We use the Parallel Ice Sheet Model (development version e9d2d1f), an open-source, finite-difference, shallow ice sheet model . The model requires input on initial bedrock and glacier topographies, geothermal heat flux, and climate forcing. It computes the evolution of ice extent and thickness over time, the thermal and dynamic states of the ice sheet, and the associated lithospheric response. The model set-up used here was previously developed and tested on the Cordilleran ice sheet and subsequently adapted for a steady climate and regional applications to the European Alps.

Ice deformation follows a temperature- and water-content-dependent creep formulation (Sect. ). Basal sliding follows a pseudo-plastic law where the yield stress accounts for till dilatation under high water pressure (Sect. ). Bedrock topography is deflected under the ice load (Sect. ). Surface mass balance is computed using a positive degree-day (PDD) model (Sect. ). Climate forcing is provided by a monthly climatology from interpolated observational data WorldClim; and the European Centre for Medium-Range Weather Forecasts Reanalysis Interim ERA-Interim;, which is amended with temperature lapse-rate corrections (Sect. ), time-dependent temperature offsets, and in some cases, time-dependent palaeo-precipitation reductions (Sect. ).

Each simulation starts from assumed present-day ice thickness and equilibrium temperature in the ice and bedrock at 120 ka and runs to the present. Our modelling domain of 900 by 600 km encompasses the entire Alpine range (Fig. ). The simulations were run on two different grids, using horizontal resolutions of 2 and 1 km. All parameter values are summarized in Table .

Ice sheet dynamics are typically modelled using a combination of internal deformation and basal sliding. PISM is a shallow ice sheet model, which implies that the balance of stresses is approximated based on their predominant components. The shallow shelf approximation (SSA) is combined with the shallow ice approximation (SIA) by adding velocity solutions of the two approximations Eqs. 7–9 and 15. In the SIA, topographic roughness is parametrized using a bed smoother range of 5 km . Although the SSA–SIA heuristic is inaccurate in the transition zone where both vertical shear and longitudinal stretch are significant, it was shown to effectively approximate complex ice sheet flow dynamics over mountainous topographies similar to that of the European Alps .

Ice deformation is governed by the constitutive law for ice . Ice softness depends on ice temperature, pressure, and water content through an enthalpy scheme Table ;. It follows a piecewise Arrhenius-type rheology representative of Holocene polar ice Table ;p. 77 and increases with liquid water fractions up to 0.01 p. 65–66, an arbitrary threshold above which new ice deformation measurements are critically needed .

Surface air temperature derived from climate forcing (Sects. , ) provides the upper boundary condition to the ice enthalpy model. Temperature is computed in the ice and in the bedrock down to a depth of 3 km below the glacier base, where it is conditioned by a lower boundary geothermal heat flux estimate from multiple geothermal proxies (similarity method; Fig. a).

A pseudo-plastic sliding law relates the bed-parallel shear stresses to the sliding velocity. The yield stress is modelled using the Mohr–Coulomb criterion Table ;Eq. 18, using a constant basal friction angle corresponding to the average of available measurements p. 268. Effective pressure is related to the ice overburden stress and the modelled amount of subglacial water, using a formula derived from laboratory experiments with till extracted from the base of Ice Stream B in West Antarctica Table ;Eqs. 23 and 24. Basal meltwater is accumulated locally without horizontal transport. When the till becomes saturated, additional meltwater is assumed to drain instantaneously and is removed from the system in an accountable way. Other parameters (Table ) follow simulations of the Greenland ice sheet or benchmarks when other data are missing .

The initial basal topography is bilinearly interpolated from the hole-filled Shuttle Radar Topography Mission (SRTM) data with a resolution of 30 arcsec Fig. b;. These data include post-glacial sediment fills and lake surface topography. However, they were corrected for an estimation of present-day ice thickness based on modern glacier outlines, surface topography, and simplified ice physics Fig. c;.

Basal topography responds to ice load following a bedrock deformation model that includes local isostasy, elastic lithosphere flexure, and viscous asthenosphere deformation in an infinite half-space . Model parameters were set according to results from glacial isostatic adjustment modelling of deglacial rebound in the Alps, most closely reproducing observed modern uplift rates Supplementary Fig. 7.

Lithosphere rigidity was computed using the erroneous formula, D=YE3/12(1-ν), in . The correct formula is D=YE3/12(1-ν2) p. 443. Using Young's modulus, Y=100 GPa, and the Poisson ratio, ν=0.25 , the consequence of this error is that the simulations effectively use an elastic thickness of the lithosphere of E=53.8 km instead of 50 km, which is well within uncertainties , thus introducing a 1+ν4-1=5.7 % change in the length scale of bedrock deformation .

Ice surface accumulation and ablation are computed from monthly mean near-surface air temperature, Tm, monthly standard deviation of near-surface air temperature, σ, and monthly precipitation, Pm, using a temperature-index model e.g.. Accumulation is equal to precipitation when air temperatures are below 0 ∘C and decreases to zero linearly with temperatures between 0 and 2 ∘C. Ablation is computed from PDD, the integral of temperatures above 0 ∘C.

The PDD computation accounts for stochastic temperature variations by assuming a normal temperature distribution with standard deviation, σ, around the expected value, Tm. It is expressed by an error-function formulation , which is numerically approximated using week-long subintervals. In order to account for the effects of spatial and seasonal variations in temperature variability , σ is computed from ERA-Interim daily temperature values from 1979 to 2012 , including variability associated with the seasonal cycle , and bilinearly interpolated to the model grids (Fig. d and e). Degree-day factors for snow and ice melt are set to values used in the European Ice Sheet Modelling INiTiative Table ; EISMINT,.

The climate forcing driving the ice sheet simulations consists spatially of a present-day monthly mean climatology, {Tm0,Pm0}, modified by a temperature lapse-rate correction, ΔTLR, temperature offset time series, ΔTTS, and time-dependent palaeo-precipitation corrections, ΨPP: Tm(t,x,y)=Tm0(x,y)+ΔTLR(t,x,y)+ΔTTS(t),Pm(t,x,y)=Pm0(x,y)⋅ΨPP(t). The present-day monthly mean climatology was bilinearly interpolated from near-surface air temperature and precipitation rate fields from WorldClim , which is representative of the period 1960 to 1990. The modern climate of the European Alps is characterized by a north–south gradient in winter and summer air temperatures (Fig. f and g) and an east–west gradient in winter precipitation (Fig. h), which is reversed in summer (Fig. i). WorldClim data were selected as an input to the ice sheet model because they incorporate observations from the dense weather station network of central Europe Fig. 1. Besides, WorldClim data were previously used as climate forcing for PISM to model the LGM extent of the former Cordilleran ice sheet in good agreement with geological evidence along the southern margin where weather station density is lower than in the Alps. Finally, the last glacial cycle Alpine glaciers did not extend over marine areas where WorldClim data are missing.

The temperature lapse-rate corrections, ΔTLR, are computed as a function of ice surface elevation, s, using the SRTM-based topography data provided with WorldClim as a reference, bref: ΔTLR(t,x,y)=-γst,x,y-bref,=-γht,x,y+bt,x,y-bref, thus accounting for the evolution of ice thickness, h=s-b, on the one hand, and for differences between the basal topography of the ice flow model, b, and the WorldClim reference topography, bref, on the other hand. All simulations use an annual temperature lapse rate of γ=6Kkm-1, which is slightly above annual lapse rates measured in the Alps but more representative of summer months when surface melt occurs Fig. 3.

Only few regional proxy records exist that extend over periods when the Alps were glaciated . These include lake sediment records in the north and west of the Alps e.g. and cave speleothems in the Eastern e.g. and Western Alps. Due to the scarcity of vegetation north of the Alps during glacial periods, varying sources for moisture advection, and the limited duration of the records, quantitative palaeoclimatic interpretation will require both combining multiple proxies in space and time, and comparing them against regional circulation model output .

In our simulations, temperature offset time series, ΔTTS, are derived from distal palaeo-temperature proxy records from the Greenland Ice Core Project GRIP;, the European Project for Ice Coring in Antarctica EPICA;, and an oceanic sediment core from the Iberian margin MD01-2444;. Palaeo-temperature anomalies from the GRIP record are calculated from the oxygen isotope (δ18O) measurements using a quadratic equation , ΔTTS(t)=-11.88δ18O(t)-δ18O(0)-0.1925δ18O(t)2-δ18O(0)2, while temperature reconstructions from the EPICA and MD01-2444 records are provided as such. For each proxy record used and each of the parameter set-ups used in the sensitivity tests, palaeo-temperature anomalies are scaled linearly (Table , Fig. a) so that, within a 150×100 km rectangular domain covering the Rhine glacier piedmont lobe (Fig. a, black rectangle), the modelled cumulative glaciated area during MIS 2 (29–14 ka) is consistent with the glaciated area of the geological reconstruction .

Finally, in some simulations (hereafter labelled PP), precipitation was reduced with air temperature in order to simulate the potential rarefaction of atmospheric moisture in colder climates. This was done using an empirical relationship derived from observed accumulation rates and oxygen isotope concentrations in the GRIP ice core , ΨPP(t)=exp⁡ψΔTTS(t), with ψ=0.169/2.4=0.0704 . The control simulations use constant precipitation, corresponding to ψ=0. This simple relationship does not reflect the complexity of atmospheric circulation changes that governed moisture availability over the Alps during the last glacial cycle. Palaeo-climate proxies indicate slightly reduced LGM precipitation in western Europe with anomalies diminishing eastwards . Regional circulation models indicate generally dryer conditions during MIS 3 but more precipitation south of the Alps during MIS 2 . Besides, changes in the ice surface topography may have redistributed orographic precipitation.

For the three palaeo-temperature records and the two palaeo-precipitation parametrizations used, the model yields significant ice volume build-up during MIS 4 and 2 (Fig. b), corresponding to documented glaciation periods in the Alps . All simulations also yield important glaciations during MIS 5 and 3, but their timing and amplitude varies significantly depending on the climate forcing used. All six simulations overestimate the late-glacial re-advance during the Younger Dryas 12.9–11.7 ka; cf. e.g.. This is partly because the 2 km resolution used in these simulations is too coarse to resolve Younger Dryas Alpine glaciers. However, the large total ice volume overestimation resulting from the GRIP temperature forcing (Fig. b, blue curves) certainly indicates that the vigorous cooling experienced during the Younger Dryas in Greenland is not representative of the Alps.

Geological data from the best documented Alpine piedmont lobes indicate that their maximum extension during MIS 2 occurred around 25.5–24.0 ka , after which Alpine glaciers remained or potentially re-advanced to within close reach of this maximum extent, until as late as 22 to 17 ka Fig. 5. The EPICA simulations yield an early maximum ice volume at 25.2/24.6 (without/with palaeo-precipitation reductions), followed by a retreat and then a standstill until 17.3 ka (Fig. b, red curves) in a very good agreement with these geological data. On the other hand, the simulations forced by the GRIP palaeo-temperature record yield two distinct total ice volume maxima at 27.3/27.0 and 21.8/21.7 ka followed by an early deglaciation of the foreland by 21.4 ka. The MD01-2444 palaeo-temperature forcing yields a late LGM peak ice volume at 16.5/15.7 ka followed by a rapid retreat, in contrast to dated records of large-scale ice retreat by 17 to 16 ka .

Finally, all simulations yield very strong ice volume variability over the last 120 kyr (Fig. b), indicative of many more than the two or three documented cycles of glacier advance and retreats onto the foreland for this time period . Palaeo-precipitation reductions dampen some of the small-scale variability, but they have little effect on the millennial scales that characterize those cycles (Fig. b, light colour curves). In particular, strong millennial variability recorded in GRIP δ18O (Dansgaard–Oeschger events) has large repercussions on the modelled Alpine ice volume, including around six glaciations of LGM magnitude (Fig. b, blue curves), which are not supported by geologic evidence . Dansgaard–Oeschger events, however, have been recorded in European lake sediments and cave speleothems . Our results indicate that their magnitude in Europe was likely smaller than that recorded in GRIP δ18O, which either implies smaller associated palaeo-temperature fluctuations in Europe than in Greenland, or temperature-independent controls on GRIP δ18O. In contrast, the EPICA palaeo-temperature record has the smallest variability among the records used (Fig. a, red curves). This results in smaller variability in the total ice volume and more restrictive glaciations during MIS 5 and 3 (Fig. b, red curves).

During MIS 2, all six simulations yield comparative modelled maximum extents (Fig. a). This result is inherent to the palaeo-climate forcing approach used, which involves a linear scaling of each palaeo-temperature anomaly record to a geomorphological reconstruction of the area glaciated by the Rhine Glacier piedmont lobe (Sect. ; Fig. a, black rectangle). Outside this benchmark area, all simulations underestimate ice extent in the western part of the model domain and overestimate it in the eastern part, relative to the geomorphological ice limits (Fig. a). This result might indicate that the LGM temperature depression, here taken as homogeneous, was actually lower in the Eastern Alps than in the Western Alps, as previously shown by positive degree-day modelling of the central European palaeo-ice caps . Alternatively, the east–west gradient in winter precipitation existing today (Fig. h) was perhaps enhanced during the LGM, as indicated by pollen reconstructions . The LGM extent of Alpine glaciers might have been affected by an east–west gradient in variables not accounted for by the PDD model, such as cloud cover or dust deposition. Finally, modern precipitation data from WorldClim also bear uncertainties and exhibit local disagreement with other regional data .

Besides this general pattern, there exist differences in glaciated area depending on the palaeo-climate forcing used. The MD01-2444, and to a greater extent the GRIP palaeo-temperature records, tend to overestimate MIS 2 ice cover on all peripheral ranges that surround the Alps (Fig. a and c). This is particularly true when no precipitation corrections are applied (bright colours). In fact, both palaeo-temperature records have a larger temperature variability and contain brief spells of cold temperatures (Fig. b, blue and green curves), which are too short to develop a fully grown Alpine ice sheet, but long enough to build up ice cover on a smaller scale in these peripheral ranges. On the other hand, peripheral glaciation modelled using the EPICA record (Fig. b), which has smaller temperature variability, is in relatively good agreement with the geomorphological reconstructions.

The modelled extent of glaciation during MIS 4 depicts a more pronounced sensitivity to the choice of palaeo-climate forcing used. Using the GRIP and MD01-2444 palaeo-temperature records, Alpine glaciers are modelled to extend well beyond the reach of documented LGM end-moraines (Fig. d and f). Because, in both cases, the modelled glaciation extent corresponds to brief cold spells in the palaeo-temperature forcing (Fig. a), palaeo-precipitation reductions greatly reduce the excessive modelled ice cover (Figs. b and d and f, light colours). However, with or without palaeo-precipitation corrections, the GRIP and MD01-2444 forcing yield modelled ice extents (Fig. d and f) and volumes (Fig. b) considerably larger during MIS 4 than during MIS 2, which is not supported by geological evidence . Gravel deposits from the Rhine Glacier indicate an early last glacial cycle foreland glaciation, but the extent reached is less than that in MIS 2 . Dropstones in lake sediments indicate a late MIS 4 advance restricted to the upper Inn Valley . For the Linth Glacier, no evidence for a pre-LGM glacier advance during the last glacial cycle is available .

The EPICA palaeo-temperature forcing, on the other hand, yields a MIS 4 glaciation that is only slightly less expansive than the MIS 2 glaciation (Fig. e). The sensitivity of the glaciated area to palaeo-precipitation reductions is mostly limited to the southern terminal lobes of central-Alpine glaciers, where reduced precipitation results in a slightly lesser extent during both MIS 2 and 4 (Fig. b and e).

Based on the above considerations on timing of the LGM during MIS 2, and modelled ice extent during MIS 4, we select EPICA as our optimal palaeo-temperature record for a more detailed and higher-resolution comparison of modelled glacier dynamics to available geological evidence (Sect. ). As a conservative approach in regard to the rapid ice volume fluctuations, we choose to include palaeo-precipitation corrections in the following higher-resolution run.

The LGM extent of Alpine glaciers has been mapped with varying level of detail across the Western , and Eastern Alps . Some of these maps have been compiled into a reconstruction, covering the entire Alps , and reproduced here (Fig. , red line) for comparison against model results.

The modelled total ice volume reaches a maximum of 123×103 km3, or 300 mm of sea-level equivalent, at 24.56 ka (Fig. b). A maximum glacierized area of 163×103 km2 is attained shortly afterwards at 24.57 ka (Fig. a). Although the modelled timing of the LGM varies across the mountain range (Sect. ), at 24.57 ka nearly all outlet glaciers extend to within a few kilometres from their modelled maximum stage (Fig. a, dashed orange line).

The palaeo-climate forcing was adapted (Sect. ) to model the maximum configuration of the Rhine Glacier in broad agreement with the geological reconstruction of the LGM extent (Fig. a, solid red line), yet discrepancies remain elsewhere. As already outlined for low resolution runs (Sect. ), the extent of glaciation in the north-western Alps – including the Rhône Glacier complex, the Jura ice cap, and the Lyon Lobe – is underestimated in the model results Fig. a, cf.. On the other hand, the model yields excessive ice cover in the Eastern Alps, where the Mur, Drava, and Sava glaciers are modelled to extend tens of kilometres beyond the mapped ice limits Fig. a, cf.. As previously discussed, these discrepancies might indicate that the LGM climate was characterized by an east–west gradient in temperature cf. or precipitation cf., anomalies relative to present, or an east–west gradient in variables that are not accounted for by the PDD model.

On the other hand, while atmospheric circulation models and palaeo-climate proxies both support differential precipitation changes north and south of the Alps, our model results show no obvious north–south bias as compared to the mapped LGM margins (Fig. a) despite the homogeneous temperature and precipitation anomalies applied relative to present. Although this may appear as a contradiction with previous, constant-climate modelling results , it follows from introducing a time-dependent palaeo-temperature forcing. Without introducing differential precipitation change north and south of the Alps, constant climate forcing systematically resulted in extraneous Fig. 3 and premature Fig. 4 glaciation on the north slope of the Alps. Using time-dependent palaeo-temperature forcing, progressive atmospheric cooling over several thousand years allows for warmer temperatures – closer to the thermodynamic equilibrium – within the ice and the bedrock. This results in thinner glaciers and limits overshoot of the equilibrium state (discussed further in Sect. ).

On a more regional level, the maximum extent of the Adda, Oglio, Adige, and Piave glaciers in the central southern Alps is modelled several kilometres within the mapped LGM moraines (Fig. a). This appears to result from the palaeo-precipitation reduction used to force the model (Fig. b), which is likely unrealistic in at least this part of the model domain. On the other hand, the Durance Glacier in the south-western Alps is modelled to extend several kilometres beyond the mapped limit (Fig. a), indicating that the LGM temperature depression was likely dampened by the Mediterranean climate in this part of the model domain. However, geochronological data from the south-western Alps are sparse, and the LGM extent compilation is inconsistent with more recent regional reconstructions .

The model reproduces peripheral ice caps where documented by geological evidence on the Vercors, Chartreuse, and Bauge Prealpine reliefs, the Jura Mountains, the Vosges Mountains, the Black Forest, the Bohemian Forest and the Dinaric Alps (Fig. a). An independent ice cap also covers the Hochwart massif during most of the simulation, yet it is engulfed by Alpine glaciers during the LGM to become a peripheral ice dome Fig. a; cf.Fig. 2.5. The LGM extent of peripheral ice caps is underestimated in the Vosges and Jura mountains, but it is overestimated for the more meridional Vercors Massif Fig. a; cf.Figs. 4.28, 4.32, and 4.33, p. 322–321. Thus, there exists a regional conformity between the model–data discrepancies obtained for the main Alpine ice sheet and those obtained for peripheral ice caps, including too extensive modelled ice cover to the east and extreme south-west, and too restrictive modelled ice cover to the north-west. Because peripheral ice caps have very different glacier dynamics than the main ice sheet, this conformity most likely indicates a climatic cause, rather than an ice dynamics cause, for these discrepancies.

The LGM Alpine ice flow pattern is traditionally described as that of a network of interconnected valley glaciers, which are primarily controlled by subglacial topography. However, geomorphology shows that ice was thick enough to flow across high mountain passes throughout the mountain range e.g., and even perhaps to form self-sustained ice domes .

The modelled flow pattern at 24.57 ka is complex (Fig. a, blue colour mapping). Ice velocities of several hundred metres per year – characteristic of basal sliding – generally occur along the main river valleys, while ice domes and ice divides are predominantly located over major relief areas, where ice moves only by a few metres per year, characteristic of internal deformation without basal sliding (Fig. a). Nevertheless, the model results depict exceptions to this general pattern as occasional ice flow across the modern water divides, i.e. transfluences.

In the Western Alps, major transfluences occur for instance across Col de Montgenèvre, Col du Mont-Cenis, Simplon Pass and Brünig Pass (Fig. a). Although a transfluence across Col de Montgenèvre was previously questioned Fig. 2, evidence for southerly ice flow across Col du Mont-Cenis has been recognized Fig. 3.18, p. 284. Similarly, transfluences have been previously identified from the geomorphology across Simplon Pass and from the distribution of glacial erratics across Brünig Pass . On the other hand, the simplified climate forcing used in this simulation could not reproduce the transport of glacial erratics from southern Valais to observed deposition sites in the Solothurn region cf. .

In the Eastern Alps, major transfluences are modelled to have occurred for instance across Fern Pass, the Seefeld Saddle, the Gailberg Saddle, the Kreuzberg Saddle, Kronhof Pass, Studenbodenalm, and Pyhrn Pass (Fig. a). Transfluences across Fern Pass and the Seefeld Saddle are known from the geomorphology Fig. 2.4. It is also known that ice flowed across the Gailberg and Kreuzberg saddles as well as Pyhrn Pass Fig. 2.5. On the other hand, no transfluence has previously been documented over the Kronhof Pass, Studenbodenalm or elsewhere over the Carnic Alps. The Tagliamento catchment is usually assumed to not have received transfluences from the Drava catchment , but this would not be incompatible with reconstructed ice surface elevations in this area .

Except for too extensive ice cover in the easternmost part of the model domain (Sect. ), there is generally a good agreement between transfluences observed in the geomorphology and the model results. Both depict the LGM Alpine ice flow pattern as an intermediate between that of a topography-controlled ice field, and that of a self-sustained ice cap.

The timing of the LGM has been documented by radiocarbon and cosmogenic isotope dating techniques at multiple locations around the Alps (cf. reviews by , and , and the more recent publications by , , , and ). These data indicate that Alpine glaciers reached their maximum extent between 26 and 20 ka (calibrated 14C and 10Be ages), but also that terminal lobes stayed in the foreland until 22 ka to 17 ka, when they experienced a rapid retreat synchronously with the lowering of the ice surface in the mountains (; Fig. 5; Fig. 3).

In our simulation, the maximum areal cover is reached at 24.57 ka (Fig. a), yet individual glacier lobes reach their maximum extent at different ages (Fig. ). For instance, the Dora Riparia, Dora Baltea, and Tagliamento glaciers reach an early maximum before 27 ka; the Durance, Rhône, Inn, Enns, Ticino, and Adige glaciers reach their maximum thickness in phase with the overall Alpine areal maximum around 25 ka; while the Isère, Adda and Oglio glaciers reach a late maximum after 24 ka (Fig. ). Remarkably, peripheral ice caps on the Vercors, Jura Mountains, Black Forest and Hochschwab reach their maximum extent even later and locally after 21 ka (Fig. ).

These differences in timing of the LGM occur in the model results despite the homogeneous temperature and precipitation anomalies supplied as palaeo-climate forcing. The LGM timing differences modelled here are thus not related to climate, but are inherent to modelled glacier dynamics. They result from differences in the subglacial topography, in particular catchment sizes and hypsometry, of the different outlet glaciers.

For large Alpine glaciers flowing in deep valleys, such as the Rhône, Rhine and Adige glaciers, several thousand model years are needed to attain a thermodynamical equilibrium between cold-ice advection from the high accumulation areas, upward diffusion of geothermal heat, and heat release from strong basal shear strain. Temperature evolution is, in part, limited by the slow warming of the subglacial bedrock, that has been previously cooled downed by subfreezing air temperature before glacier advance. For larger glaciers, this thermodynamical equilibrium is typically not yet attained by around 25 ka. Several Alpine lobes surge and overshoot their balanced extent before thinning and receding towards the mountains as they warm towards thermodynamical equilibrium (supplementary animation in ). In contrast, peripheral ice caps with basal topography restricted to high elevations experience very low shear strain and virtually no basal sliding. They advance regularly on a frozen bed during the entire cold period, resulting in a late maximum stage (Fig. ).

Our results indicate that even for a relatively small ice sheet like that found in the Alps, the LGM glacier extent corresponds to a transient stage, while millennial-scale spatial differences in its timing can result not only from climate variability but also from complex glacier dynamics. Heterogeneous climatic anomalies, such as temperature or precipitation anomaly gradients, which are not included in our model set-up, could perhaps induce further spatial differences in the timing of the LGM Alpine ice sheet, which may counterbalance or enhance those modelled here.

Following the general assumption that trimlines, which mark the transition between the steep frost-shattered ridges and the more gentle glacially sculpted valley troughs, represent the maximum elevation of the LGM ice surface, the maximum ice thickness in the Alps has been reconstructed in several areas . However, this assumption is challenged by geomorphological and cosmogenic nuclide dating evidence from Scandinavia e.g., the British Isles e.g., and North America e.g., that pre-glacial landscapes located well above the trimlines have been glaciated and preserved under cold-based ice, sometimes for several glacial cycles . Thus, the trimline could also mark the maximum elevation of the transition from temperate to cold-based ice or a late-glacial ice surface elevation Fig. 1, p. 403. However, no such evidence has been reported in the Alps. Instead, tree and fire remains record the presence of nunataks in the south-western Alps by 21 ka .

In the upper Rhône Valley, the maximum ice surface elevation reached during MIS 2 in our simulation, corrected for bedrock deformation, is consistently modelled to have occurred several hundred metres above the observed trimline elevations (Fig. a and b), with a mean difference of 861 m (and a standard deviation 197 m, Fig. b), which is significant in respect to the modelled maximum ice thickness in the Rhône Valley (and the Alps) of 2586 m. This result depends on uncertain basal sliding and ice rheological parameters, the model sensitivity to which was not tested here. However, regional model sensitivity tests show that a modelled surface compatible with the trimline elevations is incompatible with the documented overspilling flow across the Simplon Pass and LGM palaeo-climate proxy records . Unfortunately, validation through the bedrock uplift rate cf. is not doable in the Alps due its lower values, active tectonics, and uncertainties in geological properties cf..

Our model results depict a polythermal LGM Alpine ice cover. Due to sub-freezing temperatures applied in the climate forcing of the model, the entire Alpine ice sheet is capped by an upper layer of cold ice. In major Alpine valleys, such as the upper Rhône Valley, important ice thickness and strain heating contribute to form a layer of temperate ice near the glacier base, allowing basal melt, sliding, and potentially erosion (Fig. c, white areas). On the other hand, on the highest mountains, ice cover is too thin and too static to form temperate ice, resulting in cold ice down to the bed and preventing potential erosion (Fig. c, hatched areas).

In the upper Rhône Valley, observed trimlines are often located near the LGM cold–temperate basal thermal transition or within cold-based areas (Fig. c). The remaining discrepancies between the trimline locations and the modelled basal thermal boundary may relate to temporal migrations of the basal thermal boundary, an absence of sliding in warm-based areas, and levelling of small-scale topographic features in the 1 km horizontal grid. They call for more detailed comparisons spanning the entire Alpine range and specific sensitivity studies to relevant basal sliding and ice rheological parameters, as well as to the uncertain subglacial topography. However, the presence of an upper layer of cold ice during the LGM, already found at high altitudes in the much warmer climate of today e.g., is inevitable . The bedrock thermal model yields significant volumes of frozen ground periglacially north of the Alps and subglacially under the highest mountains, which is also in agreement with previous studies .

In this context, our results challenge the assumption that Alpine trimlines mark the maximum upper ice surface elevation of the LGM ice cover and call for a more accurate estimation of the thickness of the upper layer of cold ice in the Alps.

Glacial history of the Alps prior to the LGM remains poorly constrained. Although the four glaciations model has long been used, it is now known that glaciers advanced onto the foreland at least fifteen times since the beginning of the Quaternary at 2.58 Ma . Sparse geological data indicate that the last glacial cycle may have comprised two or even three periods of glacier growth and decay . Luminescence dating from two sites in the central northern foreland indicate an early last glacial advance of Alpine glaciers onto the near foreland from around 107±9 to 101±5 ka . Evidence for a MIS 4 glaciation is equally sparse Sect. ; and its timing remains uncertain e.g.. During MIS 3, major Eastern Alpine valleys hosted woolly mammoths and open vegetation , indicating more restricted glaciation.

As previously mentioned (Sect. ), independent of the palaeo-temperature (GRIP, EPICA, or MD01-2444) and palaeo-precipitation (with or without corrections) applied, all simulations presented here result in a high temporal variability in the total modelled ice volume (Fig. ). Using the optimal EPICA palaeo-temperature record with least variability (Sect. ) and conservatively including palaeo-precipitation reductions (Sect. ), the 1 km resolution simulation also results in strong total ice volume variability throughout the last glacial cycle (Fig. b). Two major glaciations occur during MIS 4 and 2 (Fig. b). However, several minor glaciations occur during MIS 5 and 3, as well as a minor late-glacial re-advance at the onset of MIS 1 (Fig. b). These episodes are the result of synchronous advances of several Alpine glaciers well into the major valleys and sometimes even onto the foreland (Fig. ; supplementary animation).

For instance the Rhine Glacier (Fig. a) extends beyond the outermost limestone reliefs six times during the simulation, and sometimes retreats almost completely between two advances (Fig. b). The Rhône Glacier, fed by several high-altitude accumulation areas, advances eight times onto modern Lake Geneva (Fig. c and d). The Dora Baltea Glacier, characterized by a very steep catchment, and multiple tributaries, shows even a higher variability and reaches close to its maximum position six times throughout the simulation (Fig. e and f). The Isère (Fig. g and h) and Inn (Fig. i and j) glaciers, with their complex system of confluences and diffluences, need longer time to build up and reach the foreland only two or three times (Fig. g–j). Finally, the Tagliamento Glacier, distant from the major ice-dispersal centres of the inner Alps, is absent for most of the modelled glacial cycle (Fig. k and l)

Despite the low temperature variability of the palaeo-climate forcing and reduced precipitation dampening glacier response, our simulation depicts the Alpine ice complex as highly dynamic, with many more than two or three cf. glaciations and regional glacier dynamics controlled by spatial variations in catchment size and hypsometry. Importantly, Dansgaard–Oeschger events, recorded in Europe but absent from the EPICA palaeo-temperature forcing, may have induced an even more dynamic glacier response than that modelled here.