A hierarchy of approximations of the force balance for the flow of
grounded ice exists, ranging from the most sophisticated full Stokes (FS)
formulation to the most simplified shallow ice approximation (SIA). Both are
implemented in the ice flow model Elmer/Ice, and we compare them by applying
the model to the East Antarctic Shirase drainage basin. First, we apply the
control inverse method to infer the distribution of basal friction with FS.
We then compare FS and SIA by simulating the flow of the drainage basin under
present-day conditions and for three scenarios 100 years into the future
defined by the SeaRISE (Sea-level Response to Ice Sheet Evolution) project.
FS reproduces the observed flow pattern of the drainage basin well, in
particular the zone of fast flow near the grounding line, while SIA generally
overpredicts the surface velocities. As for the transient scenarios, the ice
volume change (relative to the constant-climate control run) of the surface
climate experiment is nearly the same for FS and SIA, while for the basal
sliding experiment (halved basal friction), the ice volume change is

The Shirase drainage basin (SDB) in East Antarctica covers an area of

Due to the nature of the ice flow, the SDB is an
interesting location for comparing different formulations of ice
dynamics. Such comparisons have already been the subject of several
studies in the past. In an early work by

Comparisons of the impact of different formulations of ice dynamics on
simulation results have also been carried out for real-world problems.

Here, we apply the model Elmer/Ice

Since ice is an (almost) incompressible material, conservation of
mass entails that the velocity field is solenoidal. Furthermore, the
acceleration (inertial force) is negligible. The FS
equations are thus

The temperature equation follows from the general balance equation
of internal energy and is

Standard physical parameters used for the simulations
with both the FS and SIA dynamics
(following SeaRISE Antarctica with SICOPOLIS;

In contrast to the FS formulation, which neglects none of the stress
components, SIA assumes that grounded ice flow is governed only by
ice pressure and the vertical shear stresses. This yields the following
equations for the velocities

The SIA formulation of the temperature equation (Eq.

For both FS and SIA, we assumed a stress-free ice surface (atmospheric
pressure and wind stress neglected). In the case of FS, the evolution
of the upper surface

The mean annual surface temperature

The surface mass balance

On the short timescales of our simulations, the bed topography

Finally, the temperature equations (Eqs.

Our computational domain (see Sect.

For the SIA case, neither of the dynamic conditions of Eqs. (

The control inverse method, introduced by

The Elmer/Ice model was applied to the SDB. The
present-day surface and bed topographies were extracted from the Bedmap2
data set by

Shirase drainage basin:

For this domain, we solved Eqs. (

The nonlinearity of the model equations was dealt with by Picard iteration as
in

In order to infer an initial spatial distribution of the temperature
field that contains a historical footprint of ice sheet evolution
during glacial cycles, a spin-up of the whole ice sheet is generally
needed. However, this procedure does not always produce a distribution
in satisfactory agreement with present conditions, particularly at the
ice base

Using the temperature field computed by solving Eq. (

The obtained present-day state of the SDB served as
initial condition for runs into the future. We used a subset of the
SeaRISE experiments defined in

Experiment CTL (“constant-climate control run”) starts at the present (more precisely, the epoch 1 January 2004 0:00
corresponding to

Experiment S1 (“basal sliding experiment”) is constant-climate forcing with increased basal lubrication. This was implemented in Elmer/Ice (for both FS and SIA) by halving the basal friction coefficient (approximately doubling the basal sliding) everywhere in the domain.

Experiment C2 (“surface climate experiment”) is

As described previously (Sect.

L curve obtained with the control inverse method: cost function

Figure

The spatial pattern of the basal friction coefficient obtained by the
optimization (Fig.

An initial velocity distribution was also computed with SIA, using the basal
friction coefficient inferred from FS and the control inverse method
(Fig.

These experiments with evolving ice surface were carried out with
the previously computed present-day state as the initial condition.
For both FS and SIA, the distribution of the basal friction coefficient
obtained by FS and the control inverse method was used
(Fig.

Figure

For both FS and SIA, all scenarios produce a volume loss over the 100 years of model time. However, it is much smaller for FS than for SIA, so that the results are more dependent on the used model dynamics than on the scenario. The more rapid volume loss in the SIA experiments is certainly related to the larger flow velocities produced by the SIA for the initial state (see above).

The difference between FS and SIA becomes much smaller if we consider volume
changes relative to the respective control run (

S1 (basal sliding exp.) – CTL (control):

C2 (surface climate exp.) – CTL (control):

Computed as

Surface velocities

Figure

Surface velocities

The surface velocities and slip ratios obtained with SIA dynamics
are shown in Fig.

Our finding that the volume change of the Shirase drainage basin relative to
CTL is small for the surface climate experiment C2 is consistent with the
SeaRISE findings. For both the entire AIS

Especially in the fast-flowing region near the grounding line, the
Shirase Drainage Basin is characterized by a complex stress regime

Stress ratio

Figure

When starting transient numerical simulations of ice sheets, depending
on the employed initial conditions, spurious noise in the computed
velocity field

As described previously (Sect.

The alternative would have been to invert the SIA problem for basal friction
separately and run the SIA experiments with the result of this inversion. We
also attempted to do so. Due to the prescribed ice geometry, temperature
field and the local nature of the SIA flow field (Eq. 6), this is a
straightforward exercise that does not require the control inverse method or
anything similarly sophisticated. However, the inversion failed to produce
meaningful results. For most parts of the domain, even for no-slip conditions
at the base the SIA produces surface velocities that exceed the observed ones
(Fig.

We have seen in Sect.

Further, we compared only the flow dynamics for grounded ice and assumed a fixed grounding line for Shirase Glacier. Therefore, we did not account for the potentially important impacts of grounding line migration and ice shelf buttressing on the dynamics of the system. Such effects are beyond the scope of the SIA and require at least some flavor of higher-order dynamics.

We compared two approaches to represent ice flow dynamics for the Shirase Drainage Basin, namely the FS formulation and the SIA, implemented within the same dynamic/thermodynamic ice flow model (Elmer/Ice). The complex nature of the stress regime in the drainage basin allows a good characterization of the differences in the evolution and dynamics of the area resulting from the two approaches. In the first step, we applied an inverse method to infer the distribution of the basal friction coefficient with FS. We then compared FS and SIA by assessing the respective response of the drainage basin to different climatic and dynamic and forcings.

There were evident differences in the computed surface velocities between the two approaches. The surface velocities computed with FS showed a distinct, well-defined fast-flowing area near the grounding line. A similar flow feature is observed in current surface velocities, essentially coinciding with Shirase Glacier. In contrast, the SIA produced a less well-defined contrast between the narrow fast-flowing region of Shirase Glacier and the surrounding, slower-flowing ice; the zone of fast flow is distributed over a larger area. In general, the SIA overpredicted the surface velocities everywhere in the domain, which is a consequence of the neglected longitudinal stresses and horizontal shear stress that can generate an efficient resistance to ice flow. Consequently, in transient scenarios, SIA runs consistently produced smaller ice volumes than FS runs. However, when considering ice volume evolution relative to a control run, the difference between the FS and SIA results was not overly large. Nevertheless, our findings show clearly that FS is superior to the SIA in modeling the ice flow in the area, in particular in fast-flowing regions with high slip ratios.

In this study, we considered grounded ice only and kept the grounding line fixed. A desirable extension would be to include floating ice (the small ice shelf attached to Shirase Glacier) and compare FS to coupled SIA and shallow shelf approximation dynamics within the same model. This would reveal whether the complex interactions between grounded and floating ice, including grounding line dynamics, lead to further differences in the response of the system to external forcings.

The model Elmer/Ice is part of the open-source multiphysical simulation software Elmer and accessible via Elmer/Ice Project (2017). The data produced by Elmer/Ice for this study are available from the corresponding author upon request.

The SIA is implemented into the finite element method by solving the
degenerated Poisson equation

The authors declare that they have no conflict of interest.

We wish to thank Ayako Abe-Ouchi (Univ. Tokyo) and Fuyuki Saito (JAMSTEC Yokohama) for helpful discussions, and Olivier Gagliardini (Univ. Grenoble Alpes) for his contributions to the SIA solver in Elmer/Ice. Further, we are grateful for helpful suggestions from the scientific editor Eric Larour and the anonymous reviewers.

Hakime Seddik, Ralf Greve and Shin Sugiyama were supported by Japan Society for the Promotion of Science (JSPS) KAKENHI grant number 22244058. Hakime Seddik and Ralf Greve were supported by JSPS KAKENHI grant number 25241005. Further, Hakime Seddik was supported by a JSPS Postdoctoral Fellowship for Overseas Researchers (Pathway to University Positions in Japan) (no. PU15902, associated JSPS KAKENHI grant number 15F15902), and Ralf Greve was supported by JSPS KAKENHI grant number 17H06323. Edited by: Eric Larour Reviewed by: four anonymous referees