ALBERT

Note: Contents 1 Slow Viscous Flow 1.1 Introduction 1.2 Coordinate Systems and the Material Derivative 1.2.1 Eulerian and Lagrangian Coordinates 1.2.2 The Material Derivative 1.3 Mass Conservation 1.4 The Stress Tensor and Momentum Conservation 1.4.1 The Stress Tensor 1.4.2 Momentum Conservation 1.4.3 Rheology 1.4.4 The Navier-Stokes Equations 1.4.5 Stokes Flow 1.5 Boundary Conditions 1.5.1 The No-Slip Condition and the Sliding Law 1.5.2 Dynamic Boundary Conditions 1.5.3 Kinematic Boundary Conditions 1.6 Temperature and Energy Conservation 1.7 Glacier and Ice Sheet Flow 1.8 Examples 1.8.1 Uniform Flow on a Slope 1.8.2 Spreading Flow at an Ice Divide 1.8.3 Small-Amplitude Perturbations 1.9 The Shallow Ice Approximation 1.10 Conclusions and Outlook 1.11 Appendix: Non-dimensionalisation Exercises 2 Thermal Structure 2.1 Temperature Profiles 2.2 Boundary Conditions 2.2.1 The Thermal Near-Surface Wave 2.3 Models: Simple to Complicated 2.4 Basal Conditions 2.4.1 Polythermal Ice 2.5 Modelling Issues 2.5.1 Non-dimensionalisation 2.5.2 Thermomechanical Coupling 2.5.3 Thermal Runaway Exercises 3 Sliding, Drainage and Subglacial Geomorphology 3.1 Introduction 3.2 Sliding Over Hard Beds 3.2.1 Weertman Sliding 3.2.2 Nye-Kamb Theory 3.2.3 Sub-temperate Sliding 3.2.4 Nonlinear Sliding Laws 3.2.5 Cavitation 3.2.6 Comparison with Experiment 3.3 Subglacial Drainage Theory 3.3.1 Weertman Films 3.3.2 Röthlisberger Channels (or ‘R-Channels’) 3.3.3 Jökulhlaups 3.3.4 Subglacial Lakes 3.3.5 Linked Cavities 3.3.6 Drainage Transitions and Glacier Surges 3.3.7 Ongoing Developments 3.4 Basal Processes and Geomorphology 3.4.1 Soft Glacier Beds 3.4.2 Drainage Over Till 3.4.3 Geomorphological Processes Exercises 4 Tidewater Glaciers 4.1 Introduction 4.2 Calving 4.3 Tidewater Glacier Dynamics 4.3.1 Tidewater Glacier Retreat and Instability 4.3.2 Tidewater Glacier Advance 4.3.3 Flow Variability of Tidewater Glaciers 4.4 The Link to Climate: Triggers for Retreat 4.4.1 Ice Shelf Collapse and Backstress 4.4.2 Grounded Calving Fronts 4.5 Outlook 5 Interaction of Ice Shelves with the Ocean 5.1 Introduction 5.2 Impact of Melting Ice on the Ocean 5.3 Processes at the Ice-Ocean Interface 5.4 Buoyancy-Driven Flow on Geophysical Scales 5.5 Sensitivity to Ocean Temperature 5.6 Impact of Meltwater Outflow at the Grounding Line 5.7 Fundamentals of the Three-Dimensional Ocean Circulation 5.8 Some Properties and Limitations of the Geostrophic Equations 5.9 Effects of Stratification 5.10 Three-Dimensional Circulation in Sub-Ice-Shelf Cavities Exercises 6 Polar Meteorology 6.1 Introduction 6.2 Shortwave and Longwave Radiation 6.3 Radiation Climate at the Top of the Atmosphere 6.4 Large Scale Circulation 6.5 Surface Energy Balance 6.5.1 Shortwave Radiation 6.5.2 Surface Albedo 6.5.3 Longwave Radiation 6.5.4 Turbulent Fluxes 6.6 Temperature Inversion and Katabatic Winds 6.6.1 Surface Temperature Inversion and Deficit 6.6.2 Katabatic Winds 6.7 Precipitation 6.8 Notes and References Exercises 7 Mass Balance 7.1 Introduction 7.2 Definitions 7.3 Methods 7.3.1 In Situ Observations 7.3.2 Satellite/Airborne Altimetry 7.3.3 Satellite Gravimetry 7.3.4 Mass Budget Method 7.4 Valley Glaciers and Ice Caps 7.4.1 In Situ Observations 7.4.2 Modelling 7.4.3 Dynamical Response 7.4.4 Remote Sensing 7.5 Antarctic Ice Sheet 7.5.1 Spatial SSMB Variability 7.5.2 Blue Ice Areas 7.5.3 Temporal SSMB Variability 7.6 Greenland Ice Sheet 7.6.1 Spatial SSMB Variability 7.6.2 Temporal SSMB Variability 7.6.3 Role of the Liquid Water Balance 8 Numerical Modelling of Ice Sheets, Streams, and Shelves 8.1 Introduction 8.2 Ice Flow Equations 8.2.1 The Shallow Ice Approximation 8.2.2 Analogy with the Heat Equation 8.3 Finite Difference Numerics 8.3.1 Explicit Scheme for the Heat Equation 8.3.2 A First Implemented Scheme 8.3.3 Stability Criteria and Adaptive Time Stepping 8.3.4 Implicit Schemes 8.3.5 Numerical Solution of Diffusion Equations 8.4 Numerically Solving the SIA 8.5 Exact Solutions and Verification 8.5.1 Exact Solution of the Heat Equation 8.5.2 Halfar’s Exact Similarity Solution to the SIA 8.5.3 Using Halfar’s Solution 8.5.4 A Test of Robustness 8.6 Applying Our Numerical Ice Sheet Model 8.7 Shelves and Streams 8.7.1 The Shallow Shelf Approximation (SSA) 8.7.2 Numerical Solution of the SSA 8.7.3 Numerics of the Linear Boundary Value Problem 8.7.4 Solving the Stress Balance for an Ice Shelf 8.7.5 Realistic Ice Shelf Modelling 8.8 A Summary of Numerical Ice Flow Modelling 8.9 Notes Exercises 9 Least-Squares Data Inversion in Glaciology 9.1 Preamble 9.2 Introduction 9.3 The Roots of GPS in Glaciology 9.4 Introduction to GPS 9.4.1 History 9.4.2 Coarse Acquisition (C/A) Code 9.5 The Equations of Pseudorange 9.6 Least-Squares Solution of an Overdetermined System of Linear Equations 9.7 Observational Techniques to Improve GPS Accuracy 9.7.1 The Ionosphere-Free Combination 9.7.2 Carrier-Phase Determined Range and Integer Wavelength Ambiguity 9.7.3 Resolving Range Ambiguity by Phase Tracking 9.7.4 Differential GPS Exercises 10 Analytical Models of Ice Sheets and Ice Shelves 10.1 Introduction 10.2 Perfectly-Plastic Ice Sheet Model 10.3 The Height–Mass Balance Feedback 10.4 Ice-Sheet Profile for Plane Shear with Glen’s Law 10.5 Ice Shelves Exercise 11 Firn 11.1 Introduction 11.2 Firn Densification 11.2.1 Mechanisms of Firn Densification 11.2.2 Firn Densification Models 11.2.3 Firn Layering and Microstructure 11.3 Applications of Firn Models 11.3.1 Ice Sheet Surface Mass Balance from Altimetry 11.3.2 Delta Age Calculations in Deep Ice Cores 11.4 Summary and Conclusions 12 Ice Cores: Archive of the Climate System 12.1 Introduction 12.2 Dating Ice Cores 12.3 Stable Water Isotopes 12.3.1 Basics and Nomenclature 12.3.2 The Isotope Proxy Thermometer 12.3.3 Examples of Isotope Records 12.3.4 Isotope Diffusion in Firn and Ice 12.3.5 Diffusion Thermometry 12.4 Aerosols in Ice 12.4.1 Introduction and Origin of Aerosols in Ice 12.4.2 Aerosol Sources and Transport 12.4.3 Post-depositional Modification 12.4.4 Seasonal Cycles in Aerosol and Particle Constituents in Ice 12.4.5 The Volcanic Signal in Ice and Its Use for Chronological Control 12.4.6 Marine Biogenic MSA and Sea Salt as Sea-Ice Proxies 12.4.7 The Record of Anthropogenic Pollution 12.4.8 Long Aerosol Records from Greenland and Antarctica 12.4.9 Electrical Properties of Ice and Their Relationship to Chemistry 12.5 Gases Enclosed in Ice 12.5.1 Firn Gas and Gas Occlusion 12.5.2 Trace Gases 12.6 Timing of Climate Events Exercises 13 Satellite Remote Sensing of Glaciers and Ice Sheets 13.1 Introduction 13.2 Optical Sensors and Applications 13.2.1 Sensors and Satellites 13.2.2 Applications 13.3 SAR Methods and Applications 13.3.1 Radar Signal Interaction with Snow and Ice 13.3.2 SAR Sensor and Image Characteristics 13.3.3 InSAR Measurement Principles and Applications 13.4 Satellite Altimetry 13.4.1 Altimetry Missions 13.4.2 Measuring Elevation Change 14 Geophysics 14.1 Geophysical Methods: Overview 14.2 Passive Methods 14.2.1 Gravimetry 14.2.2 Magnetics 14.2.3 Seismology 14.3 Active Methods: Basics 14.3.1 Propagation Properties and Reflection Origin 14.3.2 Seismic System Set-Up 14.3.3 Radar System Set-Up 14.4 Data Acquisition and Processing 14.5 Seismic Applications in Ice 14.5.1 Ice Thickness and Basal Topography 14.5.2 Subglacial Structure and Properties 14.5.3 Rheological and Other Englacial Properties 14.6 Radar Applications in Ice 14.6.1 Internal Layer Architecture and Ice Dynamics 14.6.2 Subglacial Conditions 14.6.3 Englacial Conditions 14.7 Notes and References 14.7.1 Further Reading 14.7.2 Gravimetry 14.7.3 General Wave Equation and Solution 14.7.4 Seismic Waves 14.7.5 Electromagnetic Waves Exercises 15 Glacial Isostatic Adjustment 15.1 Introduction 15.2 Earth Response to Loading 15.2.1 Rheology of the Earth 15.2.2 Building an Earth Model 15.2.3 Earth Models Used in Glaciology and Glacial Isostatic Adjustment 15.3 The Cryosphere and Sea Level 15.3.1 Factors Affecting Sea-Level Change 15.3.2 Eu

Note: Contents 1 Slow Viscous Flow 1.1 Introduction 1.2 Coordinate Systems and the Material Derivative 1.2.1 Eulerian and Lagrangian Coordinates 1.2.2 The Material Derivative 1.3 Mass Conservation 1.4 The Stress Tensor and Momentum Conservation 1.4.1 The Stress Tensor 1.4.2 Momentum Conservation 1.4.3 Rheology 1.4.4 The Navier-Stokes Equations 1.4.5 Stokes Flow 1.5 Boundary Conditions 1.5.1 The No-Slip Condition and the Sliding Law 1.5.2 Dynamic Boundary Conditions 1.5.3 Kinematic Boundary Conditions 1.6 Temperature and Energy Conservation 1.7 Glacier and Ice Sheet Flow 1.8 Examples 1.8.1 Uniform Flow on a Slope 1.8.2 Spreading Flow at an Ice Divide 1.8.3 Small-Amplitude Perturbations 1.9 The Shallow Ice Approximation 1.10 Conclusions and Outlook 1.11 Appendix: Non-dimensionalisation Exercises 2 Thermal Structure 2.1 Temperature Profiles 2.2 Boundary Conditions 2.2.1 The Thermal Near-Surface Wave 2.3 Models: Simple to Complicated 2.4 Basal Conditions 2.4.1 Polythermal Ice 2.5 Modelling Issues 2.5.1 Non-dimensionalisation 2.5.2 Thermomechanical Coupling 2.5.3 Thermal Runaway Exercises 3 Sliding, Drainage and Subglacial Geomorphology 3.1 Introduction 3.2 Sliding Over Hard Beds 3.2.1 Weertman Sliding 3.2.2 Nye-Kamb Theory 3.2.3 Sub-temperate Sliding 3.2.4 Nonlinear Sliding Laws 3.2.5 Cavitation 3.2.6 Comparison with Experiment 3.3 Subglacial Drainage Theory 3.3.1 Weertman Films 3.3.2 Röthlisberger Channels (or ‘R-Channels’) 3.3.3 Jökulhlaups 3.3.4 Subglacial Lakes 3.3.5 Linked Cavities 3.3.6 Drainage Transitions and Glacier Surges 3.3.7 Ongoing Developments 3.4 Basal Processes and Geomorphology 3.4.1 Soft Glacier Beds 3.4.2 Drainage Over Till 3.4.3 Geomorphological Processes Exercises 4 Tidewater Glaciers 4.1 Introduction 4.2 Calving 4.3 Tidewater Glacier Dynamics 4.3.1 Tidewater Glacier Retreat and Instability 4.3.2 Tidewater Glacier Advance 4.3.3 Flow Variability of Tidewater Glaciers 4.4 The Link to Climate: Triggers for Retreat 4.4.1 Ice Shelf Collapse and Backstress 4.4.2 Grounded Calving Fronts 4.5 Outlook 5 Interaction of Ice Shelves with the Ocean 5.1 Introduction 5.2 Impact of Melting Ice on the Ocean 5.3 Processes at the Ice-Ocean Interface 5.4 Buoyancy-Driven Flow on Geophysical Scales 5.5 Sensitivity to Ocean Temperature 5.6 Impact of Meltwater Outflow at the Grounding Line 5.7 Fundamentals of the Three-Dimensional Ocean Circulation 5.8 Some Properties and Limitations of the Geostrophic Equations 5.9 Effects of Stratification 5.10 Three-Dimensional Circulation in Sub-Ice-Shelf Cavities Exercises 6 Polar Meteorology 6.1 Introduction 6.2 Shortwave and Longwave Radiation 6.3 Radiation Climate at the Top of the Atmosphere 6.4 Large Scale Circulation 6.5 Surface Energy Balance 6.5.1 Shortwave Radiation 6.5.2 Surface Albedo 6.5.3 Longwave Radiation 6.5.4 Turbulent Fluxes 6.6 Temperature Inversion and Katabatic Winds 6.6.1 Surface Temperature Inversion and Deficit 6.6.2 Katabatic Winds 6.7 Precipitation 6.8 Notes and References Exercises 7 Mass Balance 7.1 Introduction 7.2 Definitions 7.3 Methods 7.3.1 In Situ Observations 7.3.2 Satellite/Airborne Altimetry 7.3.3 Satellite Gravimetry 7.3.4 Mass Budget Method 7.4 Valley Glaciers and Ice Caps 7.4.1 In Situ Observations 7.4.2 Modelling 7.4.3 Dynamical Response 7.4.4 Remote Sensing 7.5 Antarctic Ice Sheet 7.5.1 Spatial SSMB Variability 7.5.2 Blue Ice Areas 7.5.3 Temporal SSMB Variability 7.6 Greenland Ice Sheet 7.6.1 Spatial SSMB Variability 7.6.2 Temporal SSMB Variability 7.6.3 Role of the Liquid Water Balance 8 Numerical Modelling of Ice Sheets, Streams, and Shelves 8.1 Introduction 8.2 Ice Flow Equations 8.2.1 The Shallow Ice Approximation 8.2.2 Analogy with the Heat Equation 8.3 Finite Difference Numerics 8.3.1 Explicit Scheme for the Heat Equation 8.3.2 A First Implemented Scheme 8.3.3 Stability Criteria and Adaptive Time Stepping 8.3.4 Implicit Schemes 8.3.5 Numerical Solution of Diffusion Equations 8.4 Numerically Solving the SIA 8.5 Exact Solutions and Verification 8.5.1 Exact Solution of the Heat Equation 8.5.2 Halfar’s Exact Similarity Solution to the SIA 8.5.3 Using Halfar’s Solution 8.5.4 A Test of Robustness 8.6 Applying Our Numerical Ice Sheet Model 8.7 Shelves and Streams 8.7.1 The Shallow Shelf Approximation (SSA) 8.7.2 Numerical Solution of the SSA 8.7.3 Numerics of the Linear Boundary Value Problem 8.7.4 Solving the Stress Balance for an Ice Shelf 8.7.5 Realistic Ice Shelf Modelling 8.8 A Summary of Numerical Ice Flow Modelling 8.9 Notes Exercises 9 Least-Squares Data Inversion in Glaciology 9.1 Preamble 9.2 Introduction 9.3 The Roots of GPS in Glaciology 9.4 Introduction to GPS 9.4.1 History 9.4.2 Coarse Acquisition (C/A) Code 9.5 The Equations of Pseudorange 9.6 Least-Squares Solution of an Overdetermined System of Linear Equations 9.7 Observational Techniques to Improve GPS Accuracy 9.7.1 The Ionosphere-Free Combination 9.7.2 Carrier-Phase Determined Range and Integer Wavelength Ambiguity 9.7.3 Resolving Range Ambiguity by Phase Tracking 9.7.4 Differential GPS Exercises 10 Analytical Models of Ice Sheets and Ice Shelves 10.1 Introduction 10.2 Perfectly-Plastic Ice Sheet Model 10.3 The Height–Mass Balance Feedback 10.4 Ice-Sheet Profile for Plane Shear with Glen’s Law 10.5 Ice Shelves Exercise 11 Firn 11.1 Introduction 11.2 Firn Densification 11.2.1 Mechanisms of Firn Densification 11.2.2 Firn Densification Models 11.2.3 Firn Layering and Microstructure 11.3 Applications of Firn Models 11.3.1 Ice Sheet Surface Mass Balance from Altimetry 11.3.2 Delta Age Calculations in Deep Ice Cores 11.4 Summary and Conclusions 12 Ice Cores: Archive of the Climate System 12.1 Introduction 12.2 Dating Ice Cores 12.3 Stable Water Isotopes 12.3.1 Basics and Nomenclature 12.3.2 The Isotope Proxy Thermometer 12.3.3 Examples of Isotope Records 12.3.4 Isotope Diffusion in Firn and Ice 12.3.5 Diffusion Thermometry 12.4 Aerosols in Ice 12.4.1 Introduction and Origin of Aerosols in Ice 12.4.2 Aerosol Sources and Transport 12.4.3 Post-depositional Modification 12.4.4 Seasonal Cycles in Aerosol and Particle Constituents in Ice 12.4.5 The Volcanic Signal in Ice and Its Use for Chronological Control 12.4.6 Marine Biogenic MSA and Sea Salt as Sea-Ice Proxies 12.4.7 The Record of Anthropogenic Pollution 12.4.8 Long Aerosol Records from Greenland and Antarctica 12.4.9 Electrical Properties of Ice and Their Relationship to Chemistry 12.5 Gases Enclosed in Ice 12.5.1 Firn Gas and Gas Occlusion 12.5.2 Trace Gases 12.6 Timing of Climate Events Exercises 13 Satellite Remote Sensing of Glaciers and Ice Sheets 13.1 Introduction 13.2 Optical Sensors and Applications 13.2.1 Sensors and Satellites 13.2.2 Applications 13.3 SAR Methods and Applications 13.3.1 Radar Signal Interaction with Snow and Ice 13.3.2 SAR Sensor and Image Characteristics 13.3.3 InSAR Measurement Principles and Applications 13.4 Satellite Altimetry 13.4.1 Altimetry Missions 13.4.2 Measuring Elevation Change 14 Geophysics 14.1 Geophysical Methods: Overview 14.2 Passive Methods 14.2.1 Gravimetry 14.2.2 Magnetics 14.2.3 Seismology 14.3 Active Methods: Basics 14.3.1 Propagation Properties and Reflection Origin 14.3.2 Seismic System Set-Up 14.3.3 Radar System Set-Up 14.4 Data Acquisition and Processing 14.5 Seismic Applications in Ice 14.5.1 Ice Thickness and Basal Topography 14.5.2 Subglacial Structure and Properties 14.5.3 Rheological and Other Englacial Properties 14.6 Radar Applications in Ice 14.6.1 Internal Layer Architecture and Ice Dynamics 14.6.2 Subglacial Conditions 14.6.3 Englacial Conditions 14.7 Notes and References 14.7.1 Further Reading 14.7.2 Gravimetry 14.7.3 General Wave Equation and Solution 14.7.4 Seismic Waves 14.7.5 Electromagnetic Waves Exercises 15 Glacial Isostatic Adjustment 15.1 Introduction 15.2 Earth Response to Loading 15.2.1 Rheology of the Earth 15.2.2 Building an Earth Model 15.2.3 Earth Models Used in Glaciology and Glacial Isostatic Adjustment 15.3 The Cryosphere and Sea Level 15.3.1 Factors Affecting Sea-Level Change 15