The model domain consists of a 300 × 200 km rectangular section of an idealised ice sheet. The domain coordinates extend from x=-150 km to x=150 km and y=-100 km to y=100 km. We create an ice cap mirrored along the x direction, i.e. where the centre of the domain (x=0) represents the ice divide. We apply all changes symmetrically with respect to x but focus our analysis on the positive side of the domain (i.e. x>0; thus our effective domain is 150 km long). This is done in order to ensure a natural boundary at the upstream side (the ice divide). The downstream side is kept as an open boundary, allowing the ice front to retreat. On the lateral limits, a free-slip boundary condition is applied (i.e. velocities are zero perpendicular to the boundary but unconstrained parallel to it), and unless stated otherwise, we only consider the region within 50 km of the centreline (y=0) to avoid boundary effects.

We set the bed elevation at the divide (B0) to 750 m above sea level (m a.s.l.), which we keep constant in all experiments. We prescribe a prograde sloping bed (B) with an inclination β=0.9 % (Eq. ), which results in the bed sloping below sea level at x≈83 km. 1B(x)=B0-β⋅|x|

Because nunataks close to the Antarctic coast or along the margins of palaeo ice streams are often situated in proximity to over-deepened troughs, which can trigger marine ice sheet instability, a retrograde-slope case was also tested. In this case, we set B0 to -750 m and β to -0.1 % so that the bedrock at the downstream end of the domain is at the same elevation (-600 m a.s.l.) in both types of experiments, which is representative of continental shelf depths of Antarctica . The grounding line in Úa is defined by the flotation condition, and although its migration is not the focus of our study, we allow it to move freely, refining the mesh elements around the grounding line as it evolves (as described at the end of this section).

Nunataks in this domain are represented as three-dimensional Gaussian surfaces, which are superimposed on B with their centre (i.e. the nunatak summit) at |x|=50 km and y=0 km. All generated nunataks have an outcrop size (i.e. exposed area above the ice surface) after spin-up of approximately 12 × 5 km along their main axes. Depending on the experiment, the nunatak was elongated transversely to flow (i.e. along the y axis) or parallel to flow (i.e. along the x axis; Fig. ). The adopted idealised nunatak dimensions and the value of β are within the interval constrained by 33 real nunatak topographic profiles across Antarctica (Figs. S1–S3).

The effects of ice rheology (including temperature) in Úa are accounted for by the strain-rate factor A in Glen's flow law. To compute A, which we treat as spatially uniform and constant over time for the purpose of our experiments, we assume an ice temperature (T) of -20 ∘C. This yields a value similar to that found for regions surrounding nunatak escarpments in Antarctica, as obtained in when inverting for A and basal slipperiness (C) based on satellite-derived ice surface velocities. Following the same reasoning for A, we assume basal sliding to follow Weertman's sliding law and use a constant value for C of log10(C)≈-4.5 (C=2.9×10-5 m kPa-1a-1). The main model parameters are summarised in Table , and sensitivity analyses of values for A and C are presented in the Supplement (Figs. S4, S5).

The surface mass balance (SMB; in metres per year, ma-1) parameterisation applied to the spin-up and subsequent experiments is given by a(x,t) in Eq. (): 2a(x,t)=a0-a0-aeLx⋅|x|+b(t), where a0 is the SMB at the ice divide (x=0), ae is the SMB at the edge of the domain (x=150 km), Lx=150 km, and b(t) is a time-dependent SMB term through which perturbations to the total SMB are applied. We prescribe a0=1.3 ma-1, and ae=-0.3 ma-1, which results in no ablation occurring over our region of interest (black line in Fig. a), as is usual for Antarctic settings .

The model spin-up starts from a 1200 m thick uniform ice distribution, to which a constant (but spatially variable) SMB is applied (i.e. b(t)=0 ma-1 in Eq. ) for 20 000 years (20 kyr). This period is long enough for the system to reach equilibrium with the SMB forcing. The average ice surface slope between the ice divide and the grounding line (which after spin-up was located at x=136 km; Fig. b) is ∼1.3 %. This inclination is representative of measured profiles along nunataks for various regions of the Antarctic Ice Sheet (Figs. d, S1, S2). The resulting surface velocity varies from 0 ma-1 at the divide to 121 ma-1 downstream of the grounding line, with median and mean velocities of 29 and 33 ma-1, respectively (see Fig. S9 for a comparison of velocity profiles along the centreline). The ice flow velocity increases along the nunatak flanks, with a maximum of ∼53 m a-1, consistent with observed values in . Although velocities at the floating end are slower than observed for ice shelves in Antarctica, they are far from our region of interest, and the setup reproduces key features observed along the analysed nunatak profiles, such as ice surface gradients across nunataks and ice acceleration upstream and downstream of the nunatak (e.g. Figs. , S9). Our idealised setup focuses on capturing the key components mentioned above, while excluding unnecessary complex features that could prevent identifying the ice surface response signal to the ice flow interaction with the obstacles.

In order to understand whether ice thinning occurs uniformly up- and downstream of a nunatak, we impose three different degrees of ice thinning uniformly over the domain. The perturbations to SMB that induce ice thinning are applied through b(t) in Eq. () and only evolve in one direction over the entire domain, towards increased ablation. The evolution of b(t) is based on a smoothed step curve that applies a weight, evolving from 8 % to 100 %, mimicking the deglaciation progression recorded in ocean sediment and ice cores for the past 20 kyr e.g.. This “weight curve” is then multiplied by a constant chosen in order to match Last Glacial Maximum-to-present ice thinning at three contrasting regions as inferred from a continent-wide transient modelling experiment . These regions are Thwaites Glacier, in West Antarctica (“thw”, -0.75 m a-1), and the Transantarctic Mountains (“tam”, -0.60 m a-1) and Dronning Maud Land (“dml”, -0.45 m a-1) in East Antarctica (Fig. c). Using the thw scenario as an example, the value for b(0) is -0.75⋅0.08 m a-1 and for b(20kyr) is -0.75⋅1.0 m a-1. These three different thinning scenarios were applied to nunataks that were elongated along and transversely to ice flow and to reference experiments without a nunatak for comparison purposes.

Across much of Antarctica and Greenland, nunataks are more common in groups than in isolated cases. The aim of this set of experiments is therefore to test whether three nunataks, separated by narrow glaciers, yield ice surface elevations and gradients that differ from the control run due to a combined effect of all the nunataks on ice flow. For this test, we perform sets of experiments with the same SMB as applied in the thw experiment with one nunatak (control run) but using three nunataks aligned transversely to flow at |x|=50 km and of the same size as in the ice-thinning experiments. The spacing between nunataks differs in each experiment, resulting in glacier widths of 0 (i.e. the three nunataks form a single, wider barrier), 5, 10, and 15 km. The range of applied widths reflects realistic values observed around Antarctica Fig. S3c;.

In a final series of sensitivity experiments, we assess how well regularly spaced grids of coarser resolutions typically used in continental-scale ice sheet models (5, 10, and 20 km) resolve the ice surface elevation pattern around nunataks compared to the solution using an unstructured, locally refined mesh. We do so by repeating the full set of three-nunatak experiments and the one-nunatak thw scenario (as control) for these regular meshes (i.e. without refinement).

All sets of experiments, their respective surface mass balance (SMB) perturbations, and the number of nunataks in each set are summarised in Table .

Our experiments clearly demonstrate that the presence of a nunatak impacts ice surface elevations and that the response magnitude depends on its orientation relative to ice flow. The flow barrier created by the nunatak causes ice to accumulate upstream, depleting the downstream side of ice and causing the ice surface there to lower (Fig. a, b). At the start of the thinning experiments where the nunatak is placed transversely to flow, the effect of the nunatak on the ice surface elevation is seen up to 30 km away from its summit along flow and 15 km perpendicularly to flow. The ice surface directly upstream of the nunatak is 360 m above the general ice sheet surface (considered to be the elevation 15 km away from the centreline), while downstream the lowest ice surface elevation (located 3 km downstream of the summit) is 100 m lower than the general ice surface elevation. Along the nunatak flanks, the ice surface elevation falls about 300 m. In summary, while the general ice surface slope over the grounded part of the domain is 1.3 %, the slope is about 2–3 times steeper around the nunataks. For experiments where the nunatak is elongated parallel to ice flow, the impact of the nunatak is smaller, producing a lower ice surface elevation gradient and relative elevation-change effects that can be discerned 20 km up- and downstream along the centreline and up to 5 km transversely to the centreline (Fig. b). In both cases, changes in ice surface elevation perpendicular to ice flow, caused by the nunatak presence, stand out from the general ice surface elevation for the entire extent of the subglacial nunatak obstruction (cf. Figs. a, b and a, b).

Both ice-thinning experiments detailed in Fig. a and b reveal an overall steepening of the ice surface gradient during the 20 kyr thinning period (profiles from red to green). The evolution of this steepening is illustrated in Fig. c and d, where points equidistant from the nunatak summit (upstream and downstream) show ice surface elevation differences that increase through time. For equidistant locations closer to the nunatak summit (< 2.5 and < 7.5 km for Fig. c and d, respectively), the increase in ice surface steepening (i.e. in elevation difference) is disrupted when the downstream location becomes exposed, since its elevation no longer changes while the equivalent point upstream is still ice covered and thinning. This pattern of increased steepening around nunataks is potentially important for cosmogenic-nuclide studies to consider because from these modelling experiments, significant differences in the time of exposure of up- and downstream faces of nunataks become apparent (Fig. ).

To analyse the difference in timing of ice surface evolution and nunatak surface exposure up- and downstream of the nunatak, we select five pairs of points equidistant from the nunatak summit. They span a distance from where the downstream side is already exposed at the start of the experiment (1.5 km) to the closest element to the nunatak where the ice continues to thin normally (i.e. it does not become exposed or stops thinning; 2.5 km) in any ice-thinning scenario. The chosen points (Fig. ) are also separated in the model mesh by one element size from one another. In our analysis, we consider that nunatak surface exposure commences when ice thickness falls below 10 m. A thickness threshold larger than the minimum thickness in the model is used for a consistent identification of the timing of surface exposure between the different experiments and different points analysed, and 10 m yielded the best results among the thresholds tested. For the purpose of cosmogenic exposure dating, most cosmogenic-nuclide production occurs when ice is thinner than ∼10 m . At the upstream side, if the bedrock surface remains ice covered according to the previous criterion, we then determine when the ice surface stabilises and thus reaches its minimum elevation. The implications of comparing the time of exposure (thickness <10 m) with time of stabilisation (thinning <5 cm/100 years) are discussed further in Sect. . For visualisation purposes, only the experiments with nunataks elongated transversely to ice flow are shown in Fig. , but the patterns shown also hold true for nunataks elongated parallel to ice flow. Because of the elongation of the nunatak, its initial area exposed along the centreline is larger, and thus the pair of points compared for the latter are located further from the nunatak summit.

The differences in the ice surface elevation between the up- and downstream sides of the nunatak result in differences in the time of nunatak surface exposure. This occurs for all scenarios but with different lags in the time of exposure depending on the degree of thinning and distance from the summit. For some scenarios and locations, only the downstream side is exposed, while its upstream counterpart is still thinning at the end of the simulation time. Within 2 km of the nunatak summit (or 6 km for the nunatak elongated parallel to flow) and for those scenarios where both up- and downstream sides are exposed (or meet the stabilisation criterion), the upstream side lags its downstream counterpart by 2 to 14 kyr (Fig. b). In a retrograde-slope setting, however, this effect is not observed. Rapid and uniform thinning happens once accelerated grounding line retreat is triggered, akin to marine ice sheet instability, yielding a similar adjustment up- and downstream of the nunatak (Fig. S10).

The experiments with three nunataks show how different glacier widths produce dissimilar responses of ice surface elevation (Fig. ), mimicking situations where multiple nunataks are separated by narrow glaciers. After 20 kyr of ice thinning, the 0 km width experiment (i.e. where the nunataks merge into a single, 36 km wide nunatak; Fig. a) yields the highest ice retention upstream (further delaying surface lowering at the upstream side), with its surface up to 250 m higher compared to the control one-nunatak case (Fig. e). For the experiments with the smallest glacier widths (0, 5 km; Fig. a, b), the constructive interference between the nunataks results in an ice retention that is strong enough to instigate a strong deficit response downstream. Compared to the control scenario, this response reaches as far as 50 km downstream of the nunataks, also impacting the grounding line position, displacing it further inland on the lee side of the nunatak range. For the experiments where glacier widths are 10 and 15 km (Fig. c, d), the influence of multiple nunataks decreases but similar patterns arise, albeit with smaller differences compared with the control experiment.

The differences in glacier width also result in different organisations of ice flow around the nunataks. The wider barriers formed by the 0 and 5 km wide glaciers yield a different pattern of ice flux upstream and downstream of the nunataks (cf. Figs. a, b and ), where most of the flux is concentrated 13–30 km away from the centreline. In the case of the two experiments that have wider glaciers (10 and 15 km), ice flux also peaks further away from the centreline than in the control experiment but is more uniformly distributed across the domain. Their largest flux peaks are closer to those of the control experiment, and distinct smaller peaks occur exactly where the glaciers are located. The difference between the flux downstream and upstream is inversely proportional to glacier width (i.e. the narrower the glacier, the larger the difference), which points to an increased retention of ice upstream as the cause for the increase in relative heightening/lowering of the ice surface.

The coarser regular-mesh experiments (5, 10, and 20 km horizontal resolution) applied to the series of three-nunatak configurations show important differences from the refined-mesh experiments (Fig. ; cf. Fig. ). The 5 km mesh experiments deviate the least from the refined-mesh experiments and capture the relationship between glacier width and the ice surface elevation gradient. Still, this mesh does not capture the same magnitude of ice surface lowering downstream, underestimating it by as much as 80 % (100–200 m) relative to the refined mesh, an effect that is particularly pronounced close to the nunataks (cf. Figs. a, b and a, b).

The 10 km mesh captures, to some degree, the original nunatak shape and consequent surface elevation increases upstream and decreases downstream of the nunataks (Fig. f–j). However, there are problems with both signal strength (under-/overestimations of at most 250 m) and location (not coinciding with the pattern seen in the refined-mesh experiments). In the 20 km mesh, the original nunatak shape is not properly captured (Fig. k–o), and the large element sizes cause increased heightening/lowering of ice surface elevation to occur much farther away up- and downstream of the obstacles compared to their respective refined-mesh counterparts (Fig. ). Finally, the 20 km mesh yields similar results for the 0, 5, and 10 km glacier-width experiments (Fig. k–m) but yields a very different response for the 15 km width experiment (Fig. n), which is much more in tune with the results of the refined mesh. The differences between the regular-mesh and the refined-mesh experiments also result in different thinning rates between them (Fig. ). The 20 km resolution model run overestimates the rate of thinning under the same forcing (Fig. c), while the 5 km resolution model run shows values much closer to those of the refined-mesh experiment, despite the underestimated thinning (Fig. b).

The finest regular mesh tested (5 km) performs the best among the regular meshes, since it partially captures the ice flow through the widest glaciers tested (15 km wide) and best represents the effect of glacier width on ice flow constriction (Fig. ). The differences between ice flow downstream and upstream shown in Fig.  evolve similarly in the refined and 5 km meshes but are not well represented in the 10 or 20 km meshes. The coarsest mesh (20 km) shows similar results for the 5 and 15 km wide glacier experiments and for the wider nunatak (0 km) and the 10 km wide glacier experiments. This is because the coarse mesh resolution misses two nunataks summits, which lie between two nodes. As a result, the lower topography that is captured by this coarse mesh becomes a subglacial continuation of the single central nunatak. This also explains why the 15 km width experiment shows an ice surface elevation pattern similar to that of the control run. Further tests indicate that the 20 km mesh only captures the existence of three nunatak summits when they are spaced by glaciers that are at least 20 km wide (not shown). Still, in these tests the glaciers are not wide enough for the mesh to capture their existence, and thus the results only reflect the effect of a wider obstacle.

Our modelling experiments demonstrate that the magnitude of the ice surface elevation response due to the presence of nunataks is proportional to their ability to obstruct or constrict ice flow. The nunatak orientation relative to ice flow or, in the experiments where three nunataks are present, the width of the glaciers formed between them modulates the advection of ice downstream and consequently the ice surface elevation difference. showed that much wider channels (∼40 km), more characteristic of ice streams, can already provide lateral drag capable of decreasing advection of ice downstream, slowing down unstable grounding line retreat. In their fjord experiment, used widths that are more similar to our narrower glaciers (∼5 km) and showed that advection of ice from wider to narrower passages, as happens in our experiments, can be greatly slowed down by the lateral drag provided by the nunatak flanks. The magnitude of this response in our experiments could have been influenced by our use of uniform and constant ice rheology, which results in more rigid ice in the narrow glaciers, where flow is faster e.g.. The experiments indicate that, across ice flow (i.e. along the y direction), the extent of the ice surface that is impacted is more likely related to the interaction of ice flow with the subglacial extension of the nunatak, as commonly reported for different spatial and temporal scales and modelling setups of different complexities e.g.. A relative increase in the ice surface elevation was observed immediately downstream of the nunatak, while the lowest elevation attained was located 3 km downstream of the nunatak summit at the start of the simulations. Although a similar effect can be observed on the lee side of isolated nunataks in Dronning Maud Land, Mac.Robertson Land, and the Transantarctic Mountains , we cannot rule out that this is a numerical effect, caused by artificial advection of ice downstream of the regions where the minimum thickness constraint is violated and thus numerically modified e.g..

The ice surface steepening and consequent mismatch between the up- and downstream sides of the nunatak increases as the ice thins, until the downstream side becomes exposed. Exposure happens earlier downstream, as expected due to lower ice surface elevation, and an equidistant point upstream becomes exposed (or has its thinning stabilised) up to 14 kyr later than its downstream counterpart. The rates of thinning and consequently the timing of bedrock exposure are dependent on the choice of the basal sliding coefficient. An increase of ca. 50 % in ice thinning was observed between the higher sliding and control experiments, as well as between the control and lower sliding experiments (not shown). This pattern is expected given the influence of basal sliding on the initial ice sheet geometry (Fig. S5) and highlights that the exposure lags between up- and downstream sides of a nunatak observed in the real world will be site dependent. The choice of minimum ice thickness used in our model (1 m) also influences the timing of exposure at a given point on the nunatak (Fig. S7). Although the lag times are still comparable to the range observed (2–16 and 2–14 kyr; cf. Figs. b and S7b), a lower minimum thickness increased such lags, while a higher minimum thickness reduced them. The surroundings of nunataks are often crevassed, which results in a change in ice rheology. Hence, another test was carried out where the prescribed ice rheology becomes progressively softer towards the nunataks by a factor of 10 (Figs. S6 and S7). In this test, the most notable effect from the choice of ice rheology was a delay in the timing of exposure downstream of the nunatak relative to the control case, by 0.5–5 kyr, but the lag times were still of the same magnitude as the control case (i.e. between 2 and 10 kyr). While we use stabilisation of thinning upstream to determine whether the ice surface attained its minimum elevation before reaching the minimum thickness criterion, such stabilisation does not happen because of an equilibrium of the modelled upstream surface with the applied SMB. If the upstream surface had attained equilibrium, stabilisation would have happened earlier when the imposed thinning was lower, which is the opposite of what was observed when comparing the three thinning scenarios. The fact that equidistant points up- and downstream of the nunatak are in some cases not both exposed also implies that important changes in ice surface elevation might not be recorded upstream of a nunatak. The difference in time of exposure up- and downstream is also higher in the experiments when ice flow is further constricted by the narrow glaciers (Fig. S11).

Idealised experiments have the advantage that certain interactions can be easily isolated and studied, but they also simplify certain phenomena that are observed in the real world, which could have influenced our results. For example, our setup is largely based on Antarctic settings, but the idealised SMB forcing deviates from what is more commonly observed in Antarctica in two ways. First, we prescribe ice thinning through surface melting, which is more characteristic of Greenland e.g., while Antarctica's main source of mass loss is through dynamic thinning . Since mass loss only occurs downstream of the nunataks and is highest at the downstream end of the model domain, we believe that the interaction between ice flow and nunataks and the consequent ice surface elevation response were not significantly impacted by how ice thinning was imposed. Second, the temporal perturbations to SMB are spatially uniform, which is indeed different from what was observed in areas of complex topography e.g.. In regions where high mountains act as a barrier to the advection of moisture in the atmosphere, total variations in SMB are significantly lower inland of the mountain ranges compared to in coastal regions, which means that the surface elevation gradients found here and the lags in surface exposure/stabilisation could be higher in non-idealised settings. A final set of sensitivity experiments (Fig. S8) shows that the ice surface elevation difference between locations equidistant up- and downstream of a nunatak is more sensitive to the spatial gradient of prescribed SMB than to the absolute values of SMB. We do not consider glacial isostatic adjustment (GIA) in our experiments. Bedrock topography evolves through time due to changes in ice loading, which can influence the sensitivity of the ice sheet to sea-level forcing, potentially impacting the grounding line position , and the cosmogenic production rate, possibly impacting the calculated exposure ages . In terms of patterns of exposure up- and downstream of nunataks, the influence of differential isostatic bedrock uplift rates on each side of the nunatak is much smaller than the changes in ice surface elevation considered here, since bedrock elevation changes due to GIA happen at a spatial scale larger than the ∼ 60 km observed in our experiments (e.g. ). Despite the simplifications mentioned above, important insights can be drawn from our idealised experiments regarding the impact of nunataks on ice flow patterns and how the differential response between the upstream and downstream surfaces introduces a bias into the timing of bedrock surface exposure around the nunatak. These effects are important for the interpretation of cosmogenic-nuclide exposure ages and for comparing such ages with results from ice sheet models and are discussed in more detail next.

The steepening of the ice surface around nunataks has important implications for the interpretation of in situ constraints on past ice thickness changes from surface exposure dating. A commonly adopted practice is to assume that regional ice surface elevations are directly reflected by the absolute elevations of samples, but this is likely to yield inaccurate results for past ice sheet reconstructions. Our study demonstrates that this assumption would often yield an error of up to almost 400 m (see Fig. ), which is of the same magnitude as many reported thickness-change estimates in regions of significant ice surface relief e.g.. A different practice, which allows for improved comparisons between sites, uses sample elevations relative to the modern ice surface elevation to infer past ice sheet thickness changes e.g.. A key assumption in this approach is that the ice surface gradient and organisation of ice flow around a nunatak remained the same through time. This assumption contradicts our modelling results, which show increasing ice surface gradients around nunataks during ice sheet thinning. Our experiments further indicate that when samples are taken upstream (downstream) of a nunatak, estimates of past regional ice surface elevation will be overestimated (underestimated). In directions transverse to ice flow, the ice surface is typically at a lower (higher) elevation than directly upstream (downstream) of the nunatak summit. The interpretation of thickness evolution is further complicated considering that the direction of ice flow could have changed as the ice thinned e.g.. This means that using the present-day ice surface as reference elevation could lead to misleading results.

An alternative practice for inferring ice thickness changes is to determine minimum and maximum estimates. In a recent study by , three estimates were provided to put bounds on ice thickness changes, recognising that ice covering the sampled sites could have been sourced locally (from nearby higher terrain), regionally, or distally (the latter two referring to a thickening of the nearby ice stream Jutulstraumen). For minimum estimates of required thickening to reach the sampled elevation, two reference points on the present-day ice surface were determined. A “local” reference point was defined by the lowest elevation in locally sourced ice surrounding a nunatak. A “regional” reference point was defined by the major “break-in-slope” between the adjacent ice stream surface and more stagnant ice flanking the sampled nunatak (normally within 1–2 km of the nunatak). For a maximum estimate of thickening, the lowest point on the ice stream served as reference (lowest point along a 100 km profile perpendicular to ice flow in the ice stream and across the sampled site). We find that these estimates of minimum and maximum thickening are qualitatively comparable to our modelling results. Minimum estimates are similar to the difference between sample elevation and reference ice surface elevation for samples on the downstream side of nunatak summits. Conversely, maximum estimates could yield overestimated thickness changes of several hundreds of metres – in this sense it is comparable to samples from the upstream side of the nunatak summits in our experiments, when referenced to the lowest-elevation present-day ice surfaces, usually downstream of the nunatak or along the ice stream.

Our modelling results can also be used to guide the collection of samples for cosmogenic dating. Typically, nunatak flanks are likely to provide the most accurate estimates of regional ice sheet thickness change, as these areas are least impacted by differences in ice surface steepening (cf. dashed coloured lines and dotted black line in Fig. a, b and their respective locations in Fig. ). Also, sampling at the nunatak flanks should diminish the ice surface gradient effect on exposure ages, while samples taken from the up- or downstream sides would yield increased age differences (see Sect. ). While nunatak flanks are commonly sampled, there has been no strong preference for such locations (Fig. ). We assess the degree to which our model results are reflected in empirical data by analysing all current cosmogenic-nuclide exposure ages from nunataks in Antarctica that indicate ice recession since the Last Glacial Maximum (last 21 kyr). Choosing this time interval minimises the effect of inherited concentrations from prior exposure and confirms that samples taken upstream and downstream at the same elevation interval show significant age differences. This pattern exists regardless of whether exposure ages from 10Be (which are most ubiquitous) or 14C (which are less susceptible to inherited concentrations) are considered (Fig. S12). Age gradients are not as pronounced for samples taken from nunatak flanks (Fig. S13). However, complex exposure histories and different sampling strategies among studies, which target specific thinning histories, introduce a spatial bias into the data, and to our knowledge, no sampling strategy has been designed to test for lags in exposure up- and downstream of nunataks. Such experiments would aid the validation and interpretation of our findings. Regardless, we recommend that in addition to reporting cosmogenic-nuclide concentrations, apparent ages, and uncertainties, the sample location relative to ice flow near the nunatak should be specified. This additional information would be crucial in the interpretation of the local ice sheet history and would aid a domain-wide comparison of field constraints and modelling results.

The recommendations for sample collection based on our results also apply to subglacial bedrock locations that are targeted to test for past ice sheet collapse (e.g. ). In particular, sampled subglacial bedrock ridges on the downstream side of a nunatak may record past exposure indicative of a thinner-than-present ice sheet, but an equivalent subglacial ridge on the upstream side may record no such exposure. High-resolution ice flow modelling around a sampled nunatak is therefore necessary to understand how representative the sample location is of regional-scale ice loss. Irrespective of the sampling strategy and application, it is important to keep in mind that not only modern ice flow direction but also past variations in flow patterns should be considered. Local signs of palaeo ice flow (e.g. glacial striations) can help with such interpretations.

Model resolution plays a substantial role in how modelled ice flow interacts with nunataks. Regular-mesh experiments with resolutions typical of ice sheet models that simulate multi-millennial changes (5–20 km) show a more pronounced elevation gradient between the ice surface up- and downstream of the nunataks than experiments with a variable mesh resolution. These resolutions do not properly capture the prescribed nunatak shape, patterns of ice flow, or thinning rates, which are different in each experiment despite the same applied forcing. Palaeo ice sheet models are run at relatively coarse horizontal resolutions 5–40 km; e.g. to keep computational times reasonable and often use cosmogenic exposure dates as constraints (i.e. to define model parameters) or benchmarks (i.e. to assess the model's ability to reproduce the geological record). Experiments using a regular grid at these resolutions do not resolve site-specific ice surface elevation responses, which contributes to mismatches relative to reconstructions from cosmogenic exposure dating . In our experiments, a grid-cell size smaller than the glacier width manages to capture the drainage effect to some degree. Still, a higher number of grid cells is needed to properly resolve ice flow through the narrow glaciers between nunataks and over large topographic relief, thus resolving the observed differences in deglaciation age up- and downstream of nunatak summits. A better representation of ice flow diminishes the overestimation of the ice surface elevation gradient. To overcome this limitation in palaeo ice sheet models, the model grid could be refined around nunataks so that it properly resolves this pattern of ice flow. The use of adaptive meshes in ice sheet models e.g., nested regional models, or downscaled setups, which take a lower-resolution ice sheet model state as boundary/initial conditions, could be potential solutions. Furthermore, due to their appropriate representation of regional ice flow patterns, higher-resolution simulations could also help when designing sampling strategies and reconstructing regional thickness changes, thus diminishing the mismatch between modelled and reconstructed ice surface elevation. In summary, we advocate for a closer collaboration between ice sheet modellers and field scientists. This would allow for common goals to be defined and for both disciplines to help each other overcome their respective inherent limitations.