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Accurate modelling of glacier flow Waddington, Edwin Donald 1981-04-15

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ACCURATE MODELLING OF GLACIER FLOW by EDWIN DONALD WADDINGTON B.Sc, University of Toronto, 1971 M.Sc, University of Alberta, 1973 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Geophysics and Astronomy) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1981 © Edwin Donald Waddington, 1981 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Geophysics and Astronomy The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Da te February 20, 1982, /7cn i i ABSTRACT Recent interest in climatic change and ice .sheet variations points out the need for accurate and numerically stable models of time-dependent ice masses. Little attention has been paid to this topic by the glaciological community, and there is good reason to believe that much of the published literature on numerical modelling of the flow of glaciers and ice sheets is quantitatively incorrect. In particular, the importance of the nonlinear instability has not been widely recognized. The purposes of this thesis are to develop and to verify a new numerical model for glacier flow, compare the model to another widely accepted model, and to demonstrate the model in several glaciologically interesting applications. As in earlier work, the computer model solves the continuity equation together with a flow law for ice. Thickness profiles along flow lines are obtained as a function of time for a temperate ice mass with arbitrary bed topography and mass balance. A set of necessary tests to be satisfied by any numerical model of glacier flow is presented. The numerical solutions are compared with analytical solutions; these include a simple thickness-velocity relation to check terminus mobility, and Burgers* equation to check continuity and dynamic behaviour with full nonlinearity. An attempt has been made to verify the accuracy of the computer model of Budd and Mclnnes (1974), Rudd (1975) and Mclnnes (unpublished). These authors have reported problems with numerical instability. If the existing documentation is iii accurate, the Budd-Mclnnes model appears to suffer from mass conservation violations both locally and globally. The new numerical model developed in this thesis can be used to reconstruct the velocity field within the glacier at each time step; this velocity field satisfies continuity and Glen's flow law for ice. Integration of this velocity field yields the trajectories of individual ice elements flowing through the time-varying ice mass. The trajectories and velocity field are checked by comparison with an analytical solution for a steady state ice sheet (Nagata, 1977). The model in this thesis is not restricted to steady state, and it avoids the violations of mass conservation, and the approximations about the velocity field found in some published trajectory models. The'feasibility of using stable isotopes to investigate prehistoric surging of valley glaciers has been studied with a model simulating the Steele Glacier, Yukon Territory. A sliding < velocity and surge duration were specified, based on the observations of the 1966-67 surge. A surge period of roughly 100 years gave the most realistic ice thickness throughout the surge cycle. By calculating ice trajectories and using two plausible relationships between 6(01B/016) and position or height, longitudinal sections and surface profiles of 6 were constructed for times before, during, and after a surge. Discontinuities of up to 0.8°/Oo were found across several surfaces dipping upstream into the glacier. Each of these surfaces is the present location of the ice which formed the ice-air interface at the time a previous surge began. It may be difficult to observe these surfaces on the Steele Glacier due to iv the large and poorly-understood background variability of 6. The generation of wave ogives has been examined following the theory of Nye (I958[b])r wherein waves are caused by a combination of seasonal variation in mass balance and plastic deformation in an icefall. The wave train generated on a glacier is shown in this thesis to be a convolution of the velocity gradient with an integral of the mass balance function. This integral is the impulse response of the glacier surface to a step in the velocity function. Spatial variations in the glacier width and mass balance also contribute to the wave train. This formulation is used to explain why many icefalls do not generate wave ogives in spite of large seasonal balance variations and large plastic deformations. V TABLE OF CONTENTS Abstract ii List Of Tables xiv List Of Figures xAcknowledgements xix Chapter 1: Beginnings 3 1.1 Introduction1.1.1 Ends And Means1.1.2 Conventions Used 9 1.2 Previous Work 11 1.2.1 Four Centuries Of Glacier Flow Theory 11 1.2.2 Previous Ice Profile Models 13 1.2.3 Previous Ice Trajectory Models 5 1.3 The Continuity Equation For An Ice Mass 17 1.3.1 The General Glacier Flow Problem1.3.2 Rectangular Cross-section Flow Model 18 1.3.3 The Continuity Equation 20 1.3.4 Physical Interpretation 2 1.4 Physics Of Ice Deformation 23 1.4.1 Introduction 21.4.2 Stress Equilibrium Equations 24 1.4.3 Constitutive Relation For Ice Deformation 26 1.4.4 Shear Stress And Ice Flux 30 1.5 Basal Sliding 35 1.5.1 Introduction1.5.2 Basal Ice Temperature And Sliding 35 1.5.3 Philosophy Of Sliding In This Study 36 vi Chapter 2: Models And Tests ; 38 2.1 Introduction 32.1.1 Outline2.1.2 Importance Of Model Testing 38 2.2 Continuity Equation Profile Model 42 2.2.1 Introduction 42.2.2 The Numerical Scheme 43 2.2.3 Boundary Conditions 6 2.2.4 Numerical Stability 48 2.2.5 Accuracy 55 2.3 Testing The Continuity Model 57 2.3.1 Introduction2.3.2 Continuity Test With Terminus Motion 57 2.3.3 Continuity Test With Burgers' Equation 62 2.4 The Ice Trajectory Computer Model . .' 67 2.4.1 Introduction 62.4.2 The Velocity And Displacement Fields 67 2.4.3 The Ice Particle Trajectories 70 2.4.4 Accuracy Of The Trajectory Model 72.5 Testing The Trajectory Model 74 2.5.1 Introduction 72.5.2 Nagata Ice Sheet Test 74 2.5.3 Surface Mass Conservation Test 79 Chapter 3: Can Stable Isotopes Reveal A History Of Surging? 83 3.1 Introduction 83.2 Steele Glacier 85 3.2.1 General Description 85 vii 3.2.2 Glacier Surges 86 3.2.3 Observations Of The Steele Glacier 88 3.2.4 Period Of Steele Surges 91 3.3 Numerical Model 1 93 3.3.1 Flow Law Constants And Shape Factor 93 3.3.2 Bed Topography 94 3.3.3 Channel Width 6 3.3.4 Mass Balance 7 3.3.5 Cyclic Surge Pattern For The Model 100 3.4 Steele Glacier Model 2 103 3.4.1 Problems With Steele Model 1 103.4.2 Simplifications 108 3.5 Stable Isotopes In Glaciology 103.5.1 Definition Of The Del Scale 108 3.5.2 Factors Affecting Del 110 3.5.3 Previous Isotopic Studies 112 3.5.4 Del Relations For The Model 116 3.6 Model Results: Surge Period And Trajectories 119 3.6.1 Introduction 113.6.2 Periodically Repeating State 119 3.6.3 Ice Trajectories 126 3.7 Model Results: Distribution Of Isotopes — 127 3.7.1 Introduction 123.7.2 Model 61: Longitudinal 6 Sections 129 3.7.3 Model 61: Surface 6 Profiles 133 3.7.4 Model 62: Sections And Surface Profiles 135 3.7.5 Are The Predicted Effects Observable? 138 3.7.6 Conclusions 140 vi i i Chapter 4: Wave Ogives 142 4.1. Introduction4.1.1 Description Of Ogive Systems 142 4.1.2 Theories Of Wave Ogive Formation 145 4.1.3 Disappearance Of The Waves 146 4.2 Nye's Theory Of Wave Ogives 148 4.2.1 Outline Of The Nye Theory 144.2.2 Nye's Annually Repeating State Solution 148 4.2.3 Ogives Above The Firn Line 151 4.2.4 An Unanswered Question 152 4.3 A New Solution For Ogives 3 4.3.1 Using Method Of Characteristics 154.3.2 Separable Mass Balance 155 4.3.3 The Generalized Velocity 157 4.3.4 The Upstream Boundary Condition 154.3.5 The Terms Of The Flux Solution 158 4.3.6 Physical Interpretation 160 4.3.7 The Green's Function For Ogives 167 4.3.8 A Convolution Formulation For Ogives 167 4.4 Some Simple Examples 168 4.4.1 Introduction4.4.2 Example: Linear Velocity Gradient 169 4.4.3 Example: Double Step Icefall Model 174 4.5 Austerdalsbreen 176 4.5.1 Introduction4.5.2 Estimated Wave Generation 177 4.5.3 Ogive Solution For Austerdalsbreen 178 4.5.4 Finding The Wave Generating Region 181 ix 4.6 Conclusions 184 List Of Symbols 6 Literature Cited 201 Appendix 1: Continuity Model 235 A1.1 The Numerical SchemeA1.1.1 The Continuity Equation 235 A1 .1.2 A Matrix Formulation 8 A1.2 Nonlinearity 239 A1.3 Boundary Conditions 243 A 1.3.1 The Upper BoundaryA1.3.2 The Downstream Boundary 245 A1.3.3 Nonzero Flux Leaves Downstream Boundary 245 A1.3.4 Moving Wedge Terminus 247 A1.4 Numerical Stability 254 A1.4.1 IntroductionA1.4.2 The Linear Computational Instability 255 A1.4.3 The-Nonlinear Instability 257 A1.4.4 Velocity Smoothing 260 A1.4.5 Numerical Dissipation 2 A1.4.6 Dissipation From The Velocity Equation 265 A1.4.7 Wavenumber Spectral Truncation 267 A1.5 Accuracy 269 A1.5.1 6 Parameter: Accuracy 26A1.5.2 Phase Errors 271 A1.5.3 Truncation Error 274 Al.5.4 Interpolation Error 8 Appendix 2: Ice Trajectory Model 282 A2.1 Introduction 28X A2.2 The Velocity Field 284 A2.2.1 The Rectangular Flow Model 28A2.2.2 The Downslope Velocity 285 A2.2.3 The Longitudinal Strain Rate 286 A2.2.4 Velocity Normal To The Bed 288 A2.3 Ice Displacement Field 290 A2.3.1 Four Point Interpolation 29A2.3.2 Displacements At Meshpoints 291 A2.4 Ice Particle Trajectories 295 A2.4.1 Tracking ProcedureA2.4.2 Particles Which Reach Ice Surface 296 A2.4.3 Tracking Backwards In Time 29A2.4.4 Boundary Condition At Upstream End 297 Appendix 3: Aspects Of Discrete Data Series 298 A3.1 The Z Transform 29A3.2 Aliasing 299 Appendix 4: Density Of Glacier Ice 302 A4.1 Firn As Equivalent Ice Thickness 30A4.2 Constant Density Assumption 303 Appendix 5: Continuity Equation For An Ice Mass 306 A5.1 Mass Conservation In A Moving Continuum 306 A5.2 In A Stationary Glacier Cross-section 307 A5.3 In An Arbitrary Channel 310 A5.4 In A Rectangular Channel 1 A5.5 In Bed-normal Coordinates 313 Appendix 6: Equations For Perturbations 316 Appendix 7: Velocity Equation For An Ice Mass 318 A7.1 Introduction 31A7.2 The Shear Stress Equation 319 A7.3 Approximations 322 A7.4 Shape Factors 8 A7.5 Shear Strain Rate 330 A7.6 Ice Flux And Average Velocity 332 Appendix 8: Glacier Sliding 334 A8.1 Measurements . 33A8.2 Physical Processes In Sliding 335 A8.3 Computer Models Of Sliding 338 A8.3.1 Using Weertman Sliding 33A8.3.2 Budd-Mclnnes Model 338 A8.3.3 Sliding In This Study 342 Appendix 9: Burgers' Equation 3 Appendix 10: Matrix Coefficients 346 Appendix 11: Convergence Criteria 350 Appendix 12: Machine Roundoff Errors 354 Appendix 13: Differencing Scheme For The Flux Gradient .... 356 A13.1 Introduction 35A13.2 Perturbation Equations 357 A13.3 Space Differencing Scheme 358 A13.4 Transfer Function 360 A13.5 Conditions On The Mesh Interval 361 Appendix 14: Ice Surface Elevations 364 Appendix 15: Analytic Models Of Ice Sheets 366 A15.1 Introduction 36A15.2 Nye Ice Sheet Model 366 A15.3 Nagata Ice Sheet Model 7 A15.3.1 Basic Equations 36xi i Al5.3.2 Ice Depth, Mass Balance, And Velocity 368 A15.3.3 Streamlines 370 A15.4 Haefeli-Paterson Ice Sheet Model 374 Appendix 16: Tests Of The Budd-Mclnnes Model 375 A16.1 Introduction 37A16.2 Ice Flow In The Budd-Mclnnes Model 376 A16.3 Vatnajokull Model: Nonlinear Instability 377 A16.4 Vatnajokull (Model 1): Mass Conservation 381 A16.5 BruarjSkull (Model 2): Steady State Flux 386 A16.6 Fedchenko Glacier: Steady State Flux 392 A16.7 Fedchenko Glacier: Nonsliding Model 395 A16.8 Fedchenko Glacier: Dynamic Behaviour 398 A16.9 Conclusions 411 Appendix 17.: Four Centuries Of Glacier Flow Theory . 414 A17.1 Introduction 41A17.2 The Years 1570 To 1840 415 A17.2.1 Earliest PioneersA17.2.2 H. B. De Saussure 416 A17.2.3 Rendu 418 A17.3 1840 To 1915 9 A17.3.1 Louis Agassiz 41A17.3.2 J. D. Forbes 421 A17.3.3 John Tyndall 4 A17.3.4 Many Wondrous Theories 427 A17.3.5 Ice Deformation Experiments 429 A17.3.6 Mathematical Glaciology ...- 431 A17.3.7 The Briny Depths Of Glaciers 434 A17.4 1915 To 1953 436 xi i i A17.4.1 Introduction 436 A17.4.2 Shear Plane Slip Or Viscous Flow? 437 A17.4.3 Continuum Mechanics For Glaciers 439 A17.4.4 The Glacier Anticyclone 441 A17.4.5 Field Studies: 1934 Spitzbergen Expedition 444 A17.4.6 Field Studies: Jungfraujoch Research Party .... 446 A17.4.7 Extrusion Flow 449 Appendix 18: Stability Condition For A Surge Bulge 454 Appendix 19: Steele Glacier Tributaries 457 xiv LIST OF TABLES 2.1. Parameters For Vatnajokull (Figure 2.3) 53 2.2. Parameters For Test With Moving Terminus 58 2.3. Model Parameters For Burgers' Equation Test 66 2.4. Parameters For Nagata Ice Sheet 78 2.5. Residence Times In Nagata Ice Sheet 80 2.6. Nagata Ice Sheet Surface Boundary Test 82 3.1. Parameters For Steele Steady State 104 3.2. Velocity Pattern For Steele Surge 120 3.3. Numerical Time Steps For Steele Surge 124 4.1. Odinsbreen: Linear Velocity Approximations 177 A1 5.1 . Parameters For Nagata Ice Sheet 374 A16.1. Parameters For Vatnajokull Model 379 A16.2. Parameters For Fedchenko Model 398 A16.3. Parameters For Sliding Models 405 A19.1. Ice Flux From Steele Glacier Tributaries (a) 457 A19.2. Ice Flux From Steele Glacier Tributaries (b) 459 A19.3. Tributaries: Effect On Mass Balance 460 XV LIST OF FIGURES Frontispiece .. 2 1.1. Rectangular Cross-section Flow 20 1.2. Vertical Prism For Continuity Interpretation 23 2.1. Numerical Scheme At Ice Divide 47 2.2. The Wedge Terminus 48 2.3. Example Of The Nonlinear Instability 52 2.4. Filter To Suppress The Nonlinear Instability 55 2.5. Continuity Test with Moving Terminus 59 2.6. Continuity Test With Moving Terminus 61 2.7. Nonlinear Test With Burgers' Equation 5 2.8. Meshpoints For Ice Velocity Calculations 68 2.9. Nagata Steady Ice Sheet 75 2.10. Growth Of Nagata Ice Sheet 6 2.11. Velocity Field For Nagata Model 77 2.12. Trajectories In Nagata Model 9 2.13. Surface Mass Conservation Test 81 3.1. Icefield Ranges Location Map 6 3.2. Steele Glacier And Tributaries 89 3.3. Model 1 For Steele Glacier 98 3.4. Sliding Model For Steele Glacier 101 3.5. Steele Model 1 Growth To Steady State 104 3.6. Steady State Streamlines For Model 1 105 3.7. Model 2 For Steele Glacier 106 3.8. Reference Surface For 6-x Function 118 3.9. Sliding Velocity: Steele Glacier Model 121 3.10. Pre- And Post-surge Profiles: 47 Year Period 122 xvi 3.11. Pre- And Post-surge Profiles: 97 Year Period 123 3.12. Pre- And Post-surge Profiles: 147 Year Period 124 3.13. Steele Glacier Thickness: One Surge Cycle 125 3.14. Ice Trajectories For 97 Year Surge Period 126 3.15. Longitudinal 6 Sections: Model 61 130 3.16. Model 61: Surface 6 Profiles 134 3.17. Model 62: Longitudinal 6 Sections 136 3.18. Model 62: Surface 6 Profiles 137 4.1. Austerdalsbreen Velocity And Mass Balance 151 4.2. The Characteristics In T-t Space 154 4.3. Double Step Icefall Model 162 4.4. Single Velocity Step Model 4 4.5. Three Factors Generating Waves 166 4.6. Ogives From A Velocity Gradient 170 4.7. Flow Past A Velocity Gradient: Numerical Solution .... 173 4.8. Double Step Icefall Model 174 4.9. Odinsbreen: Generalized Velocity Per Unit Width 176 4.10. Austerdalsbreen Wave Ogives 179 4.11. Austerdalsbreen Ice Thickness 180 4.12. Variations On Odinsbreen Icefall 183 A1.1. Mesh Increment On Bed 236 A1.2. Model Terminus 248 A1.3. Aliasing And The Nonlinear Instability 260 A1.4. Transfer Functions Of Smoothing Schemes 261 A1.5. Transfer Functions: Slope-dependent Damping 266 A1.6. Filter To Suppress Nonlinear Instability 267 A1.7. Transfer Function Modulus For Various 6 270 A1.8. Transfer Function Phase Comparison 272 xvi i A2.1. Mesh For Ice Displacement Calculations 283 A2.2. The Rectangular Flow Model 284 A2.3. Four Point Interpolation Scheme 290 A2.4. Interpolation Surface f(P) 292 A2.5. Cell Vertex Notation For Downward Velocity 294 A2.6. Displacement Field In A Steady State 295 A2.7. Cell Vertex Notation For Negative Time 297 A3.1 . The Z Plane 299 A3.2. Signals With Wavelengths 1.5Ax And 3.0AX 300 A4.1. Force Balance On An Ice Element 303 A5.1. Surfaces for Derivation of Continuity Equation 307 A5.2. The Thin Cross-section Limit 309 A5.3. Coordinates And Variables In Rectangular Channel .... 313 A10.1. Quantities In Slope Calculation 348 A13.1. Damping Using The Ice Surface Slope 359 A14.1 . Ice Surface Elevation 364 A15.1. Nagata Steady Ice Sheet 373 A16.1. Vatnajokull: Instability And Growth Rate 378 A16.2. Glacier Mass As A Function Of Time 383 A16.3. Rate Of Growth As A Function Of Length L 384 A16.4. Bruarjdkull Ice Profiles (Model 2) 387 A16.5. Bruarjdkull Flux Test 389 A16.6. Fedchenko Glacier Flux Test 393 A16.7. Fedchenko Steady State Ice Profiles 396 A16.8. Fedchenko Nonsliding Model 397 A16.9. Fedchenko nonsliding model with n=3 399 A16.10. Growth Of Fedchenko Glacier 400 A16.11. Fedchenko Growth: Other Nonsliding Models 402 xvi i i A16.12 . Fedchenko Growth: Moderate Sliding 404 A16.13. Fedchenko Glacier: 0-V Plane 406 A16.14. Fedchenko Growth: Sliding Model (2) 407 A16.15. Fedchenko Growth: Sliding Model (3) 409 A16.16. Fedchenko Growth: Sliding Model (4) 410 A18.1. Advancing Surge Bulge 454 xix ACKNOWLEDGEMENTS In the early stages of this work, encouragement and support from my advisor G. K. C. Clarke helped me keep my research going, especially during the periods when the computer programs were uncooperative. His enthusiasm is contagious, and it is delightful to work with him. My appreciation of him grows along with my knowledge of glaciology. Discussions with W. S. B. Paterson, B. B. Narod, W. H. Mathews, D. W. Oldenburg, C. F. Raymond, R. D. Russell, M. C. Quick, R. A. Bindschadler, and W. D. Hibler III have shed light on aspects of glacier dynamics and numerical methods. During the late stages of this project, P. K. Fullagar, completing his Ph.D. thesis in the adjacent office, has been a frequent companion on "the night shift" exchanging ideas and mutual encouragement. B. B. Narod put in long hours proofreading the manuscript and offering helpful suggestions, and he, together with a group of colleagues and friends, helped me through that last hectic night when the manuscript was produced. Julia Forbes, through her encouragement, interest, and support, has helped me to formulate and to work toward my personal goals. I have made good friends at U.B.C. In particular, Barry Narod, J.G. Napoleoni, Bo Chandra, and Peter Fullagar have shared many happy non-academic experiences with me. I was supported in part by a National Research Council of Canada postgraduate scholarship, and by an H. R. MacMillan Family Fellowship from the University of British Columbia. The computations have been carried out at the Computing Centre at U.B.C. Austin Post, of the U. S. Geological Survey, provided the frontispiece photograph of the Trimble Glacier. FRONTISPIECE: Wave ogives on the North Branch, Trimble Glacier, Alaska Range, 61°40'N, 152°18'W. Photo by Austin Post, U. S. Geological Survey, 1965. 2 3 CHAPTER j_: BEGINNINGS '"Its the job that's never started as takes longest to finish" as my old gaffer used to say.'1 1.1 INTRODUCTION 1.1.1 ENDS AND MEANS Although glaciers and ice sheets often appear to be far-removed from most day-to-day matters, there are a number of compelling reasons to study their behaviour. Advances of some glaciers would threaten roads, dams and mines. Berendon Glacier, British Columbia was studied by Untersteiner and Nye (1968) and by Fisher and Jones (1971) for this reason. Ice avalanches from glaciers have caused a long history of destruction. For example, a series of four ice avalanches from the Randa Glacier, Switzerland (Agassiz, 1840, p. 158) between 1636 and 1819 destroyed many buildings and fields and killed dozens of citizens. Ice avalanches into moraine-dammed lakes in Peru caused damaging floods (Lliboutry and others, 1977). Advancing glaciers can dam streams or rivers; the resulting lakes often drain catastrophically (Clarke and Mathews 1981; Clarke, in press) when the ice dam fails. Cunningham (1854 (reprinted 1970), p. 100) reported damaging floods on the Indus River in the nineteennth century, and Forbes (1845, p. 262) described the 1818 disaster when the Getroz glacier dammed the 1 Sam Gamgee, in The Fellowship of the Ring. J. R. R. Tolkien. 4 Dranse in the Val, de Bagnes in Switzerland. Several glacier-dammed lakes threaten a proposed pipeline. route in the Yukon Territory (Canada, unpublished). There is still no universally accepted theory on the cause of ice ages and continental glaciation; an understanding of ice sheet dynamics helps to select and test hypotheses. To correctly interpret the geomorphological record of P.leistocene ice sheets, we must understand the processes of glacial erosion and deposition. This requires a knowledge of glacier mechanics (e.c>. Boulton, 1979; Hallet, 1979). The volume, distribution, and rate of growth and decay of the Pleistocene ice sheets (§_.£. Paterson, 1972) are important data for the determination of the viscosity of the upper mantle, vertical crustal movements, and sea level changes (e.£. Andrews, 1974). The isotopic composition of polar ice sheets has been used to reconstruct temperature changes and climate over the past 10s years (Dansgaard and others, 1969). To correctly date deep cores, it is necessary to determine the flow pattern within the ice sheet (Dansgaard and Johnsen, I969[a]; Philberth and Federer, 1971; Hammer and others, 1978). A current question of some concern is the possibility of global atmospheric warming due to combustion of fossil fuels and clearing of temperate forests (e«g_« SMIC, 1971). A multidisciplinary study (NOAA, unpublished) is underway in Boulder, Colorado to investigate the effect increased atmospheric C02 would have on the Antarctic ice sheets. Disintegration of the East Antarctic Ice Sheet could raise sea level by 75 m and substantially reduce the albedo of the earth 5 (Wilson, 1969). Of more immediate concern is the possibility of a surge and disintegration of the West Antarctic ice sheet; this could raise sea level by seven metres in less than 100 years (Thomas and others, 1979). A group at NASA (NASA, unpublished) is using satellite radiometry, altimetry and radar imaging to monitor and to help model variations of the Greenland ice sheet. Nye (1951, I952[a], 1953, 1957) made the first quantitative studies of the steady flow of glaciers and ice sheets using analytical models, and Weertman (1958), Lliboutry (I958[b]), and Nye (1960, 1961, I963[a], I963[b], 1963[C], I965[a], I965[b]) developed the theory of glacier variations, kinematic waves, and response to climate, by using perturbation methods. Many interesting glaciological problems have large temporal variations or complicated boundary conditions; the analytical solutions cannot be used. Answers to some of these more complicated problems can be found by numerical methods using finite differences on digital computers. However, numerical solutions have their own special pitfalls. A numerical solution of a differential equation may differ from the correct solution for many reasons (e.<j. Richtmyer and Morton, 1967; Gary, 1975). It is extremely difficult to prove that a numerical model has correctly solved a particular differential equation with complicated boundaries if no analytical check is available; yet this is precisely the type of problem for which numerical models are necessary. It is essential to first check numerical models against analytical solutions for a variety of simpler problems. The glaciological literature contains very little discussion of 6 model verification in spite of its obvious importance. There is some indication that much of the published literature on numerical modelling may be quantitatively incorrect. The major thrust of my work has been aimed at understanding the problems of numerical models, finding ways to avoid the problems, and devising tests to verify the accuracy of the model results. With this in mind, I have developed a new computer model of glacier flow (Appendix 1, and Chapter 2, Section 2.2). Like several previous models (Budd and Jenssen, 1975; Bindschadler, unpublished), this model uses finite differences to solve the mass conservation equation together with a flow law for ice, to give the time-dependent glacier surface for a temperate ice mass in a channel of arbitrary width and bed topography, with an arbitrary mass balance, assuming the flow is driven by gravitational stresses. I have analyzed the numerical stability and accuracy of this model as thoroughly as is possible for nonlinear equations (Appendix 1). In Chapter 2 I present a set of tests comparing the numerical solutions to analytical solutions to check terminus mobility and both local and global mass conservation, including a case with a nonlinear flow law. If, in a computer model, the glacier terminus moves incorrectly, it can seriously affect the ice thickness and the velocity throughout the glacier (Section A1.3.4). The physics of the deformation of a glacier snout is complicated (Nye, 1967) for any realistic ice rheology. In the standard numerical approximation (e.S. Budd and Jenssen, 1975; Bindschadler, unpublished, p. 105), the terminus is simply a wedge-shaped 7 volume with slope and apex chosen so as to conserve mass (see Section A1.3.4). The error in using this kinematic approximation cannot be determined? the correct general solution for the motion of a glacier terminus on an arbitrary slope with Glen's flow in tensor form is still an unsolved problem. However, I have tested the numerical implementation of the wedge terminus by comparing the computed solution to a time-dependent analytical solution with a simpler "rheology" (Section 2.3.2). Many standard numerical schemes for linear equations break down when applied to nonlinear equations. It is important to test a numerical model with a nonlinear problem. Burgers' equation (Section 2.3.3) is a nonlinear hyperbolic equation with an analytical solution; it is also related to the mass conservation equation. I have compared numerical results with the analytical solution to Burgers' equation to show that my model correctly solves nonlinear problems. For some glacier flow problems, such as dating ice cores (Dansgaard and Johnsen, 1969[a]) and finding the temperature distribution of cold ice masses (e.£. Jenssen, 1977), it is necessary to know the trajectories of ice particles. My computer model can calculate the velocity field on a vertical longitudinal mesh for a time-dependent glacier, by using Glen's flow law to find the horizontal velocity, and using the continuity equation to then derive the vertical velocity. Particle trajectories are found by a numerical integration of the time-dependent velocity. I tested this part of the computer model against an analytical solution by Nagata (1977) for particle paths in a steady ice sheet. 8 This numerical model is probably the most thoroughly and accurately tested of its type. The set of tests which I have assembled, or others similar to them, should be used to verify any numerical model of glacier flow. Only then can the models be used with confidence to solve more complicated problems. I have used this new computer model in two studies. Previous efforts have concentrated on using variations in isotopic ratios in ice cores to investigate climate change, assuming steady state flow. In Chapter 3 I have evaluated the feasibility of using stable isotope measurements to study the surge history of valley glaciers, assuming a constant climate and unsteady flow. The example I considered was the Steele Glacier, Yukon Territory. I found that surging leaves a characteristic pattern in the isotope distribution, but preliminary measurements of 6(018/016) suggest that this pattern may be masked by other effects. Finally, in Chapter 4 I have derived a linear convolution relating the amplitude of wave ogives to the velocity, channel width and mass balance in icefalls. This work is an extension of studies by Nye (I958[b]). I used the computer model to verify the convolution formulation and to determine which features of the icefall on Austerdalsbreen, Norway are most critical to the formulation of its large wave ogives. 9 1.1.2 CONVENTIONS USED There are many diverse views on the most appropriate style for a Ph.D. thesis. My aim has been to produce a document which fully describes my work, and which can be understood on its own by those with a basic knowledge of physics or physical glaciology. I have documented all my numerical methods in detail, and summarized the relevant work of others; references substantiate the text rather than substitute for it. This results in a lengthy manuscript. To keep the main text as short as possible, I have placed the numerical methods and the background material in appendices. The work of others should be clearly identifiable. I hope that this level of detail will be appreciated by some readers, since brevity will be required in the version of this work in preparation for publication. Unless stated otherwise, I have used a righthanded locally orthonormal coordinate system such that the x axis lies along the glacier bed down the centreline of the channel. The y axis is transverse and horizontal, and the z axis is normal to the bed and positive upward in the vertical plane containing the centreline. The velocity components are (u,w,v) along the (x,y,z) axes. This notation differs from the standard convention (i..e. (u,v,w) ) due to historical developments in the thesis. Underscores are used to indicate tensors. The number of underscores indicates the rank of the tensor, e.£. v is the velocity vector, and A is a coefficient matrix. The dot symbol when located above a variable, indicates 10 a time derivative. When located between two vectors, it represents the standard scalar inner product or dot product, e.cj. Malvern ( 1969, p. 17). A bar above a variable indicates an average value, usually over a depth range (z direction), or over time (§_.£. annual averages). A list of symbols, together with their meanings and the section in which each is introduced, can be found following Chapter 4. Equation numbers, and textual references to equation numbers, are enclosed in round parentheses, e.£. (2.2.5), or (A5.6), or (A1.1.3). The characters preceding the first decimal point are the chapter or appendix number. The middle number (if present) identifies the chapter subsection where the equation is given, and the final number is the consecutive equation number in that subsection. References to chapter sections are always identified as such, and the numbers are not enclosed in parentheses. The LITERATURE CITED is in the style of The Journal of  Glaciology. I was not able to obtain the use of some of the very early literature, and some of the literature in languages other than English. In those cases, where I could not verify the citations of others, I have included the citing author in square brackets. 11 1.2 PREVIOUS WORK 1.2.1 FOUR CENTURIES OF GLACIER FLOW THEORY The framework within which we currently understand and investigate glacier flow has been assembled in the past three decades. Deep coring techniques (Hansen and Langway, 1966) and radio-echo sounders (Evans, 1963), coming shortly after important experiments on the deformation of ice (Glen, 1952, 1955) and mathematical treatment of the flow of ice sheets and glaciers (Nye, 1951, I952[a], 1957) started the rapid growth of glaciological research. However, investigation of the flow of glaciers goes back hundreds of years. In Appendix 17, I review the development of ideas on glacier flow in that early period. Some early works were based on fertile imagination and limited observations; others were concise and lucid statements on topics which are subjects of research today. Some misconceptions about glacier flow were raised, debated, and resolved more than once during the period. The main thrust of research during the second half of the nineteenth century was directed by physicists; in the early part of this century, geologists dominated the field (with a few notable exceptions), and research priorities and theories reflected this difference. Contemporary reviews of glacier flow theory were given by Croll (1875, Chapter XXX, p. 495), by Geikie (1894, Chapter 3, p. 25), by Russell (1897, Chapter 9, p. 160), by Hawkes (1930), by Matthes (1942), by Perutz (1947), and by Orowan (1949). Since 1950, research on glacier flow has progressed 12 rapidly. The flow law for ice was established for many practical purposes by Glen (1952, 1955) and Nye (1953). Weertman (1957), Lliboutry d 968[a], I968[b]), Kamb (1970), Nye d969[b], 1970) and Morland d 976[a], 1976[b3) contributed to the theory of glacier sliding; some aspects of this question are still unresolved. Papers by Nye (1951, I952[a], I952[b], 1952[C], 1957, 1959[C]) established realistic analytical solutions for steady glacier and ice sheet profiles, velocities, and stress fields, while identifying many useful approximations. Vialov (1957) used Glen's flow law to derive a steady ice sheet profile which matched the flow line through Mirny, Antarctica. Weertman (1961[b]) examined the effects of longitudinal strain rates on steady ice sheet profiles, and included isostatic depression of the bed. Weertman (1963) considered the effects of fringing mountain ranges on steady ice sheets, and (1966) the effect of a basal water layer. Although temporal variations of ice masses are difficult to study fully, some useful analytical results have been derived. Kinematic waves on glaciers were observed by Vallot (1900) and were studied by Lliboutry d958[b]), Nye (1958[a])f and by Weertman (1958) using perturbation methods. Nye, in a series of papers, (1960, 1961, I963[a], I963[b], 1963[C], I965[a], I965[b]) extended the method to analyze the response of glaciers to climatic change, and to estimate past climate from the record of advance and retreat of glaciers. Bodvarsson (1955) derived equations for a thin ice sheet and analyzed its stability to climatic change. This model is not widely used due to its assumed relation between basal stress and ice flux. Weertman 13 (1961[a]) performed a similar stability analysis assuming Weertman (1957) sliding. He also derived (I964[a]) the time scales for the growth or decay of a perfectly plastic ice sheet. Jenssen and Radok (1963) obtained a numerical solution for the temperature field in the central region of an ice sheet undergoing thinning. The study of fully time dependent ice masses with arbitrary boundaries and source terms is a recent development made possible by high speed computers. Numerical solutions of the equations governing ice masses have been obtained for a range of problems by, e.g_. Shumskiy (1963), Budd and Jenssen (1975), Mahaffy (1976), Jenssen (1977), Bindschadler (unpublished) and Clarke (1976). The complete solution of the equations of motion, the constitutive equations and the equations of state for a time varying ice mass with arbitrary sources and boundaries is an outstanding problem. 1.2.2 PREVIOUS ICE PROFILE MODELS Computer models which find the surface height of time-varying glaciers and ice sheets are a relatively new research tool. Campbell and Rasmussen (1969, 1970) and Rasmussen and Campbell (1973) developed a model which found ice depth at points on a horizontal mesh. They assumed that the ice was a viscous material- with a basal friction coefficient determined by mass flux. By arbitrarily lowering the basal friction coefficient they simulated glacier surges. 14 Budd and others (1971) and Budd and Jenssen (1975) developed a finite difference model to solve the continuity equation for the glacier thickness profile along a flowline. While the equations are similar to those I have used, our numerical methods differ in some respects (see Appendix 16). These authors have included additional important physical properties of ice flow (e.g_. the effect of longitudinal stress deviators) and I have devoted more effort toward obtaining an accurate and stable numerical scheme; the additional physical properties will be included later. This model was later developed by Budd (1975) and Budd and Mclnnes (1974, 1978, 1979) to generate periodic surges. Working from the assumption that basal meltwater can cause sliding, they used the strain energy dissipation to redistribute the basal shear stress, causing large longitudinal strain rates and rapid flow. The sliding behaviour of the model, viewed as a qualitative phenomenon, may be its most important contribution to our ideas on surges. Bindschadler (unpublished) developed another finite difference profile model similar to the one I describe in this thesis. Bindschadler also did a careful analysis of numerical stability and used a numerical scheme similar to the one I discuss in Appendix 1. He used this model to investigate the changes in the surge-type Variegated Glacier, Alaska, during its quiescent phase. Mahaffy (Mahaffy, unpublished; Mahaffy 1976; Mahaffy and Andrews 1976; Andrews and Mahaffy, 1976) used a two-dimensional finite difference model to study the ice thickness and lateral extent of the Laurentide ice sheet and the Barnes ice cap. 15 Jenssen (1977) published the only fully three dimensional model of ice sheets. This model calculated flowlines and temperature as well as ice surface height. Its accuracy was limited by severe computer size limits (Jenssen used a mesh of 12 x 12 x 10 points to represent the whole Greenland Ice Sheet). However, this model is an ambitious development, and probably will be followed by other models of this kind. 1.2.3 PREVIOUS ICE TRAJECTORY MODELS Quantitative attempts to calculate streamlines date back to the late nineteenth, century. Nansen (cited by Shumskiy, 1978, p. 133) calculated flow lines near the ice divide in central Greenland by assuming (1) steady state, (2) constant ice thickness, density and mass balance, and (3) horizontal velocity independent of depth. Haefeli (I963[b]) independently derived the same solution. Reid (1894) and Finsterwalder (1897) independently developed a method of calculating streamlines in steady glaciers using the concept of flow tubes and properties of smooth vector fields. All these methods were only qualitative; none made any use of the constitutive properties of ice. Haefeli (1961) derived the velocity field in the central portion of a steady isothermal ice sheet, assuming (1) no sliding, and (2) deformation by shear parallel to the bed, using Glen's flow law for ice (Glen, 1955). Nagata (1977) developed an analytic steady state ice sheet model assuming no horizontal shear deformation; the flow was all 16 basal sliding following a Weertman-type relation (see A8.3.1). By assuming that the vertical velocity was constant along the ice sheet surface, Nagata also derived the streamlines, and used the results to model the concentration of meteorites by glacier flow in Antarctica (Nagata, 1978). I use this model as a test for my numerical streamline calculations in Section 2.5.2. Nielson and Stockton ( 1956) derived^ the flow field in valley glaciers of constant valley cross-section assuming steady plastic flow, and Shumskiy (1967) found a solution for stress and velocity in a steady glacier with nonlinear viscosity. Several trajectory models have been derived for regions near ice divides on steady ice sheets in order to date ice cores (Dansgaard and Johnsen, I969[a]; Philberth and Federer, 1971; Hammer and others, 1978), and to model temperature with depth (Weertman, 1968). All these models make some assumptions about vertical strain rates or temperature gradients. Budd and others (1971) calculated trajectories along steady state Antarctic flow lines assuming that the vertical strain rate was constant in any vertical column, (p. 51) or weighted by the horizontal velocity variation with depth (p. 55). This process does not appear to satisfy continuity locally. The ice surface elevation was calculated by the numerical model described in Section 1.2.2. Jenssen (1977) calculated trajectories in a finite difference three dimensional time-dependent ice sheet model, in order to solve for the temperature field. Jenssen (1978) also found the trajectories of ice particles for a surging model of a flowline through Mirny, Antarctica by 1 7 Budd and Mclnnes (1978). He did not describe the method used to obtain the flowlines. This is the only previous model, of which I am aware, to calculate the trajectories in a time-varying ice mass. 1.3 THE CONTINUITY EQUATION FOR AN ICE MASS 1.3.1 THE GENERAL GLACIER FLOW PROBLEM The equation of continuity expresses the manner in which the ice mass changes its shape over time, in response to mass input (accumulation or ablation), subject to the physics of deformation and sliding of ice (Sections 1.4 and 1.5), and with the assumptions about the flow field imposed below. I have derived the continuity equation for an ice mass from first principles in Appendix 5, using standard methods of continuum mechanics (e.c[. Truesdell and Toupin, 1960; Malvern, 1969; Prager, 1973). In this Section, I will give the resulting equation, the assumptions involved in its derivation, and the physical interpretation of its terms. The coordinate system I have used in this study is described in Section 1.1.2. The position vector x is the triplet [x,y,z], and the velocity vector v(x) is the triplet [u(x),w(x),v(x)]. Three basic assumptions of the glacier model are: 1. matter is conserved. 2. momentum is conserved, i.e. acceleration negligible. 3. ice is incompressible (see Appendix 4). 18 To solve the general set of conservation equations, constitutive equations, and equations of state for the temperature distribution, internal energy content, and all the components of the stress tensor and the velocity field, with boundary conditions on a possibly arbitrary boundary, is a problem to instill a sense of humility in even the most enthusiastic and optimistic glaciologist or- numerical analyst. All attempts, of which I am aware, to find solutions to ice flow problems start by making some additional assumptions about the channel geometry (boundaries and symmetry), or about the temperature distribution (e.g_. isothermal), and/or about the flow field itself (e.£. plane strain, simple shear, uniform strain rates, etc.). The model I describe in this study is no exception. 1.3.2 RECTANGULAR CROSS-SECTION FLOW MODEL The glacier flow volume being modelled (see Figure 1.1) is assumed to have a rectangular cross-section, and a width W(x). The two dimensional model includes variations in the third dimension in an approximate way by the assumptions that the velocity components u and v are independent of y, and that the lateral component w varies linearly with y (lateral strain rate independent of y), such that the net velocity y_(x) at the margins is parallel to the margins, i..e. 19 dv = 0 dy (1.3.2) dv = u(x,z,t) dW dy W(x) dx (1.3.3) The glacier thickness h(x,t) is also independent of y. dh(x,t) = 0 dy (1.3.4) Nye (1959[C], equation (33)) suggested this method of including lateral variations, and Budd and Jenssen (1975, equation 3.34) incorporated it into their model. Figure 1.1 illustrates the form of this model. For an ice sheet, W(x) can be the distance between two possibly nonparallel flowlines; the 'walls' of the channel are a mathematical fiction, and the assumptions (1.3.1) through (1.3.4) are reasonable. For a valley glacier, drag from the valley walls is important, and the ice thickness and glacier bed vary with y. If I attempt to identify W(x) with the valley width, the assumptions (1.3.1) to (1.3.4) may be grossly violated. If, however, I let W(x) be the distance between two flowlines near the glacier centreline, §_.£. W(x) may be a few percent of the valley width at the level of the ice surface, then all the assumptions are reasonable, and I obtain a central flowline solution, but with the lateral variation in valley width included to a good approximation. The effect of the valley sidewall drag can be included in an approximate way by using shape factors (Nye, 1965[C]) to modify the shear stress (see equation (1.4.25) below). 20 FIGURE 1.1. Rectangular Cross-section Flow. The triad x-y-z shows the coordinate axes, and the bold arrows u, v, and w are the vector components of the velocity v. The example shows the velocity in the accumulation zone (v is negative). 1.3.3 THE CONTINUITY EQUATION With these assumptions, the well-known continuity equation (the derivation of which I show in Appendix 5) is dh(x,t) + 1 g_Q(x,t) = A(x,t) at wTx) dx. (1.3.5) where h(x,t) is the ice thickness normal to the bed, Q(x,t) is the ice flux through a cross-section from bed to surface, and the source term A(x,t) is the net mass balance normal to the bed, the net accumulation or ablation rate in ice equivalent thickness units per unit time, including snowfall, surface melting, and basal melting and refreezing. The last two 21 'contributions are usually negligibly small (e.g.. Rothlisberger, 1972). The ice flux Q(x,t) is defined by /%h(x,t) Q(x,t) = W(x) I u(x,z,t) dz (1.3.6) J 0 where u(x,z,t) is the velocity component parallel to the bed. It will be derived in the next section. Q(x,t) can also be written Q(x,t) = V(x,t) h(x,t) W(x) (1.3.7) where V(x,t) is u(x,t), i..e. the downslope velocity u(x,z,t) averaged between the bed and the surface. Jh(x,t) u(x,z,t) dz (1.3.8) ... ... 0 Assuming that the upstream end of the glacier section under consideration is at x=0, the boundary condition is Q(0,t) = Q (t) 0 (1.3.9) If x=0 actually represents the physical upper extent of the ice mass, Q0(t) is identically zero. For the case of an ice divide, this is achieved by setting V(0,t) =0 (1.3.10) by a vanishing ice surface slope angle (see (1.4.38)), and letting ice thickness h(0,t) vary. For the case of a valley glacier originating on a slope, the boundary condition (1.3.9) is achieved by setting h(0,t) =0 (1.3.11) If the lower end x=L(t) is the glacier terminus, then L(t) is defined implicitly by 22 h(L(t),t) =0 (1.3.12) (This is not a mathematical boundary condition, but a physical condition defining the limits of the ice mass.) An initial condition of the form h(x,0) = H (x) 0 (1.3.13) is also required. For example, one simple initial condition is Ho(x)=0, i_.e. unglacierized ground. 1.3.4 PHYSICAL INTERPRETATION Equation (1.3.5) may be interpreted in the following manner. Consider a vertical prism of ice as shown in Figure 1.2, extending from the bed to the surface h(x,t), with width W(x) in the y direction, and thickness 6x in the x direction. Let p be the constant density of glacier ice. When (1.3.5) is multiplied by the constant />W(x)6x6t, the first term is the net change in mass in the prism in a time 6t (the prism is then taller or shorter). The second term on the left is the difference in mass between that which flowed out of the prism through the downslope face, and that which flowed into the prism through the upstream face, in the time 6t. This is the net loss of mass from the prism into the downstream flow. The term on the right side is just the total mass added to the prism in time 6t by snowfall or melting. Thus, (1.3.5) states that the total layer of mass added to the top of the prism at any position x is the sum of the snowfall onto the prism and the net mass left inside the prism by spatial flow variations. 23 1.4 PHYSICS OF ICE DEFORMATION 1.4.1 INTRODUCTION In this Section, I will outline the derivation of a second equation relating glacier thickness and ice flux so that the continuity equation (1.3.5) can be solved. There are three steps in this derivation. This is a standard procedure in modelling glacier flow (e.g.. Paterson, 1980; Raymond, 1980). Newton's second law establishes relationships between the surface and body forces and the accelerations in any continuum. The stress equilibrium equations are outlined in Section 1.4.2. Second, observations and theory of the deformation of glacier ice establish constitutive relationships between the stresses applied to ice, and the resulting deformation. Glen's flow law 24 for ice (Glen, 1955, 1958) is presented in Section 1.4.3. Finally, in Section 1.4.4, after making simplifying assumptions, substituting the stress equations of Section 1.4.2 into Glen's flow law to get the strain rate, and integrating over appropriate coordinates, I express the downslope component of the ice velocity in terms of the ice thickness. Since ice flux, thickness, and velocity are related through (1.3.7), this will complete the derivation of a second equation needed to solve the continuity equation (1..3.5) for ice thickness and flux. 1.4.2 STRESS EQUILIBRIUM EQUATIONS Since glacier ice deforms slowly, the acceleration term in Newton's second law can be neglected, leaving the result that, for any volume V of a slowly deforming continuous medium, B + T = 0 (1.4.1) where*B is the total body force found by integrating the specific body forces (force per unit mass) f.(r) at position £ over all r_ throughout the volume V. Its components are B V (1.4.2) and T is the total surface traction on the surface S enclosing the volume V with surface normal vector n. The components of T are T =|| n (r) c (r) dS jk (1.4.3) S tf.k is the stress tensor, i_.e. the force per unit area on a surface normal to the Xj axis acting in the xk direction. The 25 Einstein convention, whereby repeated indices are summed, is used in this section. Orthogonal axes are assumed. Applying Gauss' Theorem (e.g.. Prager, 1973, p. 29) to (1.4.3) and substituting (1.4.2) and (1.4.3) into (1.4.1) by components gives d3r = 0 dx V 1 (1.4.4) Since the volume V is arbitrary, the global equation (1.4.4) has a local counterpart Ic (r) (r) f (r) + ki = 0 dx (1.4.5) k Assuming that the net angular acceleration is zero, and setting to zero the sum of moments acting on the volume V, gives the result c = <t (1.4.6) ij Ji .i.e. the stress tensor is symmetric. The development is very similar to (1.4.1) through (1.4.5), and is given in Prager (1973, p. 47). For a glacier, the only body force is gravity, so f(r) = g (1.4.7) When the x, axis (to be called x) is taken along the glacier bed which is at an angle fix) to the horizontal, the x3 axis (z) is normal to it and upward, and the x2 axis (y) is horizontal, then the stress equations (1.4.5) are 26 off <^ff he xx + xy + xz + pgsin(p)= 0 (1.4.8) dx dy Tz ^ff <5ff dff xz + yz + zz - />gcos(*) = 0 (1.4.9) <^x by £z do 5ff ^ff xy_ + Yl + vz = 0 (1.4.10) dx dy Tz 1.4.3 CONSTITUTIVE RELATION FOR ICE DEFORMATION The constitutive equations for deformation relate the stresses applied to ice to the resulting deformation rate. The components of the strain rate tensor are 1D —i + —j dx dx • j i (1.4.11) where u^ is the ith component of the ice velocity. Glen (1958) showed that the most general relation between the stress tensor and the strain rate tensor for a nonlinear, originally isotropic material had the form € = A(T ,T ,T )6 + B(T ,T ,T )c ij 1 2 3 ij 1 2 3 ij + C(T ,T ,T )<s a 1 2 3 ik kj (1.4.12) where 6-tj is the Kroenecker delta 27 6 =0 i^j i j = 1 i=j (1.4.13) and T1f T2, and T3 are the first, second, and third scalar invariants of the stress tensor (e.g.. Prager, 1973, p. 22) T = c 1 ii (1.4.14) T = J_( <r e ~ c a ) 2 2 ij ji ii jj (1.4.15) T - M2<y a c - 3* c e +tf*tf) 3 6 ij jk ki ij ji kk ii jj kk (1.4.16) Terms in higher powers of • can be eliminated by the Hamilton-Cayley equation (§_.g_. Prager, 1973, p. 25). The stress-dependent coefficients must be functions of, at most, the scalar invariants, because the relation (1.4.12) is independent of the choice of axes. The coefficients A, B, and C may be dependent on temperature. Rigsby (1958) showed that, to a good approximation, the deformation rate of ice crystals was independent of the hydrostatic pressure p, where P = 1 « 3 ii (1.4.17) when the ice temperature was measured relative to the pressure-melting point. This result means that the constitutive relation can be written more simply in terms of the deviatoric stress *|j 28 cr' = e - _1_ 6 cr ij ij 3 ij kk (1.4.18) By definition, the first scalar invariant T', of cr'j is zero. By further assumptions, first that the density of glacier ice is constant (see Appendix 4), second, that, for a given stress, the components of strain rate are proportional to the components of the stress deviator tenspr, and third, that the second invariant of the strain rate tensor is a function of T2 only (see Glen, 1958) the flow law presented by Nye (1953) reduced the general relation (1.4.12) to € = B(T' ) <r' ij 2 ij (1.4.19) When € and T are the square roots of the second scalar invariants of and (T = JT2 ) , Nye (1953) showed that a plausible relationship was n t = A T (1.4.20) Using the case of a simple slab deforming by shear parallel to the x axis, combining (1.4.20) with (1.4.19) implies that n-1 B(T') = A T (1.4.21) 2 and n-1 e =AT *' (1.4.22) i j i j The exponent n in (1.4.20) is independent of temperature. Values in the literature vary from 1.5 (Gerrard and others, 1952) to 4.2 (Glen, 1955), and the value usually used for glacier modelling is n=3 (e.cj. Paterson, 1980). The factor A 29 follows the exponential Arrhenius temperature dependence A = A exp(-Q/RT) 0 (1.4.23) where A0 is a constant, R is the gas constant (8.314 J K^mol-1), Q is the activation energy for creep, and T is the temperature (°K). Values of Q for secondary creep of polycrystalline ice are 60 kj mol"1 for T < -10°C, and approximately 139 kJ mol"1 for T > -10°C (Paterson, 1981, p. 34). The presence of small amounts of water causes grain boundary sliding (Barnes and others, 1971? Jones and Brunet, 1978) above -10°C. The deformation below -8°C is dominated by basal glide (Barnes and others, 1971). In this study, the ice is assumed to be isothermal at 0°C, and the flow law parameters used are (Paterson, 1981, Table 3.3, P- 39) n=3 A = 5.3 10"15 s"1 kPa"3 (1.4.24) These values apply only for secondary creep, after the initial transient response to loading has died away. Other constitutive relations have been proposed for glacier ice, such as a hyperbolic sine relation (Barnes and others, 1971), or a polynomial with odd order stress terms (Meier, 1958, 1960; Lliboutry, I969[a]; Colbeck and Evans, 1973). However, laboratory experiments (e.g.. Glen, 1952, 1955; Steinemann, 1958) and field measurements of closure of boreholes and tunnels and deformation of boreholes (e.cj. Gerrard and others, 1952; Nye, 1953; Mathews, 1959; Meier, 1960; Paterson and Savage, I963[a], 1963[b]; Haefeli, 1963[a]; Shreve and Sharp, 1970; Raymond, 1971; Paterson, 1977) and the flow of ice shelves (Thomas, 1973) 30 indicate that (1.4.22), known as Glen's flow law, is a useful and satisfactory constitutive relation for glacier ice. 1.4.4 SHEAR STRESS AND ICE FLUX In this section, I will outline the derivation of the shear stress parallel to the bed, and how it can be integrated to give the downslope velocity component and the ice flux. The errors and assumptions are explicitly shown. The details are given in Appendix 7. The stress equilibrium equation (1.4.8) can be integrated from the surface to a height z above the glacier bed to give the shear stress c (x,z) = s/>g(h-z)sin« xz 1 + 0 9*' 3*' 2h xx + h yy dx dx pqha (1.4.25) which is derived from (A7.3.21) in Appendix 7. The leading factor is the standard formula for shear stress in a parallel-sided slab deforming by simple shear parallel to the bed (e.c[. Paterson, 1969, p. 91) when the shape factor s is unity (see Appendix 7, Section A7.4). The surface slope a is an effective slope averaged over a distance of at least the order of 4h (Budd,l968; I970[a]). The average stress deviators in the second term are defined by 31 h(x) be' = _1_ r d<r' z<h xx (h-z)l xx dz' dx J dx (1.4.26) z = 0 z=h with a corresponding definition for the yy component. If I' consider the x- and y-directed forces on an ice column from the ice surface to the bed, the correction terms in (1.4.25) are, very roughly speaking, ratios of the normal forces to the basal shear force. These ratios are usually very small for glaciers and ice sheets. Lliboutry (I958[b]), Shumskiy (1961), and Robin (1967) used a correction term similar to this to account for longitudinal strain, and Collins (1968) published a mathematical justification of it. Nye (I969[a]) simplified the analytical formulation by an appropriate choice of axes. My formulation differs in some details, partly because I use the axes of the numerical model (see Section 1.3). Budd (1968; 1970[a]; I970[b]; 1971) gave a detailed discussion of stress variations in glaciers, including correction terms and the wavelength ranges for which they may be important. Hutter (in press) gives the most recent and rigorous treatment of stress in glaciers. The shape factor is an approximate correction between zero and unity for the reduction in the shear stress «rxz along the channel centreline when some of the weight of the glacier is supported by the valley sidewalls. It was first used quantitatively in work on rectilinear flow in rectangular, parabolic, and elliptical channels by Nye (I965[c]). I describe shape factors in more detail in Appendix 7, Section A7.4. 32 In arriving at (1.4.24) in Appendix 7, I assumed that the slope angles o of the ice surface, and p of the glacier bed were small, i.e. |o(x)| « 1 ( 1 .4.27) |f(x) | « 1 (1.4.28I also assumed that a was never negligibly small compared to p. This may not be true near an ice divide. Although (1.4.8) contains sin$, the final result (1.4.25) for the shear stress depends only on sine. Nye (l952[b]) first pointed out this result. When the small angle assumptions (1.4.27) and (1.4.28) hold, the longitudinal stress gradient term d*^ /dx in (1.4.8) introduces a term in io-p) which cancels the bed slope dependence, leaving only the surface slope dependence of (1.4.25). With the assumption that the major shear deformation occurs parallel to the bed, .i.e. iv /bu "57/ bz « 1 (1.4.29) the shear strain rate * = 1 xz 2 du + ^v dz dx (1.4.30) is approximately € = 1 lu xz 2 bz (1.4.31) which can be integrated directly, from the bed to height z, to give 33 u(x,z) = u (x) + z e dz' xz 0 (1.4.32) where u_(x) is the basal sliding velocity discussed in Section 1.5. If I further assume that T, the square root of the second invariant of the stress deviator tensor (1.4.19) is approximately equal to the shear stress exz , i_.e. shear stress parallel to the bed is by far the largest stress deviator component, or 6T T - e e = xz xz e xz << 1 (1.4.33) then I can substitute Glen's flow law (1.4.22) for the strain rate «xZ in (1.4.32), using <rxz from (1.4.25) for both T and * , to get the velocity component u(x,z) parallel to the bed. u(x,z) - u (x) = n n+1 n+1 2A[s(x)Pqsin(g(x))] [h - (h-z) ] [1 + e(x)] (n+T5 (1.4.34) where the error e(x), i.e. the terms not included in the computer model, has the form e(x) = 0 he' de' 2h xx + h yy dx dx + (n-1) 6T e - b_v /fru XZ bx/ bz pgho (1.4.35) where the symbol 0[x] means "is of the order of x", i.e. the function goes to zero at the same rate as x. I have assumed that 34 the glacier is isothermal, i.e. temperate, so that the coefficient A of Glen's flow law(1.4.22) can be treated as a constant. If the ice temperature varies with z, the integral can be evaluated numerically. The downslope ice flux for use in the continuity equation (1.3.5) is h(x,t) Q(x,t) =1 u(x,z,t) dz ^0 n n+2 = u (x,t) h(x,t) + 2A[s/pgsinq] [h(x,t)] [1+ e(x)] s (n+2) (1.4.36) With the assumptions discussed above, the error term involving e(x) is small; it is neglected in the computer model in its present form. The average velocity V(x,t) used in Section 1.3 and Appendix 1 is defined as V(x,t) = Q(x,t)/h(x,t) (1.4.37) which is n n+1 V(x,t) = u (x,t) + 2A[spgsino] [h(x,t)] [1+ e(x)] s (n+2) (1.4.38) The term on the right due to the internal deformation is just (n+1)/(n+2) times the downslope velocity component at the ice surface u(x,h(x),t) from (1.4.34). 35 1.5 BASAL SLIDING 1.5.1 INTRODUCTION The one quantity still required to complete the derivation of the downslope velocity u(x,z) and the flux Q(x) is the basal sliding velocity u5(x) which appeared as an integration constant in (1.4.32). Raymond (1980), in a recent review, gave a summary of sliding behaviour, measurements, and the physical processes possibly involved, and pointed out some difficulties of quantitative modelling of sliding. In Appendix 8, I summarize current ideas on the physics of sliding, and review the use of sliding in computer models. In this section, I discuss the importance of sliding, the way I treat sliding in Chapter 3, and the reason for my choice. 1.5.2 BASAL ICE TEMPERATURE AND SLIDING Ice masses which are cold at the base, have temperatures below the pressure melting point, do not appear to slide. The basal ice is effectively frozen to the glacier bed, and u (x) = 0 s (1.5.1) Ice masses which have temperatures at the pressure melting point at the ice-rock interface exhibit sliding velocities which range from zero to values much greater than the velocities due to internal deformation. The model I describe in this study assumes an isothermal ice mass. Due to the existence of the 36 geothermal heat flux, and the resulting geothermal temperature gradient, the only possible essentially isothermal ice mass is one at the pressure melting point throughout its volume (neglecting a possibly cold surface layer caused by diffusive penetration of the winter cold wave), because only then can the geothermal flux be absorbed at the base by being transformed into energy of fusion. This means that sliding velocities can be an important component of motion for my modelling situations. 1.5.3 PHILOSOPHY OF SLIDING IN THIS STUDY Correctly modelling the physical processes of glacier sliding is, at the present, very difficult, due to inadequate observations, and the large number of uncontrolled physical variables possibly involved in sliding processes. In Appendix 8, I have outlined the problems of measurements, the physical complications of sliding processes, the present state of sliding theory and its quantitative application in computer models. In the models presented in this study, I do not attempt to investigate or to simulate the physics of glacier sliding. My aim, in Chapter 3, is to investigate the consequences of surging (defined by a periodic sliding history) on structures within a glacier, given that periodic surging occurs in the defined manner. I do not attempt to induce surges in the model by any particular physical mechanism. This approach to investigating effects of surging was also used by Campbell and Rasmussen (1969) and by Clarke (1976). I could easily incorporate the theories of Weertman (1957), 37 Nye (I969[b]y 1970), Kamb (1970), Morland (1976[al; I976[b]), or Budd (1975) to calculate the sliding velocities, but the results would be numerically suspect, due to the problems discussed in Appendix 8, and would add nothing to my investigation of the consequences of surging. My approach is, instead, to use a predetermined sliding function u5(x,t) (based as closely as possible on the reasonably well inferred sliding history of a surging glacier such as the Steele) as a driving function for periodic surges in the computer model. I calculate the response of the glacier model to this driving function by using continuity and Glen's flow law to find the internal deformation. For my purpose of finding the effects of surging on the internal structure, this approach is no worse than using a numerically inadequate sliding theory, and it has the distinct advantage that I can control the sliding at will while I relate sliding patterns to resulting changes within the ice mass. 38 CHAPTER 2: MODELS AND TESTS "Things are seldom what they seem; Skim milk masquerades as cream."1 2.1 INTRODUCTION 2.1.1 OUTLINE In this chapter, I outline the operation of the computer models and I describe tests used to verify their correct operation. In this introductory section, I explain why I think tests are important. In Section 2.2, I describe the continuity equation glacier profile model, and in Section 2.3, how I have tested it. In section 2.4 I describe the particle trajectory calculations, and in Section 2.5, how they were tested. 2.1.2 IMPORTANCE OF MODEL TESTING Analytical solutions of initial value problems are most desirable, because the correctness of the solution can be verified for all space and time simply by substituting the solution into the differential equation. Unfortunately, analytical solutions to ice flow problems are restricted to a few cases with simple boundary conditions, uncomplicated rheologies, and, often, steady states. A finite difference numerical model uses a set of algebraic 1 H.M.S. Pinafore. Gilbert and Sullivan. 39 equations whose solution closely approximates a digitized version of the true solution of the differential equation, to within a truncation error (see Appendix 1, Section A1.5.3). Numerical solutions can extend the domain of solvable problems to include those with quite general boundary conditions, ice rheology, and temporal variations. The price which is paid for this increased generality, however, is a new inherent uncertainty in the validity of the solution obtained. Substitution of a numerical solution into the finite difference equations, or into the differential equation, can give, at most, the residual errors in the most recent time step. The long term integrated error is unknown. The danger is that a numerical solution will behave in a qualitatively reasonable manner, yet quantitatively may be, over some time scales, grossly wrong. For instance, ice velocities and thicknesses may be in error by tens of percent, and the phase of cyclic phenomena such as surging may become totally unrelated to the phase of the true solution. These possibilities should make us quite cautious about any predictive claims made for numerical models. We could obtain a result no more accurate, at best, than an educated guess, yet be lulled into believing that it was an quantitative prediction of glacier behaviour. There are two possible sources of error in numerical solutions. First, the computer program may not correctly solve the set of algebraic equations. Programming errors, such as incorrect constants or missing minus signs, sometimes go undetected. Spurious numerical "solutions" of this kind have on occasion found their way into the scientific literature. By 40 careful program design and the use of consistency checks, these errors can be eliminated, although not all programmers have taken the time to do so. Assuming then, that the computer program works correctly, there is still another source of error. The numerical scheme used in the computer model may not adequately represent the differential equation at all time scales. -One example of this is the linear computational instability (Appendix 1, Section A1.4.2). The solution of the finite difference equations may include an exponentially growing high wavenumber oscillation completely unrelated to the differential equation. Fortunately, this error is usually easy to recognize! The other cause for concern is the possibility that the numerical solution may drift away from the true solution, yet still look "physically reasonable". This could be caused by any of several factors. For example, inappropriate mesh intervals (Appendix 1, Section A1.5.2) may cause incorrect dispersion and spectral attenuation, distorting the solution to an unacceptable degree. Introducing smoothing schemes in attempts to suppress numerical instabilities can cause similar distortions (and may still fail to totally remove the instabilities). For example, the widely reported ice sheet model of the Melbourne group (Budd and Jenssen, 1975; Budd and Radok, 1971) was used by Mclnnes (unpublished), who appears to have encountered all of the above difficulties. In an attempt to model surging glaciers in the presence of growing numerical instability, Mclnnes (p. 64) reported: 41 " Different time steps give slightly different surge times and periods and therefore at the same growth time, profiles are not directly comparable. Also due to the different number of steps, and the smoothing scheme not being perfect, the lower the time step the more iterations, which leads to more smoothing which tends to lower either the depth or the base velocity, and therefore affects the surging times." and further, discussing a criterion to introduce automatic smoothing at the appearance of instability problems (p. 64): "Using this test before automatic smoothing, lessens the number of times smoothing is used, and therefore decreases the effect smoothing has on the exact profiles." There is obviously little cause for optimism in the belief that this numerical model, for instance, was providing a solution that closely matched the true solution at all times. The results of the Mclnnes study were published by Budd (1975) and by Budd and Mclnnes (1974; 1978; 1979). I mention this example, not to criticize any particular individuals, but to illustrate the lack of attention paid to this serious question by most members of the glaciological modelling community. Even a major glacier modelling program has apparently had serious difficulties with accuracy, consistency, and mass conservation (see Appendix 16), yet model verification has not been given priority discussion in the published literature. Because numerical model results cannot be verified for the complicated problems the numerical models are created to solve, it is imperative that numerical schemes be verified by comparing their numerical solutions with analytical solutions for simpler problems before the numerical models are used for new problems. There is always a temptation with a new model to rush into the solution of complicated glaciological problems. This urge to 42 break new ground in a hurry must be controlled until the numerical model has been demonstrated to work accurately on known ground. Only in this way can there be any confidence in the results. 2.2 CONTINUITY EQUATION PROFILE MODEL 2.2.1 INTRODUCTION I have written a FORTRAN IV computer program which numerically solves the continuity equation (1.3.5), when provided with a subroutine to calculate the velocity V(x,t) averaged through the ice thickness. The flow equation (1.4.38) based on Glen's flow law (1.4.22) is used for glacier simulations (Chapter 3). I describe the computational scheme in detail in Appendix 1. I summarize the computational aspects in this Section, including the numerical scheme (Section 2.2.2), the boundary conditions (Section 2.2.3), numerical stability (Section 2.2.4), and accuracy (Section 2.2.5). . • Inputs to the model are bedrock elevation b(x), mass balance A(x,t), the initial ice thickness H0(x), the ice flux Q0(t) through the upslope boundary, and parameters to specify the sliding and flow properties of the deforming medium. Output from the model is the time-varying ice thickness profile. 43 2.2.2 THE NUMERICAL SCHEME Complete details of the numerical scheme are given in Appendix 1, Sections A1.1 and A1.2. I solve the continuity equation (1.3.5) by a finite difference method (e.g_. Richtmyer and Morton, 1967). The partial differential equation (1.3.5) is approximated by a set of algebraic equations for the ice thickness (h-|j=1,J} at a set of mesh points at equal horizontal intervals of Ax. Starting from an initial condition {h^lj^jj}, the solution is obtained by time marching, using a possibly variable time step At. The finite difference equations are n+1 n n+1 n+1 n n h -h + 6 (Q - Q ) + (1-6) (Q - Q ) _j i " j + 1/2 j-1/2 j + 1/2 j-1/2 At W Ax W Ax j j j j n+1 n = 6A + (1-6)A (2.2.1) j j 1 £ j < J 1 < n < N where superscripts indicate the time step, and subscripts indicate the spatial mesh point. Mass balance A- and ice thickness h- are measured normal to the bed, and the mesh w increments AXj are measured along the bed. The weight factor 6 is a constant between zero and unity, used to stabilize the scheme. I discuss 6 further in Section 2.2.4. The ice flux Q j +1/2 between the meshpoints is related to the ice thickness hj at the meshpoints, the channel width Wj+1/z and the vertically averaged downslope velocity vj • 1/2 between the meshpoints by 44 Q j±l/2 = V W (h J±1/2 J±1/2 j±1 + h ) i (2.2.2) 2 Most of the variables in (2.2.2) are shown in Figure 2.1. The flux is calculated midway between mesh points because of numerical stability considerations. The set of equations (2.2.1) can be written in matrix form as where the components of the vector H are the unknown ice thicknesses at the mesh points at the future time step {h-* 1|j = 1,J}, the right side vector D contains quantities from J the previous time step, and the matrix M contains coefficients involving velocity at the future time step. Since the velocity is related to the ice thickness (Sections 1.3 and 1.4), this makes (2.2.3) nonlinear, and the system must be solved iteratively. For the first iteration, the velocity profile at the previous time step {V? |j=3/2, 5/2,...J+1/2} is used as an estimate of {0VP* 1|j = 3/2, 5/2,...J+1/2} , to calculate the first estimate of the ice thickness {0h?* 1|j=1,J} . Prescripts indicate iteration number. These ice thickness estimates are then used to calculate an estimate { , Vp+ 1 | j = 3/2 , 5/2, .. . J+1/2} of the longitudinal velocity profile at the future step. The residuals (2.2.4) M H = D (2.2.3) 45 n+1 n+1 n+1 n n r = 2p 6[Q - Q ] + 2p (1-6)[Q - Q ] j " j j+1/2 j-1/2 j 3+1/2 j-1/2 n+1 n n+1 n + h - h - 6A At - (1-6)A At j j j j 1^j<J 1 < n < N (2.2.4) where p = At/(2AxW ) (2.2.5) j j then measure the degree to which the current estimates of ice thickness and velocity fail to satisfy the continuity equation (1.3.5). By using a linearized equation (2.2.6) to relate residuals to ice thickness corrections 6h- required to make the residuals go to zero, J n+1 \-~\ br** 1 r = > —j 6h (2.2.6) j *—* dh* +1 k k=1 k or A 6h = r (2.2.7) I obtain essentially a multi-dimensional formulation of the Newton-Raphson method (e.cj. Carnahan and others, 1969, p. 319) to solve (2.2.4). The iterations terminate when the largest residual in absolute value is smaller than a preset test criterion. I discuss the choice of a criterion in Appendix 11. 46 2.2.3 BOUNDARY CONDITIONS The set of equations (2.2.1) with (2.2.2) consists of J equations in J+2 unknowns { h?* 1|j = 0,J+1}. Since equation (1.3.5) is a first order differential equation, it requires one boundary condition selected from (1.3.10) through (1.3.12). This gives one of the required two extra equations. At an ice divide, zero input flux is modelled by including an image point h0 at a distance Ax outside the boundary j=1, with h = h O 2 V = -v 1'2 3/2 (2.2.8) This forces the surface slope to be zero at the boundary, but the ice thickness can vary with time. This situation is illustrated in Figure 2.1. For a valley glacier originating on a bedrock slope, h = 0 1 (2.2.9) and the ice surface slope can adjust to any appropriate value. To model only a portion of an ice mass such that the upstream end of the model is some distance downslope from the bergschrund or divide (e.£. Chapter 4, where an icefall is modelled), the ice flux must be specified at 1/2 mesh increment above the boundary, i..e. n Q = Q (nAt) 1/2 0 (2.2.10) The second extra equation required to balance the number of 47 FIGURE 2.1. Numerical Scheme At Ice Divide. equations and unknowns comes from the treatment of the downslope boundary. If I treat the boundary as fixed in space, and simply allow ice to flow through it and out of the model, I can get the equation by writing hj+1 as an appropriate extrapolation of the ice thickness at the meshpoints. I use the second order Newton's divided differences polynomial (e.g.. Carnahan and others, 1969, p. 12). If I choose to follow the actual motion of the glacier snout, defined by x=L(t), I must keep track of L(t) when it falls between the meshpoints, and be able to add or subtract points to or from the mesh as the terminus moves. I assume that the terminus is wedge shaped from the last mesh point J to the snout at L(t), as shown in Figure 2.2. Then, I apply the principle of conservation of mass to the shaded section of the wedge, obtaining the required final equation by balancing the 48 4.— AX/2—H FIGURE 2.2. The Wedge Terminus. volume change of the wedge against ice flux into the wedge from upslope plus net mass balance on the upper surface, during each time step. The details may be found in Appendix 1, Section A1.3.4. As the terminus moves, the ice thickness h^. at the upstream end, the snout position L(t), and the slope of the ice surface are all free to adjust to changes in the flow. This terminus model is similar in many respects to that used by Bindschadler (unpublished). 2.2.4 NUMERICAL STABILITY If the finite difference numerical scheme and the mesh increments are chosen unwisely, the finite difference equations may admit a solution quite different from the solution of the differential equation. This usually involves a spurious exponential growth of some wavenumber component which quickly 49 dominates the desired bounded, physically reasonable solution. Rigorous stability analysis of nonlinear equations is usually not feasible, but stability criteria for linearized analogues generally give useful guidelines and insights into stability of the nonlinear forms. The first type of stability problem, the linear computational instability, involves the choice of the mesh increments At and Ax. For many systems of finite difference equations, instability can occur when the time step At is too large relative to the space step Ax. If Ax/At is much greater than the material velocity V, material travels many mesh intervals per time step, and the system tends to 'forget' the physical solution. The von Neumann method (e.g^. Richtmyer and Morton, 1967, p. 70), is one standard stability analysis for linear or linearized equations. The method involves finding the transfer function T(m) in the wavenumber domain which multiplies the Fourier transform of the solution at the previous time step n, to give the Fourier transform of the solution at the future t ime step n+1. If T(m) < 1 (2.2.11) at all wavenumbers m, no wavenumber can grow, so no instability can exist. After linearizing (1.3.5) by setting the velocity V to a constant in (1.3.7), I find that by choosing 0^1/2 (2.2.12) I obtain unconditional linear computational stability for any choice of Ax and At. The second type of numerical instability is called 'the nonlinear instability' (e.£. Phillips, 1959; Mesinger and 50 Arakawa, 1976, p. 35). This is a problem which arises in any numerical solution of a differential equation using a discrete mesh, and having terms which are nonlinear in some combination of the dependent and the independent variables. In (1.3.5), dQ/dx has this form. The nonlinearity pumps energy (squared amplitude of the wavenumber spectrum) from the low wavenumber end to the high wavenumber part of the wavenumber spectrum of the dependent variable, and the aliasing (Appendix 3) due to discrete sampling folds this energy back to the lower wavenumbers, where it distorts the solution. Since the nonlinear instability has an importance which is not widely recognized in the glacier modelling community, I discuss it in detail in Appendix 1, Sect ion A1.4.3. If a function is known only at discrete intervals Ax, a wellknown result from sampling theory (see Appendix 3) is the fact that its Fourier spectrum can be found only up to a wavenumber m^, called the Nyquist wavenumber, m = rr N Ax (2.2.13) This is a sampling rate of two samples per cycle. Wavenumbers above this limit are misinterpreted as lower wavenumbers (see Figure A3.2), by being 'folded' back into the spectrum symmetrically about mN (see Figure A1.3). It is easy to show (see (A1.4.7) through (A1.4.9)) that multiplying two band-limited Fourier series together gives a product bandlimited to the sum of the bandwidths of the two signals. This happens with the ice flux Q=hV. If both h and V 51 are always bandlimited to 2/3m^, their product is bandlimited to 4/3m^, which is aliased back onto the interval from 2/3m^ to m^, but the lower 2/3 of the spectrum remains correct (see Figure A1.3). It is evident, then, that to avoid the nonlinear instability, the aliased signal at high wavenumbers, i.e. above 2/3m^, must be heavily attenuated. At the same time, the attenuation must not distort the low wavenumbers which contain information about the glacier. Budd and Jenssen (1975) encountered an instability which they attributed to machine roundoff error. I think it was actually the nonlinear instability. Budd and Jenssen attempted to cope with the instability by smoothing the velocity profile whenever it began to oscillate spatially. Mclnnes (unpublished, p. 58; p. 102) used the same computer programs to simulate surging at Bruarjokull, Iceland. The broken curves in Figure 2.3 (redrawn from Mclnnes, (unpublished), p. 58) show the instability which arose as he attempted to build up the glacier to a steady state with no sliding, and no smoothing of the profiles. In Appendix 1 Section A1.4.4, I discuss the velocity smoothing method used by Budd and Jenssen (1975), and also, the addition of a purely numerical dissipation term to (1.3.5) to preferentially damp high wavenumbers. I conclude that the use of either of these methods is hard to justify. The two methods I use in this numerical model are superior on physical grounds. First, if the flow equation for the continuum (e.g.. (1.4.34)) depends on the local ice surface slope o(x), then large amplitude bumps in the solution profile should tend to 52 1500 o UJ X 1000 500 Vatnajokull model Plots at 50 years 200 years 250 years 10 20 DISTANCE (km) 30 AO FIGURE 2.3. Example Of The Nonlinear Instability. The dashed curves redrawn from Mclnnes (unpublished, p. 58) show the onset of the nonlinear instability on ice surface profiles at 50 year intervals for a flowline on Vatnajokull (Iceland). The Budd-Mclnnes model did not calculate flux at the midpoints of the mesh intervals for h(x), and therefore was vulnerable to the nonlinear instability. The solid curves are the 50 year profiles from my computer model using parameters in Table 2.1. The Budd-Mclnnes model also appears to create mass (see Appendix 16). diffuse out rapidly, due to the physical properties of the medium (See Appendix 6, which relates perturbations of (1.3.5) to a diffusion equation following Nye (1960)). This same process should also efficiently smooth out high wavenumber instabilities. In Appendix 13, I have shown that, when the ice 53 flux is calculated at the midpoints of the mesh intervals, .i.e. at J±1/2, as shown in Figure 2.2, this, physical diffusion process is incorporated into the numerical model and controls the nonlinear instability at all wavenumbers. For example, the broken curves in Figure 2.3 show the onset of the nonlinear instability in the Budd-Mclnnes (1974) computer model (Budd, 1975; Mclnnes, unpublished). The original caption on Figure 4.3 of Mclnnes, from which these profiles are redrawn, was "Profiles from the Vatnajokull model at fifty year intervals, showing the increase in the magnitude of the oscillations with time, due to the two point finite difference approximation. In this case, no smoothing was used." These authors did not calculate the ice flux at the midpoints of their mesh intervals. As a result, the diffusive mechanism of Glen's flow law was unable to prevent serious aliasing at the Nyquist wavelength (in this case, 2 km). The trigger for the instability could have been a large truncation error (Appendix 1, Section A1.5.3) resulting from the use of a forward difference at the boundary (Budd and Jenssen, 1975). The solid n A s g p Ax At -n -1 bar a ms"2 kg m"3 m a 2 .225 1.0 9.8 910. 1000. 1.0 TABLE 2.1. Parameters for Vatnajokull (Figure 2.3) curves are the 50 year profiles using my computer model with the parameters in Table 2.1. As far as I can tell, Mclnnes also used 54 these values. The nonlinear instability is removed at all wavelengths due to the choice of numerical scheme. As well as the high wavenumber oscillation, the Budd-Mclnnes model appears to suffer from a mass conservation error. I discuss this further in Appendix 16. A second method must be used when the ice flux is not a function of the local ice slope. For example, the gravitational stress may be calculated using an intermediate or large scale slope (e.a.. Bindschadler, unpublished, p. 92). In this case, I remove the nonlinear instability without distorting the low wavenumber signal at all, by, at the completion of each time step, taking the Fourier transform of both the velocity profile and the ice thickness profile, and multiplying by the lowpass filter in Figure 2.4; this filter has a cutoff at 2/3m^. I then perform the inverse Fourier transform to obtain the profiles which are bandlimited at 2/3 m^, and unaffected by aliasing. The nonlinear instability cannot grow, and cannot affect the solution below 2/3mv. By a suitably small choice of the mesh increment Ax, m.. can be made large enough so that all physically interesting wavenumbers in the glacier profile are well below the cutoff wavenumber. Phillips (1959), who originally identified the nonlinear instability, suppressed it in this manner. However, the procedure was quite costly to implement, because his work predated the Fast Fourier Transform algorithm (Cooley and Tukey, 1965). In my work, using the filter of Figure 2.4 with the Fast Fourier Transform method did not dramatically increase the computer execution time for the tests I performed. 55 IT(m)l m AX TT/2 2TT/3 FIGURE 2.4. Filter To Suppress The Nonlinear Instability, The Nyquist wavenumber is at mAx=ir. 2.2.5 ACCURACY Having achieved a stable scheme, I must now ask how accurately it solves the partial differential equation. In Section A1.5, I examine the accuracy of the numerical scheme by two somewhat complementary methods. The first method follows from the von Neumann stability analysis in Section 2.2.4. At all wavenumbers m, I compare both amplitude and phase of the transfer function T(m) of the the numerical scheme with those of the partial differential eqation. Errors in amplitude represent incorrect attenuation, and errors in phase represent incorrect propagation speeds and dispersion. For the linear equation, the conclusion is to use 6=1/2 (2.2.14) for the most accurate amplitudes (see Figure A1.7), and to take Ax as small as feasibly possible to get accurate phase (see 56 Figure A1.8). I use these conclusions as a guide when selecting parameters for the nonlinear model. The second method is an estimation of the 'truncation error'. This is the difference between an exact solution h(x,t) of the differential equation, and an exact solution {hM|j=l,J} of the finite difference equations. It is a spatial domain error estimate, i.e. it estimates the total error at each x position, rather than estimating the error in each sine wave component. After assuming that ice thickness h(x,t) and ice flux Q(x,t) are infinitely differentiable, the finite difference solution can be expressed as truncated Taylor expansions about the solution h(x,t) and Q(x,t) of the differential equation. The error is expressed in terms of the first neglected derivatives. The result is that the truncation error has the form J n € = At j (1-26) *2h 2 at2 1 a3h n 1 b3Q n + At2 — + Ax2 — 6 at3 j 24 dx3 j (2.2.15) The leading term vanishes with the choice 6=1/2 (2.2.16) This is the same result obtained from the wavenumber analysis (2.2.14). The truncation error is also minimized by keeping the mesh increments as small as possible. The coefficients can be estimated from the third derivatives of h and Q in the numerical model to get a quantitative estimate of the error. 57 2.3 TESTING THE CONTINUITY MODEL 2.3.1 INTRODUCTION Two tests are described in this subchapter. The first verifies that the model satisfies continuity with nonconstant width, balance, bed, velocity, and ice thickness, and that the terminus moves correctly. The second test verifies that the programs accurately solve a realistic nonlinear problem in which the flow velocity depends on both h(x,t) and its gradient in x. Although other tests of numerical models are available (e.£. Waddington, 1979), the tests presented here are a simple, reasonably comprehensive, and stringent trial of any numerical flowline model based on the continuity equation (1.3.5). I think that it is a reasonable proposal that these tests be used as a minimum standard for verification and comparison of all numerical models of glacier flow prior to attempts to use them to model complicated ice masses. 2.3.2 CONTINUITY TEST WITH TERMINUS MOTION The ability of the glacier model terminus to move correctly, i..e. at a rate consistent with the flow law being used, is critical to accurate simulation of glacier flow (e.£. Nye(l963[a], I963[b]); Nye in discussion of a paper by Mahaffy and Andrews (1976)). If the model terminus advances too slowly, it will act as a dam, resulting in a glacier solution which is too thick, slow, and short, even though continuity is satisfied everywhere. If the terminus of the model advances more rapidly 58 h s • s w w c AX At 0 0 0 0.1 -1 .0 0.01 1 .0 1 .0 -0.02 0.01 0.5 0.1 -0.1 -0.01 1.0 1 .0 -0.02 0.01 0.5 TABLE 2.2. Parameters for continuity test with moving terminus. The first line gives the advancing model, and the second line gives a retreating model. See Figure 2.5 and Figure 2.6. than the correct rate, the glacier solution will be too thin, fast flowing, and extended. No simple approximations are available to simplify the stress equations (1.4.8) through (1.4.10) near the glacier terminus. Nye (1967) published a solution for the shape of a glacier terminus. Since this solution assumed steady state, a horizontal bed, and perfectly plastic ice, it is not suitable for inclusion in my numerical model. The terminus model being tested is described in Section 2.2.3 and A1.3. To check that the numerical model satisfies continuity everywhere and advances or retreats correctly when it is allowed to choose its own terminus position, I use a channel, mass balance, and a flow law giving an analytical solution to the continuity equation (A1.1.1) and show that the model reproduces this solution accurately. My choice is to some extent arbitrary. Other solutions could easily be found. However, the one used below is a good one because it is evident at a glance whether the model results are correct, and it tests the model with a number of nonconstant and nontrivial input functions. Let h0, So* s, W0, W , and c be constants, let the channel width W(x) be 59 Distance FIGURE 2.5. Continuity Test with Moving Terminus. Flow is to the right. The solution profiles are shown at intervals of 10 time units up to 40 units, then at 5 units. The model is described by equations (2.3.1) through (2.3.5) with the constants in Table 2.2. W(x) = W +" W x o (2.3.1) and the mass balance A(x) be A(x) = sx + c/W(x) (2.3.2) I use a velocity (averaged over depth) given by (2.3.3). This has no physical significance; it is merely a numerical test. 60 V(x,t) = cx/[W(x)(h(x,t)-h )] 0 x*0 = c/[s(t )W.(x) ] x = 0 (2.3.3) where the surface slope s is given by s(t) = s + s t 0 (2.3.4) (During the numerical solution procedure, care should be taken to avoid the singularity in (2.3.3) when h(x,t) approaches h0; this can be done by approximating (2.3.3) by a suitably smooth analytical function in the region of h(x,t) near h0 and h(x,t)>h0.) Substitution of (2.3.1) through (2.3.5) into (A1.1.1) verifies that the thickness solution h(x,t) is h(x,t) = h + s(t) x 0 (2.3.5) Distance x is measured along the glacier bed. Any bed profile may be used in the numerical model. The thickness h0 at x=0 is constant for all time, and h(x,t) varies linearly with x at all times, resulting in a wedge-shaped "glacier" with slope s(t). This slope s(t) changes linearly with time at the constant rate s. By solving (2.3.5) for the value of L such that h(L(t),t)=0, the correct glacier terminus position is found to be at L(t)=-h/s(t) (2.3.6) o Figure 2.5 shows the numerical results for the advancing model using the constants in the first line of Table 2.2, and the curvilinear bed shown in Figure 2.5. The wedge solution advances to the right and is shown at equal time intervals of 10 nondimensionalized units for the first 40 units, and at 5 units 61 FIGURE 2.6. Continuity Test With Moving Terminus. Results in Figure 2.5 with bed elevation removed and distance measured along the bed. thereafter. Figure 2.6 shows the same results with the bed elevation subtracted, and distance x measured along the model bed rather than horizontally, i.e. it should be the solution h(x,t) in (2.3.5). In fact, it reproduces (2.3.5) to within one part in 103. The same degree of accuracy is obtained with the retreating model (Table 2.2) which duplicates the curves of Figure 2.6, but in the reverse order. This verifies that the numerical model satisfies continuity everywhere and moves the terminus correctly under quite general conditions; the width 62 varies with x, the mass balance varies with x, and the velocity varies with h, x, and t. 2.3.3 CONTINUITY TEST WITH BURGERS' EQUATION The test in Section 2.3.2 verifies that the model works correctly with general geometrical input and a moving terminus. In this section I show that the iterative procedure in the numerical model works accurately by correctly solving a fully nonlinear equation with a realistic form of velocity depending on both ice thickness and slope. The problem solved also includes kinematic waves. The theory of propagation of shock waves in a gas with diffusion has been investigated by many authors. Contributors to the literature of gas dynamics have included Stokes (1848), Rankine (1870) and Taylor (1910). Methods discussed in the text on nonlinear waves by Whitham (1974) are closely followed here. One standard approach to find the gas density as a function space and time in a shock front is to solve a continuity equation analogous to (1.3.5) (but with no source term on the right hand side) together with an equation analogous to a flow law relating gas flux Q to gas density H. One of the simplest flux relations is Q = Q (H) - v oH (2.3.7) The first term gives the tendency of flux to increase with thickness, and the second term is diffusive damping with diffusion coefficient i/>0. When the variable change 63 dQ (H) (2.3.8) c(H) = — dH is introduced to the continuity equation (1.3.5) with mass balance A(x,t) set to zero and width W(x) set to unity, the result is a nonlinear diffusion equation known as Burgers' equation (Burgers, 1948). When the flux law (2.3.7) has the form the nonlinear Cole-Hopf transformation (Cole, 1951; Hopf, 1950), reduces (2.3.9) to a linear diffusion equation, to which the analytical solution is well-known. Applying the inverse transformation to the solution gives the analytical solution to Burgers' equation, and thus to (1.3.5) with the flux relation (2.3.10). The details of the solution are given in Appendix 9. Since it is rare to find nontrivial analytical solutions to nonlinear partial differential equations, this result is remarkable. It provides an excellent opportunity for an exact test of the model with a nonlinear flow law. Burgers' equation has appeared previously in the glaciological literature in papers on kinematic waves of finite amplitude (Johnson, 1968; Lick, 1970; Hutter, 1980). When the initial condition is = v a2c(x,t) bx7 (2.3.9) Q(H) = oH2 + pH + r - vdH dx (2.3.10) c(x,0) = A 6(x) (2.3.11) and the boundary conditions are 64 c( co,t) = c(- oo,t) = 0 (2.3.12) the solution is (see Appendix .9) V X2/(4l/t) R t e (e - 1 ) c(x,t) = (2.3.13) J? + (eR - 1) J OO _z2 e dz xA/TTTF The parameter R is equal to h/2v. This -solution (2.3.13) is a single asymmetric decaying hump which propagates in the positive x direction. A shock (vertical front) tends to form on the leading edge, but is prevented from doing so by diffusion. When o=1/2 and 0=7=0, we obtain from (2.3.8) c(x,t) = H(x,t) (2.3.14) so that the solution (2.3.13) is also the solution for H(x,t). It is worth noting that equation (2.3.9), as well as being a nonlinear test, is also a test of • kinematic wave behaviour (Lighthill and Whitham, 1955). For application to waves on glaciers, see Nye (1960). It is readily apparent from (2.3.9) that the diffusive property of c(x,t) dc(x,t) = i/d2c(x,t) (2.3.15) dt dx"2" is carried as a kinematic wave at velocity dx = c(x,t) (2.3.16) dt which is thus the kinematic wave velocity. (The left side of (2.3.15) is a total derivative.) this kinematic wave behaviour is included in the complete solution (2.3.13). To test the numerical solution of (1.4.1) against the analytical solution (2.3.13) and (2.3.14), I set the material 65 1 .0 FIGURE 2.7. Nonlinear Test With Burgers' Equation. Dashed lines indicating the analytical solution (2.3.13) have been superimposed on the numerical model solution for a single hump. Because of the close agreement (one part in 103), the curves are indistinguishable in this Figure. Plots are at intervals of 2 time units. The parameters of the model are given in Table 2.3. velocity V(x,t) to be, using (2.3.10), V(x,t) = Q(x,t) H(x,t) H*0 = 0 H=0 (2.3.17) I solve the finite difference equations on the interval [-L,L], choosing L sufficiently large that H(-L,t)=0 adequately approximates (2.3.12). The condition at +co is not needed since 66 A V L to o e y Ax At 1.0 0.1 7.5 2.0 1/2 0.0 0.0 0. 125 0.05 TABLE 2.3. Parameters for Burgers' equation test. (1.4.1) is first order. Since finite difference equations cannot represent the impulse initial condition (2.3.11), I start instead with an initial condition (2.3.13) c(x,t0) at some small to>0. In Figure 2.7, the analytical solution (2.3.13) is superimposed as dashed lines on the numerical model results. The parameters are given in Table 2.3. The agreement is so close (generally to better than three figures) that the dashed lines cannot be distinguished. This test shows that the numerical model conserves mass with fully nonlinear equations. The iterative scheme for nonlinearities works correctly. Since the numerical solution finds the correct time response for the hump, the behaviour of kinematic waves in the numerical solution is also correct. 67 2.4 THE ICE TRAJECTORY COMPUTER MODEL 2.4.1 INTRODUCTION I have written a FORTRAN IV computer program which locates the trajectories of specified particles of ice, as they flow through a time-varying glacier. I have described the model in detail in Appendix 2. The inputs to the model are the bedrock topography, the constants for Glen's flow law (1.4.22), and the number N, and the initial positions, of the ice particles to be tracked. The trajectory model uses the same mesh increments Ax and At as the continuity model (Section 2.2), and the same assumptions about the channel geometry. At each time step, the ice thickness profile {h-|j=1,J} and the basal sliding profile { us. |j=1,J} J J used by the" continuity model are also used as input to this model. 2.4.2 THE VELOCITY AND DISPLACEMENT FIELDS The trajectory of an ice particle is given by P(t), its displacement vector as a function of time. For a particle at position P0 at time t0, where v(x,t) is the ice velocity field (u,w,v). To solve (2.4.1), I start at each time step by finding the velocity field t P(t) P (2.4.1 ) o 68 FIGURE 2.8. Meshpoints For Ice Velocity Calculations. v(x,t) throughout the vertical plane in the glacier centreline, on the x-z mesh shown in Figure 2.8, and described in Appendix 2, Section A2.1. First, the downslope velocity component u(x,z) is found at each meshpoint using the integrated form (1.4.34) of Glen's flow law, with h(x), o(x), and ug(x) given by the continuity model solution. The longitudinal strain rate du/dx is estimated by a finite difference u(i+1,j) - u(i-1,j) ou(i,j) = dx DX + DX i-1,j i,j (2.4.2) of the downslope velocity from (1.4.34), and the lateral strain rate dw/^y is given in terms of u(x,z) and the channel width W(x) by (1.3.3). The incompressibility condition with mass conservation 69 gives dv = -51 du - dw d"x 5y (2.4.3) Using the approximation (usually very good, e.g_. Rothlisberger, 1972) that basal melting is negligible as far as mass balance is concerned, i.e. (2.4.3) is integrated numerically by Simpson's rule (e.c[. Carnahan and others, 1969, p. 73) from the bed to level z to give v(x,z,t). The lateral velocity component w(x,y,z) is exactly zero on the flowline down the centre of the channel, and is very small in a narrow flow volume (Figure 1.1) centred on this flowline. The average value of w is zero in this volume. The computer model uses w=0. When the velocity field has been completely determined at the meshpoints in Figure 2.8, the displacement field of the ice leaving each meshpoint P0 and going to a point P in a time interval At is found by estimating the integral (2.4.1) by i.e., the average of the velocities at the beginning and end of the time interval At. The velocity v(P,t+At) is estimated from the values of v at the four surrounding meshpoints at time t+At using an interpolation scheme described in Appendix 2, Section A2.3.1. v(x,0,t) = 0 (2.4.4) P(t+At) - P0(t) = 1 v(P0,t) 2 L At (2.4.5) 70 2.4.3 THE ICE PARTICLE TRAJECTORIES At each time step, and for each of the N particles being tracked, the coordinates (6x,6z) relative to a meshpoint (i,j) are recorded. The displacements of the particles in the time At are interpolated from the displacements of ice at the four surrounding meshpoints, and the new coordinates of the particles are then saved. The program checks whether the new positions are still within the glacier mass, and saves the interpolated times and positions at which ice particles reach the ice surface. By using At<0, the program is easily adapted to track particles backward in time and upslope (e.g.. from a borehole), to find where and when they entered the ice mass as precipitation. 2.4.4 ACCURACY OF THE TRAJECTORY MODEL The velocity field in this model is obtained by assuming that shearing parallel to the glacier bed is the dominant component of deformation. The downslope velocity u(x,z) is then found using the stress equations and the mechanical properties of ice. This is the only place where the force equations are used. The other velocity components are determined purely kinematically, using continuity and geometric assumptions about the lateral variation of the flow field. When the assumption of predominantly shear deformation holds, this approach works very well. The fractional error in longitudinal strain rate, and in the vertical velocity, is quite 71 small. The leading term is approximately the ratio of the unbalanced longitudinal forces on a vertical column, to the basal shear force. The details are given in Appendix 7, equation (A7.5.9), which is repeated here as (2.4.6). e(x) = 0 dtf' dtf' 2h xx + h yy 6x 6x />gho + (n-1) 6T a xz d v /ou o~x/ dz max (2.4.6) When the assumption of predominantly shear deformation is not accurate, the downslope velocity component u(x,z) is relatively inaccurate. The final term in the error estimate (2.4.6) may be of order unity, or larger, when du/oz is small. The second term in the same equation may also be of order unity, or larger, when stress deviators other than the shear <JXZ contribute to the effective stress. Furthermore, the presence of any additional stress components other than *xZ always softens the ice, so there is no possibility of compensating approximations in this term. This situation may arise at an ice divide, where the predominant motions may be vertical sinking, and longitudinal extension as the ice flows downslope in both directions. Fortunately, the ice velocity is very small at an ice divide, so that the total error in the trajectories is not large. To be safe, the trajectories near an ice divide should be interpreted only qualitatively. Longitudinal stresses are also important in icefalls. The" use of Glen's flow law, and the. inclusion of 72 longitudinal strain rates (to the approximation of (2.4.6)) is an improvement over the trajectory models of Nagata (1977), who assumed that du/dz was zero, and Dansgaard and Johnsen (I969[a]) who set du/dz to a constant at each x in the lowest 400 metres, and zero above that level, and assumed that du/dx was constant for a given z, in a model for the Camp Century, Greenland borehole. The fact that my ' model is time dependent is also an improvement over these models. One advantage of this model is that the mass conservation law is still obeyed globally and locally at all times. This has not been the case with some other trajectory models. For instance, the Weertman (1968) analytical model for ice velocity and temperature at Camp Century, Greenland, assumed a constant vertical strain rate, but used a horizontal velocity given independently by integrating Glen's flow law (1.4.22) to get a result like (1.4.34). Dansgaard and Johnsen (1969[b]) showed that this violation of continuity was the likely cause of a discrepancy of 2°C between predicted and measured temperature at the bed at Camp Century, Greenland. The flow model used by Dansgaard and Johnsen (1969[a]) satisfied continuity everywhere, but assumed the form of the horizontal velocity component, instead of using Glen's flow law. While the model I described in Section 2.2 does not yet include temperature variations, that is a simple development that will not violate mass conservation. Budd and others, (1971, p. 42) assumed a constant vertical strain rate dv/dz independent of depth, at each position x, 73 regardless of the values of the other terms in (2.4.3). They also described another model (p. 43) in which the vertical strain rate was weighted by the downslope velocity u(x,z) at each point. While this makes the vertical strain rate curves more closely resemble the expected shape in a real ice mass, it still does not satisfy continuity locally through (2.4.3). An incompressible continuum flowing with such a velocity field would have to locally create or destroy mass. Constant vertical strain rate (or a strain rate specified a priori) may be a reasonable approximation when deriving analytical solutions to some flow problems to illustrate the physics involved (e.cj. Robin, 1955; for the effect of advection on temperature profiles), and it arises naturally in some simple kinds of flow, i_.e. horizontal shearing restricted to the basal layer (e.c=. Hill, 1950, p. 233; Nye, 1951; or Nye, 1957). However, given the sophistication of current computer models accepting otherwise quite general inputs, this assumption now appears to be a needless limitation. While the errors involved may turn out, in some situations, to be small, it is preferable to avoid them at no additional complication. The only model of which I am aware which attempts to use the continuity equation in a manner similar to the way I described in Section 2.4.2 to find the vertical velocity component, and also makes no a priori assumptions about the flow field other than those described in Section 2.4.2, is the preliminary three-dimensional model of Jenssen (1977) for the Greenland ice cap. This model also solves for the temperature distribution, for use in the temperature dependent flow law. The 74 Jenssen model is still under development to improve the boundary treatment and and to reduce inaccuracy resulting from the coarse grid imposed by computer limitations (Greenland was modelled by a 12x12 map grid, 10 points deep); however, the Jenssen approach is the logical next step that must be taken towards the numerical solution of the complete set of equations for ice sheets. 2.5 TESTING THE TRAJECTORY MODEL 2.5.1 INTRODUCTION In Section 2.5, I describe two tests of the trajectory model. In Section 2.5.2, I compare the steady state trajectories and velocity field calculated by my computer model to an analytical solution for a steady state ice sheet (Nagata,1977). In Section 2.5.3, I use the balance condition (A5.8) at the ice surface to show that continuity is satisfied by the velocity field. 2.5.2 NAGATA ICE SHEET TEST Nagata (1977) derived an analytical solution for the surface profile, velocity field, and streamlines of a steady two-dimensional ice sheet resting on a flat bed. For the steady state case, streamlines and ice trajectories coincide. I describe the Nagata model in detail in Appendix 15, Section A15.3. Nagata (1978) used this ice sheet model to 75 E CD O c o o CD 1.0 0.5 Nagata (1977) Model Steady state ice sheet Basal sliding m=3 Velocity fielS and streamlines •6723 0.5 x/L 0 -2 1.0 FIGURE 2.9. Nagata Steady Ice Sheet. The analytical solution for the ice thickness, mass balance, particle paths, and velocity field using the parameters in equation (2.5.2). The velocities have been multiplied by 250 years. The numbers are the time in years for ice to flow the length of each particle path. explain the concentration of meteorites at the Meteorite Ice Field in Antarctica. The Nagata model assumes that the forward ice velocity u(x) is constant throughout a vertical column and depends on the basal shear stress (1.4.25) through a Weertman-type relation (see Appendix 8, Section A8.3.1) of the 76 T X (km) FIGURE 2.10. Growth Of Nagata Ice Sheet. The ice surface elevation is shown at intervals of 500 years, starting from ice-free conditions. The flow parameters, are given in Table 2.4 and the mass balance in Figure 2.9. The steady state ice thickness agrees with the analytical solution by Nagata (1977) to one part in 103. form m u(x) = A T (2.5.1) Although it is not clear in the original paper, the Nagata model also assumes that the vertical velocity component v(x,z) is equal to a constant b along the upper surface of the ice sheet. The scale of the ice sheet model is set by H, the thickness at the ice divide. While the form of the mass balance and the velocity field are not a close representation of real ice sheets, the Nagata model is well suited to testing the accuracy of my numerical 77 Distance (km) FIGURE 2.11. Velocity Field For Nagata Model. The velocity vectors calculated by the Waddington trajectory model for the steady state profile in Figure 2.10 and using flow parameters in Table 2.4. The velocities have been multiplied by 250 years (The units of the vectors are displacement). trajectory program. I compare my numerical results to the Nagata model shown in Figure 2.9 for the constants in Table 2.4. Figure 2.10 shows the growth of the Nagata ice sheet to steady state using the profile model described in Section 2.2). The steady state profile agrees with the analytical model in Figure 2.9 to within one part in 103. The solid vectors in Figure 2.11 show the steady state velocity field calculated by my numerical model. The velocities have been multiplied by a factor of 250 years. The velocity 78 m A b H L p g bar"2 a'1 m a" 1 m km kg m"3 m s"2 2 100. 1.0 3000 454.6 910. 9.8 AX km At a 7.215 10.0 TABLE 2.4. Parameters for Nagata ice sheet. field agrees with the analytical solution (Figure 2.9) to within a few parts in 103 (except near the terminus where the agreement is only to two parts in 102, because the mass balance in the numerical model remains finite while the analytical mass balance, goes to -oo ). Figure 2.12 shows the trajectories for ice entering the ice sheet at time t=0.; these are the same five points shown for the streamlines in Figure 2.9. The arrowheads indicate 250 year intervals. The total residence times along each of the five streamlines are compared to the analytical values in Table 2.5. The good agreement between the numerical results and the analytical solution indicates that the velocity field is reconstructed accurately and the integration of the velocity field to get flowlines is correct and accurate. 79 0 100 200 300 X (km) 400 500 FIGURE 2.12. Trajectories In Nagata Model. The solid curves are the particle paths calculated by the Waddington trajectory model by integrating the velocity field in Figure 2.11. The arrowheads indicate 250 year intervals. In Table 2.5 the total residence times in the ice sheet are compared to values for the analytical solution. 2.5.3 SURFACE MASS CONSERVATION TEST Conservation of mass at the glacier surface with normal vector n implies that the normal velocity of the ice-air interface is equal to the sum of the normal ice velocity plus the surface accumulation rate, i..e. — • n = v • n + a • n (2.5.2) dt This equation is derived in Appendix 5, Section A5.2. When ice 80 Streamline Number Numerical model years Analytical model years 1 6746. 6723. 2 4625. 4606. 3 3335. 3322. 4 2350. 2346. -> 5 1477. 1466. TABLE. 2.5. Residence times in Nagata ice sheet. thickness change rate dh/dt and mass balance vector a are measured normal to the bed, and the velocity components (u,v) are parallel to and normal to the bed, (2.5.2) becomes oh = v(x,h) - u(x,h) djh + A(x) St 6x (2.5.3) where A(x) is now a scalar giving the mass balance. My derivation (Section 2.4.2 and Appendix 2, Section A2.2) of the velocity field uses only incompressibility (2.4.3) and the basal boundary condition (2.4.4); (2.5.3) can- be used as an independent check of the degree to which the velocity field conserves mass. Models which use (2.5.3) to derive the vertical velocity (e.£. Weertman, 1968; Budd and others, 1971, p. 42) do not have this consistency check. Figure 2.13 (a) shows the mass balance A(x) and the surface rise dh/dt for a typical time step, with parameters described by Table 2.6, during the growth of the Nagata ice sheet model. I 'apply the test at a time (2500 years) when the ice sheet is growing vigorously because conditions at that time impose the most stringent conditions on accuracy. Figure 2.13 (b) shows the residual of (2.5.3), i..e. the difference between the left and right sides of the equation on substituting the values of h, A, 81 FIGURE 2.13. Surface Mass Conservation Test. Time=2500 years during the growth of the Nagata ice sheet model to a steady state, (a) shows the mass balance (solid curve) and the rate of surface rise dh/dt (broken curve), (b) shows the residual of (2.5.3). It is very small (note scale change of 10~2, indicating that the velocity field satisfies incompressibility. The terminus is at x=37.8 km. The residual increases near the terminus because there are few meshpoints in each vertical column; vertical integration is inaccurate. u, and v from the numerical solution. The residual error is three orders of magnitude less than the average magnitude of the mass balance, except near the terminus where there are insufficient mesh points in any vertical column to guarantee accurate vertical integration of (2.4*3). This region has no 82 Time Ax DZ At a km m a 2500. 7.215 150. 10. TABLE 2.6. Nagata ice sheet surface boundary test. effect on the trajectories investigated in this work. Similar tests (not shown) on the Steele Glacier Model 1 (Figure 3.3) routinely give residuals of the order of one part in 102 of the average mass balance, even though the mass balance, is discontinuous (tributaries), the width is variable, and the flow law includes a height-dependent longitudinal strain rate. Results of these tests indicate that the velocity field satisfies the continuity equation (2.3.3) to a very good approximation. This test verifies the accuracy of the trajectory model under time-varying conditions. 83 CHAPTER 3: CAN STABLE ISOTOPES REVEAL A HISTORY OF SURGING? 3.1 INTRODUCTION The stable heavy isotopes O18 of oxygen and D (deuterium) of hydrogen in glacier ice have been widely used as indicators of climatic change (e.ci. Dansgaard and others, 1969; 1971). This procedure requires assumptions about the pattern of glacier flow. In this chapter, I investigate a related problem; assuming that the past climate is known, can the stable isotope distribution be used to reveal the flow history of time-varying glaciers? The example I consider in this chapter is the Steele Glacier, Yukon Territory. This glacier was observed to surge in 1966-1967 (Stanley, 1969). I used the computer models described in Chapter 2 to find the ice surface and the velocity field throughout the surge cycle, and to calculate the ice trajectories. Knowing the trajectories and the isotopic composition (from climate) at the time and place the material was precipitated as snow on the glacier surface, has allowed me to construct longitudinal cross-sections and surface profiles showing the isotopic distribution at a series of times during the surge cycle; this pattern would facilitate the selection of optimum borehole and surface sampling locations for an isotopic study of past flow patterns. In Section 3.2, I describe the Steele Glacier and its surge history. In Sections 3.3 and 3.4 I describe the computer models I used to simulate the Steele Glacier. Model 1 in Section 3.3 is 84 a detailed model based on all the available data. Because of limited sliding data and an approximation in the mass balance, it is not at present the best model for particle trajectory calculations. With more complete data and some computer model refinements, it may become so. Model 2 in Section 3.4 is a simplified version better matched to the resolution of the sliding observations. Model 2 is used for the trajectory and isotopic calculations in Sections 3.6 and 3.7. In Section 3.5, I describe the use of stable isotopes in glaciology, and present two possible isotopic relations for precipitation at Steele Glacier. The ice surface profile calculations in Section 3.6 indicate that a surge period of approximately 100 years is appropriate for the Steele Glacier if the present mass balance and the velocity of the 1966-67 surge are representative of the average long-term climate and of the glacier flow pattern. In Section 3.7, I present computed longitudinal cross-sections and surface profiles of the isotopic distribution 6(018/016) for a model with a surge periodicity of 97 years. Isotopic discontinuities occur in the ice along surfaces which were at the ice-air interface in the accumulation region when a surge began. Even for the isotopic-precipitation model most favourable to the formation of discontinuities, the discontinuities do not exceed one DEL unit on the Steele Glacier; this amount may be hidden by noise in the Steele Glacier environment. 85 3.2 STEELE GLACIER 3.2.1 GENERAL DESCRIPTION Steele Glacier (61°10'N, 140°15'W) is a surging valley glacier on the northeast slopes of the Icefield Ranges (see Figure 3.1) of the St. Elias Mountains, Yukon Territory, Canada. Prior to 1963 it was called the Wolf Glacier. (Sharp (1951) and Bostock (1948, p. 99) state that this is more proper than the often-used notation Wolf Creek Glacier.). The glacier length varies from 34 to 44 km, and the width of the main channel is one to two km. The main channel flows down from 3000 m elevation on the north side of Mt. Steele (Figure 3.2) to 1200 m elevation in Steele Creek, where the ice is stagnant and moraine-covered between surges. The continental slope of the Icefield Ranges is semi-arid. Annual precipitation drops from 300 cm a"1 at Yakutat on the Gulf of Alaska (see Figure 3.1), to about 35 cm at Kluane Lake (Wood, 1972). The firn line on the Steele Glacier is presently very high (2400 m or more). A major source of accumulation is avalanches from the north face (labelled (0) in Figure 3.2) of Mt. Steele (5070 m). In its upper reaches, the Steele is an extensive system of accumulation basins and converging tributaries. These tributary ice streams (see (1) through (5), Figure 3.2) also contribute a substantial fraction of the Steele Glacier mass balance. In its lower reaches, the Steele Glacier makes a 90° bend to the east, and enters a straight and narrow valley formed by the Wolf Creek monocline (Sharp, 1943). 86 FIGURE 3.1. Icefield Ranges Location Map. Nonstippled areas are major glaciers and icefields. The triangles indicate major summits of the Icefield Ranges (St. Elias Mountains). The Steele Glacier is on the north-east (continental) slope (top centre). 3.2.2 GLACIER SURGES A glacier surge (e.cj. Meier and Post, 1969) is a short period of very rapid flow, during which ice is transferred from an ice reservoir area to an ice receiving area downstream. A surge is followed by a longer period of stagnation and ablation 87 in the receiving area, with renewed ice buildup in the ice reservoir; these areas do not necessarily correspond to the accumulation and ablation zones defined by mass balance. For a large valley glacier like the Steele, the maximum velocity during a surge may be 500 ma"1 to 10 km a"1, with downstream ice displacement of one to 10 km. Surging appears to be a periodic phenomenon. For glaciers like, the Steele, the surge duration is typically one to two years, with a quiescent phase lasting from 20 to 150 years. Glaciers of all sizes have been observed to surge. Examples are the Trapridge Glacier (in the Steele Creek watershed) which is only three km long, and the Muldrow Glacier on Mt. McKinley, Alaska which is over 50 km long. There has been speculation that the Antarctic ice sheet may surge (Hollin, 1969; Wilson, 1969). Surging glaciers are found in many parts of the world, e«2. Alaska (Tarr and Martin, 1914, p. 168), British Columbia and the Yukon Territory. (Post, 1969), the Arctic Islands (Hattersley-Smith, 1964; L0ken, 1969), Iceland (Thorarinsson, 1969), the Karakoram (Hewitt, 1969), the Pamirs, Tien Shan, Caucasus, and Kamchatka (Dolgoushin and Osipova, 1975). Surges occur in both temperate and cold or subpolar glaciers. Surging does not appear to be triggered by climate variations. The high velocity during surging is generally attributed to rapid basal sliding. Various hypotheses on the mechanism of surging have been put forward. The more plausible ones include thermal regulation (Robin, 1955; Clarke, 1976; Lliboutry, I969[b]), stress instabilities (e.g. Post, 1960; Robin, 1969) and basal water film instabilities (Weertman, 1962, 1969; Robin 88 and Weertman, 1973; Budd, 1975). A concensus on the cause of glacier surging has yet to be reached. In this chapter, I investigate the effect of periodic surging on the stable isotope distribution in the Steele Glacier. The amount of basal sliding is important to this question; the mechanism which causes it is not. For this reason, I specify a basal sliding velocity us(x,t) consistent with the observations of the Steele Glacier surge of 1966-1967, and I use this as a boundary condition for the computer model. 3.2.3 OBSERVATIONS OF THE STEELE GLACIER The flow pattern before and during a surge is better known for the Steele Glacier than for most other surging glaciers. Scientific expeditions sponsored by the American Geographical Society and led by W. A. Wood explored the Steele Creek watershed in 1935, 1936, 1939, and 1941. Work included the experimental use of oblique aerial photography for mapping areas of high relief. These air photos and the panoramas from ground control points give information on the ice surface elevation and state of flow during the quiescent phase. Sharp (1943) described the geology of Steele Creek, observed the glaciers of the Steele Creek basin (Sharp, 1947), and interpreted the. glacial history (Sharp, 1951). The Surveys and Mapping Branch of the Department of Energy, Mines and Resources, Government of Canada, obtained vertical aerial photography of the Icefield Ranges from 10 700 m in the summer of 1951. Flight lines A13232/33 covered the Steele 89 61°20 61°05 140° 30 U0° FIGURE 3.2. Steele Glacier And Tributaries. The central flowline used in the computer model is marked at 2.5 km intervals. The Hodgson Glacier is included in the channel width function (Figure 3.3 (b)) for the main ice stream. For Model 1, the mass contributions from the minor tributaries (0) through (5) are included in the mass balance function (Figure 3.3 (a)). Glacier. This photography was used to prepare the government map 115F at the scale 1:250,000; it was also used to prepare a map of the Steele Glacier at the scale 1:25,000 (Topographical Survey, 1967). Oblique aerial photography of the Steele Glacier was obtained periodically, between 1960 and 1965 by W. A. Wood (American Geographical Society) and by A. Post (U.S. Geological Survey). In 1960, Post predicted an imminent surge for the 90 Steele Glacier (Wood, 1972), and in 1965, noted the first signs of increased flow on the upper Steele (Wood, 1972). During the 1966-1967 surge, high oblique aerial photos were obtained by Wood, pre-1941 survey stations were re-occupied, and the Surveys and Mapping Branch obtained vertical air photo coverage on August 13 and September 15, 1966. Wood (1972) and Stanley (1969) measured displacements of identifiable surface markings. Features originally between the Hodgson confluence and the 90° bend (Figure 3.2) all advanced at roughly the same velocity of 5 km a"1 during 1966, with a total displacement of roughly 8 km when the surge ended late in 1967. There was no significant lateral variation of velocity beyond 200 m from the glacier margins. Bayrock (1967) observed the details of the terminus advance, and the reactivation of stagnant ice. A bulge of active ice 30 m high moved forward at about 10 m d"1 (3650 m a"1). This bulge sometimes has been called the terminus in the literature on the 1966-1967 surge. Alford from the Whitehorse office of the Water Survey of Canada obtained monthly air photographs of the advancing bulge during the winter of 1966-1967. He observed an advance of 6000 feet (1830 m) between September 10, 1966 and January 15, 1967 (Roots, 1967). By August 1967, the active bulge had slowed to 2 m d"1 (730 ma-1) (Thomson, 1972). Stanley (1969) identified three zones based on surface elevation changes during the surge. An upper zone, entirely in the accumulation area, apparently was not involved in the 1966-1967 surge. In a middle zone (the ice reservoir zone) from a point above the firn line (based on Stanley's description, I 91 locate this point at about x=7 km) to a point about 3 km above the 90° bend (at x=27 km in my model), there was a net lowering of the ice surface, with a maximum value of 130 m above the Hodgson-Steele confluence. In the lower remaining zone (ice receiving area) the surface rose by up to 100 m. During the winter of 1966-1967, the Hodgson Glacier began a year-long surge during which it pushed the still-surging Steele ice stream to one third of its normal width at the ice surface. The effect of the Hodgson surge on deep ice is unknown. The Hodgson ice formed a large lobe extending three km down the main Steele channel. This tributary surge may have been triggered by a reduced confining stress resulting from the rapid lowering of the Steele Glacier at their confluence. 3.2.4 PERIOD OF STEELE SURGES Sharp (1951) showed that, based on 1941 observations, the Steele Glacier below the bend had been stagnant since 1916 or earlier. Based on biological recolonization rates inside the most recent trim line, he estimated that the last advance ended in the period 1840-1890, j..e. 115 to 75 years before the 1966-1967 surge. Wood (1972) pointed out evidence for ice displacements of over two km between 1935 and 1941 near the Steele-Hodgson confluence. The previously smooth ice in that region was heavily crevassed in 1941, and the crevasses looked several years old. The evidence suggests that the Steele Glacier had a minor surge 92 or a failed attempt at a surge in 1938 or 1939. Sharp's (1951) estimate of the time of the previous terminus advance would then imply a buildup time of 50 to 90 years. Sharp (1951) pointed out moraines 100 m to 150 m above the 1941 ice surface below the bend. An ablation estimate of 2 m a"1 implies the maximum ice elevation occurred 50 to 75 years prior to 1941 . There are no direct observations on the Steele Glacier of regular looped moraines, or of disrupted ogive patterns which could reveal pre-historical surge episodes. The assumption of periodicity is based on observation of surging glaciers elsewhere (e.£. Variegated Glacier, Alaska, (Bindschadler and  others, 1977)). The assumption of exact periodicity for the computer model is, at best, an approximation; significant climate variability on the time scale of glacier surges has been widely documented (e.£. Bryson and Goodman, 1980; Gribbon, 1979). If the Steele Glacier surges periodically, the period is in the range 50 to 150 years. 93 3.3 NUMERICAL MODEL 1 3.3.1 FLOW LAW CONSTANTS AND SHAPE FACTOR The computer model in its present form assumes that the ice is isothermal. Temperatures in the upper 100 metres of the Steele Glacier (Jarvis and Clarke, 1974; Clarke and Jarvis, 1976) following the 1966-1967 surge were in the range -1°C to -7°C. It is likely that the basal ice, within which most of the shear deformation takes place, is at or near the pressure melting point. The assumption of temperate ice is not unreasonable. I use the constants (Paterson, 1981, p. 39) A = 5.3 10-15 s"1 kPa'3 n=3 (3.3.1) in Glen's flow law (1.4.22) for the Steele Glacier simulation. Since most of the motion in a surge is due to sliding, changes in the flow law constants have little effect on the ice movement. The shape of the valley cross-section is unknown. I use a shape factor (see Section A7.4) of s = 0.8 independent of position x. Since the shape factor is raised to the nth power (see equation (1.4.34)), uncertainty in s introduces substantial uncertainty into the deformation velocity. However, as pointed out above, the internal deformation of the Steele Glacier is a small component of the total motion. 94 3.3.2 BED TOPOGRAPHY There are no published ice depth data for the Steele Glacier. Jarvis and Clarke (1974) and Clarke and Jarvis (1976) reached depths exceeding 100 metres with a hot-point drill without any indication of bottom; the glacier is likely much thicker than this. Typical depths of large valley glaciers can exceed 600 metres (e.g.. Lowell Glacier, St. Elias Mountains, based on monopulse radio-echo sounding, 1977; G. K. C. Clarke, personal communication). A rough estimate of the depth e.cj. above the confluence of the Steele and the Hodgson can be obtained by assuming a basal shear stress of one bar and measuring the ice surface slope to be c=0.03 from the 1951 map (Topographical Survey, 1969). Assuming a shape factor of s=0.8, the stress relation (1.4.25) gives the depth as (approximately) h = xz = 472 metres s/>go (3.3.2) Alternatively, assuming flow by simple shear with no sliding, and using the 1951 velocity of 25 m a"1 measured at this point by Wood (1972) (1.4.38) with the flow law constants (3.3.1) gives the depth estimate l/(n+D = 445 metres h = (n+2)V n 2A( Syoga) (3.3.3) Since the motion of the Steele Glacier is obviously not an example of steady nonslip flow in a cylindrical channel, these are merely rough estimates. The presence of basal stress of less than one bar, or of nonzero basal sliding would result in an 95 overestimate of the true thickness. Neglecting the stress perturbations and lateral asymmetry caused by bends in the channel, I assume the central flowline (x axis) follows the broken curve shown in Figure 3.2. For the Steele Glacier bed topography in Figure 3.3 (c), I use an exponential function having the form -bx h(x) = ae + c (3.3.4) where a, b, and c are constants determined by fitting the three points: (1) 2900 metres elevation at the bergschrund of the main ice stream (x=0). (2) 1200 metres at the 1978 terminus position (x=42 km). (3) 1650 metres at the confluence of the Hodgson and Steele Glaciers (X=18 km); this value would give an ice depth of about 400 metres at this point in 1951 (Topographical Survey, 1969)., The 1951 longitudinal surface profile is shown in Figure 3.3 (c). The actual topography beneath the Steele Glacier is undoubtedly more complicated. This approximation means that the computer model cannot be expected to quantitatively reproduce the observed ice surface elevations. 96 3.3.3 CHANNEL WIDTH The width of the main ice stream of Steele Glacier can be estimated by measuring the separation of the lateral moraine ridges on the topographic map at the scale 1:25,000 (Topographical Survey, 1969). The published government mapsheet 115G and 115F(El/2), at the scale 1:250,000, (from which Figure 3.2 is drawn), is not a reliable indication of the channel width. Ideally, tributaries should be included in the numerical model as separate ice streams with their own bed and mass balance functions, and coupled to the main ice stream by thickness and flux conditions at the junctions; this option is not available in the computer model in its present form. Instead, I can include the effects of tributaries in approximate ways through the width or mass balance functions. Since the discharge and depth of the Hodgson Glacier are probably comparable to those of the Steele at their confluence, I use the sum of their widths above this point. This is a simple approximation. In fact, their surface gradients are not equal everywhere in this region, their mass balance functions probably differ, and the two glaciers do not surge together (the Hodgson last surged in 1967-68, one year after the Steele). Because of their much shallower depths, to include the minor tributaries (1) through (5) (Figure 3.2) in this way would tend to grossly reduce the thickness and velocity of the main channel. The mass contribution of these tributaries is included in Model 1 by an addition to the mass balance function (Figure 3.3 (a)) at the regions of confluence. 97 3.3.4 MASS BALANCE No comprehensive mass balance measurements have been made on the Steele Glacier. Between an average date of July 19, 1974, and an average date of July 2, 1975, Collins (unpublished) measured ablation at eight survey targets near the right angle bend (x=30 km). The measurements ranged from 1.57 m to 2.77 m of ice, with a mean of 2.15 m. All other evidence is indirect. On the lower Steele Glacier, Stanley (1969) estimated the ablation to be 1.5 m a"1 based on downwasting of ice shown to be stagnant by Sharp (1951). Since snowfall can depend strongly on local conditions, extrapolating mass balance information even from adjacent valleys is at best a risky procedure. However, the qualitative patterns of mass balance, and its order of magnitude throughout the Icefield Ranges give some control on reasonable estimates for the Steele Glacier. Marcus and Ragle (1970) reported winter accumulation measurements on a traverse across the Icefield Ranges from the lower Seward Glacier to the Kaskawulsh Glacier. The values for the Kaskawulsh are instructive because both Kaskawulsh and Steele Glaciers occupy similar positions on the east side of the Icefield Ranges (Figure 3.1). The precipitation on the Kaskawulsh increases with elevation. The 1964-65 winter accumulation was 1.7 metres of ice equivalent at 2640 metres elevation, decreasing to 0.35 metres of ice equivalent at 1615 metres elevation. These authors also reported negligible summer melting on the ice plateau at Divide Station 30 km west of the Kaskawulsh Glacier. Divide Station at 2620 metres 98 1 1 r ol 1 i L 10001 1 1 0 10 20 30 40 50 x (km) FIGURE 3.3..Model 1 For Steele Glacier. (a) Mass balance. The dashed curve is balance for the main ice stream. The solid curve includes mass contributions from the tributaries (0) through (5) identified in Figure 3.2. (b) Glacier width. The broken curve is the width of the main ice stream only. The solid curve includes the width of the Hodgson Glacier. (c) Bed topography. The ice surface profile in 1951 is also shown. 99 elevation receives about 2.2 metres (ice equivalent) net accumulation each year; the record there indicates that 1965-66 was a low snowfall year by about 40% in the Kaskawulsh region (Marcus and Ragle, (1970, Figure 7)). Keeler (1969) reported that, on Mount Logan, 60 km south of the Steele Glacier, elevation has little effect on precipitation above 2500 metres elevation. The net .accumulation is about 0.8 metres of ice annually. Stanley (1969, Figure 1) located the firn line at about 2400 metres on the Steele Glacier based on the 1951 aerial photography. Wood (1972) put the firn line in the higher elevation range of 2750 to 2900 metres. This would leave almost no accumulation area. Sharp (1947) gave the estimate of 8000 to 9100 feet (2440 to 2775 metres). The broken line in Figure 3.3 (a) shows an estimated mass balance function for the main ice stream of the Steele Glacier consistent with Collins (unpublished) and with the indirect observations. The ice flux from the ith small tributary glacier is included through a perturbation 6b^ to the mass balance function. In Appendix 19 I describe two methods of estimating the ice flux from each tributary and how I use this to estimate the 6b£. The solid curve in Figure 3.3 (a) shows the mass balance with the tributary terms 6b^ included. While Figure 3.3 (a) represents the best available estimate of the Steele Glacier mass balance, it should be kept in mind that mass balance may change significantly with longterm climatic change. The decades 1930-1950, upon which much of the data for the Steele Glacier is based, appear to have been an 100 exceptionally warm period (§_.£. Hansen and others, 1981; Schneider and Mesirow, 1976, Chapter 3). 3.3.5 CYCLIC SURGE PATTERN FOR THE MODEL Because I am investigating consequences of surging, rather than surge mechanisms, I specify a priori the sliding velocity u5(x,t). This aspect of surge modelling is discussed in Section 1.5.3. I have chosen to use a sliding velocity having the form u (x,t) = X(x,t) T(t) (3.3.5) s because it can represent the observed sliding of the Steele Glacier reasonably well. Other functional forms of us(x,t) could equally well fit the observations of Stanley (1969) and Wood (1972). The time dependent term T(t) is a nondimensional weighting factor between zero and unity. Figure 3.4 (b) shows the form of T(t) for a surge cycle of length t«. During the quiescent stage, T has some small constant value f (£.£. f=0 gives no sliding). The surge starts at time t0 and the velocity rises to the peak value by time t1f remains at the maximum until t2, then falls back to the nonsurge level by time t3. The term X(x,t) shown in Figure 3.4 (a) gives the normalized spatial distribution of the sliding velocity at a time (t-t0). Observations on the Steele Glacier (Stanley, 1969) and on other surging glaciers, and some theoretical work on surging (e.g. Robin, 1969; Robin and Weertman, 1973) suggest that rapid sliding starts in a small region, and the boundaries of this 101 FIGURE 3.4. Sliding Model For Steele Glacier. (a) X(x,t) gives the normalized spatial dependence of the sliding velocity at time t. (b) T(t) is the temporal weighting function for the sliding velocity in (a) . zone of rapid sliding then propagate down (and possibly up) the glacier. U0 is the maximum sliding velocity during the surge. In this model, each transition from a zone of rapid sliding to a zone of no sliding is given by one half cycle of a cosine. The distribution of sliding can change during the surge as the four points x.2(t), x.,(t), x,(t), and x2(t) move at velocities c.2(t), c.,(t), c,(t), and c2(t) respectively. The data from the surge of the Steele Glacier are not sufficiently complete to allow detailed estimation of the c-(t). I use constant values for these four velocities, so that the velocity transition points move according to 1 02 x (t) = x (t ) + (t-t ) c 2 20 02 (3.3.6) with similar equations for the other three points. The constants x.2(t0), x.,(t0), x,(t0), and x2(t0) define the extent of the trigger zone. Observations by Raymond and others (unpublished, Figure 9) on the Variegated Glacier, Alaska, indicate a regular increase of sliding velocity, from 0.05 m d"1 (18 m a"1) in 1973, to 0.3 m d"1 (110 m a"1) in 1979 in the upper reaches of the glacier. The Variegated Glacier appears to have a surge period of about 20 years, and is expected to surge sometime in the mid-1980's. The simulations which I have carried out for this chapter do not include a pre-surge increase in basal sliding, although it probably could be modelled satisfactorily by the Weertman (1957) sliding mechanism; this option is available in the computer program. However, for the Steele Glacier, the observed pre-surge velocities (Wood, 1972) are low, and are not sufficiently detailed to warrant this additional complication. The leading edge of the zone of rapid sliding (i.e. x,(t) to x2(t)) is a region of compressive flow. The instability of regions of compressive flow is widely recognized (e.g. Paterson, 1969, p. 207). For nonsurging glaciers, the surface rise resulting from compressive flow in the ablation area is largely balanced by surface melting. A surge lasting only one or two years occurs too quickly for melting to have any appreciable control on surface elevation; to avoid a growing bulge or shock wave moving with the leading edge of the zone of rapid sliding, the velocities c, and c2 of the velocity disturbance must be 103 sufficiently greater than the material velocity U0 that the velocity change can move ahead of any bulge forming in the ice. For the elementary case of ice with sliding velocity U0 and thickness h0 advancing into thinner stagnant ice of thickness h, in a channel of constant width, a simple continuity argument in Appendix 18 shows that when c,=c2, the velocity disturbance must move at least at the speed c £ U i o to prevent the development of a shock front in the advancing velocity transition. Equation (3.3.7) can be used to estimate the ice thickness h0 and h,; for the Steele Glacier, I have used it only to choose reasonable values of c, and c2. 3.4 STEELE GLACIER MODEL 2 3.4.1 PROBLEMS WITH STEELE MODEL 1 I ran the Steele Glacier model 1 (Figure 3.3) using the flow law parameters (3.3.1) with no sliding. The numerical parameters are given in Table 3.1. The surface profiles at 50 year intervals starting from ice-free conditions are shown in Figure 3.5. The individual tributaries coalesced by 250 years. The maximum velocity (averaged over depth) in steady state is 33 m a"1 at x=12.5 km. The corresponding velocity at the ice surface is (see Section 1.4.4) (n + 2)/(n+l) times this, j..e. 41 ma"1. The final steady state length is 35.5 km, and the h -h o 1 (7. 7. 7 ) 104 n A 5 g fi Ax At -n -1 kPa s ms"2 kg irr 3 m a 3 5.3 10"15 0.8 9.8 900. 500. 1 .0 TABLE 3.1. Parameters for Steele steady state. AOOOl T Steele Glacier Growth to steady state No sliding Profiles every 50 years 3000 2000h 1000 0 10 20 30 x (km) FIGURE 3.5. Steele Model 1 Growth To Steady State. There was no sliding. The model parameters are given in Table 3.1. The ice surface profiles are shown at 50 year intervals starting from ice-free conditions. The ice lobes at 11 km and 13 km are caused by ice from tributaries (1) and (2). maximum depth is 444 m at x=25 km. The surface profiles from Model 1 appear to be reasonable, 105 and the model coped with the rapidly-varying mass balance and width functions in Figure 3.3. However, I calculated some of the steady state streamlines for Model 1; these are shown in Figure 3.6 (note that the abscissa is ice thickness rather than elevation). The vertical mesh increment for the velocity 1 1 r Steele Glacier Model 1 Steady streamlines No sliding Arrows every 100 years FIGURE 3.6. Steady State Streamlines For Model 1. This cross-section shows ice depth. Arrows on the streamlines indicate each 100 years of flow. There was no sliding. The perturbations to the smooth streamlines are due to the addition of ice flux from tributaries through mass balance terms. calculations (see Figure 2.8) was DZ = 15m The perturbations in the streamlines are caused by the additions of tributary ice flux through the mass balance function. I am looking for irregularities in streamline shape and in isotope distribution resulting from surging; to have features of this 106 £ co o c a a CO. u) a 3 2 1 0 -1 •2 3000h £ 2000h •g £ 1000 3000 2000 CXI X 1000 Steele Glacier Model 2 Growth to steady state No sliding Profiles every 50 years 0 10 20 30 AO 50 (km) FIGURE 3.7. Model 2 For Steele Glacier. (a) Mass balance. (b) Glacier width. (c) Bed topography and ice surface profiles at 50 year intervals during growth to steady state with no sliding. form introduced through the mass balance function is 107 undesirable. Adding the tributary flux at the glacier surface rather than at the channel margins is an adequate approximation if only the ice surface shape is desired. If particle trajectories are also desired, this approximation is unacceptable. An additional problem arises with Model 1 when sliding is added. The width function (Figure 3.3 (b)) is a detailed representation of the valley of Steele Creek. The spatial distribution of the surging velocity of the Steele Glacier is not known with the same resolution; Wood (1972) and Stanley (1969) obtained only spatially and temporally averaged velocities. Forcing the ice to slide at nearly constant velocity through a channel of highly variable width can result in some unrealistic surface configurations. For instance, in regions where the channel width gradient is large and negative, the ice can thicken rapidly and obtain a reversed surface slope. More detailed data on how the glacier actually changes speed to prevent this situation are not available for the 1966-67 surge. It is necessary to use a width function which has the same degree of spatial detail as the sliding data. When a network computer model for tributaries is developed, and when more detailed sliding observations are available, the amount of detail in Model 1 can be justified. 108 3.4.2 SIMPLIFICATIONS Figure 3.7 (a) and (b) show a simplified model for the Steele Glacier. This model resembles Model 1 ' in its gross features, yet avoids the difficulties I described in the previous section. Figure 3.7(c) shows the growth of Model 2 to steady state with no sliding, using the parameters in Table 3.1, and starting from ice-free conditions. The steady state length is 36.5 km, the maximum ice thickness is 452 m at x=21 km, and the maximum velocity (averaged over depth) is 30.4 ma"1 at x=11 km. These values are close to the values for Model 1 (see Section 3.4.1). 3.5 STABLE ISOTOPES IN GLACIOLOGY 3.5.1 DEFINITION OF THE DEL SCALE The standard method of describing the isotopic composition of oxygen and hydrogen in water is the DEL scale (6). The ratio R of the concentrations of the heavy and light isotopes b18/016 and D/H can be measured with a mass spectrometer; a practical concentration scale should be based on R. The 6 value (3.5.1) of an ice sample is the relative difference between R^ of the sample and R of a reference sample known as Standard Mean Ocean Water (SMOW) (Craig, 1961). 109 R - R 6 = S SMOW 103 R SMOW -I (3.5. 1 ) A drawback of true SMOW is the fact that no samples are available. Samples of other 018/016 isotopic standards from the U.S. National Bureau of Standards have been distributed in the past by the International Atomic Energy Agency, Vienna. In September 1976, at Vienna, the Consultants' Meeting on Stable Isotope Standards and Intercalibration in Hydrology and Geochemistry set up a standard sample called Vienna SMOW. The difference between true SMOW (Craig, 1961) and Vienna SMOW is -0.05°/oo- This difference, is not significant for my glaciological applications. The reports of isotopic data for the Icefield Ranges (West and Krouse, 1972; West, unpublished; Ahern, unpublished [b]) predate this change. Dansgaard (1969) reported a reproducibility of ±0.12°/oo (per mil) for routine mass spectrometer measurements of 6. Ahern (unpublished [b], p. 158) reported ±0.03°/Oo for samples from the Steele Glacier. Dansgaard's laboratory in Copenhagen also now achieves this level. This is adequate for work on glaciers, since 6 may vary by several parts°/00 to tens of parts°/0o for samples from any one glacier. 1 10 3.5.2 FACTORS AFFECTING DEL The nonzero value of DEL (6) for an ice sample from a glacier is the result of a long series of processes in the hydrological cycle since the water left the well-mixed ocean (where 6 is close to zero). At 0°C, the vapour pressures of the three major isotopic forms of water have the approximate ratios (e.g. Dansgaard, 1964) H O16 : H O18 : HDO = 1.000 : 0.989 : 0.904 2 2 (3.5.2) and the differences increase with decreasing temperature. The resulting differences in volatility lead to temperature dependent isotopic fractionation in evaporation and condensation processes. Under fast evaporation or condensation conditions (i_.e. equilibrium conditions do not exist between the vapour and the liquid phases) the fractionation factor (the ratio of the concentrations of a particular isotopic species in the two phases) is complicated. The process which tends to control the 6 values in glacier precipitation is the condensation of droplets from cloud vapour; fortunately, this can, in most cases, be adequately modelled by a Rayleigh condensation process, .i.e. a slow condensation (quasi-equilibrium of the vapour and liquid phases) with immediate removal of the condensate (Dansgaard, 1964). For slow condensation or evaporation, the fractionation factor is just a ratio of the vapour pressures of the different isotopic species of water at the ambient temperature. These ratios are well known above 0°C from laboratory measurements, and have been extrapolated to -20°C (Dansgaard, 1964, Table 1) using a formula of Zhavoronkov and others, (1955). When the 111 Rayleigh condensation model is applicable, the 6 value of the precipitation is primarily an indication of the condensation temperature. In general terms, 6 values tend to decrease with altitude and with latitude, and, at any one site, to be more negative in winter than in summer. A continental effect is also sometimes observed (Dansgaard, 1964); the 6 values of precipitation at constant condensation tejnperature may decrease with distance from the ocean due to depletion of heavy isotopes from the storm systems through precipitation, and due to dilution with isotopically light vapour from freshwater sources. Factors other than temperature can influence the 6 value of precipitation. Dansgaard (1964) discussed the effects of evaporation from falling droplets, isotopic exchange between drops and air through which they fall, non-equilibrium phase changes, and variations in the frequency and isotopic composition of the source storms. While these processes can cause variations in 6 from storm to storm, their effect on the average summer or winter 6 value tends to be constant from year to year. 6(018/016) and 6(D/H) are linearly related under Rayleigh conditions (Dansgaard, 1964); simultaneous measurement of 6(018/016) and 6(D/H) can be used to reveal the presence of non-equilibrium condensation. • Several processes in snow and firn tend to homogenize the isotopic distribution, obliterating differences between individual storms, an"d sometimes the summer to winter difference. In regions with summer melting, recrystallization in the presence of percolating meltwater can bring the whole vertical column of snowpack to the average 6-value (Dansgaard 1 12 and others, 1973). However, the effects of meltwater are not always simple; Ahern (unpublished [a], p. 109) found that percolating meltwater could enhance rather than smooth the isotopic variations in a cold snowpack with variable density. In regions with no summer melting, some smoothing of the isotopic profile occurs due to sublimation and recrystallization in the firn. Vertical vapour motion is most pronounced in regions of high vertical gradients of temperature in the firn (e.£. due to large seasonal temperature variations), or in stormy regions with frequent barometric pressure changes (Dansgaard, and  others, 1973). Below the depth at which firn reaches a density of 550 kg nr3, the " vapour spaces in the firn are isolated. The isotopic profile is smoothed only by solid diffusion. This process is too slow to have an appreciable effect on ice in the Steele Glacier. 3.5.3 PREVIOUS ISOTOPIC STUDIES Assuming (1) that the precipitating air masses follow similar tracks with similar frequency the year around and from year to year, (2) that Rayleigh condensation occurs, (3) that surface and condensation temperatures can be simply related, (4) that the 6-temperature relationship is constant with time, and (5) that the flow pattern of the ice mass can be calculated, then 6 values in ice cores can be related to climate at the time of precipitation. This was first pointed out by Dansgaard (1954). A thorough discussion of ice core studies can be found 1 13 in Chapter 15 of Paterson (1981). The Greenland ice sheet provides the most suitable ice flow and meteorological conditions for a simple climatic interpretation of a deep ice core (Dansgaard and others, 1973). The first major drilling program was undertaken in 1956 by S.I.P.R.E. (U.S. Army Snow, Ice, and Permafrost Research Establishment, now called CRREL, Cold Regions Research and Engineering Laboratory); a 411 m core was recovered at Site 2, in northwest Greenland. This was followed by deep cores at Camp Century, Greenland in 1966 (1387 m), and at Byrd Station, Antarctica, in 1968 (2164 m). The Camp Century core has been used to derive climate variations over the past 100,000 years (Dansgaard and Johnsen, I969[a], I969[b]; Dansgaard and others, 1969, 1971). The Byrd Station core also shows long term climate variations (Epstein and others, 1970; Johnsen and others, 1972), but is more difficult to date absolutely, because the annual variations of 6 were not preserved during the ice formation process. The length of the flowline, and the time scale for this hole are in dispute (Robin, 1977). Coring programs and climatic interpretations have also been undertaken at other sites in Greenland (Dansgaard and others, 1973) , in Antarctica, including Vostok (Barkov and others, 1974, 1975, 1977), Dome C (Lorius and others, 1979), Little America V (Dansgaard and others, 1977), and Terre Adelie (Lorius and Merlivat, 1977), and at sites in the Canadian Arctic, including Meighen Ice Cap (Koerner and others, 1973; Koerner and Paterson, 1974) , Devon Ice Cap (Paterson and others, 1977; Paterson and Clarke, 1978; Fisher, 1979), and Agassiz Ice Cap, Ellesmere 1 1 4 Island (D. Fisher, personal communication). Ice cores for isotopic analysis have been obtained from the plateau at 5400 m on Mount Logan, Yukon Territory by G. Holdsworth. During periods of extensive glaciation, deep sea sediments are enriched in O18; the isotopic composition of sea water is altered because of the large volume of 018-depleted ice on land. Measurements of the isotopic composition of deep sea sediments (e.g. Hayes and others, 1976) have complemented the climatic studies of deep ice cores. The validity of the climate interpretation of ice cores has also been supported by other studies. Robin (1976) and Johnsen (1977) found that the temperature history derived from the isotopic records was compatible with the present vertical distribution of temperature in boreholes. Paterson and Clarke (1978) used the isotopically-derived temperature history as a boundary condition for a time-dependent heat flow model for the Devon Island boreholes. Picciotto and others, (i960) demonstrated the existence of a linear relationship between cloud temperature and 6 on the coast of East Antarctica, and Lorius and Delmas (1975) found a linear relationship between 6(D/H) and ten metre firn temperatures. Picciotto and others, (1968), and Lorius and  others, (1970) used the annual variation of 6 in near-surface samples to measure the recent accumulation rate at Antarctic sites. Lorius and others, (1969) measured regional variations of 6(D/H) in Antarctic precipitation. West and Krouse (1972) measured isotopic ratios at several sites in the St. Elias Mountains, Yukon Territory, including 115 Mount logan, Divide Station, and Rusty Glacier, obtaining mass balance estimates and relating isotopic composition to weather patterns. A longitudinal surface sampling of Rusty Glacier (a small surge-type glacier in the Steele Creek basin) showed a systematic increase of 6 with height in the ablation zone, demonstrating that the general isotopic pattern of the glacier was not destroyed by surging. Ahern (unpublished [b], p. 162) obtained an isotopic profile to a depth of 36 m in a borehole at x=15 km (Figure 3.2) on the Steele Glacier. This profile appeared to show periodic oscillations of wavelength 7 metres and amplitude ±1.5°/0o in 6(018/016). Isotopic studies on glaciers and ice sheets are closely related to studies of ice flow. The age of the ice must be determined, and annual layers cannot always be detected isotopically. Time scales for ice cores have been derived by Dansgaard and Johnsen (I969[a]), by Philberth and Federer (1971), and by Hammer and others (1978), using assumptions of steady state, proximity to an ice divide, and specific forms of vertical strain rate or temperature gradient. Paterson and  others (1977) measured the vertical deformation in the Devon boreholes; this allowed them to eliminate the strain rate assumptions when deriving a time scale from the flow. For boreholes at large distances from ice divides, the horizontal velocity influences the time scale calculation, and two-dimensional flow simulations, such as given by Budd and  others (1971, p. 148) for the Byrd station flowline, must be used to date the ice. 1 16 A conceptual inconsistency can arise with the use of steady state flow models; the existence of the climatic variations, which the isotopic profiles attempt to determine, could preclude the existence of steady state conditions. In addition, the ice surface elevation may vary in the absence of climatic variations. For this reason, time-dependent flow models are essential to the interpretation of some isotopic data. Jenssen (1978) investigated the effect of ice sheet elevation changes on the isotopic profile in the Vostok core. He calculated ice trajectories and simulated isotopic profiles on the Antarctic ice sheet flowline from Vostok to Mirny, assuming periodic surges; the time-dependent ice surface profiles were computed by Budd and Mclnnes (1978, 1979). This is the only previous work of which I am .aware in which trajectories and isotopic distribution are calculated in a time-varying ice mass. Jenssen (1978) was interested in the effect of surging on the climatic interpretation; in this chapter, I investigate the feasibility of using isotopic information to reveal the surge history. Jenssen (1978) used a linear isotopic-elevation relation similar to equation (3.5.3) below. The details of the flowline calculations were not described. 3.5.4 DEL RELATIONS FOR THE MODEL In the computer model, I assume that: (1) all precipitation has the annual average value for the given elevation or location, (!•£_• rapid isotopic homogenization in the firn), and 1 17 (2) isotopic diffusion is negligible in the solid ice. I use two 6-precipitation models, representing opposite extremes of topographic control of precipitation. The first, which I call Model 61, is based on the assumption that the isotopic composition of snowfall is completely controlled by the surface elevation of the glacier h(x,t), and can be approximated by a linear relationship over the range of elevations considered, i.e. 6(x,t) = 60 + c h(x,t) (3.5.3) Values of the constants 60 and c consistent with the few measurements in the Steele Creek basin (West and Krouse, 1972) and on the Steele Glacier (Ahern, unpublished [b]) are 60 = -15.0 °/oo c = -0.004 m- 1 %o (3.5.4) The gradient c is close to the value of -0.005 m"1 °/oo found by Sharp and others, (1960) for c at Blue Glacier, Washington, U.S.A. It is within a factor of two of the value of -0.002 nr 1 °/00 reported by Dansgaard (1961). Jenssen (1978) used c = -0.006 irr1 °/oo in a modelling study of an Antarctic flowline. Isotopic values on ice sheets and ice caps can be described by Model 61, because the ice sheet surface height strongly influences the vertical motion of incoming air masses. The second model, called Model 62, assumes that the isotopic composition of the snowfall is completely controlled by the regional mountain topography and by distance from the storm source in the Gulf of Alaska; 6 is assumed to be a function only of position x. Approximating the 1951 Steele Glacier surface (Topographical Survey, 1969) by a straight line in the 1 18 FIGURE 3.8. Reference Surface For 6-x Function. The accumulation area of the Steele Glacier is assumed to extend (see Figure 3.7 (a)) to x=13 km (vertical broken line). The curved line is the observed ice surface elevation in 1951. Applying the 6-elevation relation (3.5.3) to the linear approximation (dashed line) to this surface, and using the constants (3.5.4) gives a linear 6-x relationship with the constants in (3.5.6). accumulation area, as shown in Figure 3.8, and applying the 6-h relationship to this line, with the constants (3.5.4), gives the x-6 relationship 6(x,t) = 60 + k x (3.5.5) with the constants 6o = —26.6 °/oo k = 0.00022 m-1 °/oo (3.5.6) Model 62 is more appropriate than Model 61 for small valley glaciers. The Steele Glacier, being a large valley glacier, probably falls between the two extremes. Because Model 61 allows the isotopic composition of snowfall to vary with both position and time, while Model 62 allows only variation with position, Model 61 can produce larger variations or discontinuities in 6(018/016). Models 61 and 62 119 can be considered to give the maximum and minimum structure respectively to the isotopic distribution within the Steele Glacier. 3.6 MODEL RESULTS: SURGE PERIOD AND TRAJECTORIES 3.6.1 INTRODUCTION The surge periodicity of the Steele Glacier is unknown. In Section 3.6.2, I show three computer simulations of the Steele Glacier using surge periods spanning the range of 50 to 150 years suggested by field observations. I used the same velocity pattern during the surge in all three cases. The period (97 years) which gave the most reasonable ice thickness profiles at all times was selected for trajectory and isotopic calculations. In Section 3.6.3, I present typical trajectories for ice particles in this model. These trajectories show periodic abrupt changes of direction and speed. 3.6.2 PERIODICALLY REPEATING STATE To obtain a periodically repeating surge cycle, I used the constants in Table 3.2 to generate the sliding velocity. This sliding pattern is shown at intervals of 0.25 years in Figure 3.9. The surge duration was two years. I used three surge periods; 47, 97, and 147 years span the range of possible periods described in Section 3.2.4. Starting from the nonsliding steady state in Figure 3.6, I ran the Steele Glacier Model 2 through fifteen surge cycles of length t4. By that time, the 120 to (a) 0.0 x. 2<t0) (km) 8.0 t, (a) 0.75 X . i(t0) (km) 18.0 t2 (a) 1.5 Xi (t0) (km) 19.0 t3 (a) 2.0 X2 (t0) (km) 26.0 t« (a) 1 ) 47. 0 2) 97. 0 c . :(t0) (m a- 1) -1000.0 3) 147. 0 c. i(t0) (m a' 1) 7500.0 f 0.0 C i (t0) (m a' 1) 15000.0 U0 (m a-1) 5000. c2 (t0) (m a" 1) • 15000.0 TABLE 3.2. Velocity pattern for Steele surge. The three values of surge period t« were used for Figures 3.10, 3.11, and 3.12 respectively. model no longer "remembered" the initial steady state condition; it repeated the same surface profile sequence to one part in 10' with each new cycle for 97 and 147 year periods, and to a few parts in 103 "with a 47 year period. The time steps in the computer model must be very small when the glacier is surging in order to maintain accuracy (see Section A1.5). Table 3.3 summarizes the numerical time step sequence used. The other numerical and physical constants had the values shown in Table 3.1. Figures 3.10 through 3.12 show the pre-surge glacier surface (solid line) at t0, and the post-surge surface (broken line) at t3 for the three surge periods of 47, 97, and 147 years respectively. The model with a 47 year surge period (figure 3.10) has a pre-surge profile (solid line) which is less than 100 m thick beyond x=20 km. The post-surge profile is less than 100 m thick beyond x=25 km. It seems unlikely that a surge of ice always less than 100 m thick could advance 12 km in two years. The profiles shown in this diagram are unrealistic. This conclusion indicates that the Steele Glacier accumulation area cannot provide sufficient mass in just 47 years to generate 121 8000 6000 o "54000 > cn c CO 2000 T Steele Glacier Model 2 Surge period 97 year* Sliding velocity at intervals of 0.25 years t=1.0 0 10 20 30 40 x (km) FIGURE 3.9. Sliding Velocity: Steele Glacier Model. The sliding velocity profile is shown at intervals of 0.25 years during the surge of two years duration. The end of each curve indicates the position of the advancing terminus (for the model with 97 year surge period.) surges which move as quickly or as far as the 1966-67 example. Either some surges must be less vigorous, or the surge period must be substantially longer than 47 years. The model with a period of 147 years (Figure 3.12) is reasonably thick at all times. It could be criticized on the basis of the exceptionally thick ice lobe below x=25 km following the surge. Radio echo sounding (e.£. Narod and Clarke, 1980) of the Steele Glacier would resolve the validity of this criticism. This model could be acceptable for the trajectory analysis, but I did not use it because the observations of Stanley (1969), Wood (1972) and 1 22 40001 | 1 | 1 1 Steele Glacier Model 8 Surge period 47 years Pre—surge profile Post-surge profile 3000 J -4-» cn 2000 1000 I I I I I J 0 10 20 30 40 50 x (km) FIGURE 3.10. Pre- And Post-surge Profiles: 47 Year Period. The solid curve shows the glacier as the surge begins. The broken curve shows the surface elevation as the surge ends two years later. The sliding function is given in Table 3.2 and Figure 3.9. The ice depths of the lower glacier are unreasonably thin; the Steele has insufficient accumulation to surge as vigorously as the 1966-67 event as frequently as every 47 years. Sharp (1947), described in Section 3.2.4, indicate that 150 years is an upper limit for the time between surges. The surge period of 97 years is the most acceptable of the three. There is little change in the ice surface elevation above x=8 km. Between 8 km and 20 km, the surface drops by up to 110 m, while below 20 km, the surface rises during the surge by up to 120 m. This pattern agrees well with the zones described by Stanley (1969). I used this model for the trajectory and 123 4000 3000 CD X 2000 1000 T T T Steele Glacier Model 2 Surge period 97 years Pre-surge profile Post—surge profile - -0 10 20 (km) 30 40 50 FIGURE 3.11. Pre- And Post-surge Profiles: 97 Year Period. The solid curve shows the glacier as the surge begins. The broken curve shows the surface elevation as the surge ends two years later. The sliding function is given in Table 3.2 and Figure 3.9. This model is used in Section 3.7. isotopic calculations. Figure 3.13 shows an orthographic view of the ice thickness h(x,t) during one complete surge cycle for the case with a 97 year period. The rapid decrease in surface elevation between x=8 km and x=20 km and between time zero and two years is hidden by the pre-surge profile, but the rapid surface rise of the lower glacier can be seen. The subsequent slow ablation of the terminus region, and the buildup and advance of the ice in the source region are evident. Three steps or waves can be seen in the ablation zone. One wave forms during each surge. The ice 124 4000 3000 JC Ol X 2000 1000 (km) FIGURE 3.12. Pre- And Post-surge Profiles: 147 Year Period. The solid curve shows the glacier as the surge begins. The broken curve shows the surface elevation as the surge ends two years later. The sliding function is given in Table 3.2 and Figure 3.9. This model is physically reasonable, but 147 years is an upper limit on the surge period based on the observations of Stanley (1969), Wood (1972), and Sharp (1947). Interval Time (a) At (a) t0-t, 0.0 - 0.75 0.01 1 ti-t2 0.75 - 1.5 0.01 t2-t3 1.5 - 2.0 0.01 t 3 ~ t a 2.0 - 7.0 0.1 7.0 - 97.0 1.0 TABLE 3.3. Numerical time steps for Steele surge. 125 FIGURE 3.13. Steele Glacier Thickness: One Surge Cycle. Ice thickness h(x,t) in orthographic view from a point 45° above the x axis, and 45° around the thickness axis from the time axis. The model used the constants in Tables 3.1, 3.2, and 3.3. The surge started at time=0.0 and ended at 2 years. The surge period was 97 years. The transverse lines indicate 1 km intervals, and the longitudinal lines are spaced at intervals of 5 years. thickens as a result of being forced into the converging channel (see Figure 3.7 (b)). The observed profile of the Steele Glacier in 1951 (Figure 3.3 (c)) has long surface undulations, but these could result from bed topography. Repeated surface mapping and a radio-echo depth survey would resolve this question. The undulations in Figure 3.13 may be an artifice of the model due to the poorly-known spatial variation of the surge velocity, and the inclusion of the Hodgson Glacier through the width function 126 1 1 1 1 Steele Glacier Model 2 x (km) FIGURE 3.14. Ice Trajectories .For 97 Year Surge Period. Trajectories for 5 particles starting on the ice surface as a surge begins. The dashed profile is the ice thickness at t0. The dashed-dotted profile is the post-surge ice thickness at t3=2 years. The arrows on the trajectories indicate time intervals of 48.5 years, !•£. the midpoint and end of each 97 year surge cycle. Beyond x=22 km, the ice is virtually stagnant between surges, and the arrows for mid-period and for the end of the period are superimposed. shown in Figure 3.3 (b). 3.6.3 ICE TRAJECTORIES Figure 3.14 shows the trajectories of particles starting on the ice surface at t0 as a surge begins. The arrows on the trajectories show the particle locations each 48.5 years, i.e. at the midpoint and at the termination of each 97 year surge cycle. The pre-surge (dotted curve) and the post-surge 1 27 (dotted-dashed curve) ice thickness are also shown. The ice beyond x=22 km is virtually stagnant between surges, and the arrows indicating the end of each surge cycle are superimposed on the mid-cycle arrow. Between x=8 km and x=20 km, the trajectories descend abruptly during the surge; the high longitudinal strain rate resulting from the gradient of the sliding velocity in Figure 3.9 causes a general thinning in this zone. The trajectories rise during the surge in the region below x=20 km because the velocity gradient is small and the ice thickens as it flows into a channel of decreasing width. The endpoint of each trajectory indicates the position of ice surface at the time the ice particle came to the surface. 3.7 MODEL RESULTS: DISTRIBUTION OF ISOTOPES 3.7.1 INTRODUCTION In this section, I show the isotopic distributions found by calculating ice trajectories for the surge model with 97 year period (Figures 3.11 and 3.13) and the precipitation-6 models 61 and 62. I have chosen to look at the isotope distribution at four times during the cycle: (1) when the surge begins at t=0.0, (2) at mid-surge, t=1.0 years, (3) at t=1u.O years, when it is again possible to walk over most of the glacier surface, and (4) at t=50.0 years, near the midpoint of the quiescent period. Since the velocity field is smoothly-varying even during a surge (Stanley, 1969), discontinuities cannot be introduced into the isotope distribution within the ice by the rapid flow alone; 1 28 this would require shear fracturing and dislocation on a large scale. Discontinuities or sharp changes in the gradient of 6 can be introduced only by abrupt changes in the 6-value of precipitation falling onto a given element of ice at the glacier surface. When 6 is considered to be a function of position and surface elevation (Section 3.5.2), the 6-value of new precipitation falling on an element of ice at the surface can change abruptly only if (1) the surface elevation changes abruptly, or (2) the surface element is rapidly moved to a new position at which snowfall has a different 6-value. Thus, isotopic discontinuities can be formed only in ice which (1) is at the ice-air interface at the moment a surge begins, (2) is in the accumulation region, and (3) participates in the surge through forward motion or a decrease in ice thickness. The region above x=8 km on the Steele Glacier did not take part in the 1966-67 surge (Stanley, 1969); this feature is incorporated into the computer model. Since all the ice downstream from x=25, and the deep ice throughout the glacier comes from the region upstream from x=8 km, there cannot be any discontinuities or abrupt variations in 6 in this ice. (The surges do, of course, alter the positions and distort the gradients of the isodel lines, but all quantities remain slowly-varying; this would be a difficult and subtle matter to interpret in a flow regime as complicated as that of a surging valley glacier). In Section 3.7.2, I show longitudinal vertical sections of the Steele Glacier between x=5 km and x=25 km at the four times metioned above using the height-dependent precipitation model 61. I present detailed surface profiles of 6 at the same 129 four times in Section 3.7.3. In Section 3.7.4, I show sections and profiles at the same times using the the precipitation model 62. 3.7.2 MODEL 61: LONGITUDINAL 6 SECTIONS Would a well-placed borehole, or series of boreholes in the Steele Glacier recover convincing isotop'ic evidence of past surges? In Figure 3.15, I show the expected isotopic distributions in a longitudinal section at the four times 0, 1, 10, and 50 years measured from the initiation of a surge. The region from x=5 km to x=25 km contains all the ice which is capable of containing discontinuities in 6 and its gradient (see Section 3.7.1). A 6-value was assigned to each point on a mesh (indicated by dots) with a horizontal spacing of 500 m, and a vertical spacing of 5 m. The 6-value was calculated by applying the isotopic model 61 to the starting coordinates of the ice Overleaf: FIGURE 3.15. Longitudinal 6 sections: Model 61. The 6-values within the Steele Glacier are contoured at intervals of 0.1°/oo« Tne dots show the 5x500 m mesh at which 6 was evaluated using Model 61. (Section 3.5.4). The surge model (Figures 3.11 and 3.13) had a period of 97 years. Each line of steep 6 gradient which intersects the ice surface and is attenuated with distance upstream and into the ice outlines the relict ice-air interface at the start of a previous surge. a) t=0 year; pre-surge profile. b) t=1 year; mid-surge. c) t=l0 years; post-surge profile. Foot travel would again be possible. d) t=50 years; this is approximately the midpoint of the quiescent period. In Model 61, precipitation is a linear function of ice surface elevation; this tends to over-estimate the magnitude of isotopic discontinuities due to surging. 130 -C cn "5 100 10 15 x (km) 131 trajectories which passed through the meshpoints at the time of the cross-section. The Model 61 (Section 3.5.4) assumes that the isotopic ratio of precipitation is a linear function of the ice surface elevation. I calculated the trajectories using the velocity evaluated on a mesh (Figure 2.8) with DZ = 15 m Ax = 500 m (3.7.1) (The velocity mesh can be coarser than the isotopic mesh because there are no discontinuities in the velocity field). The 6-values for the isotopic mesh were contoured automatically with an interval of 0.1°/oo« The only features in the otherwise smooth isotopic distributions are curved lines which intersect the glacier surface and dip upstream into the ice. Crossing one of these lines from above to below, 6 decreases by an amount which varies from 0.8°/oo °r more at the surface near the firn line (X=13 km) to near zero at depth. Each of these lines reveals the present location of ice which was at the surface in the accumulation region at the time a previous surge began. The line generated by the surge cycle shown in Figure 3.15 is not visible unti& (d) at 50 years. From zero to ten years, the discontinuity is too close to the ice surface and to the edge of the 5 m mesh to be displayed in the contour plots. I discuss the ice surface further in the next section. To understand the creation and evolution of one of these relict pre-surge surface lines, consider the ice at x=12 km where the -24°/0o contour intersects the ice surface in Figure 3.15 (a). During the surge, this element of ice moves rapidly downstream and to a lower elevation. In (b), it is at 1 32 x=8 km. The combined effect of bed slope and ice thinning has lowered this ice element by approximately 100 m, so, using the value of c from (3.5.4), the snow falling on this element of ice has a 6-value of -23.6°/oo» At the end of the surge, this ice element is further downstream and still lower; the 6-value of new snow on this element is near -23°/oo. A 6 discontinuity of the order of 1°/oo has been created. As snow accumulates during the quiescent phase, the discontinuity is buried; it is at a depth of 20 m in (d). The magnitude of the 6 transition across the relict surface decreases with distance up the glacier, because the sliding velocity during the surge and the amount of surface-lowering also decrease in this direction (see Figures 3.9 and 3.11). After one complete surge, this line is clearly visible in Figure 3.15 (a); it intersects the ice surface at X=15 km. During the subsequent surge (b) and (c), this line is seen to move down the glacier. Since it then intersects the glacier surface in the ablation zone (at x=20 km in (c)), the downstream part, with the largest 6 contrast, is rapidly destroyed by melting (d), until, at the start of the third surge cycle (a), all that remains is a slight perturbation of the isodel lines near x=17.5 km. An isotopic profile in a single borehole would show, at best, a single abrupt decrease of about 0.8°/Oo to 0.5°/oo at the most recent or second most recent relict pre-surge surface. Because this ice is displaced by three to five km during each surge, it is not possible to see more than one discontinuity in a single borehole. A large number of boreholes (5-30) would be required to interpret the 6 pattern with reliability and 1 33 precision. This would be an expensive field project. 3.7.3 MODEL 61: SURFACE 6 PROFILES A cheap alternative to borehole drilling is detailed surface sampling in the ablation zone. In Figure 3.16, I show the isotopic 6-values along the glacier surface for the same models and the same times as for the cross-sections in Section 3.7.2. The 6-values were poorly determined by the computer model where the profiles are broken. When an ice particle is very near the surface, and its velocity is nearly parallel to the surface, small truncation errors in the normal velocity component (see Section 2.5.3) can cause the trajectory to intersect the ice surface at a large distance from the correct position; this gives an incorrect value for 6. I estimate the error in 6 to be ±0.1°/Oo in this region. Elsewhere, the trajectories intersect the ice surface at larger angles, giving reliable intersection positions and 6-values. At any given time, isotopic discontinuities due to three previous surges can be seen, diminishing in amplitude from 0.8°/oo to 0.2°/oo with age. As in the previous section, it is possible to follow the motion of any relict pre-surge surface, this time seeing only where it intersects ' the present ice surface. The four isotopic peaks and discontinuities labelled t=0, t=1, t=lO, and t=50 in Figure 3.16 show the position and the amplitude change across a relict surface during the cycle in which it forms. The discontinuities labelled simply 1, 10, and 50 show how the pattern moves downstream with the ice in the 134 -23 to O \ GO -27 t=50 Steele Glacier Model 2 Surface isotopic profiles Precipitation Model <J 1 Surge ' Surge eriod 97 years uration 2 years 10 15 20 25 (km) 30 FIGURE 3.16. Model 61: Surface 6 Profiles. 6 was calculated at 50 m intervals along the surface of the Steele Glacier model with 97 year surge period (Figures 3.11 and 3.13). The profiles correspond to the longitudinal sections in Figure 3.15. The discontinuities in 6 occur where the relict pre-surge surfaces in Figure 3.15 intersect the ice surface. The 6-values may be in error by ±0.1°/Oo where the lines are broken, due to small vertical velocity errors in trajectories nearly parallel to the surface. The firn line is at x=13 km. second surge (1 and 10), then how the intersection point retreats back upstream (50) as the shallow downstream portion of the relict surface is melted off. Information on either surge periodicity, or on ice displacement during previous surges could be derived from a surface isotopic sampling program if Model 61 is appropriate for 1 35 the Steele Glacier, and if the background noise in 6(01B/016) is small. Some shallow coring could help to control the uncertainties in surface intersection points of the relict surfaces; these points may be difficult to locate precisely due to the small angles between ice surfaces and relict pre-surge surfaces. • 3.7.4 MODEL 62: SECTIONS AND SURFACE PROFILES Model 61 assumes that the 6-value of snowfall is determined by the surface height of the ice. In fact, cloud height (and thus 6) is probably strongly influenced by the height of the mountains in the vicinity of the Steele Glacier, and is thus also a function of location x. Figure 3.17 shows isotopic cross-sections at times t=0 and t=lO years using the 6-precipitation Model 62 (Section 3.5.4). The surge period, surface height, velocity field, and trajectories are identical to those used for Figure 3.15; only the relationship between 6 and the point of origin of the snow has been changed. Compared to Figure 3.15 (a) and (c), the amplitude of the 6 change across each relict pre-surge surface is reduced by a factor of approximately one half, and only the most recent pre-surge surface can be seen plainly. Figure 3.18 shows the surface 6 profiles at the same for times as the profiles in Figure 3.16, using the same flow models, but isotope model 62. The same discontinuities are visible in both figures, but the amplitudes are smaller by a factor of two to three using Model 62. During a surge, ice 1 36 5 10 15 20 25 x (km) FIGURE 3.17. Model 62: Longitudinal 6 Sections. The isotopic content of precipitation is assumed to depend only on location x. The surge period, surface height, and trajectories are identical to those used for Figure 3.15 (a) and (c). The amplitude of the 6 discontinuities is reduced by a factor of 1/3 to 1/2 by the change in precipitation-6 model. (a) t=0; pre-surge cross-section (b) t=lO years; 8 years after the end of a surge. particles on the surface a few kilometres above the firn line both (1) move downstream approximately 3 km, losing roughly 150-200 m of elevation due to the slope of the glacier bed, and (2) are lowered a further 100-120 m by the thinning of the glacier. Both these factors are used to calculate 6 with 137 -23 \ oo -25 -27 T Steele Glacier Model 2 Surface isotopic profiles Precipitation Model 0*2 Surge period 97 years Surge duration 2 years _L 10 15 20 x (km) 25 30 FIGURE 3.18. Model 62: Surface 6 Profiles. Isotopic variations expected on the surface along the central flowline of the Steele Glacier if the 6-value of snow depends only on x. The flow model is identical to that used for Figure 3.16. The amplitude of the 6 discontinuities is reduced by approximately 1/2 from the amplitudes in Figure 3.16. Model 61. With Model 62, the variation of 6 with x is based on the elevation drop of an average slope (see Figure 3.8), but 6 is considered to be independent of the second factor; hence the 6 contrast at the buried pre-surge surfaces is smaller by a factor given, approximately, by the ratio of the height changes considered relevant in the two cases. Since the Steele is a large glacier, its surface height may influence the flow of local air masses. The amplitude of 6 discontinuities on the 138 Steele should fall between the limits set by models 61 and 62. 3.7.5 ARE THE PREDICTED EFFECTS OBSERVABLE? The computational results in Figures 3.15 through 3.18 predict that isotopic features with amplitudes in the range 0.2°/oo to 0.8°/oo should outline several relict pre-surge surfaces now buried in the ice. This amplitude is an order of magnitude larger than the measurement repeatability error of mass spectrometers (Ahern, unpublished [b], p. 158). The principle obstacle to a successful match of observed 6 profiles and numerical predictions is the unknown level of natural background noise in the isotopic distribution. How large are the isotopic effects of (1) meltwater percolat ion, (2) wind scour and wind-blown deposition of snow, (3) accumulation from avalanches, (4) annual and year-to-year variations of precipitation and temperature, (5) crevassing and shear displacement of seracs during surges, and (6) surging tributaries? The effect of these processes can be evaluated only after systematic data collection on and around the Steele Glac ier. A crude estimate of the background noise level can be obtained from the isotopic profile to a depth of 35 m at X=15 km on the Steele Glacier by Ahern (unpublished [b], p. 162) in 1975, nine years after the start of the 1966-67 surge. Ahern measured 6-values for water samples obtained at intervals of approximately two metres by periodically bailing a hole made by a hot point drill. The data points varied from -29.8°/00 to 139 -26.5°/oo' From the surface to 10 metres depth, the points fall close to a line with slope -0.1°/oo m~ 1 . Below 10 metres, Ahern fit the data with a sawtooth wave with amplitude ±1.5°/00 and wavelength 7 metres, and a small amount of scatter. Ahern (unpublished [b], p. 165) suggested that the wave pattern resulted from periodic surging; the computed longitudinal section (Figure 3.15 (c)) at the time and location of this borehole does not support this interpretation. I also calculated the isotopic distribution (not shown) using a tighter mesh (2x100 m) to look for fine structure not found in Figure 3.15 (c); none was found. This same simulation found that the approximate thickness of annual layers in the vicinity of the borehole was 0.5 m. The wavelength seen by Ahern is an order of magnitude too large to be an annual effect, and an order of magnitude too small to be a surge period effect. With the present limited understanding of the isotopes on the Steele Glacier, I can only say that the variations seen by Ahern are a source of noise which could easily obscure the presence of buried pre-surge surfaces. If Ahern's observations can be explained by a simple physical mechanism which allows a prediction of their spatial distribution, amplitude, and phase, then it might be possible to recover the signals due to surges by data filtering methods. 140 3.7.6 CONCLUSIONS I have carried out a computer simulation of the flow of the Steele Glacier using a simplified model based on observations of the 1966-67 surge, and on limited surveys and mass balance estimates extending back to the late 1930's. The numerical model is accurate, but due to uncertainties in the data (mass balance, bed topography, flow law constants, shape factor, ice temperature, surge period and surge velocity), I do not expect the model profiles to give more than qualitative information; one figure accuracy in ice depth is the most that can be claimed (no more should be claimed of any simulations of complicated glacier flow). By comparing surface profiles for a suite of models with differing surge periods, I chose a surge period of 97 years. My calculations of ice trajectories and isotope distributions in this model indicate that it is possible to locate the buried pre-surge surface ice from the accumulation area for the previous one to three surges, by drilling a series of boreholes to a depth of 50 to 100 m in the region from X=10 to x=20 km. Detailed surface sampling along the centreline can also reveal surge-created isotopic discontinuities. If the background isotopic noise is large everywhere on the glacier, then the signal due to surges may be overwhelmed; Ahern's (unpublished [b], p. 162) data show a high noise level. I conclude that the isotopic signal due to surges of the Steele Glacier would be difficult to measure without a much better understanding of the precipitation-6 relationship and the causes 141 of noise in the isotopic record. The technique may be more feasible on glaciers for which the surge reservoir region (Stanley, 1969) and the accumulation region have a larger overlap. 1 42 CHAPTER 4: WAVE OGIVES 4.1. INTRODUCTION 4.1.1 DESCRIPTION OF OGIVE SYSTEMS Ogives are transverse surface features which form at the rate of one per year at icefalls on some alpine glaciers. Ogives travel downglacier at the surface velocity of the ice, so that the wavelength of an ogive pattern is the distance ice flows in one year. There are two related types of ogives; topographic waves, called "wave ogives", and pairs of alternating light and coloured bands, called "Forbes bands". There are some types of bands on glaciers which do not repeat annually, and are not associated with icefalls. These bands are not considered to be true ogives (e.g. Lliboutry, 1965, p. 386). Ogives occur in many parts of the world e.g. the Alps (Forbes, 1845), Norway (King and Lewis, 1961), Iceland (Ives and King, 1954), Greenland (Atherton, 1963), the Canadian Rockies (Sherzer, 1907, p. 50), the Andes (Lliboutry, 1957; I958[a]), the Karakoram (Yafeng and Wenying, 1980), the Himalayas (Haefeli, 1957) and Alaska (Leighton, 1951). Typical wave ogives may have a crest-to-trough amplitude of 10 metres right below the icefall. The amplitude usually decays with distance travelled down the glacier, so that often only 10 to 20 waves are seen. Some wave ogive trains persist, however, for many more years, e.g. Trimble Glacier North Branch, Alaska 1 43 Range (frontispiece, p. 2). Forbes bands were first described by Forbes (1845, p. 162) on the Mer de Glace, although Agassiz (1840, p. 121) may have seen them earlier, in 1833. In French nomenclature, Forbes bands are also called "chevrons", because, due to differential flow, they become convex downglacier. The name "bandes brunes" (e.g.. Lliboutry, 1965, p. 338) has also been used. It probably refers to the colour of morainic material often found on the dark bands. The bands often become more visible with distance down the glacier from the icefall. King and Lewis (1961) who worked on Odinsbreen icefall at Austerdalsbreen, an outlet glacier of Jostedalsbre in western Norway, gave a complete description of the origins of the coloured bands. In their view, crevasses near the top of the Odinsbreen icefall collect dust in summer, and snow in winter. Years later, when these closed and compressed crevasses are below the icefall, they are seen as narrow structural bands (1 - 100 cm thick) of dirty and bubbly ice respectively, extending to a large depth in the glacier. The Forbes bands are the result of variations in the numbers of these narrow structural bands per unit distance down the glacier. Leighton (1951), Lliboutry (1957), and Fisher (1951; 1962) also discussed the likelihood that the structure of ogives extended deep into the ice. On the Mer de Glace, Vallon (unpublished), Reynaud (1979), and Lliboutry and Reynaud (1981) indicate that the colour and structural differences extend to a depth of several hundred metres in the ice immediately downstream of the icefall Seracs du Geant. However, Lliboutry d958[a]; 1965) reported 1 44 that in some circumstances the colouration appeared to be purely surficial rather than continuing into the body of the glacier. It has long been known that there is some connection between wave ogives and Forbes bands. Tyndall (1874, p. 131) and King and Lewis (1961) associated the dark bands with the troughs of the wave ogives, and the light bands with the crests. Atherton (1963) and Elliston (1957), who worked in a variety of different climatic regimes, associated the dark bands with the leading slopes of the waves. Since the earliest observations, there has been a tradition of controversy in the literature concerning the origin of the layered structure in ogives. Agassiz (1840, p. 40) thought that all layering and foliation in glaciers was sedimentary in origin, whereas Forbes (1845) realized that this was not true for the bubbly ice and dirt layers of ogives. The controversy persisted into this century. Vareschi (1942), (and reported by Godwin, 1949), using a careful pollen analysis on Grosser Aletschgletscher, attempted to correlate ogives with annual layering in the accumulation zone, but King and Lewis (1961) were able to trace the ogives back to a steep avalanching icefall. Fisher (1947) suggested that ogives in northern climates were sedimentary in origin, and proposed a category called "Alaskan bands". This interpretation is probably incorrect, and Alaskan ogives are similar to ogives elsewhere. I propose in this chapter to consider only wave ogives, and these can be modelled on the large scale using mass conservation. Therefore, the controversy about the small scale 145 processes of crevassing and dirt accumulation does not affect my conclusions. 4.1.2 THEORIES OF WAVE OGIVE FORMATION Early theories of ogive wave formation (Forbes, 1845; Streiff-Becker, 1952; Haefeli, I95l[a]; 1951[b]; 1957) favoured a seasonally varying longitudinal stress at the foot of the icefall; these variations were assumed to be caused by seasonal changes in the sliding velocity. In fact, in early terminology, wave ogives were called "pressure waves", an unfortunate choice of words that may have biased future thinking on ogive origins. The pressure mechanism is still a topic of research. Williams (1979) [abstract only] reported a theoretical derivation of wave trains on glaciers as a second order creep effect. Pressure does appear to be the cause of some wave trains on cold glaciers, in particular on Meserve Glacier in Wright Valley, Antarctica (Holdsworth, 1969; Hughes, 1971, 1975). The waves on Meserve Glacier are associated with the glacier terminus rather than with an icefall, so they are not ogives in the classical sense (Lliboutry, 1965, p. 386). From observations on Austerdalsbreen, however, Nye (I959[a]; I959[b]) calculated the distribution of stress and found that the ogives there could not be explained by pressure. Nye (1958[b]) proposed another mechanism for creating wave ogives. He showed that annual waves should be expected below an icefall due to the annually periodic nature of the seasonal mass balance, and the large plastic deformations taking place in the 146 icefall, even when the velocity was independent of time. Vallon (unpublished, p. 51) applied Nye's theory to the generation of ogives by les Seracs du Geant on the Mer de Glace. Martin (1977) showed that this mechanism could also generate kinematic waves at icefalls in response to fluctuations of climate. The Nye theory is the basis of the developments reported in this chapter. Some authors (Atherton, 1961; Elliston, 1957; Ives and King, 1954; King and Ives, 1956) reported multiple systems of ogives with more than one wave or pair of bands per year. Sharp (1960) reported that the ogive bands on the Blue Glacier, Washington State, USA, were not annual. He attributed the periodicity to the regular passage of serac blocks over the icefall, rather than to annual balance variations. Waves due to serac blocks are seen on the Mer de Glace below Seracs du Geant (Tyndall, 1874, p. 180), and Vallon (unpublished, p.80) found four new waves in a six month period. Washburn (1935) and Fisher (1947) also suggested this mechanism for forming waves. There may be a range of periodic sources or a combination of sources which generate waves, but I will consider only the annual ablation-stretching mechanism. 4.1.3 DISAPPEARANCE OF THE WAVES The wave crests are free of snow earlier in the melt season (e.£. North Trimble Glacier, frontispiece, p. 2; Gilkey Glacier, Post and LaChapelle, 1971, Plate 66, p. 56).and tend to receive less shadowing than the troughs. Nye (I958[*b]) reported 30% 147 higher ablation on the crests than in the troughs at Austerdalsbreen. Kamb (1964) favoured this mechanism of wave decay. It appears likely, however, that, for some glaciers, there is enough dirt or meltwater in the troughs to cause the ablation there to be just as high as on the crests, due to lowered albedo and increased absorption of solar radiation. A compressive longitudinal strain rate as the ice slows and thickens below the icefall can lead to amplification of the waves (Nye, I958[b]). Glen (1958) pointed out that the associated longitudinal stress could effectively soften the ice in the region downstream from an icefall, allowing the waves to decay by flow under their own weight. The presence of the waves themselves also causes a perturbation of the stress field near the ice surface, and Vallon (unpublished, p. 54) suggested that, on the Mer de Glace, the waves may decay in amplitude due to viscous relaxation of the stress perturbation. However, due to the nonlinear rheology of ice, the effective viscosity increases as the waves (and the stress perturbation) decrease. Lick (1970) indicated that this effect increased the relaxation time by an order of magnitude in the case of the Vaughn Lewis Glacier. He concluded that the wave decay was due almost entirely to differential ablation. The relative importance of these processes on a particular glacier will determine how long the wave train remains visible. I do not model any of these processes in this thesis. 148 4.2 NYE'S THEORY OF WAVE OGIVES 4.2.1 OUTLINE OF THE NYE THEORY To give a concise and lucid description of Nye's wave producing mechanism, I quote from the abstract to Nye 0 958[b]): "The widely held theory that the waves are the result of pressure requires that the forward velocity of the ice U, depends both on distance x down the glacier and on the time t. The simpler case where U depends only on x is treated analytically, and it is found that, owing to the essentially periodic nature of the ablation, even this case gives waves. All elements are stretched out as they pass through the icefall, owing to the high local velocity, and they therefore present greater surface area. Those passing through the icefall in the summer therefore lose more ice by ablation than those which spend the summer in regions of lower velocity. Waves are thus produced by a combination of plastic deformation and ablation." Washburn (1935) also suggested that the annual mass balance cycle could cause wave ogives, but he did not attempt to formulate the principle mathematically. He attributed the troughs to increased summer melt made possible by an increased exposed surface area due to shattering in the icefall, whereas Nye (I958[b]) attributed the greater surface area to thinning in regions of rapid flow. 4.2.2 NYE'S ANNUALLY REPEATING STATE SOLUTION Since the wavelength of wave ogives where they become visible below an icefall rapidly becomes less than the ice thickness (due to compressive flow), the waveforms are too short to propagate as kinematic waves (Nye (I958[b]) discussed this possibility), and essentially the waves do not perturb the flow. This is consistent with the assumption that U(x) is independent 149 of time, i_.e. dU = 0 <Tt (4.2.1) The waves are simply carried forward at the velocity of the ice surface, which is treated strictly as a "conveyor belt" in this theory. Nye defined an Annually Repeating State (A.R.S.) by h(x,t) = h(x,t+1) (4.2.2) for time measured in years, and used the continuity equation (1.3.5) repeated here as (4.2.3) d_h(x,t) + 1 d_Q(x,t) = A(x,t) dt wTx) dx (4.2.3) where h is ice thickness, W(x) is the transverse width, A(x,t) is the mass balance rate, and Q is the ice flux given by Q(x,t) = h(x,t) U(x) W(x) (4.2.4) U(x) is the forward velocity of the ice. Because the theory concerns regions of thin and rapidly sliding ice, the difference between V(x) of (1.4.38), the average velocity through a column, and u(x,h), the surface velocity from (1.4.34), is very small. U(x) may be identified with the ice surface velocity. For unit width following a section of ice moving at U(x), introducing the total derivative DQ/Dx (e.g. Malvern, 1969, p. 211) and using the assumption (4.2.1), Nye found DQ(x,t) = A(x,t) Dx (4.2.5) Nye assumed that, at Austerdalsbreen, where Odinsbreen icefall is in the ablation zone, the amount of winter snowfall was unimportant to the wave generation process, being merely a protective covering for the true glacier surface. To simplify the. analysis, he also assumed that the net annual ablation of 150 ice occurred instantaneously each year at time t0, CO A(x,t) = b(x) £j6(t-t +n) (4.2.6) where 6(t) is the Dirac delta function (e.£. Morse and Feshbach, 1953, p. 122), n is an integer, and b(x) is the net annual balance given by He then derived a recursion relation for the glacier thickness profile h(x) in an A.R.S. immediately after the ablation at t0 for ice that was at position (x-X.) one year previously. Assuming an input ice thickness at the origin x=0 (13.8 m and constant with time for Odinsbreen icefall), Nye used (4.2.8) to predict the wave pattern at Austerdalsbreen. The spatial balance b(x) and the surface velocity U(x) shown in Figure 4.1 (redrawn from Nye, I958[b]), were a fit to the observations of the Cambridge Austerdalsbreen Expedition. The agreement between the observed and the predicted waves was good, except for the decay of the waves (which the theory does not attempt to predict), and the sharpness of the troughs, which resulted from the assumption (4.2.6). The ablation season at Austerdalsbreen actually lasts about three months (Nye, I958[b]). b(x) = A(x,t) dt (4.2.7) h(x) = U(x-X) h(x-X) - b(x) "uTxT (4.2.8) 151 0. .2 A .6 T (years) .8 1.0 2.0 5.0 2000 U(x) m yr"1 1000 1000 2000 x (metres) 3000 FIGURE 4.1. Austerdalsbreen Velocity And Mass Balance. Odinsbreen icefall: curves redrawn from Nye (I958[b]). T(x) is the time in years for ice to flow from the origin to position x. 4.2.3 OGIVES ABOVE THE FIRN LINE Fisher (1962) thought that the head of an icefall that generated ogives had to be at or below the firn line. This is now known to be incorrect. Post and LaChapelle (1971, plate 68, p. 57) showed distinct wave ogives in firn on the Grand Pacific Glacier, Fairweather Range, British Columbia, and Atherton (1963) reported wave ogives in the accumulation zone of Eldridge Glacier, Alaska. Atherton thought that Nye's d958[b]) theory was inadequate above the firn line, perhaps because he thought that the instantaneous mass balance form (4.2.6) was essential to the theory. In fact, it is not, and Nye d959[a]) pointed out that a related process would produce waves above the firn line. (Atherton also thought that, by Nye's theory, the wavelength of ogives should increase going down an icefall above the firn 152 line. He must have misunderstood some aspect of the theory, because the wavelength is related to the velocity U(x) and not to the mass balance.) Above the firn line, the ratio of winter accumulation to summer melt is important to the wave generation process, and the mass balance in the vicinity of the icefall might be better expressed, to a first approximation, as the sum of the net annual mass balance x(x) and an annual variation T(t) independent of x, so that A(x,t) = x(x) + T(t) (4.2.9) This simple function will generate wave ogives above the firn line by Nye's mechanism. 4.2.4 AN UNANSWERED QUESTION The ablation-plastic deformation mechanism is capable of producing very large waves. The question which is often asked (e.g^. Post and LaChapelle, 1971, p. 57) is why large ogives are not present below many active icefalls? In particular (Nye,1958[b], p. 153), why are waves from the ablation-plastic stretching mechanism absent? In Section 4.3, I solve the continuity equation (4.2.3) in a manner that answers this question. 153 4.3 A NEW SOLUTION FOR OGIVES 4.3.1 USING METHOD OF CHARACTERISTICS In this section, I solve the continuity equation (4.2.3) with velocity U(x) independent of time (4.2.1), in a more general form. The solution includes the wave ogives caused by the mechanism Nye described. Multiplying the continuity equation (4.2.3) by U(x) and by W(x) gives d_Q(x,t) + U(x) d_Q(x,t) = A(x,t) U(x) W(x) dt dx (4.3.1) I used (4.2.1) and (4.2.4) to take all the factors inside the time derivative. Now I change the distance variable x to a . new variable T(x), the time required for ice to flow from the origin (x=0) to position x, travelling at the ice velocity U(x). x T(x) = (4.3.2) Nye also used this transformation when evaluating (4.2.8). (When the variable T is used, X. becomes unity.) Using the chain rule gives U(x) dQ(x,t) = dx dQ(x,t) = d_Q_(T,t) dx dT "die dT (4.3.3) so that (4.3.1) becomes o\Q(T,t) + d_Q(T,t) = A(T,t) U(T) W(T) dt dT (4.3.4) This equation is readily solved by the method of characteristics (e.g. Whitham, 1974, p. 19) to give 154 dQ = A(T,t) U(T) W(T) dT (4.3.5) along the characteristic curves dT = 1 dt (4.3.6) or T = t - 0 (4.3.7) These characteristics are straight lines at 45° to both axes in the T-t plane, Figure 4.2. They are the space-time trajectories FIGURE 4.2. The Characteristics In T-t Space. T(x) is the time taken by ice to flow from the origin to x, so it measures position on the glacier, and t is time. Each characteristic, representing the trajectory of an ice column,is parameterized by 0, the time the ice passed the origin T=0. of the vertical ice columns. Each characteristic is labelled by a value of 0, the time when that ice passed the origin x=0. 0 T0 T position 155 The equation (4.3.5) is a single total derivative in one variable T. It is quite similar to equation (4.2.5) used by Nye (I958[b]). It is easily integrated to give T I Q(T,T+0) - Q(T ,T +0) = I A(s,S+0) U(s) W(s) ds o o T (4.3.8) o where T0 is a reference position where the boundary condition is applied. Now I want to show that.the solution (4.3.8) contains terms of the form Q(t-T), which are waveforms travelling down the glacier at the speed of the ice, i..e. at one year's flow distance per year. I also need to show that this propagating solution has a spatial periodicity of one year's flow distance below the icefall. 4.3.2 SEPARABLE MASS BALANCE To proceed further, I will assume that the mass balance A(T,t) is separable into the summation N 1 x (T) T (t) (4.3.9) A(T,t) = ^ x.(T) T.(t) 1 = 1 This form includes, as a special case, A(T,t) = x(T) T(t) (4.3.10) where the annual cycle T(t) is weighted by the annual net balance x(T) at each position T. The mass balance (4.2.6) that Nye used for Austerdalsbreen had this form. It also includes the special case (4.2.9) which I discussed 156 in connection with waves above the firn line. In this case, an annual cycle T(t) is added to the annual net balance. It is possible to introduce travelling waveforms into the solution through propagating mass balance waves having the form when N is larger than unity in (4.3.9). In fact, the disappearance of the wave ogives (see Section 4.1.3) can be represented by such a term; zones of excessive ablation rate may propagate down the glacier so as to remain on wave crests. However, I have not included this effect in the examples I show. Any propagating waves formed between T0 and T in the examples arise from the ablation-stretching mechanism. Because (4.3.8) is linear in the mass balance, the summation in (4.3.9) will carry through all the linear operations which follow. To keep the expressions as simple as possible, I can consider the case N=1, and drop the subscripts, without loss of generality. It will be useful, subsequently, to define a function B(t) which is an integral of the temporal variation T(t) of the mass balance. A(T,t) = b(T-t) (4.3.11) t B(t) T(s) ds (4.3.12) 157 4.3.3 THE GENERALIZED VELOCITY To simplify the appearance of the equations, I will define a 'generalized velocity function' v(T) v(T) = U(T) W(T) x(T) (4.3.13) which comprises the total spatial dependence of the integrand in (4.3.8). I have called it a velocity because, of the three factors, U(T) is likely to have the largest relative changes in an icefall. 4.3.4 THE UPSTREAM BOUNDARY CONDITION In this section, I derive an expression for the input flux at the upstream boundary T=T0. By substituting the mass balance (4.3.9) into the flux continuity equation (4.3.4) evaluated at the boundary T0, I get dQ(T ,t) = v(T ) T(t) - d_Q(T ,t) dt 0 0 dT 0 (4.3.14) Integrating this from time zero to time t = T + 0 0 (4.3.15) i.e. up the vertical boundary line at T0 in the T-t plane (Figure 4.2), gives T +0 Q(T ,T +0) = Q(T ,0) + v(T )B(T +0) - ( dQ(T ,s)ds O O 0 0 0 I £~tp 0 0 0 (4.3.16) This is the input flux at the boundary T0 for the characteristic 0. It is composed of three terms. The first is a constant which may be thought of as a datum flux, the flux which " was crossing 158 the boundary T0 at time t=0. The second term is the change in flux at the boundary due to the synchronous rise and fall B(t) of the whole surface in response to the seasonal changes. The final term is a function of 0=[t-T], so it represents a waveform travelling at the speed U(x) of the ice. It gives the flux changes at the boundary due to advection of spatial variations across the boundary. 4.3.5 THE TERMS OF THE FLUX SOLUTION When I substitute the mass balance (4.3.9) into the flux solution (4.3.8) using the generalized velocity (4.3.13), I get T Q(T,T+0) - Q(T ,T +0) = I T(s+0) v(s) ds 0 0 J ip 0 (4.3.17) Using integration by parts on the right side, Q(T,T+0) - Q(T ,T +0) o o s v(s) T(s+0)ds T r S Jdv(s) \ T(r+0)dr dT KT js=T T O 0 ds (4.3.18) Evaluating the first expression on the right at its limits, and recalling the definition (4.3.12) of B(t), 1 59 Q(T,T+0) - Q(T ,T +0) o o T = v(T)[B(T+J2f) - B(T +0)3 - f dv(s)[B(s+0) - B(T +0) ]ds dT ' 0 T 0 (4.3.19) vtn - C Taking the second term in the integrand outside the integral, substituting the boundary condition (4.3.16), and setting T+0=t from (4.3.7), two terms cancel, and T +(t-T) Q(T,t) = Q(T ,0) - I bQ(T ,S) ds o I dT 0 ^0 T + v(T) B(t) - f dv(s) B(s+[t-T]) ds T /dvl dT 0 (4.3.20) If the summation is carried through from (4.3.9), T +(t-T) Q(T,t) = Q(T ,0) - I dQ(T ,s) ds o I 5T 0 ^0 T N N /* + 2 v.{T) B.(t) " X* I dv. i=l i=l ^ dT (s) B (s+[t-Tl) ds i T 0 (4. 3*. 21) To find the waves in the ice thickness profile h(T,t), divide through by U(T). To see the waves as a function of distance x rather than the travel time T, stretch the T axis by the inverse transformation to (4.3.2), i.e. 160 x(T) = U(s) ds (4.3.22) 4.3.6 PHYSICAL INTERPRETATION The equation (4.3.21) is the complete solution for the flux at all times t and positions T>T0 with the assumptions 1) velocity is independent of time (4.2.1), and 2) mass balance is separable (4.3.9). The first two terms were discussed in Section 4.3.4. The first term is the datum flux that existed at the origin at time t=0. It has no effect on the generation of waves. The second term is the advection of spatial flux variations dQ/3T across the boundary at T0. If waves exist above T0, this term will represent their propagation through the region of interest. Nye's assumption of constant velocity and ice thickness above the Austerdalsbreen icefall eliminated this effect from his analysis. I will also avoid introducing waves through the boundary condition in the examples I show. The third term represents the synchronous rise and fall of the whole glacier surface due to the changing seasons. There can be no propagating wave of the form Q(t-T) from this term. The final term contains the ogive waves. It is a function of the limits T0 and T, and of [t-T], the phase shift of the balance integral B(t). 161 T P(T ,T,[t-T]) = o dv(s) B(s+[t-T]) ds dT o (4.3.23) The dependence on [t-T] indicates that this term contains a disturbance propagating in the positive direction at the same speed as the ice. If the annual balance variation T(t) is cyclic with a period of one year, then so is its integral B(t) (4.3.12), to within a linear trend which is made zero by a suitable choice of the terms in (4.3.9), i.e. This suggests that the propagating disturbance (4.3.23) also has a strong periodic component at one year. For example, if I consider a region below the icefall where the generalized velocity is constant, then the upper integration limit of (4.3.23) can be put to oo , and substituting (4.3.24) into (4.3.23) shows that j_.e. that the propagating disturbance is a wave that repeats with a period of one year's flow distance. The wave term (4.3.23) has the physical interpretation that any incremental step change 6v in the generalized velocity at position T,, or, equivalently, an impulse of strength 6v in the velocity gradient, generates a set of waves downstream given by B(t) = B(t+1) (4.3.24) P(T ,T+1,[t-T+1]) = P(T ,T,[t-T]) o o (4.3.25) G([t-T];T ) = 6v B(T +[t-T]) (4.3.26) Nye (I958[b]) illustrated the wave generating mechanism by 162 an,example of an A.R.S. shown in Figure 4.3 (redrawn from Nye (I958[b]). The ice velocity (curve(a)) is doubled from U0 to U(x) U0. a B T c b(x) FIGURE 4.3. Double Step Icefall Model. The ice travels the distance BC in 6 months. The mass balance is applied instantaneously each year. (a) the ice velocity. (b) the average annual mass balance. (c) solid curve: ice thickness just before ablation, broken curve: immediately after ablation. U,=2U0 between B and C. The ice travels this distance in six months. The mass balance function (curve (b)) is spatially \ 163 constant, and is applied instantaneously each year at the same time. The resulting ice thickness (Figure 4.3 (c)) immediately before the ablation occurs is shown by the solid line. The ice thickness immediately after ablation is shown by the broken line. The volume elements S0 and W0 are nearly equal before ablation. S0 is on the fast section BC during the summer ablation, and W0 is not. The square waves result from the fact that the ablation removes approximately twice as much mass from the element S0 as from the elements W0 and W, on either side, because S0 has approximately twice as much surface area exposed to ablation. Downstream, the volumes S- which were in the icefall in summer are shorter than the elements Wj , and so form troughs. However, the result (4.3.26) indicates that there is an even simpler wave generating model. If the velocity merely increases or decreases, but not both, waves are still generated. This model is shown in Figure 4.4 for a decreasing velocity step from U0 to U^UQ/2 , with an instantaneous mass balance. As before, the solid curve in (c) is the ice thickness just before ablation, and the broken curve is immediately after. In this case, the ablation removes twice the volume from the column A0, immediately upstream from the velocity decrease, as from the column B0 of equal volume immediately below. Later, when both A0 and B0 have moved downstream, and are travelling at the same velocity, B0 will be higher than A0. A new discontinuity is generated in this way every year, giving the sawtooth pattern. From the definition (4.3.13) of the generalized velocity 164 U(x) U, 0 a b(x) FIGURE 4.4. Single Velocity Step Model. (a) the ice velocity. (b) the average annual mass balance. (c) solid curve: ice thickness just before ablation, broken curve: immediately after ablation. The mass balance is applied instantaneously each year, gradient, it is apparent that spatial changes in the mass balance and in the channel width contribute annual waves in the same manner as do velocity changes. However, the relative changes in these factors on actual glaciers are usually less than the relative velocity changes in an icefall. Nye (I958[b]) mentioned waves due to changes in mass balance with x. A theory was presented at a meeting of the British Glaciological Society, 165 November 1, 1957. These three factors would be expected to influence the wave amplitude on simple physical grounds. If I consider the ice in terms of vertical prisms, then the waves arise because the mass balance removes or adds a different amount to prisms immediately above and below the position x(T,) (To relate x to T, see the transformation (4.3.22)). As illustrated in Figure 4.5 (a), a change of velocity achieves this by stretching the ice prism in the downstream direction to expose a different surface area. A change in glacier width achieves this by stretching the ice prism laterally (Figure 4.5 (b).) to expose a different surface area. A change in mass balance achieves this by removing ice to a different depth in prisms presenting equal surface area (Figure 4.5(c)). Of course, a longer ablation season would generate smoother waves. Overleaf: FIGURE 4.5. Three factors generating waves. (a) a change of velocity U(x). W(x) and x(x) are constant. (b) a change of width W(x). U(x) and x(x) are constant. (c) a change of mass balance x(x). U(x) and W(x) are constant. The mass balance is assumed to be applied instantaneously for the purpose of illustration. The A.R.S. profiles are shown immediately before the ablation is applied. The shaded volumes indicate the mass about to be ablated from previously equal volumes of ice above and below the transition point x(T,). 166 167 4.3.7 THE GREEN'S FUNCTION FOR OGIVES The function (4.3.26) is the Green's function (e.g. Morse and Feshbach, 1953, p. 791) or impulse solution for wave ogives. The total resultant wave pattern (4.3.23) is the integral of the Green's function over the whole upstream region where the generalized velocity varies. The reason that annual waves are not seen on all glaciers is that all the small waves due to spatial changes in velocity, mass balance, or channel width tend to have differing phase, which makes them interfere destructively. Only on glaciers where the spatial changes are large and localized, such as in an icefall, can these waves add together constructively to give visible ogives. 4.3.8 A CONVOLUTION FORMULATION FOR OGIVES If I define a reversed cumulative balance function B_(t) by r B (t) = B(-t) r (4.3.27) and if the generalized velocity v(T) is constant above the boundary T0 and below the observation point T, the ogive term (4.3.23) takes the standard form of a simple linear convolution (with variable [t-T]), of the time-dependent term Br with the spatial velocity gradient term dv/dT. oo P(t-T) = I dv(s) B ([t-T]-s) ds J dT r (4.3.28) -oo The theory of convolutions, and algorithms to do convolutions, are widely known. For example, the velocity gradient dv/dT may 168 be treated as a smoothing filter applied to the wave term Br. If the velocity gradient is nonzero downstream from T, the truncated convolution with finite limits must be used for the exact solution. If the gradient below T is small, or is uniform for a large distance, however, the convolution (4.3.28) is a good approximation, because a small, uniform gradient has little effect on wave generation. I shall illustrate this in the next section. The only effect of a small negative velocity gradient is to cause longitudinal compression. This amplifies existing waves expressed in terms of ice thickness and has no effect on waves expressed in terms of ice flux. 4.4 SOME SIMPLE EXAMPLES 4.4.1 INTRODUCTION I shall consider two very simple types of generalized velocity changes, and use the convolution equation (4.3.28) to illustrate how destructive interference can occur, even on active icefalls, to prevent the formation of observable waves. Combinations of these simple velocity patterns can be applied to any icefall to get a rough but easy estimate of the expected ogive wave amplitude. 169 4.4.2 EXAMPLE: LINEAR VELOCITY GRADIENT Equation (4.3.26) showed that the simplest feature generating wave ogives is an impulsive velocity change from v0 to v,. A generalization of this step change is a constant velocity gradient from v0 at T0 to v, at T,, a distance I will call T, as shown in Figure 4.6 (b). The velocity gradient dv/dT in Figure 4.6 (b) is a "boxcar" function of length r. v -v T <T<T dv = _J °_ 0 1 dT T = 0 T<T or T>T o 1 (4.4.1) Some general results for this velocity gradient are immediately apparent. First, the wave amplitude will, in general, tend to decrease as the gradient decreases, i..e. as T lengthens, or as v0 approaches v,. Second, because convolution using (4.4.1) is (except for a constant factor) just a running average over a distance T, (4.3.28) must be identically zero whenever r is an integer and the mass balance integral B(t) is Overleaf: FIGURE 4.6. Ogives from a velocity gradient. T is a measure of distance downglacier, and t is time. (a) Generalized velocity. (b) Generalized velocity gradient. (c) Mass balance. (d) Normalized crest-to-trough wave amplitude as function of T, the spatial extent of the gradient. The solid triangles are the amplitude of waves in numerical solutions (Figure 4.7). 170 171 an annually repeating function with zero mean. If, for example, the mass balance (4.3.9) is a simple harmonic function T(t) = A cos(2»rt) (4.4.2) as shown in Figure 4.6 (c), then B (t) = -A_sin(2fft) r 2TT (4.4.3) Performing the convolution (4.3.28) using (4.4.1) and (4.4.3), and using the standard addition formula for cosines (e.cj. Abramowitz and Stegun, 1965, formula (4.3.17), p. 72) gives P(t-T) = - A(v -v ) 2n 1 0 sin ( ITT ) If T Sin(2ir[t-T] + ir[T +T ]) 1 o (4.4.4) The final factor is the propagating annual wave train. The crest-to-trough amplitude of this wave train is modulated by A(v -v ) sin(n-r) M(T) = ITT (4.4.5) tr which is shown in Figure 4.6 (d). A velocity gradient over an integer number of years generates no waves at all, and the amplitude of the ogive waves falls rapidly with increasing length of the gradient region between zero and one year. It is always small for lower gradients, i.e. larger T. Because of the processes which can destroy wave ogives (Section 4.1.3), waves formed by gradients longer than 6 months may, in most cases, be too small to be observable. Figure 4.7 shows the numerical solution to the continuity equation (4.3.4) for a suite of models with velocity gradient sections of varying lengths r as shown in Figure 4.6 (a), using v^Vo/4 and the mass balance (4.4.2). The constant A is 0.1ho 172 per year, where h0 is the average input ice thickness at the boundary. I used the numerical model described in Chapter 2. The numerical solutions are presented as orthographic projections of the ice flux Q(T,t), seen from a position rotated 45° up about the t axis, and 30° forward about the Q axis. In these numerical solutions, the terms in the analytical solution (4.3.20) can be identified. The first term Q(To,0)=Qo is the amplitude at the point T=0, as the ice surface passes through its average level in the middle of the accumulation season. The second term is zero, because there are no input waveforms at the boundary T=0. The annual variations on lines parallel to the time axis are the third term, the whole glacier surface going up and down with the changing seasons. This is clearest on Figure 4.7 (c). Finally, the fourth term, the wave ogives, describes the annual disturbances which propagate at a velocity of unity (one year per year). The triangular data points on Figure 4.6 (d) are the wave amplitudes from these numerical solutions. Figure 4.7 is a graphic illustration of the differences in wave amplitude due to slightly different geometrical situations. Overleaf: FIGURE 4.7. Flow past a velocity gradient: numerical solution. Orthographic projections of ice flux Q(T,t) for various gradient lengths T. T is a measure of distance downglacier, and t is time. Surface profiles are at intervals of 1.5 months. The velocity and the mass balance have the form shown in Figure 4.6. (a) T = 0.2 (b) T = 0.5 (c) T = 1.0 (d) T = 1.5 173 174 V(T) dV dT T I T FIGURE 4.8. Double Step Icefall Model. T is a measure of distance downglacier, and t is time. The mass balance is given in Figure 4.7 (c). (a) Generalized velocity. (b) Generalized velocity gradient. (c) Normalized crest-to-trough amplitude of waves in flux as a function of icefall length T. 4.4.3 EXAMPLE: DOUBLE STEP ICEFALL MODEL The second simple example (Figure 4.8) of a velocity distribution and the resulting ogives is a generalization of 175 Nye's (I958[b]) simple illustrative model previously shown in Figure 4.3 (a). This time, I will look at the amplitude of the waves as a function of T, the length of the 'icefall', as shown in Figure 4.8 (a) In this case, the velocity gradient dv/dT is two Dirac delta functions (e.g. Morse and Feshbach, 1953, p. 122) of opposite polarity, shown in Figure 4.8 (b). If Br is annually repeating, and if r is an integer, these two delta functions will contribute equal and opposite amounts to the convolution (4.3.28), i_.e. no waves are formed. For example, using the harmonic mass balance (4.4.2), the convolution (4.3.28) becomes P(t-T) = -A(v -v ) sin(nT) sin(2ir[t-T] + n[T +T ]) - 1 o 10 (4.4.6) The peak-to-trough amplitude is given by M(r) = 2A(v -v ) sin ( ITT ) — 1 o (4.4.7) which is shown in Figure 4.8 (c). Even when the velocity changes are abrupt, the wave interference from the speed-up phase and from the slowdown phase modulates the ogive amplitude, depending on the length of the icefall. 176 4.5 AUSTERDALSBREEN 4.5.1 INTRODUCTION King and Lewis (1961) attributed the Forbes bands on Austerdalsbreen to seasonal differences in dust accumulation, melting, and snow accumulation in the crevasses formed in the upper part of the Odinsbreen icefall, where the ice undergoes a longitudinal extension. This region is between x=500 metres and X=1000 metres in Figure 4.1. Since wave ogives are often associated with Forbes bands, it is interesting to see which sections of the Odinsbreen icefall are most important for forming the waves. -6000 T (years) FIGURE 4.9. Odinsbreen: Generalized Velocity Per Unit Width. The arrows indicate the ends of. the approximately linear sections in Table 4.1. 177 4.5.2 ESTIMATED WAVE GENERATION If I assume that the width variations are unimportant (Nye (I958[b]) also assumed this), then, by multiplying the curves U(x) and b(x) in Figure 4.1, and using the transformation (4.3.2), I get the generalized velocity v(T) for unit width on Odinsbre. This is shown in Figure 4.9. In rough terms, this curve can be approximated by four sections of constant gradient, as described in Table 4.1. The wave amplitude function M(T) in T T T (v -v ) M(r) 0 i 1 0 years years years m3a'2 normalized 0.5 0.9 0.4 -5500. 0.76 0.9 1 .6 0.7 2200. 0.15 1 .6 2.6 1.0 1 100. 0.0 2.6 5.0 2.4 400. 0.01 TABLE 4.1. Approximation to Odinsbreen by linear velocity sections. The endpoints are shown by arrows on Figure 4.9. Values of generalized velocity v are given per metre width. The wave amplitude factor M has been normalized using T=0 and v1-vo=5500. m3a"2. (Note that the units of generalized velocity are not m a"1). Table 4.1 is the value of M(T) from equation (4.4.5) normalized to T=0 and v1-vo=5500. m3 a"2. Although (4.4.5) is exact only for the mass balance (4.4.2), other annually varying mass balance functions would show similar rapid falloff of the wave amplitude with T. The M(T) column in Table 4.1 suggests that the largest contribution to the generation of the waves comes from the initial section of the generalized velocity as the ice accelerates to maximum speed. This is the same section that King 1 78 and Lewis (1961) identified as the region controlling the formation of the Forbes bands. 4.5.3 OGIVE SOLUTION FOR AUSTERDALSBREEN To test this idea, I have used the Odinsbreen icefall profile (Figure 4.9) to solve the continuity equation (4.3.4) with the numerical model described in Chapter 2. The upstream boundary condition was satisfied by a constant input flux. Using the values of ice thickness (h =13.8 m) and ice velocity (U =1375. ma-1) at the upstream boundary from Nye (I958[b]), the input flux per metre width was Q = h U = 18975. m3 a' 1  000 (4.5.1) (For Nye's model of Austerdalsbreen, the input flux was constant in time because the mass balance at the top of the icefall was zero.) Because the finite difference model cannot accurately represent an instantaneous mass balance of the form (4.2.6), I used a constant ablation rate for three months each year, i_.e. T(t) = 4[H(t-t ) - H(t-t -1/4)] 0 0 (4.5.2) where H(t) is the unit Heaviside step function (e.£. Morse and Feshbach, 1953, p. 123). Nye d958[b]) indicated this was Overleaf: FIGURE 4.10. Austerdalsbreen wave ogives. The numerical solution using the generalized velocity in Figure 4.9, with constant input flux at T=0, and a 3 month constant ablation season. Profiles at 1.5 month intervals. T is a measure of distance downglacier, and t is time. (a) ice flux Q(T,t). (b) ice thickness h(x,t). 180 actually a more realistic representation of the Austerdalsbreen mass balance. Figure 4.10 (a) shows the computed ice flux Q(T,t) in an orthographic view from an angle 30° forward about the Q axis, and 45° up about the t axis. Figure 4.10 (b) shows the transformation to ice thickness h(x,t). The view is from 20° back about the h axis, and 30° up about the t axis. The wave ogives are evident in both plots. The amplification of the waves due to compressive flow can be seen in Figure 4.10 (b). 150 h(T) 100 metres 50 0 2 4 6 8 T years FIGURE 4.11. Austerdalsbreen Ice Thickness. Ice thickness h(T) at the midpoint of the ablation season. T(x) is a measure of distance downglacier (the time to flow from the origin to position x). solid curve: numerical solution with 3 month ablation season. broken curve: Nye (I958[b]) with instantaneous ablation. The solid curve in Figure 4.11 shows the ice thickness profile h(T) in the middle of the ablation season. For comparison, the broken line is the wave pattern found by Nye 181 using an instantaneous ablation season. The longer ablation season smooths out the sharp troughs on the waves. To test whether the steep gradient of v(T) in the upper icefall essentially causes the wave ogives, I then used the numerical model with two modified velocity profiles. To generate the annual repeating state in Figure 4.12 (a), I brought the ice into the icefall already travelling at the high generalized velocity of -5500. m3a~2, i..e. using the dotted horizontal curve in Figure 4.9, then let the ice slow down on the standard Odinsbreen curve (solid line). The upstream boundary condition for this A.R.S. model is time-dependent, because b(x) cannot be zero at the boundary. The boundary flux is the waves generated in this model, have only 15%-20% of the amplitude of those in Figure 4.10 (a). The prediction in Table 4.1 was 15%. Next, I ran the numerical model using the standard Odinsbreen curve (solid line in Figure 4.9) up to the peak of the generalized velocity curve, but letting the ice leave the icefall and travel down Austerdalsbreen without slowing down, i_.e. using the broken horizontal curve. The boundary condition was (4.5.1). The average slope of the flux surface is much 4.5.4 FINDING THE WAVE GENERATING REGION t T(s) ds (4.5.3) 182 larger in this case, because the ice remains thin, and ablation takes a larger proportion of the mass each year, but the amplitude of the waves in this case, Figure 4.12 (b), is approximately 80% of the amplitude in Figure 4.10 (a). The prediction in Table 4.1, based on the approximating linear velocity gradient segments, was 76%. This agreement suggests that the simple estimates are quite adequate. These results also substantiate the idea that the rapid velocity increase in the upper region of Odinsbreen causes the waves at Austerdalsbreen, and the subsequent slowdown of the ice only amplifies the waves in ice thickness by compressive flow. It appears that the rapid extension, which is responsible for the crevassing controlling Forbes bands, is also responsible for the formation of wave ogives. The velocity and ablation data for Seracs du Geant in Vallon (unpublished, p. 52) suggest that the slowdown phase at Mer de Glace also contributes very little to the wave ogives on that glacier. Overleaf: FIGURE 4.12. Variations on Odinsbreen icefall. T is a measure of distance downglacier, and t is time, the vertical arrows at T=5 show the wave amplitude. (a) The ice enters the icefall already travelling at the maximum generalized velocity (dotted curve in Figure 4.9). It slows down following the solid curve. The wave amplitude is small. (b) The ice reaches peak generalized velocity and maintains it as it descends the icefall (broken curve in Figure 4.9). The wave amplitude is comparable to the amplitude in Figure 4.10 (a). OO 184 4.6 CONCLUSIONS The ablation-plastic stretching mechanism of Nye (I958[b]) has been re-examined through a solution of the continuity equation for ice flux by the method of characteristics. The total wave ogive pattern on a glacier can be written as a convolution of a spatial term, the velocity-width-mass balance gradient, with a temporal term, the time-integrated mass balance. Nye's plastic stretching is in the spatial term, and the annual mass balance is in the time term. The convolution describes their interaction. The integrated mass balance has an annual periodicity. The velocity gradient can be viewed as a filter applied to this periodic function. Wave ogives appear in the filter output (the glacier flux profile), only if the filter does not heavily attenuate the annual component. The theory predicts that factors such as the icefall length, and the magnitude and spatial extent of velocity changes in the icefall modulate the amplitude of the resulting wave ogives. The modulation factor may go to zero. Since small annual waves may go unnoticed, or be rapidly obliterated by differential ablation, this modulation effect may be why some icefalls with rapid ice velocities and large annual balance variations do not generate observable wave ogives by the ablation-stretching mechanism. Several points of physical interest can also be made: 1. Longitudinal variation of ice velocity, channel width, and mass balance all can generate annual waves in the same way. The waves due to velocity changes are usually the largest. 185 Every incremental change with x of any of these three factors generates a wave train downglacier. This annually periodic wave train is the Green's function for the total wave pattern. Waves are not observed on all glaciers, because the velocity gradients are small, and waves generated over a large spatial range tend to be out of phase and to interfere destructively. Only large and localized gradients traversed by the ice in six months or less can generate waves sufficiently coherent to form large wave ogives. Wave ogives and Forbes bands often are found together, because, while the physical processes causing them are different, they both depend on the occurrence of a short zone of rapid ice acceleration, such as the upper stretch of some icefalls. 186 LIST OF SYMBOLS SYMBOL SECTION1 MEANING A 1.4.3 temperature dependent coefficient in Glen's flow law A 1.4.3 coefficient in general stress-strain rate relationship A 2.5.2 coefficient in Weertman sliding law A 4.4.2 half amplitude of seasonal mass balance A A2.3.2 coefficient in quadratic equation for 6x A(m) A13.4 real term common to numerator and denominator of T(m) A'(x,t) A5.4 mass balance normal to glacier surface A A1.3.4 mass balance averaged over wedge terminus A A1.1.2 sub-diagonal element of M row j j A0 1.4.3 constant in Arrhenius temperature relation for A in Glen's flow law A0 4.3.6 a vertical prism of ice at lower edge of icefall during ablation A, 4.3.6 a vertical prism of ice which occupied position A0 one year previously A 2.2.2 tridiagonal matrix in iterations for 6h A(x,t) 1.3.3 mass balance normal to bed n A 2.2.2 mass balance normal to bed at mesh point j, j time step n 1The SECTION column indicates the section in the text where this particular definition of the symbol is first used. 187 A0(x) A6 mass balance for a datum (steady) state a(x,t) A6 perturbation to mass balance A0(x) a A5.2 a vector in a set whose endpoints define mass balance as a function of position everywhere on the glacier surface a A1.2 element in row j column k of matrix A a A1.2 sub-diagonal element in row j matrix A j equal to a -J + 1 , j B 1.4.3 coefficient in general stress-strain rate relat ionship B A2.3.2 coefficient in quadratic equation for 6x B0 4.3.6 a vertical prism of ice which left icefall just prior to ablation B, 4.3.6 a vertical prism of ice which occupied position B0 one year previously B 1.4.2 total body force on a deforming continuum B 1.4.2 ith component of body force B i B A1.1.2 row j main diagonal element of matrix M j B(t) 4.3.2 temporal integral of bt(t) B (t) 4.3.8 time reversal of B(t) i.e. B(-t) r B(m) A13.4 real factor common to imaginary parts of numerator and denominator of T(m) B(x) A15.3.1 mass balance for Nagata ice sheet model b 2.5.2 constant term in mass balance of Nagata ice sheet model b A15.2 constant mass balance for Nye ice sheet model b A5.2 vector in a set whose endpoints define basal melting rate over bottom surface 188 b A11 glacier bed elevation at gridpoint j j b A1.2 diagonal element a of matrix A j jj b(x) 4.2.2 total depth of accumulation in one year C 1.4.3 coefficient in general stress-strain rate relationship C A2.3.2 coefficient in quadratic equation for 6x C A1.1.2 super-diagonal element in row j matrix M j c 2.3.2 scale constant in analytical solution for terminus motion test c 3.5.4 rate of change of 6(018/016) with height c A15.3.3 streamline parameter for Nagata model c A1.2 super-diagonal element in row j matrix A j c(H) 2.3.3 rate of change of flux Q, with H (variable for Burgers' equation) c(x,t) 2.3.3 same as c(H) c.2(t) 3.3.5 velocity of transition region edge x.2(t) c.,(t) 3.3.5 velocity of transition region edge x.,(t) c,(t) 3.3.5 velocity of transition region edge x,(t) c2(t) 3.3.5 velocity of transition region edge x2(t) c0(x) A6 rate of change of flux Q with depth H in datum state c A13.4 value of c0(x) at gridpoint j °j D 4.2.2 material derivative or total derivative of a function of both space and time D 2.2.2 right side column vector containing known terms in equation for H D A1.1.2 element j of vector D j 189 D0(x) A6 rate of change of ice flux Q with surface slope in datum state D A13.4 value of D0(x) at gridpoint j °j DX A1.3.4 length of wedge terminus region T DX 2.4.2 grid interval in x direction at ij position i and height j DZ 2.4.2 grid interval in z di-rection DX0 A2.3.1 DX between meshpoints P0 and P2 i j DX, A2.3.1 DX between meshpoints P, and P3 i j dP A2.3.1 displacement vector (6x,6z) of an ice particle relative to a meshpoint P0 E A1.5.4 fractional interpolation error in ice h thickness h at midpoint of mesh interval e A1.5.4 error in dQ/dx resulting from E h e(x) 1.4.4 fractional error in ice velocity due to assumption of simple shear deformation e A11 an error in h at some meshpoint j h j F(m) A3.1 Fourier transform of a function f(x) F(Z) A3.1 Z transform of a sampled function f j F A5.2 a length vector tangent to glacier surface F A7.3 gravitational term in expression (A7.2.20) 9 F A7.3 longitudinal stress term in (A7.2.20) 1 F A7.3 shear stress-dependent term in (A7.2.20) s f 3.3.5 ratio of sliding velocity between surges to peak sliding velocity during surges 190 f A1.5.4 fractional error in flux Q due to error E h f A3.1 jth point of a sampled arbitrary function f j £(r_) 1.4.2 specific body forces at position r_ f (r) 1.4.2 ith component of .f(r) i f(P) A2.3.1 value of an arbitrary scalar function f at a point P in a vertical longitudinal section f0,...f3 A2.3.1 values of arbitrary function f(P) at meshpoints P0,...P3 G A5.2 arbitrary vector tangent to ice surface (units of mass balance) g 1.4.2 acceleration due to gravity 9.8 m s-2 H 2.3.3 gas density H 2.5.2 thickness of Nagata ice sheet at x=0 n+1 H 2.2.2 column vector with jth element h j H0(x) 1.3.3 initial condition for ice thickness H(t) 4.5.3 Heaviside step function =0 t<0 = 1 t>0 n H (z) A1.4.2 polynomial Z transform of {h |j=1,J} j H0(x) A6 ice thickness in datum (steady) state H(x,t) A6 total ice thickness H0(x)+h(x,t) h(x,t) A6 perturbation to ice thickness H0(x) h(x,t) 1.3.2 ice thickness normal to glacier bed n h 2.2.2 ice thickness at meshpoint j, time step n j 191 h A5.2 a position vector from a set whose endpoints define the ice surface h'(x,t) A5.4 ice thickness measured normal to glacier surface h(m,t) A1.5.1 Fourier transform of h(x,t) h0 2.3.2 ice thickness at x=0 for analytical solution testing terminus motion i A1.4.2 i2 = -1 J 2.2.2 number of finite difference meshpoints between bergschrund and terminus K(m) A1.5.2 a real term in the transfer function T(m) K(Z) A1.4.2 a real term in the transfer function T(Z) k 3.5.4 rate of change of 6(01B/016) with x k A1.3.1 nonzero element in last row of matrix M k A16.2 flow law constant used by Budd and Mclnnes (related to Glen's flow law through k=2A/[n+1] ) L(t) 1.3.3 length of glacier M 2.2.2 coefficient matrix in equation for H M(T) 4.4.2 amplitude of wave ogives generated by velocity changes over a distance T M(t) A5.1 mass inside a volume V of a continuum M A19 width of ith tributary of Steele Glacier i m 2.2.4 wavenumber m 2.5.2 exponent in Weertman sliding relation m 2.2.4 Nyquist wavenumber; highest wavenumber N which can be detected on a discrete grid m A7.2 ice surface downslope unit tangent vector N 2.2.2 total number of time steps At 192 n 1.4.3 exponent in Glen's flow law for ice n 1.4.2 unit vector normal to a surface n 1.4.2 jth component of n j 0[x] 1.4.4 f=0[x] means f is of the order of x It is used in the approximate sense: |10-1x j <|f|<|1 Ox| Technically it means lim (f) is bounded x->0 P0 2.4.2 initial position of an ice particle P0,...P3 A2.3.1 four meshpoints at the vertices of a quadrilateral mesh cell in vertical longitudinal section (used for 4-point interpolation scheme) P(t) 2.4.2 trajectory of an ice particle P(T0,T,[t-T]) 4.3.6 term containing wave ogives in solution of continuity equation p 1.4.3 pressure p 2.2.2 constant At/(2AxW ) at gridpoint j j j Q 1.4.3 activation energy for creep Q(x,t) 1.3.3 ice flux through a vertical transverse cross-section n Q 2.2.2 ice flux at time step n, midway j+1/2 between gridpoints j and j+1 n n Q' A11 estimate of Q when h has error e j+1/2 j+1/2 j h Q0(t) 2.2.1 ice flux at x=0 (boundary condition) Q0(x) A6 ice flux in datum (steady) state Q, (H) 2.3.3 flux term depending on H (derivation of Burgers' equation) A5.2 ice flux through surface S, 193 0.2 A5.2 ice flux through surface S2 Q (x) A16.4 balance flux for steady glacier bal Q A19 discharge of ith tributary of i Steele Glacier q(x,t) A6 perturbation to datum flux Q0(x) R 1.4.3 gas constant 8.314 J °R-1 mol"1 R 2.3.3 Reynold's number for Burgers' equation R 3.5.1 isotopic concentration ratio of a sample S R 3.5.1 isotopic concentration ratio of SMOW Standard Mean Ocean Water n+1 n+1 r 2.2.2 residual in iterative solution for h j at gridpoint j, time step n+1 j n+1 r 2.2.2 residual vector with jth element r j r 1.4.2 position vector of a point in a continuum S 1.4.2 surface enclosing a deforming continuum V S, A5.2 upstream surface of a thin transverse vertical prism S2 A5.2 downstream surface of a thin transverse vertical prism S0 4.3.6 prism of ice in icefall during ablation (Summer ice) S, 4.3.6 prism of ice which occupied position S0 one year previously S(x',t) A5.3 transverse cross-sectional area of glacier channel s 1.4.4 shape factor (may be x dependent) s(t) 2.3.2 slope of analytical solution to continuity equation (used to test terminus motion) 194 s0 2.3.2 initial value of s(t) • s 2.3.2 rate of change of s(t) T 1.4.3 temperature (°K) T 1.4.2 surface traction vector for a deforming volume V of a continuum T 1.4.2 kth component of T k Ti,T2,T3 1.4.3 scalar invariants of stress tensor a jk Ti,T2,T3 1.4.3 invariants of stress deviator tensor a' jk T(m) 2.2.4 transfer function for numerical scheme when using Fourier transform T(Z) A1.4.2 transfer function for numerical scheme when using Z transform T(t) 3.3.5 amplitude function used to model surge velocity of Steele Glacier T(x) 4.3.1 time for ice to flow from origin to x (For ogive problem, this is a nonlinear measure of distance) T0 4.3.1 reference position at which boundary condition Q0(t) is applied T0 4.4.2 upstream end of a linear gradient in generalized velocity v T, 4.4.2 downstream end of a linear gradient in generalized velocity v t 1.3.3 time t0 2.3.3 initial time t, A2.3.2 time subsequent to t0: t^to+At t0 3.3.5 time at which a surge starts t, 3.3.5 time at which a surge reaches peak sliding velocity 195 t2 3.3.5 time at which a surge begins to slow t3 3.3.5 time at which a surge ends t, 3.3.5 surge period U0 3.3.5 maximum sliding velocity in Steele Glacier surge simulations U(x) 4.2.2 surface velocity of ice for ogive models U0 4.3.6 velocity of ice at upstream edge of transition zone for s»imple ogive models U, 4.3.6 velocity of ice at downstream edge of transition zone for simple ogive models u(x) 1.3.1 velocity component parallel to bed when x is the triplet (x,y,z) u 1.4.3 velocity component in direction x i when x is (x,,x2,x3) i u A2.2.3 maximum value of u(x) on [0,L(t)] max u (x,t) 1.4.4 basal sliding velocity s u 2.4.1 u (x) at gridpoint j s j s u0,...u3 A2.3.2 velocity component u at mesh points Ed P 1 I £2 1 P 3 u A2.3.2 first difference of u in x direction x between P0 and P2 u A2.3.2 first difference of u in z direction z between P0 and P, u A2.3.2 first difference of u in z direction xz X V 1.4.2 volume of a deforming continuum V(x,t) 1.3.3 downslope velocity component u(x) averaged over width and depth n V 2.2.2 V(x,t) at time step n, midway between j+1/2 gridpoints j and j+1 196 V0(x) A15.3.1 vertical velocity component at ice surface for Nagata ice sheet model V A1.5.2 phase velocity of wavenumber component m phase VOL A1.3.4 volume of wedge terminus T VW A10 ice velocity V times width W j-1/2 j-1/2 j-1/2 v(x) 1.3.1 velocity vector (u,w,v) v(x) 1.3.1 z component of v(x) v(T) 4.3.3 generalized velocity function for ogives v0,...v3 A2.3.2 velocity component v at mesh points P_0 I £l I P 2 I P_3 v A2.3.2 first difference of v in x direction x between P0 and P2 v A2.3.2 first difference of v in z direction z between P0 and P, v A2.3.2 first difference of v in z direction XZ X W(x) 1.3.1 channel width W 2.2.2 width W(x) at gridpoint j j W0 2.3.2 channel width at x=0 for analytical solution used to test terminus W 2.3.2 x gradient of channel width for analytical solution used to test terminus W0 4.3.6 a prism of ice above icefall during ablation (in icefall in Winter) W, 4.3.6 ice which occupied volume W0 one year previously W A1.3.4 average width of wedge terminus W A19 width of Steele Glacier at confluence i with ith tributary 197 w(x) 1.3.1 velocity component in y direction X A1.3.4 nondimensional length of wedge terminus X(x,t) 3.3.5 spatial pattern of sliding velocity for Steele Glacier surge simulations x 1.3.1 position vector (x,y,z) x 1.1.2 axis along glacier bed on a flowline (positive downstream) x, 1.4.2 equivalent to x axis x2 1.4.2 equivalent to y axis x3 1.4.2 equivalent to z axis x' A5.2 axis along glacier surface (positive downstream) x A1.3.4 position of glacier terminus T (equivalent to L(t) ) x A15.3.2 equilibrium line for Nagata steady e ice sheet model x.2(t) 3.3.5 points defining zone in x.,(t) which sliding takes place x,(t) in simulations of surges of x2(t) the Steele Glacier y 1.1.2 transverse horizontal axis imAx Z A1.4.1 Z transform variable Z=e z 1.1.2 axis in vertical plane and normal to x (positive upward) z' A5.2 axis in vertical plane and normal to x' f\ A6 inclination of ice surface slope in datum (steady) state o 1.4.4 inclination of ice surface slope c A6 perturbation to -P\ a 2.3.3 coefficient in quadratic relation between gas flux QT and gas density H (Burgers' equation) 198 p 1.4.2 inclination angle of glacier bed p 2.3.3 coefficient in quadratic relation between gas flux Q, and gas density H y 2.3.3 coefficient in quadratic relation between gas flux Q, and gas density H At 2.2.2 time increment for finite difference solution of continuity equation Ax 2.2.2 horizontal grid interval for finite difference scheme Ax 2.2.2 grid interval along bed at grid point j j for finite difference scheme 6(018/016) 3.5.1 isotopic composition of oxygen 6t 1.3.4 a small time interval 6x 1.3.4 a small increment in x direction (6x,6z) 2.4.3 position of an ice particle relative to a meshpoint P0 6 1.4.3 Kroenecker delta function: =1 if i=j ij =0 if i*j 6(x) 2.3.3 Dirac delta distribution 60 3.5.4 isotopic composition at a reference location 6V A1.3.4 volume change of wedge terminus in bal time At due to surface melting 6V A1.3.4 volume change of wedge terminus in flux time At due to ice flow through upstream boundary 6b 3.3.4 perturbation to Steele Glacier mass i balance to simulate ice discharge from ith tributary n+1 6h 2.2.2 iterative correction to h j j 6h 2.2.2 correction vector: ith element is 6h j 199 6v 4.3.6 magnitude of a step change in generalized velocity v 6T 1.4.4 difference between effective shear stress T and shear stress c xz c A1.4.5 numerical viscosity (Lax-Wendroff scheme) n e 2.2.5 truncation error at gridpoint j j at time step n e 1.4.3 strain rate tensor i j e 1.4.3 effective strain rate (square root of second invariant of e ) i j n A8.3.1 Newtonian viscosity for longitudinal strain (Budd, 1975) 0 2.2.2 weighting parameter between 0 and 1, used to mix past and future time steps in finite difference scheme X. 4.2.2 displacement of surface ice in one year v 2.3.3 diffusion coefficient fi 1.3.4 density of glacier ice <r 1.4.2 stress tensor a' 1.4.3 deviatoric stress tensor jk T 1.4.3 effective shear stress (square root of T') 2 T 4.4.2 time for ice to travel the length of a channel section with a velocity gradient T A8.3.1 simple gravitational shear stress c given by spgh sine * T A8.3.1 local lubrication-lowered stress c (Budd, 1975) 200 T A8.3.1 'actual' basal shear stress b (Budd, 1975) T(t) 4.2.3 temporal term in mass balance for ogives 0 4.3.1 time when ice passes origin (this is also the label for characteristic curves) 0 A8.3.1 basal lubrication factor (Budd, 1975) 0 A13.3 baseline halflength for finite difference estimate of dQ/dx 0 (m) A1.5.2 phase angle of transfer function T(m) pde for partial differential equation at wavenumber m 0 (m) A1.5.2 phase angle of transfer function T(m) fd for finite difference scheme at wavenumber m 0Or...09 A1.5.3 real numbers between 0 and 1/2 giving x locations of remainder terms in truncated Taylor expansions of h(x) and Q(x) x(x) 4.2.3 spatial term in mass balance for ogives * A13.3 baseline halflength for finite difference estimate of slope a *2 A1.5.4 real numbers between 0 and 1 giving x locations of remainder terms for Taylor expansions for interpolation error analysis underscore indicating a vector, double underscore indicates a matrix A1.4.2 convolution operator superior dot indicating time derivative superior bar indicating spatial average 201 LITERATURE CITED Abramowitz, M. , and Stegun, I. 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But not without a plan; A wild, where weeds and flowers promiscuous shoot, Or garden, tempting with forbidden fruit."1 A 1.1 THE NUMERICAL SCHEME A 1.1.1 THE CONTINUITY EQUATION The solution of the partial differential equation (A1.1.1) dh(x,t) + 1 d_Q(x,t) = A(x,t) dt Wlx) dx 0 < x < L 0 < t < T (A1.1.1) is approximated by the solution of a corresponding set of algebraic equations (A1.1.4) for {h?|j=1,J;n=1,N], the values of h(x,t) at a set of mesh points in x-t space. i-1 Ax , nAt)|j=1,J; n=1,N} i = 1 i J-1 Ax n =At/T (A1.1.2) j = 1 j The x axis runs along the glacier bed, and the thickness h and the source term A are measured normal to it. Q is the ice flux through the cross section normal to the bed. W is the width of the glacier channel. The mesh points are chosen with a uniform spacing Ax in the horizontal direction. The intervals along the x axis on the glacier bed are then 1 An essay on man. Alexander Pope. 236 Ax = Ax/cosU ) (A 1.1.3) j j The slope of the glacier bed is p. This is illustrated in Figure A1 .1 . n+1 n n+1 n+1 n n h -h + 6 (Q - Q ) + (1-6) (Q - Q ) _J i " i + i-1/2 i + 1/2 i-1/2 At W Ax W Ax j j j j n+1 n = OA + (1-6)A (A1.1.4) j j 1 < j < J 1 < n < N The weight factor 6 is a constant between 0 and 1 used to FIGURE A1.1. Mesh Increment On Bed. stabilize the numerical scheme by mixing estimates of the spatial partial derivative at successive time steps. This is discussed further in Section A1.4 below. The ice flux Qj + 1/2 can be written in terms of the ice thickness hj at the mesh points by 237 Q = V W (h + h ) (Al.1.5) J±1/2 j±l/2 j±l/2 j±1 j 2 where vj +1/2 is the ice velocity midway between the mesh points and averaged over the depth hj + 1/2 normal to the glacier bed. wj • 1/2 is the channel width. After substituting (A1.1.5) into (A1.1.4), separating past and future values of thickness h and setting p =At/(2Ax W ) j j j (A1.1.6) (A1.1.4) gives n+1 n+1 n+1 [1 + p e(V W - V w ) ] h j j+1/2 j+1/2 j-1/2 j-1/2 j n+1 n+1 n+1 n+1 + t-ep w V ] h +[6pW V ] h j j-1/2 j-1/2 j-1 j j+1/2 j+1/2 j+1 n n n = [ 1 - p (1-6)(V w - V W ) ] h j j+V2 j + V2 j-V2 j-1/2 j n n n n + [d-e)p w v ] h - [d-e)p w v ] h j j-1/2 j-V2 j-1 j j + 1/2 j + 1/2 j + 1 n+1 n + [0A + (1-6)A 3 At j j 1 <: j < J 1 < n < N (A1.1.7) Starting from an initial condition {h? |j=1,J}, this system of equations is solved for {h! | j = 1 ,»J} . This time stepping method of solution is continued until n=N. Since the left side of (A1.1.7) contains more than one of the unknowns i.e. h!?:],h!J*1, and h^-1 ], this is an implicit numerical scheme for 238 A1.1.2 A MATRIX FORMULATION The system of equations (A1.1.7) can be written as a matrix equat ion B 1 c 1 0 0 • hn* i 1 D 1 A 2 B 2 C 2 0 • 0 • • • 0 A 3 B 3 C 3 . 0 • • • • 6 A B C • 0 0 . D j j D j j • • 0 • • • A J-1 . 0 B J-A J • c 1 J-1 B J • hn+1 J • D J or M H = D ( where D- is the expression on the right side of (A1.1.7) containing only known quantities. B^C,, AJf Bj., D, , and DT are determined by the choice of boundary conditions discussed below in Section A1.3. The matrix M is tridiagonal and is diagonally dominant because the diagonal terms Bj , the coefficients of h?*1 in (A1.1.7), are of order unity, while the off-diagonal terms are of the order of pQ, which is usually less than unity for reasonable glacier models, and choices of mesh. This means that the matrix equation is easily solved by Gaussian elimination without using any special pivoting strategy (Carnahan and  others, 1969, p. 272). 239 A1.2 NONLINEARITY If the velocities {VP* 1 | j = 3/2, 5/2, .. .J+1/2} were perfectly J known, the complete solution would be obtained by simply solving the matrix equation (A1.1.8). Since the {VP* 1 | j = 3/2, 5/2, .. .J+1/2} depend in general on {h*?* 1 | j= 1 ,J} through some flow and sliding law, (A1.1.8) only appears to be linear in {h£*1|j=1,J} . The standard technique to solve nonlinear equations of the form of (A1.1.8) is to use an iterative method to improve the coefficient values (here, the {VP* 1[j = 3/2, 5/2,...J+1/2} ) based on the values of the unknowns {hj1 + 1 | j = 1 , J} at the previous iteration. Specifically, letting the sublevel prescripts indicate the iteration number, UVP*1|j=3/2, 5/2,...J+1/2} at the new time step are approximated by (V? |j=3/2, 5/2, J+1/2}, the velocity at the previous time step. Using these values for the coefficients, (A1.1.8) is solved for {0h?+1|j=1,J} . This thickness profile estimate is then put into the flow and sliding law to find a better velocity profile {,VP *1|j = 3/2, 5/2,...J+1/2} . The residuals { r?+l|j=1,J} are then calculated. n+1 n+1 n+1 n n r = 2p 6[Q - Q ] + 2p (l-e)[Q -Q ] j j j+1/2 j-1/2 j j+1/2 j-1/2 n+1 n n+1 n + h - h - [6A - (1-0)A ]At j j j j 1 £ j < J 1 < n < N (A1.2.1) and 240 p = At/(2Ax W ) (A1.1.6) j j j The residuals are a measure of the degree to which the continuity equation is satisfied by the current values of {h?*1|j=1,J} and {VT*1|j=3/2, 5/2,...J+1/2} . The quantity r-is the volume of ice created or destroyed in one time step per unit length and width at mesh point j, through error either in the surface elevation, or in the amount of ice flowing in and out from the adjacent mesh points. It is then necessary to calculate the corrections {6h^|j=1,J} to the thicknesses such that the residuals will be reduced to zero using n+1 n+1 h h - 6h (A1.2.2) k+1 j k j j This can be done using a multidimensional Newton-Raphson technique. A variation of this method was used by Bindschadler (unpublished, p. 84), who did not solve the matrix equation (A1.1.8) for the first estimate {0h*?*l|j=1,J} , but instead set {0hV1|J = 1^} = {h-|j«1,J} i_.e. used a steady state starting estimate. The procedure described in this study gives a better starting estimate of (onj + 1M = 1rJ} when the profile is changing with time, but requires the extra computing effort of solving (A1.1.8). The residuals { r?+1|j=1,J} are a function of the thickness values {h?*1|j=1,J} either directly, or through the ice surface slope or higher derivatives. Dropping iteration prescripts for the moment, the first order Taylor expansion of r?+1 about zero is 241 J r n+1 1 6h k or, in matrix form, A 6h = r (A1.2.3) This is a set of algebraic equations wjiich can be solved for {6h-|j=1,J} . The elements of matrix A can be found by differentiating the residuals in (A1.2.3) This is another implicit system of equations that can be solved by Gaussian elimination (e.g. Carnahan and others, 1969, p. 272) for the corrections to the ice thickness. A common modelling case is that in which the flux at x is a function of the thickness at x and the slope at x. In this case, the matrix A is also tridiagonal. For models with a flow law such that the flux depends on thickness and slope over a range of x, the diagonal band of A becomes wider than three. These models are no more difficult in theory, but merely require more computational time. For the case that flux Q(x) depends only on thickness and slope at x, dr a (A1.2.4) 242 ^3 3-1 D+1 dQ = -2p 6—j-1/2 j dh j-1 j + 1 b = c = dr dr 5h" j-1 = i+2p e h j j+1 ao dQ —j + 1/2 - —j-1/2 dh dh j j j-1 i j + 1 i dQ j-1 = 2p 0—j+1/2 i j dh j + 1 j-1 j+1 j +1 j (A1.2.5) where each partial derivative is evaluated with all the other hj held constant. The bj's are the main diagonal elements and the a-'s, and c•'s are the sub- and super-diagonal elements J J a j+ and a^ j+, . For computations it is more convenient to express the coefficients in the matrix A as partials with respect to thickness between mesh points hj t ,/2 and the slope between mesh points OJ + !/2 , since these are the quantities from which the flux is directly calculated. This form of the coefficients is derived in Appendix 10. As with the matrix equation (A1.1.8) for the full thickness for the first iteration at a new time step, the coefficients in rows 1 and J are determined by the boundary conditions. The details are described below in Section A1.3. If the residuals were truly a linear function of the {h?+1|j=1,J} , the solution would be obtained exactly after solving the residual equation (A1.2.3) once. Since the matrix equation (A1.2.3) was obtained by linearization through a Taylor series, the iterations must be continued until the the largest 243 residual in absolute value is smaller than some preset test criterion. The choice of such a criterion is discussed in Appendix 11. A 1.3 BOUNDARY CONDITIONS A1.3.1 THE UPPER BOUNDARY The first order partial differential equation (A1.1.1) requires one boundary condition. The condition for the finite difference system (A1.1.4) is applied at the upstream end j=1 in complete analogy to the condition on equation (A1.1.1). The various possible types of condition are described below. 1) Zero flux input: wedge type The thickness h, is zero. This models alpine glaciers starting on a steep slope. In the equations (A1.1.8) for the first iteration, B =1.0 C =0.0 D =1.0 (A1.3.1) 1 1 1 For subsequent iterations to find {6hj|j=1,J} , the same values are used. 2) Zero flux input: zero slope The surface slope is zero at x=0. This is the model for a stable ice divide on an ice sheet. The boundary condition is implemented by reflection at j=1 by using an image point in the equations (A1.1.7) with subscript 0 such that 244 h = h 0 2 w = w 1/2 3/2 V = -v 1/2 3/2 (A1.3.2) In the matrix equation (A1.1.8) n+1 B =1 + 2p 0V W 1 1 3/2 3/2 n+1 c =2p ev w 1 1 3/2 3/2 n n n D= h -2(l-6)pV [h +h ]W (A1.3.3) 1 1 13/2213/2 n+1 n + 0A At +(1-6)A At 1 1 3) Flux input Q (t) o The flux Q0(t) at x=-Ax/2, or j=l/2, is substituted into (A1.1.7). This models a section of an ice mass starting some distance below the bergschrund or ice divide. For example this model is used to investigate the generation of wave ogives at an icefall when the input flux from the upper glacier is Q0(t). Note that in the limit of Q0(t) going to zero, this is not equivalent to boundary conditions 1) or 2) above, since in this case neither the slope nor the thickness is specifically set to zero at the boundary, and the boundary is located at x=-Ax/2 rather than at x=0. In the matrix (A1.1.8) 245 n+1 B =1 + p ev W 1 13/23/2 n+1 c = p ev w 1 1 3/2 3/2 n+1 n n n n D = - 2p [OQ +(1-6)Q ] - p (1-6)V [h +h ] W 1 o 3/2 2 3 /2 n n+1 n +(1-0)A At +0 A At + h i i i (A1.3.4) A 1.3.2 THE DOWNSTREAM BOUNDARY Since the partial differential equation (A1.1.1) is first order, it requires only one boundary condition. However, to implement the numerical scheme (A1.1.7) it is necessary to impose some condition at the lower boundary j=J so that the the matrix M can be terminated i..e. to eliminate h£* , . A1.3.3 NONZERO FLUX LEAVES DOWNSTREAM BOUNDARY This treatment is useful when modelling a short section of an ice mass which does not include the terminus region. No mesh points are added to or removed from the grid as time advances, and a nonzero ice flux exists at Ax/2 beyond the last mesh point j=J. This ice flux is given by n+1 n+1 n+1 Q = V h W J+1 J+1/2 J+1/2 J+1/2 where h]/2 is estimated by the second order Newton's divided 246 difference polynomial (e, Carnahan and others, 1969, p. 12) h J+1/2 h -h h -h J J-1 - J-1 J-2 h + Ax h - h J J-1 + Ax 3 Ax J 2 AX 2 2 Ax AX 2 Ax (A1.3.5) This quadratic extrapolation ignores the third order term (d3h/dx3)Ax3, so that the extrapolated flux has an error that is 0(V [d3h/dx3]Ax3), and the flux gradient has an error term that is 0(V [d3h/dx3]Ax2), which is the same order as the truncation error described in Section A1.5. To use any extrapolation with a larger error than the truncation error would reduce the accuracy of the total solution. This was done by Budd and Jenssen (1975) who used a linear extrapolation. The 'shocks' at the boundaries described by these authors are possibly a result of this treatment. However, this extrapolation introduces a nonzero coefficient k for hjt2 in the last row J of the matrix (A1.1.8), so that the matrix is no longer tridiagonal, but has the form 0 . A B C 0 = D J-2 J-2 J-2 J-2 J-2 0 A B C h"+ 1 D J-1 J-1 J-1 J-1 J-1 k A B h*+1 D J J J J (A1.3.6) in the last three rows. When k is nonzero, the matrix can be restored to tridiagonal form by subtracting (k/A^) times the 247 (J-1)st row from the Jth row to give . A J-B C 2 J-2 J-2 0 hn* 1 J-2 — D J-2 . 0 A B J-1 J-1 C J-1 hn+ 1 J-1 D J-1 0 A -kB J J-1 A J B -kC J J-1 A J • hn+ 1 J D -kD J J A L J (A1.3.7) A 1.3.4 MOVING WEDGE TERMINUS For some applications it is essential to include the terminus in the glacier model, and allow it to move so as to satisfy mass conservation and the flow law for ice. For example, the terminus of a surging glacier may advance and retreat by 10% or more of the glacier length during one cycle. This requires a procedure to add meshpoints to the grid, or to remove them, based on a calculation of the terminus position at each time step. To calculate the terminus position when it lies between mesh points, it is necessary to make some assumptions. The velocity (1.4.34) based on an integration of Glen's flow law (Glen, 1955) for simple shear cannot be used near the terminus (Nye, 1967). The scheme described below is based on the one used by Bindschadler (unpublished, p. 105), with some modifications to the x increments. Referring to Figure A1.2, the terminus is assumed to be wedge-shaped i.e. the ice thickness h(x) is linear with distance from the last mesh point xT to the terminus 248 FIGURE A1.2. Model Terminus. position xT, at which the thickness is zero. Mass is conserved in the shaded wedge beyond xT + ,/2 in Figure A1.2. The volume of this terminus section is DX f I h(x) W(x, J 0 VOL = | h(x) W(x) dx (A 1.3.8) T J 0 where x is measured parallel to the bed, and h(x) is normal to the bed. Neglecting the possibility that dW/dx may change slightly below xT+1, h(x) = h + h'x h' = -h /DX j J+1/2/ T (A1.3.9) W(x) = W + W'x W = (W -W )/DX J J+1 V/1 Substituting (A1.3.9) into (A1.3.8) gives 249 h W DX W h DX2 VOL = J+1/2 J+1/2 T + J+1/2 T 2 (A1.3.10) T 2 6 and, using the similar triangles of height hj and hJ+1/2 shown in Figure A1.2, DX = T h + h J J+1 h - h J J+1 Ax (A1.3. 1 1 ) Changes in the volume VOLT occur due to the mass balance acting on the surface, and ice flowing into VOLT through the cross-section at xT+1/2. The average mass balance (normal to the bed) on the surface of the wedge, to the same accuracy as used elsewhere on the grid, is A(x ) + A(x ) A = J+1/2 T (A1.3.12) The surface area exposed normal to this ablation is W DX = T W(x ) + W(x ) J+1/2 T DX 2 (A1.3.13) Bars over variables indicate average values. W(xT) and A(xT) can be interpolated linearly between their values at xT and xT+1. Setting 2 DX X = T AX J (Al.3.14) then the change in volume of the wedge terminus due to melting on the top surface in a time interval At is 250 6V = A W DX = bal T = At DX 16 T (2-X)W +(2+X)W J J+1 (2-X)A +(2+X)A J J+1 (A1.3.15) The change in volume of the wedge due to influx of ice through the cross-section at xJ+1/2 during the same time interval At is 6V = W h V At flux J+1/2 J+1/2 J+1/2 (A1.3.16) Conservation of mass in the terminus from time step n to time step (n+1) requires n+,1 n 6V + 6V bal bal n+1 n + 6V + 6V flux flux n+1 n VOL - VOL = T T (A1.3.17) On substituting (A1.3.10) through (A1.3.16), the unknowns in this equation are hJ+1 and hjl\. Since it is not linear in these unknowns, it cannot be included directly in the matrix equation (Al.1.8) at the first iteration for (h?+1|j=1,J+1}. Instead, an initial guess at hjti must be used. For instance, I can assume that the terminus has the same slope as at the previous step, i.e. n n+1 n+1 n h - h = h - h J+1 J J+1 J and use this in (A1.1.8) by setting A =-1.0 J+1 B =1.0 J+1 (A1.3.18) D = h - h (A1.3.19) J+1 J+1 J After dividing through (A1 .3.17) by a surface area WJ+1/2AXj to give the same units as the other residuals in (A1.2.1), the 251 residual at xT+, is n+1 J+1 n+1 n VOL -VOL - J_ T T 2 n+1 n 6V +6V • bal balJ - j_ 2 n+1 n 6 V +6V flux fluxJ W Ax J+1/2 J (Al .-3.20) Using the Taylor expansion (A1.2.3) on (Al.3.20) gives a (J+l)th equation for the corrections {6hj |j=1,J+l} . The coefficients in (A1.2.3) are a dr j+1, J = a~H DX T + 4Ax 12W J W' DX2 At V At h ~6v T - J - J+1/2 J + Ax 4Ax * 2 Ax dh J+1/2 J J J J c3DX dh 1 h h W'DX J+1/2 + J+1/2 T - 6V ~2AX 3W Ax bal Ax DX W j J+1/2 J J T J+1/2 At DX T 16AX*W J J+1/2 [W -W ] [(2-X)A +(2+X)A ] J+1 J J J+1 + [A -A ] [(2-X)W +(2+X)W ] J+1 J J J+1 (A1.3.21) The coefficient at J+1 has the same form, with the changes 252 t^DX 0DX T —> T oh"- dh J J+1 gV dV J —> J (A1 .3.22) W sir J J+1 The derivatives of V^. can be found in equation (A10.5). The derivatives of DXT are d(DX ) 2DX h T = - T J oh h2 ^h7 (A1.3.23) J J+1 J d(DX ) 2DX h T = T J+1 oh Iv2 rh^ (A1.3.24) J+1 J+1 J At each iteration for the thickness, DXT is calculated from (A1.3.11). When the residuals have been reduced sufficiently at the (n+1)th time step, the value of DXT determines whether the number J+1 of mesh points must be changed. The terminus region is a critical region for any numerical model. If the model terminus advances too slowly due to incorrect use of the flow law at the terminus, it tends to dam up the ice behind. This creates a model solution which is too thick and flows too slowly. If the model terminus advances too quickly, it can stretch the whole model profile beyond the correct length, even though continuity is satisfied everywhere. There are two points to consider: The equation (A1.3.16) uses the flow law only in the flux term 6Vflux through the velocity factor Vj+1/2. If the terminus is allowed to advance too far beyond xj-+1/2, the terminus will 253 satisfy continuity, but will bear little similarity to the expected terminus shape of a real ice mass governed by the flow law. This can lead to grossly inaccurate behaviour of the whole model. On the other hand, integrated forms of Glen's flow law (Glen, 1955), as discussed in Appendix 7, assume small slopes and thick ice with the major deformation b,eing shearing parallel to the glacier bed. This is not the case near the terminus. The sliding velocity near the terminus is also difficult to model. This means that xT must not be allowed to get too close to x.T^V2 either, because then the value of VJ+1/2 would be in question. A reasonable compromise is to keep the length DXT of the wedge terminus of the order of the average ice thickness. If maximum and minimum acceptable lengths are DXmax and DXmin respectively, then a new meshpoint is added when xT exceeds Xj+1/2 +DXmax. Similarily, if xT is less than x^ 1/2 +DXmin, then the meshpoint J+1 is discarded at the subsequent time step. 254 A 1.4 NUMERICAL STABILITY A 1.4.1 INTRODUCTION Physically realizable solutions of the partial differential equation (A1.1.1) are bounded for all time t. Solutions of the set of finite difference equations (A1.1.4) must be bounded also. For what values of At, Ax, and 0 is this true? Assuming that the source term Aj will not affect stability, it can be set to zero without loss of generality. A large positive mass balance will result in faster flow of a bounded solution, while a large negative mass balance will terminate the computations in a bounded time by eliminating the ice mass. Rigourous stability analysis of nonlinear equations is in general not possible. However, the stability conditions for the linearized system of equations used in the iterative procedure are a helpful guide. The approach I take in the next few sections, to find stability conditions involving 0, is a variation on the von Neumann, or Fourier series method. (e.3., Richtmyer and Morton, 1967, p. 70). I use Z transform notation (Claerbout, 1976, p. 2) because it is equivalent to Fourier series, but more direct and notationally simpler, (see Appendix 3). The method in brief is to find the transfer function T(Z) in the wavenumber domain which takes the transform Hn(Z) of the ice profile {hj|j=1,J} at time n to the transform Hn+1(Z) at time (n+1). If |T(Z)|<1 for all Z or wavenumbers that can be sensed by the grid, then the profile (h?+1|j=1,J} is also bounded for all n, 255 .i.e. the scheme is stable. Accuracy of the scheme can be estimated by comparing the amplitude and phase of the transfer functions for the partial differential equation and the finite difference equations. There are two questions to ask about numerical stability. First, how should the mesh increments Ax and At be chosen for stability of the linearized equations? Second, what effects are introduced by the nonlinearity, and how should they be handled? The scheme used for-the flux gradient is a critical factor for both questions. A1.4.2 THE LINEAR COMPUTATIONAL INSTABILITY If 6 is set to zero giving the explicit numerical scheme, no effect can propogate through the mesh faster than the characteristic speed Ax/At. This is unrealistic for a system with diffusive characteristics (see Appendix 6). In this model diffusion arises from the dependence of the ice flux on the surface slope. Explicit schemes for diffusion equations are usually numerically unstable unless the time steps are very small, giving a high characteristic speed Ax/At (e.g. Richtmyer and Morton, 1967, p. 18). Using an implicit scheme, .i.e. 6>0 lets the domain of dependence (e.g., Mesinger and Arakawa, 1976, p. 5) for each mesh point be the entire mesh at the subsequent time step. For diffusion equations, this tends to alleviate stability problems (e.g. Richtmyer and Morton, 1967, p. 18). The velocities (V?+1|j=3/2, 5/2,...J+1/2} are treated as constants at each iteration in Section A1.2. Here, they are all 256 set equal to one .constant value V to get an approximate stability criterion involving 6. The channel width W(x) is also held constant. Similarily, the differences in the {AXj | j = 1 , J} due to the bed slope are not expected to be important. All AXj are set equal to Ax. The actual variation with x will presumably alter the criterion in a minor way. The signal h is assumed to be periodic with period 2JAx in the x direction, so that Fourier series can be used. (This is also a form of boundary condition.) With zero mass balance, the finite difference system of equations can be written as convolutions (e.g. Claerbout, 1976, p. 5) n+1 n+1 n+1 {h ,h ,...h } * (pvwe,1,-pvwe} 1 2 J n n n = {h ,h ,...h } * {-pVW(1-6),1,pVW(1-G)} (Al.4.1) 1 2 J After taking the Z transform the convolutions become multiplications in the Z domain, and the series become sums. n+1 n H (z)[pvwe+i-pvwez] = H (z)[-pvw(i-e)+i+pvw(i-e)z] z z (A1.4.2) Since Z is a complex number of the form e,mAX, the Z transform is equivalent to a discrete Fourier transform with wavenumber m. The factor (Z-1/Z) is purely imaginary and is equal to i[2 sin(mAx)] where "i" is the square root of minus one. Setting i K(Z) = WpV(Z-1/Z) = i AtVsi n(mAx) Ax (A1.4.3) where K(Z) is real, the transfer function is 257 1 + (1-e)iK(Z) T(Z) = (A1.4.4) 1 - 0iK(Z) Setting |T(Z)| < 1 gives K2 (K262+1) (1-20) < 0 (A1.4.5) Since the first two factors are nonnegative for all Z, the stability criterion is ©>l/2 for all At and Ax. (A1.4.6) The case 0=1/2 gives |T(Z)|=1 for all Z and thus is marginally stable at all wavenumbers. The presence of nonlinearity could alter the stability criterion (A1.4.6). A velocity dependent on ice slope tends to stabilize the scheme (Section A1.4.6 and stability analysis of Budd and Jenssen, 1975). However, a velocity dependent on ice thickness tends to form shocks, and is a destabilizing factor. To guarantee stability, I could be tempted to use a larger value of 0. However, the accuracy of the scheme also depends on 0. This is discussed in Section A1.5.1 and Section A1.5.3. A 1.4.3 THE NONLINEAR INSTABILITY In this Section, I will discuss the nonlinear instability (NLI) (e.£. Haltiner, 1971, p. 199; Mesinger and Arakawa, 1976, p. 35; Gary, 1975, p. 8.41; Haltiner and Williams, 1980, p. 170), a problem which arises in any numerical solution of a differential equation using a discrete mesh, and having terms which are nonlinear in some combination of the dependent and the independent variables. In (A1.1.1), dQ/dx is such a term. 258 Basically, the nonlinearity pumps energy (squared amplitude of the wavenumber spectrum) into the high wavenumber part of the wavenumber spectrum of the dependent variable, and the aliasing (Appendix 3) due to discrete sampling folds this energy back to the lower wavenumbers, where it distorts the solution. Assuming that the initial profile {h°|j=1,J} and the velocity (V?|j = 3/2, 5/2,...J+1/2} derived from it are periodic with period 2JAx, they can be expressed as Fourier series with wavenumbers (1»T)/(JAX) from -mN to the Nyquist wavenumber mN (see Appendix 3, Section A3.2). m = n/Ax (A1.4.7 ) N J/2 i2rfl EJAx H e (A1.4.8) j 1=-J 1 2 J/2 i2irk EJAX V e (A1.4.9) j k=-J k 2 The flux . Q(x,t) in a channel of unit width, however, is the product of these two series. J/2 J/2 i2ir(l + k) E V^ JAx LJ H V e (A1.4.10) j 1=-J k=-J 1 k 2 2 This flux has harmonic components up to 2ir/Ax, or twice the Nyquist wavenumber. The energy in these high wavenumbers should not appear in the solution because the solution is bandlimited; however, due to the discrete mesh this energy is aliased (see 259 Appendix 3, Section A3.2) back into the wavenumber interval [0,TT/AX] and thus distorts the solution. Most important, however, is the fact that more energy is aliased into the solution at each time step. This numerical effect can increase without bound and will dominate the true solution of the partial differential equation (A1.1.1) in a finite time. The situation can be avoided if the numerical scheme, allows damping of the higher wavenumbers, so that aliased energy is attenuated; this also leaves little energy available at the wavenumbers from which it can be pumped up past the Nyquist wavenumber by the nonlinearity. In principle, the nonlinear instability can be completely eliminated if, at every time step, the wavenumber spectra of the velocity profile and of the ice thickness profile contain no energy above 2/3 of the Nyquist wavenumber mN. Then, in the following time step, the highest wavenumber component activated by the nonlinearity (see (A1.4.10)) is 4/3 mN . The spectral band from m., to 4/3 m is folded back onto the band from N N 2/3 mN to mN , making the spectrum of Q in this latter band incorrect. This is illustrated schematically in Figure A1.3. The important point to note, however, is that the band from 0 to 2/3 mN, to which I originally band limited h and V, is unaffected by the aliasing. If the energy above 2/3 mN can be eliminated at every time step, the signal in 0 to 2/3 mN will always be correct. Choosing Ax small enough can always push the cutoff (2»r)/(3Ax) out beyond the wavenumber of any feature of interest in the glacier spectrum, no matter how short its wavelength. 260 FIGURE A1.3. Aliasing And The Nonlinear Instability. The spectrum of Q(m)=v(m)h(m) beyond the Nyquist wavenumber at mAx = ir is folded back (lower dotted curve) as an aliasing error between 2ir/3 and TT. The upper dotted line is the spectrum of the sampled Q(m) with aliasing. There are several ways to damp the instability in the aliased part of the spectrum. Some are more satisfactory than others. I will discuss four methods. The first two have conceptual drawbacks. I have used the third and fourth methods in this study. A1.4.4 VELOCITY SMOOTHING Budd and Jenssen (1975), who attributed the instability to machine roundoff, replaced the velocity profile V(x) by a version smoothed by a second derivative operator V(x) <— V(x) + 1 d2V Ax2 4 oT3" (A1 .4.1 1 ) whenever they saw high wavenumber components in the velocity profile. To see what (Al.4.11) does to the spectrum of V(x), 261 T i i FIGURE A1.4. Transfer Functions Of Smoothing Schemes. Solid curve: Budd and Jenssen smoothing applied occasionally to velocity profile. Broken curve: Lax-Wendroff numerical scheme with maximum dissipation. This would be applied to h(x) at each time step. take the Z transform (Appendix 3) of the right side using the standard finite second difference estimate dfy V - 2V + V dx2 = j+i j j-1 Ax"5 (A1.4.12) to get V(Z) <— V(Z) + ]_ V(Z) (Z-2+1) 4 Z (A1.4.13) or, in terms of wavenumber m, V(m) <— V(m) [1 + cos(mAx)3 2 (A1.4.14) since Z=e'mAX (Section A1.4.2). The smoothing filter (the second factor on the right side of (A1.4.14)) is shown in Figure A1.4. Although it substantially reduces the signal beyond 2/3 mN, it does not totally eliminate it. Since Budd and Jenssen (1975) 262 applied it only to the velocity, and not to the ice thickness profile, and only at infrequent times when the solution was already seriously in error, it could not completely or correctly remove the nonlinear instability. It could only keep it from totally dominating the physical solution. In addition, the filter attenuates the amplitude at all wavenumbers except zero. Both these effects distort the shape and total energy content (integral of the squared amplitude) of the velocity profile. This can cause errors in mass conservation through the flux gradient term. The effect in many cases may be small. However, better methods are available. A1.4.5 NUMERICAL DISSIPATION A second method to stabilize the equations (A1.1.4) is to add a dissipative diffusion term of the form where e is small, directly into the differential equation (A1.1.1). Kreiss (1964) showed that a wide class of difference schemes for linear hyperbolic equations could be stabilized this way, rather than by using implicit schemes such as (A1.1.4). Dissipative terms control instabilities by preferentially damping the high wavenumbers at which instabilities typically arise. They can also be used to control the nonlinear instability. For example, consider the transfer function which results from adding the dissipative term to the linear form of the 263 continuity equation, a simple special case of (A1.1.1) with constant width and velocity. dh + Vdh - ib2h = 0 dt bx Jx7 (Al.4.15) The simple finite difference scheme for (Al.4.15) n+1 n n n n n n h -h V(h - h ) (h - 2h + h ) At 2AX AX2 (A1.4.16) has the transfer function T(m) = 1 + 2eAt[1 - cos(mAx)] + iVAt sin(mAx) Ax2 Ax (A1.4.17) The derivation parallels Section A1.4.2. When c = V2At 2 (Al.4.18) Gary (1975, p. 3.69) showed that (A1.4.16) is the standard Lax-Wendroff dissipative formulation (Lax and Wendroff, 1960) which is stable for VAt < 1 Ax (A1 . 4 . 1 9) and approaches the solution of oh + Vdh =0 . dt "ox (A1.4.20) in the limit At-»0. Using (A1.4.18), the modulus of the transfer function is |T(m)| = [l ~ 4cd-c) sin"(mAx/2)J where c is given by 1/2 (A1.4.21) c = 2 c At Ax = V2At2 Ap (A1.4.22) The magnitude of the damping term is largest when c=1/2, i.e. 264 when material advances (1/2) 1/2 mesh intervals per time step. |T(m)| is shown for that case in Figure A1.4. For other values of At, there is less damping of the high wavenumber components. This scheme would be preferable to the method of Budd and Jenssen (1975). Because it is applied at every time step, it is more likely to keep the nonlinear instability in check at all times. It nevertheless suffers from some of the same drawbacks. It attenuates the low wavenumbers. The damping term also changes the equation from first order to second order. This means a second boundary condition is needed. Because the new term has no physical meaning, there is no immediately obvious physical boundary condition to apply, and the amplitude of the solution to partial differential equations can often be very sensitive to the boundary conditions. The reason that these two methods appear to work in practice is that the nonlinear instability tends to grow at least exponentially. It either totally dominates the solution, or it is insignificantly small. It seldom exists undetected with a magnitude comparable to that of the physical solution, although this is always a possibility with these second derivative schemes. 265 A1.4.6 DISSIPATION FROM THE VELOCITY EQUATION A more aesthetically pleasing way to avoid the nonlinear instability is to utilize the damping inherent in the flow properties of ice, rather than to introduce an artificial diffusive term, as in the previous section. For any realistic flow model of glacier ice,Jthe flow velocity of ice increases with increasing ice surface slope. By using the linearized perturbation form (Appendix 6) of the continuity equation with a velocity dependent on ice thickness and surface slope, I illustrate, in Appendix 13, the main features of this method of suppressing the nonlinear instability. The modulus |T(m)| of the transfer function (A13.2.5) of the differential equation, with parameters in (A13.2.6), is shown by a dotted curve in Figure A1.5 . The slope dependence leads to substantial damping at high wavenumbers. In Appendix 13, I show that, in order to achieve acceptable damping with the numerical scheme, it is necessary to calculate the ice surface slope over at most one mesh interval Ax, and to evaluate the ice flux between the mesh points, so that the flux gradient is also evaluated over a distance of at most -Ax. The solid curve in Figure A1.5 shows the transfer function modulus for the difference scheme when these restrictions are met. Like the previous two methods, this transfer function does not completely eliminate the components above 2/3 m^. However, in practice, the attenuation is adequate to prevent growth of the nonlinear instability. There are advantages to this method. Although it attenuates wave numbers below 2/3 m , this 266 FIGURE A1.5. Transfer Functions: Slope-dependent Damping. Solid curve: transfer function modulus for the finite difference scheme with space increment Ax. Broken curve: transfer function modulus for the linear partial differential equation. The Nyquist wavenumber is at mAx=n. attenuation has a physical basis, i..e. the diffusive nature of ice flow. The two curves in Figure A1.5 are similar at low wavenumber, indicating that the numerical scheme models this diffusive property quite well. Since no artificial damping term is introduced, the order of the equation is unchanged, and no extra boundary conditions are required. This method was used by Mahaffy (unpublished), Mahaffy (1976), Mahaffy and Andrews (1976) and Bindschadler (unpublished) to control the nonlinear instability. It is also used in this study for the Steele Glacier model. 267 . A 1.4.7 WAVENUMBER SPECTRAL TRUNCATION The most straightforward way to eliminate energy above 2/3 m,, without altering the spectrum up to that wavenumber is to N apply a lowpass filter with cutoff at 2/3 mN to the Fourier transforms of the ice thickness and velocity in the wavenumber domain. Phillips (1959) did this in the original paper identifying the nonlinear instability. At that time, to calculate the discrete Fourier series of a long profile was an 1.0 IT(m)l 0.5 0 max "/2 2TT/3 TT FIGURE A1.6. Filter To Suppress Nonlinear Instability. The length of the taper at 2n/3 is exaggerated for the purpose of illustration. The Nyquist wavenumber is at mAx=jr. expensive procedure, so the method was rarely used. Since the introduction of the Fast Fourier Transform (FFT) by Cooley and Tukey (1965), however, it is quite feasible to use the lowpass filter in Figure A1.6 at each time step with only a modest increase in cost. The filter is smoothed at the corner at 2/3 m 268 with a raised cosine to prevent spurious ripple or sidelobe formation in the spatial domain (e.g_. Kanasewich, 1 975, p. 96). The length of the cosine taper is exaggerated in Figure A1.6. Briefly, the procedure is to remove the mean value and trend from the glacier profile, take its FFT to get the wavenumber spectrum, multiply it by the filter in Figure A1.6 to remove the high wavenumbers, then take the inverse FFT to get back to the spatial domain, then finally add back the trends and mean value originally removed. This method is the most appropriate one when the velocity of the ice does not depend on the local surface slope. Some observations (Meier and others, 1974; Bindschadler and  others, 1977; Budd, 1968) and theory (Budd, 1968; I970[a]; I970[b]) indicate that glacier flow responds to an effective slope averaged over several times the ice thickness, rather than to the local slope, due to the influence of longitudinal stress gradients. Bindschadler (unpublished) used a weighted average of a long-scale slope, to match observations, and the local slope to get numerical stability. Using the lowpass filter eliminates the need to include an amount of local slope in the effective slope definition. The effective slope can be based strictly on the physics of glacier flow, and not on the numerical difficulties. 269 A1.5 ACCURACY A1.5.1 6 PARAMETER: ACCURACY In Section A1.4.2, I concluded that for numerical stability, 6^1/2, and that using a value strictly greater than 1/2 would help ensure stability when nonlinearity is included. However, the accuracy with which the finite difference scheme represents the partial differential equation also depends on 6, leading to a tradeoff situation. I will now compare the transfer function (A1.4.4) with the transfer function for the corresponding partial differential equation, to find conditions on 6 for an accurate solution. When the velocity V is a constant, and mass balance is zero, (A1.1.1) for unit width reduces to Taking the Fourier transform represented by tildes, with respect to x (e.£. Morse and Feshbach, 1953, p. 453), and using the derivative property Bh + Vdh = 0 at dx (A1.5.1) dh = i mh(m) dx (A1.5.2) where m is the wavenumber, and i2=-1, gives dh(m,t) + iVmh(m,t) = 0 dt (A1.5.3) The solution of (A1.5.3) is imVt h(m,t) = h e (Al.5.4) o In a time interval At, the transfer function is then 270 imVAt T(m) = e (A1.5.5) The transfer function T(m) for the partial differential equation has unit modulus at all wavenumbers. Figure A1.7 shows the modulus of the numerical transfer function for several stable values of 6, using VAt = 1 Ax (A 1.5.6) which means that waveforms travel one mesh interval in one time step. It is evident that I must keep 6 near 1/2 to minimize discrepancies between the two transfer functions. Systematic differences in modulus can spuriously create or destroy mass in the glacier. Figure A1.7 indicates that this would be most evident as a nonphysical decay of features at one half the Nyquist wavenumber, or at a wavelength of four meshpoints. FIGURE A1.7. Transfer Function Modulus For Various ©. The analysis of the truncation error by a spatial domain method with different assumptions in Section AT.5.3 also 271 indicates that 6=1/2 is the best choice for an accurate scheme. To compromise between the accuracy problem and the marginal stability at 6=1/2, I usually used values of 6 in the range 0.50 to 0.55 . A1.5.2 PHASE ERRORS Any differences in phase between the transfer functions (A1.5.5) of the partial differential equation and (A1.4.4) of the numerical scheme will distort the glacier thickness, by causing errors in phase velocity, leading to incorrect dispersion. The phase speed at which the component at wavenumber m propagates is 0 V (m) = phase mAt (A 1.5.7) where 0 is the phase of the transfer function. The phase of the transfer function (A1.5.5) of the partial differential equation is 0 (m) = mVAt pde (A1.5.8) Its phase velocity is V, a constant for all wavenumbers. The phase increases linearly with m. The straight line in Figure A1.8 is the phase (A1.5.8) when VAt=Ax/4. The phase of the finite difference transfer function (A1.4.4) using 6=1/2, is 0 (m) = tan-1 fd K(m) 1 - K2(m)/2 (A1.5.9) By using the multiple angle formula for arctangent (Abramowitz 272 and Stegun, 1965, (4.3.26), p. 73) 0 (m) = 2tan"1[K(m)/2] fd (A 1.5.10) K(m) is given by (A1.4.3). The phase (A1.5.10) is also shown in 0(m) 0.5 1 i -7 V"0pde / -\ •.•"*"** r " (for AX/2) -0 1 TT/2 n 3n/2 m AX FIGURE A1.8. Transfer Function Phase Comparison. Solid curve: phase for solution to partial differential equation. Broken curve: phase for the finite difference scheme with mesh interval Ax. Dotted curve: phase for the finite difference scheme with mesh interval Ax/2. Figure A1.8 (broken line) for the same time step At as in (A1.5.8). The finite difference phase obviously diverges badly from the correct value at wavenumbers above one half the Nyquist wavenumber n/Ax, giving large errors in phase speed. In fact, at the Nyquist wavenumber, the phase velocity goes to zero. Any waves sampled at two points per wavelength do not propagate at 273 all! Gary (1975, p. 2.21) discussed this behaviour for a mixed differential-difference hyperbolic system. The numerical scheme causes gross dispersion of the high wavenumber components. The difficulty is due almost entirely to inadequate spatial sampling. When the single time step At is replaced by 100 time steps of lO~2At, there is almost no change in the cumulative phase. The differences cannot be distinguished on the scale of Figure A1.8. If, on the other hand, the spatial mesh increment Ax is halved, keeping the same large time step, the phase is given by the dotted curve in Figure A1.8. The agreement with the correct phase is greatly improved out to higher wavenumbers (note that the Nyquist wavenumber has also been increased. It is now at mAx = 2JT when the mesh increment is Ax/2). It is evident that, to maintain phase accuracy, I must v choose Ax sufficiently small so that the spectrum of the glacier profile h(x) is essentially bandlimited to the region mAx « 1 (A 1.5.11) which means that there are many mesh points per wavelength for all wavelengths having significant amplitude in the spectrum. In the region (A1.5.11), I can evaluate the error in the phase (A1.5.10) relative to (A1.5.8) by expanding sin(mAx) as a Taylor series to third order (Abramowitz and Stegun, 1965, (4.3.65), p. 74) in the small parameter mAx. When K(m)/2 < 1 (A1.5.12) or 274 Ax > V At 2 (Al.5.13) i..e. material moves less than two mesh increments per time step, the power series expansion (4.4.12) of Abramowitz and Stegun (1965, p. 81) can be used for the arctangent. With the restrictions (A1.5.11) and (A1.5.13), the phase is 0 (m) = mVAt fd 1 - (mAx)2 1 + V2At2 6 12Ax2 + 0(mAx)* (A1.5.14) No matter how small I choose the time step At, the fractional error in the phase is still at least (mAx)2/6 \ (Al.5.15) This error can be reduced by smaller mesh intervals Ax subject to (Al.5.13). The results in this section are based on the analysis of the linear equation (A1.5.1), but similar phenomena occur in numerical solutions of the nonlinear analogue, and the restrictions derived here give excellent guidance for the nonlinear case. A1.5.3 TRUNCATION ERROR In the previous sections, I examined the accuracy of the finite difference scheme (A1.1.4) as a function of wavenumber, assuming constant velocity. In this section, I will examine the accuracy of the scheme starting from different assumptions. The truncation error is the difference between an exact solution h(x,t), indicated by tildes, of the partial differential equation (A1.1.1) and an exact solution {h?|j=1,J} of the 275 corresponding system of algebraic equations (A1.1.4). The name arises because the finite differences can be represented by truncated Taylor expansions of the solutions h(x,t) and Q(x,t) of the partial differential equation. The assumption in this analysis is that h(x,t) and Q(x,t) are infinitely differentiable in x and t. This is reasonable since glacier profiles are so smooth. The accuracy is expressed in terms of neglected derivatives at each spatial position, rather than in terms of wavenumber. To derive an expression for the truncation error, first expand all the quantities in the finite difference equations (A1.1.4) as Taylor series with remainder about the point ((j-1)Ax,(n+1/2)At)) in terms of the exact solution h(x,t) and its derivatives there. n+1 n+1/2 At 9h h = h - + j j 2 at n+1/2 At2d2h 8 at: n+1/2 At3 a3h 48 3t3 n+1/2+0 n = h n+1/2 At oh 2 at n+1/2 At2d2h 8 dt: n+1/2 At3 33h 48 at3 n+1/2-0 3 n+1 n+1 Ax dQ Q = Q ± j+1/2 j 2 dx j-1/2 n+1 Ax2 d2Q 8 dx2 n+1 Ax3 a3Q 48 dx3 n+1 j+0 4 j-0 (cont'd) 276 n n Ax dQ Q = Q ± j+1/2 j 2 dx j-1/2 n Ax2 d2 Q 8 dx2 Ax dJQ 48 dx3 n j+0 j-0 (A1.5.16) The derivatives in the expressions for the flux Q are expanded in the same way. d_Q dT d2Q dx2 n+1 n n+1/2 n+1/2 dQ dx" At2 d2Q 2 dx2 At3d3Q 8 dx: n+1/2+0 n+1/2-0 n+1 n n+1/2 d2Q bx At b3Q 2 dx3 n+1/2+0 i n+1/2-0 (A1.5.17) The bars followed by subscripts and superscripts indicate the mesh indices at which the derivatives are evaluated. The 0£ in the remainder terms are real numbers between zero and 1/2. The existence of remainder terms of this form is guaranteed by the Taylor Formula with Remainder Theorem, (e.£. Kaplan, 1952, Theorem 41, p. 357). After substituting these expressions into the finite difference equations (A1.1.4), assuming unit width, and cancelling the terms which identically satisfy the partial differential equation (A1.1.1), the truncation error which remains is 277 € = At j (26-1) d2Q 2 dxdt n+1/2 At' 48 60 c3Q axot2 n+1/2+0 + 6(1-0) a3h at3 n+1/2+0 0 + dx>dt: n+1/2-0 33h ae n-1/2-0 Ax' 48 d3Q © n+1/2 j+0 © + 0-e)-33Q ax3 n+1/2 + (1-©) j+0 a3Q ax3 a3Q ax3 n+1/2 3-0 3 n+1/2 j-0 In terms of dependence on the mesh increments, € = (26-1) O(At) + 0(At2) + 0(Ax2) j (A1.5.18) (A1.5.19) It appears that using the scheme with 6=1/2 results in a minimum truncation error, as the term O(At) goes to zero. This advantage must be traded off against the marginal numerical stability at this value. The coefficients of Ax2, At, and At2 cannot be evaluated exactly because the derivatives of the exact solution h(x,t) are not known exactly, and the shifts 0^ at which they are to be evaluated are also unknown. However, useful estimates of the 278 coefficients can be obtained for order of magnitude effects. First, assume that the derivatives change slowly with time and space, so that the 0£ can be ignored without large error. Then note that for time-independent mass balance, 6 dt Bx = -d2h aT2" (A1.5.20) by using the continuity equation (A1.1.1). The truncation error is then approximately At 3 (1-29)d2h 2 dt2 1 d3h n 1 d3Q n + At2 — + Ax2 — 6 at3 j 24 dx3 j (A1.5.21) and the coefficients can be estimated by the finite difference analogues of these simple partial derivatives, using the numerical solution. A1.5.4 INTERPOLATION ERROR There is an error introduced by representing the thickness hj «. ,/2 at the midpoints of the mesh intervals by a linear interpolation from the primary grid, .i.e. by using h = (h +h )/2 j + 1/2 j + 1 j (A1.5.22)' To estimate this error, write the thickness values at the mesh points as Taylor series with remainder about the true value at a point j+0 between j and j+1. 279 dh h = h + (1-0) Ax — j+1 j+0 dx Ax2 d2h + (1-0)2 — j+0 2 dx2 j + * 0<*,<1 h = h + 0 Ax — j j+0 3x j+0 0: AX2 a2h 2 dx2 j + * O<*2<0 (A1.5.23) The interpolation error E at j+0 is then E = h - (0h + d-0)h ) 0 j+0 j+1 j = 0(1-0) AX (1-0) d2h dx2 + 0 j+* d2h dx2 j+* The interpolation error at 0=1/2 is approximately Ax2 d2h j + 1/2 1/2 8 dx2 when d2h/dx2 is slowly varying with x. The fractional interpolation error is E = E r32h Ax2 h 1/2 = — h dx2 8h (A1.5.24) (A1.5.25) (A1.5.26) For example, on a glacier of unit length, with a mesh chosen so that Ax and h are 0(10"2) and thickness and slope change by of the order of 10% between adjacent mesh points, the fractional interpolation error is E = 10"5 h (A1.5.27) which is quite acceptable. The error is largest where the curvature is largest. The interpolation error also propagates into the dQ/dx term. For instance, if flux Q is proportional to 280 m h , as in the simple case of slab flow with Glen's flow law, (Glen, 1955), the fractional error in Q is f = E = m E Q h (A1.5.28) How does this error in Q affect the flux gradient estimates? Let (Q • |j = 1,J} be the true values of the ice flux, and {Q'. |j = 1,J} with primes be the estimates including the interpolation error. Then Q' = Q (1 + f ) j+1/2 j+1/2 and the error in the representation of dQ/dx is Q' - Q' Q - Q e = j+1/2 j-1/2 - j+1/2 j-1/2 Ax Ax (A1.5.29) Q f " Q f j+1/2 j+1/2 j-1/2 j-1/2 AX Q - Q j+1/2 j-1/2 AX f + f 2 Q + Q i+1/2 i-1/2 2 f - f i+1/2 i-1/2 Ax = d_Q f + Q df dx dx (A1.5.30) Substituting (A1.5.28) for f, 281 e = mAx' 8 d_Q + d2h 1 + Q + a3h - Q d3h d_h dx SP h h cVp h7 dp" dx = mAx • 1 d2h h a~P ao -Lbx Q dh h dx-Q d3h h dx7 using Q=hV. Thus mAx' 8 d2h dv + V d3h dx2 dx dx3 mAx2 V 8 dx (A1.5.31 ) (A1.5.32) which can be kept small by suitably small choice of Ax. .This is the same requirement as found previously for the truncation error and transfer function error. The coefficient of Ax2 can be estimated from the solution profile. It is generally small, because velocity and ice thickness usually vary slowly with x. APPENDIX 2: ICE TRAJECTORY MODEL 282 A2.1 INTRODUCTION This Appendix describes the procedures I have used in the computer model which calculates the trajectories of individual particles of ice, as they travel through the time-varying glac ier. At each time step, the glacier profile obtained with the continuity equation model (Appendix 1) is used to determine the velocity field within the glacier on a two dimensional vertical surface through the glacier centreline, as shown in Figure A2.1. The x axis is along the glacier bed, z is up and normal to x, and y is transverse and horizontal making a right handed system. The gridpoints lie at equal intervals of DZ along lines normal to the bed. These lines are rooted on the bed at the midpoints of the intervals Ax used in the continuity model (Appendix 1, Section A1.1.1, i«e. at equal horizontal increments of Ax, starting at Ax/2 from the origin). The meshpoint (i,j) is the jth point above the bed over X^. The meshpoints divide the glacier section into quadrilateral cells (see Figure A2.1). The top and bottom of each cell are parallel, but the sides may diverge slightly due to the curvature of the glacier bed. This makes necessary some minor geometric corrections, but has definite advantages over a rectangular Cartesian grid when I calculate the velocity components. Jenssen (1977) used a grid with a vertical mesh increment 283 FIGURE A2.1. Mesh For Ice Displacement Calculations. which varied with the ice thickness, based on a atmospheric modelling scheme of Phillips (1957). This scheme simplifies the boundary treatment, at the expense of more complicated derivat ives. 284 A2.2 THE VELOCITY FIELD A2.2 . 1 THE RECTANGULAR FLOW MODEL The velocity vector y_(x) at a point x within the glacier has components v(x) = (u,w,v) (A2.2.1) on the x,y,z axes. The glacier channel (Figure A2.2) is rectangular in cross-section, with width W(x). The ice thickness FIGURE A2.2. The Rectangular Flow Model. The triad x-y-z is the coordinate system, and the bold arrows u, v, and w show the vector components of the velocity field v. h(x,t) and the velocity components u(x,z,t) and v(x,z,t) are assumed to be independent of y, the lateral position in the channel. Only the lateral velocity w(x,y,z,t) varies with y. At 285 the sidewalls of the flow tube (Figure A2.2), the total velocity vector v(x,t) must be in the local plane of the sidewalls, i_.e. w(x,±W,z,t) = ±u(x,z,t) dW 2 2 dx (A2.2.2) and assuming constant lateral strain rate across the channel, dw(x,z,t) = u(x,z,t) dW dy W(x) dx (A2.2.3) A2.2.2 THE DOWNSLOPE VELOCITY The component of velocity u(x,z,t) parallel to the glacier bed at height z where the ice thickness normal to the bed is h(x,t) and the ice surface slope angle is a, is given in Appendix 7 (A7.5.8) as r n+1 n+1-i n u(x,z,t)-u (x,t) = 2A h - (h-z) [s(x)pgsinc] s L I (A2.2.4) n+1 plus some correction terms for stresses and strain rates other than shear parallel to the bed. A and n are the constants in Glen's flow law (Glen, 1955) (see 1.4.22)), p is the density of glacier ice, g is the acceleration due to gravity at the surface of the earth, and us(x,t) is the basal sliding velocity which I discussed in Section 1.4. 286 A2.2.3 THE LONGITUDINAL STRAIN RATE After u(i,j) is obtained at each mesh point (i,j) as in Figure A2.1, the gradient du/dx is estimated by the first term of the finite difference u(i + 1 , j) - u(i-1,j) du(i,j) =  0(d3uAx2) (A2.2.5) c5x DX + DX i " 1 r j i , j If any of the points (i±1,j) on the right side of (A2.2.5) are above the glacier surface, the velocities there are estimated by extrapolation from within the ice mass (purely for the numerical procedure). The error term in (A2.2.5) due to the use of the finite difference is made small by a suitably small choice of Ax. The factor d3u/dx3 is also small, since glacier flow tends to be smooth. Using (A2.2.4) for the velocity gradient (A2.2.5) neglects all the stress and strain rate components in the error terms in (A7.5.9), repeated below as (A2.2.8), yet purports to give the longitudinal strain rate, which is related to the longitudinal stress deviator by Glen's flow law (Glen, 1955). Is this inconsistent? I will show in the following pages that the estimate of the velocity gradient is in fact accurate to within an error term which is usually small. The error term contains stress and strain rate terms other than those parallel to the bed. If u(x,z) is the velocity due to internal deformation by simple shearing from (A7.5.8), without the correction terms (A7.5.9) 287 u(x,z) = 2A n+1 L n+1 n+1 -(h-z) ][s(x )pgsina] n (A2.2.6) then the total velocity component parallel to the bed is u(x,z) = u (x) + U(X,Z)[1+ e(x)] s (A2.2.7) where e(x) is the error term containing other stress and strain rate components subject to the assumptions in Appendix 7. e(x) = 0 2h xx + h yy dx dx J ^gho 6T + (n-1) e - dv /du xz dx/ dz 1 maxJ (A2.2.8) Taking the x derivative of (A2.2.7) gives du(x,z) = du (x) + du(x,z) + d[u(x,z) e(x)] dx —s dx dx dx (A2.2.9) The procedure I described in (A2.2.5) to estimate the longitudinal velocity used the first two terms of (A2.2.9), but neglected the third. In the most favourable case, i..e. e(x) does not vary with x, (A2.2.9) reduces to du(x,z) = eu (x) + du(x,z)[l + e(x)] d~x —s ox; dx (A2.2.10) and the error in my method is always a small fraction of the longitudinal velocity gradient. In the more general case, both e(x) and u(x,z) can vary with x, and the relative error in du/dx in (A2.2.9) is I /du( / dx (A2.2.11) which can be large when u(x,z) changes slowly with x. This may be the case if the glacier near position x behaves like a parallel-sided slab in simple shear. This implies that where the d[u(x,z) e(x)] /du(x,z) dx 288 longitudinal gradient du/dx is very small, i.e. U / I max/ L| du ( X , z ) 57 << where L is the glacier length, and u J max downslope velocity in the glacier, or (A2.2.12) is the greatest du(x,z) dx v(x,h(x) ) ~~hfxl (A2.2.13) then estimates of du/dx in the computer model are unreliable. However, in absolute terms, the error is likely always small, because e(x) is small and presumably slowly varying with x. Unless the glacier has pronounced icefalls, or other steep bed gradient changes, the error is likely to satisfy £[u e] dx U / max/ L (A2.2.14) and the gross flow pattern of the glacier model will be essentially correct. A2.2.4 VELOCITY NORMAL TO THE BED Because ice is incompressible, the divergence of the velocity field v(x,t) is zero, i.e. du + dv + dw = 0 die dz d"y (A2.2.15) The first term is calculated in (A2.2.5), and the third term is given by (A2.2.3) with (A2.2.4). Neglecting basal melt, which is usually less than a few centimetres per year (except during surging, when it may be comparable to surface melting for short periods), gives a boundary condition v(x,0,t) = 0 (A2.2.16) 289 so that integrating (A2.2.7) from the bed to level z gives v(x,z,t) = -I du(x,z,t) u(x,z,t) dW dx W(x) dx dz (A2.2.17) This integral is evaluated at each meshpoint using Simpson's Rule (e.g_. Carnahan and others, 1969, p. 73). This completes the solution for the velocity field y_(x,t) at-each time t. At the glacier surface with normal vector n, the condition OH v • n - —• n - a • n = 0 dt (A2.2.18) must be satisfied. The terms a«n and (dh/dt)«n are the mass balance and the rate of change of ice thickness measured normal to the ice surface. Input values of h and a, and v derived by the numerical procedure described above, when substituted into (A2.2.11) will leave a residual, the size of which indicates the accuracy achieved in determining v. This is used as an independent test of this model in Chapter 2. 290 A2.3 ICE DISPLACEMENT FIELD A2.3.1 FOUR POINT INTERPOLATION Since ice particles may be located anywhere in the cross-section at the beginning of any time step, I must be able to relate any arbitrary quantity f(P) at an arbitrary point P inside a mesh cell to the values of f at the four vertices labelled P0 through P3 as shown in Figure A2.3. Let P be displaced by an amount dP from P0. FIGURE A2.3. Four Point Interpolation Scheme. dP = (6x,6z) (A2.3.1) To find f(P), first find f(A) and f(B) on the boundaries of the mesh cell on the line through P parallel to P0Pi using a linear interpolation. Letting subscripts j on f- indicate the meshpoint (see Figure A2.4), 291 f(A) = f + o f(B) = f + 1 6x 6x Then, interpolating between f(A) and f(B), f(P) = f(A) + f(B)-f(A) DZ 6z (A2.3.2) (A2.3.3) f -f f -f f -f f -f f + 2 0 6x + 1 0 6Z + 3 1 _ 2 0 6x6z 0 DX DZ DX DX 0 . _ 1 0 L DZ -J (A2 The interpolated values f(P) then lie on the surface sketched in Figure A2.4. This surface is linear along any line parallel to the x or z axes. . A2.3.2 DISPLACEMENTS AT MESHPOINTS The next step is to find the displacement from time t0 to time t,=t0+At of the ice which was at meshpoint P at time t0. This is given by t dP = I v(t) dt J t 0 (A2.3.5) This can be approximated by using the arithmetic mean of the velocities at its position at the beginning and at the end of the time step, so that 292 dP = v(P(t0),t0)+v(P(t,),t1) At (A2.3.6) After using the interpolation scheme (A2.3.4) to express the velocity vector yJPtt^ft,) in terms of the components 6x and 6z of dP, letting integer subscripts on the velocity component u indicate the vertex number, and introducing the notation (A2.3.7) for the partial differences of the component u, u (t ) - u (t ) u = X 2 1 0 1 DX u = z u (t ) - u (t ) 11 0 1 DZ u (t ) - u (t ) u (t ) - u (t ) 3 1 1 1 _ 2 1 0 1 XZ DX DX DZ (A2.3.7) together with the obvious equivalent definitions for v, the two 293 components of the vector equation (A2.3.6) yield two coupled nonlinear equations (A2.3.8) and (A2.3.9) for 6x and 6z. 6x = =Atu(t)+u(t)+u6x+u6z+u 6x6z 2L00 01 x z xz-(A2.3.8) 6z = At 2L v (t ) + v (t ) + v 6x + v 6z + v 6X6Z 00 01 X z xz (A2.3.9) Solving (A2.3.9) for 6z and substituting it into (A2.3.8) gives A 6x2 + B 6x + C = 0 (A2.3.10) where A = u v -u v 2 x xz xz x At B=v [u(t)+u(t)]-u [v (t )+v (t )] xz 00 01 xz 0 0 0 1 + u v - u v - _2[u + v ] + _4 xz zx Atx z At2 (A2.3.11) C = v [u (t ) + u (t )] - u [v (t ) + v (t )] 00 02 + 2[u (t ) + u (t )] At 0 0 01 00 01 The solution of (A2.3.10) is given by the standard quadratic formula taking the positive square root. The other displacement component 6z is found by substituting 6x into (A2.3.8). If the point P is in a region where v is positive (upward flow), the procedure described above is used. This is the usual situation in the ablation region of a glacier. However, if v<0, 294 FIGURE A2.5. Cell Vertex Notation For Downward Velocity. the ice at P0 would flow into the cell below the cell illustrated in Figure A2.3. In this case P0 is chosen to be the upper left vertex of the cell, DZ is negative, and similar equations are derived for 6x and 6z. The vertex notation for this case is illustrated in Figure A2.5. This is the usual situation in the accumulation zone. This procedure is repeated for each mesh point (i, j) to find the displacement field throughout the longitudinal section of the glacier. Figure A2.6 shows the displacement field calculated by this method for a glacier in steady state. 295 4000 Steele Glacier Velocity field Steady state with no sliding 3000 CT) CD X 2000k 1000 10 20 30 x (km) 40 50 FIGURE A2.6. Displacement Field In A Steady State. The mass balance and the width for this Steele Glacier model are shown in Figure 3.3. The displacement vectors indicate the flow for 25'years in a steady state with no sliding. A2.4 ICE PARTICLE TRAJECTORIES A2.4.1 TRACKING PROCEDURE At time t0, the cell occupied by each particle P currently being tracked is identified, and the coordinates (6x,6z) of .the particle relative to the vertex P0 are determined. Then, using the interpolation scheme (A2.3.4) where f(P) is a Cartesian displacement component, the displacement components f(P) at P are found using the known displacements at the surrounding 296 vertices. The new Cartesian coordinates of the particle at time t, are then saved. The program then checks to see whether the particle has moved into a new cell, and calculates the new (6x,6z) for the next time step. This procedure is repeated at each time step for each particle being tracked. A2.4.2 PARTICLES WHICH REACH ICE SURFACE If the calculated position of a particle at the end of a time step is above the glacier surface, it has obviously gone too far! The position and time at which it actually reached the glacier surface are interpolated from the surface position and particle position at the two times involved, and this information is saved. A2.4.3 TRACKING BACKWARDS IN TIME For some applications, I do not want to know where the ice is going, but from where it came. For instance, given a sample of ice from a position P' in a borehole at time t', I may wish to know where and when it was precipitated at the glacier surface as snow. To do this I run the model described above backwards through time from t', using At<0. Then (6x,6z) in (A2.3.8) and (A2.3.9) are the displacements of ice particles which arrived at mesh point P0 at time t0, rather than displacements of ice which left P0 at time t0. When At<0, P0 is chosen to be on the downstream boundary of the mesh cell, as in Figure A2.7. 297 FIGURE A2.7. Cell Vertex Notation For Negative Time. A2.4.4 BOUNDARY CONDITION AT UPSTREAM END If the boundary condition at x=0 is zero input flux with zero surface slope (ice divide model), then image points at -Ax/2 are assigned the velocities u(-Ax,z,t) = -u(Ax,z,t) 2 2 v(-Ax,z,t) = v(Ax,z,t) 2 2 (A2.4.1) This prevents the ice particles from flowing across the transverse section at x=0 . If there is a nonzero input flux Q0(t) into the model at x=0, then the fictitious points at -Ax/2 are assigned velocities derived by extrapolation from the mesh points at x>0, and ice particles tracked back across the section at x=0 disappear from the model. 298 APPENDIX 3: ASPECTS OF DISCRETE DATA SERIES A3.1 THE Z TRANSFORM A digitized function on a grid with spacing AX can be represented as a series f(x) = [f ,f ,f , .. ,f ] (A3.1.1) 0 12 J It can also be represented by a polynomial F(Z), where the coefficients are the {fj|J=1,J}, and Z is the unit space shift operator. J F(Z) = 2 f 1 (A3.1.2) j = 0 j (A3.1.2) is the Z transform of (A3.1.1). If the substitution imAx Z = e (A3.1.3) is made into (A3.1.2), the Fourier transform of the digitized function is obtained. J EimjAx f e (A3.1.4) j = 0 j As the wavenumber m goes from zero to 2»r/Ax, Z moves counterclockwise around the unit circle in the complex plane, as shown in Figure A3.1. Convolution of time series is equivalent to multiplying the Z polynomials. The forward Fourier transform is equivalent to summing the terms in the Z polynomial, and the inverse Fourier transform is equivalent to identifying the coefficients of each 299 Alm(Z) -i FIGURE A3.1. The Z Plane, power of Z in the polynomial. A3.2 ALIASING A mesh with spacing Ax cannot resolve a sinusoidal variation with a wavelength less than 2Ax, i..e. with less than two mesh points per wavelength. A signal with a wavelength less than 2AX will be misinterpreted by the mesh as a signal at a longer wavelength. Figure A3.2 shows how a signal with the wavelength 1.5Ax, or wavenumber 4ir/(3Ax), is indistinguishable from a signal with a wavelength 3Ax, or wavenumber 2ir/(3Ax). The limiting wavenumber which is detectable, i..e. ir/Ax is called the Nyquist wavenumber mN. It is also called the folding wavenumber. In general, energy in a signal at the wavenumber (2mkl -m) with 0<m<m,„, will be 'folded' back to the wavenumber m N N within the 'principal alias' [0,mN ]. This phenomenon is called 300 } 1 1 1 1 K\ / \ ./ J| \i ' » / ' »»  i /V < t . » * ' i/ ' * 'VI A -* \ w i A i \ l : \ i / .* » / »*. •y \ ' Wi i FIGURE A3.2. Signal's With Wavelengths 1 . 5AX And 3.0Ax. The sampled signal (spikes) is identical for both the continuous signals. aliasing because the energy at high wavenumbers is disguised as energy at lower wavenumbers due to the discrete mesh. To demonstrate this, consider the signal f = sin[(2ir -m)jAx] j = 1,J (A3.1.1) j Ax which has a wavenumber beyond mN. Basic trigonometric manipulation gives sin[(2jr - m)jAx] = -sinfmjAx] Ax (A3.1.2) This shows that the high wavenumber (2mN -m) signal takes on the same values at the mesh points as a signal at the lower wavenumber m. Similarily, signals at wavenumbers higher than 2mN are folded back into the principal alias. All the energy at wavenumbers (2nmN±m) for integer n in the Fourier spectrum of a signal appears in the mesh at the wavenumber m. This aliasing causes signal distortion. The aliasing problem arises in this study because of the 301 nonlinearity of the flux gradient term in the continuity equation. The nonlinearity pumps energy into wavenumbers above mN at each time step. This energy is then aliased back into the principal alias by the discrete nature of the mesh. This misplaced energy can grow with time and dominate the true solution. This nonlinear instability is discussed in Appendix 1 Section (A1.4.3). 302 APPENDIX 4: DENSITY OF GLACIER ICE A4.1 FIRN AS EQUIVALENT ICE THICKNESS While the firn is generally restricted to the upper 10% or less of an ice mass, the deformation by shearing is concentrated near the bed. In a region of laminar flow parallel to the bed, the ice velocity depends only on e%x , the shear stress parallel to the x axis. Balancing forces on an ice element above a bed with slope c as in Figure A4.1, c (z+6z) -e (z) = p{z) g sin(o) 6z (A4.1.1) xz xz where p(z) is ice density, g is the acceleration due to gravity, and o is ice surface slope. In the limit as 6z goes to zero, a* xz = p{z) g sin(o) (A4.1.2) Sz Integrating (A4.1.2) gives c (z) = g sin(o) f />(y) dy (A4.1.3) xz JO where z is positive downward and the free surface is at z=0. (A4.1.3) shows that the shear stress parallel to the bed depends only on the integral of the density above level z, i.e. the total mass above, and not on its distribution. For z below the firn, the shear stress is not affected by representing the firn by an ice layer of equal mass. Within the firn itself, the stresses are generally too small to cause significant deformation other than compaction. When comparing model results 303 FIGURE A4.1. Force Balance On An Ice Element. with field observations, it must be remembered, however, to use the elevation of the equivalent ice layer, not the observed glacier surface. A4.2 CONSTANT DENSITY ASSUMPTION New snow may have a density as low as 50 kg nr 3 (Seligman, 1936) on falling. The density increases with depth of burial due to compression and metamorphism of snow and firn crystals, until at a density of about 850 kg nr3, the interconnecting air passages between grains are sealed off. The depth at which this occurs can vary widely. Paterson (1969, p. 16) gives two examples. On the upper Seward Glacier (a wet snow regime) the transition to ice occurs at a depth of 13 metres, while at Site 2, Greenland, (a dry snow regime) the transition is at 80 304 metres. In climates with significant summer ablation, ice can also form by the freezing of meltwater percolating down into soaked firn. This is called superimposed ice (e.cj. Paterson, 1969, p. 9). The density of glacier ice itself varies with temperature, pressure, bubble content and debris content. The coefficient of cubical expansion of ice in the range -10°C to -50°C is of the order of 1.5 x 10"4 deg"1 (Hobbs, 1974, p. 350). Thus the total variation in density of ice that could be expected due to a temperature difference of 50°C is of the order of 10"2, or 1%. This is negligible given the other uncertainties in glacier parameters. Values of the bulk modulus of polycrystalline ice at -5°C (Hobbs, 1974, p. 258) are of the order of l01oPa (10s bar). Since maximum basal pressures (overburden load) in ice sheets are of the order of 107Pa, the maximum variation in density to be expected due to pressure contraction is 10~3, or 0.1%. The assumption that ice is incompressible is included explicitly in Glen's formulation of the flow law for ice (Glen, 1955). The presence of gas bubbles within glacier ice can have a larger effect on density. Seligman (1936, p. 119) gives a variation of 38 kg irr3 or about 3% between measurements on white bubbly ice and blue bubble-free ice. The bubbly ice could be compressed by pressure on flowing to depth by up to this 3% by merely compressing the gas in the bubbles. Even a 3% variation is negligible, however, given other assumptions of modelling. The inclusion of debris in the basal layers of a glacier may increase the local density considerably. For example, 50% 305 debris by volume (a large amount) with a density of 2100 kg nr3 would raise the density of the ice-debris mixture to 1500 kg nr3, or by over 50%. It would presumably also alter the flow law parameters in the basal region where a large part of the deformation of a glacier takes place. There is very little that one can do about this source of error in the numerical model. The only salvation is the observation that extensive debris is usually restricted to within a few metres of the glacier bed. For most computations carried out in this study, the density of glacier ice was taken as 900 kg nr3. 306 APPENDIX 5: CONTINUITY EQUATION FOR AN ICE MASS A5.1 MASS CONSERVATION IN A MOVING CONTINUUM Let M(t) be the volume integral of a continuously differentiable function />(x,t) defined in a volume V(x,t) enclosed by a surface S(x,t) moving with the velocity v(x,t) of the continuum. The position vector is x and t is time. Underscores indicate vector quantities. Then M(t) = fff />(x,t) d3r (A5. 1 ) V When p(x,t) is the material density, M(t) is the mass contained within V. The material derivative DM/Dt is the rate of change of M(t) with respect to time 't' (e.g. Malvern, 1969, p. 211). CCC d/»(x,t) rr DM = /fi d3r + / / ,(x,t) v(x,t)-n ds Dt JJJ 6t JJ (A5.2) V S where n is the outward unit normal to S. By the law of conservation of mass, DM = 0 (A5.3) Dt For ice masses in this study, the firn and the mass balance are expressed as equivalent ice thickness, and the density of ice is taken constant. These assumptions are examined in Appendix 4. Under these assumptions, (A5.2) reduces to v(x,t) - n ds =0 (A5.-4) S c _ _ _. tflU-307 A5.2 IN A STATIONARY GLACIER CROSS-SECTION Although (A5.4) was derived assuming the surface S moved with the continuum, (A5.4) is also true for a stationary surface that corresponds to S at the instant considered. Since />y_*n is the mass flux density, (A5.4) states that the net ice flux across the surface S is zero. To put this into a usable form, let S comprise two transverse sections S, and S2 through the ice FIGURE A5.1. Surfaces for Derivation of Continuity Equation. mass in Figure A5.1, plus the annular surface joining them, i_.e. the sections TOP and BED. Then (A5.4) is J*J*v• n ds + j!^*v*n ds + j*J*v • n ds + v«n ds =0 S S TOP BED 1 2 (A5.5) Now consider the motion of the free surface TOP defined as the endpoints of position vectors h. As the surface TOP moves, dh/dt«n is the normal velocity of the free surface. The vector 308 dh/dt is determined only to within an arbitrary constant vector F normal to n, or tangent to the surface TOP, i_.e. dt dh — • n dt " n + F F-n = 0 (A5.6) Similarily, the mass balance a(x,t) at the surface is arbitrary to within a constant vector G tangent to the surface TOP, but the component a«n is the normal velocity of the melting surface or accumulation surface with respect to the material. a(x,t) = (a•n) n + G G-n = 0 (A5.7) It is apparent that the net normal velocity of the surface must be the sum of the normal velocity y(x,t)«n of the material at the surface plus the normal velocity into the material of the melting surface, i_.e. , dh —«n = v«n + a*n (A5.8) dt ~ " ~ where y(x,t) is the material velocity. Similarily, on the constrained surface BED y-n + b-n = 0 (A5.9) where b(x,t) is the rate of melting or freezing at the bed. Letting Q, and Q2 be the total fluxes through S, and S2 using the downslope unit normal, (A5.5) becomes Q2~ °i+ SS """*-ds ~ SS~'~ds ~ SS~'~ds = ° TOP TOP BED (A5.10) To proceed further, it is necessary to choose some axes. Let the x' axis run along the glacier surface down the "centre" of the channel (how this is defined is not crucial). The z' axis is 309 orthogonal to x' and positive upward in the vertical surface containing the centreline. The horizontal y' axis completes a right-handed orthogonal coordinate system. These axes are shown in Figure A5.1. Next, let the surface S become thin such that and S2 intersect the glacier surface at x, and x2 separated by a small amount 6x' (see Figure A5.2). Then, when W(x',Z') is the channel '^.^datum FIGURE A5.2. The Thin Cross-section Limit. width at the level Z'(x',y') of the ice surface above some datum level, (A5.10) becomes 310 0 = Q (x' ) - Q (x' ) 2 2 11 + 6x' rW(x',Z') W(x',Z') W(x',Z') -i r an r r I —• n dy - I a• n dy -I b«n dy / ot J J 0 0 o J (A5. 1 1 ) A5.3 IN AN ARBITRARY CHANNEL At this point there are two possible lines of development, dependent on the form of channel cross-section to be modelled. Letting (A5.12) define a scalar B(x',t) which is mass balance plus basal melting per unit channel width normal to the ice surface at height z'=0 and averaged across the glacier channel width W(x',0) , W(x',0) B(x',t) W(x',0) = I (a-n + b-n) dy' (A5.12) *0 and recognizing oh dS(x',t) —•n dy = — dt dt (A5.13) where S(x',t) is the transverse cross sectional area of the glacier in the y'-z' plane, (A5.11) becomes Q (x') - Q (x') + dS(x',t) 22 11 — = B(x',t) W(x',0) (A5.14) 6x' ot In the limit as 6x'-» 0, 31 1 dS(x',t) + dCj(x',t) = A(x',t) W(x',0) (A5.15) d~t dx' This formulation (A5.15) is useful when the channel width varies with depth z' (e.g. Bindschadler, unpublished; Raymond, 1980). A5.4 IN A RECTANGULAR CHANNEL In this study, however, I use models .with rectangular cross section, so that the cross sectional area S need not be introduced. In such a channel (Figure A5.3), the thickness and the x' and z' velocity components are assumed to be independent of y'. The transverse velocity component is such that at every point on the channel wall, the total velocity vector is in the plane of the wall, i.e. there are no voids along the margins. The transverse strain rate is then assumed to be constant across the channel. I have described this model in Section A2.2.1. This approach was suggested by Nye (I959[c]). It is. quite a good model for nonparallel flowlines on ice sheets, where the "walls" of the "channel" are fictitious. Even for valley glaciers, the same approach may be used. W(x) may be thought of as the width of a narrow band of flow lines near the glacier centreline, e.g. a few percent of the valley widt