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Modelling subglacial erosion and englacial sediment transport of the North American ice sheets Hildes, Dave Henry Degast 2002

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Modelling Subglacial Erosion and Englacial Sediment Transport of the North American Ice Sheets by Dave Henry Degast Hildes B.Sc. (Honours), Queen's University, 1991  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Department of Earth and Ocean Sciences) We accept this thesis as conforming to the required standard  The University of British Columbia December 2001 © Dave Henry Degast Hildes, 2001  In  presenting  degree freely  at  the  available  copying  of  department publication  this  in  partial  fulfilment  University  of  British  Columbia,  for  this or of  thesis  reference  thesis by  this  for  his thesis  and  scholarly  or for  her  g? ^> 0  of  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  '  °&  I  I further  purposes  gain  the  shall  requirements  agree  that  agree  may  representatives.  financial  permission.  Department  study.  of  be  It not  is be  that  the  for  Library  an  advanced  shall  permission for  granted  by  understood allowed  the  make  extensive  head  that  without  it  of  copying my  my or  written  ?  Abstract T h e glacial geology of N o r t h A m e r i c a is a rich resource u p o n which reconstructions of ice sheets are m a d e .  N u m e r i c a l modelling of ice sheets based o n ice physics is a n alternate avenue for  cryospheric reconstruction.  However, such algorithms are unable to draw f r o m the full wealth  of geomorphic d a t a because a large-scale forward m o d e l of basal processes, necessary to link the ice a n d the b e d , is not available. I develop a process-based m o d e l of sediment p r o d u c t i o n , entrainment, deposition a n d transport to fill this conspicuous gap. Subglacial abrasion is m o d e l l e d following  Hallet  constructed.  Semmens, 1995] deposited.  [1979, 1981] a n d a quarrying m o d e l , dependent o n s u b c r i t i c a l crack growth, is  E n t r a i n m e n t proceeds predominantly b y intrusion into the sediment  [Iverson and  a n d when basal melt exceeds the rate of entrainment, englacial sediment  is  B o t h vertical redistribution of englacial sediment w i t h i n a n ice c o l u m n a n d l a t e r a l  transport of debris is considered.  Ice entrains loose debris f r o m the bed, transports it downstream a n d deposits a n allochthonous debris t r a i n , observable on the m o d e r n landscape.  T h i s can be m o d e l l e d when a n accurate geo-  logic representation of the b e d is used, allowing the exploitation of distinct lithologies as n a t u r a l tracers of ice m o t i o n . A l s o , i n c o r p o r a t i o n of the differences i n physical properties between various lithologies is possible w i t h such a lithologically realistic description of the b e d . Several processes are functions of b e d topography a n d a description of small-scale topography w i t h i n a large-scale g r i d is required; a downscaling m e t h o d is therefore developed.  Basal-ice processes must be coupled to b o t h ice sheet conditions a n d subglacial hydrology. T h e M a r s h a l l - C l a r k e thermo-mechanicalice sheet m o d e l  [Marshall,  1996;  Marshall and Clarke, 1997a,b;  Marshall et al., 2000] gives the necessary ice sheet fields a n d provides the forcing for the hydrology m o d e l of Flowers [2000] which i n t u r n delivers subglacial water pressure, used for b o t h the quarryi n g a n d entrainment simulations. C o m p a r i s o n s of m o d e l results w i t h the d o c u m e n t e d large-scale debris trains of H u d s o n B a y Paleozoic sedimentary rocks a n d D u b a w n t G r o u p detritus are used to test the veracity of the transport m o d e l while estimates of paleo-erosion are used to assess the erosion m o d e l .  ii  Contents Abstract  ii  Contents  in  List of Tables  vi  List of Figures  vii  List of Symbols  ix  Acknowledgements  xvi  Dedication 1  Introduction 1.1 1.2 1.3 1.4  2  3  xviii ^ 1  Evidence of an ice age Modelling an ice age Elements of the basal processes model Thesis objectives and structure 1.4.1 Scales, weights and measures 1.4.2 Outline of chapters  1 3 5 7 8 8  Downscaling Bed Slopes 2.1 Direct bed slope calculation from D E M data  10 11  2.2 2.3 2.4  15 16 22  Fractal parameters of bed slope Downscaling methodology Application to North America  Erosion 3.1 Abrasion 3.1.1 Adhesive and abrasive wear  3.2  24 24 24  3.1.2 Glaciological abrasion 3.1.3 Parameter determination—geological 3.1.4 Parameter determination—non-geological Quarrying  iii  25 29 32 35  4  3.2.1  Subcritical crack propagation  35  3.2.2  Subcritical crack parameter determination  37  3.2.3  Effect of topography  41  3.2.4  Cavity size  47  3.2.5  A quarrying model  48  3.2.6  Quarrying sensitivity  51  Englacial Processes 4.1  4.2  4.3  53  Entrainment  53  4.1.1  55  Overriding of pre-existing ice  4.1.2  Freeze-on  55  4.1.3  Intrusion by regelation  57  4.1.4  Englacial tectonics  59  4.1.5  A n entrainment model  61  Englacial mixing  67  4.2.1  68  Englacial tectonics — mixing  4.2.2  Particles in a shear flow  69  4.2.3  A mixing model  69  Advection  71  5 Model Inputs 5.1  5.2  5.3  7  Ice model  74  5.1.1  Theory  74  5.1.2  Examples of the ice sheet model  76  Hydrological model  78  5.2.1  Sheet flow  78  5.2.2  Groundwater flow  80  5.2.3  The aquitard  81  5.2.4  Examples of hydrological results  83  Geological inputs  84  5.3.1  Bedrock geology  84  5.3.2  Surficial geology  88  6 Modelled Simulations of North American Ice Sheets 6.1  6.2  91  Transport results  91  6.1.1  Southward carbonate transport  91  6.1.2  Eastward Dubawnt Group transport  98  Erosion results  103  6.2.1  Cordilleran erosion  103  6.2.2  Laurentide erosion  107  7 Conclusions 7.1  4  HO  Summary of work  HO  iv  7.2  Outlook  112  References  114  Appendices  133  Appendix A 2D Stress  133  A.1 Evaluation of state of stress A.1.1 Normal loading A.1.2 Shear loading A . 2 Evaluation of failure criterion  133 133 137 139  Appendix B Single Clast Rotational Mixing B . l Perturbation due to a cylinder B.2 Steady state rotation B.3 Resultant mixing  v  143 143 147 148  List of Tables 2.1 2.2  Details of sub-grid analysis areas Coefficients for downscaling  12 17  3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9  Hardness of geological materials Thermal resistivity of geological materials Parameters for erosion sensitivity test for Z Parameters for erosion sensitivity tests for R and 6 Kj for various geological material nj for various geological materials Case details for F R A N C 2 D stress analysis Parameters for stress cross section Parameters for quarrying sensitivity tests  29 30 31 34 37 40 41 51 52  4.1  Parameters for vertical grid  63  5.1 5.2 5.3  Parameters for ice sheet model Parameters for hydrology model Bedrock types of North America  77 83 86  5.4  Surficial materials of North America  89  6.1 6.2  Parameters for model runs Correlation coefficients  T  94 108  vi  List of Figures 1.1 1.2 1.3  Stable oxygen isotope record for the past 2.5 Ma Conceptual diagram of inversion and forward modelling Medial moraines  2 4 6  2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10  Abrasion sensitivity to 9 Sub-grid analysis areas A sketch showing sub-grid discretization Slope histograms Mean slope scaling Fractal dimension Variance downscaling Fourth moment downscaling Histograms for sampled and downscaled data Corrected and uncorrected dip angles of North America  10 11 13 14 15 16 18 19 21 23  3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11  Sketch of local bed dip and velocity components Abrasion sensitivity test for Z Grain-size distribution Erosion sensitivity tests Mesh for FRANC2D stress analysis <r results for the FRANC2D stress analysis (T results for the FRANC2D stress analysis Sketch of topography simplification Maximum shear stress results for the FRANC2D stress analysis 2-D stress cross sections Quarrying sensitivity tests  26 31 32 34 42 44 45 46 46 50 51  4.1 4.2 4.3 4.4 4.5 4.6 4.7  Debris-laden basal ice A sketch of intrusion Entrainment processes Grid for basal ice and sediment Sketch of energy density used in freeze-on calculation Dependence of l on C A conceptual picture of dispersion by collision  54 58 62 64 66 67 70  T  xx  yy  a  vii  4.8 4.9  A sketch the of Prather [1986] advection scheme A n example of the Prather [1986] advection scheme  72 73  5.1 5.2 5.3 5.4  L G M ice sheet reconstruction L G M hydrology reconstruction Bedrock geology Initial sediment depth  77 82 85 90  6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11  Bedrock lithology and observed carbonate dispersal in the Hudson Bay area . . . . Simulation results of the Hudson Bay carbonate dispersal train Simulation results of the Hudson Bay carbonate dispersal train Simulation results of the Hudson Bay carbonate dispersal train Bedrock lithology and observed Dubawnt Group dispersal i n western Nunavut . . Simulation results of the Dubawnt Group dispersal train Simulation results of the Dubawnt Group dispersal train Simulation results of the Dubawnt Group dispersal train Volume of Cordilleran ice Model results of Cordilleran erosion Model results of continental erosion  93 95 96 97 99 100 101 102 104 106 109  A.l A.2 A.3 A.4 A. 5  Sketch of an irifinite Une of force normal to the plane Sketch of discretization for normal loading convolution integral Sketch of an infinite Une of force tangential to the plane Sketch of discretization for tangential loading convolution integral A sketch of a' versus cr5 showing a Mohr circle and failure envelope  134 135 137 138 141  B. l  Geometry of rotating cyUnder problem  144  B.2  Vertical component of total velocity field  149  N  viii  List of Symbols Symbol  Description  Units  a, b  indices of stress tensor unit area  —  A A  areal abrasion rate  m s  a,b,c™ '  downscaling coef.  4.  grid transformation parameter  — —  1  m  25  2  10, 25, 106, 108  _ 1  17-19 63-64  -3,-4,-B  ac  advection polynomial coefficient  A  G  Glen flow law parameter  a-  A  L  subcritical crack growth parameter  m s  m  Page reference  Pa"  1  71-72 47  3  36, 48-49  _ 1  —  am7 b-m ratios of cell size in mesh segment volumetric abrasive abrasion rate  42  m s 3  25  _ 1  A?  volumetric adhesive abrasion rate  m s-  b bf  mass balance rate  m s  fraction of uncovered bed  ^melt  —  basal melting rate  m s  B  Glen flow law parameter  Pa"  3  B  sliding law parameter  m s  _ 1  C  debris fraction max. C for Weertman regelation  — —  cohesion  Pa  specific heat capacity of ice  J kg" K -  d  depth of sediment  m  28, 48, 67,81-82, 88-90  V  general data space  3-4  d  characteristic depth of sediment  — m  28, 49  do  coefficient of sediment depth  m  28  di  coefficient of sediment depth  m  28  observed data  3-4 67  0  3  C  reg  CO  3  predicted data  — —  tf  max. Weertman regelation layer  m  e g  74-75  _ 1  88-89 27, 65, 75, 76,78  _ 1  s-  ix  47, 75, 77  1  Pa  - 1  76, 77, 102 25, 28, 54, 66, 67, 69-73 67 141  1  d° d p  25  1  1  56, 65, 75, 77  3-4  aquitard thickness  m  minimum aquifer thickness  m  Dco\ D  collisional diffusion parameter  m s  debris eddy-diffusion parameter  Ded  81-83 81, 83 2  _ 1  69, 92  m^-  1  68  characteristic diffusion parameter  m  _ 1  68, 92, 98  ^max  maximum initial sediment depth  m  ^min  minimum initial sediment depth  m  -Crot  ed  s  2  88 88  rotational-diffusion parameter  2  m s-  1  D'rot  single rotational-diffusion parameter  m s  _ 1  -Dthrust  thrust diffusion parameter  m s  Act  total diffusion parameter  m s"  excess stress constant  — —  expected value  T  energy available for freeze-on inverse model activated forward model  ^  unactivated forward model  X  N fN fN Jbed F  normal load on erosional surface  2  2  depth of freeze-on  69  15  Jm-  2  65-66  — — —  3-4 3-4 3-4  N  25-27  —  viscous drag modifier  68, 92, 98  48  viscous drag modifier  f Jbed Fd  69  1  Pam"  2-D tangential boundary forcing  69, 92, 148  _ 1  2-D normal boundary forcing  fs T  2  1  49-50,132 26  Pa m "  1  49-50,136  —  27  m  65  crushing strength modifier  49  g  acceleration of gravity  ms  eL  Green's function tangential force  m-  1  137  s i  Green's function normal force  m"  1  132  h  slope and dip class index  —  12  groundwater thickness  m  80-83  thickness of ice  m  65, 74-76, 132  subglacial sheet thickness  m  78-80  critical sheet thickness  m  79-80, 8 3 , 1 0 8  p  abrading particle hardness  Pa  25, 28, 29, 84-87  s  abraded surface hardness  Pa  25, 28, 2 9 , 8 4 - 8 7  model-grid indices  —  H H  ice  w TJW  •"crit  h h K  apparent conductivity of sediment  - 2  m Pa- s2  1  aquifer conductivity  m s  _ 1  aquitard conductivity  m s  _ 1  1  58-59,65 81, 83 81-83  conductivity of sheet  m s  _ 1  78-80  max. sheet conductivity  m s  _ 1  79-80, 83  K min  min. sheet conductivity  m s  _ 1  79-80, 83  &a  hydraulic conductivity parameter  ^abr  abrasive wear coefficient  &ad  adhesive wear coefficient  h  hydraulic conductivity parameter  •^•frozen  frozen ground conductivity  — — — —  K  w  max  w  m s  79, 83 25, 28-30 25 79, 83 80, 81, 83  _ 1  thermal conductivity of ice  W m- K -  K  stress intensity factor  Kl Kt  L  L  1  1  56, 58-59, 65  Pa m /  2  35-37, 48-4£  rmnimum stress intensity factor  Pa m /  2  36  critical stress intensity factor  Pa m /  2  35-38, 48-4£  confined critical stress intensity  Pam /  thermal conductivity of clast  W m" K  crack mode index  —  36  1  1  1  1  38  2  1  lithology index  _  1  58-59  'a  effective depth of basal ice  m  58, 65-67  'a  array depth modifier  66, 67  *a  equilibrium l  — m  65-67  /max  max. intrusion per At  m  67  L  cavity size  m  47-48  half crack length  m  36, 38, 48  maximum normalized cavity size  — —  47  jeq  a  a C  la jJ  max  L\  normalized cavity size  Lfus  latent heat of fusion  Jkg-  47-48 26, 58, 65  1  model layer thickness  m  63-64  L  topographical wavelength  m  41, 47-48  M  general model space  3-4  Mi M m m  Model 1  — — —  m  slope of tangential fine forcing  m m, n  geometric modifier of KL  n n" n  aquifer porosity  t  2  N  r  s  g  gw  G  Model 2 slope of normal fine forcing  Pa m ~  recovered model  —  98-102,108 94, 105-108 2  3-4  P a rn"  137  — — — — —  36  2  sub-grid indices porosity of sediment Glen flow law exponent  XI  133  81, 83 58-59 47, 75, 77  N.  number of ice or sediment layer  m  63-64  TIL  subcritical crack growth exponent  36-39, 48-49, 84-87  Nrr,  # of longitudinal sub-grid cells  •-Yrnesri  number of cells i n mesh segment  N  # of latitudinal sub-grid cells  — — — —  P  perturbation velocity field  ms  V  general property  —  it  n  pressure melting coefficient p  88 42 88 143-146  - 1  15  KPa"  26, 58  1  gw Pi P  groundwater pressure  Pa  80-83  ice overburden pressure  Pa  41, 47-51, 58, 65, 79, 83  subglacial water pressure  Pa  41, 47-51, 58, 65, 78-83,106  q  distribution moment index  Q  area! quarrying rate  —  Qgw  groundwater flux  Q  A  15-19  m s  48, 106,108  _ 1  m  2  s-  subglacial sheet flux  m  2  s  Qi  creep activation energy of ice  Jmor  qi  heat flux through ice  Wm-  R  radius of englacial particle  m  r  grain size index  r,9,z  polar coordinates  R&™  ideal gas constant  — —  R+  transition radius  Re  radius of the Earth  S  W  1  80  _ 1  78 47,75, 77  1  56  2  10,26-27, 32-33, 69, 143-148  J mol" K 1  1  47, 75  m  26-27, 30  crushing strength of ice  Pa  49  Ss  shear strength of ice  Pa  49  t  time  T  characteristic adfreezing time  s  65  Ti T^bed i T •*pmp U  temperature of ice  K  47, 56, 65-76  basal ice temperature  K  75, 76  pressure melting point  K  fluid velocity field  m s  "ext  longitudinal extension velocity  Vi  c  75 _ 1  143-148  m s  _ 1  27  ice velocity  m s  _ 1  69, 71-73, 75, 76,142-144  Vi  vertically averaged ice velocity  m s  - 1  74  L  mode L crack speed  m s  _ 1  36, 48-49  Vn  ice velocity normal to bed  m s  _ 1  26-27  particle sliding velocity  ms  speed of intrusion  m s  V  V  P  V  r  xu  25,27-28  - 1  - 1  58-59  v  sliding velocity of ice  m s  characteristic shding speed  m s  w  vorticity field  s-  X  coordinate in direction of shding  x'  convolution integration variable  —  s  68  _ 1  144  1  48  m  Cartesian coordinates  — —  general coordinate  x,y,z z  27, 33, 47, 68,  1  133  characteristic depth  m  68  ^bed  bed elevation  m  12, 79  z  normalized standard deviation  —  s d  17  Z  thermal resistivity of particle  mKW-  OL°  aquifer compressibility  Pa"  ice-surface dip  —  r W  a  ice  velocity gradient  s-  1  1  26, 30-31 81, 83 137 69, 142-148  1  compressibility of water  Pa"  convolution integration variable  m  135  elevation difference  m  15  Kronecker delta  — — —  139  m  132  Cartesian coordinate interval  m  132  Cartesian coordinate interval  m  132  mixed layer depth  m  k  shear strain rate in ice  •bed  Pv, X Az  hed  Sab Aj> AO Ax  longitude interval latitude interval Cartesian coordinate interval  Ay Az A z  mix  1  80, 83  69  s"  1  27  shear strain rate of basal ice  s-  1  76  iab  shear strain rate in ice  s-  1  75  e  slide enabling modifier  K  65, 75, 77  e  cut-off length for 2-D stress  m  138  V  effective viscosity  Pa s  26, 33, 143  1  arbitrary angle  —  139  li A(-)  fractal dimension of q  —  15-16  % €  T  x  th  principal stress rock friction coefficient  /*  -1X10"  moment  5  parameter for downscaling internal friction  Mice  internal angle of friction of ice  Mint  KJ internal coefficient of friction  xiii  Pa  48-51, 139  — — —  25, 27, 30 20 141  —  50  —  38  4  g  §  t h  — —  moment of distribution  Va  principal axis direction  ft  angular velocity  4>  longitude  s-  <j>  139 143-148  1  —  internal angle of friction r:a  15-22  141  runoff and groundwater exchange  m s  _ 1  sheet and groundwater exchange  m s  _ 1  $ J.mean ^grain  strain heating  J m-  mean logarithmic particle size  <b . ™ grain  standard dev. of log particle size  — —  d  sd  vertical exchange of sediment  80, 82 78, 80-81 s-  3  1  75 33 33  m s  71  _ 1  stream function  m  hydraulic head  m  density of ice  kg m ~  density of englacial particle  kg m  density of water  kg m ~  <7ab  alternate stress tensor  Pa  N  modified normal stress  Pa  *7  2  Pa  0~ab  stress tensor  Pa  36, 43-46, 74, 132-139  deviatoric stress tensor  Pa  75  excess boundary stress  Pa  48  normal stress  Pa  38,139  °s  shear stress  Pa  139  _max °S 'max S  maximum shear stress  Pa  45  modified maximum shear stress  Pa  45, 48-51  r  basal shear forcing  N  76  r  basal shear stress  Pa  41, 47-51  shear stress i n the basal ice  Pa  27  t9 freq. count  — — — — —  17-21  Pi Pr Pw -1  a  <l  ao  a  T  ice  0  n d  of deviatoric stress  bed dip 0  modified i? freq. count  0  modified freq. count of bed slope  9  characteristic bed dip  9  latitude  ^mean  mean bed dip  — — —  bed slope parameter for downscaling  xiv  s  2  144-146  _ 1  78-79 3  - 3  3  74, 77 26-27 78, 80 139 141  2  75  10-23, 26-28, 33, 45-47, 68 20-21 33 68 22-23, 28 11-23 20  i?  e w  east-west bed slope  i?  n s  north-south bed slope  — —  downscaled bed slope  —  17  transformed grid  —  63-64  i?  D S  xv  12 12  Acknowledgements I have been fortunate in the course of my graduate career. I came to U B C to study glaciology on somewhat of a whim and have often wondered at my luck of landing where I did. Glaciology attracts all types from wild to wonderful and it has been a pleasure to work alongside Urs Fischer, Shawn Marshall, Doug LaPrairie, A l u n Hubbard, Jeff Kavanaugh, Gwenn Flowers, Dori Kovanen, T i m Creyts, Fern Webb and Nicholas Lhomme. Garry Clarke, my research supervisor, of course has had an important hand to play and deserves many thanks. He has always given me plenty of Une to run with resulting in an eclectic and varied (if not short) graduate experience. While it was evident that Trapridge would not haunt the pages of this thesis, Garry nevertheless gave me the opportunity to participate i n four Yukon field seasons. Trapridge is a magical place that has indeUbly marked me. Garry, Alun, Jeff, Gwenn, Kuan-Neng, Thomas and the fine folk at Kluane Base Camp all played a part in brightening the experience. It was the nature of the people involved in earth sciences that wooed me away from my undergraduate beginnings in chemical physics. I cannot imagine a more stimulating and enjoyable work atmosphere than that which exists in our department. In particular I would Uke to thank John Amor and Gerry Grieve for their extensive computer support; Chris Wijns and Dave MacMiUan for late night jams; Stephane Rondenay, CamiUe L i and Lara Fletcher for being excellent housemates; Eldad Haber, P h i l Hammer, Mathieu Dumberry, Len Pasion and Yuval for being amiable coffee room occupants and ski partners. For the company of Shawn and Gwenn i n conferences (inclusive of late night preparations), road- and ski-trips I am grateful. It is not clear whether A l u n helped or hindered my doctorate, but there is no doubt he contributed a dose of adventure to my time at U B C which, along with his continued friendship, I value greatly. I am appreciative that my family have always been accepting of my choices with surprisingly few sniggers along the way. Tracey WaUace has been patient and supportive during my times of thesis angst for which I am thankful. M y daughter Micah has helped immensely by being obUvious and always ready to play. I would Uke to thank my committee for their input to this thesis. Garry i n particular has helped enormously with the poUshing of rough drafts. Thanks also to T i m Creyts, Gwenn Flowers and xvi  K i m Goodliffe whose generous editing has improved the final product. Any errors are no doubt last minute inclusions, bypassing all my careful readers. The National Science and Engineering Research Council ( N S E R C ) provided funding for my studies and the Northern Scientific Training Program ( N S T P ) provided partial support for field research at Trapridge Glacier.  D A V E HENRY D E G A S T HILDES  The University of British Columbia October 2001  xvii  To the memory of my father who piqued my interest i n the North, who instilled a wonderment of wild places and who cultured an interest in the nature of things. M y path to glaciology is a direct result of all three.  XVlll  Chapter 1  Introduction 1.1  Evidence of an ice age In my opinion, the only way to account for all these facts and relate them to known geological phenomena is to assume that ... the earth was covered by a huge ice sheet which buried the Siberian mammoths, and reached just as far south as did the phenomenon of erratic boulders. ... Springs dried up, streams ceased to Row, and sunrays riding over that frozen shore (if they ever reached it) were met only by the whistling of northern winds and rumbling of the crevasses as they opened across the surface of that huge ocean of ice. [Agassiz, 1840, translated by Carozzi [1967]]  W i t h his assertion that great ice sheets stretched from the Pole to the Mediterranean, Agassiz, a great intellect of his day, established the idea that glacial periods and large climate oscillations occurred i n Earth's history (not to mention the eponymous M t . Agassiz, Manitoba, the ski area of my childhood). He was not alone i n recognizing that glaciers alter the landscape, as his peers applied this principle to infer a greater past extent of Alpine glaciers. . However it was Agassiz who pushed these ideas forward to invoke large-scale, continental glaciation ; the new theory was not well received. Nevertheless, in addition to data provided by his predecessors, Agassiz amassed a large body of corroborating evidence from the Jura mountains and the Alps, the British Isles and North America. This substantial collection of observations comprising erratics and striated 1  2  T h e " g l a c i a l t h e o r y " e x p l a i n e d the existence o f a l l o c h t h o n o u s b o u l d e r s a n d s t r i a t e d rocks d i s t a n t f r o m the observed glacier t e r m i n u s b y a s s u m i n g greater ice extent i n the geologic past. It was developed b y I. Venetz a n d J. G . de C h a r p e n t i e r s t a r t i n g i n 1818 w i t h a large a m o u n t o f field evidence. A g a s s i z was a s u m m e r guest o f de C h a r p e n t i e r i n 1836, where he o b t a i n e d a n i n t r o d u c t i o n t o g l a c i o l o g y a n d de C h a r p e n t i e r ' s theories. 1  K . S c h i m p e r , A g a s s i z ' s c o l l a b o r a t o r o n the theory o f c o n t i n e n t a l g l a c i a t i o n , was the first t o c o i n the t e r m "ice-age". 2  1  CHAPTER 1. INTRODUCTION  2500  1500  Time (ka BP)  500  140  100  50  0  Figure 1.1. The stable oxygen isotope record of benthic foraminifera from Ocean Drilling Program site ODP-677 (1.2°N, 83.73°W) [Shackleton et al, 1990]. As a proxy for global ice volume, the 6 0 record suggests an oscillating climate over the past « 2.5 M a coinciding with the period of northern, mid-latitude Pleistocene glaciation. The most recent data, which includes the period modelled i n this study, is expanded i n scale from 140 ka B P to present. 18  rocks gave enough credence to the theory that it was eventually accepted as mainstream geology. Evidence that Earth has undergone many glacial oscillations is plentiful. Deep ocean cores are a rich resource because they contain records of global ice volume, through proxies such as the SlsO stratigraphy (Figure 1.1) obtained by analysing benthic foraminifera. The character of the time series i n Figure 1.1 has a marked shift at 1 M a before present ( B P ) where the oscillation amplitude increases and a 100 ka cyclicity becomes apparent. In Chapter 6, simulation results are presented from 120 ka B P to present day, encompassing a full 100 ka glacial cycle. 3  In addition to the wealth of paleoclimatic information recoverable from ocean basins, cores from modern ice sheets provide high resolution coverage of the most recent glacial cycle, the Wisconsin Glaciation. While these data represent the time evolution of quantities at one point i n space, the terrestrial evidence for past glaciations has good spatial coverage but poor temporal extent. The terrestrial database, which inspired Agassiz's early ideas, continues to be refined and expanded today. 3  ,  <5 0 = water. L d  18  - (H Q/H 0)SMOW1 , « • , —7 , ^ x 1000 where SMOW is standard mean ocean (H 0/H 0 ) S M O W J  r( 2 Q/ 2 Q) H  1 8 n  18  H  18  16  2  1g  18  2  1 6  2  16  2  N  i n n n  c  n  w  A  A  CHAPTER  1.2  1.  INTRODUCTION  3  Modelling an ice age  To examine landforms on the ground and infer the glaciological conditions that created them is a fundamental task of glacial geology. The data — observable landforms — are inverted to give a reconstruction of paleo-ice conditions. This style of qualitative continental-scale reconstruction, pioneered by Agassiz [1840], has yielded considerable insight regarding the ultimate Pleistocene glaciation; a sample of Laurentide Ice Sheet reconstructions derived from geomorphic data include those of Dyke et al. [1982], Dyke and Prest [1987] and Boulton and Clark [1990]. Continued debate over the form and evolution of the Laurentide Ice Sheet highlights the nonuniqueness of these reconstructions arising from an ill-posed problem. Inversion of ill-posed problems is well studied in geophysics, and some of the formalisms developed in this field may be used to identify shortcomings of ice sheet reconstruction. Figure 1.2 is a schematic illustration of the concepts discussed below. Consider a model space M with each element represented by a vector m and a data space consisting of geomorphological observations V with each element represented by a vector d° of glacial geology observations. The inverse operator maps data space onto model space while the forward operator T maps model 4  r  space onto data space. If the observed data d° are inverted by the operator T~ (not generally formalized i n glacial geology) to result i n two recovered models , rrii and m^, the question must be asked: which model is better? x  5  The most basic test of goodness is whether the predicted data d , calculated from a recovered model, satisfactorily match the observations d°. To apply this test, a forward operator !F must be defined such that ^[m ] = d . Herein lies the problem for glacial geology: a rigorous and satisfactory forward model has not been developed. p  1  p  That an inverse formulation exists with no established forward counterpart may appear surprising; however, note that the processes are hidden from view (by time in the geological sense and by ice in the glaciological sense) and only end results are visible. The absence of a forward algorithm does not diminish the accomplishments of glacial geologists whose work has provided the best picture of ancient ice sheets to date. Agassiz knew little of glaciology when developing the ice-age theory and presented it with a far-fetched forward model. Nevertheless, his reconstruction was based on copious observations, which eventually convinced mainstream geologists of the theory's merits. From experience then, we know an insightful reconstruction can be made without a detailed knowledge of the processes involved. Sophisticated forward models do exist i n glaciology and cryospheric numerical modelling is an integral tool for exploring the complex relationship between ocean, ice and atmosphere. The ice  Each element of M. represents one realization of time-evolving paleoclimatic conditions. The inversion operator T~ is not a 1 : 1 mapping if the dimension of M. is greater than the dimension of V. 4  5  x  CHAPTER  1.  INTRODUCTION  4  Figure 1.2. A conceptual diagram of inversion and forward modelling. M. represents model space which is of greater dimension than the data space T>. is a forward operator that operates on an element of M. to predict a data vector dj such that dj € V. T operates on an element of M. to predict a data vector d!j such that $ V. The inverse operator T~ maps an element of data space onto a non-unique element of model space. L  x  and hydrology models described i n Chapter 5 are examples of algorithms based on first principles driven by paleoclimatic records and from an element of model space m j , associated predicted data are generated. However, these algorithms predict subglacial water pressure, ice thickness and ice temperature rather than Quaternary geology. The mapping T that represents these forward models produce data largely outside the denned data space , therefore glacial geologic observations cannot be directly predicted or used as constraints (Figure 1.2). This view of V, comprising only glacial geology data, is selective and other types of observables can be used to partially ascertain the veracity of the models. For example, growing and disintegrating ice sheets produce a measurable visco-elastic response that can be used to impose bounds on paleo-ice thickness [e.g. Peltier, 1994, 1996]. Nevertheless, most of our knowledge of Quaternary ice sheets comes from the present-day landscape. L  6  Thus there is a conspicuous gap in the current method of ice sheet reconstruction. A great effort has been made to map glacial features and infer the ice history that produced them. However, this approach is hampered by a shortage of verification tools and a limited temporal extent. Although numerical ice sheet models exist, prediction of glacial landforms is not within their scope; I address this problem by presenting a forward model of basal processes, enabling geological predictions from ice sheet modelling. 6  C o u n t e r e x a m p l e s exist. T h e geographical l i m i t s o f the W i s c o n s i n G l a c i a t i o n are g e o m o r p h i c d a t a t h a t  c a n be p r e d i c t e d by ice physics a l g o r i t h m s .  CHAPTER  1.3  1.  5  INTRODUCTION  Elements of the basal processes model  The erosive prowess of glaciers is well-documented, though widely debated, i n the literature [e.g. Laine, 1980; Bell and Laine, 1985; Nesje and Whillans, 1994; Hallet et al., 1996; Elverh0i et al., 1998]. Although without the detail or quantitative weight of the aforementioned glacial erosion studies, none match the imagery of Ball [1906]:  An ice-sheet, ponderous of almost illimitable  in mass and irresistible  capacity;  from their bed, crushed  by its influence  to pieces, reduced  to  in power, forms a destructive  fragments  engine  of the living rock are  ripped  mud.  While avoiding the question of animate rocks, I seek to defuse the illimitable erosion through a model that predicts (most often) finite results.  capacity oi glacial  Glacial erosion is known to be selective rather than irresistible because it depends on the thermal basal regime. Cold-bedded glaciers are poor excavators compared to warm-bedded ones [Sugden, 1978]. Preglacial landscapes that survive ice coverage relatively unchanged have primarily supported cold-bedded ice [Dyke, 1993; Kleman, 1994]. In addition to the thermal regime, erosive potential is also controlled by englacial debris content; glaciers need tools to abrade. This was apparent to the eminent glaciologist James Forbes who wrote, 7  It is not the wheel of a lapidary primed.  which slits a pebble,  The gravel, sand and impalpable  but the emery with which it is  mud are the emery of the glacier.  [Forbes,  1845]  I develop a process-based forward model, consistent with these controls (among others), to calculate erosion. Model results are compared to geological estimates of erosion. After the modelled sediment is produced, there are several possibilities for its evolution. It may • remain at the glacier bed, protecting the underlying bedrock from further erosive assault, or it may be entrained into the ice. Englacial debris is subject to redistribution processes that reorganize the sediment i n the ice column and deposit it as till. Lateral advection with the ice flow yields modelled dispersal trains that are compared to observed debris trains. In general I use Occam's razor sparingly, and consider a great number of processes and associated parameters. This approach is appropriate in a study where there is little prior work to act as guidance on process significance. In addition, the subglacial conditions varied wildly over the : 7  A glacier that is warm-bedded is at the pressure melting point at its bed.  CHAPTER  1.  INTRODUCTION  6  Figure 1.3. Medial moraines on the Kaskawalsh Glacier, St. Elias Range, Yukon Territory. Under present ice conditions, supraglacial moraines are a significant debris transport mechanism. In a climate that produces a large enough ice volume to drown the exposed rock, this mechanism would become unimportant. modelled domain and period; to determine a priori that a process will remain insignificant over the entire domain for a complete glacial cycle is not a trivial task. I therefore favour a complete, if slightly bloated, model over a sparse one.  Excluded processes  This thesis is by no means exhaustive i n the analysis of geomorphic features. I present a forward model for sediment transport arising from basal-ice processes, though there are other means of sediment dispersion. On modern glaciers, medial moraines carrying debris of subaerial provenance are ubiquitous; a model of debris transport by medial moraines is presented by Anderson [2000]. Figure 1.3 illustrates that this process can be effective only when valley walls and nunataks are exposed. As ice volume grows and engulfs mountaintops, debris generated by mass wasting will be increasingly insignificant, especially in areas of low relief. Therefore it is not a serious limit ation to assume that supraglacial debris transport is unimportant i n this study. A notable exception is the Cordillera, characterized by high relief and variable climate throughout the Wisconsinan, where this assumption is suspect . 8  9  A more serious omission in this thesis is the debris transport by sediment deformation. Subglacial sediment deformation accounts for an estimated 90% of the total motion of Breidamerkurjokull, Iceland [Boulton and Jones, 1979] and is thought to be the cause of fast flow at Ice Stream B , Antarctica [Blankenship et al., 1986; Alley et al., 1986, 1987]. Clarke [1987], Jenson et al. [1995, distinct from the supraglacial debris originating from thrust faults, discussed in Section 4.1.4 A well-documented example of a debris train of subaerial provenance is the belt of angular erratics extending in excess of 1000 km from their source area, the Athabasca Valley [Prest and Nielsen, 1987]. 8 9  CHAPTER  1.  INTRODUCTION  7  1996] and Boulton [1996] present models of deforming-sediment transport and several recent continental-scale simulations include fast flow caused by basal sediment deformation [Clark et al., 1996; Marshall and Clarke, 1997a,b]. Despite the acknowledged occurence of sediment deformation in some cases, neglecting it does not necessarily compromise the sediment transport model or the results. Sliding and sediment deformation compete with each other and depend on the ice-bed coupling strength at the ice-bed interface. A well-coupled ice-bed interface promotes sediment deformation while a decoupled interface serves to enhance shding. These two modes of glacier motion are generally observed to be out of phase at Trapridge Glacier, Yukon Territory [Kavanaugh, 2000]. Since both mechanisms are dependent on basal shear stress but only one is active at a time, it is not disastrous i n most instances to assume all basal transport occurs by shding. Following Marshall et al. [2000] I do include a sediment enhancement factor for shding, designed to represent ice motion from basal deformation (Section 5.1.1). The most serious errors arising from this omission will occur when shding and sediment deformation do not act in concert. For example, high subglacial water pressure inhibits sediment entrainment and by consequence basal-ice transport, but enables sediment failure and subglacial transport. A n accurate partitioning of shding versus sediment deformation would require knowledge of the basal boundary condition which is not generally available on a large scale. In areas where there is strong geological evidence for prolonged fast flow enabled by sediment deformation, an advective model such as that presented by Jenson et al. [1995, 1996] would be a more suitable choice than the sediment transport model presented here.  1.4  Thesis objectives and structure  The subglacial landscape is hidden from view and direct observation is both costly and logistically challenging; this has limited our understanding of subglacial operation. The primary aim of this thesis is to advance the theoretical description of subglacial processes at the continental scale. As part of this work I refine relevant parameter values through numerical experiments. Through accurate lithological representation of the bed and careful tracking of each lithological component, the basal transport model is predictive of glacial erratics data — long recognized as one of glacial geology's most tractable archives. Mapped subglacially derived debris trains can now be more rigorously exploited to constrain numerical reconstructions of ice sheets.  :  CHAPTER 1. INTRODUCTION 1.4.1  8  Scales, weights and measures  The Late Pleistocene ice examined in this study was extensive, covering almost all of Canada; Figure 5.1 on page 77 shows the modelled ice extent at LGM (Last Glacial Maximum). In order to simulate a continental-scale problem, current computing power demands a relatively coarse spatial grid. The grid size chosen for this project is 120 X 1° meridional cells and 100 X 0.5° zonal cells, corresponding to ss 55 km zonal resolution. This coarse grid creates a problem of disparate scales. To include physical processes that operate on much smaller scales than the grid, I develop methods to incorporate sub-grid scale information. Throughout this thesis, the indices i,j are reserved for model cell longitude and latitude while m, n are used for sub-grid indexing. System Internationale (SI) units are mostly used in the text and these are the internal units of the algorithms developed to produce the results presented throughout this thesis. However these are not always the most intuitive units for the reader and often non-SI units (particularly annum) appear in the text where their use is justified by familiarity and tradition.  1.4.2  Outline of chapters  Between the introductory and concluding chapters, this thesis consists of three model development chapters, one chapter describing model inputs and one chapter summarizing simulation results. Details of a 2D stress analysis and the fluid velocity solution of a rotating cylinder in a viscous shear flow are appended to the body of this thesis. The main chapters are outlined below. Scaling is a theme that threads through the thesis, as small-scale processes must be accounted for within the framework of the continental-scale study area. Chapter 2 explores the scaling of topography; a downscaling method is developed to transform calculated bed slopes from coarsescale sampling to an estimate of the result that fine-scale sampling would yield. It is calibrated with six areas where both coarse and fine data are available and then applied to the complete study domain. Erosion of the glacier bed is the topic of Chapter 3. There are two distinct mechanisms that cause subglacial erosion: abrasion and quarrying, addressed in Sections 3.1 and 3.2 respectively. Current glacial abrasion theory is reviewed and extended in Section 3.1.2, followed by a discussion of parameter determination. Subcritical crack growth, the controlling process of quarrying, is examined as are its associated physical parameters. For both abrasion and quarrying, sub-grid information is extracted from geological data sets. The effect of topography on quarrying is ; explored and the role of cavities is quantified. A quarrying model is presented at the end of the  CHAPTER  1.  INTRODUCTION  9  chapter. In Chapter 4, the discussion moves upward from the bed to address processes that occur i n basal ice. Modes of entrainment are reviewed and quantified in Section 4.1, followed by an examination of englacial debris redistribution mechanisms. Section 4.2 explores vertical redistribution through mixing while lateral transport by advection is considered in Section 4.3. A large amount of debris may be an island but the debris model is not and cannot exist i n isolation but must be connected with auxiliary models. These are the ice and hydrology input models, which are both physics-based, predictive, forward models developed by Marshall [1996] and Flowers [2000] respectively; a brief overview of these is given i n Sections 5.1 and 5.2. Geological Survey of Canada (GSC) maps [Fulton, 1996; Wheeler et al., 1997] are used to cast the geology of Canada into a form compatible with the terrestrial model. This facilitates the sub-grid parameterizations of Chapter 3 as well as enabling lithological tracking i n the englacial transport model developed in Chapter 4. Chapter 6 contains simulation results for the Wisconsin Glaciation (120 ka B P to present). The eastward transport of the Dubawnt red beds in western Nunavut and the southern transport of Paleozoic carbonates of the Hudson Bay area are contrasted with simulated debris transport i n Section 6.1. Finally, geological estimates of paleo-erosion come from a number of sources and are compared to modelled results in Section 6.2. Modelling a system as complex as the cryosphere, comprising ice, climate, water and terrain components, requires an integrated approach. The final result of this thesis is a union of the individual state-of-the-art models into a single, self-consistent forward model of the North American cryosphere through the last glacial cycle.  Chapter 2  Downscaling Bed Slopes Scaling issues are endemic i n ice sheet modelling; variables that control important physical processes change at scales much smaller than those that are computationally tractable for large-scale models. One problematic variable is the bed dip 6 appearing in the basic erosion laws detailed i n Chapter 3. Note that 6 is the surface dip of the bed, not the dip of sub-bed internal strata; this is illustrated in Figure 3.1 on page 26. As seen from Figure 2.1, the variation of 6 gives rise to over an order of magnitude change of the abrasion rate.  0  5  10  15 20 B e d dip (deg)  25  30  35  Figure 2.1. Abrasion rate versus bed dip for various grain sizes R. The abrasion is calculated by Equation (3.12) using the method developed in Section 3.1. The effective viscosity 77 is taken as 1 x 1 0 P a a and the thermal resistivity Z as 0.3 m K W with other parameters identical to those in Tables 3.3 and 3.4 on pages 31 and 34. 6  - 1  r  CHAPTER 2. DOWNSCALING  - 1 2 0 W  BED SLOPES  - 1 1 0 W  11  - 1 0 0 W  Figure 2.2. Areas (denoted by small boxes) on which sub-grid bed slopes are analysed to determine a slope downscaling relationship. Table 2.1 gives further details.  Existing meso- and large-scale erosion models rely on two static values of 6 (one for abrasion, the other for quarrying) regardless of cell topography [Tulley, 1995] or use simplified erosion laws with no 9 dependence [e.g. Oerlemans, 1985; MacGregor et al., 2000]. A n accurate description of the bed dip without having to resort to untenably fine grids is desirable. This is achieved through direct calculation of bed slope i? (related to the bed dip through 6 = t a n 1 ? ) from a digital -1  elevation model ( D E M ) followed by a downscaling procedure.  2.1  Direct bed slope calculation from D E M data  To calculate bed slope, a digital elevation model is needed. The finest comprehensive data set available for North America is a 1000 m D E M [Geophysical Data Center, 1998]. However, erosion processes act at scales much less than 1000 m; the equations for erosion are appropriate on scales of 1 -» 10 m . A method to downscale the calculated bed slopes from the 1000 m sampling to an appropriate scale is paramount. To elucidate such a downscaling relationship, elevation sampling and direct bed slope calculation is done at a variety of sub-grid cell sizes i n select areas where 30 and 10 m D E M s [EROS Data Center, 2000] are available to extend the analysis. Six areas are used, shown i n Figure 2.2, each 0.375° X 0.375°; Table 2.1 details the selected areas. The 1000 m data uses a Lambert conformal conic projection, while the 30 and 10 m data use universal transverse Mercator ( U T M ) coordinates (Zones 10, 11 and 14). These are projected  CHAPTER 2. DOWNSCALING BED SLOPES Area  State  Longitude  1 2 3 4 5 6  OR WA WA WA WA ND  117.750° W 122.375° W 121.625° W 122.750° W 122.000° W 98.625° W  12  Latitude 45.125° 48.500° 48.500° 48.500° 48.500° 48.000°  Data  N N N N N N  1000, 30 1000, 30 & 10 1000, 30 & 10 1000, 30 & 10 1000, 30 & 10 1000, 30  Table 2.1. Details of the selected areas used to determine a slope downscaling relationship. Longitude and latitude are given for the south-west corner of each 0.375° X 0.375° area. The data sets used are a 1000 m D E M [Geophysical Data Center, 1998], 30 m D E M [EROS Data Center, 2000] and 10 m D E M [EROS Data Center, 2000]; the 10 m data are not universally available. Figure 2.2 shows a location map.  to geographical coordinates and resampled to a specified cell size using standard A R C - I N F O packages with cubic interpolation. The elevation z is sampled at the cell centre of each (m, n) sub-grid cell and north-south and east-west slopes are calculated on the nodes using the following 4-point stencil with second order accuracy, b  ^bed  02  _ ^bed  Z  e  — z' ^  d  3 6  A0„  A0_ _bed 2' ~2 m +  e  e  d  (2.1a).  2R  2R  b  e  sin(7r/2 -  n  ,,bed ~2' ~2 m  n  _|_  .-bed _ -bed +2'"+2 ~2'" 2 TO  m  +  (2.1b)  0 ,„) m  where R is the radius of the Earth. Discrete slopes categories are defined and frequency histograms are made of all i?J£ and i ? ^ that fall within the 0.375° square. Figure 2.3 is a sketch illustrating the above. e  n  The histograms are made with slope divisions of 0.012. There are 200 slope categories indexed by h from —1.2 to +1-2 (corresponding to ±50° for the discrete dip classes); this even distribution in i? space achieves the desired concentration of dip divisions close to zero i n 9 space. Histograms for areas 3, 4 and 5 using a sample spacing of 0.01875° from the 1000 m data and 0.003° from the 30 m data are shown in Figure 2.4. A cell size of 0.01875° gives 20 X 20 sub-grid blocks within the 0.375° test area which is enough for a reasonable statistical analysis, while 0.003° sampling gives 125 x 125 sub-grid blocks. The most obvious difference between the two sets of histograms is that finer sampling yields more extreme slope representation: the coarse sampling of data results i n underestimating the variability of the slope distribution. This effect must be quantified i n order to downscale.  CHAPTER 2. DOWNSCALING BED SLOPES  13  m-1/2 i m+1/2 i mi  ; - n+1/2 n  n-1/2  <l>m ,n-1/2  A  Elevation defined on the 1/2 grid  o  Slopes defined on the (m,n) integer sub-grid  ©  Binned slopes defined on the (i,j) model grid  Figure 2.3. A sketch (in latitude 0 and longitude <f>) showing the sub-grid scale in light dashed lines on which the elevations are sampled and solid lines where the slopes are calculated. The heavy dashed Une denotes a 0.375° x 0.375° area which is comparable to the model grid cell — frequency histograms are summed within this block.  CHAPTER 2. DOWNSCALING BED SLOPES  14  Histogram of slopes using 30 m DEM  -0.6  -0.4  -0.2  0  Slope  0.2  0.4  0.6  (a) Cell size = 0.003°, giving 15625 (125 x 125) cells in the 0.375° test block. Histogram of slopes using 1000 m DEM  (b) Cell size = 0.01875°, giving 400 (20 x 20) cells in the 0.375° test block. Figure 2.4. Slope (average north-south and east-west) histograms from a 30 m D E M i n (a) and a 1000 m D E M i n (b). The histograms are formed with discrete slope bins of 0.012. Positive slopes have southern and western aspects.  CHAPTER 2. DOWNSCALING Mean of A z 600  b e d  BED SLOPES  15  versus cell size for 30 m data  Mean of slope versus cell size for all data 0.03  r  0.02  -oo eeoo-cee^  CD QL  £CO 0.01 O  area 1 area 2 area 3 10 m data areas 2 - 3 1000 m data areas 1-3  c  CO CD  -0.01  10  10  10"  10"  Cell size (decimal degrees)  II  -0.02 10  I I II  io"  j  10  i<r  Cell size (decimal degrees)  (a) M e a n N S elevation difference for areas 1,2 a n d 3 for 30 m d a t a o n l y .  (b) M e a n N S slope for areas 1,2 a n d 3 for a l l d a t a sets.  Figure 2.5. The mean slope is approximately stationary with scale while the mean elevation difference is not; therefore bed slope is the preferred variable over elevation difference.  2.2  Fractal parameters of bed slope  Downscaling is often quantized through the assumption of exponential, or fractal behaviour [e.g. Dubayah et al., 1997]). For example i f the distribution of a property V measured with a cell size Ad has an expected variance E^g[vax(P)] then a fractal scaling relationship would be  E  (Ac5)> (?>)] a r  \(M) J 2  where 7 is the fractal dimension for the variance. This can be extended to any g the distribution (for q > 2) by var  4  t h  moment of  = £ #)[(7>-mean)«]  §  (2.3)  (A  :  which would have a fractal dimension 7 . Note that when q — 2, (2.3) is equivalent to the definition of the variance, thus 72 = 7 - A simple scaling relationship is sought and it is desirable to work with a distribution where the mean of V is constant with scale (i.e. = 0) such that V is the only non-stationary variable i n (2.3). The gradient of the elevation, or bed slope, rather than direct elevation difference satisfies this criterion as shown by Figure 2.5. 9  var  To test the appropriateness of (2.2), each area is sampled with a variety of sub-grid cell sizes: the 1000 m data are sampled at 0.075 -»• 0.015 deg, the 30 m data at 0.075 ->• 3.75 X 1 0  - 4  deg and ,  CHAPTER 2. DOWNSCALING  BED SLOPES  16  Cell size  Cell size  (a) A log-log plot of variance against cell size to show fractal dimension.  (b) A log-log plot of the fourth moment against cell size to show fractal dimension.  Figure 2.6. The break in slope at cell size ss 10~ deg shows that (2.2) does not hold over the range of scales considered. 2  the 10 m data at 0.075 —» 1 X 10 deg. A log-log plot of the g moment of bed slope versus the cell size should result in a straight Une of slope 7 if (2.2) holds. 4  t h  g  Figure 2.6 indicates that the scaUng of ^2 (the variance) and ^4 do not have a constant j over the range of sample scales; there is a break in slope at a ceU size of approximately 0.01° (corresponding to 1.1 k m of latitude). This is suggestive of the break between "short" and "long" scales at ss 0.6 k m seen by a number of authors in topographic data [e.g. Mark and Aronson, 1984; Chase, 1992; Lavallee et al., 1993]. The fractal dimension plot for area 6 has a markedly different form from the other curves i n Figure 2.6: the others are translations of a master curve while area 6 marches to its own tune. This is not surprising as it is the only area i n the group i n the plains and completely covered by sediments. q  2.3  Downscaling methodology  Because of the difficulties introduced by the scale-break in the data, (2.2) is not assumed to hold and I take an empirical approach. Plots of \i against ceU size, extracted from the finest resolution D E M (either 10 or 30 m), are extrapolated to 1 x 1 0 deg; this is ss 1 m i n latitude, an estimate of the scale on which erosion acts. For each test area ^ ° from the 1000 m data set is plotted against Pq ° from the extrapolation; a second order polynomial is fit and results q  - 5  0 1 8 7 5  Xl  CHAPTER  2. DOWNSCALING  BED  SLOPES  -scale  g= 2 g= 4  1.797 x 10 1.171 x IO  2  5  17  Jscale  -scale  4.102 -27.966  6.025 X 10~ 4.243 X I O  3  - 3  Table 2.2. Coefficients for equation (2.4). They are the coefficients for best fit 2 mials from Figures 2.7(c) and 2.8(c) for \ii and /z respectively.  n d  order polyno-  4  in a downscaling relationship ,,1X10  „scale /,,0.01875\  - 5  2  , iscale 0.01875 , „scale  (n  A  \  The 2 and 4 moments of the 10 and 30 m data are plotted in Figures 2.7(a) and 2.8(a) along with the extrapolation to 1 X 1 0 deg; the 1000 m data are shown in Figures 2.7(b) and 2.8(b). The shaded points (1 X 1 0 and 0.01875 deg) are plotted against each other i n Figures 2.7(c) and 2.8(c) with the best fit second order polynomial; the coefficients are listed i n Table 2.2. A similar analysis is not shown for /X3 because there is wider disagreement between the different data sets and a second order polynomial does not satisfactorily describe the downscaling relationship. A possible explanation is that pz describes a directional property. This would be strongly influenced by factors that are specific to each area (e.g. predominant geological strike) and therefore no global rule would be applicable. n d  t h  - 5  - 5  Once it is known how the moments change with scale, the distribution can be downscaled. In every model grid cell (i, j), each slope class h has a normalized standard deviation z associated with 'dh through d  h  Z  ~  h  ^0.01875-  ^•°>  Recall from Figure 2.5 that the mean /xi is independent of cell size. The frequency count for each component of z remains constant through the downscaling procedure; however the slope classes are modified through p\ ° , the downscaled variance from (2.4). The downscaled slope classes are given by d  h  xl  v  = h V^2 z  h  +Mi  ( -6) 2  where i ? ^ are unique to every (i, j ' ) model cell; therefore the frequency counts 0»j,/i are interpolated back to the original slope classes 1?^ for consistency between model cells. s  t h  Comparison of sampled and downscaled data to the same cell size show that this methodology tends to overestimate both /x and /x . The sharp central peak i n the finely sampled data has no counterpart in data from the 1000 m D E M (see Figure 2.4) and this is the root of the downscaled overestimation. A further empirical relationship is used to modify the downscaled frequency 2  4  CHAPTER 2. DOWNSCALING BED SLOPES  0.35  0.25 CD  g-  0.05  -*- area 1 - e - area 2 -e- area 3 -B- area 4 -#- area 5 area 6 10 m data areas 2 - 5 extrapolated to 1 x 10"^  0.3  0.2  C/3  O 0.15  !8  CM  zi. 0.1 0.05 0  10"' 10"' Cell size (decimal degrees)  (a) V a r i a n c e using the 30 a n d 10 m D E M s s u b - g r i d cell sizes f r o m 1 x 1 0 —• 7.5 x deg. T h e curves are e x t r a p o l a t e d to 1 x deg u s i n g 10 m d a t a w h e n available, or data. - 4  Cell size (decimal degrees) (b) V a r i a n c e using the 1000 m D E M w i t h subg r i d cell sizes f r o m 1.5 x 1 0 —» 7.5 x 1 0 deg.  with 10 IO 30 m  - 2  - 2  - 2  - 5  Relationship for downscaling variance  0.4  I  1  1  1  1  1  1  r  (c) T h e variance o f the shaded d a t a f r o m (a) a n d (b). T h e 1000 m D E M d a t a s a m p l e d at 0.01875 deg are p l o t t e d against the 10 m D E M d a t a e x t r a p o l a t e d to 1 x 1 0 deg to result i n a d o w n s c a l i n g r e l a t i o n s h i p w h i c h transforms the 1000 m D E M d a t a to 1 x 10 deg corrected d a t a . - 5  - 5  Figure 2.7. Variance of bed slope (averaged north-south and east-west) for a large range of subgrid cell sizes. The shaded data are plotted against each other in (c) and are fit with a second order polynomial / x " = a| (/x° ) + 6| /x° + c| where a | = 1.797 X 10 , . scaie _ 4 1Q17 and c | = 6.0245 X 1 0 . This defines the downscaling relationship for the variance. l x 1 0  5  c a l e  6  c a l e  0 1 8 7 5  - 3  2  c a l e  0 1 8 7 5  c a l e  c a l e  2  CHAPTER 2. DOWNSCALING  BED SLOPES  19  .x 10"  10"  CD CL O CO  10"  area 1 area 2 area 3 area 4 area 5 area 6 10 m data areas 2 - 5 extrapolated to 1 x 1 0 "  -B-  10"  V  10"* 10"' Cell size (decimal degrees)  Cell size (decimal degrees)  (b) The fourth moment using the 1000 m D E M with sub-grid cell sizes from 1.5 x 1 0 —• 7.5 x l O " deg.  (a) The fourth moment using the 30 and 10 m DEMs with sub-grid cell sizes from 1 x 10~ —» 7.5 x 1 0 deg. Extrapolation is made to 1 x 10~ deg using 10 m data when available, or the 30 m data. 4  - 2  - 2  2  5  Relationship for downscaling fourth moment  0  1  0.5  2  1.5  3  2.5  |j. from 1000 m data 4  x  1 0  -3  (c) The fourth moment of the shaded data from (a) and (b). The 1000 m DEM data sampled at 0.01875 deg are plotted against the 10 m DEM data extrapolated to 1 x 10~ deg to result in a downscaling relationship which transforms the 1000 m DEM data to 1 x 10~ deg corrected data. 5  5  Figure 2.8. Fourth moment of bed slope (averaged north-south and east-west) for a large range of sub-grid cell sizes. The shaded data are plotted against each other i n (c) and are fit with a second order polynomial so that / x = a\ (/i° ) + b\ n™ + c\ where a\ = 1.1709 X 10 , 6= = -2.7966 X 10 and c* = 4.2430 X 10~ . This defines the downscaling relationship for the fourth moment. l x l 0 _ 5  5  cale  1  cele  cale  0 1 8 7 5  2  cale  3  1875  cale  cale  CHAPTER 2. DOWNSCALING BED SLOPES  20  distribution to ameliorate the disagreement; it is lxio ijh  - 5  (2.7)  1 + exp P2  where •d and fll ~ are parameters taken as 0.012 (which is At?) and 0.2 (an approximate median value for the set of downscaled variances) respectively. Examples of applying the methodology are shown in Figure 2.9. These examples are downscaled to 3 X 10 deg to allow for comparison to a directly calculated histogram from the 30 m DEM. xl0  b  -4  The fourth moment is not explicitly used to downscale the bed dip. Nevertheless, similarities in the analysis of /Z2 and /z and agreement of // between downscaled and sampled distributions bolster the method. 4  4  CHAPTER  2.  DOWNSCALING  -0.5 0.1  -0.3  -0.2  21  SLOPES  -0.1  0  0.1  0.2  0.3  0.4  0.5  r  0.08 -  — — -  Area 4  o c 0.06 CD CT CD  -0.4  BED  30 m data 1000 m data downscaled  0.04 -  LL 0.02 0  -0.6  -0.4  0.06 30 m data 1000 m data downscaled  -0.5  -0.4  -0.3  -0.2  -0.1  0  0.1  Bed slope  0.2  0.3  0.4  0.5  Figure 2.9. Slope (average north-south and east-west) histograms for areas 3,4 and 5 using the modified frequency Q'^ from (2.7). The 1000 m data are downscaled to 3 X 1 0 deg and plotted with a sampled histogram of the same cell size. - 4  h  CHAPTER 2. DOWNSCALING  2.4  BED SLOPES  22  Application to North America  The topography of the study area, which covers a wide extent of North America, can now be downscaled. The input data are from a 1000 m D E M from Geophysical Data Center [1998] for land data and 5 m i n D E M from NOAA [1988] for marine areas. Figure 2.10 (upper panel) is a map showing contours of the mean absolute slope (north, south, east and west) for blocks of 0.5° X 0.5°. They are calculated from the raw (unsealed) discretized slopes with 0.02° sampling. The lower panel of Figure 2.10 shows the corrected version of the upper panel; the methodology of Section 2.3 is applied to downscale the slopes to 1 x 1 0 deg sampling i n each 0 . 5 ° X 0.5° cell. The maximum corrected mean slope, which is off the scale i n Figure 2.10(b), is « 25°. - 5  The end result of this analysis is a normalized, downscaled histogram for each 0.5° X 1.0° model cell. Each process that is dependent on bed slope (e.g. Equation (3.7)) is calculated by a linear superposition of weighted results of each slope class.  CHAPTER 2. DOWNSCALING  BED SLOPES  23  mean uncorrected bed dip  0  1  2  3  4  5  6  7  8  9  8  9  10  mean corrected bed dip  0  1  2  3  4  5  6  7  10  Figure 2.10. Discretized mean of uncorrected and corrected absolute value of bed dip for North America (0.5° cell size). Sub-grid data are sampled at 0.02° from the 1000 m land DEM [Geophysical Data Center, 1998] and 5 minute marine DEM [NOAA, 1988] then binned into 200 slope classes from -50° —> +50° for the uncorrected figure. The correction is made using the methodology outlined in Section 2.3 to a 1 X 10 deg sample size. Filled contours of 2.5, 5, 7.5 and 10° are plotted. -5  Chapter 3  Erosion 3.1  Abrasion  The study of friction and wear, tribology, hails from before the early days of scientific investigation; Assyrian villages were skilled in the use of bearings before 6000 B P . It was not until the 1 5 century that the effect of shding objects over each other was treated i n a truly scientific manner by Leonardo da Vinci. One of the earliest applications of tribology to glaciology is the description by Gilbert [1910] of the reduction of bedrock to fine fragments via shding of debris-laden basal ice. t h  To date, subglacial erosion has mostly been treated crudely i n large-scale models; simplified erosion laws dependent only on shding velocity or globally prescribed erosion rates have been used [e.g. Oerlemans, 1985; Puranen, 1988]. A similar approach has been taken i n valley-scale modelling of erosion: only a linear dependence on shding velocity is assumed [e.g. Harbor, 1992; MacGregor et al., 2000], albeit with results that successfully reproduce observed landforms. Tulley [1995] presents a process-driven approach to abrasion modelling; I extend this by recognizing and honouring both inter-cell and intra-cell parameter variability.  3.1.1  Adhesive and abrasive wear  There are two types of wear: adhesive wear and abrasive wear [see Rabinowicz, 1995]. Adhesive wear occurs when two solids are pressed together with microscopic attraction between the surfaces. If the force required to break the surface-to-surface junction is greater than the force required to create an internal fracture, the internal fissure forms: this is wear. The process is largely independent of the roughness and occurs on both surfaces even i f the junction is made of dissimilar  24  CHAPTER 3.  EROSION  25  solids. The process was quantified by both Holm [1946] and Archard [1953] as jad _  (3.1)  v —  A  where A* ( m s ) is the volumetric abrasion rate, & j is the probability that an adhesive junction will form for any given contact, F (N) is the normal load, v (m s ) is the particle shding speed and h (Pa) is the hardness of the surface. d  3  _ 1  a(  N  _ 1  p  s  Abrasive wear occurs when a particle of hardness h slides against a softer surface of hardness h„ and creates a groove. Although the physical process is not analogous to adhesion wear at all, the quantitative form is almost identical p  if  r  =  k  ^f  N  v  P.  (3.2)  In general & b is a few orders of magnitude greater than & a. Rabinowicz [1995, Figure 6.21] shows a relationship between the friction coefficient p and & a that results i n & a ~ 2 X 1 0 i f fi is taken as 0.85 (see Section 3.1.3) while typical geological values for & b are much higher than this (see Section 3.1.3). It is clear that abrasion wear will dominate and adhesion wear will not be discussed further. a  r  a  - 5  a  a  a  r  The resultant wear of abrasion is a function of the hardness ratio between the abrading particle h and the wear surface h [Rabinowicz, 1995, Figure 7.10], which is approximated to a step function; i f the abrading particle is softer than the wear surface, the wear is zero. Applying the step function to (3.2) over a unit area gives p  s  ^abr  A= I  E  p  N  VpC  r  if hp > h , s  (3.3)  if hp < h ,  0  g  where C is the areal fraction of debris (assumed equivalent to the volume fraction) with radius R and A is the abrasion rate i n m/s. Note that v C/A gives the particle flux. The grain-size distribution, indexed by r, is discussed in Section 3.1.4. r  r  3.1.2  p  Glaciological abrasion  Modern treatments of glaciological abrasion follow the form of (3.3) but differ i n the analysis of F. Boulton [1979] asserts an F dependence on effective pressure while Hallet [1979, 1981] champions no such dependence. Laboratory experiments generally support the theory of Hallet [1979, 1981] although there appears to be a weak dependence on effective pressure due to a presumed thinning of a microscopic water film [Iverson, 1990]. Nevertheless, the dependence is weak and the analysis of Hallet [1979, 1981] is used for the calculation of F i n this study. N  N  N  CHAPTER 3. EROSION  26  Vext .melt Vp Vs  Figure 3.1. A sketch showing the local bed dip 9, the sliding and particle velocities v ,v and the melting and extensional velocities 6 ,u . 3  p  m e l t  e x t  The rate of abrasion is identical to (3.3) when the constant of proportionality of Hallet [1979, 1981] is identified with the more fundamental parameters fc b //i». F has two components, a buoyancy term and a viscous drag term, N  a  r  F» = | x J * U - )g cos 9 + Ca^f^v , Pi  (3.4)  n  buoyancy  viscous drag  where R (m) is the radius of the particle, p and pi ( k g m ) are the densities of the particle and ice, g ( m s ) is the vertical acceleration of gravity and 9 is local bed dip angle (see Figure 3.1). The variables i n the drag term / ^ , n, v and R+ are described below. Typically the buoyancy term is overshadowed by the drag term. - 3  r  - 2  d  n  The second term is the viscous drag, derived by Watts [1974] and modified by Hallet [1981] with the inclusion of the bed modification factor f^ . Without the expression gives the viscous drag of a sphere i n ice, moving at a relative speed v . The modifier is necessary due to the proximity of the inclusion to the bed; an appropriate empirically-determined value for / j ^ is 2.4 [Goran, 1970]. ed  n  d  R* (m) is a parameter analogous to the transition radius of regelation versus enhanced creep [Weertman, 1964] and is calculated by [Watts, 1974]  *> = {frzY  <-> 3  where the pressure melting coefficient V and the latent heat of fusion Z f for pure ice are taken as 7.42 x 1 0 K P a and 3.34 x 10 J k g respectively [Paterson, 1994]; Z (m K W ) is the thermal resistivity of the inclusion. The effective viscosity of the ice n (Pa s) is calculated by c  _ 8  _ 1  5  us  - 1  _ 1  r  5  CHAPTER  3. EROSION  where the shear stress r 5.1).  lce  27 and the shear strain rate k are given by the ice sheet model (see Section  The viscous drag term is also proportional to v , the component normal to the bed of the relative velocity of ice over the particle, in (3.4). A particle in contact with the bed must be zero normal velocity and the relative velocity is due solely to the ice. v has three components: a melting • component, a shding component and a component from longitudinal extension of the ice u such that n  n  ext  imelt+ v„ sin 6 + v  exi  cos 0.  (3-7)  Figure 3.1 illustrates the relative directions of the velocities. t> is derived from applying the continuity equation to the shding velocity (V • v = 0). Taking the appropriate length scale as the diameter of the particle, ext  s  v  ext  1  dv J- H 8  = 2R ,  . \  R sm<f>d<t>  *  y  e  e  where R is the radius of the Earth. As R <C R , v e  (3.8)  3  s  _R sm(j> d0 e  d,  . — (v . sw.d>)  is negligible.  eKt  The particles embedded in the ice generally move at a different rate than the ice; for a particle speed v and a shding speed v , the relative speed of the ice over the particle is v — v . If the particle is in steady state, the tangential forces of viscous drag of the ice, gravitational buoyancy and friction against the bed are in balance as p  s  f£l  g  = ^f-(Pr  frictional  ~ Pi)9 sing + /bed  ^ buoyancy  s  (v. - v ) p  p  (3.9)  v viscous drag  or in terms of the particle speed (which is needed to calculate A in equation 3.3) ATR 3 1 Rl + R 2  — (p -  fiF  /bed  N  L'  T  r  pi)gsin0  (3.10)  where the friction coefficient p is discussed in Section 3.1.3 and an appropriate tangential bed modifier for viscous /bed - [O'Neill, 1968]. i s1  7  Force balance in (3.9) is only vahd if there is frictional contact between bed and particle. If v < 0, the particle is moving away from the interface and erosion is zero. n  When v from (3.10) is less than zero, the particle lodges itself into the bed and becomes part of the subglacial sediment. Typically the frictional force overcomes any buoyancy contribution and v can be estimated as p  p  vp = v (l - pf» sm6/g ) 3  d  ed  -  W/x/bV/bed)  (3-11)  CHAPTER 3. EROSION  28  by using (3.4) and (3.7). W i t h this simplified expression it is clear that, for beds of moderate slope, v can only become less than zero i n conditions where 6 i is the same order of magnitude oftv p  m e  t  The laboratory experiments of Iverson [1990] show there are a number of factors complicating the scenario, such as clast rotation, angular clasts and dense packing of particles. Nevertheless, the experiments found good agreement with Hallet's theory and confirmed the most important variable as the normal velocity of the particle, which translates into contact force by (3.4). The final step towards a complete abrasion law is to describe the shielding power that existing sediment has on further abrasion. Cuffey and Alley [1996] argue that there is no significant erosion resulting from the abrasion of a deforming till on the bed except i n instances of exceptional bed softness or strong till divergence. However a model depth of sediment d is unlikely to represent a case of uniform thickness throughout the cell. Sediment depth has spatial variability within a model cell, so typically there will be some area of the cell with negligible cover and thus subject to erosion. To account for this (3.3) is modified to ^abr  A=l  ^ p VpC N  r  ~7 2_( *s r  (3.12)  P  0  i f hp < h , s  where d is a characteristic sediment depth finked to the mean surface slope to represent increased spatial variability of sediment i n regions of rough topography. The characteristic depth is calculated by d = demean + ^0-  (3.13)  The parameters cii and do are chosen as 10.0 m and 2.0 m respectively. Using values of # an shown i n Figure 2.10 , d has a range of 2.0 - 6 . 5 m with this parameter choice, which is reasonable. me  1  In summary, the abrasion is calculated as follows: particles of radius R with concentration C , present a force F on the bed calculated by (3.4) due to a normal velocity v from (3.7) for each Oh (see Chapter 2). The particle moves across the bed at speed v given by (3.10). These processes depend on the viscosity of the ice 77 and the transition radius R+, which is a function of the thermal resistivity of the clast Z and n. Finally, the erosion rate is given by (3.12) which is dependent on the hardness of the bed and the clast and summed for all particle radii. The dependence of R+ on n and Z through (3.5), the inclusion of the sediment shielding term e~ ^ and the explicit dependence on hardness / i are expansions on the model of Hallet [1979, 1981]. r  r  N  n  p  T  d  r  s p  E q u a t i o n (3.13) requires the mean bed dip t 9 i n radians, as do all formulae i n this thesis. Figure 2.10 shows #mean in degrees, a more standard unit for bed dip. 1  m e a n  CHAPTER 3. EROSION 3.1.3  Parameter determination — geological  Hardness  The hardness h or h is not a fundamental material property [Chandler, 1999]; it is determined by arbitrary engineering tests. A n appropriate test for hardness (in this study) is one that mimics the geometry of an englacial particle being dragged over a bed. In Vicker's hardness test a loaded diamond tip is pressed into the substrate and a length scale of the resulting indentation is measured; this is an acceptable measure of hardness for the purpose of calculating abrasion. Table 3.1 is a compilation from Mott [1956] of Vicker hardness numbers that have been modified to be dimensionally consistent with (3.3). The common scale for hardness used by mineralogists (Mohs' hardness number) is also listed i n Table 3.1 as this can be used as a guide to the hardness of other geologic materials for which no Vicker hardness data are available. a  p  As the quantification of hardness is somewhat arbitrary, relative values are more instructive than their absolute value. Typically the geological materials that are considered can be split up into two main groups: the silicates which are of hardness « 7 X 10~ P a and the calcareous rocks which are of hardness w 1.5 x 1 0 Pa. Low-grade metamorphic minerals such as chlorite (Mohs' hardness = 2.5) also fall within the softer group. Evaporites are softer still, but do not play a significant role in the geological makeup of North America. Rocks do not favour sweeping generalizations and there is a wide range of published values for hardness; for example Boulton [1979] fists metamorphic rocks such as marble and slate approximately 2.5 and 3 times harder than limestone. Nevertheless, a bimodal hardness grouping of 7 X 1 0 and 1.5 X 1 0 P a is expedient and used i n this study. 9  - 9  - 9  Mineral  Hardness (Pa)  XlO" Quartz (SiC^) Orthoclase ( K A l S i 0 ) Magnetite ( F e 0 ) Limestone (CaCOa) Marble ( C a C 0 ) Halite (NaCl) 3  3  4  3  8  _ 9  Mohs' Hardness  9  11 - 12 7.0-7.8 4.7-7.3 1.9-2.6 1.2 - 1.6 0.26 - 0.46  7 6 5.5 - 6.5 3 3 2.5  Table 3.1. Vicker and Mohs' hardness as reported by Mott [1956]  Abrasive wear coefficient  Finding appropriate values for &  aDr  i n the literature is difficult as the vast majority of research i n  tribology concerns itself with metal upon metal friction rather than geological materials. Never-  CHAPTER 3. EROSION  30  Rock Type  Z range" mKW" 0.27-4.0 0.23 - 2.0 0.23 - 1.1 0.25-0.77 0.20 - 0.77 0.15-0.63 0.20 - 0.28 r  1  clays shales basalt limestones marble granite dolomite ultrabasic intrusive  Z median mKW" 2.11.1 0.67 0.51 0.49 0.39 0.24  R+ range for n = 1 x 10 P a a 0.09-0.02 0.10-0.03 0.10 - 0.05 0.010 - 0.06 1.08 - 0.06 1.12 - 0.06 1.08 - 0.09  r  1  5  6  "from Kobranova [1989] °This value is the same as used by Hallet [1979] and Iverson [1990] and is in the range produced by the ice sheet model (see Section 5.1). Table 3.2. Values of thermal resistivity and corresponding R* for a variety of geologic materials. Hallet [1979, 1981] uses R± — 0.11 m which corresponds to Z = 0 . 5 6 m K W for limestone and Iverson [1990] uses R+ = 0.13 m corresponding to Z = 0.475mK W for granite. - 1  T  _ 1  r  theless there are experiments for a sandstone [Scholz, 1987] and a granite [Yoshioka, 1986] that both give an estimate of fc as 0.3; this is consistent with Hallet [1979] where a value of & b = 0.2 is implied from the attritivity coefficient for limestone (= kgh /h ) of 1 X I O using h = 2 X 10 Pa. However, a value of fc ~ 15 is calculated by Metcalf [1979] . In an experiment specific to debris-laden ice abrading its bed, fragment rotation is frequent [Iverson, 1990]: this changes the problem from 2-body to 3-body wear which typically decreases fc b by a factor of 10-100 [Rabinowicz, 1995]. Thus there is a range of possibilities for & b spanning 2 orders of magnitude; an appropriate choice is discussed in Section 6.2. aDr  a  r  - 1 0  T  s  9  s  2  aDr  a  a  r  r  Thermal resistivity, coefficient of friction Values of the thermal resistivity Z and the corresponding R± (for a particular value of n) are shown i n Table 3.2 for a variety of geologic materials. A sensitivity test using the parameters in Table 3.3 is shown in Figure 3.2; the erosion rates differ by up to two orders of magnitude between cases of maximum and minimum Z . However the intra-rock type variability of Z is large compared to the inter-rock type variability and no obvious patterns emerge from the data to guide assignment of thermal resistivities (apart from clays which tend to be good heat conductors). Therefore a global value of Z = 0.35 m K W will be used for all lithologies. This value is i n the range of almost all rock types reported in Kobranova [1989]. r  r  r  _ 1  r  Although several authors report different values for /z the coefficient of friction for different rock types [see Ohnaka, 1975; Jaeger and Cook, 1979], in a compilation of friction results Byerlee [1978] Metcalf [1979] uses attritivity ( k g j ) and does not explore fc b whatsoever. Attritivity is defined in terms of indentation hardness not Vicker's hardness, which may account for the difference in abrasive wear coefficients. 2  - 1  a  r  CHAPTER 3. EROSION  31  CO  lio«  Z =2 mK W  - 1  Z = 0.16 m K W "  c  1  o 'co co  <  R = 0.01 m 10" 0.2  0.1  0.3 B e d dip (rad)  0.4  0.6  0.5  Figure 3.2. Sensitivity tests for various values of grain size R, bed dip 0 and thermal resistivity Z . See Table 3.3 for parameter details. r  Parameter ^abr / hg  Pi 9 ° ^melt V  fext  A*  Value 1  x 1(T  Units  900 9.81 7.42 X 10~ 0.5 0 0.85  Pa" kgm" ms KPa ma ma  Parameter  1  10  3  - 2  8  - 1  - 1  - 1  C Pr «a V v  6  5  s  /bed r  "This value o f T? leads to R* = 0.38 m for Z = 0.16 m K W T  Units  1.0 2100 1 X 10 3.34 x 10 100 1.7 0.16 / 2.0  Z  —  Value  -  1  —  kgm" Pa a Jkg" m a  3  1  - 1  —  mKW"  1  and .R* = 0.11 m for Z = 2.0 m K W  -  1  r  Table 3.3. Parameters used for sensitivity tests of Figure 3.2. Non-SI units are used for some variables as this is standard i n the literature and facilitates comparisons. However, internally the model code uses strictly SI.  CHAPTER 3. EROSION  32  Figure 3.3. The basal ice grain-size distribution.  finds no significant and reproducible dependence on rock type for low to intermediate pressure and concludes that a universal designation of // = 0.85 is appropriate. I adhere to this and, as for the thermal resistivity, globally set fi — 0.85 for all lithologies.  3.1.4  Parameter determination — non-geological  Englacial particle radius  The radius of the englacial clast R, assumed to be non-lithologicalfy controlled, must be determined. The most expedient method is to choose a representative clast size; i n the recent process-driven large-scale erosion model of Tulley [1995], a single clast size of 0.1 m is used with the argument that all other clast sizes are washed away by subglacial waterflow. A n appropriate distribution of R is preferable to this approach. Many subglacial tills appear to show fractal distributions of particle sizes [Hooke and Iverson, 1995; Fischer, 1995], though this result is not universally accepted [Hubbard et al., 1996]. There are also arguments against this distribution holding within the ice; Alley et al. [1997] suggest that  CHAPTER  3. EROSION  33  the high surface energy associated with fine particles could impede their inclusion into the ice by intrusion (Section 4.1.3). Sedimentological analyses of basal ice from Matanuska Glacier, a large (ss 400km ) Alaskan coastal glacier [Lawson, 1979], several outlet glaciers of West Greenland [Knight, 1994] and eleven Alpine glaciers [Hubbard et al., 1996] may be used as examples for grain-size distribution within basal ice. Many facies of sediment-laden ice are identifiable and different facies may have distinct grain distributions. For example, the clotted-dispersed facies of Knight [1994] and the suspended facies of Lawson [1979] both comprise predominantly silt-size particles. However, these facies do not make up a large fraction of the total debris. A mean grain size* </^jjj£ = 0.75 and a standard deviation oi ^>^ = 2.5 is consistent with all three analyses, excluding the englacial and dispersed facies. This distribution, shown in Figure 3.3 is what is used i n the erosion model. N size classes are evenly distributed in c^grain-space from R ss 0.05 m to R ss 2 X 1 0 m . Processes dependent on R are calculated for each discrete R , weighted by the concentration C and then summed over all r radii classes. 2  &in  T  - 6  r  r  Bed dip, viscosity and sensitivity tests  The determination of the frequency distribution for north-south and east-west bed dip 0 and 0 is described in Chapter 2. A directional frequency distribution of bed dip 0 is derived by ns  ew  A  *  taking the projection of the bed dip distribution i n the direction of shding. Therefore 0 is given by  , v .Q™ + v .Q™ 0 = —1 . s  s  (3.14)  v.. + v  s  The effective viscosity of the ice n is determined by the ice model (see Section 5.1) and exhibits a range of several orders of magnitude (from 2 x 10 - 3 X 10 Pa a); as 77 has such a large range, it becomes an important parameter for determining erosion rates. 4  7  Sensitivity tests to show the effects of 6, R and n on the erosion rate are shown i n Figure 3.4. The parameters and constants used in the sensitivity calculations are given i n Table 3.4.  *The logarithmic ci in scale is common in sedimentology. It is denned as where R is in metres. E a c h pair ( © ^ , ©ft ) is considered a vector. gra  4  w  8  ^ in = gra  — l°g2(2  x  10 R) , 3  CHAPTER 3. EROSION  34  Contours of abrasion in mm/a  Bed dip (rad) Figure 3.4. Sensitivity tests for various values of grain size R, bed dip 6 and ice effective viscosity 77. See Table 3.4 for parameter details. Contours of 1, 10, 100 and 1000 m m / a are plotted.  Parameter  ^abr /  Value  Units  1 XlO 900 9.81 7.42 X 10~ 0.5 0 0.85  pi 9 Vc  ^metl ^ext P-  Pa" kg m " ms KPa ma ma  Parameter  1  - 1 0  - 2  8  3  C Pr V  - 1  —  Units  1.0 2100 1 X 10 /1 X 10 3.34 x 10 100 1.7 0.3 6  - 1  - 1  Value  5  v,  /bed  — 7  kgm" Pa a Jkgma  3  1  - 1  —  mKW"  1  "This value of Z leads to .R* = 0.28 m for 77 = 1 x 10 Pa a and R* = 0.88 m for 77 = 1 x 10 Pa a 6  T  Table 3.4. Parameters used for sensitivity tests of Figure 3.4  7  CHAPTER 3. EROSION  3.2  35  Quarrying  Quarrying, or plucking, is when part of the bed is pulled off through the action of shding ice. It is less readily accepted (and studied) than abrasion; some contend that quarrying is merely the removal of previously loose parts of the bed and cannot lead to sustained erosion [see Lindstom, 1988]. Calculations i n support of this minimalist view of quarrying indicate that the stress induced by a shding glacier over a bedrock hump is unable to meet the Coulomb failure (shear) criterion  [Morland and Boulton, 1975; Morland and Morris, 1977]. Nevertheless asymmetric subglacial landforms such as roches moutonnees (bedrock mounds with a smooth and gentle upstream surface and a steep, broken downstream flank) challenge the notion that abrasion is the only significant agent of erosion. The above-noted problem of glacier-induced stress being insufficient to instigate bedrock failure can be circumvented by process-based theories of quarrying that assume the existence of water-filled cavities [Iverson, 1991; Hallet, 1996]. Fastvarying water pressure within fissures can facilitate and enable rock fracture through fatigue [Iverson, 1991]. However, these conditions of rapid change are unlikely to exist under large ice sheets and do not play a lead role in ice sheet erosion. Hallet [1996] develops a model for quarrying based on subcritical mode I (tensile) crack growth due to differential normal loading of the ice and a water-filled cavity; this forms a basis for the treatment of quarrying i n the present study. I begin with a discussion of subcritical fracture and associated parameters. The state of stress in the bed is required to solve for the rate of subcritical fracture and the effect of topography is examined using a 2-D stress solver. The existence of a cavity is crucial to quarrying; without cavities, subglacial cracks cannot grow. I therefore briefly examine a solution for the extent of cavitation and its effect on the state of stress. Lastly the salient points are compiled into a comprehensive quarrying model.  3.2.1  Subcritical crack propagation  Stress intensity factor  There are three modes of crack propagation: a mode I crack is formed under tension while modes II and III are shear propagated. Crack growth in any mode can proceed very quickly (approaching the acoustic velocity) when the stress intensity factor KL exceeds the critical stress intensity (synonymous with fracture toughness) K\ for a mode L crack. The stress intensity near the tip of a crack is given by [Lawn and Wilshaw, 1975] KL = m aL\/irTc g  (3.15)  CHAPTER 3. EROSION  36  where <TT, is the far-field stress (for a crack with its major axis oriented i n the x direction, 07 = Cyy, o~u = o~ and o~m = cr ), l is the half-crack length and m is a modification factor to account for crack geometry and loading conditions. For example m — 2 / x for a favorably oriented penny-shaped crack and m = 1.12 when the crack is single ended i n a semi-infinite space. xy  zy  c  g  g  g  Kachanov and Laures [1989] examine the effect of interacting cracks within a network, which is a plausible scenario for quarrying because a crack is unlikely to exist in isolation. Their results show that both stress amplification and shielding is possible depending on the orientation of the interacting cracks. Some conclusions that can be drawn from their work: Mode III stress intensity factors are small compared to mode II except in certain exceptional geometric situations and both amplification and shielding of mode I and mode II cracks is negligible i f the inter-crack distance is more than 15% of the crack length. In this study a single-ended penny-shaped crack i n a favorable orientation is assumed to exist with no significant interactions with other cracks. Mode III cracks are not considered i n this thesis.  Crack growth When the stress intensity factor reaches the critical value, the crack propagates catastrophically; however subcritical crack growth can occur for KT, < K£ through a number of micro-processes and the most important at low temperatures is stress corrosion. A non-negative entropy condition imphes a minimum stress intensity K\ below which no subcritical crack growth can occur i n brittle materials [Rice, 1978] although this does not necessarily apply to heterogeneous, multiphase solids such as rocks [Segall, 1984]. Theoretical and heuristic arguments suggest that if£ « 0.2-K^ O.SK^ [Atkinson, 1984; Atkinson and Meredith, 1987b] despite the experimental failure to observe a subcritical crack growth hmit i n rocks. The simplest equation for subcritical crack growth is based on the power law of Charles [1958] (3.16) where VT, is the speed of propagation and nr, is the subcritical crack growth exponent for a mode L crack. The parameter AL has large uncertainty associated with it and has rarely been published for geological materials [Atkinson and Meredith, 1987b, page 136] therefore it is used as a tuning parameter and a single value will be used for all geological materials. There are other empirical formulae that honour the thermodynamic limit of however the difference between results is miniscule as UL is large ( > 10) for geological materials. Aside from being a simple relationship, (3.16) has the advantage of being widely used, therefore there are some compilations of the parameters K\ and njr,. There is a satisfactory fit of (3.16) to empirical data as long as stress corrosion is the rate hmiting step — this occurs at KL < 0.75if£ [Atkinson, 1984]. A t  CHAPTER  3.  37  EROSION  values of Ki beyond this point there is very little increase i n vr, (until critical KT, is reached); VT, will be taken as constant for KL > 0.75JK"£ i n this study.  3.2.2  Subcritical crack parameter determination  The following sections discuss the subcritical crack growth parameters and list compilations gathered from the literature of these parameters. Assignment of Kj, KJJ, ni andrc/jto the lithologies identified and used i n the transport model is left until Section 5.3.1.  Critical stress intensity  A selection of published values for K\ is presented i n Table 3.5. Experiments that did not meet the criteria set out i n Atkinson and Meredith [1987a] for true critical stress intensities are not fisted. These values form the basis for assignment of Kj values to the lithologies of North America i n Section 5.3. Table 3.5: Compiled values for mode I critical stress intensity of various geological material Rock Type  Reference MPam / 0.34, 0.79, 0.92, 2.66 0.45 1.18 0.34 1  sandstone mean minimum marble  mean minimum limestone  2  0.64 0.87 1.40 1.49 1.10 0.64  [Meredith et al, 1984] [Atkinson, 1979]  [Atkinson, 1979] [Meredith et al, 1984] [Costin and Mecholsky, 1983] [Atkinson et al, 1985]  0.87, 1.01 1.14,1.65 1.17 0.87  [Atkinson, 1984] [Meredith et al, 1984]  [Atkinson, 1980] [Meredith et al, 1984]  mean minimum  1.34 1.60 1.47 1.34  quartzite  2.10  [Atkinson, 1984] continued on next page  mean minimum novaculite  CHAPTER  3.  EROSION  38 Table 3.5: continued  Rock Type  K\ MPam / 1  granite  mean minimum  1.66, 2.48 2.43, 2.45, 3.52,3.50 2.28 1.74 2.17 1.80 2.00 1.85 2.32 1.66  [Atkinson, 1984] [Atkinson et al., 1984] [Meredith et ai, 1984] [Atkinson and Rawlings, 1981] [Der et al, 1978] [Swanson, 1980] [Ingraffea, 1981] [Costin and Mecholsky, 1983]  2.54 2.58 2.56 2.54  [Meredith et al., 1984] [Meredith and Atkinson, 1983]  basalt mean minimum gabbro  2.88 2.72, 3.03 3.75 3.10 2.72  mean minimum diabase mean minimum Table 3.5: of various for values 5.3. From  Reference 2  .  [Atkinson and Rawlings, 1981] [Meredith and Atkinson, 1985] [Atkinson et al., 1985]  [Meredith et al., 1984] 3.28 [Meredith and Atkinson, 1983] 3.32 3.30 3.28 Compiled values for mode I critical stress intensity geological material. This table provides the basis assigned to North American lithologies i n Section Atkinson and Meredith [1987a].  The lion's share of data are for mode I cracks, yet subcritical growth can occur i n all three modes [Atkinson and Meredith, 1987a]. Mode III fracture toughness is shown to be 3-4 times the mode I fracture toughness for granite and limestone at atmospheric pressure [Cox and Scholz, 1985]; it is assumed that values of KJJ are comparable to Kj . To obtain the fracture toughness under compression, a Coulomb-like (linear) relationship is assumed with a coefficient of internal friction pi taken as 0.6 (from examination of results reported in Cox and Scholz [1985] and consistent with the general trend indicated by Jaeger and Cook [1979]). Therefore KJJ™, the fracture toughness with a normal stress acting on the plane of the crack CT/V, is given by TI  nt  III  K  III  K  . III K  + Mint^iW ^ 7  °~N > 0, (TN < 0.  (3.17)  The linear form (3.17) is motivated by the direct dependence of the stress intensity factor on the far-field stress and the success of the Coulomb fracture criterion for undamaged rock.  CHAPTER 3. EROSION  39  Subcritical crack growth exponent A physical interpretation of raj, relates it to multiple fractures and branching: TIL tends to increase with increasing tortuosity of the forming crack [Swanson, 1984]. When a crack encounters a heterogeneity it is more likely to branch, thereby explaining the general trend that more complex material exhibit higher values of raj (corresponding to lower susceptibility to subcritical cracking). Linking of pre-existing cracks can decrease tortuosity; this can explain the large variability in published values of raj for similar materials. Values for mode I subcritical crack growth indices are given in Table 3.6. Due to a lack of experimental data for either mode II or mode III, it will be assumed that raj = nu = nm- A l l values reported in Table 3.6 were determined under similar experimental conditions; measurement of raj is a challenging experimental task and as discussed above there is large natural variability present i n geologic materials. However the general trend of higher raj associated with higher heterogeneity is reflected in Table 3.6. The lowest values of ra/ are the homogeneous samples of marble and sandstone; the highest values of raj correspond to the more complex igneous rock types.  CHAPTER 3. EROSION  Rock Type  40  Mode I Crack Growth Index  Reference  ni sandstone marble  14  mean minimum  9 26-29° 15 17.2 9  novaculite  25  andesite  26  micrite  26  limestone  27* 26  minimum  [Atkinson, 1984] [Costin and Mecholsky, 1983] [Henry and Paquet, 1976] [Atkinson, 1984]  [Atkinson, 1980] [Waza et al, 1980] [Henry and Paquet, 1976] [Atkinson, 1984]  24 33 28.5 24  [Meredith and Atkinson, 1983] [Waza et al, 1980]  diabase  29  gabbro  32.5 29  [Meredith and Atkinson, 1983] [Atkinson and Rawlings, 1981]  basalt  mean minimum  minimum quart zite  minimum granite  mean minimum  C  [Atkinson, 1984]  34 30  d  50 34 35 35-46 24 53 39.4 24  e  [Atkinson et al, 1984] [Atkinson, 1980] [Atkinson and Rawlings, 1981] [Atkinson, 1984] [Costin and Mecholsky, 1983] [Swanson, 1980]  Table 3.6. Compiled values for ni the subcritical crack growth exponent for mode I cracks. This table provides the basis for values assigned to North American lithologies i n Section 5.3. From  Atkinson and Meredith [1987a]. "27.5 t a k e n as representative value range of 2 6 - 2 8 was r e p o r t e d by the a u t h o r A range o f 2 9 - 3 6 was r e p o r t e d b y the authors A range o f 3 0 - 3 8 was reported b y the a u t h o r 4 0 . 5 t a k e n as a representative value C  d  e  CHAPTER 3. EROSION  41  Topography Case Case Case Case Case  1 2 3 4  Radius of Curvature 18.03 29.15 42.72 57.01  m m m m  Amplitude (crest —> trough) 16.06 m 8.31 m 5.44 m 4.02 m  Table 3.7. Four different cases for topographical amplitude are imposed on the mesh template shown i n Figure 3.5. 3.2.3  Effect of topography  B y (3.15) KT, is directly proportional to the state of stress. Local topography affects the state of stress i n the subsurface [see Morland and Boulton, 1975; Morland and Morris, 1977] and is of primary importance i n cavity formation. How significant is topography i n terms of a stress enhancing role given the existence of a cavity? To resolve this question F R A N C 2 D , a two-dimensional, finite-difference package to analyse stress [Ingraffea et al., 1997; Ingraffea and Wawrzynek, 1995], is used to simulate a subglacial bed with topography. The package uses a linear elastic stress solver for an isotropic material to solve two dimensional stress problems.  F R A N C 2 D setup  A mesh, shown schematically i n Figure 3.5, is created to define the domain of the problem. It is designed to maximize cell density in the upper centre of the domain, specifically at the trough of the topography as this is the presumed area of maximal crack growth. The mesh is large enough that boundary conditions of the sides and bottom do not affect the solution; the domain is effectively infinite. Four different topographies (consisting of circular arcs) with topographical wavelength L used. Specifics of each case are detailed in Table 3.7.  t  are  Boundary conditions ( B C ) drive the problem. The boundary is fixed on the sides and the bottom of the domain; this is an identical condition to the familiar requirement that all stresses and strains of a point source vanish as r —> oo. The B C on the upper boundary defines the nature of the imposed load. The goal is to elucidate patterns of stress i n the bed and because the bed response is linear with the surface loading, absolute B C values are irrelevant and dimensionless loads are used. A body of ice approximately 200 m in length is represented by a vertical load Pi of 10 and a tangential load r of 1. Where a cavity is inserted, there is a normal load P of 5 with zero shear. The ratios of ice-overburden to shear (10:1) and ice-overburden to water pressure (2:1) are representative of observed values. w  CHAPTER 3. EROSION  42  Mesh for F R A N C 2 D Analysis  Figure 3.5. Mesh used for the FRANC2D stress analysis. Each segment A <—> B is labelled with -^mesh(«m : bm) where i V h is the number of cells assigned for AB and the ratio (a : bm) indicates the relative cell size at the point A and point B. Only two segments have the points A and B explicitly labelled. The cell size varies linearly through the segment. The mesh is designed to concentrate nodes at the downstream edge of the cavity where fracture is most likely to occur. m e s  TO  CHAPTER 3.  EROSION  43  A cavity of length L /3 is assumed to exist on the downstream slope of the bump; this cavity size is large for the normalized effective pressure chosen (0.5) and equation 3.19 requires the ice to be thin (on the order of 100 m) when v = 200 m a . t  - 1  a  F R A N C 2 D results  Figure 3.6 shows contours of the horizontal stress cr for each of the four cases outlined i n Table 3.7. The points in Figure 3.6 represent mesh points where stress is calculated, concentrated at the downstream cavity-ice junction. The thick contour lines represent integral contour values, while the thin lines are half-integral values. The grey shaded regions have cracks with a net tensile (T ; they are under the cavity and thus have water pressure acting against the compressive load of the ice. This is where vertical mode I cracks can grow subcritically. xx  xx  This analysis shows that, contrary to the notion that topography enhances local stress regimes, increased topography inhibits vertical mode I crack growth for this geometry. This is due to the increased compressive stress o added through vertical loading of the following downstream bump (extending from x = 0 m to x = 30m in Figure 3.6). Caution should be exercised i n the interpretation of Figure 3.6 as such simple-minded tests do not capture important points: for example there is no feedback from topography to cavity size. Nevertheless as a test of the effect of topography on the stress regime it is instructive. xx  Figure 3.6 indicates that the most likely location for vertical mode I crack growth is i n the vicinity of the downstream cavity-ice junction, i n agreement with the model of Hallet [1996]. However Iverson [1991] calculates stress on only the upstream cavity-ice junction, justified by field observations of the asymmetry of roches moutonnees and "bullet boulders" [Iverson, personal communication]. Whether or not downstream cavity-ice quarrying can reproduce the steep leeward sides of these landforms is dependent on complex feedbacks between the position of the cavity, topography and quarrying. This simple model cannot capture such complexity. Vertical stress o results are shown i n Figure 3.7; they show similar behaviour to the o~ . Interestingly, there is a more balanced likelihood of upstream/downstream crack growth. In general, assuming the absence of regional stresses that may exert significant influence i n mountainous regions, Vyy is more compressive than cr : horizontal mode I cracks are more difficult to propagate than vertical mode I cracks. yy  xx  xx  These results are dependent on the specific geometry chosen and a wide variety of subglacial geometries are possible ranging from the smooth topography examined i n this section to jagged steps. Nevertheless a general result can be extracted: tensile crack growth is not significantly affected by topography (aside from the role that topography plays in cavity formation). This allows the transformation of a complicated topographic stress regime to a simple one as shown  CHAPTER  3.  44  EROSION  Figure 3.6. cr (horizontal) results for the F R A N C 2 D test of topography. A l l axes are i n metres, and contours are in normalized units of stress where the surface loading is Pi = 10, P = 5 and r = 1. See text for further details. xx  w  CHAPTER 3. EROSION  45  Figure 3.7. cT (vertical) results for the F R A N C 2 D test of topography. A l l axes are i n metres, and contours are i n normalized units of stress where the surface loading is Pi = 10, P = 5 and r = 1. See text for further details. yy  w  by Figure 3.8. The maximum shear stress results shown in Figure 3.9 have a different response to topography. The most likely location for shear fracture occurs at the upstream cavity-ice junction and tends to increase with increasing topography; this is more akin to the results of Morland and Boulton [1975]; Morland and Morris [1977]. This observed topographic enhancement is heuristicaUy modelled as a function of local bed dip 6 through <r  max  = (2(9 + l ) < r  max  .  (3.18)  This linear enhancement satisfies two requirements: there is no modification for 9 = 0 and, based on the examination of Figure 3.9, an enhancement of  2.5 results from 6 = 7r/4.  CHAPTER 3.  46  EROSION  Ozz >"*  *  i  -x  PW_J<  Ozz  . /  ,  ;  I  rw.  ^gA \  CJzz  Pw T  Gxx  — 1  T  T  »  * 1 (-*•  T  T  »  Pw T  T  Gxx  _. __  Figure 3.8. The assumption that the topography does not significantly affect the cavity dominated stress field allows for considerable simplification of the top panel to the bottom panel.  Figure 3.9. Maximum shear stress results for the FRANC2D test of topography. All axes are in metres, and contours are in normalized units of stress where the surface loading is either lO(vertical), 5(normal) or l(tangential). See text for further details.  CHAPTER  3.2.4  3.  EROSION  47  Cavity size  T h e existence of a subglacial cavity is a prerequisite for quarrying; the stress regime without cavitation is generally insufficient to achieve the m i n i m u m stress intensity  needed to propagate  cracks. Cavities arise when ice separates f r o m the b e d on the downstream side of bedrock knobs [Agassiz,  1876; Lliboutry,  1958, 1968]. Shding, which generates the cavity, competes w i t h closure  by ice creep; balancing shding a n d creep closure allows the prediction of steady state c a v i t a t i o n [Walder,  1986].  T h e size of the cavity L  can range f r o m zero to the topographical length scale L .  c  t  n o r m a l i z e d cavity extent L' = L /L , c  used to determine L' .  c  D e n n i n g the  I assume linear interpolation between two endpoints can be  t  T h e upper endpoint L' = I represents complete decoupling of ice a n d bed;  c  c  this can be expected when subglacial water pressure equals ice overburden pressure a n d there is no creep closure. T h e lower endpoint L' , modified f r o m Walder c  [1986], is when the creep closure  balances shding. T h i s is written as P  w  P  w  = Pi  L' = 1,  (3.19a)  < P i - UQBQ  L' = 0,  (3.19b)  c  e  where BQ is the G l e n flow law parameter a n d TIQ is the flow exponent, taken as 4.19 X 1 0 P a s / 7  1  a n d 3 respectively f r o m Table 5.1 a n d E q u a t i o n (5.3) . 5  T h e endpoint (3.19a) is the notation condition where complete decoupling is expected, (3.19b) is where shding equals creep closure velocity [Nye, the ice surface taken as the topographic wavelength L t  of Walder  while  1953] w i t h the radius of curvature of  (3.19b) is a modification of the analysis  [1986] where a single cavity size is calculated for a bedrock step. W i t h the geometric  inclusion of s i n g , I generalize to a variety of b e d topography a n d the present f o r m u l a t i o n is appropriate to find the areal extent of cavitation over the entire b e d . E q u a t i o n (3.19) reproduces the f u n d a m e n t a l features that L'  c  of low b e d dip 6.  —> 1 as P  w  — > P, a n d that cavitation is suppressed i n regions  In the h m i t i n g case of a flat b e d , no cavitation can occur u n t i l P  w  reaches  flotation. T o avoid the non-physical solution of L' > 1 when the water pressure exceeds n o t a t i o n , c  L'  c  is capped at L '  m a K c  , which is taken as 0.9.  Excess stress due to cavity  T h e integrated stress o n the b e d is determined by the ice height (for the a  zz  a n d surface slope (for the a  b o u n d a r y condition).  xz  b o u n d a r y condition)  T h e existence of a cavity will not change  the integrated stress, but will redistribute it, concentrating stress on the p o r t i o n of the b e d i n B  5  G  =  [B exp(-Q /RS™T )}-  1/3  0  i  i  3  CHAPTER  3.  EROSION  48  direct contact with ice. The excess stress on bed with ice contact must satisfy the following conservation equations of the integrated stress  j Lt  of {x) dx = L (Pi z  (3.20a)  - P ),  c  w  L  c  I Lit  Lt  af {x)dx z  = Lcr,  (3.20b)  Lc  where x is the coordinate coinciding with the direction of v . The right-hand side of (3.20) represents the "missing" stress because of the presence of a cavity with length L . The left-hand side is the integrated redistribution of the excess stress over the portion of the bed in direct contact with the ice. s  c  The form of the stress excess is taken as <l = U ^ - ^ )  (3.21)  2  which reproduces the required property of greater stress concentration towards cavity-ice junctions [Iverson, 1991]. The constants £ , determined by satisfying (3.20), are ao  c  _  2AL' (Pi - P ) c  w  (  3 2  2  a  )  L\ [(1 - L' f - (L' - if] ' c  c  24lZr L\ [(1 - L' f  3.2.5  (3.22b)  - (L' - If  c  c  '  A q u a r r y n ig m o d e l  Any process-based quarrying law must be proportional to the rate of crack propagation; (3.16) is used for mode I and mode II crack growth. Unlike the model of Hallet [1996] there is no direct geometrical dependence on cavity extent. I do not attempt to explain the full process of quarrying and ignore the quandaries of how cracks connect and how blocks are evacuated; these details are geometry-specific and defy modelling efforts at this level. The important variable is crack growth velocity and therefore this is the only process variable used. The quarrying rate Q is given by Q = [vi + v ]e- f* d  n  (3.23)  CHAPTER 3. EROSION  49  where the rationale for the exponential term is identical to that given i n Section 3.1.2 and d is calculated by (3.13). vj and VJJ are given by 0  Ki < 0 . 2 5 i q ,  m v^i{-\w+pwy K* g  ni  (3.24)  0.25KJ < Ki < 0.75KJ,  T  Ai0.75  K > Q.75K$,  ni  T  and  K <0.25Kf-  0  \  xW+Fw  n  njj  vn = <An  0.25KII  II  (_ (l)  <K  n  A  < 0.75K  n  +  p ) w  (3.25)  K  AnO.  75 "  K  n  > 0.75Kf  u  r  where A^ ) is the minimum principal stress. The calculation of A^ ) and <r , which is modified to cr by (3.18), is outhned i n Appendix A . The normal loading F and tangential loading F (N m ) used in Appendix A as boundary conditions are driven by the cavity water pressure P (both as a direct B C and for determination of cavity extent through (3.19)), the ice overburden pressure P{ and the basal shear stress r , modified by the effect of the cavity as described i n Section 23. P is determined by the hydrological model (see Section 5.2) and Pj and r by the ice model (see Section 5.1). The parameters ni,nu,K\ and K have a lithological dependence and are assigned in Section 5.3. AIJI are tunable parameters and are explored i n Section 6.2. 1  1  m a x  6  max  N  s  _ 1  w  7  w  n  There are limitations on F and F because ice has finite strength; ice fracture puts a maximal limit on the interface stress. Hallet [1996] argues that Pi — P cannot exert more than 10 M P a on the bed based on the tensile strength of ice; however the ice never exists i n a state of tension. A more appropriate measure is the crushing strength S , which is 2 M P a for ice at 0° [Butkovich, 1954]. The crushing strength is a function of many parameters and can vary by a factor of « 3 with different loading rates [Butkovich, 1955]. For slow (order of minutes) loading conditions that exist under ice sheets and glaciers, a crushing strength of 6.5 M P a is expected for samples with loading area > 0.25 m . This is multiplied by a factor F to account for the confinement of the ice. Therefore the condition on the normal loading will be N  s  w  8  c  2  g  Pi +  v'ex  FgS '  (3.26)  C  The limit for the boundary condition for shear stress will necessarily involve S the shear strength of ice, taken as 1.2 M P a [Butkovich, 1954]. This value is corrected to a confining pressure of Pj s  Note that cr™ is not the maximum shear stress but rather the shear stress on the plane of failure. T h e linear boundary conditions F and F are divided by a unit length to get the pressures Pi +arf£,P and shear stress r + . T h e crushing strength is the maximum compressive pressure on an unconfined sample of ice before fracture. 6  7  w  8  ax  N  s  CHAPTER 3. EROSION  50  Contours of minimum principal stress in M P a  Figure 3.10. Cross sections of the minimum principal stress A^) and the shear stress on the failure plane <r . The cavity length is 1.7 m (while the topographical length scale is 10.0 m) 1  max  through /Xj the coefficient of internal friction for ice ce  T +  where p  lCK  (Tf  <  /X Pi + ice  S,  (3.27)  is assumed to be 0.3.  A realization of the calculated stress due to a P; = 9.00 MPa, P = 8.10 MPa, r = 1.00 MPa and v, = 150ma is shown in Figure 3.10 (other parameters listed in Table 3.8). The shaded region is the area under the cavity (which has L = 1.7 m) where the water pressure exceeds compressive forces acting on the crack: the crack is in tension. The area is not as extensive in the bottom panel because the orientation for maximum probability of shearing is not coincident with the minimum principal stress, i.e. on the failure surface, cr^ > A^ ). The most likely location of fracture (both mode I and II) are again predicted to be on the downstream cavity-ice junction (ice flows from left to right in Figure 3.10). w  -1  c  1  CHAPTER  3.  EROSION  Parameter  no S Ss e c  51  Value 3  x x  6.5 1.2  10 lO  45°  6  6  Units  Parameter  —  B  Value 4.19  G  Pa Pa deg 6  X  lO  Mice  0.3  L  10  t  7  Pas / 1  8  a  — —  3 c  Units  m  Table 3.8. Parameters used to generate the stress cross-section i n Figure 3.10 and the sensitivity tests of Figure 3.11 "From Paterson [1994] F r o m Butkovich [1954, 1955] F r o m Butkovich [1954] 6  c  Quarrying rate for a variety of effective pressures 10'  I  10°  -e-  CO  R  -4  Pe = 0.6 MPa Pe = 1 MPa Pe = 0.8 MPa Soft rock (0.8 MPa) Long crack (0.8 MPa)  O 10  10" 6  8 Basal shear (Pa)  10  12  14 5  x 10  Figure 3.11. Quarrying sensitivity tests using A ^ and crg extracted from many two dimensional cross sections as shown i n Figure 3.10. Effective pressure P is defined as Pj — P . iax  e  3.2.6  w  Quarrying sensitivity  Many analyses such as the example shown in Figure 3.10 are calculated for a range of basal shear stress and water pressure, erg" * and A^ ) are extracted from each realization; the quarrying rate is then determined by ( 3 . 2 3 ) , ( 3 . 2 4 ) and ( 3 . 2 5 ) using parameters from Table 3 . 9 . This assemblage of results is plotted i n Figure 3.11 . 1  1  Fairly low effective pressure (defined by P = Pi - P ) is required for quarrying i n this model. e  w  CHAPTER 3. EROSION Parameter  52 Value  1.76/0.34 X 10 20.0 150 0.7 0.6  ni,n  n  m.•9 Mint  Units 6  Parameter  Pam / 1  Value  Units  7.0/1.2 X 10' 1 x 10~ 1000 0.02 / 0.1  2  6  ms"  1  Pam / 1  2  ms m  Table 3.9. Parameters used for sensitivity tests of Figure 3.11. Dual values are given for standard and soft rock for Kj and K , standard and long crack length for l . n  " w i t h pi = 9 0 0 k g m ~  3  a n d g = 9.812 m s -  c  2  B y (3.19b) a 2000 m thick ice sheet shding at 1 0 0 m a with 10 m bed topography of 45° bed dip, water pressure that is 95% of flotation is needed to form a cavity. Cavities occur at a lower fraction of flotation water pressure for thinner ice; it is the absolute effective pressure that controls the cavity size and the quarrying rate. For a given effective pressure the quarrying rate is very insensitive to the height of the ice column. - 1  In general the results are also insensitive to the strength of the ice as the quarrying is limited by the 0.75-fifj cap rather than the crushing strength of the ice (similarily for mode II cracking). Mode II subcritical cracking is only possible i n very soft rocks. Even i n these cases, it mostly remains insignificant compared to the mode I cracking. The only exception to this is at high values of r when the mode II crack growth catches up to the mode I crack growth, already curtailed at the 0.75Kj limit. This is seen in the upper right hand corner of Figure 3.11. This model produces much lower quarrying rates than the model of Hallet [1996]. The main reason for this discrepancy lies in the calculation of stress i n the bed. Hallet [1996] contends that tensile stresses, 2/3 the magnitude of the loading stress, are induced i n the bed. From the stress analysis of Hetenyi [1960] (upon which the claim of Hallet [1996] is made) this can only happen under very specific conditions: when the corner of an infinite quarter plane is loaded. Possibly this situation could manifest itself subglaciaUy when a very large proportion of the stepped bed is covered by cavities; I assume that the realization of this possibility does not play a significant role and take the more conventional view that the bed is in a state of compression due to ice loading.  Chapter 4 Englacial Processes The basal layer of glaciers and ice sheets is of immense significance not only to glaciology, but to the whole of Quaternary science. [Knight, 1997]  This chapter deals with the mechanics and modelling of englacial activity. Entrainment (and complementary deposition) are those processes that directly exchange sediment with the bed through the bottom of the ice; they are discussed i n Section 4.1. Englacial mixing (Section 4.2) controls the vertical motion of the sediment after entrainment (and prior to deposition). Lastly the lateral motion of sediment is considered in Section 4.3.  4.1  Entrainment  The margins of glaciers can have debris within the basal portion of the ice (see Figure 4.1); this is typical especially of cold or polythermal glaciers and surge-type glaciers [Boulton, 1970; Hambrey et al., 1999]. Bore holes from modern ice sheets show that this phenomenon is not exclusive to the margins, and the central portions of ice sheets also have "dirty" basal ice [e.g. Gemmell et al., 1986; Gow and Meese, 1996]. This debris is unquestionably of subglacial rather than supraglacial origin [Lawson and Kulla, 1978; Gemmell et al., 1986; Strasser et al., 1996] and the mechanism of its entrainment has captured the attention of glaciologists for over a century. Chamberlain [1895] asks:  How does a glacier take up its material? How does it carry it forward? How does it put it down? What functions do water and topography play in the process? A glacier's methods in respect to lateral and medial moraines, which are superficial, are simple  53  CHAPTER 4. ENGLACIAL  PROCESSES  54  Figure 4.1. A n example of debris-laden basal ice. This unnamed surge-type glacier flanks M t . Wood i n the St. Elias Range, Yukon Territory.  and well known. Its methods respecting basal material, which is largely concealed, constitute the problem.  A century of glaciological research has yet to fully answer these questions. The concealed nature of entrainment that beset the study of basal ice in the era of Chamberlain [1895] continues to limit workers today. Nevertheless advances from theory and experiment (both laboratory and ice tunnel) have contributed greatly to the modern discussion on entrainment of debris and our understanding of the subglacial processes responsible for basal ice is substantial, though not complete. The subject of basal ice has direct consequence on the present study. I use englacial sediments as lithological tracers of the bed and the need for a process-driven model of entrainment is selfevident. There are also feedbacks to other modelled components; the debris volume fraction C affects the erosive power of ice (Equation (3.12) on page 28). To determine the time evolution of the volume fraction of hthology I in the (i, j, fc) cell of ice Cijki is the objective of the englacial model. th  CHAPTER  4. ENGLACIAL  PROCESSES  55  Basal ice differs rheologically from pure ice although the relationship is complex [Hubbard and Sharp, 1989]. Despite basal ice typically representing a small volumetric fraction of the ice mass, deformation is concentrated towards the lower ice and differences in basal rheology may assume a substantial importance in ice sheet models. This represents an avenue of potential feedback to the ice sheet model. A number of mechanisms can entrain debris into ice (see Alley et al. [1997]; Knight [1997] for comprehensive reviews); these include incorporation of pre-existing ice, freeze-on, regelation and englacial tectonics. Each is discussed below followed by a description of the modelling strategy used i n the present study.  4.1.1  Overriding of pre-existing ice  The simplest method by which ice sheets entrain debris is the cannibalisation of pro-glacial aprons  1  or peri-glacial frozen soils . During an advance the ice can incorporate this debris-rich material 2  instead of overrunning it as exhibited by the advance of Trapridge Glacier over its stagnant apron [Clarke and Blake, 1991]. Isotopic, gas, solute and sedimentological analysis suggest that this method accounts for dirty basal ice at Law Dome, Antarctica [Goodwin, 1993], GISP2 [Gow and Meese, 1996] and G R I P [Souchez et al., 1994] stations in Greenland.  4.1.2  Freeze-on  The entrapment of debris through the freezing-on of water at the sole of the glacier has several guises. In appropriate thermal conditions net basal freezing indiscriminantly adfreeezes both ice and sediment. In the absence of a downward migrating freezing front, melting and refreezing cycles of Weertman regelation and glaciohydraulic supercooling (both discussed below on pages 56 and 57) can result in accretion of basal debris by freeze-on. Stable isotope analysis has been instrumental in the identification of basal ice accreted by freezeon; Gow et al. [1979] provides the first concrete evidence using S 0 18  analysis. During freezing,  fractionation between heavy and light isotopes takes place giving frozen-on basal ice an isotopic signature. The co-analysis of S 0  and 8D* can extend the stable isotope conclusions by offering  18  information concerning the origin of the initial liquid [Jouzel and Souchez, 1982; Souchez and Jouzel, 1984]. Glaciers can terminate by a steep margin with an apron of fallen-ice blocks, snow and sediment at the  1  toe. Frozen ground on the perimeter of an ice sheet. HB = [(HD 0/H2 0 ] l -^ (HD 0 / H 0 ) S M O W  2  16  16  16  1 6  2  L  (HD  1 6  0/H  1 6 2  0) MOW S  j  x  J  S  M  Q  W  s t a n d a r d  watgr  _  CHAPTER 4. ENGLACIAL  PROCESSES  56  Net basal freezing  The thermodynamic evolution of the ice is calculated by the Marshall-Clarke ice sheet model, described in Section 5.1.1 on page 74. B y momentarily ignoring the advective and strain heating terms, the energy balance equation (5.4) can be rewritten as PiCi^T  = V -qi  (4.1)  where q i is the heat flux through the ice. Using Fourier's Law, with the assumption that vertical gradients dominate their lateral counterparts, the vertical heat flux is given by (4.2)  Under a warm-bedded glacier with available water, the temperature is fixed at Tpmp the pressure melting point therefore (4.1) demands that q ; be solenoidal and the vertical heat flux must balance the basal boundary conditions. If the ice thins and the thermal gradient dTi/dz steepens, the increased heat flux from (4.2) must be accompanied by an extra boundary condition: the freezing front migrates downward, releasing latent heat of freezing. This mechanism was first proposed by Weertman [1961] and has been invoked as the entrainment mechanism for many glaciers and ice  sheets [e.g. Gow et ai, 1979; Herron and Langway, 1979; Sugden et al., 1987; Tison et al., 1989; Knight, 1994; Zdanowicz et al., 1996].  Weertman regelation  The shding of glaciers partly relies on regelation around small topography of the bed [Weertman, 1957]; this involves local melting of ice on the upstream side of an asperity and freezing on the downstream side (see Section 4.1.3). The downstream refreezing has the propensity to freeze-on fine particles suspended i n the subglacial water. Generally this mechanism does not produce an appreciable net gain of ice because the newly formed basal ice on the lee side of an asperity is mostly melted on the upstream side of the following bump. Net accretion can form i n bed troughs although the thickness, which is a function of bed roughness, is limited to ss 10 cm [Hubbard and Sharp, 1993]. This thickness is based on a model of multiple melting-refreezing cycles and is i n accordance with observational data. Lliboutry [1993] offers an alternate theory for Weertman regelation i n which the temperature of the ice is completely determined by its stress regime and the water source for the accretionary ice is englacial. Although the theory differs from the classical theory used by Hubbard and Sharp [1993] where the water originates and travels along the bedrock surface, the results are similar: a layer of accreted ice ss 3.5 cm thick. Laboratory work supports the theory of englacial water  CHAPTER  4. ENGLACIAL  PROCESSES  57  transport and this vein flow is observed to entrain fine grained debris [Knight and Knight, 1994, 1999]. T h i n bands of debris are expected from the regelation mechanism of either Weertman [1957] or Lliboutry [1993] and laminated facies of basal ice are commonly attributed to this process [Hubbard and Sharp, 1993; Sharp et al., 1994; Hubbard and Sharp, 1995]. Regelation is also the foremost entrainment hypothesis for a dispersed clotted facies observed in Arctic outlet glaciers [Sugden et al., 1987; Souchez et al, 1988; Knight, 1994].  Glaciohydraulic supercooling Water flowing into a region of lower ice-overburden pressure may become supercooled i f it cannot warm quickly enough to accommodate the changing melting point; this supercooled water can freeze-on debris [Lawson et al., 1998; Alley et ai, 1998]. However the opposite effect occurs when the water descends a bedrock slope and no net accretion would normally be expected via this mechanism. Also, large volumes of subglacial water flow are required for glaciohydraulic supercooling to be effective as an agent of entrainment [Alley et al., 1997]; this may exist i n alpine glaciers where surface melt can reach the bed via crevasses and moulins but is unlikely i n ice sheets where such surface-bed communication does not exist (Section 5.2).  4.1.3  Intrusion by regelation  Background  The first modern definition of regelation is given by Thomson [1859/60] as "melting by pressure and freezing again on relief of pressure" ten years after Faraday first described the phenomenon as a curiosity i n a lecture for the Royal Society. It is the process by which ice melts from higher pressure on one side of a grain (or asperity) and refreezes on the opposite side with heat flow through the grain driving the process. A n early quantitative experiment describing a wire mesh passing through a block of ice [Bottomley, 1871/72] anchored the qualitative regelation theory of Thomson [1859/60]; the possible importance to subglacial processes was noted by Bottomley [1871/72]. A comprehensive regelation theory was outlined by Ornstein [1906] and Deeley and Parr [1914] developed an early glacial sliding mechanism reliant on regelation. Experiments of regelation around geologic materials [Deeley and Parr, 1914] buoyed the early theory (which is remarkably similar to the modern theory of Weertman [1957]) by allowing predictions of glacial shding speed, which were i n good accordance with observation. Refinements to regelation theory and  CHAPTER 4. ENGLACIAL  PROCESSES  58  quantitative experiments continued over the following half-century [e.g. Drake and Shreve, 1973]. The next step towards the development of regelation as an entrainment mechanism is the work of Phtiip [1980] with a theoretical treatise of ice flow past an array of cylinders by regelation. Experimental work supports the theory [Iverson, 1993; Iverson and Semmens, 1995] and demonstrates that regelation can occur around an array of englacial, cobble-sized, angular clasts and also into coarse-grained sediment. This unequivocably marks intrusion by regelation as a potential entrainment mechanism. Furthermore, isotopic analysis [Iverson and Souchez, 1996] demonstrates that the mechanism is consistent with existing basal isotopic data. Fine-grained sediment may not behave the same as coarse-grained sediment because silt to clay-sized particles have large surface-to-volume ratios and the correspondingly large surface energies may impede intrusion by regelation [Alley et al., 1997].  Theory  The steady intrusion speed v for ice at the pressure melting point moving into an array of clasts with a macroscopic pressure difference between the ice overburden and subglacial water pressure Pi P across it is [Phttip, 1980] r  w  P - P  (4.3)  where K (m P a s ) is the apparent conductivity of the array to the ice and l is the effective depth of grain-supported basal ice. Figure 4.2 illustrates the variables involved. _ 1  _ 1  a  The conductivity K is written in terms of the thermal conductivity of the clast and ice k and ki, the porosity n°, the pressure melting Figure 4.2. A sketch of intrucoefficient V , ice density pi and the latent heat of fusion for water sion showing ice overburden pressure Pi, subglacial water Z f as [Phdip, 1980; Iverson and Semmens, 1995] pressure P , effective depth of grain-supported basal ice Z -j~ k V ir(ki + K + h) and intrusion speed v . K r  c  us  w  a  r  fc{  ~f~  &r  c  £fus/>;(l -  n")  7r + (1 - n>)  r  - 4xe- ^ 2  (4.4)  CHAPTER 4. ENGLACIAL PROCESSES  59  where the conductivity k\ is k = (k - fe;) j l - y/ir(l - n") coth v M t  r  1  - n")  •2y/(l - n»)/7rcoth (ir-y/ir(l - n )) - (1-2^(1 - n )/rr) a  s  j (4.5)  For pure ice (at 0° C ) , k = 2.10 W m K , V = 7.42 x 1 0 K P a ,L = 3.34 X 10 J k g " and pi = 9 1 0 k g m . The porosity n" is taken as 0.40 (consistent with Section 5.2.1 on page 78) and a value of 2.86 W m K is used for k as determined in Section 11 on page 30. These parameters result i n K = 2.01 X 1 0 m Pa s . -  {  - 1  1  - 8  c  - 1  5  1  {us  - 3  -  1  _  1  r  _ 1 5  2  _ 1  - 1  Classical regelation theory requires ice to be at the pressure melting point. However, there is a theoretical basis for regelation at sub-freezing temperatures [Shreve, 1984] and observational evidence of cold shding [Echelmeyer and Zhongxiang, 1987; Cuffey et al., 1999] and cold intrusion by regelation [Cuffey et al., 2000] although these processes are very sluggish in cold ice compared to their rates i n ice at the pressure melting point.  4.1.4  Englacial tectonics  Evidence for thrust faulting The argument that sediment is scraped off the bed and incorporated englacially by thrust faults in the ice was put forward by Chamberlain [1895] to explain marginal debris bands i n Greenland; it continued to be developed as the sole unchallenged entrainment mechanism over the following seven decades [e.g. Goldthwait, 1951; Bishop, 1957]. The theory was attacked as untenable by Weertman [1961] based on the observations of finely interbedded layers of clean and dirty ice and a lack of shear evidence across the debris bands; these features are a challenge to explain using the thrusting model of Chamberlain [1895]. In the same paper as the critique of the entrainment-byshearing mechanism, Weertman [1961] proposed an alternate theory of bulk freeze-on to explain the observed debris bands, described in Section 4.1.2.  Souchez [1967] hypothesizes that multiple processes are active to produce two coincident yet distinct types of debris bands (with very different orientations and grain size assemblage); one is attributed to the net freeze-on mechanism of Weertman [1961], the other to thrust faulting. Further alternative mechanisms for the formation of the marginal debris bands of Greenland and Baffin Island are provided by Hooke [1973]. Nevertheless, outside the high arctic of Canada and Greenland, observational evidence of thrust  CHAPTER 4. ENGLACIAL  PROCESSES  60  faults with associated debris continues to accumulate. Many Svalbard glaciers, both surge and non-surge type, regularly exhibit thrust faults with debris-laden ice [Hambrey et al., 1996, 1999]. Tison et al. [1989] document a case from a tunnel i n an Alpine glacier, Grubengletscher, that shows a thrust surface with frozen ground moraine being transported upward; similar evidence exists for large Alaskan glaciers [Hart, 1995]. Clarke and Blake [1991] document englacial debris emplacement by thrust faulting in Trapridge Glacier, Yukon Territory, a small surge-type glacier in a quiescent state and Sharp et al. [1988] directly observe thrust faulting at Variegated Glacier, Alaska during an active surge. Thus the evidence that thrust faults act as conveyor belts transporting material away from the bed i n alpine glaciers is irrefutable. A more subtle question remains: whether shearing at the bed can independently entrain material via basal decollement or whether the mechanism merely acts as a vehicle for previously entrained debris as suggested by Boulton [1970]. There are examples of debris bands with properties that are difficult to explain with a freeze-on paradigm. Hambrey et al. [1999] note in a review of twelve Svalbard glaciers that some bands have thick (0.5 m) zones of sediment with little interstitial ice while Clarke and Blake [1991] observe a thrust plane develop during a period of ice thickening and bed warming; neither observation is conducive to a freeze-on mechanism. However these examples could be consistent with intrusion (Section 4.1.3) and do not therefore constitute a convincing argument for entrainment by thrusting. The question of true entrainment by thrust faulting remains unanswered.  Location of thrust faulting  Supraglacial debris ridges from shearing along thrust planes tend to be marginal features occurring within « 1 k m of the toe, but ice cliff observations i n Svalbard show that many thrusts do not reach the surface [Hambrey et al., 1999] as does a direct drilling program on Trapridge Glacier [Clarke and Blake, 1991]. Surficial expression of debris-laden faults is only possible where the ice is sufficiently thin. The corollary is that the evidence that thrusting only occurs at the margins is a biased one: unseen thrusting may take place under thick ice hidden from direct observation. Thrust faults occur where there is strong longitudinal compression [Sharp et al., 1988]. For quiescent glaciers this is most likely where warm-bedded shding ice encounters relatively stationary cold-bedded ice as observed on Trapridge Glacier [Clarke and Blake, 1991] or i n the vicinity of a bedrock obstruction [Boulton, 1970]. During an active surge there is a more complex stress regime and non-thermally initiated thrusts can be expected [Sharp et ai, 1988]. There are exceptions: Tison et al. [1993] document an example of a debris-laden fault over a part of the glacier well below the pressure melting point and Echelmeyer and Zhongxiang [1987] report movement along planes within an ice-laden drift at subfreezing temperatures from direct measurement. Alley et al. [1997] suggest that the overall volumetric importance of thrust-fault debris is negligible  CHAPTER 4. ENGLACIAL  PROCESSES  61  on a whole ice sheet debris budget; indeed, as indicated above, the contribution may be zero. Regardless, thrust faults play an important role for the vertical transport of englacial debris, which has implications for total englacial residence time and thus total distance of transport. Basal melt prevents concentrated debris proximal to the sole from logging substantial advected travel; this is especially true during times of vigorous sliding when frictional heat production is high. Thrust-fault transport away from the frictional surface can mitigate this problem.  Folding Evidence of folding is common i n the basal ice of alpine glaciers [e.g. Hudleston, 1976; Lorrain et al., 1981; Hart and Waller, 1999; Waller et al., 2000] and is inferred i n the central areas of ice sheets [Paterson et al., 1977; Taylor et al., 1993]. A change i n either ice thickness or surface slope over a bedrock ridge can lead to recumbent and overturned folds [Hudleston, 1976]. This tectonic activity potentially entrains material as a recumbent fold can trap a section of the bed, thus incorporating it into an englacial position [Tison et al, 1989; Glasser et al., 1998]. The axis of a fold can also initiate the formation of a shear plane [Sharp et al., 1994]. Basal ice can have isotopic sequences that cannot be explained using known entrainment mechanisms [Lorrain et al., 1981; Knight, 1989]; englacial tectonic activity is invoked to reconcile these observations [Lorrain et al., 1981; Knight, 1994]. Melting and refreezing around bedrock bumps can typically create ?s 10 cm of basal ice (Section 33) while much thicker sequences of regelation ice is sometimes observed [Sharp et al., 1994; Hubbard and Sharp, 1995]. Folding is again used to reconcile observation and theory by post-entrainment tectonic thickening of pre-existing basal layers.  4.1.5  A n entrainment model  The full spectrum of entrainment mechanisms is wide and I will narrow the scope. Net basal freezing, Weertman (or Lliboutry) regelation and intrusion are considered significant entrainment mechanisms while thrust faulting is accepted as a vertical transport mechanism and its implementation details are deferred until Section 4.2.3. A sketch of the modelled processes is shown i n Figure 4.3. Most existing entrainment models are based on the empirical observation of Krumbein [1937] that allochthonous debris concentration exponentially decays with distance from the source; this has proved expedient for the rich field of drift prospecting [e.g. Clark, 1987; Parent et al., 1996] but a process-driven approach is required to make use of glaciological knowledge and models. Boulton [1984] describes a simple model where the basal ice thickness is held constant and entrainment and deposition are calculated using the continuity equation. Tulley [1995] offers a large-scale  CHAPTER 4. ENGLACIAL  PROCESSES  62  Figure 4.3. A sketch of entrainment processes. Both freeze-on and basal melt affect the basal ice layer; the latter competes against intrusion. Where the ice is shding over bedrock, Weertman regelation around bedrock bumps entrains a small amount of debris. Thrust faults transport previously entrained material away from the bed (along with other mixing processes detailed in Section 4.2). The vertical grid, defined by (4.6) is logarithmic to concentrate grid cells towards the ice-bed interface.  CHAPTER 4. ENGLACIAL PROCESSES Parameter Value  A\ -3  Ice k 30 m  63 Sediment A l -1 -10 m  Ni 10  a  t  3  N 5  3  Table 4.1. Parameters for the vertical grid, shown i n Figure 4.4.  entrainment model of intrusion and Iverson [2000] also explores this mechanism i n ice sheets with a channelized hydrology. To concentrate computing resources towards the ice-bed interface, an exponential transform of the vertical coordinate z is used: fc = exp (A\z/h) ,  (4.6a)  £ = ezp(Atz/l ), a  (4.6b)  a  where A^ is a grid transformation parameter and li is the thickness of the modelled layer for the sediment (s) and basal ice (i). The grid is regularly spaced i n fc with 3  t3  |S  =  ( 4  .  7 )  where Ni is the number of vertical cells. These parameter are given i n Table 4.1 and the grid is illustrated in Figure 4.4. The debris content C is defined at the cell centres from £1/2 to CJJV,_I/2! the non-vertical velocities are similarily placed, therefore lateral advection (Section 4.3) also acts on the half grid. Conversely, vertical mixing (Section 4.2) is defined at the cell interface, on the integer £ grid. The ice-bed interface at fc = 0 and fc = N is not stationary and migrates with the interface. >3  3  Vertical stratification of the lithological assemblage is maintained i n both the ice and the sediment during any motion of the ice-bed interface within the bounds of the vertical discretization . Thus the grid repositioning from entrainment or deposition leads to vertical numerical diffusion. This is not a serious issue because non-numerical mixing mechanisms do operate within the ice and sediment. Englacial mixing is discussed i n Section 4.2 and although mixing is not explicitly considered within the sediment, some physical mixing is expected. 4  The determination of net basal freezing is enigmatic. There must be an adequate source of water to ensure that the glacier's basal temperature remains at the pressure melting point, otherwise the freezing front migrates into the sediment and the glacier becomes frozen to the bed. To rigourously solve for basal freezing is therefore a non-trivial problem and to date, no ice sheet model treats it adequately. The Marshall-Clarke ( M C ) ice model (described i n Section 5.1.1 and references therein) provides the basal temperature necessary to balance the internal thermodynamics of the ice with a temporally constant geothermal heat flux. Latent heat of freezing is not included i n this 4  In contrast to the lateral distribution, no sub-grid scheme is used to describe intra-cell debris variations.  CHAPTER 4. ENGLACIAL PROCESSES  64  30 25 20 CD -i— « CO  15  c TJ i_ o 10 o o "CO  o CD >  0 -5  -10 Figure 4.4. The vertical grid used for the basal ice and sediment. The spacing is regular i n £ i-space which is defined by (4.6) with parameters A* ,UiS and Ni>3 from Table 4.1. 3|  CHAPTER 4. ENGLACIAL PROCESSES  65  calculation and the bed cools to maintain the necessary heat flux through the basal ice rather than allowing freeze-on. I use the given fields from the M C ice model to estimate potential freeze-on. Modelled adfreezing onto the bed is limited to a single time step when the basal temperature of the ice falls from the pressure melting point T . The characteristic time of water availability is taken as T = At/2 = 200 a; after this it is assumed that the sole of the ice becomes cold, as the M C model predicts. The temperature at the bed has an effect on a thermal boundary layer; the height of this layer is estimated by A / K T where / c ( m s ) is the thermal diffusivity of the ice ; using values from Table 5.1, K = 1.1 x 1 0 m s . This results in a thermal boundary layer of ss 80 m . pmp  2  _ 6  2  - 1  5  _ 1  I integrate the energy leached from this layer (or from the entire ice thickness H , whichever is less) over the grid ceU through lce  min[y/K~T,H ] ice  T=  J  AT> c dz.  (4.8)  ed  iPi  0  The energy density T (J m ) is assumed to be available for freeze-on and translates into a freezeon depth !F& (m) by - 2  ft  =  (4-9)  - 7 j -  Pi  Mus  where the specific heat of ice c; is 152.5 + 7.1222T; J k g K and I is 3.34 X 10 J k g [Paterson, 1994]. This estimate of freeze-on is not consistent with the M C ice model: I violate the conservation of energy as there is no feedback mechanism and the ice thermal regime is not altered as it rigorously should be. The method is illustrated i n Figure 4.5. - 1  _  1  5  - 1  f u s  Intrusion occurs when the ice is at the pressure melting point at a rate given by (4.3). There is competition with basal melt 6 it and an equihbrium depth of basal sediment l is expected, corresponding to a balance when basal melt rate b \ is equal to the intrusion rate v . Therefore, by (4.3) 6  eq  me  me  e  l  q  =  K  Piz3  t  r  1  (  4  1  Q  )  "melt  The response time to approach l is generally short with respect to the time step and adjustment is instantaneous on the model time scale. If l > l , the difference is melted off and t i l l is deposited. If l < l , sediment is entrained subject to availability. A maximum entrainment of q  a  e q  a  a  e q  a  a  The thermal diffusivity is defined in terms of the thermal conductivity ki, specific heat capacity Ci and density pi as Ki = ki/piCi depressed by ey (0.1 K) to account for cold entrainment. 5  6  CHAPTER  4. ENGLACIAL  PROCESSES  66  Ice temperature Figure 4.5. A sketch of the energy density T available for freeze-on.  I™ = l 1  — /„ of vl^ At is imposed for any single time step where v^  e q  2  a  2  is defined as  (4 11)"  K.{ i - w) P  1/2  P  1 /2  This amehorates the situation when v is very small, resulting i n a large It that would violate the approximation of instantaneous response time. Similarily, a maximum melt of — 6 i t A t is imposed for any single time step At. 9  r  me  Interpretation of l  a  (or 1%*) is not as straightforward as one may initially presume.  Intrusion  theory assumes that englacial debris forms a grain skeleton which takes up the pressure difference Pi — Pw\ a dispersed basal layer does not contribute to the grain skeleton and therefore does not impede intrusion. The right-hand side of Figure 4.6 illustrates the issue; i n panel ( A ) , where the debris concentration is equal to that of the sediment (1 - n ) , the effective array depth is s  equivalent to the depth of englacial debris, l = z\. However i n panel (C) the englacial debris is a  dispersed and the effective array depth l = 0. Panel (B) represents an intermediate stage where a  0 < l < z . Quantification of this effect is needed. a  2  A n array depth modifier l is defined as a  l = ^[tanh(20(7- 8) + l ] . a  (4.12)  The rationale for this function, shown in the left-hand panel of Figure 4.6, is that by assigning l a low weight when C <C 0.3, dispersed debris is prohibited from acting as an effective agent of a  CHAPTER  4. ENGLACIAL  0.2  0.4  PROCESSES  67  0.6  Debris cone.  Figure 4.6. A plot of the array depth modifier l , which is a function of debris concentration C. The sketch on the right provides the physical motivation of the need for l . Details are in the text. a  a  the grain skeleton. The effective array depth is defined as l a =  Y,  i °  A  Z  (  k  4  -  1  3  )  fc=l,2Vi  where k indexes the vertical discretization of the basal ice. In areas where the ice is shding over bedrock (defined where d < oF and c P = 0.25), Weertman eg  g  regelation can entrain a dispersed facies in a thin layer next to the bed. Where these conditions exist, a 10 cm layer of basal ice with debris fraction C  r e g  = 0.1 is entrained from the bed, subject  to sediment availability and on condition that the debris fraction i n the bottom layer Ck=i does not already exceed C  4.2  r e g  .  Englacial mixing  Mixing of basal debris within the ice has two facets of interest in this study. Firstly, mixing moves debris away from the ice-bed interface where friction and geothermal inputs concentrate the melt. This increases debris residence time, and as a corollary, transported distance. Secondly, mixing fragments the grain matrix to allow greater entrainment of debris by intrusion (see Figure 4.6 and Equations (4.13) and (4.12)). Various mixing mechanisms are summarized below followed by a  CHAPTER 4. ENGLACIAL  PROCESSES  68  description of the mixing model and details of its implementation. The vertical velocity of the ice (determined from the M C ice model) also affects the vertical distribution of englacial sediment.  4.2.1  Englacial tectonics — mixing  Englacial tectonics (shear planes and folding) have been examined with respect to their role in entrainment in Section 4.1.4; these processes also mix the ice. Small scale mixing has been attributed to englacial folding i n the Greenland ice cores [Taylor et al., 1993; Souchez et al., 1995] and folding is suggested to be a crucial process in the formation of the dispersed facies of basal ice [Waller et al., 2000]. Englacial tectonics can happen whenever steady ice flow is perturbed; basal bumps can provide this instability and produce recumbent folds [Hudleston, 1976]. The mixing from random encounters of debris-laden ice and basal topography has been conceptualized as an eddy-diffusion model with a debris diffusivity of D <i « l x 1 0 m s " ' [Alley and MacAyeal, 1994; Alley et al., 1997] and a dependence on sliding speed v and basal topography 6 can be expected by heuristic arguments. Also, mixing is expected to be strongest proximal to the basal topography that causes it and a decaying effect dependent on distance from the bed is expected. I use - 1 0  2  1  e  7  s  (4.14) where the characteristic eddy-diffusion parameter D ^ is taken as 1 X 10  1 0  e  m s 2  1  and typical  sliding speeds v of 1 0 0 m a , bed dips 6 of 10° and a decay depth scale z of 5.0 m are taken.: - 1  s  The mixing that can result from thrusting can be locally large; Clarke and Blake [1991] infer movement along a thrust plane of « 100 m over a period of 7 a. If these values are taken as characteristic lengths and times, a diffusion coefficient for this process of 4.5 X 1 0 m s is expected. Hambrey et al. [1996] find thrust planes of width « 0.5 m with spacings on the order of 100 m from a Svalbard glacier in active surge. W i t h these values, a volumetrically corrected diffusion coefficient for thrusting .Dthrust can be estimated as 2 x 1 0 m s . This must be accepted as an extreme upper limit and this frequency of thrusting would not be expected over an entire model grid cell. A more conservative estimate of 1 X 10~ m s is taken here. This mechanism is only enabled when the cell is warm and the downstream cell is cold which creates the high compressive stress needed to initiate thrusting. Small thrusts from bedrock topography are assumed to be encompassed in -D d- 5  - 7  9  2  2  2  _ 1  - 1  _ 1  e  7  T h i s m e c h a n i s m was o r i g i n a l l y considered (and rejected) as a p o t e n t i a l e n t r a i n m e n t m e c h a n i s m b y Alley  and MacAyeal  [1994]. I assume t h a t e n g l a c i a l tectonics are o n l y m i x i n g m e c h a n i s m s a n d c a n n o t e n t r a i n  debris. T h e e s t i m a t e D d f » l x 1 0 e  _ l o  m s 2  _ 1  f r o m Alley and MacAyeal  [1994] is p o o r l y c o n s t r a i n e d .  CHAPTER 4.2.2  4. ENGLACIAL  PROCESSES  69  Particles in a shear flow  Rigid particle rotation  A single particle i n a shear flow rotates and this causes mixing. Given a linear shear flow Vi = f3z • y  where (3 is the vertical strain rate (see Figure B . l ) , analysis of a freely rotating cylinder with 8  radius R gives an estimate of the rotational diffusion parameter D'  ( m s ) as 2  Iot  _ 1  = /3£ /4. 2  D'  TOt  (4.15)  This result is derived in Appendix B . The particle distribution from Section 3.1.4 is used to give a total rotational diffusion parameter as D  X>fl C /4. 2  =  Iot  r  (4.16)  r  The assumption that the solution for a single cylinder can be used for many clasts has obvious . shortcomings; for example clast interactions will affect the solution. Nevertheless, this approach provides an order of magnitude estimate of the rotational contribution to mixing. A value of  p= 5  X  1 ( T s8  1  gives D  = 2 x lCT  Iot  1 3  m  2  s" . 1  Rigid particle collision  Clast interactions and collisions can disperse particles; the tendency for grains to move away from each other i n a shear flow was first observed by Reynolds [1885] for a granular fluid and the effect was investigated i n a suspension by Bagnold [1954,1956] who coined the phrase "grain dispersion pressure".  Weertman [1968] develops a simple geometric relationship to describe the dispersion  from two-particle collisions in a linear shear flow; a conceptual schematic of this process is shown in Figure 4.7. The collision-diffusion coefficient D \ is a function of the particle radius R, the co  particle volume fraction C and the vertical velocity gradient /3. It is written as r  I>cai = X > - R C W ? / ( 5 x ) . 2  (4-17).  r  The mixing from particle collision is on the same order of magnitude as that from particle rotation. Three-body collisions are not considered in the derivation of (4.17).  4.2.3  A mixing model  Diffusion coefficients D ^, -Dthrust, D t and D \ have been outlined above and are summed to e  ro  co  give a total diffusion coefficient Dtot(k) at the cell interface. The vertical velocity gradient J3 is 8  Note that the coordinate system used in Appendix B differs from that used here.  CHAPTER 4. ENGLACIAL  PROCESSES  70  ^  Before collision j After collision Figure 4.7. A conceptual picture of dispersion by collision.  calculated at the cell interface from the ice model discretization by a cubic Hermite interpolation scheme [Kahaner et al., 1989], which guarantees a monotone interpolating function given monotone input data. A vertical mixed layer around the cell interface is denned with depth  AzfjL =  1/2) At.  y/Du> (k + t  (4.18)  This mixed layer has equal representation from Cfc and Ck+i- This leads to a diffusion-like equation for the time evolution of C as A r t C*_i(«) + A r t C {t) - C (t) [ ( A r t C (t + At) = C (t) + — ^ 1 ^1 — ^1 k+1  k  +  l  k  k  Art)] ^ 1 .  (4.19)  Mixing is prohibited through both the top and bottom of the modelled basal layers. Vertical velocity of the ice, also interpolated to the cell interfaces by the cubic Hermite scheme of Kahaner et al. [1989], advects material through the equation  which is discretized using a simple upstream scheme. Therefore when the vertical velocities are ; negative (ice moving towards the bed),  C {t)vi  , -  k  C (t)vi k+1  C (t + At) = C (t) + At k  ^ i ,  k  (4.21)  Azk and when the ice velocities are positive,  C -i(t)vi k  C (t + At) = C (t) + At k  k  t  - C (t)vi  -4-  Azk  k  (4.22)  71  CHAPTER 4. ENGLACIAL PROCESSES  4.3  Advection  Englacial sediment is not known to be ambulatory; its lateral motion is wholly derived from that of the host ice. The governing equation for the evolution of the volume fraction Ci of a lithological component I stems from the continuity equation,  ^  =  -y.((v  + V i  )C )+  •  t  advection  ver  (4-23)  '• • ex  The vertical exchange term <f>1 accounts for vertical flux through entrainment, deposition, mixing and a simple treatment of vertical advection, discussed i n Sections 4.1 and 4.2. ed  There is no term in (4.23) to describe lateral diffusion. It is not a physical process and therefore any numerical scheme must strive to minimize the introduction of numerical diffusion. This is of particular importance i n this problem as the advected lithologies are used as tracers which tend to have distributions with sharp lithological boundaries. For a numerical tracer to be useful, this sharp boundary must be preserved as accurately as possible. This is not a concern for the vertical advection as physical mixing processes are real and are modelled explicitly (Section 4.2); because vertical numerical diffusion is not a concern, it is implemented with a simpler treatment as described i n Section 4.2. This is a common situation i n atmospheric transport models and a large body of work concerning numerical solutions to (4.23) has resulted. The advection scheme of Prather [1986] is recognized to be highly accurate, though not computationally cheap [Muller, 1992; Lin and Rood, 1996; Xiaozhen and Xiaoen, 1998]. In a recent review and comparison of nine Eulerian numerical advection schemes the Prather [1986] algorithm shows exemplary performance; it is best able to maintain a narrow peak and is the most accurate of the nine schemes tested [Reames and Zapotocny, 1999a]. It also rated well against semi-Lagrangian schemes i n a companion review [Reames and Zapotocny, 1999b]. The Prather [1986] scheme is an Eulerian upstream finite-volume algorithm that conserves secondorder moments. It incorporates sub-grid information by denning a second-order polynomial function for the lithology fraction C\ i n each model cell. I use a two-dimensional version  d = a° + a x + a%y + a x + a y + a xy c  c  x  2  xx  c  2  yy  (4.24)  c  xy  where the internal distribution of the tracer lithology is described through six a coefficients. If only the zero-order term a% is used the scheme collapses to standard upstream differencing; the vertical exchange c/>j is done through the zero-order term a% . The first order terms describe the slopes of the distribution while the second order terms describe the curvature. A pseudo positive definite condition (such that the mean concentration along any axis is positive) can be imposed with little cost i n both computational time and diffusive properties. c  ed  CHAPTER 4. ENGLACIAL  PROCESSES  72  Vsr  Ue  •  At  *  a  r  e  a  1  area 2 /  \  -: VSrAt  areal  •  '—•  (i-i.j.k)  l^area^ (i,j,k)  a°(i+1,j,k)... a^l+1J,k) (i+1 J,k)  Figure 4.8. A sketch of the Prather [1986] advection scheme. Six coefficients are used for each (i,j,k) cell to define a second-order polynomial sub-grid function. When a portion is advected, the second order moments are conserved. A n extension to spherical coordinates is made by considering the changing area of a grid cell on a spherical grid. Note that for purely meridional velocities as shown above, Al = A2 = A3 and there is no modification of the original scheme of Prather [1986].  The algorithm is explicit i n time and must be initialized with values for each a . I set a% to the initial concentration and the rest of the coefficients to zero. For each time-step the grid box is divided into two parts: the portion of the cell that is advected downstream and the portion of the cell that remains. Both inherit the same distribution of (4.24) but have new moments calculated with respect to their individual centres of mass. The advected portion of the cell is then combined with the downstream non-transported part of the receiving cell to create a new set of coefficients a . A correction for non-equal cell areas is introduced i n this step to extend the method of Prather [1986] to a spherical grid. Figure 4.8 illustrates the scheme. c  c  The advection is multidimensional and this is dealt with by time splitting (synonymously alternating direction integration). Thus a complete pass is made over the model domain with meridional velocities and advection, followed by a zonal treatment. Each vertical layer k is solved separately with its own set of coefficients a . c  A n example of the scheme i n action is shown in Figure 4.9. The initial function is a box of unity height that is then transported by a series of purely zonal or meridional velocity fields and, finally, a more general velocity field. The velocities are such that approximately 40% of the grid cell is  CHAPTER 4. ENGLACIAL PROCESSES  73  Figure 4.9. A n example of an initial box function with C\ = 1.0 transported with the Prather [1986] advection scheme. The dashed lines indicate the path of the tracer with outputs after 125, 3050 and 3250 time steps. The heavy Une is a 0.01 contour to illustrate the numerical diffusion of the scheme.  advected i n any single time step. Note that the 2-D velocity field is not divergence-free, which results i n concentration at higher latitudes to honour mass conservation; this is not a physical realization for ice (where incompressibility requires V - ( v i + v ) = 0) but is useful as an illustration of the dispersive properties of the algorithm. s  Chapter 5  Model Inputs 5.1  Ice model  The numerical reconstruction of the North American ice sheets is central to this work. A n overarching goal is to provide a tool for ice sheet model validation that uses Quaternary geological data as a constraint. In addition, ice sheet output fields are crucial requirements for the model components described i n Chapters 3 and 4 and the hydrology model described i n Section 5.2.  5.1.1  Theory  I use the Marshall-Clarke U B C ice sheet model ( M C ) [Marshall, 1996; Marshall and Clarke, 1997a,b; Marshall et ai, 2000], a three-dimensional thermo-mechanicalice model driven by climate forcings and ice physics. The dynamical evolution of the ice follows the scheme of Mahaffy [1976] with governing equations of mass and momentum balance. The continuity equation of the ice thickness H with a constant density pi leads to a mass balance relationship lce  —  + V • ( iT ) = b c e  V i  (5.1)  where v is the vertically averaged velocity and b is the mass balance rate. 1  Neglecting acceleration terms, the momentum balance is =  (  5  -  2  )  vertical integration reduces the system to a function of ice thickness and surface slope. To translate the stress tensor a b into velocity, Glen's flow law [Glen, 1955] provides a constitutive relationship a  74  CHAPTER 5. MODEL INPUTS between the strain rate  ki  a h  75  and the deviatoric stress cr' tensors ab  e  iab  = 5 ex 0  P  (-Jhf.\A\  i n G  -  (5-3)  1 ) l 2  where BQ is an empirical constant, Qi is the creep activation energy, R is the gas constant, Tj is the temperature of the ice, S' is the second invariant of the deviatoric stress and no is the Glen flow law exponent. SAS  2  The three-dimensional temperature evolution follows Jenssen [1977] and the governing equation expresses conservation of internal energy. The local energy balance is  dTi ot  „  ki d Ti piCi dz  1 dh fdTi\ ci dT \dz )  2  m  - — „ — ' advection  2  d  Pi  v divergence of heat  v  $  2  P i  y  flux  Ci  ' strain heating  Internal deformation of ice gives rise to strain heating The specific heat capacity of ice c» and the thermal conductivity of ice k{ are both functions of Ti (see Table 5.1). Thermodynamic evolution is subject to boundary conditions on the surface and at the bed of the ice sheet. Basal ice receives heat from shding friction and a geothermal heat flux determined from a North American heat flux map [Blackwell and Steele, 1992]. W i t h these basal boundary conditions and the climatic surface boundary conditions (discussed below), the system can be solved to extract basal temperature T . When the bed is at the pressure melting point, latent heat of fusion enters the analysis as a modifier to the basal boundary condition to yield the basal melt rate 6 . b e d  t  m e l t  The climate forcing delivers both the upper boundary condition for the thermodynamic evolution and the mass balance b i n (5.1). A base climate from CCCv2.0, the general circulation model of the Canadian Chmate Centre for Modelling and Analysis [McFarlane et al., 1992; Vettoretti et al., 2000], is reconstructed for present day and Last Glacial Maximum ( L G M ) conditions. The difference between these results gives glacial maximum perturbation fields for temperature and precipitation which modulate present day observations [Kalnay et al., 1996; Legate and Willmott, 1990] for a given time t based on a "glacial index" from the Greenland G R I P ice core 6 0 record [Dansgaard et al., 1993]. Full details of the methodology are found i n Marshall et al. [2000]. 18  A thermally regulated shding mechanism dependent on basal shear stress is used following the hnear model of Payne [1995]. The shding velocity v is calculated through s  v  (-B gH™V{H™  s  = <  lO  + z^ )  > T^p - e  d  aPi  '  T ^ < T^ e d  where Tpmp is the pressure-corrected melting point of ice and  r  - e  .  (5.5)  T  is a modifier to initiate shding  when ice temperature is slightly below the melting point. This crudely accounts for the fact  76  CHAPTER 5. MODEL INPUTS  that glaciers can slide at subfreezing temperatures, albeit slowly [Shreve, 1984; Echelmeyer and Zhongxiang, 1987]. B is a free parameter used to restrain or enhance sliding. s  Marshall et al. [2000] modify the thermally regulated sliding of Payne [1995] through a geological dependence on soft sediment. The deformation of sediment, a mode of glacier motion commonly associated with ice streaming, serves to reduce the Laurentide ice volume to match the inferred volume from geophysical measurements [Peltier, 1994, 1996] and has strong geological corroborating evidence [Marshall et al., 1996]. Inclusion of deformable beds i n reconstructions of the Laurentide Ice Sheet has enjoyed recent numerical modelling success [Clark et al., 1996; Jenson et al., 1996; Marshall and Clarke, 1997a,b] and the sliding law of Marshall et al. [2000] is designed to represent sediment deformation without an explicit treatment. However, this approach is disastrous from an erosion modelling viewpoint. The scheme of Marshall et al. [2000] completely disables sliding on a hard bed; zero sliding translates to zero abrasion by Equation (3.12) on page 28 and zero quarrying by Equation (3.19b) on page 47. There is overwhelming geological evidence that glacial, hard-bed erosion takes place (e.g. striations), therefore the sliding model of Marshall et al. [2000] must be modified. I choose an intermediate path between Payne [1995] and Marshall et al. [2000]: B is enhanced over soft-bedded domains to account for sediment deformation and is non-zero over hard beds to allow for the reality of hard-bed glacial suding. 1  2  s  5.1.2  Examples of the ice sheet model  The M C ice sheet model is run from 120 ka B P to present with H , v , 6 ice  s  shear forcing), e  b e d  m e l t  , T> , r (the basal ed  and Ti and V J output every 400 years. Parameters given i n Table 5.1 are used.  Results at approximately L G M (21.2 ka B P ) are shown i n Figure 5.1. Note that the model domain is expansive enough that all the ice at L G M remains confined within model bounds thus disarming potential boundary condition issues. Basal melt and sliding tend to reach their maximum values later during deglaciation. The "spoking" or "fingering" pattern of the sliding speed is the direct result of the thermal regime. This type of patterning is endemic among thermally coupled ice sheet models [Payne et al., 2000]; it is initiated by the interaction of ice thickness, temperature and velocity. The patterns are strongly correlated to the underlying grid and discretization, therefore caution must be exercised i n detailed interpretation of the basal results [Payne and Baldwin, 2000]. ;  T h e p a r t i c l e v e l o c i t y v i n (3.12) requires v > 0 i n order to be non-zero as s h o w n b y (3.10) a n d the subsequent discussion. T h e existence of a c a v i t y demands t h a t v, > 0 b y (3.19). I f n o c a v i t y exists, the stress regime w i l l n o t be such t h a t vi a n d D J J > 0 i n E q u a t i o n s (3.24) a n d (3.25). 1  p  2  s  CHAPTER 5. MODEL INPUTS  77  Parameter Flow law parameter", T; < - 1 0 ° C Flow law parameter", Ti > - 1 0 ° C Sliding law parameter, soft bed Sliding law parameter, hard bed Creep activation energy, Ti < —10°C Creep activation energy, Ti > —10°C Gas constant Specific heat capacity of ice Thermal conductivity of ice Flow law exponent Density of ice Sliding temperature modifier 6  6  Variable  Value  B B B B  1.08 x l O 5.20 x 10 2.0 x l O 1.0 x l O " 6.00 x 10 1.39 x 10 8.314 152.5 + 7.122T; 9.828exp(-5.7 x 10~ T;) 3 910 0.1  Units - 1 2  0  3  0  a a  Qi Qi  -ffgas Ci ki  no Pi e T  - 1 0  1 0  4  5  s- P a " s- P a " ms^Pa ms Pa JmolJmolJmol^KJkg" IT 1  3  1  3  - 1  - 1  - 1  1  1  1  1  1  3  —  kgm K  - 3  Table 5.1. Parameters used for ice sheet model. " T h i s value has a n e m b e d d e d enhancement factor of 3 to account for the effects o f c r y s t a l a n i s o t r o p y a n d ice i m p u r i t i e s . 6  f r o m Paterson  [1994].  Ice thickness and basal sliding speed  1300  250  200  150  100  110°W  Figure 5.1. A reconstruction of the North American ice sheets at approximately L G M using parameters from Table 5.1. The grey scale contours of 50, 150 and 250 show shding speed in m a while the lines give ice thickness contours of 1, 1500, 2500, 3500 and 4500 m. - 1  CHAPTER 5. MODEL INPUTS  5.2  78  Hydrological model  The subglacial water pressure P has a lead role i n quarrying (Section 3.2.5) and is instrumental in the calculation of intrusion (Section 4.1.3); the model of Flowers [2000] is used to calculate P . This model is designed for basin-scale subglacial drainage therefore a number of changes and : simplifications are necessary and permissible to solve a continental-scale problem. w  w  Flowers [2000] incorporates a multicomponent description of glacial hydrology. A surface component calculates ablation and surface runoff, which is coupled to the bed through moulins and crevasses. While this surface-bed communication is indisputably active i n valley glaciers, any such link is unlikely on a continental ice sheet outside the marginal regions; surface-bed water exchange is prohibited to reflect this shift i n scale. Similarily, the englacial water system which routes surface melt to the bed and has storage capabilities, is disabled for continental simulations.  5.2.1  Sheet flow  A subglacial water sheet system is assumed to exist wherever the bed is unfrozen (as determined by the M C ice model described in Section 5.1). The treatment is consistent with subglacial drainage through a transmissive macroporous horizon [Clarke, 1996]. Although this does not account for the myriad of possible drainage structures under an ice sheet such as ice-walled conduits [Rothlisberger, 1972],bedrock channels [Nye, 1976], linked cavities [Walder, 1986; Kamb, 1987] and water films and sheets [Weertman, 1972; Walder, 1982; Weertman and Birchfield, 1983], a macroporous horizon fits readily into the framework of continuum mathematics. This simplification is most suspect proximal to the ice margins where alternate flow morphologies become more probable. The water balance equation for the sheet is a  TTW (JL  + V•Q  w  = 6  m e l t  - </>  (5.6)  s:a  where H is the sheet water thickness, Q is the vertically integrated sheet water flux, j> is the basal melt rate (determined by the M C thermodynamic model) and (f> is the exchange term between the sheet and an underlying aquifer, discussed i n Section 5.2.3. w  w  melt  s:a  Assuming Darcian flow allows an expression for the flux to be written as  with the density of water p assumed constant for the sheet. conductivity and the hydraulic head il> is given by w  K  w  is the isotropic hydraulic  w  V> = P /( g) w  w  Pw  +z  b e d  (5.8)  CHAPTERS. where z  MODEL  INPUTS  79  is the mean bed elevation (sub-grid information is ignored).  b e d  Finally a relationship is needed between the sheet thickness H pressure P . This is taken to be  w  and the variable of interest, water  w  H  w  TTW  -"crit  \ * ^  (5.9)  for Trapridge Glacier, a small alpine glacier in the Yukon Territory. Pi is the ice overburden pressure and H™ is the critical amount of sheet water needed for flotation, occurring when the saturation of sediment is complete; Flowers [2000] describes the theory behind (5.9) i n detail. For lack of guidance on upscaling to a continental scale, I accept the form of (5.9) although the unknown scale effect on the exponent i n (5.9) could affect model results. it  Hydraulic conductivity has temporal dependence through its relationship to sheet thickness. Drainage networks can exhibit connected (high K ) or unconnected behaviour (low R~ ) which is parameterized through H . Flowers [2000] argues for the following relationship w  w  w  log  1 0  ( ^ ) = i(log (tf 1 0  m a x  ) - log (JCin)) arctan 10  a I  (HK ~ crit  )  H  + \ (logloKax) + log (*£n))  (5-10)  1 0  with k = 15.0 and kb = 0.85. Again, I leave the form of (5.10) as well as the values for k a  a  and  k unchanged. 0  However, several parameters are changed to reflect upscaling to continental scale. Observation yields a 0.375 m macroporous layer with 40% porosity at Trapridge Glacier [Blake, 1992; Kavanaugh, 2000] giving H™ = 0.15 m with minimum and maximum sheet hydraulic conductivities as = 2.5 X 1 0 and K™ = 5.0 X 1 0 ~ m s . The transmissivity of the layer can be lt  - 2  2  _ 1  &x  interpreted as the product of H™ and K based on (5.7). Numerical testing indicates that higher values of sheet transmissivity are necessary to evacuate water from the North American ice sheets; Trapridge Glacier parameters produce an ice sheet hydrology predominantly well above flotation pressures, an unreasonable realization of the model. Designations of H™ = 4.0 m , -^nin 2-5 X 1 0 and i^™ = 5.0 X 1 0 m s are used. Numerical instabilities prevent further increase of the conductivity, which corresponds to increased water velocity; this has an adverse effect on the maximum stable time step. A thick sheet is therefore chosen as the vehicle to increase transmissivity. I do not suggest that the true thickness of an ice sheet macroporous horizon is ^ 10 m , but the strategy is useful to reconcile high transmissivity with numerical stability of the continental-scale solution. lt  w  lt  =  _ 1  _ 1  - 1  ax  The necessary inflation of sheet transmissivity to produce physically reasonable results is suggestive of two scenarios: 1) The hydrology is never in equihbrium and the ice sheet evacuates water through subglacial outburst floods [e.g. Shaw et al, 1996; Shaw, 1996] or; 2) Channels develop  CHAPTER 5. MODEL INPUTS  80  that efficiently drain the bed. Neither of these possibilities can be explicitly accounted for with the current model; enhanced values of sheet transmissivity must suffice. The conductivities are assumed to be isotropic and are universally applied to all subglacial warmbedded ground. Ground that is frozen as determined by the M C ice thermodynamic model is assigned K^ = # = K = 1 X 1 0 ~ m s" . 14  n  max  1  flozen  The model domain is large enough to always encompass the entire ice sheet therefore boundary conditions are not troublesome; P is set to zero at the margins. This is an advantage to running the full North American model; if warm ice exists at the edge of the hydrological domain, an arbitrary pressure must be prescribed to achieve mathematical closure. The appropriate choice for this boundary condition is a non-trivial task. w  5.2.2  G r o u n d w a t e r flow  Groundwater flow under Pleistocene ice sheets has been explored using the Fennoscandian Ice Sheet as a testing ground [Boulton et al., 1995; Piotrowski, 1997; van Weert et al., 1997]. The model of Flowers [2000] distinguishes itself by explicitly considering sheet flow, nevertheless it is probable that groundwater also played an important role i n the subglacial hydrology of the North American ice sheets. Flowers [2000] follows the conventional treatment of groundwater based on Darcian flow [Freeze and Cherry, 1979]. A balance equation for both saturated and unsaturated groundwater flow is described as H \ gw  do  dH  9W  +v-Q  ir  +  g w  =  saturated  ^ subglacial  unsaturated  + ^  (5.ii)  subaerial  where H is the thickness of the groundwater layer and Q is the groundwater flux, similarly defined as for the water sheet in (5.7). The density of water p is a function of pressure with the equation of state g w  gw  w  p = lOOOe^*""  (5.12)  w  where (3 = 5.04 X l O ^ P a W  - 1  .  Equation (5.11) can operate either i n unsaturated or saturated mode: only one of the pair ~dp dH ' can be significantly non-zero at any time. During unsaturated flow the water dt ' dt gw  w  table is a free surface and dp /dt is small while dH /dt is small i n saturated flow. Also one of [<f) , <t> ] must be zero; <j) ' controls subaerial/groundwater exchange while (f> covers subglacial/groundwater exchange. The exchange terms are developed below i n Section 5.2.3. gw  w  s:a  r:a  r  a  s:a  CHAPTER 5. MODEL INPUTS  81  As for the sheet flow a relationship between pressure and water thickness provides a link between (5.11) and the Darcian flux (the groundwater equivalent of (5.7)). This relationship is  {  p gH  H  9W  gw  w  <n d gw  (unsaturated),  gw _ w (5.13) p gH + H > n d (saturated). where a is the sediment compressibility, the porosity n is taken as 40% and d is the thickness of the aquifer. H  9  gw  d  gw  gw  w  gw  gw  The conductivity of the unfrozen aquifer is assumed constant throughout the study area and is taken as K = 1 X 1 0 m s . This value is consistent with some large-scale ice sheet hydrology models [Boulton et al., 1995] while others use a higher value for their aquifer [Piotrowski, 1997; van Weert et ai, 1997]. Ground that is frozen (as determined by the M C ice model) is assigned 9W  K  g  —  - 7  - 1  -^-frozen-  The thickness of the aquifer varies with sediment depth as outlined i n Section 5.3.2 (aquifer thickness = sediment depth = d). To avoid numerical strife this is modified i n areas where there is very thin or non-existent surficial cover so when d < <£"•, d = dZ- where d™- is taken as n u n '  J  m i n  m i n  0.5 m. Water is discouraged from flowing i n these virtual aquifers by assigning the hydraulic conductivities i n these areas as K = Kf . gw  TOZen  As for the sheet flow, boundary conditions pose no difficulties because the ice margin is always several model cells away from the grid boundary.  5.2.3  T h e aquitard  The subglacial sheet and the groundwater system exchange water through a thin aquitard. Vertical coupling is parameterized to reduce the full three-dimensional problem to a set of adjacent twodimensional systems. The exchange term between the sheet and the groundwater <j) ' is s  1  pg  ^[(P -P )  K  w  gw  + p gd ]  a  H >n d,  t  gw  w  gw  (5.i4).::  w  jjgw  <  n  gw ^ d  where P is calculated using only the second term from (5.13) as the pressure at the top of the aquifer is what drives the exchange. The aquitard thickness d divided by the hydraulic conductivity K gives the aquitard time constant. gw  1  l  CHAPTER  5.  MODEL  82  INPUTS  Figure 5.2. A reconstruction of the North American glacier complex hydrology at approximately L G M using parameters from Table 5.2. The ice sheet inputs are shown in Figure 5.1. The grey scale contours of 0.2 0.5 0.8 and 1.0 show sheet pressure normalized to ice overburden pressure (i.e. a value of 1.0 indicated that P = Pi). w  Similarily, the exchange between the groundwater and runoff (j> is written as r:a  1 Pw9  1 [ Pw9  K  l  &  K  '-Pgw + Pwgd*]  H  gw  > n d, gw  l  d  l  Pwgd } 1  H  (5.15)  gw < gw _ n  d  An assumption in (5.15) is that the pressure of the runoff is atmospheric ( « 0). This is suspect especially during deglaciation when large pro-glacial lakes dominate the peri-glacial landscape [Marshall and Clarke, 1999] and serve to increase the marginal subglacial water pressure. Heave the conductivity of the aquitard K* identical to the value oi Flowers [2000] at l x l 0 ~ m s . The thickness of the aquitard d is taken to be 0.2 m over significant sediments (d > dZ^ ) and 1.0 m over negligible cover. The enhanced thickness over hard beds inhibits any attempt to exchange water from the sheet to a very thin or non-existent aquifer; this eases the numerics of the problem. 9  t  a  - 1  CHAPTER  5.  MODEL  83  INPUTS  Parameter  Variable  Critical sheet thickness M i n . sheet hydraulic conductivity Max. sheet hydraulic conductivity Aquitard hydraulic conductivity Aquifer hydraulic conductivity Frozen ground sheet hydraulic conductivity Aquitard thickness (soft bed) Aquitard thickness (hard bed) Sheet hydraulic conductivity parameter Sheet hydraulic conductivity parameter Aquifer porosity Aquifer compressibility Minimum aquifer thickness Compressibility of water  TT1V  -"exit  K min K max W  w  K  gw  -^frozen  h a d-  g w  w  min  Pv,  Value 0.4 2.5 X 10"- l 5.0 X 10"- l 1.0 x 10"- 9 1.0 x 10"- 7 1.0 x 10"-14 0.2 1.0 15.0 0.85 0.40 1 X 10-8 0.5 5.04 x 10 -10  Units m m m m m m m m  s-l s-l s-l s- l s- l  — — —  P a - -1 in Pa~ l  Table 5.2. Parameters used for hydrology model.  5.2.4  Examples of hydrological results  Figure 5.2 shows results of the hydrological model at 21.2 ka B P (approximately L G M ) . The groundwater pressures (not shown in Figure 5.2) have similar patterns and magnitudes as the sheet pressures. The forcings for these plots are shown i n Figure 5.1 and the hydrological parameters used are given i n Table 5.2. The model is run for 50 a with the given static forcings; this integration time is sufficient for the hydrology to reach equihbrium if started with appropriate initial conditions (discussed below). The fingering pattern of Figure 5.2 is thermally induced and is discussed in Section 5.1.2 The hydrology model is not fully coupled to the ice: there is no feedback. The ice model is run without any regard for hydrology with a time slice of hydrologically important variables (H , 6 it and T ) written every 400 a. A single time slice is then used as a static forcing for the hydrology model with initial conditions being the results from the previous time slice (modified by setting P = 0 and H = n d) in cells where the ice had disappeared or the ground had frozen since the previous time step). Initial conditions for the start of the run at 120 ka B P is a saturated aquifer with P = 0. Between time slices, values of both sheet and aquifer pressures greater that 5P; are set equal to 5P; to prevent propagation of unphysical results. lce  me  b e d  gw  w  w  9W  CHAPTER 5. MODEL INPUTS  5.3  84  Geological inputs  There are several aims in including an accurate lithological representation of the bed. The first is to use distinct lithologies as natural tracers of ice motion; the ice entrains loose debris from the bed, then transports it downstream depositing a debris train. These deposits are observable data on the modern landscape offering a constraint for the ice sheet model. To exploit these data, lithologically labelled englacial sediment must be tracked from bedrock to ice to t i l l . The second goal is to incorporate the differences in physical properties between various lithologies. This is of particular importance in the calculation of erosion where fundamental physical parameters of the bedrock (e.g. hardness) play a lead role in the erosion equations. For example, there appears to be some correlation between the shape of glacial valleys and lithological parameters based on examination of several sites in the Southern Alps, New Zealand [Augustinus, 1992, 1995] and this effect has been successfully modelled with a simple (^-dependent) erosion law [Harbor, 1995]. To date there has been no effort at incorporating lithological parameters on a large scale with a more complete erosion law (such as that outlined in Chapter 3). The lithological assignments are done using the maps of Wheeler et al. [1997] for the bedrock lithology and Fulton [1996] for the surficial material. The bedrock map displays information at or near the surface of the land, as well as less detailed depiction of marine areas above the continental crust and offshore oceanic crust. The surficial map gives the nature of the cover encompassing broad depositional categories from which limited lithological and thickness information can be distilled.  5.3.1  Bedrock geology  There are 49 distinct lithologies identified by Wheeler et al. [1997], listed i n Table 5.3. The map domain is projected onto geographical coordinates then gridded into cells of 0.02° X 0.02° using standard A R C - I N F O packages; the resultant map is shown i n Figure 5.3. These sub-grid cells are binned into the model grid (1.0° zonal X 0.5° meridional) by assigning each lithology a summed representative area! fraction of the model cell. This preserves an accurate account of the sub-grid areal fraction of each rock type while only losing sub-grid geometrical information. The fractions are stored as a linked list; calculations are therefore not performed on non-existent lithologies. Hard-bed parameters (h , h ,njj, iT£) for a model cell are calculated as a weighted mean of the individual constituent's parameter value. s  v  CHAPTERS.  85  MODEL INPUTS  Bedrock 8 0  7 0  6 0  5 0  Geology  N  N  N  N  1 4 0 W  E x t r I n t  •  Me 1 a n g e s  e  d .  & m a f  .  .  s e d .  & e  .  c  u n d  u n d .  o  r  t  p a r  ma  S e d i me n t a r y  • • • • • • •  u n d .  s e  c o n g l  .  n o n - m a  .  f  .  .  • • •  .  g n s .  g n s  .  g n s .  g n s  s e d  .  p a r a . - o r t  . g n s  c h a r n o c k i  t e  r  a  1  s  e  u  s  1  d .  & v  o  n g  e  t  .  m a  f  a  n g e  n k n  o w n  me  i  e  t  o n  I  u  ii cl .  1  .  t  e  a l i t e  t r  t m  q t z . a s  a  1  it  .  &  .  g n s  k . e x t .  1  me  u  m e d . g r d . g n s .  & s n d s t  r  .  a .  v o l  d .  g n s  c  e x  s i  f  • • • • • • •  7. .  r  a p a k  i  2  -  a  q  7.  f  e  1.  a  1  k .  s  y n .  a  1  a  n o  mi  k  c t  .  n .  v i  .d i  & m a  g r  n  g r  n  .  o r  i  t  e  f  f  .  h s i  t  e  .  ma  g a b .  r t  g a b . c a r b n n a t  e  mi  •  s y n  .  g n s  . ma  f  .  ma  f  .  u l  i  .  v p .  m a o  e  f  r f  b -  1  e  s h  1  o g e o c l  f  . i  o a I  Figure 5.3. Bedrock geology by rock type from Wheeler et al. [1997].  &u 1  t  m a  .  f  CHAPTER 5. MODEL INPUTS Table 5.3: Bedrock types as designated by Wheeler et al. [1997] with hardness, Mode I and Mode II subcritical crack growth indices and critical stress intensities assigned. Bedrock Type  h ,h s  ni,n  p  n  Pa  K\  K*  u  Pa m / 1  Pa m /  2  1  2  Extrusive undivided alkaline bimodal bimodal alkaline intermediate felsic felsic-intermediate mafic oceanic mafic  2.2 x 10 7.0 X 10  9  7.7 x 10  6  6  25 1.5 x 10 1.85 x 10 e  2.9 x 10  6  6  5.3 x 10 6.5 x 10 1.0 x 10  e  6  7  Intrusive felsite tonalite undivided granitoid quartz monzonite, granite rapakivi granite two mica granite, diatexite granodiorite, quartz diorite felsic & mafic alkaline syenite, mozodiorite alkaline gabbro & syenite anorthosite gabbro- anorthosite mafic (diorite, gabbro) mafic & ultramafic ultramafic  1.5 x 10  2.2 X 10 7.0 X 10  9  6  6  5.3 x 10  7.7 X 10  6  6  25  2.9 x 10  6  1.0 x 10  7  Metamorphic undivided gneiss orthogneiss paragneiss mafic gneiss volcanic gneiss medium-grade gneiss paragneiss-orthogneiss charnockite syenitic gneiss  7.0 x 10  9  25  2.2 x 10  6  7.7 X 10  6  continued on next page  CHAPTER 5. MODEL INPUTS  87 Table 5.3: continued  Bedrock Type  h ,h Pa s  ni,nn  p  K\ Pa m / 1  K* Pa m / u  2  1  2  Sedimentary undivided sedimentary conglomerate & sandstone non-marine sedimentary carbonate evaporite marble offshelf miogeoclinal  1.5 x 10  6.2 x 10  15  9  5  2.2 x 10  6  Mela nge and Unknown sedimentary & mafic extrusive undivided sedimentary & extrusive coarse clastic sediments & alkaline extrusive 4.3 x 10 20 1.7 x 10 5.8 x 10 sedimentary & volcanic gneiss melange ultramafic melange unknown Table 5.3: Bedrock types as designated by Wheeler et al. [1997]. Hardness, Mode I and Mode II subcritical crack growth indices and critical stress intensities are assigned based on Section 3.2.2. 9  6  6  The critical stress intensities and crack growth indices stem from data presented i n Tables 3.5 and 3.6. Kj is divided into three categories based on Table 3.5: sedimentary (including all carbonates), felsic and mafic rocks with KJ of 0.62, 1.5 and 2.9 X 1 0 P a m / respectively. Intermediate igneous and metamorphic rocks are taken to Ue between the felsic and mafic values while the melanges and unknowns are averaged from aU three categories. The minimum value rather than the mean from Table 3.5 is used with the foUowing rationale: the natural range of K\ is quite large and as the law governing the rate of crack growth (3.16) is a power law with a large exponent (nj/jj), the quarrying wiU be dominated by rocks that fall on the low end of the Kj spectrum. 6  1  2  The assignment of ra/ is based on Table 3.6 with two categories identified: metamorphic/igneous with ra/ = 25 and sedimentary/carbonate with « / = 15. The melanges and unknowns are given ra/ = 20. As with Kj, the minimum values of ra/ wiU dominate and these parameter assignments reflect this. For mode II cracks, KJJ is taken as 3.5 X KJ while ra/j is taken as equal to ra/ (see Section 3.1.3). Hardness is described in Section 3.1.3 and is assigned as either hard or soft with unknown Uthologies and melanges given intermediate values.  CHAPTER 5. MODEL INPUTS 5.3.2  88  Surficial geology  Surficial materials are more enigmatic. While it is safe to assume that the bedrock composition : of North America has not altered significantly i n the past 120 000 years, the same cannot be said of the cover on top of the bedrock. Some deposits are a direct consequence of the last glacial period, thus formulation of initial conditions for a model to explore such consequences is necessarily suspect and circular. Nevertheless, the principle of uniformitarianism is applied and today's surficial geology is assumed to be a reasonable representation of what existed at the termination of the last inter-glacial period. Twenty five surficial materials from Fulton [1996] are similarly projected and gridded as the bedrock information, however the binning into model cells is dealt with differently as depth information is not part of the data. Each surficial material is assigned a bedrock fraction based on the descriptions of Fulton [1996]; this is summarized i n Table 5.4. Sub-grid b* values are transformed into a model grid sediment depth by summing the sub-grid bedrock fractions over the model grid cell then using the following second-order mapping  £ bltnn  1  m.n  (5.16)  NN m  n  where dmax is the maximum initial depth taken as 15.0 m, Nm and Nn are the number of longitudinal and latitudinal sub-grid cells. A minimum sediment thickness dmin of 0.01 m is also imposed to avoid the improbable situation of completely clean bedrock, which can adversely af- ' feet the erosion modelling through (3.12) on page 28. A plot of initial sediment thickness using this methodology is shown in Figure 5.4. The lithological makeup of the surficial cover is handled identically as the bedrock (except with multiple layers as described i n Section 4.1.5): a linked list of particular rock types (from Table 5.3) is maintained for each layer i n each model cell. Fresh glacial erosion produces sediment of identical lithological signature as the parent bed. However most of the initial surface cover has no guarantee of being autochthonous aside from a few categories specifically identified as locally derived (colluvial blocks, rubble, fines, sand and Quaternary volcanics). The allochthonous surficial categories are composed of transported materials and the lithology for these initial sediments is unknown.  89  CHAPTER 5. MODEL INPUTS  Surficial Category  undivided rock alpine complexes glaciers water till veneer glacio-fluvial complex lag marine colluvial blocks colluvial rubble colluvial fines colluvial sand Quaternary volcanics till blanket glacio-fluvial plain coarse-grained marine fine-grained marine coarse-grained lacustrian fine-grained lacustrian coarse-grained glacio-marine fine-grained glacio-marine coarse-grained glacio-lacustrian fine-grained glacio-lacustrian organic deposits eolian deposits alluvial deposits  bf  Surficial Lithology 1.0 = Autochthonous 0.0 = Allochthonous)  1.0  —  0.9  0.0  0.25  0.0  0.0  1.0  0.0  0.0  Bedrock Fraction  Table 5.4. Surficial materials as designated by Fulton [1996]. Bedrock fraction is the amount of uncovered bedrock taken for that surficial assignment. Surficial lithology is 1 when the cover is locally derived, 0 when it is not.  Figure 5.4. Initial sediment depths using (5.16) with d [1996]. The plot shows 2.5, 7.5 and 12.5 m contours.  m a x  = 15.0 m , based on data from Fulton  C h a p e t r 6  Modelled Simulations of North American Ice Sheets  6.1  Transport  results  The use of erratics to infer past glaciological conditions dates back to the "glacial theory" of Venetz, de Charpentier and Agassiz i n the early nineteenth century (Chapter 1). The characteristic length of a dispersal train varies over several orders of magnitude from local (a few km) to regional (tens of km) to continental (thousands of km) [Clark, 1987]. W i t h the model grid (1° X 0.5°), only continental debris trains are resolvable. Two such debris trains are used to assess model results: the southward dispersal of Hudson Bay carbonates onto Precambrian shield and the eastward transport of Dubawnt Group erratics i n Nunavut (formerly Keewatin District, Northwest Territories).  6.1.1  Southward carbonate transport  A Paleozoic platform surrounds and underlies Hudson Bay (Figure 6.1). The sedimentary rocks comprise gently dipping carbonates and extend southwest from Hudson B a y for 100-200 k m [Wheeler et al., 1997]. Cradling the Paleozoic platform is a crystalline Precambrian shield on which carbonates are glacially disseminated. These allochthonous carbonates were noted i n the early geologic mapping of the Precambrian shield i n northern Ontario [e.g. Bell, 1872]. The northern l i m i t of exotic carbonates is sharply denned near the mouth of the Seal River, i n the vicinity of Churchill, Manitoba [Shilts, 1980]. Aside from rare instances attributed to Late Wisconsinan ice rafting, the border between Ontario and Quebec (79°30'W) marks roughly the 1  1  T h i s is a local limit; carbonate dispersal trains are found in the central Arctic [e.g. Dyke et al.,  91  1992].  CHAPTER  6. MODELLED  SIMULATIONS OF NORTH AMERICAN ICE SHEETS  92  eastern boundary of carbonate erratics [Shilts, 1980]; further detail of eastern Ontario, shown i n Figure 6.1 is drawn from Karrow [1992]. Although groundmass carbonate can be derived locally from shield marbles or the weathering of feldspars, these sources can only provide 1-2% of the till matrix [Dredge, 1987], a relatively insignificant amount. Figure 6.1 shows a compilation of the carbonate dispersal data from the literature. Time slices of model results (using parameters under M\ i n Table 6.1) from 60-10 ka B P i n 10 ka intervals are shown i n Figures 6.2 and 6.3. Percentage carbonate produced during the last glacial cycle of the sediment (dark grey) and of the englacial debris (light grey) are represented by scaled semi-circles at each grid point on the left-hand panel. Contours of presently observed percentage of carbonate i n the till matrix from Dredge and Cowan [1989] are also plotted i n the left-hand panel. On the right are snapshots of the ice model showing ice thickness and basal sliding. Figure 6.4 shows the final modelled debris train. Prior to 60 ka B P , the ice is largely cold based i n this region. Experiments using the initial surface lithology as described i n Section 5.3.2 failed to produce an appreciable debris train and instead surface lithology of Model M\ is assumed identical to bedrock lithology . The model results show initial transport of the carbonate component during the ice flow reorganization from 60-40 ka B P . In the final years of the Wisconsin Glaciation, renewed southern flow firmly emplaces the carbonate erratics in greater concentration and to much greater distances than is observed i n the geological record. Setting all mixing parameters -Dthmst, -Ded, D \ and D to zero resulted i n a small decrease i n southward dispersion but did not alleviate the overzealous southward carbonate transport. Modelled basal sliding appears too vigorous i n this area. 2  co  TOt  In addition, the model produces carbonate dispersal into Quebec, east of Hudson Bay, which is absent i n the geological record. This is indicative of either the modelled Labrador ice dome being insufficiently developed or the modelled Hudson Bay ice dome being excessively prominent.  2  T h i s is the equivalent to the r i g h t - h a n d c o l u m n i n T a b l e 5.4 set t o 1.0 for a l l surficial m a t e r i a l s .  CHAPTER  6. MODELLED  SIMULATIONS OF NORTH AMERICAN ICE SHEETS  Figure 6.1. A map showing the bedrock lithology [Wheeler et al., 1997] of the Hudson Bay area (colour key is identical to that of Figure 5.3 on page 85) and carbonate dispersal data. The yellow contours show the percentage carbonate in the till matrix (after Dredge and Cowan [1989]). The dashed yellow line represents an extension of the 10% contour through eastern Ontario after Karrow and Geddes [1987] and Karrow [1992]. Zoltai [1965] notes that no carbonates are found west of the Nipigon Moraine, marked by the dashed white line. The percentage of carbonate in the coarse fraction (pebbles) tends to be slightly higher than that of the fine fraction shown above, but the pattern and magnitude are similar.  CHAPTER  6. MODELLED J  STMULATIONS  Parameter  n  3  Units —  Pi  v  KPa"  M  —  c  fN  1  .  'grain r  •i-fus d ^ 1  z  reg  d  0  di  ^max /^ice  s  c  S  3  F m a  g  Ai Au pint  ^abr  Ded 0  W m - ^ Jkg1  —  Z  thrust  1  0.85 2.4  1.7  1.7  0.75  0.75  2.5  2.5  2.10  2.10  0.5  0.5  3.34  m  2.0  m m  10.0  —  0.3  10  3.34  5  — —  1.2  —  2  - 1  1.0  — —  0.3  x X  IO IO  1.2  6  s  1  x  - 1  10~  1  x  1.0  8  IO"  6  IO  6  8  1.0  x x  10~  0.3 IO"  10~  9  1 0  8  IO"  0.6  100 _ 1  IO  0.7  x x  5.0  s  x x  3.0  10°  2  6.5  6  0.3  —  5  —  0.6 _ 1  10  —  0.7 1.0  x  — —  15.0  3.0  m  x  0.1  6.5  m  - 8  2.4  0.25  - 1  3  2  0.1  ms m s —  x IO 9.1 x 1 0 7.42 x I O " 2.7  3  0.85  Pa Pa  ICE  2  m m  ma m  Vs  0.4 2  imean ^ grain  ki k  M  0.4 2.7  3  T  6  M  X 10 9.1 x 1 0 7.42 X I O  3  Jbed f Jbed sd  AMERICAN  x  kg m ~ kg m ~  Pr  OF NORTH  — — — — —  Table 6.1. Parameters for Models M.\ and M.2-  8  8  SHEETS  CHAPTER 6 0 %  6. MODELLED SIMULATIONS OF NORTH AMERICAN ICE SHEETS # 5 0  %  # 4 0 %  # 2 0 %  •  1 0 %  95 •  5 %  55°N  5  2  N  51 °N  85W  95"W  90°W  85°W  90°W  85°W  Time = 60 000 a B P  5 2 N  Time = 50 000 a BP  Time = 40 000 a BP  90"W  85°W  Figure 6.2. Simulation results of the Hudson Bay carbonate train, from 85°W to 95°W. The left-hand panels show model results (represented by filled half-circles) and observed carbonate contours from Dredge and Cowan [1989]. The right-hand panels show modelled ice thickness and sliding velocity. Figure 6.3 is a continuation of this figure.  CHAPTER  # 6 0 %  6. MODELLED  # 5 0  %  SIMULATIONS OF NORTH AMERICAN  # 4 0 %  # 2 0 %  •  ICE SHEETS  10%  96  •  5 %  Time = 10 000 a B P  Figure 6.3. This figure continues Figure 6.2, showing simulation results of the Hudson Bay carbonate train, from 85°W to 95°W.  CHAPTER 6. MODELLED SIMULATIONS OF NORTH AMERICAN ICE SHEETS  #  6 0 %  #50  %  9 5  W  #40%  9  0  020%  w  8 5  •  1 0 %  97  • 5 %  W  Time = M O D E R N  Figure 6.4. This figure continues Figure 6.3, showing simulation results of the Hudson Bay carbonate train, from 85°W to 95°W.  CHAPTER  6.1.2  6.  MODELLED  SIMULATIONS  OF NORTH  AMERICAN  ICE  98  SHEETS  Eastward Dubawnt Group transport  The Dubawnt Group dispersal train in western Nunavut has particular importance as an argument against the sole dominance of a Hudson Bay dome throughout the Wisconsinan. The question of whether the Laurentide consisted of a single ice dome or whether it comprised a more complicated collection of centres has been debated over the past century. The extensive eastward dispersal of the distinctive red Dubawnt Group unequivocably demonstrates that a Keewatin centre must have been active for at least a portion of the Wisconsinan [Shilts et al, 1979; Shilts, 1980] . 3  The Dubawnt Group outcrops i n the region northwest from a Une connecting Dubawnt Lake (63°N, 102°W) and Baker Lake (64°N, 95°W) [Donaldson, 1965] and its debris train reaches southward to the Manitoba border (60°N) and eastward to Coats Island (62.5°N, 83°W) [Shilts et al, 1979; Shilts, 1980]. A bedrock map of the area and contours of dispersed Dubawnt erratics is shown i n Figure 6.5. The Dubawnt Group is not represented i n the bedrock map; the purple and maroon volcanic Pitz Formation is mapped specifically as alkaline volcanic and the red sandstones as nonmarine sedimentary rocks [Donaldson, 1965; Wheeler et al, 1997]. Because the Dubawnt group cannot be traced directly, the alkaline volcanics are traced i n their stead. The assumption that alkaline volcanic rock can be a proxy-tracer for the Dubawnt Group is buoyed by their similar, isolated occurrences. The proxy-tracer will however tend to underestimate the true Dubawnt dispersion as the red Dubawnt sandstones (more areally extensive than the Dubawnt volcanic rocks) are not included. Time slices of model results (using parameters under A 4 i in Table 6.1) from 60-10 ka B P i n 10 ka intervals are shown in Figures 6.6 and 6.7. Percentage Dubawnt Group erratics produced during the last glacial cycle of the sediment (dark grey) and of the englacial debris (light grey) are represented by scaled semi-circles at each grid point on the left-hand panel. Contours of presently observed percentage of Dubawnt Group erratics from Shilts et al. [1979] are also plotted in the left-hand panel. On the right are snapshots of the ice model showing ice thickness and basal shding. Figure 6.8 shows the final modelled debris train (note the scale in this figure is logarithmic i n contrast to the linear scales i n previous figures i n this chapter). Prior to 60 ka B P , the ice is largely cold based in this region. In contrast to the Hudson Bay area, the modelled response of Dubawnt Group erratics is far short of the observed debris train. Very little basal shding occurs i n the area and consequently any entrained debris is sluggish. Increasing the mixing parameters Dthrust to 1 X 1 0 m s and Z>ed to 1 x 1 0 m s had a discernable, yet ultimately insignificant effect on the modelled debris train. A logarithmic scale is used in Figure 6.8 to examine the limits of Dubawnt Group dispersion. This shows that the debris train extends too far northeast, but almost reaches the eastward observed limit on Coats Island. _ 8  _ 9  2  2  - 1  _ 1  There are other lines of evidence supporting a multi-domed Laurentide Ice Sheet. For example, the lack of eastward carbonate dispersal in Quebec suggests a well-developed Labrador dome. 3  CHAPTER  6. MODELLED  SIMULATIONS  OF NORTH AMERICAN  ICE  SHEETS  99  6 6N  6 4 N  6 2N  6 0 N  I 05W 1 0 0 w  9 5W  9 0 W  8 5 W  Figure 6.5. A map showing the bedrock lithology [ Wheeler et al., 1997] of the region west of Hudson Bay (colour key is identical to that of Figure 5.3 on page 85) and observed allochthonous Dubawnt Group dispersal. The Dubawnt Group consists mainly of pink and purple unmetamorphosed sandstones and volcanic rocks [Wright, 1957; Donaldson, 1965]; the distinctive red hue of many Dubawnt components enables visual identification of dispersed Dubawnt erratics. The black contours show the percentage of Dubawnt erratics in the coarse fraction (2-6 mm) [Shilts et ai, 1979; Kaszycki and Shilts, 1980]. Alternate size categories exhibit slightly different dispersal contours, but the pattern and magnitude remain similar to those shown. The source area of the Dubawnt Group does not appear in the bedrock map of Wheeler et al. [1997], however the mapped extent of the Pitz Formation (comprising deep purple and maroon volcanic rocks) of the Dubawnt Group [Donaldson, 1965] matches an isolated group of alkaline volcanic rocks in the compilation of Wheeler et al. [1997]. This is used as the tracer lithology.  CHAPTER  6 0 %  6. MODELLED SIMULATIONS OF NORTH AMERICAN ICE SHEETS  # 5 0  %  4 0 %  2 0 %  •  1 0 %  100  •  5 %  61 °N eo°N 105°W n  9 5 W  Time = 60 000 a BP  9 0W  60°N , 0 5  100 W  95°W  100°W  95°W  100°W  95°W  U  „-  °W  Time = 50 000 a B P  90°W  90°W  65°N  61 "N 60°N „ 105°W 9 5 W  Time = 40 000 a BP  90°W  Figure 6.6. Simulation results of the Dubawnt Group dispersal train from 90°W to 100°W. The left-hand side shows model results (represented by filled half-circles) and observed Dubawnt Group erratic contours from Shilts et al. [1979]. The right-hand panels show modelled ice thickness and shding velocity. Figure 6.7 is a continuation of this figure.  CHAPTER  6 0 %  6. MODELLED SIMULATIONS OF NORTH AMERICAN ICE SHEETS  5 0 %  4 0 %  2 0 %  •  1 0 %  60°N„  Time = 30 000 a B P  100°W  95°W  100°W  95°W  100°W  95°W  BO°N 9 5 W  9 0W  , 0 5  °W  101  5 %  90°W  90°\N  Time = 20 000 a B P 65°N  61 °N  eo°N. 5 W  9 0 W  105  Time = 10 000 a BP  Figure 6.7. This figure continues Figure 6.6, showing simulation results of the Dubawnt Group dispersal train from 90°W to 100°W.  CHAPTER  6. MODELLED  V_j 1 O'ppm ^  SIMULATIONS OF NORTH AMERICAN  1 O^ppm IP  10'ppm  ICE SHEETS  102  # 1 0 'ppm • 1 0 ' p m • 1 0 p p ni P  Figure 6.8. This figure continues Figure 6.7, showing simulation results of the Dubawnt Group debris train. The different modes of failure between the two test cases is instructive. Clearly the transport is overly vigorous in the Hudson Bay carbonate area suggesting that the shding coefficients used in the ice model (B, = 1 X l O ^ m s " Pa" for a hard bed and B = 2 x l O ^ m s - P a for a soft bed) are too high. The modelled dispersion is also not selective enough to match the observations with eastward dispersal well beyond the Ontario-Quebec border and northern transport above the Manitoba-Nunavut border. The opposite situation prevails over western Nunavut where debris dispersal is sluggish. This can be attributed to an absence of a well-formed Keewatin Dome. 1  1  1  -1  a  There is strong evidence that Hudson Strait acted as an ice stream conduit episodically during the Wisconsin Glaciation, drawing down the ice in the Hudson Bay basin and encouraging ice flow from the Keewatin District and western Quebec into the head of Hudson Strait. This would serve to ameliorate the lack of eastward dispersion of the Dubawnt Group, the excessive southward transport of carbonates and the existence of western Quebec carbonates in the model results. The results presented in this section account only for dispersion from the Wisconsin Glaciation. Clearly this is inadequate as many glaciations have taken place over the study area (Figure 1.1). The exercise is nevertheless useful and explicit polycyclic modelling of glacial cycles is an obvious future step.  CHAPTER  6.2  6. MODELLED SIMULATIONS OF NORTH AMERICAN ICE SHEETS  103  Erosion results  The measurement of paleo-erosion is problematic; proof of its existence is often ephemeral as transport agents work to disperse erosional products from their moment of creation. Nevertheless there are methods of obtaining denudation rates to compare and assess model results. If a dated paleosurface can be reconstructed, differencing the surface with modern topography results i n time-integrated erosion. Sediment volume measured in a well-defined depositional basin yields the average erosion of the source area. Cosmogenically derived isotopes are used to give point measurements of erosion where good independent dating controls exist. W i t h all methods, the problem of separating glacial and non-glacial erosion is always present. A focus of this work is the reconstruction of the Laurentide Ice Sheet. It was the largest of the Wisconsinan North American ice sheets and the relatively coarse grid used for this study was chosen to enable a domain expansive enough for the entire Laurentide Ice Sheet. However, there are smaUer ice complexes captured by the model domain (e.g. Newfoundland Ice Cap, Cordilleran Ice Sheet). The Cordillera is particularly instructive as several studies have examined paleo-erosion in Fennoscandia, an analogue of the Cordillera.  6.2.1  Cordilleran erosion  The Wisconsinan ice that encompassed the rugged accreted terrains of western North America is labeUed the Cordilleran Ice Sheet; the St. Elias ice fields and the many Coast Range ice caps are its modern remnants. The Cordilleran winter climate is dominated by a succession of cyclonic storms from the North Pacific that, when combined with the orographic effect of several northsouth trending mountain chains, lead to massive alpine precipitation. This wet climate maintains the abundant ice fields presently observed in the Coast Mountains and St. Elias Range. The Cordilleran ice history is complex, but generally follows the conceptual model of Kerr [1934]: in a colder climate, small mountain glaciers, similar to their modern counterparts, can grow to cover an extensive area in mountainous regions. As the chmate cooled, the alpine glaciers coalesced into an ice mass that blanketed the region, although ice dynamics were still largely topographically controlled. In some cycles of glaciation (likely excluding the ultimate one), the ice thickened, culminating in radially outward flowing domes, largely independent of subglacial topography. The evolution of the Cordilleran Ice Sheet over the past 120 ka has been punctuated by several glacial pulses. During the Wisconsin Glaciation there was at least one period (the Olympia non-glacial interval) where glacial limits were similar to present day configuration [Clague, 1981]. This pulsating character is partially captured by the ice model although major details, such as the Olympia non-glacial interval, are missed. A plot of model ice volume between 45-65°N, west of 120°W is shown in Figure 6.9.  CHAPTER 6. MODELLED  SIMULATIONS OF NORTH AMERICAN  ICE SHEETS  104  Figure 6.9. Modelled Wisconsinan ice volume west of 120° W and between 45° and 65° N , which roughly coincides with the bounds of the Cordilleran Ice Sheet.  Paleo-erosion data from the Cordillera are scant, but such studies have been undertaken on the western seaboard of Fennoscandia. Both regions have maritime mountain chains with extremely wet winters and their Wisconsinan ice history, both largely topographically controlled, includes episodic coastal glaciation [Clague, 2000; Elverh0i et al, 1995a]. The analogue between the two is reasonable and studies of Fennoscandia can be used to constrain Cordilleran paleo-erosion despite notable differences i n glacial landforms.  Estimates of erosion  Isotope C 1 is created by interaction of terrigenous elements with cosmogenic radiation, altering the background ratio of C 1 / C 1 to a depth of « 2 m . Where the timing of glacial advance and retreat are well constrained, the expected C 1 / C 1 ratio can be calculated and deviations from the expected value is attributable to glacial erosion. Briner and Swanson [1998] present a C 1 study where ice cover is well-constrained from 18-15 ka B P i n Washington State. A maximum erosion of 0.26-1.06 m is calculated at abrasion sites (the upstream flank of local topography) while a minimum of « 2 m is inferred i n areas of inferred quarrying (lee faces); the study area is heavily fractured, which enables aggressive quarrying. 3 6  36  36  3 6  While the above represents an important result, it is nevertheless a single datum i n space and wide-scope integrated erosion values are more directly comparable to model output. The differencing between ancient and modern topography gives an extensive spatial result and a study of Sognefjord, Norway [Nesje and Whillans, 1994; Andersen and Nesje, 1992; Nesje et al, 1992] provides such an example. A smooth Tertiary surface is interpolated from remnants of the paleosurface i n the basin highlands which have been nearly free of Quaternary erosion, with some  CHAPTER  6. MODELLED  SIMULATIONS OF NORTH AMERICAN  ICE SHEETS  105  areas retaining a Tertiary regolith [Nesje and WhUlans, 1994]. The difference between modern topography and the estimated paleosurface yields an average basinal erosion of 610 m, which is in agreement with the estimate from the marine sediment wedge [Riis and Fjeldskaar, 1992]. W i t h the assumption that the entire difference is attributable to an estimated 600 ka of ice cover since the Tertiary, an average erosion rate of 1 . 0 2 m m a is calculated [Nesje et al., 1992]. A more conservative estimate that allows for preglacial, fluvial channeling is an average of 320 m denudation over the basin [Andersen and Nesje, 1992], which yields 0.53 m m a . As Nesje and WhUlans [1994] note, neither of these estimates considers mass wastage of the headwall (where modern colluvial debris fans are observed) or a tectonic contribution to the fjord formation. _1  - 1  The assumption that the erosion rate has remained constant over the past 2.5 M a is fraught with pitfalls. Aggressive early erosion of the less-competent Tertiary regolith is probable. From 1-0.6 M a B P , there is a marked increase of ice-rafted debris (IRD) in the Norwegian Sea compared to earlier rates [Jansen and Sj0holm, 1991], indicating that glacial erosion i n the later part of the integrated period may have outstripped the earlier years. Based on Figure 1.1 and ocean cores that capture predominantly Greenland I R D [Jansen et ai, 2000], this increased trend is expected to continue throughout the most recent 0.6 M a . This suggests that the estimates of erosion based on even distribution throughout the past 2.5 M a underestimate the erosion rate of the past 120 ka. Another method to obtain integrated denudation rates over a basin is to integrate the deposited sediments; this approach has been applied to marine sediment fans i n Svalbard. The integration of the glacial sediments from Isfjorden Fan yields a total denudation over the past 120 ka of 24 m [Elverh0i et al, 1995b]. This estimate is based on seismic mapping of a "Glacial Unit" with tentative age assignment through identification of marine isotope stage 5e (RS 120 ka B P ) within the unit from ocean coring. This denudation rate is consistent with, though i n the low end of the bracket, other estimates of long term erosion rates from other Barents Sea basin sediments (Storfjorden and Bear Island Trough) [Elverh0i et ai, 1998]. Modern glaciated basins in Alaska provide a glimpse at a possible paleo-analogue of the Wisconsin Glaciation further south. Measurements show that modern day erosion from Alaskan glaciers is on the same order as the above paleo-estimates [Hallet et al., 1996]. The analogies between modern Alaskan glaciers, paleo-ice on Svalbard and Norway and the Cordilleran Ice Sheet are by no means exact, but can nevertheless serve to roughly constrain Cordilleran paleo-erosion.  Modelled Cordilleran erosion The modelled erosion results illustrated i n Figure 6.10 show that this model produces erosion rates much lower than the above estimates. The erosion results shown i n Figure 6.10 is a very favorable realization for erosion with parameters given by M.2 in Table 6.1. To maximize the erosion  CHAPTER 6. MODELLED SIMULATIONS OF NORTH AMERICAN ICE SHEETS  Abrasion output r-  30  , Point abrasion  106  Quarrying output  2h  400  Basal sliding  200  Point quarrying  Water Pressure o<>^ s  100  50  Time (ka BP)  0  0.5  1  1.5  I  Figure 6.10. Simulation results for Cordilleran erosion, using favorable conditions for erosion. Time-integrated results are shown for the entire region and time series are plotted for the points 133.5°W, 58.25°N (denoted by the star) and 125.5°W, 57.25°N (denoted by the diamond). Time series are also shown for the basal sliding speed and normalized subglacial water pressure which are well correlated to abrasion (correlation coefficient = 0.79) and quarrying (correlation coefficient = 0.50) respectively.  potential (and to minimize the effects of other aspects of the complete model), the initial cover conditions are set to zero (bare basement) and no build up of sediment is allowed. Physically, this can be thought of as perfectly effective transport in action through either complete entrainment or through vigorous subglacial water flow. The failure of the model to predict vigorous erosion stems from the lack of quarrying. In Section 5.2, the assumption was made that englacial routing of surface water is insignificant for the Wisconsinan ice sheets. However for much of the past 120 ka, the ice volume in the Cordillera would not have been sufficient to engulf the mountains and coalesce into an extended ice mass.  CHAPTER  6. MODELLED  SIMULATIONS  OF NORTH  AMERICAN  ICE  SHEETS  107  Conditions more akin to present-day would have been prevalent and i n these conditions, surface water can leak i n down the sides of nunataks or valley walls. This input is unaccounted for and could raise the modelled subglacial water pressure to levels where quarrying could be expected. Supraglacial debris production, very active in modern glaciers i n the Cordillera and St. Elias, can be a significant source of erosion when ice volume is insufficient to drown nunataks. As this was the case during much of the Wisconsin Glaciation i n the Cordillera, the exclusion of this process may lend to the underestimation of Cordilleran paleo-erosion.  6.2.2  Laurentide erosion  Estimates of erosion Estimates of the Laurentide's erosional signature have historically fluctuated by several orders of magnitude from a few tens of metres [e.g. Dyke et ai, 1989] to very deep erosion (greater than 1000 m i n Hudson Bay [White, 1972] . The deep erosion hypothesis, while an attractive explanation of broad features of both the Laurentian and Fennoscandian landscape is inconsistent with many glacial geomorphic observations [e.g. Gravenor, 1975; Sugden, 1976; Lidmar-Bergstrom, 1997]. A tectonic origin for the Hudson Bay basin is generally accepted [e.g. Sugden, 1976; LidmarBergstrom, 1997]. 4  One method for estimating the integrated erosion is to examine the amount of deposited sediment. This approach is explored for North Atlantic deposits by Laine [1980,1982] and for all the marine sediment repositories of the Laurentide by Bell and Laine [1985] to yield an average denudation from the Laurentide source area of 120 m over 3 M a , a value that begins to approach the deep erosion hypothesis of White [1972, 1988]. Bell and Laine [1985] consider this a minimum value although no allowance is made for non-glacial erosion and Hay et al. [1989] charge that the estimate of sediment with a Laurentide provenance i n the Gulf of Mexico is greatly overestimated i n the analysis of Bell and Laine [1985]. Reanalysis using the methodology of Bell and Laine [1985] with the more conservative estimate of Hay et al. [1989] for glacial sediments i n the Gulf of Mexico gives an expected mean glacial erosion of 80 m through the Laurentide source area over 3 M a . 5  6  Integration of the Dubawnt dispersal train can be linked back to a specific source area because of its distinctive colour, which makes it as useful tracer lithology exploited i n Section 6.1.2. As the entire debris train has been mapped, an estimate of the total amount of debris can be made, N o value is a c t u a l l y g i v e n b y White [1972] a n d this value is e s t i m a t e d f r o m a c o n c e p t u a l sketch b y  4  Sugden [1976]. T h e C o r d i l l e r a n Ice Sheet is not i n c l u d e d i n the analysis  5  Hay  6  [1998] cites a corrected value o f 16 m , a l t h o u g h this appears to disregard a l l g l a c i a l sediments  n o r t h o f the G u l f o f M e x i c o .  CHAPTER 6. MODELLED SIMULATIONS OF NORTH AMERICAN ICE SHEETS Model variable Mi, /  0  Mi,  /  0  Mi, /  0  M, 2  Sliding distance  Water Pressure  iAf  0M  0.24  <JAf  0.16  0.62  iA<  0.79  0.25  <5A<  0.12  0.50  1 2 0 k a  1 2 0 k a  1 2 0 k a  /o  20ka  108  Table 6.2. Correlation coefficients for total abrasion and quarrying with integrated shding distance and normalized water pressure.  and because the source area is also well denned, a value for denudation can be calculated. This is the strategy of Kaszycki and Shdts [1980], who estimate 6-20 m of erosion during the last glacial cycle.  Modelled Laurentide erosion Figure 6.11 shows the total erosion for the Models Mi and M2 (see Table 6.1). The average erosion over the entire model domain for Mi is 0.26 m and for M2 is 0.73 m. These figures underestimate the true average erosion as they take the entire model domain for the whole integration time into consideration while the estimates outlined above generally restrict the integration to glaciated area. However this error at most could alter the results by a factor of 2, leaving the modelled erosion short of observed amount of sediment produced over the past 120 ka. Table 6.2 shows that the abrasion is controlled mostly by the shding while quarrying is correlated to water pressure. It is difficult to justify increased abrasion to better match the observations. This would require either a significant change i n the physical parameters of the bed or increased shding velocities. Given that the transport model results indicate that shding velocity is already overestimated, an increase i n quarrying is the more sensible solution to deal with the erosive deficit. Quarrying results could be bolstered by two methods: either increasing the water pressures or by decreasing the resistance of the bedrock to subcritical fracture. Both solutions are feasible. The water pressure is kept at an arbitrary level through the sheet thickness -ff„ as described in Section 5.2.1. There is no strong basis to this parameter and reducing H™lt by a nominal amount would lead to higher water pressures and thereby more aggressive quarrying. On the other hand, in the assignment of subcritical crack parameters i n Section 5.3.1, no allowance is made for pre-fractured rock, which could increase quarrying significantly. it  :  CHAPTER  6. MODELLED SIMULATIONS OF NORTH AMERICAN ICE SHEETS  109  Figure 6.11. Integrated erosion (in metres) for Mi (top panel) and M2 (bottom panel), the parameters of which are listed in Table 6.1.  C h a p e t r 7  Conclusions 7.1  Summary of work  This thesis presents a large-scale physically based model of subglacial processes driven by existing ice sheet and hydrological models. It comprises subglacial erosion, sediment entrainment and deposition, englacial mixing and advective transport components. A lithologicalfy accurate description of the bed enables tracer capabilities and incorporation of geologically dependent parameters. The modelled subglacial erosion proceeds by two mechanisms: abrasion and quarrying. Abrasion is calculated by the classic wear formula, proportional to the normal force presented by the abrading particle F and the particle velocity v , and inversely proportional to the hardness of the wear surface h . The normal force is calculated, following Hallet [1981], through the viscous force of the ice against the abrading particle. The wear is modified to account for the shielding effect that sediment has on further abrasion. Hardness is hthologically dependent and a sub-grid geological representation of the bed is used [ Wheeler et al., 1997]. A distribution of particle radii, upon which the viscous force is dependent, is taken from sedimentological studies i n the literature. Bed topography is downscaled based on the scaling behavior of six test areas where both coarse and high resolution elevation data are available for calibration. N  p  s  A quarrying model is constructed as a function of subcritical crack growth. This i n turn depends on lithological parameters and the state of stress in the bedrock. As for abrasion, an accurate description of bed geology is used to assess the hthologically dependent parameters. The stress is calculated using a two dimensional convolution of the solution for a point load acting on the planar surface of an infinite half-space. Numerical experiments suggest that a topographic modifier to this planar solution is an acceptable approximation to the rigorous solution for the loading of a complex surface with topography. The stress induced by a continuous ice load is generally 110  CHAPTER  7. CONCLUSIONS  111  sufficient to induce subcritical crack growth and the existence of water-filled cavities and joints are necessary for quarrying to proceed. Therefore, I formulate an expression for subglacial cavity areal extent and its effect on the stress solution. The quarrying rate is then calculated by modifying the crack growth rate by a tunable parameter AL and a sediment shielding term. The total modelled abrasion over the Wisconsin Glaciation is highly correlated with the mean sliding speed of the ice; total quarrying is well correlated with water pressure normalized by the ice overburden. Thus for both processes, the input fields are crucial. The Marshall-Clarke ice sheet model [Marshall, 1996; Marshall and Clarke, 1997a,b; Marshall et al., 2000], driven by estimates of paleoclimate, is used to predict ice dynamics and thermodynamics; this includes a thermally regulated sliding scheme with an enhancement for soft-bedded domains. The subglacial water pressure is solved using the model of Flowers [2000], modified for a continental-scale reconstruction. Sensitivity tests show that the contribution of subcritical shear crack growth (Mode II/III) is limited in all but the most extreme cases and extensional crack growth (Mode I) is the primary method of subcritical crack propagation. Over a glacial cycle, quarrying is the more efficient excavator; even the most aggressive parameterization of abrasion cannot match the observed data and it is necessary to include quarrying to approach the estimates of paleo-erosion. It is therefore difficult to assess the appropriateness of the value used for the abrasive wear coefficient & b a  r  Quarrying, although deeper than abrasion, also does not achieve the quantity demanded by the geological record. This is suggestive of several possibilities  (a) The quarrying factor of Ai = 1 X 1 0  - 8  ms  - 1  used in this study is too low.  (b) The modelled rock competency is too great. (c) The subglacial macroporous horizon, artificially thickened to increase its transmissivity, is too thick which results in low water pressures. (d) The assumption that englacial routing of surface melt to the bed can be ignored is false..  It is unlikely that Ai should be increased by a significant amount. This does not serve to areally expand the quarrying domain, but rather to make the quarrying more aggressive. Already with Ai set to 1 X 1 0 m s , point quarrying rates reach K, 4 m m a , which is consistent with presently observed rates [Hallet et al., 1996] and some areas are subject to deep quarrying ( « 25 m). In some cases, the rock competency may be overestimated, as fractured or weathered rock is not considered. The macroporous horizon critical sheet thickness H™ is a tunable parameter taken as 4 m for this study. A thinner value is plausible and would lead to higher water pressures and more aggressive quarrying. Another method to increase water pressure is to enable surface-bed melt coupling. The prohibition of this exchange is a good assumption at L G M for the majority - 8  _ 1  - 1  it  CHAPTER  7.  CONCLUSIONS  112  of the L a u r e n t i d e , but there are areas a n d times where this component of the h y d r o l o g i c a l m o d e l of  Flowers  [2000] should be included. T h i s is the most probable reason for the u n d e r e s t i m a t i o n  of m o d e l l e d erosion i n the C o r d i l l e r a n Ice Sheet.  The  englacial processes modelled i n this thesis begin w i t h sediment entrainment a n d e n d w i t h  deposition.  T h r e e methods of entrainment are considered:  net basal adfreezing, i n t r u s i o n b y  regelation a n d freeze-on by W e e r t m a n regelation. T h e t h e r m o d y n a m i c s of the ice sheet m o d e l do not allow for freeze-on m a k i n g the calculation of net basal freezing p r o b l e m a t i c . Nevertheless, a physically based estimate is made when the b e d makes a w a r m - t o - c o l d transition. C a l c u l a t i o n of intrusion follows  Iverson and Semmens  [1995] a n d is a strong function of water pressure. L i m i t e d  entrainment can occur by W e e r t m a n regelation when the ice slides over a p r e d o m i n a n t l y h a r d bed.  Intrusion by regelation is the most significant of the three mechanisms examined. D e p o s i t i o n  occurs when basal melt overtakes the combined entrainment.  V e r t i c a l redistribution of englacial sediment is modelled t h r o u g h a n u m b e r of processes. by englacial tectonics is cast as an eddy-diffusion m o d e l following  Mixing  Alley and MacAyeal  [1994]  and a f o r m u l a t i o n is developed for m i x i n g by thrust faults. A n expression describing the m i x i n g by rigid-particle r o t a t i o n is developed, which is o n the same order as the m i x i n g b y r i g i d - b o d y collisions  [Weertman, 1968]. V e r t i c a l  advection of debris is accounted for w i t h a simple u p s t r e a m  discretization. L a t e r a l advection of sediment w i t h the ice is calculated using the scheme of  [1986],  Prather  which conserves second-order moments a n d has excellent n u m e r i c a l diffusion attributes.  A c c u r a t e tracking of the lithological signature of sediment t h r o u g h each of the above processes allows the transport of distinct lithological tracers to be predicted a n d c o m p a r e d to observed debris trains on the m o d e r n landscape. A detailed bedrock m a p of C a n a d a [ Wheeler  et al.,  1997] is used  for i n i t i a l i z a t i o n . T h e comparison of observed H u d s o n B a y Paleozoic carbonate dispersal w i t h m o d e l results shows a n overzealous transport of carbonates i n b o t h amount a n d direction. T h i s is suggestive of either an excessively vigorous shding law used i n the ice sheet m o d e l or incorrect central ice geometry which drives the shding. Conversely, the m o d e l l e d D u b a w n t G r o u p transport is m i n u t e c o m p a r e d to the observed debris train. T h e ice sheet m o d e l does not sufficiently capture the coarse structure of the Laurentide Ice Sheet to allow for this geological observation; more fully developed K e e w a t i n a n d L a b r a d o r D o m e s are necessary.  7.2  Outlook  A p r i m a r y objective of this thesis is to further the theoretical description of subglacial a n d englacial processes w i t h the intent of creating a predictive forward m o d e l for glacial geology. T h e preceding chapters represent a step towards this goal. T h i s effort is neither exhaustive nor complete. T h e r e is a n abundance of geomorphic landforms not touched u p o n a n d there are m a n y  conceivable  CHAPTER  7.  CONCLUSIONS  113  improvements to the processes described. Refined input models offer immediate potential for improvement. There is strong evidence of an ice stream i n Hudson Strait from the provenance of Heinrich Layer ice-rafted debris; such an ice stream would drain and subsequently lower Hudson Bay ice. This would lessen the southward carbonate transport and enhance the eastward Dubawnt Group transport, ameliorating both problems evidenced i n the comparisons of model results to observations. A s discussed above, the prohibition of surface-bed water routing is too general a constraint that adversely affects the erosion results. Careful application, dependent on topography and ice thickness, of this model component of Flowers [2000] could alleviate this problem. The melding of diverse models (climate, ice, hydrology and terrestrial) gives a wide scope to this study that is particularly relevant in the current atmosphere of interdisciphnary science. However, since there are no feedbacks between the models, the coupling is incomplete. This work provides an avenue of extension for existing ice sheet and hydrology models; basal conditions affect both ice and water dynamics and a large-scale description of bed evolution is a vehicle for feedback. Much of the geomorphic data is of smaller scale than what is resolvable with the model grid used. A smaller model domain has potential economic interest; lithological tracing capabilities could be applied to drift prospecting. Two and half centuries after Daniel Tilas began drift prospecting using boulder erratics i n Finland [Kauranne, 1992] a process-based forward model is at hand! Glaciologists have increasingly recognized that basal processes are of great import to ice dynamics. The wealth of Quaternary geological data cannot be used as an effective constraint i n an ad hoc manner. Predictive forward modelling of basal processes fully coupled to an ice sheet model (inclusive of subglacial hydrology) is crucial. I present a step i n this direction.  ... this thread of ink which I have let run on for page after page, swarming with cancellations, corrections, doodles, blots and gaps, bursting at times into clear big berries, coagulating at others into piles of tiny starry seeds, then twisting away, forking off, surrounding buds of phrases with frameworks of leaves and clouds, then interweaving again, and so running on and on and on until it sputters and bursts into a last senseless cluster of words, ideas, dreams, and so ends. [Calvino, 1959, from The Baron in the Trees]  References References annoted with ' are included to acknowledge the original authors. works have not been examined due to inaccessibility of the original documents.  The referenced  The number(s) within the double square brackets following each reference indicate the page(s) on which that reference is cited. Agassiz, L . Etudes sur les glaciers. Neuchatel, Jent et Gassmann, 1840.  [[1, 3, 117]]  Agassiz, L . Geological Sketches. Second Series. James R . Osgood and Co., Boston, 1876. [[47]] Alley, R . B . , D . D . Blankenship, C . R . Bentley, and S. T . Rooney. Deformation of till beneath Ice Stream B , West Antarctica. Nature, 322(6074):57-59, 1986. [[6]] Alley, R . B . , D . D . Blankenship, C . R. Bentley, and S. T . Rooney. T i l l beneath Ice Stream B . 3. T i l l deformation: Evidence and implications. Journal of Geophysical Research, 92(B9):8921-8928, 1987. [[6]] Alley, R. B . , K . M . Cuffey, E . B . Evenson, J . C . Strasser, D . E . Lawson, and G . J . Larson. How glaciers entrain and transport basal sediment: Physical constraints. Quaternary Science Reviews, 16(9):1017-1038, 1997. [[32, 55, 57, 58, 60, 68]] Alley, R. B . , D . E . Lawson, E . B . Evenson, J . C. Strasser, and G . J . Larson. Glaciohydraulic supercooling: A freeze-on mechanism to create stratified, debris-rich basal ice: II. Theory. Journal of Glaciology, 44(148):562-568, 1998. [[57]] Alley, R . B . and D . R. MacAyeal. Ice-rafted debris associated with binge/purge oscillations of the Laurentide Ice Sheet. Paleoceanography, 9(4):503-511, 1994. [[68, 112]] Andersen, B . G . and A . Nesje. Quantification of Late Cenozoic glacial erosion i n a fjord landscape. In Robertsson, A . - M . , editor, Quaternary Stratigraphy, Glacial Morphology and Environmental Changes, volume C a 81, pages 15-20. Geological Survey of Sweden, 1992. [[104, 105]] Anderson, R . S. A model of ablation-dominated medial moraines and the generation of debrismantled glacier snouts. Journal of Glaciology, 46(154):459-469, 2000. [[6]] Archard, J . F . Contact and rubbing of flat surfaces. 1953.  [[25]]  114  Journal of Applied Physics, 24:981-988,  115  REFERENCES  A t k i n s o n , B . K . Fracture toughness of Tennesee Sandstone a n d C a r r a r a M a r b l e using the double torsion testing m e t h o d .  International Journal of Rock Mechanics and Mining Sciences &  Geomechanics Abstracts, 16:49-53, 1979.  [[37]]  A t k i n s o n , B . K . Stress corrosion and the rate-dependent tensile failure of a fine-grained quartz  rock. Tectonophysics, 65:281-290, 1980. [[37, 40]] A t k i n s o n , B . K . S u b c r i t i c a l crack g r o w t h i n geological materials. 89(B6):4077-4114, 1984.  Journal of Geophysical Research,  [[36, 37, 38, 40]]  A t k i n s o n , B . K . , S. M . D e n n i s , P . G . M e r e d i t h , and J . Zheng. Stress corrosion a n d m i c r o s e i s m i c a c t i v i t y d u r i n g tensile, shear a n d compressive failure of rocks*. F i n a l technical report to the managers of the U . S . N a t i o n a l E a r t h q u a k e H a z a r d R e d u c t i o n P r o g r a m C o n t r a c t N o : 14-080001-21287, U . S . G e o l o g i c a l Survey, M e n l o P a r k , C A , 1984.  [[38, 40]]  A t k i n s o n , B . K . , S. M . Dennis, P . G . M e r e d i t h , a n d J . Zheng. Stress corrosion a n d m i c r o s e i s m i c a c t i v i t y d u r i n g tensile, shear a n d compressive failure of rocks*. F i n a l technical report to the managers of the U . S . N a t i o n a l E a r t h q u a k e H a z a r d R e d u c t i o n P r o g r a m C o n t r a c t N o : 14-080001-21287, U . S . G e o l o g i c a l Survey, M e n l o P a r k , C A , 1985.  [[37, 38]]  A t k i n s o n , B . K . and P . G . M e r e d i t h . E x p e r i m e n t a l fracture mechanics d a t a for rocks a n d minerals. I n A t k i n s o n , B . K . , editor,  Fracture Mechanics of Rock, chapter 11, pages 477-526. A c a d e m i c  Press Inc. (London) L t d . , 1987a.  [[37, 38, 40]]  A t k i n s o n , B . K . and P . G . M e r e d i t h . T h e theory of s u b c r i t i c a l crack g r o w t h w i t h applications to minerals a n d rocks. I n A t k i n s o n , B . K . , editor,  Fracture Mechanics of Rock, chapter 4, pages  111-166. A c a d e m i c Press Inc. (London) L t d . , 1987b.  [[36]]  A t k i n s o n , B . K . a n d R . D . R a w l i n g s . A c o u s t i c emission d u r i n g stress corrosion cracking i n rocks. I n S i m p s o n , D . W . a n d P . G . R i c h a r d s , editors,  Earthquake Prediction, an International Review,  v o l u m e 4 of  Maurice Ewing Ser., pages 605-616. A m e r i c a n G e o p h y s i c a l U n i o n , W a s h i n g t o n ,  D . C . , 1981.  [[38, 40]]  A u g u s t i n u s , P . C . T h e influence of rock mass strength o n g l a c i a l valley cross-profile m o r p h o m e t r y : A case study from the Southern A l p s , N e w Zealand. 17:39-51, 1992.  Earth Surface Processes and Landforms,  [[84]]  A u g u s t i n u s , P . C . G l a c i a l valley cross-profile development:  T h e influence of i n s i t u rock stress  a n d rock mass strength, w i t h examples from the Southern A l p s , N e w Zealand. 14:87-97, 1995.  Geomorphology,  [[84]]  B a g n o l d , R . A . E x p e r i m e n t s on a gravity-free dispersion of large solid spheres i n a N e w t o n i a n fluid under shear.  Proceedings of the Royal Society of London. Series A, 225:49-63,1954.  *See preamble on page 114 *See preamble on page 114  [[69]]  REFERENCES  116  B a g n o l d , R . A . T h e flow of cohesionless grains i n fluids.  Society of London. Series A, 249(964):235-297, 1956.  Philosophical Transactions of the Royal [[69]]  B a l l , R . The Cause of an Ice Age. K e g a n P a u l , Trench, T r i i b n e r & C o . L t d . , L o n d o n , 3 1906.  r d  edition,  [[5]]  Bell, M . and E . P. Laine. glaciofluvial processes.  E r o s i o n of the Laurentide region of N o r t h A m e r i c a b y g l a c i a l a n d  Quaternary Research, 23:154-175, 1985. [[5, 107]]  B e l l , R . R e p o r t o n the country n o r t h of L a k e Superior between the N i p i g o n a n d M i c h i p i c a t e n Rivers.  Report of Progress, 1870-1871, chapter 8, pages 322-351. G e o l o g i c a l Survey of  In  C a n a d a , 1872. Bishop, B . C . S I P R E , 1957.  [[91]]  Shear moraines i n the T h u l e area, northwest G r e e n l a n d .  Research report 17,  [[59]]  B l a c k w e l l , D . B . and J . L . Steele. G e o t h e r m a l m a p of N o r t h A m e r i c a . Decade of N o r t h A m e r i c a n geology. M a p 006, Geological Society of A m e r i c a , B o u l d e r , C o l o r a d o , 1992.  [[75]]  The Deforming Bed Beneath a Surge-Type Glacier: Measurements of Mechanical and Electrical Properties. P h . D . thesis, U n i v e r s i t y of B r i t i s h C o l u m b i a , 1992. [[79]]  Blake, E . W .  B l a n k e n s h i p , D . D . , C . R . Bentley, S. T . Rooney, a n d R . B . A l l e y . Seismic measurements reveal a saturated porous layer beneath an active A n t a r c t i c ice stream.  Nature, 322(6074):54-57, 1986.  [[6]] B o t t o m l e y , J . T . M e l t i n g a n d regelation of ice.  Nature, 5:185, 1871/72.  [[57]]  B o u l t o n , G . S. O n the o r i g i n and transport of englacial debris i n S v a l b a r d glaciers.  Glaciology, 9(56):213-229, 1970.  [[53, 60]]  B o u l t o n , G . S. Processes of glacial erosion on different substrata. 38, 1979.  Journal of  Journal of Glaciology, 23(89):15-  [[25, 29]]  B o u l t o n , G . S. Development of a theoretical m o d e l of sediment disposal b y ice sheets. I n  Prospect-  ing in Areas of Glaciated Terrain 1984, pages 213-223. T h e Institute of M i n i n g a n d M e t a l l u r g y , 1984.  [[61]]  B o u l t o n , G . S. T h e o r y of glacial erosion, transport and deposition as a consequence of s u b g l a c i a l : sediment deformation.  Journal of Glaciology, 42(140):43-62, 1996.  [[7]]  B o u l t o n , G . S., P . E . C a b a n , and K . van Gijssel. G r o u n d w a t e r flow beneath ice sheets: P a r t I large scale patterns.  Quaternary Science Reviews, 14(6):545-562, 1995. [[80, 81]]  B o u l t o n , G . S. a n d C D . C l a r k . A h i g h l y mobile Laurentide ice sheet revealed b y satellite images of g l a c i a l lineations.  Nature, 346(6287):813-817,1990.  [[3]]  B o u l t o n , G . S. a n d A . S. Jones. S t a b i l i t y of temperate ice caps and ice sheets resting o n beds of deformable sediment.  Journal of Glaciology, 24(90):29-43, 1979.  [[6]]  REFERENCES  111  Application des potentiels a I 'etude de I'equilibre et du mouvement des solides elastiques, principalement au calcul des deformations et des pressions que produisent, dans ces solides, des efforts quelconques exerces sur une petite partie de leur surface ou de leur interieur: memoire suivi de notes etendues sur divers points de physique mathematique et d'analyse. P a r i s : ;  Boussinesq, J .  G a u t h i e r - V i U a r s , 1885.  [[133]]  B r i n e r , J . P . a n d T . W . Swanson. U s i n g inherited cosmogenic  3 6  rates o f the C o r d i l l e r a n Ice Sheet. Geology, 2 6 ( l ) : 3 - 6 , 1998.  C 1 t o constrain g l a c i a l erosion [[104]]  B u t k o v i c h , T . R . T h e u l t i m a t e strength of ice. Research paper 11, S I P R E , 1954.  [[49, 51]]  B u t k o v i c h , T . R . C r u s h i n g strength of lake ice. Research paper 15, S I P R E , 1955. Byerlee, J . D . F r i c t i o n o f rocks.  I n Byerlee, J . D . a n d M . W y s s , editors,  [[49, 51]]  Rock Friction and  Earthquake Prediction, C o n t r i b u t i o n s to current research i n geophysics; 6, pages 615-626. B a s e l : B i r k h a u s e r V e r l a g , 1978. R e p r i n t e d from Pure and Applied Geophysics v. 116. [[30]] C a l v i n o , I.  The Baron in the Trees. R a n d o m House, 1959. T r a n s l a t e d b y A . C o l q u h o u n .  [[113]]  Studies on Glaciers Preceded by the Discourse of Neuchdtel. Hafner P u b l i s h i n g C o m p a n y , 1967. T r a n s l a t i o n of Agassiz [1840] w i t h a d d i t i o n a l h i s t o r i c a l notes a n d  C a r o z z i , A . V . , editor. i n t r o d u c t i o n . [[1]]  Reale Accademia dei Lincei, Memorie della Classe di Scienze Fisiche, Matematiche e Naturali, Serie 3, 1 3 : 8 1 - 1 2 3 , 1 8 8 1 / 8 2 .  C e r r u t i , V . Ricerche i n t o r n o a l l ' equilibrio de' corpi elastici i s t r o p i . [[138]] C h a m b e r l a i n , T . C . Recent glacial studies i n G r e e n l a n d . 6:199-220, 1895.  Geological Society of America Bulletin,  [[53, 54, 59]]  C h a n d l e r , H . , editor. Hardness Testing. A S M International, 2 C h a r l e s , R . Static fatigue of glass.  n d  e d i t i o n , 1999.  Journal of Applied Physics, 29:1549-1560, 1958.  Chase, C . G . F l u v i a l l a n d s c u l p t i n g a n d the fractal dimension o f topography. 5(l-2):39-57,1992.  [[29]] [[36]]  Geomorphology,  [[16]]  C l a g u e , J . J . L a t e Quaternary geology a n d geochronology of B r i t i s h C o l u m b i a . P a r t 2: S u m m a r y a n d discussion o f radiocarbon-dated Quaternary history. C a n a d a , 1981.  P a p e r 80-35, G e o l o g i c a l Survey o f  [[103]]  Clague, J . J . R e c o g n i z i n g order i n chaotic sequences o f Q u a t e r n a r y sediments i n the C a n a d i a n Cordillera.  Quaternary International, 68-71:29-38, 2000.  [[104]]  C l a r k , P . U . S u b g l a c i a l sediment dispersal and t i l l c o m p o s i t i o n . 541, 1987.  Journal of Geology, 95(4):527-  [[61, 91]]  C l a r k , P . U . , J . M . L i c c i a r d i , D . R . M a c A y e a l , a n d J . W . Jenson.  N u m e r i c a l reconstruction o f  a soft-bedded L a u r e n t i d e Ice Sheet during the last glacial m a x i m u m . Geology, 24(8):679-682, 1996.  [[7, 76]]  REFERENCES  118  C l a r k e , G . K . C . S u b g l a c i a l t i l l : A p h y s i c a l framework for its properties a n d processes.  Journal  of Geophysical Research, 92(B9):9023-9036, 1987. [[6]] C l a r k e , G . K . C . Lumped-element analysis of subglacial h y d r a u l i c circuits.  Research, 101(B8):17547-17599,1996.  Journal of Geophysical  [[78]]  C l a r k e , G . K . C . a n d E . W . B l a k e . G e o m e t r i c a n d t h e r m a l evolution o f a surge-type glacier i n its quiescent state: T r a p r i d g e G l a c i e r , Y u k o n T e r r i t o r y , C a n a d a , 1969-89. 37(125):158-169, 1991.  Journal of Glaciology,  [[55, 60, 68]]  C o s t i n , L . S. a n d J . J . Mecholsky. T i t l e unknown*. I n  Proceedings of the 24 U.S. Symposium of th  Rock Mechanics, pages 285-394. 1983. [[37, 38, 40]] C o x , S. J . D . a n d C . H . Scholz. A direct measurement of shear fracture energy i n rocks. Geophysical  Research Letters, 12(12):813-816, 1985.  [[38]]  Cuffey, K . a n d R . B . A l l e y . Is erosion b y deforming subglacial sediments significant? ( T o w a r d s till continuity).  Annals of Glaciology, 22:17-24, 1996.  [[28]]  Cuffey, K . M . , H . Conway, A . M . Gades, B . H a l l e t , R . L o r r a i n , J . P . Severinghaus, E . J . Steig, B . V a u g h n , a n d J . W . C . W h i t e . E n t r a i n m e n t at cold glacier beds. 2000.  Geology, 28(4):351-354,  [[59]]  Cuffey, K . M . , H . Conway, B . H a l l e t , A . M . Gades, a n d C . F . R a y m o n d . Interfacial water i n  p o l a r glaciers, and glacier shding at - 1 7 ° C . Geophysical Research Letters, 26(6):751-754,1999. [[59]] D a n s g a a r d , W . , S. J . H o j n s e n , H . B . C l a u s e n , D . Dahljensen, N . S. G u n d e s t r u p , C . U . H a m m e r , C . S. F v i d b e r g , N . P . Steffensen, A . E . Sveinbjornsdottir, J . J o u z e l , a n d G . B o n d . E v i d e n c e for general i n s t a b i l i t y o f past c l i m a t e f r o m a 250-kyr ice-core record. 1993.  Nature, 364(6434):218-220,  [[75]]  Deeley, R . M . a n d P . H . P a r r . 27(157):153-176, 1914.  T h e Hintereis G l a c i e r .  Philosophical Magazine. Series 6,  [[57]]  Der, V . , D . C . Holloway, a n d T . K o b a y a s h i . Techniques for d y n a m i c fracture toughness measurements*. R e p o r t to U . S . N a t i o n a l Science F o u n d a t i o n , P h o t o m e c h . L a b . , M e c h a n i c a l E n g i n e e r i n g D e p a r t m e n t , U n i v e r s i t y o f M a r y l a n d , 1978.  [[38]]  D o n a l d s o n , J . A . T h e D u b a w n t G r o u p , D i s t r i c t s of K e e w a t i n a n d M a c k e n z i e . P a p e r 64-20, G e o l o g i c a l Survey o f C a n a d a , 1965.  [[98, 99]]  D r a k e , L . D . a n d R . L . Shreve. Pressure m e l t i n g a n d regelation o f ice b y r o u n d wires. Proceedings  of the Royal Society of London. Series A, 332:51-83, 1973.  *See preamble on page 114 *See preamble on page 114  [[58]]  REFERENCES  119  Dredge, L . A . Drift carbonate o n the C a n a d i a n Shield. II: C a r b o n a t e dispersal a n d ice-flow patterns i n n o r t h e r n M a n i t o b a .  Canadian Journal of Earth Science, 25(5):783-787, 1987.  [[92]] Dredge, L . A . a n d W . R . C o w a n . Quaternary geology o f the southwestern C a n a d i a n Shield. I n F u l t o n , R . J . , editor,  Quaternary Geology of Canada and Greenland, v o l u m e 1 o f Geology of  North America, chapter 3, pages 214-235. G e o l o g i c a l Survey o f C a n a d a , 1989.  [[92, 93, 95]]  D u b a y a h , R . , E . F . W o o d , a n d D . Lavallee. M u l t i s c a l i n g analysis i n d i s t r i b u t e d m o d e l i n g a n d remote sensing: A n a p p l i c a t i o n using soil moisture. I n Q u a t t r o c h i , D . A . a n d M . F . G o o d c h i l d , editors,  Scale in Remote Sensing and GIS, chapter 5, pages 93-112. C R C Press, 1997.  [[15]]  D y k e , A . S. Landscapes of cold-centred L a t e W i s c o n s i n a n ice caps, A r c t i c C a n a d a . Progress in  Physical Geography, 17(2):223-247, 1993. [[5]] D y k e , A . S., L . A . Dredge, a n d J . - S . V i n c e n t . C o n f i g u r a t i o n a n d dynamics o f the L a u r e n t i d e Ice Sheet d u r i n g the L a t e W i s c o n s i n m a x i m u m .  Geographie physique et Quaternaire, 3 6 ( l - 2 ) : 5 - 1 4 ,  1982. [[3]] D y k e , A . S., T . F . M o r r i s , D . E . C . G r e e n , and J . E n g l a n d . Q u a t e r n a r y geology of P r i n c e o f Wales Island, A r c t i c C a n a d a . M e m o i r 433, G e o l o g i c a l Survey of C a n a d a , 1992.  [[91]]  D y k e , A . S. a n d V . K . Prest. L a t e W i s c o n s i n a n a n d Holocene h i s t o r y of the L a u r e n t i d e Ice Sheet.  Geographie physique et Quaternaire, 41:237-263, 1987. [[3]] D y k e , A . S., J . - S . V i n c e n t , J . T . A n d r e w s , L . A . Dredge, a n d W . R . C o w a n . T h e L a u r e n t i d e Ice . Sheet a n d a n i n t r o d u c t i o n t o the Quaternary geology of the C a n a d i a n Shield. I n F u l t o n , R . J . , editor,  Quaternary Geology of Canada and Greenland, volume 1 o f Geology of North America,  chapter 3, pages 178-189. G e o l o g i c a l Survey o f C a n a d a , 1989.  [[107]]  Echehneyer, K . a n d W . Z h o n g x i a n g . D i r e c t observation of b a s a l shding a n d deformation of b a s a l drift at sub-freezing temperatures.  Journal of Glaciology, 33(113):83-98, 1987.  [[59, 60, 76]]  E l v e r h 0 i , A . , E . S. A n d e r s e n , T . D o k k e n , D . H e b b e l n , R . Spielhagen, J . I. Svendsen, M . S0rflaten, A . R0rnes, M . H a l d , a n d C . F . Forsberg. T h e g r o w t h a n d decay o f the L a t e W e i c h s e l i a n ice sheet i n western S v a l b a r d and adjacent areas based o n provenance studies of m a r i n e sediments.  Quaternary Research, 55:303-316, 1995a. Elverh0i, A . , R . L . Hooke, and A . Solheim.  [[104]] L a t e Cenozoic erosion a n d sediment y i e l d f r o m  the Svalbard-Barents Sea region: Implications for understanding erosion of glacierized basins.  Quaternary Science Reviews, 1 7 ( l - 3 ) : 2 0 9 - 2 4 1 , 1 9 9 8 .  [[5, 105]]  E l v e r h 0 i , A . , J . I. Svendsen, A . S o l h e i m , E . S. A n d e r s e n , J . M i l l i m a n , J . M a n g e r u d , a n d R . L . . H o o k e . L a t e Q u a t e r n a r y sediment y i e l d from the h i g h arctic S v a l b a r d area. 103(1):1-17, 1995b.  Journal of Geology,  [[105]]  E R O S D a t a Center. U S G e o D a t a . U S G S web-site, 2000. h t t p : / / e d c . u s g s . g o v / .  [[11, 12]]  REFERENCES Fischer, U . H .  120  Mechanical Conditions Beneath a Surge-Type Glacier. P h . D . thesis, U n i v e r s i t y of  B r i t i s h C o l u m b i a , 1995. Flowers, G . E .  [[32]]  A Multicomponent Coupled Model of Glacier Hydrology. P h . D . thesis, U n i v e r s i t y  of B r i t i s h C o l u m b i a , 2000.  [[ii, 9, 78, 79, 80, 82, 111, 112, 113]]  Travels Through the Alps of Savoy and Other Parts of the Penine Chain, with Observations on the Phenomena of Glaciers. A . and C . B l a c k , E d i b u r g h , 2 e d i t i o n , 1845.  Forbes, J . D .  n d  [[5]] Freeze, R . A . a n d J . A . Cherry.  Groundwater. Prentice H a l l , E n g l e w o o d Cliffs, N . J . , 1979. [[80]]  F u l t o n , R . J . , compiler. Surficial materials of C a n a d a . M a p D 1 8 8 0 A , G e o l o g i c a l Survey of C a n a d a , 1996.  [[9, 84, 88, 89, 90]]  G e m m e l l , A . M . D . , M . J . Sharp, and D . E . Sugden. Debris f r o m the b a s a l ice of the Agassiz Ice C a p , Ellesmere Island, A r c t i c C a n a d a . 1986.  Earth Surface Processes and Landforms, 11(2):123-130,  [[53]]  G e o p h y s i c a l D a t a Center. G D C T O P O l . Geological Survey of C a n a d a , 1998. Gilbert, G . K .  [[11, 12, 22, 23]]  Glaciers and Glaciation, volume 3 of Harriman Alaska Series. S m i t h s o n i a n  Institute, W a s h i n g t o n , D C , 1910.  [[24]]  Glasser, N . F . , M . J . H a m b r e y , K . R . C r a w f o r d , M . R . B e n n e t t , a n d D . H u d d a r t . T h e s t r u c t u r a l glaciology of K o n g s v e g a n , S v a l b a r d , and its role i n l a n d f o r m genesis. 44(146):136-148, 1998.  [[61]]  G l e n , J . W . T h e creep of p o l y crystalline ice. A, 228:519-538,1955.  Journal of Glaciology,  Proceedings of the Royal Society of London. Series  [[74]]  G o l d t h w a i t , R . P . Development of end moraines i n east-central Baffin Island. 59(6):567-577,1951.  Journal of Geology,  [[59]]  G o o d w i n , I. D . B a s a l ice accretion and debris entrainment w i t h i n the coastal ice m a r g i n , L a w Dome, Antarctica.  Journal of Glaciology, 39(131):157-166, 1993.  [[55]]  G o r a n , S. L . T h e n o r m a l force exerted b y creeping flow o n a s m a l l sphere t o u c h i n g a plane.  Journal of Fluid Mechanics, 41(3):619-625, 1970.  [[26]]  G o w , A . J . , S. E p s t e i n , and W . Sheeny. O n the o r i g i n of stratified debris i n ice cores f r o m the b o t t o m of the A n t a r c t i c Ice Sheet.  Journal of Glaciology, 23(89):185-192, 1979. [[55,56]] .  G o w , A . J . and D . A . Meese. N a t u r e of basal debris i n the G I S P 2 a n d B y r d ice cores a n d its relevance to b e d processes.  Annals of Glaciology, 22:134-140, 1996. [[53, 55]]  G r a d s h t e y n , I. S. and I. M . R y z h i k . 4  t h  e d i t i o n , 1965.  [[136]]  Table of Integrals, Series, and Products. A c a d e m i c Press,  REFERENCES  121  G r a v e n o r , C . P . E r o s i o n by continental ice sheets. 1975.  American Journal of Science, 275(5):595-604,  [[107]]  H a l l e t , B . A theoretical m o d e l of glacial abrasion.  Journal of Glaciology, 23(3):22-25, 1979. [[ii,  25, 26, 28, 30]] H a l l e t , B . G l a c i a l abrasion and shding: T h e i r dependence on the debris concentration i n b a s a l ice.  Annals of Glaciology, 2:23-28, 1981. [[ii, 25, 26, 28, 30, 110]]  H a l l e t , B . G l a c i a l q u a r r y i n g : A simple theoretical m o d e l . Annals of Glaciology, 22:1-8, 1996. [[35, 43, 48, 49, 52]] H a l l e t , B . , L . H u n t e r , and J . B o g e n . Rates of erosion a n d sediment evacuation b y glaciers: A review of field d a t a a n d their i m p l i c a t i o n s .  Global and Planetary Change, 12:213-235, 1996.  [[5, 105, 111]] H a m b r e y , M . J . , M . R . Bennett, J . A . Dowdeswell, N . F . Glasser, a n d D . H u d d a r t . entrainment and transfer i n p o l y t h e r m a l valley glaciers. 1999.  Debris  Journal of Glaciology, 45(149):69-86,  [[53, 60]]  H a m b r e y , M . J . , J . A . Dowdeswell, T . M u r r a y , and P . R . P o r t e r . T h r u s t i n g a n d debris entrainment  Annals of Glaciology, 22:241-248, 1996.  i n a surging glacier: Bakaninbreen, Svalbard.  [[60,  68]] H a r b o r , J . M . N u m e r i c a l m o d e l i n g of the development of U - s h a p e d valleys b y g l a c i a l erosion.  Geological Society of America Bulletin, 104:1364-1375, 1992. [[24]] H a r b o r , J . M . Development of glacial-valley cross sections under conditions of s p a t i a l l y variable resistance to erosion.  Geomorphology, 14:99-107, 1995. [[84]]  H a r t , J . K . A n investigation of the deforming l a y e r / d e b r i s - r i c h basal-ice c o n t i n u u m , i l l u s t r a t e d f r o m three A l a s k a n glaciers. H a r t , J . K . a n d R . I. W a l l e r . Glacier, Alaska, U . S . A .  Journal of Glaciology, 41(139):619-633,1995.  [[60]]  A n investigation of the debris-rich b a s a l ice f r o m W o r t h i n g t o n  Journal of Glaciology, 45(149):54-62, 1999. [[61]]  Hay, W . W . D e t r i t a l sediment fluxes f r o m continents to oceans. Chemical Geology, 145(3-4):287323, 1998.  [[107]]  Hay, W . W . , C . A . Shaw, a n d C . N . W o l d .  Mass-balanced paleogeographic reconstructions.  Geologische Rundschau, 7 8 ( l ) : 2 0 7 - 2 4 2 , 1989.  [[107]]  Henry, J . - P . and J . Paquet. M e c a n i q u e de l a rupture de roches calcitiques.  Bulletin de la Societe  geologique de France, Serie 7, 18(6):1573-1582, 1976. [[40]] H e r r o n , S. a n d C . C . Langway, J r . T h e debris-laden ice at the b o t t o m of the G r e e n l a n d Ice Sheet.  Journal of Glaciology, 23(89):193-207,1979.  [[56]]  122  REFERENCES  Hetenyi, M. A method of solution for the elastic quarter-plane. Journal of Applied Mechanics, 27:289-296, 1960. [[52]] Ho, B. P. and L. G. Leal. Inertial migration of rigid spheres in two-dimensional unidirectional flows. Journal of Fluid Mechanics, 65:365-400, 1974. [[148]] Holm, R. Electric Contacts. Almqvist and Wiksells, Stockholm, 1946. [[25]] Hooke, R. L. Flow near the margin of the Barnes Ice Cap, and the development of ice-cored moraines. Geological Society of America Bulletin, 84:3929-3948, 1973. [[59]]  Hooke, R. L. and N. R. Iverson. Grain-size distribution in deforming subglacial tills: Role of grain fracture. Geology, 23(l):57-60, 1995. [[32]] Hubbard, B. and M. Sharp. Basal ice formation and deformation: A review. Progress in Physical Geography, 13(4):529-558, 1989. [[55]] Hubbard, B. and M. Sharp. Weertman regelation, multiple refreezing events and the isotopic evolution of the basal ice layer. Journal of Glaciology, 39(132):275-291, 1993. [[56, 57]] Hubbard, B. and M. Sharp. Basal ice facies and their formation in the western Alps. Arctic and Alpine Research, 27(4):301-310, 1995. [[57, 61]]  Hubbard, B., M. Sharp, and W. J. Lawson. On the sedimentological character of Alpine basal ice facies. Annals of Glaciology, 22:187-193, 1996. [[32, 33]] Hudleston, P. J. Recumbent folding in the base of the Barnes Ice Cap, Baffin Island, Northwest Territories, Canada. Geological Society of America Bulletin, 87(12):1684-1692, 1976. [[61,  68]] Ingraffea, A. R. Title unknown*. In Proceedings of the 22 U.S. Symposium of Rock Mechanics, pages 186-191. MIT, Cambridge, MA, 1981. [[38]] til  Ingraffea, A. R. and P. A. Wawrzynek. FRANC2D: A case study in transfer of software technology. In Colville, A. and A. Amde, editors, Research Transformed into Practice: Implementations of  NSF Research, pages 233-344. ASCE Press, New York, 1995. [[41]] Ingraffea, A. R., P. A. Wawrzynek, and B. J. Carter. FRANC2D. Cornell Fracture Group web-site, 1997. http://www.cfg.cornell.edu/software/FRANC2D.html. [[41]] Iverson, N. R. Laboratory simulations of glacial abrasion: Comparisons with theory. Journal of Glaciology, 36(124):304-313, 1990. [[25, 28, 30]] Iverson, N. R. Potential effects of subglacial water-pressure fluctuations on quarrying. Journal of Glaciology, 37(125):27-36, 1991. [[35, 43, 48]] Iverson, N. R. Regelation of ice through debris at glacier beds: Implications for sediment transport. Geology, 21(6):559-562, 1993. [[58]] tSee preamble on page 114  REFERENCES  123  Iverson, N . R . Sediment entrainment by a soft-bedded glacier: A m o d e l based o n regelation i n t o the bed.  Earth Surface Processes and Landforms, 25(8):881-893, 2000. [[63]]  Iverson, N . R . a n d D . J . Semmens. Intrusion of ice porous m e d i a b y regelation: A m e c h a n i s m of sediment entrainment by glaciers. 1995.  Journal of Geophysical Research, 100(B7):10219-10230,  [[ii, 58, 112]]  Iverson, N . R . a n d R . Souchez. Isotopic signature of debris-rich ice formed by regelation i n t o a  subglacial sediment bed. Geophysical Research Letters, 23(10):1151-1154, 1996. Jaeger, J . C . a n d N . G . W . C o o k . HaU L t d , 3  r d  e d i t i o n , 1979.  [[58]]  Fundamentals of Rock Mechanics, chapter 3. C h a p m a n a n d  [[30, 38]]  Jansen, E . , T . F r o n v a l , F . R a c k , and J . E . T . C h a n n e l l . Pliocene-Pleistocene ice rafting h i s t o r y a n d cyclicity i n the N o r d i c seas during the last 3.5 M y r .  Paleoceanography, 15(6):709-721, 2000.  [[105]] Jansen, E . and J . S j 0 h o l m . R e c o n s t r u c t i o n of g l a c i a t i o n over the past 6 M y r f r o m ice-borne deposits i n the N o r w e g i a n Sea.  Nature, 349(6310):600-603, 1991.  [[105]]  Jenson, J . W . , P . U . C l a r k , D . R . M a c A y e a l , C . H o , and J . C . V e l a . N u m e r i c a l m o d e l i n g of advective transport of saturated deforming sediment beneath the L a k e M i c h i g a n L o b e , L a u r e n t i d e Ice Sheet.  Geomorphology, 14:157-166, 1995.  [[6, 7]]  Jenson, J . W . , D . R . M a c A y e a l , P . U . C l a r k , C . L . H o , a n d J . C . V e l a .  Numerical modeling  of subglacial sediment deformation: Implications for the behavior of the L a k e M i c h i g a n L o b e , L a u r e n t i d e Ice Sheet.  Journal of Geophysical Research, 101(B4):8717-8728, 1996.  Jenssen, D . A three-dimensional p o l a r ice sheet m o d e l .  [[7, 76]]  Journal of Glaciology, 18:373-389,1977.  [[75]] J o u z e l , J . a n d R . A . Souchez. Melting-refreezing at the glacier sole a n d the isotopic c o m p o s i t i o n of the ice.  Journal of Glaciology, 28(98):35-42, 1982. [[55]]  K a c h a n o v , M . a n d J . P . Laures. Three-dimensional problems of strongly i n t e r a c t i n g a r b i t r a r i l y l o c a t e d penny-shaped cracks.  International Journal of Fracture, 41:289-331, 1989.  K a h a n e r , D . , C . M o l e r , and S. N a s h .  [[36]]  Numerical Methods and Software. P r e n t i c e H a l l , 1989.  [[70]] K a l n a y , E . , M . K a n a m i t s u , R . K i s t l e r , W . C o l l i n s , D . Deaven, L . G a n d i n , M . Iredell, S. Saha, G . W h i t e , J . Woollen, Y . Zhu, M . Chelliah, W . Ebisuzaki, W . Higgins, J . Janowiak, K . C . M o , C . R o p e l e w s k i , J . W a n g , A . L e e t m a a , R . R e y n o l d s , R . Jenne, and D . Joseph. T h e N C E P / N C A R 40-year reanalysis project. 1996.  Bulletin of the American Meteorological Society, 77(3):437-471,  [[75]]  K a m b , B . G l a c i e r surge m e c h a n i s m based o n l i n k e d cavity configuration of the b a s a l water conduit system.  Journal of Geophysical Research, 92(B10):16585-16595, 1987. [[78]]  ;  REFERENCES  124  K a r r o w , P . F . Carbonates, granulometry, and color of tills on the south-central C a n a d i a n Shield a n d their i m p l i c a t i o n s for stratigraphy and r a d i o c a r b o n dating.  Boreas, 21:379-391,1992. [[92,  93]] K a r r o w , P . F . a n d R . S. Geddes. Drift carbonate on the C a n a d i a n Shield.  Earth Science, 24(2):365-369, 1987.  Canadian Journal of  [[93]]  K a s z y c k i , C . A . a n d W . W . Shilts. G l a c i a l erosion of the C a n a d i a n S h i e l d — c a l c u l a t i o n of average depths. Technical record T R - 1 0 6 , A t o m i c E n e r g y of C a n a d a L i m i t e d , 1980.  [[99, 108]]  K a u r a n n e , L . K . H i s t o r y of soil geochemical prospecting. I n G o v e t t , G . J . S., editor, Regolith Ex-  ploration Geochemistry in Arctic and Temperate Terrains, volume 5 of Handbook of Exploration Geochemistry, chapter 1, pages 3-6. Elsevier, 1992. [[113]] Kavanaugh, J . L . Hydromechanical Behavior of a Surge-Type Glacier: Trapridge Glacier, Yukon Territory, Canada. P h . D . thesis, U n i v e r s i t y of B r i t i s h C o l u m b i a , 2000. [[7,79]] Kerr, F . A . Glaciation i n northern British Columbia.  Series 3, 28:17-31, 1934.  Transactions of the Royal Society of Canada,  [[103]]  K l e m a n , J . Preservation of landforms under ice sheets and ice caps. 1994.  Geomorphology, 9 ( l ) : 1 9 - 3 2 ,  [[5]]  K n i g h t , P . G . Stacking of basal debris layers without b u l k freezing-on: Isotopic evidence f r o m west G r e e n l a n d .  Journal of Glaciology, 35(120):214-216,1989.  [[61]]  K n i g h t , P . G . Two-facies interpretation of the basal layer of the G r e e n l a n d ice sheet contributes ; to a unified m o d e l of b a s a l ice f o r m a t i o n . Geology, 2 2 ( l l ) : 9 7 1 - 9 7 4 , 1 9 9 4 . Knight, P. G .  T h e basal ice layer of glaciers a n d ice sheets.  16(9):975-993, 1997.  [[33,56,57,61]]  Quaternary Science Reviews,  [[53, 55]]  K n i g h t , P . G . a n d D . A . K n i g h t . G l a c i e r shding, regelation water flow a n d development of b a s a l ice.  Journal of Glaciology, 40(136):600-601, 1994.  [[57]]  K n i g h t , P . G . a n d D . A . K n i g h t . E x p e r i m e n t a l observations of subglacial debris entrainment i n t o the v e i n network of p o l y c r y s t a l h n e ice.  Glacial Geology and Geomorphology, 1999.  http://boris.qub.ac.uk/ggg/papers/full/1999/rp051999/rp05.html. Kobranova, V . N . Kuznetsov.  [[57]]  Petrophysics. M i r Publishers, 1989. T r a n s l a t e d f r o m the R u s s i a n b y V . V .  [[30]]  K r u m b e i n , W . C . Sediments and exponential curves.  Journal of Geology, 45(6):577-601, 1937.  [[61]] L a i n e , E . P . N e w evidence f r o m beneath western N o r t h A t l a n t i c for the d e p t h of g l a c i a l erosion in Greenland and N o r t h America.  Quaternary Research, 14:188-198, 1980. [[5, 107]]  L a i n e , E . P . R e p l y to A n d r e w ' s comment.  Quaternary Research, 17:125-127, 1982.  [[107]]  REFERENCES  125  Lavallee, D . , S. Lovejoy, D . Schertzer, and P . Ladoy. N o n l i n e a r v a r i a b i l i t y of landscape topography: M u l t i f r a c t a l analysis a n d s i m u l a t i o n . I n L a m , N . S . - N . and L . de C o l a , editors, Fractals in  Geography, chapter 8, pages 158-192. P R T P r e n t i c e - H a l l , 1993. [[16]] Lawn, B . R . and T . R . Wilshaw.  Fracture of Brittle Solids. C a m b r i d g e U n i v e r s i t y Press, 1975.  [[35]] L a w s o n , D . E . Sedimentological analysis of the W e s t e r n terminus region of the M a t a n u s k a G l a c i e r . Technical report 79-9, C R R E L , 1979.  [[33]]  L a w s o n , D . E . a n d J . B . K u l l a . A n oxygen isotope investigation of the o r i g i n of the b a s a l zone of the M a t a n u s k a G l a c i e r , A l a s k a .  Journal of Geology, 86:673-685, 1978.  [[53]]  L a w s o n , D . E . , J . C . Strasser, E . B . E v e n s o n , R . B . A l l e y , G . J . L a r s o n , a n d S. A . A r c o n e . G l a c i o h y d r a u l i c supercooling: A freeze-on m e c h a n i s m to create stratified, debris-rich b a s a l ice: I. F i e l d evidence.  Journal of Glaciology, 44(148):547-561, 1998. [[57]]  Legate, D . R . and C . J . W i U m o t t . M e a n seasonal and s p a t i a l v a r i a b i l i t y i n gauge-corrected g l o b a l precipitation.  International Journal of Climatology, 10(2):111-127, 1990.  L i d m a r - B e r g s t r o m , K . A l o n g - t e r m perspective o n glacial erosion.  Landforms, 22(3):297-306,1997. L i n , S.-J. a n d R . B . R o o d .  [[75]]  Earth Surface Processes and  [[107]]  M u l t i d i m e n s i o n a l flux-form s e m i - L a g r a n g i a n t r a n s p o r t  schemes.  Monthly Weather Review, 124(9):2046-2070, 1996. [[71]] L i n d s t o m , E . A r e roches moutonnees m a i n l y p e r i g l a c i a l forms? Geografiska Annaler, 7 0 A ( 4 ) : 3 2 3 331, 1988.  [[35]]  L l i b o u t r y , L . C o n t r i b u t i o n a l a theorie d u frottement  d u glacier sur son l i t .  Comptes Rendus  Hebdomadaires des Seances de I'Academie des Science, 247(3):318-320, 1958.  [[47]]  L l i b o u t r y , L . G e n e r a l theory of subglacial c a v i t a t i o n a n d shding of temperate glaciers.  Journal  of Glaciology, 7(49):21-58, 1968. [[47]] L l i b o u t r y , L . Internal m e l t i n g and ice accretion at the b o t t o m of temperate glaciers.  Glaciology, 39(131):50-64, 1993.  Journal of  [[56, 57]]  L o r r a i n , R . D . , R . A . Souchez, and J . - L . T i s o n . Characteristics of b a s a l ice f r o m two outlet glaciers i n the C a n a d i a n A r c t i c — i m p l i c a t i o n s for glacier erosion. I n 8 1 - 1 B , pages 137-144. G e o l o g i c a l Survey of C a n a d a , 1981.  Current Research, Part B, P a p e r [[61]]  M a c G r e g o r , K . R . , R . S. A n d e r s o n , S. P . A n d e r s o n , and E . D . W a d d i n g t o n . N u m e r i c a l simulations of glacial-valley l o n g i t u d i n a l profile evolution. Geology, 28(11):1031-1034, 2000.  [[11,24]]  Mahaffy, M . W . A three-dimensional n u m e r i c a l m o d e l of ice sheets: Tests o n the Barnes Ice C a p , Northwest T e r r i t o r i e s .  Journal of Geophysical Research, 81:1059-1066, 1976. [[74]]  REFERENCES  126  M a r k , D . M . a n d P . B . A r o n s o n . Scale-dependent fractal dimensions o f topographic surfaces: A n e m p i r i c a l investigation, w i t h apphcations i n geomorphology a n d computer m a p p i n g .  Mathe-  matical Geology, 16, 1984. [[16]] M a r s h a l l , S. J .  Modelling Laurentide Ice Stream Thermodynamics. P h . D . thesis, U n i v e r s i t y o f  B r i t i s h C o l u m b i a , 1996.  [[ii, 9, 74, 111]]  M a r s h a l l , S. J . a n d G . K . C . C l a r k e . A c o n t i n u u m m i x t u r e m o d e l o f ice s t r e a m thermomechanics i n the L a u r e n t i d e Ice Sheet. 1. Theory. 1997a.  Journal of Geophysical Research, 102(B9):20599-20613,  [[ii, 7, 74, 76, 111]]  M a r s h a l l , S. J . a n d G . K . C . C l a r k e . A c o n t i n u u m m i x t u r e m o d e l of ice s t r e a m thermomechanics i n the L a u r e n t i d e Ice Sheet. 2. A p p l i c a t i o n to the H u d s o n Strait Ice S t r e a m .  Research, 102(B9):20615-20637, 1997b.  Journal of Geophysical  [[ii, 7, 74, 76, 111]]  M a r s h a l l , S. J . a n d G . K . C . C l a r k e . M o d e l i n g N o r t h A m e r i c a n freshwater runoff t h r o u g h the last g l a c i a l cycle.  Quaternary Research, 52(3):300-315, 1999. [[82]]  M a r s h a l l , S. J . , G . K . C . C l a r k e , A . S. D y k e , a n d D . A . F i s h e r . Topographic a n d geologic controls on fast flow i n the Laurentide a n d C o r d i l l e r a n Ice Sheets. 101(B8):17827-17840, 1996.  Journal of Geophysical Research,  [[76]]  M a r s h a l l , S. J . , L . Tarasov, G . K . C . C l a r k e , a n d W . R . P e l t i e r . G l a c i o l o g i c a l reconstruction o f the L a u r e n t i d e Ice Sheet: P h y s i c a l processes a n d m o d e l l i n g challenges.  Earth Science, 37(5):769-793, 2000.  Canadian Journal of  [[ii, 7, 74, 75, 76, 111]]  M c F a r l a n e , N . A . , G . J . B o e r , J . - P . B l a n c h e t , a n d M . L a z a r e . T h e C a n a d i a n C l i m a t e Centre second generation general c i r c u l a t i o n m o d e l a n d its e q u i h b r i u m c l i m a t e . 5(10):1013-1044,1992.  Journal of Climate,  [[75]]  M e r e d i t h , P . G . a n d B . K . A t k i n s o n . Stress corrosion a n d acoustic emission d u r i n g tensile crack p r o p a g a t i o n i n W h i n S i l l D o l e r i t e a n d other basic rocks.  Geophysical Journal of the Royal  Astronomical Society, 75(1):1-21, 1983. [[38, 40]] Meredith, P. G . and B . K . Atkinson. high-temperature  Fracture toughness a n d s u b c r i t i c a l crack g r o w t h d u r i n g  tensile deformation of W e s t e r l y G r a n i t e a n d black gabbro.  Physics of the  Earth and Planetary Interiors, 39(1):33-51, 1985. [[38]] Meredith, P . G . , B . K . Atkinson, and N . B . Hillman.  Progress in Experimental Petrology*, pages  228-232. N u m b e r 25 i n N E R C P u b s . Ser. D . N a t u r a l E n v i r o n m e n t a l Research C o u n c i l , S w i n d o n , E n g l a n d , 1984. [[37, 38]] M e t c a l f , R . C . E n e r g y dissipation during subglacial abrasion at N i s q u a l l y G l a c i e r , W a s h i n g t o n U.S.A.  Journal of Glaciology, 23(89):233-246, 1979. [[30]]  tSee preamble on page 114  REFERENCES  127  M o r l a n d , L . W . a n d G . S. B o u l t o n . Stress i n a n elastic h u m p : T h e effects o f glacier flow over elastic bedrock.  :  Proceedings of the Royal Society of London. Series A, 344(1637):157-173,1975.  [[35, 41, 45]] M o r l a n d , L . W . and E . M . M o r r i s . Stress i n a n elastic bedrock h u m p due t o glacier flow.  of Glaciology, 18(78):67-75, 1977. Mott, B . W .  Journal  [[35, 41, 45]]  Micro-indentation Hardness Testing. B u t t e r w o r t h s Scientific P u b . , L o n d o n , 1956.  [[29]] M u l l e r , R . T h e performance of classical versus m o d e r n finite-volume advection schemes for atmo-  Monthly Weather Review, 120(7):1407-1415,  spheric m o d e l i n g i n a one-dimensional test-bed. 1992.  [[71]]  Nesje, A . , S. 0 . D a h l , V . V a l e n , a n d J . 0 v s t e d a l . Q u a t e r n a r y erosion i n the Sognefjord drainage b a s i n , western Norway.  Geomorphology, 5:511-520, 1992. [[104, 105]]  Nesje, A . a n d I. M . W h i l l a n s . E r o s i o n of Sognefjord, Norway.  Geomorphology, 9:33-45,1994. [[5,  104, 105]] NOAA.  D a t a A n n o u n c e m e n t 8 8 - M G G - 0 2 , D i g i t a l relief of the surface o f the E a r t h . N a t i o n a l  G e o p h y s i c a l D a t a Center, B o u l d e r , C o l o r a d o , 1988.  [[22, 23]]  N y e , J . F . T h e flow law of ice f r o m measurements i n glacier tunnels, l a b o r a t o r y experiments a n d the jungfraufirn borehole experiment. 219(1139):477-489, 1953. Nye, J . F .  W a t e r flow i n glaciers:  17(76):181-207,1976.  Proceedings of the Royal Society of London. Series A,  [[47]] J o k u l h l a u p s , tunnels a n d veins.  [[78]]  Oerlemans, J . N u m e r i c a l experiments on large-scale glacial erosion.  und glazialgeologie, 20:107-126, 1985.  Zeitschrift fur gletscherkunde  [[11, 24]]  O h n a k a , M . F r i c t i o n a l characteristics of t y p i c a l rocks. 23:87-112,1975.  Journal of Glaciology,  Journal of the Physics of the Earth,  [[30]]  O ' N e i l l , M . E . A sphere i n contact w i t h a plane w a l l i n a slow linear shear flow.  Engineering Science, 23(11):1293-1298,1968.  Chemical  [[27]]  O r n s t e i n , L . S. O n the m o t i o n of a m e t a l wire t h r o u g h a l u m p of ice. Proceedings of the Koninklijke  Nederlandse Akademie van Wetenschappen te Amsterdam. Part 2, 8:653-659, 1906.  [[57]]  P a r e n t , M . , S. J . P a r a d i s , and A . D o i r o n . Palimpsest g l a c i a l dispersal trains a n d their significance for drift prospecting. P a t e r s o n , W . S. B . 65, 77]]  Journal of Geochemical Exploration, 56:123-140, 1996.  The Physics of Glaciers. Elsevier Science L t d , 3  r d  [[61]]  e d i t i o n , 1994.  [[26, 51,  REFERENCES  128  P a t e r s o n , W . S. B . , R . M . K o e r n e r , D . Fisher, S. J . Johnsen, H . B . C l a u s e n , W . D a n s g a a r d , P . B u c h e r , a n d H . Oeschger. A n oxygen-isotope climate record from the D e v o n I s l a n d Ice C a p , arctic C a n a d a . Payne, A . J .  Nature, 266(5602):508-511, 1977. [[61]] Journal of Geophysical  L i m i t cycles i n the basal t h e r m a l regime of ice sheets.  Research, 100(B3):4249-4263, 1995. P a y n e , A . J . and D . J . B a l d w i n . i n t e r c o m p a r i s o n exercise.  [[75, 76]]  A n a l y s i s of ice-flow instabilities identified i n the E I S M I N T  Annals of Glaciology, 30:204-210, 2000. [[76]]  P a y n e , A . J . , P . Huybrechts, A . A b e - O u c h i , R . C a l o v , J . L . Fastook, R . G r e v e , S. J . M a r s h a l l , I. M a r s i a t , C . R i t z , L . Tarasov, and M . P . A . Thomassen. Results f r o m the E I S M I N T m o d e l i n t e r c o m p a r i s o n : T h e effects of t h e r m o m e c h a m c a l c o u p l i n g . 238, 2000.  Journal of Glaciology, 46(153):227-  [[76]]  P e l t i e r , W . R . Ice-age paleotopography. Science, 265(5197):195-201, 1994.  [[4, 76]]  P e l t i e r , W . R . M a n t l e viscosity and ice-age ice sheet topography. Science, 273(5280):1359-1364, 1996.  [[4, 76]]  P h i l i p , J . R . T h e r m a l fields during regelation. 1980.  Cold Regions Science and Technology, 3:193-203,  [[58]]  Piotrowski, J . A .  Subglacial hydrology i n north-western G e r m a n y d u r i n g the last glaciation:,  G r o u n d w a t e r flow, t u n n e l valleys and h y d r o l o g i c a l cycles. 16(2):169-185, 1997.  Quaternary Science Reviews,  [[80, 81]]  P r a t h e r , M . J . N u m e r i c a l advection by conservation of second-order m o m e n t s .  Journal of Geo-  physical Research, 91(D6):6671-6681, 1986. [[viii, 71, 72, 73, 112]] P r e s t , V . K . a n d E . Nielsen. T h e Laurentide Ice Sheet and long-distance t r a n s p o r t . I n K u j a n suu, R . a n d M . Saarnisto, editors, G e o l o g i c a l Survey of F i n l a n d , 1987.  INQUA Till Symposium, Special P a p e r 3, pages 9 1 - 1 0 1 . [[6]]  P u r a n e n , R . M o d e l l i n g of glacial transport of b a s a l tills i n F i n l a n d . R e p o r t of Investigation 81, G e o l o g i c a l Survey of F i n l a n d , 1988. Rabinowicz, E .  [[24]]  Friction and Wear of Materials. J o h n W i l e y & Sons, 2  n d  e d i t i o n , 1995.  [[24, 25,  30]] Reames, F . M . a n d T . H . Zapotocny. Inert trace constituent i n s i g m a a n d h y b r i d isentropic-sigma models. P a r t I: N i n e advection algorithms.  Monthly Weather Review, 127(2):173-187, 1999a.  Reames, F . M . a n d T . H . Zapotocny. Inert trace constituent i n s i g m a a n d h y b r i d isentropic-sigma models. P a r t II: Twelve semi-Lagrangian algorithms. 1999b.  [[71]]  Monthly Weather Review, 127(2):188-200,  REFERENCES  129  R e y n o l d s , 0 . D i l a t a n c y of m e d i a composed of r i g i d particles i n contact.  Philosophical Magazine.  Series 5, 20:469-481, 1885. [[69]] R i c e , J . R . T h e r m o d y n a m i c s of the quasi-static g r o w t h of Griffith cracks.  Journal of the mechanics  and physics of solids, 26:61-78, 1978. [[36]] R i i s , F . a n d W . Fjeldskaar. O n the magnitude of the L a t e T e r t i a r y a n d Q u a t e r n a r y erosion a n d , its significance for the uplift of Scandinavia and the Barents Sea. I n L a r s e n , R . M . , H . B r e k k e , B . T . L a r s e n , and E . Talleraas, editors,  Structural and Tectonic Modelling and its Application  to Petroleum Geology, 1, pages 163-185. Elsevier, 1992. Rothlisberger, H .  [[105]]  W a t e r pressure i n i n t r a - and subglacial channels.  l l ( 6 2 ) : 1 7 7 - 2 0 3 , 1972.  Journal of Glaciology,  [[78]]  Scholz, C . H . W e a r and gouge f o r m a t i o n i n b r i t t l e faulting. Geology, 15:493-495, 1987.  [[30]]  Segall, P . Rate-dependent extensional deformation resulting f r o m crack g r o w t h i n rock.  Journal  of Geophysical Research, 89(B6):4185-4195, 1984. Shackleton, N . J . , A . Berger, and W . R . P e l t i e r .  [[36]] A n alternative a s t r o n o m i c a l c a l i b r a t i o n of  the lower Pleistocene timescale based on O D P site 677.  Transactions of the Royal Society of  Edinburgh - Earth Sciences, 81:251-261, 1990. [[2]] Sharp, M . , J . J o u z e l , B . H u b b a r d , and W . L a w s o n . T h e character, structure a n d o r i g i n of the b a s a l ice layer of a surge-type glacier.  Journal of Glaciology, 40(135):327-340,1994.  Sharp, M . , W . L a w s o n , a n d R . S. A n d e r s o n . Tectonic processes i n a surge-type glacier.  [[57, 61]]  Journal  of Structural Geology, 10(5):499-515, 1988. [[60]] Shaw, J . A meltwater m o d e l for Laurentide subglacial landscapes. I n M c C a n n , S. B . a n d D . . C . F o r d , editors,  Geomorphology sans frontieres, pages 181-239. W i l e y , 1996. [[79]]  Shaw, J . , R . B . R a i n s , R . E y t o n , a n d L . Weissling. Laurentide subglacial outburst floods: L a n d f o r m evidence f r o m d i g i t a l elevation models. 1168, 1996.  Canadian Journal of Earth Science, 8(33):1154-  [[79]]  Shilts, W . W . F l o w patterns i n the central N o r t h A m e r i c a n ice sheet.  Nature, 286:213-218,1980.  [[91, 92, 98]] Shilts, W . W . , C . M . C u n n i n g h a m , a n d C . A . K a s z y c k i . K e e w a t i n Ice Sheet—re-evaluation o f the t r a d i t i o n a l concept of the Laurentide Ice Sheet. Geology, 7:537-541, 1979. Shreve, R . L . G l a c i e r sliding at subfreezing temperatures. 1984.  [[98,99,100]]  Journal of Glaciology, 30(106):341-347,  [[59, 76]]  Souchez, R . , M . L e m m e n s , a n d J . C h a p p e l l a z . F l o w - i n d u c e d m i x i n g i n the G R I P b a s a l ice deduced  from the CO2 a n d C H records. Geophysical Research Letters, 2 2 ( l ) : 4 1 - 4 4 , 1 9 9 5 . 4  [[68]]  ,  REFERENCES  130  Souchez, R . , R . L o r r a i n , J . L . T i s o n , a n d J . J o u z e l . Co-isotopic signature o f t w o mechanisms o f basal-ice f o r m a t i o n i n A r c t i c outlet glaciers. Annals of Glaciology, 10:163-166, 1988.  [[57]]  Souchez, R . A . T h e f o r m a t i o n o f shear moraines: A n example f r o m south V i c t o r i a L a n d , A n t a r c t i c a . Journal of Glaciology, 6(48):837-843, 1967.  [[59]]  Souchez, R . A . a n d J . J o u z e l . O n the isotopic c o m p o s i t i o n i n SD a n d 8 0 18  freezing. Journal of Glaciology, 30(106):369-372, 1984.  o f water a n d ice d u r i n g  [[55]]  Souchez, R . A . , J . - L . T i s o n , R . L o r r a i n , M . L e m m e n s , L . Janssens, M . Stievenard, J . J o u z e l , A . Sveinbjornsdottir, a n d S. J . Johnsen. Stable isotopes i n the basal silty ice preserved i n the G r e e n l a n d Ice Sheet at S u m m i t : E n v i r o n m e n t a l i m p l i c a t i o n s . Geophysical Research Letters, 21(8):693-696,1994.  [[55]]  Strasser, J . C . , D . E . L a w s o n , G . J . L a r s o n , E . B . E v e n s o n , a n d R . B . A l l e y . P r e l i m i n a r y results of t r i t i u m analyses i n basal ice, M a t a n u s k a G l a c i e r , A l a s k a , U . S . A . : E v i d e n c e for subglacial ice accretion. Annals of Glaciology, 22:126-133, 1996.  [[53]]  Sugden, D . E . A case against deep erosion of shield b y ice sheets. Geology, 4(10):580-582, 1976. [[107]] Sugden, D . E . G l a c i a l erosion b y the Laurentide Ice Sheet. Journal of Glaciology, 20(83):367-391, 1978.  [[5]]  Sugden, D . E . , P . G . K n i g h t , N . Livesey, R . D . L o r r a i n , R . A . Souchez, J . - L . T i s o n , a n d J . J o u z e l . E v i d e n c e for t w o zones o f debris entrainment beneath the G r e e n l a n d Ice Sheet. 328(6127):238-241, 1987.  Nature,  [[56, 57]]  Swanson, P . L . Stress Corrosion Cracking in Westerly Granite: An Examination by the Double Torsion Method). M a s t e r ' s thesis, U n i v e r s i t y of C o l o r a d o , B o u l d e r , 1980. [[38, 40]] Swanson, P . L . S u b c r i t i c a l crack g r o w t h a n d other time- a n d environment- dependent behavior i n crustal rocks. Journal of Geophysical Research, 89(B6):4137-4152, 1984.  [[39]]  T a y l o r , K . , C . U . H a m m e r , R . B . A l l e y , H . B . Clausen, D . Dahl-Jensen, A . J . G o w , N . S. G u n d e strup, J . K i p f s t u l , J . C . M o o r e , a n d E . D . W a d d i n g t o n . E l e c t r i c a l c o n d u c t i v i t y measurements f r o m the G I S P 2 a n d G R I P G r e e n l a n d ice cores. Nature, 366(6455):549-552, 1993. Thomson, J .  [[61, 68]]  O n recent theories a n d experiments regarding ice at or near its m e l t i n g - p o i n t .  Proceedings of the Royal Society of London, 10:152-160, 1859/60. Tison, J . - L . , J . - R . Petit, J . - M . Barnola, and W . C . Mahaney.  [[57]]  Debris entrainment at the i c e -  bedrock interface i n sub-freezing temperature conditions (Terre A d e l i e , A n t a r c t i c a ) . Journal of Glaciology, 39(132):303-315, 1993. t See preamble on page 114  [[60]]  REFERENCES  131  T i s o n , J . - L . , R . Souchez, and R . L o r r a i n . i n b a s a l ice: Present-day examples. 1989.  O n the i n c o r p o r a t i o n of unconsolidated sediments  Zeitschrift fur Geomorphologie. Supplement, 72:173-183,  [[56, 60, 61]]  TuUey, M . J . C .  Numerical Modelling of Erosion and Deposition Beneath Quaternary Ice Sheets.  P h . D . thesis, U n i v e r s i t y of C a m b r i d g e , 1995.  [[11, 24, 32, 61]]  van der M e e r , J . J . M . P a r t i c l e and aggregate m o b i l i t y i n t i l l : M i c r o s c o p i c evidence of subglacial processes.  Quaternary Science Reviews, 16:827-831, 1997.  [[143]]  van Weert, F . H . A . , K . v a n Gijssel, A . Leijnse, and G . S. B o u l t o n . T h e effects of Pleistocene glaciations o n the geohydrological system of Northwest E u r o p e . 4):137-159, 1997.  Journal of Hydrology, 1 9 5 ( 1 -  [[80, 81]]  V e t t o r e t t i , G . , W . R . P e l t i e r , a n d N . A . M c F a r l a n d . G l o b a l water balance and atmospheric water vapour transport at Last G l a c i a l m a x i m u m . 2000.  Canadian Journal of Earth Science, 37(5):695-723,  [[75]]  W a l d e r , J . S. S t a b i l i t y of sheet flow of water beneath temperate glaciers a n d i m p l i c a t i o n s for glacier surging.  Journal of Glaciology, 28(99):273-293, 1982.  W a l d e r , J . S. H y d r a u l i c s of subglacial cavities.  [[78]]  Journal of Glaciology, 32(112):439-445, 1986.  [[47, 78]] W a l l e r , R . I., J . K . H a r t , and P . G . K n i g h t . v a r i a b i l i t y i n stratified debris-rich basal ice.  T h e influence of tectonic deformation o n facies  Quaternary Science Reviews, 19(8):775-786, 2000.  [[61,68]] Watts, P. A .  Inclusions in Ice. P h . D . thesis, U n i v e r s i t y of B r i s t o l , 1974.  [[26]]  W a z a , T . , K . K u r i t a , a n d H . M i z u t a n i . T h e effect of water o n the s u b c r i t i c a l crack g r o w t h i n  Tectonophysics, 67:25-34, 1980.  silicate rocks.  W e e r t m a n , J . O n the shding of glaciers.  [[40]]  Journal of Glaciology, 3(21):33-38, 1957. [[56, 57]]  W e e r t m a n , J . M e c h a n i s m for the f o r m a t i o n of inner moraines found near the edge of cold ice caps a n d ice sheets.  Journal of Glaciology, 3(30):965-978, 1961.  W e e r t m a n , J . T h e theory of glacier shding.  [[56,59]]  Journal of Glaciology, 5(39):287-303, 1964.  [[26]]  W e e r t m a n , J . Diffusion law for the dispersion of h a r d particles i n a n ice m a t r i x t h a t undergoes simple shear deformation. Weertman, J .  Journal of Glaciology, 7(50):161-165, 1968. [[69, 112]]  G e n e r a l theory of water flow at the base of a glacier or ice sheet.  Geophysics and Space Science, 1 0 ( l ) : 2 8 7 - 3 3 3 , 1972.  Reviews of  [[78]]  W e e r t m a n , J . a n d G . E . Birchfield. B a s a l water film, basal water pressure, a n d v e l o c i t y of t r a v e l l i n g waves on glaciers.  Journal of Glaciology, 29(101):20-27,1983.  [[78]]  REFERENCES  132  W h e e l e r , J. 0 . , P . F . Hoffman, K . D . C a r d , A . D a v i d s o n , B . V . Sanford, A . V . O k u l i t c h , a n d W . R . R o e s t , compilers. Geological m a p of C a n a d a . M a p D 1 8 6 0 A , G e o l o g i c a l Survey o f C a n a d a , 1997. [[9, 84, 85, 86, 87, 91, 93, 98, 99, 110, 112]] W h i t e , W . A . Deep erosion by continental ice sheets. 83:1037-1056,1972.  Geological Society of America Bulletin,  [[107]]  W h i t e , W . A . M o r e o n deep glacial erosion b y continental ice sheets a n d their tongues o f dist r i b u t a r y ice.  Quaternary Research, 30:137-150, 1988.  [[107]]  W r i g h t , G . M . G e o l o g i c a l notes o n eastern D i s t r i c t of M a c k e n z i e Northwest Territories. P a p e r 56-10, G e o l o g i c a l Survey o f C a n a d a , 1957. Xiaozhen, G . and L . Xiaoen.  [[99]]  A p p l i c a t i o n a n d n u m e r i c a l experiments w i t h a highly-accurate  advection scheme i n a regional transfer m o d e l . Meteorology and Atmospheric Physics, 6 6 ( 3 4):131-142, 1998.  [[71]]  Y o s h i o k a , N . Fracture energy a n d the v a r i a t i o n o f gouge a n d surface roughness d u r i n g f r i c t i o n a l shding of rocks.  Journal of the Physics of the Earth, 34:335-355, 1986.  [[30]]  Z d a n o w i c z , C . M . , F . A . M i c h e l , a n d W . W . Shuts. B a s a l debris entrainment a n d transport i n glaciers of southwestern B y l o t Island, C a n a d i a n A r c t i c .  Annals of Glaciology, 22:107-113,1996.  [[56]] Z o l t a i , S. C . G l a c i a l features i f the Q u e t i c o - N i p i g o n area, O n t a r i o .  Science, 2(4):247-269,1965.  [[93]]  Canadian Journal of Earth  Appendix A  2D Stress A.l A.1.1  Evaluation of state of stress Normal loading  Consider an infinite line of force F  N  with no variation in the y direction acting normally to the  surface of a solid at z = 0, with the solid extending —> oo z (see Figure A . l ) . The stress due to this force is given by the following response functions, which are the two dimensional version of the Boussinesq solution for a normal load on a half-space [Boussinesq,  1885] . These define 1  the Green's functions for a line source,  2F  N  .  xz 2  7T  r  4  = >  ^  2x z =  2F z N  7T  =>  4  s»  2F Z X N  7T  / A  - - ^ 4 - '  (  A  -  l  a  )  -H> ( ) Alb  z  r  .  2  =  7r  4 '  2  r  QL -  4  Tr  r  4  where r = x + z . 2  2  (A.lc) (A.ld)  2  Extending the above concept, consider a surface with no variation in the y direction representing an ice sheet sitting on top of the z = 0 plane defined by ice heights H[ , H ,..., ce  ce  2  for i  Although it appears that the solution of normal traction postdates that of the more complicated tangential traction (see references in Section A.1.2), this is not the case. Boussinesq published a number of notes concerning the solution of a point normal force on an infinite half space in Comptes Rendus de I'Academie des Sciences in 1878 and 1879. These notes, among others, are gathered in the more complete work cited above. 1  133  APPENDIX  A. 2D STRESS  134  X  Gab(x,z)  F i g u r e A . 1 . A n infinite line of force n o r m a l to the z = 0 plane. T h e coordinate system is positive z down i n t o the substrate, positive x to the right along the surface. T h e response at any point o n the xz plane is given by ( A . 1 ) .  intervals of ice, each of A a : ; i n w i d t h ; the forcing on the b o u n d a r y is calculated f r o m these. L e t the ice heights be denned o n the integer g r i d a n d the forcing on the h a l f integer g r i d (see F i g u r e A . 2 ) . T h e n o r m a l forcing is given b y  m + H. i±i ce  pN a n d the n o r m a l forcing slope m  N  F?  and  Pi9-  PiQH?  (A.2)  is pN  _ pN  m . ! = ——  .  A  T o consider the effect of a continuous ice surface f r o m x _i {  (A.3)  to x  i +  i , a c o n v o l u t i o n i n t e g r a l must  be solved of the f o r m  Jx.  i  1  2  2  V  2  /  J  j  [Fl + mf _(x'-x )]g^((x-x'),z)dx'. f  +l  i  (AA)  APPENDIX  A.  135  STRESS  2D  •  Ice Thickness Inputs  — Interpolated Ice Surface  $ I  F i g u r e A . 2 . D i s c r e t i z a t i o n of the b o u n d a r y n o r m a l forcing o n the z = 0 plane, due to the ice surface denned b y H\ , .... ce  APPENDIX A. 2D STRESS  136  Using (A.1) and substituting x — — x  x  '  m  (AA) (therefore dx' = —dx),  7T  m  N 1  x°  /  - (z +X ) 2  J X — Xi  2  dx  2  ,  (A.5a)  ,  (A.5b)  2z  3  x—X. 1 ( Z  (  X —X. —X. /•X  /  •  + X  2)2  1  +7  ( 2 2)2 Z  </X —Xi  t—  2  +  X  o  X-X.  1  ( Z  *  \ JX — Xi  ( 2 Z  + X  2  X dX 2)2  + X  2)2  dX  .  (A.5c)  A l l of the convolution integrals are of the f o r m /•X2  »xi  .^m  (  z2  (A.6)  +xY 2  T h e necessary integrals follow [Gradshteyn and Ryzhik, 1965, page 69] w h i c h are not v a l i d for z = 0; r L  2  2  2  Xi  X2  l — d x - (* + x ) * •2zn(z  2  + xl)  (z + x\) 2  1 2z3  tan"  1  ( ^ ) - tan"  1  ,  (A.7a)  APPENDIX  A. 2D STRESS  137  w  <3ab(x,z)  F i g u r e A . 3 . A n infinite line force tangential to the z = 0 plane. T h e response at any p o i n t i n the xz plane is given by ( A . 8 ) .  X 2  1  ~(z*  X2  f  X +  V  2 X  )  2  d  X  _ 1 2 =  2  (z*+ )  2  2  X  X  X2  J  3  (z  r  2  X  (2 Z  +  2  X  2)2  dx = -  1  Xi  ( 2  X  Z  + X  2)  2  xi ( Z  2 +  X  (z + 2  X?  2)  ( z  2  +  + 3z 2 X  )  1 +  A. 1.2  ( ? ) "  Tz  (* + xD (* + x?)  2  4  X  +  In  =  + X )  2  ( T )  ,  (A.7c)  2  , _1  X 2  Xl  (Z  2  (A.7b)  (z +xl)\  X2 2)  1  2  f  [(z' + xl)  2z  2  xl)  2  1 /  (  (A.7d)  ( z + X?)  2V( 2 2  X2 2) +  X  tan- ( * ) -  ( Z  JO \ 2 2 y  tan-  + X  }  (*)  (A.7e)  Shear loading  N o w consider an analogous case to F i g u r e A . l , but w i t h the line of force a c t i n g t a n g e n t i a l to the z = 0 plane i n the x d i r e c t i o n (see F i g u r e A . 3 ) . T h e state of stress due to such an infinite line  APPENDIX  A.  2D  STRESS  •  Ice Thickness Inputs  —-  S  138  Interpolated Ice Surface  —N  IFi-i/2=-Fi-i/2 sin  oCi-1/2  —N  —S  Fi+i/2=-Fi+i/2 sin  oCi+1/2  X T  Z  X  F i g u r e A . 4 . D i s c r e t i z a t i o n of the b o u n d a r y tangential forcing o n the z = 0 plane due to a g i v e n ice surface.  of t a n g e n t i a l force has the foUowing response functions, w h i c h are the two d i m e n s i o n a l version of the C e r r u t i s o l u t i o n for a t a n g e n t i a l l o a d o n a half-space  [Cerruti, 1881/82].  These define  Q^  b  the  Green's functions for a line source 2F*S „x 3 7T  2F  S  2F  S  r  ell =  4  xz  2  xz 2  QW  =>XZ  =  2x^_ 4 >  2  xz  x  r  2  xz  _ _ 7T  (A.8a)  2  4  (A.8b)  2  _  A '  (A.8c)  T h e t a n g e n t i a l forcing is calculated b y J+2  where a  l c e  »+2  '+2  (A.9)  is the ice-surface dip (see F i g u r e A . 4 ) . T h e tangential forcing slope is g i v e n b y  F. i - F s  m f  =  s  2  Aa;,-  ! (A.10)  APPENDIX  A. 2D  STRESS  139  Consider the effect o f continuous tangential l o a d i n g f r o m x  t o x i\  n  a convolution integral of  n+  the f o r m  \F? + mf(x' - * . i ) | gl{{x - x%z)dx>  = Jx.  i * 2~  L  2  (A.11)  J  —  must be evaluated. S u b s t i t u t i n g v  •(^  7T  +raf  above a n d using the expression for Q  ab  ("-M))(/r?-(?^?F*)  +  (  J x—x. _ i  2z  f r o m (A-8)  Z  2  +  X  2 ) 2  ,  (A.12a)  ,  (A.12b)  2  i  m  77  (  x  ~  \ Jx—x. 1 (z + X ) 2  2  2  dx) +  rri:  2z 7T  -(^ ^-M))(jr?-(?^??*) +  dx  (z + X ) 2  J x—x._ 1  2  2  +  rx+x. _ +  1  Jx—x. 1  (z + X ) 2  2  2  dx  .  (A.12c)  T h e integrals are solved using ( A . 7 ) . For each g r i d i n t e r v a l , the effect o f the tangential a n d n o r m a l l o a d i n g is calculated for every point Xi - e < X{ < X{ + e where e is chosen such that the effect of the l o a d i n g b e y o n d Xi — e a n d x  x  x  x  Xi + e is s m a l l . T h e effect o f each ice interval's n o r m a l a n d tangential l o a d i n g is calculated a n d x  s u m m e d together to get a final state o f stress (T b(m, n, z) at any a r b i t r a r i l y fine sub-grid. a  A.2  Evaluation of failure criterion  In soil science, a different sign convention is used t h a n i n elasticity; compressive forces are positive and the m a j o r p r i n c i p a l stress is the most compressive. I n order to be consistent w i t h the soil  APPENDIX  A.  2D  140  STRESS  science literature, from which the failure criterion is taken, a new stress tensor a  is used defined  ao  by  ab a  =  -<Jab.  For any state of stress, a coordinate system can be found on which there is zero shear stress on the planes associated with the coordinate directions; these are the principal axes. For a principal axis direction u , the traction vector acting on a plane normal to v is a v . a  a  ao  As the stress is  a  acting on a principal plane, it can be described by A( )j> where the scalar A( ) is the principal a  a  a  stress associated with the principal axis v . Therefore we can write a  ~0-abh  =  a u -  or  ab  b  A i>  (A.13)  (a)  a  = 0  \ 6 i>a {a)  ab  (A.14)  where S b is the Kronecker delta. Equation (A.14) has a non-trivial solution when a  ||*o6 -  = 0.  This is solved to give the two principal stresses A ~ a  1 , 2  (A.15)  where they are ordered such that A > A . 1  2  The components of the principal direction vector v are given by solving (A.14) for each A(°). a  Consider the stress acting on a plane rotated by 7 with respect to the major principal plane (the plane for which v\ is the normal). In terms of the principal coordinate system, the normal vector n  a  of this plane will be given by n  a  = cos 7 i>i + sin 7 i>2 •  (A.16)  The normal stress &N and the shear stress as on a surface with a normal vector ra are given by tt  a  N  = \Wnl  + \^n\  and a  = n (\M  s  ni  2  - A< )) 2  (A.17)  and using the trigonometric identities that , cos 2 7 - f 1 cos 7 = and  .  SU17COS7  sin 2 7 = —-—,  .  .  (A.18)  CTJV and as can be expressed in terms of functions of 2 7 as  * , = ( >  m  - « y ^ y .  ( A  .  2 0 )  The interplay between the shear stress and the normal stress acting on a plane as i n (A.19) and (A.20) describe a half circle in 0-5 versus  crjy  space when 7 varies from 0 to x / 2 . This is a Mohr  circle and every point on the circle corresponds to a plane at an angle 7 to the plane of the major  APPENDIX  A.  2D STRESS  141  as  unstable failure envelope  Mohr-Coulomb failure criterion  A  stable CO  90°+ \ \ <|> \ F R L 0  B  (2)  \ X \  Figure A.5. A sketch of a' versus &s showing a Mohr circle, a failure envelope and the MohrCoulomb failure criterion, which is a linearization of the failure envelope. N  APPENDIX  A.  2D  STRESS  142  p r i n c i p a l d i r e c t i o n (see F i g u r e A . 5 and note that the abcissa a'  N  is related to crjv b y a t r a n s l a t i o n  described below). A l s o shown i n F i g u r e A . 5 is the failure envelope — at a shear stress above the envelope, shear 2  failure w i l l occur (a tensile mode I failure can occur at point B w h i c h has zero shear). T h i s c a n be a p p r o x i m a t e d by the M o h r - C o u l o m b failure c r i t e r i o n that predicts failure w h e n a  > c + M  s  where C o , the cohesion, a n d p,  mt  of the soil.  a'  N  — tan^>  i n t  0  fric  (^>  fric  *'N  (A.21)  is the i n t e r n a l angle of friction) are properties  is a modified n o r m a l stress that takes into account the n o r m a l stress due to  the forcing at the z = 0 b o u n d a r y (given by ( A . 1 9 ) ) as well as a h y d r o s t a t i c compressive stress due to the overlying soil and a hydrostatic tensile stress due to pore water pressure.  T h e soil  overburden increases the confining pressure w h i c h shifts the M o h r circle away f r o m failure (recall that a compressive stress is positive using the current sign convention) while the pore pressure has the opposite effect and forces the M o h r circle left i n F i g u r e A . 5 . T h i s can be w r i t t e n as a'  = p gz + a -  N  where p is the density of the soil a n d P  3  s  w  N  P  w  (A.22)  is the cavity water pressure.  T h e failure w i l l occur first at the point labelled A i n F i g u r e A . 5 corresponding to a plane at an angle of 4 5 ° + <fr /2 to the plane of the m a j o r p r i n c i p a l axis. T h i s corresponds to a n angle of iTlc  45° — ^> j2 between the plane of failure and the m a j o r p r i n c i p a l axis. fric  This is the catastrophic failure envelope. Subcritical failure can occur within the stable region as discussed in Section 3.2.1. 2  Appendix B  Single Clast Rotational Mixing A particle i n a shear flow w i l l rotate to produce a p e r t u r b a t i o n velocity field, w h i c h leads to perpendicular m o t i o n not present i n the i n i t i a l field. I n the context of a shear flow at the base of a glacier, this represents v e r t i c a l m i x i n g . R o t a t i o n a l structures of this k i n d are observed i n deformed sediment [van der Meer, 1997] and analogies are expected i n b a s a l ice. I b e g i n b y seeking a p e r t u r b a t i o n flow caused by a freely r o t a t i n g cylinder i m m e r s e d i n a backg r o u n d regular shear flow. T h e steady state solution is then explored a n d the resultant m i x i n g examined.  B.l  Perturbation due to a cylinder  Consider a viscous fluid flowing i n a u n i f o r m linear shear flow. I n C a r t e s i a n coordinates, such a flow is described as (see F i g u r e B . l ) Vi = f3yx. P o l a r coordinates (r,0,z)  (B.l)  are better suited to the p r o b l e m and ( B . l ) is r e - w r i t t e n as Vi = [0r sin 0 cos 0] f + [-0r  s i n 0] 0.  (B.2)  2  It w i l l be convenient below to write the angular dependence of Vi i n terms of sin s0 a n d cos s0 a n d ,„ 1 cos2« , . „ „ sin 20 , . .„ ' the t r i g o n o m e t r i c identities sin 0 = — a n d sin0cost? = — - — are used to w r i t e ( B . 2 ) as 2 Vi  =  0r  •— cos 20 2  sin 20  143  2  0.  (B.3)-  APPENDIX  B. SINGLE CLAST ROTATIONAL MIXING  144  F i g u r e B . l . T h e geometry of the r o t a t i n g cylinder p r o b l e m . T h e c y l i n d e r is infinite i n the z direction, has a radius of R and can rotate w i t h angular velocity Clz. T h e b a c k g r o u n d shear flow described i n ( B . l ) a n d ( B . 2 ) is also shown.  A n infinitely long cylinder of radius R is introduced i n t o this b a c k g r o u n d flow aligned along the z d i r e c t i o n , r o t a t i n g about this axis w i t h an angular velocity f i (see F i g u r e B . l ) . T h e t o t a l fluid velocity u is U = Vi  +p  (B.4)  where p is the p e r t u r b a t i o n flow f r o m the cyhnder. T h e governing equations for u are conservation of mass for an incompressible fluid a n d the NavierStokes equation, V  u = 0,  (B.5)  'du  (B.6)  - V P + T?V U, 2  where n is the viscosity, P is the pressure and fluid b o d y forces are assumed negligible. T h e effective viscosity of ice 77 is large (for example, 7/ = 1 x 1 0 P a a i n T a b l e 3.3) a n d the i n e r t i a l 6  terms of ( B . 6 ) can be discarded, as is t y p i c a l for creeping flows, to o b t a i n V P = r?V u.  (B.7)  2  T h e divergence and the L a p l a c i a n i n c y l i n d r i c a l coordinates are given b y V •  A  0(rA ) 8A 8(rA dr do dA IdAn l_(PAn d^An dr r dr r d9 dz r  6  z  (B.8)  2  n  2  2  2  2  (B.9)  APPENDIX  B. SINGLE CLAST ROTATIONAL  MIXING  145  S u b s t i t u t i o n o f ( B . 2 ) or (B.3) into the above expressions shows that V • Vi = 0, V t>; 2  V t>;  T  = 0 and  = 0. L i n e a r superposition o f the background flow a n d the p e r t u r b a t i o n flow allows the  2  e  governing equations ( B . 5 ) and (B.7) t o be r e w r i t t e n i n terms of the p e r t u r b a t i o n velocity p so that V - p = 0,  (B.10)  V P = nV p.  (B.11J  2  S y m m e t r y disallows any p e r t u r b a t i o n flow i n the i direction a n d the p r o b l e m is two d i m e n s i o n a l . Therefore i t is advantageous t o introduce a stream function ip such that p = V X ipz.  (B.12)  F r o m this definition i t follows that  1 dip ^  dip *  r 66  dr  (B.13)  v — — t)  p  a n d because of the vector i d e n t i t y V • ( V X A) = 0, the f o r m of p i n (B.12) a u t o m a t i c a l l y satisfies the requirement f r o m (B.10) that V p = 0. A n o t h e r expedient m a n i p u l a t i o n is t o take the c u r l o f ( B . 7 ) w h i c h t h e n reduces t o L a p l a c e ' s equation (B.14)  V t u = 0, 2  where w is the v o r t i c i t y o f the p e r t u r b a t i o n  field,  defined as w = V X p. A s for the s t r e a m  function, the two-dimensional flow reduces the v o r t i c i t y t o a single component w =  djrpe) _ dpr_ dr 89  (B.15)  a n d using ( B . 1 3 ) this becomes  d ip _ l c t y _ 1 d ip z = dr r dr r 86 2  w  2  2  -V ip. 2  2  (B.16)  Because the v o r t i c i t y satisfies Laplace's equation i n ( B . 1 4 ) , the s t r e a m function satisfies t h e b i h a r m o n i c equation  vV =  (B.17)  0.  B y assuming that tp has a solution that is separable i n variables a n d is of the f o r m TZ(r)T(9), (B.17) can be expanded by repeated a p p l i c a t i o n of ( B . 9 ) t o  dT dT n^rzr - 2<90 86 A  2  4  2  dn dr 2  2  2  r  dn_ dr  2 n  + T  + 1r  A  dr  4  dr  3  dr  2  +r  an dr  0. ( B . 1 8 )  APPENDIX  B. SINGLE CLAST ROTATIONAL  MIXING  146  The T(6) portion of the solution must be periodic in 2-K, which prompts the search for angular solutions of the form T = e where s is restricted to integers. The fact that each term has the form r d TZ/dr suggests Euler radial solutions of the form IZ = r . By substitution of fi = e and TZ = r into (B.18) the following expression is obtained relating p and s for the basis functions %s9  p  p  p  p  ia8  p  5  4  +  2 s  2  _  l  )  +  p  _  2  ]  +  l p { p  _  _  2 ) { p  _  3 ) +  2 p ( p  _  _  2 )  _  p ( p  _ i ) + p] = o,  or ,.3 , \ 2 , .2 ,,„2 , „4 p„4 - 4p + (4 - o2s„ 2)p + 4s p - 4s + s = 0. 4  3  2  2  2  2  (B.19)  4  The particular solution for rp will be of the form ip = Y Y s  A  >  P  e  i  s  6  -  ( - °) B  2  p  The appropriate choices for s become apparent when the boundary conditions at the cylinder surface are considered. There is an impermeable boundary at r = R which requires the flow normal to the cylinder to be zero. Also at this boundary a no-slip condition is imposed. These can be expressed as  {  tt • v = u = 0, impermeable boundary, u X f = ug = ClR, no slip. r  (B.21)  This is transformed into conditions for p using (B.3) and (B.4) p =  3R  r  *tr = R{  sin 29,  impermeable boundary,  2  = - — cos 29 + — + SIR, 2 2  g  (  no slip.  B  2  2  )  By inspection of the assumed form of the angular function T and the boundary condition on pe at r = R it is clear that s must take values of 0 and 2. The 4 roots of (B.19) when s = 0 are double roots of p = 0 and 2; when s — 2, the roots of (B.19) are p = 0,2, -2 and 4. Therefore (B.20) can be expanded to tp = A°  In r + A r +*A r In r 2  0  2  0  2  2  0  + A' r~ cos 261 + A° cos 29 + A\ T C O S 29 + A\ r cos 29, 2  2  2  4  (B.23)  where the starred coefficients account for the double roots. The boundary condition far from the cylinder is that all components of the perturbation field must vanish, implying that positive  APPENDIX  B.  SINGLE  CLAST  ROTATIONAL  MIXING  147  powers of r must be d i s a l l o w e d and A\ = A\ = A\ =*A% = 0. U s i n g (B.13) the p e r t u r b a t i o n flow 1  is given b y  ldrf;  r~ sin 29 - 2A% r sin 29, 89 dtp p = - ^ - = - % r " + 2A^ r~ cos 29. or  Pr  6  (B.24a)  _ 1  1  2  (B.24b)  z  e  B y a p p l y i n g the b o u n d a r y conditions at r = R set forth i n (B.22) the r e m a i n i n g non-zero coefficients As* can be w r i t t e n i n terms of 0, f i and R as 2A~  2 (  +  ~R?  2A\  _  0R  2Az  2  _  R* _*A% _ pR ~ R ~  2  p,  (B.25a)  from cos 29 terms i n p$,  (B.25b)  from constant terms i n p$.  (B.25c)  from sin 29 terms i n  R 0R  SIR  r  B y simphfying a n d substituting (B.25b) into (B.25a), A'  (B.26a)  -  2  A  2  ~  2  (B.26b)  '  *° = - ^ - C l R A  2  .  (B.26c)  Therefore the solution for the p e r t u r b a t i o n flow from the cylinder is \ sin 29 - PR - sin 29 r + v r 2  2  V  2  2  r  3  cos 20  0,  (B.27)  a n d the complete solution is  u =  R /3sin20(-  4  R  r 3  r  2  +  r 2  PR  2  r  B.2  + /3cos20( 2 \2 H  r  R ^ 2r A  z  6.  (B.28)  Steady state rotation  T h e previous section described the velocity field i n terms of P, R a n d O . T h e flow p a r a m e t e r P a n d the radius R are b o t h free parameters, but steady-state angular fi is determined b y the T h e boundary condition is imposed on the perturbation field, not the stream function. Therefore a In r solution for if> is permissible. 1  APPENDIX  B.  SINGLE  CLAST  ROTATIONAL  148  MIXING  viscous forces o n the cylinder given by the non-zero components of the viscous stress tensor  a. ao  S y m m e t r y dictates that the t o t a l force on the cylinder be zero but the m o m e n t of the forces w i l l , i n general, be non-zero because of off-diagonal components given b y  = °~6r = f]  °~rd  1 / d(rug)  du  r \ d r  80  T  (B.29)  U s i n g (B.28) into (B.29) gives  o- e = V r  pR  f3R  V 2r  r  A  r  V  pr  2  (B.30)  2  +  a n d at r = R,  <Tr8 = V  -2(| +  ft)  (B.31)  + 2/3 cos 20  Cross the viscous force (purely i n the 9 direction) w i t h R f to get the m o m e n t M M  = nR  'P  '1  (B.32)  - 2 ( | + f i ) + 2/3 cos 20  Integrate (B.32) around the surface of the cylinder to get the t o t a l m o m e n t  as  A4 , T  r2n  M  20 (Rd9)dz,  1  2  =>f(§+") 2  (B.33)  2TTR , 2  for a unit length strip of cylinder. A steady state solution is sought, r e q u i r i n g M  T  = 0. T h u s the  steady state r o t a t i o n is  2  (B.34)  w h i c h is equivalent to the v o r t i c i t y of the background flow. T h i s is consistent w i t h the results of Ho and Leal [1974] where a s i m i l a r result is derived for an isolated sphere.  B.3  Resultant mixing  F i g u r e B . 2 shows the t o t a l field u as described b y ( B . 2 8 ) . T h e contours are of the v e r t i c a l component (entirely f r o m the p e r t u r b a t i o n field) w i t h dashed contours for negative v e r t i c a l velocities a n d solid contours for positive v e r t i c a l velocities.  APPENDIX  B. SINGLE CLAST ROTATIONAL  MIXING  149  T o examine v e r t i c a l m i x i n g , integrate the vertical component of the velocity f r o m  (B.28) along the  0 = 0° line for the steady state. T h i s gives D , an estimate of the diffusion coefficient associated TOt  w i t h the process, as f  R  D-rot  0r  /r  £ \ 4  0R  A  dr =  1  0R  2  (B.35)  


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