Subglacial water amplifies Antarctic contributions to sea-level rise

Domain: earth_science
Published: 2025-04-07
Editorial outcome: minor_revision

Abstract
Antarctica’s contribution to global sea-level rise is deeply uncertain, with subglacial water suspected to play a critical role, yet its impact remains unclear. We demonstrate that water at the base of ice sheets influences sliding behaviour and that its exclusion from models can underestimate sea-level rise projections and delay the predicted onset of tipping points. Here we use an Antarctic Ice Sheet model (Elmer/Ice) to explore how different assumptions about water pressure at the ice base affect sea-level rise projections from 2015 to 2300. Our results indicate that incorporating subglacial water can amplify ice discharge across the Antarctic Ice Sheet by up to threefold above the standard approach, potentially contributing an additional 2.2 metres to sea-level rise by 2300. Notably, a smoothly decreasing basal drag near the grounding line more than doubles grounding line flux by 2300 relative to scenarios where effective pressure is simplified into a spatially constant coefficient. Basin-specific responses vary significantly, with some scenarios advancing tipping points by up to 40 years. These findings underscore the critical need to integrate evolving subglacial hydrology into ice sheet models.

Introduction
The Antarctic Ice Sheet provides the greatest source of uncertainty in current sea-level rise projections1,2. This deep uncertainty, and the risk of potential tipping points that can cause rapid ice loss, is driven by the lack of understanding about fundamental processes influencing ice-sheet dynamics3. In the Ice Sheet Model Intercomparison for CMIP6 (ISMIP6), more than half of the total uncertainty is due to discrepancies between different ice-sheet simulations4,5. While it is challenging to model the complex interactions of Antarctic Ice Sheet with climate4,6, a key process that remains largely unconstrained by observations is basal sliding.

The widespread production of subglacial water beneath the Antarctic Ice Sheet, and the potential intrusion of warm seawater near the grounding line, plays a crucial role in modulating basal sliding and enabling rapid ice flow7–11. Subglacial water drainage beneath the ice sheet occurs through two primary pathways: distributed flow and channelized flow12. The existence of channelized drainage beneath the Antarctic Ice Sheet has been inferred from the alignment of predicted subglacial meltwater flux across the grounding line with observed ‘subglacially-sourced’ sub-ice-shelf channels13–16. These drainage channels are fed by a distributed drainage system, which consists of inter-linked cavities and serves as an initially inefficient pathway for subglacial meltwater transport12. The source of the distributed and channelised subglacial hydrologic systems is meltwater production from both frictional heating, due to basal sliding and internal ice deformation, and geothermal heating from the Earth’s interior7.

The Antarctic Ice Sheet flows through two primary mechanisms: internal ice deformation and sliding over the bed, with the latter dominating the dynamics of all fast flowing outlet glaciers. Accurately representing basal sliding is essential for projecting future ice sheet change. The processes controlling sliding broadly fall into two regimes: The ‘Weertman’ regime, represented in models by expressing basal resistance as a power law function of sliding speed17, and the ‘Coulomb’ regime, represented in models by expressing basal resistance as a function of the effective normal force at the ice-bedrock interface18,19. For ice sheets, this normal force is given by the ice overburden pressure minus the basal water pressure, henceforth referred to as ‘effective pressure’, N. The Weertman regime is expected to apply for the case of hard bedrock, low sliding speeds, and high effective pressure. The Coulomb regime is expected to apply for the case of a soft (sediment) bed, fast sliding speeds, and low effective pressure. Recent laboratory experiments20 have yielded compelling evidence that a ‘regularised Coulomb’ (hereafter ‘RC’) sliding parameterisation, capturing both Weertman and Coulomb regimes, provides a unified framework for modelling sliding across diverse bed types18,21–23. When applying RC sliding, the regime shift between Weertman and Coulomb type sliding is governed by sliding speed, effective pressure, and parameter choices.

Because the actual distribution of effective pressure under the Antarctic Ice Sheet is unknown, model-based sea-level rise projections typically either apply Weertman sliding everywhere, which neglects effective pressure entirely (e.g., refs. 24–26) or makes simple assumptions about effective pressure when applying RC sliding (e.g., refs. 27–29). A summary of sliding laws used in three major Model Intercomparison Projects (Table S1) reveals that the majority of Antarctic Ice Sheet models (e.g., 10 of 16 in ISMIP6-21002, 10 of 19 in ISMIP6-23005, and 7 of 15 in ABUMIP30) employed the Weertman sliding relation.

For models employing RC sliding, assumptions must be made about effective pressure. One such assumption is perfect hydrological connectivity between the subglacial system and the ocean27–29,31. Another one is to prescribe effective pressure as proportional to ice overburden pressure32,33. While practical, these approaches lack robust observational support and may lead to systematic over- or underestimation of basal water pressures across large regions27,31,34–36. An alternative approach with a stronger physical justification is to simulate effective pressure using more advanced hydrology models. Models such as GlaDS (Glacier Drainage System37) and SHAKTI38 offer distributed and channelized drainage systems that can capture spatial variability in effective pressure. While these models improve physical realism, they are computationally expensive and sensitive to uncertainties in basal boundary conditions (e.g., bed roughness, geothermal heat flux)35,39.

A simpler version of RC sliding has been proposed that removes explicit dependence on effective pressure21,40. In this approach, effective pressure is instead implicitly embedded within spatially optimized parameters of the sliding relation, which are calibrated to reproduce observed ice flow speeds21. While computationally efficient and practical for large-scale ice sheet simulations, this method may overlook important spatial and temporal variability in effective pressure.

A further assumption about the effective pressure and its impact on sliding near the grounding line can be imposed using a ‘height-above-floatation’ (HAF) scaling21. HAF is defined as the height of the upper ice surface above sea level minus the height required for the ice overburden pressure to match the ocean water pressure at the bed. This assumption scales the basal resistance or the effective pressure from a prescribed HAF value down to zero at the grounding line, reflecting the influence of warm seawater intrusion beneath grounded ice41. However, the inland extent of this effect is poorly constrained due to limited observations, and this approach neglects significant spatial and temporal variability driven by evolving glacier geometry42,43. Both the HAF-scaling and perfect ocean connection assumption enforce the floating condition at the grounding line. However, HAF-scaling, applied only near the grounding line, adjusts basal resistance based on ice elevation relative to flotation, potentially under- or overestimating effective pressure. In contrast, the perfect ocean connection assumes uniform hydraulic connectivity across the domain, systematically overestimating effective pressure by neglecting local hydraulic potential gradients.

Building on these approaches, we conduct six experiments (Table 1) to assess the impact of these differing assumptions about spatial and temporal variations in effective pressure on projections of future Antarctic Ice Sheet behaviour. These experiments include approaches with both implicit and explicit representations of effective pressure, including the assumption of perfect ocean connection, hydrology model-derived estimates, and height-above-flotation (HAF) scaling. For comparison against the more physically justifiable RC sliding parameterisation, we also apply the linear Weertman sliding relation (hereafter ‘LW’), which, despite its tendency to underestimate mass loss21, is still widely used for its simplicity in both numerical and computational aspects2,5,26,30. By isolating these parameterisations, we evaluate their influence on basal sliding and the resulting ice dynamics, providing insights into the role of hydrology in ice sheet modelling.

Results
Subglacial effective pressure must be included in sliding relations
Our simulations show large differences in projected total Antarctic ice-mass loss between experiments. The largest differences are between LW and the RC relation with implicit effective pressure and HAF-scaling (RC_iN_HAF; Fig. 1). By 2300, the grounding line flux using RC_iN_HAF (8453 Gt yr−1) is almost four times larger than using LW (2245 Gt yr−1) under a high emissions scenario. While LW suggests a negative sea-level contribution, the RC simulations all show a positive sea-level contribution by 2300, with RC_iN_HAF producing the largest rise of ~2 m sea-level equivalent (SLE; Fig. 1b).

The LW relation, which ignores the role of subglacial water pressure, underestimates sea-level rise contributions relative to the RC relation under both low and high emissions scenarios (Fig. 2). Under high emissions scenarios, all RC experiments produce a positive sea-level contribution, whereas this is the case for only two LW experiments. Under the two low emissions scenarios, the RC relation suggests near equilibrium by 2300, which aligns more closely with observed mass loss in recent decades44, as opposed to the substantial mass gain (UKESM) or loss (NorESM1-M) with the LW relation (see further analysis in the Supplementary Material).

Substantial variations of sea-level rise contributions are produced in each Antarctic sector depending on the choice of sliding relation used (Fig. 3). The West Antarctic Ice Sheet exhibits the largest differences. For West Antarctica, under a high emissions scenario and using the RC relation, we obtain a net mass loss of 6–24 mm SLE by 2100 for all climate models, except CCSM4, which predicts a mass gain. Conversely, with the LW relation, three of five climate models produce a mass gain. Beyond 2100, both sliding relations produce positive sea-level rise contributions from West Antarctica. However, using the RC relation results in contributions up to five times higher than using the LW relation by 2300. For the East Antarctic Ice Sheet, all climate models predict net mass gain for this century, with LW predicting larger numbers than RC. However, East Antarctica starts to lose more ice after 2100 using the RC relation, with sea-level rise reaching 17–93 mm by 2200 and 23–248 mm by 2300. In contrast, LW produces overall mass gain, with three of five the climate models producing negative sea-level rise over the period 2200–2300. The Antarctic Peninsula shows less sensitivity to the choice of sliding relations, due to the steep valley glaciers and low water pressure in this region, where the high driving stress mean the ice flow is relatively insensitive to the choice of sliding relation.

The RC simulations, incorporating either an explicit or implicit dependence on effective pressure, consistently predict higher mass loss than LW under strong warming scenarios, and exhibit a wide spread in sea-level projections due to differing assumptions about effective pressure distribution. These results highlight the importance of representing effective pressure in Antarctic Ice Sheet projections and carefully evaluating the assumptions about its spatial distribution.

Subglacial effective pressure controls ice fluxes
Our simulations reveal the treatment of effective pressure significantly affects the spatial tuning of sliding coefficients, which, in turn, impact the evolution of basal shear stress. For example, at the continental-scale, the implicit representation of effective pressure produces higher grounding line flux and mass loss than its explicit treatment (Fig. 1). A negative sea-level rise contribution before 2100 is simulated in most RC experiments (Fig. 1b), which is primarily due to high surface mass inputs from the climate models (see Supplementary Material), however, the ice dynamic loss characterised by grounding line flux is still positive in this century (~2000 Gt yr−1).

The grounding line flux in the implicit run (RC_iN) increases after 2100, peaking at 4984 Gt yr−1 around 2250. However, when HAF-scaling is introduced (RC_iN_HAF), the results diverged dramatically, showing exponential growth in grounding line flux and reaching over 8000 Gt yr−1—the highest ice discharge across the grounding line—displaying a significant disparity in magnitude. This suggests that HAF-scaling enhances basal sliding due to the progressive reduction of basal shear stress over time as the ice approaches flotation. The corresponding sea-level rise contribution reaches nearly 2 m by 2300, almost three times the increase relative to the non-HAF-scaling case (RC_iN). The divergence between the two curves beyond 2100 highlights how poorly constrained basal shear stress near the grounding line can significantly alter long-term sea-level rise projections.

Under the explicit representation of effective pressure, the perfect ocean connection assumption has been commonly used in previous regional and continental-scale studies27–29, due to limited observations and the less advanced state of hydrology models. Under such assumption, RC_eN_POC updates effective pressure over time, which is more realistic near the grounding line where the ice approaches flotation, although this approach systematically underestimates basal water pressure which is not only influenced by the sea level but also the subglacial hydrological system (Fig. S11). While this parameterisation ensures the Coulomb regime dominates near the grounding line, the calculated effective pressure based on ice geometries is consistently higher as compared with simulations using the hydrology model (see Supplementary Material), resulting in the second lowest mass loss (540 mm SLE). The grounding line flux increases from 2185 Gt yr−1 to 3698 Gt yr−1, with a peak flux of ~4245 Gt yr−1 at the year 2240.

While we adopted the simulated effective pressure from the GlaDS hydrology model, the sole experiment enforcing a constant effective pressure (RC_eN_GlaDS) produces the least multi-century ice-mass loss (183 mm SLE by 2300 in Fig. 1b), with the grounding line flux increasing from 2012 Gt yr−1 to 3209 Gt yr−1, with a later and lower peak at ~3578 Gt yr−1 around the year 2270. Similar to the comparison between RC_iN and RC_iN_HAF, with a HAF-scaling applied to the constant effective pressure from GlaDS (RC_eN_GlaDS_HAF), the grounding line flux matches RC_eN_POC by the year 2140, and subsequently undergoes a rapid increase, peaking at 5152 Gt yr−1 around the year 2200. The sea-level rise contribution reaches 682 mm SLE, nearly four times of the case with a constant effective pressure.

The assumption made about effective pressure substantially impacts the timing of tipping points (Fig. 1a), whereby peaks in grounding line flux correspond to rapid grounding line retreat along a retrograde slope. Both experiments with HAF-scaling reach a maximum grounding line flux around 2200, at least 40 years earlier than other experiments. The flux in the explicit HAF-scaled experiment with GlaDS gradually decreases after 2200, whereas the implicit treatment of effective pressure in RC_iN_HAF produces an additional maximum around 2250 which continues increasing by 2300. The modification of effective pressure through HAF-scaling substantially amplifies the grounding line flux. This underscores the significance of the magnitude and spatial distribution of effective pressure near the grounding line, and subsequent impact on ice sheet discharge and sea-level rise.

Almost all basins exhibit the highest mass loss from RC_iN_HAF and lowest from LW (Fig. 4). Four basins with reverse bed slopes, that is G-H (feeding Pine Island and Thwaites Glaciers), E-EP (Ross Ice Shelf), D-Dp (Cook Glacier) and Cp-D (Totten Glacier), are chosen for a closer analysis because these are especially vulnerable to marine ice sheet instability (MISI)45. In West Antarctica, Basin G-H, feeding Pine Island Glacier (PIG) and Thwaites Glacier, has experienced significant grounding line retreat and is particularly sensitive to the MISI25,46. Most experiments suggest a gradual increase in ice flux to 275–370 Gt yr−1 by 2300. With implicit effective pressure and HAF-scaling, the projection is nearly 10 times larger than other parameterisations. In this region, the grounding line at 2300 retreats more than 300 km inland compared to its initial position in 2015 (Fig. 5c), while other experiments show a retreat of about 60 km. Notably, the grounding line retreats over 100 km in just 10 years around 2280 (Fig. 5c). The timing of reaching a tipping point varies, with RC_eN_POC reaching the tipping point around 2095, almost 100 years earlier than RC_eN_GlaDS_HAF (2185) and LW (2215). In contrast, for Basin Ep-F, which feeds the Ross Ice Shelf, the total grounding line flux and movement is relatively invariant across all experiments.

In East Antarctica, the Wilkes Subglacial Basin (Basin D-Dp) projection with implicit effective pressure and HAF-scaling reaches its first tipping point with a peak grounding line flux of 503 Gt/yr by 2190, a factor of 1.6 larger and nearly a century earlier than the experiment with effective pressure calculated based on evolving ice geometry (RC_eN_POC). The Recovery basin (Basin Jpp-K) reaches its first tipping point around 2200 across all experiments, but another tipping point is reached around 2260 in RC_iN_HAF that is two times larger than the other RC experiments. In contrast, the Aurora Subglacial Basin (Basin Cp-D) is less sensitive, with peaks between 2170 and 2190 and a lower rate of mass loss until 2300. Separate research on this basin shows that ice-hydrology interaction increases sea-level contribution by 30% at 2100, compared to an ice-only model with constant effective pressure from a hydrology model9. This aligns with the comparison presented here between the two experiments adopting effective pressure from GlaDS, where HAF-scaling enhances the grounding line flux by approximately 20%.

Increased sensitivity to effective pressure near the grounding line
To better understand how assumptions about effective pressure can lead to such a widespread of sea-level rise projections for RC sliding, we examine the physical processes near the grounding line. As subglacial water crosses the grounding line into the ocean, it reaches pressure balance with the ocean, driving the effective pressure at the bed near the grounding line to nearly zero. This results in a Coulomb regime with low basal resistance, with an upstream regime shift toward Weertman sliding. As the grounding line retreats, this low-drag region migrates upstream, allowing further speed up of ice flow, providing a positive feedback enhancing grounding line retreat and ice sheet mass loss. Grounding line retreat also exposes more of the ice shelf to ocean-induced melting, further accelerating the process. In regions with low gradients in bedrock elevation and ice thickness, such as the Siple Coast feeding the Ross Ice Shelf, this low-drag Coulomb regime can extend far from the grounding line (Fig. S12c), forming an extensive ice plain47.

All RC implementations in this study (see Methods), except for the later stages of the RC_eN_GlaDS simulation (after significant grounding line retreat), impose a low-drag Coulomb region near the grounding line due to assumptions about the spatial distribution of effective pressure, either explicitly (e.g., the perfect ocean connection assumption) or implicitly (e.g., through the HAF-scaling). These differing assumptions lead to variations in the location of the Weertman/Coulomb regime shift and the extent of the low-drag region. For example, the threshold hT in HAF-scaling determines the extent of basal drag reduction upstream of the grounding line as a function of ice geometry48. Separate tests (see Supplementary Material) using a lower hT value (46 m) underscore the sensitivity of the low-drag region’s extent to this threshold, highlighting its importance in sea-level rise projections and raising concerns about the broader validity of this approach.

Taking Thwaites Glacier as an example, the RC_iN_HAF simulation exhibits large-scale collapse during the 23rd century, whereas the other simulations show a more modest retreat. While changes in basal drag before 2015 are relatively small (Fig. S14d), the area of low basal drag in 2270 for RC_iN_HAF extends more than 50 km inland (Fig. S15d), much farther than in the others simulations (typically 10–15 km upstream of the grounding line). This enlarged low-drag region, which represents the formation of a near-flotation ice plain, is conditioned by the hT parameter as well as the bedrock geometry of the region into which the grounding line has retreated. The existence of such a zone of low basal drag explains the significantly high mass loss for Thwaites Glacier in this simulation. A similar low-drag ice plain exists at the ice streams feeding the Ross Ice Shelf47, which is also a sensitive region in our model (Fig. S4).

The primary difference between implicit and explicit representations of effective pressure lies in the location where the shift from the Weertman regime (slower inland flow) to the Coulomb regime (faster outlet flow) occurs, which determines the extent of the low-drag region or ice plain. The Coulomb limit defines the maximum basal shear stress under a Coulomb sliding regime, where sliding is primarily controlled by effective pressure and the sliding coefficient. In simulations using implicit representations of effective pressure (RC_iN and RC_iN_HAF), fast-flowing regions where basal sliding velocity exceeds 300 m yr−1 exhibit Coulomb-plastic behaviour (see Methods for more details), effectively marking the Coulomb limit. Conversely, simulations with explicit effective pressure representation determine the Coulomb limit via the term Cm3NmA, which varies spatially and temporally with effective pressure. As a result, experiments with implicit effective pressure produce much larger Coulomb-dominated regions compared to those with explicit effective pressure, where the Coulomb regime is confined to less than 10 km upstream of the moving grounding line (Fig. S12). This distinction explains why RC_iN_HAF exhibits greater sensitivity to HAF-scaling compared to simulations that incorporate effective pressure from a hydrology model (RC_eN_GlaDS_HAF).

Methods
Model setup
Our simulations do not capture a fully coupled ice-hydrology system, as such a model is not yet available for continental-scale applications. Instead, our simulations aim to determine whether a coupling is indeed necessary, rather than predetermining what might be considered a valid future projection of ice-mass change. We report mass loss estimates from drainage basins in both East and West Antarctica, showing how the parameterisation of the effective pressure exerts control on tipping points characterized by sharp increases in grounding line flux and rapid grounding line retreat in deep basins with retrograde bed slopes. The sea-level contributions in this study are computed based on changes in total ice volume above flotation by assuming zero or very small changes in the bedrock elevation due to isostatic adjustment and a constant ocean area of typically 3.625 × 1014 m2 51.

All experiments are conducted using the Shallow Shelf Approximation (SSA52) implementation of Elmer/Ice53, an open-source finite-element ice dynamic model, to solve the vertically integrated ice dynamics. Upper and lower ice surface elevation and bedrock elevation are from the MEaSUREs BedMachine Antarctica, Version 354. An unstructured finite-element mesh is created initially using Gmsh55 for the Antarctic domain following coastline positions provided by BedMachine V3, and refined using MMG56 based on ice thickness gradients and the Hessian matrix of the observed velocity fields provided by MEaSUREs Phase-Based Antarctica Ice Velocity map (Version 1)57. The low-computation cost of SSA model allows a high-resolution mesh of 1 km in fast-flowing regions and near the grounding line, with a coarser resolution (25 km) further inland. This represents the highest resolution used in continental Antarctic ice sheet models from ISMIP6 studies2,5. Sensitivity analysis using the LW sliding relation (Fig. S7) shows consistency between model outputs generated with 1 km mesh and 500 m mesh. Given the projected ice-mass loss using the Weertman sliding relation is more grid-size sensitive than the Coulomb relation used here28,58, the mesh resolution artefact in this study is acceptable.

An inverse method is used to simulate both the basal drag coefficient β and enhancement factor η simultaneously through reducing the mismatch between the simulated and observed velocity. The inversion is performed with respect to a linear sliding coefficient:
τb = Cw ub = 10β ub (1)
where ub is the basal sliding velocity, β is the sliding coefficient. The viscosity is defined by:
η = E2 η0 (2)
where η0 is a reference viscosity field calculated based on a prescribed and vertically integrated 3-D volume temperature distribution59. Three regularisation terms are introduced in the cost function to improve the conditioning of the inverse problem and balance between fitting with observations and smoothness in β and η. An ‘L-surface’ analysis58 is conducted to determine the optimal regularisation parameters (λβ = 1e5, λEη1 = 1e6 and λEη2 = 0.02). With the optimal parameters, we simulate the basal drag (Fig. S3a) and ice viscosity (Fig. S3b). The inverted basal drag is largely influenced by the bed geometry through affecting the ice flow resistance. The RMSEs between the simulated initial state in 1995 and observed ice thickness and ice surface velocity are shown in Fig. S8, which are among the low range of the RMSEs for the participating ice-sheet models in ISMIP6-2300 projections5, except for the RMSE of velocity from RC_iN and RC_iN_HAF.

Taking the initial geometry and inverted basal drag and viscosity as the starting point at 1995, we conduct a historical run from 1995 to 2014 to relax the model as an initial adjustment. A projection run from 2015 to 2300 is followed with climate forcing derived from selected climate models under the ISMIP6-2300 protocol60. In the transient simulations, the surface mass balance (SMB) is given by the combination of a reference SMB and SMB anomalies, in which the reference SMB is the average mean SMB over 1995–2014 from the MAR product61. The basal melt rate underneath the ice shelf is parameterised following the simplified ISMIP6 standard basal melt parameterisation based on a prescribed relation between ocean thermal forcing and ice shelf melting rates2. Both SMB anomalies and thermal forcing are provided through the forcing provided by the ISMIP6 community5. We adopt the forcing using NorESM1-M/RCP8.5 model for the historical run and UKESM/ssp5-85 model for the projection run. Forcing data from other four climate models under high emission scenario (CCSM4/RCP8.5, HadGEM2/RCP8.5, CESM2/ssp5-85, NorESM1-M/RCP8.5) and two model under low emissions scenario (UKESM/ssp1-26, NorESM1-M/RCP2.6) are applied in the projection run to explore the sensitivity to various climate forcing. For partially floating elements, the representation of basal friction follows the sub-element parameterisation 3 (SEP3) with 20 integration points per element60, while the basal melt only is applied on fully floating cells recommended by ref. 28,58. The ice front is fixed with a minimum ice thickness of 40 m applied, which may result in a vast and thin future Ice Shelf and could potentially lead to an overestimated buttressing effect on the grounded ice, though the effect might be small.

Experiment design
Two different sliding relations, linear Weertman and RC, are chosen in the transient simulations. For the RC relation, the friction coefficients are converted based on the inverted basal shear stress, τb = 10β ub, to ensure that all experiments begin with the same initial basal shear stress (see details below). Importantly, the diverging transient behaviours arise from differences in how sliding evolves under the respective assumptions. To address uncertainties associated with the effective pressure, two formats of the RC sliding equations are considered (Table 1): one with explicit representation of effective pressure (hereafter ‘RC_eN’) and another with implicit representation of effective pressure through a spatially tuned parameter (hereafter ‘RC_iN’). The transition between the Weertman and Coulomb-plastic behaviours differs in each format, as detailed below. Notably, in the Weertman regime, basal sliding is independent of effective pressure, making the transition critical for determining the extent of areas sensitive to spatial and temporal variations in effective pressure.

Implicit representation of effective pressure
• Experiment RC_iN and RC_iN_HAF
Here we follow ref. 21’s version of RC sliding relation and subsumed the roles of effective pressure, N, into the friction coefficient C1:
τb = C1 ub / (ub + u0)m−1 (3)
where m is a flow exponent (set to 3). C1 is converted based on Eq (1) and Eq (3):
C1 = 10β ub (ub + u0)m−1 (4)
Here we use u0 = 300 m yr−1 as the velocity threshold above which a Coulomb-plastic behaviour will occur. The model performance is proved to be relatively insensitive to u0 values21,49. The Coulomb limit occurs when:
ub ≫ u0 (5)
Then Eq (3) reduces to:
τb ≃ C1 (6)
In RC_iN, we assume N, and therefore C1 obtained from inversion, do not change with time. However, this assumption may hold well for the interior of ice sheet but will breakdown in the fast flowing regions where the ice approaches flotation. To take this effect into account, a reduction on the basal shear stress τb is applied in experiment RC_iN_HAF so that τb can decrease smoothly towards zero at the grounding line, as shown below:
τb = λC2 ub / (ub + u0)m−1 (7)
λ = 1 if haf ≥ hT
λ = haf / hT if haf < hT (8)
where haf is the height-above-flotation, λ is the HAF-scaling factor, hT is the threshold. In such case, the basal shear stress near the grounding line will evolve over time with ice geometry. To choose a reasonable value of hT, we calculate λC2 for each element in our model:
λC2 = τb (ub + u0)m−1 / ub (9)
where τb and ub are simulated from inversion. The correlation between λC2 and haf (Fig. S9) indicates that λC2 reaches a relatively constant value of 0.21 MPa where haf = 75 m. When haf < 75 m, a roughly linear trend is found, which indicates the linear decrease in basal drag when haf approaches zero. So we set hT to 75 m here and a sensitivity analysis on its values is discussed. In such case, C2 is converted based on Eq (1) and Eq (8):
C2 = 10β ub (ub + u0)m−1 λ−1 (10)

Explicit representation of effective pressure
The implemented Coulomb sliding relation can be written as19:
τb = C3 N χ ub / (1 + aχq)m−1 (11)
a = (q−1)q−1 / q (12)
χ = ub / (C3 N mA s) (13)
q is a ‘post-peak’ exponent (set to 1), C3 and As are two different friction coefficients that affect the Coulomb and Weertman behaviour correspondingly. This equation can be rearranged to:
τb = C3 N (ub / (ub + Cm3NmA s))m−1 (14)
The Coulomb limit occurs when the basal sliding resistance τb is primarily determined by C3 N, with minimal influence from the basal sliding velocity. This happens in regions where the sliding relation transitions from Weertman regime to Coulomb regime when:
ub ≫ Cm3NmA s (15)
Under this condition, Eq (14) reduces to:
τb ≃ C3 N (16)
This is classic Coulomb friction law, where basal shear stress τb is directly proportional to the effective pressure N and the Coulomb coefficient C3. We now need to convert the linear Weertman coefficient CW = 10β to the unknown Coulomb coefficients C3 and As. The sliding parameter As determines the ease of basal sliding, with higher values representing more resistance to sliding, which is influenced by both physical properties of the ice-bed interface and the effective pressure as defined in ref. 18. In terms of the linear Weertman coefficient, the following relation:
As = |ub|1−m 10−mβ tanh(2N) (17)
Eq (14) is rearranged in terms of the Coulomb C parameter:
C3 = |ub|1−m 10β N−1 (1 − 10mβ |ub|m−1 As)−1/m (18)

• Experiment RC_eN_POC
In this experiment, we assume a perfect hydrological connection between the subglacial drainage system and the ocean so that N can be calculated based on ice thickness and bedrock elevation:
N = ρi g H + ρo g zb (19)
where H is ice thickness, g is acceleration due to gravity, ρi is ice density, ρo is ocean water density, zb is bedrock elevation. It allows N to evolve with ice geometry under the assumption that subglacial water pressure equals the ocean pressure at the grounding line, which is more realistic near the grounding line when ice approaches flotation.

• Experiments RC_eN_GlaDS and RC_eN_GlaDS_HAF
In experiments RC_eN_GlaDS and RC_eN_GlaDS_HAF, the basal drag is decided following Eq. (14) and N is simulated with the Glacier Drainage System model (GlaDS37), implemented in Elmer/Ice as a separate module62. The GlaDS model was selected for this study as it combines a distributed water sheet and channelized drainage system, providing a good representation of subglacial hydrology that is particularly suited to investigating how bed topography influences effective pressure and basal drag9,35,37,62. It is fully implemented within Elmer/Ice, allowing the ice sheet models to calculate the spatial distribution of effective pressure at the start of the simulation63. In this setup, frictional heating dominates in areas of significant basal sliding, where high ice velocities generate far more heat than the relatively small and more uniform geothermal flux64. To ensure that the hydrology system reaches a steady state, the total subglacial water budget is held constant throughout the spin-up period (over ~100 model years). This assumption introduces negligible variability in the final results compared to alternative assumptions on subglacial melt supply.

Bedrock geometry and ice thickness strongly influence subglacial water flow and channel distribution, as lower-elevation areas, such as valleys, accumulate water due to hydraulic potential gradients shaped by bedrock topography and ice thickness, resulting in higher water pressures and lower effective pressures. In RC_eN_GlaDS, we assume the initially computed effective pressure to remain constant with time. To account for a temporally evolving effective pressure near the grounding line, the previously described HAF-scaling is applied on effective pressure in RC_eN_GlaDS_HAF when haf < hT, defining it as N*.
τb = C4 N* ub / (ub + Cm3N*mA s)m−1 (20)
N* = λN (21)
λ = 1 if haf ≥ hT
λ = haf / hT if haf < hT (22)
As = |ub|1−m 10−mβ tanh(2N*) (23)
C4 is converted based on Eq (1):
C4 = |ub|1−m 10β N*−1 (1 − 10mβ |ub|m−1 As)−1/m (24)

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "1. What is the element sizes in the Model and how dependent is the SLE for th ediffernt szenarios on the mesh size.In\ngeneral you should in a few lines give basic information on model parameters\n2. is there a correlation between the bed topography and the distribution of effective pressures und inverted basal drag. If so\nyou should point it out\n3. I would like to point to a publication where although in a different manner the same problem is adressed and maybe you\ncan use it to further complete your argument.\nGoeller et al.:A balanced water layer concept for subglacial hydrology in\nlarge-scale ice sheet models, The Cryosphere, 7, 1095-1106,2013 doi:10.51947tc-7-1095-2013\n4. Finally - from a person which is color blind - you would make life much easier for about 10 % of your readers if you would\nuse different line styles ind addition to colors", "reviewer_2": "Subglacial Water Amplifies Antarctic Contributions to\nSea-Level Rise\nThe present manuscript investigates the effect of different variants for computing the basal pressure below the Antarctic ice\nsheet used in the basal sliding law. Basal sliding is a key component in ice sheet modeling that is hard to constrain due to a\nlack of direct observations and it has a strong influence on the predicted sea level rise and timing of tipping points.\nUsing the Ice sheet model Elmer/Ice, the authors conduct a number of different experiments each with a different treatment\nfor computing the effective pressure N (ice overburden pressure minus subglacial water pressure) and investigate the effect\nunder several climate forcing scenarios. They show that different Antarctic sectors react differently to the sliding relation\nchosen, depending mostly on the topography and dominating driving stress regime.\nThis study is original in that it focuses on comparing the effects of different sliding relations within the same ice sheet model.\nSubglacial hydrology is an important missing piece in ice sheet modeling (and especially relevant for sea level rise\npredictions), and while the paper does not examine the validity of them, it investigates the implications of the different\napproaches that are used in the wild.\nREMARKS\nDiscussion of sliding laws\nIn the introduction, the authors compare the two overarching classes of sliding laws: classical Weertman and regularized\nCoulomb and state that models often ignore or crudely implement the effective pressure N “by adopting the Weertman\nrelation as the commonplace approach” (L. 39f). While this is generally correct, I would prefer a more nuanced statement\nbecause the current formulation can imply that most models have no treatment of effective pressure at all. From skimming\nover Seroussi et al. (2020), I noticed that out of the 13 used models, only 4 to 6 seem to use classical Weertman without\nconsidering basal pressure (unfortunately, not all of them clearly state their sliding relation). Maybe the variety in the\napproaches can be mentioned in the paper. The following paragraph on HAF (L. 45ff) describes one simple approach for\nincluding N, but I think it could be better integrated with the previous part.\nSliding laws used in ISMIP6:\nAWI_PISM: Schoof/Coulomb\nDOE_MALI: linear basal friction law with spatially varying basal friction coefficient\nILTS_PIK_SICOPOLIS: Weertman–Budd-type sliding law with sub-melt sliding (Sato and Greve, 2012) and subglacial\nhydrology IMAU_IMAUICE: ?? probably Schoof/Coulomb? citing Bueler Brown 2009 JPL_ISSM: ??? probably coulomb-\nlike, citing Morlighem et al. 2010\nLCSE_GRISLI: power law basal friction (Weertman)\nNCAR_CISM: basal sliding is similar to that of Schoof (2005), combining power law and\nCoulomb behavior PIK_PISM: Mohr-Coulomb UCIJPL_ISSM: ??? ULB_FETISH: power law basal sliding UTAS_ELMERICE: linear sliding\nrelation VUB_AISMPALEO: Weertman, but with height above bouancy VUW_PISM: probably Schoof/Coulomb\n2. Explaining the experiments in the main text\nThe Method section does a great job of explaining the different experiments, but it would be helpful for the reader to get a\nshort overview already in the main text to facilitate the understanding of the results. It would be enough to briefly explain the\ndifferent treatments of N and possibly further explain the difference between HAF and POC (they seem similar in that they\nboth have the goal of enforcing floating condition at the grounding line).\nI would also appreciate a comment on why these particular treatments were chosen; are they representative of what is used\nin current models?\nIt would be good to lead the reader along from simple implementations to more complex/realistic and explain why you chose\nthe GlaDS model out of a variety of subglacial hydrology models (see e.g. De Fleurian et al. 2018).  \nI am a little bit concerned with the validity of the inversion result for future projections when the basal conditions (especially\nat the grounding line) have changed. This affects mostly the experiments where N is treated implicitly (as part of the\ninversion). Are they still a good baseline after e.g. a substantial retreat of the GL which would imply a major reconfiguration\nof the hydraulic system?\n3. Structure of results and headlines\nI believe the presentation of the results could be enhanced for better readability and comprehension. While the beginning\nthat focuses on the implicit handling of N does a good job of describing the difference between LW and RC_iN_HAF, the\nfollowing parts could be arranged in a more organized way, as I experienced them hard to follow. One idea could be to adapt\nsomething like the following structure (very similar to the current one):\nKeep the first part on implicit handling of N: LW vs RC_iN_HAF as it is (L. 59 to L. 80).\nThen have a second part that covers the explicit handling of N: RC_eN_GlaDS vs RC_eN_POC vs RC_eN_GlaDS_HAF.\nHere I would suggest to start with RC_eN_POC as the most simple approach and then introduce GlaDS and the extended\nGlaDS (RC_eN_GlaDS_HAF).\nWith this continental view established, continue with the Tipping points and the regional analysis.\nNot much change is needed, but these sections should be clearly identifiable and might deserve their own headings, also\nbecause the current in-between headings (“Subglacial effective pressure must be included in sliding relations” and\n“Subglacial effective pressure controls ice fluxes”) are very general and do not help much to navigate the text.\nThe final part before the discussion (“Increased sensitivity to effective pressure near the grounding line”) is in a good place.\nAt some point, Kazmierczak et al. (2022) should also be referenced, as they also investigate the influence of different sliding\nrelations for Antarctica.\nRemarks by lines\nL. 29: also frictional heating due to internal deformation of ice\nL. 45f: Can’t the effect of N be also approximated in the sliding coefficients for Weertman sliding laws? see also concern 1.\nL. 48: more realistic than what?\nL. 56: Why SSA, what is the implication?\nL. 57: Why is SSA used here? Why not hybrid/higher order? What are the implications?\nL. 60: Why is RC_iN_HAF more realistic? is combined effects of seawater intrusion and meltwater production is also\nconsidered in the eN variants? is this only about RC vs LW? more explanation is needed.\nL. 68: Replace “near mass balance” with “near equilibrium”. I don’t understand the second part of the sentence regarding\nobserved mass loss.\nL.71: “For WAIS…”: Sentence is unclear. Maybe “For the WAIS the experiment using the RC relation and high emission\nscenario,…”\nL. 73: produce.\nL. 74: “However, the RC…” -> “However, using the RC…”\nL. 75: “than LW” -> “than using the LW relation”\nThe whole paragraph seems to be missing words.\nL. 82: This sentence is not very clear, seems obvious.\nL 99ff: Isn’t the addition of HAF to the GlaDS N a little bit of double counting, because the effect of the ocean is already\nincluded in the GlaDS model? Though it is only appropriate at the time where N was computed in GlaDS.\nL. 106: Explain coulomb limit\nL. 115-122: Can the experiments using POC produce meaningful results for basins with reverse bed slope? Won’t POC on\nthe retrograde bed will lead to higher basal pressure in the inland than at the grounding line?\nL. 153: How valid is the distribution of N computed by GlaDS for present day conditions after more than a century? Even with\nthe HAF scaling, the whole hydraulic regime might have shifted.\nL. 163: What does “constrained future surface mass budgets estimates” mean? Improvement of the predictions of SMB?\nFigure 1:\ngood colors, bigger fonts, no box around legend\nFigure 2:\ncaption: “sea-level contributions between linear Weertman law” -> “sea-level contributions in experiment using linear\nWeertman law”.\nFigure 4:\ncaption: “with a Hydrology model” -> “from GlaDS”\nDoes channel area mean cross section of individual channels?\nwhy is a diverging colormap used? What is special about 0.5 MPa?\nsmall inset figures in G-H and D-Dp are hardly readable.\nFigure 5:\nWhy are the lines dashed?\nReferences\nDE FLEURIAN B, WERDER MA, BEYER S, et al. SHMIP The subglacial hydrology model intercomparison Project. Journal\nof Glaciology. 2018;64(248):897-916. doi:10.1017/jog.2018.78\nKazmierczak, E., Sun, S., Coulon, V., and Pattyn, F.: Subglacial hydrology modulates basal sliding response of the Antarctic\nice sheet to climate forcing, The Cryosphere, 16, 4537–4552, https://doi.org/10.5194/tc-16-4537-2022, 2022.\nBueler, E., and J. Brown (2009), Shallow shelf approximation as a “sliding law” in a thermomechanically coupled ice sheet\nmodel, J. Geophys. Res., 114, F03008, doi:10.1029/2008JF001179.\nMorlighem, M., E. Rignot, H. Seroussi, E. Larour, H. Ben Dhia, and D. Aubry (2010), Spatial patterns of basal drag inferred\nusing control methods from a full-Stokes and simpler models for Pine Island Glacier, West Antarctica, Geophys. Res. Lett.,\n37, L14502, doi:10.1029/2010GL043853.", "reviewer_3": "Review of “Subglacial Water Amplifies Antarctic Contributions to Sea-Level Rise”, for Nature Communications, by Zhao,\nGladstone, Zwinger, Gillet-Chaulet, Wang, Caillet, Mathiot, Saraste, Jager, Galton-Fenzi, Christoffersen and King\nThis manuscript describes the importance of incorporating subglacial hydrology parameterizations into whole Antarctic ice\nsheet models, by varying parameterization of basal water effective pressure in a basal slip relation. The authors conclude that\neffective pressure is required in the slip relationship, and that ice flux is sensitive to effective pressure model, especially\nnear the grounding line.\nWe believe the topic of this manuscript is important and can be of interest to a wide range of readers. However, we believe\nthat this manuscript can be significantly improved by considering the comments listed below.\nFor now, the manuscript is “expert-facing”, i.e., it is assumed that the readers have a substantial understanding of the\nbackground and methods of this study. The manuscript leaps over many (essential) details that could have been helpful for\nthe readers to follow along.\n• For an important study that can be of interest to the general public, especially in the journal of Nature Communication, we\nargue this is not appropriate. We recommend the authors to provide more detailed elaborations on the background, methods,\nfindings, and implications of this study. A few examples can be found in the comments below.\n• In the section “Subglacial effective pressure must be included in sliding relations”; its implied the observed mass gain of\nthe linear Weertman models demonstrate the need for subglacial effective pressure. The authors need to make a stronger\ncase - ie demonstrate that mass gain is not plausible over these time frames.\n• We recommend the authors to elaborate on the differences between model setups. We appreciate the balance between the\namount of work presented in this manuscript and the space limitations. But the current manuscript makes it really difficult to\nfollow the different model setups, especially considering the number of runs that are being presented and discussed.\n• It is not clear why/how including different subglacial hydrology components can change the AIS SLR projection by such a\nlarge degree. What is the physical process behind the differences? We recommend the authors to elaborate on this and\nmake it more clear.\n• Line 47-52: HAF is a very important concept for this study. However, the authors didn’t provide any explanation/elaboration\non the concept, other than providing one reference. (This is also true for some other important concepts throughout the\nmanuscript.) This approach makes it difficult for non-expert readers to understand the work and follow the logic flow. We\nrecommend providing a more detailed explanation of the concept, and/or maybe even a conceptual diagram.\n• It is not clear what assumption is made about the subglacial meltwater budget. Does the model run only require a certain\nand finite amount of subglacial water? Or do the authors assume an “unlimited” supply of subglacial water generated by\nbasal melting upstream? And does a limited subglacial meltwater budget change the model output?\n* The authors need to address the limitation of the study in the Discussion section. We understand that the authors need to\nconsider the space limitations per journal guideline, but we strongly recommend the authors to reconsider the balance\nbetween (1) further elaboration and discussion of the study findings, (2) connection with other studies, (3) broader\nimplications, and (4) limitation of the study, for the Discussion section.\nminor comments:\n* This manuscript uses a lot of acronyms - are they really necessary? Like, “GL” for “grounding line”? Having too many\nacronyms can be counterproductive. Also, it may help to spell out the run names in Figure 1.\n* We noticed that no doi/web-link is provided for the references. We encourage the authors to consider including such\ninformation.\nUnsure of what is meant by the second and third authors contributed equally.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts."}, "author_response": "Response to Reviewers\nMany thanks to the reviewers for their valuable comments. Our responses are highlighted\nin blue, and the line numbers refer to the revised manuscript.\nReviewer #1 (Remarks to the Author):\n1. What is the element sizes in the Model and how dependent is the SLE for the different\nscenarios on the mesh size. In general you should in a few lines give basic information\non model parameters.\nThanks for the comments. We have included the element size on the Method Setup\nSection in the original manuscript “The low-computation cost of SSA model allows a\nhigh-resolution mesh of 1 km in fast-flowing regions and near the GL and a coarser\nresolution (25 km) further inland.”\nWe have done sensitivity tests to different mesh resolutions (500 m, 1 km, 2 km) near the\ngrounding line with the linear Weertman relation (LW). See the figure below. It suggests\nthat higher resolution would predict less mass loss than coarser resolution, which is\nconsistent with similar sensitivity tests done on Wilkes Subglacial Basin (Wang et al.\n2024) and an idealised domain (Seroussi et al., 2018). It can be seen that the 1 km mesh\nused in this study suggests a more consistent result with 500 m mesh compared with the\n2 km mesh. Previous studies (Wang et al., 2024; Seroussi et al., 2018) indicate that the\nRegularised Coulomb sliding relation with effective pressure included is less sensitive to\nthe mesh resolution compared with LW. So we think the sensitivity of sea-level rise\nprojections to the mesh size is little compared with the dependence on the different\ntreatment of effective pressure.\nTo make it clearer and include the information above, we added the figure below into the\nsupplementary material and a few sentences in the Methods section (Line 426-431).\n“The low-computation cost of SSA model allows a\nhigh-resolution mesh of 1 km in\nfast-flowing regions and near the grounding line, with a coarser resolution (25 km)\nfurther inland. This represents the highest resolution used in continental Antarctic\nice sheet models from ISMIP6 studies 8,9. Sensitivity analysis using the LW sliding\nrelation (Fig. S7) shows consistency between model outputs generated with 1 km\nand 500 m meshes. Given the projected ice mass loss using the Weertman sliding\nrelation is more grid-size sensitive than the Coulomb relation used here 10,11, the\nmesh resolution artefact in this study is acceptable.”\nFigure S7. Sensitivity of sea-level contributions with linear Weertman law to various\nmesh resolutions.\n2. is there a correlation between the bed topography and the distribution of effective\npressures und inverted basal drag. If so you should point it out\nThank you for your comment. There is indeed a correlation between the bed topography\nand the distribution of effective pressure and inverted basal drag. Low-lying areas tend to\nhave lower effective pressure and basal drag, while elevated regions exhibit higher values.\nTo include this, we add one sentence in the Methods section (Line 520-523):\n“Bedrock geometry and ice thickness strongly influences subglacial water flow and\nchannel distribution, as lower-elevation areas, such as valleys, accumulate water\ndue to hydraulic potential gradients shaped by bedrock topography and ice\nthickness, resulting in higher water pressures and lower effective pressures.”\nThe correlation between bed topography and inverted basal drag is mediated by how bed\ngeometry influences ice flow resistance, with high basal drag generally associated with\nelevated or rough areas and low basal drag linked to smooth, low-lying regions. To include\nthis, we add one sentence in the inversion section (Line 440-441):\n“The inverted basal drag is largely influenced by the bed geometry through affecting\nthe ice flow resistance.”\n3. I would like to point to a publication where although in a different manner the same\nproblem is addressed and maybe you can use it to further complete your argument.\nGoeller et al.:A balanced water layer concept for subglacial hydrology in\nlarge-scale ice sheet models, The Cryosphere, 7, 1095-1106,2013 doi:10.51947tc-7-\n1095-2013\nMany thanks for recommending this valuable paper. This paper introduces the balanced\nwater layer concept, which provides a simplified representation of subglacial hydrology\nsuitable for large-scale ice sheet models. We have cited this paper in our Introduction:\n“The widespread production of subglacial water beneath the Antarctic Ice Sheet,\nand the potential intrusion of warm seawater near the grounding line, plays a crucial\nrole in modulating basal sliding and enabling rapid ice flow 7-11.” (Line 26-27)\n4. Finally - from a person which is color blind - you would make life much easier for\nabout 10 % of your readers if you would use different line styles ind addition to colors\nThank you for highlighting this important point. Although Reviewer 2 noted that Figure 1\nuses good colors, we have updated all figures to ensure they are more color-blind-friendly\nby incorporating both improved color schemes and distinct line styles.\nReviewer #2 (Remarks to the Author):\nSubglacial Water Amplifies Antarctic Contributions to Sea-Level Rise\nThe present manuscript investigates the effect of different variants for computing the\nbasal pressure below the Antarctic ice sheet used in the basal sliding law. Basal sliding\nis a key component in ice sheet modeling that is hard to constrain due to a lack of direct\nobservations and it has a strong influence on the predicted sea level rise and timing of\ntipping points.\nUsing the Ice sheet model Elmer/Ice, the authors conduct a number of different\nexperiments each with a different treatment for computing the effective pressure N (ice\noverburden pressure minus subglacial water pressure) and investigate the effect under\nseveral climate forcing scenarios. They show that different Antarctic sectors react\ndifferently to the sliding relation chosen, depending mostly on the topography and\ndominating driving stress regime.\nThis study is original in that it focuses on comparing the effects of different sliding\nrelations within the same ice sheet model. Subglacial hydrology is an important missing\npiece in ice sheet modeling (and especially relevant for sea level rise predictions), and\nwhile the paper does not examine the validity of them, it investigates the implications of\nthe different approaches that are used in the wild.\nREMARKS\nDiscussion of sliding laws\nIn the introduction, the authors compare the two overarching classes of sliding laws:\nclassical Weertman and regularized Coulomb and state that models often ignore or\ncrudely implement the effective pressure N “by adopting the Weertman relation as the\ncommonplace approach” (L. 39f). While this is generally correct, I would prefer a more\nnuanced statement because the current formulation can imply that most models have\nno treatment of effective pressure at all. From skimming over Seroussi et al. (2020), I\nnoticed that out of the 13 used models, only 4 to 6 seem to use classical Weertman\nwithout considering basal pressure (unfortunately, not all of them clearly state their\nsliding relation). Maybe the variety in the approaches can be mentioned in the paper.\nSliding laws used in ISMIP6:\nAWI_PISM: Schoof/Coulomb\nDOE_MALI: linear basal friction law with spatially varying basal friction coefficient\nILTS_PIK_SICOPOLIS: Weertman–Budd-type sliding law with sub-melt sliding (Sato and\nGreve, 2012) and subglacial hydrology IMAU_IMAUICE: ?? probably Schoof/Coulomb?\nciting Bueler Brown 2009 JPL_ISSM: ??? probably coulomb-like, citing Morlighem et al.\nNCAR_CISM: basal sliding is similar to that of Schoof (2005), combining power law and\nCoulomb behavior PIK_PISM: Mohr-Coulomb UCIJPL_ISSM: ??? ULB_FETISH:\npower law basal sliding UTAS_ELMERICE: linear sliding relation VUB_AISMPALEO:\nWeertman, but with height above bouancy VUW_PISM: probably Schoof/Coulomb\nThanks for the suggestions and providing the sliding law used in the ISMIP6-2100\nprojections. However, I would like to clarify that, based on Seroussi et al. 2020, 13 out of\n16 models in ISMIP6-2100 have used Weerman law.\nTo clarify and respond to this comment we have included a table that summarises the\nsliding laws used in all recent model inter-comparison projects (ISMIP6-2100, ISMIP6-\n2300, and ABUMIP) in Table S1 (Supplementary Material).\nIn Line 47-51, we added the text:\n“Because the actual distribution of effective pressure under the Antarctic Ice Sheet\nis unknown, model-based sea-level rise projections typically either use Weertman\nsliding everywhere, which does not include effective pressure (e.g., refs.24–26), or\nmake simple assumptions about effective pressure when applying RC sliding, (e.g.,\nrefs.27–29). A summary of sliding laws used in three major Model Intercomparison\nProjects reveals that the majority of Antarctic Ice Sheet models (e.g., 13 out of 16\nmodels in ISMIP6-21002, 9 out of 16 models in ISMIP6-23005, and 7 out of 15 models\nin ABUMIP30) employed the Weertman sliding relation.”\nThe following paragraph on HAF (L. 45ff) describes one simple approach for including N,\nbut I think it could be better integrated with the previous part.\nTo better illustrate the differences in how effective pressure is treated across experiments,\nwe have reorganized the Introduction as suggested by Reviewer 2. We now introduce the\nexplicit representation of effective pressure first (Line 52-59), followed by the implicit\nrepresentation (Line 60-63). Additionally, we have added a separate paragraph (Line 64-\n73) specifically discussing HAF-scaling. To further clarify the experimental design, we\nhave also moved Table 1 from the Methods section to the Introduction.\n2. Explaining the experiments in the main text\nThe Method section does a great job of explaining the different experiments, but it would\nbe helpful for the reader to get a short overview already in the main text to facilitate the\nunderstanding of the results. It would be enough to briefly explain the different\ntreatments of N and possibly further explain the difference between HAF and POC (they\nseem similar in that they both have the goal of enforcing floating condition at the\ngrounding line).\nThanks for the helpful suggestions. As mentioned in our response to the comment above,\nwe reorganized the Introduction to better explain the difference between experiments\nand added a summary paragraph at the end (Line 74-81). A modified Table 1 is moved to\nthe Introduction to help better interpret the differences.\nTo explain the differences between HAF and POC, we added two sentences at Line 70-73:\n“Both the HAF-scaling and perfect ocean connection assumption enforce the\nfloating condition at the grounding line. However, HAF-scaling, applied only near the\ngrounding line, adjusts basal resistance based on ice elevation relative to flotation,\npotentially under- or overestimating effective pressure. In contrast, the perfect\nocean connection assumes uniform hydraulic connectivity across the domain,\nsystematically overestimating effective pressure by neglecting hydraulic potential\ngradients.”\nI would also appreciate a comment on why these particular treatments were chosen; are\nthey representative of what is used in current models?\nIn the reorganized Introduction section, we now provide a clearer and more logical\nexplanation for why these specific treatments were chosen, with appropriate citations as\nevidence for their use in current models. To be consistent with the logical structure of the\nResults section suggested by Reviewer 2, we begin with the explicit representation of\neffective pressure, employing widely recognized approaches: the perfect ocean\nconnection assumption, which is commonly used to enforce the floating condition at the\ngrounding line (Line 52-55), and an alternative approach that derives effective pressure\nusing a subglacial hydrology model (GlaDS) to account for spatial variability (Line 56-59).\nWe then include the implicit representation of effective pressure, as used in Pine Island\nGlacier simulations by Joughin et al. (2019). This approach, chosen for its computational\nefficiency and ability to reproduce observed ice flow speeds through spatially optimized\nparameters, remains a practical and widely adopted method for large-scale ice sheet\nsimulations (Line 60-63). Then a separate paragraph (Line 64-73) about Height Above\nFlotation scaling (HAF-scaling) is added to manually scale the basal resistance or the\neffective pressure from a prescribed HAF value down to zero at the grounding line. The\nlast paragraph is a summary of experiment design in this study with a modified Table 1.\nIt would be good to lead the reader along from simple implementations to more\ncomplex/realistic and explain why you chose the GlaDS model out of a variety of\nsubglacial hydrology models (see e.g. De Fleurian et al. 2018).\nThank you for your comment. We use the GlaDS model because it was designed for\nglacial applications (Werder et al., 2013) and has been successfully utilized in previous\nice sheet studies (Gagliardini et al., 2018, Dow et al. 2022, Pelle et al. 2024, etc).\nIntroducing an alternative model would be impractical and time-consuming.\nGlaDS is in fact particularly well-suited for our study as it represents a continuum\napproach that combines distributed and channelized drainage systems, making it ideal\nfor capturing the diverse hydrological processes that occur beneath glaciers and ice\nsheets. Its ability to simulate both water-filled channels and sheet-like flows aligns well\nwith the goals of our research, particularly in exploring how bed topography influences\neffective pressure and basal drag.\nWhile other hydrology models exist, such as those focused exclusively on either\ndistributed or channelized systems, GlaDS offers a balance of physical realism and\ncomputational feasibility, making it a robust choice for studying subglacial hydrology in\ncoupled ice sheet modeling frameworks. This combination of practical implementation\nwithin Elmer/Ice and its theoretical strengths guided our decision to use GlaDS in this\nstudy.\nTo address this comment in the manuscript, we include a brief explanation in the\nMethods section (Line 511-515):\n\"The GlaDS model was selected for this study as it combines a distributed water\nsheet and channelized drainage system, providing a good representation of\nsubglacial hydrology that is particularly suited to investigating how bed topography\ninfluences effective pressure and basal drag 19-22. It is fully implemented within\nElmer/Ice, allowing for seamless spin-up of the hydrology model with a restart from\nthe ice sheet model.\"\nI am a little bit concerned with the validity of the inversion result for future projections\nwhen the basal conditions (especially at the grounding line) have changed. This affects\nmostly the experiments where N is treated implicitly (as part of the inversion). Are they\nstill a good baseline after e.g. a substantial retreat of the GL which would imply a major\nreconfiguration of the hydraulic system?\nWe share the reviewer’s concern, which is precisely what this study aims to emphasize.\nWe agree that basal drag should vary over time, especially near the grounding line.\nAs the reviewer pointed out, in experiment with an implicit effective pressure (RC_iN),\nassuming effective pressure into a constant parameter fails to account for the temporal\nvariability of effective pressure in fast-flowing regions, which is why we use HAF-scaling\nto manually reduce basal drag near the grounding line in RC_iN_HAF. Similarly, in the\nexperiment with simulated effective pressure from a hydrology model (RC_eN_GlaDS),\nwe assumed a constant effective pressure, neglecting changes in basal drag over time.\nTo address this, we conducted a separate experiment, RC_eN_GlaDS_HAF, by applying\nHAF-scaling to the simulated effective pressure.\nOur results show that ice dynamics are highly sensitive to how temporally varying\neffective pressure is treated, particularly near the grounding line. This underlines the\nimportance of using a coupled ice sheet–hydrology model, which would allow for a more\nrealistic representation of the evolving hydraulic system, particularly under future\nprojections when significant changes to basal conditions are expected. This has been\nwell addressed in the Discussion (Line 212-219).\n3. Structure of results and headlines\nI believe the presentation of the results could be enhanced for better readability and\ncomprehension. While the beginning that focuses on the implicit handling of N does a\ngood job of describing the difference between LW and RC_iN_HAF, the following parts\ncould be arranged in a more organized way, as I experienced them hard to follow. One\nidea could be to adapt something like the following structure (very similar to the current\none):\nKeep the first part on implicit handling of N: LW vs RC_iN_HAF as it is (L. 59 to L. 80).\nThen have a second part that covers the explicit handling of N: RC_eN_GlaDS vs\nRC_eN_POC vs RC_eN_GlaDS_HAF. Here I would suggest to start with RC_eN_POC as\nthe most simple approach and then introduce GlaDS and the extended GlaDS\n(RC_eN_GlaDS_HAF).\nWith this continental view established, continue with the Tipping points and the regional\nanalysis.\nNot much change is needed, but these sections should be clearly identifiable and might\ndeserve their own headings, also because the current in-between headings (“Subglacial\neffective pressure must be included in sliding relations” and “Subglacial effective\npressure controls ice fluxes”) are very general and do not help much to navigate the text.\nThe final part before the discussion (“Increased sensitivity to effective pressure near the\ngrounding line”) is in a good place.\nMany thanks for the suggestions, which are super helpful. We have reorganised the\nResults section as recommended:\nSubglacial effective pressure must be included in sliding relations\nThis section compares the experiments using linear Weertman and Regularised\nCoulomb sliding relations and indicates that effective pressure is necessary in\nsliding relations.\nSubglacial effective pressure controls ice fluxes\nThis section indicates the influence of different treatments of effective pressure\non the projected mass loss. To maintain consistency with the structure of the\nIntroduction section and adhere to Reviewer 2's suggestions, we presented the\nresults using three bullet points:\n• Implicit representation of effective pressure N\n• Explicit representation of effective pressure N\n• Basin-scale response\nIncreased sensitivity to effective pressure near the grounding line\nThis section emphasizes the critical role of effective pressure near the grounding\nline in affecting the low-drag area and projected ice mass loss.\nWe believe that the original subtitles effectively convey the key messages when\nsupplemented by these key bullets. Please see the modified main text for detailed\nchanges.\nAt some point, Kazmierczak et al. (2022) should also be referenced, as they also\ninvestigate the influence of different sliding relations for Antarctica.\nThanks for sharing this paper. We have cited this paper in our Introduction (Line 26-27)\nand Discussion (Line 213-215), especially about the finding that decreased effective\npressure near the grounding line amplifies the ice sheet's sensitivity to climatic forcing,\nparticularly for a given sliding relation power.\nRemarks by lines\nL. 29: also frictional heating due to internal deformation of ice\nThanks. Added.\nL. 45f: Can’t the effect of N be also approximated in the sliding coefficients for Weertman\nsliding laws? see also concern 1.\nThe short answer is “no”, except for the present-day state of the ice sheet. The effect of\neffective pressure N can be viewed as being implicitly represented in the spatial pattern\nof the Weertman coefficient, but this varies by many orders of magnitude because the\nWeertman sliding laws do not explicitly capture anything about the role of subglacial\nhydrology in sliding nor its evolution over time; all the relevant physics are hidden in one\nspatially tuned parameter. So we have no confidence in the contribution of N to\nprojected evolution of the ice sheet when using Weertman sliding.\nL. 48: more realistic than what?\nHere we mean the HAF-scaling is more realistic than using a constant coefficient in\nRC_iN or a constant simulated effective pressure in RC_eN_GlaDS. This is because the\neffective pressure is expected to vary with time, especially in regions where the ice\napproaches flotation with a moving grounding line. To address this and incorporate other\nrelated comments regarding the HAF-scaling, we have rewritten this entire paragraph as\nfollows (Line 64-73):\n“A further assumption about the effective pressure and its impact on sliding can be\nimposed using on \"height above floatation\" (HAF) scaling21. HAF is defined as the\nheight of the upper ice surface above sea level minus the height required for the ice\noverburden pressure to match the ocean water pressure at the bed. This assumption\nscales the basal resistance or the effective pressure from a prescribed HAF value\ndown to zero at the grounding line, reflecting the influence of warm seawater\nintrusion beneath grounded ice41. However, the inland extent of this effect is poorly\nconstrained due to limited observations, and this approach neglects significant\nspatial and temporal variability driven by evolving glacier geometry 42,43. Both the\nHAF-scaling and perfect ocean connection assumption enforce the floating\ncondition at the grounding line. However, HAF-scaling, applied only near the\ngrounding line, adjusts basal resistance based on ice elevation relative to flotation,\npotentially under- or overestimating effective pressure. In contrast, the perfect\nocean connection assumes uniform hydraulic connectivity across the domain,\nsystematically overestimating effective pressure by neglecting hydraulic potential\ngradients.”\nL. 56: Why SSA, what is the implication?\nThe SSA simplifies the full-Stokes equations by neglecting vertical shear stresses,\nfocusing only on the horizontal stress balance. This reduction significantly decreases the\ncomputational cost, making SSA suitable for large-scale and long-term simulations, i.e.\ncentury-scale Antarctic Ice Sheet projections. The SSA is well-suited to simulate the\nbehavior of fast-flowing ice streams, outlet glaciers, and floating ice shelves, which are\ncritical regions influencing ice discharge from Antarctica. However, the neglect of vertical\nshear can lead to inaccuracies in simulating inland ice dynamics and interactions with\nthe grounding zone, which might underestimate or oversimplify the response of grounded\nice to changes in basal conditions or surface mass balance.\nTo address this limitation, we expanded the Discussion section to provide a more detailed\nexplanation of the study's limitations (Line ***:\n“Our simulations explore the sensitivity to basal sliding paramerisations under\nvarious climate scenarios. However, limitations in our approach highlight areas for\nimprovement. The use of SSA assumes vertical hydrostatic equilibrium and neglects\nvertical shear stresses, which restricts its ability to fully capture the ice dynamics at\nthe grounding line. This is particularly relevant in regions with pronounced bedrock\nslopes, where non-hydrostatic effects and deviatoric (bridging) stresses—ignored by\nSSA become significant. While a fully resolved full-Stokes model, combined with a\ncontact problem formulation for the lower ice surface, would be needed to\nconsistently resolve these dynamics 51, we argue that the sliding relation itself is less\nsensitive to the choice of stress representation in the ice-flow model. For instance,\na similar sensitivity to driving stress has been found between an SSA model and one\nincorporating simple vertical shear, particularly under a regularized Coulomb law or\na Weertman sliding parameterisation 21.\nNevertheless, unresolved processes may further alter projected sea-level\ncontributions. Given the importance of the grounding line region to our results, we\nexpect further changes when considering three-dimensional shear fields,\nhydrology-ocean-bedrock-sediment interactions, iceberg calving, and\nuncertainties in predicted future surface mass balance. While these processes are\nunlikely to negate the effects we identify—given that effective pressure is largely\nisolated from external forcings—they underscore the complexity of grounding line\ndynamics and the need for urgent model development to advance the fidelity of sea-\nlevel projections.”\nL. 57: Why is SSA used here? Why not hybrid/higher order? What are the implications?\nSee my response to L.56 above.\nL. 60: Why is RC_iN_HAF more realistic? is combined effects of seawater intrusion and\nmeltwater production is also considered in the eN variants? is this only about RC vs LW?\nmore explanation is needed.\nFor the 1st and 3rd question here, Yes, the comparison here is only about RC vs LW. We\nsuggest that RC relation is more physically realistic because it can capture both\nWeertman and Coulomb regimes, provides a unified framework for modelling sliding\nacross diverse bed types as explained in the modified Introduction (Line 43-46). The HAF-\nscaling in RC_iN_HAF can well represent the combined effects of seawater intrusion and\nmeltwater production near the grounding line as explained in Line 66-68. However,\nconsidering the restructured Introduction and Results sections, we deleted this sentence\nhere and added a similar sentence in the Introduction (Line 77-80):\n“For comparison against the more physically justifiable RC sliding parameterisation,\nwe also apply the linear Weertman sliding relation (hereafter “LW”'), which, despite\nits tendency to underestimate mass loss 21, is still widely used for its simplicity in\nboth numerical and computational aspects 2, 5, 26, 30.”\nFor the second question, while we agree that RC_eN_POC also considers the combined\neffects of seawater intrusion and meltwater production, it is important to note that\nRC_eN_GlaDS uses a constant effective pressure simulated from the GlaDS hydrology\nmodel. To account for a temporally evolving effective pressure, we applied HAF-scaling\nto the simulated effective pressure in RC_eN_GlaDS_HAF. We have further clarified the\nHAF scaling, along with its similarities and differences with POC, in the revised\nIntroduction (Line 70-73).\nL. 68: Replace “near mass balance” with “near equilibrium”. I don’t understand the\nsecond part of the sentence regarding observed mass loss.\nModified.\nThe second part of the sentence refers to how the RC relation suggests a near-zero mass\nloss by 2300, which aligns more closely with the observed mass balance trends in recent\ndecades. We have revised the sentence for greater clarity, as follows (Line 91-93):\n“Under the two low emission scenarios, the RC relation suggests near equilibrium by\n2300, which aligns more closely with observed mass loss in recent decades 44, …”\nL.71: “For WAIS…”: Sentence is unclear. Maybe “For the WAIS the experiment using the\nRC relation and high emission scenario,…”\nModified.\nL. 73: produce.\nModified.\nL. 74: “However, the RC…” -> “However, using the RC…”\nModified.\nL. 75: “than LW” -> “than using the LW relation”. The whole paragraph seems to be missing\nwords.\nThanks for the suggestions. The whole paragraph has been modified accordingly (Line 95-\n105).\nL. 82: This sentence is not very clear, seems obvious.\nWe modified the sentence to increase the clarity but we’re not sure what the reviewer is\nrequesting here.\n“Our simulations reveal the treatment of effective pressure significantly affects the\nspatial tuning of sliding coefficients, which, in turn, impacts the evolution of basal\nshear stress” (Line 111-112)\nL 99ff: Isn’t the addition of HAF to the GlaDS N a little bit of double counting, because the\neffect of the ocean is already included in the GlaDS model? Though it is only appropriate\nat the time where N was computed in GlaDS.\nNo, as we mentioned in response to an earlier comment, RC_eN_GlaDS uses a constant\neffective pressure simulated by the GlaDS hydrology model. While the GlaDS model does\naccount for ocean effects, the constant simulated effective pressure becomes\nunsuitable for regions experiencing grounding line retreat. To address this time-varying\neffect, we apply HAF-scaling to the simulated effective pressure.\nL. 106: Explain coulomb limit\nThanks for your comment. The Coulomb limit defines the maximum basal shear stress\nunder a Coulomb sliding regime, where sliding is primarily controlled by effective\npressure and the sliding coefficient. We decide to move all the sentences related with\nCoulomb limit into a separate paragraph in the subsection “Increased sensitivity to\neffective pressure near the grounding line” (Line 199-210). In the modified Methods\nsection, we further explained the shift between Weertman and Coulomb regime under\ndifferent assumptions of effective pressure (Line 475, 492).\nWe modified the sentence here into: “While this parameterisation ensures the Coulomb\nregime dominates near the grounding line, …” (Line 131-132)\nL. 115-122: Can the experiments using POC produce meaningful results for basins with\nreverse bed slope? Won’t POC on the retrograde bed will lead to higher basal pressure in\nthe inland than at the grounding line?\nNo, the POC assumption does not produce meaningful results, as it consistently\nunderestimates water pressure inland. We included it in our study because it is one of\nthe commonly used assumptions (see modified Introduction, Line 52-55, Line 72-73),\nnot because we believe it to be a good assumption.\nYes, on a retrograde bed, POC will lead to higher basal water pressures inland compared\nto the grounding line, but these pressures are still insufficient. Given that basal friction\nand geothermal heat provide a basal water source nearly everywhere under the ice, and\nbasal water sinks are rare, there is always a hydraulic gradient driving water toward the\ngrounding line. In other words, subglacial water pressure is expected to build up\nsufficiently to drive flow toward the grounding line under normal conditions.\nL. 153: How valid is the distribution of N computed by GlaDS for present day conditions\nafter more than a century? Even with the HAF scaling, the whole hydraulic regime might\nhave shifted.\nThe distribution of N computed by GlaDS for present-day conditions is based on the initial\ngeometry and the friction heating calculated from the inverted basal shear stress. We\nacknowledge that over more than a century, changes in the ice sheet and surrounding\nenvironment—such as variations in oceanic conditions, climate, and ice dynamics—\ncould have altered the hydraulic regime, especially near the moving grounding line. While\nthe HAF-scaling helps account for some of these time-varying effects, it is true that the\nentire hydraulic regime may have shifted, particularly in regions where significant\nchanges in the ice flow or grounding line dynamics have occurred. However, in this study,\nwe’re not aiming to provide an accurate representation of the evolving hydrology over time,\nwhich can only be achieved by applying a coupled ice sheet-hydrology model. We have\nmade it clear at the start of the Methods section (Line 414-416):\n“Our simulations do not capture a fully coupled ice-hydrology system, as such a\nmodel is not yet available for continental-scale applications. Instead, our\nsimulations aim to determine whether a coupling is indeed necessary, rather than\npredetermining what might be considered a valid future projection of ice mass\nchange.”\nMoreover, we emphasized the limitation of HAF-scaling and the importance of having a\ncoupled ice-hydrology model in the modified Discussion (Line 221-229):\n“Among these RC experiments, we place greater confidence in the results from\nexperiment RC_eN_GlaDS_HAF, as it uses a more physically realistic subglacial\nhydrology system where HAF-scaling adjusts effective pressure as the grounding\nline retreats. However, while the HAF-scaling accounts for some of these time-\nvarying effects, the entire hydraulic regime may have shifted, particularly in regions\nexperiencing significant changes in ice flow or grounding line dynamics12, 39.\nMoreover, the GlaDS model, originally designed to simulate subglacial hydrology on\nhard bedrock, may predict more efficient drainage systems than those occurring on\na soft bed, potentially leading to lower modelled subglacial water pressures12,37,39.\nThis highlights the importance of accurately predicting subglacial water pressure\nand developing a more comprehensive ice sheet-hydrology model that can account\nfor the temporal co-evolution of ice sheet dynamics and the subglacial hydrological\nsystem.”\nL. 163: What does “constrained future surface mass budgets estimates” mean?\nImprovement of the predictions of SMB?\nYes, by \"constrained future surface mass budget estimates,\" we mean the uncertainties\nin predicting the future surface mass balance (SMB). To clarify, we have revised the\nsentence to: “…and uncertainties in predicted future surface mass balance.” (Line\n241)\nFigure 1: good colors, bigger fonts, no box around legend\nThanks for the suggestions. Based on comments from Reviewer 1, we have modified\nFigure 1 to be more colour-blind friendly with bigger fonts and no box around legend.\nFigure 2:\ncaption: “sea-level contributions between linear Weertman law” -> “sea-level\ncontributions in experiment using linear Weertman law”.\nModified.\nFigure 4:\ncaption: “with a Hydrology model” -> “from GlaDS”\nModified.\nDoes channel area mean cross section of individual channels?\nYes, the channel area refers to the cross section of each channel. To clarify this, we have\nupdated the caption of Fig. 4 to read: “The background image shows the simulated\neffective pressure and channel area (cross-sectional area of each channel) generated\nfrom GlaDS.”\nwhy is a diverging colormap used? What is special about 0.5 MPa?\nThe diverging colormap and the value of 0.5 MPa do not have any special significance. We\nchose this color scale to emphasize regions of low effective pressure, such as the Siple\nCoast and Lake Vostok.\nsmall inset figures in G-H and D-Dp are hardly readable.\nWe have zoomed in the inset figures for G-H and D-Dp.\nFigure 5:\nWhy are the lines dashed?\nThere is no special meaning here. We have modified them into solid lines.\nReferences\nDE FLEURIAN B, WERDER MA, BEYER S, et al. SHMIP The subglacial hydrology model\nintercomparison Project. Journal of Glaciology. 2018;64(248):897-916.\ndoi:10.1017/jog.2018.78\nKazmierczak, E., Sun, S., Coulon, V., and Pattyn, F.: Subglacial hydrology modulates\nbasal sliding response of the Antarctic ice sheet to climate forcing, The Cryosphere, 16,\n4537–4552, https://doi.org/10.5194/tc-16-4537-2022, 2022.\nBueler, E., and J. Brown (2009), Shallow shelf approximation as a “sliding law” in a\nthermomechanically coupled ice sheet model, J. Geophys. Res., 114, F03008,\ndoi:10.1029/2008JF001179.\nMorlighem, M., E. Rignot, H. Seroussi, E. Larour, H. Ben Dhia, and D. Aubry (2010),\nSpatial patterns of basal drag inferred using control methods from a full-Stokes and\nsimpler models for Pine Island Glacier, West Antarctica, Geophys. Res. Lett., 37,\nL14502, doi:10.1029/2010GL043853.\nReviewer #3 (Remarks to the Author):\nReview of “Subglacial Water Amplifies Antarctic Contributions to Sea-Level Rise”, for\nNature Communications, by Zhao, Gladstone, Zwinger, Gillet-Chaulet, Wang, Caillet,\nMathiot, Saraste, Jager, Galton-Fenzi, Christoffersen and King\nThis manuscript describes the importance of incorporating subglacial hydrology\nparameterizations into whole Antarctic ice sheet models, by varying parameterization of\nbasal water effective pressure in a basal slip relation. The authors conclude that\neffective pressure is required in the slip relationship, and that ice flux is sensitive to\neffective pressure model, especially near the grounding line.\nWe believe the topic of this manuscript is important and can be of interest to a wide\nrange of readers. However, we believe that this manuscript can be significantly\nimproved by considering the comments listed below.\nFor now, the manuscript is “expert-facing”, i.e., it is assumed that the readers have a\nsubstantial understanding of the background and methods of this study. The\nmanuscript leaps over many (essential) details that could have been helpful for the\nreaders to follow along.\nWe have gone through the manuscript line by line with a view to providing clarity for non-\nexperts. This is most noticeable in the modified Introduction. See also our specific\nresponses to further comments below.\n• For an important study that can be of interest to the general public, especially in the\njournal of Nature Communication, we argue this is not appropriate. We recommend the\nauthors to provide more detailed elaborations on the background, methods, findings, and\nimplications of this study. A few examples can be found in the comments below.\nThanks for the comments. We have further elaborated on these sections. See our specific\nresponses to further comments below.\n• In the section “Subglacial effective pressure must be included in sliding relations”; its\nimplied the observed mass gain of the linear Weertman models demonstrate the need\nfor subglacial effective pressure. The authors need to make a stronger case - ie\ndemonstrate that mass gain is not plausible over these time frames.\nThe comparison between linear Weertman (LW) and regularised Coulomb (RC) sliding\nrelations suggests that inclusion of effective pressure in the sliding relation largely\nenhances the basal sliding and projected mass loss. Whether the mass gain simulated\nin LW is plausible is not the focus here. We have clarified this in the modified Methods\nsection (Line 414-416) that we do not aim to predetermine what might be considered a\nvalid future projection in terms of mass change. We have also made minor modifications\nto the Introduction section (Line 77-80) to make it clear that the existing literature already\nprovides strong evidence for favouring RC over LW. The existence of a wide spread\nhydrological network under the Antarctic Ice Sheet can lubricate the ice base and thus\nlead to increased ice velocities, which should be considered in the basal sliding relations.\nOur aim in the current study is not to convince the reader that RC is better than LW (this\nhas already been done), but rather to show just how large an impact this choice can have.\n• We recommend the authors to elaborate on the differences between model setups. We\nappreciate the balance between the amount of work presented in this manuscript and\nthe space limitations. But the current manuscript makes it really difficult to follow the\ndifferent model setups, especially considering the number of runs that are being\npresented and discussed.\nThanks for the suggestions. We have provided a more accessible conceptual description\nof the different approaches in the modified Introduction section, i.e. the different\nassumptions implied by the different versions of sliding laws. We have also moved the\nmodified Table 1 to the Introduction to help better interpret the differences. See our\nresponse to Reviewer 2 for more details.\n• It is not clear why/how including different subglacial hydrology components can change\nthe AIS SLR projection by such a large degree. What is the physical process behind the\ndifferences? We recommend the authors to elaborate on this and make it more clear.\nThanks for the comment. We have added a description of the most important physical\nprocesses at the start of the subsection “Increased sensitivity to effective pressure near\nthe grounding line” (Line 174-182). The key concept, which we now believe comes across\nclearly to the non-expert, is that of a low friction “ice plain” immediately upstream of the\ngrounding line. This section has also been expanded to include a discussion of how\nassumptions about effective pressure influence the extent of this low-friction “ice plain”\nregion (Line 262-273), which can be widely different depending on the bedrock and ice\ngeometry.\n• Line 47-52: HAF is a very important concept for this study. However, the authors didn’t\nprovide any explanation/elaboration on the concept, other than providing one reference.\n(This is also true for some other important concepts throughout the manuscript.) This\napproach makes it difficult for non-expert readers to understand the work and follow the\nlogic flow. We recommend providing a more detailed explanation of the concept, and/or\nmaybe even a conceptual diagram.\nThanks for the comment. We have provided an accessible description of the concept of\nHAF in the modified Introduction (Line 65-68):\n“HAF is defined as the height of the upper ice surface above sea level minus the\nheight required for the ice overburden pressure to match the ocean water pressure\nat the bed. This assumption scales the basal resistance or the effective pressure\nfrom a prescribed HAF value down to zero at the grounding line, reflecting the\ninfluence of warm seawater intrusion beneath grounded ice 41.”\nWe have also provided a more detailed explanation for other concepts like the Coulomb\nlimit (Line 201-202)\n• It is not clear what assumption is made about the subglacial meltwater budget. Does\nthe model run only require a certain and finite amount of subglacial water? Or do the\nauthors assume an “unlimited” supply of subglacial water generated by basal melting\nupstream? And does a limited subglacial meltwater budget change the model output?\nThanks for the comments. We have expanded our description of the hydrology model in\nthe Methods section (Line 511-519). The hydrology model spin-up does make specific\nassumptions about the basal water source and the other parameters on which basal\nwater pressure depends, most important of which are parameters describing\nconductivity of both basal channels and the distributed basal water system. A brief\ndescription is given in the Methods section (Line 515-516), and the citation to the recent\npaper (Zhang et al., 2024) is the most directly relevant as their use of GlaDS is almost\nidentical to the current study.\nThe ice dynamic simulations have no direct dependence on water budget; instead, they\ndepend on basal water pressure. This pressure is influenced by various factors,\nparticularly the conductivities mentioned above, with basal water production being\ndriven solely by frictional heating. Consequently, the only simulations where\nassumptions about the basal water budget are directly relevant are those utilizing the\nspatial distribution of basal water from the hydrology model. Even in those cases, the\ndependence remains weak due to the influence of other factors.\nTo answer the question literally (though the fact that our hydrology model is in steady\nstate probably makes this clarification unnecessary), the subglacial water supply is\neffectively unlimited, but any other answer would result in no hydrology at all at steady\nstate.\nThe feedback between ice dynamics and basal water generation, through friction\nheating due to sliding, may be important for future evolution of the coupled system, but\nthis is beyond the scope of the current study.\n* The authors need to address the limitation of the study in the Discussion section. We\nunderstand that the authors need to consider the space limitations per journal\nguideline, but we strongly recommend the authors to reconsider the balance between\n(1) further elaboration and discussion of the study findings, (2) connection with other\nstudies, (3) broader implications, and (4) limitation of the study, for the Discussion\nsection.\nThanks for your comments. We have carefully considered your suggestions and revised\nthe Discussion section, which we hope will satisfy this request. Specifically:\n1. Elaboration and discussion of findings: We have added a paragraph\nemphasizing the critical role of subglacial water pressure and the significant\ninfluence of a small region near the grounding line on multi-century ice sheet\nmass loss. This discussion highlights the connection between effective pressure\nand sliding relations, as well as the significance of model resolution in capturing\nsmall-scale transitions near the grounding line (Lines 212–219).\n2. Connection with other studies: We have incorporated relevant citations in the\nDiscussion to place our findings in the context of previous research. For example,\nwe cite Kazmierczak et al. (2022) to emphasize the influence of different sliding\nrelations for Antarctica (Lines 213–215), suggested by Reviewer 2.\n3. Broader implications: In the final paragraph of the Discussion (Lines 245–251),\nwe elaborate on the implications of our findings for understanding ice sheet\ndynamics and their influence on sea-level projections. This paragraph\nemphasizes the importance of integrating realistic subglacial processes into\nfuture models to improve predictions.\n4. Study limitations: In response to your comments and those of Reviewer 2, we\nhave added a paragraph discussing the limitations of using the Shallow Shelf\nApproximation (SSA) in this study (Line 230-238). Additionally, the original\nsubmission already addressed other limitations, such as hydrology-ocean-\nbedrock-sediment interactions, iceberg calving, and uncertainties in future\nsurface mass balance projections (Line 239-241).\nminor comments:\n* This manuscript uses a lot of acronyms - are they really necessary? Like, “GL” for\n“grounding line”? Having too many acronyms can be counterproductive. Also, it may\nhelp to spell out the run names in Figure 1.\nThanks for the comments. We have removed the acronyms including GL (grounding line),\nN (effective pressure), SLR (sea-level rise), AIS (Antarctic Ice Sheet), WAIS (West\nAntarctic Ice Sheet), EAIS (East Antarctic Ice Sheet). We now only keep three acronyms:\nHAF (height above flotation), LW (linear Weertman), RC (regularised Coulomb).\nWe have moved the modified Table 1 from Methods to Introduction, which can well\nexplain the difference across these experiments in Figure 1.\n* We noticed that no doi/web-link is provided for the references. We encourage the\nauthors to consider including such information.\nOur manuscript is available in LaTeX format, which includes the DOI and web links in the\nreferences. If the manuscript is accepted, we hope the editing team can assist with\nensuring the DOI and web links are properly included in the final references.\nUnsure of what is meant by the second and third authors contributed equally.\nWe mean that the second and third authors contributed to this paper equally. To reduce\nthe confusion, we removed this statement.\nReviewer #4 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports.\nThis is part of the Nature Communications initiative to facilitate training in peer review\nand to provide appropriate recognition for Early Career Researchers who co-review\nmanuscripts.\nMany thanks for your contribution to provide the valuable comments.\nReferences:\nDow, C.F., Ross, N., Jeofry, H. et al. Antarctic basal environment shaped by high-\npressure flow through a subglacial river system. Nat. Geosci. 15, 892–898 (2022).\nhttps://doi.org/10.1038/s41561-022-01059-1\nGAGLIARDINI, O. and WERDER, M.A. (2018) ‘Influence of increasing surface melt over\ndecadal timescales on land-terminating Greenland-type outlet glaciers’, Journal of\nGlaciology, 64(247), pp. 700–710. doi:10.1017/jog.2018.59.\nJoughin, I., Smith, B. E., & Schoof, C. G. (2019). Regularized Coulomb friction laws for\nice sheet sliding: Application to Pine Island Glacier, Antarctica. Geophysical\nResearch Letters, 46, 4764–4771. https://doi.org/10.1029/2019GL082526\nPelle, T., Greenbaum, J. S., Ehrenfeucht, S., Dow, C. F., & McCormack, F. S. (2024).\nSubglacial discharge accelerates dynamic retreat of aurora subglacial basin outlet\nglaciers, East Antarctica, over the 21st century. Journal of Geophysical Research:\nEarth Surface, 129, e2023JF007513. https://doi.org/10.1029/2023JF007513\nSeroussi, H. and Morlighem, M.: Representation of basal melting at the grounding line in\nice flow models, The Cryosphere, 12, 3085–3096, https://doi.org/10.5194/tc-12-\n3085-2018, 2018.\nSeroussi, H., Pelle, T., Lipscomb, W. H., Abe-Ouchi, A., Albrecht, T., Alvarez-Solas, J., et\nal. (2024). Evolution of the Antarctic Ice Sheet over the next three centuries from an\nISMIP6 model ensemble. Earth's Future, 12,\ne2024EF004561. https://doi.org/10.1029/2024EF004561\nWang, Y., Zhao, C., Gladstone, R., Zwinger, T., Galton-Fenzi, B. K., and Christoffersen, P.:\nSensitivity of the future evolution of the Wilkes Subglacial Basin ice sheet to\ngrounding-line melt parameterizations, The Cryosphere, 18, 5117–5137,\nhttps://doi.org/10.5194/tc-18-5117-2024, 2024.\nWeertman, J. (1972). General theory of ice-sheet flow. Journal of Glaciology, 11(62),\n157-170. DOI: 10.1017/S002214300002302X\nWerder, M. A., I. J. Hewitt, C. G. Schoof, and G. E. Flowers (2013), Modeling channelized\nand distributed subglacial drainage in two dimensions, J. Geophys. Res. Earth Surf.,\n118, doi:10.1002/jgrf.20146.\nYufang Zhang, John C Moore, Liyun Zhao, Mauro A Werder, Rupert Gladstone, and\nMichael Wolovick. The role of hydraulic conductivity in the Pine Island Glacier’s\nsubglacial water distribution. Science of The Total Environment, 927:172144, 2024."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The paper is the result of a numerical study investigating the effect of basal pressure under the Antarctic ice sheet and the\nrole of the pressurized water layer in basal slip. While for a direct prognostic application there is lack of data, the paper\ndefines the range of solutions for Ice Sheet/iceshelf dynamics depending on pressure in aprticular at he icesheet-iceshelf\nboundary.\nThe authors have responded to all reviewers comments and i am satisfied with the answers to my specific comments.", "reviewer_2": "The authors have addressed the reviewers’ concerns effectively. I’m pleased with the improved paper structure and\nextended description. I have only one minor comment:\nI appreciate the more detailed description of sliding laws used in the ISMIP/ABUMIP experiments, but I would like to point\nout that the PISM model uses a Mohr-Coulomb type sliding law and not a Weertman type. It is specifically mentioned in the\nmanual that Weertman should not be used with PISM: https://www.pism.io/docs/manual/modeling-\nchoices/dynamics/weertman.html and also in the model description it is pretty clear:\nSeroussi et al. 2020\nPIK_PISM\nWe apply a power law for sliding with a Mohr–Coulomb criterion relating the yield stress to parameterized till material\nproperties and the effective pressure of the overlaying ice on the saturated till (Bueler and van Pelt, 2015).\nSeroussi et al. 2024:\nPIK_PISM\nA generalized power law (Schoof & Hindmarsh, 2010) is applied to parameterize basal sliding. The basal friction coefficient\ndepends on the effective pressure and till friction angle, that is parameterized using a heuristic, piecewise linear function of\nthe bed elevation (Martin et al., 2011).\nI know from personal communication that AWI_PISM uses the same setup in that regard.\nThis does of course not change the overall conclusion of the authors that Weertman type sliding laws are dominant in the\nbenchmarks.", "reviewer_3": "This work illustrates the need for improved knowledge of the evolution of subglacial hydrology, and we recommend\npublication.\nThe revised manuscript reads much better with acronym expansion and with more detailed concept explanations and\ndiscussions. Our only minor follow-up suggestion is to explicitly state in the Method section that the subglacial water budget\nis held constant because (1) the model requires such assumption to reach steady state, and (2) different assumptions on this\ndo not cause significant variation in the modeling results.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts."}, "author_response": "Response to Reviewers\nMany thanks to the reviewers for their valuable comments. Our responses are highlighted\nin blue, and the line numbers refer to the revised manuscript.\nReviewer #1 (Remarks to the Author):\nThe paper is the result of a numerical study investigating the effect of basal pressure\nunder the Antarctic ice sheet and the role of the pressurized water layer in basal slip.\nWhile for a direct prognostic application there is lack of data, the paper defines the range\nof solutions for Ice Sheet/iceshelf dynamics depending on pressure in aprticular at he\nicesheet-iceshelf boundary.\nThe authors have responded to all reviewers comments and i am satisfied with the\nanswers to my specific comments.\nThanks again for your comments.\nReviewer #2 (Remarks to the Author):\nThe authors have addressed the reviewers’ concerns effectively. I’m pleased with the\nimproved paper structure and extended description. I have only one minor comment:\nI appreciate the more detailed description of sliding laws used in the ISMIP/ABUMIP\nexperiments, but I would like to point out that the PISM model uses a Mohr-Coulomb\ntype sliding law and not a Weertman type. It is specifically mentioned in the manual that\nWeertman should not be used with PISM: https://www.pism.io/docs/manual/modeling-\nchoices/dynamics/weertman.html and also in the model description it is pretty clear:\nSeroussi et al. 2020\nPIK_PISM\nWe apply a power law for sliding with a Mohr–Coulomb criterion relating the yield stress\nto parameterized till material properties and the effective pressure of the overlaying ice\non the saturated till (Bueler and van Pelt, 2015).\nSeroussi et al. 2024:\nPIK_PISM\nA generalized power law (Schoof & Hindmarsh, 2010) is applied to parameterize basal\nsliding. The basal friction coefficient depends on the effective pressure and till friction\nangle, that is parameterized using a heuristic, piecewise linear function of the bed\nelevation (Martin et al., 2011).\nI know from personal communication that AWI_PISM uses the same setup in that\nregard.\nThis does of course not change the overall conclusion of the authors that Weertman\ntype sliding laws are dominant in the benchmarks.\nMany thanks for the correction and providing the detailed information. I have modified\nthe Table S1 in the supplementary material accordingly. See the modified table below.\nAs the reviewer noted, this does not change our conclusion stated in the Introduction\n“the majority of Antarctic Ice Sheet models employed the Weertman sliding relation”. To\nensure statistical accuracy, we have revised this sentence to:\n“A summary of sliding laws used in three major Model Intercomparison Projects\n(Table S1) reveals that the majority of Antarctic Ice Sheet models (e.g., 10 of 16 in\nISMIP6-21002, 8 of 16 in ISMIP6-23005, and 7 of 15 in ABUMIP30) employed the\nWeertman sliding relation.” (Line 49-51)\nReviewer #3 (Remarks to the Author):\nThis work illustrates the need for improved knowledge of the evolution of subglacial\nhydrology, and we recommend publication.\nThe revised manuscript reads much better with acronym expansion and with more\ndetailed concept explanations and discussions. Our only minor follow-up suggestion is\nto explicitly state in the Method section that the subglacial water budget is held\nconstant because (1) the model requires such assumption to reach steady state, and (2)\ndifferent assumptions on this do not cause significant variation in the modeling results.\nMany thanks for the suggestions. We have added this statement in the Methods section:\n“To ensure that the hydrology system reaches a steady state, the total subglacial\nwater budget is held constant throughout the spin-up period (over approximately\n100 model years). This assumption introduces negligible variability in the final\nresults compared to alternative assumptions on subglacial melt supply.” (Line 519-\n521)\nReviewer #4 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports.\nThis is part of the Nature Communications initiative to facilitate training in peer review\nand to provide appropriate recognition for Early Career Researchers who co-review\nmanuscripts.\nMany thanks for your contribution to provide the valuable comments."}]

Decision letter