Three-dimensional ice flow modelling requires a large number of computing resources and observation data, such that 2-D simulations are often preferable. However, when there is significant lateral divergence, this must be accounted for (2.5-D models), and a flow tube is considered (volume between two horizontal flowlines). In the absence of velocity observations, this flow tube can be derived assuming that the flowlines follow the steepest slope of the surface, under a few flow assumptions. This method typically consists of scanning a digital elevation model (DEM) with a moving window and computing the curvature at the centre of this window. The ability of the 2.5-D models to account properly for a 3-D state of strain and stress has not clearly been established, nor their sensitivity to the size of the scanning window and to the geometry of the ice surface, for example in the cases of sharp ridges. Here, we study the applicability of a 2.5-D ice flow model around a dome, typical of the East Antarctic plateau conditions. A twin experiment is carried out, comparing 3-D and 2.5-D computed velocities, on three dome geometries, for several scanning windows and thermal conditions. The chosen scanning window used to evaluate the ice surface curvature should be comparable to the typical radius of this curvature. For isothermal ice, the error made by the 2.5-D model is in the range 0–10 % for weakly diverging flows, but is 2 or 3 times higher for highly diverging flows and could lead to a non-physical ice surface at the dome. For non-isothermal ice, assuming a linear temperature profile, the presence of a sharp ridge makes the 2.5-D velocity field unrealistic. In such cases, the basal ice is warmer and more easily laterally strained than the upper one, the walls of the flow tube are not vertical, and the assumptions of the 2.5-D model are no longer valid.

Computing performance has continued to increase through the last decade, and
3-D numerical simulations that could not be performed a few years ago are
nowadays affordable. In particular, the Stokes equations can be solved on
large 3-D data sets

In this study, we describe the volume between theoretical stream lines as a
“flow tube” (Fig.

The

A particularly simple case is that of an axisymmetric circular dome, where
the width of the flow tube increases linearly along the flowline. In any
other case, there are several possible methods for determining the width of
the flow tube. At a large scale, the ice velocity can be determined via
interferometric synthetic-aperture radar data

The 2-D models that account for the width of the flow tube vary in the
complexity of their physics and their underlying assumptions

Different authors have used the modelling approach proposed by

No assumptions of the 2.5-D model specifically prohibit its use for a highly
diverging tube. However,

As a consequence, the applicability of the 2.5-D model should particularly be
examined on dome geometries, where simple 2-D models would be unable to
account for flow divergence. The flow tubes may widen by several orders of
magnitude on a few tens of kilometres, especially on the sharpest ridge of
the dome. The goal of this study is to perform several comparisons between
2.5-D and 3-D models, for various dome geometries and temperature conditions,
to answer the following questions.

What is the error associated with the computation of

How well can a 3-D state of stress be accounted for by the single
parameter

In future work, we hope to investigate a small dome on the East Antarctic plateau. As such, we presently consider a synthetic case with similar geometric and thermal conditions.

In order to investigate the influence of flow divergence, we model a ridge of
a dome, as this results in significant divergence. We perform the present
model comparison on a synthetic geometry which consists of a 15 km-radius
domain, whose shape is a quarter of cylinder only, for reasons of symmetry.
The initial thickness of the ice is 3239 m at the summit, the mean surface
slope is around 0.6/1000 and the underlying bed is flat. The space coordinate
is a

The 3-D mesh is horizontally unstructured and vertically extruded on 10
levels. The horizontal mean spacing between the nodes is 1 km
(Fig.

View of the two meshes used in this study. For each run, the 2-D mesh is extracted from the geometry of the steady-state solution of the 3-D simulation. BC1 to BC5 refer to the five boundary conditions of the 3-D case.

We denote the velocity vector

Description and values of the model parameters.

The three domes stabilized at steady state: surface contour lines
(spacing: 1

Since the 3-D mesh is a quarter of a cylinder, the conditions have to be set
on five different boundaries, numbered from BC1 to BC5
(Fig.

The coordinate system used by

The flowlines are perpendicular to the surface contour lines.

The directions of the horizontal velocity components are constant with depth, which implies that the walls of the flow tube are vertical.

There are no shear stresses on the vertical boundaries defined by the flow tube.

The ice deforms according to Glen's flow law.

These assumptions together mean that the surface horizontal strain is
transferred to the bottom, so that the surface contour lines and the
horizontal velocity in the flow direction impose the transverse stresses.
Such assumptions are reasonable in the centre of an ice sheet for a slowly
varying bed

The 2-D domain is taken as a vertical slice of the 3-D domain, on one of its
lateral boundaries (Fig.

We denote the width of the considered flow tube

An axisymmetric dome leads to the simple relations

The following sections present the equations of mass and momentum
conservation, modified to account for the divergence of the tube. As the
velocity field mainly depends on the input/output balance, some authors only
conserve the mass

At the ice divide, the velocity component

If the flow tube has a constant width, the value of

For a straight flowline, the horizontal shear strain rate is written as

Along a divide between drainage basins,

The boundary conditions are inherited from the 3-D case: no sliding, free
surface, vanishing velocity at the ice divide and an imposed horizontal
velocity profile downstream. Note that the value of the mean output velocity

The modified mechanical equations are implemented in the Elmer/Ice finite
element software

To determine the radius of curvature of the surface contour lines, we first
export a DEM from the surface nodes of the 3-D model. These nodes are fitted
using an inverse distance weighting, with a power of 4 in order to ensure a
good smoothing of the computed surface, representative of a real ice sheet.
For comparability with a real case, the spatial resolution of the DEM is
taken equal to 400

We first run the 3-D transient isothermal simulation, and stop when a steady
state is reached (

Horizontal velocity at the ice surface (

The absolute error in the ice velocity for an axisymmetric 2.5-D model
(

The computation of the radius of the surface contour lines is strongly
influenced by the size of the scanning window. For a circular geometry, the
variation of

Relative width of the flow tube

For elongated domes, we consider both the flowline along the sharpest ridge
(

Along a ridge, the flow tube is non-linearly diverging. For a given output width, the accumulation area is smaller than in the axisymmetric case, thus leading to lower output velocities.

With fixed geometries, it clearly appears that the velocity is underestimated
for elongated domes (Fig.

The downstream velocities are always quite accurate (10 % error), since they mainly depend on the tube surface calculation, incorporated into the velocity boundary condition. When releasing the surface, the surface slope slightly increases to accommodate the velocity boundary condition, and the computed velocity field is then closer to the 3-D reference. The relative error made in the downstream part of the flow is comparatively higher near the divide since the velocities are very small.

In the case of a sharp ridge, the ice surface has the shape of a circus tent
(Fig.

Height of the ice surface (m) for the 2.5-D free surface model,
along the ridge. Solid line: circular geometry,

For

Horizontal velocity field (m a

As the divergence is much smaller perpendicular to the sharpest ridge, the
velocity field is much smoother, and its spatial evolution closer to the 3-D
reference than the along-ridge case. The RMSE is 11.9 and 7.5 % for the
intermediate and large window size respectively, for

A supplementary comparison is carried out on the sharp ridge (

Width of the flow tube (m), for the isothermal (red) and
non-isothermal case (blue), for

This result also suggests that, on sharp ridges and with non-isothermal ice,
working with a fixed vertical profile of velocity will prevent such
unintended behaviour. This artefact may affect the results of

For reasons of continuity, the walls of the flow tube cannot be vertical in the direction perpendicular to the ridge either, but the effect is too weak to impact the computed velocity field.

A systematic comparison between 2.5-D and 3-D models has been presented in order to evaluate the ability of the former to accurately compute the velocity field on a small dome of an ice sheet. The error made when estimating the value of the radius of the surface contour lines is of the order of 10 % if the computation window is well chosen, though it can be comparatively higher close to the divide. The radius of curvature of the surface elevation contour lines should be determined with a sufficiently large computation window, but choosing the optimum size is not completely straightforward; in any case we suggest it should not be less than one-third of the maximum measured radius, and several windows should be tested to ensure the robustness of the results.

The 2.5-D model can be used without any specific restriction for tubes
diverging less than and up to an axisymmetric flow. For isothermal ice, the
model can be used with tubes diverging more than an axisymmetric flow, if the
divergence is not too high. For very high divergence, the ice is in our study
mainly pulled by the output boundary condition, and the resulted velocity
field may be somewhat irregular, and surface geometry unphysical close to the
divide. This means that, in the case of a sharp ridge, accounting for a
certain state of stress via a single parameter

For non-isothermal ice, the tube should not diverge more than axisymmetry, because the softer basal ice would be much more easily laterally strained in the case of an elongated dome. The walls of the flow tube are therefore not vertical, which violates the model assumptions, and the corresponding horizontal velocity profile may be not physical near the divide. This has significant consequences for dating purposes: as the computed velocity field shows too small values in the upper layers, the age of the ice is overestimated, and the modelled isochrone layers are too high. In the absence of a reference 3-D solution as produced here, the velocity field may not necessarily appear to be unphysical, but this does not mean that the numerical artefact does not significantly affect the age calculation.

This study shows that the use of a 2.5-D model, which is a trade-off between
2-D and 3-D, must avoid several pitfalls, and its applicability domain
appears to be significantly narrower than initially thought. The geometric
error, resulting from miscalculation of

The presented simulations were performed using the Elmer/Ice v.7.0 rev. 7016
finite element model. The source code of the 2.5-D model has been available
in the distribution since v.8.0 rev. d9d4a2f, implemented in the AIFlow
solver. The link to the source code is:

To check the correct implementation of the mass conservation in the 2.5-D
model, we hereafter compute the height of a Vialov profile corresponding to a
regularly diverging flow tube. Note that such a surface is only
representative of a single flowline, and not of a whole surface, as is
usually done for a Vialov profile in plane strain

Figure

The following reasoning is then similar to that of

This expression is consistent with the one previously derived for
axisymmetry. We then use this expression to control the 2.5-D model by
comparing the value of

We consider the same flow as in Appendix

As

The experiments were designed by Olivier Gagliardini, Frédéric Parrenin and Joe Todd. Olivier Passalacqua carried them out, helped by Catherine Ritz for analytical developments, and by Fabien Gillet-Chaulet for the Elmer/Ice implementation. Olivier Passalacqua prepared the manuscript with contributions from all co-authors.

We would like to thank the Editor D. Goldberg as well as D. Brinkerhoff and an anonymous referee for their frank comments which greatly improved the initial version of the manuscript. This project is supported by the Université Grenoble Alpes in the framework of the proposal called Grenoble Innovation Recherche AGIR. Edited by: D. Goldberg