Since the last decade of the 20th century, the Greenland ice sheet (GrIS) has lost mass with accelerating speed e. g., shaping one of the most important contributors to sea level rise . This mass loss is not only driven by decreasing surface mass balance (SMB), but also by increasing ice discharge via outlet glaciers. The partition between these two contributions to GrIS mass loss is about equal . Understanding the processes determining the GrIS ice loss is vital for estimates of its contribution to future sea level rise.

Nowadays, the scientific community recognizes Greenland as a complex system mainly composed of the ice sheet and numerous outlet glaciers , in subtle interaction with the surrounding ocean via fjord circulation , and uprising meltwater plumes in an interplay with the calving outlet glaciers . In our paper, we introduce the model IGLOO (Ice sheet model for Greenland including Ocean and Outlet glaciers, Fig. ) intended to represent the major processes important for the future mass changes of the GrIS on the timescale of a few centuries. The contribution to future sea level rise of the several glaciers and ice caps detached from the GrIS is small compared to the ice sheet and its attached outlet glaciers . However, these glacial bodies are taken into account by SICOPOLIS in our approach. The idea of IGLOO is to capture the complexity of the system by its involved model components and, at the same time, to make the description of all single components as detailed as necessary . We aim to have a tool with sufficient computational efficiency to enable large ensemble simulations on timescales important for future climate change.

Knowledge of the present-day state of the GrIS has been improving considerably. Not only are there reliable data from numerous observations e.g., but several modelling studies also exist. Present-day GrIS velocities are resolved by ice sheet models in horizontal resolutions as high as 600 m, including flow patterns of outlet glaciers . explicitly demonstrated the multistable-hysteresis behaviour of the GrIS with a threshold of 1.6 ∘C above present-day global temperature for the decay of the GrIS; although such a decay will last at least about 1000 years. The past climate is an important element for GrIS ice sheet modelling as well, as it serves as a constraint for parameters particularly capturing the present-day GrIS , and as it provides the history of the temperature field inside the present-day GrIS , which is important for the initialization of the GrIS in future warming simulations. However, palaeo-simulations with a free surface have the drawback that their resulting present-day ice thickness can differ considerably from observations e. g.. Such a simulated ice thickness is an unfavourable initial condition for projections because, in this case, the future simulation would start with ice which resides at the wrong locations or is absent at positions where it should reside according to observations. This leads to an erroneous drift in projected ice volume evolution. Therefore, we opt for a fixed domain approach in our palaeo-spin-ups or, more precisely, for a scheme that relaxes the simulated surface elevation towards the observed one . This approach has the advantage that it provides a good approximation of the present-day temperature–velocity field for initialization and at the same time prevents a spurious response in volume during future simulations of several hundred years. Different initialization methods are discussed by .

There are several approaches for projecting future ice mass change of the GrIS, often with a special focus on a certain component of the Greenland glacial system. Classical SMB approaches assume a passive ice sheet, but resolve the atmosphere with general circulation models of the atmosphere e.g. or additionally with a regional model . Several pioneering studies used three-dimensional dynamic ice sheet models in the shallow ice approximation (SIA) for projections of GrIS sea level contribution e.g.. Later, higher-order or even full-Stokes ice dynamics were included for GrIS future projections. In a higher-order ice sheet model, parameterizes ice sliding via ocean-temperature rise, due to future climate change, to investigate the impact of ocean warming on future projections of GrIS sea level contribution. Studies with an atmosphere–ocean general circulation model coupled to a SIA ice sheet model via surface-energy fluxes were undertaken by . Inspections of GrIS sea level contribution with a special focus on outlet glaciers were accomplished with a 3-D ice sheet model by or with a 1-D shallow shelf model .

Here, we opt for the new version of SICOPOLIS v3.3 . This version includes hybrid dynamics, incorporating longitudinal and lateral stresses via the shelfy-stream approximation SStA; longitudinal and lateral stresses, which are important for nearer-margin fast flow areas, along with horizontal plane shear via the shallow ice approximation (SIA), important for the slow-flow regions in the more central regions of the ice sheet. Hybrid models have been developed before by , , , , , and . They are a compromise between the shallow ice approximation and the full-Stokes approach. Key to these hybrid models is that SIA and SStA operate on a common domain, although there are other approaches to treat longitudinal and lateral stresses . Compared to the SIA, the hybrid dynamics is more promising for reproducing the velocity field of the GrIS in the catchment area of ice streams, where there is already fast flow . We do not employ ice-shelf dynamics in SICOPOLIS because the dynamics of outlet glaciers, which can have a floating ice tongue, are part of the outlet glacier component of IGLOO. We investigate the response of GrIS outlet glaciers to global warming (including ocean warming) with IGLOO in a separate paper .

Models assuming a basal water layer for treatment of subglacial hydrology were often applied to the Antarctic ice sheet . Here, we apply such a model to the GrIS, because it captures the major pathways of basal water toward the outlet glaciers , i.e. the model resolves in a good approximation the partition of basal water for the main GrIS outlet glaciers. This is important for reproducing the subglacial discharge of outlet glaciers, which is fed into our model of meltwater plumes. Further, our model for basal hydrology simulates a thickening of the basal water layer toward the major GrIS outlet glaciers, regions over which the ice velocity becomes higher . Therefore, we couple the ice velocities to the basal water layer, while the basal melt rate of the ice sheet model provides the input to the model of basal hydrology. We expect this approach to be suitable for large-scale modelling of ice sheets on decadal timescales.

Simulating submarine melt rates at tidewater glaciers has been accomplished with different models that all share the core of the buoyant plume theory . Recent studies show that the line plume model by is an adequate tool to determine submarine melt rates for tidewater glaciers. In our paper, we apply the recently developed line plume model by , based on the equations by , to two outlet glaciers, the Store and Helheim glaciers (Fig. ), of the Greenland ice sheet. We have chosen the Helheim and Store glaciers for investigating the impact of future warming on glacier melt and for testing our methods because they are well examined glaciers. Numerous studies on these glaciers and their connecting fjord systems to the open ocean exist . Some provide data on temperature and salinity profiles inside the fjord from conductivity–temperature–depth (CTD) measurements or moorings.

We start with a description of the elements of the glacial system model IGLOO 1.0, including the future and past forcings utilized in our paper (Sect. ). In Sect. , we describe our initialization method, while Sect.  compares the simulated present-day surface elevation and velocity with observations. Further on, modelled basal properties are compared with findings of other works. In Sect. , we present projections of the GrIS sea level contribution, the GrIS total basal and surface runoff and the submarine melt rates for two GrIS outlet glaciers (Store and Helheim glaciers). The paper closes with a discussion (Sect. ) and conclusions (Sect. ).

For the initialization of the ice sheet model, we use the forcings for the surface temperature and the SMB as described in Sect. . Using a standard setting of SICOPOLIS for palaeo runs, isostatic depression and rebound of the lithosphere due to changing ice load is modelled assuming a local lithosphere with relaxing asthenosphere with an isostatic time lag LLRA approach,. For the geothermal heat, we use the spatial dependent data by . In order to cover one full glacial cycle, we run the model over 135 kyrs. Initial conditions of these runs are the present-day ice thickness and elevation by . The original data with 1 km horizontal resolution are downsampled to 5 and 10 km grid spacings. To perform a simulation in 5 km horizontal resolution over the entire glacial cycle with the hybrid model is illusive, as it takes 1 day for 100 model years on one HLRS2015 Lenovo NeXtScale nx360M5 processor. Therefore, we opt to perform the first 130 kyrs of the glacial cycle in 10 km horizontal resolution with the classical shallow ice approximation (SIA) employing the diffusivity method with an over-implicit ice-thickness solver. The last 5 kyr of the palaeo-run are performed in 5 km horizontal resolution. As we use different model hierarchies and settings, we devote more of our explanation to these last 5 kyr.

During the last 5 kyrs of the run, we have three switches: one for refining the horizontal resolution, one for switching from SIA mode to hybrid mode, and a further one for switching from relaxing ice surface to free ice surface. The first switch at 5 kyr BP refines the horizontal resolution of the model from 10 to 5 km. The second switch at 500 years BP changes from SIA to hybrid mode and additionally applies the mass conservation scheme for the evolution equation of ice thickness (Eq. ). The third switch, which releases the relaxing ice surface to free development, is imposed at 100 years BP (year 1900). We introduced the three switches at different times because they represent different regime changes.

We chose the time of the resolution switch furthest back in time (5 kyr BP) to allow the ice temperature to adapt to the refined basal topography. Indeed, by comparing the simulated present-day basal temperature in 10 km resolution with that in 5 km resolution from our simulation with the switches, one can observe that the 5 km basal temperature shows a much finer structure of the temperate basal regions at locations of outlet glaciers and their catchment areas; see also Fig. S1c and d in . As these temperate regions determine areas with basal sliding, enabling fast flow important for ice dynamics, we regard a time of 5 kyr as sufficient for resolving the major characteristics of the ice temperature field in the 5 km resolution.

Another important aspect for the choice of an adequate timing of the switches is the rate of sea level change, which quantifies the model drift and shows directly how the model recovers from the transition shock. We demonstrate this by inspecting different switching times and comparing our favourite initialization run with a test simulation (Fig. ). While our initialization run has the switch from SIA to hybrid at the year 1500 and the switch from relaxing surface to free surface at the year 1900, the test simulation has both of these switches at the year 1900. Both curves of the rate of sea level change first rise fast to a maximum and then drop slowly towards smaller values. For the test simulation, the maximum is larger (0.06 cm yr-1) compared to that of our initialization run (0.05 cm yr-1). More importantly, the test simulation has not enough time to recover. While the rate of sea level change of our initialization run at the year 2000 amounts only to -0.0007 cm yr-1, it is much larger (0.03 cm yr-1) for the test simulation. In future projections, a drift of 0.03 cm yr-1 would yield a non-negligible contribution to modelled sea level rise of 3 cm in 100 years. Therefore, our initialization with separate switches for regime changes from SIA to hybrid and from relaxed to free surface is much more favourable and thus used for our projections (Sect. ).

The choice of the relaxation constant rests on numerous simulations in 10 km horizontal resolution in SIA mode, running the model over one glacial cycle until the present day. Figure  shows the root mean square error (RMSE) in surface elevation and the total difference in SMB (the total implied SMB over the GrIS minus the total SMB simulated by MAR). With increasing relaxation constant, the RMSE in surface elevation increases moderately, while the total difference in SMB decreases strongly, i.e. there is a tradeoff between the RMSE in elevation and the total difference in SMB.

Figure  shows the spatial differences between the observed and modelled surface elevation and SMB for different relaxation constants. Again, the tradeoff for representing both surface elevation and SMB is visible. While the simulated elevation is very close to the observation for small relaxation constants, the SMB deviation is very high, even in the interior of the ice sheet, where the deviations reach the amount of magnitude of the accumulation rate. Therefore, too small relaxation constants should be excluded. For larger relaxation constants, both difference fields become smoother, but rather high deviations in surface elevation appear over vast areas of the GrIS. Therefore, too high relaxation constants should be excluded too. The spatial differences in Fig. b, e are from the medium relaxation constant of τrelax=100 years. One can see that the simulated surface elevation is too low over the north-western and south-eastern marginal regions (Fig. b). Over these regions, the simulated velocities are too high, which is compensated by an increased implied SMB (Fig. e) compared to the SMB from MAR. For most regions of the Northeast Greenland Ice Stream (NEGIS), the opposite situation occurs. The simulated surface elevation is too high, while the implied SMB is mostly negative over that region, which compensates the too low simulated velocities; see also Fig. . Again, the choice of the relaxation constant τrelax is a tradeoff between the errors in the surface elevation and the errors in the implied SMB.

Here, we present our optimal simulation of the GrIS using the SICOPOLIS model version 3.3 with hybrid dynamics and the model for basal hydrology (HYDRO). Both models are fully coupled (see Sect. ), and the horizontal resolution is always 5 km from now on. In the hybrid mode, a threshold of rthr=0 applies to the slip ratio (Eq. 8 in ), i. e., the SStA equations are solved over the entire ice sheet, and the ice velocities are the weighted sum from the SIA and SStA velocities with the slip ratio as weight. The boundary conditions and initialization method to yield the present-day GrIS are described in Sects.  and , respectively. As relaxation constant for the surface elevation we use τrelax=100 years. Optimal values for the sliding parameters are found by minimizing the error of simulated horizontal surface velocities for values >50 m yr-1, using observations by . For such velocities, we expect basal sliding and hybrid ice dynamics to be relevant. We found Cb=25 m Pa-1 yr-1 to be optimal for the hybrid model with basal hydrology.

By design of the initialization, the simulated surface elevation compares overall well with the observed one; see Fig. a, b. However, as our surface relaxation method leaves the ice sheet's surface a certain degree of freedom (see also Fig. ), the simulated ice surface over Summit and South Dome as well as on the ridge in between them is slightly lower. The simulated surface velocities (<2 m yr-1) over the slow flow regions in the vicinity of the ridges are somewhat smaller compared to the observed surface velocities. Such (small) mismatches also appear with other higher-order models, even in higher resolution . As we adjusted the sliding parameter Cb to match velocities higher than 50 m yr-1 with observations, it is obvious that the model cannot resolve every detail over the slow flow regions. Still, the model resolves the major flow patterns of the GrIS, including the flow over the catchment area of the outlet glaciers and the fast flow of the major outlet glaciers and ice streams. Only the smaller-scale outlet glaciers, e.g. in north-west Greenland, are not fully resolved. As we have excluded ice shelves in SICOPOLIS, we cannot reproduce outlet glaciers with floating tongues, such as Petermann, Nioghalvfjerdsbræ and Zachariæ Isström. The NEGIS is the only larger scale feature which we cannot reproduce properly. This feature cannot be simulated without additional assumptions (see the Discussion section).

Fast flow mainly appears over regions with a temperate ice bed. The simulated basal temperature in Fig. a shows a pattern which agrees basically with the reconstruction by . Regions where there is basal melt, mainly caused by basal friction, exhibit a 1 to 5 mm thick water layer (Fig. b). There is a pronounced thickening of the water layer with our Shreve-flow modelling toward major ice streams and outlet glaciers, which is most visible for NEGIS, Jakobshavn Isbræ and Helheim Glacier. Moreover, smaller outlet glaciers like Store Glacier and Daugaard–Jensen Glacier receive intensified basal water supply too. The red dots over central-east Greenland correspond to sinks in the hydrological potential. These sinks are interpreted as subglacial lakes, following who reported similar results. In our simulations, these sinks can appear over a frozen base too, because we operate HYDRO with an option that allows computation of the hydrological potential over the entire ice area. This has the advantage that all basal water can safely reach the ocean. Allowing a water layer over a frozen bed for such sinks in the hydrological potential affects ice dynamics only very little because fully developed sliding appears only over temperate basal areas, while sub-melt sliding decays rapidly with decreasing temperature (Sect. ).

Figure  zooms in Jakobshavn Isbræ and the two major outlet glaciers Helheim and Kangerlussuaq. Here, the ability of the model to resolve the catchment areas of these outlet glaciers in a 50 to 500 m yr-1 range can be seen in more detail. However, the high-velocity patterns near the glacier termini do not fully match the simulations. In particular, the tributaries of Helheim and Kangerlussuaq glaciers and the tip of Jakobshavn Isbræ appear rather smooth compared to the observation.

In Sect. , we investigated the role of the relaxation constant for initialization. For very small relaxation constants, i.e. an essentially fixed ice surface, the difference between implied and observed SMB at present day becomes very large (more than 2000 Gt yr-1, compared to an insignificant amount for τrelax=100 years). Note that the present-day magnitude of observed total SMB is only about 500 Gt yr-1 e .g.. This means that computations with fully fixed surface should be treated with care, as the required total implied SMB is very high. A further advantage of the relaxation of the ice surface is that this smoothes the ice surface because it has a certain degree of freedom due to the relaxation constant while solving the ice thickness equation. A similar smoothing effect when running an ice sheet model with free surface evolution over 100 years was already observed by . Furthermore, they demonstrated that a smooth ice surface avoids irregular variations in the vertical velocity field.

For our initialization method, we made a number of simplifications. We ignored changes in surface elevation and spatial extent of the GrIS during the past glacial cycle. However, the elevation changes during the last glacial cycle over central parts of Greenland like Summit about 100 m, or Dye 3 some 100 m, were rather small compared to the ice thickness. These rather small differences in elevation limit the inaccuracies in the ice temperature field caused by our relaxation approach, also because the maxima of these changes appeared about 10 000 years ago during the Holocene. Furthermore, due to slowly flowing ice there, a slightly different temperature–velocity field will not play an important role in the overall ice dynamic. Also, ignoring the larger areal extension at the Last Glacial Maximum (LGM) should not affect the ice temperature field too strongly because over the more peripheral regions of the ice sheet the velocities are rather high and the LGM is 21 000 years ago. Here, the ice temperature will adapt faster and has more time to adapt compared to the central regions of the ice sheet. The assumption that the derived present-day elevation correction is valid over the entire glacial cycle is reasonable. We have examined this by evaluating the temporal dependence of the vertical temperature gradient from the MAR data. Indeed, for the future, this temperature gradient does not change too strongly in time, which does not mean that the elevation correction itself can be neglected here. Regardless, for the palaeo-run, our ice surface is rather constant in time due to our relaxation approach, which makes the elevation correction small in this case. As a further simplification, we applied the temperature anomaly from the GRIP record to the entire GrIS. This is a standard approach; however, we are aware that this anomaly is spatially dependent in reality. Possible errors in this anomaly will become less and less important the nearer the palaeo-simulation approaches the present day. That our present-day temperature–velocity field is reasonable is supported by the fact that it lies in the range found by . In this context, we would like to repeat that our initialization serves the purpose of having the simulated surface elevation in the present day as close as possible to the observed one to minimize the drift in the future projections.

In our simulations, we cannot reproduce the NEGIS correctly. Certainly, one reason is that we do not optimize the surface velocity by a spatially dependent basal sliding coefficient. With spatially dependent basal sliding coefficients, other studies such as and more recently simulated the NEGIS in better agreement with observations. Nowadays, there are process-oriented approaches to capture effects important for basal sliding. For example, stronger basal melting at the onset of the NEGIS caused by increased geothermal heat due to a palaeo-hotspot could be a factor speeding up the simulated NEGIS velocity.

For our 300-year sea level projections, which reach beyond the 21st century, we prolong the forcing data of the MAR model until the year 2300. Because we merely held the forcing constant between 2101 and 2300, the real RCP 8.5 forcing could be larger, i.e. we expect our simulations with the RCP 8.5 scenario to be a lower estimate of sea level contribution of the GrIS, i.e. the estimate is a rather conservative one. Most certainly, even all our projections including RCP 4.5 are a conservative estimate because a full coupling with ice–ocean interactions is missing in our model yet, and found that ocean warming caused additional mass loss of the GrIS in his projections by applying a parameterization of ocean warming.

Our additional sea level rise for the year 2100 due to elevation SMB feedback is somewhat higher than that by , who used the regional model MAR actively coupled to an ice sheet model for their simulations. For the year 2100, give an additional contribution to sea level rise of the GrIS due to this feedback, ranging between 0.25 and 0.32 cm under the SRES A1B scenario; after all, we excluded their outlier of 0.1 cm. Still, their estimates of this feedback are rather small compared to our estimates lying between 0.6 and 1.7 cm, which we produced with the RCP 8.5 scenario. The SRES A1B scenario is somewhat more moderate that the RCP 8.5 scenario. However, this cannot explain such low numbers for the elevation SMB feedback. As demonstrated to be of importance by with fully interactive two-way coupling, this feedback deserves a detailed inspection in the future.

Our presented projections for the GrIS contribution to global sea level rise in the 21st century (1.9–13.0 cm) are consistent with previous publications. However, they do not account for the dynamic response of Greenland outlet glaciers to ocean warming and increase of subglacial discharge. This effect will be accounted for in a forthcoming paper. We also intend to couple the 3-D ice sheet model SICOPOLIS with the 1-D model for many outlet glaciers.

We introduced the coupled Greenland glacial system model IGLOO 1.0, designed to describe the most important parts of the Greenland glacial system: the ice sheet, the subglacial hydrological system, the outlet glaciers and the ice–ocean interaction in the Greenland fjords. The applicability of the hybrid mode of the ice sheet model SICOPOLIS 3.3 to the Greenland ice sheet was demonstrated. Full coupling between the ice sheet model and the model of subglacial water HYDRO has been accomplished, while the coupling between HYDRO and the meltwater plume only works offline so far.

As initialization, we used a relaxation method similar to , but with a higher relaxation constant of 100 years. For this choice of the relaxation constant, we varied it systematically and investigated the resulting model behaviour by inspecting the RMSE in surface elevation as well as the difference between total simulated SMB and total SMB from the MAR regional climate model. It showed that, for a relaxation constant of 100 years, the relative error of our simulated total SMB is only 2 % when using the total MAR SMB as reference, while – at the same time – the root mean square of the simulated error in surface elevation stays with a value of 63 m reasonably small. Additionally, we showed that medium-value relaxation times lead to smooth 2-D fields of the implied SMB, while for small relaxation times the fields become noisier, and for large relaxation times regional deviations of the simulated elevation from the observed one become relatively large (RMSE of 95 m for τrelax=300 years; see Fig. a).

Furthermore, we performed projections of the contribution of the GrIS to sea level rise until the year 2300 with hybrid ice dynamics forced by SMB anomalies from the MAR regional model. For the RCP 4.5 and 8.5 scenarios generated by MAR, three CMIP5 GCMs with different climate sensitivity were applied. Altogether, our projected GrIS sea level contribution for the year 2100, obtained with elevation SMB feedback, ranges from 1.9 to 13.0 cm, and for the year 2300 from 3.5 to 76.4 cm. The effect of elevation SMB feedback contributes clearly to our simulated contribution of the GrIS to sea level rise. Generally, its impact increases in the long run with decreasing surface elevation (see Table ).

Moreover, we demonstrated the importance of the different factors determining the increase of the melt rate of Greenland outlet glaciers under the extreme RCP 8.5 scenario, using Store and Helheim Glaciers as examples. It showed that the knowledge of near-terminus temperature and subglacial discharge in the fjord are both about equally important to determine the future melt of these two outlet glaciers. This underlines the importance of our approach with the Greenland system model IGLOO 1.0.