PUBLICATIONS. Journal of Geophysical Research: Earth Surface

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1 PUBLICATIONS Journal of Geophysical Research: Earth Surface RESEARCH ARTICLE Key Points: 3-D thermal regime model forced by daily air temperature Fully coupled thermo-mechanical model including firn densification process Take into account melt water percolation and refreezing processes A 3-D thermal regime model suitable for cold accumulation zones of polythermal mountain glaciers A. Gilbert 1,2, O. Gagliardini 1,2,3, C. Vincent 1,2, and P. Wagnon 2,4,5,6 1 CNRS, LGGE (UMR5183), Grenoble, France, 2 Univ. Grenoble Alpes, LGGE (UMR5183), Grenoble, France, 3 IUF, Paris, France, 4 IRD, LGGE (UMR5183), Grenoble, France, 5 IRD, LTHE (UMR5564), Grenoble, France, 6 ICIMOD, Kathmandu, Nepal Correspondence to: A. Gilbert, gilbert@lgge.obs.ujf-grenoble.fr Citation: Gilbert, A., O. Gagliardini, C. Vincent, and P. Wagnon (2014), A 3-D thermal regime model suitable for cold accumulation zones of polythermal mountain glaciers, J. Geophys. Res. Earth Surf., 119, , doi:. Received 30 APR 2014 Accepted 16 AUG 2014 Accepted article online 23 AUG 2014 Published online 19 SEP 2014 Abstract Analysis of the thermal and mechanical response of high altitude glaciers to climate change is crucial to assess future glacier hazards associated with thermal regime changes. This paper presents a new fully thermo-mechanically coupled transient thermal regime model including enthalpy transport, firn densification, full-stokes porous flow, free surface evolution, strain heating, surface meltwater percolation, and refreezing. The model is forced by daily air temperature data and can therefore be used to perform prognostic simulations for different future climate scenarios. The set of equations is solved using the finite element ice sheet/ice flow model Elmer/Ice. This model is applied to the Col du Dôme glacier (Mont Blanc area, 4250 m a.s.l., France) where a comprehensive data set is available. The results show that the model is capable of reproducing observed density and velocity fields as well as borehole temperature evolution. The strong spatial variability of englacial temperature change observed at Col du Dôme is well reproduced. This spatial variability is mainly a result of the variability of the slope aspect of the glacier surface and snow accumulation. Results support the use of this model to study the influence of climate change on cold accumulation zones, in particular to estimate where and under what conditions glaciers will become temperate in the future. 1. Introduction The thermal regime of glaciers influences both their dynamics and their hydrology. In particular, many glacier characteristics such as basal sliding, ice viscosity, and liquid water permeability depend on temperature [Cuffey and Paterson, 2010]. Although the relationship between glacier mass balance and climate has been widely investigated, fewer studies have focused on the response of the thermal regime of glaciers to climate change. This is, however, a key to understanding dynamical and hydrological changes in glaciers and improving glacier hazard analysis [Gilbert et al., 2012; Faillettaz et al., 2011; Huggel et al., 2004; Haeberli et al., 1989] and the reconstruction of paleoclimatic records [Gilbert et al., 2010; Gilbert and Vincent, 2013]. The destabilization of hanging mountain glaciers is a great threat to life and property in populated mountain areas. Hanging glaciers are quite common in the Alps (Taconnaz glacier, South Face of Grande Jorasse, Weisshorn, Altels glacier ) and therefore represent an important glacial hazard. For example, in 1965, a large part of Allalin glacier (Switzerland) broke off and killed 88 employees of the Mattmark dam construction site [Röthlisberger, 1981; Raymond et al., 2003]. Ice falls on Taconnaz glacier were responsible for important infrastructure damage in the Chamonix valley (France) in February One of the causes of hanging glacier destabilization is linked to thermal regime changes due to climatic variations. For example, thermal changes are suspected to be the cause of the gigantic break-off of Altels hanging glacier in 1895 following the development of a temperate basal layer below the glacier [Faillettaz et al., 2011]. Cold hanging glaciers are generally located in cold accumulation zones which are particularly sensitive to climate change due to the influence of surface melting [Gilbert et al., 2014]. The glacier response to these changes is complex: glacier velocity, temperature, density, and thickness are all linked, forming a coupled system. For instance, velocity and temperature interact through the dependence of viscosity on temperature, strain heating, and advective energy transport [Cuffey and Paterson, 2010]. Density and velocity fields are linked through mass conservation and firn rheology [Gagliardini and Meyssonnier, 1997]. Glacier thickness depends on the velocity field and the surface mass balance [Cuffey and Paterson, 2010]. Investigation of the impact of climate change on a cold firn dominated glacier therefore needs to take into account every component of this coupled system and carefully define the boundary processes. GILBERT ET AL American Geophysical Union. All Rights Reserved. 1876

2 For this study, we developed a new model that was applied to the Col du Dôme glacier (Mont Blanc area, France), already studied by Gagliardini and Meyssonnier [1997]. These authors focused only on the flow law for firn along a two-dimensional flow line and assumed isothermal conditions. They validated the firn rheology by comparing modeled surface velocities with measurements obtained from a stake network and modeled density profiles and firn dating with measurements obtained from ice core analysis. Lüthi and Funk [2001] extended this work by including thermal model coupling in a study at Col Gnifetti (Swiss Alps). They demonstrated the importance of taking into account the past thermal history of the bedrock below the glacier to adequately model heat flux at the glacier base. Transient simulations were carried out assuming a parameterized surface temperature increase and a constant velocity field. They concluded that their model simulated the observed englacial temperature well. In our model, the full-stokes thermo-mechanical model including firn rheology [Gagliardini and Meyssonnier, 1997; Lüthi and Funk, 2000; Zwinger et al., 2007] is coupled with a firn model accounting for near-surface processes [Gilbert et al.,2014].thisisthefirst time such a model has been forced by physical surface boundary conditions including meltwater percolation and refreezing processes. Boundary conditions are therefore determined by both surface temperature and liquid water flux. The thermal model is based on the work of Aschwanden et al. [2012] using the enthalpy variable instead of temperature. This makes it possible to properly account for the presence of temperate ice. In the present study, four main improvements are made to allow reliable modeling of future transient thermal regimes: (i) surface meltwater and its spatial variability are included as model inputs, (ii) the transient response of velocity, density, free surface, and temperature are solved in a fully coupled manner, (iii) the enthalpy method is used to make it possible to account for temperate ice in an energy conservative manner by estimating water moisture [Hutter, 1982], and (iv) water percolation and refreezing are modeled to account for latent heat released into the firn. Including meltwater as input is essential because the thermal regime of a cold glacier is extremely dependent on surface melt intensity that induces strong spatial variability in firn temperature via the latent heat released through water percolation and refreezing. This induces nonlinear englacial temperature warming in comparison with air temperature rise [Suter, 2002; Gilbert et al., 2014]. The next section describes the model. Section 3 presents the application of the model to Col du Dôme, and section 4 investigates glacier evolution over the 20th century and in the future. Conclusions are drawninsection5. 2. Model Description The thermo-mechanical model is mainly based on the work of Gagliardini and Meyssonnier [1997], Lüthi and Funk [2000, 2001], and Zwinger et al. [2007] with two main improvements in the mathematical formulation, i.e., (i) instead of glacier temperature, the enthalpy variable is used [Aschwanden and Blatter, 2009] to specifically take into account the temperate ice (i.e., ice at the pressure melting point) and (ii) a surface water percolation scheme has been implemented to account for meltwater transport and refreezing in firn [Gilbert et al., 2012, 2014], making it possible to carry out reliable transient simulations in cold firn using meteorological data [Gilbertetal., 2014]. All variables and parameters are listed in Table 1. The mathematical model has been solved using the finite element code Elmer/Ice [e.g., Gagliardini et al., 2013] based on the Elmer open-source multi-physics package (see for details) Field Equations Water Percolation Water saturation S is calculated from the pure-advection equation. We define the effective water saturation S * [Colbeck and Davidson, 1973]: S S Sr ¼ ð Þ ð1 SrÞ ; (1) where S (dimensionless) is the water saturation in the firn and Sr is the irreducible water saturation that is permanently retained by capillary forces. If S < Sr, there is no flow of water (S * =0)andifS > Sr, S * is gravitationally advected according to the following equation: S t þ v S w z ¼ 0 ; (2) GILBERT ET AL American Geophysical Union. All Rights Reserved. 1877

3 Table 1. Parameters and Variables Used in the Model With Their Respective Values When Available Symbol Values and Units Clausius-Clapeyron constant β KPa 1 Function a(d) of firn rheological law a Flow rate factor A(T) Pa s 1 Surface accumulation a c myr 1 Function b(d) of firn rheological law b Ice heat capacity c p Jkg 1 K 1 Bedrock heat capacity Cp rock Kkg 1 K 1 [Lüthi and Funk, 2001] Relative density D - Heat flux at domain base f b Wm 2 Acceleration due to gravity g 9.81 m s 2 Lapse rate g T Km 1 Enthalpy H J kg 1 Enthalpy of fusion Hf J kg 1 Firn/ice conductivity k W K 1 m 1 Bedrock conductivity k rock 3.2 W m 1 K 1 [Lüthi and Funk, 2001] Latent heat of fusion L Jkg 1 Melting factor m f kg m 2 day 1 Glen s low exponent n 3 Pressure p Pa Pressure of water triple point p ptr Pa Latent heat Q lat Wm 3 Free surface elevation s m Water saturation S Effective water saturation S * Residual saturation Sr [Gilbert et al., 2014] Firn/ice temperature T K Reference temperature for enthalpy T K Temperature of water triple point T ptr K Pressure-melting point temperature T m K Surface temperature T s K Steady state temperature at level z 0 T steady K Surface velocity u s,v s,w s myr 1 Water percolation velocity v w ms 1 [Gilbert et al., 2014] Air temperature measurement elevation z m Strain rate tensor ε s 1 Enthalpy diffusivity κ kg m 1 s 1 Moisture diffusivity κ kg m 1 s 1 [Aschwanden et al., 2012] Snow surface density ρ s 380 kg m 3 Firn density ρ kg m 3 Bedrock density ρ rock 2800 kg m 3 [Lüthi and Funk, 2001] Water density ρ w 1000 kg m 3 Stress tensor σ Pa Velocity field v ms 1 Porosity Φ Water content ω kg m 3 where v w is the water percolation velocity (m s 1 ). We assume that the water percolation velocity is uniform and constant within the firn (see Table 1). We further assume here that meltwater percolation is limited and does not reach the impermeable ice. Without this assumption, it would be necessary to model slush formation and runoff Enthalpy Transport Among polythermal schemes, a main advantage of the enthalpy method is that it describes the Cold- Temperate Transition Surface (CTS) as a set level of the enthalpy variable. No explicit surface representation scheme is required and no a priori restrictions apply to CTS shape [Aschwanden et al., 2012]. This makes it possible to take moisture content in temperate ice into account in a relatively simple manner. It can moreover be implemented relatively easily in a numerical model. Enthalpy is calculated as proposed by Aschwanden et al. [2012], neglecting heat flux arising from the temperature gradient in the temperate ice: ρ H t þ v H ¼ ðκ HÞþtr ðσϵ ÞþQ lat ; (3) GILBERT ET AL American Geophysical Union. All Rights Reserved. 1878

4 where ρ is firn density (kg m 3 ), H enthalpy (J kg 1 ), v the ice velocity field, κ enthalpy diffusivity (kg m 1 s 1 ), tr(σϵ ) the strain heating (J m 3 s 1 ), and Q lat (J m 3 s 1 ) a source term coming from meltwater refreezing Stokes Equations and Mass Conservation Neglecting the acceleration terms in the momentum balance yields the Stokes equations: divðσþþρg ¼ 0 ; (4) where σ is the Cauchy stress tensor (Pa) and g acceleration due to gravity (m s 2 ). Mass conservation reads: ρ t þ ðρvþ ¼ Q lat ; (5) L where Q lat is the latent heat released by meltwater refreezing (equation 3) and L is the latent heat of fusion (J kg 1 ) Free Surface Changes in the upper surface elevation are computed by solving: s t þ u s s x þ v s s y w s ¼ ρ w a c ; (6) ρ s where s(x,y,t) is the free surface elevation (m), u s, v s, and w s are the surface velocity (m s 1 ) components on axes x, y, and z, respectively, a c the surface accumulation (m w.e. s 1 ), ρ w the density of water (kg m 3 ), and ρ s the density of snow at the surface (kg m 3 ) Constitutive Equations Flow Law We adopt the flow law for compressible firn initially proposed by Gagliardini and Meyssonnier [1997] and later applied by Zwinger et al. [2007] and Lüthi and Funk [2000]. This rheological law expresses the relationship between the deviatoric parts (τ ¼ σ trðσþ tr σ 3 I and ė ¼ ϵ trðþi) ϵ and the isotropic parts (p ¼ ð Þ 3 and ϵ ṁ¼ trðþ) ϵ of the stress tensor and those of the strain rate tensor, such as: ϵ ṁ ¼ 2Abσ n 1 D p ; (7) τ ¼ 2 a ð2a 1 n Þ 1 n ϵḋ ė ; (8) n where a = a(d) and b = b(d) are only functions of the relative density (D=ρ/ρ ice where ρ is the firn density), A=A(T)is the flow rate factor in Glen s flow law, n is Glen s law exponent, and ϵ Ḋ and σ D are invariants of the strain rate tensor and stress tensor, respectively: ϵ 2D ¼ γ2 e a þ b ; γ2 e ¼ 2 trðþ2 ė ; (9) σ 2 D ¼ aτ2 þ bp 2 : (10) The functions a and b are calculated using the analytical solution proposed by Duva and Crow [1994] for high relative densities (0.81 < D < 1.0): 0 1 a 0 ðdþ ¼ 1 þ 21 ð DÞ=3 ; b D 2n= ðn þ 1Þ 0 ðdþ ¼ 3 ð1 DÞ 1=n 2n= h ia : (11) 4 n 1 ð1 DÞ 1=n In this way, the limiting case where D = 1.0 (ice density) gives a = 1 and b = 0 and the previous flow law (equations (7) and (8)) becomes simply the classical incompressible Glen s flow law. For smaller relative densities, we use the parameterization for a and b proposed by Zwinger et al. [2007] that reads: ad ð Þ ¼ bd ð Þ ¼ expð13: :78652DÞ; 0:4 D < 0:81 a 0 ðdþ; 0:81 < D 1:0 expð15: :46489DÞ; 0:4 D < 0:81 b 0 ðdþ; 0:81 < D 1:0 (12) (13) GILBERT ET AL American Geophysical Union. All Rights Reserved. 1879

5 These relations (equations (12) and (13)) give acceptable results for simulated density profiles and surface velocities at Col du Dôme (see next section). Numerical experiments at Col du Dôme show that the parameterization used by Lüthi and Funk [2000] leads to high values of a(d) and b(d) for relative densities less than 0.45, making surface velocities extremely sensitive to surface density and producing excessively fast densification in comparison with measurements. The Zwinger et al. [2007] parameterization was therefore adopted. Flow rate dependence on temperature is calculated using the Arrhenius law with the activation energy and pre-exponential factors values as in Cuffey and Paterson [2010]. For the temperate ice, flow rate dependence on water content is calculated according to Lliboutry and Duval [1985] Thermodynamic Settings Enthalpy is defined as a function of water content ω and temperature T (K) by: HT; ð ωþ ¼ 8 >< >: T m ðpþ T 0 T C p ðtþdt; H < H f ðpþ T 0 C p ðtþdt þ ωl; H H f ðpþ ; (14) where C p is the heat capacity (J K 1 kg 1 ), T 0 the reference temperature for enthalpy (set to 200 K), L the latent heat of fusion, and p the pressure (Pa). The enthalpy of fusion H f (J kg 1 ) is the enthalpy of temperate ice with zero liquid water content and defined from the pressure-melting point temperature T m (K) according to the Clausius-Clapeyron relationship: T m ¼ T ptr β p p ptr ; (15) H f ðpþ ¼ T m ðpþ T 0 C p ðtþdt ; (16) where β is the Clausius-Clapeyron constant, p ptr (Pa) the triple point water pressure, and T ptr (K) the triple point water temperature. The heat capacity Cp is calculated as a function of temperature [Yen, 1981]: C p ¼ 152:5 þ 7:122 T : (17) The enthalpy diffusivity is expressed as: 8 < kðρ; TÞ κ ¼ C p ðtþ ; H < H f ðpþ ; (18) : κ 0 ; H H f ðpþ where k is the thermal conductivity and κ 0 the moisture diffusivity in temperate ice. The conductivity is calculated using the relationship proposed by Calonne et al. [2011] (ρ in kg m 3 ): k ¼ 2:510 6 ρ 2 1: ρ þ 0:024 : (19) The temperature dependence of the thermal conductivity is given by Cuffey and Paterson [2010] (T in K): k ice ¼ 9:828 exp 5: T : (20) It is assumed that the thermal conductivity of firn shows the same temperature dependence, which is accounted for by a multiplicative factor in equation (19): kt ð Þ ¼ k iceðtþ k : (21) k ice T ptr In the percolation zone, enthalpy is limited to enthalpy of fusion H f, and water content is taken into account by the water saturation term from the percolation equation (equations (1) and (2)). As all meltwater refreezes in the first few meters of the firn, the density itself does not put an additional limit on the amount of refreezing here (the density cannot exceed the density of ice after refreezing). At each time step, the maximum liquid water that can refreeze (S max )iscalculatedas: ð S max ¼ H f HÞρ ; (22) LΦρ w GILBERT ET AL American Geophysical Union. All Rights Reserved. 1880

6 where Φ is the porosity. This gives: 8 SΦρ w L >< ; S < S max Q lat ¼ dt >: S max Φρ w L ; S S max dt 0; S < S max S ¼ ðs S max Þ; S S max (23) ; (24) where dt is the time step Boundary Conditions Surface We assume a Dirichlet condition for the enthalpy H = H(T s ) and density (ρ = ρ s ), where the subscript s indicates the snow surface. A melt flux f w (kg m 2 s 1 ) is specified to solve the percolation equation (equation (2)). The surface is treated as a stress-free boundary for the Stokes equation (equation (4)) Bedrock As pointed out by Lüthi and Funk [2001], the thermal state of the bedrock below the glacier must be taken into account. A bedrock model is therefore included, and a basal heat flux f b (W m 2 )isdefined several tens of meters (depending on the simulation duration) below the glacier base. Sliding is neglected at the bedrock interface due to cold conditions. The enthalpy equation is solved over the whole domain including the bedrock. To avoid creating an enthalpy field discontinuity due to a heat capacity discontinuity, we use the heat capacity of ice for the bedrock. We compensate for the error introduced by adopting this incorrect heat capacity value by using an apparent density ρ a for the bedrock: ρ a ¼ ρ rock C p rock ; (25) C p where ρ rock and Cp rock are, respectively, the density and heat capacity of bedrock. Diffusivity κ rock is set to: κ rock ¼ k rock C p ; (26) The bedrock temperature at the rock/ice interface calculated from the enthalpy field is then used as a Dirichlet condition to solve a purely diffusive heat equation in the bedrock and obtain the correct temperature field in the bedrock. 3. Application: Investigation of Climate Change Impact on a Cold Accumulation Zone (Col du Dôme, Mont Blanc Area, France) 3.1. Study Site Col du Dôme is a cold accumulation zone located at 4250 m above sea level (a.s.l.) near the Mount Blanc summit (Figure 1). This site has been instrumented for many years and provides a valuable opportunity to both calibrate and validate our model Field Data Temperature and Density Englacial temperature measurements were performed from the surface to bedrock in seven boreholes drilled between 1994 and 2011 at three different sites located between 4240 and 4300 m a.s.l. (Figure 1). Ice thicknesses were 40, 126, and 103 m at sites 1, 2, and 3, respectively (see Gilbert and Vincent [2013] for more details). Density profiles were measured along ice cores extracted in 1994 (site 2) and 1999 (sites 1 and 3). Note that the density profile reported for site 2, where temperature was measured, comes from another 140 m drilling performed also in 1994 but 30 m apart from site Velocity Fields and DEMs Surface velocity fields were determined from a stake network surveyed by D-GPS since 1994 [Vincent et al., 2007]. Subsidence velocities are calculated by removing the vertical velocities due to slope from the total vertical velocities measured by D-GPS. For a steady state mass balance, subsidence velocities are equal to the GILBERT ET AL American Geophysical Union. All Rights Reserved. 1881

7 Latitude ( ) Longitude ( ) Figure 1. Map of the study area. Stars are the locations of the drilling sites. The blue line is the periphery of the three-dimensional model. The red dashed line shows the area of measured bedrock elevations. mean snow accumulation [Vincentetal., 2007]. Comparison between mean snow accumulation and subsidence velocities confirms that the surface mass balance is close to a steady state [Vincent et al., 2007]. Therefore, subsidence velocities can be considered as the mean surface mass balance. The surface DEM was measured in 2005 by D-GPS, and the bedrock DEM was determined by GPR measurements in 1993 and 1994 [Vincent et al., 1997, 2007]. Bedrock measurements (red dashed line in Figure 1) do not cover the entire modeling zone. Bedrock elevations were therefore extrapolated in the outflow zone by maintaining a constant slope (inset in Figure 1) Radioactive Dating The radioactive fallout from atmospheric thermonuclear tests [Picciotto and Wilgain,1963],conductedin1954 and , and from the Chernobyl accident in 1986 [Pourchet et al., 1988] provide well-known radioactive horizons in glaciers that can be used for absolute dating. Gamma radioactivity analysis in ice cores drilled in the Col du Dôme has been used to locate these events and date the corresponding ice core levels [Pinglot and Pourchet, 1995]. Dating was carried out on the 1994 ice core from site 2 and the 1999 ice core from site Meteorological Data Daily maximum and minimum air temperatures used to perform transient simulations come from the Lyon-Bron meteorological station located ~200 km west of the glacier. These data are available from 1907 to present and are well correlated with air temperature variations at Col du Dôme [Gilbert and Vincent, 2013]. Precipitation data from Besse-en-Oisans meteorological station located ~100 km south of the Mont Blanc area in the Grandes Rousses range at 1525 m a.s.l. were used to reconstruct Col du Dôme snow accumulation since Vincent et al. [2007] have shown a good correlation between Besse-en-Oisans precipitation and snow accumulation at site 2 at Col du Dôme. We use this correlation to reconstruct snow accumulation at site 2. Snow accumulation is then spatialized over the whole study area using the measured ratio between snow accumulation at site 2 and snow accumulation at other places. Future climate scenarios come from the ENSEMBLES climate scenario database ( [Van der Linden and Mitchell, 2009]. From this database, we selected three realizations: CNRM-RM5.1_ARPEGE (ARPEGE), METNOHIRHAM_HadCM3Q0 (HadCM3Q0), and MPI-M-REMO_ECHAM5 (ECHAM5) (with the short label used hereafter indicated in brackets). Climate scenarios were produced by Regional Climate Models (RCM) GILBERT ET AL American Geophysical Union. All Rights Reserved. 1882

8 forced by Global Circulation Models (GCM) using the A1B scenario [Nakicenovic et al., 2000] and downscaled using the empirical-statistical error correction method (quantile mapping [Themessl et al., 2011]) at Col du Grand Saint Bernard meteorological station located 50 km east of the glacier at 2469 m a.s.l.. For our study, we used a 100 year time series ( ) of daily mean, minimum, and maximum air temperatures Model Setup Geometry and Meshing Periphery lines for meshing are shown in the inset in Figure 1 (blue lines) and follow the surface flow lines except at the front (dashed blue line in the inset in Figure 1) and the south-east part of the periphery. A threedimensional mesh with 4515 nodes and 35 vertical layers is obtained by extrusion of an unstructured triangular 2D mesh (Figure 2, 50 m resolution). Extrusion makes it possible to obtain nodes that are aligned in the vertical direction, which is required in particular to solve the water percolation equation. The mesh is refined up to 50 cm along the vertical axis close to the surface to solve the heat and water percolation equations on a daily time scale (see Figure 2) Boundary Conditions Our domain is constrained by five boundaries which are the glacier surface, the lateral boundaries, the outflow limit, the ice-bedrock interface, and the base of the bedrock model (Figure 1 inset and Figure 2) Outflow and Lateral Limits The outflow limit is shown in Figures 1 and 2. The boundary condition for the Stokes equation is given as a pressure set to the hydro-static pressure. A zero-flux boundary condition is assumed for the other equations on the outflow boundary. Given that lateral boundary no. 1 is perpendicular to the elevation contours (Figure 1) and the horizontal velocity is very close to zero (<3myr 1 ) on boundaries no. 2 and 3, a zero-flux boundary condition is assumed for all equations on these boundaries. However, temperature simulations show that lateral boundary no. 1 near drilling site 1 seems to be influenced by a negative lateral heat flux coming from the north-facing slope beyond boundary no. 1. For the transient simulations, a comparison with temperature measurements in the deeper part of borehole no. 1 suggests that englacial temperatures are influenced by excessively strong bedrock warming although surface temperature is well modeled. To avoid extending the simulation domain in order to model this heat flux, we therefore introduce a negative normal heat flux, called F lat (set to Wm 2 ) as a boundary condition on this side (throughout the whole thickness) Surface and Basal Conditions Spatial variability of the surface temperature is calculated using: T s ðx; yþ ¼ T steady g T ðx; yþ ðsx; ð yþ z 0 Þ ; (27) where T s (x,y) is the surface temperature, T steady the steady state temperature at altitude z 0, z(x,y) the surface elevation, and g T (x,y) thelocallapseratedefining the gradient between air temperature at altitude z 0 and surface temperature at Col du Dôme. T steady is set to 11.4 C (mean air temperature recorded at Lyon-Bron meteorological station over the period ), and z 0 is set to 200 m a.s.l. (altitude of Lyon-Bron station). Gilbert et al. [2014] found that g T was equal to 5.95, 5.70, and Km 1 at sites 1, 2, and 3, respectively. Assuming that these values depend on the mean local wind speed, which is the main driver controlling local snow accumulation, g T was correlated to mean snow accumulation using measured accumulation at the three drilling sites. The obtained linear relationship between mean snow accumulation and g T (R 2 = 0.97, n = 3) was used to extrapolate g T over the entire study area. Assuming that the glacier surface remained approximately unchanged over the last century [Vincentetal., 2007], the surface mass balance is set to the measured subsidence velocities (Figure 7B) to solve the free surface equation. Where no measurements are available, the surface mass balance is set to the modeled steady state subsidence velocities using the measured surface elevation and assuming steady state. For transient simulations, only the enthalpy and water percolation equations have time-dependent boundary conditions. Surface temperature is variable as a function of time: T s ðx; y; tþ ¼ T air ðþ t g T ðx; yþ ðsx; ð yþ z 0 Þ; (28) where T air (t) is the daily mean air temperature measured at altitude z 0. Liquid water input comes from a meltwater flux calculated according to Gilbert et al. [2014] using a degree-day model: ( T max ðx; y; tþ T ptr mf ðx; yþ; T max ðx; y; tþ > T f Mx; ð y; tþ ¼ ; (29) 0; T max ðx; y; tþ T f GILBERT ET AL American Geophysical Union. All Rights Reserved. 1883

9 Figure 2. Glacier topography and mesh used in the 3D model. The three drilling sites from Figure 1 are shown. Ice and firn are shown in color and bedrock in gray. where M is the daily amount of surface melt (m w.e. d 1 ), m f (x,y) the melt factor (m w.e. d 1 K 1 ), and T max the daily maximum temperature. The melt factor m f is calculated as a function of potential solar radiation using the relationship given by Gilbert et al. [2014]. Daily maximum temperature T max (x,y,t) is calculated using the lapse rate g T and the measured daily maximum air temperature at Lyon-Bron station. Aconstantheatflux is specified at an elevation of 3980 m a.s.l., at the base of the bedrock model (see Figure 2). Bedrock thickness ranges consequently from 40 m (near the outflow limit) to 280 m (at drilling site 1). The heat flux is set to Wm 2 in order to match the measured basal temperature at the three drilling sites. This value is close to the one used by Lüthi and Funk [2001] at Col Gnifetti. A no-sliding condition is assumed at the bedrock interface Numerical Solution Using Elmer/Ice Steady State Simulations A steady state is calculated using the following method. An initial diagnostic simulation is performed using a constant initial density field and surface elevation, solving the steady state temperature and velocity fields. The initial density field for this first step is estimated as a function of depth using the 1-D densification model of Herron and Langway [1980], and the measured DEM is used to initialize the surface elevation. This first diagnostic run is then used as an initial condition for a transient simulation where density, surface elevation, and velocity evolve until all variables reach a quasi-equilibrium. This equilibrium is reached after 50 years when both density and free surface no longer change.the time step for this transient simulation is set to one year. The resulting density field and surface elevation are then used to run a new diagnostic simulation (similar to the first one) to obtain a new temperature field. This process is repeated until all variables reach equilibrium (see diagram in Figure 3) Transient Simulations Time-dependent boundary conditions are the surface temperature and the meltwater input, varying on a daily timescale. Enthalpy and water percolation equations are therefore solved for a daily time step. Every 100 days, the new temperature field is used to calculate a new velocity field and update the free surface and density field (see Figure 4). Given the high computational-time cost of solving the Stokes equations, the velocity field cannot be calculated for a daily time step. 4. Results 4.1. Steady State Simulations Free Surface and Bedrock Adjustment For given surface and bedrock topographies, the modeled subsidence velocities in a steady state simulation indicate the required accumulation to maintain constant geometry. However, the bedrock topography is not GILBERT ET AL American Geophysical Union. All Rights Reserved. 1884

10 Figure 3. Glacier topography and mesh used in the 3D model. The three drilling sites from Figure 1 are shown. Ice and firn are shown in color and bedrock in gray. known as accurately as the surface topography or the accumulation. Therefore, the bedrock topography is deformed progressively so that the accumulation needed to keep the surface geometry constant matches the observed accumulation. The bedrock topography is adjusted using the following method: at each iteration a new surface elevation is computed from the free surface equation using the modeled velocity field and the measured mean accumulation. The modeled surface variations are then transferred the bedrock elevation in order to keep the surface elevation constant. The process is repeated until the glacier thickness reaches equilibrium. As shown in Figure 5, the modified glacier thickness is in better agreement with borehole measurements than was the initial measured thickness. This new bedrock DEM will be used for all the following applications Density and Velocity Fields Due to the very high altitudes involved, little meltwater refreezing occurs, and we therefore assume that refreezing water does not significantly influence the density field. The source term in equation (5) is therefore neglected. If lower altitudes were involved, densification by percolation and refreezing of meltwater would Figure 4. Diagram of the transient solution method. In red: boundary condition inputs. In blue: solved equations. In green: solved variables. Note that the only time-dependent boundary conditions are the surface temperature and the surfacemelt flux. GILBERT ET AL American Geophysical Union. All Rights Reserved. 1885

11 Mesh thickness (m) Borehole thickness (m) Figure 5. Comparison between measured glacier thickness in the three boreholes (borehole thickness) and the mesh thickness in the model before (red stars) and after (blue stars) the bedrock correction assuming an unchanged surface elevation. be an important process [Vallon et al., 1976], and the source term would have to be included in equation (5). Starting from the diagnostic run using parameterized density fields (see section 2.4.1), a steady state for density and velocity fields was reached after 50 years. Results are shown in Figures 6 and 7. Comparisons between measured and modeled density profiles at the three drilling sites show good agreement. This confirms that the rheology of firn is adequately taken into account by the firn rheological model as already noted by Lüthi and Funk [2000] and Gagliardini and Meyssonnier [1997]. Parameterization of a and b (equations (12) and (13)) as function of the relative density proposed by Zwinger et al. [2007] gives satisfactory results for the snow and firn conditions prevailing at Col du Dôme. This good agreement with measurements also confirms that densification due to meltwater refreezing can be neglected at Col du Dôme. Note that the mean surface melting over the period does not exceed 12 cm w.e. yr 1 at Col du Dôme (deduced from our degree-day model, section ), meaning that densification due to the refreezing of meltwater could only have a significant impact on the part of the accumulation area experiencing very low snow accumulation (see Figure 7B) as well as high surface melting. The annual ratio between mean surface melting and yearly snow accumulation does not exceed 5% except along the southern limit of the modeled area (limit BC2 in Figure 1) where the ratio can reach 15 to 20% due to low snow accumulation. This ratio leads to an increase in the nearsurface density from <20 kg m 3 to 50 or 80 kg m 3 near the southern limit of our study area. AcomparisonwiththeHerron and Langway [1980] model is shown in Figure 6 and reveals that this model is not able to reproduce the data at this site. The firn densification is not only due to cryostatic pressure butalsotostraininducedbytheflow. This makes the Herron and Langway [1980] model inapplicable for this kind of site. Modeled subsidence velocities are in good agreement with the measurements because the bedrock has been adjusted accordingly (see Figure 7). This comparison cannot therefore be used as a validation of the model. However, the fact that modeled glacier thicknesses are consistent with measured thicknesses at the drilling sites means that the subsidence velocities are modeled with a realistic glacier thickness and validate the flow model. Comparison between measured and modeled horizontal surface velocities Depth (m) Density Density Density Figure 6. Measured (circles) and modeled (solid lines) firn density at drilling sites 1, 2, and 3 (from left to right, respectively). Dashed lines show modeled density using the Herron and Langway [1980] model. GILBERT ET AL American Geophysical Union. All Rights Reserved. 1886

12 Figure 7. Mean measured surface flow velocity fields over the period (A and B) compared to modeled surface flow velocities (C and D). Horizontal and subsidence velocities are plotted on the left (A and C) and right (B and D) panels, respectively. Inset E shows modeled horizontal velocities as a function of the measured values. Note that measured subsidence velocities (B) can be considered as representative of the mean snow accumulation due to the steady state surface elevations and very low surface melting. Depth (m) Age (years) Figure 8. Modeled depth/age relationship at site 2 (black dashed line) and site 3 (green dashed line) compared to radioactive dating (black and green stars, respectively). (see Figure 7E) reveals a mean difference of 0.71 m yr 1 with a standard deviation of 1 m yr 1, also supporting the model. Consistency between modeled and measured firn dating at sites 2 and 3 (Figure 8) confirms the validity of the modeled three-dimensional velocity field and the corrected bedrock depth, as dating is extremely dependent on bedrock depth Strain Heating Influence To assess the influence of strain heating on the temperature field, we performed steady state simulations with and without incorporating strain heating. Temperature differences between the two simulations reveal that strain heating is far from negligible (Figure 9), inducing temperature warming greater than 1 C near the glacier base in the highest velocity zones. However, strain heating is localized at the bottom of the glacier, and numerical experiments show that it can be taken into account by artificially increasing the basal heat flux (from to Wm 2 here). This explains the stronger basal heat flux inferred by Gilbert and Vincent [2013] using one-dimensional heat flowsimulation and neglecting strain heating. Heat production GILBERT ET AL American Geophysical Union. All Rights Reserved. 1887

13 Elevation (m) Drilling site Bedrock Distance (m) 0.7 Drilling site 3 Figure 9. Difference in steady state temperature between two simulations with and without taking into account strain heating (plotted on a 2D section located between sites 1 and 3) from strain heating is estimated to be between and Wm 3 at the bottom of the glacier in the first 10 m above the bedrock. Fifty percent of the strain heating is localized in the first 20 m above bedrock Transient Simulation From 1907 to 2012 We assume that at the beginning of the 20th century, the thermal regime at Col du Dôme was close to equilibrium [Gilbert and Vincent, 2013]. We therefore used the previously calculated steady state as the initial condition for transient modeling. This assumption does not influence the modeled temperature for the present. The surface mass balance is assumed to be constant since surface elevation has remained unchanged over the 20th century [Vincent et al., 2007]. We apply here the same surface mass balance as used for steady state calculations. Transient simulations have been performed over the period for a daily time step with updating of the density, velocity, and free surface every 100 days (see section 2.4.2). Melting occurs each summer, and meltwater refreezing leads to a supplementary energy input into the firnpack [Gilbert et al., 2014] during summer. The value of Sr was set to according to Gilbert et al. [2014]. It is worth noting that this value of Sr in our study was one order of magnitude lower than previously published values (Sr = 0.03 to 0.07) [Illangasekare et al., 1990]. It might be explained by the fact that water does not percolate within the cold firn in a uniform manner but more locally using small preferential pathways, explaining why less water is retained by capillarity. Water percolates, in this case, 1 to 4 m deep each summer depending on the amount of meltwater. This is in agreement with on-site percolation events already observed by Gilbert et al. [2014] during summer The results are in good agreement with measured temperature profiles at each date and site (Figure 10) and show that the physical processes are well accounted for by the model. This indicates that the strong spatial variability of englacial temperature response to climate change is due to flow and surface melting spatial variability, which leads to a high variability in surface temperature conditions and heat advection velocities that are in turn responsible for the modeled englacial temperature spatial variability. So far, the bedrock temperature has been almost unaffected by climate warming except in areas of low glacier thickness such as on Dôme du Goûter summit in the Mount Blanc range (temperature increase of 1 C since the beginning of the 20th century) or in areas experiencing strong vertical flow advection. The modeling results show that basal temperature increased by 0.5 C at site 2 (Figure 10) Simulations With Future Climate Scenarios (until 2050) To investigate future thermal regime changes at Col du Dôme, we perform three transient simulations using the climate scenarios described in section The resulting future daily air temperatures are then used to force the model between 2013 and 2050 (see Figure 11). Future temperatures are adjusted to Lyon-Bron temperature so that the mean temperatures for 2012 are equal for both time series, thereby avoiding a discontinuity. The surface mass balance is assumed to remain the same over the period and the effect of refreezing on the density field is once again neglected. Note that a sensitivity test including refreezing as a source term in the mass conservation equation (equation (5)) was carried out and showed that, even in the future (until 2050), the density field will be very little influenced by refreezing at this elevation Temperature Changes Expected englacial temperature changes at sites 1, 2, and 3 for the three scenarios are plotted in Figure 12. The three scenarios give similar results and show that the temperature at a depth of 40 m will reach 6 C to 5 C by 2050 at site 2 but will remain below 10 C at the two other sites. Warming near site 2 could reach 3 C below Temperature difference (K) GILBERT ET AL American Geophysical Union. All Rights Reserved. 1888

14 Depth (m) Temperature ( C ) Figure 10. Modeled (solid lines) and measured (dots) temperature profiles at drilling sites 1, 2, and 3 for the different dates indicated in the figure. The modeled steady state profile is shown by the black dashed lines. a depth of 40 m whereas it is not expected to exceed 1 C around site 3. Stronger englacial warming simulated using the HADCM3Q scenario is likely due to a warmer period around 2040 (see Figure 11) that is not present in the other scenarios. Site 2 is subject to stronger englacial warming due to its location where ice/firn flow is the fastest because surface temperature anomalies can be quickly advected deep into the glacier. Enhanced melt percolation and refreezing also lead to higher near-surface temperatures and faster englacial warming. Basal temperatures increase slowly in response to rising surface temperatures, the warming signal reaching the base not until 2050 except around Dôme du Goûter summit (site 1) where the ice thickness is small. These simulations assume constant surface accumulation in the future. A change in surface accumulation could change the vertical advection profile and affect the simulated temperature profiles. However, this effect would be significant only if surface accumulation at Col du Dôme experiences strong changes, which is unlikely since snow accumulation at this altitude depends only on precipitation (and not on air temperature) and precipitation is not expected to change significantly [IPCC, 2007] Surface Elevation, Density, and Velocity Field Changes We use the simulation with the ARPEGE climate scenario to compare surface velocity, glacier thickness, and density changes between 1907 and 2050 (Figures 13 and 14). Air temperature ( C ) ARPEGE HADCM3Q ECHAM5 Lyon Bron meteorological station Years Figure 11. Annual mean air temperature from Lyon-Bron meteorological station extrapolated to Col du Dôme (black thick line) and future climate scenarios (see section ). GILBERT ET AL American Geophysical Union. All Rights Reserved. 1889

15 BEDROCK Depth (m) Steady state 2030 HADCM HADCM ARPEGE 2050 ARPEGE 2030 ECHAM ECHAM5 BEDROCK 100 BEDROCK SITE 1 SITE 2 SITE Temperature ( C ) Figure 12. Measured temperature profiles (black lines with dots, see Figure 9 caption for dates) and modeled future temperature profiles in 2030 and 2050 at sites 1, 2, and 3 for the different climate scenarios. These results reveal that decreasing ice viscosity due to increasing englacial temperature has an impact on the flow velocity (Figure 14). However, this effect is partially compensated by a density increase and a glacier thickness decrease (Figure 14) that both tend to reduce glacier flow. The surface velocity is expected to increase by 12 to 15% (corresponding to 1 to 2 m yr 1 ) in the area that is most affected by temperature change. The surface elevation variation associated with these changes is expected to reach 8 to 15 m and could be measured using a D-GPS survey of the zone. Tracking surface elevation changes at high spatial and temporal resolution could provide a suitable indicator of englacial warming in cold accumulation zones if the mean surface balance can be considered constant (which is the case at Col du Dôme since 1900 [Vincent et al., 2007]). However, inter-annual variability of snow accumulation introduces strong surface elevation interannual variability. Glacier thickness simulations performed by taking into account a surface accumulation variability reconstructed using precipitation records in the valley (see section ) are shown for site 2 by the dashed green line in Figure 14 (bottom panel). These simulations reveal that surface elevation changes due to snow accumulation inter-annual variability amount to about 2 to 4 m at site 2. Very long-term elevation measurements (up to 40 years) are therefore needed to identify a surface elevation trend related to englacial temperature warming. 5. Conclusion In this study, we introduce a new three-dimensional polythermal glacier thermal regime model especially adapted to transient simulation in cold accumulation zones of mountain glaciers. The model is based on the conclusions of previous studies that have shown the importance of taking into account firn rheology Figure 13. Expected horizontal velocity (left panel) and surface elevation changes (right panel) between 1980 and 2050 due to englacial temperature warming modeled using the ARPEGE climate scenario. GILBERT ET AL American Geophysical Union. All Rights Reserved. 1890

16 Figure 14. (A) Air temperature, (B) englacial temperature, (C) density, and (D) surface velocity and thickness evolution modeled from 1907 to 2050 at site 2 using the ARPEGE climate scenario for the period Borehole measurements are indicated by black lines. The vertical dashed line shows the transition between the past and the future with respect to the simulation. In the bottom panel, the green dashed line shows the evolution of modeled glacier thickness taking into account the inter-annual variability of snow accumulation. [Gagliardini and Meyssonnier, 1997; Lüthi and Funk, 2000, 2001; Zwinger et al., 2007], bedrock thermal regime [Lüthi and Funk, 2001], and water content in the temperate ice [Hutter, 1982; Aschwanden et al., 2012; Wilson and Flowers, 2013]. It therefore includes enthalpy transport, firn densification, full-stokes porous flow, free surface evolution, and strain heating. However, as pointed out by Lüthi and Funk [2001] and Zwinger et al. [2007], surface boundary conditions were poorly determined in previous studies, and it was largely unclear how climate forcing could be used as input to a thermal regime model. The implementation of meltwater percolation and refreezing in our model makes it possible to apply physically based surface boundary conditions in order to use climate forcing as input to the model. Using the formulation developed by Gilbert et al. [2014], climatic forcing is inferred from daily air temperature to provide surface temperature and meltwater input to the thermal regime model. For the first time, the response of glaciers to climate change is addressed in a physical model by coupling temperature, density, flow velocity, and free surface elevation in transient simulations. We have successfully applied the model to simulate the response of Col du Dôme glacier to climatic changes. Col du Dôme constitutes a very good validation site due to a comprehensive data set that includes surface velocities, ice core dating, deep borehole temperatures, and density profiles as well as radar measurements. In this way, thermal regime variations over the past 100 years and for the coming 40 years were inferred in 3D. This application has demonstrated the importance of taking into account surface melt spatial variability and strain heating. Col du Dôme observations and model results have revealed the strong spatial variability of englacial temperature change that can be observed over short distances in a cold glacier (several tens of meters). This variability is due to different slope aspects and high spatial variability of snow accumulation inducing high variability in surface temperature and vertical advection velocities. Future climate scenarios until 2050 show that Col du Dôme will remain entirely cold although some parts of the glacier below a depth of 40 m could experience a 3 C temperature rise. The glacier flow response to rising englacial temperature is partially compensated by increasing density and decreasing glacier thickness, assuming an unchanged surface mass balance. The impact on glacier dynamics is expected to be small, with velocity changes within observational uncertainties. Expected surface elevation changes associated with englacial temperature GILBERT ET AL American Geophysical Union. All Rights Reserved. 1891