Upstream flow effects revealed in the EastGRIP ice core using Monte Carlo inversion of a two-dimensional ice-flow model

. The Northeast Greenland Ice Stream (NEGIS) is the largest active ice stream on the Greenland Ice Sheet (GrIS) and a crucial contributor to the ice-sheet mass balance. To investigate the ice-stream dynamics and to gain information about the past climate, a deep ice core is drilled in the upstream part of the NEGIS, termed the East Green-land Ice-core Project (EastGRIP). Upstream ﬂow can introduce climatic bias into ice cores through the advection of ice deposited under different conditions further upstream. This is particularly true for EastGRIP due to its location inside an ice stream on the eastern ﬂank of the GrIS. Understanding and ultimately correcting for such effects requires information on the atmospheric conditions at the time and location of snow deposition. We use a two-dimensional Dansgaard–Johnsen model to simulate ice ﬂow along three approximated ﬂow lines between the summit of the ice sheet (GRIP) and East-GRIP. Isochrones are traced in radio-echo-sounding images along these ﬂow lines and dated with the GRIP and EastGRIP ice-core chronologies. The observed depth–age relationship constrains the Monte Carlo method which is used to determine unknown model parameters. We calculate backward-in-time particle trajectories to determine the source location of ice found in the EastGRIP ice core and present estimates of surface elevation and past accumulation rates at the deposition site. Our results indicate that increased snow accumulation with increasing upstream distance is predominantly responsible for the constant annual layer thicknesses observed in the upper part of the ice column at EastGRIP, and the inverted model parameters suggest that basal melting and sliding are important factors determining ice ﬂow in the NEGIS. The results of this study form a basis for applying upstream corrections to a variety of ice-core measurements, and the inverted model parameters are useful constraints for more sophisticated modelling approaches in the future

1 (a) Overview of past and ongoing deep ice-core drilling projects on the GrIS (surface elevation and Greenland contour lines by Simonsen and Sørensen, 2017;Greene et al., 2017) and the outline of the study area. The NEGIS appears as a distinct feature in the surface velocities (Joughin et al., 2018). It extends from the central ice divide to the northeastern coast, where it splits up into the three marine-terminating glaciers 79N Glacier, Zachariae Isbrae and Storstrømmen Glacier. (b) The present-day EastGRIP flow line is derived from the DTU_SPACE surface velocity product (Andersen et al., 2020). Due to the limited availability of radar data along the flow line, we construct three approximate flow lines through a combination of various radar products (profile A-C) between GRIP and EastGRIP. Flow line B and C lack data in the centre of the profiles, marked as a dashed line.
The total depth uncertainty ::: (z t ) : was calculated as : where the depth uncertainty introduced during the picking process , : (z p , : ) is estimated to be 10 m, andz rr is : . ::: The :::::::::: uncertainty 205 ::::: related :: to : the radar range resolution :::: (z rr ) of the corresponding RES image and :: is defined as where k is the window widening factor of 1.53, c is the speed of light, B is the radar bandwidth and 3.15 is the dielectric permittivity of ice.

Ice flow model
A full simulation of ice flow in the catchment area of the NEGIS is a highly under-determined problem , 250 lacking geophysical, climatic and ice-core data, some of which will later become available ::::::: become ::::::: available ::: in ::: the ::::: future.
Here, we use a two-dimensional Dansgaard-Johnsen model (Dansgaard and Johnsen, 1969) to simulate the propagation 255 ::: and :::::::::: deformation : of internal layers along approximated flow lines between the ice-sheet summit (GRIP) and EastGRIP. The simplicity of the model makes it well suited for the Monte Carlo method due to its few model parameters, the allowance for large time steps, and because it has an analytical solution (Grinsted and Dahl-Jensen, 2002). The model assumes ice incompressibility and a constant vertical strain rate down to the so-called kink height , : (h, : ) below which the strain rate decreases linearly. Basal sliding and melting are included in the model, and the ice-sheet thickness , : (H, ) : is assumed to be constant in time.

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We consider a coordinate system where the x-axis points along the approximated flow line, the y-axis is horizontal and perpendicular to the flow line, and the z-axis indicates the height above the bed. The horizontal velocities parallel (u ) and perpendicular (u ⊥ ) to the profiles are described by Grinsted and Dahl-Jensen (2002) as: where u ,sur and u ⊥,sur are the surface velocities :::::: parallel ::: and :::::::::::: perpendicular :: to :: the :::::: profile, and the basal sliding factor , f B ::: f bed , is the ratio between the ice velocity at the bed and at the surface.
Reflectors 1-9 were deposited during the Holocene. The remaining reflectors are found in ice from the Last Glacial Period from which reflector 10 and 11 can be attributed to the onset of the Younger Dryas and the Bølling-Allerød. Due to computational reasons, we did not use all 15 layers for the Monte Carlo inversion but picked eight isochrones with approximately equal vertical spacing (Table 3), and used the EastGRIP ages for our simulation of layer propagation. The relation between the GRIP and EastGRIP depths of the traced IRHs fits well with the GICC05 time scale (Mojtabavi et al., 2020;Rasmussen et al., 2014), and the ages obtained from the two drill sites agree within the uncertainties. We note that the layer dating at EastGRIP consistently leads to younger ages than the dating at GRIP, which is a likely consequence of inaccuracies related to the transformation between ice-core and radar depths.
The flow-line characteristics and model parameters for each flow line are summarized in Fig. 4. The radar profiles with ::: the observed and modelled isochrones are displayed as a function of the distance from the EastGRIP borehole :: ice :::: core ::::::: location.

Ice origin and ice-flow history
From the modelled velocity field, we calculated ::::::: calculate : particle trajectories backwards in time (Fig. 4) , which give insight into the source location and flow history of an ice particle :: ice : found at a certain depth in the EastGRIP ice core, and allow us 445 to determine the accumulation rate during its deposition (Fig7 : . :: 7e). Due to the higher velocities in the ice stream, the source area of ice :: ice :::::: source ::::::: location : in the upper 1,600 m m : of ::: the ::: ice :::: core : lies further upstream for flow line C compared to flow line A and B. For deeper ice, this trend is reversed, as the velocity along flow line C drops below the velocity of line A and B (Fig. 7a). A similar effect manifests itself in the upstream elevation, where higher velocities along flow line C result in higher elevations in the upper part of the ice column, which is compensated by a flatter topographic profile for ice deeper than 1,300 450 m ::: 400 : m (Fig. 7b).
From the model-inferred in situ accumulation rates, λ H,m ::: a m , and annual layer thicknesses, λ m , we calculate the ice-core  .
The bed topography and bed lubrication have a considerable effect on ice-flow parameters. Flow over bed undulations affect the elevation of internal layers due to variations in the longitudinal stresses within the ice (Hvidberg et al., 1997) and is often reflected in the surface topography (Cuffey and Paterson, 2010). If the bed is 'sticky', i.e. the basal sliding is small, the ice is 500 compressed along the flow direction while vertically extended (Weertman, 1976), and IRHs are pushed upwards. At a slippery bed, the opposite is the case, resulting in along-flow extension of IRHs which leads to thinning and thus decreasing distance between the IRHs. Keisling et al. (2014) argued that major fold trains existing independently of bed undulations can be explained by variations in the basal sliding conditions. This is, for instance, observed across shear margins, where local, steady state folds are formed as a response to the basal conditions Holschuh et al., 2014). In flow line A, we observe 505 similar 'fold-trains' on a larger scale downstream of a substantial bed undulation (100-200 km km upstream of EastGRIP) .
While it is commonly accepted that the NEGIS is initiated by a locally enhanced geothermal heat flux (e.g. Fahnestock et al., 2001;Alley et al., 2019), the magnitude thereof and the resulting hydrological conditions of the bed are still highly debated.
However, the accuracy of these findings is limited since the local layer approximation (Waddington et al., 2007) is not valid in the surrounding of the NEGIS MacGregor et al., 2016). Remarkably high basal melt rates of 0.16-0.22 ma −1 are also suggested by a recent study (Zeising and Humbert, 2021) using an autonomous phase-sensitive radio-echo sounder (ApRES) at EastGRIP. Melt rates in these order of magnitudes would ::::: either require an unusual high geothermal heat 540 flux , immensely exceeding the continental background Bons et al., 2021) .
4.2 EastGRIP source area ::::::: location and upstream effects 555 The source region of ice in the EastGRIP ice core extends over ∼ :::: more :::: than 300 km km upstream. Holocene ice characterizes the upper 1,244 m m of the ice core and has been advected up to 189 km ::: 197 : km. The climatic conditions during the last 8 kyr remained nearly constant with similar accumulation rates as today (Table 5). However, due to increasing precipitation towards the central ice divide, ice from the past 8 kyr was deposited under increasingly higher accumulation rates with increasing age . ::::: (Table ::: 5). : Our results indicate that this upstream effect happens to compensate for the vertical layer thinning and results in 560 the constant annual layer thicknesses observed in the upper 900 m m of the EastGRIP ice core (Mojtabavi et al., 2020). One possible conclusion of this peculiar observation is that snow depositions must have been advected from far enough upstream to allow the compensation of vertical thinning by increased accumulation rates in the source area ::::::: location. This gives reason to the hypothesis that ice flow velocities in the past 8 ka ::: kyr : must have been similarly fast as today, and that, therefore, the NEGIS has likely been active during this time. However, we believe that RES images and estimates of present-day accumulation rates 565 along the EastGRIP flow line are necessary to evaluate this hypothesis further.
Ice which ::: that is entering the NEGIS must somehow propagate through ::::::: penetrate : the shear margin, which is an important characteristic of ice flow in ice streams and might have left an imprint on the crystal fabric and texture of ice extracted at has passed the shear margin 82 km km from EastGRIP around 1,810 years ago :: 1.8 ::: ka ::: b2k. Slightly enhanced annual layer 580 thicknesses observed :: in ::: the :: ice :::: core : at a depth of 230 m m ::: (Fig. ::: 3) seem unrelated to short-term warmer and wetter climate and might thus be an effect of enhanced accumulation across the shear margin, supporting our results. However, flow lines derived from various surface velocity products show quite a large spread with shear-margin crossings between 97 and 152 km from EastGRIP, corresponding to depths between 324 and 826 m in the ice core.
Our model shows ::::: results ::::: show surface elevations at the deposition site which are up to 459 m ::: 500 :: m higher than EastGRIP 585 at the corresponding time. Assuming a normal thermal and pressure gradient, this implies that ice was deposited under up to 2.9 • C ::::: ∼3.25 • C colder temperatures and up to 41 hPa :: 45 : hPa lower pressure than conditions found at the bore-hole ::::::: borehole location at the time of deposition.
Our estimates on the surface elevation of the source area :::::: location : must thus not be interpreted as absolute values but rather as relative changes with respect to the surface elevation of EastGRIP ::: the :::::::: EastGRIP ::: site : at the corresponding time.
Lacking data and a general understanding of ice-sheet flow far back in time put up additional constraints, and due to the relatively recent discovery of the NEGIS (Fahnestock et al., 1993), little is known about its evolution in the past. Observations 615 of surface elevation and ice-flow velocities imply that the downstream end of the NEGIS has entered a state of dynamic thinning after at least 25 years of stability (Khan et al., 2014). However, it is not clear for how long the NEGIS has been active and how its catchment geometry changed over time.
The assumption of a constant flow field throughout the past 50 ::: 100 kyr is thus the best currently available, but potentially inaccurate, estimate of the past flow regime.
Our results do not give clear evidence on which of the flow lines gives the best results for upstream corrections. Since the 620 present-day EastGRIP flow line is likely located somewhere between flow line A and C, our results can be interpreted as the outer boundaries and we consider the average over the three flow lines the best estimate for the upstream flow characteristics with the corresponding model spread as uncertainties.

Conclusions
We traced isochrones in RES images along three approximated EastGRIP flow lines connecting the EastGRIP and GRIP drill 625 sites. A two-dimensional Dansgaard-Johnsen model was used to simulate the propagation of isochrones along these flow lines.
The simplicity of our model allowed :: the :::::: model ::::::: allowed :: us : to invert for the ice-flow parameters accumulation rate, basal melt rate, kink height and basal sliding fraction. The flow parameters obtained from the Monte Carlo inversion give , :::::: which :::: give :::::: limited ::: but helpful insight into basal properties and ice-flow dynamics and can be used to constrain large-scale ice-sheet models.

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On the basis of our modelled two-dimensional velocity field, we calculated particle trajectories backwards in time to determine the deposition site of ice found in the EastGRIP ice core. We present estimates of the upstream distance, surface elevation and accumulation rate at the time and location of ice deposition. This is valuable and necessary information for interpreting ice-core parameters ::::::::::: measurements, and to separate past climate variability from non-climatic bias :::::::: non-local ::::::: imprints introduced by upstream effects. Our studies show that spatially increasing accumulation rates ::: with ::::::::: increasing :::::::: upstream ::::::: distance along the 635 flow line in the upstream area are mainly responsible for the constant annual layer thicknesses observed for the last 8 kyr in the EastGRIP ice core.
The lack of radar data along the EastGRIP flow line is the biggest limitation of this study. None of the three simulated flow lines accurately represents the present-day flow field but can be regarded as upper and lower limits framing the upstream effects. The acquisition of further radar data along the NEGIS flow line :::::: NEGIS :::: flow ::::: lines in the future would ::: thus : provide 640 more accurate and valuable insights into the flow history of the EastGRIP ice and the NEGIS.