Glacier Mechanics Research Articles

Crossover scaling phenomena for glaciers and ice caps

ABSTRACTWhile the terms ‘glacier’ and ‘ice cap’ have distinct morphological meanings, no easily defined boundary or transition distinguishes one from the other. Despite this, the exponent of the power law function relating volume to surface area differs sharply for glaciers and ice caps, suggesting a fundamental distinction beyond a smoothly transitioning morphology. A standard percolation technique from statistical physics is used to show that valley glaciers are in fact differentiated from ice caps by an abrupt geometric transition. The crossover is a function of increasing glacier thickness, but it owes its existence more to the nature of the underlying bedrock topography than to specifics of glacier mechanics: the crossover is caused by a switch from directed flow that is constrained by surrounding bedrock topography to unconstrained radial flow of thicker ice that has subsumed the topography. The crossover phenomenon is nonlinear and rapid so that few if any glaciers will have geometries or dynamics that blend the two extremes. The exponents of scaling relationships change abruptly at the crossover from one regime to another; in particular, the volume/area scaling exponent will switch from γ = 1.375 for glaciers to γ = 1.25 for ice caps, with few, if any, ice bodies having exponents that fall between these values.

Glacier volume‐area relation for high‐order mechanics and transient glacier states

Glacier volume is known for less than 0.1% of the world's glaciers, but this information is needed to quantify the impacts of glacier changes on global sea level and regional water resources. Observations indicate a power‐law relation between glacier area and volume, with an exponentγ≈ 1.36. Through numerical simulations of 3D, high‐order glacier mechanics, we demonstrate how different topographic and climatic settings, glacier flow dynamics, and the degree of disequilibrium with climate systematically affect the volume‐area relation. We recommend more accurate scaling relations through characterization of individual glacier shape, slope and size. An ensemble of 280 randomly‐generated valley glaciers spanning a spectrum of plausible glaciological conditions yields a steady‐state exponentγ= 1.46. This declines to 1.38 for glaciers that are 100 years into a sustained retreat, which corresponds exceptionally well with the observed value for present‐day glaciers.

A short history of the thermomechanical theory and modeling of glaciers and ice sheets

AbstractObservations of glacier flow and explanations of its origin started as early as the 18th century. Several mechanisms were suggested before gravity-driven viscous flow became the accepted theory of glacier flow in the 1950s, the early years of the Journal of Glaciology. Since the viscosity of ice is strongly temperature-dependent, the topic of glacier and ice-sheet dynamics became essentially a fluid-dynamical problem. The availability of growing computing power turned the field of glacier mechanics and thermodynamics into a field of numerical modelling with increasing sophistication.

Dielectric Permittivity Measurements on Ice Cores: Implications for Interpretation of Radar to Yield Glacial Unfrozen Water Content

The dielectric permittivity of glaciers and ice caps can be measured in field or airborne surveys using ice-penetrating radar. Permittivity contrasts in polar ice caps indicate ice stratigraphy and age, whereas those in temperate glaciers have been interpreted as changes in unfrozen water content, which is an important control on glacier mechanics. Many previous workers have assumed simple relationships between permittivity (inferred from radar velocity) and unfrozen water content, but these relationships have never been verified in the laboratory. Here, we present measurements of the dielectric properties of ice cores from a temperate glacier in Switzerland, using the Time Domain Reflectometry (TDR) technique, which has a measurement frequency close to that of radar. The objectives of the measurement were to quantify the effects of intercrystalline unfrozen water and air content on ice dielectric permittivity. TDR probes were specially designed and built for ice core measurement to allow them to be pressed onto the ice core surface, and to maximize the signal travel time. The dielectric permittivity of the ice cores was primarily dependent on air content. Most cores had dielectric permittivities of between 3.1 and 3.3, which correspond with literature values for air-free ice above the Debye dispersion frequency (which is typically in the kHz range). Ice cores from accumulation areas of the glacier, from ice which had not been deeply buried, showed lower dielectric permittivity values owing to significant air content. The effects of unfrozen water present within the intercrystalline vein system on dielectric permittivity were too small to detect at temperatures below [Formula: see text]. This is in accord with theoretical predictions, which suggest unfrozen water contents of [Formula: see text] for freshwater ice at such temperatures. Significant water contents inferred from radar surveys of glaciers must indicate well rotted ice at the pressure melting point or the presence macroscale water bodies.

Reversal of ice motion during the outburst of a glacier-dammed lake on Gornergletscher, Switzerland

AbstractDuring the outburst flood of a glacier-dammed lake on Gornergletscher, Switzerland, in July 2004, the drained lake water triggered anomalous glacier motion. At the onset of the outburst, the ice-flow direction in the vicinity of the lake became closer to the central flowline. When the lake discharge magnitude decreased, the flow direction altered such that the ice moved back to the azimuth of the initial motion. At one of the survey points, where the ice flows parallel to the central flowline, the ice accelerated along the pre-event flow direction followed by a 180° backward motion that lasted over 2 days. These observations indicate the impact of the lake outburst on the subglacial and englacial stress conditions; however, the reversal in the flow direction is difficult to explain by drawing on our current understanding of glacier mechanics. The timing and the timescale of the flow-direction changes suggest that the elastic glacier motion and its rebound played a role under the rapidly changing stress conditions, but the Young’s modulus of ice is too large to cause the observed ice motion. Other processes, including basal separation and subglacial sediment deformation, are discussed as possible mechanisms for the reversal of the ice motion.

Variational methods for glacier flow over plastic till

We investigate the mechanics of ice streams and glaciers flowing over a bed consisting of Coulomb-plastic subglacial sediment, or more generally, of channel flows with Coulomb or ‘solid’ friction laws at the boundary. Sliding is assumed to occur if shear stress at the glacier bed attains a prescribed, locally defined yield stress, while no sliding is assumed possible below that yield stress. Importantly, the location of regions of slip and no slip at the bed is not known a priori, but forms part of the solution. By analogy with friction problems in elasticity, we derive a weak formulation as a semi-coercive variational inequality, which admits a unique solution provided a solvability condition ensuring force balance is satisfied. The variational formulation is then exploited to calculate numerical solutions, and we investigate the effect of variations in subglacial water pressure, ice thickness and surface slope on the discharge of a valley glacier with a plastic bed. Significant differences are found between the behaviour of wide and narrow as well as steep and shallow-angled glaciers, and our results further indicate the need to develop models capable of accounting for longitudinal stresses.

Paradoxes and problems with the longitudinal stress approximation used in glacier mechanics

The biharmonic equation in two dimensions is the governing equation for the slow plane flow of a Newtonian fluid. Where the flow domain is thin, and the aspect ratio small, the biharmonic equation can be expanded using the aspect ratio as the expansion parameter. The one-term approximation to the biharmonic equation is the lubrication approximation. In glaciology, the lubrication approximation is extended to deal with special glaciological features, and is known as the shallow ice approximation. The next order term in glaciology is known as the longitudinal stress approximation, but this is not a higher-order expansion (two-term representation) of the biharmonic equation. The first part of this paper deals with this paradox. It shows how that longitudinal stress approximation relates to the truncated biharmonic equation, and carries out a perturbation expansion in the free surface, comparing the decay spectra for surface perturbations of the shallow ice approximation, the longitudinal stress approximation, the truncated biharmonic expansion and the full system of equations. In most glaciological situations, the longitudinal stress approximation is a better approximation to the truncated biharmonic equation, and even where it is not, the loss in accuracy is relatively small. Moreover, the truncated biharmonic equation introduces spurious instabilities, while the longitudinal stress approximation does not. The second part of this paper deals with the perturbation expansion in terms of the widthto-length ratio of a perfectly slippery streaming flow of ice, where flow is retarded at the lateral margins. The case of a linear rheology is considered. The zeroth order equation for the thickness is a non-linear diffusion equation, whereas the first-order equation is linear, even though a finite amplitude perturbation is being considered. The equation is second-order differential in the flow direction, and integral in the transverse direction. Some numerical examples show finite amplitude stability. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Principles of glacier mechanics

Why study glaciers? some basic concepts mass balance flow of a crystalline material the velocity field in a glacier temperature distribution in polar ice sheets the coupling between a glacier and its bed water flow in and under glaciers - geomorphic implications stress and deformation stress and velocity distribution in an idealized glacier applications of stress and deformation principles to classical problems response of glaciers to changes in mass balance problems.

Book Reviews

Books reviewed in this article: Worlds of production: the action frameworks of the economy. Michael Storper and Robert Salais. Digital places: living with geographic information technologies. Michael R Curry. Social formation and symbolic landscape (with a new introduction). Denis E Cosgrove. Principles of glacier mechanics. Roger LeB Hooke. A theory of shopping. Daniel Miller. Contested natures. Phil Macnaghten and John Urry. A world of difference: society, nature, development. Philip W Porter and Eric S Sheppard. Geomorphology: a systematic analysis of late Cenozoic landforms. 3rd edition. Arthur L Bloom. Globalising food: agrarian questions and global restructuring. David Goodman and Michael Watts (eds).

Principles of Glacier Mechanics

Principles of Glacier Mechanics

Principles of Glacier Mechanics

Glaciers are awesome in size and move at a majestic pace, and they frequently occupy spectacular mountainous terrain. Naturally, many Earth scientists are attracted to glaciers. Some of us are even fortunate enough to make a career of studying glacier flow. Many others work on the large, flat polar ice sheets where there is no scenery. As a leader of one of the foremost research projects now studying the flow of mountain glaciers (Storglaciaren, Norway), Roger Hooke is well qualified to describe the principles of glacier mechanics. Principles of Glacier Mechanics is written for upper‐level undergraduate students and graduate students with an interest in glaciers and the landforms that glaciers produce. While most of the examples in the text are drawn from valley glacier studies, much of the material is also relevant to “glacier flatland” on the polar ice sheets.

Review of Principles of Glacier Mechanics by Roger Leb. Hooke

Review of <i>Principles of Glacier Mechanics</i> by Roger Leb. Hooke

Edge function analysis of glacier mechanics problems

The edge function method is considered for the analysis of plane strain problems in glacier mechanics. The essence of the approach is the approximation of the solution by a linear combination of analytical solutions (based on the complex variable formulation of anisotropic elasticity) of the field equations. The unknowns in the linear combination are obtained from a system of equations which follows from the approximation of the boundary conditions by a boundary Galerkin energy method. A range of representative glacier geometries are examined and the presence of areas of tension on the surface is evident. The introduction of small crevasses in the form of widely spaced notches does not seem to have a significant effect in the reduction of that tension, away from the notches. A comparison with a finite element approach indicates that an accurate solution is obtained from the edge function method with fewer degrees of freedom and reduced setup effort. The method is well suited to capturing the singularity at the tip of the crevasse but is limited to wide crevasses and the examination of realistic narrow crevasses would be handled more optimally by a combination of the edge function method with its local accuracy and a conventional method with more robust general geometry capabilities.

Fabric analysis of rock glacier debris mantles, La Sal Mountains, Utah

AbstractMacrofabrics of rock glacier debris mantles display significant variation even among similar microtopographic features. Fabrics of longitudinal ridges of La Sal rock glaciers are equally divided between cluster and girdle distributions. In some cases, fabric patterns resemble those of lahars, implying that the flow mechanics of rock glaciers bears some similarity to that of large‐scale debris flows. Shear zones are found at the margins of many longitudinal ridges, and in at least one case these zones separate the rock glacier from a pair of bounding ridges that resemble debris‐flow levees. Other sites display fabric patterns that indicate that the dominant processes are dry creep of the open‐matrix debris mantle. Weak fabrics are found at sites where rockfall inputs occur on a low‐angle surface, or where a debris mantle is undergoing passive transport by a moving core.

Stress-gradient Coupling in Glacier Flow: IV. Effects of the “T” Term

AbstractThe “Tterm” in the longitudinal stress-equilibrium equation for glacier mechanics, a doubley-integral of ∂2τxy/∂x2wherexis a longitudinal coordinate andyis roughly normal to the ice surface, can be evaluated within the framework of longitudinal flow-coupling theory by linking the local shear stressτxyat any depth to the local shear stressτBat the base, which is determined by the theory. This approach leads to a modified longitudinal flow-coupling equation, in which the modifications deriving from theTterm are as follows: 1. The longitudinal coupling lengthis increased by about 5%. 2. The asymmetry parameterσis altered by a variable but small amount depending on longitudinal gradients in ice thicknesshand surface slopeα. 3. There is a significant direct modification of the influence of localhandαon flow, which represents a distinct “driving force” in glacier mechanics, whose origin is in pressure gradients linked to stress gradients of the type ∂τxy/∂x.For longitudinal variations inh, the “Tforce” varies as d2h/dx2and results in an in-phase enhancement of the flow response to the variations inh, describable (for sinusoidal variations) by a wavelength-dependent enhancement factor. For longitudinal variations in α, the “force” varies as dα/dxand gives a phase-shifted flow response. Although the “Tforce” is not negligible, its actual effect onτBand on ice flow proves to be small, because it is attenuated by longitudinal stress coupling. The greatest effect is at shortest wavelengths (λ2.5h), where the flow response to variations inhdoes not tend to zero as it would otherwise do because of longitudinal coupling, but instead, because of the effect of the “Tforce”, tends to a response about 4% of what would occur in the absence of longitudinal coupling. If an effect of this small size can be considered negligible, then the influence of theTterm can be disregarded. It is then unnecessary to distinguish in glacier mechanics between two length scales for longitudinal averaging ofτb, one over which theTterm is negligible and one over which it is not.Longitudinal flow-coupling theory also provides a basis for evaluating the additional datum-state effects of theTterm on the flow perturbations Δuthat result from perturbations Δhand Δα from a datum state with longitudinal stress gradients. Although there are many small effects at the ~1% level, none of them seems to stand out significantly, and at the 10% level all can be neglected.The foregoing conclusions apply for long wavelengths λh.For short wavelengths (λh), effects of theTterm become important in longitudinal coupling, as will be shown in a later paper in this series.

Stress-gradient Coupling in Glacier Flow: IV. Effects of the “T” Term

AbstractThe “T term” in the longitudinal stress-equilibrium equation for glacier mechanics, a double y-integral of ∂2τxy/∂x2 where x is a longitudinal coordinate and y is roughly normal to the ice surface, can be evaluated within the framework of longitudinal flow-coupling theory by linking the local shear stress τxy at any depth to the local shear stress τB at the base, which is determined by the theory. This approach leads to a modified longitudinal flow-coupling equation, in which the modifications deriving from the T term are as follows: 1. The longitudinal coupling length is increased by about 5%. 2. The asymmetry parameter σ is altered by a variable but small amount depending on longitudinal gradients in ice thickness h and surface slope α. 3. There is a significant direct modification of the influence of local h and α on flow, which represents a distinct “driving force” in glacier mechanics, whose origin is in pressure gradients linked to stress gradients of the type ∂τxy/∂x. For longitudinal variations in h, the “T force” varies as d2h/dx2 and results in an in-phase enhancement of the flow response to the variations in h, describable (for sinusoidal variations) by a wavelength-dependent enhancement factor. For longitudinal variations in α, the “force” varies as dα/dx and gives a phase-shifted flow response. Although the “T force” is not negligible, its actual effect on τB and on ice flow proves to be small, because it is attenuated by longitudinal stress coupling. The greatest effect is at shortest wavelengths (λ 2.5h), where the flow response to variations in h does not tend to zero as it would otherwise do because of longitudinal coupling, but instead, because of the effect of the “T force”, tends to a response about 4% of what would occur in the absence of longitudinal coupling. If an effect of this small size can be considered negligible, then the influence of the T term can be disregarded. It is then unnecessary to distinguish in glacier mechanics between two length scales for longitudinal averaging of τb, one over which the T term is negligible and one over which it is not.Longitudinal flow-coupling theory also provides a basis for evaluating the additional datum-state effects of the T term on the flow perturbations Δu that result from perturbations Δh and Δα from a datum state with longitudinal stress gradients. Although there are many small effects at the ~1% level, none of them seems to stand out significantly, and at the 10% level all can be neglected.The foregoing conclusions apply for long wavelengths λh. For short wavelengths (λ h), effects of the T term become important in longitudinal coupling, as will be shown in a later paper in this series.

On the Mechanics of Surging Glaciers

AbstractObservations of surging glaciers indicate that the flow regime is one dominated by extensional flow. The stress state has substantial longitudinal deviatoric stress. This regime is very different from the conventional model for glacier dynamics which is dominated by shearing flow. In addition, the initiation of surging often involves a compression front which travels down the glacier. The compression front seems to divide an up-stream region of high drag at the base of the glacier from one of low drag which allows the rapid sliding. We develop a framework for the mechanics of glaciers undergoing surging. Relevant issues are the extensional and compression flows, high longitudinal deviatoric stress, and the stress state near the basal discontinuity. We find that some of the down-slope component of glacier weight is borne by longitudinal stress in the rapidly sliding region. This stress thrusts against the slowly moving parts of the glacier. We hypothesize that this effect causes the rapidly sliding part to spread and causes the compression front to travel down the glacier. A criterion for spreading of the rapidly sliding part is developed. The mechanics outlined above are used to develop a highly idealized model for glacier surging. We propose that regions of low drag are relatively common features of glaciers. The surge initiates when conditions are met which allow the surge nucleus to spread. The rapidly sliding region of low drag spreads to a large part of the glacier. Surging ends when the low-drag conditions terminate. Because of the changed state of the glacier, surge nuclei are now stable against spreading. Several years of rebuilding must occur before nuclei are once more unstable. Calculations are performed for the evolution of the shape of Medvezhy Glacier during the surge of 1963. We find a remarkable similarity between the data and our computations.

On the Mechanics of Surging Glaciers

AbstractObservations of surging glaciers indicate that the flow regime is one dominated by extensional flow. The stress state has substantial longitudinal deviatoric stress. This regime is very different from the conventional model for glacier dynamics which is dominated by shearing flow. In addition, the initiation of surging often involves a compression front which travels down the glacier. The compression front seems to divide an up-stream region of high drag at the base of the glacier from one of low drag which allows the rapid sliding. We develop a framework for the mechanics of glaciers undergoing surging. Relevant issues are the extensional and compression flows, high longitudinal deviatoric stress, and the stress state near the basal discontinuity. We find that some of the down-slope component of glacier weight is borne by longitudinal stress in the rapidly sliding region. This stress thrusts against the slowly moving parts of the glacier. We hypothesize that this effect causes the rapidly sliding part to spread and causes the compression front to travel down the glacier. A criterion for spreading of the rapidly sliding part is developed. The mechanics outlined above are used to develop a highly idealized model for glacier surging. We propose that regions of low drag are relatively common features of glaciers. The surge initiates when conditions are met which allow the surge nucleus to spread. The rapidly sliding region of low drag spreads to a large part of the glacier. Surging ends when the low-drag conditions terminate. Because of the changed state of the glacier, surge nuclei are now stable against spreading. Several years of rebuilding must occur before nuclei are once more unstable. Calculations are performed for the evolution of the shape of Medvezhy Glacier during the surge of 1963. We find a remarkable similarity between the data and our computations.

Theoretical Glaciology: Material Science of Ice and the Mechanics of Glaciers and Ice Sheets

Theoretical Glaciology: Material Science of Ice and the Mechanics of Glaciers and Ice Sheets

Kolumban Hutter. Theoretical glaciology; material science of ice and the mechanics of glaciers and ice sheets. Dordrecht, etc., D. Reidel Publishing Co.; Tokyo, Terra Scientific Publishing Co., [1983]. xxxii, 510 p. (Mathematical Approaches to Geophysics.) Guilders 240, $104.

Kolumban Hutter. Theoretical glaciology; material science of ice and the mechanics of glaciers and ice sheets. Dordrecht, etc., D. Reidel Publishing Co.; Tokyo, Terra Scientific Publishing Co., [1983]. xxxii, 510 p. (Mathematical Approaches to Geophysics.) Guilders 240, $104. - Volume 30 Issue 105