On the nonlinear viscosity of the orthotropic bulk rheology

Abstract

We compare different ways the bulk flow nonlinearity of glacier ice can be captured in an orthotropic rheology. Specifically, we compare the unapproximated orthotropic rheology, derived from plastic potential theory, to existing approximations that assume either the nonlinear viscosity or fluidity is identical to that of Glen's isotropic flow law. We find, overall, a reasonable agreement between the three orthotropic rheologies, and with existing Dye 3 ice-core deformation tests, although assuming Glen's viscosity provides the best approximation to the unapproximated rheology. Our results therefore suggest that previous studies based on either approximation to the orthotropic rheology are on relatively safe ground in the sense that both approximations generally agree with the unapproximated rheology and experimental data. Finally, we provide the forward and inverse analytical forms of all three rheologies for use in future numerical ice-flow modelling.

Simulating higher-order fabric structure in a coupled, anisotropic ice-flow model: application to Dome C

“…Some fundamental differences have been clear before this work; for example, laboratory ice-deformation experiments do not seem to produce strong girdles, despite widespread observations of girdles in natural ice. This modeling, and the recent work on which it relies (Richards and others, 2021;Rathmann and Lilien, 2022b), allows direct comparison of the fabric evolution at laboratory scales/rates and ice-core scales/rates, potentially providing a path forward for understanding these differences.…”

“…Large-scale models of fabric development rely on the distribution of preferred grain orientations in order to construct grain-weighted-average quantities such as directional viscosities (e.g. Rathmann and Lilien, 2022b). Different weights can be used to construct average quantities, such as the distribution of number of grains or their mass in orientation space, S 2 (surface of the unit sphere).…”

“…Including this effect is particularly important for understanding the implications of the fabric upon ice flow, since the directional viscosities are thought to depend on both a (2) and a (4) (e.g. Gillet-Chaulet and others, 2006;Rathmann and Lilien, 2022b). The gray contours in Figure 1 show the modeled compressional/extensional enhancement factor, E zz , along the z-axis (the viscosity model is introduced below), indicating fabric-induced hardness may be overestimated by 40% if DDRX is active (red and gray markers) but a correlation based on lattice rotation is assumed (green model line).…”

“…Here, we incorporate the above spectral description of fabric as a new module in Elmer/Ice (Zwinger and others, 2007;Gagliardini and others, 2013), a state-of-the-art ice-flow model, providing an alternative to the current structuretensor-based fabric representation. In coupling to flow, we also implement the recently derived, unapproximated nonlinear orthotropic rheology, validated against Dye-3 deformation tests (Rathmann and Lilien, 2022b), which is expected to be important when fabric and stress are misaligned.…”

“…For a constitutive relation, we adopt an unapproximated nonlinear extension of the orthotropic flow law (Rathmann and Lilien, 2022b), which, following plastic potential theory, has a nonlinear viscosity that depends on all orthotropic strain-rate tensor invariants:…”