Basal sliding in Antarctic glaciers is often modeled using a friction law that relates basal shear stresses to the effective pressure. As few ice sheet models are dynamically coupled to subglacial hydrology models, variability in subglacial hydrology associated with the effective pressure is often implicitly captured in the basal friction coefficient – an unknown parameter in the basal friction law. We investigate the impact of using effective pressures calculated from the Glacier Drainage System (GlaDS) model on basal friction coefficients calculated using inverse methods in the Ice-sheet and Sea-level System Model (ISSM) at Denman Glacier, East Antarctica, for the Schoof and Budd friction laws. For the Schoof friction law, a positive correlation emerges between the GlaDS effective pressure and basal friction coefficient in regions of fast ice flow. Using GlaDS effective pressures generally leads to smoother basal friction coefficients and basal shear stresses, and larger differences between the simulated and observed ice surface velocities compared with using an effective pressure equal to the ice overburden pressure plus the gravitational potential energy of the water. Compared with the Budd friction law, the Schoof friction law offers improved capabilities in capturing the spatial variations associated with known physics of the subglacial hydrology. Our results indicate that ice sheet model representation of basal sliding is more realistic when using direct outputs from a subglacial hydrology model, demonstrating the importance of coupling between ice sheet and subglacial hydrological systems. However, using our outputs we have also developed an empirical parameterization of effective pressure that improves the application of the Schoof friction law without requiring explicit hydrological modeling.

The health of Antarctic glaciers and their future susceptibility to climate-driven change is often assessed by the retreat rates of their grounding lines and melt rates of adjacent ice shelves. In the East Antarctic, Denman Glacier of the Denman–Scott catchment (Fig.

Denman–Scott catchment.

Determining the future of Denman Glacier, and others in the Antarctic, largely relies on ice dynamics models to capture the evolution of ice flow under changing climates. These models frequently use inversion techniques to constrain important parameters that control ice velocity, including those related to the basal environment beneath the ice sheet

Antarctic subglacial hydrology is increasingly being shown to be varied and dynamic, with large channels that discharge into ice shelf cavities

Despite the key role of basal boundary conditions, and in particular subglacial water pressure, in ice dynamics, there has not yet been a systematic investigation of the impact of the effective pressure on basal sliding in different friction laws applied in the Denman–Scott catchment. Given the critical role of the friction law for accurate ice dynamics and sea level predictions in the future

GlaDS is a 2D finite-element subglacial hydrology model that calculates the development of a distributed water system on the elements and channel growth, fed by the distributed system on the element edges

As discussed in

The domain for the GlaDS model is a subset of the grounded portion of the Denman–Scott catchment based on the hydraulic potential (at overburden) catchment for this region (Fig.

GlaDS model parameters described in detail in

ISSM is a finite-element model that uses a non-uniform mesh to simulate ice dynamics. We employ the inverse capabilities within ISSM to estimate basal friction coefficients in the Denman–Scott catchment using various basal friction laws, as described below. The inverse model uses the shallow-shelf approximation

The domain for the ISSM model is the full Denman–Scott catchment, extending the boundary used for the GlaDS simulations to include the floating ice shelves (Fig.

We investigate the impact of the prescription of the effective pressure in the Budd and Schoof friction laws by calculating basal friction coefficients using inversion. The scalar form of the Budd friction law is given by the following expression:

For each friction law, we calculate an inversion for the basal friction coefficients by minimizing a cost function that includes the contributions of absolute and logarithmic misfits between the observed and simulated ice surface speeds

Though it is not the primary focus of this work, we invert for ice rigidity as well, while initializing our model for the inversion for the basal friction coefficients. By inverting for the ice rigidity we capture ice rheological processes which are not explicitly accounted for in the model such as damage, anisotropy, chemical impurities, and liquid water.

We perform the following inversion procedure. First, we invert for the ice rigidity over the floating portion of the domain. Next, we invert for the Budd basal friction coefficient over the grounded portion of the domain, using an ice rigidity on grounded ice specified by the Paterson function from

We compare the difference in the basal friction coefficients when we use two different prescriptions for the effective pressure. (1) The first is an effective pressure given by assuming water pressure equals the ice overburden pressure plus the gravitational potential energy of the water:

We present the results of the GlaDS modeling followed by the inversion results using ISSM and compare the impact of using

The GlaDS modeling indicates that major subglacial hydrology channels form in the Denman–Scott catchment as seen in Fig.

GlaDS simulation results.

As effective pressure is the ice overburden pressure minus the subglacial water pressure, low effective pressure implies high subglacial water pressure, with negative effective pressure implying that the subglacial water pressure is greater than the ice overburden pressure. A low effective pressure corresponds to a high fraction of flotation, which is defined to be the subglacial water pressure divided by the ice overburden pressure. When the subglacial water pressure is equal to the ice overburden pressure; flotation is reached and the fraction of flotation is 100 %. Effective pressure in the GlaDS outputs is lowest in the basin feature (Fig.

We first consider the results for the Schoof friction law (Eq.

Effective pressure inputs (note the color map scales are not all the same).

Ice dynamics outputs (note the color map limits are not all the same). Panels

The Schoof basal friction coefficient estimated using

Median, mean, and root mean square error (RMSE) of the differences between the simulated and observed ice surface speeds for the Schoof and the Budd friction laws using

Relationship between the effective pressure and basal friction coefficient for

We next compare the difference between basal friction coefficients with the Schoof and Budd friction laws and using

The GlaDS-calculated effective pressure in this region of the Antarctic reflects variability in ice thickness and basal topography, with regions of negative effective pressure concentrated in deep basins and subglacial valleys and troughs. The effective pressure in the remainder of the domain, including the interior of the catchment, is close to zero with minimal spatial variation. A region of high effective pressure is found to the north of the Denman trough. In this area, the bed topography is steep and the ice thin; this causes water to drain rapidly and lower the local water pressure. It is unclear whether this is a realistic representation of the basal hydrology in this region or whether the low pressures indicate a limitation of the hydrology model – in situ data collection is necessary to investigate this further.

Using the Schoof friction law in the ice dynamics inversions, the resulting basal friction coefficient is smoother when

Our parameterization for

Our finding of a positive correlation between

The relationship between low effective pressures and low basal friction did not hold for the Budd friction law (Figs.

The choice of friction law plays an important role in grounding line migration and mass loss, with studies showing that friction laws that incorporate effective pressure yield more realistic representation of ice sheet dynamics, grounding line retreat, and mass loss than those that do not

In this work we have used an SSA ice flow model, which fails to capture bed-parallel vertical shear deformations. This may affect the results of our inversions for the basal friction coefficient in areas of non-negligible vertical shear, such as at the onset of fast-flowing ice streams of the Denman and Scott troughs. However, the use of the Glen flow relation may also impact the capacity of even higher-order models to accurately capture bed-parallel vertical shear deformations. For example,

Initializing our temperature field with surface temperatures from RACMOv2.3

The GlaDS model used here has been demonstrated to accurately represent observed properties of the Antarctic subglacial hydrologic system

Effective pressures as a fraction of ice overburden pressure for

We suggest here an alternative parameterization of effective pressure,

The proposed empirical effective pressure (

Though this parameterization is not derived from physical principles and it lacks complete hydrological connectivity to the ocean, it produces physically realizable effective pressures

The Denman–Scott catchment in East Antarctica is a region that is undergoing active glacial retreat and is vulnerable to ongoing mass loss as the climate warms. We have investigated the coupled interactions between the subglacial hydrological system and the ice sheet through the basal friction coefficient – an important tuning parameter used in basal friction laws – and the dependence of the basal friction coefficient on the form of the effective pressure. We find that when the Schoof friction law is used, there is a smoother basal friction coefficient, and a slightly improved match between the simulated and observed ice surface velocities is found when

In their application of the GlaDS model to the Aurora Subglacial Basin,

We proposed a new parameterization for the effective pressure based on empirical data of the Denman–Scott catchment and the GlaDS results that more closely matches simulated effective pressures than suggestions from previous literature and which shows promise in capturing key features of the subglacial system in the absence of a complete subglacial hydrology model output. However, given the empirical nature of this parameterization, our results highlight the importance of simulating coupled interactions between the subglacial hydrology and ice sheet systems to accurately represent ice sheet flow, and future work should focus on the coupling of these two systems in transient simulations of the Antarctic Ice Sheet.

The domains of the ISSM and GlaDS models differ, with the GlaDS domain being a subset of the ISSM domain due to the differing subglacial hydrological and ice catchments and limits to the GlaDS domain size requiring restriction to the primary hydrological outlets of Denman and Scott glaciers. This means that there are regions within the ISSM domain for which the GlaDS effective pressure does not exist. We test the impact of two different methods to extrapolate the effective pressure in regions outside the GlaDS domain.

The first method relates the GlaDS effective pressure to the bed topography and ice surface speed. We partition the domain into three regions based on ice surface speed, with the first encompassing areas where

The second method uses

Linear regression coefficients

Normalized (to 3500 kg

In general, there is no reason that the hydrology and ice dynamics domains should match exactly, and where this is the case, we would always expect regions where we will need to extrapolate boundary conditions between the models. The fact that we have had to extrapolate effective pressure is not in itself a problem, and our results show the importance of taking into consideration the method of extrapolation. There may be other methods for extrapolating the boundary conditions that we did not consider, including parameterizing the effective pressure using different physical assumptions

Depending on the degree of regularization in the inverse method, small-scale features may be present in the basal friction coefficient. We perform a sensitivity analysis to examine the extent to which such features may arise due to the location of the nodes and vertices in the mesh. We generate five different meshes by adding a random perturbation in the surface velocity field of up to 20 % of the MEaSUREs velocities and then perform non-uniform mesh refinement on the velocity magnitude, as follows: the typical edge length of the elements is 3000 m; for regions where the ice surface speed is

Using each of these five meshes, we calculate the Budd and Schoof basal friction coefficients. The probability distributions of the basal friction coefficients normalized to 3500 kg

For regions of very low or negative effective pressure, the basal shear stresses are also very low or almost vanish. In these regions, the inverse method compensates by increasing the basal friction coefficient upstream and around the region of the anomalously low effective pressure, leading to an underestimate of ice surface speeds there compared with the observations. The ice surface speeds are also generally overestimated in the region of vanishing shear stresses.

To account for this, we adjust the effective pressure in regions of low or negative effective pressure, capping the minimum as a percentage of the ice overburden pressure. We test caps of 0.2 %, 0.4 %, 1 %, 2 %, and 4 %. For a cap of 0.4 %, approximately 2 % of the domain has an effective pressure that is linearly proportional to the ice overburden pressure, that is, 98 % of the effective pressure in the ISSM simulation is derived directly from the GlaDS simulated effective pressure. Increasing the cap to 1 % – which is used in the Budd runs – decreases the area over which the GlaDS effective pressure is used to 96 % and increasing the cap to 4 % decreases the area to 48 %. Hence, we use a cap of 0.4 % for our Schoof simulations and 1 % for our Budd simulations.

Here, we compare the impact of using a nonlinear exponent

The

Despite the smaller normalized variance in the basal friction coefficients for the

Budd basal friction coefficients (

Relationship between the Budd basal friction coefficient

Iken's bound,

Schoof basal friction coefficient mean

Schoof basal friction coefficient (

Ratio between the driving stress and the maximum stress that the bed can support (

We arrive at the same qualitative conclusions as in the main text for all four values of

The Schoof friction law is a regularized Coulomb friction law which tends towards a Weertman sliding regime when

L-curve analysis for the Budd basal friction coefficient,

L-curve analysis for the Schoof basal friction coefficient,

Cost function coefficients for each inversion. I1: ice rigidity over ice shelf; I2: Budd basal friction coefficient over grounded domain; I3: ice rigidity over entire domain; I4 – Budd: Budd basal friction coefficient over the grounded domain; I4 – Schoof: Schoof basal friction coefficient over the grounded domain; I5: ice rigidity over the entire domain.

Basal Friction coefficients from inversion (note that the units and the color bar are not the same in each subplot).

The inversion procedure described in Sect.

Probability distributions of the normalized basal friction coefficients for the Budd (

Ice Rigidities from inversion (

The linear and logarithmic cost function coefficients (

We propose a new effective pressure parameterization (

We rearrange Eq. (

Delving further into the implications of these effective pressure parameterizations, we see that the hydraulic potential is given by

Except for the lack of complete hydrological connectivity to the ocean, the proposed parameterization in Eq. (

We use version 4.21 of the open-source ISSM software, which is freely available for download from

KM designed and completed the ISSM model runs and produced figures; CFD ran the GlaDS model; FSM and CFD provided project direction. All authors wrote the paper.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We thank the Digital Research Alliance of Canada for access to supercomputer resources. We thank the editor – Nanna Bjørnholt Karlsson – and two reviewers – Thomas Zwinger and Elise Kazmierczak – for their constructive comments on this paper.

This research has been supported by the Australian Research Council (ARC) Discovery Early Career Research Award (grant no. DE210101433) and the ARC Special Research Initiative Securing Antarctica's Environmental Future (grant no. SR200100005).

This paper was edited by Nanna Bjørnholt Karlsson and reviewed by Elise Kazmierczak and Thomas Zwinger.