Indication of high basal melting at the EastGRIP drill site on the Northeast Greenland Ice Stream

. The accelerated ice ﬂow of ice streams that reach far into the interior of the ice sheets, is associated with lubrication of the ice sheet base by basal melt water. However, the amount of basal melting under the large ice streams – such as the Northeast Greenland Ice Stream (NEGIS) – is largely unknown. In-situ measurements of basal melt rates are important from various perspectives as they indicate the heat budget, the hydrological regime and the relative importance of sliding in glacier motion. The few previous estimates of basal melt rates in the NEGIS region were 0 . 1ma − 1 and more, based on radiostratigraphy 5 methods. These ﬁndings raised the question of the heat source, since even an increased geothermal heat ﬂux could not deliver the necessary amount of heat. Here, we present basal melt rates at the recent deep drill site EastGRIP, located in the center of NEGIS. Within two subsequent years, we found basal melt rates of 0 . 19 ± 0 . 04ma − 1 that are based on analysis of repeated phase-sensitive radar measurements. In order to quantify the contribution of processes that contribute to melting, we carried out an assessment of the energy balance at the interface and found the subglacial water system to play a key role in facilitating 10 such high melt rates.


Introduction
Ice sheet models are used to quantify the contribution of the Greenland Ice Sheet (GrIS) to future sea-level rise under different climatic scenarios. In these simulations, the distinctive extent of Greenland's largest ice stream -the Northeast Greenland Ice Stream (NEGIS, Fig. 1) -can only be reproduced well if a higher-order approximation is considered for the momentum balance 15 and initial states are based on inversion (Goelzer et al., 2018) or involve subglacial hydrological models (Smith-Johnsen et al., 2020a). Primarily, this is due to the model's inability to accurately represent lubrication and thus the subsequent sliding at the ice stream base that occurs.
The NEGIS is the only large ice stream in Greenland, extending from a distance of 100 km from the ice divide over a length of about 700 km towards the coast (Fahnestock et al., 1993(Fahnestock et al., , 2001bJoughin et al., 2001). It drains about 12 % of 20 Greenland's ice through three major outlet glaciers Nioghalvfjerdsbrae, Zachariae Isstrøm and Storstrømmen Glacier (Rignot and Mouginot, 2012). Loss of the floating tongue of Zachariae Isstrøm has already led to ice flow acceleration and increased mass loss (Mouginot et al., 2015). Consequently, it is expected and projected that NEGIS will contribute significantly to sea-1 level rise in the future (Khan et al., 2014), highlighting the importance to understand :: of :::::::::::: understanding : the general ice flow dynamics and its driving mechanisms. 25 One hypothesis for the genesis of NEGIS is locally increased basal melting at the onset area that enables and enhances basal sliding (Fahnestock et al., 2001a;Christianson et al., 2014;Franke et al., 2021) and forms a subglacial hydrological system.
The coupling with basal sliding is facilitated via the water pressure, so that the sliding velocity rises with increasing water pressure (e.g., Beyer et al., 2018;Smith-Johnsen et al., 2020a). However, little is known about the amount of subglacial water below the up to ∼ 3300 m thick ice sheet. 30 First estimates of basal melt rates by Fahnestock et al. (2001a) and later by Keisling et al. (2014) and MacGregor et al. (2016) are based on the interpretation of chronology in radiostratigraphy. All three studies found melt rates of 0.1 m a −1 and morewhich is extremely large for inland ice. However, these estimates may be prone to limited validity given the assumptions about the flow regime and constant accumulation rate. The cause for such intensive melt was attributed to a high geothermal heat flux which possibly originates from the passage of Greenland over the Icelandic hot spot (Fahnestock et al., 2001a;Rogozhina 35 et al., 2016;Martos et al., 2018;Alley et al., 2019).
In order to directly observe, among other things, flow regimes and basal conditions of ice streams, an ice core is drilled in the course :::: being :::::: drilled :: as :::: part of the East Greenland Ice-Core Project (EastGRIP) near the onset of the NEGIS. Here, surface velocities reach about 57 m a −1 (Hvidberg et al., 2020)  However, measurements with an adequate accuracy are still required to narrow down the basal melt rates further. Here, we 45 present the first estimates of basal melt rates from repeated in-situ phase-sensitive radar measurements from the EastGRIP drill site and consider the contribution of different heat sources at the ice base.

Ice thickness evolution
The method we use to derive a basal melt rate is based on the ice thickness evolution equation that is valid in both, the Eulerian 70 and Lagrangian reference system with the ice thickness H, the time t, the volume flux Q, the surface mass balance a s and the basal melt rate a b (positive for melting) (e.g., Greve and Blatter, 2009). Equation (1) states that a temporal change in ice thickness is caused by a changing volume flux arising from deformation and accumulation or ablation at the ice surface and base. It is worth to note that a basal 75 melt rate larger than the accumulation rate only leads to thinning of the glacier, if the volume flux cannot supply sufficient amount of ice to balance this out. The volume flux Q is defined as the vertically integrated horizontal velocities v x , v y (x, y, z, t) (2) (Greve and Blatter, 2009) and represents the ice thickness change due to deformation and sliding, thus stretching or compres-80 sion in the horizontal direction. This may, for example, be due to changes in basal velocities or ice creeping across a bedrock undulation. Using the continuity equation for incompressible materials, div v = 0, and Leibniz's integral rule we can rewrite div Q as withε zz the vertical strain rateε zz = ∂v z /∂z.

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The recorded ApRES time series allows for a precise estimation of changes in ice thickness ∆H from the vertical displacement of the basal reflector and of internal layers from consecutive measurements. However, applying the ice thickness evolution equation (Eq. (1)) to the ApRES measurements requires some modifications. Since the ApRES is located within a trench below the surface, the 'measured ice thickness' H is defined as the range between the ApRES and the ice base. The total ice thickness -the range from the surface to the ice base -is about 7 to 8 m thicker and includes the upper firn and snow layers. Thus,

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∆H is independent on the surface mass balance, a s = 0 m a −1 , but influenced by firn densification that significantly affects the vertical displacement in the upper ∼ 100 m. As this is not considered in Eq.
(1), we add the term ∆H f /∆t to correct for the densification process below the ApRES.
Equation ( with the change in ice thickness due to vertical strain ∆H ε . Finally, the modified ice thickness evolution equation can be written as

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All three quantities ∆H, ∆H f and ∆H ε , which are needed to derive a b , are described by vertical displacements and hence by the :::: radar : measurement itself in a consistent manner: -∆H is derived from the vertical displacement of the basal return.
-In order to estimate ε zz and thus ∆H ε , a regression analysis of the vertical displacements needs to be calculated and the displacement extrapolated to the surface (the location of the ApRES) and to the ice base.

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-∆H f is the intercept of the regression function at the surface.
Negative values of all three quantities contribute to the thinning of the ice thickness. . : 2.3 ApRES processing ::::::::: Derivation ::: of ::::: basal :::: melt ::::: rates We use an ApRES time series to achieve a reliable estimation of the annual mean basal melt rate.

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Firstly, we divided the depth profile into 6 m segments with a 3 m overlap from a depth of 20 m below the antennas to 20 m above the ice base and a wider segment of 10 m (-9 to +1 m) around the basal return, characterized by a strong increase in amplitude. In order to derive vertical displacements, each depth segment of the first measurement (t 1 ) was cross-correlated with the same segment of each repeated measurement (t i ). This is in contrast to Vaňková et al. (2020), who derived displacements from pairwise time-consecutive measurements (t i−1t i ). The lag of the minimum mean phase difference obtained from the 120 cross-correlation gives the cumulative displacement at the given depth. The range of expected lag was limited by the estimation to the previous measurement (t 1t i−1 ). This results in a time series of displacements for each segment individually. The vertical displacement of the basal segment is the change in the measured ice thickness ∆H.
Next, we estimate the vertical strain ε obs zz and quantify ∆H f as well as ∆H ε :::: based ::: on : a ::::::::: regression ::::::: analysis :: of ::: the ::::::: vertical ::::::::::: displacements. To avoid influences of firn densification on the determination of ε obs zz , we excluded all segments above a depth 125 of 250 m (∼ 9 % of all segments). In addition, segments below the noise-level depth limit (depth at which the noise-level of the ApRES measurement : of :::::::::: h ≈ 1450 m :::::: (where ::::: noise prevents an unambiguous estimation) of h ≈ 1450 m were excluded (∼ 45 % of all segments). Furthermore, outliers were filtered out (∼ 7 %). We found a linear fit u z (z) of that best matches the cumulative vertical displacements of the remaining ∼ 400 segments within the ice. The gradient of this fit is ε obs zz and the offset at the surface :::: shift ::::::: between ::: the ::::::: intercept :: at ::: the ::::: depth ::: of ::: the :::::: ApRES :::: and :::: ∆H : is ∆H f . However, ε zz for z ≥ h is unknown. Here, we used two scenarios to estimate ∆H ε (Fig. 2, Appendix Fig. A1). First, we assumed that ε zz is constant with depth: As a second scenario, we used a vertical strain distribution (ε sim zz ) obtained from an ice sheet model based on inverse surface 135 flow velocities (Rückamp et al., 2020). Here, ε sim zz increases with depth and reaches values of roughly twice ε obs zz at the base. In order to be less dependent on a single measurement, we compute for each of the last 65 days (records; roughly 25 % of the measurements) of a year an annual melt rate and compute from these 65 melt rate estimates a mean annual value by averaging.
Finally, ∆H ε was derived from Eq. (4) for the two vertical strain distributions (∆H const ε , ∆H sim ε ), and the basal melt rate a b from Eq. (5). Given errors are based on the standard deviation of the estimates based on the considered 65 measurements and 140 a 1 % uncertainty in the signal propagation speed in ice (Fujita et al., 2000). For visualization, we calculated the cumulative vertical displacement referenced to the ice base ( Fig. 2).

Discussion
We used estimated vertical displacements from the upper half of the ice column to estimate the dynamic thinning, since noise 155 prevents an unambiguous estimation of the vertical strain for the lower half. To cover a range of variations in the dynamic thinning, we used two different scenarios for vertical strain distribution. The resulting dynamic thinning of the simulated vertical strain and the constant strain differs only slightly. However, we cannot exclude ::: the ::::::::: possibility that larger strain values are reached at the base, which would lead to an overestimation of the basal melt rates. In case of a non-existing melt rate, the dynamic thinning of the lower half of the ice column would bein average : , :: on :::::::: average, more than four times as large as the 160 one of the upper half. However, a strong increase is not found in higher-order ice sheet simulations (Rückamp et al., 2020).
As this model assumes a linearly decreasing strain in the shear zone that reaches zero at the ice base, the resulting basal melt rate at EastGRIP would be even larger. However, the Dansgaard-Johnsen model represents a no-slip boundary condition 165 at the ice base. As this is an unrealistic assumption in an ice stream, we did not consider the Dansgaard-Johnsen model further. The derived vertical strain is based on more than 300 vertical displacements estimated between the firn-ice transition and about 1450 m. In contrast, the estimation of the displacement of the basal return is based on the phase shift of only one segment around the basal return, slightly above the noise-level. This makes the determination more prone to errors. Instead of comparing the first measurement (t 1 ) with all repeated measurements (t i ), the pairwise comparison of time-consecutive

Considerations of the energy balance at the ice base
In order to constrain the heat flux required to sustain the basal melt rates a b derived in this study, we consider the energy balance at the ice base. As for any surface across which a physical quantity may not be continuous, a jump condition is formulated.

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In typical continuum mechanical formulation, the jump ([[ψ]]) of a quantity ψ is defined as [[ψ]] = ψ + − ψ − , meaning the difference in the quantity ψ across the interface (Greve and Blatter, 2009). The jump condition of the energy at the ice base reads as with the heat flux q, the velocity v, the velocity of the singular surface w, the normal vector n pointing outwards of the ice 190 body, the Cauchy stress t, the ice density ρ i and the internal energy u (Greve and Blatter, 2009). The jump of the heat flux [[q · n]] becomes (q geo + q sw ) · n − κ(T ) grad T , with q geo the geothermal heat flux and q sw the heat flux from subglacial water with a temperature above pressure melting point, T temperature and κ thermal conductivity. For the jump in work of surface forces we find with t sw the Cauchy stress of the subglacial water side of the singular surface, v i b the ice velocity and t i the stress field of the ice at the base.
We split the traction vector of the subglacial water in a normal and tangential component, with the water pressure p sw and the stress in the normal direction. Following the same approach as at an ice shelf base (Greve and Blatter, 2009), we employ an empirical relation with e t = v sw /|v sw | and e t ⊥ n. The drag coefficient at the underside of the ice is C i/sw , similar as a Manning roughness is taken into account in subglacial conduits. So that the part of the subglacial water becomes with v sw ⊥ , v sw the normal and tangential velocity of the subglacial water, respectively. This formulation is quite similar to the 205 treatment of the jump condition at an ice shelf base. For the traction vector at the ice base, we follow the same procedure and find with N the normal component and τ b the component in the tangential plane. For The tangential components C i/sw ρ sw |v sw | 3 and τ b v i b · e t are frictional heating and are dominating the contribution of heat arising from work of surface forces. They need to be seen as two end members of the system: either the ice is only in contact with a thick subglacial hydrological system, then C i/sw ρ sw |v sw | 3 is at place ::::: active, or the subglacial hydrological system is 215 permanently in contact with a lubricated base, then the second term τ b v i b · e t is governing. The components are visualized in Fig. 3.
Next, we aim at constraining the individual terms for which we use the following material parameters: ρ i = 910 kg m −3 , the latent heat of fusion, L = 335 kJ kg −1 , and the thermal conductivity for ice at the pressure melting point of 270.81 K κ(270.81 K) = 2.10 W m −1 K −1 (Greve and Blatter, 2009). 220 We consider three scenarios: (i) there is only temperate ice that is melting, (ii) heat is required to warm the ice to the pressure melting point and (iii) friction at the base is contributing significantly to basal melting. (i) For temperate ice and no heat arising from work of surface forces, we find a melt rates of at least 0.19 m a −1 to correspond to a heat flux of 1.84 W m −2 . (ii) Considering grad T to be less than 10 −1 K m −1 , this increases the required heat flux from scenario (i) by up to 0.21 W m −2 , as this additional heat is required to warm the ice to the pressure melting point.

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(iii) Heat arising from work of the surface forces may, however, reduce the required heat flux into the ice to melt this amount of ice. To this end, we need to estimate the magnitude of the components of the stress tensors.
We assume that the normal stress component N is hydrostatic and bridging stresses to be negligible. With a mean density of ice of 910 kg m −3 we find p i = 23.8 MPa. The normal velocity is of the order of the basal melt rate v ⊥ b ≈ −0.2 m a −1 by assuming the velocity of the interface (w) to be zero. The normal component of the ice side is then in the order of 0.15 W m −2 .

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For the tangential components of the ice side, we consider the shear stress at the base to be τ b ≈ 1 to 100 kPa. This compares to basal shear stress found by Rückamp et al. (2020) of 50 kPa. To constrain the sliding velocity, we assume it to be maximum the surface velocity of 57 m a −1 and minimum half of the surface velocity. This leads to a tangential component on the ice side to be up to 0.15 W m −2 (Fig. 4).
Next, we constrain the normal component of the subglacial water p sw v sw ⊥ . A water pressure of 10 to 23 MPa is consistent 235 with subglacial hydrological modelling (Beyer et al., 2018;Smith-Johnsen et al., 2020a). Assuming the normal velocity to be at most as large as the basal melt rate, we find the range of this term to be between 0.05 to 0.12 W m −2 (Fig. 4). The tangential component C i/sw ρ w |v sw | 3 needs an assumption on the roughness C i/sw , for which we consider a range from the roughness of the ice shelf base :: of ::::: 10 −3 to a maximum roughness ten times larger 10 −3 to 10 −2 . :: as ::::: large. : The motivation for this is that ice shelf roughness is governed by convection cells at the interface, whereas in the inland ice, 240 the interaction with the bedrock may lead to a larger roughness. As nothing is known about the shape of the subglacial conduit, the range of velocity cannot be constrained well. We consider a speed similar to the one of the ocean 0.1 m s −1 , but as surface rivers easily reach 1.0 m s −1 , we take this as an upper limit (Fig. 4). Thus, the contribution of friction to the energy available for basal melting may account for at least ∼ 0.20 W m −2 , with the potential to be far larger based on the assumptions we made.
with t sw the Cauchy stress of the subglacial water side of the singular surface, v i b the ice velocity and t i ice at the base.
We split the traction vector of the subglacial water in a normal and tangential component, with the water p 175 in the normal direction and use for the tangential component an empirical relation t sw · n = p w n + C i/sw ⇢ w |v sw | 2 e t with e t = v sw /|v sw | and e t ? n. The roughness at the underside of the ice is C i/sw , similar as a Mannin into account in subglacial conduits. So that the part of the subglacial water becomes v sw · t sw · n = p sw v sw · n + v sw · C i/sw ⇢ w |v sw | 2 e t = p sw v sw with v sw ? , v sw q the normal and tangential velocity of the subglacial water, respectively. This formulation treatment of the jump condition at an ice shelf base. For the traction vector at the ice base, we follow the find with t sw the Cauchy stress of the subglacial water side of the singular surface, v i b the ice velocity and t i the stress field of the ice at the base.
We split the traction vector of the subglacial water in a normal and tangential component, with the water pressure p sw the stress 175 in the normal direction and use for the tangential component an empirical relation with e t = v sw /|v sw | and e t ? n. The roughness at the underside of the ice is C i/sw , similar as a Manning roughness is taken into account in subglacial conduits. So that the part of the subglacial water becomes v sw · t sw · n = p sw v sw · n + v sw · C i/sw ⇢ w |v sw | 2 e t = p sw v sw with v sw ? , v sw q the normal and tangential velocity of the subglacial water, respectively. This formulation is quite similar to the treatment of the jump condition at an ice shelf base. For the traction vector at the ice base, we follow the same procedure and find with N the normal component and ⌧ b the component in the tangential plane. For v i b · t i · n we find 185 forces we find with t sw the Cauchy stress of the subglacial water side of the singular surface, v i b the ice veloc ice at the base.
We split the traction vector of the subglacial water in a normal and tangential component, with t 175 in the normal direction and use for the tangential component an empirical relation t sw · n = p w n + C i/sw ⇢ w |v sw | 2 e t with e t = v sw /|v sw | and e t ? n. The roughness at the underside of the ice is C i/sw , similar as into account in subglacial conduits. So that the part of the subglacial water becomes v sw · t sw · n = p sw v sw · n + v sw · C i/sw ⇢ w |v sw | 2 e t = p sw v sw with v sw ? , v sw q the normal and tangential velocity of the subglacial water, respectively. This for treatment of the jump condition at an ice shelf base. For the traction vector at the ice base, we find with t sw the Cauchy stress of the subglacial water side of the singular surface, v i b the i ice at the base.
We split the traction vector of the subglacial water in a normal and tangential componen 175 in the normal direction and use for the tangential component an empirical relation t sw · n = p w n + C i/sw ⇢ w |v sw | 2 e t with e t = v sw /|v sw | and e t ? n. The roughness at the underside of the ice is C i/sw , si into account in subglacial conduits. So that the part of the subglacial water becomes v sw · t sw · n = p sw v sw · n + v sw · C i/sw ⇢ w |v sw | 2 e t = p sw v sw with v sw ? , v sw q the normal and tangential velocity of the subglacial water, respectively. treatment of the jump condition at an ice shelf base. For the traction vector at the ice b find t i · n = N n + ⌧ b e t with N the normal component and ⌧ b the component in the tangential plane.
With the jump of the internal energy [[u]] = L, we can reformulate Eq. (8) to The tangential components C i/sw ⇢ sw |v sw q | 3 and ⌧ b v i b · e t are frictional heating and a arising from work of surface forces. They need to be seen as two end members of the s 190 with a thick subglacial hydrological system, then C i/sw ⇢ sw |v sw q | 3 is at place, or the su nently in contact with a lubricated base, then the second term ⌧ b v i b · e t is governing. Next, we aim at constraining the individual terms for which we use the following materi tent heat of fusion, L = 335 kJ kg 1 , and the thermal conductivity for ice at 273.15 K  and Blatter, 2009). 195 We consider three scenarios: (i) there is only temperate ice that is melting, (ii) heat is a shear zone is the Dansgaard-Johnsen distribution model (Dansgaard and Johnsen, 1969). As this model decreasing strain that reaches zero at the ice base, the resulting basal melt rate at EastGRIP would be even la 140 Dansgaard-Johnsen model represents a no-slip boundary condition at the ice base. As this is an unrealisti ice stream, we did not consider the Dansgaard-Johnsen model further.
The derived vertical strain is based on more than 300 vertical displacements estimated between the firn about 1450 m. In contrast, the estimation of the displacement of the basal return is based on the phase segment around the basal return, slightly above the noise-level. This makes the determination more pron 145 of comparing the first measurement (t 1 ) with all repeated measurements (t i ), the pairwise comparison o measurements (t i 1 and t i ), as it is shown by Vaňková et al. (2020), leads to a lower thinning rate of H 2018/19 ( 0.441 ± 0.004 m a 1 in 2017/18, 0.467 ± 0.009 m a 1 in 2018/19). Thus, the variability foun a variability of the ice sheet system but can rather be influenced by the methodology.
A variation in the selected depth limit of densification, to exclude segments affected by densification, ca with the heat flux q, the velocity v, the velocity of the singular surface w, the normal vector n pointing o body, the Cauchy stress t, the ice density ⇢ i and the internal energy u (Greve and Blatter, 2009). The jum [[q · n]] becomes (q geo + q sw ) · n (T ) grad T , with q geo the geothermal heat flux and q sw the heat flux fro with a temperature above pressure melting point, T temperature and  thermal conductivity. For the jump 170 8 a shear zone is the Dansgaard-Johnsen distribution model (Dansgaard and Johnsen, 1969). As this model assumes a l decreasing strain that reaches zero at the ice base, the resulting basal melt rate at EastGRIP would be even larger. Howev 140 Dansgaard-Johnsen model represents a no-slip boundary condition at the ice base. As this is an unrealistic assumptio ice stream, we did not consider the Dansgaard-Johnsen model further.
The derived vertical strain is based on more than 300 vertical displacements estimated between the firn-ice transiti about 1450 m. In contrast, the estimation of the displacement of the basal return is based on the phase shift of on segment around the basal return, slightly above the noise-level. This makes the determination more prone to errors.
with the heat flux q, the velocity v, the velocity of the singular surface w, the normal vector n pointing outwards of body, the Cauchy stress t, the ice density ⇢ i and the internal energy u (Greve and Blatter, 2009). The jump of the he [[q · n]] becomes (q geo + q sw ) · n (T ) grad T , with q geo the geothermal heat flux and q sw the heat flux from subglacia with a temperature above pressure melting point, T temperature and  thermal conductivity. For the jump in work of 170 8 a shear zone is the Dansgaard-Johnsen distribution model (Dansgaard and Johnsen, 1969). As this model assumes decreasing strain that reaches zero at the ice base, the resulting basal melt rate at EastGRIP would be even larger. How 140 Dansgaard-Johnsen model represents a no-slip boundary condition at the ice base. As this is an unrealistic assump ice stream, we did not consider the Dansgaard-Johnsen model further.
The derived vertical strain is based on more than 300 vertical displacements estimated between the firn-ice trans about 1450 m. In contrast, the estimation of the displacement of the basal return is based on the phase shift of segment around the basal return, slightly above the noise-level. This makes the determination more prone to error 145 of comparing the first measurement (t 1 ) with all repeated measurements (t i ), the pairwise comparison of time-co measurements (t i 1 and t i ), as it is shown by Vaňková et al. (2020), leads to a lower thinning rate of H in 2017/1 2018/19 ( 0.441 ± 0.004 m a 1 in 2017/18, 0.467 ± 0.009 m a 1 in 2018/19). Thus, the variability found is not ne a variability of the ice sheet system but can rather be influenced by the methodology.
A variation in the selected depth limit of densification, to exclude segments affected by densification, causes sligh with the heat flux q, the velocity v, the velocity of the singular surface w, the normal vector n pointing outwards body, the Cauchy stress t, the ice density ⇢ i and the internal energy u (Greve and Blatter, 2009). The jump of the [[q · n]] becomes (q geo + q sw ) · n (T ) grad T , with q geo the geothermal heat flux and q sw the heat flux from subgla with a temperature above pressure melting point, T temperature and  thermal conductivity. For the jump in work o 170 8 segment around the basal return, slightly above the noise-level. This makes the determination more prone to errors. Instead 145 of comparing the first measurement (t 1 ) with all repeated measurements (t i ), the pairwise comparison of time-consecutive measurements (t i 1 and t i ), as it is shown by Vaňková et al. (2020), leads to a lower thinning rate of H in 2017/18 than in 2018/19 ( 0.441 ± 0.004 m a 1 in 2017/18, 0.467 ± 0.009 m a 1 in 2018/19). Thus, the variability found is not necessarily a variability of the ice sheet system but can rather be influenced by the methodology.
A variation in the selected depth limit of densification, to exclude segments affected by densification, causes slight changes 150 in vertical strain and thus in basal melt rate in the order of millimeters. However, we observed an increased densification rate within the considered 65 records. The increased densification can possibly be a result of increased load from the camp at the surface.
Our derived basal melt rate of 0.19 ± 0.04 m a 1 is above previous estimates from airborne radar measurements. In order to constrain the heat flux required to sustain the basal melt rates a b derived in this study, we consider the energy balance at the ice base. As for any surface across which a physical quantity may not be continuous, a jump condition is formulated.
with the heat flux q, the velocity v, the velocity of the singular surface w, the normal vector n pointing outwards of the ice body, the Cauchy stress t, the ice density ⇢ i and the internal energy u (Greve and Blatter, 2009). The jump of the heat flux [[q · n]] becomes (q geo + q sw ) · n (T ) grad T , with q geo the geothermal heat flux and q sw the heat flux from subglacial water with a temperature above pressure melting point, T temperature and  thermal conductivity. For the jump in work of surface with t sw the Cauchy stress of the subglacial water side of the singular surface, v i b the ice velocity and t i the ice at the base.
We split the traction vector of the subglacial water in a normal and tangential component, with the water press 175 in the normal direction and use for the tangential component an empirical relation t sw · n = p w n + C i/sw ⇢ w |v sw | 2 e t with e t = v sw /|v sw | and e t ? n. The roughness at the underside of the ice is C i/sw , similar as a Manning r into account in subglacial conduits. So that the part of the subglacial water becomes v sw · t sw · n = p sw v sw · n + v sw · C i/sw ⇢ w |v sw | 2 e t = p sw v sw with v sw ? , v sw q the normal and tangential velocity of the subglacial water, respectively. This formulation is q treatment of the jump condition at an ice shelf base. For the traction vector at the ice base, we follow the sam find t i · n = N n + ⌧ b e t with N the normal component and ⌧ b the component in the tangential plane. For v i b · t i · n we find To summarize, the jump in the tangential component (friction) has the potential to govern the heat budget, depending on flow speeds in subglacial water and roughness of the ice base, as can be seen in Fig. 4. However, assuming the geothermal heat flux to be in the order of O(q geo ) ≈ 0.25 W m −2 makes evident that the key player in facilitating such high melt rates is the subglacial water system, that may supply the ice base with an additional heat flux.
We have focused our consideration onto the interface between a subglacial water layer and the ice, as this drives the basal 250 melt rate. However, observations of Christianson et al. (2014) highlight the existence of a wet till layer beneath the ice stream.
Depending on the thickness of the water layer, the velocity and pressure of the water and the porosity of the till layer, complex interaction between the till and water may arise, too. Kutscher et al. (2019) present high resolution simulations of a comparable system that highlight the importance of studying this interface as well. To date, it is unclear which vertical extent of the water layer is required to decouple the interaction of a water-till interface from the ice-water interface and thus the influence on the 255 basal melt rate.

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Future measurements at EastGRIP after successful completion :: of ::: the :::::: drilling :: to :::: the :: ice ::::: base will shed more light onto the sliding speed and may also provide more information on the characteristics of the subglacial hydrological system. This will enable the community to put our melt rate estimates into further context.

Conclusions
We estimated annual mean basal melt rates at the EastGRIP drill site from time series of high-precision phase-sensitive radar 265 measurements. We derived the change in the measured ice thickness, thinning from firn densification occurring below the 11 instrument and the vertical strain in the upper 1450 m of the roughly 2668 m thick ice. Two different scenarios for vertical strain distribution were used to to quantify a plausible range of dynamic thinning. Thus, we derived an averaged melt rate of 0.19 ± 0.04 m a −1 . We are aware that these melt rates require an extremely large amount of heat that we suggest to arise from the subglacial water system and the geothermal heat flux. However, these melt rates are based on measurements with a modern 270 ice penetrating radar whose penetration depth is limited due to transmitting power. Thus, no assumptions on past accumulation rates or other uncertainties in age reconstruction are involved. Our major uncertainty is the vertical strain in the lower part of the ice stream. This could be overcome if a more powerful radar with a similar vertical resolution could be operated autonomously over several monthswhich we want to encourage herewith.
Code availability. Own developed MATLAB routines for ApRES processing are available from the corresponding author on request.