In view of the observed strong warming of the polar regions, the future evolution of arctic ice masses is of great concern. Glacier retreat has a huge impact on regional ecosystems, and it contributes significantly to sea-level rise. The large ice sheets of Greenland and Antarctica are losing mass (IPCC, 2013; Van den Broeke et al., 2016; Martín-Moreno et al., 2017), and glaciers have retreated far behind their Little Ice Age (LIA) maximum stands almost everywhere (Leclercq et al., 2014; Zemp et al., 2015). Even the large tidewater glaciers of the Arctic region have retreated over large distances (kilometres) during the past 100 years. Although tidewater and surging glaciers have a strong internal component to their dynamics, it has become clear that climatic forcing will now determine to a large extent their future evolution.

Different approaches have been taken to model the future behaviour of glaciers. Individual glaciers have been studied with flowline models based on approximate formulations of the mechanics of ice flow (e.g. Kruss, 1984; Huybrechts et al., 1989; Oerlemans, 1997; Nick et al., 2007). In more recent years, three-dimensional higher-order models have been employed, for instance for the Rhône glacier (Jouvet et al., 2009) and for the Morteratsch glacier (Zekkolari et al., 2013). Smaller ice caps are also being studied with such models (e.g. Zekkolari et al., 2017; Schäfer et al., 2015).

The projection of future glacier changes depends strongly on the definition of the initial state used to start the integration. Glaciers have response times of decades to centuries, implying that a reliable projection of future evolution can only be achieved when changes over a past timespan can first of all be reproduced (termed “dynamic calibration” in Oerlemans, 1997). This timespan should have a duration of at least the response time, preferably longer. In Oerlemans et al. (1998) this approach was applied to a sample of 12 glaciers, and a rather simple attempt was made to apply the results to the entire glacier population on the globe.

Along a different line, models have been constructed to treat the entire glacier population on the globe (e.g. Radić et al., 2014; Huss and Hock, 2015). In these models the focus is on estimating global glacier mass changes as a response to warming and its effect on sea-level rise. Processes like accumulation, melting, calving, and dynamic adjustments of glacier geometries are treated in a schematic way. The models have a large number of empirical parameters and are tuned by comparison with observed mass balance data over the past decades. Clearly, when questions of a global nature have to be dealt with, global models are needed in which not every single glacier can be treated in detail. Nevertheless, it seems that in studies of this kind historical data on glacier fluctuations over the past centuries have not yet been fully exploited.

The modelling strategies discussed above have their advantages as well as their limitations. The three-dimensional models require a large amount of geometric input data, and the issue of boundary conditions in steep terrain has not yet been resolved in a satisfactory way (dynamic simulation of steep ice-free slopes and mass conservation). There appears to be room for an intermediate approach with simpler models for individual glaciers that still take into account a number of essential feedback mechanisms (altitude–mass balance feedback, effect of overdeepenings, variable calving rates, effect of regular surging on the long-term mass budget, inclusion of tributary glaciers and basins).

Minimal glacier models (MGMs, Oerlemans, 2011) offer the possibility to model individual glaciers in a relatively simple way, while still dealing with mechanisms like altitude–mass balance feedback, the effect of overdeepenings in the bed, variable calving rates, the effect of regular surging on the long-term mass budget, etc. MGMs require limited computational resources and can therefore be used efficiently in control methods where many integrations have to be carried out. MGMs have been applied to McCall glacier in Alaska, (Oerlemans, 2011), to Hansbreen, a calving glacier in southern Spitsbergen (Oerlemans et al., 2011), and to Abrahamsenbreen, a surging glacier in northern Spitsbergen (Oerlemans and Van Pelt, 2015).

In this paper, a MGM is applied to Monacobreen, a 40 km long glacier in northern Spitsbergen that flows from the plateau Isachsenfonna (∼1000 m a.s.l.) into the Liefdefjord (glacier coordinates 79∘23′ N, 12∘38′ E; see the interactive topographic thematic map operated by the Norsk Polarinstitutt: http://toposvalbard.npolar.no, last access: 14 September 2018). The glacier drains an area of about 400 km2, and the main stream has an average slope of only 0.027. Modelling Monacobreen is difficult, because it is a surge-type calving glacier with many tributary glaciers and basins. Some information is available on former positions of the glacier front (Fig. 1). From 1991 to 1997 the glacier surged, leading to a 2 km advance of the glacier front (Mansell et al., 2012). Røthe et al. (2015) made a reconstruction of the equilibrium-line altitude (ELA) for the Holocene, derived from a thorough analysis of lake sediments in the catchment of the small glacier Karlbreen in north-west Spitsbergen. Karlbreen is located only about 25 km to the west of Monacobreen, and the ELA reconstruction thus offers a unique possibility to simulate the evolution of Monacobreen through Holocene times.

Monacobreen, with a distinct calving main stream and a large number of tributaries, represents a glacier system that is typical for Spitsbergen. The complexity of the geometry as well as the limited amount of data make it a real challenge for a modelling study. Nevertheless, the question of how the mass of such glacier systems will change in the near future has to be considered, and the approach taken in this study is an attempt to do this in a meaningful way. A MGM provides a reasonable match between the paucity of data and an integrated mass budget approach, in which glacier mechanics are parameterized in a simple way. The larger glacier systems on Svalbard presumably have long response times, because they have small slopes and are subject to relatively dry conditions. The strategy of using observations on former glacier stands for calibration before integrating the model into the future is tested in this paper. It is envisaged that the methodology can be applied to other complex glacier systems in Svalbard and the Arctic.

With respect to Monacobreen, the following more specific questions will be addressed: (i) is it possible to simulate the broad characteristics of the late Holocene evolution of Monacobreen?; (ii) to what extent does regular surging affect the mass budget and long-term evolution of the glacier?; and (iii) what is the likely range of mass loss in the coming centuries for different scenarios of climate change?

Monacobreen is considered to be a stream (flowband) of length L and constant width W, to which 10 tributary glaciers and basins supply mass if the equilibrium line is sufficiently low (Fig. 2). The length is measured along the x-axis, which follows the centerline of the flowband. With a width of 5 km and a length of 40 km, the area of the main stream is about 200 km2. Currently the area of the tributary basins is 191 km2 (not including Seligerbreen), and since these basins have a higher mean surface elevation than the main stream, they make a major contribution to the total mass budget. The number of basins that actually feed the main stream depends on L.

In the following sections a number of parameterizations are introduced concerning the global ice mechanics, geometry, calving, and climate forcing. An overview of the parameters and their values (including the sources) is given in Table 1.

According to the map in Hagen et al. (1993), the estimated equilibrium-line altitude in the region of Monacobreen is 500 to 600 m a.s.l., with a tendency to lower values when going in northwesterly direction. The estimate is based on an assumption of balance, i.e. a situation in which the glaciers would be in equilibrium. This is currently not the case, and the equilibrium line over the past decades has certainly been higher. In the model the value of E is taken as the same for the entire domain, except for basins 1, 2, and 3, where E is perturbed by -100, -100, and -50 m, respectively. This is done to take the decline of the ELA as mapped in Hagen et al. (1993) into account. In fact, without the lowering of the ELA the net mass budget of basin 1 (Seligerbreen) would be negative, and the tributary could never supply mass to Monacobreen (as it did until recently). Altogether, the ELA map of Hagen et al. (1993) appears to be consistent with the mass budgets of the tributaries.

The evolution of Monacobreen for stepwise forcing of the equilibrium line is shown in Fig. 5. As can be expected in view of the very small slope, it turns out that the climate sensitivity of the glacier is very large. Changing E from 775 to 575 m makes the glacier grow by 24 km. Bringing back the equilibrium line to 600 m (in two steps) does not reveal a sign of hysteresis (as would probably occur when there would be an overdeepening in the bed). The adjustment to a changing climate apparently takes a long time; the e-folding response time is about 250 a.

The corresponding components of the mass budget are shown in Fig. 5. To facilitate a comparison, all components are expressed as a specific balance. The important role played by the tributary glaciers and basins becomes clear immediately. Only at t=500 CE, when E drops instantaneously by 200 m, does the main stream have a positive balance. This does not last very long, however, because the increasing glacier length soon leads to a much larger ablation zone. When the glacier length is of the order of 40 km, as it is today, the negative net surface balance of the main stream and the calving flux contribute roughly equally to the compensation of the net input from the tributaries.

The timescale of about 250 years can be compared with an estimate of the much used volume timescale τJ defined by Jóhannesson et al. (1989): τJ=-H∗/b˙x=L, where H∗ is the maximum ice thickness, and b˙x=L is the balance rate at the glacier front. Using 350 m as a maximum ice thickness and -2.5ma-1 as a typical balance rate yields a value of about 140 a. However, as demonstrated by Oerlemans (2001) and confirmed by Leysinger Vieli and Gudmundsson (2004) with a higher-order numerical glacier model, for glaciers with small slopes the altitude–mass balance feedback makes the timescale considerably longer. Since Monacobreen has a very small average slope (0.027), the value of ∼250 a is a plausible one.

A simulation of the Holocene evolution of Monacobreen requires an appropriate climatic forcing function. Fortunately, an ELA history for a small valley glacier on Mitrahalvøya, which is only 25 km west of the central part of Monacobreen, has been reconstructed from lake sediments (Røthe et al., 2014). This glacier, Karlbreen, currently has an area of about 2km2, and the glacier head is at about 500 m a.s.l. This is an important height because it puts a lower limit on the ELA in times when there was no glacial activity (in this case the ELA has to be above the mountain top). The glacier drains through a series of three lakes, and an extensive sediment analysis has provided a full history of glacial activity. By using former glacier stands, Røthe et al. (2014) were able to convert sediment parameters to ELA values, assuming an equilibrium between glacier size and ELA. According to Røthe et al. (2014), there was very little glacial activity during the period of 9200 to 3500 BCE, implying that the equilibrium line was mostly above 500 m a.s.l. After the Holocene climatic optimum there has been a long-term trend towards lower values of the ELA, with significant fluctuations superposed on this. Since a glacier like Karlbreen probably has a response timescale of the order of 20 to 30 a, the reconstruction is bound to be less accurate for shorter timescales. Therefore the reconstruction for Karlbreen is combined with ELAs estimated from meteorological observations at Svalbard Airport (Longyearbyen) since 1899 (Førland et al., 2011).

There is a distinct west–east gradient in the ELA in northwest Spitsbergen (Hagen et al., 1993). The ELA rises in the eastward direction mainly due to lower precipitation rates. Therefore, the absolute reconstructed ELA values cannot be used to force the model for Monacobreen, so the ELA perturbation ER relative to a reference value E0 is used. The ELA is now formulated as fort<1899:E(t)=E0+ER(t),fort≥1899:E(t)=E1+EM(t). Here EM(t) is the ELA perturbation obtained from the meteorological data. The constants E0 and E1 need not necessarily be the same because it is unknown how precisely the meteorological observations connect to the past. These constants will therefore be used as tuning parameters.

The relation between the ELA and temperature and precipitation is based on calculations with an energy balance model, as described in Van Pelt et al. (2012) and used in Oerlemans and Van Pelt (2015). The sensitivities are ∂E/∂T=35mK-1 and ∂E/∂P=-2.25m%-1, where T and P are perturbations of the annual mean temperature and precipitation. It should be noted that the value of ∂E/∂T is rather small compared to values found for glaciers in a midlatitude alpine setting, which are of the order of 100mK-1. This stems from the fact that summer temperature anomalies over Spitsbergen (and in general over the Arctic region) are much smaller than mean annual temperature anomalies. Since summer temperature determines to a large extent the ELA perturbation, the net effect is that the sensitivity to an annual temperature anomaly is relatively small (for a further discussion on this point, see Van Pelt et al., 2012).

As shown in Fig. 6a, the variation of reconstructed ELA values from mid-Holocene times until today have a typical range of 200 m. If this would solely be a temperature effect, the drop in ELA since the mid-Holocene would correspond to a 5.7 K decrease in temperature (according to the sensitivity referred to above). This is more than the reconstructions of mid-Holocene warmth in the Arctic actually suggest, which are in the 2 to 4 K range (e.g. CAPE, 2006; Axford et al., 2017). However, there is also a direct effect of changes in orbitally driven insolation variations. The differences in summer insolation between mid-Holocene and present day are between 5 % and 10 %, depending on the precise location and definition of the summer season (Berger and Loutre, 1991). The increased insolation certainly caused higher melt rates in the mid-Holocene, and thereby a higher equilibrium line.

The minimal glacier model constructed in the previous sections has been forced with the ELA reconstruction discussed above. Information on former stands of Monacobreen is limited, but the existing data points can nevertheless be used to calibrate and test the model. For a successful simulation, in which the simulated glacier length matches the observed record, two conditions have to be fulfilled: (i) the MGM should have sufficient dynamics to respond in a realistic way to climate forcing, and (ii) the forcing function (the ELA reconstruction) is accurate enough to make the glacier grow and shrink at the right times. It is not at all trivial, even with optimal values of E0 and E1, that the maximum glacier stand comes at the right time.

In Sect. 5.1 the best possible simulation will be discussed (the reference simulation). In Sect. 5.2, sensitivity tests are described in which parameters are varied, or processes are switched off.

Recent trends and future climate change in Svalbard has been examined in detail by Fjørland et al. (2011). There is broad agreement between temperature trends obtained by downscaling results from global climate models (forced with greenhouse gas emissions) and observations at Svalbard Airport for the period 1912–2010. Projections for the Svalbard region indicate a future warming rate up to year 2100 that is three times larger than the observed rate during the past 100 years. For precipitation, the long-term observational series show a modest increase over the past 100 years, and projections indicate a further increase up to year 2100. In the present study the projected changes in temperature and precipitation for northwest Spitsbergen are used to estimate the future trend in the ELA.

For the B2 emission scenario (IPCC, 2013), the rate at which annual temperature and precipitation increases in the ensemble–mean projection is fairly linear until about 2090 and levels off somewhat afterwards (Fjørland et al., 2011). Combining the mid-B2 annual temperature and precipitation trends with the ELA sensitivities discussed in Sect. 3 yields dE/dt=1.86ma-1. Against this reference the possible future evolution of Monacobreen was studied by imposing values of 0, 2, 4, and 6 ma-1 for dE/dt. It should be noted that these values have also been used for glaciers in other parts of the world (e.g. in the Alps; Oerlemans et al., 2017). Although warming is larger in the polar regions (the “polar amplification”), this is not so much reflected in the rise of the equilibrium line. The reason is that summer temperatures increase much less than annual mean temperatures. In the calculation of the ELA sensitivities this has been taken into account. However, since the climate change scenarios are formulated as a change in the ELA, the possibility remains to convert meteorological variables into changes in the ELA for different sensitivities.

When making projections of future climate change scenarios, the outcome depends on the choice of the reference period. Starting from a warm year (e.g. 2015, ELA = 809 m) and increasing the ELA by a certain amount will give a very different result from starting in a cold year (e.g. 2014, ELA 668 m). Therefore the reference ELA should be a mean value over a longer period. Moreover, it is unclear to what extent the very high ELA values since 2000 represent an expression of natural variability on the decadal timescale or whether they are a direct response to greenhouse-gas-induced warming. To deal with this uncertainty, two 30 a reference periods were used to define the ELA perturbation associated with the projected climate change: (i) 1987–2016, i.e. a recent 30 a period; and (ii) 1961–1990 as the last official period to define the climatology. The resulting eight projections of glacier length are shown in Fig. 10a. The integrations are extended until 2200, and the ELA perturbation is kept fixed after 2100. The curves immediately make it clear that typically half of the response to 21st century warming will come after 2100.

For reference period (ii) and no climate change (dE/dt=0), Monacobreen would hardly retreat. Note that this scenario actually implies that ELA values return to significantly lower values than observed over the last two decades. In contrast, for reference period (i) and no climate change, in the year 2100 the glacier front would have retreated by 3.5 km.

For the strong-warming scenario with dE/dt=4ma-1 and reference period (i), the model predicts a glacier retreat of about 9 km by the year 2100. At this point the glacier would be grossly out of balance, since the ELA is then at 1038 m a.s.l., which is above most of the accumulation basin. In 2100 the volume of Monacobreen would be about 60 % of the present-day volume (Fig. 10b). For the extreme-warming scenario dE/dt=6ma-1 and reference period (i), Monacobreen would have virtually disappeared within two centuries.

If the Paris Agreement becomes reality, the mid-B2 scenario with dE/dt=2ma-1 is perhaps the most likely one. In this case the front of Monacobreen would retreat by 3.5 to 5.5 km in the coming 80 years, depending on the choice of reference period. The glacier volume would have been reduced by 20 % to 30 % with respect to the current volume.

In this paper it has been demonstrated that the conceptual simplicity of the MGM makes it possible to study the response of a large tidewater glacier to climate change in a transparent way. Calibration turned out to be a straightforward procedure, in which three characteristic glacier front positions could be reproduced accurately (LIA maximum stand, front position at the end of the recent surge, and present-day front position). Here the quality of the reconstructed ELA history from Røthe at al. (2014) certainly plays an important role. The approach of defining a main stream that is fed by tributary glaciers and basins seems to work well. The tributary basins were included in a passive way. In fact, the basins were assumed to have a fixed geometry and to be in balance with the prevailing ELA. This should work well as long as the characteristic timescales of the basins are shorter than those of the main stream (perhaps for Louetbreen this is not quite the case). In principle all individual basins can be modelled with a MGM as well, implying the introduction of a response time and the local height–mass balance feedback. However, for the present study this was not considered appropriate given the little information available for these basins.

A significant result from this study is that surging has a limited effect on the long-term evolution of Monacobreen. The relative surge amplitude As, loosely defined as the maximum advance divided by the glacier length, is quite small (≈0.05). As determines to a large extent by how much the mean surface elevation drops. The combined effect of a lower surface elevation and a larger area determines the mass budget perturbation implied by the surge. It was found that for Monacobreen, this perturbation amounts to -0.13mw.e.a-1 (when expressed as a change in the net balance rate of the main stream, see Fig. 4). It is interesting to compare these numbers with those of Abrahamsenbreen (Oerlemans and Van Pelt, 2015). For Abrahamsenbreen the surge amplitude is 0.14, and the perturbation of the mass budget is about -0.5mw.e.a-1.

It should be noted that the perturbation of the mass budget is solely determined by the redistribution of mass and not by the details of how this distribution actually takes place. When a glacier increases its length during a surge, the change in mean surface elevation is entirely dictated by the conservation of mass and not by the details of the surging mechanism. This implies that conclusions about the effect of surging on the long-term mass budget can be drawn even when the surging process itself is not fully understood.

Calving has a significant effect on the total mass budget of Monacobreen, but different values of the calving parameter do not change the qualitative evolution of the glacier during the Holocene very much. The range over which Monacobreen fluctuates is somewhat smaller for a larger calving parameter (the green curve in Fig. 9). This is understandable for a bed profile that slopes downward along the flowline, because the front of a growing glacier comes into deeper water and the mass loss by calving increases.

It has been observed that on shorter timescales details of the bathymetry may have significant effects on the calving rate and thereby on the position of a tidewater glacier front (e.g. Vieli et al., 2002). According to the measured bathymetry in the Liefdefjorden, these variations, with an amplitude of 10 to 50 m, are irregularly spaced and consist mainly of deposited moraines. It is unlikely that a similar bed would currently be present under Monacobreen with its very smooth surface, or have existed in the fjord before the glacier started to advance in late Holocene times. Therefore it does not seem meaningful to include a map of the present-day bathymetry of the Liefdefjorden in one way or another. Probably, the smaller features of the bed profile do not matter too much for the glacier evolution on longer timescales, unless there are very marked jumps in bed or side geometry that could serve a pinning points. However, this does not seem to be the case.

The MGM has very simple dynamics: there is no spatial resolution and the mechanics are contained in a relation between length, mean ice thickness, and mean bed slope, with simple formulations of the calving and surging processes. These formulations do not shed further light on the nature of the calving and surging processes, but the effects on the mass budget, and thus on the long-term evolution of the glacier, are dealt with. At this point one may ask whether it would be possible to study the Monacobreen glacier system with a three-dimensional higher-order model and make a comparison with the simple approach taken here. Modelling the main stream would probably be feasible, but to deal with all the tributaries would require a large amount of data on the bed geometry, which are not available and thus would have to be generated indirectly. Generating the surge cycle in a higher-order model, involving a coupling of ice mechanics and glacier hydrology, is another major difficulty to deal with. Perhaps at the present state of the art, it would be more realistic to strive for a hybrid model, in which a comprehensive model for the main stream is combined with MGMs for the tributary glaciers and basins. However, on longer timescales glacier mechanics are always slaved by the mass budget. The results from the recalibrated runs with different parameter values for balance gradient, surge amplitude, and calving parameters support this view. It thus seems likely that a model with a comprehensive treatment of ice mechanics, when driven with the same climatic forcing and calibrated with the observed glacier stands, will produce a similar evolution of Monacobreen during the late Holocene.