TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-12-3215-2018Processes influencing heat transfer in the near-surface ice of Greenland's
ablation zoneNear-surface heat transferHillsBenjamin H.bhills@uw.eduhttps://orcid.org/0000-0003-4490-7416HarperJoel T.MeierbachtolToby W.https://orcid.org/0000-0002-8487-7920JohnsonJesse V.HumphreyNeil F.WrightPatrick J.https://orcid.org/0000-0003-2999-9076Department of Earth and Space Sciences, University of Washington,
Seattle, Washington, USADepartment of Geosciences, University of Montana, Missoula, Montana,
USADepartment of Computer Science, University of Montana, Missoula,
Montana, USADepartment of Geology and Geophysics, University of Wyoming, Laramie,
Wyoming, USAInversion Labs LLC, Wilson, Wyoming, USABenjamin H. Hills (bhills@uw.edu)8October20181210321532279March20182May201816September201821September2018This 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/12/3215/2018/tc-12-3215-2018.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/12/3215/2018/tc-12-3215-2018.pdf
To assess the influence of various heat transfer processes on the
thermal structure of near-surface ice in Greenland's ablation zone, we
compare in situ measurements with thermal modeling experiments. A total of
seven temperature strings were installed at three different field sites,
each with between 17 and 32 sensors and extending up to 21 m below the
ice surface. In one string, temperatures were measured every 30 min, and
the record is continuous for more than 3 years. We use these measured
ice temperatures to constrain our modeling experiments, focusing on four
isolated processes and assessing the relative importance of each for the
near-surface ice temperature: (1) the moving boundary of an ablating surface,
(2) thermal insulation by snow, (3) radiative energy input, and (4) subsurface
ice temperature gradients below the seasonally active near-surface layer. In
addition to these four processes, transient heating events were observed in
two of the temperature strings. Despite no observations of meltwater
pathways to the subsurface, these heating events are likely the refreezing
of liquid water below 5–10 m of cold ice. Together with subsurface
refreezing, the five heat transfer mechanisms presented here account for
measured differences of up to 3 ∘C between the mean annual air
temperature and the ice temperature at the depth where annual temperature
variability is dissipated. Thus, in Greenland's ablation zone, the mean
annual air temperature is not a reliable predictor of the near-surface ice
temperature, as is commonly assumed.
Introduction
Bare ice regions of the Greenland ice sheet have high summer melt rates.
Here, the surface ice temperature is important to ablation processes such as
melt, water storage, runoff, and albedo modifications associated with the
surface cryoconite layer. The ice surface temperature also acts as an
essential boundary condition for the transfer of heat into deeper ice below
and is therefore important for ice flow modeling
(e.g. Meierbachtol et al., 2015) as well as
interpretation of borehole temperature measurements
(Harrington et al., 2015; Hills et al., 2017; Lüthi et al., 2015). In order to constrain the
rate of ice melting and more generally to understand the mechanisms which
move energy between the ice and the atmosphere above, we must understand the
processes that control near-surface heat transfer in bare ice.
Heat transfer at the ice surface is dominated by thermal diffusion from the
overlying air (Cuffey and Paterson, 2010). Seasonal air temperature
oscillations are diminished with depth in the ice until they are
negligible (i.e. ∼1 %) at a “depth of zero annual amplitude” (van
Everdingen, 1998). The exact location of this depth is dependent on the
thermal diffusivity of the material through which heat is conducted as well
as the period of oscillation (Carslaw and Jaeger, 1959; pp. 64–70). In
theory, the temperature at the depth of zero annual amplitude, a value we
will call T0, is approximately constant and equal to the mean annual air
temperature. In snow and ice, the depths of zero annual amplitude are
approximately 10 and 15 m, respectively (Hooke, 1976). For this reason,
studies in the cryosphere often use T0 as a proxy for the mean air
temperature, drilling to 10 m or more to measure the snow or ice
temperature at that depth (Loewe, 1970; Mock and Weeks, 1966).
In places where heat transfer is purely diffusive, the snow or ice is
homogeneous, and interannual climate variations are minimal, T0 is a
good approximation for the mean air temperature. However, prior studies have
shown that, in many areas of glaciers and ice sheets, the relationship
between air and ice temperatures can be substantially altered by additional
heat transfer processes. For example, in the percolation zone, infiltration
and refreezing of surface meltwater warm the subsurface (Humphrey et al.,
2012; Müller, 1976). Studies have also revealed ice anomalously warmed by
5 ∘C or more in the ablation zone (Hooke et al., 1983; Meierbachtol
et al., 2015), but the mechanisms for this are unclear.
Hooke et al. (1983) explored the impacts of several heat transfer processes
within near-surface ice at Storglaciären and the Barnes Ice Cap. They
focused on the wintertime snowpack, which acts as insulation to cold air
temperatures but is permeable to meltwater percolation. Their results showed
that the average ice temperature at and below the equilibrium line of those
glaciers tends to be higher than the mean annual air temperature. They
attributed the observed difference mainly to snow insulation because the
strength of their measured offset was correlated with the thickness of the
snowpack.
In this study, we expand the analysis of Hooke et al. (1983) and turn our focus
to the GrIS ablation zone with near-surface temperature profiles from seven
locations. We use our temperature measurements in conjunction with a
one-dimensional model to assess heat transfer processes in this area. The
processes which make the ablation zone different from other areas of a
glacier or ice sheet are, first, that the ice surface spends much of the
summer period pinned at the melting point, despite slightly warmer air
temperatures. Next, high ablation rates counter emergent ice flow, removing
the ice surface and exposing deeper ice, along with its heat content, to the
surface. The contrast of a wintertime snowpack to bare ice in the summer
enables an insulating effect during winter months. The deep penetration of
solar radiation into bare ice results in subsurface heating and melting
(Brandt and Warren, 1993; Liston et al., 1999). Finally,
surface melt can move through open fractures, carrying latent heat with it to
deeper and colder ice, and upon refreezing, the meltwater warms that ice
below the surface (Jarvis and Clarke, 1974; Phillips et al., 2010).
Our near-surface temperature observations represent an aggregated sum of the
processes mentioned above. A numerical model can be used to partition the
relative importance of those processes, but only with measurements in hand
as validation. Therefore, confidently constraining the role of near-surface
heat transfer processes requires temperature measurements with both high
temporal and spatial resolution, and records that span hours to seasons.
Field site and instrumentation
Field observations used in this study are from three sites in western
Greenland (Fig. 1). Each site is named by its location with respect to the
terminus of Isunnguata Sermia, a land-terminating outlet glacier. The
equilibrium line altitude is at about 1500 m elevation in this area
(van de Wal et al., 2012), which is 400 m above
the furthest inland site, 46km-11, so all sites are well within the ablation
zone and ablation rates are high (2–3 m yr-1). Solar radiation in the
summer creates a layer of interconnected cryoconite holes at the ice surface,
and water moving through that cryoconite layer converges into surface
streams. There are no large supraglacial lakes in the immediate area of any
site; all streams eventually drain from the surface through moulins. A series
of dark folded layers emerge at the ice surface in this region of the ice
sheet (Wientjes and Oerlemans, 2010).
A site map from western Greenland with field sites (red) named by
their location with respect to the outlet terminus of Isunnguata Sermia and the two-digit year the site was first visited (i.e. 27km-11). The
inset shows locations of near-surface temperature strings (black) named by
the year they were installed and an automated weather station (blue). Surface
elevation contours are shown at 200 m spacing (Howat et al., 2014).
At each field site, boreholes for temperature instrumentation were drilled
from the surface to between 10 and 21 m depth using hot-water methods. In
total, seven strings of temperature sensors were installed – one at both
27km-11 and 46km-11 in 2011, followed by five at 33km-14 between 2014 and
2016. Strings are named by the year they were installed. Each consists of
between 17 and 32 sensors spaced at 0.5–3.0 m along the cable (Table 1). In
2011 and 2014, thermistors were used as temperature sensors. The thermistors
have a measurement resolution of 0.02 ∘C and accuracy of about
0.5 ∘C after accounting for drift (Humphrey et al., 2012). In
subsequent years, we used a digital temperature sensor (model DS18B20 from
Maxim Integrated Products, Inc.). This sensor has a resolution of
0.0625 ∘C and about the same accuracy as the thermistors. To
increase accuracy, each sensor was lab calibrated in a 0 ∘C bath
and field calibrated with a temperature measurement during freeze-in
(borehole water is exactly 0 ∘C). Because each temperature sensor is
in a black casing, the measurement error surely increases due to solar heating as
sensors move into the rotten cryoconite layer (∼0.2 m depth), and we
completely discard any measurement taken after the sensor is exposed at the
surface.
Temperature strings.
StringTime periodTime stepSensorNo. ofSensor spacingLatitudeLongitudeElevationname(mm/dd/yy)(h)Sensors(m)(m)T-11a07/05/11–07/15/133Thermistor320.667.195175-49.719515848T-11b07/11/11–12/17/113Thermistor320.667.201553-49.2890581095T-1407/18/14–06/23/170.5Thermistor31<11 m deep–0.5 >11 m deep–1.067.18127-49.56982956T-15a08/17/16–05/20/170.5DS18B2017<15 m deep–1.0 >15 m deep–3.067.18211-49.568272954T-15b08/17/16–05/20/170.5DS18B2017<15 m deep–1.0 >15 m deep–3.067.182054-49.568059954T-15c08/17/16–05/20/170.5DS18B2017<15 m deep–1.0 >15 m deep–3.067.182114-49.568484954T-1608/17/16–07/22/170.5DS18B20180.567.18147-49.57025951
Meteorological variables were measured at each field site as well. In this
study, we use the near-surface (∼2 m) air temperature (Vaisala HMP60
with a radiation shield), the net radiative heat flux over all wavelengths
shorter than 100 µm (Kipp and Zonen NR Lite), and the change in
surface elevation measured with a sonic distance sensor (Campbell SR50A).
Data from the sonic distance sensor are filtered manually, removing any
obvious outliers (more than 0.5 m from the surrounding measurements). The
filtered data are then partitioned into two variables, cumulative ablation
during the melt season and changes in snow depth during the winter. An
automated weather station with all the above instrumentation was mounted on a
fixed pole frozen in the ice, with segments being removed from the mounting
pole each summer so the instrumentation remains close to the surface and does
not extend significantly into the air temperature inversion (Miller et al.,
2013). Out of concern for error in our air temperature measurement, we offer
a comparison (Fig. S4 in the Supplement) to the nearby weather station
monitored by the Programme for Monitoring of the Greenland Ice Sheet
(PROMICE) (van As et al., 2012). The measurement frequency for meteorological
data varies from 10 min to an hour, but all data are collapsed to a
daily mean for input to a heat transfer model.
In addition to ice temperature and meteorological measurements,
investigations of the subsurface were completed at 33km-14 with a
borehole video camera and a high-frequency ground-penetrating radar survey
(see Supplement). These investigations were carried out in pursuit of what we
think may have been subsurface fractures that are not expressed at the ice
surface (described in Sect. 5.2). With five temperature sensor strings, an
automated weather station, and the subsurface investigations, 33km-14 is
by far the most thoroughly studied of the three sites. For that reason,
measurements from this site serve as the foundation for the model case study
presented in Sect. 4.
ResultsObserved ice temperature
Near-surface ice temperatures were measured through time in seven shallow
boreholes at three different field sites (Fig. 2). Although hot-water
drilling methods temporarily warm ice near the instrumentation, the ice
around these shallow boreholes cools to its original temperature within days
to weeks. Measured temperatures are spatially variable between sites. The
mean value from the lowermost sensor (analogous to T0) is -3.2∘C at
27km-11, -8.6∘C at 46km-11, and from -9.7 to -8.1∘C at 33km-14. In
all cases, measured T0 values are warmer than the mean annual air
temperature. Temperature gradients are calculated by fitting a line to the
mean temperature of the four lowermost sensors for each string. These
gradients are also variable, typically being between -0.15 and
0.0 ∘C m-1 but
+0.16∘C m-1 at 27km-11 (positive being
increasing temperature with depth below the surface). As expected, the
direction of the temperature gradients measured here correlate with those
measured in the uppermost ∼100 m for full-thickness temperature
profiles (Harrington et al., 2015; Hills et al., 2017).
Near-surface ice temperature measurements from seven strings: T-11a,
T-11b, T-14, T-15a, T-15b, T-15c, and T-16. For each, the shaded region shows
the range of measured temperatures over the entire measurement period, and
the solid line indicates the mean temperature profile. Depths are plotted
with respect to the surface at the time of measurement, so sensor locations
move toward the surface as ice melts. Strings with less than 11 months of
data are slightly more transparent. For field sites at which the air
temperature was measured for at least a full year, a dashed line shows the
mean air temperature.
Even the five temperature profiles measured at 33km-14 exhibit some
amount of spatial variability. Three temperature strings, T-15a, b, and c,
are all similar, having strong negative temperature gradients (approximately
-0.14∘C m-1) and cold T0 temperatures
(-9.6∘C). Close to the surface, these three temperature strings
are cold compared to the others. However, those strings stopped collecting
measurements in May 2017 and did not yield a full year of data. The missing
summer period explains the strong positive temperature gradient near the
surface for those three strings. T-16 is the shortest string, extending to
only 9.5 m depth. This short string exhibits the smallest range in
temperatures throughout a season with the coldest surface temperatures not
even reaching -15∘C. In terms of mean temperature, T-16 is
similar to T-14, having a small negative temperature gradient and warm
temperatures in comparison to those of T-15a, b, and c. Based on our
observations, spatial variability in near-surface ice temperature at
33km-14 is controlled on the scale of hundreds of meters. Proximal
observations from the nearby T-15a, b, and c strings are similar to one
another, but greater variability is observed when including the more distant
strings, T-14 and T-16.
Closer inspection of the measured temperature record through time reveals the
transient nature of near-surface ice temperature (Fig. 3). As expected, these
data show a strong seasonal oscillation near the surface. During the melt
season, the ice surface quickly drops as ice is warmed to the melting point.
Just below the surface, the winter cold wave persists for several weeks into
the summer season. In string T-14 we observe delayed freeze-in behavior in
one sensor (Fig. 3b) and transient heating events during the melt seasons
(Fig. 3c, d, e). Similar heating events were observed in string T-16
(Fig. 4) but not in any other. The events range in magnitude, but in one
instance ice is warmed from -10 to -2∘C in 2 h (Fig. 3c).
We can only speculate on the origins of these events and address this below
in Sect. 5.2.
Three years of ice temperature measurements from the T-14 string.
While this string was initially installed to 21 m depth, measurements are
plotted with reference to the moving surface, so the sensors move up
throughout the time period, revealing a gray mask. Transient features in the
data include anomalously slow freeze-in behavior in one sensor (b)
as well as heating events throughout the collection time period (c, d, e). The heating events are plotted as a series of temperature profiles
with the darker shades being later times and time steps between profiles of
2 h (c), 10 h (d), and 1 h (e).
Meteorological data
Meteorological data from 33km-14 were observed over 3 years
(Supplement Fig. S3). Air temperatures are normally at or above the melting
temperature during the summer but fall to below -30∘C in winter
months. The measured ablation rate is 2–3 m yr-1 and maximum snow
accumulation is only up to 0.5 m. Net radiation is less than zero in the
winter (net outgoing because thermal emission in the infrared wavelengths
dominates over atmospheric inputs) but over 100 W m-2 (daily mean) on
some days in the summer.
The mean air temperature over the entire measurement period at 33km-14
(-10.5∘C) is cold in comparison to measured ice temperatures at
that site (Fig. 2; T-14, T-15a, T-15b, T-15c, and T-16). This warm anomaly
between the ice and air temperature is also observed at 27km-11 and 46km-11,
where ice is warmer than the measured air temperature and significantly
warmer than the reference from a regional climate model
(Meierbachtol et al., 2015). Interestingly, we measure
almost no winter snowpack at 27km-11 and 46km-11 due to low precipitation and
strong winds during the time period over which those data were collected
(2011–2013). Our observations are thus in contradiction to the inferences
made by Hooke et al. (1983) in Arctic Canada, where the offset between air
and ice temperature appeared to be primarily a result of snow insulation.
Overall, the 3 years for which meteorological data were collected are
significantly different. The 2014–2015 winter was particularly cold,
bringing the mean air temperature of that year more than a degree lower than
the other two seasons. Snow accumulation was approximately doubled that
winter in comparison to the other two. Also, the summer melt season is longer
in 2016 than in 2015. In comparison with past trends from the nearby PROMICE
station, KAN_L, the second year is more typical for this area (van As et
al., 2012). To model a representative season, data from that second year
(July 2015 to July 2016) were chosen as annual input for the model case
study.
Analysis
Our objective is now to investigate how various processes active in
Greenland's ablation zone influence T0. In order for model results to
achieve fidelity, inputs and parameters need to be representative of actual
conditions. We therefore use the meteorological data to constrain the
modeling experiments. Our modeling is focused at 33km-14, where we
have the most data for constraining the problem.
Model formulation
The foundation for quantifying impacts of near-surface heat transfer
processes is a one-dimensional thermodynamic model. We argue that the
processes tested here are close enough to being homogeneous that they can be
adequately assessed in one dimension. The one exception is the measured
heating events, which are transient and spatially discrete; these are
discussed in Sect. 5.2 and are not included in the model analysis. Our model
uses measured meteorological variables as the surface boundary condition and
simulates ice temperature to 21 m, a depth chosen for consistency with
measured data. The ice temperature at the depth of zero annual amplitude,
T0, is output from the bottom of the domain for each model experiment
and used as a metric to compare net temperature changes between simulations.
The model, its boundary conditions, and the experiments are all designed to
test heat transfer processes within the ice itself. To maintain focus on ice
processes, we ignore any atmospheric effects above the ice surface such as
turbulent heat fluxes. The model does not, nor is it meant to, simulate the
surface mass balance.
We implement an Eulerian framework, treating the z dimension as depth from
a moving surface boundary so that emerging ice moves through the domain
and is removed when it melts at z=0. We use a finite element model with a
first-order linear element and 0.5 m mesh spacing refined to 2 cm near the
surface. For a seamless representation of energy across the water/ice phase
boundary, we implement an advection–diffusion enthalpy formulation (i.e.
Aschwanden et al., 2012; Brinkerhoff and Johnson, 2013),
(∂t+u∂z)H=∂z(α∂zH)+ϕρi.
Here, ∂ is a partial derivative, t is time, u is the vertical
ice velocity with respect to the lowering ice surface, z is depth, H is
specific enthalpy, α is thermal diffusivity, ϕ is any added
energy source, and ρi is the density of ice. The diffusivity
term is enthalpy dependent,
αH=kiρiCpνρicold,H<Hmtemperate,H>Hm,
where ki is the thermal conductivity of ice which we assume is
constant over the small temperature range in this study (∼25∘C), Cp is the specific heat capacity which is again
assumed constant, ν is the moisture diffusivity in temperate ice, and
Hm is the reference enthalpy at the melting point (all constants
are shown in Table 2). Aschwanden et al. (2012) include a thermally diffusive
component in temperate ice (i.e. ki∂z2TmP). However, since we consider only near-surface ice, where pressures
(P) are low, this term reduces to zero. Using this formulation, energy
moves by a sensible heat flux in cold ice and a latent heat flux in temperate
ice. We assume that the latent heat flux, prescribed by temperate ice
diffusivity (ν/ρi) in Eq. 2), is an order of magnitude
smaller than the cold ice diffusivity (ki/ρiCp).
We argue that this is representative of the near-surface ice when cold ice is
impermeable to meltwater.
Constants.
VariableSymbolValueUnitsReferenceReference enthalpyHm0J kg-1Ice densityρi917kg m-3Cuffey and Paterson (2010)Snow densityρs300kg m-3Water densityρw1000kg m-3Specific heat capacityCp2097J kg-1 K-1Cuffey and Paterson (2010)Latent heat of fusionLf3.335×105J kg-1Cuffey and Paterson (2010)Thermal conductivity of iceki2.1J m-1 K-1 s-1Cuffey and Paterson (2010)Thermal conductivity of snowks0.2J m-1 K-1 s-1Calonne et al. (2011)Moisture diffusivityν1×10-4kg m-1 s-1Aschwanden et al. (2012)
The desired model output is ice temperature. It has been argued that
temperature is related to enthalpy through a continuous function, where the
transition between cold and temperate ice is smooth over some
“cold-temperate transition surface” (Lüthi et al., 2002). On the other
hand, we argue that cold ice is impermeable to water except in open fractures
(which we do not include in these simulations), so we use a stepwise
transition,
TH=H-HmCp+TmTmcoldtemperate.
Additional enthalpy above the reference increases the water content in ice,
ωH=0H-HmLfcoldtemperate,
where Lf is the latent heat of fusion. If enough energy is added
to ice that its temperature would exceed the melting point, excess energy
goes to melting. In our case study, we limit the water content based on field
observations of water accumulation in the layer of rotten ice and cryoconite
holes. This rotten cryoconite layer extends to approximately 20 cm depth and
as an upper limit accumulates a maximum 50 % liquid water. Therefore, we
limit the water content in the rotten cryoconite layer,
0.0≤ω≤0.5,
with any excess melt immediately leaving the model domain as surface runoff.
The two boundary conditions are (1) fixed to the air temperature at the
surface,
Tsurface,t=Tair,
and (2) free at the bottom of the domain,
∂T∂zbottom=0.0.
Both boundary conditions are with no liquid water content, ω=0. The
surface boundary condition is updated at each time step to match the measured
air temperature. The bottom boundary condition is fixed in time. This bottom
boundary condition is also changed for some model experiments to test the
influence of a temperature gradient at the bottom of the domain
(Sect. 4.2.4).
Experiments
Four separate model experiments are run, each with a new process
incorporated into the physics and each guided by observational data. All
simulations use the enthalpy formulation above rather than temperature in
order to track the internal energy of the ice–water mixtures that are
prevalent in the ablation zone. The results from each experiment are
referenced to an initial control run, which is simple thermal diffusion of
the measured air temperature in the absence of any additional heat transfer
processes. Meteorological data are input where needed for an associated
process in the model. These data are clipped to 1 full year and input at
the surface boundary in an annual cycle. The model is run with a 1-day
time step until ice temperature at the bottom of the domain converges to a
steady temperature. A description of each of the model experiments follows
below. These experiments build on one another, so each new experiment
incorporates the physics of all previously discussed processes.
Ablation
The first experiment simulates motion of the ablating surface. While the
control run is performed with no advective transport (i.e. u=0), in this
experiment we incorporate advection by setting the vertical velocity equal to
measurements of the changing surface elevation through time. When ice melts,
the ice surface location drops. Because the vertical coordinate, z, in the
model domain is treated as a distance from the moving surface, ablation
brings simulated ice closer to the surface boundary. Hence, the simulated ice
velocity, u, is assigned to the ablation rate (except in the opposite
direction, ice moves upward) for this first model experiment. The ablation
rate is calculated as a forward difference of the measured surface lowering.
Snow insulation
The second experiment incorporates measured snow accumulation, which
thermally insulates the ice from the air. The upper boundary condition is
now assigned to the snow surface, the location of which changes in time. Diffusion
through the snowpack is then simulated as an extension of the ice domain but
with different physical properties. The thermal conductivity of snow
(Calonne et al., 2011),
ks=2.5×10-6ρs2-1.23×10-4ρs+0.024,
is dependent on snow density, ρs, for which we use a constant
value, 300 kg m-3. We treat the specific heat capacity of snow to be
the same as ice (Yen, 1981).
Radiative energy
The third model experiment incorporates an energy source from the net
radiation measured at the surface. Energy from radiation is absorbed in the
ice and is transferred to thermal energy and to ice melting
(van den Broeke et al., 2008). We assume
that all this radiative energy is absorbed in the uppermost 20 cm, the
rotten cryoconite layer, and if snow is present the melt production
immediately drains to that cryoconite layer. When the net radiation is
negative (wintertime), we assume that it is controlling the air temperature,
so it is already accommodated in our simulation; thus, the radiative energy
input is ignored in the negative case. This radiative source term is
incorporated into Eq. (1) at each time step,
ϕrad=Q0.2m, where Q is the measured
radiative flux at the surface in W m-2. All constants for the rotten
cryoconite layer are the same as that for ice.
While some models treat the absorption of radiation in snow and ice more
explicitly with a spectrally dependent Beer–Lambert law (Brandt and Warren,
1993), we argue that it is reasonable to assume all wavelengths are absorbed
near the surface over the length scales that we consider. The only documented
value that we know of for an absorption coefficient in the cryoconite layer
is 28 m-1 (Lliboutry, 1965), which is close to that of snow (Perovich,
2007). If the properties are truly similar to that of snow, about 90 % of
the energy is absorbed in the uppermost 20 cm (Warren, 1982). Moreover, we
argue that this is precisely the reason that the cryoconite layer only
extends to a limited depth; it is a result of where radiative energy causes
melting.
Subsurface temperature gradient
Finally, in the fourth model experiment we change the boundary condition at
the bottom of the domain. The free boundary is changed to a Neumann boundary
with a gradient of -0.05∘C m-1, a value that approximately
matches the measured gradient at 33km-14. Importantly, this simulated
gradient is in the same direction, although with a larger magnitude, as the
upper ∼100 m of ice in our measurements of deep temperature profiles
(Hills et al., 2017). In this case, the advective energy flux is upward, but
the temperature gradient is negative, bringing colder ice to the surface. In
addition, two limiting cases were tested, with gradients of ±0.15 ∘C m-1. This is the approximate range in the measured
gradients (Fig. 2).
Heating events from temperature string T-16. Profiles are plotted as
in Fig. 3c, d, and e. The time steps between profiles are 2 h (a)
and 4 h (b).
Model results
The control model run of simple thermal diffusion predicts that ice
temperature converges to approximately the mean annual air temperature of the
study year (-9.9∘C) by about 15 m below the ice surface. This
result is in agreement with the analytical solution (Carslaw and Jaeger,
1959) but is slightly different from the mean air temperature
(-9.6∘C) because the air can exceed the melting temperature in
the summer, while the ice cannot. Other atmospheric effects such as turbulent
heat fluxes and the thermal inversion could also cause a difference between
measured air temperature and ice surface temperature, but these are not
considered here. For each model treatment, 1–4, the incorporation of an
additional physical process changes the ice temperature. Differences between
model runs are compared using T0 at 21 m. Again, the model experiments
are progressive, so each new experiment includes the processes from all
previous experiments. Key results from each experiment are as follows
(Fig. 5):
Model results for five separate simulations. In each case, twelve
simulated temperature profiles are shown throughout the year-long
period, and control results (from a) are displayed for comparison
(gray). Differences between the simulations are analyzed quantitatively using
T0, the convergent temperature at 21 m. Processes are, from top to
bottom, (a) control simulation of pure diffusion,
(b) ablation, (c) snow insulation, (d) radiative
energy input, and finally (e) subsurface temperature gradient. The
two limiting cases for the subsurface temperature gradient are plotted with
dashed gray lines (e).
Diffusion alone results in T0=-9.9∘C, whereas
observed temperatures range from -9.7 to -8.1∘C at
33km-14.
Because the ablation rate is strongest in the summer, the effect of
incorporating ablation is to counteract the diffusion of warm summer air
temperatures. The result is a net cooling of T0 from experiment (1) by
-0.92∘C.
Snow on the ice surface insulates the ice from the air temperature. In
the winter, snow insulation keeps the ice warmer than the cold air, but with
warm air temperatures in the spring it has the opposite effect. Because snow
quickly melts in the springtime, the net effect of snow insulation is
substantially more warming than cooling. T0 for this experiment is
+0.78∘C warmer than the previous one.
Radiative energy input mainly controls melting
(van den Broeke et al., 2008), but
incorporating this process does warm T0 by +0.52∘C.
Imposing a -0.05∘C m-1 temperature gradient at the bottom
of the model domain, consistent with observations, dramatically changes
T0 by -2.5∘C.
Both ablation and the subsurface temperature gradient have a cooling effect
on near-surface ice temperature. On the other hand, snow and radiative energy
input have a warming effect. For this case study, the first three processes
together result in almost no net change, so that the modeled T0 is close
to the observed mean air temperature (Fig. 5d). However, inclusion of the
subsurface temperature gradient has a strong cooling effect on the simulated
temperatures, bringing T0 far from the mean measured air temperature.
The limiting cases show that this bottom boundary condition strongly controls
the near-surface temperature, with a range in the resulting T0 values
from -17.0 to -2.0∘C. In summary, measured ice temperatures are
consistently warmer than both the measured air temperature and simulated ice
temperature (Fig. 6), except in the case of a positive subsurface gradient,
which is discussed below.
A comparison of model output (gray) and data from 33km-14, including
mean ice temperatures (red) and mean annual air temperatures for three
seasons (black dashed). The observed ice temperatures are plotted the same as
in Fig. 2. Note that three of the temperature strings collected only
∼9 months of data (transparent red). Mean temperatures from those three
strings are cold near the surface because they collected more
measurements in wintertime than summertime.
Discussion
Our observations show that measurements of near-surface ice in the ablation
zone of western Greenland are significantly warmer than would be predicted
by diffusive heat exchange with the atmosphere. This is in agreement with
past observations collected in other ablation zones (Hooke
et al., 1983). With four experiments in a numerical model that progressively
incorporate more physical complexity, we are unable to precisely match
independent model output to observations. Our measurement and model output
point toward a disconnect between air and ice temperatures in the GrIS
ablation zone, with ice temperatures being consistently warmer than the air.
Ablation–diffusion
The strongest result from our model case study was a drop in T0 by
-2.5∘C associated with the imposed subsurface temperature
gradient. While it was important to test this scenario for one case, the
temperature gradient we used was representative but somewhat arbitrary. In
reality, the observed temperature gradients are widely variable from one site
to another and even within one site (Fig. 2). Interestingly, full ice
thickness temperature profiles show similar temperature gradients, both
positive and negative (Harrington et al., 2015; Hills et al., 2017). Hence,
the limiting cases were added to show simulation results over the range of
measured gradients from our temperature strings. The resulting T0 values span a
range of 19 ∘C.
The majority of the subsurface temperature gradients that we measure are
negative, and theoretically the gradient should be negative. Consider that
fast horizontal velocities (∼100 m yr-1) advect cold ice from the
divide to the ablation zone, and the air temperature lapse rate couples with
the relatively steep surface gradients so that the surface warms rapidly
toward the terminus. These conditions lead to a vertical temperature gradient
below the ice surface that is negative (Hooke, 2005; pp. 131–135), as in our
model example. The one exception is in the case of deep latent heating in a
crevasse field (Harrington et al., 2015; sites S3 and S4), where the deep ice
temperature is warmer than the mean air temperature rather than colder.
Overall, our results demonstrate that the effect of the subsurface
temperature gradient is coupled to that of surface lowering. With respect to
the surface, the temperature gradient below is advected upward as ice melts.
There is competition between surface lowering and diffusion of atmospheric
energy into the ice: as near-surface ice gets warmer, it can be removed
quickly and a new boundary is set. Therefore, our conceptualization of
temperature in the near-surface ice of the ablation zone should not be a
seasonally oscillating upper boundary with purely diffusive heat transfer
(Carslaw and Jaeger, 1959), but one with advection and
diffusion (Logan and Zlotnik, 1995; Paterson,
1972). This conceptualization is unique to the ablation zone because of the
rapid rate of surface lowering, whereas a diffusive model for near-surface
heat transfer is much more appropriate in the accumulation zone.
The disconnect between air and ice temperature implies that the
near-surface active layer in the ablation zone is shallow (i.e. less than
15 m) and could be skewed toward the subsurface temperature gradient.
Therefore, the surface boundary condition has weak influence on diffusion for
ice well below the surface. This is in contrast to the accumulation zone
where new snow is advected downward, so the surface temperature quickly
influences that at depth. Under these conditions, it is no surprise that we
see spatial variability in near-surface ice temperature even within one field
site. That variability is simply an expression of the deeper ice temperature
variations, which are hypothesized to exist from variations in vertical
advection (Hills et al., 2017), and do not have time to completely diffuse
away before they are exposed at the surface.
Subsurface refreezing
We observe heating events in two temperature strings, the largest case being
8 ∘C in 2 h between 3 and 8 m below the ice surface (Fig. 3c).
These events are transient, they are spatially discrete, and they are
generated at depth. All of these factors are most easily explained by the refreezing
of liquid water in cold ice. Similar refreezing events have been observed in
firn (Humphrey et al., 2012), where they are not only important for ice
temperature but could also imply a large storage reservoir for surface
meltwater (Harper et al., 2012). However, unlike firn, solid ice is
impermeable to water unless fractures are present
(Fountain et al., 2005). Two persistently warm
features are also observed between 5 and 10 m depth into the winters of 2015
and 2017 (Fig. 3a). We interpret these as a nearby latent heat source, either
with running or ponded water that does not freeze for an extended time.
In Greenland's ablation zone, prior work has demonstrated the importance of
large-scale latent heating in open crevasses (Phillips et al., 2013; Poinar
et al., 2016). Additionally, water-filled cavities have been observed in
cold, near-surface ice on mountain glaciers (Jarvis and Clarke, 1974;
Paterson and Savage, 1970). In our case, however, an explanation for
refreezing water is not obvious. While the field site has occasional
millimeter-aperture “hairline” cracks, there are no visible open crevasses at the
surface for routing water to depth. As far as we know, this work is the first
to report evidence of short-term transient latent-heating events in cold ice
that is not obviously linked to open surface fractures.
While the hairline fractures could perhaps move some water, to permit much
water to move meters through cold ice they would need to be large enough that
water moves quickly and does not instantaneously refreeze. For example, a
1 mm wide crack in ice that is -10∘C freezes shut in about 45 s
(Alley et al., 2005; Eq. 8). That amount of time could be long enough for
small volumes of water to move 5–10 m below the surface but would require a
hydropotential gradient to drive water flow. Thus, top-down hairline
crevassing does not seem a plausible explanation for the events we observe.
Importantly, several independent field observations in this area, including
hole drainage of water during hot-water drilling, ground-penetrating radar
reflections, and borehole video observations, all point to the existence of
subsurface air-filled and open fractures with apertures of up to a few
centimeters (see Supplement). That they are open at depth, but are narrow or nonexistent
at the surface, could be linked to the colder ice at depth and its stiffer
rheology. Nath and Vaughan (2003) observed similar subsurface fractures in
firn, although in their case density controls the stiffness rather than
temperature. On rare occasions, we argue that the aperture of the fractures
open wider to the surface, where there is copious water stored in the
cryoconite layer (Cooper et al., 2018) that can drain and refreeze at depth.
While the events seem to happen in the springtime and it would be tempting to
assert that fracture opening coincides with speedup, our measurements of
surface velocity at these sites show that this is not always the case. This
may be due to that fact that the spring speedup coincides with early melt
rather than peak melt and copious water in the cryoconite layer.
Energy source for the observed heating events. (a–c)
Observed energy density through time for the differenced temperature profile
calculated with Eq. (9) (black) and conductive energy density through time
calculated with Eq. (10) (red). (d–f) Percentage by volume water
refrozen for the associated source in (a–c). This value is
proportional to the difference between the black and red lines above. The
temperature string from which measurements were taken is labeled at the top.
Latent heating in the form of these subsurface refreezing events is an
obvious candidate for a source for the “extra” heat we observe in our
temperature strings relative to simulations. Our data show that refreezing in
subsurface fractures has the potential to warm ice substantially over short
periods of time, and apparently this can occur in places where open crevasses
are not readily observed at the surface. Furthermore, the difference between
measured and modeled temperatures (up to 3 ∘C) is the equivalent of
only ∼1.7 % water by volume. Our simplified one-dimensional model
would not be well suited to assess the influence of these latent heating
events. Instead, we provide a simple calculation for energy input from the
events by differencing the temperature profiles in time and integrating for
total energy density (Fig. 7a–c),
ϕmeasured=ρiCpΔz∫ΔTdz,
where Δz is the total depth of the profile, and ΔT is the
differenced temperature profile. Only sensors that are below the ice surface
for the entire time period are considered. To calculate the total water
content refrozen in the associated event, we remove the conductive energy
fluxes from the total energy density calculated above. We do this by
calculating the temperature gradients at the top and bottom of the measured
temperature profile at each time step as in Cox et al. (2015):
ϕconductive=-kiΔz∫∂T∂ztop-∂T∂zbottomdt.
The resulting energy sources are then converted to a volume fraction of water
by
ωmeasured=ϕmeasured-ϕconductiveρwLf,
where ρw is the density of water. Results show that each year some
fractions of a percentage of water are refrozen (Fig. 7). Through several
seasons that amount of refreezing could easily add up to the ∼3∘C anomaly that we observe.
Unfortunately, without a more thorough investigation, we do not have enough
evidence to show that these refreezing events are more than a local anomaly.
Of our seven near-surface temperature strings, only T-14 and T-16
demonstrated refreezing events, so we are not confident that they are
temporally or spatially ubiquitous.
The only other logical mechanism for the warm offset between measurements
and model results would be warming from below through a positive subsurface
temperature gradient. While it is tempting to associate deep warm ice with
residual heat from the exceptionally hot summers of 2010 and 2012
(Tedesco et al., 2013), this scenario is unlikely because the ablation rates are so high that any ice warmed during
those years has likely already melted. Deeper latent heating from an
upstream crevasse field is a more plausible alternative; however, full-depth temperature profiles from 33km-14 do not show deeper ice to be
anomalously warmed except in one localized case
(Hills et al., 2017).
Conclusions
We observe the temperature of ice at the depth of zero annual amplitude,
T0, in Greenland's ablation zone to be markedly warmer than the mean
annual air temperature. These findings contradict predictions from purely
diffusive heat transport but are not surprising considering the processes
which impact heat transfer in the ice of the ablation zone. High ablation rates
in this area indicate that ice temperatures below 15 m reflect the
temperature of deep ice that is emerging to the surface, confirming that the
ice does not have time to equilibrate with the atmosphere. In other words,
ice flow brings cold ice to the surface at a faster rate than heat from the
atmosphere can diffuse into the ablating surface. The coupling between rapid
ablation and the spatial variability in deep ice temperature implies there
will always be a disconnect between air and ice temperatures. Additionally,
we observe refreezing events below 5–10 m of cold ice. Meltwater is likely
moving to that depth through subsurface fractures that are not obviously
visible at the surface.
In analyzing a series of processes that control near-surface ice
temperature, we find that some lead to colder ice and others to warmer ice, but
most are strong enough to dramatically alter the ice temperature from the
purely diffusive case. With rapid ablation, a spatially variable temperature
field, and subsurface refreezing events, T0 in the ablation zone should
not be expected to match the air temperature. That our measurements are
consistently warmer could simply be due to the limited number of
observations we have, but latent heat additions are clearly measured and
could be common in near-surface ice of the western Greenland ablation zone.
All temperature measurements are available at https://doi.org/10.18739/A2G44HQ01 (Hills, 2018).
The supplement related to this article is available online at: https://doi.org/10.5194/tc-12-3215-2018-supplement.
BHH, JVJ, NFH, and PJW designed and built instrumentation for temperature measurements. BHH wrote the numerical model with JTH and TWM contributing to model experiment design. BHH led the writing with significant input from all co-authors.
The authors declare that they have no conflict of
interest.
Acknowledgements
This work was funded by the National Science Foundation (Office of Polar
Programs-Arctic Natural Sciences grant no. 1203451 and no. 0909495). We thank
the editor, Tobias Sauter, and two anonymous reviewers for their comments,
which greatly improved the original
manuscript. We also thank Stephen Warren for valuable discussions.Edited by: Tobias Sauter
Reviewed by: two anonymous referees
ReferencesAlley, R. B., Dupont, T. K., Parizek, B. R., and Anandakrishnan, S.: Access
of surface meltwater to beds of sub-freezing glaciers: Preliminary insights,
Ann. Glaciol., 40, 8–14, 10.3189/172756405781813483, 2005.Aschwanden, A., Bueler, E., Khroulev, C., and Blatter, H.: An enthalpy
formulation for glaciers and ice sheets, J. Glaciol., 58, 441–457,
10.3189/2012JoG11J088, 2012.Brandt, R. and Warren, S.: Solar-heating rates and temperature profiles in
Antarctica snow and ice, J. Glaciol., 39, 99–110, 1993.Brinkerhoff, D. J. and Johnson, J. V.: Data assimilation and prognostic whole
ice sheet modelling with the variationally derived, higher order, open
source, and fully parallel ice sheet model VarGlaS, The Cryosphere, 7,
1161–1184, 10.5194/tc-7-1161-2013, 2013.Calonne, N., Flin, F., Morin, S., Lesaffre, B., Du Roscoat, S. R., and
Geindreau, C.: Numerical and experimental investigations of the effective
thermal conductivity of snow, Geophys. Res. Lett., 38, 1–6,
10.1029/2011GL049234, 2011.Carslaw, H. S. and Jaeger, J. C.: Conduction of Heat in Solids, 2nd Edn.,
London: Oxford University Press, 1959.Cooper, M. G., Smith, L. C., Rennermalm, A. K., Miège, C., Pitcher, L. H.,
Ryan, J. C., Yang, K., and Cooley, S. W.: Meltwater storage in low-density
near-surface bare ice in the Greenland ice sheet ablation zone, The
Cryosphere, 12, 955–970, 10.5194/tc-12-955-2018, 2018.Cox, C., Humphrey, N., and Harper, J.: Quantifying meltwater refreezing along
a transect of sites on the Greenland ice sheet, The Cryosphere, 9, 691–701,
10.5194/tc-9-691-2015, 2015.Cuffey, K. and Paterson, W. S. B.: The Physics of Glaciers, 4th Edn.,
Oxford, UK, Butterworth-Heinemann, 2010.Fountain, A. G., Jacobel, R. W., Schlichting, R., and Jansson, P.: Fractures
as the main pathways of water flow in temperate glaciers, Nature, 433,
618–621, 10.1038/nature03296, 2005.Harper, J. T., Humphrey, N. F., Pfeffer, W. T., Brown, J., and Fettweis, X.:
Greenland ice-sheet contribution to sea-level rise buffered by meltwater
storage in firn, Nature, 491, 240–243, 10.1038/nature11566, 2012.Harrington, J. A., Humphrey, N. F., and Harper, J. T.: Temperature
distribution and thermal anomalies along a flowline of the Greenland Ice
Sheet, Ann. Glaciol., 56, 98–104, 10.3189/2015AoG70A945, 2015.Hills, B.: Near-surface ice temperature measurements from Western Greenland, 2011–2017, urn:node:ARCTIC, 10.18739/A2G44HQ01, 2018.Hills, B. H., Harper, J. T., Humphrey, N. F., and Meierbachtol, T. W.:
Measured horizontal temperature gradients constrain heat transfer mechanisms
in Greenland ice, Geophys. Res. Lett., 44, 1–8, 10.1002/2017GL074917, 2017.Hooke, R. L.: Near-surface temperatures in the superimposed ice zone and
lower part of the soaked zone of polar ice sheets, J. Glaciol., 16, 302–304,
1976.Hooke, R. L.: Principles of Glacier Mechanics, 2nd Edn., Cambridge
University Press, 2005.Hooke, R. L., Gould, J. E., and Brzozowski, J.: Near-surface temperatures
near and below the equilibrium line on polar and subpolar glaciers, Z.
Gletscherkd. Glazialgeol., 19, 1–25, 1983.Howat, I. M., Negrete, A., and Smith, B. E.: The Greenland Ice Mapping
Project (GIMP) land classification and surface elevation data sets, The
Cryosphere, 8, 1509–1518, 10.5194/tc-8-1509-2014, 2014.Humphrey, N. F., Harper, J. T., and Pfeffer, W. T.: Thermal tracking of
meltwater retention in Greenland's accumulation area, J. Geophys. Res., 117,
1–11, 10.1029/2011JF002083, 2012.Jarvis, G. T. and Clarke, G. K. C.: Thermal effects of crevassing on Steele
Glacier, Yukon Territory, Canada, J. Glaciol., 13, 243–254, 1974.Liston, G. E., Winther, J.-G., Bruland, O., Elvehoy, H., and Sand, K.:
Below-surface ice melt on the coastal Antarctic ice sheet, J. Glaciol., 45,
273–285, 1999.
Lliboutry, L. A.: Traité de Glaciologie, Vol. 2, Masson, Paris, 1965.Loewe, F.: Screen temperatures and 10 m temperatures, J. Glaciol., 9,
263–268, 1970.Logan, J. D. and Zlotnik, V.: The convection-diffusion equation with
periodic boundary conditions, Appl. Math. Lett., 8, 55–61,
10.1016/0893-9659(95)00030-T, 1995.Lüthi, M., Funk, M., Iken, A., Gogineni, S., and Truffer, M.: Mechanisms
of fast flow in Jakobshavn Isbræ, West Greenland: Part III, measurements
of ice deformation, temperature and cross-borehole conductivity in boreholes
to the bedrock, J. Glaciol., 48, 369–385, 2002.Lüthi, M. P., Ryser, C., Andrews, L. C., Catania, G. A., Funk, M., Hawley,
R. L., Hoffman, M. J., and Neumann, T. A.: Heat sources within the Greenland
Ice Sheet: dissipation, temperate paleo-firn and cryo-hydrologic warming, The
Cryosphere, 9, 245–253, 10.5194/tc-9-245-2015, 2015.Meierbachtol, T. W., Harper, J. T., Johnson, J. V., Humphrey, N. F., and
Brinkerhoff, D. J.: Thermal boundary conditions on western Greenland:
Observational constraints and impacts on the modeled thermomechanical state,
J. Geophys. Res.-Earth, 120, 623–636, 10.1002/2014JF003375, 2015.Miller, N. B., Turner, D. D., Bennartz, R., Shupe, M. D., Kulie, M. S.,
Cadeddu, M. P., and Walden, V. P.: Surface-based inversions above central
Greenland, J. Geophys. Res.-Atmos., 118, 495–506, 10.1029/2012JD018867,
2013.Mock, S. J. and Weeks, W. F.: The distribution of 10 meter snow
temperatures on the Greenland Ice Sheet, J. Glaciol., 6, 23–41, 1966.Müller, F.: On the thermal regime of a high-arctic valley glacier, J.
Glaciol., 16, 119–133, 1976.Nath, P. C. and Vaughan, D. G.: Subsurface crevasse formation in glaciers
and ice sheets, J. Geophys. Res., 108, 1–12, 10.1029/2001JB000453, 2003.Paterson, W. S. B.: Temperature distribution in the upper layers of the
ablation area of Athabasca Glacier, Alberta, Canada, J. Glaciol., 11, 31–41,
1972.Paterson, W. S. B. and Savage, J. C.: Excess pressure observed in a
water-filled cavity in Athabasca Glacier, Canada, J. Glaciol., 9, 103–107,
1970.Perovich, D. K.: Light reflection and transmission by a temperate snow
cover, J. Glaciol., 53, 201–210, 10.3189/172756507782202919, 2007.Phillips, T., Rajaram, H., and Steffen, K.: Cryo-hydrologic warming: A
potential mechanism for rapid thermal response of ice sheets, Geophys. Res.
Lett., 37, 1–5, 10.1029/2010GL044397, 2010.Phillips, T., Rajaram, H., Colgan, W., Steffen, K., and Abdalati, W.:
Evaluation of cryo-hydrologic warming as an explanation for increased ice
velocities in the wet snow zone, Sermeq Avannarleq, West Greenland, J.
Geophys. Res.-Earth, 118, 1241–1256, 10.1002/jgrf.20079, 2013.Poinar, K., Joughin, I., Lenaerts, J. T. M., and van den Broeke, M. R.:
Englacial latent-heat transfer has limited influence on seaward ice flux in
western Greenland, J. Glaciol., 63, 1–16, 10.1017/jog.2016.103, 2016.Tedesco, M., Fettweis, X., Mote, T., Wahr, J., Alexander, P., Box, J. E., and
Wouters, B.: Evidence and analysis of 2012 Greenland records from spaceborne
observations, a regional climate model and reanalysis data, The Cryosphere,
7, 615–630, 10.5194/tc-7-615-2013, 2013.van As, D., Hubbard, A. L., Hasholt, B., Mikkelsen, A. B., van den Broeke, M.
R., and Fausto, R. S.: Large surface meltwater discharge from the
Kangerlussuaq sector of the Greenland ice sheet during the record-warm year
2010 explained by detailed energy balance observations, The Cryosphere, 6,
199–209, 10.5194/tc-6-199-2012, 2012.van den Broeke, M., Smeets, P., Ettema, J., van der Veen, C., van de Wal, R.,
and Oerlemans, J.: Partitioning of melt energy and meltwater fluxes in the
ablation zone of the west Greenland ice sheet, The Cryosphere, 2, 179–189,
10.5194/tc-2-179-2008, 2008.
van de Wal, R. S. W., Boot, W., Smeets, C. J. P. P., Snellen, H., van den
Broeke, M. R., and Oerlemans, J.: Twenty-one years of mass balance
observations along the K-transect, West Greenland, Earth Syst. Sci. Data, 4,
31–35, 10.5194/essd-4-31-2012, 2012.van Everdingen, R. O.: Multi-language glossary of permafrost and related
ground-ice terms, Calgary, Alberta, CA, International Permafrost Association,
1998.Warren, S. G.: Optical properties of snow, Rev. Geophys., 20, 67–89,
10.1029/RG020i001p00067, 1982.Wientjes, I. G. M. and Oerlemans, J.: An explanation for the dark region in
the western melt zone of the Greenland ice sheet, The Cryosphere, 4,
261–268, 10.5194/tc-4-261-2010, 2010.Yen, Y. C.: Review of thermal properties of snow, ice, and sea ice, CRREL
Report 81-10, 1981.