Pine Island Glacier - a 3D full-Stokes model study Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich Geowissenschaften der Universit¨atHamburg vorgelegt von Nina Wilkens aus Berlin Hamburg 2014 Als Dissertation angenommen vom Fachbereich Geowissenschaften der Universit¨atHamburg auf Grund der Gutachten von Prof. Dr. Angelika Humbert und Prof. Dr. J¨orn Behrens Hamburg, den 4. April 2014 Prof. Dr. Christian Betzler Leiter des Fachbereichs Geowissenschaften Contents List of Figures VII List of Tables XI Abstract XII 1 Introduction 1 1.1 TheAntarcticIceSheet ............................. 2 1.1.1 Geologic history . 2 1.1.2 Themarineicesheetinstability. 3 1.1.3 Basal properties . 5 1.2 Icesheetmodels ................................. 7 1.2.1 Approximations . 8 1.2.2 Basal motion . 9 1.2.3 Grounding line migration . 10 1.3 PineIslandGlacier................................ 11 1.3.1 Observations . 12 1.3.2 Modelstudies............................... 13 1.4 Objectivesandstructureofthisstudy . 14 2 Theory 17 2.1 Balance equations . 17 2.1.1 Mass balance - Continuity equation . 18 2.1.2 Momentum balance - Momentum equation . 19 2.1.3 Energy balance - Heat transfer equation . 20 2.2 Constitutive relation - Rheology of ice . 22 2.2.1 Glen’sflowlaw .............................. 22 2.2.2 Rate factor . 24 2.2.3 Enhancement factor . 25 2.3 Overview of equations . 26 2.4 Boundary conditions . 26 2.4.1 Ice surface . 27 2.4.2 Ice base . 28 2.4.3 Lateral boundaries - Ice divide, calving front and inflow . 30 2.5 Finiteelementmethod-FEM. 32 2.5.1 Meshing.................................. 32 2.5.2 Approximation functions - Basis functions . 33 2.5.3 Weighted-integral form . 34 2.5.4 Boundary conditions . 35 2.5.5 Assembly . 35 III Contents 2.5.6 Solution . 36 3 The 3D full-Stokes model for Pine Island Glacier 39 3.1 Data........................................ 39 3.1.1 Ice geometry . 40 3.1.2 Grounding line position . 41 3.1.3 Icerises .................................. 42 3.1.4 Surface temperature . 43 3.1.5 Geothermal heat flux . 43 3.1.6 Surfacevelocity.............................. 45 3.2 Implementation.................................. 46 3.2.1 Model geometry . 48 3.2.2 Iceflowmodel .............................. 49 3.2.3 Thermal model . 53 3.2.4 Mesh.................................... 55 3.2.5 Solver . 57 3.3 Verification and validation . 58 3.3.1 Ice shelf ramp . 58 3.3.2 MISMIP 3D . 63 4 Identification of dominant local flow mechanisms 67 4.1 No-slip simulations . 68 4.1.1 Drivingstress............................... 69 4.1.2 Heat conduction . 71 4.1.3 Strain heating . 72 4.1.4 Internal deformation . 76 4.2 Referencesimulations .............................. 78 4.2.1 Quasi-inversion technique . 79 4.2.2 Reference simulation . 81 4.2.3 Temperate layer . 84 4.2.4 Water content . 85 4.2.5 Full-Stokesvs.SIA............................ 86 4.2.6 Sensitivity to geothermal heat flux . 88 4.3 Hydraulic potential . 90 4.4 Basal roughness . 91 4.5 Discussion..................................... 92 5 Basal sliding 97 5.1 Theory - Basal sliding . 97 5.1.1 Hard beds . 98 5.1.2 Deformable beds . 101 5.2 Evaluation method of results . 102 5.3 Constant sets of sliding parameters p, q and Cb . 104 5.3.1 Simulations . 104 5.3.2 Discussion . 107 5.4 Matching of roughness measure ξ and sliding parameter Cb . 110 5.4.1 Simulations . 112 IV Contents 5.4.2 Discussion . 116 5.5 Li-sliding .....................................117 5.5.1 The two parameter roughness index - ξ2 and η2 . 118 5.5.2 Assumptions - Controlling obstacle size - Constant CL . 119 5.5.3 Simulations . 120 5.5.4 Discussion . 123 5.6 Discussion.....................................123 6 Conclusions and outlook 127 A 131 A.1 Integration theorems . 131 A.1.1 Reynold’s transport theorem . 131 A.1.2 Integral formula of Gauss - Divergence theorem . 131 A.1.3 Integration by parts - Green-Gauss theorem . 131 Bibliography 133 Acknowledgements 147 V List of Figures 1.1 Bedrock topography of Antarctica . 3 1.2 Schematic of a marine ice sheet on a retrograde bed . 4 1.3 Bed roughness distribution below the Antarctic Ice Sheet . 6 1.4 Surface velocity and grounding line positions at Pine Island Glacier . 13 2.1 Stress-strainrelationships . 23 2.2 Theory - Boundary conditions - Ice surface . 27 2.3 Theory - Boundary conditions - Ice base - floating . 28 2.4 Theory - Boundary conditions - Ice base - grounded . 29 2.5 Theory - Boundary conditions - Ice divide . 30 2.6 Theory - Boundary conditions - Calving front . 31 2.7 Example of a non-uniform FEM mesh on a complex geometry . 33 2.8 Linear basis functions . 34 2.9 Quadratic basis functions . 34 3.1 Model region on Antarctica . 39 3.2 Model domain of Pine Island Glacier on a mosaic of satellite images . 40 3.3 Surface elevation of Pine Island Glacier . 41 3.4 Bed topography of Pine Island Glacier . 41 3.5 Different grounding line and ice rise positions at Pine Island Glacier . 42 3.6 Surface temperature of Pine Island Glacier . 43 3.7 Geothermal heat flux from Shapiro 2004 of Pine Island Glacier . 44 3.8 Geothermal heat flux from Fox Maule 2005 of Pine Island Glacier . 44 3.9 Geothermal heat flux from Purucker 2012 of Pine Island Glacier . 44 3.10 Location of volcanic center at Pine Island Glacier . 44 3.11 Observed surface velocity field of Pine Island Glacier . 46 3.12 Screenshot of the COMSOL GUI . 47 3.13 Difference between interpolated data and geometry object at surface . 49 3.14 Difference between interpolated data and geometry object at base . 49 3.15 Model - Boundary conditions - Ice surface . 51 3.16 Model - Boundary conditions - Ice base . 51 3.17 Model - Boundary conditions - Ice divide . 52 3.18 Model - Boundary conditions - Calving front . 52 3.19 Model - Boundary conditions - Inflow . 53 3.20 Function for implementation of the thermal basal boundary condition . 55 3.21 FEM mesh on the 3D Pine Island Glacier model geometry . 56 3.22 Mesh quality on the 3D Pine Island Glacier model geometry . 56 3.23 Ice shelf ramp - 3D geometry with the flow field . 60 3.24 Ice shelf ramp - Horizontal velocities . 61 3.25 Iceshelframp-Verticalvelocities . 62 VII List of Figures 3.26 Ice shelf ramp - Mass balance . 63 3.27 MISMIP 3D - Perturbed basal friction parameter . 64 3.28 MISMIP3D-Geometrywithvelocityfield . 65 3.29 MISMIP 3D - Horizontal surface velocity u at grounding line . 65 3.30 MISMIP 3D - Horizontal surface velocity v at grounding line . 65 3.31 MISMIP 3D - Horizontal surface velocity field u . 66 3.32 MISMIP 3D - Horizontal surface velocity field v . 66 3.33 MISMIP 3D - Vertical surface velocity field w . 66 4.1 Numbering of tributaries on observed surface flow field . 67 4.2 SIA basal drag on Pine Island Glacier . 70 4.3 Simulated basal drag on Pine Island Glacier . 70 4.4 Homologous basal temperature due to heat conduction . 71 4.5 Homologous basal temperature - Without strain heating term . 72 4.6 Homologous basal temperature - With strain heating term . 72 4.7 Source term at base . 73 4.8 Viscosity at base . 74 4.9 Effective strain rate at base . ..