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Friction at the Base of a Glacier

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Friction at the Base of a Glacier WXW::kTi*[[Hl*f,}H Mitteilungen 101 Friction at the base of a glacier Jürg Schweizer Zürich, 1989 Herausgeber: Prof. Dr. D. Vischer Preface Any sliding process depends on the driving and the retaining forces. If the slidlng mass is cfearfy defined and its motion is only due to gravity, the driving forces can easily be de- termined i.e. the component of weight parallel to the slidlng interface. The retaining forces, howeverf are more difficult to eval-uate, since onJ.y crude empirical rel.ations for the as- sessment of the friction exist. fn modern glaci-o]ogy a fot of work is involved in in- vestigating the conditions at the gJ-acier bed and Cetermining the sliding motion of a glacier. The nature and the physical processes 'down at the base give rise to many questlons: Which sort of friction is adequate? What is the influence of the subglacial hydraulics and what role does the debris concen- tratlon of the basal ice play? The author of the present report, Dr. Jürg Schweizer, deals with these questions. He establishes different types of friction for typical bed characteristics and, to a certain extent, derives qualitative anal.ytical solutions. The appli- cation of the friction model of Ha1let requires a numerical treatment using the finite element method. The results show that the debris concentration in the basal ice and the sub- glacial {ater pressure dominate the sliding process. Accordingly thi-s study foll-ows the extensive field observatlons, carried out during the Last years by the glaci- ologicaL section of the VAW, investigating the relation be- tween the sliding velocity and the subgfaciaf water pressure. The report helps to fill the gap between measurements and em- pirical reLationshi-ps on one hand and the physica] theories on the other. The study was performed with support from the Swiss NationaL Science Foundation and supervised by Dr. Almut lken. Prof Dr. D. Vi-scher -5- CONTTNTS Zusarunenfassunq . 9 A INTRODUCTION 11 B rcE DYNAMICS.... ....... 1,'7 B.1 Introduction.... L] 8.2 Assumptions and Boundary Condj-tions .... ....... 18 8.3 Basi6 Equations . 20 B.4 Laminar Flow . 22 C GL}CIER SLIDINC. 25 C.1 Introduction .... 25 C.2 Sliding wj-thout bed separation .. 29 C.3 Sliding with bed separatj-on ....... 4"7 C.3 . l- Subglacial water pressure . 47 C-3.2 Classic theories 49 C.3.3 Sliding over a sinusoidal bed .. 53 C.3.3.1 Stress distribution without bed separation 54 C.3.3.2 Bed separation: separation and critical pressure 5'l C.3.3.3 Bed separation: stress distribution and cavity length ....... 60 C.3.4 Possibfe sliding law .. 65 D FRICTION 6'7 D.1 Introducti.on.... 6'l D.2 Coul-omb friction 69 D.3 Sandpaper friction '75 D.3,1 Basic concepts .. '75 D.3.2 Friction without bed separation .. 16 D.3.3 Sandpaper frictj-on with bed separation ....... 77 D.3 , 3. 1 Frictional drag "l'7 D.3.3,2 Separation and critical- pressure '19 D.3.3.3 Infl-uence on the sliding motion 83 D.4 Halfet friction -...... 84 -6- f,UUERICAL IPPROICE 89 8.1 Introduction .... 89 8.2 Solution method ....... 90 E.3 General assumptions of the model . ....... 92 E.3.L Temperature 93 8.3.2 Ffow law ....... 93 8.3.3 Summary .. 100 D.4 Geometry .. 101 E.4.1 Shape of the modelled area ....... 101 8.4.2 Reat bed topography .. I02 8.5 Boundary conditions ... 107 E.5.1 Top, front and back side . .. 107 8.5.2. Bottom boundary condition .. 108 E.5.2.1 No slip .... 109 8.5.2.2 Perfect slip . '.. l-09 8.5.2.3 Sliding with friction ... ..... 110 8.5.2,4 Bed separation ... 116 8.6 Test computation . ..'.. 1-l-B 8.6.1 Geometry and material- properties .... ... L20 8.6.2 Analytical, sol-ution . L20 E. 6.3 Numericaf solution . L20 E-6.4 Discussion ..... L22 r NT'I{ERICIL SIMULATION OF GLACIER SLIDING: RESULTS .- r23 F.1 Sliding without bed separation .. .. t24 F,1.1 Lj-near viscous sliding ..... 1-24 8.1,.2 Nonlinear viscous sliding .. l^32 F.l-.3 Sliding with friction ... ... 142 8.2 Sliding with bed separation . .. .. 153 E .2.'J- Frictionless sliding with bed separation. 153 E.2.2 Sliding with frictj.on in presence of bed separation ....... 159 G CONCLUSTONS.. .....156 c.1 Summary ... 165 c.2 Conclusions ..... 169 G.3 Open questions and outlook .. 170 References ..... 171 List of Symbols ...... 1,'7'l Acknowledgements . .... 181 -'7 - Abstract The rnotion of glaciers and ice sheets due to grawity consists of two components: the flow and the sliding motion. The fLo!/ of an ice mass is the internal deformation. The sliding is the motion at the interface between ice mass and substratum existing only if the temperature at the interface is at the pressure mefting point. The roughness of the glacier bed pre- vents the ice mass from slipping ar"tay: the ice is forced to flow around the bed obstacles. The classic theories usually assume that this sort of notion occurs in a frictionless way, since a very thin water film does exist between j-ce and un- derlyingr substratum, i.e. there is no local shear stress. This assumption may be true for cl-ean ice, however' basal ice is debris loaden and friction occurs between the substratum and rock particles embedded in the basal ice, as can be seen from striaes on rock bumps. The aim of this study is to investigate the influence of deb- ris concentration on the sliding process. The actual condi- tions where certain types of friction apply are defined and the consequences for the sliding law are formulated. The classic Coulomb friction is modified accordinq to the notion that a glacier is rubbing over its bed ]ike a piece of sand- paper. Eor small debris concentrations the concept of Hallet applies where the friction depends on the sliding velocj-ty. Hence a numerical approach is required. The numerical model-L- ing of the sliding of an' ice mass over an undulating bed, in- cluding the effect of both the subglacial water pressure and the friction, is done by solvinq the problem by the finite element met.hod using an existing two-dimensionaL code. Friction bet$reen a dirt.y basal layer and the qlacier bed is a refevant process and can be seen as a reduction of the driv- ing shear stress. The frictionaf drag can therefore be in- cluded into existing sliding faws which shoul-d contain as an important variable the critical pressure. A functionaf rela- tionship between the sliding velocity, the effective basal shear stress and the subglacial water pressure j-s given. Con- sidering the seasonaf velocity variations, valIey glaciers -8- may be cfassified accordi-ng to the gl-acier bed characteris- tics and probably vice versa. A more detailed classification and the simulation of the dynamic movement of an actual g1a- cier are two possible directions of further investigations out I ined . -9- Zugamnenfassung Die Bewegung der Gl-etscher und Eisschil-der beruht auf der Schwerkraft. Man unterscheidet das Fliessen und das Gleiten. Mit Ffiessen bezeichnet man den Vorgang der .internen Verfor- mung. Ist die Temperatur am Grund des Gletschers gleich dem Druckschmelzpunkt, so beqinnt das Eis über den Untergrund zu gleiten. Die Rauhj-gkeit des cletscherbettes werhindert, dass der Gletscher abstürzt, da das Eis die Unebenheiten unflies- sen muss, so dass eine in der Regel gLeichförnige Bewegung entsteht. In den klassischen Theorien wj-rd davon ausgegangen, dass die Gfeitbe'nregung reibungsfrei sei a1s Folge eines Was- serflfms zwischen Eis und Untergrund. Somit existiert lokal gresehen keine Scherspannung/ was unter Umständen bei sauberem Eis zut.rifft. TaLsächlich aber ist das basale Eis eine ge- schichtete Mischunq aus Eis und Fefspartikefn. Schliffspuren auf Fel-sbuckefn zeugen von der Reibung zwischen dem felsigen Untergrund und im Eis eingefrorenen Steinen. Zief dieser Arbeit isL €sr den Einfluss unterschiedlicher Schuttkonzent.ratj-onen des basalen Eises auf die Gleitbewegung zu untersuchen. Verschiedene Arten der Reibunqf werden charak- terisiert, und es werden die Auswirkungen auf das Gteitgesetz besprochen. Die Coulomb-Reibung wird modj-fiziert irn Hinblick auf die Idee, dass der Gletscher wie ein Stück Sandpapier den felsigen Untergrund abschrnirgelt. Für geringe Schuttkonzen- tration wird das ModelI von Hallet verv/endet: die Reibung hängt won der Gfeitgeschwindigkeit ab. Somit drängt sich eine numerische Behandfung auf. Mit der Methode der finiten EIe- mente wird das ent.stehepde Differentialgleichungssystem zur Simulation der cl-eitbewegung qel-öst/ wobei verschiedene para- meter wie Schutt.gehalt und subglazial-er Wasserdruck, die das Gleiten beeinflussen, variiert werden können. Die Reibung erweist sich al-s ein für die Gleitbewegung mass- geblicher Faktor. Die Wirkung der Reibung kann durch die Ein- führung einer werminderten, effektiven Scherspannung berück- sichtigt werden. Auf diese Weise lassen sich di-e für reines Eis gültigen Gleittheorien formal auf den Fafl schutthaltigen Eises übertragen. Ein funktionaler Zusammenhang zwischen der _ 10 _ cfeitgeschwindigkeit, der basalen Schuttkonzentration und dem subglazialen Wasserdruck bei gegebener Schubspannung wird er- nittelt. Aus der Art der saisonalen Schbrankunqen der Oberfl-ä- chengeschwindigkeit von TaLgl-etschern lassen sich Rückschlüs- se ziehen auf die Natur des Uniergrundes und die Art der Rei- bung. Eine detail-iertere Kfassifikation und die Sinufation der dynamischen Bewegung eines bestimmten Gletschers sind mögliche zukünftige Forschungsvorhaben. - 11 - Cbapter A INTRODUCTf,ON Gl-aciers are moving. This fact, r^'el-f known by the inhabitants of the Alpine regions, was a source of scientific controversy in the past century. The mountain-dr^'e11ers exper- ienced the moving force of a glacier at the beginning of the LittIe Ice Age, when advancing Alpine glaciers passed over meador,rs and forests and destroyed huts (Vögele, 1"987).
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