## Introduction

As pervasive geological agents of erosion and deposition, ice sheets can drive the evolution of continental shields, shelves, mountain ranges and passive margins. However, the long-term erosive efficacy of ice sheets has been questioned, particularly across low-relief continental shields and high latitudes with cold subglacial regimes1,2,3. Moreover, despite half a century of thorough investigation, the drivers of, and controls on, glacial erosion are still unclear4, with — to differing degrees — assertion of the primary dominance of tectonics, lithology, climate and other proxies over ice-sheet dynamics. For example, it has been argued that rates of glacial erosion that exceed 1 mm a−1 are driven by rapid tectonic uplift, focused on active margins in maritime/temperate climatic settings where fast flow is promoted by high mass-balance turnover and temperate basal regimes that are independent of subglacial processes per se5,6,7,8,9. Likewise, a glacial-buzzsaw mechanism has also gained traction, where it is argued that climate is the critical control restricting the maximum elevations that mountain regions can attain. Here, mountain glaciers and ice caps — self-limited by their equilibrium line altitude — implicitly designate a climatic/altitudinal threshold at and above which they are highly effective erosive agents but below which their erosive capacity rapidly declines10.

In this study, we aim to address and reconcile some of these issues by reference to the time-transgressive erosional footprint of the Eurasian ice sheet (EIS) over the last ~100 ka glacial cycle. At its Last Glacial Maximum (LGM) extent, the EIS was the third largest Quaternary ice complex after Antarctica and the Laurentide11, was in excess of 2500 m thick in places, and had a considerable latitudinal range spanning 49°N to 82°N12. The impact of the EIS on long-term landscape development was profound, moulding the North Atlantic passive margin and north-western European continental shelf through kilometre-scale denudation, sediment evacuation, and related uplift during repeated Quaternary glacial cycles13. Despite this, broad swathes of cold-based inter-fjord uplands and the terrestrial hinterland, including the Baltic Shield, survived multiple glaciations with virtually no modification14,15,16. Recently, these views of a cold-based “relict” landscape have been somewhat challenged by cosmogenic exposure measurements that demonstrate significant glacial modification of high plateaus between the fjords of western Fennoscandia2,17,18. Erosion of these high plateau surfaces has been linked to long-term equilibrium line altitudes that yield a distinctive bimodal erosion/elevation distribution due to consequent zonation of cold and warm-based glacier action. This observed bimodal erosive signature is claimed to provide further, nuanced, evidence for a glacial buzzsaw mechanism19.

We apply a transient 3D thermomechanical ice-sheet model, constrained by terrestrial and offshore empirical data, to determine the spatiotemporal patterns and rates of glacial erosion across Northwest Eurasia. The model is based on a first-order solution of the Stokes equations governing ice flow to compute the free evolution of ice mass balance, thermodynamics, flow, isostatic adjustment, and geometry of the EIS across a 10 km finite-difference domain12,20. The model is coupled to a modern-day reference climate via a distributed temperature index scheme based on mean (1950–2000) seasonal temperature and precipitation patterns and is forced by perturbations in these fields scaled to the NGRIP δ18O record and eustatic sea-level change (Fig. 1). The key variables of ice thickness, thermal structure, internal deformation, basal motion and meltwater production are integrated in discrete time-steps from the last interglacial (Eemian or Marine Isotope Stage - MIS5e) at 123 ka BP through to the present day (Supplementary Fig. 1). This time-transgressive model framework enables us to investigate and contextualize the transient climatic, topographic, geological, and glaciological controls on landscape development throughout the entire last glacial cycle.

In the transient model, subglacial erosion is implemented through an ice flux relation21,22 that is scaled and calibrated against back-stripped trough mouth fan (TMF) sediments and 10Be cosmogenic nuclide measurements (see “Methods”) across three primary geological provinces that characterize the Eurasian domain: Shield, Younger Basement, and Platform, delimited by rock type and age (Fig. 2a). Shield areas are typically represented by crystalline igneous and high-grade metamorphic basement rocks, and form relatively low relief within tectonically stable areas. The Baltic Shield of Sweden, Finland and northwest Russia is composed mostly of Archean and Proterozoic plutonics (quartz-bearing) >1 Ga. The largest group of younger (<1 Ga) metamorphic rocks that comprise the Younger Basement province are within the mountains of Eurasia (Norway, Scotland, Svalbard) and belong to the Caledonian Orogeny that formed during the collision of Baltica and Laurentia. Subsequent collapse of the Caledonian and Uralian (Russia) orogenic belts has exposed windows of Precambrian-aged rocks.

The Permian-Triassic transition marked the start of progradation of a major sedimentary system into the Barents Sea basin, leading to the infilling of several kilometres of sediment and shallow-marine terrigenous deposits, facilitated by what was the largest delta plain in Earth’s history23. Early Cenozoic break-up between the Greenland and Eurasian plates formed major rift basins in the present-day Northeast Atlantic and Arctic Ocean, within which further siliclastic sediments, evaporites, and thick carbonate-rich sedimentary sequences accumulated to develop the Platform province. Across the Barents Sea shelf today, an upper regional unconformity separates variously dipping Mesozoic sedimentary rocks from an overlying sequence of glacigenic sediments. In the central part of the Barents Sea the thickness of this Quaternary sediment cover is normally less than a few tens of metres, whereas thicker sequences of 0–300 m occur near the continental slope and in the southeast24,25.

## Results

### Empirical erosion scaling constraints

We derive empirically constrained erosion scaling laws for Platform and Younger Basement provinces using volumetric calculations of glacigenic sediment deposits in the Bjørnøya and mid-Norwegian TMFs, respectively (Fig. 2c). Their glacial catchments represent the largest erosional source areas for their respective geological provinces, and as such, best represent the range of glacial dynamics relevant for scaling erosion rates more broadly.

The Bjørnøyrenna catchment dominated ice drainage from the western Barents Sea shelf, extending inland towards ice domes over Fennoscandia, the central Barents Sea and Svalbard. Outcropping strata within this catchment is largely characterized by the thick sedimentary succession (e.g., shales and sandstones) that infilled the Barents Sea basin during the Mesozoic and generally have similar mechanical properties to the Mesozoic sedimentary rocks found farther south in the Eurasian domain. Towards the southern and western Barents Sea shelf, the platform is divided into minor highs and sub-basins mainly influenced by salt tectonics (e.g., Samson Dome)26. From a synthesized 2D seismic dataset of 580 profiles covering the Bjørnøya TMF (Supplementary Fig. 2) — the largest glacigenic sediment deposit within the Arctic — we identify five seismic sequences of sediment deposition within the uppermost glacial (G) seismic unit, GIII, bounded by the R1 regional (R) unconformity at its base (nomenclatures follow the regional framework set out by Faleide et al.27). By applying previously defined velocity constraints28 we estimate a total GIII sediment volume of 62,105 km3 (Fig. 2b). Assuming the age of the R1 unconformity (GIII base) is 0.2 Ma29,30, we split this sediment volume according to the three major advances of the EIS during the last glacial cycle, to derive a total eroded bedrock volume of 37,263 km3 for the last glacial cycle. The density of the source sedimentary sediment rocks is assumed to be comparable to the erosional products (compaction ratio = 0%)31.

A comparable large-scale glacigenic sediment depocentre — the Naust Formation — can be found on the mid-Norwegian margin, which built-up over the past 2.7 Ma and allowed the continental shelf to prograde seaward by as much as 150 km. Within the youngest ‘T’ sequence (0–200 ka), 16,300 km3 of sediment has been deposited, equivalent to 13,000 km3 of bedrock given a compaction ratio of 20%32 due to the varying lithologies of the source areas (i.e., crystalline rocks). In lieu of more precise chronologies for the glacigenic deposits found offshore, we conservatively estimate 50% of this bedrock volume — 6500 km— was eroded since the Eemian, taking into account a reduced ice presence on the mid-Norwegian margin during the early stages of the last glacial cycle (Supplementary Fig. 1). The source catchment for each TMF is drawn using depth-averaged ice velocities through the last glacial cycle (Fig. 2a; Supplementary Fig. 3), yielding areas of erosion of 594,000 km2 and 216,800 km2, for the Bjørnøya and mid-Norwegian margin TMFs respectively (Supplementary Table 1), and from which scaling factors can then be determined (Fig. 2c).

With limited insight on volumes of glacigenic sediment transported from Shield areas, we use a suite of 249 cosmogenic 10Be measurements from bedrock samples (Fig. 2a) to calibrate a bulk scaling factor that best fits the duration of ice cover over the last 2.7 Ma according to the LR04 stack33 (a compilation of benthic δ18O records from 57 globally distributed sites) and patterns of modelled ice discharge (Supplementary Fig. 4a). For each sample a scaling factor is determined that yields a glacial erosion rate and sample depth history compatible with the measured 10Be concentration. A Monte Carlo approach with 10,000 iterations is used to estimate positive and negative uncertainties. The median scaling factor value of these data points is applied in the erosion scaling law, yielding a fit approximately one order-of-magnitude less than that for the Younger Basement and Platform sectors (Fig. 2c).

For all provinces we acknowledge there is no explicit accounting for the initial evacuation of pre-Weichselian sediments, which would lead to a variable overestimation of the long-term erosion rates. This is partially compensated through the implementation of a lower threshold for erosion at depth-averaged velocities <10 m a−1, which is of particular relevance for areas near the former ice-divide where pre-Weichselian sediments are widely reported34. Furthermore, the TMF sediment volumes of Weichselian age used for back-stripping in Younger Basement and Platform provinces also implicitly contain a proportion of sediments that were eroded during older glaciations but were later deposited off-shelf during the Weichselian. However, the time-scale for the initial evacuation of these shelf sediments is ambiguous, which today on the Barents Shelf range in thickness from 0 to 300 m25, and ~0 to 20 m across Shield areas35. In highly lubricated zones of fast-streaming ice in West Antarctica, contemporary geophysical observations of soft sediment erosion rates of up to 1 m a−1 36 demonstrate that the active deposition, reworking and removal of subglacial sediments overlying bedrock can be extraordinarily rapid. Contrarily, pockets of pre-Weichselian sediments preserved through the last glacial cycle in some ice-marginal environments in Svalbard37 and Denmark38 suggest that erosion was not always pervasive beyond the core areas of the ice sheet.

### The glacial erosional footprint

The empirically constrained model analysis confirms that the EIS had a colossal erosional footprint that was directly responsible for the net removal of ~126,500 km3 of bedrock during the last glacial cycle. Over 82% of this total eroded volume is sourced from sedimentary strata associated with the Platform geological province, which in total accounts for 71% of the formerly glaciated domain (Table 1), including most of the offshore sectors (Fig. 2a). Shield areas, despite composing 21% of the domain, source only 8% of erosional products (Table 1). Across all sectors, the bulk pattern of erosion is primarily concentrated along the main arterial discharge pathways of the EIS, with largest removal depths of 80–190 m within the troughs of the Barents Sea (Supplementary Fig. 4b), corresponding to rates of 1.4–5.2 mm a−1 (Fig. 3). Zones of suppressed erosion, characterized by rates less than 0.1 mm a−1, are sustained across upland and divergent flow zones of Fennoscandia, the Barents/Kara seas, Great Britain, and Ireland. Over 19% of the entire domain, and 55% of the land area above 1000 m a.s.l., experienced less than 2 m net erosion, coincident with relict zones of landscape preservation in upland areas and those of extreme cold in the Kara Sea and Taimyr Peninsula (Supplementary Fig. 4b).

When aggregated over the glacial cycle, long-term patterns of erosion correspond well with independent (bulk) empirical estimates, particularly within marine-terminating source areas (Supplementary Table 1). Comparison with a recent source-to-sink sediment budget for the Baltic Sea Basin39 reveals potential overestimation of the predicted long-term glacial erosion rate in this sector of the EIS. This could reflect model limitations in reproducing ice streaming through this basin, thereby leading to excessive ice fluxes and inferred catchment size, or possibly equally due to methodological issues in the estimation and reconstruction of the erosion rates reported. Further, the discrepancy can also be indicative of the varied protective role of previously worked sediments deposited in the source area during earlier glaciations, which our model does not account for. Lastly, the Neoproterozoic and early Palaeozoic sedimentary rocks found in the Bothnian and Baltic basins are fundamentally different in terms of their mechanical properties (hardness and fracture density) when compared to the softer Mesozoic sedimentary rocks found elsewhere across other platform areas in the domain39, highlighting the local complexities associated with geological controls when modelling potential erosion at the continental/ice-sheet scale.

Over the short-term, results derived using the model will be susceptible to some variation due to choices in parameter values that impact ice flow and discharge, such as the basal sliding scheme utilized or in modifications to the ice flow law that enhance internal deformation (see “Methods”). A comprehensive suite of sensitivity experiments has previously been conducted to investigate the impact of incremental changes in internal and external parameters on our model results12. For this study, to assess the sensitivity and impact of modelled ice-sheet dynamics on the resulting erosion patterns, we systematically perturb the primary determinant of ice flow — the rate-factor multiplier (A0) — in a series of experiments conducted during peak glaciation at the LGM (Fig. 4). During this 2000-year period, glacial erosion rates across the mid-Norwegian margin broadly follow similar trends, but with significant excursions during short-lived (sub-millennial) episodes of high ice flux. Subsequent rescaling of the Younger Basement erosion law to account for these varying ice fluxes, yields a sensitivity difference amounting to ~1 mm of net erosion during such high ice-discharge episodes, lending confidence both to the approach and the reconstructed patterns of erosion. Ultimately though, the bulk erosional products are finitely constrained over the glacial cycle, thereby effectively limiting the impact of uncertainties in model degrees of freedom and specific parameter choices on our results beyond sub-millennial timescales.

## Discussion

### Spatiotemporal controls on glacial erosion

Geology and climate are conventionally regarded as the key controls on glacial erosion. Attempts to link glacial erosion rates with precipitation or latitudinal proxies have yielded statistically robust results, though consistent comparison of glacier types is a key consideration5,7,40. Analysis of our modelled subglacial erosion patterns across the Northwest Eurasian domain fails to yield statistically significant support for a synoptic-scale link between glacial erosion and latitude (Fig. 5b). However, this negative result does not entirely preclude a latitudinal link, and may reflect the all-encompassing nature of an ice sheet’s subglacial footprint compared to that of individual glaciers, where it is possible for the most dominant glacial catchments to span multiple climatic zones or even geological provinces. Instead, our results consistently support a nuanced hierarchy of interdependent environmental controls based on lithology, topographic relief and climate, all of which are critically modulated by the continuously evolving thermomechanical configuration of the ice sheet.

On the inter-regional scale, we find that geology can provide a key control on erosion. For example, erosion rates across the Baltic Shield are an order-of-magnitude lower than adjacent provinces (Table 1), ostensibly due to rock hardness and fracture density that limit abrasion and plucking41,42. Persistent cold-based conditions coupled with high overburden loads across northern Fennoscandia also combine to protect the substrate, limiting the impact of meltwater and groundwater over-pressurization43. Similarly, restricted erosion across low-lying platforms of northern Russia and the Kara Sea reflect cold-based subglacial conditions that correspond with limited entrainment of perma-frozen clayey sediments by basal ice44.

Contrasting sedimentary and crystalline bedrock lithologies have also been implicated as a distinguishing control on bulk glacial erosion, with rates doubling between the Bjørnøyrenna catchment and the mid-Norwegian shelf, respectively (Supplementary Table 1)31. While bedrock properties associated with different lithologies can exert a primary control on the dominant erosional mechanism, such as abrasion (soft, thick-bedded, wide joint spacing) versus plucking (hard, thin-bedded, narrow joint spacing)42, generally we find no significant difference in erosional efficacy between these two provinces once our model results are normalized for the duration of ice-sheet occupancy (Fig. 2c).

Sustained ice streaming routed via the main arterial troughs across the Barents Sea reflects both the broad ice-sheet configuration and warm basal conditions, with the distinct absence of lateral barriers promoting efficient mass discharge from the ice-sheet interior. Observations of multi-phase flow trajectories of the Bjørnøyrenna ice stream since the LGM attest to the spatiotemporal variability and occurrence of significant flow switching45 that would act to limit the development of glacial overdeepenings46. In contrast, the deep, narrow fjords of western Norway invoke transverse shear zones that act to stabilize and direct ice flow, suppressing lateral development of outlet discharge from the hinterland to the shelf break. Whilst this deep linear erosion of the Norwegian coast and inner shelf has yielded significant sediment volumes, these topographic steering feedbacks can ultimately act to limit erosion as the long profiles of beds beneath highly erosive glaciers tend towards steady-state angles opposed to and steeper than the overlying ice surface47. Across sufficiently steep retrograde slopes, negative feedbacks related to channel closure through glaciohydraulic supercooling will reduce the bedload transport capacity and ultimately favour sediment deposition over bedrock erosion46,48.

On 104–5 year (ice age) timescales, climate a priori controls glacial erosion since ice-sheet mass balance, volume and dynamic activity ultimately scale to snow accumulation. We find that peak erosion of 5–7 km3 a−1 occurs during three phases: 88 ka, 61 ka and 23 ka BP, coeval with maximum EIS volume and positive mass balance (Fig. 1). Erosion patterns also deviate by ±1 mm a−1 over centennial timescales in response to abrupt warming. Such transient behaviour - when the cryospheric system is perturbed from its equilibrium state - arises from abrupt external forcing, though can also result from internal ice-dynamical related switches due to thermomechanical reconfiguration.

Following such arguments, the erosion patterns described above could, reductively, be ascribed exclusively to climatic amelioration and its impact on ice dynamics. For example, the mean EIS erosion rate by province was 6–17% higher during the last 8 ka deglaciation compared to the bulk mean over the glacial cycle (Table 1). Yet, further analysis of the spatiotemporal erosion patterns reveals that this difference of up to ±1.6 mm a−1 results from dynamical reorganization during the asynchronous build-up of the EIS to its LGM extent (Fig. 5c). The shifting eastward extent from 23-19 ka BP in response to ice-divide migration12,49 intensified ice flow and erosion within the central Barents Sea and Russian sectors later than elsewhere. Subsequent warming did promote meltwater runoff and subglacial discharge during deglaciation, which further enhanced erosion rates of these late-stage ice streams50,51, but climate itself was not the underlying control.

The uplands of Fennoscandia have traditionally been considered relict landscapes preserved by cold-based ice. However, the extent to which glacial action has been limited to superficial scouring3,52 or pervasive down-wasting2,18,19 across these upland zones is debated. Our subglacial erosion results demonstrate that high-elevation sites (≥550 m a.s.l.) within the Younger Basement province account for ~19% of the total sediment flux to offshore depocenters (Table 1). Previous partitioning of glacigenic sediments transported off the mid-Norwegian shelf broadly correspond to between 10 and 45% derived from high-elevation plateaus2,19. Our analysis indicates limited support for a bimodal signature of erosion across western Fennoscandia over the entire glacial cycle. However, we do find an internal thermomechanical control on erosion across high-elevation plateaus during phases of deglaciation. For example, mean erosion rates increase from 0.16 to 0.25 mm a−1 after the LGM at elevations above 550 m a.s.l. (Table 1; Fig. 5c). Such time-transgressive impacts, masked by spatially complex patterns of erosion and preservation, illustrate how the substrate and evolving thermodynamic regime interact to drive episodes of enhanced upland erosion. Our results also reveal a relatively modest increase in bulk erosion at ~800 m a.s.l. which might be interpreted as evidence of a bi-modal signature and support for the buzzsaw hypothesis, but rather, we find these patterns relate to the hypsometric distribution of relief, and not to any intensification of glacial erosion at this elevation (Supplementary Fig. 5).

Our analysis demonstrates that bulk patterns are principally dominated by focused and selective erosion below 550 m a.s.l. (Fig. 5a). While Baltic Shield erosion is generally modest over the last glacial cycle, deep erosion of Younger Basement and Platform provinces at rates of up to 5.2 mm a−1 during transient phases account for the bulk sediment volume and stratigraphic architecture found within TMFs bounding the shelf edge. This vigorous, deep regional incision from Shield margins, along with episodes of enhanced erosion driven by abrupt disequilibrium in the climate-cryosphere-landscape system, help contextualize contemporary process studies that support extreme rates of glacial erosion7,53 when these cryospheric systems are rapidly transitioning to new equilibrium states.

### Suspended-sediment fluxes

Glacial erosion and transport of sediments, including those suspended in proglacial runoff, loaded numerous TMFs distributed along the marine shelf edge offshore Scotland, Norway, and the Barents Sea (Fig. 3). The quantification of suspended-sediment concentrations adjacent to modern-day glaciers and ice sheets is also used as an indirect proxy to derive short-term rates and trends of subglacial erosion53,54. To examine these relatively ephemeral studies within a broader geological context we utilize our own time-transgressive framework to derive and contrast potential suspended sediment (i.e., ignoring bedload) fluxes and trends throughout the last glacial cycle.

Bulk fluxes are derived by scaling contemporary process measurements from the Greenland ice sheet (see “Methods”) to meltwater runoff discharge calculated from the mass balance algorithm of the ice-flow model. Assuming a continual source of subglacial sediments, fluxes from catchments along the North Atlantic and Arctic Ocean margins peaked during phases of rapid climate warming and deglaciation, attaining a maximum of 12.78 ± 5.1 Gt a−1 (Fig. 1). This resulted in a total delivery of ~50,000 km3 of sediment into the ocean system during the last glacial cycle, yielding 125 m of net aggradation along the continental shelf-break. The maximum suspended sediment flux, associated with the collapse of the EIS during the Bølling-Allerød warming ~15 ka BP when surface melting of the ice-sheet surface was most intense and had extended northwards to affect the Eurasian Arctic too20, is an order of magnitude greater than contemporary rates from the Greenland ice sheet, and equivalent to the present contribution from all rivers globally54. The impact on downstream marine ecosystems during this episode would have been substantial, limiting primary productivity in the pelagic zone by reducing light availability in the water column and increasing the far-field supply of bioavailable micronutrients to promote phytoplankton blooms, yielding a major carbon sink55.

While subglacial erosive efficacy broadly scales to EIS flow dynamics, linked to the progressive influence of ice-stream activity through ice-sheet evolution56, subglacial mobilization of those sediments as suspended loads is phase-lagged by millennia and driven by episodes of abrupt warming and the associated rapid increases in meltwater and runoff delivered to the bed (Fig. 1). This provides a direct analogue to the contemporary response and deglaciation of the Greenland ice sheet under abrupt 21st century warming57. Record meltwater fluxes accessing the ice-bed interface due to anomalously warm and wet summers58, coupled with enhanced dynamics as interior regions of the Greenland ice sheet transition to wet-bedded regimes, promote the mobilization of new, subglacial sequences of sediment59,60 by more extensive and intense basal drainage, with resulting suspended loads suggesting erosion rates orders-of-magnitude above the considered long-term norm53,61. Such insights into the transient nature of these processes — where increased melt runoff under abrupt warming accesses new zones of the subglacial environment with increased vigour to mobilize previously reworked and deposited glacial sediments — highlights a cautionary tale when upscaling field-measurements from specific point sources on short (contemporary/process) timespans to inform long-term landscape evolution models5,7.

In summary, our analyses demonstrate that the EIS, which at its maximum spanned over 5500 km of temperate through extreme polar conditions from 49 to 82°N, was an extremely effective agent of erosion. Despite this, its spatiotemporal signature was also highly heterogenous and transient, with its erosion potential determined by an interacting hierarchy of environmental and internal controls: climate, lithology, topographic relief, and the dynamically evolving, thermomechanical ice-sheet configuration. The time-transgressive perspective presented here reveals the complex and highly nuanced impact of glacial erosion on long-term landscape evolution, which, at a continental ice-sheet scale, rudimentary environmental proxies such as latitude or climatic regime fail to capture.

Though our analysis discounts the case for any deep, uniform glacial erosion of the Fennoscandian shield, the internal dynamic modulation demonstrated by our transient analysis provides support for the view that ice sheets can be able or feckless erosive agents in the same place at different times62. Moreover, the highly selective subglacial erosion across shield margins, along with episodes of extreme rates driven by abrupt disequilibrium within the climate-cryosphere-landscape system, provide essential context to contemporary process studies that imply exceptionally high rates of erosion — or rather the mobilization of reworked subglacial sediments — as ice sheets respond to increased meltwater and runoff fluxes to the basal environment and deglaciate under abrupt climate warming.

## Methods

### Ice-flow model description

The three-dimensional thermomechanical model applied and its associated boundary conditions are of the same derivation as those used by Patton et al.12,20 to model the Late Weichselian Eurasian ice sheet (EIS) (Supplementary Table 3), with several modifications made to make it suitable for running over full glacial timescales. In brief, the ice-flow model is a first-order approximation of the Stokes equations and includes longitudinal (membrane) stresses that become increasingly important across steep gradients in topography and motion. The model is integrated forward through time on a finite-difference grid through perturbations in climate (NGRIP δ18O) and eustatic sea level63 (Fig. 1).

Ice velocity, composed of internal deformation and basal motion, is determined through flow and sliding relations. Internal deformation uses an adaption of Glen’s flow law64 that includes an empirical flow enhancement (aka softening) coefficient, used to encompass the effects of crystal anisotropy and impurities on bulk ice deformation65. Modifying the strain rate towards softer ice tends to increase ice-flow speeds. Basal sliding is related to the basal shear stress through a Weertman-style sliding law, that is introduced at sub pressure-melting temperatures using an exponential decay function66,67: at 0.75 K below the pressure-melting point, sliding is 0.47 of its value at the pressure-melting point. While this classical sliding law is based on the assumptions of a glacier flowing over hard bedrock, its applicability can break down in subglacial environments characterised by widespread cavitation or ‘soft’/till-laden beds that induce a Coulomb-type regime where basal shear stress is proportional to the effective pressure. Coulomb friction is thus more representative of sliding near the grounding line. Real-world sensitivity analyses regarding the choice of friction law on grounding line dynamics reveals a varied transient response, with Weertman laws systematically predicting the smallest changes in ice volume above flotation68, though the exact nature of the effect on ice dynamics due to parameter choices in sliding laws can be highly region-dependent69.

Surface mass balance is determined by a positive degree-day (PDD) scheme70, and derives total melt from integrated monthly positive temperatures. Both temperature and precipitation adjust to the evolving ice-sheet surface through applied lapse rates derived from multiple-regression analyses of meteorological observations at a resolution of 1 km from the WorldClim database (Version 1.4)71. To account for the large variations in climate regime across the Eurasian domain, regional reference climates and associated forcing are tuned independently for each of the three major ice-sheet accumulation centres (from north to south: the Barents Sea, Fennoscandia, and Great Britain and Ireland). An additional mass-balance term incorporated is the net water–vapour flux to and from the ice-sheet surface — a predominant component of ablation in cold continental settings where humidity can be very low72.

Calving losses at marine-terminating margins are coupled to relative sea level using a standard empirical function relating the calving flux to ice thickness and water depth73. The sensitivity of calving to, for example, variations in ocean temperature and sea-ice buttressing has been controlled spatially and temporally through a depth-scaled calving parameterization74. In the absence of explicit calculations of such external feedbacks, this depth-related calving coefficient provides a pragmatic and computationally efficient parameterization for determining mass loss at marine terminating margins of the EIS. The model is applied to a 10 km finite-difference mesh with the inclusion of grounding-line dynamics based on the analytical boundary treatment of Schoof75 and adapted in 2D by Pollard and DeConto76, which defines the ice flux at the grounding line as a function of ice thickness linearly interpolated between the adjacent node that brackets floating and grounded ice.

Several modifications were made to the ice-flow model to make it suitable for running over full glacial timescales. The first accounts for insolation variations and their impact on the surface mass balance. This parameter can account for 20–50% of the surface melt anomaly, although its effects are not homogenous over the ice sheet since the contribution of insolation to melt is modulated by the local surface albedo77. We apply a simple and tested correction factor to the PDD melt scheme (Mcorr), making it possible to tune the insolation-independent parameters from present-day data, while relying on the insolation term to handle changes in palaeo conditions:

$${M}_{{{{{{\rm{corr}}}}}}}=M\left({S}_{0}\right)+{M}_{{{{{{\rm{insol}}}}}}}$$
(1)
$${M}_{{{{{{\rm{insol}}}}}}}=a\left(T,d\right)\left(S-{S}_{0}\right)$$
(2)

where Minsol is the insolation-related melt, S is the incoming solar radiation at the top of the atmosphere, and S0 is the spatially and seasonally explicit insolation fixed to present-day levels. The spatially and seasonally varying coefficient, a, can be considered to be representative of average local conditions of surface albedo and atmospheric transmissivity for the ice sheet, in effect acting as a substitute for a full surface-albedo model. The dependence on temperature means that a cannot be taken as a constant, but must contain some seasonal variation. To better capture this local variability, a (T,d) is parameterized as a piecewise linear function of the local mean daily temperature (T) and the day of the year (d):

$$a=\left\{\begin{array}{cc}0,& {{{{{\rm{if}}}}}}\,T\le {T}_{\min }\\ {a}_{\max }\frac{\left(T-{T}_{\min }\right)}{\left({T}_{\max }-{T}_{\min }\right)},& {{{{{{\rm{if}}}}}}\,T}_{\min }\, < \,T\, < \,{T}_{\max }\\ {a}_{{\max }},& {{{{{\rm{if}}}}}}\,T\ge {T}_{\max }.\end{array}\right.$$
(3)

For simplicity, the values of amax and Tmax were chosen to be constant77. Tmin – the minimum temperature threshold for melt corrections — was parameterized as a nonlinear sinusoidal function of the time of year between Tmax in winter and Tmin,sum in summer:

$${T}_{\min }={T}_{\max }-\left({T}_{\max }-{T}_{\min,{sum}}\right){\left(\frac{1-{{\cos }}\left(2{{{{{\rm{\pi }}}}}}\frac{d}{365}\right)}{2}\right)}^{p}.$$
(4)

The values of amax, Tmax, Tmin,sum, and the exponent p used are based on tuning to best match the diagnosed values of a across the Greenland ice sheet77.

### Ice-flow model experiment design

Ice-modelling experiments were initiated for the Eemian climatic optimum at 123 ka BP to allow small ice caps to equilibrate with Eemian climate. Two important assumptions for these experiments are that the Eemian topography was similar to that of the present-day in terms of isostatic relaxation and relief, and that Eemian interglacial glaciers had reduced to a negligible size.

Manually applied perturbations of climatic patterns, such as major rain shadow effects, were kept consistent with those applied during the optimum Late Weichselian experiments of Patton et al.12, though the magnitude of these climate gradients were tuned separately for the previous two ice-sheet advances during the last glacial cycle. One major divergence from the original setup was to add a dynamic adjustment to mimic a Kara Sea precipitation shadow by scaling the precipitation gradient with ice-surface elevation over central Novaya Zemlya. The resulting effect freely encourages ice growth over the Kara Sea during phases of ice build-up but prevents runaway ice expansion once the central ice dome thickens.

The aim of these modelling experiments is to produce a glaciologically feasible EIS simulation that fits within the broad geological framework related to ice extension during the last glacial cycle (Fig. 3; Supplementary Fig. 1). The uncertainty associated with the few geophysical constraints for glaciation over this time period means that the envelope within which a potential reconstruction could fit remains open-ended. For example, what was the initial ice topography during the Eemian, and did ice centres always fully retreat during Weichselian interstadials? Although specific questions relating to the evolution of the ice complex over multiple glacial advances cannot currently be answered, the experiment used in this study provides a framework from which insights into longer-term processes can be examined.

Sensitivity experiments of the LGM reconstruction with an essentially identical setup were carried out by Patton et al.12, highlighting the variation of the maximum ice-sheet configuration to a range of key climate and internal parameter choices. Furthermore, the pattern and rate of retreat across the Eurasian Arctic after the LGM was independently validated and compared with other model reconstructions using a suite of relative-sea level curves from around the Barents Sea20,78.

### Subglacial erosion scaling laws

The long-term mean erosion rate, $$\bar{f}$$, is derived by linearly scaling eroded bedrock volume (B) with mean ice discharge ($$\bar{q}$$) (Fig. 2c) and the duration of ice coverage (ti):

$$\bar{f}=\frac{\left(B{{{{{\rm{|}}}}}}\sum \bar{q}\cdot {t}_{i}\right)\cdot \left(\bar{q}\cdot {t}_{i}\right)}{{t}_{i}}.$$
(5)

Eroded bedrock volumes for the Platform and Younger Basement provinces are derived from volumetric calculations of the Bjørnøya and mid-Norwegian margin TMFs, respectively. To tune spatial patterns of long-term erosion to observations of pre-glacial landscape preservation, erosion was limited to areas where depth-averaged velocities are at least 10 m a−1. The resulting linear fit between the rate of erosion (mm a−1) and local ice discharge within both erosion catchments is thus described (Fig. 2c):

$${f}_{{YB}}=8.3832\times {10}^{-10}\cdot {q} - 8.8373\times {10}^{-20},$$
(6)
$${{f}}_{{{{{{\rm{Platform}}}}}}}=1.0512\times {10}^{-9} \cdot {q}+1.6817\times {10}^{-19}.$$
(7)

The widespread preservation of relict glacial-79,80,81 and non-glacial landscapes82,83,84, and of preglacial weathering remnants on the Fennoscandian Shield, highlights the minimal impact of Pleistocene glacial erosion within central areas of the former EIS15,85,86,87,88,89. Semi-quantitative estimates of total glacial erosion on valley floors within central shield areas range from ~2.5 to 50 m16,85,90,91. With limited further insight on volumes of glacigenic sediment transported from Shield areas, we use a suite of 249 cosmogenic 10Be measurements from bedrock samples to calibrate a bulk scaling factor that best fits the duration of ice cover over the last 2.6 Ma according to the LR04 stack33 and patterns of modelled ice discharge (Supplementary Data 1).

For each sample a scaling factor, F, is calculated that yields a glacial erosion rate and sample-depth history compatible with the measured 10Be concentration. The calculations are based on the glacial erosion calculator glacialE.m version 201912 from the expage cosmogenic nuclide calculators (http://expage.github.io/calculator). Cosmogenic 10Be production rates are calculated based on the nuclide-specific LSD method92 with production from muons calibrated against the Beacon Heights depth-profile data93 and using a global average reference spallation production rate of 3.98 ± 0.22 atoms g−1 a−1. The time-dependent 10Be production rate is calculated based on sample location (latitude, longitude, and elevation), topographic shielding, sample depth, bedrock density, and history of ice cover. We assume full shielding from cosmic rays and a zero cosmogenic nuclide production rate through all periods of ice cover. The local ice discharge and ice cover periods from the ice-sheet model are used for the last glacial cycle. Similar to the ice-sheet model, glacial erosion occurs only when depth-averaged velocities are at least 10 m a−1. For the time 2.6 Ma to 123 ka, we use the LR04 stack33 as a proxy for ice cover, with an ice cover cut-off value determined by the modelled total duration of ice cover in the last glacial cycle with linear interpolation of the LR04 cut-off value from 3.2‰ (123 ka ice cover) to 5.0‰ (0 ka ice cover). For a particular cosmogenic nuclide location, we use the average ice discharge from the last glacial cycle for all ice cover periods prior to 123 ka. Because the last duration of exposure to cosmic rays is commonly critical for the 10Be concentration, we adjust the timing of the last deglaciation to match the reconstructed deglaciation from DATED-194. The deglaciation adjustment is only applied to the ice cover history (shielding from cosmic rays), and it does not affect the total glacial erosion derived from the ice-sheet model. For the ice-free periods we assume no interglacial erosion. For twelve samples collected from sites below the highest shoreline in Sweden, the production rate is adjusted for the water depth calculated from the uplift model by Påsse and Daniels95, with uplift following pre-LGM ice cover periods based on the duration of ice cover and the uplift history following the last deglaciation. We then calculate the central value of F and use a Monte Carlo approach with 10,000 iterations to estimate positive and negative uncertainties, applying the reported 10Be concentration uncertainty, the reference production-rate uncertainty, 10Be decay constant uncertainty96,97, a 5% uncertainty for bedrock density, a 1 ka uncertainty for the DATED-194 deglaciation age, a 10% uncertainty for the vertical uplift for samples located under the highest shoreline, a 0.1‰ uncertainty for the LR04 ice cover break value for the glaciations prior to the last glacial cycle, and a one-sided positive uncertainty of 0.005 mm a−1 for the interglacial erosion rate. For full details of the calibration, we refer to the provided supplementary code (Supplementary Data 2). Using the median scaling factor value of these data points we produce a linear glacial-erosion rule with a fit almost one order-of-magnitude less than that for the Younger Basement and Platform sectors (Fig. 2c):

$${{f}}_{{{{{{\rm{shield}}}}}}}=1.7062\times {10}^{-10}\cdot{q}.$$
(8)

### Uncertainties in estimating glacial erosion

The reworking of older glacial and interglacial sediments during glacial advances, in addition to the volume of sediments not transported to the TMF depocenters, are potential sources of bias in our back-stacking calculations. For example, in the present interglacial, glacigenic sediments across the Barents continental shelf today range in thickness from 0 to 300 m98, and the pattern of sediment thicknesses across the Baltic Shield exhibits a pronounced spatial pattern consistent with a range of 0–20 m35. Similarly, the presumption for totally efficient removal of sediments from the subglacial system will likely compound this uncertainty52, although their evacuation can proceed rapidly once active51, at rates of up to 1 m a−1 beneath streaming ice36.

The use of cosmogenic nuclide data as a proxy for constraining erosion rates, while arguably more precise, is limited by the shallow depth of cosmogenic-nuclide production in rock and, consequently, a high sensitivity to glacial erosion depths during the last ice cover period (erosion deeper than the depth of production cannot be constrained) and durations of exposure/shielding during the post-glacial period. Modest, local variation in glacial erosion depths can also yield nearby surfaces with highly varying cosmogenic-nuclide concentrations41,99. Samples for cosmogenic nuclide analysis are often sampled away from valley/basin floors (mostly because of sediment fills), which could bias estimates of erosion. For example, the sampling of plucked bedrock surfaces to yield reliable deglaciation ages by avoiding cosmogenic-nuclide inheritance99, leads to unconstrained erosion estimates in a landscape where glacial erosion otherwise was minimal, creating a complex and inconsistent patchwork of constraining point data. Similarly, samples can also be taken to establish non-glacial erosion rates, such as on blockfields or tors where geomorphic evidence indicates negligible glacial erosion.

Modelled patterns and rates of glacial erosion correspond well with available empirical records of long-term denudation along the western seaboard of the Eurasian continent (Supplementary Table 1), and are qualitatively in-line with the inferred zonation of erosion based on the spatial pattern of Fennoscandian Quaternary deposits52. Volume-based sedimentation rate estimates derived from shield sources, which fed sink areas south and east of the EIS, are, however, poorly constrained, and generally limited by uncertainty in the initial terrestrial sediment volumes. One estimate for Quaternary sediments deposited by the EIS yields a mean thickness of ~55 m across the Eastern European Plain, increasing to 84 m within the LGM domain100. With a source area representing erosion of the Shield in Finland and the Platform in the eastern Baltic, Gulf of Finland and East European Plain, these estimated sediment volumes provide an order-of-magnitude indication of glacial erosion on the scale of up to tens of metres over the last 2.7 Ma, aligning with mean modelled estimates of glacial denudation of 6.95 m (0.06 mm a−1) across Shield areas for the last glacial cycle (Table 1).