## Effect of drifting snow on the velocity field

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...

## 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...

## 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...

## 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...

## 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,...

## A

Model of a flowline down the axis of the Green Bay lobe of the Laurentide Ice Sheet. (a) Temperature specified at the margin from 55 to 20 ka. (b) Profiles of the ice sheet at eight times between 48 and 21 ka (numbers to left of curves). (c) Location of margin as a function of time. (d) Maximum thickness of permafrost. (e) Width of submarginal frozen zone measured upglacier from margin along the flowline. (Redrawn from Cutler et al., 2000, Figures 3, 9, and 11. Used with permission...

## 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)...

## 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...

## 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...

## 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...

## 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....

## 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,...

## 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...

## 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...

## Extension to three dimensions and introduction of deviatoric stresses

It has been found empirically that, to a first approximation, deformation of ice subjected to a normal stress is independent of the hydrostatic pressure or mean stress, P (see discussion of Equation (4.9)). This might well be anticipated from the observation that ice is (nearly) incompressible. In three dimensions, the mean stress is given by P (ffxx + ffyy + ffzz) (9.6) Because deformation is independent of P, we define a new set of stresses, denoted by primes, by a 'xx axx P, a'yy ayy P, 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...

## Positive feedback processes

Before proceeding, it is appropriate to mention some feedback processes that can influence the way in which a polar glacier adjusts to climatic change, but which we will not consider in detail. One process, discussed by Lliboutry (1970), results from the fact that a change in temperature has not only an immediate effect on the mass balance, but also a delayed effect on the flow, owing to the temperature dependence of the flow law. For example, an increase in temperature may increase ablation...

## 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...

## Fe tb f Ub f tbUb fin fe tb td

(4) e f tb < ub I tbub m e frozen bed Up and down arrows represent increases or decreases. Horizontal arrows may be read as leads to or results in. further, driving the system back toward A (Case 3). Similarly, a decrease in e decreases tbub, thus decreasing m and driving the system to a frozen bed condition (Case 4). In other words, there appear to be two stable states, one in which the bed is frozen and one in which it is at the melting point, m dr, and Tb < ( )Td. Raymond (2000) has...

## Parameters describing cumulative deformation

Although yoc describes the deformation of the original sphere, it does not describe either its rotation or its final orientation. For that we need two additional parameters, 0 and < (Figure 13.2) < c is the angle through which the material line that becomes a principal axis in the strained state has rotated, and 0c is the angle that the greatest principal axis makes with the x-axis. Again, the subscript c denotes cumulative. In the pure shear of Figure 13.3a, the axis of maximum elongation,...

## Effect of diffusion

Diffusion occurs whenever fluxes are proportional to gradients. In the present case, the flux, q, is proportional to the slope (or gradient), a. Where a is largest, on the downslope side of a wave, q is highest. Conversely, q is lowest on the upslope side of the wave. Thus, the flux into the wave is diminished and that out of it is enhanced. This tends to decrease the amplitude and increase the wavelength of a wave. As in the case of c (or c0) (Equation (14.5)), an analytical expression for D0...

## Mohrs circle

A convenient way to illustrate the dependence of axx, ayy, and axy on 20 is to use a graphical construction known as Mohr's circle (Figure 9.3). To construct the figure do the following. (1) Draw a rectangular coordinate system with normal stresses (aN) on the abcissa and shear stresses (aS) on the ordinate, and plot points A and A' at (axx, axy) and (ayy, -axy), respectively. (2) Connect points A and A' with a straight line, and draw a circle with B as the center and passing through A and A'....

## Geomorphic implications

Temperature distributions such as that in Figure 6.12 have implications for glacial erosion and deposition and the origin of some glacial land-forms. Erosion rates are likely to be highest where basal melt rates are low, and particularly where meltwater is refreezing to the glacier sole. Thus, we might expect to find that erosion was most intense some distance from the divide. Conversely, the formation of lodgment till by subglacial melting should be most prevalent beneath the ablation zone....

## Info

Where z is the height of the element of ice after it has traveled a distance (xo - x) from (xo, zo). The position of the element at time ti can be determined by selecting a time t1 > 0 and solving Equation (13.10) for x and then Equation (13.11) for z. A plot of x versus z defines the path of the element. By setting z jcx in Equation (13.11) and solving for x we can determine the x-coordinate of the point where the element reaches the glacier surface. Now consider a foliation plane in this...

## Finiteelement models

The finite-element method is another way of obtaining an approximate solution to the governing equations. In both finite-element and finite-difference models, the domain of interest is broken up into a large number of small elements. In early applications of finite-element models to glaciological problems the elements were quadrilaterals, but commercial packages now in use commonly have higher-order element geometries. The corners of elements are called nodes. Unlike finite-difference models,...

## 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...

## Origin of cirques and overdeepenings

Cirques and overdeepened basins in glacier beds, such as those in Figure 8.33, are similar in form. Both have steep headwalls and both tend to have beds with adverse slopes. We will discuss the headwalls first. Headwalls have ragged surfaces, apparently resulting from fracture and removal of blocks of rock. This morphology suggests that they are eroded by glacial quarrying. As we have just discussed, quarrying appears to be a result of water-pressure fluctuations on time scales of hours to...

## General equations for transformation of stress in two dimensions

Consider a domain in a slab of material of unit thickness (measured normal to the page) as shown in Figure 9.1. Stresses are uniformly Figure 9.1. Stresses on a triangular prism of material isolated from a domain. Figure 9.1. Stresses on a triangular prism of material isolated from a domain. distributed over the domain in terms of the x-y coordinate system shown, they are axx and ayx in the x-direction, and ayy and axy in the y-direction. Shear forces on any small element of the domain of unit...

## 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...

## Fdw 1 H

Combining Equations (5.22) and (5.23) gives a result first obtained by Raymond (1983). Equations (5.21) and (5.24) are plotted in Figure 5.9. The difference between them does not appear large, but we will find that it has important consequences. Let us follow this line of inquiry somewhat further, giving special attention to conditions at an idealized divide on an ice sheet. We will assume there is no flow parallel to the divide, so v 0 and , exy, eyx, ezy, and eyz are all 0. Ice on either side...

## Initial conditions and forcing

In earlier chapters we have found that it is necessary to specify conditions on the boundaries of a problem domain in order to obtain a solution for parameters within the domain. Vertical velocities at the surface were specified in Chapter 5, temperatures and temperature gradients in Chapter 6, and stresses and velocities in Chapter 10. In all of these examples, the solutions we sought were steady-state solutions, so all time derivatives were 0. In many modeling studies, time-dependent...

## Effect of a frozen bed

When the temperature at the base of a glacier is below the pressure melting temperature and the ice is frozen to the bed, it is usually assumed that sliding cannot take place. For most purposes, this is a reasonable assumption. However, Shreve (1984) showed that a liquid-like layer, present at interfaces between ice and foreign particles (including a glacier bed), could result in regelation of ice past bumps on the bed at subfreezing temperatures. The presence of the liquid-like layer is...

## 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...

## 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...

## 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...

## 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...

## Flow field

We now have the tools needed to make a first-order estimate of the flow field in a glacier, given bn(x). In a steady-state situation, Equation (5.1) gives the depth-averaged horizontal velocity, u(x), which is probably sufficient for most applications. However, various levels of sophistication could be added Equations (5.16) (with z H) and (5.19) could be solved simultaneously for us and ub, and Equation (5.16) could then be used to estimate the variation in u with depth. This would give...

## Ratelimiting processes

The rate of deformation of a crystal or of a polycrystalline aggregate depends on how rapidly dislocations can move. This, in turn, may depend upon factors such as the effectiveness of the mechanisms resisting motion, the ability of a dislocation to move from one atomic plane to another, and the orientation of the atomic plane in which the dislocation is moving. Usually, one process is significantly more important than the others, principally because it is more effective than the others in...

## Shear stress distribution

To determine the velocity distribution at depth in a glacier, we will find, below, that we need an expression for azx as a function of z. Thus, we digress briefly from the principal objectives of this chapter to derive two similar expressions for azx that are commonly used in the literature. The derivations differ only in the orientation of the axes and of the plane on which azx operates. Consider, first, the situation in Figure 5.3a. The origin is at the surface. The x-axis is taken parallel...

## Solutions for stresses and velocities in plane strain

The coordinate system to be used for the calculation is shown in Figure 10.1 x is parallel to the glacier surface in the direction of flow and z is directed downward normal to the surface. The origin is on the surface Figure 10.1. Coordinate system used in calculating stresses and velocities in plane strain. Figure 10.1. Coordinate system used in calculating stresses and velocities in plane strain. of the slab, which has a thickness h. The velocities are u, v, and w in the x-, y-, and...

## Tests of sliding theories

The only sliding theory that can be reasonably tested with field data is Kamb's approximate nonlinear one. The sliding speed and other data used for the test were collected on Blue and Athabasca Glaciers, using boreholes to the bed and tunnels along the bed. In neither of these techniques was a large enough area of the bed exposed to permit direct measurement of the roughness. Thus, instead, Kamb calculated g and kc from the measured sliding speeds and known glacier geometry (Table 7.1). When...

## Stability of ice streams

As we have noted previously, the driving stress in ice streams is typically only 10-20 kPa, but the strength of the subglacial till layer is significantly lower so only 20 -50 of the driving stress is supported by the bed (Raymond et al., 2001). Because the stress supported by the bed is so small, one might expect that the stability of ice streams would be sensitive to conditions in the bed. Tulaczyk et al. (2000b) and Raymond (2000) have studied this question. They focus on the melt rate, m,...

## Melt rates in conduits

Let us now consider the rate of melting of conduit walls, following Shreve (1972). The total amount of energy available per unit length of conduit, As, per unit time is m3 N m2 _ N - m_ J s m s s Some of this energy must be used to warm the water to keep it at the pressure melting point as ice thins in the downglacier direction. The rest is available to melt ice, thus m As (2rnr p1 L) + pw Cw C --- pi g As Q Q As m m -f--2 -f m (8.10) s m3 kg m3 kgK N m2 m m3 s2 s Here, r is the radius of 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...

## Ramp structures and esker nets

The analysis leading to Equation (8.27) is valid as long as u m. However, near the terminus of a glacier where the ice is thin, m may exceed u as we discussed in connection with Figure 8.11b. In this situation, the flow may slip off the side of the esker and build daughter eskers parallel to the parent, forming an esker net. Alternatively, the conduit may expand leading to deposition of coarser sediment in a Figure 8.29. Sketch of small daughter esker diverging from and later rejoining parent...

## 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...

## Summary

In this chapter we have explored the coupling between glaciers and both rigid and deformable beds. In the former, the dominant processes by which ice moves past irregularities on the bed are regelation and plastic flow. As small obstacles are accommodated more readily by regelation and larger obstacles by plastic flow, there are, theoretically, obstacles of intermediate size that exert more drag on a glacier than do larger or smaller ones, at least if the roughness is constant. This...

## Temperature profiles near the surface of an ice sheet

Earlier (p. 122) we noted that although the temperature at the surface, 0s, varies seasonally, the temperature at a depth of 10 m is very close to the mean annual temperature. Let us now verify this. We adopt a coordinate system with z 0 at the surface and the z-axis pointing downward. At the surface we assume that the seasonal variation can be described by 0(z 0, t) 20r sin( t), where 0r is the annual temperature range (twice the amplitude). Beneath the surface, we expect the oscillations to...

## 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...

## 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 +...

## Temperature distribution in polar ice sheets

In this chapter, we will derive the energy balance equation for a polar ice sheet. Solutions to this equation yield the temperature distribution in an ice sheet and the rate of melting or refreezing at its base. We will study some analytical solutions of the equation for certain relatively simple situations. A solution of the full equation is possible, however, only with numerical models. This is because (1) ice sheets have irregular top and bottom surfaces (2) the boundary conditions - that...

## W a

Theoretical response of the terminus of South Cascade Glacier to a sinusoidal perturbation of amplitude b, period, T, and frequency, w. Curves shown are the phase lag, < p, the time lag < p w, and the amplitude of the response H fr . (After Nye, 1963b, p. 107, Figure 9. Reproduced with permission of the author and the Royal Society, London.) The curve of y in Figure 14.15 is the phase lag between the variation in budget and the response of the terminus. For example, for an...

## The steadystate temperature profile at the center of an ice sheet

Our next task is to solve Equation (6.13) for some relatively simple situations. The first is that at an ice divide, at the center of an ice sheet, a problem first investigated by Robin (1955). The following development follows his closely. The coordinate system we will use is shown in Figure 6.4 x is horizontal and directed down glacier, and z is vertical and positive upward z 0 is at the bed. Figure 6.4. Coordinate system used in calculating the steady-state temperature profile at the center...

## Xx

We give the name principal stresses to the remaining normal stresses, and the axes in this coordinate system are called the principal axes of stress. Similarly, if the coordinate system is oriented such that shear strain rates vanish, the remaining strain rates are called the principal strain rates and the axes are the principal axes of strain rate. Equation (2.14) shows that the octahedral shear stress is the root-mean-square of the principal stress deviators. When the coordinate axes are...

## 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...

## The second integration

To obtain the actual temperature distribution, it is necessary to integrate Equation (6.19). Separating variables as before yields Here, the integration is from some level, z h, in the glacier, where the temperature is O (h), to the surface at z H where the temperature is 0s. (Note that in this case, rather than solve Equation (6.19) as an indefinite integral and then evaluate a constant of integration by applying a boundary condition, it is more convenient to express the integrals as definite...

## 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...

## Tt

Effect of vertical advection on borehole inclination. Figure 12.16. Effect of vertical advection on borehole inclination. as one would expect and as implied by our opening discussion, du dz is one of the most important velocity derivatives. Sensitivity studies suggest that the solutions obtained in these two Barnes Ice Cap experiments do not depend strongly on the assumptions. The most important term is d i dt. In instances where the casing bends abruptly, as at joints, w dli d z...

## Z t 2 e

Where & is the period of the variation (in this case, radians per year) (Carslaw and Jaeger, 1959, p. 65). If the seasonal variation in temperature at the surface is sinusoidal, the temperature profile at any given time during the year can be calculated from Equation (6.31). Some profiles for representative times are shown in Figure 6.8. Now at any given depth, 0max(z,t) and 0min(z,t) occur when z K tt 2, respectively. Thus, using Equation (6.31) to obtain 0(z,t) at these two times, and...

## Velocity solutions in a perfectly plastic medium

We now use the stress solutions, Equations (10.18), to obtain solutions for the velocities from Equations (10.5) through (10.7). From Equations (10.5) and (10.6) we obtain T - - (axx - azz) (10.20) Let us first examine the applicable boundary conditions. The stress solutions are valid only for the thickness h k pgx. Therefore, we seek a velocity solution that will maintain this thickness. Because there is accumulation, bn (or ablation, -bn) at the surface, we know that We will now show that dw...

## 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...

## 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...

## 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...

## Calving

As we discussed in Chapter 3, a great deal of ice is lost from the Greenland and Antarctic Ice Sheets and from tidewater glaciers by calving. However, the calving process is poorly understood. In the case of tidewater glaciers, extending longitudinal strain in the last several kilometers of the glacier usually results in extensive crevassing, so the ice arrives at the calving face in a weakened condition. At the calving face, blocks ranging in size from fractions of a cubic meter to 104 m3...

## References

F. and Sharp, R. P. (1960). Structure of lower Blue Glacier, Washington. Journal of Geology, 68(6), 601-625. Alley, R. B. (1989a). Water pressure coupling of sliding and bed deformation I. Water system. Journal of Glaciology, 35(119), 108-118. (1989b). Water-pressure coupling of sliding and bed deformation II. Velocity-depth profiles. Journal of Glaciology, 35(119), 119-129. (1991). Deforming bed origin for the southern Laurentide till sheets Journal of...

## 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...

## Calculating basal shear stresses using a force balance

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,...

## Calculating cumulative strain

In order to calculate the cumulative strain in a glacier, one first must know the velocity field. One then calculates the path that a particle of ice would follow through the glacier and velocity derivatives at discrete points along the path. To obtain the incremental strain as the ice moves from one point to the next, strain rates are then calculated from the velocity derivatives and multiplied by the time needed for this movement (b) Direction of maximum cumulative extension Figure 13.5. (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...

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

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...

## Deformation of subglacial till

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...

## 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...