TCThe CryosphereTCThe Cryosphere1994-0424Copernicus GmbHGöttingen, Germany10.5194/tc-9-767-2015A model study of Abrahamsenbreen, a surging glacier in northern SpitsbergenOerlemansJ.j.oerlemans@uu.nlhttps://orcid.org/0000-0001-5701-4161van PeltW. J. J.Institute for Marine and Atmospheric Research Utrecht,
Utrecht, the NetherlandsNorsk Polarinstitutt, Tromsø, NorwayJ. Oerlemans (j.oerlemans@uu.nl)27April20159276777930September20147November20149March20152April2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://www.the-cryosphere.net/9/767/2015/tc-9-767-2015.htmlThe full text article is available as a PDF file from https://www.the-cryosphere.net/9/767/2015/tc-9-767-2015.pdf
The climate sensitivity of Abrahamsenbreen, a 20 km long surge-type glacier
in northern Spitsbergen, is studied with a simple glacier model. A scheme to
describe the surges is included, which makes it possible to account for the
effect of surges on the total mass budget of the glacier. A climate
reconstruction back to AD 1300, based on ice-core data from Lomonosovfonna
and climate records from Longyearbyen, is used to drive the model. The model
is calibrated by requesting that it produce the correct Little Ice Age
maximum glacier length and simulate the observed magnitude of the
1978 surge.
Abrahamsenbreen is strongly out of balance with the current climate. If
climatic conditions remain as they were for the period 1989–2010, the
glacier will ultimately shrink to a length of about 4 km (but this will take
hundreds of years). For a climate change scenario involving a
2 myear-1 rise of the equilibrium line from now onwards, we
predict that in the year 2100 Abrahamsenbreen will be about 12 km long.
The main effect of a surge is to lower the mean surface elevation and
thereby to increase the ablation area, causing a negative perturbation of
the mass budget. We found that the occurrence of surges leads to a faster
retreat of the glacier in a warming climate.
Because of the very small bed slope, Abrahamsenbreen is sensitive to small
perturbations in the equilibrium-line altitude. If the equilibrium line were lowered by only 160 m, the glacier would steadily grow into
Woodfjorddalen until, after 2000 years, it would reach Woodfjord and
calving would slow down the advance.
The bed topography of Abrahamsenbreen is not known and was therefore
inferred from the slope and length of the glacier. The value of the
plasticity parameter needed to do this was varied by +20 and -20 %.
After recalibration the same climate change experiments were performed,
showing that a thinner glacier (higher bedrock in this
case) in a warming climate retreats somewhat faster.
Introduction
Abrahamsenbreen is a valley glacier in the north-western part of
Svalbard (79.10∘ N, 14.25∘ E), originating at the ice field
Holtedahlfonna (for more topographic information, see the interactive map:
http://www.npolar.no/en/services/maps/). It is about 20 km long and
flows in a north-easterly direction (Fig. 1). The glacier snout terminates on
land and is only a few tens of m a.s.l. (above mean sea level). The highest
regions in the accumulation area are about 900 m a.s.l. A large part of
the accumulation area is rather flat with an altitude ranging between 600 and
750 m a.s.l. According to Hagen et al. (1993), the equilibrium-line
altitude is around 600 m. The glacial river runs through the very flat
Woodfjorddalen over a distance of about 15 km before it enters the
Woodfjord.
The glacial history of northern Spitsbergen is only broadly known (Svendsen
and Mangerud, 1997; Forman et al., 2004; Salvigsen and
Høgvard, 2005). There is
abundant evidence that the fjord areas were deglaciated by 10 kyr BP
(Before Present) and that, during most of the Holocene, glaciers were less
extensive than they are today. Abrahamsenbreen most likely reached its
maximum Holocene extent during the Little Ice Age (LIA), in line with the evidence
for many large glaciers in western and southern Spitsbergen (Hagen et
al., 1993). One of the goals of this paper is to see whether this is in agreement
with palaeoclimatic information derived from the Lomonosovfonna ice cores
(Pohjola et al., 2002; Divine et al., 2011) and the meteorological record of
Longyearbyen.
Map of Abrahamsenbreen in northern Spitsbergen (inset), originating
from the ice field Holtedahlfonna (lower left corner of the map). The red line
shows the flow line along which the length is defined; numbers in red
indicate distance from the glacier head in kilometres. Basins and tributary glaciers
delivering mass to the main stream are numbered T1–T5 (left) and T6–T5
(right). Isohypses on the glacier in metres above sea level are shown in blue. Glacier
stand in 1990, i.e. after the 1978 surge. Note that looped moraines are
actually shown on the map. The rectangle in the upper part refers to the area
covered by the aerial photograph of Fig. 2. Map is courtesy of the Norwegian Polar
Institute; Landsat image is courtesy of NASA.
Abrahamsenbreen is a surging glacier. It is well known for its fine set of
looped moraines (Fig. 2) that were formed during and following the surge
that took place around 1978 (Hagen et al., 1993). The duration of the 1978
surge and the frequency with which surges occur is not known. However, it is
likely that the surge characteristics of Abrahamsenbreen are similar to those
of other gently sloping glaciers in Svalbard. These surges are of a less
vigorous type than observed on alpine glaciers like Variegated Glacier (Kamb
et al., 1985), Medvezhiy Glacier (Osipova and Tsvetkov, 1991) or North
Gasherbrum Glacier (Mayer et al., 2011). Surge characteristics of Svalbard
glaciers vary considerably, but the common element is a relatively long
surging phase which lasts for several years (Dowdeswell et al., 1991; Melvold
and Hagen, 1998; Sund et al., 2009; Dunse et al., 2012). A
“normal surge” is an event in which enhanced ice flow transports ice from
higher regions to lower regions within a relatively short time, in the end
leading to a marked advance of the glacier front. However, in a study of
50 glaciers, Sund et al. (2009) have also documented glacier surges in which
the enhanced motion stops before the stage of an advancing front is reached.
The effect of the surge then only implies a thinning of the accumulation
region and a thickening of the ablation region. In the case of
Abrahamsenbreen there is no doubt that the 1978 surge was a full surge,
during which the glacier front advanced by at least 2 km.
After a surge, a glacier will be subject to a negative net surface mass
balance, because the mean surface elevation is lower than before the surge.
However, because the ice flow becomes (almost) stagnant, after some time the
accumulation area will thicken. This implies an increasing surface elevation,
less melt in summer and consequently the transition to a stage in which the
surface steepens and the glacier volume increases until a new surge
is initiated. It is not a priori clear at which point in the cycle
Abrahamsenbreen actually is. According to the map of the equilibrium-line
altitude over Svalbard provided in Hagen et al. (1993), E≈ 600 m in
the region of Abrahamsenbreen. For the parameterized glacier geometry used in
this study (discussed in Sect. 3), this would imply that the glacier
currently has a net balance that is slightly negative. This is in agreement
with the study of Nuth et al. (2010), who derived a net balance of
-0.67 ± 0.14 myear-1 for the period 1966–2005. It
should be noted that the surge took place within this period.
There is no general consensus about the mechanism that causes glaciers in
Svalbard to surge (Murray et al., 2003). These glaciers flow over soft
sediments, and the duration of surges is significantly longer than for
glaciers in steeper alpine terrain, which are at least partly hard-bedded.
Thermal regulation has been put forward as a likely mechanism, in which the
switch from frozen to warm bed conditions plays a central role (e.g. Fowler
et al., 2001). However, direct
evidence for this theory does not exist. Oerlemans (2013) has
suggested that the steady accumulation of dissipative meltwater in the
accumulation zone plays an important role. In recent years geometric changes
caused by surging have been documented extensively (e.g. Sund et al., 2009),
but this has not yet resulted in a major step forward in our understanding.
Since so many glaciers on Svalbard are of the surging type, the question
of to what extent surges interfere with the longer-term response of
glaciers to climate change has arisen (Hagen et al., 2005; Paasche, 2010). This question
is of importance with respect to the climatic interpretation of historical
glacier fluctuations and also needs to be considered when making projections
of glacier behaviour for scenarios of global warming. In the simple glacier
model used in this study, surges are imposed and their effect on the mass
budget is then implicitly dealt with. By comparing model experiments with and
without the surging mechanism, the potential role of surges in the evolution
of Abrahamsenbreen is evaluated.
In this study the climate sensitivity of Abrahamsenbreen is studied with a
simple glacier model. A so-called minimal glacier model is used
(Oerlemans, 2011), in which the ice mechanics are strongly parameterized and
the focus is on the total mass budget of the glacier. In fact, the ice
mechanics are reduced to a relationship between the mean ice thickness,
glacier length and mean bed slope. The surge cycle is then imposed by
making the proportionality factor between length and thickness a prescribed
function of time.
We are aware of the limitations of such a model. It does not give insight
into
why surges occur and what determines the length of the surge cycle. However,
since the mass budget of a (non-calving) glacier is mainly determined by the
mean surface elevation relative to the equilibrium-line altitude, the details
of the surface topography matter less. Therefore, useful information about
the climate sensitivity of a glacier can be obtained even without the
calculation of the spatially distributed fields of surface topography and ice
velocity.
Hardly any measurements have been carried out on Abrahamsenbreen, making the
modelling of this glacier a real challenge.
The available data consist of (references to these data sources are given
later in this paper)
topographic maps from 1966 and 2002
aerial photographs from 1969 and 1990
a high-resolution satellite image (ASTER) acquired on 26 June 2001
mass-balance observations on a nearby glacier (Kongsvegen, 25 km away)
a map of the estimated equilibrium-line altitude over Svalbard
a map of annual precipitation over Svalbard
geomorphological information about the late Holocene history of Abrahamsenbreen.
In this paper we use these data to constrain and calibrate the model in the
best possible way. We consider this exercise to be useful, because for more
than 99 % of all glaciers in the world no more information is available
than maps, satellite images and photographs.
Modelling strategy and geometric input data
The geometry of the main stream of Abrahamsenbreen is simple with a very
smooth surface profile along the flow line, indicating that the bed is also
gently sloping. Major ramps or overdeepenings are likely absent, since they
would certainly be reflected in features at the glacier surface (e.g.
Raymond and Gudmundsson, 2005). Such a regular geometry is a prerequisite
for the use of a minimal glacier model, which requires a small set of input
parameters and can be calibrated easily with the limited data available. In
a minimal glacier model the state variables are glacier length and mean ice
thickness.
Before describing this model we will first summarize some of the information
about the lower part of the glacier that is evident from the two aerial
photographs (from 1969 and 1990), two topographic maps (1966 and 2002) and a
satellite image (ASTER, 26 June 2001).
Terminal moraines from the tributary glaciers T4, T5, T9 and T10 (Fig. 1) are
schematically mapped for 3 years (Fig. 3). The distance between the
locations of the moraines in different years was calculated by projecting the
moraine tips on the central flow line and measuring the displacement. For the
displacement between 1990 and 1969 we found respectively 4.5 and 4.7 km for M1 and M3
and 5.9 and 6.3 km for M2 and M4. Using the mean values of the
paired moraines (left and right of the glacier), average corresponding ice
velocities would be 219 myear-1 for the lower region of the
glacier and about 290 myear-1 for the middle part. The
displacement between 2001 and 1990 is small, with corresponding velocities of
9 and 23 myear-1. If we think of the ice velocity as composed of
a background part and a surge part and we assume that the background part
has been constant, it follows that the displacement of surface ice due to the
surge would be 4.4 km for the lower part and 5.6 km for the middle part. It
is not straightforward to convert these data into a total advance of the
glacier front during the surge. Comparing the glacier outlines on the maps
suggests a frontal advance of 1.8 km. However, the glacier front will have
melted back from a more advanced position during the period 1978 (surge) to
2002 (map outline). Judging from the size of the moraine system (Figs. 2
and 3), retreat of the snout could have been at most 1.6 km during this
period. This would imply that the total advance of the front related to the
surge is not larger than 3.4 km.
Altitude profiles along the central flow line of Abrahamsenbreen
derived from topographic maps. The surge took place around 1978. The dashed
line indicates a possible maximum extension of the glacier immediately after
the surge. The reference bed profile is shown as b(x).
Altitudinal profiles along the central flow line are shown in Fig. 4. The
absolute error of the topographic maps in this area is not known, but the
profiles appear to be consistent with the occurrence of the surge in 1978.
The mean slope of the pre-surge profile (1966) is 0.035, while that of the
post-surge profile (2002) is 0.027. The mean difference in altitude (Δh‾) between the profiles is 51 m. It should be noted once more
that by the year 2002 part of the glacier snout has retreated. Hence, the
mean difference in altitude shortly before and after the surge probably was
somewhat larger. The value of Δh‾ cannot be taken directly
as a measure of the change in mean ice thickness ΔHm,
because the mean bed elevation is also different before and after the surge
(solely due to the change in glacier length). With the representation of the
bed chosen here (discussed shortly) we found Δb‾=13 m.
Altogether, we used a value of ΔHm=50 m to characterize
the change related to the surge.
Since the maps from which the profiles are taken are 36 years apart, the
difference in mean surface elevation can also partly be due to a non-zero
surface balance rate during this period. Unfortunately, in situ measurements
are not available to check this. Nuth et al. (2010) infer a negative mean
balance rate for the period 1966–2005 from remote sensing data. However, in
their map of elevation changes over north-west Spitsbergen (their Fig. 5), the
outlines for Abrahamsenbreen are not identical to those inferred here from
the 2002 topographic map. The difference is mainly in the size of the
accumulation area (larger in the present study), which had a slightly
positive balance rate during the period 1966–2005. In view of this, we have
not made any corrections to the value of ΔHm=50 m as
being characteristic for the surge. We also note that with a significantly
different (smaller) value, it is impossible to explain the glacier advance
during the surge in terms of mass conservation (which implies a direct
relation between change in glacier length and change in mean ice thickness).
The geometric set-up of the model is shown in Fig. 1. The main glacier is
modelled as a flow band with a constant width of 2000 m. It has its own
surface mass budget, which is definitely negative because it is almost
entirely in the ablation zone. The main stream is fed by tributary basins and
glaciers numbered T1, …, T10. The mass input from these tributaries is
parameterized in terms of a schematic geometry and depends on the climatic
state. Details on this are described in Sect. 4. We assume that the
tributaries have a considerably smaller characteristic response time than the
main glacier because they are steeper, implying that the net balance of the
tributaries is calculated as if they were in a quasi-steady state. This also
implies that tributaries having a negative net balance are simply ignored in
the total mass budget.
The bed topography is basically unknown. The surface of the glacier is smooth
and has a small slope (∼ 0.03), suggesting that a simple
formulation of the bed profile is adequate. A bed topography that can be
handled well by the minimal glacier model reads
b(x)=bhexp-x/xl.
Thus the bed elevation drops off exponentially from a value bh at
the highest point of the flow band (x=0) to sea level for large values of
x (see Fig. 1). The characteristic length scale at which the bed becomes
lower is denoted by xl. We also considered using a linear bed
profile, but this generates problems for glacier stands that are
significantly larger than today (a bed far below sea level, which is
unrealistic in this case). Here we chose xl= 12 000 m.
Admittedly, this value is not more than an educated guess based on the
general picture of valleys in northern Spitsbergen that are more deglaciated
than the Abrahamsenbreen valley (topographic map:
http://www.npolar.no/en/services/maps/). The value for bh
is discussed later.
Glacier model
The theory of minimum glacier models has been developed in Oerlemans (2011),
and the reader is referred to that work for details (freely available from
the internet; http://www.staff.science.uu.nl/~oerle102/MM2011-all.pdf).
We only give a brief description of the model version used here.
The starting point for the model formulation is the continuity equation:
dVdt=F+BA+∑i=110Bi,
where V is the volume of Abrahamsenbreen, F(<0) is the calving flux,
BA is the total surface mass budget of Abrahamsenbreen and the
last term represents the mass input from the tributary glaciers as defined in
Fig. 1. The glacier length L is measured along the central line on the
glacier (Fig. 1). Since here we do not consider states of Abrahamsenbreen
where it calves into the Woodfjord, we set F=0.
Because the glacier width w is assumed to be constant, the rate of change of
ice volume can be written as
dVdt=wddtHmL=wHmdLdt+LdHmdt=Btot,
where Btot is the right-hand side of Eq. (1), the total mass
budget of the glacier.
The mean ice thickness is parameterized as (Oerlemans, 2011)
Hm=Sαm1+νs‾L1/2,
where s‾ is the mean slope of the bed over the glacier length and
αm and ν are constants. A “surge function” S has
been introduced, which makes it possible to impose a surge cycle. S is
prescribed as a function of time. A rapid decrease of S mimics the surge,
whereas a steady increase of S represents the quiescent phase during which
the glacier thickness steadily increases. The precise form of S(t) will be
discussed later.
The parameterization of the mean ice thickness as described by Eq. (4) gives
a good fit to results from numerical flow line models. For s‾→0
the mean thickness varies with the square root of the glacier length, which
is in agreement with the perfectly plastic and Vialov solutions for a
glacier/ice cap on a flat bed (Weertman, 1961; Vialov, 1958). The minimal
glacier model was used earlier in a study of Hansbreen, southern Spitsbergen
(Oerlemans et al., 2011). The bed topography of Hansbreen is known, and it
was found that Eq. (4) matches the observed mean thickness for ν=10 and
αm=3m1/2. However, Hansbreen is a non-surging
tidewater glacier in a different geographical and geological setting; therefore
from the very few glaciers with bedrock data we have selected
Kongsvegen as a better glacier to estimate the parameter αm.
Like Abrahamsenbreen, Kongsvegen is a surging glacier, which is currently in
its quiescent phase (Melvold and Hagen, 1998). It is not located far away
from Abrahamsenbreen (about 25 km). From the bed and surface profiles of
Kongsvegen a value of αm=2.27m1/2 is found,
indicating that basal conditions here allow for a lower resistance than in
the case of Hansbreen. We have used the value of αm=2.27m1/2 as the best possible estimate for Abrahamsenbreen.
However, different values of this parameter will be used later in a
sensitivity test.
Using the chain rule for differentiation, the time rate of change of ice
thickness can be expressed as
LdHmdt=αm21+νs‾SL1/2dLdt-αmν1+νs‾2SL3/2∂s‾∂LdLdt+αm1+νs‾L3/2dSdt.
Combining with Eq. (3) then yields
Btot=w3αm21+νs‾SL1/2dLdt-αmν1+νs‾2SL3/2∂s‾∂LdLdt+αm1+νs‾L3/2dSdt.
Here Btot is the total mass budget, i.e. the right-hand side of
Eq. (3).
The prognostic equation for the length of the glacier can thus be written as
dLdt=Btotw(a+b)-c(a+b)dSdt,
where
a=32Hm;b=-νHmL1+νs‾∂s‾∂L;c=-HmLS.
From Eq. (7) it is clear that a sufficiently rapid decrease of S(dS/dt≪0) leads to a strong increase in L (but not
in V).
For the exponentially decaying bed profile described by Eq. (1) the mean bed
slope over the glacier length is easily found to be
s‾=bh1-e-L/xlL.
The term ∂s‾/∂L, needed in the coefficient b in
Eq. (8), is
∂s‾∂L=-b01-e-L/xlL2+b0xl-1e-L/xlL.
This concludes the formulation of the glacier model. When Btot is
known, Eq. (7) can be integrated in time with a simple forward time-stepping
scheme. The calculation of Btot is described in the next
section.
Formulation of the mass budgetMass budget of the main glacier
Mass-balance measurements have been carried on a number of glaciers in
Svalbard but not on Abrahamsenbreen. Glaciers with a mass-balance record of
at least 10 years, as filed at the World Glacier Monitoring Service, are
Midtre Lovénbreen, Kongsvegen, Hansbreen and Austre Brøggerbreen.
Long-term mean balance profiles are shown in Fig. 5. These profiles suggest
that a schematic representation of the balance rate can be taken as a linear
function of altitude, i.e.
b˙=β(h-E),
where β is the balance gradient and E is the equilibrium-line
altitude. Linear regression on the profiles shown in Fig. 5 yields values of
β ranging from 0.0039 to 0.0053 mw.e.m-1. The mean value
is 0.0045 mw.e.m-1, which is used in this study. It is clear
from the available observations that a higher-order formulation, e.g. with a
quadratic term in h, is not meaningful. However, according to Hagen et
al. (1993; their Fig. 8), annual precipitation decreases significantly when
going in a north-easterly direction from the Holtedahlfonna, implying an
increase in the equilibrium-line altitude of about 100–150 m over the
length of Abrahamsenbreen (assuming a sensitivity of E with respect to
changes in precipitation of -2.25 m per %, see Sect. 6.1). This is
taken into account by making E a function of x:
E=E0+γx,
where γ is the spatial gradient of the equilibrium-line altitude along
the flow line of the glacier. From the glacier length and change of E we
estimate γ=0.005.
Observed mean balance profiles for Austre Brøggerbreen
(1990–2009), Hansbreen (1991–2009), Kongsvegen (2000–2009) and Midtre
Lovénbreen (2000–2009). Data are from the World Glacier Monitoring Service
(Zurich).
The total mass gain or loss can now be found by integrating the balance rate
over the glacier surface:
Bs=β∫0LH(x)+b(x)-E0-γxdx=βHm+b‾-E0L-βγ2L2,
where b‾
is the mean bed elevation of the glacier. For the exponentially decaying bed
profile it is given by
b‾=bhL∫0Le-x/xldx=xlbhL1-e-L/xl.
Tributary glaciers
For the tributary glaciers feeding the main stream, some further analysis is
required to arrive at useful estimates of the mass input. Although the
tributary glaciers could be modelled in a similar way as the main glacier, we
take a somewhat simpler approach in which the surface geometry is fixed. This
is justified because the tributary glaciers have much larger mean slopes and
therefore a weaker altitude–mass-balance feedback.
We assume that a tributary glacier can be described as a basin with a length
Ly and a width w(y)=w0+qy. Here y is a local coordinate running
from the lowest part of the basin (y=0) to the highest part of the basin
(y=Ly). The surface elevation is taken as h(y)=h0+sy, where s is the
surface slope. The parameters q and s are constants which are different
for the individual basins. The total mass budget Bi of basin i is then
obtained from
Bi=∫0Lyβb0+sy-Ew0+qydy.
Evaluating the integral yields
Bi=βw0b0-ELy+12sw0+b0-EqLy2+13sqLy3.
The geometric characteristics of the basins have been estimated from the
topographic map and are summarized in Table 1. All basins have a trapezoidal
shape, some becoming narrower when going up (q< 0) and some wider (q> 0).
Due to the spatial gradient in E (see Eq. 11) the basins which are
located further downstream along the x axis will experience a slightly
higher equilibrium-line altitude. This is accounted for by applying a
basin-dependent correction (Table 1).
Basic experiments
For S=1 and E=587 m the model produces a steady-state glacier with a
length of 17.5 km, which is close to the pre-surge length (we cannot define
this precisely). A good match between the calculated and observed (pre-surge)
mean surface elevation is obtained with bh=323 m,
αm=2.27m1/2. The mean ice thickness then is
263 m. We refer to this state as the reference state. The corresponding mass
inputs from the tributary basins are given in the last column of Table 1. The
net balance of the main glacier stream is -0.66 m w.e., and this is then
compensated exactly by the mass input from the tributaries. Glaciers, and certainly surging glaciers, are
never in steady state. Nevertheless, it
is useful to have a steady state as a reference state, because it reveals
basic properties of the glacier model. At this point it should be noted that
the value of the bed parameter bh is determined by the value of
αm. Although we believe that the value of
αm as discussed in Sect. 3 is a good choice, we will later
discuss a few sensitivity tests to show how the value of αm
affects the results.
The next step is to introduce the surge behaviour. The surge function is
formulated as
S(t)=C-Sat-t0e-t-t0/ts+Sqt-t0.
The surge starts at t=t0 and the surge amplitude Sa determines
by how much the thickness of the glacier is reduced. The characteristic timescale of the surge is denoted by ts. The last term in Eq. (16)
represents the quiescent phase of the surge cycle, during which the glacier
steadily thickens because the mass flux is smaller than the balance flux. The
constant C should be chosen in such a way that the long-term mean value of
S(t) is close to 1.
Parameter values of the geometric characteristics of the basins that
feed the main stream. See Fig. 1 for the location of the basins. The last
column shows the mass input (ice volume) to the main glacier corresponding to
the reference state described in the next section.
We use ts=2.5 years. This value is based on the observation that
most surges of Svalbard glaciers typically last a few years (Sund et
al., 2009). The value of Sq is determined by two factors: the
rate of mass addition in the accumulation area and the degree to which the
glacier motion slows down after the surge. For the present case we have
chosen values of Sa and Sq in such a way that (i) the
frontal advance related to the surge is reproduced and (ii) the difference
in the mean ice thickness before and after the surge is in agreement with the
observations (about 50 m; Sect. 2). We thus found Sa=0.168year-1 and Sq=0.002year-1.
The duration of the surge cycle for Abrahamsenbreen is not known. For most
glaciers the duration of the quiescent phase is in the 50–500-year range
(e.g. Dowdewell et al., 1991). Because Abrahamsenbreen is a large and rather
flat glacier in a relatively dry climate, we have chosen a surge cycle of
Θ=125 years. Later we will show sensitivity tests that reveal how
the particular choice of Θ affects the results.
A model simulation in which the surge mechanism is switched on at some point
in time (after the glacier has reached the reference state defined above) is
shown in Fig. 6. As discussed above, a surge leads to a sudden decrease of
the mean ice thickness and associated negative net balance
(-0.3 m year-1 just after the surge). During the quiescent
phase the net balance gradually becomes positive and the ice thickness
increases, but this is not enough to compensate for the mass loss during and
just after the surge. Therefore the glacier length decreases until a new
equilibrium is reached after about 1000 years. Thus, the net effect of the surging
mechanism is to reduce the long-term glacier length. This is in
agreement with earlier studies (Adalgeirsdóttir et al., 2005; Oerlemans,
2011).
Surge event as imposed to the model by Eq. (16).
Response of Abrahamsenbreen to climate changeReference simulation
In this section we describe how a reference simulation, including the surging
behaviour, has been obtained. A simulation of the evolution of
Abrahamsenbreen during the late Holocene requires a plausible climatic
forcing. In the present model climate change is imposed by adjusting the
equilibrium-line altitude according to
E(t)=E0+E′(t).
The annual perturbation of the equilibrium-line altitude is denoted by
E′(t) and determined by annual temperature and precipitation anomalies,
denoted by T′ (in K) and P′ (in %) respectively. E′(t) is thus
written as
E′(t)=∂E∂TT′(t)+∂E∂PP′(t),
where the sensitivities ∂E/∂T and ∂E/∂P
are assumed to be constant. Sensitivities have been determined for
Nordenskiöldbreen with a detailed energy and mass-balance model (van Pelt
et al., 2012; Table 2), and here we use their values:
∂E∂T=35mK-1;∂E∂P=-2.25m%-1.
For many glaciers in a more alpine setting, values of ∂E/∂T
are of the order of 100 mK-1 (e.g. Oerlemans,
2011). The value for ∂E/∂T given in
Eq. (19) thus appears as rather small. This is due to the fact that in the
high Arctic summer temperature, anomalies which mainly determine the
sensitivity are much smaller than annual temperature anomalies. This has
been taken into account in the determination of the sensitivities.
The input data to calculate E′ are taken from van Pelt et al. (2013). In
this paper a climate reconstruction back to AD 1300 was made on the basis of
ice-core data from Lomonosovfonna as well as climate records from
Longyearbyen. For details the reader is referred to Divine et al. (2011) and
van Pelt et al. (2013). The temperature and precipitation anomalies, relative
to the period 1989–2010, are shown in Fig. 7, and the corresponding history
of the equilibrium-line altitude in Fig. 8. The most prominent feature in the
reconstructed temperature record is the LIA, lasting from
the late 16th century to the end of the 19th century, with long-term
temperatures typically 4 K below the medieval and present-day levels. The
reconstruction does not reveal a clear correlation between temperature and
precipitation anomalies. The variation of the equilibrium-line altitude is
substantial. During the period 1750–1850, the equilibrium line was about
200 m lower than in medieval times.
The value of E0 is optimized in such a way that the simulated maximum
glacier length in 1978 corresponds to the observed length. This yields E0=657 m. Note that E′ is defined with respect to the period 1989–2010,
implying that its mean value over the period 1300–2010 is not 0. In the
climate reconstruction used here, the value of E during the period
1989–2010 was 76 m larger than the long-term mean since AD 1300.
The simulated glacier length (Fig. 8) appears to be in good agreement with
geological evidence (distribution of moraines, strand lines and floodplains).
There is general agreement that Abrahamsenbreen reached a Holocene maximum
extent during the LIA, like most glaciers in northern Spitsbergen (e.g.
Forman et al., 2004; Salvigsen and Høgvard, 2005; Humlum et al., 2005).
According to our model, Abrahamsenbreen would have had a length of about
5 km in medieval times and started to grow in the 16th century until it
reached LIA size (between 18 and 22 km) in the second half of the 19th
century. For the calculation shown in Fig. 8, the
value of E has been kept constant to the 1989–2010 value for the period after 2010. This clearly
implies steady retreat, but the timescale at which this happens is large.
This is an implication of the very small bed slope and the related strong
altitude–mass-balance feedback (Oerlemans, 2011).
Figure 9 shows the mass inputs (in m3 of ice per year) of the
tributary basins and glaciers corresponding to the simulation shown in
Fig. 8. Although the inputs are highly correlated, there are large
differences in the absolute changes of mass input through time. The input
from tributary glaciers T4, T5 and T10 is sometimes 0. For T9 this happens
just a few times. The other basins always deliver some mass to the main
glacier, but the amounts can halve or double during high or low values of
E′.
We refer to the simulation just described as the reference simulation. The
model has been tuned in the best possible way given the limited amount of
observations. There appears to be no evident discrepancy between the
simulated glacier evolution and the geological evidence.
Climate history used to simulate the evolution of Abrahamsenbreen
during the late Holocene (Van Pelt et al., 2013). Shown are anomalies of the
annual mean temperature and the annual precipitation.
Glacier length (in black, scale at left) and equilibrium-line
altitude (in red, scale at right) from the simulation of Abrahamsenbreen. The
arrow indicates the 1978 surge of the glacier.
Mass input from the 10 tributary basins and glaciers for the
simulation shown in Fig. 8. The location of the basins is shown in Fig. 1.
The effect of surges on the evolution of Abrahamsenbreen. The
“reference” simulation is the same as in Fig. 8 (note that this simulation
has a 125-year surging period).
The effect of surging
The question of how surges interfere with the long-term response of glaciers
to climate change has been raised several times (Hagen et al., 2005; Paasche,
2010). Although the present model does not initiate surges by means of an
internal mechanism, it does include the main effect of a surge on the surface
mass budget of a glacier related to the reduction of the mean surface
elevation. Since a lower surface elevation implies a more negative mass
budget, one would expect that a regularly surging glacier would have a
smaller long-term mean glacier length.
With the present model set-up it is not possible to just switch off the
surging mechanism, because by virtue of Eq. (16) the model the glacier would be
in the quiescent phase continuously and the ice thickness would increase forever. However, a meaningful way to study the effect of surging is to vary the
duration of the surge cycle and see how this effects the long-term mean
glacier length. To make a fair comparison between runs with different surge
cycles, the constant C in Eq. (16) is adjusted in such a way that the mean
value of S(t) is equal to 1 over the integration period.
Figure 10 shows a comparison of runs with a longer (doubled, i.e. 250 years)
and shorter (halved, i.e. 62.5 years) surge cycle. The integrations have been
extended until AD 3000, with the equilibrium-line altitude equal to the mean
value over the period 1989–2010. This leads to a steady decay of the
glacier, implying that the current size of Abrahamsenbreen is far too large
for the climatic conditions that prevailed during the past few decades.
The effect of a different surge frequency is small until AD 1900 but much
more obvious afterwards. This is related to the fact that, with a glacier in
a state of decay, the mass-balance effect of surges works in the same
direction as the climatic forcing. Moreover, the glacier
surface extends into a region with anomalously high ablation rates. In contrast, for
a growing glacier the mass-balance effect of surges works against the
climatic imbalance and is therefore less visible.
Many more numerical experiments were carried out with different surge
parameters. An increased surge amplitude (larger value of Sa)
enhances the effect on the long-term glacier length because it implies a
larger drop of the mean surface elevation. When the surge takes longer
(larger value of ts) there is a similar effect.
The decay of the glacier after the year 2000 is a remarkable feature given
the relatively small climatic forcing. In the model simulation, after 1989
the equilibrium line is 76 m higher than for the period 1300–1989. The new
steady state length is about 4 km, but it takes 500 years to approach this
state.
The extreme climate sensitivity of Abrahamsenbreen is a consequence of the
small bed slope. Basic theory on the relation between E and L for a
schematic glacier geometry (constant glacier width) shows that a first-order
estimate of the sensitivity is given by (Oerlemans, 2001, 2012)
∂L∂E=-2s‾,
where s‾ is the mean bed slope. For the bed parameters used here,
a typical value of s‾ is 0.015, implying that ∂L/∂E≈-133. So a 50 m change in the equilibrium-line
altitude would imply a change in glacier length of 6.7 km. Oerlemans (2011)
also reveals that the sensitivity as defined by Eq. (20) is larger when the
accumulation zone is wider than the ablation zone. For Abrahamsenbreen this
implies that the value of 133 is probably a conservative estimate.
Sensitivity to bed elevation
The basic unknown parameters that can be adjusted to make the model produce
the correct glacier length and mean surface elevation are the bed elevation
parameter bh, the shape parameter αm and the
reference equilibrium-line altitude E0. We thus have three parameters and
two constraints, implying that a unique set of parameters cannot be found. In
Sect. 5 the problem was solved by assuming that the value of
αm is the same as for Kongsvegen. Although Kongsvegen is
also in a post-surge state and is located in a similar geological setting, it
is still possible that the value of αm for Abrahamsenbreen
differs significantly. Therefore some calculations were carried out with
perturbed values of αm, namely +20 and -20 %.
Changing the value of αm implies that a recalibration has to
be done by adjusting the values of bh and E0 to get the
correct glacier length in 1978 and the correct mean surface elevation. For a
20 % larger value of αm (2.72 m1/2) we found
bh=241 m (instead of 323 m) and E0=643 m (instead of
657 m). For a 20 % smaller value of αm
(1.82 m1/2) we found bh=412 m and E0=670 m.
Because the ice thickness is proportional to αm, it is not
surprising that the adjustments in bh are quite significant.
However, the required changes in the value of E0 are rather small.
The evolution of the glacier length for the three different tunings is shown
in Fig. 11. It is interesting to see that for the case with
αm=1.82m1/2 the surge in 1853 produces a
slightly longer glacier than in 1978. The differences among the three cases
are small for the period of glacier growth and significant for the period of
glacier retreat after 2000. This is a consequence of the fact that during the
period of glacier growth the glacier length was rather close to its
equilibrium value most of the time (because, irrespective of short-term
fluctuations, the equilibrium line drops gradually). After the year 2000, the
glacier is strongly out of balance for the imposed forcing, and the effect of
different ice thicknesses on the rate of retreat turns out to be more
pronounced.
In summary, we conclude that the simulated glacier evolution depends on the
choice of αm but not in a dramatic way. The characteristic
behaviour of Abrahamsenbreen for the imposed forcing is rather similar for
the three different tunings.
The effect of different values of αm on the
simulated glacier length. The “reference” simulation is shown in red.
Sensitivity to changes in the equilibrium-line altitude
By means of numerical modelling it has been shown that a glacier on an
isolated mountain bordered by a flat plane will grow to infinity if the
equilibrium line is lowered beyond a certain critical value (Oerlemans, 1981;
Fig. 10). This occurs because the feedback of the mean surface elevation on
the balance rate keeps the total mass budget positive. In the present model
the bed profile decays exponentially to a constant value (namely, sea level),
and the critical value of the equilibrium-line altitude described above is
very likely to be in the system. In the case of Abrahamsenbreen this would
imply that for a certain drop of the equilibrium line the glacier would grow
and grow until the front reaches the Woodfjord and only mass loss by calving
could stabilize the glacier at some point.
Critical (bifurcation) points in a dynamical system normally imply an
increasing sensitivity and response time when the critical point is
approached. Theoretically, when approaching the critical point the
sensitivity and response time go to infinity. The large response time
suggested by Fig. 10 actually suggests that Abrahamsenbreen may indeed be
close to the critical point. With these considerations in mind we have
carried out a set of integrations with different values of E′ after
AD 2010.
Figure 12 shows glacier length for different climatic perturbations, all
started with the calibrated glacier history until AD 2010. Clearly, for
E′=-160 m the glacier quickly comes in a state of runaway growth, and it
ultimately grows out of the model domain. For E′=-120 m the glacier
approaches a steady state, albeit very slowly. For further increasing values of
E′, steady states are approached more quickly. In fact, the curves in
Fig. 12 show that the response time decreases when the steady-state glacier
length is smaller. This is not a direct consequence of the glacier size but
is
related to the corresponding increase in the mean bed slope and the larger
distance (in parameter space) to the critical point.
The future of Abrahamsenbreen
It is very likely that the Arctic will be subject to further warming, which
will have a large impact on the glaciers of Svalbard. Reduced sea ice may
lead to higher precipitation rates, but it is questionable whether this could
stop the retreat of the glaciers. According to Eq. (19), a precipitation
increase of about 15 % per degree warming would be required to keep the
equilibrium line in place. A detailed analysis of the precipitation regime in
the Arctic with a comprehensive climate model suggests a sensitivity of
4.5 % increase per degree of temperature warming (Bintanja and Selten,
2014). Although this is
significantly more than the global value of 1.6–1.9 % increase per
degree, it is by no means sufficient to prevent the rise of the equilibrium
line when temperatures go up.
The future evolution of Abrahamsenbreen can be studied with the present
model, because it has been calibrated and no further assumptions are needed
to define an initial state. Nevertheless, one should be aware of the
schematic nature of the model and the limited data available for
Abrahamsenbreen, implying that the constraints are not very tight. The results
given below should therefore be considered as indicative of what is a
possible scenario rather than a prediction.
We have carried out a set of integrations until AD 2150, with an equilibrium
line that rises linearly in time, according to
E′(t)=μEt-2010.
Again, the anomaly is defined with respect to the period 1989–2010; time t
is in years AD. With the aid of Eq. (19), changes in equilibrium-line altitude
can be related to changes in temperature and precipitation. For instance,
μE=1myear-1 would correspond to a warming rate of
0.028 Kyear-1 or to a warming rate of 0.04 Kyear-1
combined with an increase in precipitation of 0.016 %year-1.
In all the integrations the surge period and amplitude have been kept
constant.
Simulated evolution of Abrahamsenbreen for some selected values of
the equilibrium-line altitude (relative to the mean value for the period
1989–2010).
Simulated evolution of Abrahamsenbreen in the case of a rise of the
equilibrium line of 1 myear-1. Glacier length in black (scale at
left). The curve labelled (a) shows the net balance over the entire
glacier system; the curve labelled (b) given the mass input from the
tributary glaciers, expressed as a mean balance rate.
Figure 13 shows the result for μE=1myear-1, which we
consider a typical value for the expected warming in the Arctic. In this
case the length of Abrahamsenbreen is predicted to be reduced to 12.7 km by
the year 2100. The corresponding reduction in volume is 66 % of the value
in 2010. The net balance rate for the entire system and the input from the
tributaries is also shown. By the year 2100 the input from the tributary
glaciers has been reduced to virtually 0, because they have a negative net
balance.
Discussion
In this paper we have applied a simple model to
study the climate sensitivity of Abrahamsenbreen. We have demonstrated that
even with a limited amount of information a meaningful calibration can be
carried out, and some conclusions can be drawn about the present state of
balance and the future of Abrahamsenbreen under conditions of climate change.
Although the main trunk of Abrahamsenbreen has a relatively simple geometry,
the total glacier system is complicated with many basins and tributary
glaciers providing mass to the central flow band. The parameterization we have
chosen to represent the geometry is effective and contains sufficient
information to quantify the overall mass budget. Admittedly, the assumption
that the tributaries are in a quasi-steady state is perhaps not always
satisfied when the climatic forcing changes rapidly. However, modelling a
glacier system like Abrahamsenbreen with a two-dimensional
(vertically integrated) or three-dimensional ice-flow model, and dealing
explicitly with the tributaries would be a complicated task requiring a large
amount of input data. We therefore believe that the method used here is
suitable to study the dynamics of complex glacier systems with many
tributaries.
We found that the effect of surges on the long-term size of the glacier is
significant but not dramatic. Since surges are imposed rather than
internally generated, only the impact of surges on the mass budget, by
lowering of the mean surface elevation, could be dealt with. On the basis of
our calculations (Fig. 11), we expect that in a warming Arctic surging
glaciers are prone to retreat somewhat faster than non-surging glaciers. It
is likely that the effect of surging is larger for glaciers with a larger
surge amplitude. However, when comparing the surge amplitude of
Abrahamsenbreen with that of some other glaciers in Svalbard (e.g. Skobreen,
Kongsvegen, Monacobreen, Nathorstbreen; Sund et al., 2009), it appears that
Abrahamsenbreen is quite typical. We therefore think that our results apply
to other surging glaciers as well.
It is encouraging that forcing the model with an independently derived
climate history leads to a glacier evolution that is in line with the geological
and geomorphological evidence. This certainly lends credibility to the
approach, and makes projections for the future more believable. If the
present climatic conditions persist, we predict that Abrahamsenbreen
will shrink considerably (to a length of about 4 km). In the case of future
warming of a few degree K, the glacier will ultimately disappear, but this
will take a few hundred years. Due to the fact that Abrahamsenbreen flows
into a valley with a very small bed slope, its sensitivity to climate change
is very large. Our calculations suggest that Abrahamsenbreen is rather close
to a critical point, marking the onset of a runaway situation in which the
glacier will grow into Woodfjorden for only a modest drop of the equilibrium
line (160 m). However, this would take a long time (a few thousand years,
Fig. 12).
The large sensitivity of Abrahamsenbreen is probably not an exception. Many
large glaciers on Spitsbergen have small slopes and are subject to similar
processes. An earlier modelling study of Hansbreen in southern Spitsbergen also
revealed a large sensitivity to climate change (Oerlemans et al., 2011). The
consequence of these findings is that a temperature increase of 1–2 K would
remove most of the ice from Spitsbergen, although it may take a long time
(hundreds to thousands of years). This is in line with the growing evidence
for an only marginally glaciated landscape in Spitsbergen during the Holocene
Climatic Optimum (e.g. Humlum et al., 2005).
Acknowledgements
Our study was only possible because the Norsk Polarinstitutt does
a great job in collecting and issuing maps and photographs through the years.
This work was carried out under the programme of the Netherlands Earth System Science Centre (NESSC).
Edited by: A. Vieli
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