Rock wall permafrost is composed of rock, ice and air in non-saturated conditions. Sass (2003) approached alternating saturated/unsaturated conditions under freezing and thawing of the near-surface pore spaces of a rock wall by mean of geophysical soundings. However, the rates of saturation of alpine rock walls remain poorly understood, and therefore the numerical models of rock wall permafrost have so far considered a saturated, homogeneous and isotropic media (Wegmann et al., 1998; Noetzli et al., 2007; Hipp et al., 2014; Myhra et al., 2015). Nonetheless, such approaches have been shown satisfactory to simulate long-term temperature changes in alpine rock masses. Shorter timescale processes are clearly a hydrogeological problem.

In the scope of this study, we used FEFLOW combined with the plug-in piFreeze 1.0 which accounts for freeze and thaw processes. As a very first use, we adopted existing approaches of long-term simulations (saturated, homogeneous and isotropic media) to simulate transient thermal processes along centennial timescales. Further developments will use the potential of FEFLOW to simulate them with various saturation rates and fluid transfers.

The conservation equation of energy for advective dispersive-diffusive transport of thermal energy depends on fluid flows (in saturated or unsaturated conditions, i.e. Darcy's law incorporated in continuity equation or Richards' equation), but works in pure conduction when flows are zero. It is usually expressed as follows (Diersch, 2002): ∂φρLcL+(1-φ)ρScST∂t=-∇⋅ρLcLqT-Λ∇T with φ being the porosity (dimensionless), ρLcL and ρScS the volumetric heat capacities (J m-3 K-1) of the liquid and solid phases respectively (obtained by combining the density ρ and the specific heat capacity c), Λ the hydrodynamic thermal dispersion tensor (J m-1 s-1 K-1), which includes thermal conductivity, T the temperature (K) and q the apparent flow velocity from Darcy or Richards equation (m s-1). Equation (1) accounts for only one fluid phase and only one solid phase (water and rock respectively), which is the default use of FEFLOW in saturated conditions such as in this study.

The piFreeze plug-in working with Feflow7.0 adds the ice fraction to the solid phase in order to modify only the parameters of solid thermal conductivity and of solid thermal heat capacity in Eq. (1). The addition of the ice in the whole modelled medium is expressed throughout the bulk volume as εa+εw+εi+εr=1, with εa, εw, εi and εr the bulk fractions of air (εa=0 in our case), water, ice and rock respectively. A relation is established between the bulk volume of ice and the bulk volume of liquid, which is the mass fraction per bulk volume of the unfrozen liquid to the total liquid mass, also called the freezing function F (Clausnitzer and Mirnyy, 2015): F=εwρwεwρw+εiρi where ρ is the density of the corresponding phase (subscript i for ice and w for water). This function F decreases with the fraction of ice. For the freezing point T0, the ice forms gradually within a predefined temperature interval of the length ΔT: T0-ΔT2,T0+ΔT2. Taking into account the expressions for the bulk volume ε and of the freezing function F described above, the thermal parameters of Eq. (1) are modified as follows: a.

The solid thermal conductivity λ (W m-1 K-1) becomes

λs=λs,0+εiλi-λs1-ε.

The thermal conductivity of the rock was set to 3 W m-1 K-1, which stands for a conservative value for saturated granitic rock (Cho et al., 2009). However, the thermal conductivity of a saturated media does not only depend on the mineral properties, but also on the liquid or solid state of water, ice being up to six times more conductive than water at 0 ∘C (Williams and Smith, 1989). Thermal conductivity variations along freeze and thaw cycles are accounted for by piFreeze. The heat capacity of the rock was set to 1.8 MJ m3 K-1.

In addition to the usual adjustable parameters of FEFLOW, piFreeze allows a user-defined freezing temperature, a customisable temperature interval for freeze/thaw combined with a linear freezing function, adaptable thermal properties of ice, a user-specified residual fluid content and a configurable latent heat. Such possibilities are highly promising for adapting the modelling approach to the natural conditions.

To account for the latent heat processes related to the freeze and thaw of the interstitial ice contained in pores and fractures, we took a 5 % porosity value following the procedure from Noetzli et al. (2007), which is the maximum value for dense crystalline rocks (Domenico and Schwartz, 1997) or lowly fissured crystalline rocks (Banton and Bangoy, 1999). Indeed, dense crystalline rock porosity without any fissure is usually below 1 %, while fractured crystalline rock porosity quickly reaches values greater than 5 %. The 5 % value chosen here accounts then for the ice contained in fractures since bedrock discontinuities are not included.

Water contained in artificial pore spaces is subject to supercooling, i.e. a temperature deviation from the equilibrium freezing point at 0 ∘C, until it reaches a spontaneous freezing point which depends on pore size and material (Alba-Simionesco et al., 2006). Geophysical experiments on various hard rock samples and under controlled laboratory settings have quantified a freezing point depression of -1.2 ∘C ± 0.2 ∘C (Krautblatter, 2009) due to the pressure and water salinity. To account for this supercooling characteristic of interstitial water, the freezing point T0 was set to -1 ∘C in the piFreeze module, while the temperature interval ΔT of the freezing function was set to 1 ∘C. The latent heat of fusion was set to 334 kJ kg-1.

We first extracted the topography and the MARST from the 4 m-resolution DEM (provided by RGD 73–74), and mapped the MARST over it to serve as upper boundary conditions along the NW–SE vertical transects (Fig. 2). The MARST map has been evaluated against 43 measurement points of RST from the multi-year time series of the nine RST loggers installed around the AdM. The modelled MARST values tend to underestimate measured MARST values of sun-exposed rock faces and to overestimate those of the shaded faces. Nevertheless, the mean bias (mean difference between measured and modelled MARST) of -0.21 ∘C (Magnin et al., 2015a) indicates a generally good approximation of the real-world MARST at this site.

The linear regression used to produce the MARST map has been formulated with the mean air temperature of the 1961–1990 reference period (homogenised by Hiebl et al., 2009), and the measured MARST were adjusted to the reference period. The MARST were adjusted by applying the difference in air temperature between the reference period and the years of the MARST measurements (Boeckli et al., 2012). In our modelling procedure, we considered that the MARST extracted from the map is representative of the year 1961 since the MARST has been mapped using the 1961–1990 MAAT. Then, we lowered this MARST by 1 ∘C to approximate the MARST at the end of the LIA (Auer et al., 2007; Böhm et al., 2010) and set up the initial RST condition at the upper boundary of the model domain (Fig. 3). MARST differences driven by topo-climatic factors clearly appear along the extracted profiles but are site specific. At the GM, the MARST difference between the SE and NW face is only 1 ∘C whereas it is 5 ∘C at the AdM and 6 ∘C at the GPA. These variable temperature differences for similar aspect differences (180∘) are attributed to two factors: the differences in slope steepness and the local shading. The GM and AdM are isolated summits with no close shading. Conversely, the GPA NW face is located right below the Mont Blanc which shades the GPA west-exposed faces and lowers their MARST. The less steep NW slope and rounded summit of the GM receive solar radiation for a larger number of daylight hours than in subvertical settings with a more perpendicular incidence of the beams.

Starting from the initial RST representative of the LIA conditions, we first initialise 2-D steady-state temperature fields for the year 1850, and then run transient simulations using reconstructed, measured and projected climate time series up to 2100 (see Sect. 3.3).

Below the topographical profiles, a box with a height of 5000 m was added to shut off the model geometry of each site. A constant geothermal heat flux of 85 mW m-2 (Medici and Rybach, 1995; Maréchal et al., 2002) was set up as a lower boundary condition. Above these boxes, a finite element mesh with triangular elements was generated to discretize the subsurface material (Fig. 3). Even though the spatial resolution of the boundary conditions is 4 m, we refined the meshes close to the surface in order to better represent the near-surface temperature gradient. The spatial resolution of the initial RST, based on the 4 m resolution map, was refined accordingly using linear interpolation. This mesh and RST refinement does not provide much information at depth, nor improve the quality of the models, but it facilitates the model evaluation. At greater depth, we kept a mesh size of 4 m, in coherence with the resolution of the input data. This approach resulted in 8548 nodes and 16 141 mesh elements at the AdM, 5844 nodes and 10 952 elements at the GM and 12 087 nodes and 23 344 elements at the GPA (Fig. 3).

On the AdM site, 37 observation points were defined between the surface and 10 m depth of the NW and SE faces at the location of the boreholes (Sect. 2.1). The mesh was refined along these observations points (Fig. 3) and simulated bedrock temperature is extracted at user-defined time steps during the transient simulations to serve for model evaluation (Sect. 4.2.1).

To define an initial 2-D temperature field in the model geometries, we ran the model with the upper boundary condition (the 1850 RST) and lower boundary condition (the geothermal heat flux) until steady-state conditions were reached, balancing the respective influence of the atmosphere and geothermal heat fluxes. Steady-state conditions were reached after 80000±10000 years. After this initialisation procedure that provides an initial condition for 1850, we ran transient simulations with air temperature forcing time series from 1850 to 2100.

The transient simulations are forced with air temperature time series created from various sources of data. The temporal resolution of these forcing data was gradually refined with the increasing quality of the available data and the periods of interest (Fig. 4). For the period 1850–1961, no continuous air temperature measurements are available for the Mont Blanc massif. Therefore, we assumed a linear increase of 1 ∘C between 1850 and 1961 (Auer et al., 2007; Böhm et al., 2010), and run the simulations at an annual time step. A sensitivity analysis using larger time steps did not change the final results for the periods of interest.

In the framework of the MARST mapping (Magnin et al., 2015a), a climate time series was created at a monthly time step based on measured temperature values in Chamonix for the period 1961–1990. We used this monthly time series and extended it up to 1993 to force the model between 1961 and 1993, which constituted a first step in temporal resolution refinement of the forcing data.

In 1993, Météo France started continuous records of air temperature at hourly time step. From these hourly records, daily air temperature can be reliably calculated. Therefore, we forced the transient models with this daily air temperature time series between 1993 and 2015.

Finally, two contrasted scenarios were retained for the 21st century. Time series consist of daily 2 m air temperature simulated by the IPSL-CM5A-MR Earth system model (Dufresne et al., 2013), which participated within the 5th Coupled Model Intercomparison Project (CMIP5, Taylor et al., 2012)/AR5 of IPCC (2013). For this study and for climate projections in future decades, we used two contrasted radiative forcing scenarios, namely the Representative Concentration Pathway (Moss et al., 2008) RCP4.5 and RCP8.5. They correspond respectively to increases of +4.5 and +8.5 W m-2 for 2100 relative to pre-industrial values, resulting in air temperature increases of +3 and +5 ∘C by the end of the 21st century according to the comparison of the measured mean air temperature of the 1980–2010 period and the projected mean air temperature for the period 2070–2100.

For RCP4.5 anthropogenic greenhouse gases emissions peak around 2040 and then decline (moderately optimistic scenario), while for RCP8.5 emissions continuously increase throughout the century (pessimistic scenario). The IPSL-CM5A-MR model was chosen because its basic state is very close to the recent observational records during the first years of the 21st century (Fig. 4), and its response to the radiative forcing throughout the century is close to the median of the CMIP5 models, ensuring a representative behaviour to estimate long-term evolutions. Forcing time series are obtained by extracting data at the closest grid-point (1336 m a.s.l.) to the Mont Blanc Massif.

Equilibrium conditions for the end of the LIA (1850) are displayed in Fig. 5a for the three sites. In 1850, the GPA and the AdM showed cold permafrost (<-2 ∘C). At the GM, a cold permafrost body was characterising the NW subsurface and extending below the SE face between 3260 and 3280 m a.s.l. Warm permafrost was already occupying most of the site, including the summit area.

The shape of the isotherms varies from one site to another, depending on the topographical settings. The steepest site (AdM) shows subvertical isotherms down to 3720 m a.s.l., where they incline downwards and towards the NW to become more oblique. In the top part of the less steep site (GM), the -2 ∘C isotherm is rather oblique while the -1 ∘C one is subhorizontal in the lower part. In the GPA, isotherms are vertical in the top part and oblique in the middle part. In the lower part of the SE face, in the >-5 ∘C area, isotherms obliquity declines to become more parallel to the upwards geothermal heat flow.

The modelled temperature gradients directly depend on the temperature difference between NW and SE flanks (Sect. 3.2.1): small temperature gradients are visible in the GM cross section in accordance with the initial RST difference, whereas they are higher in the two other sites with more severe topography and more contrasted RST.

Figure 5b displays time-dependent conditions for the early 1990s, while Fig. 5c demonstrates those in 2015 after the two past decades of strong air temperature increase (Fig. 4). During the 20th and early 21st centuries, permafrost has degraded in all the three sites. Warm permafrost has extended over the entire GM site and below the AdM SE face. This degradation shows site-specific features in terms of isotherm shapes and temperature field distributions.

At the GM, the cold permafrost body has subsisted until the early 1990s below the NW face, but has narrowed down to two small bodies. Meanwhile, the lower limit of the -1 ∘C isotherm has risen up below the SE face. Its initial subhorizontal curve has moved into a more oblique shape down to 30 m, forming a square angle 25 m below the surface and inclining to a subvertical shape more parallel to the SE surface in 1992. In 2015, the cold permafrost bodies have both totally disappeared and the -1 ∘C isotherm has retreated inside the rock mass forming a subrounded body. At that stage, the 0 ∘C isotherm is parallel 5–10 m deep below the NW surface and 15–20 m below the SE surface.

At the AdM, the isotherms kept almost the same shape for the past 160 years, but warm permafrost has already penetrated a depth of the SE face in 1992, and reaches 15–20 m depth in 2015. The -5 ∘C isotherm has narrowed to 15 m wide in the shallow layers and is parallel to the NW face in 1992. By 2015, the entire permafrost summit is >-5 ∘C and a thin <-4 ∘C permafrost body subsists down to about 20 m depth of the NW face, between 3700 and 3820 m a.s.l. At this site, the 2 ∘C warming between the end of the LIA and the recent period has affected the entire rock pillar.

Conversely, the central part of the GPA has remained unchanged along the past 160 years. However, permafrost warming is visible through the change in the -5 ∘C isotherm shape, being perpendicular to the SE face at 4130 m a.s.l. in LIA equilibrium conditions, and curving upwards to 30 m depth to become more parallel to the SE face in 1992. In 2015, this warming is visible down to 70 m, the coldest permafrost body retreating towards the NW face. In the NW face, isotherms have kept the same shape, parallel to the rock surface. The shallowest 60 m with a temperature of -11 to -5 ∘C in the LIA equilibrium conditions, has turned into a uniform body of -8 ∘C into the entire NW face by 1992. Two decades later, this body has narrowed: its width decreased by 20 m; by depth, its lower limit rose up to 4070 m a.s.l. and the highest limit decreased to 4260 m a.s.l.

The air temperature increase experienced since the end of the LIA had variable effects depending on the site geometry and initial RST. The existing permafrost data in the Mont Blanc massif allow for an evaluation of the model outputs.

FEFLOW allows for extracting model output at user-defined observations points and time steps. We therefore requested extraction of simulated temperature for each observation point of the AdM model domain (see Sect. 3.2.2), and for each first day of each month between January 2010 and January 2015. Those modelled values are then compared to measured temperature in the AdM NW and SE boreholes to evaluate the model performances. Model outputs are first analysed at a daily time step before considering annual patterns. For better visibility, only four selected modelled temperatures of the year 2010 for each borehole, encompassing the four seasons, are displayed in Fig. 6.

The model reproduces the bedrock temperature below 6 m depth very well, especially in the NW borehole (Fig. 6a, b). Measured bedrock temperatures at the shallowest depths, which are affected by snow cover, fractures and strong solar irradiation are not taken into account in the modelling approach. This explains the greater discrepancy between modelled and measured ground temperature in winter (presence of snow) whereas summer temperatures show a better fit, especially in the NW face where the effect of solar radiation is weak.

The temperature profiles recorded in the NW borehole are significantly affected by an open fracture at 2.5 m depth which locally cools the bedrock due to air ventilation, especially during winter, whereas above the fracture, the insulating effect of a snow patch accumulating on a ledge at the borehole entrance warms the temperature profile down to the fracture depth (Magnin et al., 2015c). This is visible on the profile from 1 January 2010 (Fig. 6a, b): the upper part shows a small temperature gradient due to snow insulation, while a stronger temperature gradient is visible below the fracture due to its shortcut effect between the air and the subsurface. During the years 2010–2015, the maximum difference between measured and modelled daily temperature at 10 m depth in the NW borehole is 0.5 ∘C and the mean difference of the 72 observation points (corresponding to 72 days between 2010 and 2015) is only 0.01 ∘C.

In the SE borehole, the deepest measured temperatures seem less well reproduced by the model, but remain of reasonable accuracy. We observe a maximum difference of 0.7 ∘C between measured and modelled value at 10 m depth for the observation period, with a mean difference of 0.1 ∘C between the 60 measured points (two 3–4 months interruptions of the borehole records decrease the number of available data) and simulated temperatures at the same date. On the SE face, the almost continuous snow cover from autumn to spring/early summer (Magnin et al., 2017) and the high solar irradiation both affect the rock temperatures, which is not considered in the modelling approach. The solar radiation effect is visible on the measured profile from 1 July (Fig. 6a, b).

On an annual average, differences between the measured and modelled temperature values are smaller than on a daily basis (Fig. 6c). At 10 m depth, in the worst case (2010), the modelled value is 0.2 ∘C higher than the measured one at the NW borehole, with a mean difference of 0.01 ∘C during 2010–2015. At the SE borehole, the maximal difference between measured and modelled value is 0.04 ∘C, but only two full years of observations (2010 and 2011) are available for model evaluation, the 2012–2015 time series being affected by significant data gaps. Therefore, the mean difference between observed and measured value was not calculated.

The model satisfyingly reproduces the negative temperature gradient along the SE profiles, resulting from the heat sink effect of the opposite north face (Noetzli et al., 2007; Magnin et al., 2015c). On the NW face, the model shows a significantly lower temperature gradient than the measured one. Since 3-D effects seem well reproduced on the SE face, the difference between the measured and modelled temperature gradient on the NW face may result from the cooling effect of the fracture rather than from 3-D effects.

Borehole temperatures provide information at a very fine scale and are thus not appropriate for evaluating 2-D models with a multi-metric resolution. However, they are much more suitable than RST measurements since the temperature at 10 m depth results from the heat transfer of a multi-metric area at the surface. Further evaluation is possible using an electrical resistivity transect, which has been proven to be a relevant support in improving model evaluation (Noetzli et al., 2008).

In October 2012, an ERT sounding was performed along a 160 m long profile of the GM NW face with an electrode spacing of 5 m. Field measurements were combined with laboratory testing on a Mont Blanc granite boulder in order to calibrate the resistivity–temperature relationship. Results allow for a semi-quantitative description of the permafrost state and suggested the presence of warm permafrost under the GM NW face (Magnin et al., 2015d). In Fig. 7, the ERT transect is compared to the 2-D temperature model of the GM for October 2012. The lateral extent of the ERT transect is shown on the 2-D model by a red line. The active layer is represented by the positive temperatures near the surface and the resistivity body <80 kΩm corresponding to thawed granite. A warm permafrost body is delineated by the temperatures between -2 and 0 ∘C and resistivities between 80 and 200 kΩm. These features are visible on both sources of data and corroborate the model performances in the 2-D representation of the permafrost. Further analysis is limited by the lack of quantitative data: isotherms are not directly comparable to the iso-resistivities.

Given the remarkable capability of the transient simulations to reproduce current temperature conditions at the AdM borehole locations and in the GM NW face, the model can be judged to be accurate enough to consider future long-term scenarios.

Assumptions and simplifications were necessary to generate the initial 2-D temperature field at the end of the LIA and run transient simulations. Starting with LIA equilibrium conditions ignores possible transient effects from Würm and Holocene climate variability. However a thermal perturbation in depth due to millennial timescale changes is unlikely given the geometry of the investigated sites (max. width of 350 m), and results from previous studies (Kohl, 1999; Noetzli and Gruber, 2009).

Simulated 2-D temperature fields seem to accurately reproduce real-world data as shown by the comparison of modelled temperature with the AdM borehole temperatures and the GM ERT transect. This underscores two strong points. Firstly, the similarity between measurements and model (Sect. 4.2.2) in the uppermost layer of the rock walls (i.e. the 10 and 25 uppermost metres at the AdM and the GM respectively) suggests that the strategy to run transient simulations is relevant enough to simulate the permafrost changes since the LIA up to the current period. In the meantime, it emphasises the quality of the forcing data that have the advantage of being partly constructed from direct measurements of air temperature in the Mont Blanc area. This ensures a better representation of the local climate variation compared to air temperature time series extracted from kilometre-scale climate models.

The resolution of the input data is one of the most challenging issues when forcing permafrost models in highly heterogeneous land surfaces such as mountain terrains (see next section; Fiddes et al., 2015). Downscaling methods are under development for mountain terrains (Fiddes and Gruber, 2012, 2014; Fiddes et al., 2015), but currently available homogenised air temperature data sets at a kilometre-scale remain better suited to drive numerical models on larger areas with coarser spatial resolution (Jafarov et al., 2012; Westermann et al., 2013).

More sophisticated approaches capable of simulating complex energy exchanges at the bedrock–atmosphere interface have been proposed to model surface temperature of steep mountain slopes using specific algorithms to compute solar radiations (Stocker-Mittaz et al., 2002; Gruber et al., 2004; Salzmann et al., 2007) and heterogeneous snow accumulation effects (Pogliotti, 2011; Haberkorn et al., 2015a, b, 2017; Magnin et al., 2017). However such complex approaches induce a high degree of uncertainty due to the numerous input variables owning to their respective sources of error. As a consequence, a certain proportion of the modelled RST variability between different model outputs is in the range of the model noise. Furthermore, relevant data and parameterisation techniques are still missing for long-term transient models of rock wall permafrost accounting for snow effects.

In this study, possible climate evolutions over the future decades are obtained from one single climate model (namely, IPSL-CM5A-MR) and two scenarios for greenhouse gas emissions, out of four possible RCPs, and 20 to 40 models (depending on the model versions and the type of the climate simulations) that participated in the 5th Assessment Report of the IPCC (2013). The choice of the climate model was motivated by its realistic steady-state in present-day conditions when compared to observational time series and its median response to greenhouse effect evolutions. The retained radiative forcings for the 21st century consist of the RCP4.5 and RCP8.5, that is, a moderately optimistic and a pessimistic scenario (see Sect. 3.3.2, Moss et al., 2008). These RCPs can be considered currently as reasonable estimations of the lowest and highest air temperature changes likely to happen, and thus plausible contrasted boundaries within which the permafrost could evolve. According to the Paris agreement on climate change adopted on December 2015 during the COP21, climate change should be limited “well below 2 ∘C” above pre-industrial levels with more ambitious targets updated every 5 years. To date, however, the reductions in greenhouse gas emissions planned by the participating countries lead to an estimated global warming of roughly 2.5–3 ∘C, which lead us to discard the more optimistic RCP2.6 scenario, describing a rapid decrease in emissions as soon as the early 2020s, but makes RCP4.5 the most probable pathway for the 21st century.

Precipitation and snow were neglected in our modelling approach due to (i) the marginal and local effects of precipitation on long-term rock wall permafrost changes, (ii) the complexity of snow accumulation patterns on steep slopes, which do not only depend on precipitations but also on a variety of parameters such as slope roughness and steepness, aspect (melting), and wind patterns, and (iii) the large uncertainties in precipitation projections (Heinrich et al., 2013) as well as the persistent difficulties of current climate models to simulate them.

Rock wall permafrost is highly sensitive to climate change since it is subject to multi-directional heat transfers from the different sides of a summit (Noetzli et al., 2007). In this study, we assess rock wall temperature changes in 2-D only. Therefore, the simulated changes in permafrost distribution are possibly slightly underestimated since the signals only penetrate through two sides (NW and SE). Sensitivity analyses have shown, however, that 2-D simulations show similar long-term transient temperature pathways than in 3-D situations (Noetzli and Gruber, 2009). Based on these previous findings, it can be assumed that the 2-D temperature fields are acceptable for drawing reasonable patterns of permafrost distribution and changes.

Additionally, the model resolution (4 m), defined by the initial RST resolution (Sect. 3.2.1), and later refined for spatial discretisation of the model domain (Sect. 3.2.2), is sufficient to represent the main topographical control on the rock wall temperature distribution, and to drive long-term changes. Permafrost distribution modelling often suffers from coarse topographical resolution which does not represent the natural variability of environmental parameters (Etzelmüller, 2013). Metric resolution is essential for a realistic simulation of rock wall permafrost at the summit scale (Magnin et al., 2015a). The satisfying quality of the 2-D model outputs for the current period (Sect. 4.2.1) confirms that the model resolution is accurate enough to address long-term permafrost changes at the scale of the selected sites. Simulations at shorter spatial and temporal scales, especially in the uppermost layers, would certainly require higher spatial resolution of the model domain and consistent input data.

Subsurface thermal parameters have been defined by published values for hard rock. The thermal conductivity and heat capacity are assumed to be homogeneous in the model domain whereas they in fact vary with frozen/thawed conditions in the natural environment due to the changing properties of the interstitial ice/water. The variable state of the interstitial ice or water results in seasonal variation in the thermal conductivity of the porous and saturated rock media (Wegmann et al., 1998) and longer-term permafrost changes. This is accounted for in our modelling approach throughout Eqs. (3) and (4). In natural conditions, however, this changing conductivity is heterogeneous in the rock mass due to variable fracture density and porosity, which is not taken into account in our study. Based on the model evaluation (Sect. 4.2), this lack of consideration of heterogeneous porosity and related latent heat processes is not an obvious limitation for simulating thermal fields in depth. However, to simulate near-surface thermal processes at finer spatial and temporal resolutions, heterogeneous porosity would have to be taken into account.

We selected conservative values for granitic rock but tested the model sensitivity to different thermal conductivity values (2.7 and 3 W m-1 K-1). Results confirmed findings from previous studies: a lower conductivity increased the geothermal heat flux control (Maréchal et al., 2002; Noetzli et al., 2007) but did not lead to substantial changes in the modelled temperature fields (Kukkonen and Safanda, 2001). Furthermore, the thermal conductivity is naturally anisotropic (Goy et al., 1996), whereas it is considered isotropic in our modelling approach. Increasing the conductivity in horizontal directions increases the topo-climatic control, whereas increasing the conductivity in a vertical direction gives more importance to the geothermal heat flux (Noetzli and Gruber, 2009).

The influence of the geothermal heat flux appears to be of relatively high importance for running steady-state conditions. Equilibrium conditions without the influence of the upwards and deep-seated flow only depend on climate control, which can lead to highly different conditions than when balancing the respective influences of the geothermal heat flow and the climate. In Fig. 9, an example of the geothermal heat flux effect is given for the GM site, which is the most sensitive to this parameter given its relatively low relief and wide geometry. Without geothermal heat flow, the equilibrium condition for the LIA shows almost entirely cold permafrost conditions with more vertical isotherms. However, when running a transient simulation, the influence of the geothermal heat flux is less significant on the temperature range. It only affects the shape of the isotherms leading to permafrost retreat in the core of the summit. Simulations without geothermal heat flux were also run with a lower thermal conductivity (2.7 W m-1 K-1) and a monthly time step, which had no impact on the results of this study (Fig. 9). Only the uppermost layers show different temperature ranges due to different time steps in the forcing data (daily versus monthly), but shallow thermal processes are beyond the scope of this study. Idealised test cases suggested that mountain summits are decoupled from the deep-seated geothermal flow influence (Noetzli et al., 2007), but such settings are not representative of most of the alpine study cases, which are more or less sharp and elevated.

Energy transfer inside the rock mass is mainly driven by heat conduction processes, whereas fluid flows can be, in a first approximation, neglected to simulate long-term changes (Kukkonen and Safanda, 2001). Nevertheless, advective heat transport by water circulation along fractures may locally warm the bedrock at depth (Hasler et al., 2011a). Conversely, air circulation in open clefts would instead cool the bedrock (Hasler et al., 2011b; Magnin et al., 2015c). These non-conductive heat transfers are not accounted for in our modelling approach. The evaluation with borehole temperatures clearly shows their effect in the shallow layers (the first 6 m below the surface, Sect. 4.2.1) while temperatures at deeper layers are accurately represented with consideration of 2-D heat conduction only.

The melting of ice bodies in the fractures may also be expected in the natural environments, as suggested by recent observations in rockfall scars (Ravanel et al., 2010b). Ice melting may significantly dampen and lower the rate of temperature changes by depth by the consumption of latent heat (Wegmann et al., 1998; Kukkonen and Safanda, 2001). Some ice-filled fractures can turn into thawing corridors during the thawing season (Krautblatter and Hauck, 2007), which can favour the melting of ice-filled fractures, and degrade rock wall permafrost in unexpected areas and depths (Hasler et al., 2011a). Such processes were approximated with a relatively high porosity value in the model domain (5 %), which fails to represent the anisotropic and heterogeneous character of such processes. Further developments to gain data on rock wall structure are highly encouraged.

The rate of change between equilibrium conditions in 1850 and the early 1990s was in the same range as the one experienced during the past three decades. Indeed, for the first time period, the thermal perturbation has been detected down to 30 m below the surface (the depth at which the shape of the steady-state isotherms have changed), whereas it reaches at least twice that depth in 2015. Narrow peaks such as the AdM have been entirely affected by air temperature increases since the end of the LIA due to the short distance between the sides from which the signal penetrates, and the resulting intense lateral heat fluxes, especially at the top (Noetzli et al., 2007). In wider geometries, where N and S-facing surfaces are both distant by more than 150 m, such as the GPA and the GM, the core has remained intact.

Temperature changes during 1850–2015 were not as high at the GM as for the two other sites, and were not as high as the air temperature change, because the entire summit was in the temperature range within which latent heat exchanges occur. This pattern is aligned with the global trend, showing the delaying effect of latent heat uptake in boreholes with temperatures close to 0 ∘C (Romanovsky et al., 2010). The rate of temperature change strongly increases during the 21st century when the GM temperature becomes almost entirely positive.

Between the 1990s and 2010s, cold permafrost disappeared and warm permafrost largely penetrated in the shallow layers of the GM and AdM SE faces respectively. Quantitative interpretations about the permafrost lower boundary and its changes are limited by the discontinuous character of mountain permafrost mainly governed by local conditions (Etzelmüller, 2013), topography and transient controls; these different influences are difficult to distinguish (Noetzli and Gruber, 2009). Nevertheless, results of our 2-D simulations clearly suggest that climate change in between the LIA termination and the 2010s, especially since the 1990s, have led to permafrost disappearance below the S-exposed faces at least up to 3300 m a.s.l. (top of the GM), but not above 3700 m a.s.l. (foot of the AdM SE face). Thus, lower boundaries of snow-free and hard rock wall permafrost lie within this elevation interval, but Magnin et al. (2015a) have suggested that isolated permafrost bodies could exist down to 2800 m in favourable S-exposed slopes where conduction is not the prominent heat transfer process (due to high fracturing for example). Failing to account for snow and fracturing parameters could lead to a 3 ∘C underestimation of bedrock temperature in S-exposed rock walls (Hasler et al., 2011b).

Warm permafrost has thickened below the S-exposed face at least up to 3850 m a.s.l. during the past two decades while it was extending below the N-facing slopes up to 3300 m. Although warm permafrost reaches 20 m depth below the S-exposed AdM face in 2015, this depth cannot be extrapolated to other sites due to the site-specific effects of the opposite N face.

By the end of the 21st century, even the core of the relatively wide summits such as the GPA will be strongly affected by the projected increase in air temperature. Warm permafrost in the S-exposed faces will extend with depth by at least up to 4300 m a.s.l., and in the N-facing faces according to the RCP4.5. Permafrost will certainly disappear in all aspects below 3300 m, and up to 3850 m a.s.l. at least (top of the AdM) in the S-exposed faces. Following the most pessimistic scenario, permafrost will disappear in the subsurface of the S-exposed rock walls at least up to 4300 m a.s.l. (top of the GPA), while cold permafrost will still exist at the same elevation in N-facing slopes if the S-face influence does not affect it. At lower elevation such as at the AdM, warm permafrost will still occur below the N-exposed faces, but could disappear in the narrowest sections due to the S-facing slope influence. When both mountain sides do not allow for permafrost conditions any more, such as at the GM, the permafrost body will retreat downwards, in the core of the summit.

These degradation patterns locally depend on the topographical control, especially the summit width. This either reinforces the intensity of the climatic signal in narrow geometries by mixing S-facing and N-facing slope influences, or maintains independency between the shallow layers of the S- and N-exposed slopes when it is wide enough (multiple-hm). Site-specific patterns make the concept of “lower limit” not suitable to describe rock wall permafrost distribution and changes.