In the case of glaciers, the Reynolds number is very low e.g., and therefore the inertial term in the Navier–Stokes equations can be neglected. This approximation is often referred to as Stokes flow, and the dynamics of the ice flow component is solved with the full-Stokes (FS) equation. The core of the FS equation system builds on the mass balance and momentum equation for incompressible ice and is written as 1divv=0,2divt=ϱig, with the density of ice ϱi=917 kgm-3, the three-dimensional velocity field v=(vx,vy,vz) in Cartesian coordinates, the gravitational acceleration vector pointing downward g=-gez (g=9.81 ms-2), and the Cauchy stress tensor t. We split the Cauchy stress into a deviatoric part tD and an isotropic pressure p 3t=tD+pI, with p=-13tr(t) and I the identity tensor. The constitutive equation for the non-Newtonian fluid links the stress tensor to strain rates (i.e., velocity gradients). 4tD=2ηD, where D is the strain rate tensor (D=12gradv+(gradv)T). The viscosity is given by the Glen-Steinemanns flow law 5η=12Bε˙e(1-n)/n, with the flow law exponent n=3, the ice hardness B, and the effective strain rate ε˙e being the second invariant of the strain-rate tensor. The ice hardness parameter usually covers the temperature dependence of ice deformation and occasional effects like softening due to crevasses. Here, B is inferred by an inversion approach (see Sect. ).

The glacier base is subject to basal sliding according to a friction law for the basal shear stress τb in the tangential plane 6τb=-β2vb,7v⋅n=0, with the unit normal vector n pointing out of the ice, the basal drag parameter β2, and vb is the velocity in the tangential plane at the base. Basal refreezing or melting is neglected. The basal drag parameter for both glaciers is inferred by an inversion approach (see Sect. ). The boundary conditions at the ice base are enforced by Nitsche's method weak implementation, which reveals better convergence and smoother results than the strong implementation for both glaciers,.

The glacier surface, zs(x,y,t), is updated at every time step through the free surface equation : 8∂zs∂t=vs⋅n+as, where as=as(x,y,t) is the surface mass balance. Assuming an impenetrable glacier bed, zb, the surface zs must meet the condition: zs-Hmin>zb. If zs falls below zb+Hmin the surface height is constrained to zb+Hmin. The minimum ice thickness, Hmin, is set to 5 m and ensures numerical stability. We assume that the ice base, zb, is stationary.

The horizontal extension of the glacier is calculated with a level-set method LSM,, i.e. the glacier front advances horizontally with the ice velocity (no frontal/ice-cliff melt is considered). However, a thickness constraint of Hmin = 5 m deactivates/activates elements that fall below/exceed this threshold.

Equation () needs a reliable SMB at the glacier surface. Here, we rely on an energy balance model that computes the glacier melt. Snow accumulation, Ps, is calculated from total precipitation, P. The total precipitation is separated into liquid precipitation (rainfall) and solid precipitation (snowfall): Above a temperature threshold of 2 °C precipitation is assumed to be liquid, below 0 °C precipitation is completely solid. We used a smooth cosine interpolation to retrieve the precipitation fraction from solid to liquid between the temperature thresholds of 0 and 2 °C.

For ice melt, we employ the surface energy balance model (EBM) by . The EBM was originally designed for debris-covered glaciers as it accounts for energy fluxes in a dry porous debris layer on the glacier surface. Both glaciers considered here are mostly debris-free, and therefore the EBM reduces to a clean-ice scheme, where a modulation of the surface heat flux due to a porous supraglacial surface layer is absent. For an exhaustive description, the reader is referred to . The EBM is forced with the daily near-surface temperature, incoming short- and longwave downward radiation, near-surface windspeed and near-surface specific humidity time series taken from the selected EURO-CORDEX RCMs and the ISIMIP3b GCMs (see Sect. ). The SMB scheme is run on a daily timestep, but the ice flow model (IFM) is forced with the yearly SMB, as we are interested in the long-term response and not the seasonal variations of the glaciers.

A crucial ingredient for calculating the surface mass balance is the surface albedo α. Here, we use a simple parametrization to calculate the effective albedo which follows the idea of to make the albedo dependent on the equilibrium line altitude (ELA). 9α=αice+(αsnow-αice)(1+tanh⁡(π(zs-ELA+Δ)/δ))/2, where αice is the albedo of bare ice, αsnow is the albedo of snow, Δ a tuneable value, and ELA is the spatially mean ELA of the glacier. The parameter δ ensures a smooth transition from snow to bare ice albedo. Note that the unknown albedo parameters will be tuned to resemble a realistic surface mass balance (SMB) profile over HEF and GAG (see Sect. ). The albedo is updated every year. We also compared our albedo scheme with a more sophisticated albedo parameterization that includes a snow thickness , a temperature-dependent or age-dependent snow albedo , but found that the added value is too small given the increase in model complexity and computational cost.

We remain with the same model parameters as in Table 1, but climate variables are replaced with the daily time series of the GCM/RCM inputs, and we apply the albedo parametrization (Eq. ). Note that the thickness of the debris is zero.

Numerical solutions of the described ice flow model and related modules are obtained using ISSM. The governing equations are unstable if discretized using the Galerkin finite element method. For the FS equations (Eq. ), we use condensed MINI elements to fulfil the compatibility Ladyzhenskaya-Babuška-Brezzi (LBB) condition . The computations are run in parallel using the iterative GMRES solver preconditioned using the Additive Schwartz Method with an overlap of 1.

The free surface equation (Eq. ) is hyperbolic and stabilization is achieved by streamline upwind Petrov–Galerkin diffusion SUPG,. We use triangular Lagrange P1 elements (piecewise linear). Similarly, the LSM method is stabilized with SUPG and P1 elements are employed. The reinitialization frequency is set to the ISSM default value of 5. Both equations are solved with GMRES preconditioned with Block Jacobi. However, the numerical handling of the LSM method is studied in detail by .

The mesh is generated from a regular 2D triangular grid with an edge length of 25 m, and is vertically extruded into five equally spaced layers. This results in a mesh with ≈1.11 million elements for GAG; ≈0.15 million elements for HEF. We ensure that each time step complies with the Courant-Friedrichs-Levy condition for numerical stability. Having expected maximum ice velocities of about 200 m a⁻¹ at GAG and 40 m a⁻¹ at HEF we chose a time step of 0.05 and 0.25 years for the masstransport model for the transient simulations, respectively.

Due to the different mesh sizes, the computational demand of the two glaciers is vastly different. For HEF, the future climate runs over 103 simulation years require ≈1.5 h each on 128 cores (2 MPI tasks with each 64 cores). By contrast, a GAG future climate run over 90 simulation years requires ≈ 20 h on 672 cores (7 MPI tasks with each 96 cores). One CPU consists of 2xAMD EPYC 7702 64-Core Processor with 2.0 GHz.

The simulations presented make use of observations for initialization to resemble a certain known state of the glacier. This approach requires the contemporaneity of the products to maintain that, e.g. the ice surface velocity is consistent with the ice geometry. For both glaciers, the required datasets are available (see Table ) and are bi-linear interpolated on the 25 m triangular mesh. The initialization year for GAG is 2011; for HEF it is 1998.

After initializing the geometry with observed data, an inversion approach is used to infer unknown parameters to resemble the ice dynamics. In particular, the basal drag coefficient β in Eq. (), which cannot be measured directly, is inferred using an inversion method . In addition, the ice hardness factor B (Eq. ) is unknown. Usually, it is recovered by a thermomechanical coupled model, since B depends on the temperature . For reasons of ease of calculation, we do not use such a model here and assume a constant rheology during the friction inversion. We initialize B by computing a vertical temperature profile based on the solution provided by . The computed temperature profile is transferred to B using the relation given by p. 75. However, we update B by a subsequent inversion of ice hardness that uses the inferred friction coefficient. The inferred rheology remains unchanged in the projections.

Both inversions minimize a cost function that measures the misfit between observed horizontal velocities and modelled horizontal velocities v=vx2+vy2 . Observed horizontal surface velocities are available for both glaciers with large coverage (Table ). The cost function is defined as follows: 10J(v,β)=J0+Jreg=γ12∫Γslog⁡vx2+vy2+εvvx,obs2+vy,obs2+εv2dΓs+γt12∫Γb∇(β)⋅∇(β)dΓb, where Γs and Γb are the ice surface and ice base, respectively. The cost function consists of one term that fits the velocities (J0). The second term (Jreg) is a Tikhonov regularization to avoid oscillations. The parameters γ and γt weight the contributions to the cost function. Following , γ and γt are defined as follows:

11γ=var(log⁡(|vobs|))A,12γt=λ12πσ(k or B)H‾A, where A is the area of the ice domain, H‾ is the mean ice thickness, and σ(k or B) is the standard deviation of the initial guess of k or B, and λ is a dimensionless Tikhonov regularization parameter.

We determine the optimal regularization parameter value λ using an L-curve analysis for both inversions. The L-curve is a log-log plot of the smoothness of the optimized variable, indicated by the term Jreg, and the mismatch between the model and the observations represented by the term J0. For L-curve analysis of the friction inversion, we sample the range 10-2≤λ≤103 with 21 logarithmically spaced samples. For the analysis of the L-curve inversion of ice hardness, we sample the range 10-3≤λ≤103 with 16 logarithmically spaced samples. For each sample, we run the inverse model to convergence and record each contribution to J. The results of the analysis of the L-curve of each glacier are shown in Fig. . All L-curves provide a suitable monotony for picking an optimised trade-off value of λ, although not all L-curves show a clear corner. However, for GAG, the selected lambda value of the L-curve analysis of the basal friction is λ=3.16 and λ=0.25 for the inversion of B; for HEF it is λ=0.18 and λ=1.58, respectively.

The final results of the inferred ice flow are presented in Fig. . For both glaciers, the observed surface velocity is reproduced with great fidelity (Figs. c, d and S2 in the Supplement); we obtain a root mean square error (RMSE) of 1.27 m a⁻¹ for GAG and an RMSE of 0.36 m a⁻¹ for HEF. The maximum ice surface velocity from this analysis at GAG is around 200 m a⁻¹ at a steep ice fall of ES. HEF reaches maximum ice velocities of about 30 m a⁻¹ in the accumulation area, close to the equilibrium line.

After inversion simulations, an artificial relaxation run is performed to avoid spurious noise and to allow the model to adjust to its boundary conditions. The primary aim of the relaxation run is to level out the so called dynamical shock but we also intend to preserve the glacial state (surface elevation and ice velocities) of the simulation start date. We experimented with the length of the relaxation period and with the SMB forcing. We found that a relaxation time of ten years and an average SMB forcing from the time period 1952 to 2000 showed that both glacier's glacier volume do not deviate much from the literature values at this time. After relaxation, the glacier volume of GAG and HEF is 12.7 and 0.57 km³, respectively (corresponding literature (extrapolated) values are approx 14 and 0.57 km³, respectively).

The surface mass balance profile is well known from observations over the last decades for both glaciers. Therefore, we tune unconstrained parameters in the albedo scheme to resemble a valid surface mass balance profile. We sample ranges of parameters in the albedo equation (Eq. ) αsnow={0.7,0.75,0.8,0.85,0.9}, αice={0.1,0.2,0.3,0.4,0.5}, Δ={300,400,500,600,700} m, and δ={1500,1750,2000,2250,2500} m and compare the mean modelled and mean observed SMB profiles over the periods 2001–2013 for HEF and 2011–2019 for GAG. Note, that we compare the mean over the period and not each individual year as we are interested in the long-term behaviour. The comparison periods correspond to the times of available elevation changes (Table ). The parameters finally selected to result in a satisfying RMSE and mean signed difference (MSD) for HEF are αsnow=0.8, αice=0.1, Δ=700 m, and δ=1500 m; for GAG αsnow=0.9, αice=0.2, Δ=600 m, and δ=2000 m. The results of the optimized elevation-dependent SMB for both glaciers are shown in Fig. b and d and a map view in Figs.  and S3 in the Supplement. In order to demonstrate that the tuned values are valid for another climatic period, we re-run the EBM with a mean climate for the neutral climate period 1961–1990 (Fig. a and c). For both glaciers, the SMB profile shows a similar good agreement to that period. Note that the 1961–1990 SMB calculations make use of the 2011 and 1998 geometry for GAG and HEF, respectively. However, a general feature observed is that the modelled SMB does not represent the SMB inversion in the upper (accumulation) part of the glacier, which is likely a result of snow redistribution due to wind and avalanches; such processes are not included in our EBM (see Sect. ).

Based on model initialization and SMB calibration, we run the model forward in time during the period where observations of elevation changes and glacier mass balance of both glaciers exist. Climate data is taken from ERA5, and the corresponding simulation will be termed the ERA5 reference simulation. For GAG, simulation starts at the initial time of 2011 and lasts until 2023; for HEF the start time is 1998 and extends until 2023. Although we put much effort in matching available observations by the initialization and calibration approach, a non-physical dynamical shock at the beginning of the simulation can still occur due to a possible imbalance of the ice mass flux and the SMB. This is often a numerical issue due to, e.g., numerical discretization and diffusion or unresolved processes in the IFM. In order to justify that the model is reasonably initialized for the projections, the modelled elevation changes and cumulative SMB in the observation period are compared to the respective observations. For GAG, the comparison period extends from 2011 to 2019; for HEF from 2001 to 2013 (Table ). This approach also ensures that independent model fields are used for model tuning (SMB profiles, ice velocity) and validation (long-term cumulative SMB, elevation change). Note that SMB profiles and long-term cumulative SMB are not fully independent, but we want to stress that the long term MB is matched by tuning an average SMB profile.

Modelled elevation changes are shown in Fig.  and the respective cumulative mass balance during the simulation periods in Fig. c. The modelled cumulative mass balance reveals reasonably good agreement with the observations. Despite the minor variability of the observed cumulative mass balance, the overall decreasing trend is well reproduced by the model of both glaciers. However, the modelled spatial elevation change (Fig. ) reveals patterns that do not match perfectly with the observed maps (Figs. a and S4 in the Supplement). At GAG, glacier thinning seems to extend too far into the upper parts (particularly at ES). At HEF, the modelled elevation change shows localized spots with glacier thickening in the upper part that are not observed (Fig. b). Overall, the agreement is rather satisfying, represented by reasonable RMSE (RMSE = 0.98 and 0.57 m a⁻¹ for GAG and HEF, respectively).

Figure a and b displays the simulation results for HEF's glacier volume in the 21st century, including the median, the 17th-83rd percentile range and the total range for the SSP and RCP scenarios and the ERA5 reference simulation (Sect. ). All climate scenarios show continued ice loss with a substantial loss of glacier volume at the end of the century. The median reduction in glacier volume is 99.7 % with a likely range of 97 % to 100 % at the end of the century among all scenarios.

According to the ensemble median, HEFs ice volume disappears completely in the RCP 8.5 and both SSP scenarios by the end of the century; for the RCP 2.6 an glacier volume of 0.019 km² with a likely range of 0.028 and 0.007 km² remains. In general, the evolution of glacier volume is very similar between each scenario. At the beginning of the century, the SSPs show a somewhat stronger decline in glacier volume than in the RCP scenarios. RCP 2.6 diverges from the other scenarios around 2045 with a somewhat lower volume decrease and a tendency to stabilize around 2080.

We define a completely disappeared glacier (“gone”) as the year in which less than 1 % of the initial glacier volume is left (see Table ); a “mostly gone” glacier is defined as the year in which less than 10 % of the initial glacier volume is left. For SSP5-8.5, HEF is gone in 2051 with a likely range of 2041 to 2064. Under the SSP1-2.6 pathway, HEF disappeared in 2068 [2047 and beyond 2100]. The RCP 8.5 projects a complete disappearance in 2070 [2063 to 2078]. Although the RCP 2.6 scenarios do not project a complete disappearance, the HEF can be considered to be mostly gone, since only 3.4 % [1 % to 5 %] of the 1997 glacier volume remains. See Tables S1–S3 in the Supplement for glacier projections of each GCM/RCM combination.

Figure a and b display snapshots of glacier area and glacier volume per elevation band of single models that are closest to the ensemble median for the SSP5-8.5 and RCP 2.6 scenarios, respectively. The figures do not show the full range of possible results of the entire ensemble for each scenario and are intended as a general overview. In both scenarios, a complete disintegration of the glacier tongue is expected to occur. Under RCP 2.6 scenarios, ice patches located beneath Weißkugel and Langtaufererjochferner at elevations above approximately 3200 m a.s.l. are projected to persist until the year 2100, although they could potentially disappear after this period. In the mid-century, Langtaufererjochferner and Stationsferner disconnected from the accumulation part of the glacier and become independent ice patches.

Figure  displays the simulation results for GAG’s glacier volume in the 21st century, including the median, the 17th–83rd percentile range and the total range for the SSP and RCP scenarios and the ERA5 reference simulation. Similarly as HEF's glacier volume projection, GAG shows continued ice loss with a substantial loss of the glacier volume at the end of the century.

Until ≈2040, the reduction in glacier volume is almost similar for individual climate scenarios, but diverges afterward. In 2040, the median reduction in glacier volume is approximately 31.4 % [26.8 % to 35.7 %] among all scenarios. The spread of individual climate scenarios in terms of glacier volume is considerable at the end of the century. In 2100, the median reduction in glacier volume is 88.5 % [71.8 % to 97.7 %] among all scenarios. For RCP 2.6 and SSP1-2.6 an glacier volume reduction of 67.7 % [62.2 % to 77.6 %] and 86.4 % [76.2 % to 89.4 %] is expected, respectively. Both low emission scenarios tend to stabilize towards the end of the century above 10 % of the initial glacier volume, but the decline in glacier volume seems to continue after 2100. The high emission scenarios RCP 8.5 and SSP5-8.5 lead to an glacier volume reduction of 92.6 % [87.2 % to 97.7 %] and 99.7 % [96.4 % to 100 %], respectively. The SSP5-8.5 projects a complete deglaciation in 2095 [2085 and beyond 2100] (Table ). Glacier volumes drop below 10 % in the RCP 8.5 in 2095, which is about 19 years later than in SSP5-8.5 (Table ). See Tables S4–S6 in the Supplement for glacier projections of each GCM/RCM combination.

Figure a and b displays snapshots of glacier area and glacier volume per elevation band of single models that are closest to the ensemble median for the SSP5-8.5 and RCP 2.6 scenarios, respectively. These examples, representing the upper and lower extremes, do not encompass the full range of possible results, but are intended to provide an insight of the glacier retreat. In both scenarios, a complete disintegration of the 14 km long glacier tongue is expected to occur. In SSP5-8.5 an almost complete GAG disappearance by 2100 is expected with only tiny ice patches in regions above ≈3400 m a.s.l. persisting until the year 2100 (Fig. c). For RCP 2.6 a major retreat of GAG by the end of the century is expected, but Konkordiaplatz is still connected to the three main tributaries Aletschfirn, Jungfraufirn and Ewigschneefeld. Most of the ice remains in the higher parts (Fig. d), however some ice is present in the lower-lying regions. The prominent feature is the peak of glacier volume around the 2400 m elevation band in the year 2100. This peak is associated with the Konkordiaplatz site, which initially has a surface elevation of ≈2800 m a.s.l. and an ice thickness exceeding 900 m. As the ice at this location persists but the surface elevation decreases, the initial peak at around 2800 m a.s.l. decreases and shifts to a lower elevation. Although RCP 2.6 is expected to preserve more ice at the end of the simulation period than SSP5-8.5, for example, the landscape will change significantly compared to today.

Our provided projections are based on a calibration strategy for parameters to provide a best-estimate. However, several parameters in the EBM are subject to large uncertainty that is not covered by our best-estimate simulations. For instance, the gradients used for downscaling the incoming short- and longwave radiation are highly uncertain as they are based on one study measured at one location. In order to test the sensitivity of our projections on parameter choice, we use an ad-hoc approach by simply varying uncertain parameters by ±10 %. Although this variation will likely not cover the possible range of parameters, it illustrates model outcomes by the full-Stokes model to parameter changes. The parameter's we change encompasses (i) the downscaling gradients of short- and longwave radiation, precipitation, air-temperature, (ii) the error estimate of precipitation and air-temperature between ERA5 data and meteorological station (iii) and the effective albedo. Simulations of these parameter combinations are re-run for SSP 585 and RCP 26, covering the upper and lower end-members of the climate pathway, with the GCM/RCM combination closest to the ensemble median (See Tables S1–S6 in the Supplement). This results in 14 parameter combinations for each glacier and will be termed the “parameter ensemble”.

Figure  shows the computed SMB profiles of the parameter ensemble. For almost all settings, the SMB profile is similar to the “best estimate”. Except for changing the albedo by ±10 % deviate much larger from the “best estimate” but is still, however, within the range of the maximum and minimum range (at least below the ELA). The subsequent projection runs (Fig. ) of the parameter ensemble reveal that the parameter ensemble has a minor effect on projected glacier volume loss or glacier gone times (considering the median ± percentile range). However, the change in albedo by ±10% has a larger impact (see the total range of the low emission scenario in Fig. ).

Despite the fact that we use a rather complex model in terms of FS ice dynamics compared to large-scale glacier models relying mainly on SIA e.g., and a more sophisticated SMB model employing a surface energy balance model compared to simpler models e.g. temperature index models,, our results are subject to several uncertainties, and some of them we will discuss below.

Our inversion approach makes use of a remotely sensed surface velocity product as a target to constrain unknown parameters. However, the target velocity field itself is affected by uncertainties that are transferred to the IFM. On the one hand, retrieving the velocity of the ice surface of slow flowing glaciers from satellite sensors is challenging and subject to large errors particularly in slow flowing parts (i.e. accumulation zones). The retrieved observed ice flow velocity is subject to methodological errors which leads to an uncertain observed ice velocity map. The transport of ice, particularly in the accumulation area, downstream of the glacier could be subject to this uncertainty. On the other hand, the inferred basal state resulting from the inversion approach remained unchanged during the projection period. The basal friction parameter and thus the basal slipperiness could change during the projection period due to increased availability of meltwater at the base, and therefore decreased overburden ice pressure. These effects are not captured by our model (basal drag is only dependent on velocity (Eq. )), and the basal state reflects the initial state. Furthermore, thermo-mechanics are not taken into account, which might have an impact on the overall glacier evolution .

In addition to the uncertainties related to the IFM, the external forcing data are subject to several uncertainties. We are relying on future climate data from CMIP5 and CMIP6, where each GCM/RCM has its own uncertainty that produces a large spread in glacier projections. For example, in some individual GCM/RCM projections of the GAG, low-emission projections of the glacier volume at 2100 overlap with simulations of high-emission scenarios (compare the full spread of SSP1-2.6 with the full spread of SSP5-8.5 and RCP 8.5 in Fig. ). The comparison of the overlapping projections is certainly hampered, as they stem from different GCM/RCMs but demonstrate the spread of the projections.

The study by reveals the challenges encountered when using climate model data in mountainous environments. The GCM/RCM data used have a rather coarse resolution (between 12 (EURO-CORDEX) and 60 km (ISIMIP3b)) in terms of the length scale of mountain glaciers in the European Alps. Mountainous regions are characterized by complex topography and the local climate is driven mainly by the interaction between large-scale atmospheric flows and local topography. Although the employed downscaling and bias correction ensure a reasonable level of the climate variables, the overall climate output has systematic biases. Small scale variability of atmospheric dynamics, originating from complex air flow modifications by mountainous terrain is not resolved in the rather large-scale resolution GCM data. This might introduce unquantifiable inaccuracies and biases in the climate data and has to be borne in mind, when interpreting the results. A distinctly higher spatial resolution of the climate data would be necessary to adequately represent the relevant physical processes. For instance, for better resolving the turbulent fluxes (latent and sensible), requires a more sophisticated windspeed model accounting for small scale variability. However, this is beyond the scope of this study and computationally way too expensive to be accounted for in the scope of this study given the 107 different climate projections used in the modelling process.

Uncertainties in the calculated SMB are not solely due to climate data; they also arise from potential overestimations in snow accumulation at higher elevations, as we lack a model for snow redistribution by wind transport and avalanches. We experimentally applied a snow redistribution model based on surface slope and curvature following , but it resulted in only minor improvements relative to the validated SMB gradients without incorporating a wind/avalanche redistribution model (see Fig. ). Moreover, when incorporating snow redistribution based on surface slope and curvature, our model experiences instabilities in the evolution of the ice surface (e.g., holes in the surface). Although the parametrizations of snow redistributions appear promising, further refinement for application in an IFM is required.

The rate of future glacier retreat is certainly affected by supraglacial debris that alters the SMB. A debris cover has many feedbacks on the SMB by decreasing the albedo leading to enhanced melt or due to its insulating properties leading to decreased melt . Although the glacier tongue of GAG is completely protected by a debris cover and demonstrate that a supraglacial debris cover reduces glacier melt, i.e. glacier retreat, we refrain from using such a parametrization. Although they tuned the SMB to observations, used a simplified parametrization for a debris induced SMB that did not account for future supraglacial debris transport, debris redistribution and developing ponds and cryokarst features , which we think is necessary for future projections.

The future evolution of glaciers in the European Alps has been projected with models of various complexity and based on diverse climate projections. Our results of a substantial volume loss until the end of the 21st century under CMIP5 and CMIP6 future climate scenarios are in line with previous estimates based on large-scale and detailed glacier modelling studies (Fig. ). However, the timing of deglaciation and the remaining glacier volume at the end of the 21st century vary considerably.

The study by JH19, is comparable to our research with respect to the model configuration. They utilize an individual full-Stokes model setup for GAG, with model tuning and validation based on observations (glacier volume and length of glacier tongue, in-situ point measurements of ice surface velocities, and surface mass balance). However, JH19 predicts a larger volume loss by the end of the 21st century relative to our findings. For RCP 8.5, they predict a 98.5 % reduction in glacier volume, while our model predicts a reduction of 92.6 %. Specifically, under RCP 2.6 they predict a 59.5 % reduction in glacier volume, contrasting our model prediction of 67.7 %. The discrepancies may originate from the different models employed for calculating surface glacier melt. Our study utilizes a physically-based energy balance model, while JH19 employs a temperature index model . Although both models tend to overestimate the mass budget and have a limited ability to reproduce glacier volume on decadal time scales, the mass budget computed by an energy balance model, despite its physical character, is probably too high compare HTI and EB schemes in. However, this behaviour is only applicable for the RCP 8.5 scenario and RCP 2.6 until about 2075; afterwards the RCP 2.6 of JH19 reveals declined mass loss.

The volume projections of the large-scale models are generally situated above (GloGEMflow) or, close (OGGM) to our projections. Under RCP 2.6, GloGEMflow predicts a volume reduction of 57.6 % for GAG, aligning somewhat more closely with our forecast than the JH19 study; likewise, for RCP 8.5, GloGEMflow projects a 89.4 % volume decline, which is closer to our estimate than the JH19 study. OGGM displays an irregular pattern when compared to our projections. Under the SSP5-8.5 scenario, discrepancies are pronounced in the mid-century period but diminish significantly after 2075. Conversely, for the SSP1-2.6 scenario, the differences remain minor before 2075, although OGGM projects glacier growth in the subsequent years. In the case of HEF, the OGGM model aligns well with our projections over the entire projection period for the SSP5-8.5 and SSP1-2.6 scenarios. In contrast, GloGEMflow exhibits significant discrepancies. Although GloGEMflow predicts an almost complete disappearance of GAG by the end of the century, the projected volume loss is significantly delayed (≈50 years) compared to OGGM and our study. Under the RCP 2.6 scenario, a substantial proportion of ice remains (32.5 %) compared to OGGM and our study. In line with the JH19 experiments, there is no clear trend whether a TI and EBM model predicts more mass loss or not.

Disentangling the reasons for the differences is challenging, particularly when comparing model projection outcomes based on large-scale approaches to individual glacier applications, since model techniques are designed in a different manner. In addition to inherent differences arising from model concepts and design, other contributing factors include the climate forcing datasets used, the methodologies of model calibration and initialization, as well as the choice of the mass balance schemes. Please note that the comparison of the selected studies aims to demonstrate the possible spread of the model outcomes, but is not fully exhaustive, as we did not include additional studies that present projections of GAG and/or HEF e.g.,. However, our selective comparison demonstrates the potential spread of glacier model projections with regard to model sophistication, initialisation and forcing.

The most striking feature of the volume projections is that even in low emission scenarios, the HEF (almost) disappeared. In addition, the low- and high-emission scenarios at HEF are rather identical in terms of glacier volume loss. The losses agree pretty well with the expectation of (almost) full deglaciation of HEF in the middle of the 21st century. This is an intriguing result, since limiting global warming to 1.5 to 2 °C above pre-industrial levels negotiated within the Paris Agreement would cause a (almost) complete disappearance of HEF, regardless of whether the sustainable or the highest emission pathway is considered.

GAG has a longer lifetime than HEF in the different future climate scenarios. While GAG is projected to disappear under high-emission scenarios, low-emission scenarios suggest the need for stringent mitigation measures to prevent further reduction of GAG's glacier volume. The longer lifetime of GAG is related to its larger ice thicknesses that persist even at low altitude (Fig. c, d), its larger elevation range and its higher elevation-area distribution. Figure a and b show the distribution of glacier area per elevation bands of GAG and HEF and the evolution of the ensemble-median ELA of each scenario. Notable is the increase of the ELA at HEF for each scenario reaching, or even exceeding, the ice coverage fraction at the highest elevation; merely the ELA under RCP 2.6 forcing reaches a height that is below a certain fraction of the present-day ice coverage. At GAG, the increase of the ELA for the low-emission scenarios is muted compared to the high-emission scenarios. In particular, a large fraction of glacier area above >3500 m a.s.l. would remain.

The examined sample is certainly not representative for all glaciers in the alps, but simply upscaling our projection results of HEF and GAG to all glaciers in the European Alps would cause an almost complete wastage of glaciers in the eastern Alps, while a tiny fraction in the western Alps is expected to survive (Fig. c). This estimate is supported by , where all warming-level scenarios lead to a nearly complete deglaciation in the Ötztal and Stubai Alps (western Austria) before 2100. In the western alps, glaciers covering an elevation range above ≈3500 m a.s.l. have a longer lifetime or even the potential to survive until 2100. However, in the worst scenarios, we extrapolate that most glaciers in the western alps disappear at the end of the 21st century, which is consistent with .

The simple extrapolation of two exemplary glaciers to all glaciers in the European Alps is certainly hampered by several factors. In addition to the regional setting of each glacier that probably influences the glaciers' response to increasing temperatures, the glacier sensitivity depends on glacier slope and exposition, ice thickness and area-elevation distribution, mass balance gradient and hypsometry e.g.. However, comparing the regional settings GAG receives very high amounts of precipitation in the accumulation area due to moist air masses from the north. However, these air masses are blocked by the main alpine crest and create very dry conditions in the inner-Alpine valleys (e.g., Rhone valley), which leads to considerably smaller amounts of precipitation at GAG's glacier tongue.

HEF, as well as other low-lying glaciers found in the eastern European Alps, are particularly vulnerable to increasing temperatures because they are situated at lower elevations than many of their western counterparts (Fig. c), which reside at higher altitudes. In addition, HEF experiences a somewhat more intense warming than GAG. For RCP 8.5 a stronger warming of 0.7 ± 0.9 °C is expected at HEF compared to GAG; for RCP 2.6, SSP5-8.5 and SSP1-2.6 0.3±0.6, 0.2 ± 0.2, and 0.3 ± 0.2 °C, respectively. Note that the estimated values correspond to an elevation of ≈2200 m a.s.l. beneath GAG and HEF, but temperature changes are subject to local conditions and elevation .

The generalization of our modelling results to a larger area is based on a specific model combination approach applied to two individual glaciers. Further modelling attempts based on standardized tests are necessary to infer how our modelling approach can be used to improve regional-scale glacier projections. In addition, follow-up studies focussing on glaciers with a sufficient data basis for our approach facilitate the generalization or extrapolation to a larger area.