Ice fabrics – the distribution of crystal orientations in a polycrystal – are key for understanding and predicting ice flow dynamics. Despite their importance, the characteristics and evolution of fabrics produced outside of the deformation regimes of pure and simple shear flow has largely been neglected, yet they are a common occurrence within ice sheets. Here, we use a recently developed numerical model (SpecCAF) to classify all fabrics produced over a continuous spectrum of incompressible two-dimensional deformation regimes and temperatures. The model has been shown to accurately predict ice fabrics produced in experiments, where the ice has been deformed in either uniaxial compression or simple shear. Here we use the model to reveal fabrics produced in regimes intermediate to pure and simple shear, as well as those that are more rotational than simple shear. We find that intermediate deformation regimes between pure and simple shear result in a smooth transition between a fabric characterised by a girdle and a secondary cluster pattern. Highly rotational deformation regimes are revealed to produce a weak girdle fabric. Furthermore, we provide regime diagrams to help constrain deformation conditions of measured ice fabrics. We also obtain predictions for the strain scales over which fabric evolution takes place at any given temperature. The use of our model in large-scale ice flow models and for interpreting fabrics observed in ice cores and seismic anisotropy provides new tools supporting the community in predicting and interpreting ice flow in a changing climate.

Mass loss from ice sheets is set to be the main contributor to sea level rise this century

To date, the analysis and discussion of ice fabrics has focused primarily on those formed under the specific deformation regimes of uniaxial compression and pure shear (both irrotational deformations), as well as simple shear. However, these regimes represent isolated points in a parameter space of deformation regimes that occur in nature.

The objective of the present paper is to use the fabric evolution model SpecCAF

To summarise, in this paper we seek to address a number of open questions. First, what fabrics are produced under any given (incompressible) two-dimensional deformation regime? Second, how do these fabrics change over the space of increasing vorticity and temperature, and can we use this information to aid in interpreting ice cores? Third, how do fabrics evolve at very high strains which have remained inaccessible to laboratory experiments, and at what strain does the fabric reach a steady state?

We address these questions by making use of a new continuum model SpecCAF

The distribution of crystallographic orientations within a polycrystal is called the fabric or crystallographic preferred orientation (CPO). The distribution of the

There are two main recrystallisation processes that affect the ice fabric. The first is

The second recrystallisation process is

Ice fabrics can be observed through laboratory experiments, through ice cores from real-world locations, and, more recently, inferred through radar and seismic measurements. In the laboratory, the majority of experiments are performed by compressing a block of ice, resulting in an irrotational deformation: either pure shear if the block is confined in one direction or uniaxial compression otherwise

Experiments have been performed for deformations intermediate to pure and simple shear, at temperatures close to the melting point of ice

Fabrics can also be analysed by taking ice cores in ice sheets. A detailed understanding of the fabrics produced over possible deformations and temperatures enables us to interpret the deformation regime and temperature history of ice cores. Initial studies of ice cores have concentrated on ice domes or divides

Recently, data from radar and seismic data have also been used to infer fabric properties

Illustration showing common fabrics or fabrics which develop in ice, illustrated by their pole figures, as well as the deformation regime and temperature they typically occur at. The pole figures show the distribution of

In experiments and observations a number of common fabric patterns occur (Fig.

Recent experiments in simple shear produced either a single maximum at low temperatures or a single maximum with an offset secondary cluster (Fig.

There exists a significant variety of deformation regimes in the natural world. One way to classify a deformation regime is by the vorticity number

As a note for people unfamiliar, in Eq. (

Figure

Schematics illustrating two-dimensional flow regimes at different vorticity numbers

As a first step towards exploring the fabrics produced by all possible deformation regimes, we will focus here on general incompressible two-dimensional deformations. Although deformation regimes in the natural world will be three-dimensional, exploring fabrics produced by two-dimensional deformation regimes is a natural first step away from the canonical regimes of pure and simple shear. It is also common to limit the modelling of ice sheets to two dimensions, either in the vertical cross section

In order to illustrate the range of vorticity numbers which are expected to occur in natural flows, we explore here a number of scenarios. For an ice divide, the simulation shows that the vorticity number varies smoothly between

The vorticity number for ice flowing at a divide, showing a range from

The vorticity number for ice flowing over a Gaussian bump, performed in Elmer/ICE

If we consider a 2D flow of ice over a Gaussian bump (Fig.

The above examples focused on the two-dimensional flow in vertical cross sections of ice-sheet simulations. To further explore the occurrence of vorticity numbers away from 0 or 1 in natural flows, we calculate an estimate of the vorticity number in the horizontal flow near the surface of the Antarctic Ice Sheet. To do this, we use surface velocity data from Antarctica

The resulting prediction for the vorticity number shown in the map of Fig.

Figure to illustrate the range of vorticity numbers near the surface in Antarctica. This is calculated from the surface velocity data of Antarctica

The effects of deformation on the dynamics of ice cover a vast range of scales from the order of micrometres for studying grain–grain interactions to continental scales of 1000s of kilometres when studying ice sheets. Consequently, different approaches must be used depending on the scale one seeks to work on, with micro-scale models serving to provide parameterisations of small-scale processes for use in larger-scale models. At the scale of micrometres and millimetres, there exist several approaches for modelling the microstructure directly

In this contribution, we use the SpecCAF model from

We note that, by itself, SpecCAF models fabric evolution only for given applied deformation. It does not include a viscosity formulation like the CAFFE model of

In Sect.

List of mathematical symbols used in this paper, including units and the equation if they are explicitly defined.

We begin by reviewing the SpecCAF model, which is developed, experimentally calibrated, tested and solved in

The essential continuum approach was proposed previously by

Despite the model not including the Taylor hypothesis, the term for basal- slip deformation in the equation below is similar to that which would be derived from a Taylor homogenisation of ice under a simple basal slip only model

The evolution equation for

The parameter

To apply Eq. (

Equation (

The model parameters

As a preliminary illustration of the model output and its representation, we show in Fig.

As an example comparison of the model prediction and experimental observations, we have included a pole figure from laboratory experiments

To illustrate the fabric we show a simulation in simple shear at

We explore fabric evolution across a complete, continuous range of vorticity numbers

We define the strain as

Figures S1 and S2 in the Supplement show that variations in the parameters from Fig.

We also show in Fig.

To further analyse the limit of very large vorticity numbers we show the fabric produced as

Slices of the pole figure showing the value of the orientation distribution function

Pole figure for

To distil all the complex information shown in Fig.

Figure

Figures S5 and S6 in the Supplement show Fig.

The angle between the primary cluster and the closest principal axis of deformation (i.e. the axis of compression) is shown in Fig.

Figures S7 and S8 in the Supplement show Fig.

To illustrate the difference in pole figure patterns at the same finite strain but different temperatures and deformation regimes we plot pole figures at a finite strain of

Regime diagram of the different fabric patterns which occur (defined in Fig.

Contour plots showing the angle (in degrees) of the largest cluster from the compression axis. Panels are shown for progressively increasing finite strain values. The resolution of this figure is

Pole figures overlaid onto the regimes at

A variable that is central to the interpretation of ice core fabrics is the timescale or, equivalently in the non-dimensional problem, the finite strain over which fabric evolution occurs. In Fig.

Figure

Figure

Figures S11 and S12 in the Supplement show Fig.

Properties of steady-state fabrics across the

The analysis presented here gives predictions for the fabric patterns produced over the whole range of vorticity numbers and temperatures arising for incompressible two-dimensional deformation regimes, a first for fabric modelling. We have limited the analysis here to fabrics produced under a constant deformation regime and temperature. Although ice in the natural world will undergo changing deformation regimes, our analysis is a first step to provide insights into fabrics produced for deformation regimes away from pure and simple shear. Furthermore, the fabrics analysed here are highly relevant for ice deformed in the laboratory, which is in most cases deformed at constant temperature and vorticity number.

Ice in the real world will undergo three-dimensional deformations yet it is common to model ice sheets in two dimensions: either along the vertical cross section, such as in Fig.

Previous work has focused on modelling fabrics produced at single deformation regimes

Our work generally shows a smooth transition between the two deformation regimes of pure and simple shear, as can be seen in Fig.

The weak fabric seen for highly rotational flows had not previously been studied at all to date. This result is interesting as it reveals for the first time the fabric produced by rotational deformation regimes: a weak girdle fabric. Throughout these deformation regimes we have kept the magnitude of

The regime diagram in Fig.

Experimental results from

For the interpretation of ice fabrics, it is essential that we know the timescale (or in the non-dimensional case here, total finite strain) over which fabrics evolve to steady state. In this paper we have presented the first assessment of fabric timescales (i.e. strains), by examining the “half-life” for fabrics to reach steady state (Fig.

Figure

Analysis and interpretation of ice cores remain key for understanding the processes occurring in the natural world, for both understanding the past climate history

The fabric pattern is a robust way to interpret ice cores, as it requires no assumption about the deformation regime direction. The regime diagram in Fig.

If other information about the fabric history is known, Figs.

Viscous anisotropy of ice is controlled by the fabric and is a key control of the flow field

Our analysis of fabrics here can also give insight into where anisotropy may be most important in an ice sheet. Areas with strong fabrics will be highly anisotropic, with the viscosity varying in different directions. Anisotropic flow is not well studied but initial simulations with coupled anisotropic flow show that it can explain hitherto unexplained observations such as syncline patterns observed under ice divides

Accurately predicting ice fabric evolution is pivotal for the correct interpretation of ice core fabrics as well as the reliable prediction of ice losses in a changing climate. Our work extends the ability to predict fabric evolution from the deformation regimes of pure and simple shear to general two-dimensional deformation regimes. This represents a step towards understanding fabrics in fully general conditions: key for understanding viscous anisotropy and, in turn, large-scale ice-sheet flow modelling. We have shown that deformation regimes outside of pure and simple shear are important in common flow scenarios seen in ice sheets. Future work could use the modelled fabric to predict seismic properties, to compare to real world observations.

The regime diagrams presented are a useful tool to help with the interpretation of ice core data. In combination with other information, such as the plane of deformation regime or an estimate of the temperature at which a core was deformed, these regime diagrams can be used to determine the primary deformation regime and temperature undergone by an ice core. We show that for two-dimensional deformations, the double-maximum fabric is not present at high strains when only a small amount of vorticity is present in the deformation regime

Highly rotational deformation regimes were investigated for the first time and showed a weak girdle fabric aligned to the axis of vorticity. We have also shown how the timescale for fabric evolution, shown by the halfway strain to reach steady state, changes over the parameter space. Laboratory experiments around simple shear may be able to reach the strains required to get close to steady state; however compression experiments, especially at low temperatures, cannot achieve the required strains for steady state.

Our predictions of ice fabric evolution over a wide range of deformation regimes provide insights into how and where viscous anisotropy will be important for ice-flow dynamics. Intermediate deformation regimes between pure and simple shear produce the strongest fabrics, suggesting anisotropy will be most important in these regions. Similarly, as highly rotational deformation regimes produce a weak fabric, there is likely to be a less dominant effect of anisotropy in such regions. Our understanding of these issues could be further improved by combining our model with an anisotropic viscosity formulation to model the coupled fully anisotropic flow of ice. This is an important future step for accurately predicting ice flow.

A Python implementation of SpecCAF, alongside code to reproduce the figures in the Results section, is available at

The surface velocity data of Antarctica

The supplement related to this article is available online at:

All authors designed the research and edited the manuscript. DHR developed the model code, performed the analysis and visualisation and wrote the draft.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Sérgio Faria for his email correspondence and review which helped to improve this article. We also thank Ed Waddington, Maurine Montagnat, Fabien Gillet-Chaulet, and one anonymous reviewers for their helpful and insightful reviews which also improved the manuscript. Finally we thank Kaitlin Keegan for her editorial handling.

This research has been supported by the Engineering and Physical Sciences Research Council (grant no. EP/L01615X/1).

This paper was edited by Kaitlin Keegan and reviewed by Maurine Montagnat, Fabien Gillet-Chaulet, Sérgio Henrique Faria, Edwin Waddington, and one anonymous referee.