TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-11-2799-2017A continuum model for meltwater flow through compacting snowMeyerColin R.colinrmeyer@gmail.comhttps://orcid.org/0000-0002-1209-1881HewittIan J.https://orcid.org/0000-0002-9167-6481John A. Paulson School of Engineering and Applied Sciences,
Harvard University, Cambridge, MA 02138, USAMathematical Institute, Woodstock Road, Oxford, OX2 6GG, UKColin R. Meyer (colinrmeyer@gmail.com)11December2017116279928132July201719July201713October201723October2017This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://tc.copernicus.org/articles/11/2799/2017/tc-11-2799-2017.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/11/2799/2017/tc-11-2799-2017.pdf
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.
Introduction
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
. The balance between
runoff, refreezing, and storage is controlled by the mechanics and
thermodynamics of the porous snow. These processes also underlie the rate of
compaction of firn into ice and therefore control the average temperature
and accumulation rate that provide surface boundary conditions to numerical
ice-sheet models (which typically do not include the compacting firn layer
explicitly).
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. observe that the
snow at 10 m depth in Greenland is often more than 10 ∘C warmer
than the mean annual air temperature because of the refreezing of meltwater.
If, however, this water runs off through supraglacial streams or
drains to the bed through moulins, the latent heat is carried away and
subsequent cooling of the surface in the winter means that the remaining snow
is relatively cold. Since the capacity to store and/or refreeze meltwater is
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 the firn temperature and density profiles, depends on
climatic forcing (air temperature and radiative forcing as well as
accumulation). This question is of interest even under steady climate
conditions (i.e., seasonally periodic, without any year-on-year trend), and
this forms the focus of our study. A further question of current interest is
how the firn responds transiently to year-on-year increases in melting
, but we consider the steady problem a prerequisite
to understanding such transient response.
Our approach in this paper is to construct a continuum model for meltwater
percolating through porous snow, along similar lines to . This
contrasts cell-based numerical 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
and . 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 reality).
used a similar tipping-bucket method in the SNOWPACK
model and compared the results to the IMAU-FDM.
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 driven by gravity and capillary pressure, is similar to the
model we adopt in this study. This has also been used in a number of more
theoretical models for the percolation of meltwater through snow
. and
provide detailed descriptions of this approach in the context
of mixture theory.
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.
ModelPercolation through porous ice
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 1-ϕ is
solid ice while 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 water
(see the schematic in Fig. ). Conservation of
mass for ice, water, and air is expressed as
∂∂t(1-ϕ)ρi+∇⋅(1-ϕ)ρiui=-m,∂(Sϕρw)∂t+∇⋅Sϕρwuw=m,∂∂t(1-S)ϕρa+∇⋅(1-S)ϕρaua=0,
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
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 meltwater
internally is given by m and is therefore a source in
Eq. () and a sink in Eq. ().
The three components of meltwater-infiltrated snow: air, water, and
ice. Panel (a) shows water infiltrating an accumulation area where
the snow density increases with depth and snow advects down. The water
partially saturates the snow near the surface (S<1) whereas, at depth, all
of the air is replaced by water and the snow is fully saturated (S=1).
Panel (b) shows an ablation area where the there is fully saturated
porous snow in a thin layer near the surface and the underlying ice is solid,
advecting into the domain from upstream. Ice grains make contact in the third
dimension (into the page) and similarly many of the air and water pockets are
connected in the third dimension.
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.,
ϕSuw-ui=-k(ϕ)μkr(S)∇pw+ρwgz^,
where pw is the water pressure, k(ϕ) is the permeability,
kr(S) is the relatively permeability, and μ is the viscosity
of the water. For the permeability we use a simplified Carman–Kozeny
relationship, given by
k(ϕ)=dp2180ϕ3=k0ϕ3,
where dp is a typical grain size . Table
provides the parameter values we use later.
We must now distinguish between partially saturated (S<1) and fully
saturated (S=1) flow. When the snow is partially saturated, capillary
forces drive flow along liquid bridges connecting ice crystals
. Thus, we relate the water pressure to the capillary pressure
pc by pw=pa-pc, where
pa is the air pressure (taken as zero). Both the capillary
pressure and the relative permeability are prescribed functions of the
saturation S. We take
kr(S)=Sβandpc(S)=γdpS-α,
where γ is surface tension, and we choose the exponents α and
β such that β=α+1, which avoids a singularity in
kr(S)pc′(S) at S=0.
If the snow is fully saturated, water pressure pw is constrained
by mass conservation. Combining Eqs. (),
(), and () gives
∇⋅ui-k(ϕ)μρgz^+∇pw=m1ρw-1ρi,
which is an elliptic problem for pw. Boundary conditions for this
equation are provided by the constraint that pw must be
continuous across the interfaces between partially and fully saturated
regions and the constraint of no flow across impermeable boundaries (e.g., ice
lenses).
Compaction
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
∂ϕ∂t+ui⋅∇ϕ=-C,
where C is a parametrization of the rate of compaction
. The appropriateness of such models for snow containing
meltwater is uncertain. The parametrizations represent the rearrangement and
growth of snow crystals and the accompanying closure of air voids as
functions of temperature and accumulation rate, and these processes may be
modified by the presence of liquid between crystals. In the absence of a more
developed theory for wet compaction, we take the approach of using these dry
parametrizations but modify the material density derivative to include the
rate of refreezing that is calculated from the thermodynamics. Therefore, we
have
∂ϕ∂t+ui⋅∇ϕ=mρi-cϕ,
where the specific compaction rate we choose is the model,
which is written as C=cϕ. The coefficient c in units of
yr-1 is given by
c=11aexp-1222Tifϕ>0.4575aexp-2574Tifϕ≤0.4,
where a is the accumulation rate in meters of water equivalent per year and
T is the absolute temperature. This is an empirical parametrization, and
the two forms reflect a change in dominant compaction processes at a certain
snow density. Other parametrizations for compaction that could easily be
incorporated in this framework are discussed by ,
, and . We have chosen to use the Herron and
Langway model here for simplicity; from the experiments we have conducted,
different formulations do not appear to qualitatively change our results.
Combining with Eq. (), we note that Eq. ()
is equivalent to
1-ϕ∂wi∂z=-cϕ,
where wi is the vertical component of the ice velocity.
Temperature
We assume that ice and water are at the same temperature and therefore any
region containing meltwater (S>0) is at the melting point Tm.
In regions without water, we solve the temperature evolution equation,
ρicp(1-ϕ)∂T∂t+ρicp(1-ϕ)ui⋅∇T=∇⋅K‾∇T-Lm,
where the heat capacity is cp and the thermal conductivity is
K‾=(1-ϕ)K. The latent heat term -Lm operates on
interfaces of refreezing, where it is singular and causes discontinuities to
occur in the temperate gradient.
Surface boundary conditions
Here we write boundary conditions on the surface zs(t), which we
assume is locally flat, and we write wi and ww as the
vertical velocities. The kinematic conditions are
ρi(1-ϕ)(wi-z˙s)=ρw(M-a),ρwϕS(ww-z˙s)=ρw(M-R+r),
where z˙s is the velocity of the surface, M is the
rate of melting, a is the accumulation rate, R is the rainfall rate, and
r is runoff, all expressed in units of water equivalent per year. The
compaction Eq. () requires a boundary condition, ϕ=ϕ0, when the accumulation rate is greater than the rate of melting
(i.e.,
wi-z˙s<0), where (1-ϕ0)ρi
is the bulk density of freshly deposited snow. The energy balance on the
surface provides a boundary condition for the temperature equation when the
surface temperature is less than Tm, and determines the rate of
melting M when T=Tm. These conditions are combined as
ρicp1-ϕwi-z˙s(T-Tm)-K‾∂T∂z=-Q+h(T-Tm)+ρwLM,
along with the conditions M=0 when T<Tm, and M≥0 when
T=Tm. The forcing energy flux Q(t) includes the combined
effects of radiative, turbulent, and sensible heat fluxes. We assume that
this is prescribed in order to provide a simple parametrization of the
climate forcing. However, it can be related to more specific components of
the energy balance as described in Appendix A. The heat transfer coefficient
h represents a combination of radiative and turbulent heat transfer. We
expect Q to have a typical magnitude on the order of Q0=200 W m-2 with a comparable seasonal amplitude and take h=14.8 W m-2 K-1 as a representative constant
.
Numerical method
Our complete model is given by ice and water conservation ()
and (), Darcy's law (), compaction
(), and temperature evolution (), subject to
the boundary conditions ()–(). The model is forced
by a prescribed energy flux Q, accumulation a, and precipitation R, and
it predicts the temperature, porosity, and saturation profiles as well as the
surface melt rate, runoff, refreezing, and storage of liquid water.
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 .
The second step is to change variables into a frame that moves with the ice
surface. At this stage we also simplify the model to write it in one vertical
dimension, and we make the Boussinesq approximation to ignore density
differences so that ρi=ρw=ρ.
We define the total water as W, which is the sum of liquid and
solid fractions, i.e.,
W=1-ϕ+Sϕ,
and we define the enthalpy as the sum of sensible and latent heat as
H=ρcpW(T-Tm)+ρLSϕ.
The inverse relationships that relate the enthalpy H and total
water W to the temperature T, saturation S, and porosity
ϕ are
T=Tm+min0,HW,ϕ=1-W+max0,HρL,andS=max0,HρLϕ.
We define the depth below the ice surface Z, and the relative downward ice
velocity w̃i, as
Z=zs(t)-z,w̃i=z˙s-wi.
We combine the conservation Eqs. () and
() with Darcy's law () and temperature
evolution () as
∂W∂t+∂∂Zw̃iW+q=0,∂H∂t+∂∂Zw̃iH+qρcp(T-Tm)+ρL-K‾∂T∂Z=0,
where the downward water flux is
q=k(ϕ)kr(S)μρg-∂pw∂Z.
Combining ice conservation () with compaction () gives
(1-ϕ)∂w̃i∂Z=-cϕ,
and water pressure is given by
pw=-γdpS-α(S≤1)orpw≥-γdp(S=1).
The surface boundary conditions (at Z=0), re-expressed in terms of
W and H, are given as
Ww̃i+q=a+R-r,w̃iH+qρcp(T-Tm)+ρL-K‾∂T∂Z=Q-h(T-Tm)+ρLR-ρLr,w̃i=a-M1-ϕ0(w̃i>0)orw̃i=a-M1-ϕ(w̃i≤0).
In these conditions, the runoff r is assumed to be zero unless the snow at
the surface reaches full saturation, in which case
Eqs. () and () determine r. At the
bottom of our domain, we assume that the conductive heat flux and pressure
gradients vanish to replicate effective matching conditions to the deep
interior of the ice sheet. On internal interfaces between fully saturated and
partially saturated regions we apply pw=-γ/dp to ensure
pressure continuity.
Table of physical values, derived scales, and nondimensional
parameter values (defined in Appendix ).
ρ917 kg m-3Q0200 W m-2U100cp2050 m2 s-2 K-1h14.8 W m-2 K-1S12L334 000 m2 s-2ΔT13.5 KB260K2.1 kg m s-3 K-1M6×10-4 kg m-2 s-1Pe11g9.806 m s-2ℓ20.6 mα1γ0.07 N m-1k05.6×10-11 m2β2dp10-4 mt03.15×107 sμ10-3 Pa s
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 time using explicit forward
Euler time stepping, which involves evaluating the fluxes on the cell edges
using the quantities from the previous timestep. For advection, we use an
upwind scheme where the value of the variable advected depends on the
velocity direction. For edges between partially saturated cells, we evaluate
the water fluxes using the capillary pressure for pw on the
adjacent cells. For edges between fully saturated cells, we solve
Eq. () with S=1 as an elliptic equation for pw
on the saturated cells, which we then use to evaluate the water fluxes. In
order to allow cells to switch from fully to partially saturated, we compute
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.
Test problems
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 ,
, and and has an approximate analytical
solution that provides a useful test case for the enthalpy method. We also
compare the results of the analytical solution for the propagation of the
meltwater front to temperature data from . Secondly, to test
the propagation of saturated fronts, we consider an isothermal problem in
which the porosity profile is prescribed to decrease with depth. We again
investigate how rainwater propagates into the snow, with saturation
increasing as the front propagates down. At a certain point the snow fully
saturates and a saturated front propagates up toward the snow surface.
Schematic of the test problems considered in
(a) Sect. and
(b) Sect. . In both panels, rain falls at a rate
R on the surface of the snow. White shading indicates dry snow (S=0),
grey indicates partially saturated snow (0<S<1), and dark shading indicates
fully saturated snow (S=1). In panel (a), the snow is initially
cold with T=T∞ and dry, with uniform porosity ϕ0. The rainwater percolates through the snow, refreezes at the interface
Zf(t), and releases latent heat that warms the snow. The
refreezing decreases the porosity in the upper region so that
ϕ+<ϕ0. In panel (b), the snow is temperate,
T=Tm, with a porosity profile that decays exponentially with
depth. After the snow fully saturates two saturation fronts emerge with
Zl propagating downward and Zu
upward.
Rainfall into cold snow
We consider the infiltration of rain into cold, dry snow as a test problem.
We start with a patch of dry snow (S=0) with constant porosity
(ϕ=ϕ0) and temperature (T=T∞<Tm). We assume no
accumulation and ignore compaction so that the ice is stationary.
Furthermore, the porosity is large enough that the snow never fully
saturates. At time t=0 a fixed flux of rain R with a temperature
T=Tm is applied at the surface Z=0 and a wetting front at
Z=Zf moves down at velocity Z˙f (we show a
schematic in Fig. and the numerical solutions in
Fig. ). Since the capillary pressure gradients are small and
the flow is largely driven by gravity, the wetting front behaves as a
smoothed shock front. Some of the water at the shock front refreezes, warming
the snow ahead. As shown in more detail in Appendix , the
behavior of this shock can be understood by ignoring the diffusive capillary
pressure term. This approximation relegates Eqs. () and
() to hyperbolic partial differential equations for the
porosity and saturation as well as simplifying the temperature
Eq. () so that
∂S∂t+ρgk(ϕ)kr′(S)ϕμ∂S∂Z=0,0<Z<Zf,∂ϕ∂t=0,0<Z<ZfandZ>Zf,∂T∂t=Kρcp∂2T∂Z2,Z>Zf,
where kr′(S)=dkr/dS, and with initial and
boundary conditions
S=0,ϕ=ϕ0,T=T∞att=0,T=T∞asZ→∞,T=TmonZ=Zf,ρgk(ϕ)kr(S)μ=RonZ=0.
Equations ()–() have corresponding jump conditions
across the shock which incorporate the refreezing rate -mi at
that front. These are
ρgμk(ϕ)kr(S)-ϕSZ˙f+=-mi,Z˙fϕ-+=mi,(1-ϕ)K∂T∂Z-=ρLmi,
where + refers to the region above the front (Z<Zf). Using
these jump conditions and the solutions to
Eqs. ()–()
subject to the boundary conditions ()–(), we find an
approximate expression for the front velocity:
Z˙f=RLϕ+S+L+(1-ϕ0)(Tm-T∞).
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 . The temperature is made
nondimensional by T=Tm+(T∞-Tm)T^ and
the parameters are ϕ0=0.4 and R=0.54, along with other values in
Table .
Note that if T∞=Tm, i.e., isothermal snow, the front
simply propagates at the speed of the draining rainwater R/(ϕ+S+). The effect of refreezing due to
T∞<Tm is to slow the front and to cause a decrease in the
porosity as the front passes by an amount ϕ0-ϕ+=(1-ϕ0)cp(Tm-T∞)/L. This is a mechanism by which ice lenses can
form: if the pre-existing porosity is small enough, the porosity above the
front can decrease to zero and the pores freeze shut. In this case the front
stops propagating, and a saturated region forms above the lens in a similar
way to that described in Sect. . We also determine
approximate analytical solutions for temperature and saturation, which are
compared to the numerical solutions in Fig. . The agreement
between the numerical and approximate solutions is very good. The approximate
temperature profile ahead of the refreezing front is given by
T=T∞+Tm-T∞exp-ρcpZ˙f(Z-Zf)K.
Data comparison
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. and collected
temperature data in the accumulation zone on the western flank of the
Greenland Ice Sheet and inferred the movement of meltwater by warming of the
snow due to the release of latent heat. They set up a vertical string of
thermistors to determine the temperature profile in the upper 10 m of the
ice sheet. Data from one vertical string between the dates of 5 and 25 July
2007 (days 185–203) are shown in Fig. . From these data
it is clear that the ice at depth progressively warmed, likely due to the
refreezing of liquid meltwater. Over the 12 days between day 185 and
day 197, the warming front propagated about a meter, while over the course of
the next 6 days from day 197 to day 203 the meltwater penetrated two
additional meters, showing a 4-fold increase in front velocity.
infer that the warming spike on day 199 is due to an influx
of meltwater from lateral sources. A minimum temperature is observed at
around 5 m depth and the temperature recorded on the lower thermistors is
warmer, which could be due to prior warming by meltwater pulses or a
manifestation of the seasonal thermal wave.
We now compare these data to the approximate solution for the temperature
field ahead of a refreezing front, as given in Eq. (). We fit
front speed Z˙f for the days 185–197 and a larger front
speed for days 197–203. The increase in the front speed is likely due to an
increase in surface melt. We set the melting temperature
Tm=0∘C, fit a constant far-field temperature
T∞, and use the heat diffusivity for ice K/(ρcp)=1.1×10-6 m2 s-1 (Table ). In light of the
simplified analysis, the fit between Eq. () and the
data is quite good.
Data from show the propagation of refreezing fronts
in Greenland firn. We overlay the approximate temperature solution for the
temperature ahead of a refreezing front (black lines, Eq. ).
The speed of the front varies over the 18-day record: dashed lines use the
initial speed and the dotted line uses the final speed. The far-field
temperature is assumed to be constant in the model whereas the data show a
local minimum in temperature at around 5 m, which could be due to prior
freezing fronts or the seasonal wave.
Isothermal saturation fronts
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
ϕ(Z)=ϕ0e-Z/Z0,
where Z0 is a constant. We continue to ignore compaction and accumulation,
and since the snow is isothermal the porosity is therefore constant in time.
Initially, the rain partially saturates the snow and a wetting front moves
downward, as shown in Fig. a. Then, at a certain depth, the
maximum saturation reaches unity and two saturation fronts emerge, one that
propagates up and the other down, as shown in Fig. b and c.
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=ℓ/2, where ℓ is the characteristic length scale defined in
Appendix and given in Table . Panel (a)
shows the position of the front before the firn fully saturates.
Panels (b, c) show the bidirectional motion of the fully saturated
fronts. Dashed black lines show semi-analytical solutions from solving
Eq. ().
In Appendix , we derive the locations of the upper
Zu and lower Zl fronts by neglecting flow driven by
gradients in capillary pressure. This analysis results in two differential
equations for the evolution of upper front Zu and lower front
Zl:
Z˙l=qsϕlandZ˙u=qs-Rϕu1-μRe3Zu/Z0ρgk0ϕ031/β,withqs=3k0ϕ03ρg(Zu-Zl)μZ0e3Zu/Z0-e3Zl/Z0,
subject to the initial conditions
Zu=Zl=Z1at timet=t1,
where Z1 and t1 are the location and time at which full saturation
initiates. We solve these coupled, nonlinear ordinary differential equations
(ODEs) using a numerical integrator in MATLAB and compare these
semi-analytical solutions to the full numerical solutions in
Fig. (the dashed black lines). The slight differences are
due to neglecting the gradient in capillary pressure in our approximate
solutions.
Results
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 (∼ 1 m depth) and we are
interested in the full firn column (∼ tens of meters of depth). For cold
ice, the variation of surface energy flux leads to a seasonal temperature
wave and a dry-compaction density profile. This solution
breaks down if the surface temperature reaches the melting point during
summer, at which point the surface snow melts and the meltwater can percolate
through the snow and refreeze, thereby warming the snow through the release
of latent heat. Even with a small amount of melting, the resulting
temperature profiles become very different from the thermal wave.
We apply an oscillating surface forcing in Eq. () of the form
Q(t)=Q‾-Q0cos(2πt/t0),
where Q‾ is the annual mean surface forcing, and we take the
amplitude Q0=200 W m-2 and period t0=1 year. For
simplicity, we assume a constant accumulation rate and ignore rainfall.
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
ϕ(a), saturation S(b), and temperature
T(c) as a function of time for the accumulation rate
a=1.7 m w.e. yr-1 and four values of forcing: (I) cold accumulation
zone where the mean forcing is Q‾=-Q0 and (II) accumulation area
with mean forcing Q‾=-0.707Q0. In this case, a clear perennial
aquifer develops. (III) Accumulation area with larger forcing
Q‾=-0.575Q0. (IV) Ablation zone with mean forcing
Q‾=-0.146Q0. In all simulations the porosity of the falling snow
is ϕ0=0.64 and the black lines show ice streamlines.
Each case shows a different value of Q‾ with the same
accumulation rate (1.7 m water equivalent per year) and porosity of fresh
snow ϕ0=0.64. While the ice surface moves up and down during the
simulation, we plot the quantities as a function of depth below the surface
Z=zs(t)-z and plot ice streamlines to show the relative motion
of the ice. In Fig. we show how the mean
temperature at the bottom of the domain T‾∞ and the mean
surface temperature T‾s change as the mean surface
forcing varies, for three different values of accumulation rate. Each point
in this figure corresponds to an annual average of a periodic simulation such
as those in Fig. (and which are labeled in
Fig. b).
The four simulations in Fig. represent the spectrum of
possible surface types on glaciers and ice sheets, encompassing both
accumulation and ablation regions. If we interpret increasing Q‾
as a parametrization of slow climate warming, we might expect a location that
is initially an accumulation area to transition through each of these states.
Figure I is an accumulation area where there is no melting at
any point during the year. The ice streamlines show that the ice advects
downward as more snow accumulates on the surface. The snow compaction is
visible from a convergence of the streamlines with time. The temperature
variation with depth in this case is just the thermal wave and the variations
in surface temperature are only felt around K/(ρcpω)∼6 m into the snow.
Increasing Q‾ above -Q0 leads to melting during summer.
Figure II shows an accumulation area where the temperature and
porosity profiles are significantly affected by the meltwater that drains
into the snow during the summer. Here there is water below 10 m throughout
the year fed by percolation each summer. This is a perennial aquifer, as
found in a number of field observations .
Figure III shows a region which is an accumulation area but
with more melting than in Fig. II. Interestingly, this
situation no longer has a perennial aquifer and all of the meltwater that is
produced refreezes. Although still a percolation zone it is different in
character than the region shown in Fig. II. The porosity
decreases more rapidly with depth in this case so that despite more water
being produced on the surface during the summer, this larger quantity of
water is not able to percolate as far into the snow. As a consequence, it is
not so well insulated from the cold surface during the winter and all of the
water refreezes. This greater quantity of refreezing is in turn responsible
for the more rapid decrease in porosity with depth that prevents the liquid
water percolating as deep as it does in Fig. II (more
refreezing means the pore space is filled in more effectively with ice). In
contrast, the reason a perennial aquifer is sustained in
Fig. II is because the water penetrates sufficiently far that
it is insulated from the cold surface .
Above a critical Q‾ there is too much melting for the firn to
accommodate and runoff begins (this occurs at a value of Q‾
intermediate between Fig. III and IV and is clearest to see in
Fig. b). The transition from an
accumulation area to an ablation area occurs when runoff exceeds the
accumulation. Figure IV shows an ablation area where the
surface meltwater is only able to enter a few meters into the snow and
reaches the impermeable barrier of the glacial ice surface. During the course
of the summer all of the snow is melted as well as some of the glacial ice.
The streamlines show net upward motion in this case indicating that there is
net ablation over the course of the year.
Average meltwater partition (right) and annual mean temperature at
the ice surface T‾s and bottom of the domain
T‾∞ (left) as a function 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, and
(c)a=3.4 m w.e. yr-1. 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. .
In Fig. we also calculate the total
quantity of surface melt and the partitioning of the melt between runoff,
liquid storage in the ice, and refreezing in the firn. Runoff and melt are
calculated from the model output, liquid storage is taken to be the total
water flux passing out of the bottom of the domain (the domain represents
only the surface firn layer, so this represents water that is stored within
the upper part of the ice sheet), and the amount of refreezing is computed as
the residual. As shown in Fig. b and c,
the maximum storage is 0.56 and 1.5 m w.e. yr-1 for accumulation rates of
1.7 and 3.4 m w.e. yr-1, respectively.
For Q‾<-Q0 no melting occurs and the domain top and bottom
temperatures are identical. However, as soon as the annual mean surface
forcing increases above -Q0, the domain top and bottom temperatures
diverge due to the release of latent heat which warms the snow. Depending on
the accumulation rate, the average bottom firn temperature can reach the
melting point, corresponding to a perennial firn aquifer. This does not occur
for smaller accumulation rates, i.e.,
Fig. a, but does for larger accumulation
rates, i.e., Fig. b and c. Additionally,
all three panels show that when Q‾ increases further the bottom
firn temperature decreases again. This corresponds to the second type
accumulation area shown in Fig. III, in which water only
penetrates part of the way into the domain before refreezing. When
Q‾ 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. The largest bottom temperatures
occur at intermediate values of surface forcing, considerably lower than the
value required to transition to an ablation region.
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. show that storage and
refreezing can accommodate much of the melt that occurs when the warming is
not too large. Once the forcing is sufficient for runoff to start, the amount
of refreezing decreases slightly so that an increasingly large fraction of
the melt runs off. Most of this runoff is presumably routed to the glacier
bed and then the ocean. As well as a form of mass loss, the timing and
quantity of meltwater delivery to the bed will determine the style of
subglacial drainage system that develops and the subsequent ice dynamics
.
Conclusions
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 agreement with
and we find that perennial firn aquifers occur when there is
sufficiently high accumulation and sufficient melting occurs.
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 suggest the occurrence of
“piping events”, where meltwater forms a vertical channel and breaks through
to depths where the snow is much colder. These events could be captured in a
two-dimensional framework, and it is possible that a theory allowing the
solid ice and liquid water to have different temperatures may help explain
these features. On a larger scale, the horizontal scales of the ice sheet are
much larger than the depth of the firn, so a reduced, vertically integrated
version of this theory may also be useful.
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 in Fig. is encouraging and
suggests that these parameters could be determined with detailed measurements
of surface melt and snow temperatures. Here we have interpreted the porosity
and the permeability as grain-scale properties. An alternative interpretation
that might be appropriate on larger scales would treat these as averages over
fractures, pipes, and ice lenses to give a macroscopic effective porosity
and permeability.
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 T‾∞ as the surface boundary condition
for the ice sheet.
Typical numerical values for the surface energy balance
.
Sw292 W m-2Net shortwave radiationα0.6Ice albedoϵ0.97Emissivityσ5.7×10-8 W m-2 K-4Stefan–Boltzmann constantLw279 W m-2Longwave radiationχ10.3 W m-2 K-1Turbulent transfer coefficienta09.5×10-9 m s-1AccumulationTa267 KAverage air temperature
The data associated with this paper are
contained in Humphrey et
al. (2012) or can be produced from the code attached in the Supplement.
Surface energy balance
The surface energy balance is given by
-K‾∂T∂z=-(1-α)Sw-Lw+ϵσT4-χ(Ta-T)-ρwcia(Ta-T)-ρwcwR(Ta-T)+ρwLM,
where the terms represent, in order, conduction into the ice, incoming
shortwave radiation Sw (α is the albedo), incoming
longwave radiation Lw, outgoing longwave radiation (ϵ is
the emissivity and σ is the Stefan–Boltzmann constant), turbulent
heat transfer with coefficient χ, sensible heat fluxes associated with
solid and liquid precipitation, which is assumed to fall with the air
temperature Ta, and latent heat flux associated with melting.
Linearizing this equation around the melting temperature Tm gives
Eq. () in the text, where the components of Q are given by
Q(t)=(1-α)Sw+Lw-ϵσTm4+χ(Ta-Tm)+ρwcia(Ta-Tm)+ρwcwR(Ta-Tm),
and the effective heat transfer coefficient h includes contributions from
turbulent heat transfer and outgoing longwave radiation:
h=χ+4ϵσTm3.
Using the values shown in Tables and , we
determine that a reasonable scale for Q is Q0=200 W m-2 and h=14.8 W m-2 K-1.
Nondimensional model
We nondimensionalize the lengths by ℓ=Q0t0/(ρL) and
time by the annual period t0. We write T=Tm+ΔTθ and choose the temperature scale as ΔT=Q0/h. Enthalpy is
scaled with ρiciΔT, ice velocity with
ℓ/t0, water velocity with (ρwgk0)/μ, and water
pressure with ρwgℓ. We define the parameters
U=ρgk0t0ℓμ,S=LcpΔT,Pe=ρcpℓ2Kt0,B=ρgdpℓγ,
where U is the scale for the water percolation relative to ice
motion, S is the Stefan number, Pe is the Péclet
number, and B is the Bond number. Typical parameter values are shown in
Table . Both U and B are large; this
indicates that the water percolates relatively quickly and that the
percolation is mainly driven by gravity rather than capillary pressure
gradients. Both of these could be seen as justification for
tipping-bucket-type models.
Using the change of variables Z=zs(t)-z, with
w̃i=z˙s-wi, we write the
full nondimensional equations as
W=1-ϕ+ϕS,H=Wθ+SϕS,∂W∂t+∂∂Zw̃iW+q=0,∂H∂t+∂∂Zw̃iH+qθ+S-WPeθZ=0,(1-ϕ)dw̃dZ=-cϕ,q=Uk(ϕ)kr(S)1-∂pw∂Z,pw=-1BS-α(S<1)orpw≥-1B(S=1),
subject to the boundary conditions
w̃iH+qθ+S-1PeW∂θ∂Z=SQ-θ+R-ronZ=0,w̃iW+qθ+S=a+R-ronZ=0,w̃i=a-M1-ϕ0(w̃i>0)orw̃i=a-M1-ϕ(w̃i≤0)onZ=0,-Uk(ϕ)kr(S)∂pw∂Zθ+S→0asZ→∞,-1PeW∂θ∂Z→0asZ→∞.
Refreezing front
Here we detail the approximate solution for the refreezing front considered
in Sect. . The schematic is shown in
Fig. a. We use dimensionless variables and the
equations we solve are
ϕ∂S∂t+∂q∂Z=0,(0<Z<Zf)q=Uk(ϕ)kr(S)1+1Bpc′(S)∂S∂Z,(0<Z<Zf)∂ϕ∂t=0,(0<Z<Zf)and(Z>Zf)∂θ∂t=1Pe∂2θ∂Z2,(Z>Zf).
The boundary conditions for Eqs. ()–() are
θ=θ∞asZ→∞,θ=0,S=0onZ=Zf.q=RonZ=0,
where θ∞<0 is the cold far-field temperature, and R is the
prescribed constant rainfall rate. Integrating across the front at
Zf(t) gives the nondimensional jump conditions
q+ϕSw̃i-Z˙f-+=-mi,(1-ϕ)w̃i-Z˙f-+=mi,1Pe(1-ϕ)∂θ∂Z-+=-Smi,
which states that the mass -mi that freezes from the liquid phase
enters the solid phase and that the latent heat from refreezing warms the
dry ice below. We can simplify these equations since θ=0 in the upper
portion (+), ϕ=ϕ0 and S=0 in the lower portion (-), and the
ice velocity w̃i is zero, so
q-ϕ+S+Z˙f=-mi,(ϕ+-ϕ0)Z˙f=mi,1Pe(1-ϕ0)∂θ∂Z-=Smi.
After a short initial transient, the solution approximates a traveling wave
in which the upper region 0<Z<Zf has θ=0, ϕ=ϕ+ (to be determined shortly), and q=R. Since B≫1, this means
Uk(ϕ+)kr(S+)≈R, which determines the
constant S+ in the upper region (there is a narrow boundary layer behind
the front, in which S+ changes rapidly but q-ϕ+S+Z˙f is constant; see below).
We next solve for the temperature evolution in the lower region. Assuming
that the freezing front moves quickly, i.e., |Z˙f|≫1 (this
is appropriate since U is large), we can move into a translating
frame Z̃=Z-Zf and neglect the time dependence so that
1Pe∂θ∂Z̃+Z˙fθ≈Z˙fθ∞
is constant (set by the far-field temperature), and hence θ≈θ∞1-e-PeZ˙fZ̃.
This is the approximate solution given dimensionally in
Eq. (). From the temperature field we can determine the melt
rate using Eq. () as
mi=(1-ϕ0)Z˙fθ∞S,
which is negative, corresponding to freezing, since θ∞<0, and
Eq. () therefore determines the porosity jump:
ϕ+=ϕ0+(1-ϕ0)θ∞S.
Finally, the jump condition for water conservation, Eq. (),
determines the speed of the front as
Z˙f=RSϕ+S+S-(1-ϕ0)θ∞.
This result corroborates the front velocity derived by , , and .
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 1/B, so we write Z-Zf=Z^/B and
determine the leading-order quasi-static approximation
-ϕZ˙f∂S∂Z^+U∂∂Z^k(ϕ)kr(S)pc′(S)∂S∂Z^+1=0,
with the boundary conditions
S→S+asZ^→-∞andS=0onη=0.
We can integrate this once and find
Uk(ϕ)kr(S)-ϕSZ˙f+Uk(ϕ)kr(S)pc′(S)∂S∂Z^=Uk(ϕ)kr(S+)-ϕ+S+Z˙f,
where the constant comes from the matching condition. If we now make use of
pc=S-α, kr=Sβ and take β=2,
α=1, then Eq. () becomes
∂S∂Z^=S2-S+2-ψ(S-S+),
where ψ=ϕZ˙fUk(ϕ),
which can be integrated to give
S=ψ2+2S+-ψ2tanharctanhψψ-2S+-2S+-ψ2Z^,
which is similar to the result derived by .
Saturation fronts
Here we calculate the motion of the fully saturated fronts for isothermal
conditions with fixed porosity ϕ=ϕ0e-Z/Z0, as in
Sect. . We again make use of dimensionless variables. In
the time before full saturation initiates, and in the limit B≫1,
conservation of water at the wetting front Zf(t) is given, as in
the Appendix with θ∞=0, by
R-ϕfSfZ˙f=0,
where ϕf(t)=ϕ0e-Zf/Z0 is the porosity at
the front and Sf is the saturation. Using permeability k(ϕ)=ϕ3 and relative permeability kr(S)=S2, we can calculate
the saturation induced by the rainfall as
Sf=RUϕf31/2.
Thus, the initial evolution equation for the front before full saturation is
Z˙f=URϕ0exp-Zf2Z0,
which can be integrated to give
Zf=2Z0ln1+URϕ02Z0t.
We can therefore calculate the position of the front, and the time, at which
full saturation occurs by setting Sf=1. This gives
Z1=Z03lnϕ03URandt1=2Z0URϕ0ϕ03UR1/6-1.
Now in the fully saturated region, between the upper and lower saturation
fronts Zu(t)<Z<Zl(t), we have
Uk(ϕ)1-∂pw∂Z=qs,
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-Zu=qsU∫ZuZldyk(ϕ),
which determines the flux as
qs=3ϕ03U(Zl-Zu)Z0e3Zl/Z0-e3Zu/Z0.
Since there is no melting–refreezing, water conservation across the lower
front states that
qs-ϕlZ˙l=0.
The equivalent jump condition on the upper front is
qs-ϕuZ˙u=R-ϕuSuZ˙u,
where Su=(R/Uϕu3)1/2 as before. Thus, once
full saturation is initiated, we must solve the ODEs:
Z˙l=qsϕlandZ˙u=qs-Rϕu1-RUϕu31/2withqs=3ϕ03U(Zl-Zu)Z0e3Zl/Z0-e3Zu/Z0,
subject to the initial conditions
Zu=Zl=Z1at timet=t1.
In dimensional form, these are the same as Eq. (), and
the solutions are compared to the full numerical solution using the enthalpy
method in Fig. .
The Supplement related to this article is available online at https://doi.org/10.5194/tc-11-2799-2017-supplement.
The authors declare that they have no conflict of
interest.
Acknowledgements
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
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