Abstract
Accurate representation of the viscous flow of ice is fundamental to understanding glacier dynamics and projecting sealevel rise. Ice viscosity is often described by a simple but largely untested and uncalibrated constitutive relation, Glen’s Flow Law, wherein the rate of deformation is proportional to stress raised to the power n. The value n = 3 is commonly prescribed in iceflow models, though observations and experiments support a range of values across stresses and temperatures found on Earth. Here, we leverage recent remotelysensed observations of Antarctic ice shelves to show that Glen’s Flow Law approximates the viscous flow of ice with n = 4.1 ± 0.4 in fastflowing areas. The viscosity and flow rate of ice are therefore more sensitive to changes in stress than most iceflow models allow. By calibrating the governing equation of ice deformation, our result is a pathway towards improving projections of future glacier change.
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Introduction
Mass loss from ice sheets presents both the greatest potential contribution to future sealevel rise and the largest source of uncertainty in such estimates^{1,2}. In Antarctica, mass loss occurs principally through fastflowing glaciers that flow into floating ice shelves, which provide resistive buttressing stresses that impede the seaward flow of ice and stabilize marine grounding zones^{3,4,5}. The rate at which glaciers flow is controlled by the shearthinning viscous deformation of ice^{6}. The most commonly adopted constitutive relation, known as Glen’s Flow Law, is often employed to quantify the viscous deformation of glacier ice by relating the rate of deformation, hereafter called strain rate, to the deviatoric stress^{7}. Glen’s Flow Law is most simply expressed as
where \({\dot{\epsilon }}_{e}\) is the effective strain rate, τ_{e} the effective deviatoric stress, n the stress exponent, and A the rate factor or flowlaw coefficient. Variation in parameter A can be used to represent the effects of temperature, grain size, grain orientation (fabric), impurities, and interstitial water content^{8}.
Glen’s Flow Law is routinely implemented in largescale iceflow models with the prescribed value n = 3 assumed to be constant in space and time^{9,10}. Glen’s laboratory experiments pinpointed the powerlaw rheology and extrapolated his findings to flows of natural ice^{7,8,11}. Shortly thereafter, Glen’s findings and supporting evidence were widely adopted in the glaciological literature, with the field converging on the canonical value of n = 3^{12,13,14}. However, multiple mechanisms influence the viscous deformation of ice, each with a suggested value of n: dislocation creeps (n = 4), grainboundary sliding (n ≈ 2, with slight variance dictated by the direction of motion of dislocations), and diffusion creep (n = 1) all accommodate creep at the individual grain level and, in aggregate, describe the flow of glacier ice^{15}. These mechanisms are not treated independently in Glen’s Flow Law (Eq. (1)). Rather, it serves as a lumped parameterization representing the combined effect of all mechanisms. Generalized forms of the flow law have been proposed to account for multiple creep mechanisms, fabric, and grain size, but these have not been widely tested, calibrated, nor implemented^{10,15,16}.
The simplicity of Glen’s Flow Law has proven useful and, subject to suitable calibration under different conditions, has the potential to provide a reasonably accurate general description of the flow of glacier ice^{7,8,14,17}. Glen’s Flow Law (Eq. (1)) with n = 3 shows consistency with sparse observations of natural ice flows such as borehole deformation measurements and iceflow velocities, as well as laboratory experiments on polycrystalline ice aggregates under conditions relevant for ice sheets^{7,15,18,19,20,21,22,23,24,25}. However, the broad range of conditions over which the rheological behavior of ice has been examined reveals the way in which variations in stress can influence the stress exponent and, in turn, the mechanisms of creep^{10,26,27,28}. Nearly 70 years after its introduction, the need remains to test and rigorously calibrate the parameters n and A in the natural environment.
We infer the stress exponent of Glen’s Flow Law across wide areas of Antarctic ice shelves, the floating extensions of the ice sheet. Using satellite observations, we are able to address the longstanding problem of benchmarking a flow law that can be used in iceflow models. The abundance and extent of the data allow us to investigate the creep of glacial ice on a continental scale, assembling inferences to reveal spatial coherence and patterns with statistical constraints. To do so, we require independent estimates of strain rates and (deviatoric) stresses (Eq. 1). The schematic in Fig. 1 graphically illustrates the methodology, showing how we begin with independent observations of surface velocities and ice thicknesses, apply these to evaluate strainrates \({\dot{\epsilon }}_{e}\) and stresses τ_{e}, and then conduct a regression analysis to infer the parameters in Glen’s Flow Law. This method is comparable to previously published work^{21,22,26,29}, but applied to Antarctic ice shelves using continentalscale remote sensing observations. Our results reveal that a value of n = 4.1 ± 0.4 is the most representative flowlaw exponent in fastflowing, extensional regions, where the magnitude of deviatoric stresses are comparable to those expected in other dynamic regions of the ice sheet. Making use of continentscale remote sensing observations on Antarctic ice shelves, we demonstrate how the viability of powerlaw rheology can be constrained directly using observations.
We focus on ice shelves because the underlying ocean provides negligible shear resistance to ice flow, allowing for two important simplifications in our analysis. First, we can neglect drag at the base of the ice and thus consider a stress regime that is simpler for our purposes than would be expected for grounded ice, where basal drag presents a further unknown that must be constrained. Second, the lack of drag at the base means that strain rates are approximately constant with depth. For this reason, the horizontal strain rates we calculate from observations of the surface velocity fields approximate the strain rates at all depths.
Ice shelves cover areas that are large compared with the subkilometer resolution of observations, providing ample opportunities to comprehensively observe broad regions of flow undergoing relatively simple onedimensional deformation. As a result, we can focus on regions that are close to being in pure extension, where the ice spreads under its own weight in one direction and the governing equations of flow reduce to a simple twoterm balance, detailed further in this report. This basic premise has been employed for decades to study the rheology of glacier ice^{22,24,30} but has not been systematically applied on continental scales before now.
We use measurements of ice thickness provided through the BedMachine project^{31}, and surface velocity data from the NASA Intermission Time Series of Land Ice Velocity and Elevation (ITS_LIVE) project^{32}. The surface velocity data, which encompass most of the Antarctic Ice Sheet at a grid spacing of 120 m × 120 m, are derived from Landsat 4, 5, 7, and 8 imagery using the autoRIFT feature tracking processing chain, providing reliable constraints on the two horizontal components of ice velocity^{32}. We use these to calculate the horizontal strainrates \({\dot{\epsilon }}_{ij}\) (for i, j = x, y the two horizontal coordinates) across all Antarctic ice shelves, as defined by \(2{\dot{\epsilon }}_{ij}=\big(\partial {u}_{i}/\partial {x}_{j}+\partial {u}_{j}/\partial {x}_{i}\big)\), where u_{i} represents the horizontal components of the ice velocity vector and x_{i} the horizontal coordinates. To calculate the components of the velocity gradient, we apply a twodimensional SavitzkyGolay filter with a polynomial order of one and a square window of 3720 m (31 pixels)^{33}. More detail on the strainrate calculations is found in the Supplementary Methods.
After deriving strain rates from the surface velocity fields, we determine regions flowing in approximately pure extension, with a view to simplifying the force balance governing the local ice flow. The twodimensional strainrate tensor \({\dot{\epsilon }}_{ij}\) has three unique components (the offdiagonal terms are equal by definition) and a scalar invariant representing the effective horizontal strainrate \(\dot{\epsilon }=\sqrt{{\dot{\epsilon }}_{ij}{\dot{\epsilon }}_{ij}/2}\), where the summation is implied for repeated indices. Note that the effective strainrate \({\dot{\epsilon }}_{e}\) in Eq. 1 follows the same definition as for \(\dot{\epsilon }\) but is applied to the threedimensional strainrate tensor. We focus on areas of the ice shelves that are solely confined by seaward pressure in the alongflow, or x, direction, and analyze areas in which the alongflow component of the strainrate tensor \({\dot{\epsilon }}_{xx}\) is much larger than both lateral normal and shear strain rates (\({\dot{\epsilon }}_{xx}\gg {\dot{\epsilon }}_{yy},{\dot{\epsilon }}_{xy}\)). We combine these into a single criterion \({\dot{\epsilon }}_{xx}\approx \sqrt{2}\dot{\epsilon }\), corresponding to areas of the ice shelves where longitudinal extension is dominant. The more specific criterion \({\dot{\epsilon }}_{xx} \, > \,\dot{\epsilon }\) is used to define large, spatially coherent regions where the extensional component of deformation dominates the flow (Fig. 2 and Supplementary Figs. S1 and S2). Approximately 20% of the total surface area of all Antarctic ice shelves satisfies this criterion. In these areas it follows from the incompressibility of ice and the absence of drag at the base of ice shelves that \({\dot{\epsilon }}_{xx}\approx {\dot{\epsilon }}_{e}\), the threedimensional effective strain rate in Eq. (1).
To estimate the effective deviatoric stress from remote sensing observations, we utilize a wellestablished reduced form of the Stokes equations that govern the viscous flow of glacier ice. Over the ice shelves, where negligible shear stress applies at both the upper (atmosphere) and lower (ocean) surfaces of the ice, we can adopt the depthintegrated form of the Stokes equations commonly referred to as the ShallowShelf Approximation (SSA), which contains only body forces and the horizontal gradients of the stress tensor elements. Based on the conditions described above, we can further reduce the SSA equations to a simple expression relating the (depthaveraged) alongflow deviatoric stress τ_{xx} to local ice thickness H as:
where \(g^{\prime} =g(1\rho /{\rho }_{w})\) is the reduced gravity, representing the balance between the resistive longitudinal stress and the driving buoyancy force (the full derivation is provided in the Methods). Here, we take ρ = 910 kg/m^{3} as the mass density of glacier ice and ρ_{w} = 1026 kg/m^{3} as the mass density of seawater. Where the criteria for predominantly extensional flow is met (\({\dot{\epsilon }}_{xx}\approx {\dot{\epsilon }}_{e}\)), we expect τ_{xx} ≈ τ_{e}. Thus, the criteria we apply to the strainrate fields to identify areas in primarily extensional flow allows us to calculate effective stress τ_{e} (Eq. (1)) from observations of ice thickness and independently of the surface velocity fields used to calculate \({\dot{\epsilon }}_{e}\). Before fitting a model to the data, we ensure that the gradients of horizontal shear stress transverse to flow are small compared to the gradients of longitudinal stress from the position of the ice parcel all the way to the ice shelf calving front. This supports the suitability of the derivation for effective stress over the fastflowing, extensional regions of Antarctic ice shelves of interest.
Critically, this study neither takes into account firn or marine ice, which are characteristic of all ice shelves, nor do we need to explicitly account for viscous anisotropy (fabric). Complexities caused by firn and marine ice are partially subsumed by the uniform density profile but remain a source of uncertainty in our analysis. Given that the mass densities of firn and ice are within a factor of two and firn typically comprises a thin upper layer of ice shelves, we expect the uncertainties due to firn and marine ice are small enough to not meaningfully impact our results. Our focus on a single flow regime and parcels of ice defined along and parallel to flow lines allow us to avoid the complexities that arise from viscous anisotropy in ice, which would require a nonscalar form of A to represent deformation in multiple directions, and spatial variations in characteristics like ice temperature and liquid water content.
Results
Linear regressions fitted to the values for \(\log ({\dot{\epsilon }}_{e})\) and \(\log ({\tau }_{e})\) constrain n through the slope and A in the y − intercept, divulging values of the flowlaw parameters across viable regions of Antarctic ice shelves. To determine 95% confidence intervals on the regression of strain rate on stress, we implement a nonparametric bootstrap, which allows us to estimate constraints on the determined value of n without making assumptions on the underlying structure of the distribution^{34}. Our analysis encompasses regions of both large ice shelves, such as those shown in Fig. 2, and smaller ice shelves that line the continent. We focus first on highlighted areas on the Ross and FilchnerRonne Ice Shelves in Fig. 2, which we extracted from areas along flow lines, with probable consistency between values of temperature, grain size, and fabric, and therefore A and n.
The log–log plots between strain rate and deviatoric stress shown in Fig. 2 exhibit linear trends that are consistent with a powerlaw relation. These results provide strong evidence that, for a suitable choice of n, Glen’s Flow Law is a viable approximation of the viscous flow of Antarctic ice shelves and, as discussed later, likely other dynamic regions of Antarctica. Critically, we find n ≈ 4 in the fastflowing, extensional regions of Antarctic ice shelves. This result is consistent with other evidence for a higher value of the flowlaw exponent^{7,24,26,30,35,36}, and demonstrates that this higher value is applicable to natural ice flow at the continental scale. Additional comparison with the value n = 3 and other typical values of the existing flow law can be found in Supplementary Fig. S3; it is worth noting that n = 3 provides a poor fit to the data used in this study as shown in Fig. 2. Additionally, the residuals from the linear regressions in subplots ah of Fig. 2 are shown in Supplementary Fig. S4 and demonstrate the suitability of the linear fit in these areas.
The results of our full analysis covering all viable regions of Antarctic ice shelves are shown in Fig. 3, which includes regions of both large and small ice shelves (mapped in Supplementary Figs. S1 and S2). The normalized kernel density estimates of the bootstrapped values of the flowlaw exponent (Fig. 3) indicate that n = 4.1 ± 0.4 in extensional regions of Antarctic ice shelves. Figure 3 shows the confidence with which our estimate stands across geographic areas of different sizes and represents a range of stresses. Large areas extracted for analysis, > 1000 km^{2}, have less spread in the error estimation and are centered closer to n = 4.1, whereas smaller areas exhibit a greater spread in the distribution. This is likely because the broader ranges of stresses and the greater number of observations in the larger ice shelves provide more accurate inferred trends across the data. Notably, geographic regions from West Antarctica have slightly higher values of n than regions sampled from East Antarctica. This observation could be attributed to higher subiceshelf melt rates in West Antarctic ice shelves, where the bulk of ice is created on the ice shelf by compaction of snow as opposed to being inherited from the grounded glacier^{37}. Additionally, there is a possible grain size dependence wherein warmer conditions would contribute to larger grains^{38,39}. In such regions, larger grains, strain rate, and values of the stress exponent validate a hypothesis that ice deformation is facilitated primarily by dislocation creep^{15,29}. Our results highlight further spatial variability in the precise values of the flowlaw exponent and rate factor across different ice shelves, and even different regions within single ice shelves (see Fig. 3). We reserve for future work detailed analysis and modeling of these variations.
We find values of the flowlaw rate factor, A, spanning 10^{−35}–10^{−27} Pa^{−n} s^{−1} for the range of inferred n values (see Supplement Fig. S5). Inferred values of A depend on the inferred values of n. Here, we do not attempt to provide newly calibrated values for A because proper constraints on the physical properties of the ice, like temperature and grain size, are not currently available in these areas and require work that is beyond the scope of this study. Rather, we note that the smaller values of A found here to validate our method for deriving Glen’s Flow Law and we recommend that future efforts using a value n ≈ 4 utilize standard tabulated sources for A^{40} and scale these values accordingly for the new value of n. A comparison of our results to the more commonly used n = 3 can be seen in Supplement Fig. S3, highlighting the incompatible values of A in these results, and the generally poor fit of n = 3 to the data.
Conclusion
The result that n ≈ 4 challenges the longheld practice of assuming the flowlaw exponent is n = 3 everywhere, and at all times, in largescale icesheet flow models. While our observations focus on specific regions in extensional flow regimes on ice shelves that experience stresses of order 100 kPa (Supplement Fig. S6), complementary laboratory work showing that n = 4 is suitable at higher stresses^{15} supports extending our conclusion that n ≈ 4 to other dynamic regions in Antarctica. Additionally, our conclusion complements a growing body of work advocating for the use of n > 3 in other areas of the cryosphere^{19,26}. Taken together, this work calls for a broader community effort to quantify the uncertainties in the flowlaw parameters and the consequences of these uncertainties on models of glacier dynamics. A higher value of n increases the sensitivity of viscosity to changes in stress but the impact of n = 4 on largescale iceflow models used for projections of sealevel rise and icesheet evolution remains unclear as few sensitivity analyses have been conducted^{10} and n is not a parameter explored in current ensemblemodel analyses^{1,2}. The value n = 4 has the potential to increase the sensitivity of icesheet mass loss to ongoing climate change considerably relative to n = 3 due to the stronger dependence of flow rates to changes in resistive stresses.
By applying continentalscale satellite observations to standard models in glacier dynamics, we have validated Glen’s Flow Law, a constitutive relationship that helps form the foundation of modern glaciology, and calibrated the stress exponent in Antarctic ice shelves. This work serves as a pathway towards a standard calibration framework for the community using publicly available remote sensing data. Our conclusion that n ≈ 4 across much of Antarctica’s ice shelves is a step towards reassessing the governing equations of ice flow in the satellite age, and reveals an increased sensitivity of flow rates to applied stresses relative to the commonly used n = 3. As a consequence, future sealevel rise is likely more sensitive to climate forcings than predicted by present models using common assumptions of the flow law.
Methods
Solving for effective stress
Conservation of momentum (Stokes equations) describes all forces acting on the volume of glacier ice such that
where p is the pressure, ρg_{i} is the driving gravitational force (with \({{{{{{{\boldsymbol{g}}}}}}}}=g{{{\hat{{{{{\boldsymbol{z}}}}}}}}}\)), and summation is implied for repeated indices. For a layer of ice floating on top of an ocean, we can derive depthintegrated equations to describe the balance of forces in such a system, given that the ice shelf is much larger in horizontal extent than in thickness^{41}. At scales of order the ice thickness, bending (and bridging) stresses are negligible, allowing us to simplify the equilibrium equations^{42}. As a result, we take the vertical normal stress to be equivalent to the overburden stress (weight of the ice per unit area). This can be expressed as
where H is the ice thickness, \(g^{\prime} =g({\rho }_{w}\rho )/{\rho }_{w}\) is the reduced gravity, and the second equality arises from the fact that the deviatoric stress tensor is traceless. Eq. (4) is derived by integrating the vertical component of Eq. (3) and applying the condition of continuous normal stress at the top and bottom of the layer.
Then, neglecting basal drag (due to our focus on ice shelves) and depth integrating the xcomponent of Eq. (3), we can obtain
where all deviatoric stresses are now depthaveraged. A complete derivation can be found in ref. ^{43}, which uses different notation but reveals the same outcome. A comparable derivation is found in ref. ^{30} with the notable distinction here being our omission of α = τ_{yy}/τ_{xx} because we only consider areas where α ≪ 1. In this way, we are able to look at large areas without potential complications arising from multiple stress components (e.g., viscous anisotropy).
We can simplify Eq. (5) in two steps. First, we assume that the lateral normal stresses (τ_{yy}) are negligibly small compared with the longitudinal normal stresses (τ_{xx}) due to our emphasis on areas with \({\dot{\epsilon }}_{xx}\gg {\dot{\epsilon }}_{yy}\)^{44}. Then, we apply the constitutive relation in Supplementary Eq. 6 and recall that in our areas of interest, we require that \({\dot{\epsilon }}_{xx}\approx {\dot{\epsilon }}_{e}\). Thus, Eq. (5) becomes
where \(\phi =h{\dot{\epsilon }}_{xx}^{1/n}{A}^{1/n}\) and \(\beta ={\dot{\epsilon }}_{xy}/{\dot{\epsilon }}_{xx}\). The derived strainrate data indicate that in our areas of interest, the lateral (∂/∂y) and longitudinal (∂/∂x) gradients in \(h{\dot{\epsilon }}_{xx}\) have the same order of magnitude. Assuming A and n vary slowly in space in our areas of interest, then ∂ϕ/∂y is of order ∂ϕ/∂x, placing the emphasis on the term β. Our criteria that \({\dot{\epsilon }}_{xx}\approx {\dot{\epsilon }}_{e}\) requires that β ≪ 1 everywhere in our areas of interest, which are wide enough that ∂β/∂y is negligibly small. This means that within the error in currently available data, we can assume that the lateral shear term (second on the lefthand side of Eqs. (5) and (6)) is negligible.
Vastly reduced, what began as four components—extension, lateral shear, basal drag, and buoyancy—now only requires terms for extension and buoyancy to illustrate the force balance of an unconfined ice shelf^{44}. Equation (5) is now
We can now rearrange the righthand side of Eq. (7) to an equivalent form
Integrating this equation subject to the free stress condition at the front of the ice shelf and simplifying the resulting equation results in
which we use as the basis for our analysis of extensional deviatoric stress in floating ice shelves. This derivation shows how we can use the extensional deviatoric stress as the total effective stress in our regions of interest, allowing us to use a dataset of ice thickness to determine the stress in the system parameter.
Data availability
No new data were generated in this analysis; the strainrate fields were generated using velocity data from NASA ITS_LIVE (https://itslive.jpl.nasa.gov/). The MEaSUREs ice thickness data are available at the NSIDC.
Code availability
The Python codes used to analyze the remote sensing datasets and prepare figures are available on Github (https://github.com/jdmillstein/n_equals_4).
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Acknowledgements
We benefited from discussions with Jerome Neufeld, Colin Meyer, and Andrew Ashton. We appreciate insightful reviews from Jeremy Bassis and Paul Bons. J.D.M. was partially funded through an NSF Graduate Research Fellowship. J.D.M. and B.M.M. where partially funded through NSFNERC award 1853918. B.M.M. received additional funding through NSFNERC award 1739031.
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The authors worked together to conceive and design the project. J.D.M. undertook the analysis, generated the figures, and wrote the initial version of the manuscript. B.M.M. and S.S.P. helped revise the manuscript.
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Millstein, J.D., Minchew, B.M. & Pegler, S.S. Ice viscosity is more sensitive to stress than commonly assumed. Commun Earth Environ 3, 57 (2022). https://doi.org/10.1038/s4324702200385x
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DOI: https://doi.org/10.1038/s4324702200385x
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