Ice Flow Law

Motivation

Forecasts of the contribution of basal sliding to ice-mass loss fundamentally rely on flow laws describing internal deformation. Both the formulation of the flow law and the accuracy of its parameters critically influence model outcomes. Reliable extrapolation from laboratory to natural conditions therefore demands flow laws with both the correct functional form and well-constrained parameters. However, the classical Glen flow law fails to reproduce laboratory results accurately and requires substantial modification to align with field observations.

70 Years of Experimental Data

We compiled a comprehensive database containing 566 data points from published deformation experiments along with four new experiments. From these, 305 data points were used to constrain flow laws at low strains, representing peak stresses or secondary-creep strain rates, where the microstructure remains largely isotropic and lacks CPO. An additional 160 data points were used to develop a flow law for steady-state deformation at higher strains (≳8%), corresponding to tertiary-creep strain rates, where ice has undergone significant weakening due to grain-size reduction and the development of strong CPO.

Bayesian Inference

To determine the best-fit parameters for the flow laws, we used Bayesian inference, which rigorously estimates parameters from experimental data while incorporating prior knowledge. In essence, Bayesian statistics refine our understanding as more data are added—the posterior distribution shifts from the prior and progressively reflects the evidence, yielding model estimates that better match observations (refer the illustrative movie). Because analytical solutions are intractable for complex models with many unknowns (up to nine in this study), we employed the Markov Chain Monte Carlo (MCMC) method to efficiently sample the posterior distribution and achieve robust parameter estimation.

Three-component Law Derives a Better Fit

Our flow laws reproduce low-strain experimental data more accurately than the Glen and Goldsby–Kohlstedt formulations (left). Only 4% of laboratory data show stresses deviating by more than a factor of 1.5 (≈ half an order of magnitude in strain rate) from the predictions of our three-component flow law, and fewer than 1% deviate by more than a factor of 2 (≈ one order of magnitude). In contrast, 42% and 19% of data differ by these respective factors from the Glen flow law, and 21% and 5% from the Goldsby–Kohlstedt flow law.

Three-component Law Captures Premelting

Previous experiments show a gradual increase in apparent activation energy (Q) near the melting point. Our three-component flow law reproduces this trend well, predicting Q to rise from ~50 kJ mol⁻¹ at −30°C to ~110 kJ mol⁻¹ at −3°C. In contrast, a two-component flow law fails to capture this high-temperature variation. The observed increase in Q is likely caused by enhanced premelting at grain boundaries as the temperature approaches the melting point.