Introduction
The mass balance of the Greenland Ice Sheet has been negative over the last
few decades e.g.. A common
method to assess ice-sheet imbalance is altimetry, by which elevation changes
are monitored by repeated scanning of the ice-sheet surface by active laser
or radar instruments onboard airplanes or satellites. A crucial step in
translating the observed volume change to a mass change is to determine the
density associated with the volume change.
The simplest assumption is that below the equilibrium line altitude (ELA),
all mass change is caused either by ice-dynamical thinning or thickening, or
by melting of ice. There, the ice density is used to convert from volume to
mass change. Above the ELA, it was assumed in earlier studies that ice-dynamical elevation
changes are negligible and that volume changes are caused solely by changes
in the firn layer. A fixed density was adopted to convert from volume to
mass, for example a fresh-snow density, or a representative density for the
entire firn layer (e.g. and for Antarctica, and
for Greenland). This assumption is still commonly made over
smaller ice caps and glaciers e.g.,
where the error associated with this approach is generally assumed to be
small. For the Greenland Ice Sheet, such an approach can compromise the
accuracy of the retrieved mass changes, as the current mass loss is divided
between surface mass balance (SMB) and ice-dynamical changes. Moreover, the
density of the Greenland firn layer is susceptible to change, invalidating
the choice for a constant density. In fact, elevations change in the
ice-sheet interior is primarily attributed to variability in snow
accumulation . Therefore, more recent altimetry-based mass balance
estimates of (parts of) the Greenland Ice Sheet e.g. make use of firn models that take into account firn compaction
and accumulation variability. More specifically, the empirical firn models
used by , and take into account firn compaction,
as well as the formation of ice lenses due to melt and
refreezing in annual layers . The firn model used
by was developed by and .
It is driven by satellite-observed surface temperature and a fixed,
firn-core derived relation between temperature and accumulation change.
Not directly applied to altimetry-based assessments of mass loss but
nonetheless very similar to our firn model, a firn model was compared by
to Ku-band radar observations of annual layering of the Greenland firn.
Here, we present a time series of elevation changes over the Greenland Ice Sheet
due to changes in depth and mass of the firn layer, i.e. variability
and change in the surface mass balance and associated firn processes like
compaction, percolation, and refreezing. These time series extend from 1960
up to and including 2014, at a horizontal resolution of 11km×11km. The results are obtained using a semi-empirical firn
compaction model that is forced with surface mass fluxes and
temperature from the polar-adapted regional climate model RACMO2.3
. Combining the time series of firn depth and mass from this
firn model allows one to convert satellite altimetry observations of volume
change to mass change, and to partition this mass change into an
ice-dynamical and a firn/SMB component. As an added advantage, the firn model
explicitly calculates all firn and SMB processes that modify the surface
elevation, allowing the analysis of the individual components of the surface
elevation change in greater detail.
Model, data, and methods
Elevation change of the Greenland Ice Sheet surface is simulated with a model
of the firn (Sect. ), which is forced at its upper boundary by
temperature and surface mass fluxes from the regional climate model RACMO,
version 2.3 see Sect. and. In
Sect. , we evaluate the firn model results against 62 shallow
and medium-depth firn cores (details in Sect. ), and against
airborne laser altimetry collected in areas of limited ice-dynamical activity
(Sect. ).
The firn model
The firn model IMAU-FDM v1.0; describes the temporal
evolution of firn compaction, meltwater percolation and refreezing, and
temperature in a vertical, 1-D column of firn and ice. The top of the
uppermost model layer represents the surface of the ice sheet (h), which
moves up and down in time. The vertical velocity of the ice-sheet surface due
to firn and SMB processes h˙f is given by
h˙f=vacc+vsnd+ver+vme+vice+vfc,
where we neglect vertical displacement of the surface by horizontal velocity
divergence or horizontal advection of mass in the snow and firn column. The
velocity components v represent solid precipitation (vacc),
surface sublimation (included in vacc), snowdrift sublimation
(vsnd), snowdrift erosion (ver), snowmelt
(vme), and firn compaction (vfc). The solid
precipitation vacc represents solid precipitation minus surface
sublimation, and thus it is not the accumulation rate as it is usually
understood. Snowdrift sublimation vsnd differs from surface
sublimation in the sense that it represents sublimation of drifting snow that
is whirled up from the surface by surface winds . Snowdrift
erosion represents the horizontal redistribution of surface snow by surface
winds, and it can be positive (deposition) or negative (erosion).
In a steady-state firn layer, the long-term average vertical mass flux
through the lower boundary of the firn column equals the mass flux through
the upper boundary. This is represented in the firn model as a constant
vertical velocity vice that equals the mean SMB (vacc+vsnd+ver+vme) but is of opposite sign, over
a reference period for which steady-state is assumed. The choice of this
reference period is discussed in Sect. . This model setup is
similar to earlier process-based models like .
An important upper-boundary condition is the density of fresh snow,
ρ0. While valuable observations of density in the percolation area of
the Greenland Ice Sheet are available , we choose to
construct a parameterization of ρ0 based on observations from the
dry-snow zone only. The observations from are averages of
density over the uppermost 1 m of the snow/firn layer, which includes
the effect of percolation and refreezing on the vertical density profiles. It
is impossible to isolate the initial density of snow that is truly fresh. We
use two main sources for ρ0 data from the dry-snow zone: observations
from north and central Greenland from , and from the EGIG line
Expédition Glaciologique Internationale au Groenland;
in central Greenland. We find the following correlation (r2=0.52)
between observed ρ0 and annual mean surface temperature
Ts (in degrees Celsius) simulated by RACMO2.3 (see
Sect. ):
ρ0=481.0+4.834Ts.
After deposition, the fresh snow starts to become denser. The effect of
dry-firn compaction is represented by vfc in Eq. ().
Compaction is an increase of firn density ρ in time (t), expressed by
Eq. (4) in as
dρdt=Cb˙g(ρi-ρ)exp-EcRT+EgRT.
Here, C, Ec, and Eg are constants; b˙ is the mean
annual accumulation over a reference period (Sect. ); g is
the gravitational acceleration; ρi is the ice density of
917 kgm-3; and R is the gas constant. T is the firn
temperature, which varies with depth. The rate constant C has a value of 0.07 for ρ≤550 kgm-3 and a value of 0.03 above that density. This
captures the higher densification rate near the surface due to sliding of
snow grains relative to each other at low densities .
Ratio of modelled vs. observed z550 and z830*
(MO550 (red circles and fit line) and MO830 (blue
squares and fit line), respectively) for 24 dry firn cores (8 of which are deep enough to
include z830*), as a function of mean annual
accumulation (in mmyr-1).
To evaluate Eq. (), we compare modelled depths of the 550 and
830 kgm-3 density layers (z550 and z830, respectively)
to observations at 62 firn core locations around the Greenland Ice Sheet (see
Sect. ). As we find a systematic departure of the modelled
values, we introduce a correction term MO, defined as the ratio of
modelled and observed values of z550 and z830* ,
where z830*=z830-z550. Figure shows
MO as a function of b˙. As the MO values need to represent
dry compaction, we selected 22 firn cores with little surface melt. Linear
least-squares fitting then yields the following MO relations for
Greenland:
MO550=1.042-0.0916ln(b˙)forρ≤550kg m-3,MO830=1.734-0.2039ln(b˙)forρ>550kg m-3.
The coefficients in these two equations are different from a previous application
in Antarctica . We use these updated coefficients to improve the
fit with observed density profiles. There is no physical interpretation for
these coefficients to be different. The different set of coefficients for
Greenland and Antarctica could point to a process not presently captured in
the model (that happens to correlate with accumulation rate). Alternatively,
the coefficients could be different because the range of accumulation rates
on which the Antarctic parameterization is based is biased
to lower accumulation rates than those found in Greenland.
Equation () is multiplied with the correction factors
MO in Eqs. () and (), which are not allowed to be smaller than 0.25 .
The accurate performance of these densification expressions is further
demonstrated by fully independent comparisons against in situ
and remotely sensed densification rates.
However, the use of a mean value of b˙ in these equations has an
important limitation: in reality, compaction is determined by the overburden
pressure of the overlying snow. By using a constant mean b˙ rather
than the instantaneous overburden pressure, the firn compaction variability
is dampened significantly. Following an accumulation event, the model only
takes into account the increase in compactable material, not the effect of
increased overburden pressure on the underlying firn.
Rain is added to the surface snowmelt flux. This liquid water is allowed to
percolate into the firn. Each layer has a maximum irreducible water content Wc,
depending on the density . Meltwater will refreeze as soon as
it encounters a layer that can accommodate both the space of the refreezing
water and the latent heat that is released upon refreezing. For details see
and .
Suppose a firn column with total depth zi and an observed or modelled
density profile ρ(z), consisting of a mixture of air (with density
ρa), water (with density ρw), and ice
(density ρi). Using Wc from
(which can vary with depth z), it can be shown that the firn air
content F (in m) is given as
F=∫0ziρi-ρ(z)ρi/(1-Wc(z))-ρwWc(z)/(1-Wc(z))-ρadz.
Here, we will approximate this general expression by assuming that
ρa≪ρi and that Wc(z)=0, yielding
F=∫0ziρi-ρ(z)ρidz.
As an extreme example, a uniform value of Wc=0.10
(i.e. 10 % of pore space filled with water throughout the firn column)
yields a 10 % overestimation of F.
The firn model is also used to simulate surface elevation changes
h˙f in the ablation area. Here, the prescribed RACMO2.3
mass fluxes determine the ice ablation rate assuming the ice density
ρi. Section describes the method to derive
the SMB-induced surface elevation change h˙f in the
ablation area.
Model forcing from RACMO2.3
At the upper boundary of the 1-D firn column, the firn model is forced with
surface temperature and mass fluxes from RACMO2.3 , a regional
climate model that is adapted to simulate climatic conditions over ice
sheets. The horizontal spatial resolution of RACMO2.3 is 11 km.
Forcing data are available for the period 1 January 1960–31 December 2014,
in time steps of 6 h.
RACMO2.3 supersedes RACMO2.1 . In the new version,
the cloud microphysics, surface and boundary layer turbulence, and radiation
transport have been updated . The most pronounced effect of
these updates on the SMB is an increase in summer snowfall events, decreasing
the amount of snow and ice melt in the percolation and ablation area
. The agreement between RACMO2.3 SMB and mass balance stakes
in these areas is improved. The ELA is lower and in better agreement with
observations. This is expected to improve the description of firn processes
in the percolation area of the ice sheet.
Modelling strategy
While RACMO2.3 itself contains a multi-layer snowpack with the same
compaction and meltwater routines as the firn model, the rationale for using
an offline firn model is the ability to spin up the firn layer with
a reference climate until it is in equilibrium with that reference climate.
This circumvents the difficult task of assuming an initial condition of the
firn layer at the start of the RACMO2.3 simulation that is sufficiently
accurate for correctly determining h˙f. Furthermore,
because of its computational demands, RACMO2.3 cannot be used for sensitivity
tests, in contrast to an offline firn model. Finally, the vertical resolution
is higher in the offline model.
Still, the spin-up procedure requires that we define a reference climate,
i.e. a period of time in which the properties of neither the firn nor the
reference climate forcing exhibit significant trends. Recent modelling and
observations reveal that the Greenland Ice Sheet SMB has decreased since the
beginning of the 1990s e.g.. As a result, thinning
has increased sharply since the mid-1990s along the margins
of the ice sheet. Clearly, the reference period should not include this
period and should end preferably some years before its onset. Therefore, our
modelling strategy is that we choose the first 20 years of RACMO2.3
forcing data (1 January 1960–31 December 1979) and spin up the firn column
at each location with a loop over this 20-year period until the
properties of the firn layer have converged to an equilibrium. By doing so,
we assume that the pre-1960 climate (i.e. the reference climate) can be
represented by a sequence of 20-year periods. In practice,
equilibrium is reached when all the mass in the firn layer is refreshed once
following the start of the spin-up. The duration of the spin-up is therefore
computed as the thickness of the firn layer (from the surface to the depth at
which the ice density is reached) divided by the mean annual accumulation
rate. A major uncertainty in the calculated firn depth changes in this study
stems from this assumption of reference climate. We will quantify this
uncertainty in Sect. .
As a second important assumption, we set the surface elevation change
h˙f over the reference period to zero. After all, the
modelled firn layer at the end of the reference period (31 December 1979) is
the result of the spin-up procedure that uses multiple loops over the
reference period to reach an equilibrium state, plus 20 years of
model integration using the same data as in the spin-up procedure. For the
accumulation area, it means that the amount of mass leaving the bottom of the
firn layer (with a velocity vice) is assumed equal to the total
mass added to and retained in the firn column by snowfall and refreezing,
minus run-off.
For the ablation area, the assumption of h˙f=0 during
the reference period implies that the downward velocity from a negative SMB
(ablation) is balanced by an upward and equal flux of emergent ice. The
emergent ice flux is represented by the term vice. In
Eq. (), we set vice equal to the opposite sign of the sum of all other
velocity components for the reference period. Thereafter, vice
retains the same value, but the other parameters are free to evolve. In this
framework, h˙f (as presented in Fig. ) in
the ablation area represents the surface elevation change due to the anomaly
of surface melt with respect to the reference period. To clarify, the change
in surface elevation itself does not have to be zero over the reference
period: it can change due to ice-dynamical thinning or thickening. Only the
ablation-driven surface elevation change h˙f is assumed to
be zero.
Note that our choice of the period 1960–1979 for a representative reference
climate implies that modelled surface elevation after 1980 is allowed to
evolve freely due to SMB and firn processes. It is not bounded by the
requirement that h˙f should be zero over the entire
simulation period, like in studies addressing the Antarctic Ice Sheet
. As the firn layer starts to evolve freely from 1980
onwards, we present time series starting in 1980, although the complete time
series of elevation change start in 1960.
Firn cores
To evaluate the firn model, we collected vertical profiles of firn density
from 62 shallow cores from widely varying locations across the Greenland Ice
Sheet (see the map in Fig. ), drilled between 1995 and 2012. Among the cores are those drilled
for PARCA Program for Arctic Regional Climate
Assessment;; cores from the Arctic Circle
Traverses ACT;; cores from the lower part of the EGIG line
; and Das 1 and Das 2 e.g..
Map of Greenland showing the 62 firn cores used for validation of
the firn model. Solid circles represent firn cores longer than 30 m;
open circles are firn cores shorter than 30 m. The inset in the lower
right shows cores from in the west Greenland percolation area
around 69∘42′ N. Dashed contours are surface elevation
isolines with a spacing of 500 m, the uppermost being
3000 m.
Vertical profiles of density from firn cores are usually based on the mean
density of 0.5–2 m long sections. Some researchers log the midpoint
of each section as the depth of the section; others use the top or the bottom
of the section. Here, all 62 profiles have been interpolated to give the mean
density at the midpoints of each core section. For each core, the collection
date is known, and vertical profiles of modelled firn density are extracted
from the model at the time closest to the collection date, and from the
nearest model grid cell.
Laser altimetry
Since 1992, NASA's Airborne Topographic Mapper (ATM) has carried out laser
surveys of the Greenland Ice Sheet surface. If sufficient repeat observations
are available, a time series of observed surface elevation change can be
constructed, spanning the period 1992–2013.
To do so, we use the Level 2 “Qfit” product, which provides the
waveform-fitted elevations for the centroid of each laser return. In order to
derive elevation changes, we interpolated the Qfit point cloud for each
campaign to a reference grid with 30 m spacing. We then selected
reference grid points with at least one observation in each of five epochs:
1993–1996, 1997–2000, 2001–2005, 2006–2009, and 2010–2013. Of these
points, we selected 15 within the central and northern interior of the ice
sheet, where a relatively small contribution of ice dynamics is expected.
Based on differences in elevation obtained from crossovers within a few
weeks, we estimate 1σ errors in the elevation observations to be ±10 cm, which includes interpolation error. Results of the comparison
between the firn model and the ATM data are found in Table and
Fig. .
Coordinates of analysed elevation change points labelled in
Fig. with the number of repeat observations N, observed
surface elevation trend from the laser altimetry, the best-fit trend in the
residuals between the observations and the firn model heights, and the
root mean square of the residuals between the observations and fitted model
(in cm). Trends in bold are significantly different than zero at
95 % confidence.
No.
Lat
Lon
Elev.
N
Obs.
Resid.
rms
trend
trend
fit
∘ N
∘ W
m
cmyr-1
cmyr-1
cm
1
69.90
32.4
2749
6
2.6
-0.2
16.7
2
71.22
35.7
3139
8
3.7
1.4
8.7
3
70.81
41.3
2935
7
5.1
6.9
38.5
4
74.36
35.5
2933
6
4.3
6.3
49.4
5
75.23
40.4
2910
6
3.5
1.0
7.7
6
75.91
45.2
2827
8
2.0
-1.0
6.1
7
76.53
51.1
2458
8
1.6
-0.8
17.3
8
75.90
54.2
1971
8
-3.1
-5.3
13.5
9
76.32
55.3
1923
10
-2.0
-0.5
22.9
10
76.92
56.2
2019
8
-5.8
-6.6
18.5
11
77.89
56.5
2191
7
0.4
-1.4
11.1
12
78.42
52.2
2239
7
1.6
0.5
9.1
13
78.04
43.7
2564
6
0.5
-0.5
13.7
14
79.07
44.3
2370
6
0.9
1.5
6.6
15
79.36
48.5
2170
7
3.6
2.6
21.6
Model evaluation
Vertical profiles of density
We use vertical profiles of density from 62 firn cores to assess the
performance of the firn model. This evaluation is not independent, as we used
the depths of the 550 and 830 kgm-3 density levels from these
cores to tune the densification parameterization in Eqs. ()
and (). Still, we can compare the shape of the profiles beyond these
two levels, and we can assess the impact of melt and refreezing on the
vertical density profile.
Observed (black) and modelled (red) firn density profiles for 24 of
the 62 firn cores on the Greenland Ice Sheet. The four lines of text in each
panel show (1) the core name, (2) mean annual accumulation and (3) melt (in
mm w.e. yr-1) from RACMO2.3, and (4) the ratio of these fluxes
(Rma, dimensionless). Remaining profiles are shown in Appendix A
as Figs. A1 and A2.
Figure shows the observed and modelled density profiles for
all core locations. Each panel includes the mean accumulation and melt from
RACMO2.3 (in mmyr-1) for 1960–2014, and the ratio Rma
of these melt and accumulation averages.
The vertical resolution of the firn core data is typically
0.5–2.0 m, thereby smoothing out the effect of ice lenses and
higher-density layers. The model data in Fig. are shown at
full resolution, i.e. with layers of 5–10 cm thickness. The
high-density layers usually represent thick layers of refrozen meltwater with
a density close to that of solid ice.
Modelled vs. observed firn air content F (in m) for 59 of the 62
firn cores (cores H2-1, H3-1, and H4-1 are not shown). The colour scale to the right indicates the ratio of modelled mean
melt and accumulation fluxes Rma, and the colour of each dot in
the scatter plot corresponds to the value of Rma for that firn
core location.
Up to an Rma value of 0.3–0.4, the agreement between the firn
model and the observations is good. But for higher Rma, the firn
model starts to overestimate the density throughout the firn column.
Figure compares the observed and modelled firn air
content F, showing Rma in colour. The model bias clearly
increases for higher Rma. This means that there are three possible
causes for the misfit, which are not mutually exclusive: (1) RACMO2.3
simulates too much melt in the percolation areas, causing the firn to fill up
quickly with too much refrozen meltwater; (2) RACMO2.3 simulates too little
accumulation, providing insufficient pore space to store meltwater; and (3)
the firn model should allow for more and more rapid downward percolation of
meltwater without letting it refreeze.
Regarding a possible overestimation of melt in RACMO2.3, there is limited
evidence that the amount of melt observed by an automatic weather station at
location S10 (67∘00′ N 47∘01′ W,
1850 ma.s.l.) is indeed about 20 % smaller than simulated
by RACMO2.3 . Further north, find the
equilibrium line around the EGIG line at about 1100–1200 ma.s.l.,
while the equilibrium line altitude in RACMO2.3 is at ∼1650 ma.s.l., about 45 km further inland. For firn cores
H1-1 down to H5-1 (Fig. ), it is clear that under RACMO2.3
forcing (with melt larger than accumulation) a firn layer cannot be
sustained, whereas in reality there is a shallow firnpack with infiltration
ice layers. There is very limited reliable information about melt fluxes from
other parts of the percolation area around the ice sheet, so we cannot
conclude whether the overestimation of modelled melt flux is structural.
The other possible source for the misfit is the percolation scheme in the
firn model itself. The firn model adopts a so-called “tipping
bucket” approach, where meltwater is allowed to move downward from one
discrete layer to the next whenever the first layer is saturated. In
practice, the percolation is more complex, and vertical meltwater transport
through confined channels (pipes) is known to occur .
Piping of meltwater is a way to evacuate more meltwater towards the bottom of
the firn layer, reducing the amount of refreezing in the firn itself.
Alternatively, intermediate-thick ice layers may serve as an impermeable
surface along which the water can run off. Both processes increase the run-off
and decrease refreezing and density. At present, we cannot assess the
performance of the firn model in more detail, since we cannot easily isolate
it from errors in the model forcing from RACMO2.3.
It is unclear what exactly the model bias in the percolation zone implies for
the modelled rates of surface elevation change. We speculate that, if too
much refreezing is the cause for the density overestimation, then
a prescribed increase in surface melt would underestimate the rate of surface
lowering.
Upper left panel: map of elevation change points observed by the
airborne laser altimeter ATM, with colour scale giving the 20-year
trend in elevation (myr-1). The 15 panels (numbers in the upper
left corner of each panel correspond to the numbered locations on the map)
show surface elevation change from (black dots with 1σ error bars)
ATM lidar and (blue curves) firn thickness model.
Green curves are the firn model results adjusted to the trend in residuals
between the model and the observations.
Altimetry from the high-elevation interior
In the high interior of the Greenland Ice Sheet, horizontal surface ice velocities are low
generally less than 10 myr-1;, and elevation
changes resulting from ice-dynamical effects are expected to be small.
Figure shows time series of observed surface elevation change
from the ATM lidar, along with the surface elevation change predicted by the
firn model.
Surface elevation change rates at the 15 test sites range from -6.6 to
5.1 cmyr-1 over the altimetry record (map in Fig. ,
Table ). The sites in the central east (site 1, 2 and 3) had the
highest rates of surface rise, with rates increasing inland. Sites 8, 9 and
10 near the northwest margin uniquely show decreasing elevations. The time
series of observed surface elevation change (panels in Fig. )
show the substantial variability between nearby locations in both time and
space.
The firn model provides the change in surface elevation due to only
variations in snow accumulation and firn density, assuming constant vertical
ice motion. Therefore, the difference between the observed change and the
modelled elevation change represents the elevation change due to vertical ice
motion (ice dynamics) and error. We assume that, in the ice-sheet interior,
variations in ice dynamics occur over timescales that are long (centuries)
relative to the observational record and can therefore be approximated by
a linear trend. Under this assumption, the residual between the observed and
modelled surface elevations will decrease or increase at a rate equal to the
difference between the reference and actual submergence rates. The trend in
residuals is therefore the anomaly in the submergence rate from the
reference, which is assumed to approximate steady state and provides an
estimates of the contribution of ice-dynamical change to surface elevation.
These trends in residuals are given in Table . In most cases,
these trends are not statistically significant, indicating a submergence
velocity close to the reference state. At site 3, the trend in residuals is
nearly 7 cmyr-1, which accounts for more than the
6 cmyr-1 of observed increases, indicating thickening. At site 6,
a negative trend in residuals of 1 cmyr-1 opposes the
2 cmyr-1 of observed surface raising, suggesting opposing
contributions from dynamics and accumulation. At sites 8 and 10, the strongly
negative trend in residuals is larger than the observed surface lowering,
indicating that increased accumulation is partially offsetting relatively
rapid dynamic thinning.
If the trend in residuals between the observed and modelled surface elevations
provides the linear contribution in ice dynamics plus the error, the error is
then assessed by the root mean square (rms) of the detrended residuals
(Table ). This is equivalent to adjusting the firn model time
series to the ice-dynamical trend (shown as green curves in the panels of
Fig. ) and computing the difference from the observations. The
mean rms error is 17.4 cm, which is close to the lidar observational
uncertainty (∼10 cm). Sites 3 and 4 have the largest errors,
reaching 1.7 and 2.6 standard deviations, respectively. These sites are
located at similar elevations (2930 m) on the central eastern portion
of the ice sheet, where altimetry shows steadily increasing elevations while
the firn model predicts an initial increase in firn thickness until about
2005 and then a decrease thereafter.
Elevation change due to firn and SMB
Firn air content
Figure shows the modelled firn air content F on 1 September
2014. As noted in Sect. , these values are probably realistic
in the dry interior and the upper part of the percolation area. In the lower
percolation area, where the annual melt flux exceeds ∼30 % of
the annual accumulation rate, the modelled firn air content is likely
underestimated. Around the central dome, we find the highest values of F of
about 25 m. There is a remarkable contrast between the firn in the NW
and the NE, with the NW having higher F. This can be explained by more
snowfall in the NW and higher sublimation in the NE due to a lower relative
humidity.
Modelled firn air content F on 1 September 2014 (in m). Dashed
contour lines at 500 m height intervals.
Modelled average firn thickness change h˙ (in
cmyr-1) for three periods: (a) 1980–2014,
(b) 1980–1995 and
(c) 1995–2014. The equilibrium line (according to the
RACMO2.3 SMB) is shown as a thick black line in (a). Note that
the colour scale is asymmetric around 0. In the ablation area, where no
firn layer is present, ice thickness change by surface processes is
presented.
Trends
By adding up all the velocity components in Eq. (), we find
h˙f, the firn thickness change per unit of time due to all
firn and SMB processes. We can accumulate the thickness changes over longer
periods to get multi-year maps over the ice sheet. Figure
shows h˙f (in cmyr-1) for the periods
1980–2014, 1980–1995, and 1995–2014. This surface elevation
change is with respect to the reference period 1960–1979, during which zero
surface elevation change (due to firn and SMB processes) is assumed. Again,
there is a pronounced pattern of modest thickening of the firn layer in the
interior (most notably towards the east) and moderate to strong thinning of
the firn layer around the margins. The interior thickening is of the order of
1–5 cmyr-1, or 34–170 cm, over the entire
34-year period. The marginal thinning rates are much larger; they can
be up to 20–50 cmyr-1, or 6–18 m, over the entire period,
with the highest values in the southeast. In contrast to the Summit dome firn
layer, that of the southern dome of the Greenland Ice Sheet is thinning.
It is clear that the patterns have changed over this period. The surface
elevation change map over 1980–1995 (Fig. b) shows thinning
along the southeast, south, west, and northwest coasts. Thickening is
occurring in the interior (mainly east of the divide) and along the north
and northeast coasts. Since 1995, thinning has intensified and spread over
the entire coastal margin. The thickening moved to the west of the interior.
The aggregate picture for the period 1980–2014 then shows moderate
thickening up to 5 cmyr-1 in the eastern and northern interior.
Thinning occurs all around the margins, with the smallest rates
(0–15 cmyr-1) in the northern and eastern coastal regions. The
largest rates (exceeding 40 cmyr-1) occur in the southeast and
in the western ablation area.
Decomposing the trends
The firn model allows for a decomposition of the
h˙f signal into its velocity components
(Eq. ). The upper panels in Fig. show this
decomposition for the period 1980–2014. The thickening in the eastern
interior (Fig. a) can be almost fully ascribed to a positive
accumulation anomaly (Fig. a), offset by a small increase in
firn compaction due to this extra firn (Fig. g). In the lower
accumulation area, the positive accumulation anomaly is offset by
a significant increase in surface melt, giving zero or slightly negative
h˙f in the western percolation area. In the south,
increased surface melt dominates the thinning signal, whereas accumulation,
firn compaction, and snowdrift anomalies play a minor role. In the southeast,
melt has increased and accumulation decreased significantly. As the absolute
values of both accumulation and melt are large in this region, we find here
the largest rates of firn-driven surface lowering seen in Greenland.
Anomaly of vertical velocity components with respect to the
reference period 1960–1979 (in cmyr-1). (a–c) Accumulation velocity
vacc; (d–f) melt velocity
vme; (g–i) firn compaction velocity
vfc; (j–l) snowdrift sublimation, erosion and
deposition velocities (vsnd+ver). Note that the colour
scales for vme is strongly asymmetric around 0.
For the period 1980–1995, the accumulation anomaly differed from the
1995–2014 period, as shown in panels b and c of Fig. .
A negative accumulation anomaly (partly offset by a positive firn compaction
anomaly, panel h) explains the firn-driven surface lowering in the northwest.
In the absence of significant melt anomalies (panel e), the thickening in the
eastern and northeastern interior can be almost fully ascribed to a positive
accumulation anomaly (panel b). This is mirrored in a small negative firn
compaction anomaly (panel h).
Over almost the entire ice sheet, with the exception of the southeast, the
period 1995–2013 shows a positive accumulation anomaly (panel c). At lower
elevations however, the firn thickness change is dominated by the strong melt
anomaly over this period (panel f).
The velocity components that always lead to a decrease of
h˙f, melt and firn compaction, are negative by definition.
To complement this, we can add up the snowdrift and surface sublimation velocities
whenever they lead to a surface lowering. The partitioning of the surface
lowering into these components of negative velocities is shown in
Fig. . The lowering is dominated by firn compaction (panel b)
in the interior and more and more by melt around the margins. There, the
firn layer is thinner, which reduces the compaction potential. In the dry
northeast, there is a relatively large contribution from sublimation (up to
30 %, panel d). This is caused by a combination of strong winds and
a relatively low humidity, promoting snowdrift sublimation. For another part,
the relative contribution increases as the firn compaction is small due to
lower firn temperatures, and due to the relatively small thickness of the
firn layer.
Fraction of surface lowering (negative velocities only) during
1980–2014 caused by (a) melt; (b) firn compaction;
(c) snowdrift processes; and (d) sublimation. Up to
a fraction of 0.15, the colour scale is divided in steps of 0.025. Above 0.15,
the step size is 0.05.
Error estimate
An important source of uncertainty in h˙f is the
steady state assumed for the spin-up of the firn model. As explained in
Sect. , the present model setup assumes that the climate under
which the firn was formed can be represented by a loop over the forcing data
from 1960 to 1979. A reconstruction based on firn cores and a previous
version of RACMO2 found large interdecadal accumulation variability over the
past 400 years, and an accumulation increase by 12 % over
the period 1600–2009 . The mean reconstructed, ice-sheet-wide
accumulation over this period is 782 Gtyr-1. For 1960–1979, it
is 786 Gtyr-1, i.e. 0.5 % larger than the long-term
average. Other reconstructions show the 1960–1979 solid precipitation flux
to be 0–10 % different from the period between ∼1850 and the
present day . For the error analysis, we therefore
assume that the 1960–1979 accumulation fluxes (precipitation minus
evaporation/sublimation) can differ by up to 15 % from the long-term
accumulation history. We denote this accumulation uncertainty by
σb˙ (in mmyr-1).
For snowmelt, we also assume a maximum deviation of 15 % of the
1960–1979 melt flux compared to the long-term history. There is limited
evidence for this, but according to reconstructions from the
run-off flux for 1960–1979 differs about 5–10 % from the 1870–2010
mean. This uncertainty is given as σm˙ (mmyr-1).
For the suite of 62 firn core locations, we perform four sensitivity tests, in
which we increase and decrease melt or accumulation by 15 % after
completion of the spin-up. The resulting drift in
h˙f for 1960–2014 gives an error in the firn thickness
change. The error in surface elevation change due to σb˙ is
written as σh˙,b˙. Analogously, the error in surface
elevation change due to σm˙ is given as σh˙,m˙. By relating σh˙,b˙ and σh˙,m˙ empirically to accumulation and melt at the core locations, we can
expand the error product over the entire ice sheet. These relations are
σh˙,b˙=σb˙(0.107+3.609×104b˙),σh˙,m˙=σm˙(0.225+1.064×103m˙),
with b˙ and m˙ in millimetres per year.
The uncertainties σb˙ and σm˙ cannot be
regarded as independent. The SMB module in RACMO2.3 contains interactions
between accumulation and melt. We identify the melt-albedo feedback as the
most important interaction. As an example, a negative bias in summer snowfall
could lead to an excess of summer melt because albedo will be underestimated.
To capture this dependence in the error analysis, we assume the errors in
surface elevation change due to uncertainties in the melt and accumulation
fluxes to be dependent, and we add them up linearly (σh˙=σh˙,b˙+σh˙,m˙).
The resulting total error and its components are shown in
Fig. . In the interior, the total error (panel c) is
understandably dominated by the accumulation uncertainty (panel a). Towards
the ice-sheet edge, the melt uncertainty (panel b) starts to dominate the
total uncertainty. The total uncertainty increases from
0.2–1.0 cmyr-1 in the interior to 10–20 cmyr-1 in
the lower percolation area. The largest total error (more than
40 cmyr-1) is found in the southeast, where high snowfall rates
coincide with large amounts of melt.
Estimate of errors in firn thickness change (cmyr-1).
(a) Error σh˙,b˙ due to accumulation
uncertainty σb˙; (b) error σh˙,m˙ due to melt uncertainty σm˙; and (c) total
error σh˙. Note the non-linear colour scale.
For low and medium values of b˙, the term between brackets in
Eq. () is smaller than 1. It means that an uncertainty in the
accumulation rate leads to a smaller uncertainty in the elevation change. The
explanation for this behaviour is simple: the densification rate in
Eq. () linearly depends on the mean accumulation rate, so that
an elevation increase due to more snowfall is partially offset by a more
rapid densification. For b˙>2427 mmyr-1, the term
between brackets in Eq. () becomes larger than unity. There is
no physical explanation for this behaviour: it is caused by the empirical
nature of Eq. (). However, we decided not to cap the
uncertainty, for the following reason: the densification rate in
Eq. () depends on a 20-year mean accumulation rate,
whereas true densification at a particular depth in the firn depends on the
immediate overburden pressure from overlying firn, which can be more variable
than the long-term mean. This was already shown to dampen the modelled
variability in densification rate compared to observations .
Therefore, the observed, short-term firn thickness change variability could
be of the same order of magnitude as the accumulation rate variability for
large accumulation rates.
(a) Ice-sheet-integrated volume change for 1980–2014 (in
km3) due to firn and SMB processes. The thin red line is the volume
change based on weekly output from the firn model; the thick red line is
a 1-year running average; and the pink shaded area shows the
uncertainty estimate. The blue line shows the volume change for the part of
the ice-sheet surface elevated more than 2000 ma.s.l. (b) Partitioning of the total volume change into the three main components:
accumulation vacc, melt vme, and firn compaction
vfc. (c) Like (b) but for the ice sheet higher
than 2000 ma.s.l.
Integrated volume change
Figure shows the cumulative volume change of the Greenland Ice
Sheet as a consequence of changes in firn and SMB processes. Until the late
1990s, the total volume change was small. Since 2000, the total volume has
decreased by about 3295±1030 km3 due to firn and SMB processes
alone. Averaged over the ice sheet, this is a mean surface lowering of
1.96±0.61 m. Almost all of this total volume loss
took place in the part of the
Greenland Ice Sheet where the surface is under 2000 ma.s.l.
Panels b and c of Fig. show the partitioning of the volume
change for the entire ice sheet and the part elevated above
2000 ma.s.l., respectively. Over 1980–2014, the volume loss
through melt was slightly over 4700 km3. This loss was partly
compensated for by an increase in snow accumulation of about
1500 km3. On most of the ice sheet, firn compaction has
accelerated (become more negative) due to an increase in accumulation. Above
2000 m, the effect is clearly visible (Fig. c).
Integrated over the ice sheet however, we see a small slowdown in firn
compaction, corresponding to about 500 km3 (Fig. b).
The firn compaction anomaly is dominated by the southeastern part of the ice
sheet, where snowfall has decreased strongly in an absolute sense (less so in
a relative sense) and firn compaction has slowed down (Fig. h
and i).
Up to about 2005, firn volume change above 2000 ma.s.l. was
dominated by accumulation variability (consistent with e.g.
), and below 2000 ma.s.l. the volume change was
mainly melt-driven. This is consistent with the original speculation in early
reports of what would happen to the Greenland Ice Sheet in response to global
warming. After 2005 however, the total firn volume above
2000 ma.s.l. has started to decrease, mainly because surface melt
has migrated inland e.g., but also because the
accumulation increase, clearly visible between 1980 and 2000, stagnated in
the 2000s. It remains to be seen if the paradigm of interior thickening under
atmospheric warming can stand up against the inland migration of the area of
surface melt.
The extreme melt season of 2012 is clearly visible in
the results of the firn model. A large drop in total volume of
1386 km3 is seen in the summer of 2012, of which
1150 km3 is contributed to melt. Melt in the part of the ice sheet
above 2000 m contributed almost one third (371 km3) to
this volume anomaly. For perspective, the volume loss above
2000 ma.s.l. in the summer of 2012 is equal to the volume gained by
snowfall in the interior in the 16 years between 1980 and 1996.
Altimetry correction and mean density of firn-related mass loss
The present data set of firn thickness and mass change is primarily aimed at
the correction of altimetry products, allowing for the extraction of an
ice-dynamical thinning/thickening signal. The procedure for doing so is as
follows: suppose that an altimetry sensor measures a surface elevation change
h˙(t0,t1)≡(h(t1)-h(t0))/(t1-t0) between
initial time t0 and time t1>t0. The firn model computes
a surface elevation change due to firn and SMB processes
h˙f(t0,t1)≡hf(t1)-hf(t0). The supposed ice-dynamical contribution (neglecting
vertical bed movement, but this could be included; ) is then
h˙d=h˙-h˙f. The associated mass
change m˙d is simply computed as
m˙d=ρih˙d.
The mass change m˙d is caused by horizontal convergence or
divergence of ice flux, arising from, e.g., ice-flow acceleration propagating
from the margin, long-term changes in the ice-sheet viscosity
, and transient variations in ice flow due to long-term
changes in accumulation. The mass change due to the SMB,
m˙f, is computed directly from the RACMO2.3 forcing, using
SMB anomalies with respect to the appropriate reference period (1960–1979).
This is by far the simplest approach and completely consistent with the firn
model that uses the same forcing and the same reference period. The total
mass change at the given location m˙ is then computed as
m˙=m˙d+m˙f.
Conclusions
In this study, we used a time-dependent, semi-empirical model for firn
compaction, meltwater percolation, and refreezing. We forced the model with
surface mass fluxes and temperature from the regional climate model RACMO2.3
for the period 1960–2014. By forcing the model with all mass fluxes,
including melt, the result is a data series of surface elevation change over
the entire ice sheet, due not only to firn processes but also to
anomalies in the SMB. We defined a reference period from 1960 up to and
including 1979, in which we assumed the surface elevation change to be zero
due to firn and SMB processes. In the ablation zone, the computed surface
elevation change represents the ablation anomaly with respect to the
1960–1979 mean ablation.
The firn model was calibrated against vertical profiles of firn density from
more than 60 shallow and deep firn cores collected around Greenland in the
past 2 decades. This ensured a very good agreement between observed and
modelled vertical density profiles, especially in regions where the annual
surface melt flux is small (less than about 20 %) compared to the
mean annual accumulation. In regions with higher melt, the firn model
overestimates the density below the surface. Potentially, this underestimates
the presented rates of surface lowering in the percolation area.
The computed surface elevation change was compared against lidar observations
of surface elevation change at locations where we expect the ice-dynamical
changes to be small or gradual in time. At most locations, we find a good fit
of the modelled elevation change rates to the observed ones.
Between 1980 and 2014, we see a pronounced pattern of small thickening of the
firn layer in the high interior, of 1 to 5 cmyr-1, caused
predominantly by an accumulation increase over this period. Around the
margins of the ice sheet, in the percolation and ablation areas, the surface
is lowering, at rates of up to 20–50 cmyr-1. This is mostly
caused by an increase in surface melt, augmented in the southeast by
a decrease in accumulation of snow. The thinning signal in the margins of the
ice sheet has accelerated between 1980 and 2014, in line with observations of
increased surface melt.
During the period 1980–2014, the surface elevation increase in the interior
shifted from the east and northeast towards the centre of the ice sheet and
stagnated towards the end of the time series. The contribution from surface
melt to interior surface lowering has increased markedly in this period, with
the largest firn volume decrease due to surface melting in the extreme melt
summer of 2012.
The time series of surface elevation change due to SMB and firn processes
(h˙f) is suitable to isolate ice-dynamical thinning from
altimetry-based observations of surface elevation. Combining it with the next
generation of altimetry products, e.g. from Cryosat-2, allows for further
improved assessment of the current imbalance of the Greenland Ice Sheet.