Glen's Flow Law Research Articles

Improvements on the discretisation of boundary conditions to the momentum balance for glacial ice

Abstract The flow of glacial ice is typically approximated as a nonNewtonian viscous fluid, with the momentum balance described by (an approximation to) the Stokes equations, and the nonlinear rheology described by a flow law. The most commonly used rheological law for glacial ice, Glen's flow law, yields infinite viscosity in the case of zero deformation, which can be the case at the ice surface. This poses a problem when solving the momentum balance numerically. We show that two commonly-used discretisation schemes for the boundary conditions at the ice surface and base, which yield proper numerical convergence when applied to simpler problems, produce poor numerical convergence and large errors, when used to solve the momentum balance with Glen's flow law. We show that a discretisation scheme based on the concept of ghost nodes, which substitutes the boundary conditions directly into the momentum balance equations, yields second-order numerical convergence and errors that can be up to four orders of magnitude smaller. These results are robust across different momentum balance approximations. We show that the improved boundary conditions are particularly useful for solving the 3-D higher-order Blatter-Pattyn Approximation (BPA). In general, this work underlines the importance of thoroughly verifying the numerical solvers used in ice-sheet models, before applying them to future projections of ice-sheet mass loss.

Electron microscope loading and in situ nanoindentation of water ice at cryogenic temperatures.

Interest in the technique of low temperature environmental nanoindentation has gained momentum in recent years. Low temperature indentation apparatuses can, for instance, be used for systematic measurements of the mechanical properties of ice in the laboratory, in order to accurately determine the inputs for the constitutive equations describing the rheologic behaviour of natural ice (i.e., the Glen flow law). These properties are essential to predict the movement of glaciers and ice sheets over time as a response to a changing climate. Herein, we introduce a new experimental setup and protocol for electron microscope loading and in situ nanoindentation of water ice. Preliminary testing on pure water ice yield elastic modulus and hardness measurements of 4.1 GPa and 176 MPa, respectively, which fall within the range of previously published values. Our approach demonstrates the potential of low temperature, in situ, instrumented nanoindentation of ice under controlled conditions in the SEM, opening the possibility for investigating individual structural elements and systematic studies across species and concentration of impurities to refine to constitutive equations for natural ice.

On the nonlinear viscosity of the orthotropic bulk rheology

AbstractWe 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.

Geospatial investigation on transitional (quiescence to surge initiation) phase dynamics of Monacobreen tidewater glacier, Svalbard

The change in the phase, from quiescence to surge, may perturb the glacier’s dynamical behavior. To understand these changes through geospatial technique, the fastest moving tidewater Monacobreen Glacier, Svalbard, was studied for the year 2016–2019. It was elucidated that the surge had initiated during 2017–2018 before it the glacier was in quiescence since 2001. Initially, the glacier radar zones, namely percolation refreeze, lower percolation, and clean ice, were identified using the multi-temporal SAR data. These zones were utilized to delineate the equilibrium line altitude, and it was found that it is continuously moving down during surging, unlike the quiescence phase. The accumulation area ratio was always more than 0.67 throughout the analysis. The seasonal change in glacier surface velocity for both the phases was estimated, adapting the most appropriate Offset Tracking approach using the SAR data. The mean velocity over the main trunk was found to vary from 0.5 to 4 m/day. Thereafter, Glen's flow law equation was used to estimate the ice-thickness and found that the glacier has an average thickness range of 216–326 m in quiescence and 136–244 m during the surging. The analysis depicted that the basal shear stresses are increasing with surging, and the viscosity is decreasing. It can be said that the transition of the glacier from quiescent to surging phase has entirely changed its dynamic behavior and characteristics. Further, in the absence of field observations, the geospatial technique may provide reasonable estimates of the glacier’s physical and dynamical parameters.

The role of grain size evolution in the rheology of ice: implications for reconciling laboratory creep data and the Glen flow law

Abstract. Viscous flow in ice is often described by the Glen flow law – a non-Newtonian, power-law relationship between stress and strain rate with a stress exponent n ∼ 3. The Glen law is attributed to grain-size-insensitive dislocation creep; however, laboratory and field studies demonstrate that deformation in ice can be strongly dependent on grain size. This has led to the hypothesis that at sufficiently low stresses, ice flow is controlled by grain boundary sliding, which explicitly incorporates the grain size dependence of ice rheology. Experimental studies find that neither dislocation creep (n ∼ 4) nor grain boundary sliding (n ∼ 1.8) have stress exponents that match the value of n ∼ 3 in the Glen law. Thus, although the Glen law provides an approximate description of ice flow in glaciers and ice sheets, its functional form is not explained by a single deformation mechanism. Here we seek to understand the origin of the n ∼ 3 dependence of the Glen law by using the “wattmeter” to model grain size evolution in ice. The wattmeter posits that grain size is controlled by a balance between the mechanical work required for grain growth and dynamic grain size reduction. Using the wattmeter, we calculate grain size evolution in two end-member cases: (1) a 1-D shear zone and (2) as a function of depth within an ice sheet. Calculated grain sizes match both laboratory data and ice core observations for the interior of ice sheets. Finally, we show that variations in grain size with deformation conditions result in an effective stress exponent intermediate between grain boundary sliding and dislocation creep, which is consistent with a value of n = 3 ± 0.5 over the range of strain rates found in most natural systems.

Inferred basal friction and mass flux affected by crystal-orientation fabrics

We investigate the errors caused by neglecting the crystal-orientation fabric when inferring the basal friction coefficient field, and whether such errors can be alleviated by inferring an isotropic enhancement factor field to compensate for missing fabric information. We calculate the steady states that arise from ice flowing over a sticky spot and a bedrock bump using a vertical-slab numerical ice-flow model, consisting of a Weertman sliding law and the anisotropic Johnson flow law, coupled to a spectral fabric model of lattice rotation and dynamic recrystallisation. Given the steady or transient states as input for a canonical adjoint-based inversion, we find that Glen's isotropic flow law cannot necessarily be used to infer the true basal drag or friction coefficient field, which are obscured by the orientation fabric, thus potentially affecting vertically integrated mass fluxes. By inverting for an equivalent isotropic enhancement factor, a more accurate mass flux can be recovered, suggesting that joint inversions for basal friction and the isotropic flow-rate factor may be able to compensate for mechanical anisotropies caused by the fabric. Thus, in addition to other sources of rheological uncertainty, fabric might complicate attempts to relate subglacial conditions to basal properties inferred from an inversion relying on Glen's law.

Inferring the basal sliding coefficient field for the Stokes ice sheet model under rheological uncertainty

Abstract. We consider the problem of inferring the basal sliding coefficient field for an uncertain Stokes ice sheet forward model from synthetic surface velocity measurements. The uncertainty in the forward model stems from unknown (or uncertain) auxiliary parameters (e.g., rheology parameters). This inverse problem is posed within the Bayesian framework, which provides a systematic means of quantifying uncertainty in the solution. To account for the associated model uncertainty (error), we employ the Bayesian approximation error (BAE) approach to approximately premarginalize simultaneously over both the noise in measurements and uncertainty in the forward model. We also carry out approximative posterior uncertainty quantification based on a linearization of the parameter-to-observable map centered at the maximum a posteriori (MAP) basal sliding coefficient estimate, i.e., by taking the Laplace approximation. The MAP estimate is found by minimizing the negative log posterior using an inexact Newton conjugate gradient method. The gradient and Hessian actions to vectors are efficiently computed using adjoints. Sampling from the approximate covariance is made tractable by invoking a low-rank approximation of the data misfit component of the Hessian. We study the performance of the BAE approach in the context of three numerical examples in two and three dimensions. For each example, the basal sliding coefficient field is the parameter of primary interest which we seek to infer, and the rheology parameters (e.g., the flow rate factor or the Glen's flow law exponent coefficient field) represent so-called nuisance (secondary uncertain) parameters. Our results indicate that accounting for model uncertainty stemming from the presence of nuisance parameters is crucial. Namely our findings suggest that using nominal values for these parameters, as is often done in practice, without taking into account the resulting modeling error, can lead to overconfident and heavily biased results. We also show that the BAE approach can be used to account for the additional model uncertainty at no additional cost at the online stage.

A novel tilt sensor for studying ice deformation: application to streaming ice on Jarvis Glacier, Alaska

AbstractWe developed a tilt sensor for studying ice deformation and installed our tilt sensor systems in two boreholes drilled close to the shear margin of Jarvis Glacier, Alaska to obtain kinematic measurements of streaming ice. We used the collected tilt data to calculate borehole deformation by tracking the orientation of the sensors over time. The sensors' tilts generally trended down-glacier, with an element of cross-glacier flow in the borehole closer to the shear margin. We also evaluated our results against flow dynamic parameters derived from Glen's exponential flow law and explored the parameter space of the stress exponentnand enhancement factorE. Comparison with values from ice deformation experiments shows that the ice on Jarvis is characterized by highernvalues than that is expected in regions of low stress, particularly at the shear margin (~3.4). The highernvalues could be attributed to the observed high total strains coupled with potential dynamic recrystallization, causing anisotropic development and consequently sped up ice flow. Jarvis'nvalues place the creep regime of the ice between basal slip and dislocation creep. TuningEtowards a theoretical upper limit of 10 for anisotropic ice with single-maximum fabric reduces thenvalues by 0.2.

Glacial Steady State Topography Controlled by the Coupled Influence of Tectonics and Climate.

Glaciers and rivers are the main agents of mountain erosion. While in the fluvial realm empirical relationships and their mathematical description, such as the stream power law, improved the understanding of fundamental controls on landscape evolution, simple constraints on glacial topography and governing scaling relations are widely lacking. We present a steady state solution for longitudinal profiles along eroding glaciers in a coupled system that includes tectonics and climate. We combined the shallow ice approximation and a glacial erosion rule to calculate ice surface and bed topography from prescribed glacier mass balance gradient and rock uplift rate. Our approach is inspired by the classic application of the stream power law for describing a fluvial steady state but with the striking difference that, in the glacial realm, glacier mass balance is added as an altitude‐dependent variable. From our analyses we find that ice surface slope and glacial relief scale with uplift rate with scaling exponents indicating that glacial relief is less sensitive to uplift rate than relief in most fluvial landscapes. Basic scaling relations controlled by either basal sliding or internal deformation follow a power law with the exponent depending on the exponents for the glacial erosion rule and Glen's flow law. In a mixed scenario of sliding and deformation, complicated scaling relations with variable exponents emerge. Furthermore, a cutoff in glacier mass balance or cold ice in high elevations can lead to substantially larger scaling exponents which may provide an explanation for high relief in high latitudes.

The mechanics of a glacier snout

AbstractMeasurements made on a temperate glacier within 200 m of its wedge-shaped terminus cannot be interpreted as simple laminar flow. Instead they are fully explained by a model based on the nonlinear (n ≈ 3) Glen flow law that superposes longitudinal strain rate and simple shearing.

Influence of pre-existing microstructure on mechanical properties of marine ice during compression experiments

AbstractMarine ice is an important component of ice shelves in Antarctica. It accretes in substantial amounts at weak points and below ice shelves. It is likely to exhibit peculiar rheological properties, which are crucial to understanding its potential role in stabilizing ice-shelf flow. Due to its location and consolidation processes, marine ice can present a variety of textures which are likely to influence its rheological properties. We present a new dataset of unconfined uniaxial compression experiments on folded marine ice samples that have been cut at various angles to the folds. Texture and fabric analyses are described ‘before’ and ‘after’ the deformation experiment. It is shown that, in the given stress configuration, the geometry of the anisotropy controls the rheological behaviour of the marine ice. During secondary creep, folded marine ice is harder to deform than weakly textured ice when compressed parallel or perpendicular to the folds' hinges, while the reverse is true for ice compressed at 45°. The observed range of values for the n exponent in Glen's flow law is between 2.1 and 4.1. Surprisingly, we see that tertiary creep tends to develop at a higher total strain than for randomly oriented impurity-free meteoric ice.

Accuracy of the zeroth- and second-order shallow-ice approximation – numerical and theoretical results

Abstract. In ice sheet modelling, the shallow-ice approximation (SIA) and second-order shallow-ice approximation (SOSIA) schemes are approaches to approximate the solution of the full Stokes equations governing ice sheet dynamics. This is done by writing the solution to the full Stokes equations as an asymptotic expansion in the aspect ratio ε, i.e. the quotient between a characteristic height and a characteristic length of the ice sheet. SIA retains the zeroth-order terms and SOSIA the zeroth-, first-, and second-order terms in the expansion. Here, we evaluate the order of accuracy of SIA and SOSIA by numerically solving a two-dimensional model problem for different values of ε, and comparing the solutions with afinite element solution to the full Stokes equations obtained from Elmer/Ice. The SIA and SOSIA solutions are also derived analytically for the model problem. For decreasing ε, the computed errors in SIA and SOSIA decrease, but not always in the expected way. Moreover, they depend critically on a parameter introduced to avoid singularities in Glen's flow law in the ice model. This is because the assumptions behind the SIA and SOSIA neglect a thick, high-viscosity boundary layer near the ice surface. The sensitivity to the parameter is explained by the analytical solutions. As a verification of the comparison technique, the SIA and SOSIA solutions for a fluid with Newtonian rheology are compared to the solutions by Elmer/Ice, with results agreeing very well with theory.

Data assimilation using a hybrid ice flow model

Abstract. Hybrid models, or depth-integrated flow models that include the effect of both longitudinal stresses and vertical shearing, are becoming more prevalent in dynamical ice modeling. Under a wide range of conditions they closely approximate the well-known First Order stress balance, yet are of computationally lower dimension, and thus require less intensive resources. Concomitant with the development and use of these models is the need to perform inversions of observed data. Here, an inverse control method is extended to use a hybrid flow model as a forward model. We derive an adjoint of a hybrid model and use it for inversion of ice-stream basal traction from observed surface velocities. A novel aspect of the adjoint derivation is a retention of non-linearities in Glen's flow law. Experiments show that in some cases, including those nonlinearities is advantageous in minimization of the cost function, yielding a more efficient inversion procedure.

Analysis and Finite Element Approximation of a Nonlinear Stationary Stokes Problem Arising in Glaciology

The aim of this paper is to study a nonlinear stationary Stokes problem with mixed boundary conditions that describes the ice velocity and pressure fields of grounded glaciers under Glen's flow law. Using convex analysis arguments, we prove the existence and the uniqueness of a weak solution. A finite element method is applied with approximation spaces that satisfy the inf-sup condition, and a priori error estimates are established by using a quasinorm technique. Several algorithms (including Newton's method) are proposed to solve the nonlinearity of the Stokes problem and are proved to be convergent. Our results are supported by numerical convergence studies.

Internal structure and ice content of Reichenkar rock glacier (Stubai Alps, Austria) assessed by geophysical investigations

AbstractThe Reichenkar rock glacier (Tyrol, Austria) is a typical tongue‐shaped, 1400 m long, ice‐cored active rock glacier, which connects to a debris‐free cirque glacier. Aerial photographs from 1954 and 1990 indicate its mean surface velocity to be 0.6 m/a while a photograph from 2003 and annual global positioning system (GPS) measurements since 1998 show that velocities in the past decade have increased to 3 m/a. Integration of ground‐penetrating radar (GPR), seismic and gravimetric data reveals that the Reichenkar rock glacier consists of four layers. The uppermost debris layer has an average thickness of about 5 m and is underlain by ice‐rich permafrost with an average thickness of about 25 m. A prominent reflector detected by GPR is identified as the top of an unfrozen till layer located a few metres above the bedrock. Seismic refraction data clearly indicate the boundary between till and bedrock. The geophysical interpretation shows that the ice‐rich permafrost of the rock glacier has an ice content of 45–60%, depending on assumptions concerning the air content of the ice. Creep velocities calculated from the geophysical model, ice contents and an extension of Glen's flow law are in good accordance with observed surface velocities. Copyright © 2007 John Wiley & Sons, Ltd.

Numerical studies of ice flow over subglacial geothermal heat sources at Grímsvötn, Iceland, using Full Stokes equations

The importance of studying geothermal heat sources under glaciers lies in: (1) the potential for jökulhlaups from such sites; (2) the potential for enhanced sliding of ice masses caused by meltwater lubrication of the ice‐bedrock interface; and (3) the potential to gain a deeper understanding of the interaction between the geothermal system and the overlying ice. We study the temporal evolution of an elongated depression on the eastern side of the geothermally active Grímsvötn caldera through numerical 2‐dimensional forward modeling, solving the full Stokes equation. The 100–150 m deep depression formed gradually between 1998 and 2004. The model results show that heat flux estimates based on depression volume are strongly dependent on the value of the rate factor A in Glen's flow law. If flow of ice into the depression is not taken into account, heat flux underestimates of 15–75% occur, corresponding to rate factors of 10–68 × 10−16 s−1 kPa−3. The estimated heat flux at the study site was 260–390 W m−2, with the best estimate being 280 W m−2, obtained from the best estimate of A = 23 × 10−16s−1 kPa−3. The total power of the modeled site was 250–300 MW, about one tenth of the total heat output of Grímsvötn. This heat flux is of comparable magnitude to that of other powerful subglacial geothermal areas. Finally, as the maximum modeled strain rates (1 × 10−7 s−1) are within the range of the applicability of Glen's flow law, this suggests that the full Stokes approach should be valid for the study of many subglacial geothermal areas.

Creep and recrystallization of large polycrystalline masses. III. Continuum theory of ice sheets

This work sets forth the first thermodynamically consistent constitutive theory for ice sheets undergoing strain-induced anisotropy, polygonization and recrystallization effects. It is based on the notion of a mixture with continuous diversity, by picturing the ice sheet as a ‘mixture of lattice orientations’. The fabric (texture) is described by an orientation-dependent field of mass density which is sensitive not only to lattice spin, but also to grain boundary migration. No constraint is imposed on stress or strain of individual crystallites, aside from the assumption that basal slip is the dominant deformation mechanism on the grain scale. In spite of the fact that individual ice crystallites are regarded as micropolar media, it is inferred that couples on distinct grains counteract each other, so that the ice sheet behaves on a large scale as an ordinary (non-polar) continuum. Several concepts from materials science are translated to the language of continuum theory, like, for example, lattice distortion energy , grain boundary mobility and Schmid tensor , as well as some fabric (texture) parameters like the so-called degree of orientation and spherical aperture. After choosing suitable expressions for the stored energy and entropy of dislocations, it is shown that the driving pressure for grain boundary migration can be associated to differences in the dislocation potentials ( viz . the Gibbs free energies due to dislocations) of crystallites with distinct c -axis orientations. Finally, the generic representation derived for the Cauchy stress is compared with former generalizations of Glen's flow law, namely the Svendsen–Gödert–Hutter stress law and the Azuma–Goto–Azuma flow law.

On the relationship between surface and basal properties on glaciers, ice sheets, and ice streams

The influence of basal disturbances on the steady state surface topography and on surface velocities of glaciers and ice sheets is investigated numerically. This is done for finite amplitude basal perturbations in both bed topography and basal slipperiness using a nonlinear ice rheology and a nonlinear sliding law. The effects of varying the exponent n in Glen's flow law on transfer characteristics are mainly quantitative and do not affect qualitative aspects of the transfer amplitudes and phase shifts, such as the number of maxima and inflection points when plotted as functions of wavelength. In particular, the well‐known maximum in bed‐to‐surface transfer amplitude for a Newtonian medium at sufficiently high slip ratios (ratio between mean sliding velocity and mean ice deformational velocity) also forms for n > 1. Transfer amplitudes generally become smaller with increasing n for wavelengths less than about 3 times the mean ice thickness (h). For larger wavelengths the situation is reversed and transfer amplitudes increase with n. For active ice streams, characterized by high basal slipperiness, low surface slopes (<0.5°), and n = 3, topographic transfer amplitudes are generally large (>0.5) and fairly constant for all wavelengths longer than about 3–5h. With increasing n and decreasing surface slope the lower limit of wavelengths over which horizontal stress gradients can be ignored increases markedly. Perturbation solutions for Newtonian medium are found to give an accurate description of the transfer characteristics for bedrock amplitudes up to 50% ice thickness and for fractional amplitudes of slipperiness perturbations up to 0.5.

The PSU/UofC finite-element thermomechanical flowline model of ice-sheet evolution

Ice-sheet modeling to understand past changes and project future ones remains limited by uncertainties in key parameters as well as by shortage of computational resources. A fast two-dimensional dynamic/thermodynamic flowline ice-sheet model has been developed and benchmarked to complement three-dimensional models by allowing a more-thorough exploration of parameter space. A nonlinear Glen flow law with an exponent equal to 3 is assumed for ice. A diffusion formulation of the continuity equation for mass balance of ice sheets is employed. The ice-sheet model is coupled to an elastic lithosphere/relaxed asthenosphere isostatic bedrock model. Snow and superimposed ice thicknesses are determined using advective continuity equations. These thicknesses are then used in the parameterization of the surface accumulation and ablation rates under specified meteorological conditions constrained by ice- and ocean-core data. Heat-flow continuity is maintained by the time-dependent advection/diffusion equation in the ice and time-dependent diffusion equation in the underlying rock. We conduct the first benchmarking of which we are aware of a 2-D model against the European Ice-Sheet Modeling Initiative's (EISMINT) Level 2 and Level 3 intercomparison “Greenland” experiments. Appropriate behavior is simulated, with deviations from results of a three-dimensional model that are fully explainable as arising only from the change in dimensionality. Basal temperatures are quite accurate when compared to data and results from 3-D models.

Finite element approximations of a glaciology problem

In this paper we study a model problem describing the movement of a glacier under Glen's flow law and investigated by Colinge and Rappaz [Colinge and Rappaz, ESAIM: M2AN 33 (1999) 395–406]. We establish error estimates for finite element approximation using the results of Chow [Chow, SIAM J. Numer. Analysis 29 (1992) 769–780] and Liu and Barrett [Liu and Barrett, SIAM J. Numer. Analysis 33 (1996) 98–106] and give an analysis of the convergence of the successive approximations used in [Colinge and Rappaz, ESAIM: M2AN 33 (1999) 395–406]. Supporting numerical convergence studies are carried out and we also demonstrate the numerical performance of an a posteriori error estimator in adaptive mesh refinement computation of the problem.