A constitutive law for the grain gives the deformation ε˙′ of the grain under an applied deviatoric stress τ′, or vice versa, where dashes denote grain quantities. The most complete approach to the grain rheology is to consider the contribution of all possible slip systems e.g.. However, the models we discuss consider only dislocation movement along the basal plane. Therefore, with this assumption the most general model for grain rheology is to consider the grain as a transversely isotropic medium, which was first developed for metals by and first applied to ice by . Using the notation of the grain is described by two “enhancement factors” (Fig. ): An enhancement factor for deformation in the basal plane, Eca, and an enhancement factor for deformation normal to the basal plane, Ecc: 1ε˙cc′=η-1Eccτcc′,ε˙ca′=η-1Ecaτca′ where the subscript c is the direction of the c-axis, and a is any direction perpendicular to this (Fig. ), and η represents the isotropic response of the grain (i.e. some power-law). While for an isotropic material, Ecc and Eca would both be unity, for ice Ecc≈1, and Eca≫1, reflecting the ease of deformation along the basal plane. For a linear grain behaviour, the rheology is: 2ε˙ij′=η-1τij′+3(Ecc-1)-4(Eca-1)2τkl′ckclcicj+(Eca-1)(τik′ckcj+cickτkj′)-Ecc-12τkl′ckclδij where index notation is used and repeated indices imply summation. c is the c-axis and δij is the Kronecker delta. This rheology has been used previously , and simplifies to the grain rheology used by if the further assumption of keeping only terms of O(Eca) is made. It has been extended to the non-linear case in , however this introduces terms that make it computationally infeasible to include in a large-scale model. Consequently, all large-scale models which include grain dynamics have so far been limited to linear grains, and we do the same in this work. This does not preclude the use of power-law rheology at the large scale.

Equation () gives a constitutive law for grain behaviour linking the deformation ε˙′ and the stress τ′. We must now consider how the grain's orientation c changes under an applied deformation. In reality, a grain will always be embedded inside a wider network of grains. We consider a grain experiencing a known deformation ε˙′, as well as a known rotation, described by a rotation tensor ω′, describing the grain's rotation relative to a fixed reference frame. The total change in rotation of a grain is a combination of the bulk rotation and the deformation d′: 3dcidt=(ωij′+dij′)cj This rotation due to deformation is illustrated for an extensional deformation in Fig. . The diagonal lines represent basal planes, along which the ice deforms “like a deck of cards” . This induces a rotation. By assuming basal planes remain parallel under simple shear , d′ can be expressed as : 4dij′=cickε˙kj′-ε˙ik′ckcj combining Eqs. () and () gives: 5dcidt=ωijcj-(ε˙ij′cj-ε˙jk′ckcjci), If a model of transversely isotropic grains is used then the expression for grain rheology (Eq. ) can be substituted into Eq. () to express the lattice rotation in terms of grain stress and grain enhancements with no extra assumptions: 6dcidt=ωij′cj-Ecaη-1(τij′cj-τjk′ckcjci). The derivation of this can be found in Appendix .

To express Eqs. () and (), and hence Eq. () in macroscopic quantities, there are two end members available: either all grains deform at the same rate ε˙′=〈ε˙′〉=ε˙, or all grains experience the same stress τ′=〈τ′〉=τ . These two approximations, which act as the two bounds that the true grain behaviour will lie between, are called the Taylor and Sachs bounds respectively. Polycrystalline materials which deform close to the Taylor bound have many easy slip systems, such as some metals e.g. This means that dislocations can move in different directions, and the applied deformation can be easily accommodated by the grains. However, ice has only 1 easy slip system: the basal plane. When applied to ice, the Taylor approximation predicts around 60 % of the strain is due to non basal slip, and weak fabrics . Observations from ice sheets which predominantly show strong fabrics, and the fact that slip along the basal plane is around 70 times easier than other slip systems provide evidence that ice is deforming closer to the Sachs bound than the Taylor.

and derive anisotropic rheologies not by assuming an ice grain has a certain stress or deformation, but by assuming that the rheology has certain symmetries, specifically that it is invariant under reflections along symmetry directions of the fabric. As these rheologies work from assumptions about the fabric rather than the grain, they can be called macroscopic.

We now review other rheologies which have been used to attempt to approximate the anisotropic effect.

As the above macroscopic models are not tied to either the Sachs or Taylor bound, and often aim to give a response between them, they can postulate different equations for fabric evolution. first postulated a linear blend between the Taylor and Sachs approximations: 44vi=ωijcj-(1-α)(ε˙ijcj-ε˙jkckcjci)-αEcaη-1(τijcj-τjkckcjci), where α∈[0,1], with α=0 corresponding to the Taylor bound and α=1 corresponding to the Sachs bound. Although Eq. () seems to suggest that as Eca increase the contribution to lattice rotation from the stress increases. However, if the Sachs approximation and rheology is used, Eq. () shows that rewriting Eq. () in terms of the normalised stress gives: 45vi=ωijcj-(1-α)(ε˙ijcj-ε˙jkckcjci)-αι(τ^ijcj-τ^jkckcjci), with ι given by Eq. (). and used Eq. () with the GOLF rheology (Eq. ). However it should be noted that Eq. (), i.e. Eq. () with α=1, is derived from the Sachs approximation, which results in the leading Eca term, and which is cancelled out when considering the normalised stress τ^. It is more accurate to consider the expressions for the normalised stress τ^ if one is postulating this relationship with a rheology other than Eq. (), and then use these expression in Eq. () rather than Eq. (). Despite this, multiple studies have used values of ι=10 in Eq. () e.g.. Work by has also used a free parameter in front of the contribution from ε˙. Consequently, the most general form for v which captures all previous work is: 46vi=ωijcj-αD(ε˙ijcj-ε˙jkckcjci)-αSι(τ^ijcj-τ^jkckcjci), where ι is given by Eq. () and αD,αS≥0. This is the equation we use when modelling fabrics and testing the predictions of different rheologies. The lattice rotation v and hence fabric evolution implicitly depends on the rheology through the appearance of τ^ in Eq. ().

Fabric models have commonly been compared to observations at ice divides . To predict fabrics at a specific ice divide and hence compare to observational measurements from drilled ice cores, we need to approximate the deformation ε˙ as the input to the fabric evolution model. To do this a number of assumptions must be made. Firstly, it is assumed the core is drilled at a perfect ice divide, such that it only experiences unconfined compression: 47∇u=-ε˙zz/2000-ε˙zz/2000ε˙zz Secondly, The vertical shear rate ε˙zz is assumed to follow a Dansgard-Johnson profile , being constant for the upper two-thirds of the ice divide, and then decreasing linearly to zero at the ice-bedrock interface. Once this is assumed, the value of ε˙zz in the upper two-thirds, ε˙zz′ can be calculated: 48∫0Hε˙zzdz=ε˙zz′2H3+12ε˙zz′H3=a˙ where H is the ice thickness and a˙ is the accumulation. We make a further assumption that the values for H and a˙ are unchanged over time. For GRIP, H=3029 m and a˙=0.23 a⁻¹ . As we use the same parameters throughout the entire depth of the ice divide, we are implicitly assuming that the temperature and deformation rate stay approximately constant. The temperature at GRIP is between -31 and -20 °C above a depth of 2500 m, but increases rapidly over the deepest 500 m to -8 °C at the bedrock . Therefore, we exclude this region from our analysis.

To predict fabrics at the EGRIP drill site, located in the centre of an ice stream and where ice fabric measurements are available , we must estimate the three-dimensional path and deformation of the each “ice parcel” which ends up along the vertical depth of the drilled ice core (Fig. ). The ice parcel undergoes a changing history of deformations as it moves through the ice sheet.

The same methodology as is used to estimate the deformation. To leading order, we can assume that the ice flow around EGRIP obeys the shallow stream approximation , such that the ice flows with an approximate plug flow, with negligible vertical shear. This means that the horizontal velocities (and their corresponding horizontal gradients) can be applied into the ice sheet to leading order. These horizontal velocities are found using satellite velocity data .

Because the horizontal velocities are applied into the depth of the ice sheet, all the three-dimensional particle paths sketched in Fig.  follow the same surface path. Different initial locations along this surface path create three-dimensional particle paths that intersect the EGRIP drill site at different depths. At the starting point of each path on the surface, the vertical velocity can be found through the kinematic condition: 49ws=-a˙+us∂h∂s, with a˙ the accumulation rate, us the surface velocity in the streamline direction and ∂h/∂s the change in surface height in the streamline direction, calculated from surface height data . This can then be integrated forward to give a three-dimensional path from the surface to some depth at EGRIP. For each path we iteratively choose the accumulation rate such that the final age and depth of an ice parcel at EGRIP match the measured age-depth relationship .

The measured temperature along the EGRIP core is approximately -30 °C to a depth of 1400 m, – the depth to which the fabric data in this paper extends to. This temperature is similar to what is measured at GRIP at these depths . We assume this temperature-depth relationship is also valid upstream, along the particle paths. This means we can say the temperature is approximately constant throughout.

To re-summarise, there exists one equation for lattice rotation that contains all previously described models: 50vi=ωijcj-αD(ε˙ijcj-ε˙jkckcjci)-αSι(τ^ijcj-τ^jkckcjci), The deviatoric stress τ^ depends on the rheology used. Consequently, when the lattice rotation depends on the stress the fabric evolution becomes coupled to the rheology used. This means we can compare fabric predictions to ice cores to test the accuracy of different rheologies.

We review 6 different rheologies models: The Taylor and Sachs models derived from grain dynamics, as well as the 4 different macroscopic rheologies: GOLF, NLO, CAFFE and Glen's flow law.

The rheology of a grain, and consequently the large-scale, is described by two parameters Ecc and Eca controlling the increased deformation experienced by a grain in compression or shear (Eq. ). The parameters αD and αS control the contribution from deformation and stress to lattice rotation respectively in Eq. (). There are two parameters controlling recrystallization processes: λ and β controlling rotational and migration recrystallization respectively. This describes fully most rheologies. The NLO also depends on the power-law exponent n, and the CAFFE rheology (Eq. ) has unique maximum and minimum softening parameters Emax and Emin.

We non-dimensionalise the two recrystallization parameters by the macroscopic effective strain rate γ˙=ε˙ijε˙ji/2, such that λ=λ̃γ˙, β=β̃γ˙, as in .

While there are a large number of parameters, there are a number of physical constraints available to reduce the parameter space. The Taylor model requires αD=1,αS=0 in Eq. () (and hence does not involve a rheology), while the Sachs model requires αD=0,αS=1. The other 4 rheologies place no restriction on the values of αD and αS. Furthermore, there is no evidence to suggest that compression is easier or harder in the direction of the c-axis or perpendicular to it , which means Ecc=1. For Eca, finds a value of 104 best reproduces the expected polycrystal enhancement of 10. However we use Eca=102 as this is the maximum value the GOLF rheology is constrained against, and we aim to make comparisons between rheologies. This has only a minimal effect on the results here.

The GRIP ice divide (above a depth of 2500 m) and the EGRIP ice stream are at T<-20 °C and strain rates <10-11 s⁻¹. Migration recrystallization is known to be predominantly a high temperature or potentially also a low temperature high strain rate phenomena . As both GRIP and EGRIP are at low temperature and low strain rates (relative to laboratory strain rates), we do not think migration recrystallization will be significant for either location and set it to zero.

As it is the normalised stress which appears in Eq. (), most models do not depend on the value of n used. The exception is the non-linear orthotropic rheology, for which we use n=3. Furthermore, when rheologies have internal parameters we use those suggested by their authors. The non-linear orthotropic rheology depends on a Sachs-Taylor weight, which we use 98.75:1.25 following . The CAFFE model depends on parameters Emax and Emin, for which we use the values of 10 and 0.1 respectively, as is suggested in . While the CAFFE model does not depend on Ecc and Eca, we use the same value of ι in Eq. () for ease of comparison.

For the Sachs and Taylor models, this leaves only λ̃ to be constrained. For the GOLF, non-linear orthotropic and CAFFE rheologies 3 parameters must be determined: αD, αS and λ̃. Glen's rheology has only 2 free parameters: αD and λ̃ because ε˙ij=τ^ij.

For all models/rheologies the parameter space is explored and a set of best parameters are chosen (Table ), aiming to give the best fit for both locations. There is not one unique set of parameters which gives the unambiguous best fit for each model, but the parameters do show the best possible fit for each model/rheology.

The fabric predictions from the parameters shown in Table , for each model/rheology, are compared to observations from the GRIP ice divide and the EGRIP ice stream.

At the ice divide we assume the fabric undergoes unconfined compression as it moves vertically down. The measured largest eigenvalue from different depths at GRIP is shown in Fig.  as crosses . The same data points are shown on both subplots. As the fabric at an ice divide has rotational symmetry around the vertical axis, the two smaller eigenvalues are equal. Consequently, we only plot the largest eigenvalue.

Figure also shows the results from the suite of models/rheologies tested, plotted as solid lines. The figure has been split into two subplots to aid readability. The tensor rheologies are on the right-hand subplot. Because of the way we chose parameters, first finding a set that agreed well with this case and then improving the fit for the ice stream, all the models in the right subplot agree well with observations here. There is very little difference between the best predictions of the Sachs and GOLF. The NLO rheology has marginally worse agreement with ice cores above a depth of 1500 m.

In the left subplot of Fig. , the Glen rheology with αD=2.6, αS=0 and λ̃=0.32 also gives good agreement with the observed eigenvalues. For this rheology τ^=ε˙, therefore this parameter set could equivalently be expressed as αD=0, αS=1.06, λ̃=0.32, for the same value of ι=2.46 in Eq. (), as in the other models. This is a similar parameter set to that found for the Sachs model, highlighting that for this simpler case of the ice divide, it is not necessary to have an anisotropic rheology contributing to the fabric evolution to get reasonably accurate predictions.

The CAFFE model exhibits a sharp increase of the largest eigenvalue to ≈1 at a depth of around 1500 m. This is not seen in observations for GRIP but has been observed to an extent at other ice divides . We find that this sharp increase is sensitive to the value of Emin, though detailed exploration of this is beyond the scope of this work.

The flow at an ice stream is much more complex than an ice divide. Consequently, at each depth at the EGRIP ice core the ice there has experienced changing deformation history. In Fig.  we show the observed eigenvalues at EGRIP with depth as blue dots. Each eigenvalue is shown in a separate column. The data is also reproduced across each row to aid readability. Close to the surface at a depth of around 50 m, the observed eigenvalues at EGRIP have one larger eigenvalue (the vertical) and two smaller, indicating a single-maximum fabric produced as snow compresses into ice. We do not model this effect. Below this, the fabric is produced by ice stream deformation, and is characterised by one eigenvalue ≈0, and two larger eigenvalues, corresponding to a girdle fabric. This fabric is fairly invariant with depth below 550 m, with some noise.

We also show the results from the suite of models/rheologies tested in Fig. . The first row shows results from the Taylor, Glen and CAFFE models, and the second row shows the results from the tensorial rheologies. We do not include the effect of firn in our analysis, which significantly affects the observed fabric above a depth of around 550 m. Hence, we only compare the models to observations below a depth of 550 m. Although different initial conditions for the fabric could perhaps fit the observations better, we find that the results are insensitive to initial condition below a depth of 400 m (Fig. S13).

The Taylor model does not give good agreement with the EGRIP observations, showing one much larger eigenvalue of ≈0.8 over the depth we are interested in (below 550 m). The best parameters with Glen's rheology and the GOLF rheology also do not agree, not predicting that the smallest eigenvalue is much larger than what is observed. From this figure, the CAFFE rheology, Sachs rheology, and the NLO rheology give reasonably good agreement with observations, predicting a smaller eigenvalue close to zero and two larger eigenvalues.

To provide a more detailed comparison, Fig.  shows a comparison of modelled to observed eigenvalues in a depth-averaged sense. Observed eigenvalues from EGRIP are averaged over a depth of 550 to 1425 m, and plotted as dashed lines with the standard deviation shown by the shaded region. In this figure, the colour coding simply shows which eigenvalue (largest, middle or smallest) is being referred to. Then for each model/rheology (x-axis), we average the modelled eigenvalues over the same depth and plot them as as crosses, with the same colour coding as observations. Distance of the crosses from the respective dashed lines then allows us to assess agreement of models with observations in this deeper area of the ice core.

Figure shows that the CAFFE, Sachs, and NLO rheologies all have their predictions for the largest two eigenvalues within 1 standard deviation of the mean observed value at EGRIP. They also all have the smallest eigenvalue close to zero, with the Sachs model being closest to observations. This figure also shows that while Glen's and the GOLF rheologies with their respective best parameters are less accurate, they are both significantly more accurate than the Taylor model.

Previously, we have limited our analysis to solely the eigenvalues. We now extend this to examine the pole figures – showing the complete orientation distribution function (ODF) – directly in Fig. . The ODF represents the probability of finding a grain at each orientation. Brighter regions indicate a high probability of a grain being orientated towards that orientation. The space of possible orientations exist on the surface of a sphere. There is also no possibility of distinguishing a grain's top from bottom, so the ODF is symmetrical between its north and south hemispheres. Consequently, we plot the ODF over a single hemisphere, looking vertically downwards on the fabric. Figure  shows two pole figures observed from EGRIP on the left. While we have eigenvalue data over a large depth range, we only have pole figures for these two depths. There is variance between these two depths. The corresponding eigenvalues to these EGRIP pole figures are shown in Fig.  as black crosses, showing that these two pole figures represent end members of the variance seen at EGRIP. Despite this, they both show a girdle fabric pattern, indicating most grains have c-axes pointed along the bright band.

We also show pole figures for each model in Fig. . The Taylor model had one large eigenvalue and the other two ≈0. This corresponds to what is seen in the pole figure: a cluster in only 1 direction, which is repeated due to the symmetry of the ODF. Glen's rheology and GOLF rheologies both predicted the smallest eigenvalue was larger compared to observations, and this corresponds to a generally too diffuse pole figure, which was also seen in , which was the previous state-of-the-art for modelling ice stream fabrics.

The remaining models all provide improvements on this previous state-of-the-art. The pole figures for the Sachs model and NLO rheology lie within the variance of what is seen in the EGRIP pole figures, predicting girdle fabrics with a slight bias towards grains being orientated towards the edges of the pole figure. The CAFFE model also predicts a girdle fabric, though the detail features of the pattern in Figure show it is slightly different from what is seen at EGRIP, with peaks closer to the centre of the figure. These detail features can only be examined through looking directly at the pole figures.

These results are summarised in Fig. . Here we show the summed difference between the modelled and observed eigenvalues for each model/rheology. This error is then normalised against the results for Glen's rheology (so the results for Glen are by definition 100 %). This again highlights that the Sachs model has the closest agreement for both the ice divide and ice stream. Figure  also shows how the anisotropic rheologies provide the significant improvements in fabric predictions for the ice stream case (EGRIP) only.

In our previous analysis, as explained in Sect. , we use the present-day satellite velocity data to trace ice parcels upstream. However tracing a particle path upstream also means tracing a particle path back in time. Consequently, this method implicitly assumes the ice velocity field has not changed over the past 16 000 years (the time it takes for ice to travel from the furthest point upstream to EGRIP).

Recent work has raised the possibility that the North-East Greenland Ice Stream (NEGIS) may be relatively young, existing only for 2000 years , before which no ice stream existed here. To test the robustness of our results to a “young NEGIS”, we assume that before a certain time, there was no ice stream, i.e. the ice was frozen to the bed. Consequently, the deformation would be dominated by vertical shear and a single-maximum would be produced, as is observed at locations like NEEM and NGRIP . Ice cores from these locations reveal a vertical single maximum with a largest eigenvalue of around 0.75 over the depth we examine. Therefore we assume this as the pre-existing fabric over the whole depth. The ice stream deformation is then switched on at a certain time ago, and the fabric evolution is calculated from then onwards, with the vertical single-maximum as the initial fabric.

We explore this in Fig. . This again shows how the average eigenvalues evolve as in Fig. , but this time depending on the age of the NEGIS. The results are shown only for the Sachs model with the parameters shown in Table , though the trend does not change between models. Figure  shows that if the ice stream was extremely young (e.g. 100 years) the fabric would still be a vertical single maximum. If the NEGIS is was only 2000 years old, our results change slightly: a girdle fabric is still predicted, but the eigenvalues are different, consequently the best parameters would be slightly different. However, even for a young NEGIS of only 3000 years or older, the results are unchanged. Consequently, we can say our results are robust to an ice stream age of 3000 years or older. If the ice stream is found to be only 2000 years old, our best fit parameters change, but the general conclusions and ranking of the different models do not. Reproductions of Table , and Figs. , and for a 2000 year old ice stream can be found in the Supplement.

The results we have shown support the case for ice grains deforming closer to the Sachs hypothesis – that all grains experience the same stress – than the Taylor hypothesis – that all grains experience the same deformation. We conclude this because of the two models derived directly from grain dynamics, the Sachs model predicts eigenvalues much closer to observations, for both the divide (Fig. ) and the ice stream (Fig. ), than the Taylor model. The Taylor model is also not able to predict the girdle fabric observed in the ice stream (Fig. ). Furthermore, we observed that the Sachs model had the closest agreement to observations of all the examined models in this work. This is despite the fact that the other rheologies have more free parameters, which would be expected to result in better predictions. Indeed, for the anisotropic rheologies where αD and αS (controlling lattice rotation due to deformation and stress respectively) can take any value, the best predictions come from choosing values close those required by the Sachs model.

One caveat to this is the variable amount of diffusion/Browninan motion (controlled by λ) required to give accurate predictions. If there is no diffusion acting on the fabric (λ=0), the Sachs model vastly over predicts the fabric strength at both locations. This diffusional process is meant to mimic rotational recrystallization. Observations of grain boundaries suggest some rotational recrystallization is indeed occurring in the EGRIP . However it is possible that the some amount of the modelled diffusion of the fabric is acting to ameliorate an over-prediction of lattice rotation from the Sachs hypothesis alone. , showed that including nearest neighbour interactions on grains can have a diffusional-like effect on the fabric. Consequently, we are cautious about using this model to infer rates of rotational recrystallization.

Previous work including only lattice rotation due to deformation has indeed succeeded at ice divides . Yet it is worth noting that at divides the deformation is steady over time, and the observed fabric is simple – a single maximum. This makes it fairly easy for a model with a number of free parameters to fit to observations. In contrast, ice streams have a changing deformation history and the observed fabric at EGRIP is more complex. Models which take only lattice rotation due to deformation into account have not been able to accurately model this fabric . Conversely, we observe that lattice rotation due to stress rather than deformation is key, combined with an anisotropic rheology like the Sachs or the NLO rheologies (Fig. ). The parameter fitting in Sect.  finds that for all models with free parameters controlling both lattice rotation due to deformation (αD) and due to stress (αS), the best agreement comes from lattice rotation due to stress only (Table ).

Furthermore, the fabric evolution model is only able to accurately predict the fabric observed at EGRIP when it is linked to either the Sachs or NLO rheologies. What these rheologies have in common is that they (a) have a fourth-rank tensor description of the anisotropy and (b) assume the grains deform with approximately the same stress. The other models are not able to predict the observed fabrics at EGRIP for any set of parameters chosen. The Sachs and NLO rheologies more accurately predict the form of the stress tensor, and are consequently able to accurately predict the eigenvalue evolution and correct pattern for both an ice stream and an ice divide, two very different deformation conditions. Consequently, a fabric model with either the Sachs or NLO rheology, along with the parameters shown in Table , can be applied more generally and can be used in ice sheet models, at least for low temperature conditions (T<-20 °C) which apply to GRIP and EGRIP.

As explored in Sect. , we have tested the models at only two locations in the ice sheet. Further in-situ observations across a range of deformation and temperature conditions are crucial for accurate modelling of fabrics and anisotropy. Recent advances with radar e.g. and seismics e.g. are welcome, but ice cores, which provide a full fabric dataset, are still needed. These would be particularly beneficial in areas with deformation similar to EGRIP, but at higher temperatures.

By incorporating the rheology into the fabric evolution model, we are able to compare the accuracy of different rheologies and provide a test of them for the first time. All the anisotropic rheologies provide an improvement over Glen's rheology. For each rheology, we have chosen the set of parameters which minimise the difference between the modelled and observed eigenvalues. Despite this, only the Sachs and non-linear orthotropic rheology are able to predict the observed girdle fabric at EGRIP, and have the minimum difference between the modelled and the observed eigenvalues.

Despite providing accurate fabric predictions, the Sachs rheology under-predicts the maximum possible effect of fabric on viscosity. Experimental tests , have shown a vertical single maximum fabric has approximately 10 times higher viscosity in the vertical direction compared to an isotropic fabric (i.e. a fabric with the grains randomly orientated). However, the Sachs rheology (with linear grain behaviour as used here) predicts a maximum ratio in viscosity between these two fabrics of only 2.5 . This effect does not impact fabric predictions due to our use of a normalised stress throughout (Eq. ).

Fortunately, this disadvantage does not affect the NLO rheology. To give the correct maximum enhancement with the NLO rheology, an Eca value of 103 should be used, along with the Sachs-Taylor weight used here. As is seen in Fig. S9 of the Supplement, there is only a small effect of increasing Eca above 100 on fabric predictions, as its effect on the eigenvalues asymptotically approaches some value as Eca→∞. Therefore, we recommend using the NLO rheology in ice-sheet models.

It is surprising that the GOLF rheology performs poorly. The GOLF reproduces the response (through tabulated values) of the Visco-Plastic Self-Consistent (VPSC) model, a microscopic model which includes grain interactions, and should theoretically be more accurate than the Sachs or Taylor bounds, or a linear combination of them. A possible explanation for this is that the underlying VPSC model is outdated: notes that this VPSC model cannot accurately predict fabrics in simple shear, which occurs in the ice stream. This has been improved upon , so it would be interesting to see how the GOLF performs with an updated underlying model.

Finally, the CAFFE model gives some interesting predictions. It gives good predictions for the eigenvalues at EGRIP. However, for the ice divide case it predicts a sharp increase of the largest eigenvalue at 1500 m which is not seen in other models or the ice core observations. This sharp increase is sensitive to the parameter Emin⁡ (see Fig. S12 of the Supplement) whereas claim this parameter can be freely chosen between 0 and 0.1. Nevertheless, it provides a significant increase in accuracy for fabric predictions over Glen's rheology, and is easier to incorporate into ice-sheet models than a tensor flow relation.

One limitation of this work is that we have only explored one ice divide and ice stream each. There exist other observations from different divides (see for a review). Fortunately, the observed fabrics at different ice divides are generally similar, so our results applied to GRIP should also apply elsewhere.

Unfortunately, there is only one ice core drilled from an active ice stream: EGRIP. Recent advances with radar e.g. and seismics e.g. provide some information on fabrics at other ice streams, however they do not provide the full eigenvalue and pattern information with depth that is available from an ice core. Now that we have a fabric model we believe is accurate, it can be applied to locations where radar and seismic measurements are available to see if it agrees with these observations. Ideally, there would be another full ice core available from an active ice stream.

We have also not considered the effect of temperature. In fact, while the parameters we have found give good predictions for both EGRIP and GRIP, and can be applied to low temperature regions (T<-20 °C), further work is required to constrain these parameters at higher temperatures. At the lattice scale, far too small to resolve in this model, all the processes are dependent on temperature. At the larger scale, this means the parameters controlling recrystallization processes and lattice rotation will be temperature dependent. Migration recrystallization is known to be highly temperature dependent . However, the comparison to GRIP and EGRIP only covers a temperature range of -40 to -20 °C. Consequently, the parameters shown do not cover the highest temperature seen in ice sheets. Although we have neglected migration recrystallization in this study, it is expected to be significant at higher temperatures . These high temperature regions include ice shelves and the ice-bedrock interface, both areas of critical importance to the large-scale flow.

As discussed in Sect. , as part of this analysis we have assumed the deformation around EGRIP is unchanged going back around 16 000 years. However, there is open discussion in the literature over whether the ice stream at EGRIP is a recent formation. Some recent work has suggested it may be as young as 2000 years, although a region of convergent ice flow could have been present before this which would produce a similar fabric. We have shown in Fig.  that the results used here are unchanged for an ice stream age as young as 3000 years. For an age of 2000 years, the best fit parameters are changed slightly (Table S1), but the overall conclusions and ranking of best rheologies are unchanged.