Glacier Mechanics

Basal temperatures in Antarctica comparison of solutions using the Column model and a numerical model

Where Basal Drag

The reliability and weaknesses of the Column model can be illustrated further by comparing basal temperatures in Antarctica calculated using it (Budd etal., 1971) with those calculated using a state-of-the-art numerical model (Huybrechts, 1990). First, however, it is instructive to discuss some general characteristics of the Antarctic ice sheet that affect the temperature distribution. A digital elevation model (DEM) of the ice sheet is shown in Figure 6.13 (Liu et al, 1999). By constructing...

Effect of drifting snow on the velocity field

Photos Skyline Thule Greenland

Glaciers flow over irregular beds, and thus have undulating surface profiles. Furthermore, their transverse flow patterns may be influenced by nunataks or irregular valley walls. Patterns of both accumulation and ablation thus can be uneven owing to drifting and to shading from the Figure 5.14. Effect of drifting snow on the surface profile of a glacier. Owing to the additional accumulation in the lee of the surface convexity at A, ws does not need to be as high at B as otherwise would be the...

The role of normal pressure

Another effect that is overlooked in the sliding theories discussed above is that of normal stresses. Budd et al. (1979) carried out some laboratory experiments in which ice blocks upon which a normal load, N, had been placed, were dragged across rough rock surfaces. Temperature control was achieved by immersing the ice and rock surfaces in an ice-water bath. They found that S a t 3 N. The cubic dependence on t might suggest that plastic flow was the dominant sliding process, and this may very...

Mass balance

Glaciers exist because there are areas, generally at high elevations or in polar latitudes, where snow fall during the winter exceeds melt (and other losses) during the summer. This results in net accumulation, and this part of the glacier is thus called the accumulation area (Figure 3.1). As each snow layer is buried, the pressure of the overlying snow causes compaction, and movement of molecules in the liquid and vapor phases results in snow metamorphism. Snow that is more than a year old,...

Principles of Glacier Mechanics

Research Professor Department of Earth Sciences and Climate Change Institute University of Maine, Orono Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title www.cambridge.org 9780521836098 in the Second edition R. LeB. Hooke 2005 This book is in copyright. Subject to statutory...

Summary

Sweden Geothermal Gradient

We began this chapter by deriving the energy balance equation. Given boundary conditions appropriate for a polar ice sheet, solutions to this equation yield the temperature distribution in the ice sheet. The boundary conditions most commonly used are (1) the temperature at the surface, which is approximated by the mean annual temperature, perhaps with a correction for heating by percolating melt water and (2) the temperature gradient at the bed. The latter is based on estimates of the...

A flow law for glacier ice

In the preceding sections of this chapter we have looked at details of the deformation process, and have found some uncertainty, particularly in attempts to identify the rate-limiting process. In the remainder of this book, we will frequently need a simple yet reasonably accurate expression relating stress and strain rate in ice. In general, we will use the expression which, as mentioned briefly in Chapter 2, is often referred to as Glen's flow law, as it was first suggested by John Glen (1955)...

Creep of floating ice shelves

Ice shelves around Antarctica play an important environmental role, as they act as dams, restraining the flow of ice from the interior of the continent. Were they to break up, ice levels in the interior would decrease rapidly over a period of a few centuries, and sea level would rise accordingly. Break up of ice shelves in northeastern North America may have contributed to the collapse of the Laurentide Ice Sheet at the end of the Wisconsinan. Thus, understanding the flow of ice shelves is a...

Why study glaciers

Before delving into the mathematical intricacies with which much of this book is concerned, one might well ask why we are pursuing this topic - glacier mechanics For many who would like to understand how glaciers move, how they sculpt the landscape, how they respond to climatic change, mathematics does not come easily. I assure you that all of us have to think carefully about the meaning of the expressions that seem so simple to write out but so difficult to understand. Only then do they become...

Some basic concepts

In this chapter, we will introduce a few basic concepts that will be used frequently throughout this book. First, we review some commonly used classifications of glaciers by shape and thermal characteristics. Then we consider the mathematical formulation of the concept of conservation of mass and, associated with it, the condition of incompressibility. This will appear again in Chapters 6 and 9.Finally, we discuss stress and strain rate, and lay the foundation for understanding the most...

Weaknesses of present sliding theory

There are a number of processes involved in sliding of ice over a hard bed that are not adequately described in the above theoretical models. An obvious example is the failure to consider frictional forces between rock particles in the basal ice and the underlying bedrock. To study this effect, Iverson et al. (2003) conducted an experiment at the Svartisen Subglacial Laboratory in Norway. The laboratory is situated in a tunnel system in the bedrock beneath Engabreen (the Enga Glacier), an...

Elementary kinematic wave theory

Let us now develop these ideas analytically. In this development, following an analysis by Nye (1960), we consider a slab of ice on a slope, 3(x), with thickness, h(x, t), and surface slope, a(x, t) (Figure 14.2). We assume that dh dx is small and that the slab is of infinite extent in the horizontal direction normal to the x-axis. The surface slope is related to the bed slope by Figure 14.2. Relation among surface slope, a, bed slope, p, and thickness, h. Figure 14.2. Relation among surface...

Analysis of the effect of a small change in mass balance using a perturbation approach

Let us now, following Nye (1960, pp. 561-562), use perturbation techniques to study the change in thickness with time after a small change in mass balance. Consider the situation in which the specific mass balance is shown by the solid line in Figure 14.6. We will refer to the situation represented by this solid line as the 0 or datum or equilibrium state, and analyze the effect of small perturbations from this state such as those represented by the dashed lines in the figure. For example,...

Comparison with observation

Let us now discuss some actual examples of how glaciers have responded to climatic perturbations. We have already mentioned Stor-glaciaren briefly, and noted that estimates of the response time based on Equation (14.30), on a numerical model, and on observation are reasonably consistent with each other, and suggest a time of decades to a century. As expected, tVH is longer than tV, but the magnitude of the difference between them is probably due, in part, to errors in estimating the parameters....

Subglacial water conduits

In Chapter 8 we applied Equation (12.22) to closure of subglacial water conduits. As noted there, problems arise when one attempts to estimate closure rates of semicircular conduits, owing to drag on the bed. Even more profound difficulties arise in attempting to estimate closure rates of broad low conduits, as stresses in the ice are no longer symmetrically distributed about the conduit. Here, we look into another problem of interest the normal stresses on the bed at the boundaries of a...

Submergence and emergence velocities

Earlier (Equation (5.1)), we gained insight into the magnitude of the horizontal velocity by considering a glacier in a steady state, such that its surface profile remained unchanged. Let us now use this idealization to study vertical velocities. In such a steady state, the surface in the accumulation area must everywhere be sinking at a rate that balances accumulation, and conversely in the ablation area. Thus, the vertical velocity at the surface, ws, is clearly related to the net balance...

Ice streams

In the mid 1980s, glaciologists became aware that the flow field in large ice sheets was not as homogeneous as previously believed. In particular, several linear zones of accelerated flow were found in an area of West Figure 5.20. Map of West Antarctica showing the Ross Ice Shelf and the ice streams of the Siple Coast. Compiled from Jacobel et al. (1996), Joughin et al., (1999), and Hulbe et al. (2000). (Reproduced with permission of AGU and the International Glaciological Society.) Figure...

Calculating basal shear stresses using a force balance

Storglaciaren Location

To a first approximation, the basal drag can be estimated from rb pgha (or tb Sfpgha in a valley glacier). However, if longitudinal forces are unbalanced, rb may be either greater or less than pgha. For example, in Figure 12.7, the body force, pgh, has a downslope component, pgha. In addition, there are longitudinal forces F and Fd. If Fu > Fd, as suggested by the lengths of the arrows in the figure, rb will clearly have to be greater than pgha in order to balance forces parallel to the bed,...

A yield criterion

A yield criterion is a statement of the conditions under which deformation will occur. If the condition is not met, there is no deformation, and conversely. The simplest imaginable yield criterion is that of Tresca (1864) & t - & m > t, m 1, 2, 3 or when the difference between any two principal stresses exceeds a material constant, k (determined experimentally for any given material), Figure 9.4. Variation of stain rate, s, with applied stress, a, in perfectly plastic and viscoplastic...

Collapse of a cylindrical hole

The first problem we address is that of the closure of a cylindrical hole in ice. This problem was studied by Nye (1953) in the context of using closure rates of tunnels in ice to estimate the constants in Glen's flow law, and our development is based on Nye's paper. More recently, the theory has been used to analyze two problems in water flow at the base of a glacier (1) the closure of a water conduit, and (2) leakage of water into or away from a subglacial conduit. We used the first of these...

Size and location of water conduits on eskers

It is natural to assume, as a first approximation, that the tunnel within which an esker formed was comparable in size to the esker (Figure 8.28a). This is consistent with the observation that some eskers are composed of coarse gravel with a dearth of sedimentary structures. However, this Figure 8.28. Esker of height Ah with (a) conduit comparable in size to esker (b) small conduit on top of esker and (c) small conduit low on side of esker. Figure 8.28. Esker of height Ah with (a) conduit...

Deformation mechanism maps

Our discussion so far has focused on the type of creep most commonly observed in glaciers, called power-law creep because the creep rate is proportional to the stress raised to some power > 1 (Equation (4.4)). The dominant processes in power-law creep are dislocation glide and climb. For completeness, some other types of creep should be mentioned. In recent years, scientists working on ice deformation mechanisms have found it useful to plot maps showing the deformation mechanisms operating at...

Horizontal velocity in a valley glacier

In a valley glacier, some of the resistance to flow, or drag, is provided by valley sides. To see how this alters the situation, consider first a glacier in a semicircular valley of radius R (Figure 5.6a) and slope a. Balancing forces on a cylindrical surface of radius r and of unit length parallel to the flow gives Here, nr is the area of the surface and pnr2 2 is the mass of ice inside the surface. The latter, multiplied by g sin a, is the total force parallel to the surface that must be...

The Column model

Budd et al. (1971) solved Equation (6.13) in a more general form than those we have considered so far. Calculations using their model, which they refer to as the Column model, can be done by hand. The coordinate system they use is shown in Figure 6.10. The temperature profile is to be calculated at a point a distance x from the divide. Starting again with Equation (6.13), we restrict the model to two dimensions, thus eliminating derivatives in the y-direction we assume that temperature...

Intercomparison of models

Because of the large number of ice sheet models being developed, each employing slightly different approaches and each subject to inadvertent programming errors, a group of 16 modelers developed a set of tests for comparison of models (Huybrechts et al., 1996 Payne et al., 2000). One test, for example, utilizes a square domain, 1500 km on a side, with grid points at 50 km spacing. Initially there is no ice sheet in the domain. A radially symmetric mass balance pattern is specified as are the...

Role of permafrost in ice sheet dynamics and landform evolution

For decades, glacial geologists have speculated on the effects that bed conditions have on ice sheet profiles and dynamics (see, for example, Matthews, 1974 Fisher et al., 1985) and on the relation between basal (b) a'xx in a glacier 200 m thick at the calving face, calculated with the use of a finite-element model. (Reproduced from Hanson and Hooke, 2000. Used with permission of the authors and the International Glaciological Society.) 500 400 300 200 100 Distance from calving face, m thermal...

Components of foliation

The pronounced banded character of glaciers (see, for example, Figures 5.18 and 8.8) has led to considerable confusion. Banding is most prominent in the ablation area once the winter snow has melted. However, banding may also be seen in crevasse walls in the accumulation area, although it has a very different appearance there and most people would, correctly, refer to it as annual layering or sedimentary stratification. The banding is normally subparallel to the nearest bounding surface, be it...

Recrystallization

Recrystallization Snow

Crystals of glacier ice vary in size and also in the degree to which they are interlocked. If there were no bonding across grain boundaries, for example, some polycrystalline ice samples would fall apart into a pile of roughly equant grains, up to a few millimeters in maximum dimension, while others would hang together like a three-dimensional jigsaw puzzle. We will use the term texture to refer to these characteristics of crystal size Figure 4.10. Stress-strain rate data for ice at -10 C....

The coupling between a glacier and its bed

In Chapter 4 we found that the rate of deformation of ice, ee, could be related to the applied stress, oe, by ee (ae B)n (Equation (4.5)). The rigorous basis for this flow law will not be developed until Chapter 9, but some indications of the complexities involved in applying it have already been mentioned. Despite these complexities, calculations using it are reasonably accurate. Computed deformation profiles are an example. This is, in large part, because ice is a crystalline solid with...

Character of the temperature profile

Several temperature profiles calculated from Equation (6.24) are shown in Figure 6.6a. For the conditions assumed, the ice is nearly isothermal in the upper few hundred meters and then warms rapidly near the bed. Higher vertical velocities, resulting from higher accumulation rates at the surface, increase the thickness of the isothermal zone and decrease the basal temperature. In essence, cold ice is advected downward from the surface, and the upward-moving geothermal heat warms this descending...

Threedimensional models of ice sheets

Recently, glaciologists have put considerable effort into modeling entire ice sheets like those in Greenland and Antarctica. The results of some of these models have already been presented in Figures 5.2, 6.14, and 6.15. Armed with models that closely reproduce the characteristics of these modern ice sheets, one can examine the conditions under which past ice sheets expanded to lower latitudes, or predict the behavior of present ice sheets under various scenarios for climate change in the...

Rheology of basal ice

In comparison with ice higher in a glacier, basal ice may have fewer bubbles, a different solute content, and more sediment. In addition, it is quite likely to have more interstitial water because strain heating is significant here, and there is no way to remove this heat other than by melting ice. Finally, the constant changes in stress field as the ice flows around successive bumps may result in zones of transient creep as the crystal structure adjusts to the changes. In a unique experiment...

Deformation of subglacial till

Glideslope

We have known for decades that ice moving over granular subglacial materials can deform these materials. (Herein, the term granular material should be understood to include materials with significant amounts of clay, although a distinction between granular materials and clays is usually made in the soil mechanics literature.) Commonly, the granular material is till, either formed by erosion during the present glacial cycle, or left from a previous one. Recently it has become clear that a large...

Sediment supply to eskers

Eskers form where the sediment load delivered to a subglacial stream exceeds the transport capacity of the stream. The debris-laden basal ice Figure 8.26. (a) Map of the Penobscot River and a section of the Katahdin esker near Medway, Maine. Near the middle of the map, the esker leaves the valley of the river and trends south-southwestward up a small tributary valley. (b) Map of equipotential contours beneath a glacier with a southward surface slope of 0.0048. The esker generally follows a...

Water pressure and glacier quarrying

Quarrying is an important process of glacier erosion. In quarrying, blocks of bedrock must first be loosened, either along preglacial joints or along fractures formed by subglacial processes. Then they must be entrained by the glacier. Because rock fragments that have been loosened but not removed are uncommon on deglaciated bedrock surfaces, Hallet (1996) argues that loosened blocks are readily entrained. He thus concludes that fracture must be the rate-limiting process. To analyze the...

Radar stratigraphy

Prior to World War II, pilots flying over Greenland and Antarctica found that their radar altimeters were giving unreliable data. Upon investigation, it was discovered that the radar waves were passing through the ice sheet and reflecting from the bed (Waite and Schmidt, 1961). Thus was born the tool of radio echo-sounding of glaciers (Gogineni et al., 1998). Initially, the primary objective was to determine the thickness of the ice, as previously gravity measurements, seismic profiling, and...

Horizontal velocity at depth in an ice sheet

Demorest (1941, 1942) argued that the horizontal velocity in a glacier should increase with depth. He thought that the pressure of the overlying ice would soften the deeper ice, making it flow faster. Nye (1952a), however, pointed out that this concept was physically unsound because the faster-moving deeper ice would exert a shear stress on the overlying ice, and there would be no corresponding resisting forces to oppose this shear stress. Therefore, the overlying ice must move at least as fast...

The upper part of the englacial hydraulic system

Veins and the initial development of passages Nye and Frank (1973) argued that veins should be present along boundaries where three ice crystals meet, and that at four-grain intersections these veins should join to form a network of capillary-sized tubes through which water can move. They thus concluded that temperate ice should be permeable. Such capillary passages have been observed in ice cores obtained from depths of up to 60 m on Blue Glacier, Washington (Figure 8.1a) (Raymond and...

Finitedifference models

Finite-difference modeling is basically an extension of numerical integration. The defining characteristic of the finite-difference method is that gradients in a parameter are approximated by obtaining values of the parameter at grid points and dividing by the distance between the grid points. A simple example is the calculation of a temperature profile in the ablation area of a glacier. The relevant equation is (Equation (6.29)) which, again, cannot be integrated analytically. We start out, as...

Transverse profiles of surface elevation on a valley glacier

In the ablation area of a valley glacier, transverse profiles of surface elevation are commonly convex upward (Figure 5.11a), whereas in the accumulation area they are concave upward (Figure 5.11b). This can be understood by considering the emergence and submergence velocities. In a steady-state situation, ws cannot be zero along the margins of a glacier in either the accumulation area or the ablation area because there is accumulation or ablation, respectively, in these locations. However, the...

Measurement of velocity

Before describing the velocity field, a brief overview of measurement techniques may be helpful. In the early days of glaciology, velocity measurements were commonly made by triangulation from fixed points on stable surfaces off of the glacier. I have spent many hours peering through a theodolite at stakes drilled into a glacier. In the 1970s, electronic theodolites with laser distance-ranging capabilities greatly reduced the effort needed to make a measurement. Because the distance can be...

Analysis of boreholedeformation data

Our next example is drawn from the work of Shreve and Sharp (1970) and deals with the analysis of inclinometry data collected in boreholes that are undergoing deformation. In the simplest case, we might assume that at depth d, azx Sfpgda, and that successive measurements of the inclination of a borehole would give du dz. Then zx V2(9 u d z + d w dx) and, if the deformation is entirely simple shear, d w dx 0. Thus, measurements of the change in inclination at several depths would permit a...

The invariants in plane strain

Let us now examine the relation between the invariants in plane strain (Equations (9.5)) and those in Equations (9.8). By plane strain we mean that there is no deformation in one of the coordinate directions, in this case the z-direction. As deformation is caused by deviatoric stresses, this implies that aZz, a'xz, and a are all 0. From Equation (9.7) we thus have so azz P, and then from Equation (9.6) (Note that since azz P, azz does not equal 0 even though a'zz does.) With azz 0, J1 (a'xx +...

Equipotential surfaces in a glacier

In a permeable porous medium, water flows in the direction of the negative of the maximum gradient of the potential, , where is defined by Here, o is a reference potential, Pw is the pressure in the water, pw is the density of water, g is the acceleration of gravity, and z is the elevation above some datum level such as sea level. To gain some appreciation for this concept, consider the situation in a lake (Figure 8.2). Let 1 at point 1 on the lake surface. Moving down a distance Az to point 2...