙ | 1 (this is appropriate since is large), we can move which is similar to the result derived by Gray(1996). f U into a translating frame Ze = Z − Zf and neglect the time de- pendence so that Appendix D: Saturation fronts 1 ∂θ ˙ ˙ + Zfθ ≈ Zfθ∞ (C14) Here we calculate the motion of the fully saturated fronts for Pe ∂Ze −Z/Z isothermal conditions with fixed porosity φ = φ0e 0 , as is constant (set by the far-field temperature), and hence θ ≈ in Sect. 3.3. We again make use of dimensionless variables. −PeZ˙ Z θ∞ 1 − e f e . This is the approximate solution given di- In the time before full saturation initiates, and in the limit B 1, conservation of water at the wetting front Z (t) is mensionally in Eq. (39). From the temperature field we can f given, as in the AppendixC with θ∞ = 0, by determine the melt rate using Eq. (C13) as R − φ S Z˙ = , ˙ f f f 0 (D1) Zfθ∞ mi = (1 − φ0) , (C15) −Z /Z where φf(t) = φ0e f 0 is the porosity at the front and Sf S is the saturation. Using permeability k(φ) = φ3 and relative which is negative, corresponding to freezing, since θ∞ < 0, 2 permeability kr(S) = S , we can calculate the saturation in- and Eq. (C12) therefore determines the porosity jump: duced by the rainfall as

+ θ∞ !1/2 φ = φ0 + (1 − φ0) . (C16) R S = . (D2) S f 3 φf Finally, the jump condition for water conservation, U Eq. (C11), determines the speed of the front as Thus, the initial evolution equation for the front before full saturation is R ˙ = Zf + + S . (C17) ˙ p Zf φ S − (1 − φ0)θ∞ Zf = Rφ0 exp − , (D3) S U 2Z0 This result corroborates the front velocity derived by Colbeck which can be integrated to give (1972), Gray(1996), and Durey(2014). √ To capture the smoothing of the front due to capillary pres- Rφ0 Zf = 2Z0 ln 1 + U t . (D4) sure, we can examine the narrow boundary layer behind the 2Z0

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We can therefore calculate the position of the front, and the time, at which full saturation occurs by setting Sf = 1. This gives

 1/6  ( φ3 ) φ3 ! Z0 0 2Z0 0 Z = ln U and t = √  U − 1. 1 3 R 1 Rφ R U 0 (D5)

Now in the fully saturated region, between the upper and lower saturation fronts Zu(t) < Z < Zl(t), we have ∂p k(φ) 1 − w = q , (D6) U ∂Z s where qs is the water ﬂux in the fully saturated region, which is constant since there is no compaction. Rearranging and integrating again, using pw(Zu) = pw(Zl), gives

Zl q Z dy Z − Z = s , (D7) l u k(φ) U Zu which determines the ﬂux as

3 3φ (Zl − Zu) q = 0 U . (D8) s 3Z /Z 3Z /Z Z0 e l 0 − e u 0

Since there is no melting–refreezing, water conservation across the lower front states that ˙ qs − φlZl = 0. (D9)

The equivalent jump condition on the upper front is ˙ ˙ qs − φuZu = R − φuSuZu, (D10) where S = (R/ φ3)1/2 as before. Thus, once full saturation u U u is initiated, we must solve the ODEs: q q − R Z˙ = s and Z˙ = s with l φ u 1/2 l − R φu 1 3 Uφu 3 3φ (Zl − Zu) q = 0 U , (D11) s 3Z /Z 3Z /Z Z0 e l 0 − e u 0 subject to the initial conditions

Zu = Zl = Z1 at time t = t1. (D12)

In dimensional form, these are the same as Eq. (41), and the solutions are compared to the full numerical solution using the enthalpy method in Fig.5.

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The Supplement related to this article is available online Gray, J. M. N. T. and Morland, L. W.: A dry snow pack model, Cold at https://doi.org/10.5194/tc-11-2799-2017-supplement. Reg. Sci. Technol., 22, 135–148, https://doi.org/10.1016/0165- 232X(94)90025-6, 1994. Gray, J. M. N. T. and Morland, L. W.: The compaction of polar snow packs, Cold Reg. Sci. Technol., 23, 109–119, https://doi.org/10.1016/0165-232X(94)00010-U, 1995. Competing interests. The authors declare that they have no conflict Harper, J., Humphrey, N., Pfeffer, W. T., Brown, J., and Fet- of interest. tweis, X.: Greenland ice-sheet contribution to sea-level rise buffered by meltwater storage in firn, Nature, 491, 240–243, https://doi.org/10.1038/nature11566, 2012. Acknowledgements. We wish to thank the 2016 Geophysical Fluid Herron, M. M. and Langway, C. C.: Firn densifica- Dynamics summer program at the Woods Hole Oceanographic tion: an empirical model, J. Glaciol., 25, 373–385, Institution, which is supported by the National Science Foundation https://doi.org/10.1017/S0022143000015239, 1980. and the Office of Naval Research. We also acknowledge financial Hewitt, I. J. and Schoof, C.: Models for polythermal support by NSF grants DGE1144152 and PP1341499 (CRM) as ice sheets and glaciers, The Cryosphere, 11, 541–551, well as Marie Curie FP7 Career Integration Grant within the 7th https://doi.org/10.5194/tc-11-541-2017, 2017. European Union Framework Programme (IJH). Humphrey, N. F., Harper, J. T., and Pfeffer, W. T.: Thermal tracking of meltwater retention in Greenland’s accumulation area, J. Geo- Edited by: Valentina Radic phys. Res., 117, https://doi.org/10.1029/2011JF002083, 2012. Reviewed by: two anonymous referees Hutter, K.: A mathematical model of polythermal glaciers and ice sheets, Geophys. Astro. Fluid, 21, 201–224, https://doi.org/10.1080/03091928208209013, 1982. References Koenig, L. S., Miège, C., Forster, R. 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