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Mechanical conditions beneath a surge-type glacier Fischer, Urs Heinrich 1995

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MECHANICAL CONDITIONS BENEATH A SURGE-TYPE GLACIERbyURS HEINRICH FISCHERM.Sc., Trent University, 1989B.Sc. (Honours), Trent University, 1986Bacchalaureus-Diplom, Albert-Ludwigs-Universität, 1984A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of Geophysics and AstronomyWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1995© Urs Heinrich Fischer, 1995In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)__________________Department ofThe University of British ColumbiaVancouver, CanadaDate Ap’I 2 1 ‘ ‘1DE-6 (2/88)ABSTRACTThe interaction of basal processes with the subglacial drainage system is a criticalissue in understanding glacier dynamics. This is especially true for glaciers that exhibita flow instability known as “surging”, characterized by cyclical alternation betweenslow and fast flow regimes. It is accepted that sustained high subglacial water pressurecauses glacier surging by decoupling the glacier from its bed, but how basal hydrologicalconditions control coupling at the ice—bed interface remains the subject of debate.I have applied new investigative techniques for measuring basal sliding and exploring mechanical conditions of the subglacial material at Trapridge Glacier, a smallsurge-type glacier in the St. Elias MOuntains, Yukon Territory, Canada. The data presented in this thesis are unique and important because they were obtained directly atthe glacier bed using new subglacial sensors, designed and constructed at the Universityof British Columbia. Basal sliding is measured with a “drag spool” which consists of amulti-turn potentiometer connected to a spooled string. The drag spool is suspendedwithin the borehole close to the glacier bed, and measures continuously the length ofstring payed out to an anchor in the bed. Mechanical interactions between the glacierand its bed are sensed with a device, dubbed a “ploughmeter”, which is essentially asteel rod instrumented with strain gauges. The ploughmeter is installed at the bottomof a borehole with its tip protruding into subglacial sediment, and measures the bending moment acting near the rod tip which is dragged through the sediment as a resultof glacier sliding.Field data from the drag spool instruments show that, during the melt season,basal sliding can account for up to 45—65% of the total flow observed at the glacier surface. The contribution from ice creep is known to be small, so most of the remainingsurface motion must be attributed to subglacial sediment deformation. Ploughmetermeasurements reveal a spatial variability in subglacial processes or sediment texture.Quantitative analysis of the interaction of the ploughmeter with the basal layer yields11estimates of rheological parameters. If the sediment is assumed to behave as a Newtonian viscous fluid, the estimated effective viscosity is 3.0 x 10—3.1 x 1010 Pas; if it isassumed to behave as an ideal plastic solid, the estimated yield strength is 48—57 kPa.For clast-rich sediments, the rate of collision between clasts and the ploughmeter provides us with an estimated basal sliding rate of 30—50 mm d, in good agreement withthe drag spool results.Diurnal signals recorded with both types of instruments appear to be correlatedto fluctuations in subglacial water pressure. These diurnal variations in the response ofboth instruments can be interpreted in terms of changes in sliding velocity and basalresistance as the mechanical conditions at the glacier bed vary in response to changesin the subglacial water system. I have deveioped theoretical models that describe thesliding motion of ice over a surface with variable basal drag and have demonstratedhow the models can be used in numerical simulations to generate data, which can becompared with my field observations. The results from my model calculations providestrong evidence for time-varying sticky spots and for stick—slip sliding motion, both ofwhich are linked to changes in basal lubrication in response to fluctuations in subglacialwater pressure.111TABLE OF CONTENTSAbstractList of FiguresList of TablesList of SymbolsAcknowledgementsChapter 1. INTRODUCTION1.1 Glacier surging1.2 The ice—bed interface1.2.1 Previous investigations of basal conditions1.3 Trapridge Glacier1.4 Thesis overviewChapter 2. PLOUGHING OF SUBGLACIAL11viiixxxv123456991010121518212327273032SEDIMENT2.1 Introduction2.2 Methods2.2.1 Description of the device2.2.2 Theory of the device . .2.2.3 Calibration2.3 Field observations2.4 Interpretation2.4.1 Qualitative model2.4.2 Quantitative model2.4.2.1 Viscosity estimate . Yield strength estimate2.5 Concluding discussionivChapter 3. FRACTAL DESCRIPTION OF SUBGLACIALSEDIMENT 364.1 Introduction4.2 Field observations4.3 Qualitative interpretation4.4 Quantitative interpretation4.4.1 Physical model and outline of analysis4.4.2 First order perturbation4.4.3 Variations in sliding due to perturbation effects .4.4.4 Description of drag coefficient surface4.4.5 Model results4.5 Concluding discussionChapter 5. DIRECT MEASUREMENT OF SLIDING AT THEGLACIER BED5.1 Introduction5.2 Methods3.1 Introduction3.2 Fractal mathematics3.3 Field observations3.4 Interpretation3.4.1 Fractal scaling of Trapridge till3.5 Estimation of basal sliding rate3.5.1 Discussion3.6 Fractal response of ploughmeter—till interaction . .3.7 Concluding remarksChapter 4. EVIDENCE FOR TEMPORALLY-VARYING“STICKY SPOTS”3637404143495353596060626567• . .. 68697074• . . 75• . . . 8285• . 8586v5.3 Results and discussion 875.4 Water pressure induced variations in glacier sliding 945.4.1 Slider—block model 965.4.2 Elastic block model 975.4.2.1 Ice—bed contact 975.4.2.2 Description of model 985.4.2.3 Mathematical formulation 1015.4.2.4 Model results. 1035.4.2.5 Discussion 107Chapter 6. SUMMARY AND CONCLUSIONS 112REFERENCES.. 116Appendix A. FORCES ON AN ELONGATED BODY INSTOKES FLOW 125viLIST OF FIGURES1.1. Trapridge Glacier study site2.1. Ploughmeter schematic2.2. Ploughmeter circuit2.3. Geometrical parameters of ploughmeter2.4. Ploughmeter calibration2.5. Subglacial water pressure and ploughmeter data from 19912.6. Ploughmeter data from 19912.7. Down-flow/cross-flow decomposition of ploughmeter data2.8. Model till schematic2.9. Generation of synthetic ploughmeter data2.10. Synthetic ploughmeter data2.11. Calculated viscosity of subglacial sediment3.1. Ploughmeter data from 19913.2. Down-flow/cross-flow decomposition of ploughmeter data3.3. Synthetic ploughmeter data3.4. Fractal till3.5. Fractal distribution of Trapridge till3.6. Power spectrum for ploughmeter data3.7. Power spectrum for synthetic ploughmeter data . .3.8. Sliding velocity versus till porosity3.9. Fractal model for fragmentation3.10. Fractal dimension versus slope of power spectrum .3.11. Ploughmeter response versus particle-size distribution3.12. Power spectra for three ploughmeter data sets . .4.1. Subglacial water pressure and ploughmeter data from 19926111213161920222525263141424344485051525556575862vi’4.2. Sensor locations in 19924.3. Ice flow schematic4.4. Drag coefficient surface4.5. Drag coefficient surfaces and ice flow velocity fields4.6. Ice flow velocity field4.7. Idealized synthetic ploughmeter responses4.8. Synthetic ploughmeter responses5.1. Drag spool schematic5.2. Subglacial water pressure and drag spool data from 1992 .5.3. Drag spooi operation5.4. Drag spool data from 1990—925.5. Subglacial water pressure and drag spool data from 1992 .5.6. Slider-block model5.7. Ice—bed contact beneath Trapridge Glacier5.8. Elastic block model5.9. Dimensions of elastic block model5.10. Identification of strain release events5.11. Drag coefficient versus subglacial water pressure5.12. Synthetic drag spool data6467757780818387899094959699100102106107108VII’LIST OF TABLES3.1. Fractal dimensions for fragmented materials 473.2. Fractal dimensions for fragmentation models 554.1. Parameters for “sticky spot” model 765.1. Insertion depths and displacement rates 935.2. Parameters for elastic block model 1045.3. Parameters for drag coefficient calculation 106ixLIST OF SYMBOLSa radius, minor semi-axis (m)a+, a, a± slope of drag coefficient functionA area of part of glacier bed (m2)fitted signal amplitude (V)b constant, b = D/3b+, b, b± intercept of drag coefficient functionB Glen Flow Law parameter (s’ Pa3)B fitted azimuth offset (°)c major semi-axis (m)C constant of proportionalityfitted voltage offset (V)D fractal dimensionfractal dimension (cumulative statistics)Euclidian dimensionDT topological dimensionE Young’s modulus (Pa)f drag coefficient (Pa s m1)fA drag coefficient in connected region of glacier bed (Pa s m’)maximum drag coefficient in connected region (Pas m)f minimum drag coefficient in connected region (Pasm1)fig+ drag coefficient at (upper line) (Pa s m 1)fig_ drag coefficient at p)’ (lower line) (Pa s m’)fB drag coefficient in unconnected region of glacier bed (Pa sm—1)fO background component of drag coefficient (Pasm’)f’ perturbation component of drag coefficient (Pasm’)x4, f peak perturbation drag coefficient (Pasm1)7’ Fourier transform of f’fT frequency (s’)fA spatial frequency (rn’)F force per unit length (N rn’)F0 drag force (N)FL load force (N)force coordinates (N)F8 static friction force (N)g gravitational acceleration (ms2)Gf gauge factorshear modulus of ice (Pa)shear modulus of substrate (Pa)thickness of glacier ice (m)h linear dimension of fractal cubeI second moment of cross-sectional area (m4)k, k, k wavenumber (m’)kp plastic yield strength (Pa)spring constant (Nm’)£ length of strained material (m)L characteristic length scaleload (N)strain equilibration distance in ice (m)strain equilibration distance in substrate (m)L, L linear dimensions of area A of glacier bed (m)m mass (kg)M bending moment (Nm)xin flow law exponentN, N number of objectsnumber of objects (cumulative statistics)N, N number of grid pointsp fluid pressure (Pa)pw subglacial water pressure (m(H20))maximim subglacial water pressure (m(H20))p minimim subglacial water pressure (m(H20))trig+ jg— threshold level for slip initiation (m(H20))r, r, linear dimension of object (m)characteristic distance (m)r distance (m)r* dimensionless distancestrain gauge resistance (ft)S surfaceS(fT), S(fA) spectral density functiont time (s)to characteristic time (s)t dimensionless timeT time interval (s)u distance from neutral surface (m)U translational velocity (ms1)U translational velocity field (ms1)v sliding velocity (m _1)vb basal sliding velocity (ms1)v velocity coordinates (m s’)v5 shear wave velocity (ms1)xliVb°, v background component of basal sliding velocity (ms1)perturbation component of basal sliding velocity (m s’)v, characteristic velocity (m s’)v’ perturbation velocity field (ms’)dimensionless velocity fieldV volume (m3)V’ Fourier transform of v’Fourier transform of vV0, output voltage (V)Vfl input voltage (V)wz, w, width of blocks representing sticky spots (m)x horizontal coordinate, x = x1 (m)x(t) time domain functionspatial coordinates (m)x position vector, distance (m)X Fourier transform of x(t)y horizontal coordinate, y = x2 (m)z vertical spatial coordinate, z = x3 (m)Zg vertical coordinate of strain gauge (m)a area fraction of connected region of glacier bedaT, cr spectral intercept/3 spectral slope7 rescaling factorS transition zone for ice slip (m(H20))Kronecker deltabending strainij strain rate tensor (s1)xliidynamic viscosity of ice (Pa s)fluid viscosity, effective viscosity of till (Pa s)V kinematic viscosity (m2s’)azimuth angle (°)0 surface slope of glacier (°)p1 density of ice (kgm3)Ps density of substrate (kgm3)a bending stress (Pa)shear stress in connected region of bed (Pa)shear stress in unconnected region of bed (Pa)crjj stress tensor (Pa)Fourier transform of rm, u basal shear stress (Pa)background component of basal shear stress (Pa)perturbation component of basal shear stress (Pa)xivACKNOWLEDGEMENTSTo my parents, Hildegard and Alexander,who gave me the motivation to attempt a task such as this, yet neverprecisely knew what I was doing out there in the world of ice.I find great pleasure in acknowledging the help which I received from many sources.Particular thanks are due to my supervisor, Garry Clarke, whose thoughtful guidance,creative thinking and continued encouragement have contributed immeasurably to thisstudy. His enthusiasm is contagious, and it is a delight to work with him. Perhaps oneof Garry’s greatest attributes is his open-minded support: Mens sana in corpore sano— he never discouraged me in my pursuit of other, non-academic interests, but insteadseemed rather suprised to find me working at my desk on occasional weekends (“Didyou break your skis?”).I was very fortunate in being able to share the excitement and frustrations offield oriented research with a dedicated and diverse group of people. I acknowledgeall those who assisted in the field and contributed intellectual insights to this study,especially, Brian Waddington, Shawn Marshall, Tavi Murray, Erik Blake, Dan Stone,Mathew Jull and Barry Narod. I also thank Cohn Farquharson who endured endlessdiscussions on many mathematical details (“Is a factor of 7r/2 really crucial?”). Criticalscientific advice and guidance throughout the course of this research were given bymy supervisory committee members Bob Ellis, Rosemary Knight and Don Russell. Iexpress my gratitude to Dieter Schreiber for his technical assistance in the design andconstruction of subglacial instruments.I have made good friends on my long and winding path to the completion of mythesis. I am grateful to all of them for always being ready to help when I needed itand for sharing many happy times with me. Of those not already mentioned above,I express my appreciation especially to Jane Rea (“I can’t believe you boys went tosee that movie!”), Phil Hammer, Karl Butler, Dave Butler, Martyn Unsworth andLaurent Mingo. Finally, the entire staff, faculty and student body of the Department ofGeophysics and Astronomy is second-to-none for providing an interesting and pleasantworking and social environment which makes this a truly wonderful place to be.This research has been supported by grants from the National Sciences and Engineering Research Council of Canada and the University of British Columbia. Logisticalsupport for the field work was provided by the Kluane Lake Research Station, operated by the Arctic Institute of North America. The data presented in this thesis werecollected in Kluane National Park. I thank Parks Canada and the Government of theYukon Territory for granting permission to conduct field studies in the park.2.xvChapter 1INTRODUCTION“Wanted for hazardous journey. Small wages, bitter cold, ..., constant danger,safe return doubtful. Honour and recognition in case of success.”- Ernest ShackletonThe geometry, velocity and stability of glaciers and ice sheets are, in large part, determined by how flowing ice interacts with the subglacial bed. The bed may be ruggedso that ice must flow over and around obstacles. There may be spatial or temporal variations in the subglacial drag such that at some places or times ice can slide easily and atothers hardly at all. Because of the inaccessibility of the bed, conditions and processesprevailing beneath ice masses are difficult to study and therefore poorly understood.These uncertainties plague all current numerical models of ice sheet behaviour.Soft-bedded glaciers and ice sheets flow by some combination of ice creep, basalsliding and subglacial sediment deformation (Alley, 1989). Most studies of glacier motion have tended to measure spatial and temporal variations in surface motion andinterpreted velocity variations in terms of changes in basal motion (Iken and others, 1983; Kamb and others, 1985, 1994; Iken and Bindschadler, 1986; Kamb andEngelhardt, 1987; Meier and others, 1994). There have been few studies that haveattempted to identify the relative contributions of ice deformation and basal motionto the overall surface velocity. Notable exceptions are the work of Raymond (1971)on Athabasca Glacier and of Hooke and others (1987, 1992) on Storglaciären. Theprocess’es that control coupling at the ice—bed interface, and thus partitioning betweensliding and sediment deformation, are even less completely understood. It is generally accepted that there is a strong correlation between subglacial water pressures1Chapter 1. INTRODUCTION 2and both sliding (Kamb and others, 1985; Iken and Bindschadler, 1986; Kamb andEngelhardt, 1987, Hooke and others, 1989) and sediment deformation (Boulton andHindmarsh, 1987; Iverson and others, 1995). Increased sliding rates can result fromthe submergence of roughness features since high water pressures promote separationof the ice from the bed (Weertman, 1964; Kamb, 1970; Lliboutry, 1979), whereas highpore water pressures reduce the shear strength of sediment allowing increased deformation rates (Boulton and others, 1974). Thus, the basal water system is expected to playa crucial role in controffing the mechanics of glacier and ice sheet motion. However,the complex links between changes in subglacial water pressure and variations in basaldrag, sliding and sediment deformation require further study. In this thesis we describean investigation of the interactions between hydrological and mechanical conditions atthe ice—bed interface of a surge-type glacier.1.1 Glacier surgingSurge behaviour as described by Meier and Post (1969) is characterized by a two-phasecyclic flow instability involving a period of accelerated velocities (surge) and a periodof stagnation in the lower glacier and thickening in the upper reaches (quiescence).During a surge, ice flow rates increase by more than an order of magnitude and largevolumes of ice are rapidly transported from an upglacier reservoir area to a downglacierreceiving area. The active surge phase represents only a relatively short portion of thetotal cycle, usually lasting between one and six years; the intervening quiescent phasemay typically last from 15 to 100 years. Glaciers that surge appear to do so repeatedly.Roughly four percent of contemporary glaciers are known to surge (Sharp, 1988);their geographical distribution is, however, markedly non-random (Post, 1969; Clarkeand others, 1986; Raymond, 1987). Notable concentrations of surge-type glaciers arefound in the mountain ranges of Alaska, Yukon Territory and northwestern BritishColumbia (Clarke and others, 1986), the Canadian Arctic Archipelago, the Pamirs,Chapter 1. INTRODUCTION 3the Tien-Shan, the Caucasus, the Karakoram, the Andes, Svalbard, Iceland and EastGreenland (Paterson, 1994, p. 358), whereas they are virtually absent in other areassuch as the Alps, Scandinavia and the Rocky Mountains (Sharp, 1988). The geographical clustering of surge-type glaciers suggests that environmental factors have aninfluence on surging. Post (1969) considered obvious factors such as climatic conditions, geologic setting, glacier geometry and thermal regime but failed to explain thecontrol on surging. He proposed that a likely explanation of the non-random distribution of surging glaciers involves distinctive subglacial conditions, such as the roughnessand permeability characteristics of glacier beds, a subject about which very little isknown. Clarke and others (1986) and Clarke (1991) used statistical analysis to pursuethe question of a geometrical influence and concluded that glacier length correlatesstrongly with surge behaviour.Although well-documented, surges remain incompletely explained. There is wideagreement that surges involve interaction between the subglacial hydrological systemand sliding processes. High subglacial water pressure decouples the glacier from itsbed and thereby promotes rapid flow. This high water pressure is believed to resultfrom the disabling of the normal subglacial drainage system causing increased waterstorage. Outstanding questions concern the basal processes that trigger hydrologicalchanges and control coupling at the ice—bed interface.1.2 The ice—bed interfacePhysical conditions beneath glaciers and ice sheets lying on unlithified material can becomplex. Sediment is inhomogeneous and rheologically complex; influential parameterssuch as effective pressure can vary both spatially and temporally. If the sedimentarybed is frozen to the overlying ice, slip motion is inhibited and ice is strongly coupledto the bed. Strong coupling can also result from ice infiltrating unfrozen sediment;this process is promoted by low pore water pressure within the bed (Shoemaker, 1986;Chapter 1. INTRODUCTION 4Boulton and Hindmarsh, 1987; Iverson, 1993). Alternatively, complete decoupling ofice and sediment can result from the presence of a water layer at the interface (Shoemaker, 1986; Lingle and Brown, 1987). If the bed is incompletely coupled to overlyingice, both sliding and sediment deformation can contribute to glacier motion. Whenclasts protrude across an interface between partially coupled ice and sediment, ploughing of the substrate can occur (Brown and others, 1987).1.2.1 Previous investigations of basal conditionsDespite the difficulty of making observations at the base of glaciers, numerous studiesof processes and conditions at the ice--bed interface have been performed. Initially,these studies were confined to easily accessible parts of glaciers not flooded by subglacial streams and where the overlying ice is comparatively thin. Early observationsof basal sliding were carried out in deep, marginal crevasses (Carol, 1947), in natural subglacial cavities (Vivian and Bocquet, 1973; McKenzie and Peterson, 1975) andin tunnels excavated in the marginal regions of glaciers (Haefeli, 1951; McCall, 1952;Kamb and LaChapelle, 1964, 1968). Tunnels were also used to access basal sediments,allowing the measurement of deformation rates (Boulton and Jones, 1979; Boultonand Hindmarsh, 1987). The advent of hot-water drilling permitted borehole access incentral regions of glaciers where basal conditions are probably more characteristic ofthe bed as a whole. This development has allowed photographic observations of basalsliding and conditions (Harrison and Kamb, 1970, 1973; Engelhardt and others, 1978;Kamb and others, 1979) and the retrieval of sediment samples (Engelhardt and others, 1990a; Clarke and Blake, 1991). It has also enabled the emplacement of sensors tomeasure directly the deformation of the substrate (Blake and Clarke, 1989; Kohler andProksch, 1991; Blake and others, 1992; Iverson and others, 1995), the shear strengthof sediments (Humphrey and others, 1993; Fischer and Clarke, 1992, 1994a; Iversonand others, 1994, 1995) and the sliding of ice over unlithifled material (Boulton andChapter 1. INTRODUCTION 5Hindmarsh, 1987; Blake and Clarke, 1989; Kamb and Engelhardt, 1989; Engelhardtand others, 1990b; Blake and others, 1994; Fischer and Clarke, 1993, 1994b).1.3 Trapridge GlacierOur study was performed on Trapridge Glacier, a surge-type glacier located withinthe Steele Creek drainage basin in the northern St. Elias Mountains, Yukon Territory,Canada (Fig. 1.1). The glacier last surged sometime between 1941 and 1949 and iscurrently believed to be in the late stages of its quiescent phase. Trapridge Glacier issmall (length 4 km, width 1 km), relatively thin (depth 70—80 m) and presentlyranges in elevation from roughly 2200 m to 2900 m (Clarke and Blake, 1991). Over muchof its ablation area, the surface and basal slopes are both 70 in the direction of glacierflow. Over the past decade, the mean annual flow rate has varied somewhat from yearto year, ranging from 80mmd1 to 100 mmd1 (Waddington, 1992). The directionof ice flow is nearly due east and the glacier terminus has formed a cliff that we referto as “the bulge”. The glacier rests on a deforming sediment substrate (Blake, 1992;Blake and others, 1992) which is thought to be water-saturated (Stone, 1993). Thesubstrate is believed to be up to r’.’lO m thick in places (Stone, 1993) of which the top0.5 m is assumed to be deforming (Blake and Clarke, 1989). The thermal regime ofTrapridge Glacier is sub-polar (Clarke and Blake, 1991; Clarke and others, 1984); thetemperature in the upper layers is subfreezing and warms to the melting point at thebase. The warm-based zone is bounded downslope by a margin of cold-based ice.Chapter 1. INTRODUCTION 6Fig. 1.1: Trapridge Glacier study site. Topographic map of TrapridgeGlacier showing the entire catchment area. The shaded area represents glacierice, based on 1981 photogrammetric mapping (Canadian Government photographs A25841-34,36). The glacier has advanced about 300 m since thattime. The rectangular box indicates the area in which data discussed in thisthesis were collected. Inset shows a location map of the study site in westernYukon Territory, Canada.1.4 Thesis overviewA persistent question is what processes control the coupling of ice to the underlyingsediment at the base of glaciers. Without new instruments and extensive subglacialmeasurements, it is doubtful that this question can be settled. To this end, we have developed new instruments to measure basal sliding and sense the mechanical interactionbetween a glacier and its bed. The first of these instruments, which we term a “dragspool”, was originally developed by Erik Blake (Blake, 1992; Blake and others, 1994)Chapter 1. INTRODUCTION 7and deployed during the 1990 field season. The second instrument has been named a“ploughmeter” and was designed and constructed especially for this study.In this thesis, we describe the construction, calibration, field operation and interpretation theory of the two types of instruments. Analysis of ploughmeter data fromthe 1991 field season, in which we interpret our observations in terms of a translational motion through a homogeneous, fine-grained substrate, enabled us to quantifyrheological parameters for the sediment layer beneath Trapridge Glacier. These estimates provide the first continuous in situ determinations of effective viscosity and yieldstrength of subglacial material. For a heterogeneous or clast-rich sediment, the linearviscous assumption is clearly inappropriate and we introduce the concept of clast collision frequency to describe the ploughmeter response. We further demonstrate that theparticle-size distribution of Trapridge sediment is fractal, a fact which is used to examine the self-similar character of the ploughmeter response spectrum. If the subglacialsediment texture is known and assumed to be spatially and temporally homogeneous,the collision frequency asindicated by a ploughmeter can be related to the basal slidingrate.Measurements of subglacial water pressure, basal sliding and ploughmeter responsewere simultaneously recorded during the 1992 field season. These measurements confirm that the interaction of basal processes with the subglacial hydrological system iscomplex. The drag spooi and ploughmeter records appear to be correlated to fluctuations in subglacial water pressure. This correlation suggests that mechanical conditionsat the bed vary temporally in response to changes in the basal water system. We develop a theoretical framework for the sliding motion of ice over a surface with temporaland spatial variations in basal drag. We subsequently show how we can apply ourtheoretical models by numerically generating instrument responses which we can thencompare to field data. Interpretation of our model results leads to a qualitative characterization of the basal stress system.Chapter 1. INTRODUCTION 8The original aspiration of this study was simultaneously to measure basal sliding,subglacial strain, sediment strength and subglacial water pressure. For three consecutive field seasons, we battled the difficulties associated with installing and operatingarrays of drag spools, tilt sensors, ploughmeters and pressure transducers. Recordingdata using these four types of instruments at the same location and over the same timeperiod is an important requirement for a comprehensive analysis. Due to the shortlife span and proneness to failure of some of our devices, in particular the tilt sensors,we never succeeded in making simultaneous measurements involving all four types ofinstruments for any significant length of time. We were nevertheless able to identifyfundamental aspects of the coupling process at the ice—bed interface but a comprehensive study using all four types of instruments remains on the agenda for the TrapridgeGlacier project.All of the material from Chapter 2 and portions of Chapter 5, which deals withdescription of and measurements using drag spools, appeared recently in the Journalof Glaciology (Fischer and Clarke, 1994a; Blake and others, 1994). The substance ofChapter 3 discussing clast coffision frequency and estimation of basal sliding rate iscurrently under review by that journal.Chapter 2PLOUGHING OF SUBGLACIAL SEDIMENT“They [ploughmenJ are so inattentive, as to leave good soil in some places,and turn up till in others.”- A. Dickson, 17652.1 IntroductionFor a glacier resting on a deformable bed, both sliding and subglacial sediment deformation can contribute to total basal motion (Alley, 1989). Because the operation ofthese processes depends upon contact conditions at the ice—bed interface, the mechanical and hydrological coupling between a glacier and its bed are topics of interest. Areview of observations of processes at the ice—bed interface reported previously in theliterature is given in Chapter 1. Of special relevance to the study presented iii thischapter is the work of Humphrey and others (1993) who described an in situ measurement of the strength of deforming subglacial sediment. Their analysis was based on anexamination of the plastic deformation of a drill stem that was inadvertently stuck inthe bed of Columbia Glacier, Alaska.At Trapridge Glacier, Yukon Territory, we are examining processes that controlpartitioning between bottom sliding and sediment deformation by observing how theseflow contributions vary spatially and temporally. By simultaneously measuring subglacial water pressure we seek to quantify the relation between hydrological conditionsat the glacier bed and the mode of basal motion.For the 1991 field season, we devised a new sensor, dubbed a “ploughmeter”,that complements our sensors for measuring subglacial strain (Blake and others, 1992)and glacier sliding (see Chapter 5). The device consists of a steel rod with strain9Chapter 2. PLO UGHING OF SUBGLACIAL SEDIMENT 10gauges bonded onto it. To carry out measurements, the ploughmeter is lowered down aborehole so that its tip protrudes into subglacial sediment. Similar to an ice-entrainedclast, the immersed section of the ploughmeter is dragged through subglacial sediment.By measuring the bending forces acting on the ploughmeter, we extract information onthe mechanical conditions at the ice—bed interface. Hydrological conditions are likelyto play a role because high subglacial water pressures weaken sediment and hence affectploughing. We therefore expect the mechanical properties of the bed material to varytemporally in response to changes in the subglacial water system.In this chapter, we describe experimental techniques and the analysis of datafrom two ploughmeters that were installed at the bed of Trapridge Glacier duringthe 1991 field season. We interpret our ploughmeter observations either in terms ofcollisions with clasts or as a translational motion through a homogeneous, unlithifiedsediment layer. Theoretical analysis of the forces on the ploughmeter has yielded insitu determination of the effective viscosity of the deforming subglacial material.2.2 Methods2.2.1 Description of the deviceThe ploughmeter (Fig. 2.la) consists of a 1.54 m long solid steel rod having a diameterof 1.90 cm, onto which two strain gauge networks are mounted. The lower end of therod terminates in a conical tip. The rod is sheathed in an epoxy-resin-filled, protective,vinyl tube having an inside diameter of 2.54cm and an outside diameter of 3.18 cm.Eight 20.0mm long strain gauges (Tokyo Sokki Kenkyujo Co., Ltd, TML, PLS-20-11)are bonded, using a cyanoacrylate-based adhesive, onto a polished section near the rodtip (Fig. 2.2a) and register fiexure along two axes. A metal cap covers the conical tipto protect the sandwich construction during insertion. A 10.16cm long and 0.95cmdiameter guide hole has been drilled along the axis of symmetry into the upper end ofChapter 2. PLO UGHING OF SUBGLACIAL SEDIMENT 11the rod and serves to guide the percussion hammer (Blake and others, 1992) that isused to insert the instrument.Fig. 2.1: (a) Schematic diagram of the ploughmeter used during the 1991field season. (b) Diagram of ploughmeter operation (SED indicates sediment).To install the ploughmeter at the glacier bed, the device is lowered down a borehole(-..‘5 cm diameter) and hammered into subglacial sediment (Fig. 2.lb). The insertiondepth must be sufficient to ensure that all eight strain gauges are immersed in subglacialmaterial. The upper section of the ploughmeter extends into the borehole and becomespinned by the moving glacier.a<—1.90cm— <—0.95 cmEC— GUIDE HOLEAIRFLOW b—3.18 cm0EPOXY-RESINVINYL TUBE..— STRAIN GAUGEE0E0C.,E00C.,PLOUGI-4METER.— STEEL RODMETAL CAPChapter 2. PLO UGHING OF SUBGLACIAL SEDIMENTFig. 2.2: (a) Arrangement of the eight strain gauges near the tip of thesteel rod. The two strain gauge networks register flexure along two perpendicular axes. (b) Connection of the eight strain gauges in two full Wheatstonebridge circuits.2.2.2 Theory of the device12In the following analysis, the ploughmeter is assumed to be subjected to pure bending.Effects of any internal shear force are ignored and fiexure stresses are associated onlywith the internal bending moment. This assumption is safe because beam length islarge compared to beam diameter and thus the deformation effect of the shear forceis relatively small (Byars and others, 1983, p. 272). For elastic bending of a beam,Hooke’s law states that the bending strain e, defined as the fractional change in length£ of the strained material, is equal to the bending stress cr divided by Young’s modulusE,0a v b(2.1)Chapter 2. PLO UGHING OF SUBGLACIAL SEDIMENT 13The bending stress a equals the bending moment M divided by the sectional modulusI/u, a property of the cross-sectional configuration of the beam. This relation is knownas the elastic flexure equation (Byars and others, 1983, p. 305) and has the formMuU=--(2.2)where I represents the second moment of cross-sectional area about the horizontalcentroidal axis of the beam, and u is the distance from the neutral surface (Fig. 2.3).On the neutral surface, fibers comprising the beam undergo no elongation or contractionwhen the beam is bent; if the beam has a symmetrical cross section, this surface passesthrough the horizontal centroidal axis. Combining Equations (2.1) and (2.2) yields anexpression for the bending strain e in terms of the bending moment M,Mu(2.3)The sign convention is such that strain is tensile for a positive value of u (above theneutral surface) and compressive for a negative value of u (below the neutral surface).Fig. 2.3: Section of ploughmeter showing geometrical parameters (seetext for details).x71__NEUTRAL SURFACEChapter . FLU UGHING OF SUBGLACIAL SEDIMENT 14Eight strain gauges (Fig. 2.2a) are connected as two full Wheatstone bridges(Fig. 2.2b). For each bridge circuit in the unstrained condition, the relationship betweeninput voltage and output voltage V0g takes the form (Murray and Miller, 1992,p. 158),= R3— R2 (2.4)R3+R4 R1+R2unstrainedwhere R1, R2, R3 and R4 are the unstrained values of strain gauge resistance. Definingthe change in resistance caused by strain as R, the strained value of gauge resistanceis R + zR. Thus, for the strained condition, Equation (2.4) becomes— R3+LR R2+LR25- R3 + R3 + R4 + R4 - R1 + R1 + R2 + R2strainedThe strain e is related to the change in gauge resistance R by€ = (2.6)where the constant of proportionality Gf is the gauge factor.Combining Equations (2.3) and (2.6) yields= MUGfR (2.7)The product of Young’s modulus E and the second moment of area I is known as thebending modulus and is a measure of the stiffness of the beam (Byars and others, 1983,p. 306). For the ploughmeter, the bending modulus is the sum of the individual bendingmoduli for the three materials involved (steel, epoxy-resin and vinyl). Equation (2.7)relates the change in gauge resistance /.R to the bending moment M and stiffness ofthe ploughmeter. The bending moment can vary with distance from the ploughmetertip, so Equation (2.7) is position-dependent. Substitution of Equation (2.7) into Equation (2.5) allows calculation of the output voltages of the bridge circuits if the loadingon the ploughmeter is known.Chapter . FL 0 UGHING OF SUBGLACIAL SEDIMENT 152.3 CalibrationPrior to insertion, the ploughmeter is calibrated by clamping it horizontally near theguide hole and at an intermediate point that serves as a fulcrum. Loads are then appliedto the free end by hanging weights from the tip (Fig. 2.4a). Care must be taken toensure that all eight strain gauges lie between the load at the tip and the fulcrum.This configuration is equivalent to a beam that is subject to a bending stress createdby supports at both ends and having a load in-between (Fig. 2.4b). The ploughmeteris calibrated using weights ranging between 1 and 30 kg that correspond to applyinga load ranging from 10 to 300 N. For each load, the ploughmeter is rotated about itslong axis to obtain a calibration for all possible orientations of the device with respectto the direction of glacier flow. Output voltages from each bridge circuit are fitted foreach load, using least squares, to functions of the formV(L, q) = A cos(q5 + B) + C (2.8)where L are the various loads applied during the calibration, A: are the fitted signalamplitudes at those loads, B are the azimuth offsets for each load, C1 are voltage offsets,q represents the azimuth of the load with respect to the local coordinate system of theploughmeter and V is the predicted voltage output. Two sets of functions of the form ofEquation (2.8) are associated with every ploughmeter—one for each of the two bridgecircuits. In a perfect device, the circuit pair would have values of B differing by exactly90° for a given load. In practice, the orientations of the two groups of strain gaugesare not exactly perpendicular and values of B are not orthogonal.Chapter 2. PLO UGHING OF SUBGLACIAL SEDIMENT 16Fig. 2.4: (a) Schematic diagram of ploughmeter forcing in the calibrationconfiguration. (b) Forces on a point-loaded beam which is supported at bothends. With appropriate choices of FL, F1 and F2 the bending in (a) and (b)is identical.The “three-point” load model (Fig. 2.4b) applies to a ploughmeter installed beneath a glacier, if the ploughmeter is elastically bent by a force concentrated near thetip (the glacier bed) and a force part way up the device (the base of the ice) while theupper part of the ploughmeter is held fixed in the ice. This picture is only appropriateif forces are assumed to be small except at these three pinning points. For this case,the field data can be analyzed by computing values of L and from Equation (2.8)F2P LOUG H M ET ERaBEAMF2F,bChapter . FL 0 UGHING OF SUBGLACIAL SEDIMENT 17using a simplex algorithm (Press and others, 1992, p. 402) that minimizes the difference between predicted and observed output voltages. A linear regression extrapolatesvalues of A, B, and C for loads beyond the calibration range.If, in Equation (2.7), we set u equal to the radius of the steel rod, while preservingthe sign convention of Equation (2.3), and take M(zg) = Lizg then Equation (2.5) canbe fitted to the calibration data set. Here, L are the loads used during the calibrationand Zg is the distance between the point where the load is applied and the position ofthe strain gauge (Fig. 2.3). In this way, we obtain values of Young’s moduli for steel,epoxy-resin and vinyl for individual ploughmeters, as well as precise resistance valuesR for the eight strain gauges.For a ploughmeter installed at the glacier bed, the subglacial geometry is likelyto differ substantially from that of the “three-point” calibration (Fig. 2.4), but thecalibration remains valid. Bending forces acting on the ploughmeter are in general notconcentrated near the tip but are distributed along the entire section of the ploughmeter that is immersed in subglacial sediment. We consider these distributed forces in the“Interpretation” section (below). Secondly, at the fulcrum point near the ice—bed interface, pressure melting of the borehole wall might expand the area of contact betweenthe ice and the ploughmeter. As a result, the fulcrum load could become distributedover several centimetres. However, the bending moment M(zg) at the position Zg ofany of the eight strain gauges is not affected as long as the strain gauge is fully immersed in subglacial material and is not located in the region of the distributed fulcrumload. Thirdly, once the ploughmeter is firmly jammed at the base of the borehole, thelong axis of the ploughmeter will not necessarily be normal to the applied load at thetip. In our study, any effects of non-normality are likely to be small. Near-verticalityof boreholes is confirmed by inclinometry. Ploughmeter length greatly exceeds holediameter, so within each hole a ploughmeter is closely aligned with the hole axis.Chapter . PLO UGHING OF SUBGLACIAL SEDIMENT 182.3 Field observationsIn July 1991, two ploughmeters were inserted into sediment beneath Trapridge Glaciernear the centre-line flow markers and approximately 600 m up-flow from the bulge(Fig. 1.1). The insertion sites lie on a single flow line and are approximately 10 mapart. Both boreholes were unconnected to the subglacial drainage system at the timethat ploughmeters were installed. The exact insertion depth of the ploughmeters intosubglacial sediment is uncertain. Hydraulic excavation by the hot water drill is believedto loosen subglacial material to a depth of several decimetres below the ice—bed interface(Blake and others, 1992). It is likely that a ploughmeter, once lowered to the bottom ofthe borehole, settles into this disturbed layer simply by its own weight. The measuredinsertion depth (14cm for 91PLO1 and 8 cm for 91PL02) therefore represents the addedpenetration that results from the hammering procedure. The thickness of the disturbedlayer is estimated as 15—25 cm (Blake and others, 1992), so the insertion depth for theploughmeters could range from 10—40 cm. During the course of our measurements,the surface velocity of Trapridge Glacier was about 100 mmd1. Observations of thedeformation of boreholes reveal that for this glacier no more than 10mmd1 of thesurface velocity can be accounted for by internal ice creep (Blake, 1992). The remaining90mmd1 result from some combination of glacier sliding and sediment deformationthat depends on the degree of subglacial ice—sediment coupling.Figures 2.5 and 2.6 show 13 days of observations for ploughmeters 91PLO1 and91PL02. The data are presented in calibrated, though otherwise raw, form. An obvious feature in the force records for both ploughmeters is a gradual rise during the initial3—5 days of observation. A diurnal signal is superimposed on the gradually rising forcerecorded by ploughmeter 91PLO1 during the initial three days (Fig. 2.5b). Strong diurnal fluctuations in subglacial water pressure are simultaneously observed in a boreholelocated approximately 1 m from ploughmeter 91PLO1 (Fig. 2.5a). At the same time,the azimuth record (Fig. 2.5c) indicates an apparent rotation of the ploughmeter about3500.— 3000zw 250020000U.. 1500100078Liio 76:: 74ID 7268Fig. 2.5: Data from pressure sensor 91P06 and ploughmeter 91PLO1. (a)Subglacial water pressure record. Note the diurnal signal during the initialthree days. Super-flotation pressures correspond to a water level of morethan about 63 m. (b) Force record indicating load applied to the tip of theploughmeter. The arrows show the two data points that correspond to thehighest (H) and lowest (L) loading during the observation period (see text fordetails). (c) Azimuth of the load with respect to the internal coordinates ofthe ploughmeter.its long axis by up to 9°. After the initial phase, the data of ploughmeter 91PLO1display a comparatively smooth character with near-zero ploughmeter rotation. Re-Chapter . PLO UGHING OF SUBGLACIAL SEDIMENT 199080w>U-I-J504030b17 19 21JULYI IC23199125 27 29suits obtained from ploughmeter 91PL02 show a completely different character. DuringChapter 2. FL 0 UGHING OF SUBGLACIAL SEDIMENT 20the entire observation period, the force record (Fig. 2.6a) displays a sawtooth appearance, whereas the azimuth record (Fig. 2.6b) indicates that the ploughmeter appearsto undergo a 25° back-and-forth rotation about its long axis.20001600120080040065555045N< 4017 19 21 23JULY 1991Fig. 2.6: Data from ploughmeter 91PL02. (a) Force record indicatingload applied to the tip of the ploughmeter. (b) Azimuth of the load withrespect to the internal coordinates of the ploughmeter.If we assume that the principal direction of ploughmeter motion is down-flow, wecan decompose the net force values into down-flow and cross-flow components. Theazimuthal angle of the ploughmeter relative to the glacier flow direction is a key elementin this decomposition process; any disturbance on the azimuth record indicates non-zerocross-flow motion. To effect down-flow/cross-flow decomposition, we align the averageazimuth to the glacier flow direction. There could well be partitioning between rotationof the ploughmeter about its long axis and cross-flow motion but this uncertainty does91PL02aI Ib25 27 29Chapter 2. FL 0 UGHING OF SUBGLACIAL SEDIMENT 21not have serious consequences. Any fraction of cross-flow motion that is assigned toploughmeter rotation reduces the cross-flow force component by that fraction, and thedown-flow component will increase slightly to achieve the correct net force. However,since the partitioning has only a minor effect on the down-flow component (Fig. 2.7),we constrain the down-flow/cross-flow decomposition by prohibiting rotation of theploughmeter about its long axis. Decomposition of the data of ploughmeter 91PL02shows that there is no apparent correlation between the records of down-flow and cross-flow force components (Fig. 2.7). Intervals over which a positive correlation seems toprevail are followed by intervals where the correlation seems to be negative or nonexistent.2.4 InterpretationFor three to five days after installation, the ploughmeters are believed to be interactingwith subglacial material that has been loosened by hydraulic excavation during hotwater drilling (Blake and others, 1992). Support for this speculation can be seen in thegradual rise of the force records of both ploughmeters. It is reasonable that this riseoccurs because the ploughmeters are transported from initially loosened and disturbedmaterial to undisturbed bed material. The smooth character of the data of ploughmeter91PLO1 might indicate that the ploughmeter was inserted into an area of the glacier bedthat consists of mainly fine-grained material, but we cannot confirm this conjecture.On the other hand, the sawtooth appearance of the force record for 91PL02 suggeststhat the ploughmeter is interacting with a clast-rich area of the glacier bed. Thisexplanation is supported by the fact that inserting ploughmeter 91PL02 was notablymore difficult than for 91PLO1, suggesting that clasts in the sediment hampered theinsertion process.A relationship between water pressure and ploughmeter—sediment interaction is notsurprising, but we have no complete explanation for the data of ploughmeter 91PLO1.Chapter 2. PLO UGHING OF SUB GLA CIAL SEDIMENT 222000 91PL021500d1,d2131000wC)500-500I I17 19 21 23 25 27 29JULY 1991Fig. 2.7: Down-flow/cross-flow decomposition of the data from plough-meter 91PL02 showing the effect of partitioning between ploughmeter rotationand cross-flow motion. Note that the partitioning has only a minor effect onthe down-flow component (traces d1, d2, and d3 are indistinguishable) andthat cross-flow traces x2 and X3 differ in amplitude but have closely similarform. Trace xl represents 100% ploughmeter rotation. Trace x2 represents50% ploughmeter rotation and 50% cross-flow motion. Trace x3 represents100% cross-flow motion.Vertical uplift of the glacier as a result of super-flotation water pressures could explainthe reduced forcing on the ploughmeter, if the ploughmeter was lifted out of the sediment. However, when pressure drops, the forcing increases, suggesting that re-insertionhas occured. Re-insertion would be difficult to explain unless the ploughmeter is firmlywedged in the hole.Alternatively, softening of the subglacial material during intervals of high waterpressure could account for a reduced forcing. However, a near-instantaneous responseChapter . PLO UGHING OF SUBGLACIAL SEDIMENT 23of the ploughmeter requires rapid penetration of excess water pressure into subglacialsediment. The low pressure diffusivity of typical subglacial sediment makes a fastresponse unlikely.Finally, in areas where the bed is hydraulically isolated, vertical uplift of the glacierdue to an increase in subglacial water pressure should be accompanied by an essentiallysimultaneous decrease in the contact pressure of the ice oito the sediment. This reduction in ice-contact pressure would decrease the effective pressure in the sedimentand, hence, weaken it. The apparent rotation of ploughmeter 91PLO1 during periods• of increased forcing might result from minor irregularities in ploughmeter geometrythat could cause twisting or from pressure-induced cross-flow components of sedimentdeformation.The completely different character of the ploughmeter responses inspires contrasting approaches to further analysis. Our first approach is qualitative and emphasizesthe heterogeneity of subglacial sediment; the second is quantitative and emphasizeslocalized homogeneity. To explain the results for ploughmeter 91PL02 we shall introduce a simple model that represents collisions with clasts. To interpret the results forploughmeter 91PLO1, we shall consider the ploughmeter to be dragged through a layerof homogeneous, unlithified sediment.E.4. 1 Qualitative modelIn an attempt to simulate collisions between individual clasts and a ploughmeter asit is dragged through sediment, we developed a simple numerical model to describea synthetic till created by randomly filling a volume with spheres of different sizes(Fig. 2.8). The spheres represent clasts in a matrix of fine-grained solids and waterfilled voids. We assumed a porosity of 0.3 and a clast size distribution that is basedon the analysis of grain sizes of basal till from Trapridge Glacier (Clarke, 1987). Onlyclasts of the five largest size classes were used, corresponding to spheres of diametersChapter 2. FLU UGHING OF SUBGLACIAL SEDIMENT 248—32 mm. Grains of smaller size were assumed to form the matrix of fines. Higherporosities would be justified if the sediment was fully dilated and constantly deforming.These conditions were inferred for Ice Stream B (Blankenship and others, 1986, 1987),but are not believed to be the case for Trapridge Glacier. However, it is interesting tonote that any changes in the assumptions of porosity and size distribution do not alterthe underlying idea of this simple model and have no direct influence on the qualitativenature of our conclusions.The transverse motion of a vertical line through the volume represents the plough-meter being dragged through subglacial material. Force records are synthetically generated by considering how the path of the vertical line indenter intersects spheres ofthe model till. In our analysis, we view the spheres as being attached to springs so thatthey are constrained to move only in a direction that is perpendicular to the motion ofthe vertical line. The line indenter pushes the spheres to the side as it moves throughthe volume and experiences a force which is proportional to the sideways displacementof the spheres. Depending on whether the point of collision is to the right or left of thecentre of a sphere (Fig. 2.9a), a negative or positive cross-flow component of the forceis calculated, (Fig. 2.9b, lower trace). For the case where the line indenter collideswith two or more spheres, the cross-flow force components calculated for individualspheres add if the centres of all spheres are on the same side of the path of the lineindenter. On the other hand, for spheres with centres located on opposite sides of thepath, individual cross-flow effects will cancel out. To obtain the down-flow componentof the force record (Fig. 2.9b, upper trace) we sum the absolute values of cross-flowforce contributions from individual spheres. The calculation of the force record in thisway does not include any real physics of how forces are exerted onto a ploughmeteras it collides with clasts. This simple geometrical analysis is sufficient, because we areattempting to explain the character, but not the magnitudes, of the force componentChapter . PLO UGHING OF SUBGLACIAL SEDIMENT 25records. Comparison of Figures 2.7 and 2.10 show that the records generated usingthis simple model display many of the features seen in the real data.Fig. 2.8: Schematic diagram of the model tifi. The vertical line representsthe ploughmeter moving through subglacial sediment.Fig. 2.9: Illustration of how records of down-flow and cross-flow forcecomponents are synthetically generated. (a) Plan view of a simple model tillconsisting of seven spheres. (b) Synthetically generated records of down-flow(upper trace) and cross-flow (lower trace) force components for the model tillshown in (a).I-+‘ I IL/L4Chapter 2. PLO UGHING OF SUBGLACIAL SEDIMENT 26Even though it is probable that other descriptions of ploughmeter—till interactioncould lead to similar results, we feel that the above is a simple model that explains theavailable data. Our model also includes reasonable constraints on the nature of thetill. Futhermore, it suggests directions for future effort: if till texture is assumed tobe spatially homogeneous, then the “collision frequency”, indicated by a ploughmeter,is proportional to sliding rate; on the other hand, if the sliding rate is constant, thentemporal variation in the coffision frequency reflects spatial variations in subglacialsediment texture.U)CD>CuI—4-..CuUiC)0LLFig. 2.10: Synthetically generated records of down-flow (upper trace) andcross-flow (lower trace) force components. Note the similar character to thatof Figure I I I0.1 0.2DISTANCE THROUGHI I I0.3 0.4 0.5MODEL TILL (m)Chapter 2. PLO UGHING OF SUBGLACIAL SEDIMENT 272.4.2 Quantitative modelFor the case where the ploughmeter is dragged through a homogeneous sediment layer,the ploughmeter experiences a distributed force rather than a force concentrated nearits tip. We attempt to extract information on the strength of the subglacial materialby estimating how forces are distributed along the ploughmeter. However, the lack ofa complete rheological description of subglacial sediment makes it difficult to calculatethe force distribution on the immersed section of a ploughmeter. We therefore requirea rheological model of the bed material that can provide this information. Till has beenmodelled as a visco-plastic material (Boulton and Hindmarsh, 1987; Alley, 1989) thatis taken to have a yield strength below which no deformation occurs and above whichit behaves as a viscous fluid. In our study, we separately treat the subglacial sedimentas a layer of Newtonian viscous fluid and as an ideal plastic solid. Viscosity estimateTo calculate the force distribution along the section of a ploughmeter that is immersedin sediment, we apply standard hydrodynamic theory to investigate viscous fluid motionaround a finite moving cylinder. In this case, sediment is viewed as a Newtonian fluidhaving viscosity i and the ploughmeter is represented by a cylinder experiencing Stokesflow. Exact solutions for flow around finite cylinders undergoing uniform translationthrough a viscous fluid are unavailable but, by approximating cylinders as ellipsoids,standard solutions can be employed (Happel and Brenner, 1973, p. 227).Under conditions where fluid inertia is negligible, the force distribution along anelongated body can be obtained as an asymptotic expansion involving the ratio ofcross-sectional radius to body length (Batchelor, 1970; Cox, 1970; Tillet, 1970). Fora body of length 2c and cross-sectional radius a that is aligned with the z axis andChapter . PLO UGHING OF SUBGLACIAL SEDIMENT 28moving in the x direction, the force per unit length on the body can be expressed as(see Appendix A for details)F(z) = ,uvL(z; a, c) (2.9)where1 a / z2\’”21n(2c/a)——ln —__.(1__)4 2 a(z)\ C2L(z; a, c)= In(2c/a) 2 ln(2c/a) +1 (2.10)is a characteristic length scale that incorporates the shape and dimensions of the body.In effect, the force per unit length on the body is linearly proportional to the effectivefluid viscosity p, the translational velocity v, and the geometry of the body. Furthermore, the force distribution as given by Equation (2.9) is symmetrical about z = 0.If we assume a sharp ice—bed interface, we can regard it as a symmetry plane. Withthe origin taken to lie on this plane, we need only consider the negative half of theelongated body (—c z 0), which represents the section of ploughmeter immersedin subglacial sediment.In the case of a deforming sediment layer, shear strain rate is assumed constantwith depth. Accordingly, the down-flow velocity of sediment decreases linearly withdistance from the ice—bed interface and the differential velocity between ploughmeterand surrounding sediment increases with depth. For this case, where the ploughmeteris moving through an already shearing sediment layer, the velocity v in Equation (2.9)represents the differential velocity between the ploughmeter and the sediment andbecomes a function of z. The thickness of the deformable sediment layer underneathTrapridge Glacier is assumed to be 0.5 m (Blake and Clarke, 1989).To calculate the force distribution along the section of a ploughmeter immersed insubglacial sediment (Equation (2.9)), we approximate the shape of the ploughmeter asChapter 2. PLO UGHING OF SUBGLACIAL SEDIMENT 29either an ellipsoid (Equation (A.9)) or a cylinder (Equation (A.12)). While the ellipsoidassumption likely underestimates the forces in the lower section of the ploughmeter,end effects at the fiat faces of the cylinder greatly overestimate the forces near theconical tip. We expect that the true result lies somewhere between that yielded by theellipsoidal and cylindrical approximations, and regard the ellipsoid as a better overallrepresentation of the ploughmeter.Once the force distribution on the immersed section of the ploughmeter is known,the bending moment M(zg) at the position Zg of a strain gauge can easily be calculatedaccording toZgM(zg)= j F(z)(zg — z) dz (2.lla)= L v(z)L(z; a, c)(zg — z) dz (2.llb)Here, Zg is a negative value, since the strain gauge is located below the ice—bed interface,and c corresponds to the depth to which the ploughmeter is immersed in sediment. Itis assumed that the viscosity i is constant with depth in the sediment. Equation (2.11)is substituted into Equation (2.7) for all eight strain gauges of a ploughmeter. Anestimate of can be obtained by minimizing (Press and others, 1992, p. 402) theobjective functionf() = > [Q’) — (T4ut) ]2 (2.12)‘‘ observed,i predicted,iThe sum in Equation (2.12) is taken over the two bridge circuits. The data set ofploughmeter 91PLO1 supplies the observed voltage ratios and the predicted voltageratios are calculated using Equation (2.5).By this method, the apparent viscosity of subglacial sediment has been calculatedas a function of ploughmeter insertion depth for three different styles of basal motion(Fig. 2.11). Figures 2.lla and 2.llb represent the two end-member cases of 100% glacierChapter 2. PLO UGHING OF SUBGLACIAL SEDIMENT 30sliding and 10091o bed deformation assuming that the sediment velocity varies linearlywith depth. In Figure 2.llc, we assume that 50% of the 90 mmd1 basal velocity is dueto glacier sliding and 50% is due to bed deformation. Using the record of ploughmeter91PLO1, the calculations were carried out for two data points that correspond to thehighest and lowest loading during the observation period (see arrows labelled H and L inFig. 2.5b). We have not included the initial five days of the record, because we believethese represent a start-up phase that does not characterize ploughmeter interactionwith an undisturbed subglacial bed.Results are shown for both ellipsoidal and cylindrical shapes. We note that the apparent viscosity calculated for the ellipsoid is greater than that for the cylinder, becauseforces near the end of a cylinder exceed those near the tip of an ellipsoid. The increasein calculated viscosity as a function of insertion depth that we note in Figure 2.lla isa consequence of the assumed ellipsoidal geometry of the ploughmeter. Within the uncertainties and assumptions listed above, we can limit the viscosity estimate to a spreadof one order of magnitude. The 100% deformation model yields an upper bound onapparent viscosity in the range 5.6 x io to 3.1 x 1010 Pas; the 100% sliding modelyields a lower bound of 3.0 x 10—6.6 x i0 Pa s. These estimates of Trapridge sedimentviscosity are consistent with those of Blake (1992) and Blake and Clarke (1989). Yield strength estimateFollowing the analysis by Humphrey and others (1993) we can estimate the plasticyield strength of subglacial material by considering plastic flow around a stiff cylinder.We approximate the cylinder by a fiat and long rigid indenter of width equal to thediameter of the ploughmeter. If we define the problem in plane strain, thereby ignoringChapter PLO UGHING OF SUBGLACIAL SEDIMENTC,)Cu0a)CC,,0C)C,)>I—zwaFig. 2.11: Apparent viscosity of subglacial sediment as a function of insertion depth of ploughmeter 91PLO1 for three different styles of basal motion.Solid lines indicate ellipsoidal approximation. Dashed lines indicate cylindrical approximation. Highest and lowest loading (see Fig. 2.5b) is denoted by Hand L, respectively. (a) 100% glacier sliding. (b) 100% sediment deformationin a 0.5 m thick layer. (c) 50% glacier sliding and 50% bed deformation. Alinear velocity—depth profile was assumed for the deforming sediment.31end effects, the stress that the indenter exerts on the semi-infinite plastic body ontowhich it is pressed, is uniformly distributed and can be expressed as (Hill, 1971, p. 340)91PLO1 aI I I I.-8642302520151058642%-bCHL, I I I10 15 20 25 30 35 40INSERTION DEPTH (cm)cT = (2+ ir)kp (2.13)Chapter 2. PLO UGHING OF SUBGLACIAL SEDIMENT 32where k is the plastic yield strength of the material. To close the wake left by apositive indenter, Humphrey and others (1993) introduced a negative indenter of equalstrength and obtained the expressionF = 4a(2 + ir)kp (2.14)where a is the radius of the ploughmeter. If Equation (2.14) describes the dragging ofthe ploughmeter through subglacial sediment, then F denotes the constant force perunit length acting on the immersed section of the ploughmeter.With this constant force distribution, the bending moment M(zg) can be calculatedasZgM(zg) F (zg z) dz (2.15a)4a(2 + ir)kp f (Zg — z) dz (2.15b)where Zg denotes the position of the strain gauge. The development is closely analogousto that of the previous section and proceeds by substituting Equation (2.15) into Equation (2.7) for all eight strain gauges of a ploughmeter. Again, an expression similar toEquation (2.12) is minimized, analyzing the same two data points that were used toobtain a viscosity estimate, to calculate an estimate of kp. The plastic yield strengthobtained by this method is 48—57 kPa.2.5 Concluding discussionOur analysis of the strength of subglacial material includes a number of assumptions.These are listed and briefly discussed as follows:1. Solutions for creeping motion of a viscous fluid around solid bodies are based on theassumption that the bodies are immersed in a homogeneous and incompressibleNewtonian fluid. In effect, the deforming bed is treated as a continuum thatChapter 2. PLO UGHING OF SUBGLACIAL SEDIMENT 33has no internal structure and it is assumed that a no-slip boundary exists at theploughmeter surface. Thus clast-scale interaction processes are neglected.2. The results of the slender-body theory (see Appendix A) are only applicable forlong bodies possessing large length-to-diameter ratios c/a. To get some sense of theerror introduced by applying slender-body theory to short objects, we comparedthe drag calculated using Oberbeck’s formula(Equation (A.3)) to the slender-bodyresult (Equation (A.11)). For c/a = 6.3 (corresponds to an insertion depth of theploughmeter of c = 10 cm) the discrepancy is about 0.19% which reduces to about0.009% for c/a = 25 (corresponds to c = 40 cm).3. Motion of the ploughmeter through ice has been completely neglected. This assumption is probably safe since the protective vinyl tube coating the ploughmeteralso acts as a thermal insulator and suppresses the regelation mechanism.4. Hydrological and mechanical perturbations of the subglacial environment, causedby the presence of a water-filled borehole, have been neglected.The rheology of subglacial material is doubtless more complex than our simplemodel rheologies and should probably incorporate pore water pressure, deformationhistory, and heterogeneity of the material. Nevertheless, the simple models give ussome indication of the ability of subglacial material to resist deformation. Estimates ofplastic yield strength and effective viscosity derived in this study can be compared tothe applied shear stress beneath Trapridge Glacier. Assuming a plane slab geometry,the mean basal shear stress beneath our study site is 77 kPa (based on an ice thicknessof 72 m (Blake, 1992)). If subglacial sediment were plastic, then the inferred yieldstrength is about 70% of the applied shear stress. On the other hand, for the caseof Newtonian viscous behaviour, the basal strain rate can be calculated to be about33 year. Here, glacier sliding is neglected and the entire basal motion of 90 mmd’ isassumed to result from uniform shear deformation in a 0.5 m thick sediment layer. If thedeforming layer was Newtonian viscous and resisted the basal shear stress of 77 kPa,Chapter 2. FLU UGHING OF SUBGLACIAL SEDIMENT 34then the 33 year’ strain rate represents a viscosity of 4 x 1010 Pa s. The viscosityinferred in our study is from as little as 20—25% to as much as an order of magnitudelower. Thus, for the viscous and plastic rheologies, the estimates of sediment strengthyielded by the ploughmeter analysis suggest that the deformational resistance of thebed is comparable to, but somewhat less than, that required to ensure mechanicalstability in this region of Trapridge Glacier.Results of previous work on in situ measurements of the strength of deformingsediment mostly differ from those obtained in Our study. Humphrey and others (1993)obtained estimates of sediment viscosity and plastic yield strength by analyzing theplastic deformation of a drill stem that became stuck in the bed of Columbia Glacier,Alaska. They reported viscosities (2 x 108_5 x 108 Pa s) and yield strengths (5.5—13 kPa),that are as much as one order of magnitude lower than our estimates. At the sametime, the basal shear stress beneath Columbia Glacier (100 kPa) is about 30% greaterthan that calculated for Trapridge Glacier. Results from tests of basal till from beneathIce Stream B (Kamb, 1991) indicated a plastic-like behaviour and resulted in estimatesof the yield strength (1.6—2.0 kPa) that also tend to be softer than those derived here.Ice Stream B has a significantly lower basal shear stress yet higher flow rate (Alley andothers, 1987) than Trapridge Glacier, consistent with a weaker subglacial sediment. Viscosities calculated from the results of the work done by Boulton and Hindmarsh (1987)beneath Breidamerkurjökull in Iceland are generally an order of magnitude higher(1010_lOll Pa s) than our estimates. The Breidamerkurjökull observations and thoseon Trapridge may not be directly comparable. The Breidamerkurjökull measurementwere carried out in a tunnel penetrating very thin ice near the glacier margin, wheremechanical and hydrological conditions might be unrepresentative. (The calculatedshear stress was not corrected for longitudinal stresses, large near the glacier terminus,and the observation tunnel would further disturb the stress system.) In contrast, workdone by Iverson and others (1994) on Storglaciären, Sweden, yielded results that areChapter . PLO UGHING OF SUBGLACIAL SEDIMENT 35in agreement with those derived in this study. By recording the force generated on acylinder with conical ends that was dragged through the till, they were able to obtainan estimate of the yield strength (r’-60 kPa) which suggests that the subglacial till ofStorglaciären supports on average about two thirds of the calculated winter basal shearstress. In summary, the shear strength of deforming sediment is found to vary greatlybeneath different glaciers. We are not troubled by this large range in values sincenatural variations in sediment texture and composition can easily account for differentstiffnesses.In this chapter, we have demonstrated that the ploughmeter is a useful tool forassessing physical properties of subglacial sediment. In Chapter 3 we examine the possibility that ploughmeter measurements can be used to characterize sediment texture.Chapter 3FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT“Clouds are not spheres, mountains are not cones, coastlines are not circles,and bark is not smooth, nor does lightning travel in a straight line...”- Benoit Mandeibrot3.1 IntroductionMany natural patterns are known to be self-similar, possessing characteristics whichappear to be independent of the length scale of observation. For example, it is nearlyimpossible to determine absolute sizes in a photograph of many geological features without an object determining scale (Turcotte, 1989). Fractal geometry (Mandelbrot, 1983)incorporates this idea of self-similar scaling and extends the notion of integer spatialdimension, associated with simple geometric structures, to become a parameter characterizing zigzag, rough, and heterogeneous structures of real media. Thus, fractalsprovide a framework for both the description and mathematical modelling of the irregular and seemingly complex patterns found in nature.Fractal mathematics and fractal scaling have been applied to a wide range ofnatural processes (Feder, 1988). Applications have ranged from physically realisticdescriptions of topographic relief (Turcotte, 1987; Gilbert, 1989), soil fabric (Mooreand Krepfl, 1991) and aquifer permeability (Ross, 1986; Turcotte, 1989) to studiesof solid-pore interfaces in porous media (Katz and Thompson, 1985; Thompson andothers, 1987; Krohn, 1988a, 1988b; Krohn and Thompson, 1986) and estimation of soilwater retention properties (Tyler and Wheatcraft, 1989, 1990). Of special interest tothis study is the work by Hartmann (1969), Tyler and Wheatcraft (1989) and Hooke and36Chapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 37Iverson (1995) who reported fractal scaling behaviour in the particle-size distributionsof a variety of soils and fragmented rocks.In this chapter, we first give a brief review of the fundamentals of fractal geometryand introduce the necessary notation. Then, we subject results from ploughmetermeasurements in the field and from our numerical ploughmeter model to fractal analysisand obtain an estimate of the basal glacier sliding rate. In closing this chapter, weemploy simple fractal models to establish a relationship between ploughmeter responseand the particle-size distribution of the subglacial material.3.2 Fractal mathematicsThe basic premise upon which fractal concepts are based is the notion of self-similarity.The term self-similarity implies that regular patterns appear at all scales of observation.This self-similar scaling is expressed in the power-law relation which defines a fractalset according toN = - (3.1)where N is the number of objects having a characteristic linear dimension r, C isa constant of proportionality and D is the fractal dimension. In general, the fractaldimension is not an integer but a fraction lying between the topological and Euclidiandimensions (DT < D < DE) and describes the jaggedness or degree to which thefractal pattern fills up the Euclidian space. For example, a profile of a rough surface istopologically a line (dimension 1), but is defined in Euclidian 2-space, and the fractaldimension falls between 1 and 2. In order to determine D, Equation (3.1) can bewritten asD— 1og(Ni/N)3 2— log(r/ri)Chapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 38Objects that occur in nature rarely exhibit exact self-similarity. However, they oftenpossess a related property, statistical self-similarity. A statistical fractal is definedby extending the fractal distribution (3.1) to a continuous distribution in which thenumber of objects N with a characteristic linear dimension greater than r satisfiesN=-9 (3.3)The power-law distributions (3.1) and (3.3) lack characteristic length scales and arethus applicable to scale-invariant phenomena.The fractal concept can also be applied to the spatial or temporal variations ofa single variable. There is, however, a fundamental difference between the fractal behaviour of the geometrical objects discussed above and the fractal behaviour of mostrandom functions. Geometric objects are considered to exhibit fractal behaviour if theyare self-similar over a range of scales. For most spatial or temporal random processes,the units of the dependent variable differ from those of the independent variable. In thiscase, the space/time series are statistically seif-affine, i.e., measurements taken at different resolutions have the same statistical characteristics when rescaled by factors thatare generally different for the vertical and horizontal coordinates (Mandelbrot, 1983).We can investigate the statistical features of a temporal random process by obtaining its power spectral density S(fT). As discussed by Voss (1985), a function x(t)will have a Fourier transform for the period 0 < t < T given by11•X(fT, T) =— J x(t)e2Ttdt (3.4)T0and a spectral density ofS(fT) = (X(fT, T)X(fT, T)*) (3.5)where angular brackets denote time averaging. The spectral density S(fT) gives anestimate of the mean square fluctuations at frequency fT = 1/T and, consequently, ofChapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 39 the variations over a time scale of order l / / r - For a time series that is fractal, the power spectral density has a power-law dependence on frequency S(fT) ~ frP (3.6) Similarly, for a fractal space series, we obtain power spectra of the form S(h) ~ Kp (3.7) where f\ — 1/A is the spatial frequency and A is the wavelength. A functional rela-tionship between the fractal dimension D and the spectral exponent @ was derived by Berry and Lewis (1980) as D = ^S. (3.8) Numerical results of Fox (1989) suggest, however, that the predicted linear relationship in Equation (3.8) does not hold and can only be used as a reasonable approximation for 1 < 0 < 3. The special case of /3 = 2.0 in Equation (3.6) implies that the ratio of amplitude to wavelength (aspect ratio) of all component sinusoids is constant over all frequen-cies. Mixing fractal and spectral analysis terminology, the component sinusoids are self-similar in the case of D = 1.5. At all other values, the aspect ratio of the compo-nent sinusoids changes as a function of frequency, analogous to the fractal concept of self-affinity. For values of j3 > 2.0 (D < 1.5), the aspect ratio decreases with increas-ing frequency. That is, the features of the time series appear smoother at finer scales because the amplitude of high-frequency roughness is small compared to that at low fre-quencies. Conversely, for 0 < 2.0 (D > 1.5), the aspect ratio increases with increasing frequency, causing features to appear rougher at smaller scales since high-frequency roughness is more significant. Qualitatively, the fractal dimension is a "jaggedness" parameter, indicating the proportion of high- to low-frequency roughness. Chapter 3. FRACTAL DESCRIPTION OF SUB GLACIAL SEDIMENT 40 3.3 Field observat ions During the 1991 summer field season, two ploughmeters were installed at the bed of Trapridge Glacier (see Fig. 1.1). Figure 3.1 shows roughly 60 days of observation for ploughmeter 91PL02. We estimate that the ploughmeter was inserted ~15cm into the basal sediment. Figure 3.1a indicates the bending force applied to the tip of the ploughmeter, and Figure 3.1b indicates the azimuth of the force with respect to the internal coordinates of the ploughmeter. During the entire observation period, both the force record and the azimuth record display a jagged appearance with rapid and large fluctuations. Variations in azimuth of the applied force could result either from rotation of the ploughmeter about its long axis or from translational motion in a direction perpendicular to the glacier flow. Because we believe that ploughmeters are firmly gripped by glacier ice (Chapter 2), it is unlikely that they will undergo large and rapid back-and-forth rotations within a borehole (Fig. 3.1b). Furthermore, other ploughmeter observations (see Chapter 4) indicate that the apparent rotation measured simultaneously with two ploughmeters can only be explained in terms of changes of the glacier flow direction. To further interpret the data, we postulate that azimuth variations due to changing glacier flow directions are generally long-term (cycles of one day or more) and that variations due to cross-flow motion are generally short-term (cycles of the order of hours or less). In our analysis, we used a Gaussian low-pass filter (cut-off frequency « 0.074 d - 1 ) to extract the slowly-varying component of the azimuth time series (dashed line in Fig. 3.1b) and attributed this slowly-varying component to changes in the glacier flow direction. Any remaining high-frequency disturbance on the azimuth record indicates non-zero cross-flow motion. By aligning the slowly-varying azimuth component with the glacier flow direction, we can decompose the net force values (Fig. 3.1a) into down-flow and cross-flow components (Fig. 3.2). To effect this decomposition we assume that the principal direction of ploughmeter motion is down-glacier. ChapterS. FRA CTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 4160005000400030002000100060w50I4030201991Fig. 3.1: Data from ploughmeter 91PL02. (a) Force record indicatingforce applied to the tip of the ploughmeter. (b) Azimuth of the force withrespect to the internal coordinates of the ploughmeter. Dashed line representsthe slowly-varying azimuth component (see text for details).3.4 InterpretationThe jagged appearance of the force record (Fig. 3.la) and the lack of an apparentcorrelation between the records of down-flow and cross-flow force components (Fig. 3.2)suggest that ploughmeter 91PL02 is interacting with a clast-rich area of the glacier bed.The basis for this suggestion lies in the idea that the spikes in the data record are theresult of the ploughmeter colliding with individual clasts as it is dragged through basalmaterial.To support our idea we simulated collisions between a ploughmeter and individualclasts as the ploughmeter moves through a heterogeneous subglacial sediment. Chapter 2 presents a simple numerical model to describe the response of a ploughmeter20 30 10 20 30 10AUG SEPChapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 42600050004000z3000C)200010000-100020 30 10 20 30AUG SEP1991Fig. 3.2: Decomposition into down-flow (upper trace) and cross-flow(lower trace) force components of the data from ploughmeter 91PL02 shownin Figure 3.1.to such forcing by interaction with a synthetic tifi. The model till was constructedassuming a porosity of 0.3 and a clast size distribution that is based on the analysisof grain sizes of a basal till from Trapridge Glacier (Clarke, 1987). The seven largestsize classes were included in the model with clast diameters ranging from 4—32 mm(for more details of the numerical model see Section 2.4.1). Comparison of Figures 3.2and 3.3 show that the records generated using this model display many of the featuresseen in the real data. Typical basal sliding rates at Trapridge Glacier, as indicatedby drag spools, are 40 mmd’ (see Chapter 5). With this estimate of the glaciersliding velocity, ploughmeter 91PL02 would have travelled 2.4 m through subglacialsediment during the period of observation. This conversion enables us to compare themodel results (Fig. 3.3) with the observed data (Fig. 3.2) in more detail, and we notethe lack of “broad” spikes in the synthetically generated record.91PL02I I I I I I10Chapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 435.0Cl)CD3.O. 2.0LJJ01. 0.4 0.8 1.2 1.6 2.0DISTANCE THROUGH MODEL TILL (m)Fig. 3.3: Synthetically generated records of down-flow (upper trace) andcross-flow (lower trace) force components. Note the similar character to thatof Figure Fractal scaling of T’rapridge tillWe suspect that the absence of broad spikes in the synthetically generated recordsof down-flow and cross-flow force components (Fig. 3.2) reflects the absence of clastslarger than 32 mm in diameter in our model till. Even though the largest clasts in thebasal till sample from Trapridge Glacier (Clarke, 1987) were of this size, evidence fromthe ablation till in the forefield of the glacier indicates that the basal sediment layercontains clasts as large as cobbles and even boulders. The upper limit of clast diameterof 32mm, as determined by sedimentological size analysis (Clarke, 1987), is likely to beartificial and results from a bias towards smaller clasts during the sampling procedure.The sampling involved filling a pail with roughly 0.008 m3 of “typical till”, excludingmaterial too large to shovel (Clarke, 1987). Clearly, the largest particles collected willbe a function of the sample size taken and the sampling tools used. We illustrate• SYNTHETIC PLOUGHING•IIIII’IIIChapterS. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 44this idea in Figure 3.4, in which the distribution of different sized particles displays theproperty known as self-similarity. In this conceptual diagram of a cross section througha till, the self-similar nature of the particle-size distribution has been made obvious byrecursively placing smaller particles into the voids between larger particles. If we wereto examine this diagram at the smallest size scale, new and smaller particles wouldappear. Similarly, if we were to look at Figure 3.4 through a larger window, the largestclasts seen here would fill the gaps between even larger particles.Fig. 3.4: Conceptual diagram of the self-similar particle-size distributionin a cross sectional cut through a till. Inset shows a mathematically perfectApollonian gasket (adapted from Mandelbrot, 1983, p. 170).ChapterS. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 45The cascades of smaller and smaller particles filling the gaps between larger particles is similar to the packing of circles in the mathematically perfect Apollonian gasket(Fig. 3.4, inset) (Mandeibrot, 1983, p. 170). There is an infinite number of smaller andsmaller circles in any one curvilinear triangle formed by the bigger circles. We cannottell the magnification of the overall structure of an Apollonian gasket because the wholesystem is self-similar at any scale. Thus, the size distribution of the different-sized circles packing the gasket is a fractal system. Likewise, due to the similarity, the particledistribution depicted in Figure 3.4 appears to resemble a statistically self-similar Apollonian gasket. We therefore suggest that the particle-size distribution of Trapridge tillshould be a fractal system similar to that of the theoretical Apollonian gasket system.If this is the case, fractal mathematics can be used to extrapolate our model till tograin sizes that are larger than 32 mm in diameter.Rocks may be fragmented in a variety of ways. Weathering and grinding or crushing processes can lead to fragmentation. Fragments can also be produced by impactsand explosive processes, such as volcanic eruptions. If fragments are produced with awide range of sizes and if natural scales are not associated with either the fragmentedmaterial or the fragmentation process, we would expect distributions of number versussize to be fractal.A widely used statistical description used to represent the frequency—size distribution of fragments is the power-law relationN = Cm_b (39)where N is the cumulative number of fragments with mass greater than m. Theconstants C and b are chosen to fit observed distributions. Turcotte (1986) showedthat the power-law distribution of Equation (3.9) is equivalent to the fractal distributiongiven in Equation (3.3). By defining a linear dimension r as the cube root of volume,Chapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 46r = V’/3, and assuming that the density is uniform, he noted that m r3. Now,comparing Equation (3.9) with Equation (3.3), he found thatD = 3b (3.10)The fractal dimension D defines the distribution of particles by size. For D = 0, thedistribution is composed solely of particles having equal radius. When the fractal dimension is equal to 2.0, the number of particles greater than a given radius quadruplesfor each corresponding decrease in particle radius by one-half. A fractal dimension between 0 and 2.0 therefore reflects a greater number of larger particles, while a dimensiongreater than 2.0 reflects a distribution dominated by smaller particles.Many experimental studies of the frequency—size distribution of materials fragmented by natural and industrial processes have been carried out. Table 3.1 lists morethan twenty fragmented materials whose particle-size distribution behaves according tothe fractal relation (3.3). Values of the fractal dimension are seen to vary considerablybut most lie in the range 2.0 < D < 3.0. Tyler and Wheatcraft (1989) investigatedsize distributions for 10 soils ranging in texture from sand to silty clay loam to silt.The fractal dimension of the distributions ranged from 2.7 to 3.485 and generally increased as the mean grain size decreased. Of the distributions listed in Table 3.1,those representing soils have fractal dimensions approaching 3.0, which is consistentwith the results from Tyler and Wheatcraft (1989). Hartmann (1969) stated that theconditions under which rocks are fragmented play a significant role in determining thefrequency—size distribution of the fragments. He found that values of the fractal dimension appear to be correlated with the degree of abrasion, grinding, and crushing— large D values being “associated with large expenditures of energy per particle”.This finding accords with the results from fracture experiments on tuff, granite andbasalt by Matsui and others (1982) that show that the fractal dimension of the sizedistribution of the fragments increases as the imparted energy density increases.ChapterS. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 47Material D ReferenceSingly fractured basalt blocks 1.80 1Artificially crushed quartz 1.89 1Multiply fractured basalt blocks 2.01 1Disaggregated gneiss 2.13 1Disaggregated granite 2.22 1Disintegrated igneous boulders 2.37 1Disintegrated igneous glacial boulder 2.40 1Fragmented granite (chemical explosion, 0.2 kt) 2.42 2Fragmented granite (nuclear explosion, 61 kt) 2.50 2Broken coal 2.50 3Interstellar grains 2.50 4Projectile fragmentation of quartzite 2.55 5Projectile fragmentation of basalt 2.56 6Fault gouge 2.60 7Sandy clays 2.61 1Detritus from weathered gneiss 2.67 1Terrace sands and gravels 2.82 1Glacial till (Engabreen) 2.84 8Glacial till 2.88 1Glacial till (Storglaciären) 2.92 8Glacial till (Ice Stream B) 2.96 8Ash and pumice 3.54 1Table 3.1: Fractal dimensions for a variety of fragmented materials. (References are (1) Hartmann (1969), (2) Schoutens (1979), (3) Bennett (1936), (4)Mathis (1979), (5) Curran and others (1977), (6) Fujiwara and others (1977),(7) Sammis and Biegel (1989), (8) Hooke and Iverson (1995))We studied the frequency—size distribution of a basal till sample obtained fromTrapridge Glacier by Clarke (1987) by plotting the cumulative number of particleslarger than a given sieve size versus the particle size (Fig. 3.5). If plotted on a log-logscale, we see that Trapridge till shows clear fractal or power-law scaling behaviour overthe whole range of sampled grain sizes. The fractal dimension D is equivalent to thenegative of the slope of the log particle size versus log number of particles (Fig. 3.5) andwas calculated by using a least-squares regression. A good correlation with the fractalrelation (3.3) is obtained over five orders of magnitude and gives D = 2.95. Note thatChapterS. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 48log-log plots suppress any subtle features of grain-size distributions. Thus, evidence forbimodality would be obscured by graphs of this kind.The relatively large D value for our sample of basal till is probably the result ofrepeated abrasion taking place in the deformable sediment layer beneath TrapridgeGlacier. Our finding is in agreement with the statement of Hartmann (1969) thatgeological samples, such as streambeds, alluvial fans, and glacial tills, have commonlybeen exposed to extreme grinding and crushing processes. In addition, our value of thefractal dimension for Trapridge till is consistent with those determined by Hooke andIverson (1995) for three subglacial tills (ranging between 2.84 and 2.96) that have beenshown to undergo shear deformation (see Table 3.1).I102r(m)Fig. 3.5: The number N of particles with radius greater than r as afunction of r for a sample of basal till from Trapridge Glacier. The solid linerepresents Equation (3.3) with D = 2.95.10141012101010106Trapridge TillBulk Sample # 11D=2. 951 0’ 1 0 1 0 1 O4 1 o 1 02Chapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 49 3.5 Estimation of basal sliding rate One implication of the fractal scaling of Trapridge till is that the response of a plough-meter also obeys fractal statistics. Because the interaction of a ploughmeter with the till will partly be influenced by the clast-size distribution, we expect the frequency-amplitude statistics of the force record to be fractal. For both the data of ploughmeter 91PL02 (Fig. 3.2) and the synthetically gener-ated data (Fig. 3.3), the records of the cross-flow force component were analyzed by the Fourier transform method (Press and others, 1992, p. 490), and the power spectral den-sities were estimated using the periodogram method (Press and others, 1992, p. 542). Both log-log plots — spectral energy SX(/T) versus frequency fa for the 91PL02 data (Fig. 3.6) and spectral energy Sx(f\) versus spatial frequency f\ for the synthetic data (Fig. 3.7) — show good power-law dependence over large portions of the spectra. Here, the subscript x indicates that the power spectra were calculated for the cross-flow force components (lower traces of Figs. 3.2 and 3.3). The least-squares log-log slopes from Equations (3.6) and (3.7) on the quasi-linear sections of the spectra give /? = 2.7 for both cases. The corresponding fractal dimension from Equation (3.8) is D = 1.15, which we will refer to as the fractal dimension of the ploughmeter response. Obtaining the same value of spectral exponent (3 and consequently the same fractal dimension D for the data of ploughmeter 91PL02 and for the synthetically generated data indicates that the rate of change of power with scale is the same in both cases. This finding further validates the applicability of our simple numerical model to simulate collisions of a ploughmeter with individual clasts. However, for the model results (Fig. 3.7) the power spectrum rolls off at low spatial frequencies, an effect that is less noticeable for the spectrum of the 91PL02 data (Fig. 3.6). We attribute this difference again to the lack of clasts larger than 32 mm in diameter in the model till. We will further discuss this roll-off in Section 3.6 (below). Chapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 50 4.0 3.0 55 2.0 i? 1-0 -2 0.0 -1.0 -2.0 -1.0 0.0 1.0 log[fT) <T) Fig. 3.6: Power spectral density function for the record of the cross-flow force component shown in Figure 3.2. The dashed line represents Equa-tion (3.6) with f3 = 2.7 (D = 1.15). The power-law spectra given in Figures 3.6 and 3.7 provide further information beyond simply the fractal dimension. In addition to the slope, the spectra are charac-terized by a spectral intercept which represents the power of a sinusoid of frequency / r = 1 d - 1 or spatial frequency fx — l m - 1 . The spectral intercepts of the two power spectra in Figures 3.6 and 3.7, unlike the spectral exponents, are not equal for the following reasons: (1) The amplitudes in the records of the cross-flow force component (Figs. 3.2 and 3.3) differ by more than three orders of magnitude. (2) The force record of ploughmeter 91PL02 (Fig. 3.2) represents a time series (horizontal axis in units of time) whereas units of distance along the horizontal axis in the force record of the synthetic data (Fig. 3.3) characterize a space series. Because the power-law relations (3.6) and (3.7) are independent of scale, knowledge of the actual power at a specified scale is not required and we can normalize the two power spectra. This normalization corresponds to a relative, vertical displacement of \ \ i i i i i Chapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 51 CO o -5.0 -6.0 -7.0 -8.0 -9.0 10.0 11.0 N / \ j SYNTHETIC PLOUGHING < « — 0=2.7 D=1.15 i i i iff N 1.0 2.0 log[fx, m"1] 3.0 Fig. 3.7: Power spectral density function for the record of the cross-flow force component shown in Figure 3.3. The dashed line represents Equa-tion (3.7) with P = 2.7 (D = 1.15). Notice the roll-off in power at low spatial frequencies. the power spectra in log-log space. Now we can use reason (2) to make an estimate of the glacier sliding rate. By multiplying the spatial frequency fx by the velocity v with which the ploughmeter is being dragged through the till, we obtain the frequency /y . This operation is equivalent to linearly translating one of the power spectra in log-log space along the horizontal axis, i.e., log fr = log fx + log v (3.11) If we denote the spectral intercepts of the normalized power spectra for the data of ploughmeter 91PL02 and the synthetic data with a p and a\, respectively, we can calculate the velocity that will achieve perfect overlap of the two power spectra, thus v = IQ^T-ax)//! (3.12) ChapterS. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 52The velocity v in Equation (3.12) represents the differential velocity between plough-meter and sediment and is equivalent to the basal sliding rate of the glacier if sedimentdeformation is neglected. For the synthetic data set described above and illustratedin Figures 3.3 and 3.7 we calculate a sliding velocity of 36mmd1. We recall thatthe model till was assumed to have a porosity of 0.3 (Section 2.4.1). Similar analysesfor different model tills with porosities ranging from 0.1 to 0.5 yield a basal slidingrate that is a function of porosity as shown in Figure 3.8. The glacier sliding velocities of t-.,30--50 mmd1 calculated in this way are therefore in excellent agreement withdirect measurements of sliding at the base of Trapridge Glacier using a drag spooi(i..i40—60mmd’, see Chapter 5)..— 50E45.C)a-JLJJ>0 -z300.2 0.3 0.4 0.5TILL POROSITYFig. 3.8: Summary of the relationship between sliding velocity of Trapridge Glacier and till porosity as calculated using the clast collision model.0.1I I I IChapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 533.5.1 DiscussionIn our analysis we had assumed a spatially uniform size distribution and that temporalvariations in sliding rate are negligible. If, however, the subglacial sediment texture isunknown and the clast collision frequency is observed to vary with time, two contrastinginterpretations present themselves: (1) the sliding rate is constant and sediment texturespatially inhomogeneous; (2) the sediment texture is spatially homogeneous and thesliding rate varies temporally. In the latter case, sliding rate is proportional to theclast collision frequency. Suppose that sliding occurs at some constant rate v0 andthat clast collisions yield a ploughmeter response x(t). The frequency content of thisresponse can be represented by the power spectrum S(fT) given by Equation (3.5).If, at some subsequent time, the sliding velocity changed to some new but constantvalue v’ = 7V0, the time axis of the ploughmeter response would become rescaled as= t/7 to yield a response x’ having a power spectrum72SQyfT). (This result followsfrom the Fourier transform similarity theorem; see e.g., Bracewell, 1986, p. 101). Thus,the relative change in sliding rate-y = v’/vO produces a proportional rescaling of thefrequency axis. It may therefore prove possible to use ploughmeters to observe seasonalvariations in the basal sliding rate.3.6 Fractal response of ploughmeter—till interactionRecalling that the fractal dimension of the particle-size distribution may be regardedas a measure of the till texture, we proceed by investigating the impact of changes infractal scaling of the particle sizes upon the mechanical behaviour of the ploughmeter. Using the numerical clast collision model introduced in Section 2.4.1 we seek anempirical relationship between the fractal dimension of the particle-size distributionand the fractal dimension of the ploughmeter response. Based on a discrete modelof fragmentation we present a simple method for generating different frequency—sizeChapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 54distributions of particles for our model till. The fragmentation model, which was developed by Turcotte (1992) and is illustrated in Figure 3.9a, yields a specific fractaldimension. A cube of linear dimension h is divided into eight equal-sized cubic blockswith dimensions h/2. We consider fragmentation such that some blocks are retained ateach scale but others are fragmented. In the model given in Figure 3.9a two diagonallyopposed blocks are retained at each scale. We therefore have N1 = 2 for r1 = h/2,N2 = 12 for r2 = h/4, and N3 = 72 for r3 = h/8. From Equation (3.2) we find thatD = log6/log2 = 2.585. This is the fractal distribution of a discrete set which we wantto compare with statistical fractals obtained from actual particle-size distributions. Itis therefore of interest to consider also the cumulative statistics, where denotesthe cumulative number of the fragments equal to or larger than r. For the modelillustrated in Figure 3.9a we have N1 = 2 for r1 = h/2, N2 = 14 for r2 = h/4, andN3 = 86 for r3 = h/8 (Fig. 3.9b). Excellent agreement with the fractal relation (3.3)is obtained by taking D =2.593. Thus the fractal dimensions for the discrete set (D)and the cumulative statistics (Dr) are nearly equal.After carrying out an infinite number of recursions, retaining some blocks andfragmenting others, we obtain a fractal cube which is self-similar at all scales smallerthan the initial cube (Fig. 3.9a). Once the fractal cube has been generated, the fragmentation model forms the basis for the frequency—size distribution of the model tillwith a specific fractal dimension. The numbers of the various sized blocks in a fractalcube determine the numbers of spheres of different sizes in our model till.Synthetic ploughmeter data were generated for seven different frequency—size distributions of model till. Each distribution is based on a fractal cube which has a slightlydifferent generating algorithm and therefore a different fractal dimension (Table 3.2).For every synthetically generated record of the cross-flow force component, the powerspectral density was estimated and Equation (3.7) was fitted using a linear least-squaresregression in log-log space.Chapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT10101 8) 1061_____10255Fig. 3.9: (a) Illustration of a fractal model for fragmentation. Two diagonally opposite cubes are retained at each scale. With r1 = h/2, N1 = 2 andr2 = h/4, N2 = 12 we have D = log6/log2 = 2.585 (Turcotte, 1992). (b) Cumulative statistics for the fragmentation model illustrated in (a). Correlationwith Equation (3.3) gives D = 2.593.N1 N2 N3 N D D6 12 24 6 x 2Th_1 1.0 1.0395 15 45 5 x 31 1.585 1.6054 16 64 4 x 4’’ 2.0 2.0133 15 75 3 x 5n-1 2.322 2.3312 12 72 2 x 6n_1 2.585 2.5931 7 49 7n_1 2.807 2.814Table 3.2: Fractal dimensions for a variety of fragmentation models.For a low fractal dimension of the particle-size distribution, the model till is predominantly composed of larger particles and we expect more large scale interactions.On the other hand, a high fractal dimension indicates that the till consists of a significantly increasing number of smaller size particles which would result in an increase ofsmall scale interactions. The power spectrum of the force record in the first case willtherefore contain more low spatial frequency energy while in the second case more energy will be contained in the higher spatial frequencies. As a consequence, the slope ofaFtU11bQQ.Q&I I I bI10.2nh1 01ChapterS. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 56the quasi-linear curve which characterizes the power spectrum in log-log space shoulddecrease with increasing fractal dimension of the particle-size distribution of the modeltill. We can see some evidence of this inverse relation in Figure 3.10.-1.0 0.0 1.0Iog[f, hjFig. 3.10: Illustration of the inverse relationship between fractal dimension of the particle-size distribution of the model till and the slope of the powerspectrum for the synthetically generated ploughmeter response. (a) Cumulative statistics for two fragmentation models listed in Table 3.2 with D = 1.039(long dashed line) and D 2.814 (short dashed line). (b) Composite of twopower spectra plotted on log-log scales for two synthetically generated plough-meter responses using model tills having particle-size distributions shown in(a).The fractal dimension of the ploughmeter response can be determined from the log-log slope using Equation (3.8). Figure 3.11 displays the empirical relationship betweenthe fractal dimension of the particle-size distribution and the fractal dimension of thea14‘S.’10121010108) 10811021b.88U)0-2.0-3.0-4.0-5.0-6.0-7.0-8.0i- 10 1O 101nh2.0ploughmeter response.ChapterS. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 571.16zQ 1.14o 1.12DC,)_W 1.10O 1.4 1.8 2.2 2.6 3.0FRACTAL DIMENSION OF MODEL TILLFig. 3.11: Dependence of the fractal dimension of the ploughmeter response on the fractal dimension of the particle-size distribution. The errorbars reflect the uncertainty in determining the log-log slopes of the quasi-linear sections in the power spectra.Here, we included a model till whose particle-size distribution has a fractal dimension D = 2.953 (D = 2.954). This fractal distribution is based on a fragmentationmodel in which a cube is divided into 64 cubic blocks rather than 8 as in the case of themodels listed in Table 3.2. The fractal dimension of the ploughmeter response for thiscase is D = 1.15 and agrees with that obtained for the data of ploughmeter 91PL02(Fig. 3.6). From Figure 3.11 we see that the response of the ploughmeter displays onlya weak dependence on the particle-size distribution. A change of the fractal dimensionof the model till by a factor of three is accompanied by only a 10% change in the fractaldimension of the ploughmeter response.These conclusions are based on model tills whose particles, though having a fractalsize distribution, were randomly placed in space (see Section 2.4.1). A different resultmight be expected if the model tills are constructed by distributing the particles inspace in a fractal arrangement similar to that depicted in Figure 3.4. This possibilityis currently being explored separately.Chapter 3. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 58As a second application we use the fragmentation model to investigate the roll-off in power at low spatial frequencies in the spectrum of the model results (Fig. 3.7,Section 3.5). For a model till with a particle-size distribution based on the fractalcube with fractal dimension D = 2.585 (Fig. 3.9a) we generated three sets of syntheticploughmeter data. The largest clasts in the model till for the first data set were ofdiameter h/2. Subsequently, this clast size class was removed, leaving h/4 diameterparticles to be the largest clasts for the second data set. We continued this procedurefor one further iteration to obtain a model till for the third data set where the largestclasts have a diameter of h/8. The resulting power spectra for these three data sets areshown in Figure 3.12. We notice that the roll-off in power becomes more prominent asthe size of the largest clasts in the model till decreases.-2.0-3.0-4.0‘I(I) -.0)-6.0-7.0-8.0-1.0 1.0 2.0Iog[f, h1]Fig. 3.12: Composite of three power spectra for three synthetic ploughmeter interactions with a model till whose particle-size distribution is basedon the fractal cube illustrated in Figure 3.9a. Arrows point to the roll-offpoints. Largest clasts in the model till have diameters h/2, h/4, and h/8.I I0.0ChapterS. FRACTAL DESCRIPTION OF SUBGLACIAL SEDIMENT 59Furthermore, the spatial frequency at which the power spectrum rolls off can bedirectly related to the size of the largest clasts in the model till. For the three powerspectra the roll-off points correspond roughly to wavelengths (see arrows in Figure 3.12)which are about twice the diameter of the largest clasts in the respective model tills.3.7 Concluding remarksIn this chapter, we have explored the usefulness of the ploughmeter as a tool for studying sediment texture. Analysis of the particle-size distribution of Trapridge sedimentrevealed that the size distribution forms a fractal system. This fractal scaling behaviouris not only an efficient representation of the frequency—size distribution of a sedimentsample but also provides a useful extrapolation principle for the grain-size distribution to size scales that have not been sampled. We demonstrated that the responseof ploughmeters interacting with a self-similar sediment also obeys fractal statistics.This result inspired us to examine whether we can relate the spectral characteristicsof the ploughmeter response to the size distribution of the subglacial sediment. Regrettably, the fractal dimension of the ploughrneter response is remarkably insensitiveto the fractal dimension of the particle-size distribution. We therefore concluded thatit is not feasible to infer subglacial sediment texture from the response spectrum ofploughmeters. However, if the sediment texture is known and assumed to be spatiallyuniform, we can use the spectral response of a ploughmeter to estimate the basal sliding velocity. In the next chapter we apply data recorded with two ploughmeters anda water pressure sensor to study mechanical and hydrological coupling at the ice—bedinterface.Chapter 4EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS”“I am a kind of burr; I shall stick.”- William Shakespeare4.1 IntroductionThe flow of glaciers and ice streams is driven by gravity and opposed by resistive forces.The sources of the restraining forces that act at the bed are some unknown combinationof so-called “sticky spots” at the ice—bed interface and bedrock roughnesses in the caseof a hard-bedded glacier or the rheological properties of the basal material in the caseof a soft-bedded glacier. Sticky spots are localized regions of the bed where the basalshear stress is concentrated and that balance some or all of the applied driving stress(Alley, 1993).Recent data from West Antarctica suggest the presence of sticky spots that support high basal shear stress and that are surrounded by a generally well-lubricated,low-shear-strength bed. Force-budget calculations for ice flow at the Byrd StationStrain Network (Van der Veen and Whillans, 1989), where surface measurements wereused to infer stresses at depth, showed that the basal drag is highly variable acrossthe bed and concentrated at a few distinct sticky spots. These high-drag, slow-slidingsites are not always correlated with basal topographic highs, indicating that someprocess such as basal water drainage is involved in controlling resistance at the bed.MacAyeal (1992) used control methods to invert the observed surface velocity patternof Ice Stream E for the distribution of subglacial friction. Irregularity of this inferreddistribution suggests that the ice is not underlain by a uniform layer of deformable sediment and that increased basal friction is introduced by rigid bedrock or by variations in60Chapter 4. EVIDENCE FOR TEMPORALLY- VARYING “STICKY SPOTS” 61subglacial water pressure. Alley and others (1994) suggested that basal water has beendiverted away from Ice Stream C, resulting in a disruption of the lubricating water film.They therefore hypothesized that the ice stream has slowed and stopped due to thebasal stress on a few sticky spots at the bed. Neighboring Ice Stream B flows rapidly(400—800ma1)despite its similarity to Ice Stream C in physical dimensions, accumulation, temperature (Shabtaie and others, 1987) and substrate properties (Rooneyand others, 1987; Atre and Bentley, 1993). Anandakrishnan and Alley (1994) suggest that sticky spots exist beneath Ice Stream B, but rarely manifest themselves inthe force balance at the bed since they are better lubricated than those beneath IceStream C. This conclusion is supported by observations that microséismic events are20 times more abundant at the base of Ice Stream C than at the base of Ice Stream B(Anandakrishnan and Bentley, 1993). The more frequent occurrence of microseismicevents beneath Ice Stream C points to a difference in frictional character of sticky spotsbetween the fast- and slow-moving ice streams.The foregoing discussion highlights the reasons for current interest in the characteristics of subglacial sticky spots and the larger issue of ice—bed coupling. Simultaneousmeasurements of subglacial water pressure and ploughrneter response offer a uniqueapproach to studying the mechanical and hydrological coupling between a glacier andits bed. In this chapter, we interpret data from two ploughmeters in terms of changesin basal sliding velocity in response to fluctuations of subglacial water pressure. Wedevelop a mathematical model to describe the sliding motion of glacier ice over a flatsurface having spatially and temporally variable drag. Solving for the velocity field ofthe ice immediately above the glacier bed, we can calculate the behaviour of ploughmeters as they respond to temporal and spatial evolution of basal resistance. Comparisonof our model results with field measurements yields evidence for temporal and spatialvariations of sticky spots that are linked to changes in basal lubrication.Chapter . EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 624.2 Field observationsIn July 1992, arrays of ploughmeters and subglacial water pressure transducers wereinstalled beneath Trapridge Glacier. Figure 4.1 shows 15 days of observations forploughmeters 92PL02 and 92PL05. Data from subglacial water pressure sensor 92P06are also included and plotted along the same time axis. All three instruments arelocated within a circle of diameter ‘—‘10 m (Fig. 4.2) in the general vicinity of the1991 ploughmeter positions. The two ploughmeters 92PL02 and 92PL05 were insertedapproximately 10 m apart in boreholes that connected to the basal hydrological system.The line joining the two insertion sites was at an angle of --‘8° from the direction ofglacier flow (Fig. 4.2).Fig. 4.1: Next page: Data from ploughmeters and pressure sensor. (a)Force record indicating force applied to the tip of ploughmeter 92PL02. (b)Azimuth of the forces with respect to the internal coordinates of ploughmeter92PL02. (c) Force record indicating force applied to the tip of ploughmeter92PL05. (d) Azimuth of the forces with respect to the internal coordinatesof ploughmeter 92PL05. (e) Subglacial water pressure record from pressuresensor 92P06. Super-flotation pressures correspond to a water level of morethan about 63 m.WATER LEVEL (m) AZIMUTH (DEG) FORCE (N) CO o J5> o Oi O) si o o o -»• -»• -* -* l\3 l\i ro o> -si oo co o -•• rv> o o o o o o o o o o o O) 00 o o o o o o o AZIMUTH (DEG) en A co N> -± o o o o o FORCE (N) -^ I\) U A Ol o o o o o o o o o o o o o o o o co l\3 "0 o CD en CO c_ c C |\0 ^ o co CD rv) en T—i—r i 1 r OS -i -fcv. ta O to I I o to t-l to Co O Co Oi co Chapter 4- EVIDENCE FOR TEMPORALLY-VARYING "STICKY SPOTS" 64 o o o 00 co + O z I o 805 800 -795 790 785 --92PL05 • -j c e . 2 ^ _ J 1 L 92P06 • 92PL02 • 92SM07 • 1 i 92SM02 • i 760 765 770 775 780 EASTING (+535000 m) 785 Fig. 4.2: Location map of sensors during the 1992 summer field season. Data from sensors 92SM02 and 92SM07 are presented and discussed in Chap-ter 5. The records of both ploughmeters (Figs. 4.1a-4.1d) display strong diurnal sig-nals which appear to be correlated to large and rapid fluctuations in subglacial water pressure registered with sensor 92P06 (Fig. 4.1e). This correlation suggests that me-chanical conditions at the glacier bed vary temporally in response to changes in the basal hydrological system. However, there is a fundamental difference in how the two ploughmeters respond to these changes in basal conditions. In the case of ploughmeter 92PL05 we see that variations in subglacial water pressure (Fig. 4.1e) are in phase with variations in the force response (Fig. 4.1a): high and low water pressures correspond to high and low forcings, respectively. In fact, for the time period starting 21 July, the two records look virtually identical, even at the finest detail. In contrast, peak water pressures appear to coincide with low forcings experienced by ploughmeter 92PL02 and vice versa (Figs. 4.1c and 4.1e). Thus, comparing Figures 4.1a and 4.1c, we observe Chapter 4. EVIDENCE FOR TEMPORALLY-VARYING "STICKY SPOTS" 65 that the force record of ploughmeter 92PL05 indicates a response that is 180° out-of-phase with respect to that of ploughmeter 92PL02. At the same time, however, the azimuth records (Figs. 4.1b and 4.Id) indicate ploughmeter responses that are in-phase with each other: both ploughmeters appear to be rotating in the same sense back and forth about their long axes by roughly 10-15°. 4.3 Qualitative interpretation The correlation between the azimuth responses coupled with the strong anti-correlation between the force responses of the two ploughmeters could suggest: (1) that one plough-meter is forcing the other, or (2) that both ploughmeters are responding to the same external forcing. While we do not believe that the response of one ploughmeter can force that of another, in the following analysis, we investigate the possibility that the mechanical behaviour of ploughmeters is forced by variations in subglacial water pres-sure by a mechanism that operates differently for the two ploughmeters. In Section 2.4.3 we saw that, if subglacial sediment is treated as a layer of New-tonian viscous fluid, then the force on a ploughmeter that is moving through it, is linearly proportional to the effective fluid viscosity n, the translational velocity v, and the shape and dimensions of the section of ploughmeter immersed in sediment (see Equation (2.9) in Section 2.4.3). The translational velocity v is equal to the basal glacier sliding velocity vb if subglacial sediment deformation is neglected. Thus, varia-tions in the force response of a ploughmeter could be due to any or all of the following: changes in the strength of basal material, changes in glacier sliding rate, and changes in ploughmeter insertion depth into subglacial sediment. However, as previously an-alyzed (see Section 2.4), we completely ignore the possiblity of a ploughmeter being lifted out and re-inserted into the sediment as a result of vertical uplift of the glacier driven by pressurized water between ice and bed. Furthermore, we assume that the strength of basal material is not significantly affected by fluctuating subglacial water Chapter 4- EVIDENCE FOR TEMPORALLY-VARYING "STICKY SPOTS" 66 pressures and, hence, effective pressures. This assumption is reasonable because the low hydraulic conductivity usually associated with glacial sediments means that changes in subglacial water pressure take some time to diffuse into the sediment, which makes a near-instantaneous response of a ploughmeter unlikely. For this discussion, we there-fore attribute all of the variations in the force response of a ploughmeter to changes in basal sliding rate in response to fluctuating subglacial water pressures. Our measurements of water pressure beneath Trapridge Glacier show, that at any given time, basal water pressure is not uniform over the glacier bed (Stone, 1993). In addition, large spatial pressure gradients can be observed between boreholes that are connected and those that are unconnected to the subglacial drainage system (Murray and Clarke, 1995). These observations suggest that we must treat mechanical conditions at the glacier bed on a local scale rather than globally for the glacier as a whole, because mechanical conditions should vary in response to localized changes in the subglacial hydrological system. In the following, we base our interpretation on the idea that a lubricating water film is associated with high subglacial water pressure, which effectively decouples the glacier from its bed and promotes sliding. In contrast, low pressures cause increased bottom drag. We consider a patch of glacier bed over which, at a starting time t = ti, the basal water pressure is low. There is good coupling between the glacier and the subglacial sediment since bed lubrication is poor. Basal resistance is higher than average and little slip occurs between the sedimentary material and the overlying ice (Fig. 4.3a). At a later time t = t2, subglacial water pressure has increased over the same patch. Now the ice—sediment interface is well lubricated, resulting in a strong local decoupling of the glacier from its bed, so that ice is allowed to slide freely over this patch with lower-than-average bottom drag (Fig. 4.3b). As the water pressure rises on the patch of glacier bed under consideration, some of the normal load of the overlying ice will be Chapter 4- EVIDENCE FOR TEMPORALLY-VARYING "STICKY SPOTS" 67 redistributed onto surrounding regions. This lateral transfer of loading is accompanied by an increase in basal resistance in those regions. Fig. 4.3: Schematic diagram of temporal evolution of ice flow as a function of basal resistance. Shaded areas represent regions with higher-than-average bottom drag while white areas indicate lower-than-average drag, (a) Low subglacial water pressure in centre of diagram, (b) High subglacial water pressure in centre of diagram. While this interpretation approach ensures a roughly constant mean basal resis-tance averaged over the glacier bed, it also allows for spatially and temporally varying bottom drag, as "sticky spots" are created and destroyed in response to changes in bed lubrication. If a ploughmeter was positioned within the "sticky/slippery" patch in Figure 4.3 and another in the surrounding region, we see that the two ploughmeters would respond to the same water pressure forcing in ways that are 180° out-of-phase with respect to each other. 4.4 Q u a n t i t a t i v e in t e rp re t a t ion Our success in giving a reasonable qualitative explanation for the data of ploughmeters 92PL02 and 92PL05 in conjunction with the subglacial water pressure record from pressure sensor 92P06 inspired a mathematical treatment of flow of ice over a bed Chapter 4- EVIDENCE FOR TEMP ORALLY-VARYING "STICKY SPOTS" 68 having varying basal resistance. In the following sections we compute the velocity field for the ice immediately above the glacier bed as a function of bottom drag and explore how ploughmeters respond to the creation and destruction of sticky spots. In our model, we treat the bed as a hard, flat surface over which glacier sliding is controlled by a variable drag along the ice-bed interface. At first glance, there appears to be a paradox because an effective ice viscosity of the order of 1013 Pas compared to our estimated till viscosities of O(1010Pas) (see Section implies that any shear associated with the glacier motion is confined entirely to the deformable sediment rather than the ice. We therefore have to address the question of how a soft sediment bed can offer resistance to moving ice that is 1000 times as stiff. Clearly, basal drag can be induced by rigid bedrock bumps sticking into the base of a glacier. However, because we do not believe that sticky spots are spatially permanent features at Trapridge Glacier, they are unlikely to be caused by ice in contact with rigid subglacial bedrock. On the other hand, discontinuities or thinning of the deformable layer overlying a rigid substrate can result in enhanced basal resistance which has the potential to vary in space and time. Alternatively, the temporal and spatial variations of sticky spots can be caused by the formation and failure of grain bridges. During shearing of a granular material, grains can become aligned sufficiently coaxial such that the resulting bridges pose significant resistance to shear (Hooke and Iverson, 1995). 4-4-1 Physical model and outline of analysis Ice is moving with an over-all velocity v\>, the basal sliding velocity, over a flat surface with a variable drag coefficient / . These assumptions contrast with those of standard glacier sliding theories where ice overlies a bedrock surface having a given topography. Sliding is possible because the ice is not frozen to the bed surface. Resistance to sliding is caused by patches of higher-than-average basal drag rather than by topographic Chapter 4. EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 69roughness elements that obstruct the motion. The ice responds to the increased dragon the upstream sides of sticky patches by slowing down and diverging laterally to thesides at these points and thereby permitting the ice to move forward; correspondingly,it speeds up and converges behind the sticky spots, in response to the reduced dragon the downstream sides. We describe the relationship between the sliding velocity vband the basal shear stress r by the linear sliding lawTb = fvb (4.1)where f is a drag coefficient. Equation (4.1) is in direct analogy to the approach takenby MacAyeal (1992) in his calculation to infer basal friction from the observed surfacevelocity pattern of Ice Stream E.4.4.2 First order perturbationWe consider the drag coefficient f to consist of a spatially constant background component f0, which could be a function of time, upon which is superimposed a perturbationcomponent f’ which varies with time and position across the glacier bed. For a coordinate system in which the glacier rests on a horizontal bed with the x axis directed inthe glacier flow direction and the z axis pointed positive upward, we thus writef(x, y, t) = f°(t) + f’(x, y, t) (4.2)With this assumption we note that the basal sliding velocity is also the sum of a timevarying background component and a temporally- and spatially-varying perturbationcomponent, i.e.,Vb(X,y,O,t) = v(t) + v,(x,y,O,t) (4.3)Chapter 4. EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” ‘70Substituting Equations (4.2) and (4.3) into the linear sliding law (Equation (4.1)) andignoring higher-order perturbation terms, yieldsy, 0, t) = [f°(t) + f’(x, y, t)][v(1) + v,(x, y, 0, t)j (4.4a)f°(t)vb°(t) + vb° (t)f’(x, y, t) + f°(t)v(x, y, 0, t) + ... (4.4b)We can split Equation (4.4) into two equations, one that represents linear backgroundsliding,= f°(t)v(t) (4.5)and one that describes linear sliding due to the perturbation effects,T(X, y, 0, t) = v(t)f’(x, y, t) + f°(t)v,(x, y, 0, t) (4.6)Note that in Equation (4.4) Tb varies with both time and space but its variationsare constrained by the fact that the spatial average (Tb(x, y, 0, t)) = r. Likewise inEquation (4.6) the spatial average of the perturbation stress (r(x, y, 0, t)) vanishes.4.4.3 Variations in sliding due to perturbation effectsThe ice mass can move forward with a velocity v,(x, y, 0, t) provided that lateral motions of the ice allow a continually adjusting accommodation of the ice to changes inbasal drag. We examine the case in which ice is assumed to be a linear viscous fluid.The problem is formulated as follows. For a linear viscous fluid having constantdensity p and constant viscosity i and flowing at velocities slow enough that inertialeffects can be neglected, the perturbation velocity field v’(x, y, z, t) satisfiesöv’(x,y,z,t)= ,V2f(t) (4.7)atChapter . EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 71Note that we used for the dynamic viscosity of ice in contrast to which is used todescribe the viscous properties of subglacial sediment. Gravitational terms do not appear in Equation (4.7) since perturbation effects are not driven by gravity. In addition,the assumption of a completely flat bed leads to a constant pressure distribution alongthe ice—bed interface resulting in zero pressure gradients. The flow is incompressible,so thatV•v’(x,y,z,t)=O (4.8)We turn to dimensional analysis and begin by defining dimensionless variables as follows: time t t/to; distance r* = r/ro; velocity v1 v’/v. With these definitions,Equation (4.7) can be written in dimensionless formy, z, t)=_ V2*vI*(x, y, z, t) (4.9)at PIToThe characteristic values t0 and r0 are arbitrary, but a reasonable choice involves alength scale that is representative of the actual physical system. We therefore set thecharacteristic distance r equal to the thickness of the glacier ice h1. The manner inwhich the ice velocity field v’(x, y, z, t) adjusts to perturbations at the bed is principallydetermined by the dimensionless factor on the right hand-side of Equation (4.9). Wesee that the order of magnitude of time t required for a perturbation to propagatethrough the ice to the glacier surface is given by t/pjh = 1, whencet = (4.10)i.e., t is proportional to the square of the ice thickness h1, and inversely proportionalto the kinematic viscosity i’= /pj. Indeed, v serves as a diffusivity of momentumand as such has dimensions of m2s1. Substituting h1 = 72 m, p = 900 kg m3 and= 10’s Pas into Equation (4.10) yields t = 4.7 x 107s and suggests that we canChapter . EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 72assume instantaneous adjustment to steady-state conditions throughout the ice. Withthis quasi-static assumption, Equation (4.7) reduces toV2v’(x,y,z,t) = 0 (4.11)Equations (4.11) and (4.8) are to be solved subject to the conditions (a) that v’ istangential to the bed surface at z 0, and (b) that the shear stress parallel to the icesurface at z = h1 vanishes.We follow the treatment used by Kamb (1970) on regelation sliding and solveEquation (4.11) by the Fourier-analytical method. For the perturbation velocity fieldv’(x, y, z, t) the Fourier transform pair is given as‘(k, k, z, t)= j I_oo v’(x, y, z, t) exp[—i(kx + ky)] dxdy (4.12)andv’(x, y, z, t) (2ir) i_: J_oo ‘(k, k, z, t) exp[+i(kz + ky)J dkdk (4.13)where k and k are the wavenumbers in the x and y directions, respectively. Fouriertransformation of Equation (4.11) therefore yields(_k2 + = 0 (4.14)where k2 = k + k. Equation (4.14) is a second-order linear homogeneous differentialequation whose solution is readily obtained as‘(k, k, z, t) = C1(k, k, t) exp(—kz) + C2(k, k, t) exp(+kz) (4.15)Applying the boundary conditions yields expressions for the two constants C1 and 02and Equation (4.15) becomes— V’(k, k,0,t)exp(—kz) V’(k, k,U,t) exp(+kz)“ a;, y,Z,t) +1 + exp(—2hk) 1 + exp(+2hk)Chapter . EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 73In order to relate the perturbation velocity field to the perturbation drag coefficientsurface we make use of the stress—strain relations for a linear viscous fluid,= 277E— ‘5ijP (4.17)where is the viscosity of the fluid and öjjp is the hydrostatic pressure. Substitutingthe definition of the strain rate tensor1fOv 9v\ (4.18)into Equation (4.17) we write for the basal shear stress in the x direction (see Equation (4.6))‘WIo(k,k,O,t) (4.19a)= v°(t)f’(k, k, t) + f°(t)(k, k, 0, t) (4.19b)We note that 0V/c9x = 0 since the ice remains everywhere in contact with its flat bed.We now calculate— V(k,k,0,t)kexp(+kz) — V(k,k,0,t)kexp(—kz)420ôz— 1+exp(+2hk) 1+exp(—2hk)Substituting Equation (4.20) into Equation (4.19) and re-arranging terms, we solve forthe x component of the perturbation velocity field, i.e.,v(t)l(k,k,t)k (4.21)—f°(t) + (i + exp(+2hk) — 1 + exp(_2hk))The y component of the perturbation velocity field is now easily obtained by invokingthe incompressibility condition (Equation (4.8)),= _EV(k,k,0,t) (4.22)Chapter . EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 74Description of drag coefficient surfaceWe consider a part of the bed of area A, assumed to be rectangular of dimensions L andL in the x and y directions. Within this area we now define a drag coefficient surfacef(x, y, t) to consist of a spatially-constant background component f°(t) superimposedby a spatially- and temporally-variable perturbation component f’(x, y, t) with thecondition that(f’(x,y,t)) = 0 (4.23)where (>denotes averaging over A (Fig. 4.4). The perturbation drag coefficient surfaceis described by blocks that are centred on coordinates (, ), have widths w and w,,in the x and y directions, respectively, and rise 4 above zero while the remainder ofthe surface is depressed by f below zero. The magnitude of fi.. cannot exceed f°(t)since a negative drag coefficient f(x, y, t) is not physically meaningful. In addition, themagnitudes of 4 and fL are generally not equal and depend on the areal fraction thatis covered by blocks, e.g., narrow blocks with large positive f4. values are accompaniedby small negative fL values for the remainder of the surface.We suppose the drag coefficient surface f(x, y, t) to be periodically repeated overthe entire x-y plane, so that the bed, of assumed infinite extent, consists of a checkerboard of identical areas A, across which the ice slides in the x direction. While a realglacier bed is not periodic in this strict sense, our assumption of repeat distances L andL is a necessary condition for calculating the Fourier transform of the perturbationdrag coefficient surface (see Equation (4.21)).Chapter 4. EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 75Fig. 4.4: Diagram of drag coefficient surface f(x, y, t) defined on part ofglacier bed of area A =4.4.5 Model resultsIn Table 4.1 we have listed the model parameter values that were used to obtain thecalculated solutions. Glen’s flow law for simple shear in the x-y plane= Bo (4.25)combined with Equation (4.17), yields a theoretical estimate of the effective dynamicviscosity of ice= 2Br (4.26)where o = Tb is the basal shear stress. Using a flow law parameter for temperate ice(B = 6.8 x l0—15 s1 kPa3, n = 3 (Paterson, 1994, p. 97)) and a mean basal shearstress of Tb = 77kPa (based on an ice thickness of 72m (Blake, 1992)), we calculated aTrapridge Glacier ice viscosity of i = 1.24 x 1013 Pa s. However, any debris entrainedin the ice as well as the process known as enhanced creep due to stress concentrationsnear the bed are likely to soften this basal ice. We therefore used a dynamic viscosity off(x,y, t)f(x,y,t)f°(t)0 x,y x,yChapter 4. EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 76= 5.0 x 1012 Pas in our model calculations (Table 4.1). With an average basal slidingvelocity of vg = 4Ommd’ (see Chapter 5) substituted into Equation (4.5) we cancalculate a background drag coefficient of f° = 1.66 x 1011 Pasm* The remainingparameters in Table 4.1 were determined on a trial and error basis to yield the desiredresults.Parameter Symbol Value UnitsEffective dynamic ice viscosity 5.0 x 1012 PasBackground basal sliding velocity v = 40 mm d’Background drag coefficient 1.66 x 1011 PasmPeak perturbation drag coefficient f4 2.3 x 1011 Pa sLength of area A in x direction L 20 mLength of area A in y direction L 20 mNumber of grid points in x direction N 32Number of grid points in y direction N 32Width of blocks in x direction 7.5 mWidth of blocks in y direction w, 7.5 mTable 4.1: Parameters for “sticky spot” model.For the model calculations discussed in this section, area A has been divided into a32 x 32 grid. Following the description in the previous section, positive f or negativefL values were then assigned to every grid point. The resulting drag coefficient surfacef’(x, y, t) was Fourier transformed using a two-dimensional FFT algorithm (Press andothers, 1992, p. 515) to obtain ?(k, k, t). Using Equations (4.21) and (4.22) we cannow calculate the perturbation velocity field V’(k, k, 0, t) or, after taking the inverseFourier transform, v’(x, y, 0, t).The drag coefficient surface shown in Figure 4.5a consists of square patches withhigher-than-average drag centred on grid point (17,17) and the four corners of area A,while the remainder of A has lower-than-average drag. For a peak perturbation dragcoefficient f = 2.3 x 1011 Pasm’ for the blocks and using w = = 12 grid points,we used Equation (4.23) to calculate ff... = —0.9 x 1011 Pasm for the remainder ofthe surface. The resulting flow velocity field (Fig. 4.5b) shows how the ice is beingChapter 4. EVIDENCE FOR TEMPORALLY- VARYING “STICKY SPOTS” 77slowed down and diverted around the high-resistance patch in the centre of area A.Here and in subsequent plots of flow velocity fields, the background sliding velocity vhas been added to the pertubation velocity field (note that v° 0). A flat drag surface(f = f = 0, Fig. 4.5c) yields a uniform flow field with velocity vb(X, y, 0) = Vb° = v0(Fig. 4.5d). In Figure 4.5e, the drag coefficient surface consists of a resistance low inthe centre of area A with patches of higher-than-average drag located on the midpointsof the sides of the square area. Now the ice is allowed to accelerate and be channelledtoward this low-resistance patch (Fig. 4.5f).Fig. 4.5: Next page: Drag coefficient surfaces, as defined on a 32 x 32 gridof area A, and calculated ice flow velocity fields. (a) High drag centred on gridpoint (17,17). (b) Divergent and slowed down ice flow. (c) Flat drag surface.(d) Uniform ice flow. (e) Low drag in the centre of area A. (f) Convergentand accelerated ice flow.Chapter 4. EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 78a b1814E•— 10r6?2?18?146?2-18?14?2-s.::.:s--X&X:-::. 5-:-:.tX-Itiiitiiiiiiiiiiliiiliiiliitiiiiliiilii2 6 10 14 18x(m)C&&zzo&czz<..> c55y ->z’.s-: :-:-X 5.-f- xx <-> v*t*v E&&‘5ss?s%&Y%5&&/ ‘I itil i2 6 10 14 18x(m)es::..-:•::-: -::-:•:•:1:::: ::s::-:-:-.-:s&xg:v-.-- :: s: .-:&2%Z::.:::::.::.:::;.:..::*-5-ii ilii tiii till iiii iii iii iii liii liii’ ii2 6 10 14 18x(m)E 3.96 1.66 E 0.76x 1011 Pasm’1 8 ? .-. — .— .— .—. .—. .—. —--.— — ——6? ———2? —— ——2 6 10 14 18x(m)d18?14?— 0=>.6?2?illlilllIlllilllIlllilllIlllIlllillli II2 6 10 14 18x(m)f18?—_._- —— ..— .—. .—14?Ii 11111111111 ill2 6 10 14 18x(m)40 mmd1Chapter . EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 79Combining Figures 4.5b, 4.5d and 4.5f (Fig. 4.6), we see that, as the central regionof A alternates between being sticky and slippery, some of the flow vectors rotate backand forth and change their lengths. The change in length of a flow vector correspondsto a change in basal sliding velocity of ice at that point while rotation indicates a reorientation of the ice flow direction. For further analysis, we imagine the consequenceof positioning two ploughmeters (here denoted PL1 and PL2) at grid points (7,16)and (23,18) (Fig. 4.6). With the side lengths of area A chosen to be L = =20 m, the location of these two grid points corresponds to a separation of the twoploughmeters by approximately 10 m while the line joining the two points forms anangle of -..‘7° with the x direction. This choice of ploughmeter positioning is therefore agood approximation to the field setup (the two insertion sites of ploughmeters 92PL02and 92PL05 were 40 m apart while the line joining the sites was at an angle of 8°from the direction of glacier flow). With our assumption that the force response of aploughmeter is linearly proportional to the basal sliding rate, we note that the changein length of the flow vectors at the two grid points translates into a change in forcing onploughmeters PL1 and PL2. In addition, the rotation of the flow vectors is equivalentto an apparent rotation of the ploughmeters about their long axes. We computedidealized force and azimuth responses of ploughmeters PL1 and PL2 for three completecycles of creating and destroying a sticky spot at the centre of A (Fig. 4.7). Recallingour assumption of a water-pressure-dependent basal resistance, we can associate thecycles with diurnal variations in subglacial water pressure. Figure 4.7 shows, that theresponse of ploughmeters PL1 and PL2 in our model calculations are similar to those ofploughmeters 92PL02 and 92PL05 in the field (Fig. 4.1). The force records (Figs. 4.7aand 4.7b) show strongly anti-correlated responses with diurnal variations of r.’30% whilethe azimuth records (Figs. 4.7c and 4.7d) indicate an apparent in-phase rotation by 15°.The bends in the computed force responses of both ploughmeters (Figs. 4.7a and 4.7b)Chapter 4. EVIDENCE FOR TEMPORALLY- VARYING “STICKY SPOTS” 80are the result of the drag coefficient 4 for positive perturbations not being of equalmagnitude to fL for negative perturbations.18—16——- .—.--•..-_..6—4—E---__- —-o----—liii Iii I iiii lit I ill Ii iii Iiio 2 4 6 8 10 12 14 16 18x(m)Fig. 4.6: Composite of Figures 4.5b, 4.5d and 4.5f showing the changein ice flow velocity field. The locations of the numerical ploughmeters PL1positioned at grid point (7,16) and PL2 at (23,18) are indicated.We further demonstrate the qualitative success of our model calculations by computing the force and azimuth responses of ploughmeters PL1 and PL2 for a varyingbasal drag that is based on the water pressure record shown in Figure 4.le. In ourcalculations, low subglacial water pressures correspond to high resistance in the centreof area A. The computed results (Fig. 4.8) display a striking similarity to the field datashown in Figure 4.1: the force records (Figs. 4.8a and 4.8c) indicate -‘.‘30% variationsthat are 180° out-of-phase with each other while the azimuth records (Figs 4.8b and4.8d) show an in-phase rotation by roughly 10—15°.Chapter. EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 81- PL1I I I I I ICDI—w00UwI—DN4. S..• PL1I I. PL2I I I I0.0 0.5 1.0 1.5 2.0 2.5 3.0TIME (d)Fig. 4.7: Idealized responses of two numerical ploughmeters over threecycles of creating and destroying a sticky spot on the centre of area A. (a) Synthetically generated force record for ploughmeter PL1. (b) Synthetically generated force record for ploughmeter PL2. (c) Synthetically generated azimuthrecord for ploughmeter PL1. (d) Synthetically generated azimuth record forploughmeter PL2.Chapter EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 824.5 Concluding discussionOur model calculations for the flow of ice over a flat surface with variable resistance arebased on the approximation that ice behaves as a linear viscous fluid. The purpose ofthe model is to demonstrate how local values of shear stress might vary with time andspace. The nonlinearity of Glen’s flow law would greatly complicate the model withoutcontributing additional insight.In our derivation of Equation (4.6) that describes sliding due to temporally- andspatially-varying perturbations in drag coefficient, we neglected higher-order perturbation terms (see Equation (4.4)), thereby linearizing the problem by ignoring non-lineareffects. This is a reasonable approximation as long as perturbations of the drag coefficient are small compared to the background drag coefficient. However, from Table 4.1we see that the coefficients for background drag and perturbation drag have values thatare of the same order of magnitude. Nevertheless, a more rigorous calculation usingEquation (4.6) which would include non-linear terms could not be justified in light ofthe other simplifications used in our model, i.e., linear ice rheology and a very simplifiedsliding law (Equation (4.1)). Therefore, for our purpose of only demonstrating that wecan explain our field observations in terms of changes of basal resistance in response tofluctuations of subglacial water pressure, we feel confident about our decision to ignorenon-linear effects in Equation (4.4).The results from our model calculations point to the existence of sticky spotsbeneath Trapridge Glacier. Although we have only demonstrated consistency for asingle example, it is conceivable that there are sticky spots all across the bed of ourmain study region. These sites of enhanced basal drag are ephemeral in nature becausethey are created and destroyed in response to fluctuations in subglacial water pressure.Therefore, it is unlikely that they support a large fraction of the driving stress for iceflow. Our previous result, that the deformational resistance of the sedimentary bedis of comparable magnitude to that required to balance the applied basal shear stressChapter 4. EVIDENCE FOR TEMPORALLY- VARYING “STICKY SPOTS”C,)CDaSII-.aSuJC-)0LIC,)4-C(‘S4—I.(‘SUi00U-Fig. 4.8: Computed responses of two numerical ploughmeters for a varying basal drag that is based on the water pressure record shown in Figure 4.le.(a) Synthetically generated force record for ploughmeter PL1. (b) Synthetically generated azimuth records for ploughmeter PL1. (c) Synthetically generated force record for ploughmeter PL2. (d) Synthetically generated azimuthrecords for ploughmeter PL2. Note the similarity to Figures 4.la—4Jd.83- aPL10UJzIDNI I I I I454. PL2 CI I I I I II I I(!3U-iIDN20 21 22 23JULYd24199225 26 27Chapter 4. EVIDENCE FOR TEMPORALLY-VARYING “STICKY SPOTS” 84(see Chapter 2), further strengthens our argument that sticky spots are probably notdominant in controlling the flow of Trapridge Glacier.Our findings can be compared with results from Ice Stream B. Interpretation ofdata collected near the Upstream B camp (Alley, 1993) suggests that the lubricatedregions of the bed support more than 87% of the basal shear force. This leaves <13% ofthe basal shear force to be supported on sticky spots implying that sticky spots do notdominate the force balance in this region of Ice Stream B. In contrast to our results,Whillans and others (1989) reported that Byrd Glacier, Antarctica, is held mainly bybasal drag that is concentrated at a few sites separated by about 13 km. Analysis ofwork done on Ice Stream C (Anandakrishnan and Alley, 1994; Alley and others, 1993)indicated that the base of the ice stream consists of a weak till interspersed with stickyspots of area on the order of 102 m2 with a spatial density on the order of 10 km2. Asa result of a reduced water lubrication, it is claimed that these sticky spots supportalmost all of the driving stress and account for the negligible flow of Ice Stream C.This chapter concludes our examination of ploughmeters as an instrument forstudying mechanical conditions and processes at the base of the glacier. However, theanalysis of data from this type of device cannot be complete without some means ofmeasuring sliding. In Chapter 5, we present a newly developed method to measurein situ basal sliding, and we discuss some results from our measurements beneathTrapridge Glacier.Chapter 5DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED“It would be more impressive if it flowed the other way.”- Oscar Wilde5.1 IntroductionMechanical measurements of sliding over a deforming substrate are rare. Boulton andHindmarsh (1987) screwed an auger-like anchor into the sediment beneath the terminusof Breidamerkurjökull in Iceland. The anchor was inserted through a sealed hole in thefloor of an englacial tunnel and connected by a string to a reel and chart recorder. Asthe glacier sole moved relative to the underlying sediment, the string was pulled overthe reel and ice displacement recorded. Engelhardt and others (1990b) used the sameprinciple and anchored a tethered stake in till beneath Ice Stream B. By observing thepull-in of the tether cable, they inferred a sliding rate high enough to account for thetotal motion of the ice stream.In an effort to measure basal sliding of Trapridge Glacier, we have developed aninstrument which is similar to the Boulton and Hindmarsh design in that an anchoris placed in the deformable bed and the amount of string payed out is measured.Our instrument can be installed in situ at the bottom of deep, narrow boreholes andhas been termed a “drag spool”. The device consists of a multi-turn potentiometerconnected to a spooled string. The drag spool is suspended within the borehole close tothe glacier bed, and continuously measures the length of string payed out to an anchorin the bed. In this chapter, we describe the physical characteristics and field operationof this device. We also present and discuss some results from our measurements beneathTrapridge Glacier.85Chapter 5. DIRECT MEA SUREMENT OF SLIDING AT THE GLACIER BED 865.2 MethodsThe basic components of a drag spool are an anchor, a multi-turn potentiometer, and aspooled string (Fig. 5.1). A thin nylon string is wound onto a 3.0 cm diameter spool. A25-turn 5 k1 potentiometer (Bourns, part No. 3296W-1-502) is connected to the spoolso that turns of the spool can be measured electrically from the glacier surface. Thespool—potentiometer assembly is enclosed in an acrylic casing. To prevent water thatenters the case from shorting out the potentiometer leads, the interior of the spooi isfilled with an electrically insulating grease. The free end of the string is led througha small hole that penetrates the case and then attached to a conical anchor tip. Apercussion hammer (Blake and others, 1992), fitted with a 0.635 cm diameter metal-dowel attachment, is used to insert the anchor into the soft glacier bed. The dowelfits into a 0.635 cm diameter socket drilled into the back of the anchor tip. The dowelpasses through loose-fitting loops on the drag-spool casing. After the anchor is inserted,the hammer and dowel are withdrawn, leaving the anchor and the drag-spool case atthe bottom of the borehole. Although the hammer and the wire rope supporting it aredrawn past the drag-spool cable, we have not had any problem with ropes and cablesbecoming entangled. Additionally, the dead weight of the drag-spool cable appears toexceed the frictional force between wire and cable; we have not observed any unwindingof the drag-spool string, which would indicate that the spool is being pulled, as thehammer is lifted. Once the hammer has been withdrawn, slack is removed from thedrag-spool cable until a slight tension is sensed. Even though the force required tounwind the drag spool is only 1 N, it is possible to feel this tension point in the 70—80 mlong cable that we use on Trapridge Glacier. Nevertheless, a few centimetres of stringare usually pulled off the spool. As the cable freezes into the borehole, the position ofthe drag-spool case within the borehole becomes fixed. In response to glacier sliding,the anchor will distance itself from the glacier borehole and the spool will turn as theChapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 87string is payed out. Appoximately 2.5 m of thin nylon string are stored on the spooi,which is enough to last for 3—5 weeks at the anticipated sliding rates.Fig. 5.1: Schematic diagram of the drag spool. As the string attached tothe anchor is payed out, the potentiometer screw is turned and the resistancechange can be measured.5.3 Results and discussionFigure 5.2 shows an example of data obtained from a drag spool during the 1992 fieldseason on Trapridge Glacier. The data (Fig. 5.2a) indicate that the distance betweenanchor and drag-spool case increased by approximately 260 mm during the interval21—27 July. This increase translates into an average rate of string-extension of roughlySTRNGANCHORII1cmChapter 5. DIRECT MEA S UREMENT OF SLIDING AT THE GLA CIER BED 8843mmd’. The displacement record (solid line in Fig. 5.2a) shows distinct diurnaldeviations from the general trend of increase (dashed line in Fig. 5.2a). Strong diurnalfluctuations in subglacial water pressure (Fig. 5.2b) were contemporaneously observedin a borehole located approximately 12 m up-flow from drag spool 92SM02 and appearto be correlated with the variations in the displacement record. We now present twoalternative interpretations in an attempt to explain the available data.The stepwise increase in displacement seen in the drag-spool data (Fig. 5.2a) couldindicate an increase in sliding velocity in response to a rise in subglacial water pressure. The idea of high sliding velocities resulting from smoothing of bed roughnesses byaccumulation of pressurized water between ice and bed has been proposed by Weertman (1964) and further analyzed by Lliboutry (1979). We computed the sliding velocity(rate of displacement; Fig. 5.2c) by applying a five-point first-derivative filter to thedisplacement record (Abramowitz and Stegun, 1965, p. 914), followed by a Gaussiansmoothing filter having a standard deviation of 100 mm. Comparison of Figure 5.2band 5.2c shows that peak displacement rates coincide with rises in water pressure.This result contrasts with observations by Iken and Bindschadler (1986), Kamb andothers (1985), Kamb and Engelhardt (1987) and Hooke and others (1989) where peaksubglacial water pressure and peak surface velocity (and by implication basal slidingvelocity) appear to coincide.Alternatively, the stepwise increase in displacement could be interpreted in termsof the growth of water-filled cavities at the glacier bed as a function of subglacialwater pressure. As cavities grow, the glacier is separated from its bed, which resultsin an increase in distance between the anchor and the drag-spool case. The conceptof growing and shrinking cavities at the ice—bed interface has been introduced intoglacier sliding theories (Lliboutry, 1968; Kamb, 1970; Iken, 1981) to allow for thevarying extents to which a glacier is separated from its bed. Field observations onVariegated Glacier in Alaska (Kamb and others, 1985; Kamb and Engelhardt, 1987)zww.1LU>LU-JLU EC-)-jW21 22 23 24 25 26 27JULY 199292SM02Chapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 89500•_ 400E23002007060E50403012080400Fig. 5.2: Data from drag spool 92SM02 and pressure sensor 92P06 (seetext for details). (a) General increasing trend (dashed line) obtained by linear regression, superimposed on the relative displacement between anchorand drag-spool case (solid line). The arrow indicates the time when the spoolstarted to turn. (b) Subglacial water pressure record. Flotation pressure corresponds to a water level of roughly 63 m (dashed line). (c) Rate of displacementbetween anchor and drag-spool case, obtained by numerical differentiation ofthe displacement record.and on two glaciers in the Swiss Alps— Unteraargletscher (Iken and others, 1983) andFindelengletscher (Iken and Bindschadler, 1986) — support the idea that glacier upliftCis mainly due to basal cavitation driven by high basal-water pressures. We computedChapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 90the corresponding vertical uplift for Trapridge Glacier from the general displacementtrend (dashed line in Fig. 5.2a) and the actual displacement record. The dashed line wasobtained by fitting a regression line to the data from the time the spool started to turn(see arrow in Fig. 5..2a) until the end of the record. To consider trigonometric effects,we used two different models of drag-spool operation which represent end-members ofa spectrum of possibilities (Fig. 5.3). In the first scenario the nylon string is free tomove laterally through the soft sediment (Model A), while in the second the sedimentis stiff enough to prevent the string from cutting through it (Model B). We expect thatthe position of the string is closer to Model B, since the tension required to unwindthe spool is small (about 1 N); we do not believe that this tension is sufficient to allowthe string to cut easily through the basal material. The exact depth of emplacementof the anchor in the basal sediment is uncertain. However, based on our experiencewith inserting other instruments using the same procedure (Blake and others, 1992;Chapter 2), we are able to reasonably constrain this uncertainty. For our calculationsin this chapter we assume an insertion depth of the anchor of 18 ± 3 cm.Fig. 5.3: Scenarios for drag-spool operation. Model A allows for lateralmovement of the string through soft sediment. The sediment in Model B issufficiently stiff to keep the string in place. (a) Situation immediately afterinstallation of the device at the glacier bed. (SED indicates sediment) (b)Situation after some time has elapsed. (c) Same as (b) with vertical glacieruplift.IEJDRAG SPOOLMODEL____Chapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 91Model A requires a bed separation of 8 cm and Model B one of up to 20 cm toexplain the available data. Daily surveying of marker poles on the glacier surface didnot reveal vertical displacements of this magnitude. We therefore think that bothmodels predict a vertical glacier uplift that is inconceivably high for Trapridge Glacier.However, we cannot completely dismiss the possibility of fluctuating bed separation.If increases in subglacial water pressures coincide with the growth of water-filled cavities at the glacier bed, then our field observations may correspond to model results byIken (1981) that the largest sliding velocity occurs during cavity growth and not whenthe steady-state size of cavitation is attained. This interpretation agrees with observations at Unteraargletscher (Iken and others; 1983) that the highest horizontal velocityoccurred when the rate of upward motion of the ice was largest rather than at the timewhen the uplift reached its maximum. From the foregoing discussion, we conclude thatthe diurnal character of the data recorded with drag spool 92SM02 (Fig. 5.2a) is mostlya result of variable sliding velocity. Unfortunately, we cannot settle this question usingour current survey methodologies. We would require a survey accuracy of ±10 mm toresolve diurnal fluctuations in Trapridge Glacier flow.During the 1992 field season, the surface velocity of Trapridge Glacier at ourstudy site was about 80 mmd1. Observations of lateral deformation of boreholesreveal that the velocity contribution from internal ice creep for this glacier does notexceed 10 mm d’ (Blake, 1992). Therefore, almost the entire surface motion of theglacier is attributable to sliding and deformation processes at the bed. Our estimate of40 mm d’ for glacier sliding leaves roughly 30—40mmd1 of motion to be accountedfor by subglacial sediment deformation beneath the anchor.It should be noted that in the previous paragraph we have defined “sliding” inan operational manner, as the motion that occurs between the spool suspended in theborehole and the anchor in the sediment. However, glacier sliding is commonly definedas the motion between the base of the ice and the top of the bed. We recognize thatChapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 92the two definitions are not equivalent and that measurements taken with a drag spoolwill place only an upper limit on glacier sliding because the anchor is placed withindeformable sediment; any deformation of the sediment lying between the anchor andthe ice—sediment contact will introduce additional string-extension. With an anchorinsertion depth of r-,18 cm, a typical thickness of the deforming sediment layer of r..iO.5 m(Blake and Clarke, 1989) and the assumption that the sediment velocity varies linearlywith depth, about 5—10% of the measured displacement is due to sediment deformation.This proportion can increase significantly for a velocity—depth profile that is concave-down (Boulton and Hindmarsh, 1987), and thus a large fraction of the ice motion occursby bed deformation. Alternatively, if the anchor is being pulled through the sedimentby tension on the string, all the motion observed at the glacier surface could be dueto basal sliding. We do not believe that this is the case, since the tension requiredto unwind the spool is small (.‘1 N). In other instrument-insertion experiments usingsimilar-sized anchors, the force required to dislodge a freshly inserted anchor was atleast 20 N, so it appears that the bed has a good grip on the anchor.Figure 5.4 shows drag-spool data from three consecutive field seasons. The results from our 1990 measurements indicate different displacement rates for differentpoints across the glacier bed, ranging between 40 mmd—’ (traces 90b and 90c) and8O mmd’ (trace 90a). This finding is not paradoxical, since we expect mechanicalcoupling to vary with location because of spatial differences in the subglacial water system. With uniform surface motion observed over large parts of the glacier, our resultsimply a spatially varying partitioning between basal sliding and subglacial sedimentdeformation. In addition, changes in displacement rate that we observe from year toyear (Fig. 5.4) probably reflect temporal variations of mechanical ice-bed coupling inresponse to changes in the subglacial hydrological system. As an alternative, we cannotignore the possibility that different depths to which anchors of different instrumentsChapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 93are inserted, can account for differences in measured displacement rates. From inspection of Table 5.1, we see that for the 1990 measurements, low displacement rates areindicated for drag spools that have their anchors inserted quite deeply while a shallowinsertion depth corresponds to a high displacement rate. This finding is inconsistentwith what would be expected if sediment deformation contributes significantly to thedisplacement as measured with a drag spool. The 1991 observations are inconclusive inthat respect because two instruments with their anchors inserted to almost the samedepth measured rates that are different by 50%. For the 1992 measurements, our findings are opposite to those from 1990 but are consistent with the postulate that highdisplacement rates correspond to deep anchor insertion depths and vice versa. Becauseof these differences in results from one year to the next we conclude that there appearsto be no clear correlation between insertion depth and displacement rate.Sensor ID Insertion depth Displacement rate(see Fig. 5.4) (cm) (mmd1)90a 19±390c 24±39Gb 25±391b 20±391a 21±392b 18±3 4092a 23±3Table 5.1: Anchor insertion depths and average displacement rates fordrag spools used from 1990—92 (see Fig. 5.4).Chapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 941400- 1200-.-100oFz—‘- --800,.- ‘EE0I I I I I I I I I14 16 18 20 22 24 26 28 30 1 3 5 7 9JULY AUGUSTFig. 5.4: Drag-spool data from the 1990 (solid lines), 1991 (short dashedlines), and 1992 field seasons (long dashed lines), showing the variability ofbasal sliding observed at Trapridge Glacier. Measured sliding rates rangebetween r.,40 mmd1 (traces 90b, 90c, 91b, 92b) and -.‘80 mmd1 (trace 90a);(vi60 mmd’ for traces 91a and 92a). Trace 92b shows the data from dragspool 92SM02, which is discussed in detail in the text (see also Fig. 5.2a).5.4 Water pressure induced variations in glacier slidingThe apparent 90° phase shift between water pressure and sliding rate (see Fig. 5.2)is a noteworthy feature of the data. In the previous section, we suggested that a viable interpretation of this feature corresponds to the numerical results by Iken (1981)that maximum horizontal velocities coincide with times when basal water-filled cavities are growing. Unfortunately, the heavy smoothing required to clarify the velocityrecord (Fig. 5.2c) masks finer detail that might illuminate the motion mechanism. Theunsmoothed displacement rate record (Fig. 5.5a) suggests the idea that a localizedstick—slip relaxation process is at work. As water pressure rises (Fig. 5.5b), a localstrain build-up in the ice is released and the sliding rate increases momentarily; thisChapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 95small rapid motion produces S-function-like spikes in the velocity record as shown inFigure 5.5a. Once the finite relaxation has occurred, further rises in water pressure donot produce additional enhancement of basal sliding.On a cautionary note, the 6-function like pulses in sliding velocity (Fig. 5.5a) wouldbe indistinguishable from stick—slip behaviour of the drag spool, a possibility that wecannot ignore. Such behaviour of our instrument might result from the physical setupduring operation; a “sticky” spool being suspended by an “elastic” cable. However, thesmall force (-..‘l N) required to unwind the spool is not sufficient to significantly stretchthe drag-spool cable. We therefore believe that stick—slip behaviour of the instrumentis unlikely to be the cause of the spikes in the velocity record.I—w.iwE< LI.I 1.0-JI-S2 cc 0.50.0-Jw>ILl 60-Jcc E 504030• 92SM02 a.1 L.k L• 92P06 bI I21 22 23 24 25 26 27JULY 1992Fig. 5.5: (a) Unsmoothed record of displacement rate between anchor anddrag-spool case for drag spool 92SM02, obtained by numerical differentiationof the displacement record shown in Figure 5.2a. (b) Subglacial water pressurerecord of pressure sensor 92P06 (same as in Fig. 5.2b).Chapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 965.4.1 Slider-block modelA simple model for stick—slip behaviour is a slider block pulled by a spring as illustratedin Figure 5.6. Slider-block models have been used to simulate fault behaviour, fore-shocks, aftershocks, and pre- and post-seismic slip, and to explain earthquake statistics(Burridge and Knopoff, 1967; Dieterich, 1972; Rundle and Jackson, 1977; Cohen, 1977;Cao and Aki, 1984, 1986). The block is constrained to move horizontally along a planesurface. It interacts with the surface through friction, which prevents sliding of theblock until a critical value of the pulling force is reached. The block sticks and theforce in the spring increases until it equals the frictional resistance to sliding on thesurface; then slip occurs. The extension of the spring is analogous to elastic strain inrock adjacent to a fault. The slip is analogous to an earthquake on a fault. The storedelastic strain in the spring is relieved in analogy to the elastic rebound on a fault.Fig. 5.6: Illustration of the slider-block model for fault behaviour. Theconstant velocity driver extends the spring until the pulling force k5y exceedsthe static friction force F8. A similar model has been used in this study.Because the stick—slip relaxation process postulated for the glacier bed resemblesthe behaviour of a fault, we attempted to model glacier sliding using a slider block similar to that shown in Figure 5.6. The extension in the spring is analogous to the build-upof elastic strain in the ice and the slip corresponds to the momentary enhancement ofbasal sliding.V)F;Chapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 97A quantitative study of the behaviour of this simple spring—block model immediately reveals that the periodicity of the stick—slip events depends solely on parametersthat are inherent to the system, i.e., the mass of the block, the spring constant andthe ratio of static to dynamic friction. In contrast, the stick—slip events observed withdrag spooi 92SM02 (Fig. 52a) appear to be forced by diurnal subglacial water pressurefluctuations (Fig. 5.2b). In addition, the displacement record (Fig. 5.2a) indicates thatafter an enhanced sliding event the ice flow slows down to roughly half the averagesliding rate rather than coming to a complete stop. For these reasons, the spring—blockmodel as presented above is not suitable for simulating the stick-slip relaxation pro.cess which might operate at the bed of a glacier. The analogy between a glacier and aspring attached to a block lacks precision. We therefore seek a more physically basedmodel in which we can choose model parameter values that approximate more closelyconditions on real glaciers.5.4.2 Elastic block modelIn the following sections we compute the motion of ice that is purely elastic and slidesover a flat surface with a basal resistance that varies in response to fluctuating subglacialwater pressures. We begin by looking at our visualization of the bed beneath TrapridgeGlacier. Ice—bed contactMurray and Clarke (1995) described the ice—bed contact beneath Trapridge Glacieras a thin inacroporous horizon, a layer consisting of granule- and pebble-sized clastsbetween the glacier ice and the underlying matrix-rich sediments. Depending on localconditions the intergranular pore space in this horizon is occupied by water or ice. As aresult, we can identify at least two distinct components of the subglacial water system,Chapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 98which we refer to as the connected and unconnected water systems. Meitwater thatreaches the bed through crevasses or moulins from the glacier surface or water thatoriginates at the bed by melting due to frictional or geothermal heat is evacuated fromthe glacier bed through the connected water system. We visualize this water as flowingthrough the pore space of the macroporous horizon in a drainage configuration thatconsists of patches connected by a channelized system (Fig. 5.7). The remainder of theglacier bed is covered by the unconnected water system. Here, the ice penetrates intothe pore space of the horizon, possibly interspersed with isolated pockets of water whichis not in communication with other free water in the subglacial water system. With thetwo components of the subglacial water system, we effectively divide the glacier bedinto two regions. Let a be the fractional area of bed which is covered by the connectedsystem. We shall refer to this part of the bed as region A. Consequently, the areafraction of the bed which is covered by the unconnected system, refered to as region B,is 1 — a. Despite our belief that the areal coverage of the connected region can increaseas rising water pressures cause local uplift of ice in the vicinity of a connected waterchannel, we assume a to be constant in the following analysis. Description of modelModeling ice flow over a complex glacier bed such as the one shown in Figure 5.7 isbeyond the scope of this thesis. Instead, we take a simplified approach and representthe glacier/substrate interaction by a system consisting of three ice blocks and threesubstrate blocks (Fig. 5.8a). Block A1 represents the parts of the glacier that slide overthe connected region of the bed with area fraction a while block B1 represents all the icethat slides over the unconnected region of the bed with area fraction 1 — a. At the top,block A1 and block B1 are attached to block C1. In this way the two blocks are coupledto each other, but otherwise are allowed to deform and thus move independently. WeChapter 5. DIRECT MEA S UREMENT OF SLIDING AT THE GLACIER BED 99can view blocks A1 and B1 as being hinged to block C1. The height of block A1 andblock B1 represents what we call the “strain equilibration distance”. This is the distanceabove the bed at which strain differences within the glacier disappear and all the icemoves at the same rate. Below the ice—bed interface, the ice blocks are opposed by asimilar system of substrate blocks A, B5 and C5.We first consider the basal water pressure to be low in the connected region ofthe glacier bed. In terms of our block model, a poorly lubricated ice—bed interfaceimplies a high resistance to sliding for block A1 because there is a strong couplingbetween blocks A1 and As. As a result the two blocks start to deform under theapplied shear stress imposed by block B1, which continues to slide (Fig. 5.8b). Whensubglacial water pressure rises in the connected region, block A1 becomes decoupledFig. 5.7: Conceptual diagram of our image of the ice—bed contact beneathTrapridge Glacier. Bed region A is covered by the connected water systemwhile bed region B is covered by the unconnected water system. The fractionalareas of these two regions are a and 1 - a.Chapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 100Fig. 5.8: Conceptual diagram of the elastic block model. (a) Buildingblocks of the model consisting of a system of three ice blocks and a similarsystem of three substrate blocks separated by the ice—bed interface. Insetshows a plan view indicating the area fractions that represent the connectedand unconnected regions of glacier bed. (b) Behaviour of blocks as the glacierslides along the ice—bed interface. Subglacial water pressure is assumed tobe low implying a high resistance to sliding for block A1 due to strong localice—bed coupling.1-cw REGION Bcx REGION A,,—BLOCK C1 a‘ BLOCK B1BLOCK A1—’BLOCK As—_--j/__BLOCK B5ICE-BEDINTERFACES—BLOCK CICEBEDbfrom block As due to increased lubrication of the bed. At this point, any elasticChapter 5. DIRECT MEA S UREMENT OF SLIDING AT THE GLACIER BED 101component of the deformation can be recovered, i.e., block A1 snaps forward whileblock A snaps backward. Mathematical formulationWe consider a glacier of thickness h1 that flows over a horizontal bed. The x axis isdirected in the glacier flow direction and the z axis is vertical, pointing positive upwardthrough the ice. With the two bed regions A and B as introduced in Section,the basal shear stressTb = pigh1sinO (5.1)can be unevenly distributed on the bed so thatTb = ao4(x, y, 0, t) + (1 — a)o3(x, y, 0, t) (5.2)where p is the density of ice, g is the gravitational acceleration, 6 is the surface slopeof the glacier and oA and crB represent the basal shear stresses in regions A and B,respectively. In the following we again base our analysis on the assumption that glacierflow obeys the linear sliding law as given in Equation (4.1), where the basal slidingvelocity ‘Vb is related to the basal shear stress m by a drag coefficient f. With referenceto Figure 5.9 we can then write down expressions for the shear stresses on region AaA(x y, 0, t) = fA (x2 — x1) (5.3)and region BB B X3U (x,y,0,t)=f -- (5.4)where fA and fB are the drag coefficients for the connected and unconnected regionsof glacier bed, respectively.Chapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 102Fig. 5.9: Sideview of ice and substrate blocks showing dimensions anddisplacements of blocks (see text for details).Furthermore, we consider that blocks A1 and A5 only deform elastically and completely ignore any creep deformation within the ice and viscous deformation of thesediment. As wifi be subsequently shown (see Section, this simplification isa serious shortcoming of the model. Nevertheless, with this assumption and notingthat L1 and L are the strain equilibration distances in the ice and the substrate (seeFig. 5.9), we can easily write down stress—strain relations for block A1A(x,y,o,t)= (xs_x2)GI (5.5)and block AsA(x,y,o,t) = xi G (5.6)where G1 and Cs denote the shear moduli of ice and substrate.Substitution of Equations (5.5) and (5.6) into Equation (5.3) yields an equationwhich describes the motion of block A1 with respect to block. Asd 1 x3—(x2---)IL1 LsN(5.7)L11 ICEBEDI-X1E-—x3Chapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 103To obtain the corresponding equation of motion for block B1 we substitute Equations (5.4) and (5.7) into Equation (5.2)dx3 a—2Tb x3—(x21)58dt2fB(a_1) (+ . )‘G1 Gs5.4.2.4 Model resultsTable 5.2 summarizes the model parameter values that were used to obtain the calculated solutions. Using an ice thickness of h1 = 72 m, a surface slope of S = 70(Blake, 1992; Clarke and Blake, 1991) and a density of ice of p’ = 900 kg m3 substituted into Equation (5.1), we calculated a mean basal shear stress for Trapridge Glacierof 1b = 77 kPa. The drag coefficient for the unconnected region of the glacier bed wastaken to be essentially an average constant value. In principle, it is the same as the background drag coefficient in Section 4.4.5; thus, we chose fB = 1.66 x 1011 Pasm. Theareal distribution of connected and unconnected regions of the bed beneath TrapridgeGlacier can be estimated from our drilling program. About 20—25% of holes drilled tothe glacier bed appear to connect to the subglacial water system. We therefore assigneda = 0.2 for the area fraction of the connected region in our model calculations. Elasticproperties of ice are reasonably well known. Hobbs (1974, p. 258) and Sinha (1984)list values of the shear modulus of ice in the range 3.36 x 10—3.80 x io Pa. For ourmodel calculations, we used G1 = 3.4 x i09 Pa. In contrast, elastic properties of soilsare less well constrained. The shear modulus is found to depend on stress state as wellas stress history of the particular soil sample (Yu and Richart, 1984). Typical values forsands and clays easily span one order of magnitude. An estimate of the shear modulusof Trapridge sediment can be calculated from the results of seismic reflection studiesconducted on Ice Stream B (Blankenship and others, 1986, 1989). The shear waveChapter 5. DIRECT MEA S UREMENT OF SLIDING AT THE GLACIER BED 104velocity of V8 150 ms1 measured in the till layer immediately beneath the ice andan assumed substrate density of PS = 2000 kg m3 substituted intov= () (5.9)yields a theoretical estimate of the shear modulus of the substrate G = 4.5 x iW Pa.The values for the strain equilibration distance in ice and in the substrate were determined on a .trial and error basis to yield the final results.Parameter Symbol Value UnitsBasal shear stress Tb 77 kPaDrag coefficient for region B fB .1.66 x 1011 Pasm’Area fraction of connected region a 0.2Shear modulus of ice G1 3.4 X iO PaShear modulus of substrate 4.5 x i07 PaStrain equilibration distance in ice L1 50 mStrain equilibration distance in bed L5 3 mTable 5.2: Parameters for the elastic block model.To simulate the variable resistance to sliding in the connected region of the glacierbed in response to varying subglacial water pressures, a pressure dependence was included in the calculation of the drag coefficient fA However, from inspection of Figures 5.2a and 5.2b, we note that the strain build-up in the ice is only released after acertain threshold level of subglacial water pressure has been reached. For this reason,a simple linear inverse relationship between drag coefficient and subglacial water pressure, as had been used in Section 4.4.5, is not appropriate for these model calculations.Figure 5.10 shows a composite plot of data from pressure sensor 92P06 (Fig. 5.2b) anddisplacement record from drag spooi 92SM02 (Fig. 5.2a) where we have also identifiedthe strain release events by arrows. Examination of these strain release events showsthat these occur over a range of water pressures and there is no clearly defined triggerlevel at which ice slip is initiated. We are not perplexed by this behaviour because theChapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 105slip condition is probably stochastic rather than deterministic so that each slip eventis distinct from previous ones. Due to our inability to identify an obvious conditionfor strain release, we take the trigger levels of water pressure for ice slip initiation asknown a priori. We incorporated the strain build-up followed by the slip initiation bycalculating the drag coefficient as a function of subglacial water pressure, fA(pw), asfollows (Fig. 5.11)1aPw + bfA() =apW + b(apw + btrig+PW<Pwtrig+ trig—Pw PWPwpw >where tr denotes the threshold level for slip initiation, 6 = — p+ is thetransition zone over which most of the ice slip occurs and (see Table 5.3 for parametervalues)A _A+ — Jmax Jtrig+a— i-flax mmPw PwçA—— Jtrig— Jmiuamax mmPw PwcAJmaxJmin— trig— trig+Pw Pwmax ± mm— f,ax + fig+— a+PW Pw— 2 2max ± mmb- — fig_+f— a’’Pw— 2 2—fmAax + fmAintrig—+trig+— 2Pw(5.11)(5.12)(5.13)(5.10)Chapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 106Parameter Symbol Value UnitsMaximum subglacial water pressure 76 m(H20)Minimum subglacial water pressure pJ’ 23 m(H20)Maximum drag coefficient for region A fax 3.0 x 1011 Pa s mMinimum drag coefficient for region A f 0 Pasm’Drag coefficient at pfC (upper line) fig+ 2.0 x 1011 PasmDrag coefficient at p’ (lower line) fjg_ 1.0 x 1011 PasmTransition zone for ice slip 6 1 m(H20)I—zLiiUiC.)ri.C,)Table 5.3: Parameters for calculation of drag coefficient for region A.For water pressures below the trigger level p+, the drag coefficient does notchange significantly; once the water pressure reaches the trigger level, there is a dramatic drop in drag coefficient (Fig. 5.11). The threshold level for slip initiation can bealtered by shifting the steep section (transition zone) in Figure 5.11 along the waterpressure axis.70n J‘‘ Lii>-n Ui.Ju_I4OujI3021 22 23 24 25 26 27JULY 1992Fig. 5.10: Displacement record from drag spool 92SM02 (same as Figure 5.2a) and data from water pressure sensor 92P06 (dashed line, same asFigure 5.2b). Arrows indicate our identification of the trigger levels of waterpressure for strain release events.Chapter 5. DIRECT MEA SUREMENT OF SLIDING AT THE GLACIER BED 107Fig. 5.11: Relationship between the drag coefficient for the connectedregion of glacier bed fA and subglacial water pressure Pw•Figure 5.12 shows the results from our calculations using the elastic block model.The computed displacement of block A1 (Fig. 5.12a) displays a remarkable similarityto the field data from 92SM02. At the same time, the motion of block B1 (Fig. 5.12b)appears to be characteristic of the responses that have been recorded with otherdrag spools (i.e., linearly increasing displacement without distinct diurnal signal) (seeFig. 5.4). DiscussionThe assumption that glacier ice and substrate only deform elastically is certainly a grossover-simplification because our model only takes into account the short-term responsesof ice and sediment. We essentially ignore any long-term responses such as those ofa Glen-law viscous fluid in the case of ice. The deformation behaviour of ice is foundto have a complex time-dependence. Sinha (1978, 1981) investigated and analyzed thefA(pw)1maxItrig+‘trig‘mmmm trig+ trigw w wrnaxPWpwChapter 5. DIRECT MEA S UREMENT OF SLIDING AT THE GLACIER BED 108300--::300BEEBIJULY 1992Fig. 5.12: Computed displacements of (a) block A1 and (b) block B1. Notethe similarity of the displacement record of block A1 to that as measured withdrag spool 92SM02 (included for comparision in (a), dashed line).phenomenological viscoelasticity of polycrystalline ice and showed that ice creep underuniaxial compression is composed of an instantaneous elastic response followed by adelayed elastic and viscous deformation. Therefore, the question of whether viscoelasticrelaxation of elastic strain proceeds so rapidly that elastic strain cannot be accumulatedeffectively, has to be examined.To simplify the following analysis, we approximate the transient rheological behaviours of ice and sediment by assuming that both behave as viscoelastic Maxwellmaterials. In this case, the viscoelastic relaxation time (Malvern, 1969, p. 315) for iceis given by= (5.14)Chapter 5. DIRECT MEA S UREMENT OF SLIDING AT THE GLACIER BED 109and for sediment by= (5.15)where and jt denote the effective linear viscosities of ice and sediment, respectively.For 0°C ice we had previously (see Section 4.4.5) calculated a dynamic viscosity of= 1.24 x i’ Pa s which substituted in Equation (5.14) yields a lower limit on theviscoelastic relaxation time of r = 1.0 h. For Trapridge Glacier in the region of thisstudy, the melting point is reached only near the bed and upper layers consist of cold icewith below-freezing temperatures. Taking B 1.6 x 1015 skPa3 (parameter valuefrom Paterson (1994, p. 97) for —5°C ice) in Equation (4.26) yields an upper limit onthe stiffness of the flow law and gives the dynamic viscosity as = 5.27 x iO’3 Pa s.For this colder and stiffer ice, the relaxation time increases to = 4.3 h. Similarly,we can calculate upper and lower limits on the viscoelastic relaxation time for thesubstrate by substituting our estimates of the linear viscosities for Trapridge sediment(see Chapter 2) into Equation (5.15). For viscosities JL between 3.0 x io Pas and3.1 x 1010 Pas (see Section, we calculated relaxation times rs ranging from0.02h to 0.19h.Although viscoelastic relaxation in ice is unlikely to proceed so rapidly that theelastic strain build-up is completely cancelied out, accumulation of elastic strain in thesubstrate over time scales of days cannot operate because of the short relaxation times.At first glance it might be tempting to use sediment viscosities that are an order ofmagnitude higher than our estimates (e.g., those inferred from work done by Boultonand Hindmarsh (1987) beneath Breidamerkurjökull in Iceland) to calculate viscoelasticrelaxation times TS of the order of hours. Concerns about the suitability of our model,however, remain. While the sediment layer beneath Trapridge Glacier is believed tobe up to 10m thick in places (Stone, 1993), we think that a strain equilibrationdistance of L5 = 3 m (see Table 5.2) is inconceivably high because measurements ofChapter 5. DIRECT MEASUREMENT OF SLIDING AT THE GLACIER BED 110subglacial deformation (Blake and Clarke, 1989) suggest that the typical thickness of thedeforming layer does not exceed 0.5 m. We could remove this concern by substitutinga lower shear wave velocity into Equation (5.9) to yield a softer substrate with a lowershear modulus G. However, an already very low shear wave velocity of v8 = 150 ms1is only found in very porous materials under low effective pressures. Although saturatedwith water at a high pore pressure, the porosity of the sediment layer beneath TrapridgeGlacier is not believed to exceed that of Ice Stream B.We note that our estimate of the shear modulus for the substrate Gs was obtainedusing a method based on the propagation of shear waves and therefore represents thedynamic value. However, in our case of a deforming subglacial sediment, a static shearmodulus would be more appropriate. Nevertheless, dynamic methods that are based onseismic wave propagation remain applicable in estimating elastic properties providedwe know how to relate the dynamic moduli to the moduli from static measurements.In laboratory studies, Brace (1965) and Simmons and Brace (1965) investigated therelationship between static and dynamic Young’s moduli in diabase and granite andfound that the dynamic modulus is about 20% greater than the static modulus for bothrocks. Results from investigations on sandstones and shales (Cheng and Johnston, 1981;Jizba, 1991) show that the static moduli are generally lower than the dynamic ones.The discrepancy in moduli from static versus dynamic methods were found to be aslarge as a factor of five.Despite the obvious shortcomings discussed above, the good fit that we were able toachieve between model results and field observations (Fig. 5.12) suggests that the elasticblock model is successful in explaining the data that have been recorded with drag spool92SM02 (Fig. 5.2a) and pressure sensor 92P06 (Fig. 5.2b). An attractive feature of ourmodel is the potential ability to explain negative subglacial shear-strain rates observedat Trapridge Glacier (Blake, 1992) and Storglaciären (Iverson and others, 1995) duringperiods of high subglacial water pressures. The release of accumulated elastic strainChapter 5. DIRECT MEA S UREMENT OF SLIDING AT THE GLACIER BED 111in the sediment as the ice becomes decoupled from the bed due to increased waterlubrication (corresponding to the backwards snapping of block As (Fig. 5.8) in ourmodel) could account for the observed up-glacier rotation of tilt sensors.Chapter 6SUMMARY AND CONCLUSIONS“Every great scientific truth goes through three stages. First, people sayit conflicts with the Bible. Next they say it had been discovered before.Lastly they say they always believed it.”- Louis AgassizThis thesis describes an investigation of mechanical conditions at the ice—bed interface of Trapridge Glacier, a surge-type glacier currently in the late quiescent phase.The study of basal conditions and processes that control the coupling between a glacierand its bed is of critical importance in understanding the dynamics of surging glaciers.We have documented the development of two novel techniques for exploring thesubglacial environment: measurements of sliding and of mechanical phenomena at thebase of the glacier. Using these new techniques, we have recorded unique subglacialprocesses that had not been observed before. Our instruments enabled continuous andlong-term measurements of basal sliding over a soft substrate and observation of mechanical processes at the ice-bed interface. Both instruments were devised bearing inmind sizes and shapes that permit their in situ installation at the bottom of narrowboreholes (r5 cm diameter). Other important features include their simplicity in design and low costs. It should therefore prove useful to deploy arrays of both types ofinstruments at various locations across the bed in order to study evolving conditionsbeneath normal and surge-type glaciers.The 1991 ploughmeter measurements suggest that across the bed of TrapridgeGlacier, either the texture of subglacial sediment is not uniform or ice-sediment couplingvaries significantly. If one assumes that, at the scale of 10 m, lateral variations in112Chapter 6. SUMMARY A ND CONCL USIONS 113sliding velocity are insignificant, ploughmeter 91PLO1 appears to have been immersedin a region of the bed that predominantly consists of fine-grained material, whereasploughmeter 91PL02 seems to interact with a clast-rich sediment. We showed thatby assuming simple model rheologies, we can use results from ploughmeter 91PLO1 tocalculate estimates of rheological parameters for Trapridge sediment: linear effectiveviscosity and plastic yield strength. We have also demonstrated that if the subglacialsediment texture is known, the collision frequency as indicated by ploughmeter 91PL02can be used to obtain an estimate of the basal sliding velocity. Our analysis assumedtemporal variations in sliding rate to be negligible and a spatially uniform sedimenttexture. In the case of Trapridge Glacier we were able to confirm our velocity estimatesby comparing them with drag spool measurements.We found that the particle-size distribution for Trapridge sediment obeys fractalstatistics and shows clear power-law scaling behaviour over five orders of magnitude.The fractal dimension of D = 2.95 is consistent with those reported for other geological samples and suggests that Trapridge sediment is repeatedly subjected to abrasivegrinding as it deforms under the applied shear stress of the glacier. This fractal scalingbehaviour was used to extrapolate the grain-size distribution to larger size scales thanthose sampled and to examine the self-similar character of the ploughmeter responsespectrum. Furthermore, using a fractal analysis, we have explored the possibility thatthe spectral characteristics of the ploughmeter response can be used to infer subglacialsediment texture, specifically the fractal dimension of the particle-size distribution.Regrettably, it was found that the mechanical response of a ploughmeter is only to asmall extent a function of the size distribution in the sediment, as evidenced by theweak dependence of the fractal dimension of the ploughmeter response on the fractaldimension of the particle-size distribution.Field data from our drag spool instruments suggest that, during the period ofobservation, up to 45—65% of the total flow observed at the glacier surface can be dueChapter 6. SUMMARY AND CONCLUSIONS 114to basal sliding. With the contribution from ice creep known to be small, most of theremaining surface motion must be attributed to bed deformation processes.We presented data recorded simultaneously during the 1992 field season with awater pressure sensor, a drag spooi and two ploughmeters. The measurements indicatethat the basal resistance beneath Trapridge Glacier is spatially and temporally variablebut also suggest that the role of the subglacial water system appears to be complexand not always intuitive. We therefore developed a theoretical framework for thesliding motion of ice over a surface with basal resistance which we have allowed to vary.Based on our model calculations, we have shown that the strongly anti-correlated forceresponses in conjunction with in-phase azimuth responses of the two ploughmeters canbe interpreted in terms of sticky spots being created and destroyed. These spatialand temporal variations of sticky spots are believed to be linked to changes in basallubrication in response to fluctuations in subglacial water pressure. We have alsodemonstrated that the 90° phase shift between water pressure and sliding rate mightbe indicative of a stick—slip sliding process at the glacier base, whereby accumulatedelastic strain in the ice is released as the rising water pressure decouples the ice from thebed. However, such an interpretation seems to require unrealistic assumptions aboutmaterial properties, and so remains speculative.Even though we have only demonstrated consistency on a scale of ‘.1O m for asingle example, it is conceivable that high-drag, slow sliding sites exist all across thebed of Trapridge Glacier. Sticky patches appear to be created and destroyed in responseto fluctuations in subglacial water pressure leading to time-varying lateral transfer ofshear stress beneath the glacier. By associating high water pressures with slipperypatches and low water pressures with sticky patches, we expect the stresses at the bedto form a patchwork distribution which is similar to the pressure distribution in thesubglacial water system.Chapter 6. SUMMARY AND CONCLUSIONS 115 Based on our estimates of sediment strength as calculated from the ploughmeter results, we have suggested that the deformation resistance of the bed is comparable to but less than that needed to balance the applied driving stress. Because stresses are continuously redistributed on the bed, we also proposed that sticky patches do not dominate the force balance in our main study area of Trapridge Glacier and are thus likely candidates for supporting the remainder of the basal shear force required to ensure mechanical stability in this region of the glacier. In summary, we have developed inexpensive and reliable sensors for directly mon-itoring mechanical conditions in a subglacial environment. 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Eng., 110(3), 331—345.Appendix AFORCES ON AN ELONGATED BODY IN STOKES FLOWFor very small Reynolds numbers, inertial forces are negligible compared withthe pressure and viscous forces. Thus, for a body having surface S at rest in anincompressible fluid which at infinity is undergoing a uniform translational motionat velocity U, the velocity v and pressure p in the fluid satisfy the creeping motionequations (Schlichting, 1979, p. 112)iV2v = Vp (A.la)V•v=O (A.lb)with the boundary conditionsv=O on S (A.2a)v—U as x—*oo (A.2b)where is the fluid viscosity and x the position vector of a general point. Oberbeck (1876) solved this problem in the case of an ellipsoid and obtained the total dragexerted on such a body. In particular, for an elongated ellipsoid of revolution havingsemi-axes a and c with c> a, Oberbeck’s formula for fluid motion perpendicular to thesymmetry axis leads to a drag of magnitude16iruaUFD[(C,::_ 1 [(c/a)2_113/2 ln( + (c/a)2 1)](A.3)For large length-to-diameter ratios c/a, Equation (A.3) reduces toF 87rtcU (A4)D— ln(2c/a) + 0.5125Appendix A. FORCES ON AN ELONGATED BODY IN STOKES FLOW 126Also, in the limit, as the length-to-diameter ratio c/a approaches unity, Oberbeck’ssolution (A.3) yields Stokes’ lawFD = 6ir1uaU (A.5)which describes the total drag of a sphere of radius a undergoing uniform translationthrough a viscous fluid (Stokes, 1851).To obtain the force distribution along an elongated axisymmetric body movingperpendicularly through a viscous fluid, we follow the treatment by Batchelor (1970),which is based on slender-body theory for Stokes flow with negligible inertial forces.The key idea is that it is possible to approximate the body by a suitably chosen systemof forces, known as “Stokeslets”, distributed along the symmetry axis. The effect ofthese Stokeslets is to disturb the uniform motion of the surrounding fluid. For a forceF applied at the origin in an infinite body of fluid, the velocity at the point x in flowwith negligible inertial forces is (Oseen, 1927)(A6)87rpxj IxIIwhere p. is the fluid viscosity. Thus, if Stokeslets are distributed over the portion—c < x3 < c of the x3 axis so that the line density of the applied force is F(x3), theresulting fluid velocity at point x is— 1 F(x) (x — x)(x— x)F(x)V,X)—----— + x38irp.—c ((x3— x)2 + ((x3 — x)2 +where x’1 = x = 0 and r2 = x + x. This distribution of Stokeslets represents thedisturbance motion of an axisymmetric body of length 2c, and points on the bodysurface are given by r = a(x3), where a(x3) is the radius of the body as a function ofx3 and a(xs)/c is small compared to unity. If we choose the axes of reference so thatthe body is stationary, the undisturbed velocity is —U. This case corresponds to thetranslational motion of the body with velocity U through a fluid at rest at infinity.Appendix A. FORCES ON AN ELONGATED BODY IN STOKES FLOW 127In order to satisfy the no-slip condition of Equation (A.2), the resultant fluidvelocity v — U must be zero at the body surface. Thus, the unknown function F(x3),representing the Stokeslet strength distribution, must be chosen so that the velocitydue to the line of Stokeslets cancels the undisturbed velocity at all points on the surfaceof the body. It is possible to use a line distribution of Stokeslets as a means of satisfyingthe no-slip condition at points on the surface only if the body is slender. Thus, theintegral Equation (A.7) with v = U at r = a(x3)can be approximated by an asymptoticexpansion in terms of the ratio of cross-sectional radius to body length (a(x3)/c —* 0)(Batchelor, 1970; Cox, 1970; Tillet, 1970). If the undisturbed velocity U is in the x1direction, the force per unit length on the body can be expressed as1 / 2\1/22 ln(2c/a) — — in a (1 —F” ‘— 4irtU2 a(x3) cAX3) ln(2c/a) 21n(2c/a)+1 .8where a is the radius of the body at x3 = 0.In the particular case of an ellipsoidal body (Batchelor, 1970)a(x3),‘ for—cx3c (A.9)a \ C)and Equation (A8) reduces toF(x3)= ln(2c/a)+ 0.5 (A.10)To obtain the total drag of the ellipsoid, we integrate Equation (A.10) over the lengthof the body. This yields[C 87r,ucUFD= j F(x3)dx3 = ln(2c/a) + 0.5 (A.11)which corresponds to Oberbeck’s result given in Equation (A.4). In the case of acylindrical body, we substitute (Cox, 1970)a(x3)= 1 for— c x3 <c (A.12)into Equation (A.8).


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