The microstructure of polycrystalline ice evolves under prolonged deformation, leading to anisotropic patterns of crystal orientations. The response of this material to applied stresses is not adequately described by the ice flow relation most commonly used in large-scale ice sheet models – the Glen flow relation. We present a preliminary assessment of the implementation in the Ice Sheet System Model (ISSM) of a computationally efficient, empirical, scalar, constitutive relation which addresses the influence of the dynamically steady-state flow-compatible induced anisotropic crystal orientation patterns that develop when ice is subjected to the same stress regime for a prolonged period – sometimes termed tertiary flow. We call this the ESTAR flow relation. The effect on ice flow dynamics is investigated by comparing idealised simulations using ESTAR and Glen flow relations, where we include in the latter an overall flow enhancement factor. For an idealised embayed ice shelf, the Glen flow relation overestimates velocities by up to 17 % when using an enhancement factor equivalent to the maximum value prescribed in the ESTAR relation. Importantly, no single Glen enhancement factor can accurately capture the spatial variations in flow across the ice shelf generated by the ESTAR flow relation. For flow line studies of idealised grounded flow over varying topography or variable basal friction – both scenarios dominated at depth by bed-parallel shear – the differences between simulated velocities using ESTAR and Glen flow relations depend on the value of the enhancement factor used to calibrate the Glen flow relation. These results demonstrate the importance of describing the deformation of anisotropic ice in a physically realistic manner, and have implications for simulations of ice sheet evolution used to reconstruct paleo-ice sheet extent and predict future ice sheet contributions to sea level.

An essential component of any ice sheet model is the constitutive relation
(or flow relation), which connects ice deformation rates and applied
stresses. Under prolonged deformation polycrystalline ice aggregates develop
material anisotropy, patterns of preferred orientations of individual crystal

There are complex flow relations that explicitly include material anisotropy,
and models that track the evolution of crystal fabrics, as discussed briefly
below, but the Glen flow relation

Here,

To account for the increased deformability associated with steady-state
creep, a common adaptation of the Glen flow relation is the inclusion of a
constant flow enhancement factor,

Such a parameter is included in most large-scale ice sheet models

Here, as a first step towards exploring the implications of this description
of anisotropic ice in large-scale ice sheet models, we describe how to
implement the ESTAR flow relation and apply the required changes to the Ice
Sheet System Model

In this section we outline the development of anisotropic fabrics in polycrystalline ice, including the tertiary flow regime and its connection with enhanced deformation rates and the development of compatible anisotropy, and discuss the expected occurrence of anisotropy and tertiary flow conditions in polar ice sheets.

Individual ice crystals have a strong mechanical anisotropy, owing to high
levels of deformability due to slip on the crystallographic basal plane,
whose normal is the crystallographic

As discussed by

There are exceptional locations where this concept of compatibility is likely
to break down – where the stress regime experienced by the flowing ice
alters rapidly. Examples include transitions from tributary glacier or sheet
flow into the shear margins of ice streams or ice shelves

In their review,

Schematic illustrating strain rate characteristics of
polycrystalline ice undergoing deformation driven by single stresses as
measured in laboratory experiments. The part of the curve corresponding to
tertiary (steady-state) anisotropic creep is relevant to the deformation of
ice masses in typical ice sheets and glaciers. The red (blue) curve
illustrates the result of simple shear-alone (compression-alone) stress
configurations. The ratio of the shear enhancement factor

Experimental observations for pure polycrystalline ice, demonstrate that an
accumulated strain of

There are a variety of microdeformation and recovery processes that lead to
the development of anisotropic fabrics

Information about the effects of anisotropy on in situ deformation rates in
polar ice sheets is limited. Analyses of the shear strain rate profiles
inferred from borehole inclination measurements on Law Dome

While the relevant temperature regime and the amount of strain that needs to
be accumulated remains uncertain, tertiary creep, with the associated
development of polycrystalline anisotropy, may be common in polar ice sheets,
particularly in regions controlling the large scale dynamics, as discussed
further in Sect.

A range of constitutive relations have been proposed to
account for polycrystalline anisotropy. They can be broadly grouped in two
categories

Experiments on single crystals of ice demonstrate that
deformation occurs predominantly by slip on the crystallographic basal plane
(perpendicular to the

The complexity of resulting flow relations varies according to the extent to
which a physically realistic description of microdeformation and recovery
processes, or a parameterisation of these, enters into the relationship
between strain rates and the stresses driving deformation. Many of these
constitutive models are more complicated than a collinear flow relation and
involve a tensor coupling in place of Eq. (

Including any fabric-based description of material anisotropy in the flow
relation for an ice sheet model requires either a prescription of anisotropy
or an additional set of equations governing the fabric evolution. A
complication of such an approach is the computational overhead and
uncertainty associated with defining the spatial distribution of fabric
within ice sheets, which is poorly constrained by observations. Sometimes, as
a simplification, restricted forms of the ODF or orientation tensor are
specified, which may not adequately describe all fabrics likely to be
encountered in an ice sheet. To date, flow relations utilising a fabric
description that relies on fabric evolution equations or that is imposed as a
function of location within the ice sheet have been restricted to regional
simulations

As indicated in Sect.

An empirical approach to the deformation properties of ice with a tertiary
polycrystalline anisotropy has developed, comprising experimental and
observational studies

A flow line model by

The generalised tertiary flow relation proposed by

Here, we summarise the generalised constitutive relation
for ice in tertiary flow (the ESTAR flow relation) proposed by

The main features of

Recasting Eqs. (62) and (63) of

Assuming

Here,

The collinear nature of the ESTAR flow relation
Eq. (

Analysis of tertiary creep rates for experiments conducted in simple
shear-alone and compression-alone suggests that a suitable ratio of

If the enhancement parameters are selected so that

Comparisons of simulations of ice sheet dynamics using the ESTAR and Glen
flow relations will be influenced by the choice of the Glen enhancement
parameter,

A caveat is that as stated earlier, for the ESTAR flow relation to hold, the assumption of the tertiary state (i.e. crystallography and deformation rates being compatible with the instantaneous stress/deformation regime) requires that this does not change too rapidly along the flow. That is to say, for a compatible (tertiary) anisotropy to be present, the present deformation regime needs to be a suitable indicator of the recent strain history of the flowing ice.

As discussed in Sect.

Within an ice sheet there will be zones where the assumption of compatible
tertiary flow will not apply; however, these zones will be restricted in
extent

Regions where rapid transitions in dynamic conditions can lead to abrupt
changes in the pattern of applied stresses and a potential breakdown in
tertiary flow compatibility include ice shelf grounding zones and other
locations where basal traction is lost or abruptly changes, e.g. where ice
flows over a subglacial lake, or with the onset of basal sliding in ice
streams. The convergence zones where tributary glaciers or ice streams merge
with a larger flow unit at a high angle

Under the very cold and low stress conditions occurring in the uppermost
layers of the polar ice sheets, particularly towards the interior at high
elevations, any increase in the accumulated strain necessary to develop a
compatible fabric may lead to a near-surface zone in which the assumption of
tertiary creep is not valid. Since the development of anisotropic fabrics
provides an indication of the existence of, or the approach towards tertiary
flow, the observation of evolving anisotropic fabrics at modest depths,

We conclude this section with some remarks about the seeming paradox of using
an isotropic constitutive relation to describe the deformation of ice that
has an anisotropic pattern of

Anisotropy in broad terms describes differences in physical systems
associated with different directions. The various flow relations in

In materials science, anisotropy is used to refer to material properties
which have different values when measured along different directions. Indeed,
the term is often introduced

Microstructural approaches to ice deformation, such as those discussed in
Sect.

In contrast, the applicability of the flow relation we are implementing from

The general flow relation constructed by

As

The magnitude of the shear strain rate defined on the
local non-rotating shear plane,

The last projection term in Eq. (

The vorticity of a flow, whether viewed as the anti-symmetrised part of the
velocity gradient tensor or as the usual vector

From Eq. (

This vector contains the correct perpendicular component

In the present implementation of the ESTAR flow relation, we assume that
swirling effects are small for flows with the relevant spatial scales, aspect
ratios etc., which can be verified from the modelled flow-fields in our test
cases, and hence

The description of the ESTAR flow relation above is implemented in ISSM for
the full-Stokes model of flow. We also extended the implementation to ISSM
versions of the higher-order three-dimensional model of

We perform convergence tests in order to verify the
implementation of the ESTAR flow relation within the ISSM full-Stokes and
higher-order models. The objective of these tests is to compare the model
results to analytical solutions for different mesh resolutions. As the mesh
becomes finer, the error between the model and the analytical solution (i.e.

We designed our analytical solutions by considering a three-dimensional,
grounded, isothermal ice slab of unit dimension lying on a flat bed
topography, with cartesian coordinates

In the case of the higher-order model,

Here, the deviatoric stress fields are calculated using the ESTAR flow
relation as specified in Eq. (

Convergence rates of the simulated

To test the numerical implementation ISSM is forced using the analytical
expressions for

The ESTAR flow relation was applied to a suite of test
cases. The first case we present simulates flow in an embayed ice shelf; the
second two are based on experiments from the Ice Sheet Model Intercomparison
Project for Higher Order Models

The ISMIP-HOM experiments were diagnostic. In contrast, we have taken the
same geometries and boundary conditions, which are already familiar to the
modelling community, but allowed the velocity, surface, and thickness fields
to evolve to steady state, as defined in the corresponding sections below. We
choose to present steady-state results from prognostic simulations based on
ISMIP-HOM experiments (supplemented by prescribing zero accumulation or loss
of ice) using the Glen and ESTAR flow relations because in this situation the
ice sheets, the flow fields, and stress regimes are steady in time. This is
more in keeping with the assumptions underlying the ESTAR flow relation than
a simple diagnostic experiment for a prescribed geometry. It is also of
interest to see the differences in the dynamic evolution of the systems
resulting from the different material constitutive relations. While our focus
is on the differences between the results for the ESTAR and Glen flow
relations, the latter results provide a direct extension to the original
ISMIP-HOM experiments presented in

As mentioned above, we use shear and compression enhancement factors of

The first prognostic experiment simulates
three-dimensional flow through a rectangular embayed ice shelf. The
experiment was carried out for model domains with transverse spans

Rectangular ice shelf higher-order steady-state surface fields.

We run the higher-order ice flow model for each of the ESTAR and Glen flow
relations to steady state, which we define to be reached when the mean
velocity change over the surface mesh points is less than

The Glen and ESTAR higher-order steady-state surface velocity magnitudes are
compared in Fig.

The steady-state thickness patterns for each flow relation, and their ratio
are shown in Fig.

Rectangular ice shelf ESTAR higher-order steady-state surface strain
rates (s

The ESTAR strain rate components are presented in
Fig.

Rectangular ice shelf ESTAR higher-order steady-state surface
fields.

Computation times for
simulations of the higher-order embayed ice shelf model
(Sect.

Computation times for serial and parallel simulations with the higher-order
model for the embayed ice shelf, using each flow relation and for increasing
number of processors, are summarised in Table

To check the computational demands of the full ISSM model with the ESTAR flow
relation, the full-Stokes ice flow model was computed for one model year
(i.e. steady state had not yet been reached) and the results compared with
the higher-order simulation results for the same model period (results not
shown). Use of the ESTAR flow relation increased wall times by

ISMIP-HOM experiment B (ISMIPB) describes two-dimensional
flow (

ISMIPBp full-Stokes steady-state results with horizontal extent

The ESTAR and Glen full-Stokes prognostic steady-state horizontal velocities
(

The steady-state surface elevations for the Glen and ESTAR flow relations are
everywhere within

Figure

In order to examine the dynamics giving rise to the high shear-dominance
peaks in Fig.

As for Fig.

Away from the transition curves

We also investigated the impact of reducing the horizontal extent to

ISMIP-HOM experiment D (ISMIPD) describes a
two-dimensional domain over which the basal friction coefficient

The bed topography and the initial ice surface are inclined planes with a
slope of 0.1

ISMIPDp full-Stokes steady-state results with horizontal extent

The steady-state Glen and ESTAR results when

The ISMIPDp steady-state surface elevations for each flow relation are shown
in Fig.

The ESTAR component strain rates and shear fraction

As for Fig.

Steady-state simulation results for the smaller aspect ratio
(

The pattern of deformation regimes mapped by

The spatial variations in the viscosity ratio (Fig.

In this study we conducted various ice flow simulations, comparing the ESTAR and Glen flow relations. The ESTAR flow relation incorporates the observed differences between tertiary deformation rates for shear dominated and normal stress dominated stress regimes.

Our simulations of embayed ice shelf flow showed that no single Glen
enhancement factor (

These results highlight one of the key failures of the Glen flow relation: an
inability to account for complex, spatially varying stress regimes in its
prescription of ice flow. The addition of an enhancement factor

The modified ISMIP-HOM experiments B and D simulated scenarios in which the
dominant control of flow was bed-parallel simple shear. In the prognostic
runs with the larger aspect ratio (

For more rapidly varying bed topography in ISMIPBp, with

These results suggest that if major bed topography varied only on scales much
longer than the ice thickness, close agreement between simulations using the
ESTAR and Glen flow relations might be achieved more generally by choosing
the tertiary shear enhancement factor as the Glen enhancement factor
(

Our idealised test cases also provide some insights into the validity of the
tertiary flow assumption underlying the ESTAR flow relation, and the
development of anisotropic crystal fabrics compatible with the current
deformation regime. In the embayed ice shelf test the most significant change
in the deformation regime is clearly the transition to extensional flow on
approach to the ice shelf front. The contours of

The ISMIP-HOM experiments reveal potential violations of the tertiary flow
assumption, although the significance for the flow field of these apparent
short-comings needs to be assessed with regard to the somewhat artificial
nature of the tests. Indeed as we saw, the difference between the results of
the ESTAR and Glen flow relations (provided

The ISMIP-HOM experiments have a spatial periodicity, which could allow one portion of the repetitive basal conditions to dominate the overall flow. Also, there is no surface mass budget in these experiments so that, as remarked earlier, the ice surface is a streamline, whereas in a system with surface accumulation fresh snow is always being added and advected down into the ice sheet where it makes the transition to solid ice. Accordingly, in the flow regime of these prognostic experiments even the surface layers would be regarded as having developed some anisotropy just as the lower layers would, since they have in principle been deforming over an arbitrarily long time.

The main issue about the establishment of tertiary flow conditions in the periodic environment of our ISMIP-HOM experiments concerns the possible cycling of the flowing ice through a variety of stress regimes. This leads to transition regions where the stress regime and presumably the crystal anisotropy would be evolving, and the compatibility assumptions behind the ESTAR flow relation would locally be violated. Clearly the spatial extent of transitional flow and the delay in attaining any new tertiary state depends on the magnitudes of the strain rates and the velocity of the ice. By combining these with a threshold for accumulated strain as the criterion for development of a compatible (tertiary) fabric under a persistent flow regime, the extent of a transition zone can be estimated. This scale can then be compared to the horizontal variation of the stress regime. Selecting the 10 % strain required to develop a compatible anisotropy from initially randomly oriented ice should provide a conservative yardstick, when applied to gradual changes in stress regime.

The patterns of stress regimes revealed by the distributions of

For ISMIPBp with

In the ISMIPDp case, for

For the last test, ISMIPDp with

The focus of this study was to explore the effect on the dynamic response of
ice sheets of using a constitutive relation appropriate to the tertiary flow
regime, i.e. sensitive to the varying proportions of simple shear and normal
stresses, compared to using the standard (Glen) flow relation. Our results,
particularly with respect to the differences between the Glen and ESTAR
simulations, are sensitive to the choice of

In order to examine the impact of a flow relation appropriate to ice with a
compatible flow-induced anisotropic crystal fabric on simulated ice dynamics,
the ISMIP-HOM and embayed ice shelf experiments were carried out assuming
isothermal conditions. However, as discussed earlier, real ice sheets and ice
shelves typically have cold, upper layers and strong vertical gradients in
temperature, and these will often be stronger controls on vertical contrasts
in deformation rates, through

We have investigated some consequences of
incorporating the flow properties of anisotropic ice into modelling flow in
ice sheets and ice shelves. Specifically, we have investigated the flow
response to prolonged deformation under a constant or slowly changing stress
regime and the associated development of an anisotropic crystal orientation
fabric compatible with that deformation, as represented by the empirical,
scalar, tertiary constitutive relation for ice with a compatible anisotropic
crystal fabric of

Our embayed ice shelf results have significant implications for ice sheet
model simulations that rely on the Glen flow relation to simulate past,
present, and future ice flow, which are used to constrain uncertainty in
reconstructions and projections of sea levels. In particular, the effect of
unrealistically fast thinning ice near the calving front, as simulated with
the Glen flow relation, is to deform the ice shelf, which could lead to
unrealistic ice shelf geometries and affect buttressing if it were to spread
beyond the “passive ice” sector

With the implementation of the ESTAR flow relation into ISSM completed, further investigation into its capacity to replicate real-world ice sheet flow in Antarctic outlet glaciers is currently underway.

The results from this work are reproducible using ISSM
(from version 4.11). The current version of ISSM is available for download at

The authors declare that they have no conflict of interest.

The authors thank the editor, Oliver Gagliardini, and each of the three reviewers for their comments that resulted in an improved manuscript. This work was supported under the Australian Research Council's Special Research Initiative for Antarctic Gateway Partnership (Project ID SR140300001), and the Australian Government's Cooperative Research Centres Programme through the Antarctic Climate and Ecosystems Cooperative Research Centre (ACE CRC). The University of Tasmania supported the visit of Mathieu Morlighem to Hobart. This research was undertaken with the assistance of resources from the National Computational Infrastructure (NCI), which is supported by the Australian Government. Edited by: Olivier Gagliardini Reviewed by: Edwin Waddington and two anonymous referees