The PDD runoff–retention model determines the PDD sum from monthly air temperatures assuming a variability of daily near-surface (2 m) temperatures around the monthly mean. Melt rates are proportional to this melt potential. In snow-covered regions the meltwater from surface melting is initially stored as capillary water within the snowpack, until the snowpack becomes saturated, typically when melt reaches around 60 % of the annual precipitation (Janssens and Huybrechts, 2000) and runoff occurs. The formation of superimposed ice occurs when water-saturated snow survives above the impermeable ice layer until the end of the season and subsequently refreezes (Janssens and Huybrechts, 2000).

The daily variability in temperatures is expressed as a standard deviation (σ) around the monthly mean. A value of 3 ∘C is used, lower than the widely used value of 4.2 ∘C for the whole of Greenland (e.g. Fürst et al., 2015) but consistent with 1994–1995 observations of temperatures, SMB and refreezing for six locations on Hare glacier (Reeh et al., 2001). The melt rates are determined with separate degree-day factors for snow and ice, which are respectively equal to 0.0027 and 0.0065 m ice equivalent/degree day based on detailed stake observations on the ice cap (Braithwaite et al., 1995; Reeh, 1995).

The PDD sum is calculated from a parameterization of the TMA and the mean July temperature (TMJ), assuming a sinusoidal annual march. A temperature parameterization is preferred over lapse-rate-corrected RCM temperatures to remove the bias from the present-day imprint of the ice cap on its own temperature field in a different geometric setting. Furthermore the temperature parameterization is flexible and allows for a direct application at different resolutions, without the need for complex downscaling methods (needed for RCM data). The TMJ has the largest influence on the SMB as this field largely determines the amount of summer melt, while the TMA sets the amplitude of the annual sinusoidal signal and thereby determines the temperatures of the other seasons when little to no melt occurs. The rain–snow temperature threshold is set at 1 ∘C. TMA and TMJ are parameterized for the mass balance year 1994/95 based on detailed field measurements (Reeh, 1995; Reeh et al., 2001), and this parameterization is subsequently extended to the period 1961–1990 by comparing both periods in the regional climate model RACMO2.3 (with ERA-40 and ERA-Interim boundary conditions): the 1994–1995 TMA was 0.53 ∘C higher than the 1961–1990 average, while the 1995 TMJ was about 0.27 ∘C lower than the 1961–1990 average. The temperatures are parameterized as a function of latitude and elevation, and they can therefore be determined at any model resolution.

The measured temperature lapse rate in July is -0.0056 ∘Cm-1 (Reeh et al., 2001), and as a latitudinal gradient we adopt the value of -0.1681 ∘C∘N-1 from Fausto et al. (2009). Altogether this results in the following parameterization for the 1961–1990 mean July temperature (in ∘C) (see Fig. 3a): TMJ=19.47-0.1681×LAT-0.0056×ELEV, where LAT is the latitude (∘) and ELEV the elevation (m). Notice that given the limited latitudinal range of the ice cap (0.8∘) the influence of the latitudinal gradient is limited. The parameterized 1961–1990 TMJ agrees well with output from RACMO2.3 for the period 1961–1990 (Fig. 3b), with mean July temperatures around 5.5–6 ∘C at sea level and around -1 to -1.5 ∘C at the domes (1200–1300 m a.s.l.).

The mean annual temperature has not been measured on the ice cap directly. The latitudinal and elevation gradients are therefore also taken from Fausto et al. (2009). A parameterization is derived to fit with measurements from the nearby meteorological station Kap Harald Moltke (Reeh et al., 2001). For areas lower than 300 m, no elevation gradient is applied in order to represent the temperature inversion that occurs here following Reeh et al. (2001). For 1961–1990 the TMA (in ∘C) is parameterized as follows (see Fig. 3c): forELEV<300m:TMA=45.07-0.734×LAT;forELEV>300m:TMA=46.97-0.734×LAT-0.00638×ELEV. With this parameterization the 1961–1990 TMA at sea level is around -15 ∘C, while at north dome this is -22.2 ∘C and at the central dome -21.7 ∘C. The TMA at the domes agree well with the local measurements of 10 m depth temperature, which are respectively -21.0 and -20.8 ∘C (Reeh, 1995; Reeh et al., 2001). The slight difference between parameterized and observed values may be linked to firn warming due to refreezing, although this process is generally limited at those high elevations. Furthermore the parameterized TMA is in fairly good agreement with RACMO2.3 output for this period (see Fig. 3d).

The accumulation has been derived from field measurements and four shallow cores that cover most of the 20th century (Reeh et al., 2001). The accumulation is the highest in the north(-west), which is related to the proximity to the ocean, acting as a moisture source (see Fig. 3e). Due to a topographic shielding the precipitation is much lower in the central part. In the south(east) the precipitation is also very low, but generally a bit higher than in the centre. Based on accumulation measurements from the shallow cores Reeh et al. (2001) tried to derive a parameterization to describe this pattern as a function of latitude and elevation, but this resulted in an unrealistic field with very low (in some cases even negative) precipitation at low elevation. We therefore opt to use the precipitation field from the regional climate model simulation RACMO2.3, which is run at 11 km and was bilinearly downscaled to a 1 km resolution (see Fig. 3e for 1961–1990 average field) (Noël et al., 2016). For all simulations this precipitation field is further interpolated to the model resolution. This regional climate model is able to reproduce the observed precipitation patterns closely as is shown by a comparison with accumulation from four sites for the period 1975–1995 (Table 1). Field observations and model output (both from our SMB model and from RACMO2.3) indicate that for the four high-elevation sites the solid precipitation is nearly equal to the accumulation (annual rain fraction varies between 0 and 5 %).

The steady-state ice cap obtained from the 250 m HO run (Fig. 5) is close in extent to the observed ice cap in 1995 (Fig. 2). A few discrepancies exist in the south-western part, where the steady-state ice cap is somewhat smaller in extent, and for a few outlet glaciers in the central northern part, which are a bit shorter than in the observations. Notice that the latter outlet glaciers are very thin (Starzer and Reeh, 2001), and satellite-derived surface velocities indicate that these areas are almost stagnant (Joughin et al., 2010, 2015). In the south-eastern and north-western part the modelled steady-state ice cap extends slightly further than the observations, and some of the present-day ice-free ridges between the outlet glaciers are ice-covered. Except for this, the agreement is overall relatively good, especially given that there is no imposed constraint on the ice cap extent.

The modelled limited areal changes under the 1961–1990 average conditions are supported by the RCM output that indicates a near-zero average integrated SMB over this period (see Sect. 4.2). Furthermore the limited geometrical changes under the 1961–1990 climatic conditions are also in line with field evidence. While a maximum extent was reached around 1900, which is known from Little Ice Age (LIA) moraines (Koch, 1928, 1940), and a slight retreat occurred in the first part of the 20th century (Davies and Krinsley, 1962), aerial photography from the 1970s and 1990s (Weidick, 2001) suggests that the second part of the 20th century is characterized by a slower recession (limited to tens of metres), standstill or even slight readvances. A recent study by Kjeldsen et al. (2015), where aerial photography and SMB modelling are combined, also suggests limited mass changes in northern Greenland for the period 1900–1983 and an ice sheet near balance during the 1970s and 1980s.

The measured temperature profiles at central dome and at Hare glacier (Reeh, 1995; Thomsen et al., 1996) are used to tune the geothermal heat flux component and the heating component related to infiltrating meltwater in the ablation area (Fig. 6). To reproduce the observed englacial temperatures at the central dome (Reeh, 1995) (see Fig. 6c), a geothermal heat flux of 45 mWm-2 is applied. Here the modelled steady-state ice thickness (318 m) is close to the observed one (345 m). The measured almost-linear decrease in temperature, from -21 ∘C at 10 m depth (-21.7 ∘C in our model, corresponding to the TMA at the surface) to -16 ∘C, is closely reproduced. A geothermal heat flux of 45 mWm-2 is lower than the 60 mWm-2 interpolated from Shapiro and Ritzwoller (2002) to the location of Hans Tausen Iskappe. However with 60 mWm-2 the modelled basal temperature is equal to -13.8 ∘C, and the local ice thickness is 295 m. For the ablation area an additional basal-water heat flux of 150 mWm-2 is adopted to reproduce the englacial temperatures measured in the ablation area of Hare glacier (Reeh, 1995). With this additional basal heating the observed temperatures, ranging from -18.5 ∘C (at 10 m depth) to about -1.5 ∘C at the bottom (see Fig. 6), are well reproduced. Despite the fact that the modelled ice thickness (269 m) is close to the measured one (289 m), a direct comparison is difficult to make as the imposed surface temperature (-15.3 ∘C, corresponding to the local TMA) is slightly higher than the observed one. The modelled basal temperatures for Hare glacier are close to the pressure melting point (see Fig. 6b), but nowhere does basal sliding occurs. The pressure melting point is only reached for a few larger outlet glaciers and only very locally (see Fig. 6a), and the modelled contribution of basal sliding is therefore very limited. Our value of 150 mWm-2 differs significantly from the the 350 mWm-2 found by Wohlleben et al. (2009), which is expected given the very different setting (location, SMB, meltwater production and infiltration mechanisms) and the different methodological approach. Here we tune based on an evolving/modelled geometry, while Wohlleben et al. (2009) model the thermodynamics for a fixed geometry. With a value of 350 mWm-2 almost the entire ablation area of the ice cap would be at the pressure melting point, and basal sliding would have an important role, which is not supported by the field evidence. Notice that on the other hand, without any additional basal heating component for the ablation area, the basal temperatures would be severely underestimated (e.g. -10.2 ∘C at the base of the Hare glacier drill site), which would also strongly affect the ice cap geometry.

The 1961–1990 modelled steady-state geometry is generally in good agreement with the observed geometry. The observed ice thickness (Fig. 2b) is well reproduced in the model (Fig. 5), and so is the surface elevation (as the observed bedrock elevation is used in the model). The root mean square error (RMSE) between the observed and modelled ice thickness (and surface elevation) is 55.6 m. For the interior of the ice cap the regions with high ice thickness are generally well reproduced, despite some differences in the north, where the ice cap is generally thicker in the model. The steady-state outlet glaciers agree reasonably well with observations (RMSE of 67.1 m), but some modelled outlet glaciers, especially in the north, have a tendency to be slightly thicker. This difference is partly linked to the complexity of ice flow in the outlet glaciers, which may not be fully captured by the model. The ice flow in the outlet glaciers is strongly influenced by thermodynamics as the ice temperature determines the stiffness, through the rate factor, and potentially also through basal sliding. Without the additional heat source in the ablation area, which was needed to reproduce the observed temperatures, the modelled outlet glaciers are thicker, and the discrepancy between observations and model substantially increases (RMSE of 74.7 m for the ablation area). Notice that, given the limited amount of direct ice thickness measurements in the outlet glaciers (Starzer and Reeh, 2001), part of the model–observation discrepancy may be related to local errors in the bedrock DEM.

The surface velocity patterns derived from InSAR data (Joughin et al., 2010, 2015) are well reproduced in the modelled steady-state ice cap (Fig. 7). The low velocities in the interior and the ice flow direction are very similar, which indicates that the modelled position of the ice divides corresponds well with the observations. For the outlet glaciers, many of the observed velocity patterns are closely reproduced. This is for instance illustrated for the main outlet glaciers at the eastern side of the ice cap (Fig. 7c, d), for which the modelled geometry is in relatively good agreement with the observations (see Figs. 2b and 5).

The main differences between the modelled and observed ice velocities occur along the south-western edge of the ice cap, where the geometrical differences are the largest, and for two high-velocity floating tongues, which we do not model explicitly as this is of limited importance to the larger-scale dynamics of the ice cap. For a few outlet glaciers in the north the modelled surface velocities are slightly higher than the observed ones, which can partly be linked to the differences in geometry. Notice that, as the surface velocities and the modelled geometry are related, the surface velocity discrepancy may be a consequence of the geometry discrepancy. The inverse may however also be true: i.e. the surface velocity discrepancy is the cause for the geometry discrepancy.

Given its long response time, the ice cap in 1995 is not expected to be in steady state with the 1961–1990 conditions. Our model simulations however suggest that the ice cap changes only little under these conditions, and this is supported by field and SMB modelling evidence, but in reality the ice cap cannot have been in a full dynamic equilibrium with these climatic conditions. In order to calibrate (for thermodynamics) and validate (geometry, surface velocities) our model we however need to rely on a steady-state geometry. A part of the described differences between observations and modelling is therefore not only linked to the model errors and uncertainties in the input data but may also be attributed to the fact that the observed ice cap was not in steady state in 1995. This is particularly the case for the outlet glaciers, which are the most dynamic parts of the ice cap and where the model–observation discrepancies in geometry and surface velocities are the largest. In order to fully investigate the transient behaviour of the ice cap, which would be needed to reproduce the recently observed changes and to make accurate projections for the near future (coming decades), simulations encompassing the long-term ice cap evolution are needed. This is however not the focus of this study as we are aiming to understand the large-scale dynamics, response time and climatic sensitivity of this ice cap, and their implications for its long-term evolution.

As the previous experiments point out, the ice cap is very sensitive to a change in climatic conditions. For a cooling of only 0.5 ∘C compared to the 1961–1990 conditions the ice cap strongly expands (21 % area increase) (see Fig. 9b) and the volume increases by 26 %. These are the coldest conditions for which the ice cap can be modelled explicitly, as for lower temperatures the ice cap starts to connect to the GrIS and other nearby ice masses and expands beyond the domain boundaries.

Under the 1981–2010 average climatic conditions (ca. +0.6 ∘C vs. 1961–1990, 4 % higher precipitation than the 1961–1990 mean) the SMB of the southern part of the present-day ice cap is still above the collapse threshold (cf. case 3; see Fig. 9a). While the northern part of the ice cap and the local domes change little, an overall slight decrease in surface elevation occurs at lower elevations, and a frontal retreat of the southern part of the ice cap occurs, but then the ice cap quickly stabilizes. This agrees with observations from airborne surveys that indicate that between 1994 and 2004 limited changes in surface elevation occurred around the central dome (Dalå et al., 2005). About one-fifth of the ice mass is lost under these conditions (Fig. 11a). Also the output from the RCMs, MAR3.5.2 (-0.15 mw.e.a-1) and RACMO2.3 (-0.11 mw.e.a-1), and from our PDD melt–retention approach (-0.08 mw.e.a-1 based on the GIMP topography) suggests a limited negative SMB over the ice cap for this period.

For slightly warmer conditions the collapse threshold of the southern part of the ice cap is crossed (cf. case 4, Fig. 9a), and eventually, after thousands of years, the entire southern part of the ice cap is lost. This is for instance the case under the 2005–2014 climatic conditions, which are around 1.6 ∘C warmer than the 1961–1990 average conditions over Hans Tausen Iskappe (6 % higher precipitation than 1961–1990 mean) and for which the specific SMB of the present-day ice cap is very negative (-0.39, -0.32 and -0.32 mw.e.a-1 in MAR3.5.2, RACMO2.3 and PDD melt–retention approach respectively). At first the elevation changes at the southern domes are limited (local SMB is almost at the 1961–1990 level), but as a large mass loss occurs at low elevations the geometry adapts, and the rate of ice loss subsequently increases as a result of the SMB–elevation feedback (Fig. 11b) until after 1500 years all ice has disappeared. Due to this feedback the volume response time of the ice cap is very short and only amounts to 616 years. The northern part of the ice cap also changes (see Fig. 9e), but the domes remain stable (Fig. 11b), and most of the remaining ice mass, corresponding to 19 % of the initial mass, is stored here. This is in line with elevation changes derived from ICESat for the period 2003–2008 (Bolch et al., 2013), which indicate that the domes are stable during this time period and even tend to slightly gain mass (typically elevation change up to 0.2 ma-1), while the lowest regions are losing mass at a high rate (typically more than 0.5 ma-1). A more in-depth comparison between these observations and our model results is difficult given our initial steady-state assumption and the role of the ice cap response time, but the large-scale features are in reasonable agreement.

For a high-emission scenario (IPCC RCP8.5) the 2100 global average surface temperature is projected to rise by 3–5 ∘C compared to the 1961–1990 average. Over high Arctic regions such as Peary Land the temperature could potentially increase by up to 7–11 ∘C due to the polar amplification (Collins et al., 2013). This warming is most pronounced in winter, and summer temperatures (June–July–August) are projected to rise up to 8 ∘C over northern Greenland by 2100 (vs. 1961–1990) (van Oldenborgh et al., 2013). To simulate the evolution of the ice cap in a warming climate, we consider a +4 ∘C warming, which broadly represents an intermediate-emission scenario (in the line of RCP4.5), and a +8 ∘C warming, representing a high-emission scenario (cf. RCP8.5) (both vs. 1961–1990). For the +4 ∘C scenario and by maintaining the precipitation at the 1961–1990 level, the ice cap entirely disappears within 350–400 years (i.e. before 2400) (Fig. 11a), disregarding whether the forcing is immediately applied at present or incrementally until 2100. Under the high-emission scenario (+8 ∘C) it takes about 140 years for all ice to be gone or 180 years when the forcing is applied linearly; i.e. the last ice disappears in the second half of the 22nd century (Fig. 11a). Different model simulations indicate that under such warm conditions the large-scale ice cap evolution is not much affected by its initial state, whether starting from the observed geometry, the 1961–1990 steady-state geometry or a somewhat similar geometry. Modelling the transient evolution of the ice cap over the last centuries to millennia is therefore of relatively limited interest when it comes to simulating the future mid- to long-term ice cap evolution in a (much) warmer climate as the evolution is almost fully driven by the SMB rather than by the ice dynamics.

Precipitation changes influence the SMB and have the potential to (partly) attenuate the ice loss in the case of warming. In order to prevent the modelled 20 % total mass loss under 1981–2010 conditions, the precipitation has to increase by about 25 %: i.e. under these conditions the integrated SMB of the present-day ice cap is again close to 0 mw.e.a-1 (see Fig. 11a). For 2005–2014 conditions a considerably higher precipitation increase, around 75 %, is needed for the present-day 1961–1990 steady-state ice cap volume to be maintained, while for the intermediate future warming (+4 ∘C vs. 1961–1990) the precipitation needs a dramatic increase, by about around 340 % (Fig. 11a). Based on five simulations (1961–1990 +0/+1/+2/+3/+4 ∘C) this non-linear relationship is approximated as a 2nd-order polynomial (Fig. 11c): P=0.132ΔT2+0.316ΔT+1, where ΔT is the temperature forcing and P the corresponding precipitation forcing (scaling factor) (both vs. 1961–1990) needed to prevent a mass loss (vs. 1961–1990 steady state).

A future increase in precipitation over the ice cap is projected as the surrounding ocean is to become ice-free in a warmer climate and to act as an important moisture source (e.g. Braithwaite, 2005). The 20th-century conditions are at the borderline between ice-free fjords and fjords with semi-permanent ice cover (Weidick, 2001) and from palaeorecords it is known that in the warmer mid-Holocene period, when the Arctic ocean was seasonally ice-free, the precipitation was up to twice as high as at present (Madsen and Thorsteinsson, 2001). Based on this the future precipitation increase could be well above the one that would follow from a Clausius–Clapeyron relationship.

It is therefore expected that for a moderately warmer climate, up to 2–2.5 ∘C warmer than the 1961–1990 conditions, the ice loss as a result of a temperature increase may be partly attenuated (cf. Machguth et al., 2013). Under climatic conditions needed to preserve the steady-state volume (e.g. 1981–2010 (P+25 %) and 2005–2014 (P+75 %)) the ice cap total SMB would change only little compared to 1961–1990, but the SMB spatial distribution is more affected. Whereas the temperature increase mainly decreases the SMB in the present-day lower areas, the temperature increase in the higher areas leads to higher precipitation and a higher SMB. As a result the steady-state ice cap margin retreats (smaller steady-state area), while the interior thickens, resulting in a steeper ice cap (Fig. 11d). This is in agreement with recent ICESat observations on Arctic ice caps, which indicate a marginal ice loss and local thickening (for the interior). This is the case for Hans Tausen Iskappe (Bolch et al., 2013), and for instance also for the Austfonna ice cap (Svalbard) (Moholdt et al., 2010) and the Flade Isblink ice cap (Greenland) (Rinne et al., 2011; Bolch et al., 2013). An in-depth comparison between our modelling study and these observations is again not possible given the differences in timing and the model setup, but our simulations show the potential to reproduce the observed trends and the implications this can have on the future ice cap evolution.

For even warmer conditions (> 3 ∘C) the required precipitation increase of more than 200 % needed to counteract the mass loss is much higher than expected from climate modelling and palaeoclimatic records, and as a consequence the ice cap will in all cases (largely) disappear. This is for instance clear from the 1961–1990 (+4 ∘C, P+100 %) simulation, where only very little ice survives, corresponding to around 2.5–3 % of the present-day volume (see Fig. 11a). For the high-emission scenario (+8 ∘C vs. 1961–1990), all ice disappears, and even under an extreme precipitation increase of for instance 200 % it would only take 10–15 years longer for all ice to be gone compared to a constant-precipitation scenario.