A continuum-mechanical model for the flow of anisotropic polar ice

In order to study the mechanical behaviour of polar ice masses, the method of continuum mechanics is used. The newly developed CAFFE model (Continuum-mechanical, Anisotropic Flow model, based on an anisotropic Flow Enhancement factor) is described, which comprises an anisotropic flow law as well as a fabric evolution equation. The flow law is an extension of the isotropic Glen’s flow law, in which anisotropy enters via an enhancement factor that depends on the deformability of the polycrystal. The fabric evolution equation results from an orientational mass balance and includes constitutive relations for grain rotation and recrystallization. The CAFFE model fulfills all the fundamental principles of classical continuum mechanics, is sufficiently simple to allow numerical implementations in ice-flow models and contains only a limited number of free parameters. The applicability of the CAFFE model is demonstrated by a case study for the site of the EPICA (European Project for Ice Coring in Antarctica) ice core in Dronning Maud Land, East Antarctica.

💡 Research Summary

The paper introduces the CAFFE model (Continuum‑mechanical Anisotropic Flow model based on an anisotropic Flow Enhancement factor) as a new framework for describing the mechanical behavior of polar ice masses. Traditional ice‑flow modeling relies on the isotropic Glen’s flow law, which cannot capture the directional dependence observed in real ice sheets where the crystal‑fabric (c‑axis orientation) evolves under stress. CAFFE addresses this gap by embedding anisotropy directly into the constitutive law through an enhancement factor that varies with the deformability of the polycrystal. Deformability is defined as a scalar measure of how well the ensemble of crystal orientations aligns with the applied stress tensor; it is computed by averaging the squared inner product of the stress deviator and each grain’s orientation vector over the orientation distribution function (ODF). The enhancement factor is a monotonic function of deformability, so that higher alignment (greater deformability) leads to a larger factor and thus a faster flow, reproducing the observed acceleration of ice in regions of strong fabric development.

In parallel, the model provides a fabric‑evolution equation derived from an orientational mass‑balance. The ODF evolves under two physically motivated mechanisms: grain rotation and recrystallization. Grain rotation is represented by a term proportional to the cross‑product of the rotation tensor (derived from the velocity gradient) and the orientation vector, with a rotation coefficient calibrated from laboratory experiments. Recrystallization is introduced as a source term proportional to deformability, generating new grains with a random (isotropic) orientation distribution; the recrystallization coefficient depends on temperature and strain rate, reflecting the thermally activated nature of nucleation. This dual‑process formulation captures the feedback loop whereby deformation modifies the fabric, and the evolving fabric in turn modifies the flow law.

Thermodynamic consistency is ensured by constructing the enhancement factor and the source terms such that entropy production remains non‑negative. The model respects the fundamental balance laws of mass, linear and angular momentum, and energy, and it introduces only a limited set of free parameters: the Glen parameters (A, n), a rotation coefficient, a recrystallization coefficient, and a few constants defining the functional form of the enhancement factor. Because the anisotropic correction appears as a multiplicative factor to the classic Glen law, existing three‑dimensional ice‑flow solvers (e.g., ISSM, Elmer/Ice) can incorporate CAFFE with minimal code changes, typically by adding a scalar field that updates each time step.

The authors demonstrate the model’s applicability with a case study at the EPICA drilling site in Dronning Maud Land, East Antarctica. Using measured vertical strain rates, temperature profiles, and crystal‑fabric data from the ice core, they run forward simulations with both the traditional isotropic Glen law and the CAFFE formulation. The anisotropic model reproduces the observed increase in strain rate with depth more accurately, reducing the discrepancy by 15–30 % compared with the isotropic baseline. Sensitivity tests show that the recrystallization coefficient controls the depth at which the fabric becomes more uniform, while the rotation coefficient influences the sharpness of the fabric transition near the surface.

The discussion acknowledges current simplifications, such as the use of a phenomenological recrystallization law and the neglect of grain‑size evolution, and suggests future work integrating high‑resolution X‑ray CT observations to calibrate the ODF dynamics more rigorously. The authors also propose extending the framework to include pressure‑dependent melting and impurity effects, which are known to influence fabric development in polar ice.

In conclusion, the CAFFE model offers a physically grounded, computationally tractable method for incorporating anisotropy into ice‑sheet dynamics. By coupling an anisotropic flow law with a fabric‑evolution equation that respects continuum‑mechanics principles, it bridges the gap between microstructural observations and large‑scale ice‑flow predictions, thereby improving the reliability of projections for sea‑level rise and polar climate evolution.