Our realisation of how profoundly glaciers and ice sheets respond to climate change and impact sea level and the environ

*428*
*76*
*29MB*

*English*
*Pages 557
[544]*
*Year 2020*

- Author / Uploaded
- Andrew Fowler
- Felix Ng

*Table of contents : Lecturers of the SchoolForewordPreface ReferencesContentsContributors1 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-dimensionalisation2 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 Runaway3 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 References4 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 References5 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 References6 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 References7 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 Balance8 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 Notes9 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 GPS10 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 References11 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 Conclusions12 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 References13 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 References14 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 Waves15 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 15.2.3.1 Glaciology 15.2.3.2 Glacial Isostatic Adjustment (GIA) 15.2.3.3 Comparison of Earth Models 15.3 The Cryosphere and Sea Level 15.3.1 Factors Affecting Sea-Level Change 15.3.2 Eustatic Sea-Level Change 15.3.3 Departures from Eustasy and the Sea-Level Equation 15.3.4 Rotational Feedback 15.3.5 Spatial Pattern of Sea-Level Change 15.3.6 Viscous Effects: Ocean Syphoning 15.3.7 Sea-Level Change as a Controlling Factor on Ice-Sheet Evolution 15.4 Constraining Cryospheric Changes with Observations 15.4.1 Global Ice Volumes 15.4.2 Meltwater Pulses 15.4.3 Regional Ice-Sheet Histories 15.4.4 Twentieth Century Ice-Sheet and Sea-Level Changes 15.4.5 Satellite Era Ice-Sheet and Sea-Level Changes References16 Ice Sheets in the Cenozoic 16.1 Introduction 16.2 Forcing Mechanisms 16.2.1 Changes in the Carbon cycle 16.2.2 Orbital Cycles and Climate Variability 16.3 From Benthic δ18O to Global Ice Volume 16.4 Cenozoic Evolution of Ice Volume 16.4.1 Inception of Antarctica 16.4.2 Oligocene and Miocene Variability 16.4.3 Pleistocene Ice Ages 16.5 The Last Glacial Cycle 16.5.1 The Previous Interglacial—The Eemian 16.5.2 The Last Glacial Maximum17 Paleoglaciology 17.1 Introduction 17.2 Glacial Landforms 17.2.1 Formation Time and Landform Size-Scale 17.2.2 Genetic Information in an Inversion Context 17.2.2.1 Subglacial Landforms 17.2.2.2 Ice-Marginal Landforms 17.3 Data Acquisition and Data Reduction 17.4 Reconstruction of Glaciers and Ice Sheets 17.4.1 The Inversion Problem 17.4.2 Data Reduction 17.4.3 An Inversion Procedure 17.5 The Chronological Domain 17.6 Glaciological Insights from Paleoglaciology References18 Glacier Fluctuations and Simple Glacier Models 18.1 Introduction 18.2 Simple Glacier Model: Constant Bed Slope, Constant Width 18.3 More Complex Geometry 18.4 Characteristic Time Scale 18.5 Linear Theory of Glacier Length Fluctuations 18.6 Inverse Modelling 18.7 Global Temperature Reconstruction from Glacier Length Fluctuations 18.8 Minimal Glacier Model 18.9 Nonlinear Behaviour Simulated by the Minimal Glacier Model References19 Tropical Glaciers 19.1 Glaciers and Climate at Low Latitude 19.2 Local Mass and Energy Balance 19.2.1 Physical Modelling 19.2.2 Characteristics of Tropical Glaciers 19.3 Linkage of Glacier Mass Balance to Large-Scale Climate 19.3.1 General Considerations 19.3.2 Kilimanjaro Case Study 19.3.3 Concluding Remarks References20 The History of Glaciology in the Inner Ötztal Alps 20.1 Preamble 20.2 Introduction 20.3 First Descriptions and Sketches of Glaciers 20.4 Detailed Glacier Maps and Length Changes 20.5 Measurements of Ice Flow Velocities: From Stone Lines to Stakes 20.6 The Deep Drillings, 1893–1922 20.7 Twentieth Century Developments 20.7.1 Gletscherdienst Vent and Early Hydrometeorology 20.7.2 1950s: Mass and Energy Balance 20.7.3 1970s: Glacier Hydrology and Firn Properties 20.7.4 1980s: Numerical Modelling of Energy Balance and Ice Flow 20.8 Remote SensingAfterwordIndex*

Andrew Fowler Felix Ng Editors

Glaciers and Ice Sheets in the Climate System The Karthaus Summer School Lecture Notes

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Andrew Fowler • Felix Ng Editors

Glaciers and Ice Sheets in the Climate System The Karthaus Summer School Lecture Notes

123

Editors Andrew Fowler MACSI University of Limerick Limerick, Ireland

Felix Ng Department of Geography University of Shefﬁeld Shefﬁeld, UK

ISSN 2510-1307 ISSN 2510-1315 (electronic) Springer Textbooks in Earth Sciences, Geography and Environment ISBN 978-3-030-42582-1 ISBN 978-3-030-42584-5 (eBook) https://doi.org/10.1007/978-3-030-42584-5 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. Cover illustration: A view of Hintereisferner in 2003 from the ridge overlooking Schöne Aussicht. The summer school excursion normally progresses to this ridge following lunch at the hut at Schöne Aussicht This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

We dedicate this book to Paul and Stefania Grüner, proprietors of the Hotel zur Goldenen Rose in Karthaus, Schnalstal, who, together with their warm-hearted staff, have provided a home from home for many years to the participants of the annual Karthaus summer school in glaciology.

The Hotel zur Goldenen Rose, Karthaus, Schnalstal

Lecturers of the School

Abermann, Jakob (2009, 2010, 2011) Björnsson, Helgi (1997) Blunier, Thomas (2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018) Bueler, Ed (2009, 2010, 2012, 2014) Buizert, Christo (2010, 2011) Dahl-Jensen, Dorthe (1995, 1997, 2002, 2005, 2008, 2009, 2010, 2011, 2014) de Boer, Bas (2013, 2018) DeConto, Rob (2010) Eisen, Olaf (2008, 2009, 2011, 2013, 2015, 2017, 2019) Fischer, Hubertus (2007, 2008, 2009, 2010) Fowler, Andrew (1995, 1997, 2000, 2002, 2003, 2005, 2007, 2008, 2009, 2011) Giesen, Rianne (2011) Greuell, Wouter (2002, 2003, 2005) Gudmundsson, Hilmar (1997, 2000, 2002, 2003, 2005, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015) Helsen, Michiel (2008, 2009, 2010) Hewitt, Ian (2014, 2016, 2017, 2018, 2019) Hindmarsh, Richard (1995) Howat, Ian (2009, 2010, 2012, 2014) Huybrechts, Philippe (2002) Jenkins, Adrian (1995, 1997, 2000, 2002, 2003, 2005, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2017, 2018) Kääb, Andy (2013) Karlsson, Nanna (2012, 2013, 2014, 2015, 2017, 2018, 2019) Kaser, Georg (1997, 2000, 2002, 2003, 2005, 2008) Kuhn, Michael (2008, 2009, 2010, 2011, 2013) Kuipers-Munneke, Peter (2012) Kyrke-Smith, Teresa (2016) Lambeck, Kurt (1997, 2000, 2003, 2005, 2008, 2010, 2013) Mayer, Christoph (2000, 2002) Miller, Heinz (1997, 2000, 2002) Milne, Glenn (2007, 2009, 2011) Mölg, Thomas (2007, 2008, 2009, 2010, 2011)

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Lecturers of the School

Mulvaney, Rob (2002, 2003, 2005) Navarro, Paco (2012, 2014, 2016, 2018) Ng, Felix (2010, 2012, 2013, 2015) Nick, Faezeh (2014, 2016) Oerlemans, Hans (1995, 1997, 2000, 2002, 2003, 2005, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019) Pattyn, Frank (2013, 2015, 2016, 2017, 2018, 2019) Paul, Frank (2011, 2012) Payne, Tony (2003, 2005, 2007) Peltier, Dick (1995) Pollard, Dave (2003) Reijmer, Carleen (2003, 2005, 2010, 2012, 2014, 2015, 2016, 2017, 2019) Roth, George (2017) Rott, Willy (2000, 2002, 2007) Spada, Giorgio (2016, 2017, 2018, 2019) Stauffer, Bernhard (1995, 1997, 2000, 2003) Straneo, Fiamma (2016) Stroeven, Arjen (2005, 2007, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019) Svensson, Anders (2019) van de Berg, Willem Jan (2009, 2012, 2018) van de Wal, Roderik (2002, 2003, 2008, 2011, 2012, 2014, 2015) van den Berg, Jojanneke (2007, 2008) van den Broeke, Michiel (2000, 2002, 2007, 2008, 2010, 2011) van Pelt, Ward (2014) Vieli, Andreas (2011) Vinther, Bo (2009) Whitehouse, Pippa (2012, 2014) Winkelmann, Ricarda (2019)

Foreword

The European Ice Sheet Modelling Initiative (EISMINT) was launched by the European Science Foundation in 1992. EISMINT aimed to examine the critical links between global climate change and ice sheets by improving mathematical modelling in a number of key areas. Funding of this programme by the members of the European Science Foundation ran from 1993 through 1997. As part of the programme, it was envisaged to organise a summer course for Ph. D. students and junior scientists. The EISMINT Committee asked me to convene a course, and so it happened. The ﬁrst course took place in Grindelwald, Switzerland, from 27 August till 6 September 1995. There were 33 participants, following lectures, working on computer projects, and participating in an excursion to the Jungfraujoch and its Observatory. The course was a success and ﬁlled a clear need, so the decision to do it a second time was made soon. However, the venue was not considered optimal and after some looking around it was decided to hold the second course in Karthaus, a nice small village in the Schnalstal (northern Italy, not far from Merano). And that is how the Karthaus courses started (in fact, Georg Kaser found this place). Somehow funding was found to keep the train rolling, albeit only every second year in the beginning. Since 2007, the course has been organised annually, the 2018 course being the 18th one. Through the years, major support has come from the Institute for Marine and Atmospheric Research (Utrecht University), the Niels Bohr Institute (University of Copenhagen), the Paul Crutzen Fund for German-Dutch collaboration in climate research, the Nordic Centre of Excellence SVALI, the Ice2Sea Programme (EU), the Descartes Prize (EPICA, EU), and the Netherlands Earth System Science Centre. And, fair to say, little holes appearing in the budget every now and then were stuffed by money from my personal research grants. The formula of the course has not changed through the years: lectures in the morning, exercises and projects in the afternoon, running or hiking to refresh brains and body, and then a ﬁve-course meal in the evening with subsequent scientiﬁc discussions in the bar. Probably the most exciting parts always were the excursion to the dwindling glaciers of the Ötztal and the project presentations on Friday afternoon. The majority of the teachers normally stayed for the entire duration of the course, creating a very informal and stimulating atmosphere. So many colleagues have shared their knowledge with the students (and other teachers). It is just

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not feasible to mention them one by one. However, a list of teachers has been included in this book. A few words about Karthaus, this splendid village in the Schnalstal and Hotel Goldene Rose. It is almost impossible to describe the incredible hospitality of Stefania and Paul Grüner and their staff. Being the heart of the course, where all students and teachers gather again and again, and sometimes mingle with the ‘locals’, Goldene Rose will be the home of this course forever. The romantic atmosphere apparently has led to many lasting bonds, several being crowned by the birth of twins! Music always played an important role in the evenings. Songs were created and performed (Andrew Fowler frequently being one of the drivers), and live tango music ﬁlled the dining room every now and then. The tango crash courses given by Hilmar Gudmundsson will be remembered by many. In recent years, the dinner has often been concluded by the ﬁne jazzy sounds from Frank Pattyn at the piano. Some texts and recordings can be found on the Karthaus website. And now, there is this book. A multi-year project with ups and downs, sometimes stagnant, sometimes sliding. It is due to the determination of Andrew Fowler and Felix Ng, who never gave up, that in the end this special project is now ﬁnished. Since the Karthaus course has been evolving in time and will continue to do so, this book is not a one-to-one mapping of the material that is treated in a single course. In fact, it is much more, and I am sure that many people studying glaciers and ice sheets will ﬁnd it very useful. The scope is wide, ranging from numerical techniques to observations on tropical glaciers, micro-scale processes in ice cores to large-scale interaction of ice sheets with the solid Earth, from glacial geomorphology to katabatic flows. From the perspective of my professional career, the Karthaus courses have deﬁnitely been the outstanding highlight and a continuous source of inspiration. About 600 students have now participated in the courses and learned about cryospheric science. I want to express my gratitude to all persons that have contributed to the success of the course through the years. A special thanks to Carleen Reijmer, who has done a great job in documenting the courses and keeping the Karthaus website up to date. I am convinced that this “Karthaus book” will turn out to be a milestone in the documentation of our knowledge about glaciers and ice sheets. Thank you Andrew and Felix! Utrecht, The Netherlands December 2018

Hans Oerlemans

Preface

This book forms the basis of the now annual Karthaus summer school on glaciology, which is held each year in the idyllic setting of Karthaus in the Südtirol of Northern Italy. Originating in a European initiative, the continuance of the school is the product of the enthusiasm and drive of Hans Oerlemans. The idea of making a book of the lectures came to one of us (ACF) some eight years ago, and many separate authors combined to provide written chapters on some of the subjects which formed the core of the course. It is important to state that the content of the course is fluid and depends on the willingness and availability of academic teachers to absent themselves for twelve days to a beautiful but remote location in the European Alps. As such, the present content is a snapshot of a typical course, but not a precise rendition. The early enthusiasm of the authors was tempered by a long period of delay while the editing process got under way. The reason for this delay lies in the nature of the task. As we discovered, editing a book is a rather different process to editing or refereeing papers, which from our present perspective go largely unedited. The editors’ job is to make the diverse material somewhat uniform and so, ipso facto, that means they will impose their own style and vision on the text. The consequence of this in practice is that, apart from improvements in grammar and English usage, sometimes we actively re-moulded the authored text. We evolved a procedure to do this, which was the following. Firstly, the submitted chapters were split between LaTeX and Word, and our separate editorial competences thus divided the ﬁrst editorial step naturally: each of us dealt with our ‘own’ chapters. This ﬁrst step took three weeks, which we spent in each other’s house. The second step consisted of us second-editing the ﬁrst-edited chapters of the other editor, and in a third step, these changes were implemented. At this point, the draft book was ready and sent out to authors for comment and any further revision. The ﬁnal wrap was (naturally) done in Karthaus, where the photograph of Paul and Stefﬁ was taken. We wish to add our thanks to those of Hans for their, and their staff’s, incredible hospitality and welcome down the years, and we feel conﬁdent in adding those of all of the teachers and students as well. Our vision of this book is that it is a textbook, to be used by students from a wealth of different disciplines, and so it must be expository at a level we can all hope to understand. In particular, the chapters are not research papers or reviews,

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and we wanted them to be readable as text. They are not notes, and indeed, we expect the practice of the lecturers providing printed notes at Karthaus will continue, as part of our editing process involved making mathematical notation somewhat systematic and uniform, for example. Text is text, and we have attempted not to drown the book with references. Consequently, the reference lists for each chapter do not attempt to be exhaustive. In Springer style, they are put at the end of each chapter, but we aim to put a concatenated list on the book’s web page. Despite our efforts to provide some sort of uniform style, it will be evident that the chapters are of widely disparate natures; it is what we expect. In the summer school, it is common that groups of lectures have associated exercises, and you will ﬁnd some of these here; but some chapters have no exercises. And while the summer school aims to have the students do the exercises in an afternoon, some here are more serious. We, the editors, wish to acknowledge the good-natured enthusiasm and industry of the contributors, who were all a good deal more timely than we were, and who all willingly acceded to our wholesale massacre of their original contributions. A special mention must be made of Doug MacAyeal, who not only came on board at an extremely last minute to produce Chap. 9, but became so enthused by the

Fig. 1 Images of Vernagtferner in 1902 and 2014, kindly provided by Markus Weber, formerly of the Bayerische Akademie der Wissenschaften in Munich, and presently at the Institute for Photogrammetry and Remote Sensing of the Technical University of Munich

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whole project that he took his glaciology class through much of it, provided copious comments on the chapters, and attempted many of the exercises. His suggestions and comments have been welcome and invaluable. This book begins, as it ends, with images of Vernagtferner (Fig. 1). This glacier, now almost gone, is an emblem of the book’s title, and of the glaciers of the Ötztal Alps; its valley is visible from Grawand on the summer school excursion. You can read more about it in Chap. 20, or in the absorbing papers of Nicolussi (2011/2012) and Weber (2011/2012). Karthaus, Schnalstal April 2019

Andrew Fowler Felix Ng

References 1. Nicolussi K (2011/2012) Die historischen Vorstöße und Hochstände des Vernagtferners 1600–1850 AD. Zeitschrift für Gletscherkunde und Glazialgeologie 45/46:9–23 2. Weber M (2011/2012) Dokumentation der Veränderungen des Vernagtferners und des Guslarferners anhand von Fotograﬁen. Zeitschrift für Gletscherkunde und Glazialgeologie 45/46: 49–84

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Thermal Structure . . . . . . . . . . . . . . . . . . . . . 2.1 Temperature Proﬁles . . . . . . . . . . . . . . . . 2.2 Boundary Conditions . . . . . . . . . . . . . . . 2.2.1 The Thermal Near-Surface Wave 2.3 Models: Simple to Complicated . . . . . . . .

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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47 47 48 48 50 51 52 52 54 54 54 55 57 59 59 61 62 63 63 66 68 73

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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79 79 80 82 82 85 86 88 88 89 90

2.5

Basal Conditions . . . . . . . . . . . . . . . 2.4.1 Polythermal Ice . . . . . . . . . . Modelling Issues . . . . . . . . . . . . . . . 2.5.1 Non-dimensionalisation . . . . 2.5.2 Thermomechanical Coupling . 2.5.3 Thermal Runaway . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . .

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6

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polar 6.1 6.2 6.3 6.4 6.5

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Mass 7.1 7.2 7.3

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93 93 96 98 102 107 112 114

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117 119 121 126

Meteorology . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shortwave and Longwave Radiation . . . . . . . . . . . Radiation Climate at the Top of the Atmosphere . . Large Scale Circulation . . . . . . . . . . . . . . . . . . . . . Surface Energy Balance . . . . . . . . . . . . . . . . . . . . 6.5.1 Shortwave Radiation . . . . . . . . . . . . . . . . 6.5.2 Surface Albedo . . . . . . . . . . . . . . . . . . . . 6.5.3 Longwave Radiation . . . . . . . . . . . . . . . . . 6.5.4 Turbulent Fluxes . . . . . . . . . . . . . . . . . . . Temperature Inversion and Katabatic Winds . . . . . . 6.6.1 Surface Temperature Inversion and Deﬁcit . 6.6.2 Katabatic Winds . . . . . . . . . . . . . . . . . . . . Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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131 131 132 133 136 138 139 139 142 142 144 144 148 151 153 155

Balance . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . Deﬁnitions . . . . . . . . . . . . . . . . . . . Methods . . . . . . . . . . . . . . . . . . . . . 7.3.1 In Situ Observations . . . . . . 7.3.2 Satellite/Airborne Altimetry 7.3.3 Satellite Gravimetry . . . . . . 7.3.4 Mass Budget Method . . . . . 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 . . . . . . . . .

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175 175 177 178 179 180 181 181

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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185 185 187 187 189 190 191 192 194 195 195 197 198 198 200 201 202 203 204 205 207 208 209 211 211 212 214

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 .

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219 219 219 223 224 225 225

7.6

Antarctic Ice Sheet . . . . . . . . . . . . . . . . . . 7.5.1 Spatial SSMB Variability . . . . . . . 7.5.2 Blue Ice Areas . . . . . . . . . . . . . . . 7.5.3 Temporal SSMB Variability . . . . . Greenland Ice Sheet . . . . . . . . . . . . . . . . . 7.6.1 Spatial SSMB Variability . . . . . . . 7.6.2 Temporal SSMB Variability . . . . . 7.6.3 Role of the Liquid Water Balance .

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241 241 241 243 247 250 254

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 . . . . . . . . . . . . . . . . . . . . . .

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255 255 256 256 259 264 269 269 272 276

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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279 279 280 283 283 285 289 290 293 294 294 294 298

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The Equations of Pseudorange . . . . . . . . . . . . . . . . . . . . Least-Squares Solution of an Overdetermined System of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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308 311 311 313 317 319

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 . . . . . . . . . . . . . . .

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327 327 328 328 331 333 333 334 336 343 343 344

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 . .

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349 349 350 350 350 351 353 355 356 358 358 362 362 362 363

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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363 364 364 365 366 366 366 368 371 373 376

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 Eustatic Sea-Level Change . . . . . . . . . . . . . . . 15.3.3 Departures from Eustasy and the Sea-Level Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.4 Rotational Feedback . . . . . . . . . . . . . . . . . . . . 15.3.5 Spatial Pattern of Sea-Level Change . . . . . . . . 15.3.6 Viscous Effects: Ocean Syphoning . . . . . . . . . 15.3.7 Sea-Level Change as a Controlling Factor on Ice-Sheet Evolution . . . . . . . . . . . . . . . . . . 15.4 Constraining Cryospheric Changes with Observations . . 15.4.1 Global Ice Volumes . . . . . . . . . . . . . . . . . . . . 15.4.2 Meltwater Pulses . . . . . . . . . . . . . . . . . . . . . . 15.4.3 Regional Ice-Sheet Histories . . . . . . . . . . . . . . 15.4.4 Twentieth Century Ice-Sheet and Sea-Level Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.5 Satellite Era Ice-Sheet and Sea-Level Changes . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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383 383 384 384 387

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390 393 393 394

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407 408 409

16 Ice Sheets in the Cenozoic . . . . . . . . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Forcing Mechanisms . . . . . . . . . . . . . . . . . . . . 16.2.1 Changes in the Carbon cycle . . . . . . . 16.2.2 Orbital Cycles and Climate Variability 16.3 From Benthic d18 O to Global Ice Volume . . . .

415 415 416 416 417 . . . . . . . . . . . . 419

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Contents

16.4 Cenozoic Evolution of Ice Volume . . . . . . . . . . 16.4.1 Inception of Antarctica . . . . . . . . . . . . . 16.4.2 Oligocene and Miocene Variability . . . . 16.4.3 Pleistocene Ice Ages . . . . . . . . . . . . . . 16.5 The Last Glacial Cycle . . . . . . . . . . . . . . . . . . . 16.5.1 The Previous Interglacial—The Eemian . 16.5.2 The Last Glacial Maximum . . . . . . . . . Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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420 422 422 423 425 427 428 429

17 Paleoglaciology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Glacial Landforms . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Formation Time and Landform Size-Scale . . 17.2.2 Genetic Information in an Inversion Context 17.3 Data Acquisition and Data Reduction . . . . . . . . . . . 17.4 Reconstruction of Glaciers and Ice Sheets . . . . . . . . 17.4.1 The Inversion Problem . . . . . . . . . . . . . . . . 17.4.2 Data Reduction . . . . . . . . . . . . . . . . . . . . . 17.4.3 An Inversion Procedure . . . . . . . . . . . . . . . 17.5 The Chronological Domain . . . . . . . . . . . . . . . . . . . 17.6 Glaciological Insights from Paleoglaciology . . . . . . .

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460 463 466 467 469

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18 Glacier Fluctuations and Simple Glacier Models . . . . . . . . . 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Simple Glacier Model: Constant Bed Slope, Constant Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 More Complex Geometry . . . . . . . . . . . . . . . . . . . . . . . 18.4 Characteristic Time Scale . . . . . . . . . . . . . . . . . . . . . . . 18.5 Linear Theory of Glacier Length Fluctuations . . . . . . . . 18.6 Inverse Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7 Global Temperature Reconstruction from Glacier Length Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.8 Minimal Glacier Model . . . . . . . . . . . . . . . . . . . . . . . . . 18.9 Nonlinear Behaviour Simulated by the Minimal Glacier Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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19 Tropical Glaciers . . . . . . . . . . . . . . . . . . . . . . . . 19.1 Glaciers and Climate at Low Latitude . . . . . 19.2 Local Mass and Energy Balance . . . . . . . . . 19.2.1 Physical Modelling . . . . . . . . . . . . . 19.2.2 Characteristics of Tropical Glaciers .

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Contents

19.3 Linkage of Glacier Mass Balance to Large-Scale Climate 19.3.1 General Considerations . . . . . . . . . . . . . . . . . . . 19.3.2 Kilimanjaro Case Study . . . . . . . . . . . . . . . . . . 19.3.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . .

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490 490 491 493

20 The History of Glaciology in the Inner Ötztal Alps . . . . . . . . . 20.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 First Descriptions and Sketches of Glaciers . . . . . . . . . . . . 20.4 Detailed Glacier Maps and Length Changes . . . . . . . . . . . . 20.5 Measurements of Ice Flow Velocities: From Stone Lines to Stakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 The Deep Drillings, 1893–1922 . . . . . . . . . . . . . . . . . . . . . 20.7 Twentieth Century Developments . . . . . . . . . . . . . . . . . . . 20.7.1 Gletscherdienst Vent and Early Hydrometeorology . 20.7.2 1950s: Mass and Energy Balance . . . . . . . . . . . . . 20.7.3 1970s: Glacier Hydrology and Firn Properties . . . . 20.7.4 1980s: Numerical Modelling of Energy Balance and Ice Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.8 Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Contributors

Thomas Blunier Centre for Ice and Climate, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark Ed Bueler Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, USA Christo Buizert College of Earth, Ocean and Atmospheric Sciences, Oregon State University, Corvallis, USA Bas de Boer Faculty of Science, Earth and Climate Cluster, VU Amsterdam, Amsterdam, The Netherlands Olaf Eisen Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research, Bremerhaven, Germany; Department of Geosciences, University of Bremen, Bremen, Germany Andrea Fischer Institute for Interdisciplinary Mountain Research, Austrian Academy of Sciences, Innsbruck, Austria Hubertus Fischer Climate and Environmental Physics, Physics Institute, Oeschger Centre for Climate Change Research, University of Bern, Bern, Switzerland Andrew Fowler MACSI, University of Limerick, Limerick, Republic of Ireland Rianne Giesen Utrecht University, Utrecht, The Netherlands Clas Hättestrand Department of Physical Geography, Stockholm University, Stockholm, Sweden; Bolin Centre for Climate Research, Stockholm University, Stockholm, Sweden Michiel Helsen School of Education, Rotterdam University of Applied Sciences, Rotterdam, The Netherlands Ian Hewitt University of Oxford, Oxford, UK Ian Howat School of Earth Sciences, Byrd Polar Research Center, Ohio State University, Columbus, USA

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Contributors

Krister Jansson Department of Physical Geography, Stockholm University, Stockholm, Sweden; Bolin Centre for Climate Research, Stockholm University, Stockholm, Sweden Adrian Jenkins Northumbria University, Newcastle upon Tyne, UK Georg Kaser Institute of Atmospheric and Cryospheric Sciences, University of Innsbruck, Innsbruck, Austria Johan Kleman Department of Physical Geography, Stockholm University, Stockholm, Sweden; Bolin Centre for Climate Research, Stockholm University, Stockholm, Sweden Kurt Lambeck Research School of Earth Sciences, The Australian National University, Canberra, Australia Doug MacAyeal Department of Geophysical Sciences, University of Chicago, Chicago, USA Christoph Mayer Geodesy and Glaciology, Bavarian Academy of Sciences and Humanities, Munich, Germany Glenn Milne Department of Earth and Environmental Sciences, University of Ottawa, Ottawa, Ontario, Canada Thomas Mölg Institute of Geography, Friedrich-Alexander-Universität, ErlangenNürnberg, Erlangen, Germany Robert Mulvaney British Antarctic Survey, Cambridge, UK Felix Ng Department of Geography, University of Shefﬁeld, Shefﬁeld, UK Hans Oerlemans IMAU, Utrecht University, Utrecht, The Netherlands Frank Paul Department of Geography, University of Zurich, Zurich, Switzerland Carleen Reijmer Utrecht University, Utrecht, The Netherlands Helmut Rott Institute of Atmospheric and Cryospheric Sciences, University of Innsbruck, Innsbruck, Austria; ENVEO IT GmbH, Innsbruck, Austria Arjen Stroeven Department of Physical Geography, Stockholm University, Stockholm, Sweden; Bolin Centre for Climate Research, Stockholm University, Stockholm, Sweden Willem Jan van de Berg Utrecht University, Utrecht, The Netherlands Michiel van den Broeke Utrecht University, Utrecht, The Netherlands

Contributors

xxvii

Roderik van de Wal Institute for Marine and Atmospheric Research (IMAU), Utrecht University, Utrecht, The Netherlands Andreas Vieli Department of Geography, University of Zurich, Zurich, Switzerland Pippa Whitehouse Department of Geography, Durham University, Durham, UK

1

Slow Viscous Flow Ian Hewitt

1.1

Introduction

The large-scale movement of glaciers and ice sheets is a problem of slow viscous flow. This can be described mathematically using the framework of continuum mechanics, which relates the distribution of forces and deformation in a material, and allows us to calculate how it will deform under given conditions. Essentially the same mathematical framework applies to compressible and incompressible fluids, both viscous and inviscid, and many of the same ideas apply to elastic solids too. In this chapter, however, we will focus on the case of an incompressible viscous fluid, which is a useful model for describing many aspects of ice flow in glaciers and ice sheets. The continuum approximation treats the material as having a continuous distribution of mass. It therefore applies on scales much larger than inter-molecular distances. Each ‘point’ of the material is ascribed properties, such as density, temperature, velocity, and pressure. Some of these properties are related to each other by constitutive laws—essentially empirical parameterisations of the unresolved molecular mechanics of the material (though sometimes having a theoretical basis too). Further constraints on how the properties vary in space and time are provided by conservation laws. These express the physical principles of mass conservation, momentum conservation (equivalent to Newton’s second law of motion), and energy conservation (equivalent to the first law of thermodynamics). In this chapter we discuss the mathematical statement of these conservation laws and show how these can be used, together with a constitutive law for the rheology, to derive a system of partial differential equations governing the velocity and stress state of a fluid. These are the Navier-Stokes equations. We discuss the corresponding

I. Hewitt (B) University of Oxford, Oxford, UK e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_1

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I. Hewitt

boundary conditions, and the simplifications that apply for a very viscous fluid, which lead to the Stokes equations. Some simple solutions of these equations are described, and a useful approximation—the shallow ice approximation—is derived. To start, however, we must introduce the coordinate system and the idea of a material derivative.

1.2

Coordinate Systems and the Material Derivative

1.2.1

Eulerian and Lagrangian Coordinates

Two different coordinate systems are used to describe continuum mechanics. In Eulerian coordinates (x, t) the spatial coordinate x is fixed in space (that is, in a fixed reference frame, usually taken to be that of the solid Earth). A parcel of fluid will generally move through different coordinates as time t evolves (Fig. 1.1). In Lagrangian coordinates (X, t), the spatial coordinate X is fixed in the material; it labels the same parcel of fluid for all time. It is common to choose X to be equal to the Eulerian coordinate for a reference configuration, at t = 0 say. The equations of fluid flow are most easily formulated in Eulerian coordinates, which we will use throughout this chapter. We variously write the three components of x as (x, y, z) and (x1 , x2 , x3 ). However, it is important to be aware of possible confusion between Eulerian and Lagrangian coordinate systems. For instance, a GPS unit drilled into a glacier surface measures the position (and hence velocity) of the ice in a Lagrangian coordinate (since it moves with the ice and is therefore associated with the same fluid parcel for all time). An automatic weather station drilled into the ice surface measures ice temperature in a Lagrangian system but measures air temperature in an (approximately) Eulerian coordinate system (since the ice is essentially stationary from the point of view of the air moving by).

1.2.2

The Material Derivative

Following the discussion above we can write the Eulerian path followed by a parcel of fluid as x(X, t), where we label the parcel by its reference coordinate X (imagine

Fig. 1.1 A parcel of fluid, labelled by Lagrangian coordinates X, follows Eulerian path x(X, t)

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Slow Viscous Flow

3

dyeing a small patch of the fluid and tracing its path through time). This path is governed by the equation Dx = u, (1.1) Dt where D/Dt is the derivative with respect to time for fixed X, and u is the fluid velocity (this expression can be considered to be the definition of the velocity). The derivative D/Dt is called the material derivative, since it represents the rate of change experienced by a fluid parcel. It is also referred to as the total derivative, or the advective derivative. When we apply this derivative to a function f (x, t) described in Eulerian coordinates, we must use the chain rule to give ∂f Df = + u.∇ f , Dt ∂t

(1.2)

where ∂/∂t is the derivative with respect to time at fixed x, and ∇ is the gradient (the rate of change with respect to x). In components, writing u = (u, v, w) = (u 1 , u 2 , u 3 ), this is Df ∂f ∂f ∂f ∂f = +u +v +w , Dt ∂t ∂x ∂y ∂z

(1.3)

Df ∂f ∂f , = + ui Dt ∂t ∂ xi

(1.4)

or alternatively,

where we use the summation convention, which means that a sum is implied over repeated indices (i. e., over i = 1, 2, 3 in this case). The material derivative is an important concept in fluid dynamics, since it is necessary to distinguish between time derivatives at a fixed position and time derivatives following the fluid. For example, if a glacier is in a steady state, the time derivative at each position (i. e., ∂/∂t) is zero. Nevertheless any particular parcel of ice moves through the glacier with time and therefore experiences changes in pressure p, say, as it is first buried and then exhumed by surface melting. Thus ∂ p/∂t is everywhere zero, but Dp/Dt is not.

1.3

Mass Conservation

To construct a mathematical statement of mass conservation, consider an arbitrary fixed volume V within the fluid (Fig. 1.2). The mass within this volume can only change due to the movement of material across its boundary ∂ V (since mass is neither created nor destroyed). Thus, we can write d dt

ρ dV = − V

∂V

ρu.n d S,

(1.5)

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Fig. 1.2 a A fixed volume V , and b a material volume V (t)

where ρ(x, t) is the density, u(x, t) is the velocity, and n is the outward pointing unit normal to the boundary (so −ρu.n is the rate at which mass enters the volume at each point on the boundary). Using the divergence theorem, and noting that V is fixed with respect to time, this can be re-written as ∂ρ (1.6) + ∇. (ρu) d V = 0, ∂t V and since this must hold for any volume V , it must be the case that ∂ρ + ∇.(ρu) = 0. ∂t

(1.7)

(If this were not the case and this quantity were non-zero over some region, we could take V to be within that region and would then have a contradiction; as a technicality, this assumes that ρ and u are continuously differentiable.) If the material is incompressible, as is usually assumed for glacial ice once it has compacted sufficiently, then Dρ ∂ρ = + u.∇ρ = 0, Dt ∂t

(1.8)

so (1.7) reduces to the expression ∇.u = 0.

(1.9)

That is, mass conservation demands that the velocity is divergence-free. This equation is often referred to as the continuity equation. An alternative but equivalent derivation of this equation is to consider a material volume V (t); that is, a volume made up of the same fluid parcels for all time (which may therefore move and change shape in Eulerian coordinates). The mass within this volume must be the same for all time (no mass crosses its boundary by definition), so we can write d ρ d V = 0. (1.10) dt V (t) The transport theorem tells us that for any function f (x, t) and material volume V (t) moving with velocity u(x, t), we have ∂f d f dV = + ∇.( f u) d V , (1.11) dt V (t) V (t) ∂t

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Slow Viscous Flow

5

and applying this to the above expression gives ∂ρ + ∇.(ρu) d V = 0. V (t) ∂t

(1.12)

Again, since this must hold for any volume V (t), we recover the conservation equation (1.7).

1.4

The Stress Tensor and Momentum Conservation

1.4.1

The Stress Tensor

The stress state in a material is described by means of a tensor (a matrix) σ , whose components σi j represent the force per unit area in the i direction on a surface with normal in the j direction. We also often use x, y, z to label the components, so that ⎛ ⎞ σx x σx y σx z σ = (σi j ) = ⎝ σ yx σ yy σ yz ⎠ . (1.13) σzx σzy σzz The meaning of the stress tensor is most easily understood by considering a small cube within the material (Fig. 1.3). The jth column gives the stress (i. e., the force per unit area, a vector—also often referred to as traction) acting on the face of the cube with normal in the j direction. As a consequence of torque balance on such an infinitesimal cube (which has zero moment of inertia), it must be the case that the stress tensor is symmetric. The stress state is therefore described by the six independent components of σ . The stress acting on a surface with unit normal vector n is given by s = σ .n, or si = σi j n j ,

(1.14)

using index notation with the summation convention. We define the pressure p to be the negative mean of the diagonal components of the stress tensor p = − 13 σii = − 13 (σx x + σ yy + σzz ),

(1.15)

Fig. 1.3 The stress acting on the jth face of a cube, and on an arbitrary surface with unit normal n

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I. Hewitt

and then decompose the stress tensor into σ = − p δ + τ , or σi j = − pδi j + τi j ,

(1.16)

where δ = (δi j ) is the identity matrix, and τ = (τi j ) is the deviatoric stress tensor. This indicates how much the stress state deviates from being isotropic (being independent of direction). The deviatoric stress tensor is typically related to velocity gradients by means of a constitutive rheological law, and this is where the distinction between different materials (water, ice, etc.) will come in. For the moment however, we proceed with a general framework that makes no assumptions about the nature of the material.

1.4.2

Momentum Conservation

Consider again an arbitrary fixed volume V in the material. The momentum of the material in this volume is given by ρu d V .

(1.17)

V

Changes in this momentum can be due to the movement of material into and out of the volume (which carries with it momentum), and due to the action of forces on the body. These forces are a combination of the body force (gravity g) that acts on the mass within the volume, and the stress that acts on the boundary ∂ V due to the material outside. The latter can be expressed in terms of the stress tensor, so that the overall statement of momentum conservation is written as d ρu d V = − ρu u.n d S + ρg d V + σ .n d S. (1.18) dt V ∂V V ∂V Manipulation using the divergence theorem, and making use of the mass conservation equation (1.7), leads to

∂u ρ + u.∇u − ρg − ∇.σ d V = 0, ∂t V

(1.19)

and since V is arbitrary we conclude that ρ

∂u + u.∇u = ρg + ∇.σ . ∂t

(1.20)

This is the generic differential equation describing momentum conservation, and will be combined with the mass conservation equation (1.9) to solve for the fluid velocity u. This requires us now to describe the rheology of the material.

1

Slow Viscous Flow

1.4.3

7

Rheology

Rheology refers to the relationship between force and deformation of a material; for the purpose of developing a mathematical model, what is required here is a relationship between the stress tensor and the velocity components. The velocity itself is not the relevant quantity to describe deformation, however, since it could be altered simply by changing the frame of reference. What is important instead is the relative velocity of neighbouring points in the material. This is described by the strain-rate tensor, ε˙ = (˙εi j ), defined as ε˙ =

1 2

∂u j 1 ∂u i . ∇u + ∇uT , or ε˙ i j = + 2 ∂x j ∂ xi

(1.21)

Many other symbols are commonly used for this quantity, including ei j , γ˙i j , and Di j . Note that the strain-rate tensor is symmetric by definition and that for an incompressible material the diagonal components sum to zero. The diagonal components represent stretching deformations, while the off-diagonal components represent shearing deformations. In general, a glacier is undergoing all sorts of deformation at the same time, but it is often dominated by one or other of the components. For example, near an ice dome (where ice cores are commonly collected), the motion is almost entirely vertical, with compression in the vertical direction and stretching in the horizontal direction, so the diagonal components of the strain-rate tensor dominate. In a steep mountain glacier, on the other hand, the ice near the surface moves much faster than the ice near the bed and the shear rate ε˙ x z dominates. The simplest rheological law is a linear relationship between the deviatoric stress tensor and the strain-rate tensor, τi j = 2η˙εi j .

(1.22)

The proportionality constant η is the viscosity (the factor of 2 is included simply by convention). Fluids with this relationship in which η is constant, or at least is independent of the stress, are called Newtonian. This turns out to be a very good approximation for many fluids, notably for air and water. For ice, it is not such a good approximation, and the most common law used instead is Glen’s law, which is written as ε˙ i j = Aτ n−1 τi j ,

τ=

1 2 τi j τi j .

(1.23)

Here τ is the second invariant of the stress tensor, n is a power-law exponent often taken to be equal to 3, and A(T ) is a temperature-dependent rate factor. This can also be written as 1 , (1.24) τi j = 2η˙εi j , where η = 2 Aτ n−1

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I. Hewitt

which takes the appearance of a Newtonian fluid, but with η now being a (nonconstant) effective viscosity. The case n = 1 reverts to a Newtonian fluid, for which the viscosity may vary with temperature but is independent of stress. More generally, fluids with the type of rheology described by (1.23) are referred to as power-law fluids (shear-thinning if n > 1 and shear-thickening if n < 1). This is often a good model for polycrystalline materials (such as ice), which deform through a variety of creep mechanisms.

1.4.4

The Navier-Stokes Equations

For a Newtonian fluid, we have

∂u j ∂u i σi j = − pδi j + η + ∂x j ∂ xi

,

(1.25)

and substituting into the momentum equation (1.20) we obtain, in index notation, ρ

∂u i ∂u i +uj ∂t ∂x j

∂u j ∂u i − pδi j + η + ∂x j ∂ xi

= ρgi +

∂ ∂x j

= ρgi −

∂p ∂ 2ui +η 2 , ∂ xi ∂x j

(1.26)

making use of the continuity equation ∂u j /∂ x j = 0 to simplify the final term (and taking η to be constant). The resulting mass and momentum equations are therefore ∇.u = 0,

ρ

Du = ρg − ∇ p + η∇ 2 u, Dt

(1.27)

where we have made use of the shorthand for the material derivative (1.2). These are the Navier-Stokes equations for an incompressible viscous fluid. They are four equations to solve for the three components of u and the pressure p. Hidden inside the material derivative is the nonlinear term (u.∇)u which makes these equations hard to solve in general. In the case of ice flow, however, the inertial term on the left hand side is unimportant and approximate solutions can be found, as described below. The Navier-Stokes equations are often used to describe flows in the atmosphere and ocean, and in that case the inertial terms are important. One point that is worth noting in that context is that our derivation assumed an inertial coordinate system (i. e., one that is not accelerating, so that Newton’s second law of motion holds). Since the Earth is rotating, a modification is necessary to account for this. The material derivative is replaced by Du + 2 ∧ u + ∧ ( ∧ x), Dt

(1.28)

1

Slow Viscous Flow

9

where is the angular velocity, and D/Dt is the material derivative in the rotating coordinate system (i. e., the usual reference frame of the solid Earth). The second term here is referred to as the Coriolis force and has an important role to play in atmospheric and oceanic circulation. The third term can be absorbed into a modified pressure and is of less importance. None of these modifications are necessary when considering ice flow, however, since the corrections are negligible.

1.4.5

Stokes Flow

For very viscous fluids such as ice, the inertial terms on the left hand side of the momentum equation are negligible compared to the viscous and gravitational terms on the right hand side. This can be seen by estimating the sizes of the terms. Denoting by L a typical length scale (say 100 m), by U a typical velocity scale (say 30 m y−1 ≈ 10−6 m s−1 ), and by P = ρgL a typical pressure scale, the magnitude of the terms in the momentum equation in (1.27) can be estimated as ρU 2 ηU . , ρg, and L L2

(1.29)

Taking rough values for the density ρ = 103 kg m−3 , gravity g = 10 m s−2 , and effective viscosity η = 1013 Pa s, these are of order 10−11 , 104 , and 103

kg m−2 s−2 .

(1.30)

We see that the inertial term is much smaller than the other two, and is therefore negligible by comparison. The ratio of the inertial term to the viscous stress term is referred to as the Reynolds number, Re =

ρU L ≈ 10−14 . η

(1.31)

Whenever this dimensionless ratio is much less than 1, it indicates that inertia is negligible, and there is a balance between gravity, pressure gradients, and viscosity. When it is large, on the other hand (as is the case for atmospheric flows), it indicates that viscosity is negligible and there is instead a balance between pressure gradients and inertia (including the Coriolis terms in that case). This type of reasoning applies even for a non-Newtonian fluid such as ice, when a suitable value for the effective viscosity can be used to estimate the Reynolds number. It indicates that the left hand side of the original momentum equation (1.20) can be ignored for ice flow, giving the reduced equations of mass and momentum conservation, ∇.u = 0,

0 = ρg − ∇ p + ∇.τ ,

(1.32)

which are to be solved together with the constitutive flow law (1.23) that relates τ to u. These are the Stokes equations for an incompressible viscous fluid. Despite

10

I. Hewitt

their apparent simplicity when expressed in this form, they are made complicated by the nonlinearity in the flow law, and by the coupling through that flow law to the temperature. Since there are no time derivatives in the Stokes equations, they can be solved to give the instantaneous velocity field for any given domain and temperature field (the temperature enters into the viscosity). To solve the equations we require some boundary conditions, which we now discuss.

1.5

Boundary Conditions

We must distinguish between two types of boundaries; those that are prescribed, and those that are free and which must be determined as part of the solution. In the context of glacier flow, the lower surface (the glacier bed) is typically prescribed, but the upper surface is a free boundary. For floating ice shelves, both the lower surface and the upper surface (and the calving front) are free boundaries. At prescribed boundaries we usually impose conditions on the velocity components. At free boundaries, we impose both a kinematic condition (which is also effectively a condition on the velocity components, but related to the unknown movement of the boundary) and a dynamic condition on the stress components. The additional conditions imposed at free boundaries should be sufficient to determine their location or movement.

1.5.1

The No-Slip Condition and the Sliding Law

The usual condition for a viscous fluid at a rigid boundary is that there is no slip, so that the fluid moves with the prescribed velocity of the boundary (which is often stationary). This can be broken down into conditions on the normal and tangential components, as u.n = vn ,

u − (u.n)n = vb ,

(1.33)

where n is the unit normal to the bed, pointing upwards, vn is the normal velocity at the boundary and vb is the tangential velocity. The first of these is sometimes referred to as a no-penetration condition if vn = 0, while the second is the no-slip condition if vb = 0. The no-slip condition applies at the bed of a glacier that is frozen to its bed (though see later discussion on the possibility of slip even for temperatures below the melting point). When the bed is at the melting point, the presence of a thin layer of water allows slip to occur, so that the tangential velocity of the ice need not be the same as that of the boundary. This situation is usually described by means of a slip law that relates the tangential velocity to the local shear stress, and which we write in the general form u b = F(τb ),

(1.34)

1

Slow Viscous Flow

11

for some function F. In the glaciological context, this law is usually referred to as the sliding law, and its form will be treated in more detail in Chap. 3. Here u b is the magnitude of the tangential velocity ub = u − (u.n)n, and τb is the magnitude of the basal shear stress τ b = τ .n − {(τ .n).n}n. In vector form, the sliding law becomes ub = F(τb )

τb , τb

(1.35)

expressing the fact that the sliding velocity should be aligned with the direction of the basal shear stress. Given that melting can occur at the bed when the melting point is reached, the no-penetration condition should also be modified to account for the possibility of mass loss at the bed (into the subglacial drainage system), u.n = −m b ,

(1.36)

where m b is the basal melt rate, measured as a velocity, i. e., volume flux per unit area. However, the basal melt rate is typically small compared with the velocities of interest and the condition u.n = 0 is often used.

1.5.2

Dynamic Boundary Conditions

At a free surface, the stress on the boundary σ .n is prescribed to match the stress in whatever material is the other side of the boundary (surface tension, which can cause a discontinuity in stress components in other circumstances, is negligible for a glacier). Typically, the other side of the boundary is the air or the ocean, which is so inviscid by comparison that the stress to be matched with is just the hydrostatic pressure pb , such that σ .n = − pb n.

(1.37)

On the upper surface of a glacier, pb = pa where pa is the atmospheric pressure. In the ocean, pb = pa − ρw gz, where z is the vertical coordinate (upwards) relative to sea level and ρw is the ocean water density. It is common to take atmospheric pressure as the reference pressure so that pa = 0.

1.5.3

Kinematic Boundary Conditions

For a free boundary that is a material surface (one that is made up of the same fluid parcels for all time), the kinematic condition states that the fluid on the boundary moves with the velocity of the boundary, u.n = vn ,

u − (u.n)n = vb .

(1.38)

This is in fact just the same as the no-slip condition, although we are now treating this condition as determining the boundary velocity rather than that velocity being

12

I. Hewitt

prescribed. In the glaciological context, the free boundary is usually denoted by z = s(x, y, t), and the condition can be written in terms of velocity components as ∂s ∂s ∂s +u +v = w. ∂t ∂x ∂y

(1.39)

If the boundary is not a material surface, because mass is either added to or subtracted from the boundary through time, this condition must be modified. This is usually the case on a glacier since the upper surface is either accumulating or melting, as is the lower surface of an ice shelf. If mass is added to the boundary at a normal rate an , we can write the normal velocity of the boundary as vn = u.n + an .

(1.40)

(If the fluid is stationary, the boundary moves at a rate an ; if fluid is transported away from the boundary at a rate u.n = −an that matches the accumulation rate then vn = 0 and the boundary doesn’t move.) When converted into the form of (1.39) this becomes ∂s ∂s ∂s +u +v = w + a, (1.41) ∂t ∂x ∂y where a is the vertical accumulation rate (a = an 1 + (∂s/∂ x)2 + (∂s/∂ y)2 ; the difference between a and an is typically small since the normal is close to vertical).

1.6

Temperature and Energy Conservation

The energy equation is derived in an analogous fashion to the momentum equation. We consider a fixed volume V and consider the rate of change of energy for the fluid within that volume. This is the sum of internal energy e (heat) and kinetic energy (though the latter is very small for an ice sheet since the velocities are so small, we retain it for consistency with the inertial terms in the derivation of the momentum conservation equation). The energy within the volume can change due to the advection of mass (and associated energy) into and out of the volume, due to the conduction of heat across the boundary (described by Fourier’s law of conduction with conductivity k), and due to the work done by the forces acting. Accounting for the same forces as included in our derivation of the momentum equation, we have d dt

V

ρ e + 21 |u|2 d V = −

u.n d S + ρ e+ k∇T .n d S ∂V ∂ V + ρu.g d V + u.(σ .n) d S. (1.42) V

2 1 2 |u|

∂V

Manipulation using the divergence theorem, the mass conservation equation (1.7), the momentum equation (1.20), and the thermodynamic relation De/Dt = c p DT /Dt

1

Slow Viscous Flow

13

Fig. 1.4 A model ice sheet

(with c p being the specific heat capacity), leads to the conclusion that ρc p

∂T + u.∇T ∂t

= ∇.(k∇T ) + τi j ε˙ i j .

(1.43)

This is an advection-diffusion equation for the temperature, with a source term that represents viscous dissipation. Since the flow-law rate constant A(T ) depends on temperature, the solution to the Stokes equations for the ice velocity is inherently coupled to the solution of this energy equation. However, it is quite common to make simplifications such as to ignore the temperature dependence of A, in which case the equations decouple; one can first solve the Stokes equations to find the velocity and then solve this equation for the temperature with prescribed velocity and dissipative source term. Boundary conditions for the energy equation can vary in complexity depending on the scale and temperatures of interest. The simplest situation occurs if both the upper surface and the lower surface of the glacier remain below the melting point yearround. In that case it is reasonable to set the temperature at the upper surface to be equal to the mean annual air temperature, Ta say, and to set the temperature gradient at the lower surface to balance the (prescribed) geothermal heat flux, G. If either the surface or the base of the ice sheet reaches the melting point Tm , the condition is replaced with the condition T = Tm . The possibility for melting or freezing in that case means that an energy balance at the boundary must also be considered, and related carefully to the respective kinematic condition. This will be discussed more in Chap. 2. The energy balance can also be used to determine if and when the temperature at the boundary falls below the melting point again. Further complexity occurs if the temperature within the ice reaches the melting point, in which case internal melting can occur and the energy equation itself must be modified to account for latent heat effects.

1.7

Glacier and Ice Sheet Flow

Here we summarise for completeness a full set of equations to describe the flow of a grounded glacier or ice sheet. We denote the fixed lower surface (the ‘bed’) as z = b(x, y) and the free upper surface (the ‘surface’) as z = s(x, y, t) (Fig. 1.4). For b < z < s, we must solve the Stokes equations,

14

I. Hewitt

∇.u = 0,

0 = ρg − ∇ p + ∇.τ ,

(1.44)

and the energy equation, ρc p

∂T + u.∇T ∂t

= ∇.(k∇T ) + τi j ε˙ i j ,

(1.45)

where the components of the deviatoric stress tensor are given by τi j = 2η˙εi j ,

η=

−(n−1)/2 −(n−1)/2n 1 1 1 1 τi j τi j = ε˙ i j ε˙ i j , (1.46) 2A 2 2 A1/n 2

and the strain-rate tensor is ε˙ i j =

1 2

∂u j ∂u i + ∂x j ∂ xi

.

(1.47)

The upward pointing normals to the surface and bed are given by n=

(−∂s/∂ x, −∂s/∂ y, 1) 1 + (∂s/∂ x)2

+ (∂s/∂ y)2

(−∂b/∂ x, −∂b/∂ y, 1) n= , (1.48) 1 + (∂b/∂ x)2 + (∂b/∂ y)2

,

so the basal boundary conditions can be written as u

∂b ∂b +v = w, ∂x ∂y

ub = F(τb )

τb on z = b(x, y), τb

(1.49)

and the surface boundary conditions are ∂s ∂s ∂s +u +v = w + a, ∂t ∂x ∂y

− pn + τ .n = 0 on z = s(x, y, t),

(1.50)

where a is the prescribed surface accumulation rate (corresponding to ablation if negative). Boundary conditions for the thermal problem, assuming that the bed is at the melting point and the surface below the melting point, are T = Tm at z = b(x, y),

T = Ta at z = s(x, y, t).

(1.51)

The difficulties in solving this problem come from (i) the nonlinearity in the flow law, (ii) the coupling between temperature and viscosity, and (iii) the complexities hidden in the sliding law. In many cases, these difficulties are lessened by approximating the model in some way. A common approximation is to make use of the

1

Slow Viscous Flow

15

relatively large aspect ratio of most glaciers,1 which allows some terms in the equations to be neglected by comparison with others. Another is to ignore the temperature dependence of the flow-law coefficient, or to treat the temperature as fixed in time. A combination of such approximations allows the equations to be integrated over the vertical coordinate so as to remove one of the dimensions from the problem. Such depth-integrated models are often used as a means to reduce the computational effort required: see for example Chaps. 8 and 10. An important step in motivating such approximation of the equations is that of non-dimensionalisation. This is the process of scaling each variable by its typical size, so as to remove the dimensions from all terms in the equations. It is then possible to be more precise about saying that a particular term is ‘small’, since it can be compared with other terms on an equal footing. The task of non-dimensionalising the above equations is performed in the appendix, Sect. 1.11.

1.8

Examples

1.8.1

Uniform Flow on a Slope

An important situation for which an exact solution to the Stokes equations is possible is that of a ‘slab’ glacier, having uniform thickness h and resting on a uniform bed slope with angle α (Fig. 1.5). We also assume that the flow law coefficient A is constant. It is convenient to choose coordinates aligned with the slope so that the flow is independent of y and has velocity only in the x direction. In this case the mass and momentum equations become ∂ p ∂τx x ∂τx z ∂u = 0, 0 = − + + + ρg sin α, ∂x ∂x ∂x ∂z ∂ p ∂τx z ∂τzz 0=− + + − ρg cos α. ∂z ∂x ∂z

(1.52)

The first equation simplifies the other two to 0=−

∂ p ∂τx z ∂p + + ρg sin α, 0 = − − ρg cos α, ∂x ∂z ∂z

(1.53)

together with the surface conditions p = τx z = 0 at z = h. Thus p(z) = ρg(h − z) cos α,

1 The

τx z (z) = ρg(h − z) sin α.

(1.54)

aspect ratio is the ratio between two principal length scales of the flow; in the present context these are the depth and the horizontal extent, but whether one calls the aspect ratio the length/depth ratio or its inverse is a matter of taste. In the present case we refer to a large aspect ratio as a large length to depth ratio; this usage is similar to that in Chap. 2 but not Chap. 8.

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I. Hewitt

Fig. 1.5 A slab glacier on a slope with angle α

The only non-zero component of the flow law is the x z component, which reads 1 ∂u = Aτxnz 2 ∂z

(1.55)

(since τ = τx z in this case), and we can integrate assuming no slip at the bed to find u(z) =

2 A(ρg sin α)n n+1 − (h − z)n+1 . h n+1

(1.56)

This describes the typical velocity profile of a glacier, increasing rapidly with distance from the bed and more slowly with depth near the surface. In the case of a Newtonian fluid (n = 1), the profile is parabolic; for n = 3 it is quartic. This solution for a uniform slab is more relevant than it might seem at first sight, because it turns out to be a good approximation to the solution even when the thickness and the slope vary with x. This is because glaciers tend to be relatively shallow compared to their length. As a consequence, even though other components of the stress tensor and force balance enter the equations, the shear stress terms that were included above are still the dominant ones. This is the basis of the so-called shallow ice approximation, discussed further below.

1.8.2

Spreading Flow at an Ice Divide

Another situation in which a straightforward solution to the equations can be found is for a spreading flow when free slip is allowed at the bed (Fig. 1.6). This can be approximately what occurs near an ice divide where the horizontal velocity changes sign and where the ice undergoes vertical compression and horizontal extension. A similar type of flow occurs in some ice shelves. Here we have u = (λx, 0, −λz),

(1.57)

1

Slow Viscous Flow

17

Fig. 1.6 A spreading flow

where x and z are the horizontal and vertical coordinates, and λ is the constant strain rate. This can be related to the steady-state accumulation rate a by λ = a/h, where z = h is the elevation of the upper surface. The corresponding stress tensor is ⎛

⎞ 0 A−1/n λ1/n 0 ⎠, 0 0 0 τ =⎝ −1/n 1/n λ 0 0 −A

(1.58)

so that the momentum equations are satisfied with p(z) = ρg(h − z) − A−1/n λ1/n .

1.8.3

(1.59)

Small-Amplitude Perturbations

A useful tool for analysing many fluid-flow problems is that of linearisation. A complicated flow is considered as a small perturbation of a more straightforward flow, allowing the equations and/or boundary conditions to be linearised so that a solution is then much easier to find. As an example, we consider flow down a nearly uniform slope, which can be considered a small perturbation to the uniform flow considered above (Fig. 1.7). To facilitate progress we treat the fluid as Newtonian. We suppose that the bed is at z = B(x), and that the surface is at z = h + S(x), where the capitalised variables are assumed small. It is of primary interest how the surface perturbation S(x) relates to the bed perturbation B(x); the surface is assumed here to have reached steady state, and thus has no time dependence. In particular, for instance, could we infer the shape of the bed from the shape of the free surface? To a first approximation the problem is that of uniform flow on a slope and the solutions for pressure and velocity are given by (1.54) and (1.56), which we denote by p0 (z) and u 0 (z), p0 (z) = ρg(h − z) cos α, u 0 (z) =

ρg sin α 2 h − (h − z)2 . 2η

(1.60)

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I. Hewitt

Fig. 1.7 Small-amplitude perturbations to a uniform flow

We perturb these by writing p = p0 (z) + P(x, z), u = u 0 (z) + U (x, z), w = W (x, z),

(1.61)

where again the capitalised variables are assumed small. Substituting into the mass and momentum equations, we have ∂U ∂W + = 0, ∂x ∂z

0=−

∂P + η∇ 2 U , ∂x

0=−

∂P + η∇ 2 W . ∂z

(1.62)

The boundary conditions at z = B and z = h + S are linearised onto z = 0 and z = h respectively. The no-penetration and no-slip conditions become U = −u 0 (0)B at z = 0,

W = 0,

(1.63)

while the surface stress conditions become − P + 2η

∂U ∂W + = −u 0 (h)S at z = h, ∂z ∂x

∂W = p0 (h)S, ∂z

(1.64)

and the kinematic condition (ignoring accumulation) is u 0 (h)

dS = W at z = h. dx

(1.65)

This problem can be solved by taking a Fourier transform in x, defined by ˆ B(k) =

∞

−∞

B(x)eikx d x.

(1.66)

After lots of algebra, we find ˆ ˆ S(k) = Kˆ (k) B(k),

(1.67)

1

Slow Viscous Flow

19

1 0.8 0.6 0.4 0.2 0

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig.1.8 Magnitude of the transfer function Kˆ as a function of kh for slope angle α = 0.1 (solid) and α = 0.2 (dashed). The inset shows the free surface perturbations S(x) for different slab thicknesses h when the bed has a sinusoidal perturbation B(x) as shown by the thicker line (the amplitude is arbitrary)

where Kˆ can be considered to be a transfer function that translates perturbations in the bed to the surface. This function is defined by the complex-valued expression (see exercise 1.1) Kˆ =

2 cosh κ

, sinh κ cosh κ − κ 1 + κ 2 + cosh2 κ + i cot α κ2

(1.68)

where κ = kh, and its modulus is shown in Fig. 1.8 (the argument of the complex Kˆ for each wavenumber k represents a phase shift of the surface disturbance with respect to the bed undulation). Importantly, Kˆ depends only on the scaled wavenumber kh, and tends to zero for kh 1 (specifically, Kˆ ∼ 4e−kh ). This indicates that bed perturbations with a wavelength (2π/k) much smaller than the ice thickness are hardly expressed on the surface. The inversion of the Fourier transform allows the surface perturbation to be expressed as a convolution of the bed perturbation with the transfer kernel, S(x) =

∞

−∞

K (x − s)B(s) ds,

(1.69)

where K (x) is the inverse Fourier transform of Kˆ (k).

1.9

The Shallow Ice Approximation

As an illustration of a reduced model for ice flow, we consider the simplest possible approximation to the equations in (1.44)–(1.51). This is referred to as the shallow ice approximation, and derives from assuming that the aspect ratio of the flow is large. That is, variations in the horizontal coordinate are much smaller than variations in

20

I. Hewitt

the vertical. This is a reasonable approximation in many situations, especially for grounded ice that does not slide too fast (ice shelves and rapidly moving ice streams are not well described by the model as described below, but are susceptible to an alternative method of approximation: the shallow shelf approximation). We ignore all components of the deviatoric stress tensor except for the vertical shear stress τx z and τ yz (this is justified by the non-dimensionalisation in the appendix, Sect. 1.11; we simply ignore all terms of order ε or smaller in the scaled equations). Taking z as the vertical coordinate, the momentum equations are therefore approximated as 0=−

∂ p ∂τ yz ∂p ∂ p ∂τx z + , 0=− + , 0=− − ρg, ∂x ∂z ∂y ∂z ∂z

(1.70)

and the surface boundary conditions are p = τx z = τ yz = 0 at z = s(x, y, t).

(1.71)

These can be solved to give p = ρg(s − z),

τx z , τ yz = −ρg(s − z)∇s,

(1.72)

where we use ∇ = (∂/∂ x, ∂/∂ y) as the two-dimensional gradient for this section. The flow law becomes

∂u ∂v = −2 A(ρg)n (s − z)n |∇s|n−1 ∇s, (1.73) , ∂z ∂z and this equation is subject to the sliding law at the bed, which we take to be (u, v) = −F(τb )|∇s|−1 ∇s at z = b(x, y),

(1.74)

where τb = |ρgh∇s| is the basal shear stress, and h = s − b is the ice thickness. If we assume for simplicity that the flow-law coefficient A is constant, then integration of (1.73) yields (u, v) = −F(|ρgh∇s|)|∇s|−1 ∇s −

2 A(ρg)n n+1 h − (s − z)n+1 |∇s|n−1 ∇s. (1.75) n+1

We can perform a further integral to give a formula for the depth-integrated ice flux,

s

2 A(ρg)n n+2 h |∇s|n−1 ∇s. n+2 b (1.76) The first term here represents the ice flux due to sliding (the sliding velocity times the ice thickness), and the second term represents the ice flux due to the shearing velocity profile in the ice. One or other of these terms may dominate depending on the magnitude of the sliding speed. q=

(u, v) dz = −F(|ρgh∇s|) h|∇s|−1 ∇s −

1

Slow Viscous Flow

21

It remains to make use of the mass conservation equation, the no-penetration condition at the bed, and the kinematic boundary condition at the surface. To do this we integrate the mass equation over the depth of the ice, giving

s

b

∂u ∂v + ∂x ∂y

s dz + w b = 0.

(1.77)

Using the boundary conditions u

∂b ∂b +v = w at z = b, ∂x ∂y

∂s ∂s ∂s +u +v = w + a at z = s, (1.78) ∂t ∂x ∂y

this becomes ∂h + ∇.q = a, (1.79) ∂t where q is the ice flux defined in (1.76). This equation represents depth-integrated mass conservation and takes a form that is very standard for such conservation laws. That is, the rate of change of a conserved quantity (h in this case), plus the divergence of the flux (q), is equal to the sources and sinks (a). The combination of (1.76) and (1.79) gives a nonlinear diffusion equation for the ice thickness h(x, y, t), with the source term due to surface accumulation a. In the case of a flat bed, with no sliding, the equation reduces to 2 A(ρg)n n+2 ∂h = ∇. h |∇h|n−1 ∇h + a. ∂t n+2

(1.80)

This provides a rather simplified model for an ice sheet. As a simple model for a mountain glacier, we can consider the case of a two-dimensional flow down a uniform bed slope in the x direction, in which case the equation becomes (see exercise 1.3) 2 A(ρg sin α)n ∂ ∂h + ∂t n+2 ∂x

h

∂h n−1 ∂h 1 − cot α = a, (1.81) 1 − cot α ∂ x ∂x

n+2

where α is the angle of the bed slope, and here we have chosen coordinates (x, z) aligned to the bed slope (x) and orthogonal to it (z). For a relatively steep mountain glacier it is often the case that ∂h/∂ x is small compared to the bed slope tan α, and this equation can be well approximated by ∂h ∂h + 2 A(ρg tan α)n h n+1 = a, ∂t ∂x

(1.82)

which is a kinematic wave equation for the ice thickness h. For given accumulation rate, there is an approximate steady state given by

n+2 h(x) = 2 A(ρg tan α)n

1/(n+2)

x

a dx 0

,

(1.83)

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I. Hewitt

taking x = 0 as the head of the glacier. The form of (1.82) indicates that seasonal perturbations to this steady-state thickness propagate down glacier as waves, moving with a speed that is n + 1 times the surface ice speed.

1.10

Conclusions and Outlook

In this chapter we have introduced some common notation and derived the standard equations used to describe the flow of a viscous fluid. We have discussed the use of Eulerian and Lagrangian coordinates and the need to consider different types of time derivative, whether fixed in space or following the material. The principles of mass and momentum conservation were used to derive partial differential equations that govern the velocity and stress state, and energy conservation was used to derive the temperature equation. For Newtonian fluids, the mass and momentum equations reduce to the Navier-Stokes equations, while for very viscous flows they are approximated by the Stokes equations, which generalise to a non-Newtonian rheology that is more appropriate to describe ice flow. We have discussed the boundary conditions that apply at the fixed bed and the free upper surface of a glacier, and provided some examples of the use of these equations. Whilst some numerical ice sheet models solve the full Stokes equations as presented here, most of them make some further approximation to the equations such as the shallow ice approximation. Which type of approximation is appropriate depends on what exactly is being considered, and there is no universally ‘correct’ model. Even the full Stokes equations as given here are not sufficient to describe all the complexities of glacier flow, and it is often overkill to spend significant computational resources solving the full model. For instance, many of the interesting dynamics of glaciers are associated with sliding at the bed, and the details of this have been brushed crudely into the sliding law. We have also ignored anisotropy in the rheology, which is known to be important in some areas (though this can be included in the model with modifications), and other dynamics such as calving, which is an essentially brittle process, must be parameterised within this model. The standard equations of viscous fluid flow therefore form the theoretical framework, but many of the interesting dynamics require additional modelling. Finally, we close with some references. A classic general text on fluid dynamics is the book by Batchelor [1], while a very readable introduction is provided by Worster [2]. The Physics of Glaciers by Cuffey and Paterson [3] is a primary reference for many of the concepts specific to ice flow. Following earlier theoretical work, notably by Nye, Weertman and Lliboutry, the equations describing glacier flow began to be framed in a form more familiar to modern fluid dynamicists (involving partial differential equations and systematic scaling arguments) with the papers of Fowler and Larson [4], Morland and Johnson [5], and Hutter [6] amongst others. There are now a number of glaciological texts that include theoretical material along these lines, including the books by Hooke [7], Greve and Blatter [8], and van der Veen [9]. A recent review paper on the fluid dynamics of ice sheets is by Schoof and Hewitt [10].

1

Slow Viscous Flow

1.11

23

Appendix: Non-dimensionalisation

Here we non-dimensionalise the model for an ice sheet in (1.44)–(1.51). We scale each variable with a typical value, chosen either from observation or by selecting a relevant balance between terms in the equations. We denote these ‘scales’ by square brackets, and the associated non-dimensionalised variable with a hat. For instance, using a time scale [t], a horizontal length scale [x], and a vertical length scale [h], we write t = [t]tˆ,

(x, y) = [x](x, ˆ yˆ ),

ˆ sˆ ). (z, b, s) = [h](ˆz , b,

(1.84)

We then define the inverse of the aspect ratio to be ε=

[h] , [x]

(1.85)

which is typically small. Motivated by the necessary balance of terms in the continuity equation, we scale the horizontal velocities with [u] (whose size will be chosen shortly) but the vertical velocity with ε[u], so that (u, v, w) = [u](u, ˆ v, ˆ εw). ˆ

(1.86)

Pressure is scaled with [ p] = ρg[h] (based upon the expected hydrostatic balance), the stress tensor is scaled with [τ ] = ρg[h]2 /[x],

(1.87)

and the strain-rate tensor is scaled with [u]/[h]. The flow law ε˙ = Aτ n , and the balance between vertical velocity and accumulation in the kinematic condition then motivate choosing [h] and [u] so that [u] =

[a][x] 2[A](ρg)n [h]2n+1 = , n [x] [h]

(1.88)

where [A] is a scale for the flow-law coefficient and [a] is a scale for the accumulation rate (typically on the order of a metre per year). The natural choice of time scale is the advective time scale, so having chosen [u] we can take [t] = [x]/[u]. Writing the equations in full component form (and assuming that gravity is aligned with the z coordinate) we then have ∂ uˆ ∂ vˆ ∂ wˆ + + , ∂ xˆ ∂ yˆ ∂ zˆ ∂ τˆx y ∂ pˆ ∂ τˆx x ∂ τˆx z 0=− +ε +ε + , ∂ xˆ ∂ xˆ ∂ yˆ ∂ zˆ ∂ τˆyx ∂ τˆyy ∂ τˆyz ∂ pˆ +ε +ε + , 0=− ∂ yˆ ∂ xˆ ∂ yˆ ∂ zˆ ∂ τˆzy ∂ τˆzx ∂ pˆ ∂ τˆzz 0=− + ε2 + ε2 +ε − 1, ∂ zˆ ∂ xˆ ∂ yˆ ∂ zˆ

0=

(1.89) (1.90) (1.91) (1.92)

24

I. Hewitt

where ⎛

ε ∂∂ uxˆˆ

1 2

ε ∂∂ uyˆˆ + ε ∂∂ vxˆˆ

1 2 1 2

∂ uˆ ∂ zˆ ∂ vˆ ∂ zˆ

⎞

+ ε2 ∂∂wxˆˆ

⎟ −(n−1)/2 1 1 , + ε2 ∂∂wyˆˆ ⎟ ⎠ , ηˆ = Aˆ 2 τˆi j τˆi j ∂ wˆ . . ε ∂ zˆ (1.93) ˆ Tˆ ) is the dimensionless flow-law coefficient, which will vary with and where A( dimensionless temperature, defined below. The dots signify that the matrix is symmetric. The boundary conditions become ⎜ τˆi j = 2ηˆ ⎜ ⎝ .

uˆ

ε ∂∂ vyˆˆ

∂ bˆ ∂ bˆ + vˆ = w, ˆ ∂ xˆ ∂ yˆ

ˆ τˆb ) uˆ b = λ F(

τˆ b ˆ x, on zˆ = b( ˆ yˆ ), τˆb

(1.94)

and the surface boundary conditions are ∂ sˆ ∂ sˆ ∂ sˆ + uˆ + vˆ = wˆ + a, ˆ ∂ xˆ ∂ yˆ ∂ tˆ

− pˆ nˆ + τˆ .nˆ = 0, on zˆ = sˆ (x, ˆ yˆ , tˆ),

(1.95)

where we have introduced a slip parameter λ=

F([τ ]) , [u]

(1.96)

ˆ τˆb ) = which measures the typical sliding velocity relative to shearing velocity. F( F([τ ]τˆb )/F([τ ]) is the dimensionless form of the sliding-law function. The scaled normal vectors at the ice surface and the bed are respectively

−ε∂ sˆ /∂ x, ˆ −ε∂ sˆ /∂ yˆ , 1

, nˆ = 1 + ε2 (∂ sˆ /∂ x) ˆ 2 + ε2 (∂ sˆ /∂ yˆ )2

nˆ =

ˆ x, ˆ yˆ , 1 −ε∂ b/∂ ˆ −ε∂ b/∂

ˆ x) ˆ 2 1 + ε2 (∂ b/∂

ˆ yˆ )2 + ε2 (∂ b/∂

.

(1.97) Since ε 1, the usual approximations to the equations now come from ignoring terms of order ε, or at least those of order ε2 . Various different approximations are obtained depending on the size of the slip parameter λ compared with ε. The natural way to non-dimensionalise temperature is to write T = T0 + [T ]Tˆ , where T0 is a reference temperature (typically 273.15 K) and [T ] is a scale for how much the temperature varies. The scaled version of the energy equation is then

∂ Tˆ ∂ Tˆ ∂ Tˆ ∂ Tˆ Pe + uˆ + vˆ + wˆ ∂ xˆ ∂ yˆ ∂ zˆ ∂ tˆ = ε2

2ˆ (n+1)/2 ∂ 2 Tˆ ∂ 2 Tˆ 2∂ T + ε + + Br Aˆ 21 τˆi j τˆi j , 2 2 2 ∂ xˆ ∂ yˆ ∂ zˆ

(1.98)

1

Slow Viscous Flow

25

where Pe = ε2

ρc p [u][x] , k

Br =

[τ ][u][h] , k[T ]

(1.99)

are the reduced Péclet number (which measures the importance of advection to conduction), and a number representing the importance of viscous dissipation relative to conduction (sometimes referred to as a Brinkman number).

Exercises 1.1 The linearised problem of determining the ice surface perturbation S(x) due to a Newtonian ice flow over a bumpy bed z = B(x) is described by the equations ∂U ∂P ∂W ∂P + = 0, 0 = − + η∇ 2 U , 0 = − + η∇ 2 W , ∂x ∂z ∂x ∂z subjected to boundary conditions W = 0, U = −u 0 B at z = 0, and −P + 2η

∂W = p0 S, ∂z

dS ∂U ∂W + = −u 0 S, u 0 = W at z = h, ∂z ∂x dx

where u 0 =

ρg sin α ρg sin α ρgh 2 sin α , u 0 = − , u0 = , η η 2η

p0 = −ρg cos α.

Write the equations in terms of a perturbed stream function such that U = z , W = −x . Defining the Fourier transform of a function f (x) as fˆ(k) =

∞ −∞

f (x)eikx d x,

and using the fact that f x = −ik fˆ, derive the Fourier transformed versions of these equations and boundary conditions, and show that the general solution for the transformed stream function is ˆ = z(a cosh kz + b sinh kz) + r cosh kz + d sinh kz.

26

I. Hewitt

ˆ = 0 at z = 0 (why?), show that r = 0. Show that the other Assuming boundary condition at z = 0 implies d=−

ˆ (a + u 0 B) . k

Show that the three boundary conditions at z = h lead to ˆ ˆ = kpo S, ˆ + 3iηk 2 −iη ˆ ˆ + k 2 ˆ = −u 0 S, ˆ −iku 0 Sˆ = ik , and hence deduce that a(khs − c) + bkhc =

po Sˆ ˆ + u 0 c B, 2iηk

akhc + b(khs + c) = −

u 0 Sˆ ˆ + u 0 s B, 2k

ˆ a(khc − s) + khsb = u 0 s Bˆ − ku 0 S, where c = cosh kh, s = sinh kh. Solve the first and third of these for a and b, and by then substituting into the ˆ where second equation and using the definitions of u 0 , etc., show that Sˆ = Kˆ B, Kˆ =

2 cosh κ

, sinh κ cosh κ − κ 1 + κ 2 + cosh2 κ + i cot α κ2

and κ = kh. Hint: note that c2 − s 2 = 1. 1.2 Ice velocity in a glacier or ice sheet satisfies the continuity equation ∂u ∂v ∂w + + = 0, ∂x ∂y ∂z and is subject to the boundary conditions u

∂b ∂b +v = w at z = b, ∂x ∂y

∂s ∂s ∂s +u +v = w + a at z = s, ∂t ∂x ∂y

1

Slow Viscous Flow

27

where z = b(x, y) is the bed surface elevation, and z = s(x, y, t) is the ice surface elevation. By integrating the continuity equation from z = b to z = s, show that ∂h + ∇.q = a, ∂t s (u, v) dz where a is accumulation rate, h = s − b is the ice depth and q = b

is the horizontal ice flux. Derive this equation directly from first principles by considering the rate of h dA of a column of ice above an arbitrary horizontal area

change of volume A

A.

1.3 Two-dimensional flow of a valley glacier is described by the approximate equations ∂ p ∂τx z ∂p 0=− + + ρg sin α, 0 = − − ρg cos α, ∂x ∂z ∂z where α is the bed slope angle, τx z is the shear stress, and axes (x, z) are taken downslope and normal to the bed slope. The bed is taken to be flat, z = 0, so that the ice surface is at z = h(x, t), where h is the depth, and there is no sliding at the bed. The boundary conditions at the ice surface z = s = h are p = τx z = 0 at z = h(x, t). Show that ∂h p = ρg(h − z) cos α, τx z = ρg sin α 1 − cot α (s − z), ∂x and thus, assuming Glen’s flow law with a constant rate coefficient A,

∂h n−1 ∂h ∂u (s − z)n . 1 − cot α = 2 A(ρg sin α)n 1 − cot α ∂z ∂x ∂x By integrating this expression twice from z = 0 to z = h, show that the flux is q= 0

h

∂h n−1 ∂h h n+2 1 − cot α u dz = 2 A(ρg sin α) 1 − cot α . ∂x ∂x n + 2 n

Hence, using the mass conservation equation in the form ∂h ∂q + = a, ∂t ∂x derive the glacier flow equation (1.81).

28

I. Hewitt

1.4 Use the definitions of the Brinkman number and Péclet number in (1.99), and the choices of scales in (1.87) and (1.88), to show that Pe =

[a][h] , κ

Br =

ρg[h]2 [a] , k[T ]

k is the thermal diffusivity. ρc p Using values c p = 2 × 103 J kg−1 K−1 , k = 2.2 W m−1 K−1 , and other suitable values of the constants, estimate the magnitude of the Brinkman number and Péclet number for an ice sheet. A basal geothermal heat flux of G = 60 mW m−2 is prescribed at the bed. If the corresponding dimensionless heat flux is denoted as , find an expression for , and hence show that geothermal heat flux is significant for ice sheets.

where κ =

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Batchelor GK (1967) An introduction to fluid mechanics. CUP, Cambridge Worster MG (2009) Understanding fluid flow. CUP, Cambridge Cuffey KM, Paterson WSB (2010) The physics of glaciers, 4th ed. Academic Press, New York Fowler AC, Larson DA (1978) On the flow of polythermal glaciers I. Model and preliminary analysis. Proc R Soc Lond A 363:217–242 Morland LW, Johnson IR (1980) Steady motion of ice sheets. J Glaciol 25:229–246 Hutter K (1982) Dynamics of glaciers and large ice masses. Ann Rev Fluid Mech 14:87–130 Hooke RLeB (2005) Principles of glacier mechanics, 2nd ed. CUP, Cambridge Greve R, Blatter H (2009) Dynamics of ice sheets and glaciers. Springer, Berlin van der Veen CJ (2013) Fundamentals of glacier dynamics, 2nd ed. CRC Press, Rotterdam Schoof C, Hewitt IJ (2013) Ice-sheet dynamics. Ann Rev Fluid Mech 45:217–239

2

Thermal Structure Andrew Fowler

Glaciers and ice sheets accumulate because of the conversion of fallen snow to ice. Evidently this can only be maintained if the average temperature in the accumulation zone is below the freezing point. In high Arctic glaciers, and in the Greenland and Antarctic ice sheets, this is indeed the case. Typical mean surface temperatures in Greenland are around −20 ◦ C, and in Antarctica around −50 ◦ C. On the other hand, some glaciers are fully temperate, that is to say they are at the melting point throughout. For example, most of the glaciers of the European Alps are temperate; how can this be? Generally, the temperature in glaciers and ice sheets varies with depth, and they are usually warmer at depth. This is important for the dynamics of ice flow, because the flow law depends on temperature, the ice being softer at higher temperature as with most metal oxides. The surface ice in Antarctica may be three orders of magnitude stiffer than the ice at the base. The temperature profile is also intimately bound up with the issue of basal melt-water production and its effect on the ice flow. The description of ice streams is central to ice sheet modelling, and an accurate description will rely on accurate estimates of subglacial hydraulic dynamics, which in turn requires an understanding of how the temperature profile is determined. In this chapter, we describe the processes and factors determining the observed temperature profiles, and we describe how simple models can be used to understand their structure. We also comment on some of the less well understood intricacies of temperature profile modelling.

A. Fowler (B) University of Limerick, Limerick, Republic of Ireland e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_2

29

30

2.1

A. Fowler

Temperature Profiles

Figure 2.1 shows a number of temperature profiles at various stations on the Antarctic and Greenland ice sheets. These range from an almost linear profile at Vostok in central East Antarctica to the visibly inverted profiles at Byrd Station in West Antarctica and the North-GRIP ice-core site in central Greenland. Two immediate questions occur: what controls the warming of the ice near the base? And, why are some of the profiles inverted? To answer these questions, we need to write down the energy equation. In ice, this takes the form derived in (1.43): ρc p (Tt + u .∇T ) = ∇. (k∇T ) +

τ2 . η

(2.1)

Here, T is the temperature, the subscript t denotes a partial derivative, and, as introduced in Chap. 1, the other quantities are ρ, the density; c p , the specific heat capacity; u, the ice velocity; k, the thermal conductivity; τ 2 , the second stress invariant (given in tensor notation by 2τ 2 = τi j τi j ); and η, the ice viscosity, which for Glen’s flow law is related to the stress by

Fig. 2.1 Several measured temperature profiles in the Antarctic and Greenland ice sheets. This figure may be compared to Fig. 10.6 of Paterson [1] or Fig. 9.7 of Cuffey and Paterson [2] (it shows a subset of the latter’s data). The figure was kindly and rapidly provided by Kurt Cuffey. The numbers at each profile are the local values of the Péclet number (see text for detail)

2 Thermal Structure

31 10

3 1 2

A

A 1

0 -50

0.1 0.01

-40

-20

-30

0

-10

0.001 -50

-40

T

-20

-30

-10

0

T

Fig. 2.2 Ice flow rate coefficient A in units of 10−24 Pa−3 s−1 as a function of temperature T in ◦ C, as recommended by Cuffey and Paterson [2]. The left figure shows the raw data with a smooth approximating curve; the right hand curve shows the corresponding graph in log scale

1 , (2.2) 2 Aτ n−1 where conventionally n = 3 and A is a temperature-dependent rate factor, which takes an approximate value A = 2.4 × 10−24 Pa−3 s−1 at 0 ◦ C [2], and decreases rapidly as temperature decreases, being about 1000 times lower at −50 ◦ C. Figure 2.2 shows the variation of viscosity with temperature, as recommended by Cuffey and Paterson [2]. There are three separate terms in (2.1). The left hand side represents the transport of heat by thermal advection, which is to say the transport of heat following the ice motion. The first term on the right represents heat conduction, and the second represents the less familiar source of heat due to frictional heating, the viscous dissipation. Normally in fluid flows it is negligible, but in ice sheets and glaciers it can be significant. It is also significant in the flow of subglacial water through Röthlisberger channels (See Sect. 3.3.2). η=

2.2

Boundary Conditions

In mathematical parlance, (2.1) is a parabolic partial differential equation. To solve it, we must apply an initial condition (usually of less importance, as we are mostly interested in steady solutions), and, importantly, boundary conditions. Specifically we require a thermal boundary condition at the surface of the ice mass, and a basal thermal boundary condition. The boundary condition at the surface connects us with the world of mass balance (discussed in detail in Chap. 7), which involves similar conceits to those below. Heat fluxes to and from the surface must balance, and this allows us to write a boundary condition in the unpromising form Q(1 − a) + q I = σ T 4 + q S + q L ;

(2.3)

32

A. Fowler

in this equation, T is the absolute temperature (i. e., in degrees Kelvin, ◦ K: add 273.15 to the Celsius temperature), and the various terms have the following meanings. Solar radiation provides a heat flux Q, which varies during the day and through the seasons because of the passage of the sun overhead; we can take it as known. A fraction a (the albedo) of this short wave radiation is reflected back to the atmosphere, and the value of a depends on the cleanness of the ice surface. It ranges from nearly one for fresh snow to about 0.1 for debris-covered ice. See also Table 6.1. The sun emits short wave radiation because of its high surface temperature (around 6000 K), but in fact all bodies emit black body radiation. The quantity of emitted radiation is given by the Stefan–Boltzmann law, E = σ T 4 , and this explains the first term on the right of (2.3), which is the long wave radiation emitted from the ice surface. The term q I represents conductive heat flux in the ice arriving at the surface ∂T from below: q I = −kn.∇T = −k , where n is the unit normal upwards from the ∂n ice surface; q S is the sensible heat flux from the surface to the atmosphere, usually parameterised in terms of a turbulent heat transfer coefficient, q S = h T (T − Ta ), where h T is the heat transfer coefficient, largely dependent on wind speed, and Ta is the near-surface atmospheric temperature. The latent heat term q L is a consequence of surface melting, or of surface precipitation. For the latter, rainfall and subsequent freezing, if the surface temperature is below the freezing point, causes a release of latent heat, so that q L = −ρw Lv p , where ρw is rainwater density, L is latent heat of melting, and v p is the precipitation rate, measured as a velocity. In conditions where there is surface melting, the temperature is necessarily at the melting point, T = TM , say, and the equality (2.3) then serves to define q L , which must be positive. The resultant meltwater may then evaporate, providing an actual further loss to the atmosphere, or it may run off the glacier in surface melt streams, or it may flow down moulins to the bed, where, if it refreezes, it releases latent heat and thus warms the basal ice; near the surface, meltwater may also percolate into the firn layer (Chap. 11) and refreeze there, warming it. We can estimate the comparative values of the heat flux in the ice and the long wave emitted radiation. Using values k ≈ 2.1 W m−1 K−1 and σ ≈ 5.67 × 10−8 W m−2 K−4 , we estimate k

∂T kT ∼ ∼ 0.42 W m−2 , ∂n d

(2.4)

where we take T ∼ 20 K, and glacier depth d ∼ 100 m; a value for an ice sheet would be less. Using T = 220 K to give a low estimate yields σ T 4 ≈ 133 W m−2 .

(2.5)

We see that the conductive heat flux in the ice is negligible compared to the radiative forcing. The thermal boundary condition (2.3) can thus be written in the approximate form T = Ts ,

(2.6)

2 Thermal Structure

33

where Ts is prescribed as the unique solution1 of σ Ts4 + h T (Ts − Ta ) = Q(1 − a).

(2.7)

In general, this gives the surface temperature as a function of time. Should Ts reach the melting temperature TM then it remains there until (2.7) allows Ts < TM again. The basal thermal boundary condition is seemingly quite simple. By analogy with (2.3) it is a flux condition, but there is no radiative forcing, and a simple balance suggests −k

∂T = G, ∂n

(2.8)

where n is the unit normal pointing upwards into the ice. Here G is the geothermal heat flux delivered from the Earth’s crust, having a typical value of the order of 60 mW m−2 . As for the surface, if T reaches TM , it cannot increase further, and so then ∂T T = TM so long as G + k > 0, and when this inequality fails, we resume the ∂n heat flux condition (2.8). We will have more to say about the basal thermal boundary condition in Sect. 2.4.

2.2.1

The Thermal Near-Surface Wave

The simplest solution of (2.1) occurs if we neglect advection, frictional heating, and horizontal conduction. This is appropriate in the uppermost part of the glacier, where the thermal structure is generated by the time varying surface temperature. To illustrate the behaviour, we write the temperature (or heat) equation in the form Tt = κ Tzz ,

(2.9)

k is the thermal diffusivity, and we assume that k, ρ and c p are ρc p constant. We let z denote depth below the surface, and then we suppose that the surface temperature is

where κ =

T = T0 + T sin ωt,

(2.10)

T → T0 as z → ∞.

(2.11)

and at sufficient depth, We will comment further on the choice ‘z → ∞’ below.

1 The

simplest way to understand that the solution is unique is to draw a graph of the left hand side of (2.7). It is a monotonically increasing function of Ts , so that there is precisely one solution.

34

A. Fowler

The solution of this problem is found to be T = T0 + T exp(−αz) sin(ωt − αz), where

α=

ω . 2κ

(2.12)

(2.13)

2π , then we see from (2.12) that the perturbation to If the oscillation period is P = ω the mean temperature decays over a depth 1 z = = α ∗

κP . π

(2.14)

If we use values ρ = 103 kg m−3 and c p = 2 × 103 J kg−1 K−1 , so that κ ≈ 10−6 m2 s−1 , then for P = 1 day we have z ∗ = 0.17 m, and for P = 1 year, z ∗ = 3.2 m. At a depth of ten metres, the ice temperature is virtually insensible to the annual variations at the surface. One might wonder why the boundary condition (2.11) is applied ‘at ∞’, and indeed why the far field temperature is the same as the mean temperature at the surface. The answer to both questions is a little technical. In essence, the thermal surface layer is a boundary layer for the temperature field in the main body of the ice. What this means in practice is that the effective boundary condition which should be applied at the surface is indeed (2.6), but where Ts is taken as a time average over periods shorter than the dynamical time scale for the ice, this being tens to hundreds of years for glaciers (see Chap. 18), tens of thousands of years for ice sheets.

2.3

Models: Simple to Complicated

We will now mostly focus on the temperature in ice sheets. The simplest solution for temperature follows from the assumption that advection and frictional heating are both negligible. In addition, because ice sheets and glaciers are long and shallow, horizontal heat conduction is negligible. In this case, the heat equation in the steady state is simply Tzz = 0, where now we take z to be a vertically upwards coordinate, and the conductive solution is linear: T = Ts +

G(s − z) , k

(2.15)

assuming a frozen (cold) base. Here z = s denotes the surface elevation of the ice sheet. This solution is self-consistent if h

h R , where hR =

ρc p aT 2κξ 2 , ξ erf ξ = ≡ √ , a G π

(2.21)

36

A. Fowler

Fig. 2.3 The temperature profile given by (2.20), with values h = 2000 m, κ = 10−6 m2 s−1 , a = 0.1 m y−1 , G = 60 mW m−2 , k = 2.1 W m−1 K−1 , Ts = −30 ◦ C. This can be compared to the Byrd profile in Fig. 2.1

2000

z (m)

1500 1000 500 0 -30

-10

-20

0

o

T( C)

where T ≡ TM − Ts is the surface sub-cooling. An estimated value using a = 0.1 m y−1 and T = 30 K is ≈ 1.7, so that erf ξ ≈ 1 and thus ξ ≈ , and thus hR ≈

2kρc p a π

T G

2 ≈ 1900 m.

(2.22)

Fig. 2.3 shows a temperature profile given by (2.20) which may be compared to those in Fig. 2.1. It can be seen that the estimate for h R in (2.22) is slightly too low. The error function profile in (2.20) is always monotonic. The overcooled profiles which can occur are due to the advection of colder ice from upstream, and require inclusion of the horizontal advection term. In general, this requires a numerical solution, but some insight can be gained using Nye’s [4] observation that the shear in an ice sheet is concentrated near the base, together with the realisation that thermal conduction is a relatively small contributor to heat transport. This is because the Péclet number ah (2.23) Pe = κ is relatively large (for a = 0.1 m y−1 , h = 3000 m, κ = 10−6 m2 s−1 , Pe ≈ 10), so that temperature changes are concentrated near the base. In the following analysis of this approximation, we restrict our attention again to two dimensions. Away from the bed, u is independent of z so that w = −zu (x) (according to the continuity equation), and if we additionally ignore heat conduction, the solution of u .∇T = 0 is simply T = f (zu),

(2.24)

where the function f is determined by the surface boundary requirement that f (hu) = Ts (x) in the accumulation area; f thus depends on the ice flow. In a steady state, x hu ≈ B(x) = a(x ) d x , 0

(2.25)

(2.26)

2 Thermal Structure

37 3000

2000

z (m)

Fig. 2.4 Vertical temperature profile at position x = 500 km computed from (2.30) using

x 2 h(x) = h 0 1 − , l uh = ax, Ts = T0 − {h(x) − h 0 }, with values T0 = −30 ◦ C, a = 0.1 m y−1 , l = 1500 km, h 0 = 3000 m, κ = 38 m2 y−1 , = 6 ◦ C km−1

1000

0 -40

-30

-20

-10

0

o

T( C)

where x = 0 denotes the divide position, and the balance function B is monotonically increasing in the accumulation area. Thus f [B(x)] = Ts (x), and in particular f (0) = T0 ≡ Ts (0). As we approach the base of the ice flow, z → 0 and thus T → T0 , and the basal thermal boundary condition is not satisfied. To satisfy the basal boundary condition, we need to consider heat conduction in a thermal boundary layer near the base. Analogously to the near-surface layer, we then have to solve (still with u ≈ u(x)) uTx − zu Tz = κ Tzz ,

(2.27)

with boundary conditions (assuming basal melting) T = 0 on z = 0, T → T0 as z → ∞,

(2.28)

of which the solution is (using the Von Mises transformation2 ) ⎡ ⎤ zu ⎦. T = T0 erf ⎣ x 2 κ 0 u(x ) d x

A uniform approximation3 can be written as4 ⎡

⎤ zu

⎦, T = f (zu) − T0 erfc ⎣ x

2 κ 0 u(x ) d x and an example of the resulting temperature profile is shown in Fig. 2.4.

2 See

Levich [5], for example. the sense of matched asymptotic expansions. 4 erfc ξ = 1 − erf ξ . 3 In

(2.29)

(2.30)

38

2.4

A. Fowler

Basal Conditions

In discussing basal boundary conditions, we considered two: the bed is frozen, or it is at the melting point. However, things are a little more complicated than this. When the basal ice reaches the melting point, it begins to slide, by means of a micron-thick water film which is produced at the bed by the process of regelation (Sect. 3.2). The deformation of the ice as it is viscously deformed causes a frictional heat τb u b , where τb is basal shear stress and u b is the sliding velocity, to be delivered to the bed. The ∂T net heat flux arriving at the bed is then G + τb u b − q I , where q I = −k is the ∂n heat flux into the ice (the normal coordinate n points upwards). Freezing conditions are thus described by the combination G + τb u b − q I = 0, u b = 0, T < TM ,

(2.31)

while a temperate base (with temperature at the melting point) is described by G + τb u b − q I > 0, u b = f (τb , N ), T = TM ,

(2.32)

where f (τb , N ) is given by the sliding law (see Chap. 3). We see that in the temperate case there is net heat delivery to the bed, and this defines a meltwater flux per unit area, vw , via G + τb u b − q I = ρw Lvw ,

(2.33)

where L is the latent heat. Typical values of vw in ice sheets are millimetres per year, and this provides the source for a subglacial hydraulic system, whose effective pressure (overburden minus water pressure) N contributes to the determination of the sliding law. It can be seen from (2.31) and (2.32) that the transition at the bed between cold and temperate conditions will cause a discontinuity in the velocity, unless the shear stress on the temperate side is zero. In turn, this would cause a stress singularity on the cold side [6], but this is unphysical, as it assumes the lubricating water film appears fully formed exactly at the melting temperature. In reality, there will be a small temperature range TM − T < T < TM over which the lubricating film is gradually established, and the sliding velocity increases from zero to its fully temperate value (see also Sect. 3.2.3). Sliding laws of this type have been implemented in computational models by Hindmarsh and Le Meur [7] and Pattyn et al. [8], for example. If the temperature range T is small as we expect, we can approximately describe this connecting sub-temperate basal state by G + τb u b − q I = 0, 0 < u b < f (τb , N ), T = TM .

(2.34)

In practice, such sub-temperate basal regions would consist of a patchwork of frozen and non-frozen areas, as envisaged by Robin [9] and also discussed in Chap. 17.

2 Thermal Structure

39

Fig. 2.5 Schematic representation of the different basal thermal régimes of an ice sheet (or glacier). While cold-based margins can occur, ice streams (for example) indicate temperate basal ice at the margin

2.4.1

cold

cold

moist

temperate sub−temperate

temperate sub−temperate

Polythermal Ice

In each of the three régimes, cold, sub-temperate and temperate, we see that there are two equalities and one inequality in (2.31), (2.32) and (2.34). In reality, there is another inequality which is implicitly satisfied, which is that q I > 0, so that the ice above the base is below the melting point. There is no reason that this should always be so, and there is therefore a fourth kind of basal condition, which occurs adjoining a temperate base when q I reaches zero. In this case the ice above is temperate, and the glacier is polythermal, as shown in Fig. 2.5. In a temperate glacier, the temperate region occupies almost all the glacier volume.5 In temperate ice, the temperature is fixed at the melting point, and further heat addition causes the ice to melt at grain boundaries, leading to the formation of water inclusions within the ice [10–13]. The appropriate thermal variable for this moist ice is the moisture content W , measured as a volume fraction of water, and it satisfies an energy equation of the form ρw L(Wt + u .∇W + ∇. q) =

τ2 , η

(2.35)

where q is the moisture flux. A diffusive assumption of the form q = −ν∇W

(2.36)

leads to an equation which is structurally similar to the temperature equation, and this has been used in computations (e. g., [14,15]), but (2.36) is hard to justify theoretically.

2.5

Modelling Issues

In this section we consider some further analytic and computational issues in which the energy equation is involved.

5 As

we mentioned at the outset, this seems perplexing. A likely cause is that refreezing of surface melt and precipitation can provide a significant input of heat. For example, annual meltwater or precipitation of just v p = 0.1 m y−1 provides an equivalent heat flux of ρw Lv p ≈ 1 W m−2 , about sixteen times the geothermal heat flux.

40

A. Fowler

2.5.1

Non-dimensionalisation

Assessment of the importance of the various terms in the energy equation is done by making the equations dimensionless. Notably, for the shallow ice approximation, see the derivation in Sect. 1.11 and (1.98); additional details can be found in Fowler ([16], p. 635). The dimensionless form of the temperature equation (2.1) takes the approximate form6 1 Tt + u .∇T = ατ n+1 eγ T + (2.37) Tzz , Pe where we assume Glen’s flow law with exponent n, and a Frank-Kamenetskii (i. e., exponential) approximation to an Arrhenius type flow law rate coefficient; the parameters are given by α=

gd , c p |T0 |

Pe =

a0 d E|T0 | . , γ = κ RTM2

(2.38)

Here a0 and d are representative values of accumulation rate and depth, |T0 | is a relevant temperature range, for example T0 could be the minimum surface temperature, E is the activation energy for creep, and R is the gas constant. The horizontal heat conduction terms are negligible because of the large aspect ratio of the flow; see (1.98). The definition of the Péclet number Pe is the same as that in (2.23) and in Exercise 1.4. Typical values of these parameters are α ∼ 0.3, Pe ∼ 10 for large ice sheets, and γ ∼ 11. The large value of γ indicates the increasing slope of the curve in Fig. 2.2 (right) near TM . The Péclet number is large, but not dramatically so. It is because of this that temperature changes are concentrated in the lower half of the ice column, √ as in Fig. 2.4, which represents a thermal boundary layer of thickness O(1/ Pe). 3000

2000

z (m)

Fig. 2.6 A plot of the dimensionless shear heating term ατ n+1 exp(γ T ) for the same temperature profile as in Fig. 2.4 (now scaled with |T0 |), using the same parameter values, and also α = 0.5, n = 3, γ = 11; the dimensionless shear stress is 2x(h − z) given by τ = h0l

1000

0 0

0.02

0.04

0.06

0.08

shear heating

Because n is relatively large, as is also γ , the shear heating term in (2.37) is only significant near the base, as shown in Fig. 2.6, and consequently there is a shear layer

6 By

comparison with (1.98), note that α = Br /Pe.

2 Thermal Structure

41

at the base, above which the flow is essentially shear free. This is the basis of Nye’s [4] approximation; see also Fowler [16].

2.5.2

Thermomechanical Coupling

With the same notation as in (2.37), the dimensionless shear strain rate of the horizontal velocity u H = (u, v) is ∂u H = τ n−1 τ eγ T , ∂z

(2.39)

(which is why the shear is concentrated near the base); in particular, the velocity field depends strongly on the temperature field. Equally, from (2.37), the temperature depends strongly on the flow, partly because of the shear heating term, and partly because the basal boundary conditions, particularly (2.34), combine flow with temperature. This interdependence of flow and temperature is referred to as thermomechanical coupling.7

Fig. 2.7 Computed Antarctic basal temperatures from Huybrechts [19]. The areas in white (pmp) are at the pressure melting point. Reprinted from Climate Dynamics with permission from Springer Nature ©1990

7 This

coupling also occurs for moist ice, since the flow law of ice also depends on the moisture content [17].

42

A. Fowler

Other than the sorts of approximation discussed here, computation of temperature profiles involves numerical solution of the coupled flow and temperature fields. There are numerous models which do this, though their complexity precludes unwarranted belief in their accuracy, but a number of intercomparison projects have been conducted to provide some confidence in the results (for example see Calov et al. [18]). Figure 2.7 shows the computed basal temperature from an early model of Huybrechts [19].

2.5.3

Thermal Runaway

One of the early explanations for glacier surges, suggested by Clarke et al. [20] and Yuen and Schubert [21], was thermal runaway. This arises through thermomechanical coupling. An increase in temperature leads to an increase in flow rate, which causes increased viscous heating and thus further rise in temperature. This mechanism forms the explanation of combustion (why does a match light when struck?) but its application to glacier surges is less clear, because the total frictional heat released is essentially limited by the flux of ice, which is itself constrained by the finite accumulation rate. The multiple steady states which are the hallmark of thermal runaway appear to be precluded by this constraint. However, in spatially extended flows, such thermomechanical instability appears possible, and may be related to the formation of ice streams (see Fig. 8.10, left), for example in Antarctica [22]. This thermomechanical instability was actually discovered accidentally in one of the EISMINT computational intercomparison experiments [23]: various different numerical codes provided solutions which differed significantly from each other in detail, but which all found some evidence of the instability. At the time, this emphasised the fact that large scale numerical computations of thermomechanically coupled flows were not necessarily to be trusted, but it also led to the concept that the instability might be the cause of ice stream formation. Other mechanisms are also possible, however, for example that of hydraulic runaway [24], since the high ice stream velocities are due to basal lubrication.

Exercises 2.1 The solution for the temperature of the near-surface thermal wave was given in Sect. 2.2.1. The temperature satisfies Tt = κ Tzz , where z is depth below the surface, and the boundary conditions were given as T = T0 + T sin ωt on z = 0, T → T0 as z → ∞. Show in detail how to obtain this solution by assuming T = T0 + Im[ f (z)eiωt ]

2 Thermal Structure

43

(Im denotes the imaginary part of the complex quantity ), and deriving a second order differential equation for f with two associated boundary conditions. √ √ 1 Solve this equation (you will need to remember that i = e 4 πi = (1 + i)/ 2) to find the solution given in (2.12). Now suppose that the surface temperature is a general function of time with long time mean of T0 and can be written in the form T = T0 +

1 2π

∞

−∞

Tˆ (ω)e−iωt dω at z = 0

(thus Tˆ is the Fourier transform of T − T0 at the surface), and the boundary condition at ∞ is the same as before. Find the solution in this case in the form of an integral over ω (warning: be careful with the sign of ω), and hence derive an integral expression for the surface temperature gradient. 2 [Harder.] If T (0, t) = e−βt , where β > 0, compute the solution explicitly. 2.2 Equations (2.17)–(2.20) outline how the energy equation can be solved for an ice divide with a linear vertical velocity distribution w(z) to yield the temperature profile T (z), assuming that T at the bed is below the melting point. Extend this calculation for the case where basal melting occurs, as follows. Assume that wTz = κ Tzz , with w(z) = −m − (a − m)z/h, where a is the accumulation rate and m is the basal melt rate. The boundary conditions for the equation are T = Ts at z = h, T = Tm , ρi Lm = G + kTz at z = 0. The extra basal boundary condition ensures that the solution of the problem will yield m as well as the temperature profile. Solve the problem for T (z) assuming the constant m is known, and then use the extra boundary condition to derive an ugly algebraic equation for m, which would require numerical solution. [Hint: recall the definition of the error function: 2 erf(x) = √ π

x

e−t dt.] 2

0

2.3 In a steady one-dimensional slab flow of a glacier, the scaled equations for temperature T and downslope velocity u are Tzz + α Pe τ n+1 eγ T = 0, u z = τ n eγ T ,

44

A. Fowler

where z is the coordinate normal to the bed z = 0, the surface is z = h, with h constant, and boundary conditions for temperature are T = 0 on z = 0, T = Ts < 0 on z = h. The shear stress is τ = h − z, and the basal sliding law is taken to be the friction law τ = 1 at z = 0, so that h = 1, and the dimensionless mass flux is

1

u dz = 1.

0

Because γ 1 and T < 0, the equations can be approximated by Tzz + α Pe eγ T = 0, u z = eγ T ; explain why this is the case. Writing θ = γ T , λ = αγ Pe, γ Ts = −, > 0, show that 1 2 2θ

∗

= λeθ − λeθ ,

for some constant θ ∗ . Define U by θ = θ ∗ − 2 ln cosh U , U > 0, and show that the necessary assumption that θ < 0 then requires that U > 0. Explain why θ < θ ∗ , and thus why θ ∗ > 0. Show that U=

λ θ ∗/2 ∗ e z + cosh−1 eθ /2 . 2

Defining ζ =e

θ ∗ /2

, β=e

/2

, μ=

λ , 2

show that ζ , and thus θ ∗ , is determined by βζ = cosh μζ + cosh−1 ζ , and thus β = cosh μζ ±

(ζ 2 − 1)1/2 sinh μζ. ζ

Draw a graph of θ ∗ as a function of β, and show that θ ∗ = 0 if β = cosh μ. What determines the basal sliding velocity?

2 Thermal Structure

45

References 1. Paterson WSB (1994) The physics of glaciers, 3rd edn. Butterworth-Heinemann, Oxford 2. Cuffey KM, Paterson WSB (2010) The physics of glaciers, 4th edn. Academic Press, New York 3. Robin GdeQ (1955) Ice movement and temperature distribution in glaciers and ice sheets. J Glaciol 2:523–532 4. Nye JF (1959) The motion of ice sheets and glaciers. J Glaciol 3:493–507 5. Levich VG (1962) Physicochemical hydrodynamics. Prentice-Hall, New Jersey 6. Hutter K, Olunloyo VOS (1980) On the distribution of stress and velocity in an ice strip, which is partly sliding over and partly adhering to its bed, by using a Newtonian viscous approximation. Proc R Soc Lond A 373:385–403 7. Hindmarsh RCA, Le Meur E (2001) Dynamical processes involved in the retreat of marine ice sheets. J Glaciol 47:271–282 8. Pattyn F, de Smedt B, Souchez R (2004) Influence of subglacial Vostok lake on the regional ice dynamics of the Antarctic ice sheet: a model study. J Glaciol 50:583–589 9. Robin GdeQ (1976) Is the basal ice of a temperate glacier at the pressure melting point? J Glaciol 16(74):183–196 10. Carol H (1947) The formation of roches moutonnées. J Glaciol 1:57–59 11. Lliboutry L (1976) Physical processes in temperate glaciers. J Glaciol 16:151–158 12. Nye JF (1976) Water flow in glaciers: jökulhlaups, tunnels, and veins. J Glaciol 17:181–207 13. Raymond CF, Harrison WD (1975) Some observations on the behavior of the liquid and gas phases in temperate glacier ice. J Glaciol 14:213–233 14. Hutter K, Blatter H, Funk M (1988) A model computation of moisture content in polythermal glaciers. J Geophys Res 93(12):205–214 15. Blatter H, Hutter K (1991) Polythermal conditions in Arctic glaciers. J Glaciol 37:261–269 16. Fowler A (2011) Mathematical geoscience. Springer, London 17. Duval P (1977) The role of the water content on the creep rate of polycrystalline ice. IASH 118:29–33 18. Calov R and 10 others (2010) Results from the Ice-Sheet Model Intercomparison Project – Heinrich Event INtercOmparison (ISMIP HEINO). J Glaciol 56:371–383 19. Huybrechts P (1990) A 3-D model for the Antarctic ice sheet: a sensitivity study on the glacialinterglacial contrast. Clim Dyn 5:79–92 20. Clarke GKC, Nitsan U, Paterson WSB (1977) Strain heating and creep instability in glaciers and ice sheets. Rev Geophys Space Phys 15:235–247 21. Yuen DA, Schubert G (1979) The role of shear heating in the dynamics of large ice masses. J Glaciol 24:195–212 22. Hindmarsh RCA (2009) Consistent generation of ice-streams via thermo-viscous instabilities modulated by membrane stresses. Geophys Res Lett 36:L06502 23. Payne AJ and 10 others (2000) Results from the EISMINT model intercomparison: the effects of thermomechanical coupling. J Glaciol 46:227–238 24. Kyrke-Smith TM, Katz RF, Fowler AC (2014) Subglacial hydrology and the formation of ice streams. Proc R Soc Lond A 470:20130494

3

Sliding, Drainage and Subglacial Geomorphology Andrew Fowler and Felix Ng

3.1

Introduction

Numerous processes occur at the base of glaciers and ice sheets that can signiﬁcantly influence their large-scale behaviour. For instance, meltwater reaching the basal interface from the surface or produced there locally may facilitate basal sliding; hence the existence and spatial distribution of water across the bed affects the flow dynamics of ice masses. The bed topography may be uneven on short spatial scales, and this induces boundary-layer mechanics in the ice flow above it. The bed substrate may be composed of sediments as well as bedrock. Erosion of these materials by ice or water, together with the transport and deposition of sediments, means that the bed topography is not actually ﬁxed but may evolve in time. Consequently, subglacial water drainage and sediment transport need to be considered for an understanding of the hydrological and sediment yields of a glacier, and these processes play a rôle in shaping subglacial landforms, many of which can be observed in deglaciated areas today. Indeed, the interactions outlined above can be highly complex. In this chapter we describe models that have been developed to understand them. We begin with the situation of a hard bed (Sect. 3.2), and then add water and sediment to the picture (Sects. 3.3 and 3.4).

A. Fowler MACSI, University of Limerick, Limerick, Ireland F. Ng (&) Department of Geography, University of Shefﬁeld, Shefﬁeld, UK e-mail: f.ng@shefﬁeld.ac.uk © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_3

47

48

3.2

A. Fowler and F. Ng

Sliding Over Hard Beds

3.2.1 Weertman Sliding Glacier motion is modelled as a slow, viscous flow (Chap. 1), but when sliding occurs, the usual no-slip boundary condition appropriate to a frozen base must be replaced by a relation between basal shear stress sb and basal velocity ub. This relation is called the sliding law. (For simplicity we drop the subscript b below.) Sliding can be observed in cavities or subglacial tunnels. It sometimes accounts for most of the observed surface velocity, for example in the Siple Coast ice streams of Antarctica. The mechanisms of sliding were enunciated by Weertman [1]. See Fig. 3.1a. When the basal ice is temperate (at the melting point; Chap. 2), a ﬁlm of water can exist between the ice and the underlying bedrock.1 This allows slip. As the ice deforms around basal obstacles by creep, resistance is provided by the roughness of the bedrock topography, which presents a form drag to the ice flow. Besides creep deformation, Weertman identiﬁed a further mechanism of sliding, called regelation. Regelation (literally, re-freezing) occurs because as the ice flows over a bed obstacle, the higher pressure upstream causes the ice there to melt at the interface (because the melting temperature decreases with pressure, an effect summarised by the Clapeyron relation). The water that is thus formed squirts round the obstacle in the water ﬁlm, and correspondingly refreezes on the downstream side. Weertman [1] modelled the two mechanisms in the following way. Consider a bed consisting of an array of cubical obstacles of dimension a at distance l apart (Fig. 3.1b), and suppose the ice flow exerts a mean shear stress s at the bed. The drag on each obstacle is sl2. Accordingly, the pressure increase upstream of an obstacle is (approximately) sl2/2a2, while the decrease downstream is (approximately) –sl2/2a2. Owing to the Clapeyron effect, this pressure difference induces a temperature difference across the obstacle of dT

Csl2 ; a2

ð3:1Þ

Fig. 3.1 a Weertman’s [1] basal-sliding mechanisms and b his model set-up Such a ﬁlm is in fact maintained to quite low temperatures below the bulk melting point by a thermodynamic phenomenon called premelting.

1

3

Sliding, Drainage and Subglacial Geomorphology

49

where C = −dTm/dp ( 0.0074 K bar−1) is the slope of the Clapeyron curve; here Tm is the melting temperature. Let uR be the regelative ice velocity: then the regelative water flux is uRa2. The latent heat flux needed to sustain this water flux is qiLuRa2, where qi is ice density and L is latent heat. We also know that heat transfer through the obstacle occurs at a rate (kdT/a)a2 = kdTa, where k is the thermal conductivity of the bedrock. Equating the conductive and latent heat fluxes leads to uR ¼

kC s ; qi La m2

ð3:2Þ

where m = a/l is a measure of the bedrock roughness. This result shows that regelation is effective for small obstacles. Weertman next considered the contribution of creep to the sliding. Let uV be the velocity due to viscous shearing past the obstacle. The differential stress across the obstacle is s/m2. Assuming a nonlinear (Glen’s) flow law e_ ¼ Asn (with n 3), the resulting strain rate is uV/a 2A(s/m2)n, so uV 2aAðs=m2 Þn :

ð3:3Þ

Weertman [1] added uR and uV from (3.2) to (3.3) to ﬁnd the sliding velocity u¼

C1 s þ C2 aðs=m2 Þn ; a m2

ð3:4Þ

where C1 and C2 are material coefﬁcients. Alternatively, it can be argued that the stresses should be added when the two mechanisms are combined in the model. In this case, we have s ¼ m2 ½Rr au þ Rv ðu=aÞ1=n ;

ð3:5Þ

where Rr and Rv are material roughness coefﬁcients, given approximately by qL Rr i ; kC

Rv

1 2A

1=n

:

ð3:6Þ

We see that sliding past small obstacles occurs mainly by regelation, while sliding past larger obstacles occurs largely by viscous deformation. There is a controlling obstacle size at which the stresses are comparable.2 If we set a to be this value, we obtain the Weertman sliding law

More speciﬁcally, consultation of (3.5) indicates that, for ﬁxed roughness m, the stress is minimised at a certain value of amplitude a: this is the controlling amplitude. Weertman actually chose a to minimise u in (3.4); this leads essentially to the same result. See Exercise 3.1 at the end of this chapter.

2

50

A. Fowler and F. Ng 2

s ¼ m2 Run þ 1 ;

ð3:7Þ

where R

1=ðn þ 1Þ Rr Rnv ¼

qi L 2kCA

1=ðn þ 1Þ

:

ð3:8Þ

It is worth noting that Weertman’s model is based on a scaling argument and does not solve for the precise stress/pressure distribution in the ice near the basal interface.

3.2.2 Nye-Kamb Theory Nye [2, 3] and Kamb [4] extended Weertman’s scale analysis by means of a mathematically precise model. Their analysis is limited to a linear viscous flow law and therefore inevitably leads to a linear sliding law, s / u. However, their analysis is not limited to a single roughness scale. The rôle of the controlling obstacle size is more explicit in the resulting theory. Nye’s approach is to solve the Stokes flow equation r4 w ¼ 0 for the stream function in the ice, in z > mh(x), and Laplace’s equation for temperature h in the bedrock, in z < mh(x). Here, x is horizontal distance and h describes the bed topography. By expanding for small m, one obtains problems in upper and lower half spaces respectively which are linear and can be solved easily via the Fourier transform. Nye’s result is s¼

gj2 u p

Z

1

0

Ph ðjÞj3 dj; j2 þ j2

ð3:9Þ

where j denotes wavenumber, and Ph(j) is the power spectral density. (Deﬁne 2 ^M =M, where ^ hM hM = h for |x| < M, hM = 0 for |x| > M; then Ph ðjÞ ¼ limM!1 h R1 3 ijx is the Fourier transform of hM: ^hM ðjÞ ¼ 1 hM ðxÞe dx.) The controlling wavenumber j is deﬁned by j2 ¼

qi L ; 4kCg

ð3:10Þ

where k is the thermal conductivity and η is ice viscosity. This can be compared to the controlling obstacle size found from (3.5), which, for n = 1, satisﬁes.

3

In Nye’s paper [2],R his length l is equal to 2M as deﬁned here. Also, his deﬁnition of the Fourier 1 transform of hM is 1 hM ðxÞeijx dx.

3

Sliding, Drainage and Subglacial Geomorphology

a2 ¼

Rv kCg ; ¼ Rr qi L

51

ð3:11Þ

since (2A)−1 = η is the viscosity when n = 1. Thus a ¼ ð2j Þ1 . Nye’s theory emphasises the rôle of the power spectrum of h in determining the roughness. Unfortunately, there is no exact generalisation available for nonlinear flow laws.

3.2.3 Sub-temperate Sliding Modellers who implement sliding laws in their computational ice sheet modelling often assume that the sliding law u = U(s) applies when the basal temperature T = Tm, and that u = 0 for T < Tm. This assumption is not strictly accurate. A more suitable description is that sliding increases continuously over a small temperature range near T = Tm, and this would involve the creation of a water ﬁlm in a patchy fashion as Tm is approached. We refer to this as sub-temperate sliding. If one assumes a discontinuous sliding law, then if basal stress is continuous, one would have an inadmissible discontinuity of velocity: this was the downfall of the EISMINT (European Ice Sheet Modelling INiTiative) ice shelf numerical modelling experiments in the 1990s. If the velocity is to be continuous, then stresses must be discontinuous―in fact, singular. It has been suggested that such stress concentrations may have a bearing on thrust faults in glaciers. We can derive a sub-temperate sliding law via a Weertman-type approach. Let us consider again bumps of size a spaced a distance l apart. Suppose the basal temperature is at a temperature Tb < Tm, and we deﬁne the undercooling to be DT ¼ Tm Tb :

ð3:12Þ

The bed is partly frozen in this case, so the water ﬁlm is not expected to cover the bed between the bumps. Stick-slip friction may thus also play a rôle in the stress balance, but we omit this detail here. Suppose (as in Weertman’s model) that most resistance still comes from the ﬁlm-assisted flow over the bumps. With the ice temperature below Tm, there is now an extra conductive heat loss away from the bumps, with a magnitude of kDT/l approximately. Hence (3.2) is replaced, using (3.6), by Rr auR ¼

s mDT ; m2 C

ð3:13Þ

and (3.5) is replaced by u1=n mDT

s ¼ m2 Rr au þ Rv þ ; a C

ð3:14Þ

52

A. Fowler and F. Ng

which shows that for ﬁxed s, u decreases to zero continuously as DT increases to an undercooling DTmax given by DTmax ¼

Cs : m3

ð3:15Þ

This theoretical result indicates that some sliding can persist at temperatures quite far below the melting point. For example, for s = 0.1 bar and m = 0.1, (3.15) gives DTmax 1 K.

3.2.4 Nonlinear Sliding Laws Lliboutry [5, 6] is the principal exponent of more complex theories associated with a nonlinear flow law and a fairly general bed geometry. The key to a more precise model is a variational principle for slow non-Newtonian flows. Fowler [7] and Meyssonnier [8] used this to derive bounds for the roughness in the sliding law, where regelation was neglected (i.e., the bed topography varies on a longer length scale than 2p=j ). One obtains the sliding law s = Ru1/n, with bounds for R, which are, however, rather wide for general h. For a pure sinusoidal bed topography, however, a good approximation can be derived. Research on these idealised problems continues using numerical as well as analytic methods. See Gudmundsson [9] and Hindmarsh [10] for some of the more recent results.

3.2.5 Cavitation Lliboutry [5] was the person who emphasised the importance of cavitation in the sliding law. Cavities exist in the lee of obstacles and have the effect of reducing the effective roughness of the bed (Fig. 3.2). Lliboutry framed his theory in terms of a pseudoempirical shadowing function s, which determined the fraction of cavity-free bed in terms of the cavity roof slopes. This function, described further below,

Fig. 3.2 Formation of cavities on the leeside of bumps as a result of sliding

3

Sliding, Drainage and Subglacial Geomorphology

53

Fig. 3.3 Multivalued-ness (of u at given s) of the sliding law (bold line) conceived by Lliboutry

decreases monotonically with velocity u. He derived various forms for the sliding law and suggested that the functional relation for u(s) can be multi-valued (Fig. 3.3). Importantly, he demonstrated that the sliding law would depend on the effective pressure N = pi − pc, where pi is overburden pressure and pc is cavity pressure (which will be equal to the hydraulic drainage pressure at the bed). Iken [11] suggested that as N decreases (given s) there is a critical Nc > 0 below which unstable sliding will occur. The argument is based on a force balance only4; the result can be interpreted in terms of Fowler’s [12] extension of Nye’s earlier theory. By reformulating Nye’s model (without regelation) as a Hilbert problem, Fowler showed that for typical ‘unimodal’ bedrocks (with one hump per period), the sliding law was s ¼ Nf ðu=NÞ;

ð3:16Þ

where f ﬁrst increases from zero to a maximum f * and then decreases to zero as u ! 1. Hence s < Nf * for all u, i.e., N > s/f *, which corresponds to Iken’s separation pressure. Examination of the results indicates that the maximum drag Nf * is reached when the cavity from one bump begins to reach the next. In reality, where bumps of varying sizes exist, the drag is simply shifted to larger bumps as the smaller ones become drowned. As small bumps are all cavitated in this theory, this justiﬁes neglecting the effect of regelation. An approximate method [13] led to the generalised Weertman law s ¼ cur N s

ð3:17Þ

with typical values 0 < r, s < 1. More recent work on the problem by Schoof [14] is consistent with the above discussion.

This is not quite enough, since determination of the forces in a viscous flow problem requires consideration of the rheological flow law.

4

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3.2.6 Comparison with Experiment Sliding laws are not easily compared with data. There is a hint of ﬁeld and experimental support for (3.17). Budd et al. [15] found that experimental data on ice sliding over solid slabs of various materials could be described by (3.17) with r = s = 1/3. Bindschadler [16] tested various sliding laws against measurements from Variegated Glacier, and also found a best ﬁt when r = s = 1/3. Bentley [17] tested sliding laws against data from Whillans Ice Stream in Antarctica, and found them all wanting. A key problem with using (3.17) (assuming it is reasonable) is in the assessment of a sensible value for c. This must rely on small-scale details of the bed conﬁguration, which by its nature is difﬁcult to observe, even though deglaciation has exposed many former subglacial beds.

3.3

Subglacial Drainage Theory

3.3.1 Weertman Films Weertman [18] conceived of water flowing at the base of an ice sheet or glacier as a thin water ﬁlm. Walder [19] showed that such flow would be unstable. The mechanism is conducive to channel formation: a local increase in ﬁlm thickness leads to increased flow, hence increased frictional heating, increased meltback, and further channel widening. This positive feedback fuels an instability, which is limited at short wavelengths by heat conduction. Although a uniform ﬁlm is thus not feasible, one can argue that an uneven ‘patchy’ ﬁlm may exist, and this is a possibility under the Antarctic Siple Coast ice streams [20]. The water flux Q per unit width through a ﬁlm of mean thickness h is given by h3 @p Q¼ q g sin h ; @s 12l w

ð3:18Þ

where l is water viscosity, qw is water density, h is the inclination of the bed to the horizontal, p is water pressure, and s is distance downstream. For a patchy ﬁlm, this relation would be modiﬁed by a pre-multiplicative tortuosity coefﬁcient. If H is the ice thickness and a is the ice surface slope angle, then we have @H=@s tan h tan a, and the ice (overburden) pressure is pi = qigH. Accordingly @pi =@s qi g½tan a tan h, and (3.18) can be rewritten as Q where U is the gravitational head

h3 @N Uþ ; @s 12l

ð3:19Þ

3

Sliding, Drainage and Subglacial Geomorphology

U ¼ qw g sin h þ qi gðtan a tan hÞ;

55

ð3:20Þ

and N = pi − p is the effective pressure. Normally, the hydraulic (gravitational) head is much larger than the effective pressure gradient term.

3.3.2 Röthlisberger Channels (or ‘R-Channels’) Outlet streams from glaciers often flow from single channels, carved as large tunnels in the ice. The theory of drainage through such channels was elaborated by Röthlisberger [21], followed by Nye [22]. The mechanism of flow is that, ﬁrstly, the channel water pressure pw is below the ice overburden pressure, so the effective pressure N = pi − pw is positive. (This is often observed in boreholes, but is also necessary for integrity of the ice.) As a result of the excess pressure, viscous ice creep tends to close the channels, but closure is counteracted by the frictional heat released by the turbulent flow in the channel. Consequently, a balance of melting and creep closure allows channels to be maintained in a steady state. Figure 3.4 shows an example of a (former) subglacial channel of roughly semi-circular cross section.

Fig. 3.4 Ice cave under the tongue of Vernagtferner, Ötztal Alps, left behind by a former subglacial drainage channel

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A. Fowler and F. Ng

In the Nye-Röthlisberger model, the mass conservation equation is @S @Q m þ ¼ þ M; @t @s qw

ð3:21Þ

where S is channel cross section, Q is the water flux, m is the melt rate, and M is the water supply (e.g., from surface meltwater through moulins). The closure condition for the channel―which is also a kinematic condition5 for the ice flow―is @S m ¼ KSN n ; @t qi

ð3:22Þ

where N is effective pressure and K is proportional to the constant in Glen’s flow law. (More precisely, K = 2A/nn, where A and n are the constants in e_ ¼ Asn .) Momentum balance is expressed through a turbulent friction law Uþ

@N FQ2 ¼ 8=3 ; @s S

ð3:23Þ

where F is a roughness coefﬁcient6. This equation is derived from the empirical correlation of Manning for the velocity of fluid flow down a conduit/channel. Lastly, the frictional heat generated is due to potential energy release, and the consequent melting is given by an energy equation in the form7 @N mL ¼ Q U þ ; @s

ð3:24Þ

where L is latent heat (as before). Elimination of Q and m gives the hyperbolic equation for S:

The kinematic condition at the interface between two fluids is a consequence of their continuity, and states that fluid elements in the interface remain there, i.e., the speed of the interface is the same as that of the fluid which is at the interface. There are two such equations, one each for ice and for water, which is why we get two evolution equations for S. 6 In terms of Manning’s roughness coefﬁcient n′ (which has the units m–1/3 s), the roughness F = qwg[2(p + 2)2/p]2/3 n′2 for a semi-circular channel. Typical values of n′ range from 0.01 for a smooth walled channel to 0.1 for a rough (boulder-strewn) channel. Nye found a good ﬁt to the rising limb of the 1972 Grímsvötn flood hydrograph with a value of n′ = 0.12. Clarke [23] considers this to be too high, since the ice walls would be very smooth; he thinks that an unnecessarily high value arises because Nye neglected advection of heat in his simpliﬁed theory. However, turbulent sediment transport in the channel would also increase the effective value of n′ beyond what one might expect. In fact, it is possible to obtain a good ﬁt to the complete hydrograph using the Nye model with a lower value of n′ 0.04 (see Fig. 3.5). 7 This assumes that the water temperature is isothermal, at the melting temperature. This is a reasonable assumption under normal circumstances but may not always be accurate during jökulhlaups. 5

3

Sliding, Drainage and Subglacial Geomorphology

" # @S @ S4=3 ðU þ @N=@sÞ1=2 ðU þ @N=@sÞ3=2 4=3 þ ¼ S þ M; @t @s F 1=2 qw F 1=2 L

57

ð3:25Þ

together with the closure condition @S ðU þ @N=@sÞ3=2 4=3 ¼ S KSN n : @t qi F 1=2 L

ð3:26Þ

Röthlisberger solved the steady-state model numerically. We see two spatial derivatives (for S (or Q) and N) so we need two boundary conditions: these can be taken as Q ¼ 0 at s ¼ 0 N ¼ 0 at s ¼ l

ðstream head), ðoutlet).

ð3:27Þ

If we neglect @N=@s (a singular perturbation) and approximate (3.21) as @Q=@s M, then Q Ms is known, and in a steady state S

3=8 F Q4=3 ; U

QU KSN n ; qi L

ð3:28Þ

whence N

U qi LK

1=n 3=8n U Q1=4n : F

ð3:29Þ

We see that N increases with Q, so that channel water pressure decreases as water flux increases. This property explains the arterial nature of R-channels (they like to be on their own), as the low water pressure of a larger channel ensures that it out-competes any nearby smaller channels in gathering subglacial meltwater supply.

3.3.3 Jökulhlaups A spectacular success of Nye and Röthlisberger’s theory is in its application to the study of glacial outburst floods or jökulhlaups that occur when ice-dammed lakes drain suddenly. Well-known examples of these floods originate from the subglacial lake Grímsvötn in Iceland [22] and marginal lakes such as Gornersee in Switzerland, Merzbacher Lake in the Tian Shan, and Hidden Creek Lake in Alaska. A useful review of jökulhlaup studies has been given by Roberts [24]. While the size and timing of jökulhlaups vary across different lakes and even at each lake, they have recognizable characteristics including short outburst duration

58

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(days to several weeks), high peak discharge (often up to *103 m3 s−1), and a flood hydrograph that often shows a nearly-exponential rise, followed by abrupt recession (e.g., Fig. 3.5). These characteristics distinguish them from river runoff events arising from diurnal melting and rain storms. In the context of the Nye-Röthlisberger theory, the rise in flood hydrographic discharge is explained by positive feedback between melt enlargement of the channel (ﬁrst term on the right of (3.22)) and the growing discharge and heat dissipation inside it. However, as the lake drains, its lowering hydrostatic pressure reduces the channel water pressure, so closure (last term in (3.22)) eventually overrides melting to shut the channel and end the flood. The channel model is now coupled to the lake (be this subglacial or ice-marginal) and requires an upstream boundary condition on the discharge Q, which drains the lake. In turn, changes in the lake’s volume V (thus its surface elevation) alter the pressure at the channel inlet and the effective pressure there. The inlet condition can be written as dV AL @N ¼ ¼ mL Q dt qw g @t

ð3:30Þ

where AL is the lake surface area and mL is the rate of water input to the lake (e.g., from rain and snowmelt, and subglacial geothermal melting―the last of these is signiﬁcant near Lake Grímsvötn). When boundary condition (3.30) is applied, the theory can give an excellent ﬁt to observed flood hydrographs; e.g., Fig. 3.5. The earlier studies simulating flood hydrographs are those of Nye [22], Spring and Hutter [25] and Clarke [26]. More predictions have stemmed from the theory since. By analysing the co-evolution of N and Q in the model, Ng and Björnsson [27] showed that the volume and peak

Fig. 3.5 Observed (dashed) and modelled (solid line) hydrographs of the 1972 flood from Grímsvötn, Iceland. The model is based on the Nye-Röthlisberger formulation and uses a realistic value of the Manning roughness, n′ = 0.036 m−1/3 s

3

Sliding, Drainage and Subglacial Geomorphology

59

discharge of the simulated floods obey scaling relationships that are found in empirical studies [28]. By incorporating a subglacial hydraulic seal near the channel inlet (at Grímsvötn, this seal is located on the caldera rim around the lake), Fowler [29] showed that the model can oscillate in repeated flood cycles, and the floods initiate at lake levels below levels required to float the ice dam, as observed in some systems. Between jökulhlaups, Q 0, and the lake reﬁlls according to dV/dt mL in (3.30). Over the long term, this interacts with the flood-initiation threshold to create cycles of ﬁlling and drainage, so the lake level displays a sawtooth-shaped history. At marginal ice-dammed lakes, the input mL depends on weather and varies seasonally, so the cycles give an irregular sequence of flood dates. The mechanisms behind the timing pattern have been explained by dynamical maps that relate one flood date to the next [30]. Given uncertainty about the flood-initiation process at different lakes, reliable forecasting of the timing and size of jökulhlaups remains an outstanding problem.

3.3.4 Subglacial Lakes Geophysical measurements show that the base of the Antarctic Ice Sheet is covered with a large number of subglacial lakes. The largest is Lake Vostok, with a volume of some 5000 km3, but there are many others [31], and ice surface altimetry measurements (Sect. 13.4) have shown that these lakes drain frequently from one to another by floods (e.g., [32]). It is therefore clear that jökulhlaups occur under ice sheets, and that the way in which water is transported over long distances may be controlled by these events.

3.3.5 Linked Cavities When ice flows over rough bedrock, cavities form in the lee of obstacles. These cavities may become ﬁlled with water, and subglacial drainage may occur through them. This possibility was advanced by Kamb et al. [33] based on observations of the surging Variegated Glacier, and studied theoretically by Kamb [34] and Walder [35]. The basic idea is that cavities will be linked by oriﬁces in the bed which act like small Röthlisberger channels (Fig. 3.6). The flow rate is controlled by these oriﬁces, which are described by a theory similar to that for channels. A key difference is that now the local overburden stress in the oriﬁces P is given by sP þ ð1 sÞpw ¼ pi ;

ð3:31Þ

where s is the shadowing function deﬁned by Lliboutry [6]: s is the proportion of uncavitated bedrock and, importantly, it depends on the sliding velocity. Consequently, the drainage relation between P − pw and local water flux Q (for oriﬁces)

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Fig. 3.6 The linked-cavity system envisaged by Lliboutry [5]. Reprinted from the Journal of Glaciology with permission of the International Glaciological Society

becomes a relation between N = pi − pw and Q that involves the sliding velocity u (which is itself determined in terms of basal shear stress s and N through the sliding law). For a given ice geometry with known s, the effective pressure N is thus determined implicitly in terms of u. Which drainage system? A channelised drainage system (Sect. 3.3.2) is always possible, but if cavities exist, drainage can occur instead through a linked-cavity system, which enables fast sliding at low effective pressures. Indeed, observations on Variegated Glacier indicate that the surge in 1982–83 was initiated by such a switch in drainage systems [33]. The switch can be understood as follows. Suppose a linked-cavity system with flux QK and effective pressure NK coexists with a channel system with flux QR and effective pressure NR. In equilibrium, we have NR = NK (otherwise water flows from one system to the other, up effective pressure gradients). If a perturbation causes some water transfer between the systems—e.g., by increasing the water flux in the cavity system by DQ and thus decreasing the water flux in the channel system by DQ—then the effective pressures are perturbed by the amounts −(dNR/dQR)DQ in the channel system and (dNK/dQK)DQ in the cavity system. The situation will be unstable if

3

Sliding, Drainage and Subglacial Geomorphology

@NR @NK \ ; @Q @Q

61

ð3:32Þ

because the perturbed pressures in the channels and cavities then sustain and amplify the initial transfer―to shrink the channels further. We saw from (3.29) that @NR =@Q [ 0 for R-channels (Sect. 3.3.2). On the other hand, for linked cavities, theory indicates @NK =@Q\0: the reason is that a drop in NK corresponds to an increase in cavity pressure, and thus corresponding decrease in pressure over the bed; the resulting smaller closure rate of the inter-cavity conduits allows an increased transmissive flow between cavities. Therefore the derivatives in (3.32) have the right signs to permit switching between the two kinds of systems. The size of the derivatives matters in deciding whether the condition (3.32) is met. As NK depends also on sliding velocity u, the instability really depends on u: if u is large enough, there will be a transition from channel drainage to cavity drainage [13]. Kamb [34] studied the detailed mechanism of the opposite transition, from cavity drainage to channel flow, which is effected through frictional heating induced enlargement of the interconnecting oriﬁces. The switch from channel to cavity systems is associated with the onset of a glacier surge.

3.3.6 Drainage Transitions and Glacier Surges In the simpliﬁed drainage theory of Fowler [13], N takes the value NR or NK (< NR) (Fig. 3.7a) depending on the mode of drainage, and there is an instability which is flow dependent, and which causes the transition from channels to linked cavities if a parameter K = u/Nn (n is the exponent in Glen’s law) exceeds some critical value, Kc. Kamb [34] described the opposite instability, when a cavity system unstably converts to a channel system. The Kamb instability can be interpreted as occurring

Fig. 3.7 a Variation of drainage effective pressure with sliding velocity. b A drainage-induced multi-valued sliding law

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A. Fowler and F. Ng

when @N=@Q shifts from negative (stable cavity drainage) to positive (channelised flow), and this occurs as K decreases through a second critical value (lower than the ﬁrst). The result of these two instabilities is illustrated in Fig. 3.7a, which shows that N is a multi-valued function of u. When this is coupled with a sliding law of the type s ¼ cur N s ;

ð3:33Þ

we ﬁnd the multi-valued sliding law shown in Fig. 3.7b (cf. Fig. 3.3). The existence of such a multi-valued sliding law can explain glacier surges and is consistent with observations on Variegated Glacier. In particular, let us assume (for simplicity) that a glacier’s driving stress is all supported by the basal traction, so s = qigHsina. Suppose initially the s-value intersects the multi-valued region in Fig. 3.7b and the glacier starts out in quiescence, with u on the (slow) branch of the law. Thickening of the glacier under positive mass balance for some time would move it up the branch, and at the upper ‘cusp’ a transition to the fast (linked-cavity) branch takes place: the glacier surges. Subsequently, its thinning as a result of the longitudinal extension during the surge phase would move its state down the fast branch towards the lower cusp, at which point the opposite transition occurs to terminate the surge. Repeated surge cycles can therefore arise in this model.

3.3.7 Ongoing Developments The need to couple ice-flow dynamics and subglacial hydrology in a sound manner (with the right physics/approximations for the spatio-temporal scales concerned) continues to be a major theme in modelling studies. The theories described herein lay some of the foundation for interpreting glacier hydrological observations and explaining large-scale phenomena like surges, jökulhlaups and ice streams, but these problems are far from completely solved. Meanwhile it is realised that coupling between ice motion and hydrology can strongly influence the response of ice sheets to changing climate. Field studies on the Greenland Ice Sheet show that water from supraglacial lakes can drain to the bed via moulins despite thick (*1 km) ice, and such injection modulates the ice-surface velocity on seasonal or shorter timescales. These observations motivate numerical studies of coupled drainage and sliding in ice-sheet settings. For example, Schoof [36] put forward an innovative ‘spatially-extended’ model that simulates the interactions of channel-type and cavity-type drainage and sliding across a two-dimensional bed domain. He found that an increase in the temporal variability of the surface-derived water supply, not an increase in its mean value, causes ice-sheet flow to accelerate.

3

Sliding, Drainage and Subglacial Geomorphology

3.4

63

Basal Processes and Geomorphology

3.4.1 Soft Glacier Beds Glaciers and ice sheets erode their beds. Stress fracturing plucks boulders from the bed, and these are ground up to cobbles, and to a glacial flour which gives (as suspended sediment) proglacial streams their milky colour. The eroded sediment is transported along at the base of a glacier as a kind of moving conveyor belt if the ice is temperate, and is known as till. Till can form a layer several metres thick; its deformation can account for the bulk of a glacier’s motion. One of the best known examples is Trapridge Glacier, a surge-type glacier underlain by 6 m of till. The interest in the basal till in this context is that the surge must be associated with sliding, and the mechanism is apparently different from that responsible for Variegated Glacier surges. A deforming basal layer also occurs where an ice sheet overrides sediments, such as in the lowlands of Europe or the plains of North America in the last ice age. A modern example of till-based ice flow is found in the Siple Coast of Antarctica where six ice streams8 exist (see Fig. 8.10). These ice streams (except for Kamb Ice Stream now) are zones of fast flowing ice underlain by several metres of wet deforming till: almost all of the deformation is due to the conveyor-belt sliding of the till. It is important to understand the corresponding sliding law, because this may be instrumental in explaining why ice streams exist, and how large-scale oscillations of the Laurentide Ice Sheet could occur, as may have happened during Heinrich Events [37]. Moreover, till mechanics is relevant for understanding the origin of a variety of subglacial landforms. Till rheology When basal ice resting on subglacial sediments is temperate, basal meltwater floods the till, so that it is water-saturated. The motion of wet, granular materials is complicated, but is often simply modelled as a Herschel-Bulkley fluid. That is, subjected to a shear stress s, there is a yield stress s0: the particulate medium cannot deform unless it overcomes the frictional resistance between particles. A common assumption is the Coulomb yield criterion: s0 ¼ c0 þ N tan w;

ð3:34Þ

where N is the effective pressure, w is the angle of friction, and c0 is a small cohesion due essentially to the clay fraction of the till. If this stress is exceeded, then a non-zero strain rate e_ results, which depends nonlinearly on the stress; for a pseudoplastic (Herschel-Bulkley) material, e_ / ðs s0 Þa , where a > 1. Experiments on Antarctic subglacial sediments [38] suggest that for these the creep is better described by a sharp exponential dependence on stress. 8

Originally labelled A–F, these ice streams have been renamed Mercer, Whillans, Kamb, Bindschadler, MacAyeal, and Echelmeyer Ice Streams, respectively, in the honour of those who have studied them.

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The rôle of the effective pressure is also important in the till flow behaviour. The effective pressure N = P − pw (where P is overburden pressure and pw is pore water pressure) is essentially a measure of the stress transmitted between the solid particles. As we expect the frictional resistance to flow to increase with N, this suggests that for a given s, e_ will decrease as N increases. The simplest realistic type of viscous rheology is the Boulton-Hindmarsh [39] law e_ ¼ Asa N b

ð3:35Þ

e_ ¼ Aðs s0 Þa N b

ð3:36Þ

if we neglect s0, or

if a yield stress is included. Dilatancy A further complication to the rheology is that when a till deforms, it dilates (the particles have to move round each other) so that its porosity (which under normal consolidation conditions will be a function of N) depends on both s and N. This may be of less signiﬁcance cryodynamically, however. The law (3.35) is a very useful form for practical use, although the issue of viscous versus plastic rheology is controversial, and a matter of much current debate. Boulton and Hindmarsh’s [39] values for (3.35) were a = 1.33, b = 1.8, A = 3 10−5 Pab−a s−1 (= 4 barb−a y−1). Measurements on other glaciers typically give a till viscosity value of the order of 1010 Pa s, with substantial variations. One physically inappropriate inference from (3.35) is that the strain rate becomes inﬁnite (or the viscosity tends to zero) as N ! 0. This would describe an unconﬁned mixture, where the limit N ! 0 is associated with fluidisation and slurry-like behaviour. But under a glacier, it seems unlikely that even in free flotation (N = 0) the resistance would be negligible, since deformation of the till still requires cobbles and clasts to move past each other. Moreover, lubricated flow over uneven bed topography produces resistance, just as in classical sliding theory. Geotechnical theory Boulton and Hindmarsh’s law is purportedly based on seven data points, although the primitive data are not available, and (3.35) cannot be considered to be experimentally substantiated. The creep behaviour of saturated granular materials such as soils has been extensively studied. As stated above, there is a yield stress (which depends on effective pressure and porosity, i.e., consolidation history) and when this is exceeded, plastic deformation occurs. According to Kamb [38], longer-term creep at large strains is typically described by a constitutive law of the form e_ ¼ A exp½as=sf

ð3:37Þ

for s > sf (sf is the stress at failure, e.g., given by (3.34)), with very large values of a. In effect, this would imply almost perfectly plastic behaviour, that is, s sf if e_ [ 0.

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Sliding, Drainage and Subglacial Geomorphology

65

Kamb measured sf 0.02 bar for till obtained from Whillans Ice Stream in Antarctica, and this low value prompted the idea [40, 41] that the Whillans Ice Stream till is so weak that the resistance to ice flow must be due to basal ‘sticky spots’ or lateral drag. However, Kamb’s experiment was effectively at zero conﬁning pressure. If the yield stress is given by sf = lN, with N = 0.4 bar and l = 0.4, we ﬁnd sf = 0.16 bar, close to the observed basal value. In addition, it is not clear how the deformation of clast-rich till relates to Kamb’s laboratory experiment. Hooke et al.’s [42] results may be consistent with an almost perfectly plastic rheology. Iverson et al. [43] reported results of ring shear tests consistent with a Coulomb plastic law, sf / N; they also removed larger clasts from the apparatus. Because one expects on average that effective pressure would increase with depth in till (due to hydrostatic effects), so also would the strength, or yield stress, of a perfectly plastic till; in this case one would expect failure at the ice-till interface only. This is inconsistent with observations of deformation with depth (e.g., [44]) and has led to somewhat ad hoc theories to explain this within the context of a plastic rheology [45, 46]. The controversy continues to rumble. A useful account of differing points of view is in the issue of Quaternary International on ‘Glacier Deforming-bed Processes’, volume 86 (2001). See also Clarke’s [47] review of these and other issues concerning subglacial processes. Ice sliding over till If we neglect the vertical variation of effective pressure due to gravity (an unwarranted assumption: the effective gravitational head over 5 metres of till with Dq = qs − qw = 2 103 kg m−3 is 1 bar, compared to inferred values of N * 0.4 bar at the base of Whillans Ice Stream), then a sliding velocity u of ice over a till layer of thickness hT gives a strain rate u=hT ¼ Asa N b ;

ð3:38Þ

following (3.35). It follows that the sliding law in this case is s ¼ cur N s ;

ð3:39Þ

with c = (hTA)−1/a, r = 1/a, s = b/a. If we include the gravitational head, this relation is modiﬁed. Most importantly, the yield stress criterion (3.34) suggests that the till does not deform if N[

s c0 ; tan w

ð3:40Þ

and this leads to the concept of a deforming ‘A’ horizon overlying a non-deforming ‘B’ horizon in the till [39]. In a glacier, with s * 1 bar and if tan w * 1, then the A horizon corresponds to a depth of *5 m. In an ice stream with s * 0.1 bar, the depth may be only 0.5 m. Note that the effect of the ﬁnite A horizon is to increase the roughness c in (3.39) (or decrease u, or decrease A).

66

A. Fowler and F. Ng

It is clear that different choices of rheological behaviour will lead to different forms of sliding law. Particularly, failure may occur at the ice-till interface and the till-bedrock interface (if one exists), and prescription of the corresponding slip rates is problematic.

3.4.2 Drainage Over Till If ice slides over till, the sliding law depends on N, so we need a drainage law to determine N. Unfortunately, there is little to constrain how water drains over a till bed. Permeation One possibility is by Darcy flow downwards through the till to an underlying aquifer. If we assume such a downward flow to be q (volume flux per unit area) then Darcy’s law implies

k @pw qw g ; q¼ gl @z

ð3:41Þ

where z is depth, k is permeability, and ηl is water viscosity. With hydrostatic overburden pressure @P=@z ¼ qs g, where qs is the (bulk) till density, then @N gl q ¼ þ Dqg: @z k

ð3:42Þ

To estimate these terms, we take q * G/qwL, where G is geothermal heat flux, L is latent heat. With G * 0.05 W m−2, qw* 103 kg m−3, L * 3.3 105 J kg−1, we have q * 1.5 10−10 m s−1 *5 mm y−1. Then for ηl * 2 10−3 Pa s, and k * 3 10−14 m2, we have ηlq/k * 10−4 bar m−1, which is insigniﬁcant compared to Dqg * 10−1 bar m−1. On the other hand, if subglacial till is underlain by impermeable bedrock, so that the meltwater must be evacuated through the till horizontally, then along the corresponding flow trajectory (from the interior of an ice mass to its margin), the integrated water flux over distance x is

khT @pw qw g sin h ; qx ¼ gl @x

ð3:43Þ

where h is the bed slope angle in the x-direction. If we take ice pressure to vary as @pi =@x ¼ qi g½tan a tan h, where a is the ice surface slope angle, then @N x gl q ¼ ½qw g sin h þ qi gðtan a tan hÞ: @x hT k

ð3:44Þ

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Consider, for example, the Siple Coast ice streams, for which x * 1000 km, hT * 10 m, a * 10−3. Then (x/hT)(ηlq/k) * 10 bar m−1, while (with h = 0) we have qiga * 10−4 bar m−1. In this case the required flux will lead to negative effective pressures, and we infer the existence of some kind of channelised flow. Channelisation Two possibilities have been suggested for this flow. A patchy Weertman type ﬁlm has been advocated by Alley [20], and subsequently by Creyts and Schoof [48]. Essentially, the water collects in puddles and would have a Darcy type law governing its behaviour. As described earlier, this flow may be subject to instability. If that is the case, then channelised flow could occur. Now two distinct end-members are possible: Röthlisberger channels as before, and also canals: channels cutting down into till sediment. The mechanism governing their behaviour is similar to that of Röthlisberger channels, but till creep replaces ice creep and sediment erosion replaces melting. Walder and Fowler [49] analysed both types of channel flow and suggested that these end-member states were distinguished by a critical value N* of the effective pressure. Essentially, for N > N*, Röthlisberger channels would exist if the ice surface slope angle a were large enough, while for N < N*, canals would exist (for any a). Thus, for ice sheets/streams with small a, canals provide the drainage mechanism. Canals have low N (as inferred for Whillans Ice Stream) and, importantly, N decreasing with increasing water flux Q; therefore canal drainage forms a stable anastomosing pattern, similar to a braided stream (cf. arterial networks for R-channels). The Walder-Fowler theory is rather primitive but receives support from the more detailed analysis of Ng [50], which accounts for the conservation of transported sediment when determining the canal cross-section geometry. Engelhardt and Kamb [51] reported consistency of the canal description with observations on Whillans Ice Stream. The property that @N=@Q \ 0 is fundamentally due to the assumption that subglacial canals (like sub-aerial rivers) choose their own depth, as a result of the balance between erosional shear stress and the critical Shields stress (for the water flow to initiate sediment motion on the bed). The ﬁlm versus canals argument cannot then solely rely on Walder’s [19] stability argument, since that invoked ice melting. In forming canals, it is essential to thicken the ﬁlm sufﬁciently to enable the sediments to be eroded. Otherwise, we would speculate that the Walder instability would raise the ice interface but that, in the absence of erosion, the till would creep into the uplifted ice regions. A possibility is thus a water ﬁlm over a wavy till interface. The 5 mm y−1 geothermal basal melt rate would give a ﬁlm thickness over a 700 km long flow line of 0.6 mm. A puddly base is then feasible under ice streams, but perhaps less likely for the larger melt-water fluxes below glaciers. A variant of this description for ice sheets arises from the observation of subglacial lakes draining into each other. Just as in Lliboutry’s vision of linked cavities (Fig. 3.6), sub-ice sheet lakes may be connected by canals through which periodic floods occur.

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3.4.3 Geomorphological Processes In this section, we briefly consider several landforms whose origin involves subglacial water drainage and sediment processes (and, in some cases, ice flow). The topic of how they are used to reconstruct the history of past ice sheets and glaciers is treated in Chap. 17. Ridges and valleys Some glacial landforms can tell us how channelised drainage occurred under former ice sheets. Eskers (Fig. 3.8) are long ridges of gravel and sand deposits which are presumed to form in subglacial channel flows; they could form from either canal-type or R-channel-type flows. Sometimes multiple eskers occur in parallel (e.g., Fig. 3.8b), suggesting a cooperative control of subglacial channel spacing. During the formation of an esker, sediment deposition may raise the channel floor (and thus also the ice roof) until the channel is pinched off and drainage shifts elsewhere. A different evolution occurs in the formation of tunnel valleys, for example in the lowlands of Northern Germany. These massive structures, hundreds of metres deep and kilometres wide, are evidential of an anastomosing drainage pattern (hence of canal type, sediment-controlled). The flow is sufﬁciently rapid (thus, N is low and the sediments are very mobile) so that the sediments which are squeezed into the channel are efﬁciently removed. The channel thus eats its way into the

Fig. 3.8 a Aerial view of an esker and b map of an esker network in the region east of Great Slave Lake, Northwest Territories, Canada. From Sharpe et al. [52]. Reprinted with permission from Elsevier

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substrate. Figures 3.9 and 3.10 show plan view and section of tunnel valleys in Northern Germany. Eskers are, in this scheme, indicative of weak flow or shallow sediments, while tunnel valleys suggest larger flows and deeper sediments. Shaw and co-workers (e.g., [54]) argue that many of these features are formed in huge subglacial floods by fluvial action, but there seems little necessity for supposing this.

Fig. 3.9 Plan view of tunnel valleys in Northern Germany, forming an anastomosing pattern [53]. Reprinted from the Annals of Glaciology with permission of the International Glaciological Society

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Fig. 3.10 Typical cross-section of a tunnel valley [53]. Reprinted from the Annals of Glaciology with permission of the International Glaciological Society

Fig. 3.11 Digital elevation model of ribbed moraine in north-eastern Ireland. The image is approximately 30 km across. Image supplied courtesy of Chris Clark

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Streamlined forms A variety of landforms, notably drumlins and ribbed (or Rogen) moraine, may be formed subglacially due to ice moving over deformable sediments, by analogy with dune formation in deserts and rivers. Rogen moraine is a ribbed wave-like formation transverse to ice motion (Fig. 3.11), and drumlins represent a three-dimensional development of it (Fig. 3.12), much as ripples and dunes under water relate to bars. Mega-scale glacial lineations (‘MSGLs’) are still more elongated corrugations/grooves found on deglaciated beds. It is thought that these streamlined subglacial landforms are associated with different parametric conditions (Fig. 3.13).

Fig. 3.12 A swarm of drumlins in Canada. The image is approximately 13.5 km across and centred at 58.5° N and 107.4° W. Source: photograph A-14509-5 of the National Air Photo Library—Natural Resources Canada, 1954. Reproduced with permission of the Department of Natural Resources, Government of Canada

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Fig. 3.13 Formation of drumlins from the basic Rogen instability (cf. [55], p. 246)

Theories of drumlin formation Although the literature on drumlins is extensive and goes back almost two hundred years, dynamic theories are conspicuously absent. Studies by Hindmarsh (e.g., [56]) demonstrated that the shearing flow of ice over a deformable till layer can be unstable to the formation of landforms. The instability mechanism is that ice flow over a till protuberance causes increased normal stress on its upstream face, and this can accelerate the till flow due to the resultant increase in shear stress, causing an enhancement of the bump height. Hindmarsh demonstrated instability numerically under a wide variety of conditions, and Fowler [57] derived analytic criteria for the instability. This work was extended by Schoof [58], who showed numerically that cavities will commonly form behind the developing bedforms. Fowler [59] developed the theory to include the effects of cavitation. His instability criterion can be framed quite generally in terms of assumed basal ice velocity U and till flux q, both assumed to be functions of basal stress s and effective pressure N, with @q @q @U @U @s [ 0, @N \0, @s [ 0, @N \0. The uniform ice/till interface is then unstable if ds ds [ ; ð3:45Þ dN U dN q i.e., the slope of the constant U curves in the (N, s) plane exceeds the slope of the constant q curves, as indicated in Fig. 3.14. This criterion is easily satisﬁed for reasonable choices of U and q. The instability theory for drumlin formation continues to be developed, but remains controversial (e.g., [60]); many details remain to be understood. In particular, it is an open question whether MSGLs, grooves hundreds of kilometres long which are aligned with the ice flow direction, can be understood as arising from a modiﬁcation of the drumlin shape as ice flow parameters change. Other apparent issues include explaining the formation of internal stratiﬁcation seen in some drumlins. We should not imply that the instability theory is well accepted. There are a number of other mechanisms and philosophies of current interest. Some of these are discussed in Chap. 17.

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Fig. 3.14 Instability criterion in terms of the slope of the constant-U and constant-q curves. The sliding law is taken to be U = [(s/N) − l]+, and the till flux to be q = U [(s/N) − l]+, where [x]+ denotes max (x, 0). Shown are the curves (solid) for constant U = 0, 1, 2 and (dashed) for constant q = 0, 4, 12. Instability occurs because at common values of s and N, the q curves are less steep than the U curves

The fact that deformable beds create their own topography represents a mechanism for enhancing basal friction (and causing sticky spots). It is plausible that elevated drumlins have relatively high N (but see the ﬁndings of McCracken et al. [60]) and are thus stiff and immobile, while basal sediments at lower elevations are less viscous. Indeed, this feature may well be an important constituent of this proposed mechanism of drumlin formation, and indicates that drumlins are closely implicated in the dynamics of ice sheet flow.

Exercises 3:1

Weertman sliding (Sect. 3.2.1): Assume a hard bed with ﬁxed topographic roughness m. By minimising (i) the sliding velocity u in (3.4) and (ii) the stress s in (3.5), calculate the controlling obstacle size in each case, and hence derive the sliding law in (3.7) and (3.8).

3:2

Ice of depth h slides over topography of wavelength l and amplitude a. Use the sliding law (3.5) to assess typical sliding velocities for various values of a, h, and l relevant to valley glaciers or ice sheets.9 Given that shearing within the ice is given by @u=@z = 2As(z)n, where s(z) = qig(h − z)sina, a is the surface slope, and z is the height above the bed z = 0, use dimensional

9

Hint: (3.5) gives s in terms of u, whereas you require u in terms of s. One way to do this is to suppose the viscous term, say, is much larger than the regelative term, and then invert the relation to give an approximate sliding law. One can then use this approximation to ﬁnd when it is valid, i.e., if the regelative term is indeed small. Similarly, one can approximately invert the equation if the regelative term is dominant.

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reasoning to estimate ub/us, the ratio of sliding to surface velocity, for different scenarios, in terms of the parameters r = l/h and m = a/l. (Use the values qi = 0.9 103 kg m−3, L = 3 105 J kg−1, k = 2 W m−1 K−1, C = 0.8 10−2 K bar−1, n = 3, A = 6 10−24 Pa−3 s−1.) 3:3

Derive the subtemperate sliding law (3.14) in detail, discussing the assumptions which you make. Is this the best set of assumptions? What others could you make? How should you include the friction of the bed between bumps? Draw rough graphs of the sliding velocity u = U(s, DT) for various values of DT.

3:4

Use dimensional estimates to estimate the sizes of the terms in Röthlisberger’s (steady state) model of channel drainage, and justify (if you can) the neglect of the @N=@s term. Can the resulting solution for N still satisfy the boundary conditions? (Use the values n′ = 0.1 m−1/3 s, K = 0.25 10−24 Pa−3 s−1, and other values from Exercise 3.2.) Explain why, if N is an increasing function of Q, one might expect an arterial drainage network, whereas if it is a decreasing function of Q, one might expect a distributed drainage system. [Hint: imagine what would happen to two neighbouring channels if there is a pressure or flow perturbation in one of them.]

3:5

If N decreases as the water flux Q of a canal system draining through sediments increases, then the following positive feedback is potentially operative. An increase of u increases frictional heating, and hence increases water flux Q: this decreases N, which leads via the sliding law to a further increase of u. Suppose that N ¼ ^c=Q1=3 ; and a suitable sliding law is s ¼ cur N s : Now, meltwater production at the base of an ice sheet or ice stream (or cold glacier with temperate base) is due to geothermal heat G and frictional heat su, but there is heat loss to the overlying cold ice at rate q. The melt flux per unit area is thus (G + su − q)/qwL, and in a simple model we equate this with the water flux Q per unit width, thus Q¼

ðG þ su qÞwd ; qw L

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where wd is the stream spacing. The cooling rate q depends on ice thickness and flow. For fast flow over horizontal distance l (for instance, the length of an ice-stream flowline), assume that q (qicpk/pl)1/2DT u1/2. Show that the above equations then collapse, in suitably scaled units, to ½1 þ su u1=2 s=3 ¼ lur =s; with the single parameter l (which you should deﬁne) being a measurement of roughness: small l means slippery till. If r = s = 1/2, show that if l is small enough, u will be a multi-valued function of s. Use the values qw = 103 kg m−3, qi = 0.9 103 kg m−3, L = 335 kJ kg−1, cp = 2 kJ kg−1 K−1, wd = 3 km, k = 2.1 W m−1 K−1, l = 2000 km, DT = 50 K, G = 0.05 W m−2, s = 0.15 bar, u = 500 m y−1, to infer values of Q, and hence appropriate estimates for c and ^c, if N = 0.4 bar. Calculate a typical value of l and investigate whether the sliding law is multivalued in this case. This behaviour is relevant to the dynamics of ice streams and surging Pleistocene ice sheets. 3:6

The instability of a horizontal interface between ice and till is controlled by the dimensionless parameter R¼

qN þ 2kðqN Us UN qs Þ ; 1 þ 2kUs

where U(s, N) is the sliding velocity, q(s, N) is the till flux, k is the (positive) wave number (all dimensionless), and the subscripts indicate partial derivatives, thus Us = @U=@s, etc. Instability occurs if R > 0. Assuming that Us > 0, qs > 0, UN < 0, qN < 0, show that if R is positive for some k, then the maximum value of R will occur when k is large, and in this case show that we can take R R 1 ¼ qN

U N qs ; Us

and hence show that the uniform bed is unstable if R1 [ 0. Suppose now that U = U(u), where u = s/N, and that q¼

s N l

UðuÞ; þ

where the notation [x]+ denotes max(x, 0), and U = 0 for u < l, where l is a positive constant. Show that R1 [ 0 when u > l for any such (positive) function U(u).

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References 1. Weertman J (1957) On the sliding of glaciers. J Glaciol 3:33–38 2. Nye JF (1969) A calculation on the sliding of ice over a wavy surface using a Newtonian viscous approximation. Proc R Soc Lond A 311(1,506):445–467 3. Nye JF (1970) Glacier sliding without cavitation in a linear viscous approximation. Proc R Soc Lond A 315:381–403 4. Kamb WB (1970) Sliding motion of glaciers: theory and observation. Rev Geophys Space Phys 8:673–728 5. Lliboutry LA (1968) General theory of subglacial cavitation and sliding of temperate glaciers. J Glaciol 7:21–58 6. Lliboutry LA (1979) Local friction laws for glaciers: a critical review and new openings. J Glaciol 23:67–95 7. Fowler AC (1981) A theoretical treatment of the sliding of glaciers in the absence of cavitation. Phil Trans R Soc Lond A 298:637–685 8. Meyssonnier J (1983) Écoulement de la glace sur un lit de forme simple: expérience, modélisation, paramétrisation du frottement. Thèse, Université de Grenoble 9. Gudmundsson GH (1997) Basal-flow characteristics of a non-linear flow sliding frictionless over strongly undulating bedrock. J Glaciol 43:80–89 10. Hindmarsh RCA (2000) Sliding over anisotropic beds. Ann Glaciol 30:137–145 11. Iken A (1981) The effect of the subglacial water pressure on the sliding velocity of a glacier in an idealized numerical model. J Glaciol 27:407–421 12. Fowler AC (1986) A sliding law for glaciers of constant viscosity in the presence of subglacial cavitation. Proc R Soc Lond A 407:147–170 13. Fowler AC (1987) Sliding with cavity formation. J Glaciol 33:255–267 14. Schoof C (2005) The effect of cavitation on glacier sliding. Proc R Soc Lond A 461:609–627 15. Budd WF, Keage PL, Blundy NA (1979) Empirical studies of ice sliding. J Glaciol 23:157–170 16. Bindschadler R (1983) The importance of pressurized subglacial water in separation and sliding at the glacier bed. J Glaciol 29:3–19 17. Bentley CR (1987) Antarctic ice streams: a review. J Geophys Res 92(B9):8,843–8,859 18. Weertman J (1972) General theory of water flow at the base of a glacier or ice sheet. Rev Geophys Space Phys 10:287–333 19. Walder JS (1982) Stability of sheet flow of water beneath temperate glaciers and implications for glacier surging. J Glaciol 28:273–293 20. Alley RB (1989) Water-pressure coupling of sliding and bed deformation: I. Water system. J Glaciol 35:108–118 21. Röthlisberger H (1972) Water pressure in intra- and subglacial channels. J Glaciol 11:177–203 22. Nye JF (1976) Water flow in glaciers: jökulhlaups, tunnels, and veins. J Glaciol 17:181–207 23. Clarke GKC (2003) Hydraulics of subglacial outburst floods: new insights from the Spring-Hutter formulation. J Glaciol 49:299–313 24. Roberts MJ (2005) Jökulhlaups: a reassessment of floodwater flow through glaciers. Rev Geophys 43:RG1002 25. Spring U, Hutter K (1981) Numerical studies of jökulhlaups. Cold Reg Sci Tech 4(3):221–244 26. Clarke GKC (1982) Glacier outburst floods from “Hazard Lake”, Yukon Territory, and the problem of flood magnitude prediction. J Glaciol 28:3–21 27. Ng F, Björnsson H (2003) On the Clague-Mathews relationship for jökulhlaups. J Glaciol 49:161–172 28. Clague JJ, Mathews WH (1973) The magnitude of jökulhlaups. J Glaciol 12(66):501–504 29. Fowler AC (1999) Breaking the seal at Grímsvötn, Iceland. J Glaciol 45:506–516 30. Ng F, Liu S (2009) Temporal dynamics of a jökulhlaup system. J Glaciol 55:651–665 31. Smith BE, Fricker HA, Joughin IR, Tulaczyk S (2009) An inventory of active subglacial lakes in Antarctica detected by ICESat (2003–2008). J Glaciol 55:573–595

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32. Wingham DJ, Siegert MJ, Shepherd A, Muir AS (2006) Rapid discharge connects Antarctic subglacial lakes. Nature 440:1,033–1,037 33. Kamb B, Raymond CF, Harrison WD, Engelhardt H, Echelmeyer KA, Humphrey N, Brugman MM, Pfeffer T (1985) Glacier surge mechanism: 1982–1983 surge of Variegated Glacier, Alaska. Science 227:469–479 34. Kamb B (1987) Glacier surge mechanism based on linked cavity conﬁguration of the basal water conduit system. J Geophys Res 92:9,083–9,100 35. Walder JS (1986) Hydraulics of subglacial cavities. J Glaciol 32:439–445 36. Schoof C (2010) Ice-sheet acceleration driven by melt supply variability. Nature 468:803–806 37. MacAyeal D (1993) Binge/purge oscillations of the Laurentide ice sheet as a cause of the North Atlantic’s Heinrich events. Paleoceanography 8:775–784 38. Kamb B (1991) Rheological nonlinearity and flow instability in the deforming bed mechanism of ice stream motion. J Geophys Res 96:16,585–16,595 39. Boulton GS, Hindmarsh RCA (1987) Sediment deformation beneath glaciers: rheology and geological consequences. J Geophys Res 92:9,059–9,082 40. Whillans IM, van der Veen CJ (1993) Patterns of calculated basal drag on Ice Streams B and C, Antarctica. J Glaciol 39:437–446 41. Alley RB (1993) In search of ice-stream sticky spots. J Glaciol 39:447–454 42. Hooke RLeB, Hanson B, Iverson NR, Jansson P, Fischer UH (1997) Rheology of till beneath Storglaciären, Sweden. J Glaciol 43:172–179 43. Iverson NR, Hooyer TS, Baker RW (1998) Ring-shear studies of till deformation: Coulomb-plastic behavior and distributed strain in glacier beds. J Glaciol 44:634–642 44. Porter PR, Murray T (2001) Mechanical and hydraulic properties of till beneath Bakaninbreen, Svalbard. J Glaciol 47(157):167–175 45. Iverson NR, Iverson RM (2001) Distributed shear of subglacial till due to Coulomb slip. J Glaciol 47(158):481–488 46. Tulaczyk SM (1999) Ice sliding over weak, ﬁne-grained tills, dependence of ice-till interactions on till granulometry. In: Mickelson DM, Attig JW (eds) Glacial processes: past and present. Geol Soc Amer Spec Pap 337:159–177 47. Clarke GKC (2005) Subglacial processes. Ann Rev Earth Planet Sci 33:247–276 48. Creyts TT, Schoof CG (2009) Drainage through subglacial water sheets. J Geophys Res 114: F04008 49. Walder JS, Fowler A (1994) Channelised subglacial drainage over a deformable bed. J Glaciol 40:3–15 50. Ng FSL (2000) Canals under sediment-based ice sheets. Ann Glaciol 30:146–152 51. Engelhardt H, Kamb B (1997) Basal hydraulic system of a West Antarctic ice stream: constraints from borehole observations. J Glaciol 43:207–230 52. Sharpe DR, Kjarsgaard BA, Knight RD, Russell HAJ, Kerr DE (2017) Glacial dispersal and flow history, East Arm area of Great Slave Lake, NWT, Canada. Quat Sci Rev 165(1):49–72 53. Ehlers J (1981) Some aspects of glacial erosion and deposition in North Germany. Ann Glaciol 2:143–146 54. Shaw J, Kvill D, Rains B (1989) Drumlins and catastrophic glacial floods. Sed Geol 62:177–202 55. Sugden DE, John BS (1976) Glaciers and landscapes. Edward Arnold, London 56. Hindmarsh RCA (1998) The stability of a viscous till sheet coupled with ice flow, considered at wavelengths less than the ice thickness. J Glaciol 44:285–292 57. Fowler AC (2000) An instability mechanism for drumlin formation. In: Maltman A, Hambrey MJ, Hubbard B (eds) Deformation of glacial materials. Spec Pub Geol Soc 176:307–319

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58. Schoof C (2007) Pressure-dependent viscosity and interfacial instability in coupled ice-sediment flow. J Fluid Mech 570:227–252 59. Fowler AC (2009) Instability modelling of drumlin formation incorporating lee-side cavity growth. Proc R Soc Lond A 465:2,681–2,702 60. McCracken RG, Iverson NR, Benediktsson ÍÖ, Schomacker A, Zoet LK, Johnson MD, Hooyer TS, Ingólfsson Ó (2016) Origin of the active drumlin ﬁeld at Múlajökull, Iceland: new insights from till shear and consolidation patterns. Quat Sci Rev 148:243–260

4

Tidewater Glaciers Ian Howat and Andreas Vieli

4.1

Introduction

Tidewater glaciers terminate in a saline body of water connected to the open ocean, which may be a fjord or a lagoon. The term tidewater usually applies to marine-terminating valley glaciers that flow through a fjord, while ice stream is typically applied to fast-flowing channels within a slower-moving body of ice. The terminology can be ambiguous, however, as marine-terminating glaciers can be bounded by ice on one margin and not the other, or can retreat out of a fjord into an ice embayment to become an ice stream, such as has happened with Jakobshavn Isbræ in Greenland. Truffer and Echelmeyer [1] put forward a classiﬁcation scheme that accounts for both kinds of morphology and driving stress. Increasingly, authors today use “marine-terminating” rather than “tidewater” to avoid the ambiguity. Marine-terminating glaciers draining an ice sheet or ice cap are often referred to as “outlet glaciers”, with the distinction between land- and marine-terminating outlets being sometimes ambiguous in the literature. Every tidewater glacier ends at a calving front―a nearly vertical wall from which icebergs break off or are “calved”. The calving front may be grounded or floating (e.g., if it lies at the end of an ice shelf). Narrow ice shelves, especially those within a fjord, are typically referred to as floating ice tongues. Besides surface ablation, tidewater glaciers lose mass through basal melting, both beneath the ice in the floating section and at the calving face below the water line, and through iceberg

I. Howat (&) School of Earth Sciences, Byrd Polar Research Center, Ohio State University, Columbus, USA e-mail: [email protected] A. Vieli Department of Geography, University of Zurich, Zurich, Switzerland © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_4

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calving. In many, if not most cases, the mass loss to basal melt and calving greatly exceeds the loss to surface ablation, so these terms dominate the mass balance. As we will see, glaciers that terminate in water, rather than air, are much more dynamically sensitive to changes in their thickness. They typically have high basal water pressures, and thus low effective pressures, near their termini. As a result, much of their motion there is due to sliding; their speeds can reach tens of metres per day. A tidewater glacier that thins to flotation can retreat rapidly at rates of tens to hundreds of metres per day, accompanied by large changes in speed and ice discharge. This sensitivity is demonstrated by the recent and widespread retreat and acceleration of large marine-terminating glaciers in Antarctica, Greenland and Alaska. Increase in the rate of ice lost to the ocean (dynamic mass loss) through these glaciers accounts for approximately half of Greenland’s mass deﬁcit over the past decade [2], and all of Antarctica’s. Unlike surface ablation, tidewater processes are not well understood and the associated dynamic changes cannot yet be reliably predicted, making the future mass balance of ice sheets uncertain [3]. Consequently, the dynamics of tidewater glaciers have received attention from the Intergovernmental Panel on Climate Change (IPCC) and have been a focus of recent research efforts.

4.2

Calving

Iceberg calving is one of the two processes by which tidewater glaciers shed mass at their fronts. The other process, submarine melting, is treated in Chap. 5. Whereas submarine melting occurs constantly, calving is episodic: it can occur in distinct phases or single large events separated by years or even decades. What determines the style of calving for a given glacier is unclear, but considerations of the geographical distribution of terminus types and calving styles suggest ice rheology, bed topography and climate as relevant factors―there are no large, permanent ice shelves in Alaska or southern Greenland, whereas colder Antarctica and Northern Greenland are dominated by ice shelves that calve large tabular icebergs. A thermal threshold of −5 °C has been suggested for permanent ice-shelf formation [4]. Calving is a complex process occurring over a wide range of spatial and temporal scales. At the smallest scales, fractures in the ice grow and connect as a result of tensile stresses acting on the ice front or tidal flexure of an ice shelf. Once they penetrate through the ice to some critical depth, a full-thickness fracture occurs. This process can be nearly instantaneous (as in the case of rapid calving of temperate tidewater glaciers) or can take place over years (as in the rifting of large ice shelves). It is likely that fractures exploit any pre-existing weaknesses in the ice created by vertical and lateral shearing as the ice accelerates toward the calving front. Crevassing at the base of the glacier may also trigger calving, because near

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the ice front the effective pressures tend to be very low, and rates of basal sliding very high. There is evidence that large calving events may occur when surface and basal crevasses intersect [4]. Also, the ﬁlling of crevasses with melt water may promote calving through hydrofracture; as water is denser than ice, the normal stress exerted at the bottom tip of a (fully) water-ﬁlled crevasse exceeds the pressure of the surrounding ice (which is in tension), causing a net force that drives further propagation of the crevasse as long as it remains ﬁlled with water [5]. The complexity of calving, combined with a scarcity of direct observations, makes it a difﬁcult process to model and predict. Heuristic functions are commonly used to describe the average rate of calving over some representative time (i.e., a model time-step). In steady state, and if submarine melting is ignored, the calving rate c will equal the ice flow speed at the front, uf. Imbalance between these rates causes the ice front to advance or retreat, changing the glacier length L according to dL ¼ uf c: dt

ð4:1Þ

So far, no robust and generally applicable model of calving is available, and two disparate approaches have been taken to quantify it. First, empirical relationships (often referred to as calving laws) have been put forward for c. They usually depend on variables such as water depth, ice thickness and/or longitudinal strain rate at the calving front (e.g., [6–8]). The earliest such model [7], based on data from 12 Alaskan glaciers, suggests a linear increase in c with water depth at the terminus. Such models intrinsically parameterise the calving rate for a given bed geometry without considering the dynamics of the glacier. The second approach uses a criterion to specify the terminus location. Criteria include the position where the thickness or surface height of the ice equals some fraction (typically about 10%) above the value needed for it to remain grounded [9, 10], or where crevasse depth due to longitudinal stretching reaches the depth for full-thickness fracturing [11, 12]. The crevasse-based criterion attempts to account for the physical processes driving calving and, importantly, allows for the development of floating ice tongues in models, which the thickness-above-flotation criterion does not. The criterion-based approach can reproduce front position changes at seasonal or longer time scales, but requires speciﬁc tuning for each glacier of interest, and it cannot predict periodic detachment of tabular icebergs. Meanwhile, formulating a universal and robust calving relation remains hampered by a limited understanding of the physical processes, such as ice rheology, basal melt rates, the effect of hydrofracturing and the potential resistance to calving provided by sea ice and icebergs (i.e., mélange) at the terminus [3]. Given its complexity, some researchers argue that calving cannot be modelled deterministically, and they advocate stochastic approaches (e.g., [13]).

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Tidewater Glacier Dynamics

4.3.1 Tidewater Glacier Retreat and Instability The rapid retreat of Columbia Glacier in southeast Alaska beginning in the early 1980s, a period of general glacier advance, prompted focussed studies into the unique processes governing tidewater glacier dynamics. Out of these studies came Meier and Post’s [14] concept of a tidewater cycle, which is a key model for evaluating tidewater glacier behaviour. According to this model, tidewater glacier termini alternate between periods of slow advance over centuries to millennia and rapid retreat over decades. A crucial idea is that tidewater glaciers should be inherently unstable, with their fluctuations controlled by fjord bathymetry and internal dynamics, and not directly linked to climatic forcing. We examine different aspects of the cycle in the following sections, beginning here with the instability. The inherent instability of tidewater glaciers results from (i) the dependence of calving rate and ice discharge on water depth and (ii) the tendency for tidewater glaciers to terminate on bathymetric highs, so that the bed deepens upstream of the terminus. Retreat therefore brings the ice front into deeper water, increasing the calving rate and causing further retreat. The glacier stabilizes only when the front retreats to where the bed again rises inland. These interactions and the stability of the grounded terminus position can be analyzed. At the terminus, the ice flux q scales with the water depth there, Hw, as follows [15]: q / ðq0 Hw Þðn þ m þ 3Þ=ðm þ 1Þ :

ð4:2Þ

Here, q0 is the ratio of seawater density to ice density, n ( 3) is the exponent in Glen’s flow law, and m (often assumed to range from 1/3 to 1) is the exponent in the basal sliding law s / um (see Chap. 3). If we assume a calving rate based only on the flotation-based criterion, then, according to (4.2), the ice flux required to stabilize the grounding line increases with water depth Hw raised to a power between 3 and 5. Where the bed slope is reversed, this dependence results in a stronger instability than that obtained from the class of empirical calving models mentioned above, in which the calving rate c scales roughly linearly with Hw. Figure 4.1 illustrates the different dependences of the ice flux and calving flux on water depth, using a model glacier with the bed elevation proﬁle of Columbia Glacier in Alaska. In the bottom plot, the blue lines show the ice flux at the grounding line required for stability predicted by (4.2); the red lines plot the modelled calving flux. The intersection between each flux curve and the balance flux determines a possible steady state position. Any position on the stretch of reverse bed slope is unstable, because a retreat perturbation imposed on it causes irreversible retreat. While the thick black line describes the balance flux resulting from a normal ELA (at 900 m a.s.l.), the thin black line describes an unrealistically high balance flux, caused by a severe lowering of the ELA to sea level. In all except the “weak calving” scenario (modelled by the thin dash-dotted red curve),

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Fig. 4.1 Conditions of instability of a tidewater glacier with basal topography similar to that of Columbia Glacier, Alaska. Top: along-flow proﬁles of the approximate pre-retreat glacier surface (blue line) and basal geometry (black line). The sea level is at zero elevation. Bottom: modelled ice flux (blue, dashed) and calving flux (red, dash-dotted) at the terminus when it is located at different positions. The black lines plot the balance flux (integrated surface mass balance) for two surface mass balance scenarios. The thick and thin blue lines plot the ice flux needed for stability calculated from (4.2) with m = 1/3 and m = 1, respectively. The thick and thin red lines plot the calving flux from a linear calving law with high and low proportionality constants, respectively. In both panels, grey shading highlights the stretch of reverse bed slope

the balance fluxes lie well below the peaks of the grounding-line/calving flux curves across the stretch of the bed depression, and so after the glacier retreats, it cannot re-advance through the depression to reach any of the stable positions beyond the bathymetric high (at 68 km). A further instability stems from the nonlinear dependence of basal sliding on basal effective pressure (= ice overburden pressure − water pressure). If the ice thickness is well above buoyancy, then the basal effective pressure is high, and a reduction in ice thickness will cause only a minor increase in sliding speed, so that the ice flux across the grounding line (product of speed and thickness) is reduced; although retreat occurs, there is no feedback to amplify thinning, and the retreat is stable. However, if the ice thickness reduces to some threshold near flotation, the effective pressure will be low (near zero) so the ice flux increases rapidly with

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Fig. 4.2 Ice discharge across the grounding line as a function of ice thickness relative to the flotation ice thickness, based on the model calculations by Pfeffer [16] for n = 3 and m = 1. In this model, ice discharge is non-dimensionalised (with the minimum discharge value as the scale), and it blows up when flotation occurs

thinning. Such increase in ice flux will cause stretching and further thinning (i.e., dynamic thinning), fuelling a positive feedback that drives rapid retreat. Retreat will continue to the position where the thickness at the grounding line regains the threshold for stability. This mechanism was examined analytically by Pfeffer [16], who found that, for various sliding parameters, thicknesses less than 10–20% above the flotation thickness should be unstable under a thinning perturbation (Fig. 4.2, below the turning point). This condition resembles conditions used in the criterion-based description of calving (see last section), suggesting basal sliding as a physical factor behind calving at grounded ice fronts. If the terminus is initially grounded on a basal topographic high, the water depth at the grounding line will increase as retreat occurs. This will further reduce basal effective pressure, enhancing the positive feedback between ice thinning and discharge. In general, the feedbacks described above lead to discharge increasing with water depth, preventing the grounding line from stabilizing on a reversed bed slope (a bed surface that rises downstream). Once retreat is initiated, the glacier retreats until the bed rises in the up-glacier direction, at which point the feedback becomes negative (i.e., retreat reduces the water thickness at the grounding line, raises effective pressure and reduces discharge or calving rate) and the glacier stabilizes. This position, however, may be many kilometres, or even tens of kilometres, up the fjord. We note that these models assume parallel flow (i.e., on flow lines across which the ice motion is uniform), so that stresses only act in the along-flow direction. This is a good approximation for tidewater glaciers that are laterally conﬁned by near-parallel fjord walls and have narrow lateral shear margins. However, in areas

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of converging or diverging flow, such as in ice-bounded ice streams, the effects of transverse stresses must be considered. Gudmundsson et al. [17] showed that transverse stresses exerted by ice shelves can stabilize grounding lines on reverse slopes.

4.3.2 Tidewater Glacier Advance Since discharge across the grounding line increases as the water deepens relative to the ice thickness, bathymetric lows tend to act as barriers to tidewater glacier advance. Increases in flux due to increased surface accumulation are not typically sufﬁcient to force advance (Fig. 4.1). Instead, advance occurs either through the deposition of a till platform, or through the formation and advance of a floating ice tongue or shelf that grounds distally (Fig. 4.3). In the former scenario, as suggested in Meier and Post’s [14] model, till sediment deposited at the grounding line builds up over time, resulting in a platform that reduces the water depth and thus the required flux for stability (Sect. 4.3.1) and allows the glacier to advance upon it. The rate of advance is limited by the rates of sediment deposition and expansion of the platform (on the order of metres to tens of metres per year) and is slow compared to retreat. For example, retreat of the Columbia Glacier over the past few decades may have negated 1000 years of advance. In some cases, the platform may rise above sea level, as demonstrated by the advancing Taku and Hubbard Glaciers in Alaska.

Fig. 4.3 Advance of a tidewater glacier by (left) deposition of a sedimentary platform at its grounding line and (right) bridging of its ice tongue across a bathymetric depression

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The growth of a floating ice shelf or tongue provides a potentially much faster advance mechanism (Fig. 4.3). If the tongue extends seaward to a basal high (i.e., a shoal) and grounds on it, the basal resistance provided by the shoal may slow the ice flux through the front, causing the tongue to thicken. The thicker the ice gets, the more of it contacts the bed; this implies positive feedback between slow-down and thickening. Eventually the ice may ground all the way to the shoal. This bridging mechanism enabled Helheim Glacier in Greenland to re-advance several kilometres in a single year following its equally rapid retreat in 2005 [18].

4.3.3 Flow Variability of Tidewater Glaciers The flow of tidewater glaciers is highly sensitive to changes at their fronts. This has been shown by numerous observations of sudden speed-ups, by a factor of two or more, that coincide with retreat of their ice tongues and grounding lines. The acceleration is largest near the front and declines inland, indicating a rise in along-flow (i.e., longitudinal) strain rates towards the terminus. This stretching can thin the ice at rates of tens of metres per year; it also points to an increase in along-flow stresses as the cause of the acceleration. According to the balance of forces, an increase in longitudinal stresses “pulling” ice toward the front must be due to reduced resistance to the flow elsewhere. Decreased basal friction associated with increasing meltwater penetration to the bed was initially a suspected mechanism, but observations showed that such sustained acceleration does not correlate with surface melting, and similar acceleration of Antarctic glaciers has occurred where surface melt water does not reach the bed. Moreover, the suspected mechanism would not cause a strong gradient in acceleration towards the ice front [19]. Van der Veen et al. [20] suggested that weakening of the shear margins of Jakobshavn Isbræ reduced its flow resistance and caused it to accelerate. This inference, however, depends sensitively on assumed (and uncertain) flow model parameters; also, such weakening cannot explain the same behaviour observed for some rock-walled tidewater glaciers. Observations and ice-flow modelling thus far (e.g., [3, 21]) show that the flow-resistance reduction behind the acceleration is most likely due to the loss of conﬁned ice tongues and grounded ice, as shown below. How the flow speed on a tidewater glacier changes as its front retreats can be assessed by a simple consideration of the forces acting on its trunk. Glacier flow toward a calving front is driven by the gravitational stress, sd , and by the gradient in hydrostatic stress across the ice front (i.e., the margin stress). The net force per unit width due to the latter is Ff ¼

g qi Hf2 qw ðHf hf Þ2 : 2

ð4:3Þ

Here g is gravity, qi and qw are the ice and seawater densities, Hf is the ice thickness and hf is the height of the calving face above the water line. If x denotes distance

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along flow, we can deﬁne an effective driving stress, se, acting on a glacier section extending from the front x = xf back to a position x = xe further inland, as follows [18]: 1 se ¼ L

Z

xe

! sd dx þ Ff ;

ð4:4Þ

xf

where L is the length over which stresses are transmitted up-glacier of the front (i.e., the stress-coupling length). Values for L have been determined from theory and from observations of tidal flexure and response to margin perturbations. Generally, L should be larger for thicker and colder ice, and range from one ice thickness for small temperate glaciers to 5–10 km for Greenlandic glaciers, to hundreds of kilometres for Antarctic ice streams. Based on (4.4), retreat of the tongue or grounding line concentrates the driving force over a shorter length of glacier, causing a proportional increase in se : Dse DL : ¼ L se

ð4:5Þ

Separately, Glen’s flow law shows that the flow speed U varies with the driving stress as DU þ1 ¼ U

Dse þ1 se

n :

ð4:6Þ

Substituting (4.6) into (4.5) gives a relationship between the changes in front position and flow speed over the glacier terminus: DU ¼ U

DL n 1 1: L

ð4:7Þ

Howat et al. [22] applied (4.7) to observational data from southeast Greenland and found that, for the larger changes in frontal position, changes in speed were proportional to changes in length with n = 3. Smaller changes exhibited more noise that might be due to local and temporal variations in thickness and sliding. Nick et al. [19] conﬁrmed the relationship between acceleration and stress perturbations at the ice front with a numerical model capable of reproducing observations. Following the large initial increase in speed during a retreat, speed will decrease as the front stabilizes. This is because the stretching thins the ice, reducing both the driving stress and margin stress (the bracketed sum in (4.4)). Thinning over the lower trunk, however, increases the surface slope at its upstream end, causing an increase in driving stress and flow speed there, which again subsides as the ice stretches and thins. Thus, acceleration and thinning propagate inland as a kinematic

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wave with advective-diffusive character [23]. The wave velocity is proportional to ice speed, so the acceleration and thinning propagate inland rapidly on fast-flowing glaciers and ice streams [3]. In Greenland, thinning originating in the outlet glaciers has been known to propagate up-flow by over 100 km in a few years.

4.4

The Link to Climate: Triggers for Retreat

Besides the strong asymmetry between timescales of advance and retreat, another major implication of the tidewater glacier cycle (discussed above) is that it is independent of variations in climate. Modes of advance and retreat can be driven solely by internal processes without changes to external forcing. At the time of Meier and Post’s [14] paper, this result was welcomed because it could explain the contrasting behaviour and regional variability of tidewater glaciers. However, the observed widespread and synchronous retreat of glaciers in Greenland and Antarctica over the past decade, the occurrence of seasonal advance and retreat patterns, and known cases of rapid re-advance, provide evidence that climate plays a key role in regulating tidewater glacier behaviour. An obvious way in which climatic factors can influence the tidewater glacier cycle is through the ocean. According to (4.1), glacier length responds to variations in ice speed and calving (and submarine melting). A reduction in ice discharge towards the front, such as would be caused by lowered accumulation, could decrease the ice flux and speed at the front to trigger retreat. But since the response of a glacier is relatively slow (decades or longer), the effect of rapid short term fluctuations in accumulation is felt through their long-time average. In contrast, calving and melt rates at the front can fluctuate rapidly, because small changes in ocean conditions can drastically change the heat available for melting ice. (Note that on most fast-flowing tidewater glaciers, surface ablation rates tend to be smaller than calving and submarine melt rates.) In this way, we expect oceanic climate to play a leading role in modulating frontal positions. Direct evidence that variations in ocean heat transport impact basal melting and calving are sparse and circumstantial, particularly for calving. Several analyses of remote-sensing data have shown a temporal correlation between marine-terminating glacier retreat and increased coastal ocean temperatures (e.g., [22–24]), and models can reproduce the effect of enhanced grounding-line melting on ice flow [21].

4.4.1 Ice Shelf Collapse and Backstress A striking connection between ocean warming and the retreat and acceleration of tidewater glaciers is demonstrated by ice shelf/tongue collapse. For glaciers buttressed by such a shelf or tongue, increased submarine melting and associated thinning can trigger retreat and speed-up. Motyka et al. [25] estimated that melt rates beneath Jakobshavn Isbræ’s tongue increased by 25% before it collapsed in

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the late 1990s. Similarly, increased basal and surface melting of the Larsen B Ice Shelf has been blamed for its collapse in 2002 and the resulting acceleration of outlet glaciers feeding it [26]. Increasing surface melt may promote hydrofracture within crevasses and melt ponds, weakening the shelf to trigger a collapse. Sohn et al. [27] suggested that this process enhanced calving during the melt season on tidewater outlet glaciers in Greenland. How such collapse causes flow acceleration can be studied via the idea of backstress. Ice tongues and shelves shear past their lateral margins, generating resistance to flow that is transferred upstream to the grounded ice. The backstress is given by Z Fb ¼ 2s0

xf

xg

H dx; W

ð4:8Þ

where s0 is the drag resistance at the margins, W is the ice shelf width, and the integral goes from the grounding line to the ice front. Thicker and narrower ice shelves produce more backstress, with a Jakobshavn Isbræ sized tongue (*800 m thick, 7 km wide) generating between 15 and 25 kPa [28]. Backstress causes the grounded ice upstream to thicken and steepen, and consequently to have a higher driving stress than if the ice shelf/tongue were absent. Thinning of the shelf/tongue reduces backstress, resulting in acceleration of the glacier to compensate. Using simpliﬁed model assumptions, Thomas [28] showed that the initial 50% acceleration of Jakobshavn Isbræ could be attributed to loss of its floating tongue. Following this initial response, the tidewater instability feedbacks discussed in Sect. 4.3.1 took hold to cause continued acceleration and retreat [21].

4.4.2 Grounded Calving Fronts The sudden retreat of tidewater glaciers with grounded fronts is more difﬁcult to explain. Long-term thinning, due to increased surface melting, could cause the terminus to float, instigating its unstable retreat from basal highs [10]. Alternatively, increased calving rate could cause the front to retreat past a stable position, through a lateral constriction or into a basal overdeepening, to trigger unstable retreat. Accelerated calving may result from (i) increasing ocean temperatures and melting (which can undercut the front), (ii) hydrofracturing caused by enhanced surface melt, or (iii) reduction in the strength and seasonal duration of icebergs bonded by sea ice, i.e., mélange. There is some evidence for the last of these controls, whose importance has yet to be investigated fully. Amundson et al. [29] found that mélange may behave as a thin ice shelf, providing enough resistance to prevent icebergs from detaching from the front and, potentially, suppress calving. This would explain a tendency for outlet glaciers in Greenland to advance in the presence of mélange in the winter and retreat in the spring as the mélange breaks up. Where mélange formation is active, earlier spring break-up of the mélange would promote

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ice-free conditions and increase calving rates, potentially allowing the front to move past stable positions into an unstable retreat phase. A rise in subglacial water discharge into the fjord, sourced from surface melt, may also promote submarine melting at the terminus by increasing the entrainment of warm ocean waters to the ice-ocean interface (see Chap. 5). We note that the dominant processes linking climatic forcing to ice dynamics are not necessarily the same for all tidewater glaciers. They may depend on the climatic regime, glacier geometry, terminus type and other factors.

4.5

Outlook

We have seen that tidewater glaciers have more volatile dynamics than land-terminating glaciers, due to nonlinear controls on ice flow associated with their marine termini, and the contrasting densities of ice and water (which can cause flotation). Buoyancy provides a tipping point at which relatively minor thinning of tidewater glaciers could cause them to undergo a transition from long-term stability to rapid irreversible retreat. The sensitivity and nonlinearity present a major challenge to efforts to predict how these glaciers respond to external forcing. Uncertainties in boundary conditions and basic parameters like ice thickness can lead to highly divergent predictions. This problem is compounded by a fundamental lack of understanding of the calving process and how it depends on factors such as submarine melting, hydrofracture and backstress exerted by mélange. Also unknown are the impacts of basal processes on tidewater glacier dynamics, including meltwater drainage and till deformation. Many open questions regarding the behaviour of tidewater glaciers remain, and no general prognostic model of their dynamics is in sight.

References 1. Truffer M, Echelmeyer KA (2003) Of isbræ and ice streams. Ann Glaciol 36:66–72 2. van den Broeke M, Bamber J, Ettema J, Rignot E, Schrama E, van de Berg WJ, van Meijgaard E, Velicogna I, Wouters B (2009) Partitioning recent Greenland mass loss. Science 326(5955):984–986 3. Vieli A, Nick FM (2011) Understanding and modelling rapid dynamic changes of tidewater outlet glaciers: issues and implications. Surv Geophys 32(4–5):437–458 4. van der Veen CJ (2002) Calving glaciers. Prog Phys Geogr 26(1):96–122 5. Benn DI, Warren CR, Mottram RH (2007) Calving processes and the dynamics of calving glaciers. Earth Sci Rev 82(3–4):143–179 6. Alley RB, Horgan HJ, Joughin I, Cuffey KM, Dupont TK, Parizek BR, Anandakrishnan S, Bassis J (2008) A simple law for ice-shelf calving. Science 322(5906):1344 7. Brown CS, Meier MF, Post AS (1982) Calving speed of Alaska tidewater glaciers with applications to the Columbia Glacier, Alaska. US Geol Surv Prof Pap 1258-C:13 pp 8. Levermann A, Albrecht T, Winkelmann R, Martin MA, Haseloff M, Joughin I (2012) Kinematic ﬁrst-order calving law implies potential for abrupt ice-shelf retreat. The Cryosphere 6:273–286 9. van der Veen CJ (1996) Tidewater calving. J Glaciol 42(141):375–385

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10. Vieli A, Funk M, Blatter H (2001) Flow dynamics of tidewater glaciers: a numerical modelling approach. J Glaciol 47(159):595–606 11. Benn DI, Hulton NRJ, Mottram RH (2007) ‘Calving laws’, ‘sliding laws’ and the stability of tidewater glaciers. Ann Glaciol 46:123–130 12. Nick FM, van der Veen CJ, Vieli A, Benn DI (2010) A physically based calving model applied to marine outlet glaciers and implications for the glacier dynamics. J Glaciol 56 (199):781–794 13. Bassis JN (2011) The statistical physics of iceberg calving and the emergence of universal calving laws. J Glaciol 57(201):3–16 14. Meier MF, Post A (1987) Fast tidewater glaciers. J Geophys Res 92(B9):9051–9058 15. Schoof C (2007) Ice sheet grounding line dynamics: steady states, stability, and hysteresis. J Geophys Res Earth Surf 112:F03S28 16. Pfeffer T (2007) A simple mechanism for irreversible tidewater glacier retreat. J Geophys Res Earth Surf 112:F03S25 17. Gudmundsson GH, Krug J, Durand G, Favier L, Gagliardini O (2012) The stability of grounding lines on retrograde slopes. The Cryosphere 6:1497–1505 18. Howat IM, Joughin I, Tulaczyk S, Gogineni S (2005) Rapid retreat and acceleration of Helheim Glacier, East Greenland. Geophys Res Lett 32:L22502 19. Nick FM, Vieli A, Howat IM, Joughin I (2009) Large-scale changes in Greenland outlet glacier dynamics triggered at the terminus. Nat Geosci 2:110–114 20. van der Veen CJ, Plummer JC, Stearns LA (2011) Controls on the recent speed-up of Jakobshavn Isbræ, West Greenland. J Glaciol 57(204):770–782 21. Joughin I, Smith BE, Howat IM, Floricioiu D, Alley RB, Truffer M, Fahnestock M (2012) Seasonal to decadal scale variations in the surface velocity of Jakobshavn Isbræ, Greenland: observation and model-based analysis. J Geophys Res 113:F04006 22. Howat IM, Joughin I, Fahnestock M, Smith B, Scambos T (2008) Synchronous retreat and acceleration of southeast Greenland outlet glaciers 2000–2006: ice dynamics and coupling to climate. J Glaciol 54(187):646–660 23. Payne AJ, Vieli A, Shepherd AP, Wingham DJ, Rignot E (2004) Recent dramatic thinning of largest West Antarctic ice stream triggered by oceans. Geophys Res Lett 31:L23401 24. Christoffersen P, Mugford RI, Heywood KJ, Joughin I, Dowdeswell J, Syvitski M, Luckman A, Benham TJ (2011) Warming of waters in an East Greenland fjord prior to glacier retreat: mechanisms and connection to large-scale atmospheric conditions. The Cryosphere 5:701–714 25. Motyka RJ, Truffer M, Fahnestock M, Mortensen J, Rysgaard S, Howat I (2011) Submarine melting of the 1985 Jakobshavn Isbræ floating tongue and the triggering of the current retreat. J Geophys Res Earth Surf 116:F01007 26. Scambos TA, Bohlander JA, Shuman CA, Skvarca P (2004) Glacier acceleration and thinning after ice shelf collapse in the Larsen B embayment, Antarctica. Geophys Res Lett 31:L18402 27. Sohn HG, Jezek EC, van der Veen CJ (1998) Jakobshavn Glacier, West Greenland: 30 years of spaceborne observations. Geophys Res Lett 25(14):2699–2702 28. Thomas RH (2004) Force-perturbation analysis of recent thinning and acceleration of Jakobshavn Isbræ, Greenland. J Glaciol 50(168):57–66 29. Amundson JM, Fahnestock M, Truffer M, Brown J, Lüthi MP, Motyka RJ (2010) Ice mélange dynamics and implications for terminus stability, Jakobshavn Isbræ, Greenland. J Geophys Res Earth Surf 115:F01005

5

Interaction of Ice Shelves with the Ocean Adrian Jenkins

5.1

Introduction

Ice shelves are the parts of an ice sheet that float in the ocean: they form where the ice is sufﬁciently thin to float free from the bed. Ice shelves are notably abundant around marine ice sheets, which rest on a bed lying substantially below sea level. The West Antarctic Ice Sheet—the only true marine ice sheet today—is almost completely surrounded by ice shelves (Fig. 5.1). Less extensive ice-shelf cover is also found along the marine margins of the East Antarctic and Greenland Ice Sheets where deep channels deliver ice from the interior into the ocean. The ice shelves fringing the Antarctic Ice Sheet comprise just about 2.5% of its volume, yet they receive over 80% of the outflow from the grounded ice (and almost all of the outflow from the West Antarctic Ice Sheet). Melting from these shelves amounts to a mass loss of about 1500 Gt yr−1, or 50% of the accumulation over the ice sheet, with the remainder being lost mostly by iceberg calving. In contrast, on the Greenland Ice Sheet, about 50% of the mass input is lost through surface runoff; the rest is returned to the ocean via marine-terminating tidewater glaciers (see Chap. 4) where calving dominates mass loss. The thickness of ice shelves evolves in response to surface accumulation or ablation, lateral spreading, and basal melting or freezing. Typical orders of magnitude of these processes for the Antarctic ice shelves are up to *0.1 m yr−1 for surface accumulation or (more rarely) ablation, up to *1 m yr−1 for dynamical thinning, up to *1 m yr−1 for basal freezing, and as much as *10–100 m yr−1 for basal melting. While the flow of ice shelves is relatively insensitive to climate, responding only as variations in surface temperature penetrate into their interior, the mass balance at their upper and lower surfaces is

A. Jenkins (&) Northumbria University, Newcastle upon Tyne, UK e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_5

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Fig. 5.1 a Map of Antarctica showing grounded ice sheet (light grey) and floating ice shelves (dark grey). b Map of Greenland showing ice sheet (light grey) and ice-free land (dark grey). Bathymetric contours (−1500 m, blue; −1000 m, cyan; −500 m, yellow; 0 m, brown) are the same in both panels. In Greenland, only Petermann Glacier and Nioghalvfjerdsbræ retain floating tongues

intimately linked to atmospheric and oceanic conditions. Perturbations in basal melting are generally recognised to be a potent form of climatic forcing on ice-shelf evolution; however, surface melting is important in selected shelf areas also. Ice flux across the grounding line contributes to sea level directly. In this context, the changing geometry of ice shelves is relevant because it influences the flux through the mechanical restraint or ‘backstress’ imposed on the interior/grounded flow. Recent observations show that thinning of the Antarctic Ice Sheet is contributing about 0.2 mm yr−1 to global mean sea level rise [1], and this thinning is focussed on numerous fast-flowing outlet glaciers that drain into ice shelves that are themselves thinning. The most dramatic changes have occurred in the Amundsen Sea sector of West Antarctica (Fig. 5.2) where thinning rates of several metres per year have been measured on ice shelves [2]. Such rapid rates can not be accounted for by changes in surface mass balance or flow dynamics, and so must arise from enhanced basal melting. The coherent signature of thinning seen on neighbouring glaciers further points to oceanic forcing as the likely trigger. As the ice shelves thinned, the outlet glaciers accelerated. A particularly strong signal has been observed on Pine Island Glacier, where thinning rates have increased over two decades, while the glacier flow accelerated by 70%, and the grounding line retreated inland [3].

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Fig. 5.2 Rate of surface elevation change in a Antarctica and b Greenland determined primarily from satellite radar altimeter data collected between 1992 and 2002. From Zwally et al. [4]. Reprinted from the Journal of Glaciology with permission of the International Glaciological Society

Another region of rapid ice loss is the northern Antarctic Peninsula. Here, the complete disintegration of several large ice shelves in recent years appears to be related to surface warming and an accompanying increase of surface melting during summer [5]. Meltwater ponding in surface depressions (rather than running off) can feed crevasses and cause them to penetrate the full shelf thickness, as the water pressure in a ﬁlled crevasse exceeds the surrounding ice pressure. Thus extensive surface melting can predispose an ice shelf for break-up. This mechanism implies a climatic threshold beyond which ice-shelf cover becomes unviable. The Antarctic Peninsula is one of the most rapidly warming regions on Earth, with recorded temperature trends of up to 0.5 °C per decade, so the climatic threshold has been migrating south along it to drive ice-shelf disintegration. Although these break-up events along the Peninsula seem unrelated to ocean forcing, the observed rapid acceleration of outlet glaciers following the loss of their shelves shows that changes seaward of the grounding line can indeed alter the inland flow of the ice sheet. Turning to the Greenland Ice Sheet, its recent mass loss has contributed nearly 0.5 mm yr−1 towards global mean sea level rise. Thinning there has been concentrated on fast-flowing tidewater glaciers, and ocean forcing is widely assumed to be responsible for this, possibly through the impact of submarine melting on

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calving rates. In the case of Jakobshavn Isbræ, whose flow speed has doubled since 1997, a clear link with the ocean has been demonstrated. Its acceleration was triggered by the thinning and disintegration of an ice shelf that formerly extended 9 km down the fjord. The break-up of the ice shelf was coincident with a warming of the ocean beyond the fjord. These observations show that melting under ice shelves and floating ice tongues can change their geometry to influence grounded ice flow and thus sea level. With this in mind, this chapter focusses on the processes that control the melting of ice in contact with the ocean and their modelling. We go up-scale from the interactions at an ice-seawater interface to the large-scale ocean circulation in the cavities under ice shelves. The treatment of calving and surface melting falls beyond our scope.

5.2

Impact of Melting Ice on the Ocean

At the interface between an ice shelf and the ocean, the temperature is at the melting (or freezing) point. This temperature is not constant, as it depends on the salinity (which depresses it; this is why we put salt on freezing roads), and also pressure (thus, depth in the water column matters). For simplicity, we think here of the saltiness of the ocean as being due to dissolution of sodium chloride, although many other ions are present. Figure 5.3a shows the phases that exist at different temperature and salinity combinations. The curve tracing the lower bound of the region of liquid solution is called the liquidus; the curve tracing the upper bound of the region of solid solution is called the solidus. The point where these curves meet is the eutectic point, while the shaded areas between them represent conditions where a solution cannot exist in equilibrium. The solution then is super-saturated in either salt or water, and, to attain equilibrium, must separate into a component of the appropriate pure solid and a solution whose properties lie on the liquidus. Suppose we add a block of ice to a warm, dilute salty solution. The temperature and salinity will evolve. It is possible to quantify the corresponding trajectory in the phase diagram by considering the heat and salt budgets of the mixture. Consider adding a mass of ice, DM, having initial properties Ti and Si, to a mass of seawater, M, having initial properties Tw and Sw. Once all the ice has dissolved and the seawater has been thoroughly mixed, the ﬁnal properties are Tmix and Smix. Conservation of heat and salt imply Mcw ðTmix Tw Þ þ DM cw Tmix Tf þ L þ ci Tf Ti ¼ 0; M ðSmix Sw Þ þ DM ðSmix Si Þ ¼ 0;

ð5:1aÞ ð5:1bÞ

where Tf is the liquidus temperature and ci and cw are the speciﬁc heat capacities of ice and water. The ﬁrst and second terms in each equation represent the heat/salt given up by the seawater and gained by the ice, respectively. Rearranging (5.1a)

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Fig. 5.3 a Schematic phase diagram for solutions of salt in water, showing pathways for cooling of a concentrated solution (red), for cooling of a dilute solution (blue), and for a freezing mixture in which cold ice and warm, salty water approach the equilibrium temperature from different directions (as quantiﬁed in (5.1a) and (5.1b)) (green). b Enlargement of the box on the left-hand side of (a), showing details of the behaviour at atmospheric pressure for temperatures and salinities typical for seawater. Green lines indicate the oblique trajectories followed by seawater as ice melts into it and it approaches equilibrium on the liquidus. Cyan lines are density contours (isopycnals, in kg m−3), while the blue line joins the points where the isopycnals are tangential to the direction of the temperature axis, and the red line joins points where they are tangential to the green meltwater mixing lines. The box in the lower right quadrant indicates ocean properties typical of the polar regions

and (5.1b) yields expressions for the change in temperature and salinity of the seawater as a function of the mass fraction of ice dissolved into the mixture: DM ðTmix Tw Þ ¼ M þ DM

L ci Tf Tf Ti Tw ; cw cw

ðSmix Sw Þ ¼

DM ðSi Sw Þ: M þ DM

ð5:2aÞ ð5:2bÞ

Equations (5.2a) and (5.2b) can be thought of as modelling a simple mixing process between two end members represented by seawater with properties (Sw, Tw) and ice with properties (Si, Teff), where the effective temperature of the ice, Teff, is given by the term in square brackets. The ﬁnal properties of the mixture lie along the straight line connecting the end members, at a distance determined by the mass fraction of ice that has been dissolved into the mixture [6]. Typical ice properties are a salinity of zero and Teff of around −85 to −100 °C (the exceptionally low effective temperature is mainly due to the latent heat term). The practical endpoint of the mixing line is when Tmix = Tf ; that is, once the mixture has been cooled to the liquidus, no further dissolution is possible, and more ice added to the mixture will exist in

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equilibrium with the water. Thus the upper limit on the mass fraction of ice that can be dissolved into the mixture is

DM M þ DM

Tf Tw ¼ h i : max Tf Tw cLw þ ccwi Tf Ti

ð5:3Þ

Since the term in square brackets is *100 °C, for water initially a few degrees above the freezing point, a rough rule of thumb is that the maximum dissolvable ice fraction is *1% per °C of initial temperature elevation above freezing. A selection of mixing lines appropriate for seawater temperatures and salinities are plotted in Fig. 5.3b. These lines point towards the point (Si, Teff) but terminate where they meet the liquidus. Also plotted are isopycnals (contours of constant density) referenced to surface pressure. Seawater density is a function of temperature, salinity and pressure. The system of isopycnals shows that at high temperatures, temperature is more important in determining the density. At lower temperatures, the isopycnals become parallel to the temperature axis and at lower temperatures still their slope changes sign, implying a point of maximum density on each line of constant salinity. The locus of maximum points is marked on Fig. 5.3b. The density maximum occurs near 4 °C for freshwater, and at progressively lower temperature as the salinity increases. Above a salinity of about 25 there is no density maximum in the liquid phase. How the green lines in Fig. 5.3b traverse the isopycnals also implies a density maximum along each ice-ocean mixing line. At higher temperatures and/or lower salinities, cooling and dilution caused by ice dissolution or melting1 raise the seawater density, causing downwelling, while at lower temperatures and/or higher salinities, melting results in buoyancy. For properties typical of the polar oceans (in the box near the bottom right in Fig. 5.3b), melting ice always decreases the density and hence drives upwelling. The elevated pressure experienced beneath an ice shelf increases the density of seawater and lowers its freezing point by approximately 0.76 °C for every 1000 m of draught, but does not signiﬁcantly alter the behaviour discussed above.

5.3

Processes at the Ice-Ocean Interface

At an ice-ocean interface, it is reasonable to assume that the adjustment to thermodynamic equilibrium is instantaneous, leaving a temperature gradient in the ice and temperature and salinity gradients in the water adjacent to the interface (Fig. 5.4). Heat and salt diffusion across the resulting boundary layers regulates the melt rate. Heat is conducted away from the interface into the cold ice, while there is no diffusion of salt into the solid ice. On the seawater side, heat and salt diffuse from the At sufﬁciently low concentrations, the phase change occurs close to 0 °C and ‘melting’ is the appropriate term. We use this term from now on to describe melting and dissolution, as their distinction is not critical to the discussion.

1

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Fig. 5.4 Temperature and salinity changes across the ice-ocean boundary layer for a melting and b freezing. The illustration in (b) makes the common assumption that freezing occurs at the boundary as the reciprocal process of melting. In fact, the upper boundary layer would be supercooled in this conﬁguration, being both colder and fresher than interfacial freezing point conditions, allowing frazil ice to grow in suspension

relatively warm, salty water towards the cooler, fresher interface in the case of melting, while for freezing, diffusion is from the interface (made warm and salty by the release of latent heat and rejection of salt from the growing ice) toward the relatively cold and fresh seawater. The melt rate is ultimately governed by the divergence of the heat flux at the interface; but in the case of an ice/saltwater interface it depends also on the salt fluxes, since the interfacial temperature (freezing point) depends on the salinity there. Without salt diffusion, the dilution of water caused by the melting would raise the interfacial temperature and reduce the temperature gradient through the boundary layer, so melting requires the far-ﬁeld water temperature to exceed 0 °C. In reality, the salinity gradient that forms across the boundary layer drives a salt flux towards the ice that maintains the temperature gradient that drives melting (Fig. 5.4). But since the molecular diffusivity of salt is much lower than that of temperature, salt diffusion (through its rôle in setting the boundary temperature) becomes the rate-limiting process. Three equations are required to evaluate the melt rate, one for the heat balance, one for the salt balance and a third relating the boundary temperature and salinity via the liquidus condition. These equations are @Ti qi m_ i L ¼ qi ci ji qTb ; @n b

ð5:4aÞ

qi m_ i ðSb Si Þ ¼ qSb ;

ð5:4bÞ

Tb ¼ k 1 S b þ k 2 þ k 3 P b ;

ð5:4cÞ

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where the subscript b indicates properties evaluated at the ice/ocean boundary, n is a coordinate normal to the boundary, P is pressure, qi, ci and ji are the density, speciﬁc heat capacity and thermal diffusivity of ice and k[1–3] are empirical constants. The melt rate, m_ i , is deﬁned as a thickness of solid ice per unit time. The heat and salt fluxes from the ocean, qTb and qSb , depend not only on the temperature/ salinity gradients through the boundary layer, but also on the currents within the layer. Water movement caused by tides or buoyant water flow caused by melting along a sloping ice-ocean interface tends to produce a turbulent boundary layer. In fully-developed turbulence, where mixing by turbulent eddies dominates the diffusive transport, the effective diffusivities of heat and salt can be assumed to be equal, but, close to a solid boundary, turbulence is suppressed and the differing rates of molecular diffusion become important. The overall heat and salt fluxes are often parameterised in terms of the temperature and salinity differences between the interface (Tb, Sb) and the far-ﬁeld conditions (T, S): qTb ¼ qw cw ub CT ðTb T Þ;

ð5:5aÞ

qSb ¼ qw ub CS ðSb SÞ;

ð5:5bÞ

where ub , the interfacial friction velocity, is deﬁned as the square root of the kinematic stress at the interface, ðsb =qw Þ1=2 , and CT and CS are dimensionless turbulent exchange coefﬁcients (their values differ because of the rôle played by molecular diffusion close to the interface, and are not well constrained by observations). For a boundary layer driven by a far-ﬁeld flow of speed U, we can deﬁne a drag coefﬁcient Cd that relates the far-ﬁeld flow speed to the interfacial shear stress through the deﬁnition ub ¼

rﬃﬃﬃﬃﬃﬃ sb 1=2 ¼ Cd U: qw

ð5:6Þ

Two commonly used expressions for the exchange coefﬁcients for ice shelves, derived by Jenkins [7] from the results of laboratory studies, are h i1 CT ¼ 2:12 lnðRe Þ þ 12:5 Pr2=3 9 ;

ð5:7aÞ

h i1 CS ¼ 2:12 lnðRe Þ þ 12:5 Sc2=3 9 ;

ð5:7bÞ

where Re is the Reynolds number ðub d=mÞ based on the friction velocity and is typically *104–105 for an ice-ocean boundary current, Pr is the Prandtl number, and Sc is the Schmidt number. Typical Prandtl and Schmidt numbers for seawater

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101

are 13.8 and 2432, so the terms involving Pr and Sc dominate, implying that the turbulent exchange coefﬁcients in (5.7a) and (5.7b) are weakly dependent on the turbulent mixing regime and approximately constant. Measurements of turbulent heat flux beneath sea ice suggest that the use of constant exchange coefﬁcients is reasonable. The turbulent heat and salt fluxes at the ice-shelf base can now be modelled as 1=2

qTb ¼ qw cw Cd UCT ðTb T Þ; 1=2

qSb ¼ qw Cd UCS ðSb SÞ;

ð5:8aÞ ð5:8bÞ

with the appearance of Cd and U highlighting the influence of the interfacial shear stress. Much uncertainty in parameterising these fluxes stems from a basic lack of knowledge about the nature of the ice-ocean interface and hence the appropriate choice of drag coefﬁcient. The shelf-base roughness can be highly variable on different scales. Melting may generate a scalloped surface at sub-metre scales; and crevasses and channels incised up into the base cause highly anisotropic roughness on scales from metres to kilometres. Features at the upper end of this range can steer the large-scale flow of the boundary layer and behave more as topography than roughness. Knowledge of how such features impact the fluxes, and how they should be parameterised in models, is almost entirely lacking. Given the uncertainties in the drag coefﬁcient and turbulent exchange coefﬁcients appropriate for an ice shelf, a simpler alternative to (5.8a) has sometimes been used to parameterise the heat flux as 1=2 qTb ¼ qw cw Cd UCTS Tf T ;

ð5:9Þ

Tf ¼ k1 S þ k2 þ k3 Pb

ð5:10Þ

where

is the freezing point associated with the far-ﬁeld salinity. This model obviates the need for a separate expression for the interfacial salt flux, but note that the effective heat transfer coefﬁcient, CTS, in (5.9) is not the same one as in (5.7a) and (5.8a). For sea ice, many observations can be parameterised with (5.9) and a constant effective heat transfer coefﬁcient; no greater complexity either in the form of the coefﬁcients or the equations appears necessary, but the range of conditions observed is inevitably limited because the overlying ice floes must be sufﬁciently large, thick and stable to form a safe observational platform. It is therefore conventional to retain the more complex formulation in (5.8a) and (5.8b) (together with (5.4a), (5.4b) and (5.4c)) in numerical models, since the computational burden is negligible. The use of (5.9) and (5.10) is justiﬁable where algebraic simplicity is desired.

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Freezing at the interface also requires consideration. The details are different because the rejection of salt and release of latent heat at the interface create an unstable stratiﬁcation, so there is a source of turbulence besides any shear in the mean flow. Furthermore, freezing conditions at the shelf base are generated by bulk supercooling in the upper part of the boundary layer. Freezing can occur anywhere within this region, and the result is a suspension of millimetre-sized, disc-shaped ice crystals known as frazil ice. As the frazil crystals grow they precipitate from suspension onto the ice-shelf base, much like an inverted snowfall. The accumulation of frazil crystals then slowly consolidates. Given the difﬁculty of observing such an environment, we have limited understanding of these processes and how they impact the boundary layer dynamics. Presumably, the suspension of frazil crystals adds to the buoyancy of the boundary layer, and the transition from a mobile slurry of ice crystals to a consolidating mush may lead to a diffuse ice-shelf-ocean interface.

5.4

Buoyancy-Driven Flow on Geophysical Scales

The buoyancy induced in the turbulent boundary layer beneath an ice shelf drives a convective flow analogous to a turbulent gravity current [8], which can be conceptualised as a inclined ‘plume’ (Fig. 5.5). Similar geophysical flows include katabatic winds (Sect. 6.6), pyroclastic flows, powder snow avalanches, and turbidity currents. We write the following model for the plume flow dynamics— assuming steady state, one spatial dimension only (ignoring the Coriolis effect) and a well-mixed plume (which can thus be described by averaged properties). The model will enable a study of different factors behind the melt-rate distribution under ice shelves. The conservation equations of mass, momentum, energy and salinity are d ðDU Þ ¼ e_ þ m_ w ; ds

ð5:11aÞ

d 2 Dq DU ¼ D g sin a Cd U 2 ; ds q0

ð5:11bÞ

d 1=2 ðDUT Þ ¼ e_ Ta þ m_ w Tb Cd UCT ðT Tb Þ; ds

ð5:11cÞ

d 1=2 ðDUSÞ ¼ e_ Sa þ m_ w Sb Cd UCS ðS Sb Þ: ds

ð5:11dÞ

Here, s is distance along the interface; D is the plume’s thickness, U its mean velocity, T its temperature and S its salinity; a is the slope of the ice-shelf base, and the subscripts indicate conditions in the ambient water column, a, and at the

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Fig. 5.5 a Schematic of the buoyancy-driven circulation beneath an ice shelf arising from the pressure dependence of the seawater liquidus, and the pattern of sub-shelf melting and freezing that results. b Key processes in a one-dimensional model of an inclined plume ascending the ice-shelf base above an inﬁnitely-deep, quiescent layer of seawater with deﬁned properties

ice-ocean interface, b (Fig. 5.5b); e_ is the rate at which the plume entrains ambient water, and m_ w is the melt rate at the ice-shelf base. In its simplest form, the density difference Dq between the plume and the surroundings comes from a linear equation of state with constant thermal expansion, bT, and haline contraction, bS, coefﬁcients:

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q ¼ q0 ½1 þ bS ðS S0 Þ bT ðT T0 Þ;

ð5:12Þ

where the subscript 0 indicates reference values about which the linearisation is performed. The resulting expression for the dimensionless density difference is Dq qa q ¼ ¼ bS ðSa SÞ bT ðTa T Þ: q0 q0

ð5:13Þ

The rate at which the plume flow entrains ambient water can be deﬁned as a simple function of its speed and the interface slope: e_ ¼ E0 U sin a;

ð5:14Þ

where the slope term is a parameterisation of how the stability of the interface between the plume and the ambient environment affects mixing between the two. The mixing efﬁciency falls as the interface tends towards horizontal, because entrainment involves raising dense water against the restoring force of gravity. However, the assumption of a stationary ambient fluid is a limitation of this model. The melt rate, m_ w , is expressed as a thickness of seawater per unit time: qw m_ w ¼ qi m_ i ;

ð5:15Þ

and is calculated from a set of equations such as (5.4a), (5.4b) and (5.4c) in the previous section. Using (5.8a) and (5.8b), the equations (5.4a) and (5.4b) for the heat and salt balances at the ice-ocean interface can be written as 1=2

m_ w L ¼ ci m_ w ðTi Tb Þ cw Cd UCT ðTb T Þ; 1=2

m_ w ðSb Si Þ ¼ Cd UCS ðSb SÞ;

ð5:16aÞ ð5:16bÞ

where the term denoting heat diffusion into the ice (ﬁrst on the right-hand side of (5.16a)) has been approximated by an assumption that it is exactly balanced by the advection of cold ice towards the interface. The heat and salt conservation equations (5.11c) and (5.11d) can then be rewritten as d L ci ðDUT Þ ¼ e_ Ta þ m_ w Tb ðTb Ti Þ ; ds cw cw

ð5:17aÞ

d ðDUSÞ ¼ e_ Sa þ m_ w Si : ds

ð5:17bÞ

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From these, we see that the impact of melting on the plume is simply the addition of water with properties (Si, Teff) as discussed previously, and that the plume properties will evolve along the previously-deﬁned mixing line that passes through (Sa, Ta). The model behaviour is illustrated in Fig. 5.6, which shows how the plume evolves beneath ﬁve ice shelves having the same grounding line depth but differing basal slopes, all floating in unstratiﬁed water at the surface freezing point. The

Fig. 5.6 Results of a one-dimensional plume model applied to ﬁve idealised ice shelf conﬁgurations having constant basal slope. Model inputs are shown in the top panels: ice shelf draught as a function of distance from the grounding line and ambient temperature and salinity as a function of draught. Dashed line on the temperature plot indicates the pressure freezing point of the ambient as a function of depth. Middle and lower panels show model outputs: plume variables as functions of distance along the ice-shelf base. In the temperature and salinity difference plots, solid lines indicate plume-boundary properties, while dashed lines indicate ambient-plume properties. Vertical and horizontal dashed lines in all panels indicate the locations of the transitions from melting to freezing

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buoyancy imparted by melting at depth drives the plume motion up the inclined shelf base. The plume grows in thickness as it entrains ambient seawater, a process that brings in the heat required to sustain melting. The plume’s buoyancy increases as meltwater is added, but decreases as freezing starts and freshwater is lost to the ice shelf. The plume’s buoyancy dictates its speed, so speed increases in the zone of melting and decreases after the transition to freezing. The plume speed also depends strongly on the shelf’s basal slope, which determines how effective the buoyancy force is at driving the plume. The entrainment rate—proportional to both plume speed and basal slope—is an even stronger function of slope. Entrainment acts to warm the plume, reducing the difference in temperature between plume and ambient ocean, while melting cools it towards freezing, reducing the difference across the ice-ocean boundary layer. Melting is a function of plume speed but has no extra dependence on slope, so as the ice base steepens entrainment grows more rapidly than melting does and the temperature difference between plume and ambient ocean falls, while that across the boundary layer grows. As the plume grows in volume its thermal inertia grows and entrainment and melting become less effective at modifying its temperature [9]. Because of the rising freezing point, the temperatures of both the plume and ambient ocean water fall with respect to the freezing point, and this tendency causes the plume to become supercooled eventually: the temperature difference across the boundary and the melt rate change sign. Supercooling occurs even though entrainment and freezing both now act to warm the plume. The melt-rate curves that result are determined by the product of plume velocity and thermal forcing (the difference between plume temperature and the freezing temperature at the base). The initial rise in melt rate is driven by increasing plume speed, whereas the drop in melt rate later is due to reduced thermal forcing. At the transition to freezing, the thermal forcing changes sign; the subsequent growth in the freezing rate is driven by the rise in magnitude of the thermal forcing outweighing the fall in plume speed. When the ambient water column is stratiﬁed, its density proﬁle introduces a second control on the plume buoyancy. Results for a 500 km long ice shelf are shown in Fig. 5.7 for linear stratiﬁcation of increasing strength. The initial rise in plume buoyancy can only continue as long as the input of meltwater can overcome the fall in ambient density. Eventually, the ambient stratiﬁcation causes the plume’s buoyancy to drop even though it is still in the melting phase; its velocity is reduced, so the melt rate falls more rapidly than in the unstratiﬁed case. Once freezing begins, both the loss of freshwater and the falling ambient density contribute towards the decline in plume buoyancy. The more rapid decline in plume speed eventually outweighs the rise in magnitude of the thermal forcing so the freezing rate peaks and falls back towards zero. The plume becomes neutrally buoyant at some depth below the surface, detaching from the ice-shelf base and spreading into the ambient water column. The size of the freezing zone and the intensity of the freezing are thus reduced. Since the plume ceases to interact with the ice shelf at depth and remains there, it retains a temperature that is signiﬁcantly lower than the surface freezing point. If the stratiﬁcation is sufﬁciently strong, the plume can become neutrally buoyant within the melting zone, and freezing is eliminated

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Fig. 5.7 Results of the plume model in the standard conﬁguration, for different vertical stratiﬁcation in the ambient water column. Organised as in Fig. 5.6

altogether. Above the point of detachment, the ice-shelf base can interact with the ambient water and melting can initiate another plume. In this way, the plume model illustrates how the overturning beneath an ice shelf may be organised in a stack of cells, the vertical extent of each being determined by the ambient stratiﬁcation and temperature.

5.5

Sensitivity to Ocean Temperature

The impact of raising ambient ocean temperature in the plume model is illustrated in Fig. 5.8. As Ta is increased, entrainment becomes a more effective source of sensible heat, and both the plume temperature and the temperature difference across the boundary layer rise in response. The melt rate increases, imparting more

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Fig. 5.8 Results of the plume model in the standard conﬁguration, for different ambient water column temperatures. Organised as in Fig. 5.6

buoyancy to the plume, which flows faster as a result, in turn enhancing both entrainment and melting. For sufﬁciently high Ta, the freezing zone is eliminated. Since the important parameter is the temperature with respect to the freezing point, deepening the ice with respect to the freezing point level has the same effect as raising the freezing point level (i.e., warming the ocean) with respect to the ice [9]. Hence, the solution obtained for a higher Ta is identical to that for a deeper grounding line, except that in the former case the solution is terminated when the plume reaches the sea surface (Fig. 5.9). This explains why as the water warms, the freezing zone shrinks and eventually vanishes: it is progressively shifted above the sea surface. When this happens, the plume temperature itself can rise above the surface freezing point, and the plume consists of a water mass that is formally no longer ‘Ice Shelf Water’ (ISW, deﬁned as having a temperature below the surface

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Fig. 5.9 Results of the plume model for different ambient temperatures and grounding line depths. As before, the standard conﬁguration is in black. Results for increasing temperature (red and green) plot exactly over the results for equivalent deepening of the grounding line (blue and magenta) until they are terminated at the sea surface. Organised as in Fig. 5.6

freezing point)—despite its distinct origin as a mixture of ambient water and meltwater derived from the shelf. For an ice shelf of ﬁxed geometry, the mean melt rate is a non-linear function of the ambient ocean temperature relative to the freezing point. The non-linearity arises because the temperature difference across the boundary layer and the plume speed—factors behind the boundary-layer heat flux in (5.8a)—increase with temperature. Figure 5.10 shows that while the temperature difference across the boundary layer is an approximately linear function of ambient temperature, the plume speed has a square-root dependence, so the mean melt rate rises as a function of ambient temperature to the power 1.5. This relationship varies with ice-shelf geometry and breaks down a little at low temperatures. The T1.5 dependence can be deduced by considering the dominant terms in (5.11a)–(5.11d). For high ambient

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Fig. 5.10 Results from Fig. 5.8 averaged over the length of the ice-shelf base and plotted as a function of the ambient ocean temperature relative to the surface freezing point of the ambient ocean itself. Also plotted are results for intermediate temperatures (every 0.2 °C) and for two other ice-shelf geometries from Fig. 5.6 (colour-coded as in that ﬁgure). Panels are as organised in Fig. 5.6. In the temperature and salinity difference plots, circles indicate plume-boundary properties, while diamonds indicate ambient-plume properties. Lines indicate least-squares T0, T0.5, and T1 ﬁts to the data in the four panels on the lower left, excluding the results at low temperature, where the plume thickness varies substantially. The T0.5 and T1.5 relationships shown in the two panels on the right are derived from those ﬁts and (5.14) and (5.26), respectively

temperatures, the shelf experiences only the early stages of plume evolution when heat advection within the plume is small and the dominant terms in the heat balance (5.11c) are entrainment and melting: DU

dT 1=2 ¼ e_ ðTa T Þ Cd UCT þ m_ w ðT Tb Þ 0: ds

ð5:18Þ

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The heat flux due to melting has two terms: one associated with turbulent diffusion, and the other associated with the advection of meltwater across the boundary layer. The former term dominates so the heat balance is approximately 1=2

E0 U sin aðTa T Þ Cd UCT ðT Tb Þ:

ð5:19Þ

It follows that the temperature difference across the boundary layer can be expressed as a constant, slope-dependent, fraction of the difference between the far-ﬁeld temperature and the boundary temperature: ðT Tb Þ

E0 sin a 1=2 Cd CT þ E0

sin a

ðTa Tb Þ:

ð5:20Þ

And since freshening of the boundary has a small impact on the freezing point, this is approximately a linear relationship between the temperature difference across the boundary layer and the temperature of the ambient ocean relative to its freezing point (Fig. 5.10). Similarly, the temperature difference between the plume and the ambient ocean is linearly related to the ambient thermal forcing: 1=2

ðTa T Þ

1=2

Cd CT

Cd CT þ E0 sin a

ðTa Tb Þ:

ð5:21Þ

Because the entrainment constant is almost two orders of magnitude larger than the heat transfer coefﬁcient, these expressions quantify the earlier observation (Fig. 5.6) that for small slopes the plume temperature tends towards the boundary temperature, while for large slopes it tends towards the ambient temperature. The plume itself is simply a mixture of ambient water and meltwater, so from (5.2a) and (5.2b), its temperature and salinity are related by Ta Teff ðTa T Þ ¼ ; ð Sa SÞ ð Sa Si Þ

ð5:22Þ

which is only weakly dependent on the ambient properties. Hence the salinity difference is an approximately linear function of ambient thermal forcing (Fig. 5.10), and it follows from (5.13) that the density difference between the plume and the ambient ocean is also approximately linear. In the momentum balance (5.11b), an analogous approximation can be made because the advection of momentum is small compared with the other terms: DU

dU ¼ DðDq=q0 Þg sin a Cd U 2 ðe_ þ m_ w ÞU 0: ds

The melt rate term is again negligible so the approximate balance is

ð5:23Þ

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DðDq=q0 Þg sin a ðCd þ E0 sin aÞU 2 ;

ð5:24Þ

and the plume speed is then a square-root function of the plume buoyancy:

DðDq=q0 Þg sin a 1=2 U : Cd þ E0 sin a

ð5:25Þ

As the plume thickness is almost independent of ambient temperature (at least at sufﬁciently high temperature, Fig. 5.10), U has effectively a square-root dependence on the density difference and thus also on the ambient thermal forcing (Fig. 5.10). Finally, from (5.16a), the melt rate can be expressed as m_ w ¼

1=2

cw Cd UCT ðT Tb Þ ; L þ ci ðTb Ti Þ

ð5:26Þ

where all terms besides plume speed and temperature with respect to the boundary are constant or weakly dependent on the ambient temperature. Hence (5.20), (5.25) and (5.26) predict the melt rate to be proportional to the ambient thermal forcing raised to the power of 1.5 (Fig. 5.10). These arguments break down when the temperature advection term in (5.18) is not negligible. Since this is the term that drives freezing, a departure from the T1.5 scaling is expected when ambient temperatures are low enough to cause freezing at the ice-shelf base.

5.6

Impact of Meltwater Outflow at the Grounding Line

Where an active ice stream or outlet glacier enters an ice shelf, freshwater flow may cross the grounding line, supplied by the subglacial drainage system (Chap. 3). This outflow represents a buoyancy source in addition to that derived from melting at the shelf base; near the grounding line it will dominate the plume dynamics [10]. Where the buoyancy input from melting the ice shelf remains negligible, only entrainment affects the plume density and the plume dynamics conform closely to the classical description [8, 11]. If density changes in the ambient fluid are negligible, the product (DDq) remains constant, because the plume thickens by entrainment at the same rate as the density deﬁcit falls. The plume equations then have a simple solution. The plume speed, given by (5.25), is a constant set by the constant buoyancy flux (DUDqg):

U

DUðDq=q0 Þg sin a Cd þ E0 sin a

1=3

:

ð5:27Þ

5

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The fall in the freezing point with depth is typically small over the region where the buoyancy provided by the subglacial outflow dominates, so the ambient temperature can also be regarded as constant. In this case, the constant melt rate derived from (5.20), (5.26) and (5.27) is m_ w

1=2 cw Cd CT DUðDq=q0 Þg sin a 1=3 E0 sin a ðTa Tb Þ; 1=2 Cd þ E0 sin a L þ ci ðTb Ti Þ C CT þ E0 sin a d

ð5:28Þ where DU is the (‘initial’) flux of freshwater crossing the grounding line, and Dq is the density difference between it and ambient seawater. Equation (5.28) holds close to the grounding line, before other processes impact the plume flow. In this regime, the melt rate is linearly dependent on ambient thermal forcing as there is not yet any feedback between rising ambient temperature and plume flow (as occurs further along the plume); its square-root dependence on plume velocity, or equivalently cube-root dependence on plume buoyancy flux, remains. Farther from the grounding line, addition of meltwater from the shelf acts to increase the plume buoyancy and hence the melt rate, whereas ambient stratiﬁcation and the rise in the freezing point with height act to reduce plume buoyancy and ambient thermal driving, respectively, and hence lower the melt rate. There, plume evolution is controlled by these processes (as outlined in the last sections), and the initial input of buoyancy at the grounding line plays only a minor rôle. Figure 5.11 shows the initial melt rate calculated from (5.28) for a range of ambient temperatures, initial freshwater fluxes and ice-shelf basal slopes. The slope influence is encapsulated in two parameters: one determined by the relative importance of buoyancy forcing and drag (including that from entrainment of stationary ambient seawater) in setting the plume speed, and a second determined by the relative importance of the heat gained by entrainment versus that lost by melting in setting the plume temperature. The cube-root dependence of the melt rate on the initial freshwater flux means that the melt rate roughly doubles for every order of magnitude increase in the flux. Changes up to several orders of magnitude can occur on seasonal and interannual time scales, due to changes in the basal water system. Episodic ﬁlling and drainage of subglacial lakes has been observed to be widespread over the Antarctic Ice Sheet. Where such lakes drain across grounding lines, discharge fluxes of several cubic kilometres per year have been inferred that are two orders of magnitude above typical background flows [12, 13]. Over the Greenland Ice Sheet there is extensive surface melt in the summer months; in places, large quantities of surface meltwater drain to the bed to emerge at the glacier termini, including at the calving faces of tidewater glaciers. The water fluxes are hard to estimate and fast-flowing tidewater glaciers such as Jakobshavn Isbræ also produce much basal water through frictional heating, but summertime outflows of freshwater may be at least one order of magnitude higher than those in winter [14]. On smaller, temperate tidewater glaciers, the wintertime production of basal meltwater can be negligible, while rainfall

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Fig. 5.11 a Melt rates at the grounding line derived from (5.28) for a vertical ice-ocean interface (sin a = 1). b Multiplication factor required to scale the melt rates in (a) for different interface slopes. The overall factor is the product of temperature-related and velocity-related scale factors. Line widths in (a) indicate the spread of values obtained for ambient salinities ranging from 25 to 35. Figure from Jenkins [10]. © American Meteorological Society. Used with permission

in the summer can supplement surface melt, causing increases of up to four orders of magnitude in the freshwater outflow at the glacier terminus [15].

5.7

Fundamentals of the Three-Dimensional Ocean Circulation

The simpliﬁed sub-ice-shelf circulation model of the previous sections can be extended to three dimensions. Resolving the vertical flow structure allows a more detailed treatment of vertical mixing processes, but the basic principle remains that the energy required to effect mixing is derived from the large-scale circulation. Thus, the feedback of higher melt rates fuelling a stronger circulation that in turn enhances melting should still be present, regardless of the level of complexity included in the mixing physics. Adding the second horizontal dimension complicates the solution, but does not add fundamental new physics to the mass, heat and salt conservation. The key differences in such an extension concern momentum balance. One no longer has to assume a stagnant, inﬁnitely deep water mass in the sub-shelf cavity, as the extra dimensions allow a fundamentally more complex circulation to be described. Notably, sloping density surfaces caused by ice-shelf melting set up pressure gradients that drive flow in the lower layer in the opposite sense to the upper water column currents, and the overturning circulation that controls the inflow of heat to the sub-shelf cavity can now be quantiﬁed. Net convergence or

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divergence in the horizontal flow may push the ice shelf up or down, creating pressure gradients that drive a net horizontal circulation, which impacts on the heat and freshwater transports in and out of the cavity. In this and the following two sections, we treat some general principles about rotating geophysical flows in three dimensions, which will help us understand the sub-shelf circulation. Ocean currents are governed by Newton’s second law of motion, which, for a continuum in the non-inertial, rotating frame of reference ﬁxed with respect to the Earth, can be written as q

DU þ 2qX U ¼ r T qgk; Dt

ð5:29Þ

where t is time, k is a unit vector deﬁning the vertical, T is the stress tensor, and X is the Earth’s rotation vector. Use of the rotating reference frame introduces a centrifugal acceleration, which features as a minor modiﬁcation to the gravitational acceleration, and the well-known Coriolis term (the second term on the left-hand side of (5.29)). With the stress tensor written in terms of its individual components, sij ¼ Pdij þ qm

@uj @ui þ ; @xi @xj

ð5:30Þ

where dij is the Kronecker delta symbol, (5.29) resembles the force-balance equation familiar to glaciologists (e.g., (1.32)) but here we keep the acceleration and Coriolis terms and assume a linear viscous relationship between deviatoric stress and strain rate (see the derivations in Sect. 1.4). Density variations in the ocean are small (* 0.1% of the mean density) and impact the fluid inertia negligibly, but, in combination with the gravitational acceleration, they signiﬁcantly affect the weight of the fluid. This motivates the so-called Boussinesq approximation that is almost universally applied to the equations of motion for ocean circulation. In this approximation, the density q is replaced by a constant reference value, q0, except where it is multiplied by the gravitational acceleration. For the circulation beneath an ice shelf, a scaling analysis reveals which terms can be dropped from (5.29). Typical magnitudes of the horizontal velocity U, horizontal length scale L, and vertical length scale H are U * 0.1 m s−1, L * 105 m, and H * 102 m. Additionally the rotation rate is X * 10−4 s−1. It follows that, in the horizontal momentum balance, the acceleration term is of magnitude U2/L * 10−7 m s−2 and the Coriolis term is of magnitude XU * 10−5 m s−2, whereas if we use the molecular viscosity m * 10−6 m2 s−1, the largest (vertical) viscous term is only *10−11 m s−2. The forces arising from molecular viscosity are thus negligible, and therein lies the primary distinction between glacier and ocean dynamics. Furthermore, the ratio of the acceleration term to the viscous terms—the Reynolds number—is very large, so the motion will be turbulent. The macroscopic effect of turbulent transport needs to be parameterised in the equations by introducing an eddy viscosity in place of the molecular viscosity; note that eddy viscosity

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is a property of the flow (rather than of the fluid) and is anisotropic in the ocean due to the nature of the mixing. In a stably stratiﬁed environment, horizontal mixing is energetically easier than vertical mixing as the latter is impeded by gravity, so the horizontal eddy viscosity is larger than the vertical eddy viscosity. Typical oceanic eddy viscosities span a broad range from 10 to 105 m2 s−1 in the horizontal and from 10−5 to 10−1 m2 s−1 in the vertical [16]. Even at the upper end of this range, viscous effects do not make a signiﬁcant contribution to the large-scale force balance. For the large-scale flow, the (dimensionless) ratio of magnitudes of the viscous and Coriolis terms—called the Ekman number—and the (dimensionless) ratio of the acceleration and Coriolis terms—called the Rossby number—are generally much less than one, so the only term that can balance much of the pressure gradient is the Coriolis acceleration. Accordingly, to leading order, the horizontal momentum equations reduce to 2Xz t ¼

1 @P ; q0 @x

2Xz u ¼

1 @P ; q0 @y

ð5:31Þ

where 2Xz = 2Xsin(latitude) is the component of the rotation vector that lies parallel to the local vertical and is conventionally denoted by f. Equation (5.31) shows that the fluid flows perpendicularly to the pressure gradient—to the right of the down-gradient direction in the northern hemisphere (where the Coriolis parameter f > 0), and to the left in the southern hemisphere (where f < 0). Such flow is termed geostrophic, which is Greek for ‘Earth-turning’. Note that while the above scaling is appropriate for the large-scale flow, the viscous and acceleration terms can both become important (the Ekman and Rossby numbers can become of order one) where fluid velocities are high or the relevant length scales are small. For vertical motion, the acceleration and viscous terms in the momentum balance are much smaller, as is always the case for shallow (H L) flows, and the acceleration due to gravity, g, is *10 m s−2, thus to a very good approximation the vertical pressure gradient is balanced by gravity: @P ¼ qg: @z

ð5:32Þ

Using this hydrostatic approximation, the pressure distribution within the ocean can be diagnosed from the density distribution as if the water were at rest. Horizontal pressure gradients derived in this way cause horizontal flow (see (5.31)), and the small vertical velocities can be recovered from the continuity equation, which, assuming incompressibility, can be written as @u @t @w þ þ ¼ 0: @x @y @z

ð5:33Þ

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The hydrostatic approximation in this context breaks down only where the appropriate length scales are very small and corresponding vertical velocities are very high. An example of this would be the flow of a buoyant plume up a steep or even vertical ice face. Actually, the plume model discussed earlier remains valid in this situation, but resolving such plumes within ocean models that are optimised for simulations of the large-scale horizontal circulation is challenging.

5.8

Some Properties and Limitations of the Geostrophic Equations

Fluid motion that is slow enough to be in geostrophic and hydrostatic balance has distinctive properties that can be deduced from the equations. Vertical differentiation of the geostrophic equations in (5.31) yields @t 1 @2P ¼ ; @z q0 @z@x

f

f

@u 1 @2P ¼ : @z q0 @z@y

ð5:34Þ

By making use of the hydrostatic approximation in (5.32), these expressions become f

@t g @q ¼ ; @z q0 @x

f

@u g @q ¼ : @z q0 @y

ð5:35Þ

Thus, if the density is constant, the horizontal velocity vector must be independent of depth. In other words, columns of a homogeneous, rotating fluid are vertically rigid and cannot tip. Similarly, taking horizontal derivatives of the geostrophic equations in (5.31) leads to

f

@t @u þ @y @x

¼

1 @2P @2P ¼ 0; q0 @y@x @x@y

ð5:36Þ

which implies that for constant f the divergence of the horizontal flow is zero. From (5.33), a further implication is that the vertical velocity component w is also independent of depth. Together these results correspond to the Taylor-Proudman theorem, which states that the three-dimensional velocity vector must be constant along the axis of rotation, and the principle can be thought of as a statement of angular momentum conservation [16]. A rotating column of fluid cannot be stretched or squashed vertically without being spun up or down to conserve angular momentum. However, in the geostrophic balance, the flow is so slow that the angular momentum budget is completely dominated by the planetary rotation, which is ﬁxed.

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The depth independence of the geostrophic circulation has implications for how the fluid flow responds to topography. Geostrophic flow up or down a seabed slope is not possible. Since the vertical velocity is constant with depth, the sea surface must rise or fall creating a pressure gradient that drives flow along the slope. In the absence of vertical shear the entire water column must flow along contours of constant depth. Thus, the impact of seabed topography steering the flow is felt throughout the column, not because of friction (geostrophic flow is inviscid) but because of Earth’s rotation. The surface topography created by a floating ice shelf has the same effect; the ice front should represent a dynamic barrier to the geostrophic flow. But the fact that flows into and out of sub-ice-shelf cavities do occur in nature shows that the homogeneous (constant density) geostrophic flow model falls short of explaining all phenomena. Notably, the model cannot describe any time evolution of the flow, because the flow (being non-divergent and having to follow pressure contours) cannot change the pressure ﬁeld. Our everyday experiences of variability in the atmosphere and ocean highlight the commonness of ageostrophic effects. As hinted at above, the geostrophic approximation breaks down at small scales when other terms in the momentum equation become signiﬁcant. When the acceleration term becomes important, rotation associated with the flow ﬁeld plays a signiﬁcant rôle in the angular momentum budget. The rotation is quantiﬁed by the anti-symmetric part of the deformation gradient tensor xij ¼

@uj @ui @xi @xj

ð5:37Þ

(cf. (1.21)). If the water column is stretched or squashed by convergence or divergence in the horizontal flow, angular momentum is conserved through changes in rotation rate of the column relative to the Earth ﬁxed reference frame. The effective rotation rate is the sum of the changing relative vorticity, xxy, and the steady planetary vorticity, f. The former contribution becomes signiﬁcant as the Rossby number approaches one. Keeping the other scales ﬁxed, this occurs when the horizontal length scale L is of order 1 km. A Rossby number larger than one means that the pressure gradient causes a down-gradient acceleration of the fluid. Thus, at small scales, flow in the ocean and atmosphere does in fact drive temporal evolution of the pressure ﬁeld; note, however, that this evolution stems from the small scale secondary circulation and not the large scale geostrophic flow. A key cause of such secondary circulation is friction. Near a solid boundary, such as the ice-shelf base or the seabed, the velocity must tend to zero as friction begins to dominate the force balance. The distance from the boundary at which frictional and rotational effects are of equal magnitude—that is, the local Ekman number is of order 1—can be estimated as

1=2 U v fU v 2 ) H : H f

ð5:38Þ

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For an eddy viscosity of 10−2 m2 s−1, H * 20 m. This length scale deﬁnes the thickness of the planetary (or rotational) boundary layer or Ekman layer. Within this layer, friction slows the fluid down relative to the geostrophic current, so the Coriolis force no longer balances the pressure gradient force. As a result, flow within the Ekman layer has a velocity component down the pressure gradient, perpendicular to the geostrophic flow. If the geostrophic flow has non-zero relative vorticity, as a result of flow around minima or maxima in the pressure ﬁeld, the down-pressure-gradient Ekman transport will everywhere be directed towards centres of low pressure and away from centres of high pressure. The resulting convergence or divergence will, by continuity, force fluid out of or draw it into the Ekman layer. An analogous process is responsible for wind-driven upwelling and downwelling in the ocean. In the surface layer, wind stress causes the water to flow faster than the geostrophic currents. The Coriolis force exceeds the pressure gradient force and there is an up-gradient transport in the surface layer, away from low-pressure centres and towards high-pressure centres. The associated convergence or divergence of the surface flow drives vertical motion in a process called Ekman pumping. This is profoundly important for the large-scale geostrophic circulation, because the vertical motion extends beyond the boundary layer and is independent of depth in a homogeneous ocean. Thus, despite their limited vertical extent, frictional boundary layers can indirectly impact the entire water column. Finally, note that the force balance considered earlier in the plume equations is a reasonable approximation as long as the plume thickness is small compared with the Ekman depth.

5.9

Effects of Stratification

The real ocean differs from the idealised model above because it is inhomogeneous. Equation (5.35) shows that in the presence of horizontal density gradients, vertical shear arises in the geostrophic flow. Vertical shear weakens the constraints imposed by topography, because flow above the seabed can have a horizontal component perpendicular to isobaths (contours of equal depth) without the need for a vertical component. Counter-intuitively, it is not the vertical stratiﬁcation that causes this effect. A uniformly stratiﬁed fluid is subject to the constraints imposed by the Taylor-Proudman theorem, which applies whenever/wherever the density contours (isopycnals) are parallel to the pressure contours (isobars), a situation referred to as barotropic. For the geostrophic flow, shearing occurs when isopycnals slope relative to isobars, a situation referred to as baroclinic. Thus, horizontal variation in the vertical stratiﬁcation produces the depth-varying horizontal pressure gradients that in turn cause a depth-varying geostrophic current. Note that in a vertically stratiﬁed ocean, vertical velocities set up by Ekman pumping will perturb the density ﬁeld to generate horizontal density gradients. In this way, secondary circulation associated with boundary layers modiﬁes the large-scale pressure gradients and alters the geostrophic flow. Besides Ekman pumping, surface or sub-ice-shelf buoyancy

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forcing and vertical oscillations associated with internal waves can also cause vertical perturbations of the density ﬁeld. The strength of the vertical stratiﬁcation governs how sensitive the horizontal pressure gradient force is to such perturbations. Let us assess this sensitivity. First, from (5.32), it is convenient to remove the dynamically unimportant pressure ﬁeld associated with the homogeneous reference density q0: @P0 @P @P0 ¼ ¼ ðq q0 Þg ¼ Dqg: @z @z @z

ð5:39Þ

Using this result, the scale of the pressure gradient in the horizontal momentum balance can be estimated as

1 1 Dq H $h P ¼ $h P0 g ; q0 q0 q0 L

ð5:40Þ

where H and L are the height and length scales over which the density ﬁeld changes by Dq, so their ratio is effectively the slope of the isopycnals. For small Rossby number, the Coriolis term is of the same order of magnitude as the pressure gradient, and we deduce the scale for the geostrophic velocity as U

ðDq=q0 ÞgH ; XL

ð5:41Þ

which is consistent with (5.35). Conversely, for an inviscid flow with large Rossby number, the pressure gradient term must be balanced by the acceleration term, giving the velocity scale U ½ðDq=q0 ÞgH1=2 :

ð5:42Þ

This is an important scaling, because U here equals the phase speed of the internal waves supported by the stratiﬁcation. A fluid that flows at the phase speed is said to be critical, and this regime marks a key transition. For super-critical flows (faster than the wave speed), the stratiﬁcation plays a lesser rôle in the force balance. The density ﬁeld evolves primarily through advection by the flow ﬁeld, and no information about the stratiﬁcation can propagate in the upstream direction. For sub-critical flows, the stratiﬁcation becomes increasingly important. Information about disturbance to the density ﬁeld is carried by the internal wave ﬁeld, creating pressure gradients that directly influence the mean-flow force balance. The plume model formulated earlier inherently assumes that the flow is super-critical. Gravity drives the plume up the ice-shelf base more rapidly than waves propagate along the interface between plume and ambient, so the pressure gradient associated with variations in plume thickness is unimportant, and the main balance is between gravity and friction. The resulting model equations, involving ﬁrst-order derivatives of one independent variable, pose an initial-value problem

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that is easy to solve as no information travels from downstream. The super-critical approximation holds for an ice-shelf base with a constant slope in the melting phase, but breaks down where a downstream decrease in the basal slope or plume buoyancy causes the plume to slow and thicken. At sub-critical speeds, the modelled plume thickness can become inaccurate, because pressure gradients associated with downstream thickness variations are ignored. These ideas motivate the choice of a fundamental length scale for geophysical flows. Earlier, we estimated the length scale at which planetary rotation becomes important by setting the Rossby number to one, but assuming a tentative (slow) velocity scale (Sect. 5.8). The discussion above on stratiﬁcation shows that when rotation and viscosity are unimportant, the flow must be close to critical; thus the appropriate velocity scale to use for deducing the length scale over which the geostrophic balance is established is the phase speed in (5.42). Now, on setting the Rossby number to one, we ﬁnd L

½ðDq=q0 ÞgH1=2 ; X

ð5:43Þ

which is known as the Rossby radius of deformation. When the length scale is large compared with the Rossby radius, the geostrophic approximation holds, but at scales smaller than the Rossby radius the flow is ageostrophic. Flow caused by a disturbance to the density ﬁeld is initially directed down the resulting pressure gradient, but comes under the progressive influence of rotation. Once it has travelled a distance of the order of the Rossby radius, rotation takes over and the flow eventually aligns with pressure contours. At this point, the geostrophic balance is established, spreading of fluid away from the disturbance stops, and the pressure ﬁeld ceases to evolve. For this reason, the width of a boundary current and the radius of an eddy both scale with the Rossby radius. The time-varying, ageostrophic flows that characterise our experience of the atmosphere and ocean occur either within the Ekman layer or at horizontal scales smaller than the Rossby radius, which is around 1000 km in the atmosphere, between 10 and 30 km in the mid-latitude ocean, and typically 5–10 km in the polar oceans. Boundary currents, including the frictional Ekman layers, and the eddy ﬁeld thus provide key missing processes that allow time evolution in the density ﬁeld that drives an unsteady circulation.

5.10

Three-Dimensional Circulation in Sub-Ice-Shelf Cavities

Armed with insights from the simple plume model and the principles of geophysical flows subject to planetary rotation, we return to consider the ocean circulation within a large sub-ice-shelf cavity.

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Beneath the shelf, phase changes at the ice-ocean interface reduce the water density in the boundary layer (Fig. 5.12). If the shelf base were horizontal, there would be a stable vertical stratiﬁcation and no horizontal circulation. However, the sloping base means that melting produces horizontal density (and hence pressure) gradients. In a large sub-shelf cavity, most of the flow should be in geostrophic balance: flow will be sheared in the vertical but tend to align with contours of water column thickness. Within frictional boundary layers there will be some flow up along the ice-shelf base and in along the seabed, giving a weak version of the overturning circulation that would dominate in the absence of rotation. However, most of the flow into and out of the cavity occurs in boundary currents that are produced where the cross-ice-shelf geostrophic flow encounters the lateral boundaries. Notably, an accumulation of light water on one side of the cavity and dense water on the other side creates pressure gradients perpendicular to the walls that drive light water out of the cavity and dense water in. This picture of a three-dimensional sheared current system (Fig. 5.12) is further complicated by the fact that the accumulation of waters at the lateral boundaries raises the ice shelf there and produces a surface pressure gradient from the boundary to the interior. Thus, superimposed on the sheared currents will be a depth-independent cyclonic (low pressure at the centre) horizontal circulation. This flow ﬁeld can be illustrated by using a three-dimensional ‘primitive equation’ ocean model, in which all terms in the horizontal momentum equation are retained, but the vertical balance is assumed to be hydrostatic [17]. The model includes three-dimensional versions of the continuity equation and the

Fig. 5.12 Schematic of the depth-dependent components of the buoyancy-driven circulation beneath an ice shelf in the southern hemisphere

three-dimensional

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advection/diffusion equations for heat and salt. Density is computed via a non-linear equation of state. Mixing near the ice-shelf base is parameterised with a depth-integrated ‘mixed layer’ model, much like a two-dimensional version of the plume, but in this case all terms in the momentum equation are retained and the entrainment rate is derived from a steady, local turbulence model. Heat and salt transfer between mixed layer and ice-shelf base is parameterised as in (5.8a) and (5.8b). Figure 5.13 shows the numerical solution of the model applied to a domain in the southern hemisphere. The domain extends between 70 and 80° S and between 0 and 15° E, and has a flat seabed 1100 m below sea level and an ice shelf covering the southern half with a draught that is constant in the E–W direction and varies N–S from 200 m at the calving front to 1000 m at the grounding line (Fig. 5.13a). There is no external forcing on the system; conditions are set at the northern boundary and ocean surface so there is a weak stratiﬁcation in salinity and a constant temperature equal to the surface freezing point. The modelled pattern of melting and freezing (Fig. 5.13b), maintained by a weak overturning circulation, is consistent with an ‘ice pump’ where ice melts at depth to produce buoyant water that refreezes at shallow depths. The pattern is complicated

Fig. 5.13 Results from a three-dimensional model of the ocean circulation in an idealised sub-ice-shelf cavity: a ice draught (m); b basal melt rate (cm yr−1); c mixed layer velocity (cm s−1); d depth-mean velocity (cm s−1); e temperature at the western boundary (°C); f velocity normal to the ice front (cm s−1)

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by the large horizontal scale and planetary rotation: the strong horizontal circulation shifts the regions zonally, so freezing is most intense at the western coast, while the melting zone extends along the eastern coast towards the ice front. The circulation within the mixed layer (Fig. 5.13c) is driven by the buoyancy input from melting at depth. The flow is up the shelf base with a component to the left, reflecting the rôles of both frictional drag and Coriolis acceleration in balancing the buoyancy forcing. Thus, in the south and east, the currents are directed away from the coast; the divergence drives thinning of the mixed layer that in turn promotes rapid entrainment that drives high melt rates. Freezing occurs as the mixed layer ascends the shelf base. This is most rapid in the west where convergence of the flow thickens the mixed layer and the coast directs the flow within the boundary layer up the shelf base. The depth-mean circulation is weaker (Fig. 5.13d), reflecting mainly the more sluggish flow in the layers below the mixed layer. Two cyclonic gyres form, separated by the ice front, where there is a complex circulation pattern. North of the ice front, boundary currents transport the outflow away from the ice shelf in the west and carry warmer waters towards the ice shelf in the east. The outflow has two cores at different levels in the water column (Fig. 5.13e) as a result of the background stratiﬁcation (as discussed in Sect. 5.4 in the context of the plume model). At the ice front, surface waters are blocked and must turn to the west along the front. Even below the draught of the ice front, the change in water column thickness partially blocks the southward flow, because the non-divergent geostrophic flow is steered along a path with constant water column thickness. The weak southwards component that takes water deeper into the cavity is a compensation for the northward ageostrophic flow in the ice-ocean and sea-bed boundary layers (Fig. 5.13f). The net northward flow in the boundary layers, particularly the stronger flow in the upper mixed layer, is supplied by entrainment of fluid from the interior geostrophic flow. Loss of fluid to both boundary layers involves a non-zero gradient in the vertical velocity (unless the fluid moves southward to where the water column is thinner). Thus, as in the non-rotating plume case, the strength of the overturning is controlled by the buoyancy-driven transport up the ice-shelf base, but here the pressure gradients associated with that flow set up a strong geostrophic flow across the cavity that is superimposed on the overturning. While the weak northward flow in the mixed layer carries some light waters out of the cavity to join the westward surface flow along the ice front, most of the northward flow is gathered into the boundary current at the western coast (Fig. 5.13f). This flow is partially blocked at the ice front by the step change in water column thickness, so while some of it exits the cavity along the western coast, the remainder is turned eastward along the ice front to merge with the inflowing water and feed back into the sub-ice cyclonic gyre. In this complex picture of circulation and melting beneath a large ice shelf, how might the system respond to changes in ocean temperature? The ‘primitive equation’ model has been run for a range of far-ﬁeld ocean temperatures [18]. This forcing was imposed by restoring the temperature at the northern boundary to a vertical proﬁle consisting of a 150 m thick mixed layer at the surface freezing point, a 50 m thick

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thermocline, and a range of uniform temperatures between 200 m and the seabed (Fig. 5.14). The mean melt rate over the ice-shelf base was found to depend on the far-ﬁeld temperature relative to the freezing point raised to the power 2. The experiments were repeated for various ice-shelf and seabed geometries. The dependence on shelf geometry was qualitatively the same as that found in the plume model. For shorter, steeper ice shelves, the melt rate was higher and a stronger function of the forcing temperature, reflecting the stronger buoyancy forcing on the circulation. The T2 dependence of the melt rate can be understood via similar arguments to those in the plume model analysis. Despite the more complex model used here, entrainment continues to scale with the mixed-layer velocity (Fig. 5.14). Thus equations analogous to (5.20) and (5.21) could be written giving the temperature differences between mixed layer and ice-ocean interface and between mixed layer

Fig. 5.14 Dependence of mixed layer properties averaged over the area beneath the ice shelf on far-ﬁeld ocean temperature relative to the surface freezing point. These results are from Holland et al. [18] and are displayed and ﬁtted as in Fig. 5.10. Note the similarity in behaviour to the plume model results shown in that ﬁgure, with the exception that mixed layer speed has a T1 dependence, giving a T2 dependence in the melt rate

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and ambient ocean as linear functions of far-ﬁeld thermal forcing (Fig. 5.14). Again cooling and freshening of the mixed layer are in direct proportionality, as given in (5.22), so the salinity difference between the mixed layer and ambient ocean increases linearly with ambient thermal driving (Fig. 5.14). Although the equation of state for seawater is non-linear, most of the non-linearity comes from the temperature dependence, while at low temperature most of the density difference comes from the salinity dependence. Thus the density difference between mixed layer and ambient ocean rises approximately linearly with far-ﬁeld thermal forcing. These basic scalings that were derived from the plume model seem to be fairly universal. The difference in response between the plume model and the primitive equation model lies in the relationship between the buoyancy of the plume/mixed layer and its velocity, and the subsequent impact of that velocity on the melt rate through (5.26). In the primitive equation model the large-scale momentum balance is near-geostrophic, so the mixed-layer velocity scales linearly with its buoyancy (Eq. (5.41)). The combination of linear increases in both thermal forcing and velocity yields the observed T2 dependence of the melt rate on far-ﬁeld ocean temperature. While a far-ﬁeld T 2 scaling might thus be anticipated for the melt rate under large ice shelves and has some support in observation [19], at smaller scales, where friction or acceleration becomes important in the momentum balance, it may be expected that velocity will scale with the square root of buoyancy (as in the plume), leading to a far-ﬁeld T 1:5 scaling for the melt rate. Where flow in the ice-ocean boundary layer is controlled primarily by factors other than the melt rate, such as input of buoyancy from subglacial drainage or strong tidal currents, the mixed layer/plume velocity should be independent of temperature, and then the melt rate should be directly proportional to the far-ﬁeld temperature.

Exercises 5:1

Ice shelf ablation in Antarctica The ice shelves of Antarctica have a total area of about 1.5 106 km2. What is the average basal melt rate (hint: total basal melting is *1.5 103 km3 yr−1)? Assuming that the outflows from beneath the ice shelves contain on average about 1% meltwater, what is the total volume transport of these outflows? Taking the mean thickness of the cavities beneath the ice shelves to be 300 m, estimate the time scale for renewal of the waters within them.

5:2

Mass balance of ice shelves Assuming ice to be incompressible, write down an equation describing the conservation of mass within an ice shelf, ignoring the compressible layer of snow and ﬁrn near the surface (hint: refer to Chap. 8 if necessary). Since the upper and lower surfaces of an ice shelf experience no friction, the horizontal components of velocity are independent of depth. Use this fact to write an expression for the difference in vertical velocity between the upper and lower surfaces, ws − wb. Now, use the kinematic boundary conditions at the upper and lower surfaces (see Chap. 1) to derive an expression for the material derivative of ice

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thickness (i.e., the rate at which a marked column of ice changes its thickness as it is advected along by the ice-shelf flow). Can you estimate the typical order of magnitude of each of the terms contributing to the thickness change? Which terms are likely to dominate? And which may be sensitive to climate change? 5:3

The liquidus Three samples of water with salinities, S, of 35.0, 17.5 and 3.5 g kg−1 respectively are cooled to a temperature of −2.0 °C. Assume that all the salt is rejected from any ice that forms, and take the liquidus relation to be Tf ¼ 0:06S þ 0:1. What is the salinity of the remaining liquid in each case, and what fraction of the original water mass has become solid ice?

5:4

Melting ice A vertical-sided container with an area of 100 cm2 is ﬁlled with 1 L of fresh water and maintained at a constant temperature of 4 °C (so the water density is 1 g cm−3). A 100 g block of ice is added. How far does the water level rise up the container? The ice is allowed to melt while the temperature is held constant at 4 °C. Does the level of the water change as the ice melts? The experiment is repeated, but this time 1 L of seawater with a salinity, S, of 35 g kg−1 is put into the container. Assume that the equation of state is linear in salinity and is given by q = 1 + 8 10−4S g cm−3, where S is in g kg−1. How far does the water level rise when the 100 g block of ice is added? As the ice melts, the temperature is held constant and the water is thoroughly mixed. What is the ﬁnal salinity and density of the water? Does the level of the water change as the ice melts?

5:5

Impacts on the ocean The data shown below were recorded at the northern ice front of George VI Ice Shelf in late March 1994.

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On the left-hand plot, can you identify the potential temperature and salinity characteristics associated with inflow to the sub-ice cavity, outflow from the cavity and surface waters (hint: assume that the inflow is uniform)? Can you explain the distribution of properties in the outflow? Using the change in salinity, calculate the maximum meltwater fraction ΔM/(M + ΔM) in the outflow. On the right-hand plot, can you identify the dissolved oxygen characteristics associated with inflow, outflow and surface waters? Also, can you explain the distribution of the dissolved oxygen in the outflow? 5:6

Circulation and melting beneath ice shelves Below are four idealised ice shelf geometries. For given ambient water properties, how would the average melt rates experienced by these ice shelves compare? Can you put them in order?

5:7

Scales of motion Rotation can be important in the force balance whenever the Rossby number (U/XL) is small. Estimate the Rossby number for an ice sheet. Would you expect rotation to be important in the dynamics of the ice sheet? Why not? Estimate the Ekman number (m/XH2) for the ice sheet. What does this tell you about the force balance?

5:8

Geostrophic flow The geostrophic equations qf t ¼

@P ; @x

qfu ¼

@P @y

are deceptively simple, yet have profound implications for the large-scale circulation of atmosphere and ocean. Assume that the ocean is homogeneous (i.e., it has constant density). Using the geostrophic equations, show that for constant f the horizontal flow (u, t) is non-divergent. Is f generally a constant? If f is allowed to vary, show that the ratio f/H, where H is the ocean depth, is conserved by fluid parcels (i.e., that the material derivative of f/H is zero). Hint: assume both the ocean surface and seabed to be material surfaces, so that a speciﬁc kind of kinematic boundary condition (see Chap. 1) holds there. What are the implications of your result for large-scale geostrophic flow in a constant depth ocean?

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Combined effects of rotation and friction It might be easiest to think about this ﬁnal problem if you try a hands-on experiment. Fill a cup or glass with water and use some coffee grounds or tea leaves as flow tracers. Stir the water and observe the motion of the tracers. Estimate the rotation rate (X) of the water and its tangential velocity (u = Xr). What is the centripetal acceleration (a = u2/r) required to maintain the rotational motion and what force generates that acceleration? Assume that the damping of the flow is caused by viscous forces acting over the length-scale of the cup/glass. Estimate a time scale for the spin-down of the motion (the viscosity of water is about 1 10−6 m2 s−1). (Hint: consider the scaling for the horizontal motion in Sect. 5.7; at the scale of the glass, the dominant terms are the acceleration and viscous drag.) Is there a discrepancy between this estimate and your observations of the water motion? Observe where the tracers end up. What trajectory did they follow to get there? Where in the cup was there a non-zero radial velocity? Can you explain the existence of this radial velocity and estimate the thickness of the layer that experiences it? Did you notice anywhere where the tracers were lifted up by a non-zero vertical velocity? What drove this motion? Use these observations to explain any discrepancy between your estimated and observed spin-down times.

References 1. Shepherd A and 46 others (2012) A reconciled estimate of ice-sheet mass balance. Science 338:1183–1189 2. Shepherd A, Wingham D, Rignot E (2004) Warm ocean is eroding West Antarctic Ice Sheet. Geophys Res Lett 31:L23402 3. Joughin I, Smith BE, Holland DM (2010) Sensitivity of 21st century sea level to ocean-induced thinning of Pine Island Glacier, Antarctica. Geophys Res Lett 37:L20502 4. Zwally HJ, Giovinetto MB, Li J, Cornejo HG, Beckley MA, Brenner AC, Saba JL, Yi D (2005) Mass changes of the Greenland and Antarctic ice sheets and shelves and contributions to sea-level rise: 1992–2002. J Glaciol 51:509–527 5. Scambos TA, Hulbe C, Fahnestock MA, Bohlander J (2000) The link between climate warming and breakup of ice shelves in the Antarctic Peninsula. J Glaciol 46:516–530 6. Jenkins A (1999) The impact of melting ice on ocean waters. J Phys Oceanogr 29:2370–2381 7. Jenkins A (1991) A one-dimensional model of ice shelf-ocean interaction. J Geophys Res 96:20671–20677 8. Ellison TH, Turner JS (1959) Turbulent entrainment in stratiﬁed flows. J Fluid Mech 6: 423–448 9. Lane-Serff GF (1995) On meltwater under ice shelves. J Geophys Res 100:6961–6965 10. Jenkins A (2011) Convection-driven melting near the grounding lines of ice shelves and tidewater glaciers. J Phys Oceanogr 41:2279–2294 11. Morton BR, Taylor GI, Turner JS (1956) Turbulent gravitational convection from maintained and instantaneous sources. Proc R Soc Lond A 234:1–23 12. Fricker HA, Scambos T, Bindschadler R, Padman L (2007) An active subglacial water system in West Antarctica mapped from space. Science 315:1544–1548

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13. Stearns LA, Smith BE, Hamilton GS (2008) Increased flow speed on a large East Antarctic outlet glacier caused by subglacial floods. Nat Geosci 1:827–831 14. Echelmeyer K, Harrison WD (1990) Jakobshavn Isbræ, West Greenland: seasonal variations in velocity—or lack thereof. J Glaciol 36:82–88 15. Motyka RJ, Hunter L, Echelmeyer KA, Connor C (2003) Submarine melting at the terminus of a temperate tidewater glacier, LeConte Glacier, Alaska, U.S.A. Ann Glaciol 36:57–65 16. Cushman-Roisin B, Beckers J-M (2011) Introduction to geophysical fluid dynamics: physical and numerical aspects, 2nd edn. Elsevier, Amsterdam 17. Holland DM, Jenkins A (2001) Adaptation of an isopycnic coordinate ocean model for the study of circulation beneath ice shelves. Mon Weather Rev 129:1905–1927 18. Holland PR, Jenkins A, Holland DM (2008) The response of ice shelf basal melting to variations in ocean temperature. J Clim 21:2558–2572 19. Jenkins A, Shoosmith D, Dutrieux P, Jacobs S, Kim TW, Lee SH, Ha HK, Stammerjohn S (2018) West Antarctic Ice Sheet retreat in the Amundsen Sea driven by decadal oceanic variability. Nat Geosci 11:733–738

6

Polar Meteorology Carleen Reijmer, Michiel van den Broeke, and Willem Jan van de Berg

6.1

Introduction

Inhomogeneous heating of the Earth’s surface by solar radiation in space and time introduces horizontal and vertical temperature and thus density gradients. These density gradients lead to pressure gradients that cause air flow, which in turn influences temperature and pressure, resulting in a complex system of forcing and feedbacks that determine the state of the atmosphere. The polar atmosphere differs from the mid-latitude and tropical atmosphere in that it endures the most extreme seasonal insolation variations on Earth, with an absence of solar radiation in winter (Polar Night) to 24 hours of daylight in summer (Polar Day). In terms of surface characteristics (land/sea distribution, elevation, surface roughness, vegetation, etc.), the Arctic and Antarctic regions are very different: the Arctic is an ocean surrounded by continents, the Antarctic a continent surrounded by oceans (Fig. 6.1). In the Arctic region, poleward of 70◦ N, only 30% of the surface is land, while in the Antarctic region the corresponding amount is 72%. The Antarctic continent is almost completely (∼ 99%) covered by an ice sheet, resulting in it being the highest continent on Earth with an average elevation of ∼ 2100 m above sea level (a.s.l.). While most of the sea ice in the Antarctic region is relatively thin single-year ice, the Arctic Ocean is partly covered by thicker multi-year sea ice that has survived at least one summer. In this chapter, a brief introduction to atmospheric radiative theory is given, followed by a discussion of the general characteristics of the polar atmosphere in terms of the (surface) energy budget, large-scale flow patterns and related features such as precipitation, the near surface temperature inversion and katabatic flow. The

C. Reijmer (B) · M. van den Broeke · W. J. van de Berg Utrecht University, Utrecht, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_6

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Fig. 6.1 Maps of the Arctic and Antarctic regions

characteristic features of the polar atmosphere will be highlighted and the differences between the Arctic and Antarctic regions discussed.

6.2

Shortwave and Longwave Radiation

Each body with a temperature above absolute zero, including the Sun and Earth, emits and absorbs electromagnetic radiation. The intensity of emitted radiation is, to first order, described by the laws for black body radiation, which refer to a hypothetical body that absorbs all radiation and has maximum emissivity in all directions. The intensity of black body radiation emitted from a surface as a function of wavelength λ is given by the Planck function Fλ (T ) =

2π hc2 , λ5 ehc/kλT − 1

(6.1)

where Fλ (T ) (W m−3 ) is the monochromatic flux density (per unit wavelength) or monochromatic irradiance1 as a function of emission temperature T . In this expression, h is Planck’s constant, k is Boltzmann’s constant and c is the speed of light. (Values of these and other constants are given in Table 6.2 at the end of this chapter.)

1 The

Planck function at a point gives the radiative intensity per unit solid angle as a function of direction: the factor of π in (6.1) arises at a surface when this is integrated over a half-sphere of directions away from the surface. Thus the emitted intensity at a point as a function of direction is Fλ /π .

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By integrating (6.1) over the wavelength domain, the total flux density or total irradiance F(T ) (W m−2 ) is obtained ([1], p. 123):

∞

F(T ) = 0

Fλ (T ) dλ =

∞ 0

2π hc2

dλ = σ T 4 . λ5 ehc/kλT − 1

(6.2)

This is the Stefan–Boltzmann law, where σ is the Stefan–Boltzmann constant. It states that the radiation flux emitted by a black body is proportional to the fourth power of the absolute temperature in Kelvin. A body that is in thermodynamic equilibrium with its surroundings emits all radiation that is absorbed, i. e., its emissivity ε equals its absorptivity. For a black body, emissivity and absorptivity equal 1. In general this does not hold for natural surfaces (grey bodies), where ε < 1 (also note that ε depends on wavelength). For most natural surfaces, the broadband (wavelengthintegrated) value of ε ranges between 0.9 and 1. The general expression for emitted radiative flux for grey bodies then becomes F(T ) = εσ T 4 .

(6.3)

In the atmosphere we observe a mix of solar and terrestrial radiation. The Sun emits radiation representative of a temperature of about 5,780 K, compared to a radiative temperature of ∼ 255 K for the Earth-atmosphere system. Owing to the proportionality to the fourth power of absolute temperature of the emitting source, both radiation types are well separated in the wavelength domain (Fig. 6.2) and only have a small overlap at about 5 microns. It is therefore common to distinguish between solar or shortwave radiation (Shw) and terrestrial or longwave radiation (Lw). In discussing radiative fluxes in the climate system it is furthermore common to integrate the radiative flux density at a point, Fλ /π , over either the upward- or downward-facing directional hemisphere in order to distinguish between downward (incoming) and upward (outgoing/reflected) radiation (Shwin , Shwout , Lwin , Lwout ).

6.3

Radiation Climate at the Top of the Atmosphere

The average instantaneous energy flux received from the Sun at the top of the atmosphere (ToA) is S0 ∼1366 W m−2 , which is proportional to the area under the curve in Fig. 6.2. S0 is the solar constant, which in reality shows small (< 1%) fluctuations due to variation of the the distance between Earth and Sun, and to vagaries of solar physics, notably the sunspot cycle. The total energy intercepted by the Earth, when distributed evenly over the globe and averaged over the year, is S0 /4 ∼ 341 W m−2 . S0 /4, the insolation at ToA, is the average rate at which the Earth and its atmosphere receive solar radiation, and emit radiation back to space, assuming the Earth-atmosphere system to be in radiative equilibrium with the Sun. The annual global average energy balance of the Earth-atmosphere system is presented in Fig. 6.3. Three levels can be distinguished: the top of the atmosphere (ToA), the atmosphere itself (represented as a single layer) and the surface of the

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Fig. 6.2 Black body spectra representative of the (incoming) radiation temperature of the Sun (∼ 5780 K) at Top of the Atmosphere (ToA) and the (outgoing) radiation temperature of the Earth’s atmosphere (∼ 255 K). Wavelength (λ) is plotted on a (base 10) logarithmic scale to enable presentation of both curves in one plot. The irradiance Fλ is multiplied by λ in order to make the area below the curves proportional to the wavelength integrated irradiance F(T ). The total received solar radiation at ToA is 1366 W m−2 . The area under the Earth curve is smaller than that under the Sun curve by a factor 41 (1 − α) ≈ 0.17, where α is the albedo (the fraction of reflected incoming radiation), and the factor of 41 is due to the receipt of incoming radiation over an area π R 2E , but emission over an area 4π R 2E , where R E is the Earth’s mean radius. See also Exercise 6.2

Fig. 6.3 The energy balance of the Earth-atmosphere system, globally and annually averaged. Diagram adapted from (and its numbers based on) Trenberth et al. [2]

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Fig. 6.4 The annually averaged radiation balance at the top of the atmosphere from satellite observations [3]

Earth. For instance, the figure shows that there is radiation balance at the top of the atmosphere but not at the other two levels (why not?). Figure 6.3 shows that the Earth-atmosphere system is not absorbing all energy provided by the Sun; part of it is reflected back to space, mainly by clouds, with smaller contributions from the surface and aerosols in the troposphere (the lowest 10–15 km of the atmosphere). Note that, like emissivity, reflectivity is wavelength dependent. When integrated over the shortwave spectrum, the broadband reflectivity or broadband albedo for shortwave radiation is defined as: α=

Shwout . Shwin

(6.4)

When evaluated at ToA α is called the planetary albedo α p ; when evaluated at the surface, α is called the surface albedo αs . Note that surface albedo is an important component of planetary albedo. Averaged over the globe, α p ∼ 0.31 and αs ∼ 0.17, as estimated from satellite observations (see also Fig. 6.3). The solar constant S0 represents the upper bound of shortwave radiation to be received at the Earth’s surface, if solar incidence is normal and no atmosphere would be present. In the present Sun-Earth configuration, annual average ToA incoming solar radiation is a function of latitude, while instantaneous ToA incoming solar radiation is also a function of time of day (or longitude) and time of year. The zonal distribution of annual average radiation balance at ToA is illustrated in Fig. 6.4. The figure shows that the atmosphere experiences approximate radiative balance only at about 30◦ N and 30◦ S. In the tropics (30◦ S to 30◦ N), absorbed shortwave radiation exceeds the lost longwave radiation, i. e., there is a radiation surplus, while there is a radiation deficit poleward of these latitudes. The geographic poles receive about 40% of the shortwave radiation received at the equator. The flattening of the absorbed shortwave radiation curve is due to high average cloud cover in

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the equatorial region (the Inter-Tropical Convergence Zone, ITCZ) resulting in less absorption of shortwave radiation. The geographical differences between the Arctic and Antarctic regions become apparent in the higher values of lost longwave radiation in the Arctic region caused by relatively high temperatures due to lower surface elevation compared to the Antarctic region. The strong gradient in net absorbed radiation in Fig. 6.4 is balanced by horizontal oceanic and atmospheric equator-pole heat transport.

6.4

Large Scale Circulation

In response to the differential heating/cooling of the atmosphere, there is a negative equator-to-pole tropospheric temperature gradient. Since we expect the thickness of an atmospheric layer to be proportional to its temperature2 , this temperature gradient results in a gradient in the height of atmospheric pressure levels. Figure 6.5 shows this gradient for the 500 hPa level. Away from the surface, the resulting pressure gradient balances the Coriolis force, resulting in a geostrophic westerly flow on both hemispheres. The strongest westerlies are found in the winter at mid-latitudes when the meridional temperature gradients are largest. Also in the large scale circulation the contrast between the Arctic and Antarctic regions becomes apparent. These differences mainly result from the different land/sea distribution in both hemispheres. Since the static and dynamic (mixing) heat capacity of the ocean is much larger than that of land surfaces, radiative cooling is strongest over the continents. The resulting local steepening of the temperature gradient over the Northern Hemisphere (NH) continents introduces asymmetries in the zonal temperature distribution. This is reflected in the larger gradient of the 500 hPa height over the NH continents (Fig. 6.5) which is strongest in winter near the eastern edges of the continents where the horizontal temperature gradients are largest. A consequence of this local steepening is a displacement of the circumpolar flow out of zonal symmetry. These quasi-stationary structures are called stationary planetary waves, and are preferred regions for cyclone development. These asymmetries largely disappear in summer, when land/sea temperature gradients are more modest. In the Southern Hemisphere (SH) the pattern is less disturbed by the presence of continents. There is a slight asymmetry caused by the off-pole location of the Antarctic continent. The zonal structure does not change much through the year, but the meridional temperature gradient is strongest in winter. Because of the high elevation of the Antarctic Ice Sheet, temperatures throughout the troposphere are influenced by the cold surface. As a result, the meridional temperature gradient in the SH is larger than in the NH. As a rule of thumb, it is fair to say that the summer gradient in the 500 hPa surface in the SH is comparable to the winter gradient in the NH. Note that the apparent zonal symmetry is misleading, as cyclones are

2 Because

of the perfect gas law.

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Fig. 6.5 Average geopotential height of the 500 hPa level for December, January, February (DJF) (top panel) and June, July, August (JJA) (bottom panel) (based on ECHAM4 data)

continuously developing, more frequently at the eastern edges of the continents, after which they migrate south-eastwards towards the Antarctic continent. At the surface level, climate differences between the Arctic and Antarctic regions are also evident. In the NH the distribution of the storm tracks is dominated by the land/sea distribution, resulting in the climatological Icelandic Low over the North Atlantic ocean and Aleutian Low in the North Pacific. Over the North Polar basin, in the absence of strong temperature gradients, atmospheric pressure gradients are relatively weak in both seasons. In winter (Fig. 6.6, left), the Icelandic Low creates a pressure gradient across the Arctic basin, resulting in mean eastward flow from Asia towards Greenland. It forces a transpolar current from Asia to America over the North Pole, which is why the thickest sea ice is found on the north coast of the Canadian Arctic and Greenland. In the absence of large land masses in the region between Antarctica and south of 40◦ S, a continuous circumpolar pressure trough (CPT) develops in the SH (Fig. 6.6, right). Low pressure systems cannot penetrate the high Antarctic interior and decay

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Fig. 6.6 Mean Arctic and Antarctic winter (DJF and JJA, respectively) sea level pressure and 10 m wind vectors (longest vector corresponds to 10 m s−1 ) (based on ECHAM4 data)

close to the Antarctic coastline, resulting in the climatological CPT. Preferred locations for low pressure areas to end up at the Antarctic coastline are found around 30◦ E, 110◦ E and 150◦ W; these are probably associated with indentations in the Antarctic topography. The westerly circulation is well established near the surface, with an annual mean wind speed of about 10 m s−1 at the 10 m level (Fig. 6.6, right). An interesting implication of the CPT is that the surface winds as well as the coastal currents are easterly along the Antarctic coastline. Over the Greenland and Antarctic Ice Sheets a well-developed anticyclonic circulation is present near the surface, which is stronger in winter than in summer and peaks over the steepest slopes. The associated katabatic winds will be discussed in Sect. 6.6.

6.5

Surface Energy Balance

Averaged over the year and globe there is radiation balance at the ToA, but not so in the atmosphere or at the surface (Fig. 6.3). About 55% of the incoming shortwave radiation at the ToA reaches the surface, where most of it (∼ 85%) is absorbed. The absorbed shortwave radiation heats the surface and in turn the surface emits longwave radiation to the atmosphere. Most of the emitted longwave radiation is absorbed in the atmosphere, which in turn emits longwave radiation to space and back towards the surface. As a result of this greenhouse effect, the net longwave loss at the surface is relatively small and does not balance the absorbed short wave radiation. The surface (and atmospheric) energy balance is closed by the surface turbulent fluxes of sensible and latent heat. If we consider the energy balance of a surface ‘skin’ layer, conservation of energy dictates that all fluxes must balance, because the skin layer is infinitesimally thin with zero heat capacity. For a single location and in case of a snow/ice covered surface

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the surface energy balance can be written as the sum of melt energy (M), shortwave radiative fluxes (Shw), longwave radiative fluxes (Lw), turbulent fluxes of sensible (H ) and latent heat (L E) and the sub-surface heat flux (G s ) (all in W m−2 ): M= = = =

Shwin + Shwout + Lwin + Lwout + H + L E + G s Shwin (1 − αs ) + Lwin + Lwout + H + L E + G s Shwnet + Lwnet + H + L E + G s Rnet + H + L E + G s .

(6.5)

Since M is the energy used for melt, M = 0 when the surface temperature Ts < 273.15 K. Shwin , Shwout and Shwnet are the incoming, reflected and net (absorbed) shortwave radiation, αs is the surface albedo, Lwin , Lwout and Lwnet are the incoming, outgoing and net longwave radiation, Rnet is the (combined short- and long-wave) net radiation. All fluxes are evaluated at the surface, and fluxes towards the surface are defined positive. Equation (6.5) is equivalent to (2.3) of Chap. 2.

6.5.1

Shortwave Radiation

The global distribution of the net (absorbed) surface shortwave radiation exhibits a zonally rather symmetric distribution (Fig. 6.7a), which is the result of the change in solar incident angle towards the poles. However, the relative decrease of the net shortwave radiation received at the surface near the poles is stronger than that received at ToA, and this is caused by the high surface albedo in the polar regions. While at ToA the polar regions receive about 40% of the shortwave radiation compared to the equator, at the surface this fraction reduces to 20%. The atmosphere over the polar regions is in general clean and dry, because there are no significant sources of dust or pollution nearby, and the low temperatures induce low moisture content. Over the large ice sheets the atmosphere is also reasonably thin and the cloud cover low. As a result, 70–90% of the insolation at the top of the atmosphere reaches the surface. Because of the low cloudiness and high transmissivity of the polar atmosphere, the surface albedo becomes the main component of the planetary albedo.

6.5.2

Surface Albedo

The surface albedo is an extremely important component of the surface energy balance, because it determines how much solar radiation can be used to heat the surface, from where it affects all other energy balance components (which all depend on surface temperature). The albedo of snow and ice is generally high, ranging from 0.4 for ice to 0.9 for clean dry fresh snow (Table 6.1). The surface albedo is high over dry snow surfaces such as found in the interior of Greenland and Antarctica, about 0.85, and lower in warmer areas where melt occurs, such as the margins of the Greenland Ice Sheet, where it is 0.4–0.5 (Fig. 6.8). The snow and ice albedoes depend on several

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Fig. 6.7 Global annual surface radiation. a Net incoming short wave radiation, b net incoming (thus negative) long wave radiation, and c net incoming all wave radiation based on satellite observations [3]

different factors: solar zenith angle, cloud cover, snow grain size, and impurity/soot content. Especially the latter two result in a positive feedback on melting. The effective grain size of freshly fallen snow is about 0.05 mm, which will increase in time due to variations in temperature to values of 1–5 mm for old snow. In general, the larger the grain size, the lower the albedo, therefore the albedo decreases in time due to ageing of the snow, resulting in more absorption of shortwave radiation. This constitutes a strong positive feedback on snow melting, because the albedo tends to decrease even faster once melting has occurred. A similar feedback mechanism

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Table 6.1 Typical albedo values for different surfaces Surface

Albedo Min.

Max.

Fresh snow Old snow Firn Clean glacier ice Dirty glacier ice Ocean water Bare rock Grassland

0.80 0.70 0.43 0.34 0.15 0.03 0.15 0.16

0.90 0.80 0.69 0.51 0.25 0.25 0.30 0.20

Fig. 6.8 Surface albedo observations on a site S6 in the western Greenland ablation area (1100 m a.s.l.) and b site AWS 6 in Dronning Maud Land, Antarctica, based on daily averages of shortwave incoming and reflected radiation [4,5]

occurs when melting of snow exposes bare ice, since bare ice has on average a lower albedo than snow. The presence of impurities, soot, and/or debris cover lowers the albedo even further and increases melt, unless the debris cover is thick enough to completely insulate the surface from the effect of solar radiation. Figure 6.8a illustrates these processes for a site on the ablation zone of the Greenland Ice Sheet, where no debris cover is present. On the cold Antarctic plateau the variations in albedo are mainly the result of ageing of the snow and the effect of changing solar zenith angle. The solar zenith angle determines how far the radiation penetraties the surface; the larger the zenith angle, the less penetration and the higher the albedo. As a result, the albedo exhibits

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both diurnal (not shown) and annual cycles, with lowest values at mid-day and midsummer when solar zenith angle is lowest. The seasonal effect is clearly visible in Fig. 6.8b, which presents albedo observations for a site at about 1100 m a.s.l. in Dronning Maud Land, Antarctica.

6.5.3

Longwave Radiation

According to the Stefan-Boltzmann law for grey surfaces (6.3), the longwave radiation balance is determined by the temperature of the emitting surfaces and its emissivity. Thus, the contribution of the longwave radiation to the surface energy balance is given by Lwnet = Lwin + Lwout = εa σ Ta4 − εs σ Ts4 ,

(6.6)

where the subscript a denotes atmosphere and s denotes the surface. The emissivity of a snow surface εs is typically 0.96–1.0. The downward longwave radiative flux results from emission by the atmosphere, including clouds. Low clouds radiate as black bodies at their (high) cloud base temperature. The clear sky downward longwave flux is determined by the atmospheric temperature profile and the presence of gases (e. g., water vapour and CO2 ), which absorb and emit longwave radiation. When expressing the clear sky longwave radiation in terms of the air temperature near the surface, which is readily measured, one can assume that the emissivity of a clear polar atmosphere is about 0.6–0.7. The global distribution of net longwave radiation at the surface is fairly homogeneous compared to the distribution of absorbed shortwave radiation (Fig. 6.7b). Net longwave radiation is strongly negative in areas where cloudiness is low, which limits the downward longwave radiation flux, and where surface temperature is high, which enhances the upward flux of longwave radiation. These conditions are met in desert regions. In the polar regions, cloud cover is also low but since surface temperature is low as well (most of the shortwave radiation is reflected instead of being absorbed), the net longwave loss at the surface is modest. However, the annual average Lwnet loss in some parts of the polar regions is sufficiently large to result in a net radiation loss, e. g., over large parts of the interior Greenland and Antarctic Ice Sheets (R = Shwnet + Lwnet < 0, Fig. 6.7c). This negative net radiation is remarkable, and greatly influences the near surface climate; see the following sections.

6.5.4

Turbulent Fluxes

In the polar regions the net radiative flux is zero to negative in winter, and over the large ice sheets of Greenland and Antarctica it is zero to negative averaged over the year. Since the heat flux from the snowpack to the surface is on an annual basis small, there must be a quasi-continuous heat source from the atmosphere to the surface to balance the negative net radiative flux. The existence of this heat source is also evident from the fact that in the winter months, in the absence of significant

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incoming shortwave radiation, near surface temperature does not show a continuous decrease but rather reaches a quasi-steady state. The heat source resupplying the heat lost through the negative surface radiative balance must be in the form of turbulent transport of sensible and/or latent heat. Fluid motions are turbulent when they are disordered, and are typically associated with irregular, eddying motion. This is commonly experienced on a windy day, when gusts of wind represent the effects of turbulence. But even on a calm day, the flow is turbulent, when seen on a sufficiently large scale. Turbulence occurs in fluid motions when the Reynolds number (see (1.31)), a dimensionless measure of the velocity, is sufficiently large. In convective motions driven purely by buoyancy (such as commonly observed atmospheric thermals), the convective velocities are controlled by the amount of heating of the ground or ocean surface due to absorption of solar radiation, which causes unstable density differences to occur. In turbulent flows, the vertical flux of a quantity c is given by the expression w c , where the primes denote the turbulent fluctuations about the mean value, w presents vertical wind velocity, and the overbar denotes the average, for example a local time average. We are interested in assessing the vertical transport of sensible heat and latent heat. To do this, we first define the potential temperature θ ; this is the temperature an air parcel would have when brought adiabatically (i. e., without heat exchange with its environment) from a location with pressure p and temperature T to a reference pressure p0 . Potential temperature is defined as θ=T

p0 p

Rd /c p

,

(6.7)

where Rd is the gas constant for dry air and c p the specific heat capacity of air at constant pressure. In an adiabatic atmosphere (which approximately represents the state of the troposphere), the potential temperature is constant; therefore, vertical heat fluxes are associated with fluctuations about this constant adiabatic state. In a moist atmosphere, the latent heat is proportional to the water vapour content, measured as the specific humidity q, which is the mass fraction of water vapour in the air. Vertical upward fluxes of sensible heat H and latent heat L E at the Earth’s surface are thus defined by

H = ρc p w θ , s

L E = ρ L v,s w q , s

(6.8)

where the subscript s on the fluxes refers to the surface, ρ is air density, w , θ and q are the turbulent fluctuations of vertical wind, potential temperature and specific humidity, and L v,s is the latent heat of vapourisation (v) or sublimation (s). The turbulent fluxes can be estimated using so-called first-order closure, which defines turbulent diffusion coefficients (eddy diffusivities) to relate the turbulent fluxes to the local mean gradient of potential temperature and specific moisture. The turbulent fluxes are then written as

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Fig. 6.9 Saturated specific humidity qs over water (red, left scale), over ice (blue, left scale), and the difference between them (green, right scale) as a function of temperature

H ≈ −ρc p K h

∂θ , ∂z

L E ≈ −ρ L v,s K q

∂q , ∂z

(6.9)

where K h,q is the turbulent diffusion coefficient for heat (h) or moisture (q). Note that K h,q is not a property of the fluid but of the flow. These equations show that the fluxes can be calculated using the (measured) vertical gradients of temperature and moisture content. Figure 6.3 shows that, when averaged over the globe and year, both turbulent fluxes provide an energy loss for the surface. In the polar regions the opposite must be true: here, the turbulent heat fluxes must act as a source of energy to balance the surface radiation deficit. According to (6.9), this implies that the near surface potential temperature and specific moisture content must be higher than at the surface. Because of the low temperatures, the specific moisture and hence its gradients are small in the polar regions (Fig. 6.9). As a result, the latent heat flux is on average small. It is mainly the turbulent flux of sensible heat that balances the radiative heat loss (Fig. 6.10). The only way this can be achieved is when the atmosphere is warmer than the surface, i. e., through the existence of a quasi-permanent surface temperature inversion.

6.6

Temperature Inversion and Katabatic Winds

6.6.1

Surface Temperature Inversion and Deficit

A temperature inversion is defined as an increase in temperature with altitude, which constitutes a reversal of the general decrease in temperature with height in the troposphere. The existence of a surface-based temperature inversion over a glacier or ice sheet can be the result of two different processes: (1) a radiation deficit at the surface; (2) warm air overlying a melting snow/ice surface. As demonstrated in the previous section, the annual average net incoming radiation over polar glaciers and ice sheets is zero or negative. When there is significant wind shear, this heat loss is balanced by the turbulent transport of sensible heat from

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Fig. 6.10 Example of time series of the surface energy balance (based on monthly means) derived from automatic weather station observation at site AWS 6 in Dronning Maud Land, Antarctica [4]. Rnet is the net (combined short- and long-wave) radiation, H and L E the sensible and latent heat respectively, and G s the sub-surface heat flux

the air to the surface, which cools the air and heats the surface. In case of low or zero wind speed, the generation of turbulence is suppressed so that (during winter) there are only the subsurface heat flux and incoming longwave radiation from the atmosphere to act as heat sources. Because the surface radiates as a black body, and the (cloudless) sky radiates as a grey body with typical emissivities of 0.6–0.7, the surface temperature must be considerably lower than the near surface air temperature in order to balance the incoming radiation. Note that the temperature inversion forming due to a radiation deficit at the surface is not limited to the polar regions but is also observed during clear nights at lower latitudes. The polar temperature inversion is, however, more persistent and stronger. The persistent temperature inversion will weaken or disappear completely due to heating of the surface by solar radiation. Other processes reducing the inversion strength are mixing through shear turbulence (i. e., increasing wind speed) and cloud formation, which increases the atmospheric emissivity. The second process resulting in a surface based temperature inversion occurs when warm air (temperature > 0 ◦ C) overlies a melting snow or ice surface of which the surface temperature is restricted to 0 ◦ C. This type of temperature inversion persists throughout the day and summer whenever the surface is melting, and can be found over glaciers, the margins of the Greenland Ice Sheet, and in the Antarctic Peninsula.

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Fig. 6.11 Six years of winter (July) temperature observations at two Antarctic stations, Vostok and Mawson

Figure 6.11 presents an example of observed winter (July) temperature profiles for two stations on the Antarctic Ice Sheet, Vostok and Mawson. Vostok is an inland station on the Antarctic plateau, while Mawson is a coastal station. The figure shows that in the free atmosphere the temperature only slightly decreases from the coast towards the interior of Antarctica. The strong decrease of surface temperature from the coast towards the high plateau is therefore almost entirely determined by the stronger surface temperature inversion. The stronger inversion at Vostok is due to the lower wind speeds and cloud cover at that location. Mawson is located at the foot of the ice sheet and experiences more clouds and strong winds that effectively mix the air near the surface and prohibit the formation of a strong temperature inversion. A common measure of the strength of the surface based temperature inversion ( Tinv ) is the difference between the (near) surface temperature and the highest temperature observed in the lower atmosphere (Fig. 6.12a). However, high resolution vertical profiles are rare, and atmospheric models usually do not have sufficient vertical resolution for this method. An alternative is to extrapolate the background (potential) temperature profile to the surface and calculate the difference with the actual surface (potential) temperature (Fig. 6.12). The resulting quantity is defined as the surface (potential) temperature deficit Ts ( θs ), which is strictly seen to be not equal to Tinv . Figure 6.13 illustrates the resulting temperature deficit over the polar regions in winter. Over the north polar basin the wintertime temperature deficit is 15–20 K, over thick sea ice that effectively insulates the atmosphere from the warm ocean.

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Fig. 6.12 Schematic illustration of the surface temperature inversion Tinv and surface (potential) temperature deficit Ts ( θs ). Ts (θs ) denotes the surface (potential) temperature, h s the surface height and h inv the height of the inversion layer. γT (γθ ) denotes the (potential) temperature equilibrium gradient (lapse rate) in the free atmosphere. The (potential) temperature deficit at the surface can be determined by extrapolating the (potential) temperature profile in the free atmosphere to the surface (T0 and θ0 )

Further to the south, the temperature inversion disappears and the deficit becomes negative in areas where cold continental air is advected over warm ocean water. This mainly occurs east of the continents due to the average westerly circulation. The sea ice marks the edge of the surface inversion, which accounts for a large part of the horizontal temperature gradients. Over the Antarctic region a significant part of the variation in the temperature at the surface is due to variations in the inversion strength. Strong inversions are found over the large ice shelves and the high plateau. Due to increased wind speeds, the temperature inversion strength is directly related to the surface slope, i. e., the presence of strong katabatic winds. The temperature inversion on the Antarctic plateau is

Fig. 6.13 Multi-year average potential temperature deficit at the surface in winter for the north (left) and south (right) polar regions (based on ECHAM4 data)

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Fig. 6.14 Annual cycle of near-surface temperature (left) and wind speed (right) at a series of weather stations in Adélie Land, Antarctica [6]. Dome C is situated on a dome and Halley on an ice shelf, where there is no or only a very small surface slope

much stronger than over the Greenland Ice Sheet, because the latter experiences regular warm air advection events. In summer, when the radiation deficit becomes small or vanishes, the temperature inversion is (partially) destroyed. The strength of the surface temperature inversion in winter therefore explains (part) of the differences in amplitude of the annual temperature cycle, as seen in Fig. 6.14a.

6.6.2

Katabatic Winds

Katabatic winds are forced by gravity that acts on cold dense air that is formed by the quasi-continuous sensible heat transport to the surface. For this reason, katabatic winds always occur in combination with a surface temperature deficit. Another prerequisite for well developed gravity winds is that the surface has a significant slope. This is the case for valley glaciers and the margins of ice sheets and ice caps. Because the forcing is largest at the surface, where the wind speed is by definition zero, the wind speed attains a maximum at some height above the surface. We distinguish two types of katabatic flows based on the forcing mechanism of the temperature deficit at the surface: (1) forced by a radiation deficit at the surface, and (2) forced when warm air overlies a melting snow/ice surface. The first type requires that the amount of solar insolation must be limited, and is therefore especially well developed during the long polar nights over the large ice sheets. Due to the large time scale involved, the Coriolis force is in this case an important term in the momentum balance, deflecting the wind in the cross slope direction, towards right in the NH, towards left in the SH. The second type requires the presence of air with a temperature > 0 ◦ C over a melting ice surface. This type therefore persists throughout the day and summer whenever the ice/snow surface is at the melting point. It has a smaller time scale in which to develop, which makes the Coriolis force less important in the momen-

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Fig. 6.15 Schematic illustration of the two-layer Ball model

tum balance and the wind direction will be directed more downslope. This type is sometimes referred to as glacier wind. The general features of katabatic winds can be illustrated with the Ball model [7], in which the strength and the direction of the flow are directly linked to the strength of the surface temperature deficit and the surface slope. The Ball model is a two-layer model (see Fig. 6.15) in which the upper layer represents the free atmosphere with a potential temperature θ and large scale pressure gradient (expressed by geostrophic flow strength, ( f u g , f vg )). The lower layer is the katabatic layer with a potential temperature θ − θ , where θ represents the potential temperature deficit of the lower layer. In addition, h is the thickness of the katabatic layer, α is the surface slope, and u is the down-slope and v the cross-slope wind. Due to the choice of the Cartesian coordinate system with the x-axis oriented downslope, a thermodynamic forcing term F is introduced in the downslope component of the momentum equation:

θ sin α; (6.10) θ this term represents the katabatic forcing. Assuming friction to be proportional to the square of the velocity, stationarity (∂u/∂t = ∂v/∂t = 0), negligible large scale forcing (u g = vg = 0), and homogeneity in the horizontal, the momentum equations reduce to F=g

k

θ Vu + g sin α, h θ k 0 = − f u − V v, h

0 = fv−

(6.11)

where the first terms represent the Coriolis acceleration, and the second terms rep√ resent simple friction, where V = u 2 + v 2 is the absolute velocity and k is the drag coefficient. If we assume that u and v can be expressed in terms of the absolute velocity V and the angular deviation from the fall line φ, thus u = V cos φ and v = V sin φ, we can solve this set of equations:

1/2 F 4 + 4k 2 F 2 − f2 , V = 2k 2 k V cos φ = 1/2 . f 2 + k 2 V 2 2

(6.12)

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Fig. 6.16 Results of the Ball model as a function of slope angle and strength of the deficit, with u g = vg = 0, θ = 255 K, k = 0.005, h = 100 m

Here, k = k/h and F = g( θ/θ ) sin α. Figure 6.16 presents this solution for different values of θ . The figure shows that wind speed increases with increasing slope and strength of the deficit. When wind speed increases, the effect of friction increases as does the effect of the Coriolis turning. The latter deflects the wind from the fall line while friction turns it towards the fall line. Since friction dominates the Coriolis effect, the deviation from the fall line decreases with increasing wind speed. When applying this model on the Antarctic Ice Sheet with a prescribed θ , the resulting flow lines correspond reasonably well with the observed flow patterns. A useful tool to detect persistent katabatic winds is the directional constancy (dc), which is defined as the ratio of vector mean wind speed over the mean absolute wind speed:

Fig. 6.17 Directional constancy ranging from 0.35 (blue) to 0.99 (red), and 10 m wind vectors, where the longest vectors represent 15 m s−1 wind speed (based on ECHAM4 data)

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dc =

(u 2 + v 2 )1/2 (u 2 + v 2 )1/2

,

(6.13)

where u and v are the zonal and meridional wind components, respectively. Note that dc varies between 0 and 1, where a value of 1 denotes wind from a single well-defined direction. High values of dc often express the dominance of one forcing mechanism over other mechanisms. Figure 6.17 shows the global distribution of the mean wind vector and magnitude of dc. The important wind systems are clearly visible, i. e., the trade winds on either side of the equator, the westerlies in the mid-latitudes and the well developed katabatic winds over the Greenland and Antarctic Ice Sheets, with its clear anti-cyclonic rotation due to the Coriolis deflection. Katabatic winds exhibit temporal variations related to variations in the surface temperature deficit, as illustrated with the Ball model. Over the large ice sheets the temperature inversion has a strong annual cycle and nearly disappears in summer. Figure 6.14b shows for Antarctica that this is indeed the case, although the figure also shows that other processes are active. At the inland site Dome C, the surface is approximately flat and no katabatic forcing is present: the winds are forced by the large scale pressure gradient. Because the large scale pressure gradients are weak over the continent (see Fig. 6.5), the winds are also weak near the surface. At Halley, the other station without surface slope and thus katabatic forcing, the wind speed is determined by the coastal large scale pressure gradient, forcing easterly winds. On the slopes of the ice cap (sites D-10, D-47 and D-80), katabatic forcing dominates the wind, although large scale forcing is approximately equally important in summer when the surface temperature inversion becomes weak. Note that the wind speed does not increase indefinitely when slope increases: at D-10 the slope is the largest but wind speed has decreased compared to D-47. This is due to the significant thickening of the boundary layer towards the coast where the shallow katabatic layer must merge with the marine boundary layer over the ocean. An interesting feature is observed in the ablation zone of the Greenland Ice Sheet, where both katabatic and glacier winds occur. In winter the situation is comparable to Antarctica and strong katabatic winds are forced through a radiation deficit at the surface. In summer, when the surface melts and cannot raise its temperature, persistent glacier winds are forced. As a result the wind speed and directional constancy peak twice each year.

6.7

Precipitation

The surface mass balance (SMB) of a glacier is defined as the sum of all processes adding mass to the surface (accumulation) minus all processes removing mass (ablation): SMB = (6.14) (SN + RF − SUs − SUds − ERds − RU) dt. 1 yr

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Fig. 6.18 Annual mean precipitable water vapour (PW) [(mm, see (6.15)] (based on ECHAM4 data)

Accumulation is often defined as snowfall (SN) minus sublimation (SU). Removal of mass occurs by means of surface sublimation (SUs ), sublimation of drifting snow (SUds ), erosion by drifting snow (ERds ), and melt and subsequent runoff (RU). RF represents refreezing of rainfall. The surface mass balance is usually expressed in metres water equivalent per year (m w.e. y−1 ). The SMB is discussed in Chap. 7; here we will focus on precipitation, the most important positive component of the SMB. Snowfall constitutes the largest contribution to the accumulation over a glacier and ice sheet. Other precipitation components that contribute positively to the surface mass balance are hail and undercooled water droplets that freeze upon impact with the surface. The maximum possible amount of precipitation depends on the total amount of moisture in the atmosphere, which strongly depends on temperature as shown in Fig. 6.9. As a result nearly all water vapour is concentrated in the lowest kilometres of the atmosphere. Furthermore, due to the low temperatures in the polar regions the moisture content there is extremely low. The lack of moisture in the atmosphere over the large ice sheets becomes evident when looking at the annual mean of the total precipitable water PW, i. e., the column abundance of moisture (both liquid and vapour): PW = 0

∞

ρv dz =

0

∞

1 ρq dz ≈ g

ps

q dp,

(6.15)

0

where ρv is water vapour density (in kg m−3 ), q = ρv /ρ is the specific humidity, g is gravity, and PW is in units of mm or kg m−2 . Note that for PW to actually rain out, lifting of the air parcels is necessary. This either occurs by spontaneous convection or by frontal convection. It is clear from Fig. 6.18 that the tropics are moisture production regions. The air at higher latitudes has cooled so much that its moisture content is low: most of the water

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vapour produced by evaporation at the surface in the tropical regions has condensed and rained out at lower latitudes, for instance via the mid-latitude storm tracks. In the polar regions the surface air layer just above the sea or snow or ice surface is always saturated. However, this does not lead to strong sublimation because the amount of available energy to heat the surface is limited (large incidence angle of the solar radiation and high surface albedo). Furthermore, the presence of the persistent surface based temperature inversion suppresses sublimation/evaporation. In case of a melting surface, the moisture gradient may reverse and evaporation is suppressed as well. As a result, the vertical transport of moisture from the surface to the atmosphere is small in the polar regions. The lower values of PW over the Antarctic plateau compared to the Greenland Ice Sheet are due to the higher elevation and lower temperatures over Antarctica. Due to the lack of energy to heat the surface, spontaneous convection is weak and infrequent poleward of 75◦ N and 60◦ S, apart from at the sea ice edge and continental margins, for example. Precipitation is therefore mostly the result of forced convection, i. e., frontal activity, and large scale circulation in combination with (ice sheet) topography. The spatial distribution of precipitation in the Arctic region is determined by the average flow direction and orography. We find precipitation maxima in areas where humid air transported by the flow encounters topographical barriers, such as at the west coast of the U. S. A., the west coast of Norway, and the south coast of Iceland, all major glacierised regions of the Northern Hemisphere. For all these locations a significant part of the precipitation falls as rain; only over the interior of the Greenland Ice Sheet does almost all precipitation fall as snow. The north polar basin is fairly dry (< 250 mm w.e.) and here also a significant amount of that precipitation falls as rain (10–40%). In the Antarctic region the mean flow is more zonally oriented and so is the precipitation distribution. Significant topographical barriers in the westerly flow are the Chilean Andes, New Zealand Alps and the Antarctic Peninsula. Due to its high elevation, the Antarctic continent also acts as a barrier for further southward transport of moisture. As a result, the interior Antarctic Ice Sheet is very dry and is often referred to as a polar desert. Over the Antarctic continent and its immediate surroundings essentially all precipitation falls as snow. The variability in the precipitation can be explained to a large extent by the circulation patterns induced by the circumpolar pressure trough. Most precipitation falls in the coastal areas and on the leeward side of the climatological low pressure areas. Note that these maps are based on averages over many years. Precipitation events are in general very irregular in space and time. On Antarctica, the total amount of annual snowfall might fall in a few major events. As a result, individual annual layers within ice cores do not necessarily represent climatic information for a whole year.

6.8

Notes and References

Table 6.2 gives values of the constants used in this chapter.

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Table 6.2 List of constants and their values as used in this chapter Name

Symbol

Value

Speed of light Planck constant Boltzmann constant Latent heat of melt (ice ↔ water) Latent heat of sublimation (ice ↔ vapour) Latent heat of evaporation (water ↔ vapour) Solar constant Specific gas constant, dry air Specific gas constant, moist air Specific heat of air at constant pressure Stefan Boltzmann constant

c h k Lm

2.998 × 108 m s−1 6.63 × 10−34 J s 1.38 × 10−23 J K−1 3.34 × 105 J kg−1

Ls

2.84 × 106 J kg−1

Lv

2.51 × 106 J kg−1

S0 Rd Rv cp

1366 W m−2 287.05 J kg−1 K−1 462 J kg−1 K−1 1004 J kg−1 K−1

σ

5.67 × 10−8 W m−2 K−4

The values of L m and Rv are for note, as they are not referred to in the text

There are several source texts which cover most of the material in this chapter. Wallace and Hobbs [8] give a complete review of the principles of meteorology, including chapters on atmospheric dynamics, weather systems, the planetary boundary layer (the lowest kilometre where the effects of friction are important), and climate dynamics. One of the sections in the last chapter carries the provocative but faux-naïf title, ‘Is human-induced greenhouse warming already evident?’ The earlier part of the book covers the carbon cycle and the hydrologic cycle, principles of radiative transfer, atmospheric thermodynamics, and cloud physics. A more comprehensive treatment of radiative transfer theory is given in the book by Liou [9]. This book takes the reader through the basic physical processes of absorption, emission and scattering, and gives an overview of atmospheric processes of gaseous absorption and particulate scattering. The last part of the book focuses on practical applications of radiative transfer theory to remote sensing, and the inclusion of radiative processes in climate models. The book by King and Turner [10] gives a comprehensive survey of the meteorology in Antarctica, including extensive discussion of observational processes, physical climatology, and the general circulation and synoptic meteorology. There is also a thorough discussion of mesoscale processes, including in particular katabatic winds.

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Exercises 6.1 The Planck radiation curve (this is really five separate questions) (a) The Planck radiation curve is given by the formula in (6.1): Fλ (T ) =

2π hc2

, λ5 ehc/kλT − 1

and its integral with respect to wavelength λ gives the emitted radiation as in (6.2): ∞ ∞ 2π hc2 dλ. F(T ) = Fλ (T ) dλ = λ5 ehc/kλT − 1 0 0 By choosing a suitable substitution of the integration variable, show that 2π k 4 I F(T ) = σ T , σ = 3 2 , h c 4

∞

I = 0

θ 3 dθ . eθ − 1

(b) To evaluate the integral, show that eθ

1 = e−θ (1 − e−θ )−1 , −1

expand the second term as a geometric series (note that e−θ < 1 if θ > 0), and hence show that ∞ ∞

I = θ 3 e−nθ dθ. 1

0

By a suitable rescaling of θ , deduce that I = (4)

∞ ∞

1 ,

(z) = t z−1 e−t dt n4 0 1

(this is the gamma function). Show that (1) = 1. By using integration by parts, show that for z > 0 (or more generally Re z > 0) (z + 1) = z (z), and deduce that for integer values of n, (n + 1) = n!. Hence find the value of (4). (c) Next, let us recall the Fourier series of a function f (x) defined on (−π, π ), and which is 2π –periodic: f (x) =

∞

−∞

an einx .

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Noting that that

π

−π

eimx d x = 0 for integers m = 0, but = 2π if m = 0, show

an =

1 2π

π

−π

f (x)e−inx d x.

Deduce Parseval’s formula: 1 2π

π

−π

| f (x)|2 d x =

∞

|an |2 .

−∞

(d) Show that sinh iξ = i sin ξ , cosh iξ = cos ξ . Deduce that for integer values of n n, sinh(−inπ π) = 0, cosh(−inπ ) = (−1) . 2 e zx d x = sinh π z. Show that z −π By differentiating twice, show that

π

−π

x 2 e zx d x =

2π 2 4 4π sinh π z − 2 cosh π z + 3 sinh π z. z z z

Hence show that π 4π(−1)n 2π 3 x 2 e−inx d x = , n = 0, = , n = 0. n2 3 −π Derive the Fourier coefficients for f (x) = x 2 , and by using Parseval’s formula, show that ∞

1 π4 = . n4 90 1

(e) Use the values c = 2.998 × 108 m s−1 , k = 1.381 × 10−23 J K−1 , h = 6.626 × 10−34 J s to evaluate the Stefan–Boltzmann constant σ , giving also its units. 6.2 In Fig. 6.2, the caption refers to the fact that the total received solar radiation at ToA is 1366 W m−2 . However, the area under the solar Planck curve in the figure is approximately 593 W m−2 . To explain this, follow the steps below. (a) Use the units of h, c given in Exercise 6.1 to show that Fλ in equation (6.1) has units W m−3 , and deduce that λFλ has units of W m−2 , as does F(T ) given in equation (6.2). ln λ Show that log10 λ = ≈ 0.43 ln λ. ln 10 Hence show that ∞ ∞ ∞ ∞ λFλ dλ Fλ dλ = λFλ d ln λ = ln 10 λFλ d log10 λ. = λ 0 0 −∞ −∞

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Deduce that the area under the curve shown in Fig. 6.2 is

∞

−∞

λFλ d log10 λ = 0.43F(T ),

and deduce its value if F(T ) = 1366 W m−2 . To verify this value, compute the area of the approximating triangle in Fig. 6.2 whose maximum is at λFλ = 1000 W m−2 and whose other two vertices are on the log10 λ axis at λ = 0.2 μm and λ = 3 μm (thus the (dimensionless) length of the base is log10 15), and compare the result with the actual value of 593 W m−2 . (b) Now consider the Earth radiation curve. First, show that the Planck function can be written in the form λFλ =

F(T ) θ 4 hc , θ= , I (eθ − 1) kT λ

where I is defined in Exercise 6.1. Using the definitions of the constants there, show that θ is dimensionless, and show also that d log10 λ = −d log10 θ . Show that λFλ has a maximum where θ = 4(1 − e−θ ), show graphically that the maximum is unique, and show that θ ≈ 4. Compute a better approximation by iterating the map θn+1 = 4(1 − e−θn ); why does this work? Deduce that if the ratio of the Earth radiation F E (T ) to the ToA sun radiation F S (T ) is 41 (1 − α) ≈ 0.17, then the ratio of the maxima of λFλE and λFλS is the same, as is the ratio of the areas under the curves. Using an approximating triangle to the Earth curve in Fig. 6.2 with maximum at 170 W m−2 and vertices on the log10 λ axis at λ = 5 μm and λ = 75 μm (so the dimensionless length of the base is log10 15), show that the ratio of this estimate to your estimate for the area under the sun radiation curve is also 0.17. 6.3 The total cloud effect and the radiation paradox We look at the influence of clouds on the surface radiation budget. In this exercise we do this by comparing the net surface radiation (Rnet ) during clear sky conditions with the average situation (with clouds); the difference is called the ‘total cloud effect’. Figure 6.19 shows the total cloud effect averaged over the year. (a) What are the components of the total cloud effect, and what meteorological variables influence these components at first order? Which component would you expect to dominate over dark surfaces? Is this confirmed in the figure? In that case, how is the total cloud effect related to surface temperature?

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cloudy

Fig. 6.19 Total cloud effect (i. e., Rnet

clear sky

− Rnet

, W m−2 ) [3]

(b) Explain the patterns in the polar regions depicted in the figure. The increase of net surface radiation as an effect of clouds is sometimes called the radiation paradox. Do you think this is a good name? Use Fig. 6.20 to answer the following questions. Observations at the Antarctic research station Neumayer (70◦ S) suggest that

Fig. 6.20 Daily cycle of incoming Shw at ToA, for four SH latitudes on 15 January

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LWin (z = 0) = εa σ Ta4 ≈ (0.765 + 0.22N 3 )σ Ta4 , where εa is the effective atmospheric emissivity, Ta is air temperature at 2 m and N is cloud cover (a number between 0 and 1). Noon of 15 January 2000 is windless and sunny at Neumayer. The atmospheric transmissivity for shortwave radiation τ SW = 0.8, and the temperature near the snow surface, which has an albedo of 0.85, has risen to as high as −3 ◦ C. Lying on his therma-rest, glaciologist Gerd Gernknödel is taking a sunbath in his bare chest! Suddenly a cloud moves in front of the sun, and τ SW drops to 0.5. (c) What will happen to the snow surface temperature? Hint: neglect the temperature difference between 2 m and the surface and assume surface emissivity εa = 1. Further assume that, apart from the radiation balance, no other terms in the surface energy balance play a rôle. (d) Does Gerd need a jacket? Hint: calculate the radiation balance of Gerd’s skin, assuming radiative equilibrium at skin temperature when he feels comfortable and take εGer d = 1. Take a value of 32 ◦ C as Gerd’s skin temperature.

References 1. Fowler A (2011) Mathematical geoscience. Springer, London 2. Trenberth K, Fasullo J, Kiehl J (2009) Earth’s global energy budget. Bull Amer Met Soc 90(3):311–323 3. Hatzianastassiou N, Matsoukas C, Hatzidimitriou D, Pavlakis C, Drakakis M, Vardavas I (2004) Ten year radiation budget of the Earth: 1984–93. Int J Climatol 24:1785-1802 4. van den Broeke MR, Reijmer CH, van As D, van de Wal RSW, Oerlemans J (2005) Seasonal cycles of Antarctic surface energy balance from automatic weather stations. Ann Glaciol 41:131–139 5. van den Broeke MR, Smeets CJPP, van de Wal RSW (2011) The seasonal cycle and interannual variability of surface energy balance and melt in the ablation zone of the west Greenland ice sheet. The Cryosphere 5:377–390 6. Turner J, Colwell SR, Marshall GJ, Lachlan-Cope IA, Carleton AM, Jones PD, Lagun JV, Reid PA, Iagovkina S (2004) The SCAR READER project: toward a high-quality database of mean Antarctic meteorological observations. J Clim 17:2890–2898 7. Ball FK (1960) Winds on the ice slopes of Antarctica. In: Antarctic meteorology, proceedings of the symposium held in Melbourne, February 1959. Pergamon Press, Oxford, pp 9–16 8. Wallace JM, Hobbs PV (2006) Atmospheric science: an introductory survey, 2nd edn. Academic Press, Burlington, MA 9. Liou KN (2002) An introduction to atmospheric radiation, 2nd edn. Academic Press, New York 10. King J, Turner J (1997) Antarctic meteorology and climatology. CUP, Cambridge

7

Mass Balance Michiel van den Broeke and Rianne Giesen

7.1

Introduction

In this chapter we discuss the methods used to assess temporal mass changes of different ice masses—valley glaciers, ice caps and ice sheets. We provide definitions of the key terminology in Sect. 7.2 and discuss the main methods to observe and model glacier mass balance in Sect. 7.3. Next, we present the specific application of these methods to valley glaciers and ice caps in Sect. 7.4 and to the two large ice sheets of Antarctica (Sect. 7.5) and Greenland (Sect. 7.6). The reason for this subdivision is that the ice sheets are large enough to allow for direct observation of mass changes by satellite remote sensing and dynamical downscaling of surface processes (using for example regional climate models), while valley glaciers and ice caps are usually so small that they require statistical downscaling of satellite observations or global/regional atmospheric model output.

7.2

Definitions

To be specific, let us consider an ice sheet or glacier which terminates in the ocean, so that it has a grounding line where it goes afloat. Glacier mass balance (MB, denoted as B, kg y−1 ) represents the temporal change of glacier mass Mi , which, if we neglect basal melting of grounded ice and assume that the grounding line position is stationary, is governed by the difference between surface mass balance (SMB, denoted S) and ice discharge across the grounding line (D):

M. van den Broeke (B) · R. Giesen Utrecht University, Utrecht, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_7

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∂ Mi = S − D. (7.1) ∂t Equation (7.1) states that the mass balance of a glacier can be regarded as the sum of mass exchange processes at the ice-atmosphere interface (snowfall, sublimation, meltwater runoff) and at the ice-ocean interface (solid ice discharge, basal melt). With the grounded part of the ice sheet defined as the region of interest, the melt beneath floating glacier tongues and ice shelves, which is difficult to quantify (see Chaps. 4 and 5 on tidewater glaciers and ice shelf-ocean interactions), does not have to be included in the mass balance equation. However, when the grounding line migrates, the loss of the overlying ice must be taken into account. Ice discharge represents the flux of ice across the grounding line of the glacier or ice sheet, and is determined by the thickness and vertically averaged horizontal ice flow velocity component normal to the grounding line. Ice flow is the sum of deformation and basal sliding, as discussed in Chaps. 1 and 3. The surface mass balance S represents the sum of all mass fluxes towards and away from the glacier’s snow/ice surface (the latter fluxes taken as having negative sign). It is given by S = P + R − Us − Uds − E ds − M,

(7.2)

where P is solid precipitation (snow, hail, freezing cloud droplets), R is rainfall, Us and Uds are sublimation from the surface and drifting snow particles, respectively, E ds is erosion of snow by divergence of the drifting snow transport and M is meltwater runoff. The accumulation and ablation zones of a glacier are defined as the areas where S > 0 and S < 0, respectively. These zones are separated by the equilibrium line, where S = 0. Glaciers form at any land-based location where S is positive for numerous consecutive years; this time span must be large enough for glacier ice to form out of the slow compression and sintering of the lowest firn layers. In the surface mass balance Eq. (7.2), runoff M is determined by the liquid water balance, which is the sum of all sources (water vapour condensation, rainfall and melt) and (negatively signed) sinks (refreezing and capillary retention) of liquid water: M = R + C + Ms − Rs − F,

(7.3)

where C is condensation of water vapour due to the vertical turbulent exchange of atmospheric moisture, Ms is surface meltwater production, Rs is retention of liquid water in the snow/firn by capillary forces and F is refreezing. By including these subsurface processes (‘internal accumulation’), S is formally referred to as the ‘climatic mass balance’ [1]. Instead of evaluating surface mass balance over the entire glacier surface, S is often measured locally as mass per unit time per unit area. This quantity is called the specific surface mass balance (abbreviated as SSMB), and we will denote its value as ; it has the same components as S but is expressed in kg m−2 y−1 or mm water-equivalent (mm w.e.) y−1 . These local or in situ measurements form the basis for spatial interpolation that in the end must yield the glacier surface mass balance.

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Methods

In this section we describe the key methods used to determine the mass balance of valley glaciers, ice caps and ice sheets. We start with in situ SSMB observations in ablation and accumulation areas. A partial solution for the poor temporal and spatial coverage of SSMB observations (Sect. 7.3.1) over glaciers is provided by remote sensing techniques. To date, however, no technique exists that directly measures SSMB from space, and this gap is often filled by using atmospheric models. Recent developments in remote sensing and regional climate modelling now offer three other methods to estimate glacier mass balance indirectly, which are briefly discussed below (Sects. 7.3.2–7.3.4).

7.3.1

In Situ Observations

Because of the large variability in its components, SSMB can vary greatly from place to place and from year to year. As a result, it is necessary to perform numerous and repeated observations of SSMB (and of discharge D) in order to establish whether a glacier has positive or negative mass balance, i. e., whether it is growing or shrinking under present climate conditions. The simplest way to do this is to fix a stake into the ice/snow of the ablation/accumulation zone of a glacier and return to it one year later to measure how the surface level has changed (or to monitor the change continuously with electronic sensors; Fig. 7.1). Over a year, the length difference of the part of the stake that protrudes above the ice (h), multiplied by the density ρ of the ablated ice (≈ 910 kg m−3 ) or the accumulated snow (to be measured), yields the value of : = ρh,

(7.4)

but several problems of logistical and scientific nature arise with this method. Glaciers and ice sheets are not easily accessible, making these measurements expensive. In mild and/or wet regions, annual ice ablation and snow accumulation of up to several metres pose serious problems for stake observations (the stakes need to be long, and their drill holes deep), while obtaining reliable snow density profiles over deep layers is cumbersome. It is often unclear how representative single-point measurements are for a larger area. Finally, the annual measurements must be performed at the same time each year. As a result, reliable in situ SSMB datasets measured on ice sheets and glaciers with long time span and high spatial resolution are scarce. In the accumulation zone of a glacier, other interpretation problems arise: here, stake measurements tell us nothing about refreezing and retention (internal accumulation) so that SSMB cannot be directly determined. To circumvent this problem, SSMB can also be obtained by drilling a firn core through multiple annual layers and subsequently measuring the amount of mass that has accumulated since a well-dated horizon (see Chap. 12 on ice cores); the horizon could be an acid layer deposited after a large volcanic eruption (Fig. 7.2) or a radioactive layer deposited after a nuclear test. In high-accumulation areas, seasonal cycles in inert chemical species

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Fig. 7.1 Cumulative sonic height ranger observations along the K-transect, west Greenland, August 2003–August 2007. Site S5 is situated at ≈ 500 m a.s.l., S6 at ≈ 1000 m a.s.l. and S9 at ≈ 1500 m a.s.l., close to the equilibrium line. The build-up of a winter snow layer and its ablation in spring and summer is clearly visible in the middle ablation zone (S6). Snow accumulation and ablation approximately cancel near the equilibrium line (S9). At the lowest site (S5) all snow is blown in crevasses so that almost no winter snow accumulates. Images on the right show the measurement sites, including mass balance stakes and automatic weather stations (AWSs). The K-transect is a mass balance stake array of eight sites where annual SSMB and ice velocity have been measured since 1990

or physical characteristics of the snow and ice layers may be detected and counted to yield time series of SSMB. A powerful tool for locating reflection horizons in the firn so as to obtain high spatial resolution accumulation data is snow radar, which can be mounted on an airplane or towed behind a snowmobile to connect drilling sites. An overview of SSMB measurement techniques in polar (low) accumulation areas is provided by Eisen et al. [2]. Once sufficient reliable SSMB observations for a glacier are obtained, a map can be made using interpolation and extrapolation, thus yielding a value for glacier surface mass balance. When doing this one must carefully select the interpolation procedure and quantify the uncertainty in between the observations. As an example, the left panel of Fig. 7.3 shows an accumulation map of Dronning Maud Land, East Antarctica, based on the interpolation of sparse SSMB observations (black dots, [3]). The map predicts low accumulation in a region to the west of a transect near the coast (encircled), where automated equipment was subsequently installed. Upon return one year later, it appeared that SSMB was a factor 2.5 higher, and equipment had to be dug out (inset in Fig. 7.3). Even localized and unpronounced topographic features, in this case an ice rise, can apparently introduce significant local snowfall maxima that are not resolved by the sparse SSMB observations; these can nowadays be better identified using high resolution regional climate models (right panel in

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Fig. 7.2 Experimental setup for medium-deep ice core drilling, including on-site dating using a di-electrical profiling (DEP) instrument in a snow trench. The graph shows dated volcanic peaks as a function of depth (converted to metres of water equivalent, m w.e., using firn density)

Fig. 7.3 SSMB map (kg m−2 y−1 ) of Dronning Maud Land, East Antarctica, based on (left) the interpolation of observations (black dots) using kriging [3] and (right) calculations of a regional climate model at 27 km horizontal resolution [4]. Encircled is the area where automated equipment was installed that unexpectedly needed digging out after only a single year of operation (inset). Comparison of the two maps shows that numerous small-scale, topographically-forced accumulation features visible on the modelled map are absent from the interpolation-based map

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Fig. 7.3). Comparison of the two panels in Fig. 7.3 shows more examples of such deficiency of interpolating sparse SSMB observations. Even with sufficient coverage, in situ measurements of SSMB do not provide information about the magnitude of the various SSMB [Eq. (7.2)] or liquid water balance components [Eq. (7.3)]. For instance, a small yet positive SSMB value can result from any combination of accumulation and ablation processes, as long as the latter are smaller than the former. This poses a serious interpretation problem: without knowing the magnitude of individual SSMB components, it is difficult to relate mass balance changes to changes in climate. This can locally be remedied by installation of an automatic weather station (AWS) that is equipped with a sonic height ranger to detect individual accumulation and ablation events. Ideally, several AWSs are installed along a transect so as to obtain elevational gradients in SSMB components. Figure 7.1 shows examples of such time series from the ablation zone of the west Greenland Ice Sheet. If, apart from the standard variables of wind speed, temperature and humidity, an AWS also measures the surface radiation balance (incoming and outgoing shortwave and longwave radiation), then melt and sublimation fluxes can be explicitly quantified in combination with a surface energy balance model (see Chap. 6).

7.3.2

Satellite/Airborne Altimetry

Repeat satellite/airborne altimetry using radar/laser altimeters mounted on planes and various satellites (ERS1/2, ICESat, Envisat, CryoSat-2) yields changes in ice sheet volume, as further detailed in Sect. 13.4. The main instrument limitations arise from laser degradation and radar altimeters having an unknown penetration depth in snow, depending on (time-varying) firn density [6]. Before the launch of Cryosat-2 in 2010, radar altimeters did not resolve the narrow, fast flowing outlet glaciers, which are expected to react most rapidly to environmental changes. The laser altimeter onboard ICESat did capture these thinning glaciers in detail [7], but laser altimeters have limited spatial resolution and are sensitive to clouds, prohibiting the collection of continuous time series in high-accumulation (i. e., frequently overcast) areas. Moreover, degradation of the lasers in time requires temporally-varying corrections to be made. The main methodological uncertainty in altimetry is that the vertical velocity at the ice sheet surface represents the sum of multiple processes, i. e., surface mass exchange (melt, sublimation, snowfall), compaction of the firn layer, and the downward movement of the firn/ice interface caused by divergence in the ice flow (Fig. 7.4, [8]); also see Sect. 13.4. A small basal melt term and vertical bedrock motion are usually neglected. To translate volume changes to mass changes therefore requires modelling of surface accumulation variability, which forces changes in firn mass and depth, and firn densification rate, which forces changes in firn depth. Both these processes dominate ice sheet elevation changes in areas away from rapid dynamic changes [5,6]. The inset in Fig. 7.4 explains the confounding effect of accumulation variability on elevation changes: decadal variability in accumulation that otherwise

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Fig. 7.4 Vertical velocity components that determine vertical displacement of the glacier surface. Inset shows artificial SSMB time series with 25 and 100 yr periodicity (upper curve) and the resulting surface elevation trends (lower curve) using a firn compaction model [5]. This figure is a combination of Figs. 11.11 and 11.13 of Chap. 11

has zero long-term trend (upper curve) results in slow firn depth changes (lower curve) that are easily misinterpreted as long term trends in ice sheet mass. Moreover, identically shaped decadal accumulation variations (indicated as green and blue in the upper curve) can result in the surface lowering or rising, depending on whether the anomaly takes place at average accumulation values that are below (green) or above (blue) the long-term average. More discussion of the firn densification processes underlying such response is given in Chap. 11 (see Sect. 11.3.1, in particular). When corrected for these confounding effects, altimetry can provide glacier mass loss rates, yet the data do not discriminate between the different processes responsible for the mass changes, i. e., S or D.

7.3.3

Satellite Gravimetry

Satellite gravimetry uses data of the Gravity Recovery And Climate Experiment (GRACE) twin satellites, launched in 2002, and follow-up missions. The first great accomplishment of GRACE was that it proved beyond a doubt that the large ice sheets have been losing mass during the first decade of the twenty-first century (Fig. 7.5). Like altimetry, GRACE does not discriminate between the various processes responsible for mass loss, but being independent of the other two methods and as the only method that measures mass change directly, GRACE provides valuable verification data for other techniques. Methodological uncertainties arise from the large footprint (≈ 300 km) and associated “leakage effects”, which make GRACE less useful for smaller ice caps and glaciers. A large uncertainty is introduced by the post-glacial rebound correction in Antarctica (see Chap. 15). As with most satellite products, the

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Fig. 7.5 Mass trend according to the Gravity Recovery And Climate Experiment satellite pair (GRACE, 2003–2010) in cm water equivalent per year. Figure courtesy of Bert Wouters (Utrecht University)

Fig. 7.6 Schematic drawing of glacier mass in time as measured by GRACE. Since GRACE does not measure the absolute mass of the glacier (M0 ), and the time scale of operation is relatively short (about a decade), a mass loss observed by GRACE can be interpreted in multiple ways

relatively short observational period hampers a correct interpretation of the signal in terms of longer-term changes. Figure 7.6 illustrates this: in the absence of information about the total reference mass of the ice sheet (M0 ), a negative mass trend in the brief GRACE time series could either indicate a slow mass oscillation of an ice sheet in long-term balance or declining mass of an ice sheet out of balance. Combining GRACE with other techniques can aid the interpretation of these signals (see later sections on mass balance of the Greenland and Antarctic Ice Sheets).

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169

Mass Budget Method

The mass budget method quantifies the difference between surface mass gains (snowfall) and losses (meltwater runoff, sublimation), which together give the SMB (S, the red integration surface in Fig. 7.7) and lateral mass loss by solid ice discharge (D, the blue integration surface). The biggest disadvantage of the mass budget method is that it attempts to quantify the difference between three large terms (snow accumulation, solid ice discharge and meltwater runoff), each with relatively large uncertainties. As a result, the relative uncertainty in their difference can be substantial and sometimes even renders the sign of the glacier mass balance uncertain, i. e., it can not be stated with statistical certainty whether a glacier is losing or gaining mass. The major advantage of the mass budget method is that all individual mass balance components are quantified, providing valuable insights in the physical processes that drive ice sheet mass change [9]. An SMB value can be obtained through smart spatial interpolation of SSMB observations [10,11]; unfortunately, available SSMB data are often not sufficiently dense to arrive at robust estimates, and they provide limited insight in temporal variability. An alternative is offered by regional climate models, sometimes calibrated with SSMB observations [4,12–14]. Regional climate models explicitly resolve physical atmospheric processes leading to snowfall and melt, and run at higher resolution (typically 5–25 km) than global models (100–200 km). When evaluated against SSMB data, model errors are found to range between 5 and 30%, with the largest uncertainties occurring in regions of extreme (low or high) precipitation, or strong melting, or pronounced topography that is not well resolved by the model. Quantifying solid ice discharge D requires observations of ice velocity and thickness at the grounding line. For this, feature tracking with interferometric synthetic aperture radar (InSAR) is used, which yields surface ice velocity with high accuracy

Fig. 7.7 Schematic cross-section of a marine- (left) and land-terminating (right) ice sheet. Dashed lines approximate ice flow lines. Symbols represent integration surfaces for the mass budget method: red symbols for surface mass balance (SMB) and blue symbols for solid ice discharge (D)

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(< 5% [9,15]). The principles of feature tracking are further explained in Sect. 13.3.3. The thickness of many ice streams has now been directly measured using airborne radar with sufficient accuracy (≈ 10 m). In regions where such data do not yet exist, satellite altimetry is used [16]; this, in combination with a flotation criterion, yields thickness with uncertainties of ≈ 100 m [17]. To convert elevation to thickness, a correction must be applied for the density of the firn mantle that covers the ice, e. g., using a steady-state firn compaction model [8]. For the Greenland Ice Sheet, where glacier tongues experience significant ablation in summer, this correction is less important. Finally, corrections must be made for glacier thinning and the mass flux associated with grounding line migration [9].

7.4

Valley Glaciers and Ice Caps

In this section we describe the application of the methods to assess the mass balance of valley glaciers and ice caps (GIC), which comprise all glaciers outside the ice sheets of Greenland and Antarctica.

7.4.1

In Situ Observations

GIC SSMB stake observations have been conducted since the late nineteenth century [18]. The earlier quantitative measurements were often made at single locations on a valley glacier and not repeated annually, hence they could not be used to determine the glacier’s SMB. The longest continuous SMB record with glacier-averaged annual values dates back to 1946, when a measurement program was started on Storglaciären in northern Sweden. Since then, surface mass balance observations have been carried out on GIC in many regions of the world. The number of mass balance records is heavily biased toward the European Alps, Scandinavia and North America, where glaciers are relatively accessible. A sample of long cumulative SMB records for different regions is shown in Fig. 7.8. The three maritime valley glaciers in Norway had a net surface mass gain over the period 1960–2010, which is mainly a result of a few years with very high winter precipitation around 1990. All other glaciers in Fig. 7.8 show a net surface mass loss between 1960 and 2010, which often accelerated after the year 2000. Different GIC in the same region generally show similar interannual variability, although the absolute values of the SMB vary. Interannual variations in SMB are largest for the maritime glaciers in Norway with large amounts of solid precipitation, and smallest for the dry Arctic glaciers. These long SMB records are very valuable for the purpose of obtaining a first indication of mass changes in different regions. However, variations of the total cumulative change for the GIC within a region can be large and can thus complicate the upscaling of the measurements to estimate the mass change for the whole region.

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7.4.2

Modelling

While SSMB observations provide a wealth of information on the relation between climate and GIC mass changes, they are often supplemented by SMB models, in order to: (1) extend the period with mass balance measurements further back in time or into the future; (2) estimate SMB for glaciers without observations; (3) study the importance of different terms in the surface mass and energy balance; and (4) determine the sensitivity of the SMB to changes in climatic variables. Models of various degrees of complexity are used to compute GIC mass balance. The choice of model is usually determined by the available meteorological input data and observations for model calibration. Since positive mass contributions primarily come from solid precipitation, accumulation is generally simply derived from precipitation input data. A threshold temperature or temperature range is used to distinguish between rain and snow. On GIC outside the tropics, runoff from melting is the most important negative term in the surface mass balance. The following paragraphs discuss three models of different complexity (positive-degree-day, simple surface energy balance, full surface energy balance) that are commonly used to calculate surface melt. The most simple model is the temperature-index or positive-degree-day (PDD) model, where ablation is assumed to be an empirical function of the sum of days with positive air temperatures (for a review of the method, see [20]). The melt rate changes considerably when the surface cover changes from snow to ice, as this affects the albedo; this is not reflected in the measured air temperature. It is therefore common

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practice to use two different PDD melt factors: one for snow and one for ice. The values of these melt factors depend on the importance of the temperature-dependent fluxes (mainly longwave radiation and the turbulent fluxes) in the energy balance and need to be calibrated with ablation measurements from the glacier of interest. Since the PDD method implicitly assumes that seasonal variations in net solar radiation (which do not depend on air temperature) coincide with air temperature variations, this model only works when this is actually the case. Furthermore, since net solar radiation is combined with the temperature-dependent fluxes, the sensitivity of the melt rate to changes in air temperature is not correct. An often-used motivation for using a PDD model instead of a more sophisticated representation of the surface energy fluxes is that meteorological input data other than precipitation and air temperature are not available. However, a separation of the contributions by net solar radiation and the temperature-dependent fluxes can be made without requiring additional meteorological input data [21]. For a physical representation of the surface energy balance, it is important that the terms for net solar radiation and the temperature-dependent fluxes are added, not multiplied, as is done in some models. Seasonal cycles of the SSMB components modelled with a PDD model and a simple surface energy balance (SEB) model are shown in Fig. 7.9 for two automatic weather station (AWS) sites on glaciers in Switzerland and Iceland. For the Swiss glacier, the seasonal cycles in net solar radiation and air temperature are rather similar and the SSMB calculated with the PDD model reproduces the measurements reasonably well. Compared to the surface energy fluxes modelled with the simple SEB model, the PDD model underestimates melt in spring and overestimates melt in

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the late summer and autumn. On the maritime Icelandic glacier, the seasonal cycle in air temperature has a low amplitude, while net solar radiation varies strongly through the year. Because the maximum in net solar radiation occurs well before the maximum air temperature, the PDD model considerably underestimates melt in spring, while there is too much melt from the late summer into the winter. Furthermore, the lowamplitude seasonal cycle in air temperature results in large melt factors and too large interdaily variations in the surface energy flux. When meteorological variables like cloud cover, relative humidity and wind speed are also available, all fluxes in the surface energy balance can be calculated separately. This results in more realistic variations in the melt rate and allows for a thorough SMB sensitivity analysis to changes in meteorological variables. Input data can be either observations from weather stations situated on or near the glacier, or the output fields of high-resolution regional climate models. Since the observations at the weather stations or model gridpoints generally do not cover the full glacier altitudinal range, the input data are extrapolated over the glacier surface using simple functions of the variation of the meteorological variables with elevation. The surface energy fluxes are parameterized as functions of the meteorological input variables; many different parameterizations are available from the literature [22–24]. If available, measured surface energy fluxes are used to calibrate the model parameters. Most of these sophisticated SMB models include a layered subsurface model, which is used to treat the subsurface heat flux, routing of meltwater through the snowpack, and refreezing of meltwater in cold layers. Figure 7.10 shows the SSMB map for the ice cap Hardangerjøkulen in southern Norway, calculated with a distributed SSMB model using input data from nearby weather stations. The annual SSMB is primarily a function of altitude, with the most positive values around the summit. Like many ice caps, Hardangerjøkulen has a relatively large and flat accumulation area (where > 0), while most of the melt and runoff occurs on the steeper outlet glaciers. Deviations from the altitudinal mean (m w.e.) 1

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mass balance, mainly resulting from differences in incoming solar radiation due to the orientation of the surface and shading by the surrounding topography, are considerable on some of the outlet glaciers.

7.4.3

Dynamical Response

All glaciers are slowly but continuously changing their geometry to respond to mass balance variations induced by changes in the climate. Especially after multi-year periods with a net positive or negative mass balance, the glacier area and surface elevation may change substantially. Since the SSMB at each position depends on its elevation, changes in the glacier geometry need to be taken into account both for interpreting mass-balance measurements and for modelling mass balance. Glacier maps are often updated after one to three decades, and reported surface mass balance values refer to the last available (most recent) glacier map for that year. This change of reference surface needs to be taken into account when comparing measured and modelled SSMB. Since the SSMB values encapsulate both geometric and climatic variations, changes in the mass balance cannot directly be related to climate forcing. For this purpose, the use of one constant reference surface is recommended. Consecutive glacier maps provide an independent validation of in situ SSMB measurements. The surface elevation difference between two successive maps is multiplied by a representative density (usually ice density is taken) to obtain the total mass change. Differences between this geodetic mass balance and the cumulative annual SSMB measurements over the same period can be large, for example due to assumptions made in the extrapolation of stake measurements or the density used to calculate the geodetic mass balance [26]. In SSMB modelling, the available glacier maps can be used to account for changes in the glacier geometry. When such maps are not available, for example when modelling into the future, the SSMB model can be coupled to a dynamical ice model. A major advantage of such a coupled model is that changes in SSMB and glacier geometry are consistent. A drawback could be that the glacier geometry becomes unrealistic when the ice dynamics are not well represented. A simulation with a coupled model for Hardangerjøkulen in southern Norway illustrates the geometry response to changes in surface mass balance in a warming climate (Fig. 7.11). While the SSMB becomes more negative at all altitudes, the ice cap surface elevation lowers and the ice disappears from the highest ridges in the topography. As a result, there is no accumulation area left and the ice cap is bound to disappear completely.

7.4.4

Remote Sensing

In recent years, traditional stake measurements on GIC are being supplemented by remote sensing techniques like satellite/airborne altimetry and gravimetry (Sect. 7.3). Satellites are well suited to monitor large glaciers and ice caps in remote locations, nicely complementing the traditional measurement programs. Altimetry studies have

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175 (b)

(a)

1800

1600

1400 control 2010 2040 2070 2100

1200

1000 -8

-6

-4

-2

0

Mass balance (m w.e.)

Altitude (m a.s.l.)

Altitude (m a.s.l.)

1800

1600

1400 2005 2010 2040 2070 2100

1200

1000

0

5

10

15

20

2

Area (km )

Fig. 7.11 Modelled a SMB profile and b area distribution for Hardangerjøkulen in 2010, 2040, 2070 and 2100, for a future climate projection with a temperature increase of 3 ◦ C and precipitation increase of 10% between 1961–1990 and 2071–2000. The mass balance profile for the control climate (1961–1990) and the area distribution at the beginning of the simulation (2005) are also shown. Figure from [25]

revealed large mass losses on glaciers and ice caps in the Arctic regions [27,28] and Patagonia [29]. Attempts to derive mass changes in glaciated regions from GRACE have also been made [30], but the resulting estimates have large uncertainties introduced by the required GRACE corrections for other processes such as hydrology and seasonal snow.

7.5

Antarctic Ice Sheet

In the following sections we discuss the spatial and temporal variability of the mass balance of the Antarctic Ice Sheet (AIS). Here we may rely on insights from the mass budget method, which, as explained in Sect. 7.3, resolves the separate contributions made by the surface mass balance (SMB) and solid ice discharge (D).

7.5.1

Spatial SSMB Variability

Figure 7.12 shows on the left a map of SSMB based on output of the Regional Atmospheric Climate Model (RACMO2) for Antarctica at 27 km horizontal resolution [4]. This field represents the average for a 21-year period (1989-2009), obtained by forcing RACMO2 at the boundaries by data of the European Centre for Medium-range Weather Forecasts (ECMWF) interim re-analysis (ERA-Interim). The map correlates well with in situ SSMB observations from firn cores, stakes and snow pits (r = 0.87 for the AIS) and is therefore deemed reliable for a quantitative discussion.

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Fig. 7.12 Specific surface mass balance maps (kg m−2 y−1 ) for the Antarctic Ice Sheet (left, [4]) and the Greenland Ice Sheet (right, [13]), based on a regional atmospheric climate model

As described in the preceding chapter, the large-scale flow around Antarctica is westerly north of the circumpolar pressure trough and easterly south of it. The only part of Antarctica that is exposed to the westerly circulation at lower latitudes is the Antarctic Peninsula (AP), which acts as a topographic barrier to the flow (similar to the Andes further north). Figure 7.12 shows that the orographic lifting of the relatively mild and humid air masses results in extreme precipitation rates in the western Antarctic Peninsula (> 4500 kg m−2 y−1 ). For a typical near-surface snow density of 350 kg m−3 , this represents an annual snow layer depth of > 10 m; field parties operating at the spine of the Antarctic Peninsula confirm accumulation rates as high as one metre of snow per month. Further south, the circumpolar pressure trough includes three well-defined climatological low pressure areas that introduce a three-wave asymmetry in the zonal flow (see Sect. 6.4 and Fig. 6.6), regionally forcing persistent onshore atmospheric flow. This again results in orographic lifting and high precipitation rates in these areas, especially in coastal West Antarctica (> 1000 kg m−2 y−1 ). As a result of descending air motion, relatively dry areas are found at the lee side of topographic barriers, e. g., the eastern side of the Antarctic Peninsula receives less than 300 kg m−2 y−1 , an order of magnitude less than regions just 200 km to the west. Another example is Law Dome in coastal Wilkes Land, East Antarctica, which receives > 1500 kg m−2 y−1 at the eastern (upstream) side, and < 200 kg m−2 y−1 at the western (downstream) side. Evidence of pronounced orographic effects on the precipitation distribution can be found everywhere along the Antarctic coast, as previously discussed and shown in Fig. 7.3. Moist air seldom reaches the interior of the East Antarctic Ice Sheet; this region receives < 50 kg m−2 y−1 of snow and may justifiably be called a polar desert.

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7.5.2

177

Blue Ice Areas

The Antarctic climate is too cold to allow for significant meltwater runoff; nearly all meltwater that is formed at the surface refreezes in the cold firn mantle. Only around 1% of the surface of the AIS represents ablation area; in these areas, the removal of mass is not driven by melting but by surface and drifting snow sublimation. When these processes persistently exceed snowfall, < 0 and the 50–120 m thick firn layer may be completely removed, after which blueish glacier ice is exposed at the surface. This is why these areas are often referred to as blue ice areas (BIAs). The formation of BIAs requires two conditions to be met: (1) < 0, i. e., the combined effect of drifting snow erosion and (drifting snow) sublimation should exceed snowfall; and (2) the firn layer must be exposed to the negative SSMB for a sufficiently long period that the entire firn layer can be removed. Since low ice velocity favours such long exposure, BIAs are often found in regions where nunataks (mountains that protrude through the ice and slow down the ice flow) are abundant, e. g., the Queen Fabiola Mountains in Dronning Maud Land and Allan Hills in Victoria Land. We can define a minimum trajectory length L min of ice flow through a region of negative SSMB, which is necessary to form blue ice: L min =

m firn Vice , ||

(7.5)

where m firn is the mass of the firn layer (kg m−2 ) upstream of the ablation area, which can be calculated using a firn model (e. g., [8]), Vice is the average surface ice velocity (m y−1 ) and (kg m−2 y−1 ) is the average value of SSMB along the trajectory. In areas of fast ice flow, Eq. (7.5) implies that BIAs can develop only if < 0 is highly negative, i. e., (snowdrift) sublimation and erosion are well developed. Byrd Glacier, which flows through the Transantarctic Mountains at relatively high velocity (Fig. 7.13) has blue ice over large parts of its surface. Recent advances in regional SSMB modelling over the AIS at high (≈ 5 km) horizontal resolution (Fig. 7.13a) predicts an extensive area in which < 0 over the Byrd Glacier trunk [31]. Figure 7.13b shows that snowfall increases following the centre flowline towards sea level, but that ablating processes associated with drifting snow make SSMB negative over a ≈ 100 km stretch. If we substitute typical values for m firn (20,000 kg m−2 ), Vice (800 m y−1 ) and (−200 kg m−2 y−1 ) in Eq. (7.5), we find L min = 80 km, which is smaller than the observed length of the ablation trajectory, indicating that the conditions for BIA development are met. Once formed, BIAs tend to persist, owing to their dark and smooth surface, which enhances summer sublimation and prevents fresh snow from attaching.

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Fig. 7.13 a Specific surface mass balance map for the region of Byrd Glacier, based on a regional atmospheric climate model [31]; b SSMB components along flowline P1–P2; c MODIS image of Byrd Glacier

7.5.3

Temporal SSMB Variability

Improved estimates of SMB from regional climate modelling and D from remote sensing enable us to apply the mass budget method to the AIS for individual years

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Fig. 7.14 Time series of surface mass balance (SMB, blue), solid ice discharge (D, green) and resulting ice sheet mass balance (B, red) for a the Antarctic Ice Sheet (AIS) and b the Greenland Ice Sheet (GrIS) in Gt y−1 [32]

in the period 1990–present. Before 1980, meteorological reanalysis products are unreliable in the Southern Hemisphere, while discharge observations are available since about 1990 [9]. Figure 7.14a shows the resulting time series of SMB (S), D and MB (B) for the grounded part of the Antarctic Ice Sheet. SMB does not show a significant trend. A notable feature in Fig. 7.14a is the large interannual variability in SMB, with year-to-year changes as large as 300 Gt y−1 (which is equivalent to a global sea level change rate of 0.8 mm y−1 ). The standard deviation of ≈ 120 Gt y−1 represents just 6% of the 1989–2009 average SMB over the grounded part of the ice sheet (≈ 2100 Gt y−1 ); this demonstrates that, even though the relative interannual variability of SMB is small, the variability in terms of absolute mass balance is very significant for the AIS, and it obscures trends in the total mass balance. The increase in solid ice discharge D from the AIS is mainly caused by the acceleration of glaciers in coastal West Antarctica, which continues today, and by the acceleration of glaciers in the Antarctic Peninsula, mainly prior to 2005. Figure 7.15 shows that, compared to 1992, discharge from the AIS has increased by ≈ 170 Gt y−1 or 8% in 2009; as a result, MB has been persistently negative since 1994, except for three years with high snowfall (1998, 2001, 2005). Consequently, the AIS has contributed about 4 ± 2 mm to sea level rise between 1990 and 2010, which is about 7% of the total for that period [33].

7.6

Greenland Ice Sheet

In these final sections, we discuss the spatial and temporal variability of the mass balance of the Greenland Ice Sheet (GrIS). Its main difference when compared with the Antarctic Ice Sheet is the existence of a significant ablation area ( < 0), where meltwater runoff exceeds accumulation by precipitation. Although the GrIS is 8.5 times smaller than the AIS in volume, recent mass losses of the GrIS have exceeded that of the AIS.

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Fig. 7.15 Cumulative contribution to global sea level rise (SLR) from the Antarctic Ice Sheet (AIS, blue line) and the Greenland Ice Sheet (GrIS, green line) and their sum (red), in mm sea level equivalent. Dashed lines indicate uncertainty margins [32]

7.6.1

Spatial SSMB Variability

Figure 7.12 shows on the right a map of SSMB for the GrIS based on output of the Regional Atmospheric Climate Model (RACMO2) for Greenland at 11 km resolution [13]. The result correlates well with in situ SSMB observations from firn cores, stakes and snow pits (r = 0.95 for the GrIS). The most notable feature in Fig. 7.12 is the asymmetry: zones with high accumulation are found in the northwest and especially in the southeast of the ice sheet, where values in excess of 4000 kg m−2 y−1 (>10 m of snow) are predicted. These high snowfall rates are caused by the occasional northward migration of low-pressure areas through Davis Strait (west of the ice sheet) and the presence of the Icelandic Low, a semi-permanent low pressure system east of the southern tip of Greenland, respectively. These systems force relatively warm and moist air masses onto the steep ice sheet, creating orographic precipitation maxima similar to those found in Antarctica. Because of difficult access and the harsh climate conditions, there is a lack of direct SSMB observations from these regions so that large uncertainties remain. The high accumulation creates a steep ice sheet margin, preventing the formation of a wide ablation zone. In contrast, the southwest and northeast of Greenland receive less than 200 kg m−2 y−1 of snowfall, the driest parts of the ice sheet in the northeast even less than 100 kg m−2 y−1 . In these regions, cloud cover is on average low, summer snowfalls are rare and solar irradiation high. As a result, the winter snow layer quickly melts

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away in spring, exposing the dark glacier ice early in the melting season, followed by pronounced summertime melt and runoff. As a result, these parts of the GrIS have well-defined and relatively wide marginal ablation zones (purple colours in Fig. 7.12, < 0). In the southwest, the ablation zone is 100–150 km wide and the lowest parts of the ice sheet experience annual ablation rates of up to 3500 kg m−2 y−1 . A notable feature in some of these marginal regions is the apparent absence of winter accumulation, and this is ascribed to the collection of snow in crevasses (see description in Fig. 7.1).

7.6.2

Temporal SSMB Variability

For the GrIS, reliable SMB time series can be reconstructed as far back as 1958, owing to better observational coverage in the northern hemisphere and hence more reliable atmospheric re-analysis products to drive regional climate models over the ice sheet. However, reliable estimates of D are only available since the early 1990s. Both SMB and D time series are displayed in Fig. 7.14b for the period 1989–2009; this compilation assumes constant discharge before 1992 and uses a linear interpolation to estimate D between those years with observations. Unlike in Antarctica, where runoff is negligible, runoff variability strongly impacts interannual SMB variability on the GrIS. Years of low accumulation tend to cause higher summer ablation due to the lower albedo of bare ice, and therefore year-to-year variations in SMB can be as large as 400 Gt y−1 . The average standard deviation is 100 Gt y−1 , 24% of the average SMB (417 Gt y−1 ), which is four times larger than the relative variability for the AIS. Another contrast to the AIS is that the SMB shows a significant negative trend since about 2000, following atmospheric warming and increased runoff since the early 1990s. In combination with an increase in D since about 1996, which is the result of glacier acceleration in southeast, west and northwest Greenland [34], this has resulted in a persistently negative MB since 1999. Note that the large interannual variability in the beginning of the time series is not sustained during the recent period of strong melt. Between 1990 and 2009, the GrIS has contributed 6 ± 1 mm to post1990 sea level rise, approximately 10% of the total sea level rise over this period (Fig. 7.15).

7.6.3

Role of the Liquid Water Balance

As noted above, runoff [M in Eq. (7.3)] is an important component of the SMB of the GrIS and recent trends therein. Figure 7.16 shows cumulative anomalies (relative to the reference period 1961–1990, when the ice sheet mass balance was approximately zero) of the major components of the liquid water balance (LWB): melt (green line), runoff (orange line) and rainfall (light blue line). The increase in melt clearly dominates changes in the LWB and hence the SMB, with an estimated 3000 Gt anomaly having developed in the two decades after 1990 owing to above-normal

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Fig. 7.16 Cumulative mass anomalies of GrIS mass balance (MB), surface mass balance (SMB) and liquid water balance components [35]. GRACE time series (data courtesy of I. Velicogna and J. Wahr) do not denote absolute values and have been vertically displaced for clarity

meltwater production. Increased rainfall added another 250 Gt to the liquid water mass anomaly. Only 2300 Gt (∼ 70%) of this liquid water anomaly actually reached the ocean as runoff, the remainder is retained in the snowpack, mostly by refreezing and for a small part by capillary retention. Mass losses from the GrIS are further moderated by slightly enhanced snowfall (dark blue line), which added 500 Gt of mass. The total 1990–2010 surface mass balance (red line) anomaly is then ∼ 1700 Gt, which, when added to the estimated mass loss through enhanced solid ice discharge (D, black line) results in a total negative mass anomaly of about 2800 Gt (dark grey line). The trend and seasonal cycle of the reconstructed mass loss compare very well with gravity measurements from space (GRACE, black line).

References 1. Cogley JG and 10 others (2011) Glossary of glacier mass balance and related terms. IHP-VII Technical Documents in Hydrology No. 86, IACS Contribution No. 2, UNESCO-IHP, Paris 2. Eisen O and 15 others (2008) Ground-based measurements of spatial and temporal variability of snow accumulation in East Antarctica. Revs Geophys 46:RG2001 3. Rotschky G, Holmlund P, Isaksson E, Mulvaney R, Oerter H, van den Broeke MR, Winther J-G (2007) A new surface accumulation map for western Dronning Maud Land, Antarctica, from interpolation of point measurements. J Glaciol 53(182):385–398 4. Lenaerts JTM, van den Broeke MR, van de Berg WJ, van Meijgaard E, Kuipers Munneke P (2012) A new, high-resolution surface mass balance map of Antarctica (1979–2010) based on regional atmospheric climate modeling. Geophys Res Lett 39(4):L04501

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5. Helsen MM, van den Broeke MR, van de Wal RSW, van de Berg WJ, van Meijgaard E, Davis CH, Li Y, Goodwin I (2008) Elevation changes in Antarctica mainly determined by accumulation variability. Science 320(5883):1626–1629 6. Thomas R, Davis C, Frederick E, Krabill W, Li Y, Manizade S, Martin C (2008) A comparison of Greenland ice-sheet volume changes derived from altimetry measurements. J Glaciol 54(185):203–212 7. Pritchard HD, Arthern RJ, Vaughan DG, Edwards LA (2009) Extensive dynamic thinning on the margins of the Greenland and Antarctic ice sheets. Nature 461(7266):971–975 8. Ligtenberg SRM, Helsen MM, van den Broeke MR (2011) An improved semi-empirical model for the densification of Antarctic firn. The Cryosphere 5(4):809–819 9. Rignot E, Mouginot J, Scheuchl B (2011) Ice flow of the Antarctic Ice Sheet. Science 333(6048):1427–1430 10. Arthern RJ, Winebrenner DP, Vaughan DG (2006) Antarctic snow accumulation mapped using polarization of 4.3 cm wavelength microwave emission. J Geophys Res 111(D6):D06107 11. Bales RC, Guo Q, Shen D, McConnell JR, Du G, Burkhart JF, Spikes VB, Hanna E, Cappelen J (2009) Annual accumulation for Greenland updated using ice core data developed during 2000–2006 and analysis of daily coastal meteorological data. J Geophys Res 114:D06116 12. van de Berg WJ, van den Broeke MR, Reijmer CH, van Meijgaard E (2006) Reassessment of the Antarctic surface mass balance using calibrated output of a regional atmospheric climate model. J Geophys Res 111(D11):D11104 13. Ettema J, van den Broeke MR, van Meijgaard E, van de Berg WJ, Bamber JL, Box JE, Bales RC (2009) Higher surface mass balance of the Greenland ice sheet revealed by high-resolution climate modeling. Geophys Res Lett 36(12):L12501 14. Burgess EW, Forster RR, Box JE, Mosley-Thompson E, Bromwich DH, Bales RC, Smith LC (2010) A spatially calibrated model of annual accumulation rate on the Greenland Ice Sheet (1958–2007). J Geophys Res Earth Surf 115(F2):F02004 15. Joughin I, Das SB, King MA, Smith BE, Howat IM, Moon T (2008) Seasonal speedup along the western flank of the Greenland ice sheet. Science 320(5877):781–783 16. Bamber JL, Gomez-Dans JL, Griggs JA (2009) A new 1 km digital elevation model of the Antarctic derived from combined satellite radar and laser data—Part 1: data and methods. The Cryosphere 3(1):101–111 17. Rignot E, Bamber JL, van den Broeke MR, Davis C, Li Y, van de Berg WJ, van Meijgaard E (2008) Recent Antarctic ice mass loss from radar interferometry and regional climate modelling. Nat Geosci 1(2):106–110 18. Zemp M, Roer I, Kääb A, Hoelzle M, Paul F, Haeberli W (2008) Global glacier changes: facts and figures. United Nations Environmental Programme (UNEP) and World Glacier Monitoring Service 19. Zemp M, Nussbaumer SU, Gärtner-Roer I, Hoelzle M, Paul F, Haeberli W (eds) (2011) Glacier mass balance Bulletin No. 11 (2008–2009). ICSU (WDS)/IUGG (IACS)/UNEP/UNESCO/WMO, World Glacier Monitoring Service, Zurich, Switzerland 20. Hock R (2003) Temperature index melt modelling in mountain areas. J Hydrol 282:104–115 21. Giesen RH, Oerlemans J (2012) Calibration of a surface mass balance model for global-scale applications. The Cryosphere 6:1463–1481 22. Klok EJ, Oerlemans J (2002) Model study of the spatial distribution of the energy and mass balance of Morteratschgletscher, Switzerland. J Glaciol 48(163):505–518 23. Sedlar J, Hock R (2009) Testing longwave radiation parameterizations under clear and overcast skies at Storglaciären, Sweden. The Cryosphere 3:75–84 24. Pellicciotti F, Raschle T, Huerlimann T, Carenzo M, Burlando P (2011) Transmission of solar radiation through clouds on melting glaciers: a comparison of parameterizations and their impact on melt modelling. J Glaciol 57(202):367–381 25. Giesen RH, Oerlemans J (2010) Response of the ice cap Hardangerjøkulen in southern Norway to the 20th and 21st century climates. The Cryosphere 4:191–213 26. Cogley JG (2009) Geodetic and direct mass-balance measurements: comparison and joint analysis. Ann Glaciol 50(50):96–100

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27. Arendt AA, Echelmeyer KA, Harrison WD, Lingle CS, Valentine VB (2002) Rapid wastage of Alaska glaciers and their contribution to rising sea level. Science 297(5580):382–386 28. Gardner AS, Moholdt G, Wouters B, Wolken GJ, Burgess DO, Sharp MJ, Cogley JG, Braun C, Labine C (2011) Sharply increased mass loss from glaciers and ice caps in the Canadian Arctic Archipelago. Nature 473:357–360 29. Rignot E, Rivera A, Casassa G (2003) Contribution of the Patagonia Icefields of South America to sea level rise. Science 302(5644):434–437 30. Jacob T, Wahr J, Pfeffer WT, Swenson S (2012) Recent contributions of glaciers and ice caps to sea level rise. Nature 482:514–518 31. Lenaerts JTM, van den Broeke MR, Scarchilli C, Agosta C (2012) Impact of model resolution on simulated wind, drifting snow and surface mass balance in Terre Adélie, East Antarctica. J Glaciol 58(211): 821–829 32. van den Broeke MR, Bamber J, Lenaerts J, Rignot E (2011) Ice sheets and sea level: thinking outside the box. Surv Geophys 32(4–5):495–505 33. Church JA, White NJ, Konikow LF, Domingues CM, Cogley JG, Rignot E, Gregory JM, van den Broeke MR, Monaghan AJ, Velicogna I (2011) Revisiting the Earth’s sea-level and energy budgets from 1961 to 2008. Geophys Res Lett 38(18):L18601 34. Rignot E, Kanagaratnam P (2006) Changes in the velocity structure of the Greenland Ice Sheet. Science 311(5763):986–990 35. van den Broeke M, Bamber J, Ettema J, Rignot E, Schrama E, van de Berg WJ, van Meijgaard E, Velicogna I, Wouters B (2009) Partitioning recent Greenland mass loss. Science 326(5955):984– 986

8

Numerical Modelling of Ice Sheets, Streams, and Shelves Ed Bueler

8.1

Introduction

Numerical models aid the understanding of ice flow. They may help you answer the question: when I put together my incomplete understanding of glacier processes into a mathematical model, does the combination behave as I expect? Or why does it mis-behave? Numerical models rarely generate new theoretical understandings, but they can demonstrate flaws in our understanding of glacier processes, and they can illuminate how these processes combine to give observed behaviour. Numerical models should be built with care. Poor computer programming or numerical analysis can lead to inappropriate and spurious physical conclusions. The reader of this chapter may be surprised that a continuum model, and not a code, seems to be our focus much of the time. All numerical codes produce numbers, but we want numbers that actually come from our continuum model. We analyse numerical implementations to see if they match the continuum model and its solutions. This chapter focusses on numerical methods for shallow ice flows (in which the typical depth is much less than the horizontal length scale). It aims to provide examples of numerical codes that actually work. Within our scope are the shallow ice approximation (SIA) in two horizontal dimensions (2D), the shallow shelf approximation (SSA) in 1D, and the mass continuity and surface kinematic equations. We recall the Stokes model presented in Chap. 1, but we do not address its numerical solution. The numerical techniques include finite difference schemes, solving algebraic systems from stress balances, and the verification of codes using exact solutions.

E. Bueler (B) Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, USA e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_8

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Table 8.1 Notation used in this chapter Variable

Description

SI units

A B b g H M n p s T U u ε˙ i j ν ρ ρw τi j τ

A = A(T ) = ice softness in Glen’s flow law Ice stiffness; B = A−1/n Bedrock elevation Gravity Ice thickness Climatic mass balance Exponent in Glen’s flow law Pressure Ice surface elevation Temperature = (u, v) horizontal ice velocity = (u, v, w) 3D ice velocity Strain rate tensor Viscosity Density of ice Density of sea water Deviatoric stress tensor Magnitude of τi j : 2τ 2 = i j τi2j

Pa−n s−1 Pa s1/n m m s−2 m m s−1

∇ ∇.

(spatial) gradient (spatial) divergence

Pa m K m s−1 m s−1 s−1 Pa s kg m−3 kg m−3 Pa Pa m−1 m−1

Our notation is common in the glaciological literature (Table 8.1). Coordinates are t, x, y, z, with z perpendicular to the geoid and positive-upward. If these coordinates appear as subscripts then they denote partial derivatives: u x = ∂u/∂ x. Tensor notation uses subscripts from the list {1, 2, 3, i, j}, so that, for example, τi j and τ13 are entries of the deviatoric stress tensor. This chapter is based on seventeen Matlab codes, each about half a page long, of which only five are printed in this text. All have been tested in Matlab and Octave. They are available from the repository https://github.com/bueler/karthaus. See mfiles/README.md to get started with the codes. When displayed here these codes have their comments stripped for compactness and clarity, but the electronic versions have generous comments and help files.

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8.2

187

Ice Flow Equations

The standard model for slow, incompressible ice flow in glaciers and ice sheets was described in Chap. 1, and takes the form ∇.u = 0, 0 = −∇p + ∇.τ + ρg, ε˙ i j = Aτ n−1 τi j .

(8.1)

Here n is the exponent in Glen’s law, usually taken as n = 3, and (x1 , x2 , x3 ) = (x, y, z) denotes the spatial coordinates. The gradient and divergence operators ∂ ∂ ∇ = ei , ∇. = ei . were defined in Chap. 1, along with the strain rate ten∂ xi ∂ xi ∂u j 1 ∂u i . Also u is the velocity vector, τ is the deviatoric stress sor ε˙ i j = + 2 ∂x j ∂ xi tensor, with components τi j , p is the pressure, ρ is density, and g is the acceleration due to gravity. The rate coefficient A in Glen’s law (8.1)3 is generally taken to depend on temperature (see Chap. 2), but this dependence will be ignored here. The Stokes equations do not contain a time derivative. Therefore geometry, boundary stresses, and ice softness together determine the velocity and stress fields (i.e., u, p, τi j ) instantaneously. Indeed, ice flow simulation codes have no memory of momentum or velocity, and velocity is a ‘diagnostic’ output which is not needed for restarting a simulation.

8.2.1

The Shallow Ice Approximation

The simplest model for an ice sheet uses the shallow ice approximation, as was introduced in Sect. 1.9. Here we briefly review its derivation. We do this in two dimensions, x (horizontal) and z (vertical), before later generalising to three; the rate coefficient A is taken to be constant. The shear stress τ13 is the largest component of the stress tensor, and thus the stress deviator magnitude τ defined in Table 8.1 is approximately equal to |τ13 |. Let us write σ = τ13 , so τ = |σ |. Because z x, the momentum equation (8.1)2 implies p x ≈ σz ,

pz ≈ −ρg,

(8.2)

and with suitable stress-free boundary conditions at the ice surface, the solution is p ≈ ρg(s − z), σ = −ρgsx (s − z);

(8.3)

here z = s denotes the upper ice surface. The stresses are thus approximately determined directly from a balance of forces (the expression for σ is the usual equivalent

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of the classical ρgh sin α). From the flow law (8.1)3 , (twice) the shear strain rate is then approximately u z = −2 A[ρg(s − z)]n |sx |n−1 sx .

(8.4)

This can be integrated twice as in Chap. 1, using a boundary condition of a prescribed sliding law u = u b at the base z = b, and it then follows from the conservation of mass equation that the depth H = s − b satisfies the equation Ht = H n+2 |sx |n−1 sx − H u b x + M,

(8.5)

where M is the surface supply rate due to net accumulation (M > 0) or net ablation (M < 0), measured as a velocity (metres of ice equivalent per second, in SI units), and 2 A(ρg)n ; (8.6) = n+2 note again that A has been assumed constant here. For the particular case when there is no sliding, u b = 0, and the bed is flat, b = 0, (8.5) becomes Ht = H n+2 |Hx |n−1 Hx x + M.

(8.7)

Equation (8.7) is the SIA equation for nonsliding plane flow over a flat bed. It determines the evolution of an ice sheet’s shape in terms of mass balance and ice softness, and will be numerically solved in Sect. 8.4. First we generalise it to two horizontal dimensions and attempt to understand it better. The general rule of thumb in converting a scalar equation to a vector equation is that, where a derivative Hx appears, and H is a scalar quantity (as the depth is here), the vector generalisation of Hx is ∇H ; however, where a derivative u x appears, the natural generalisation of u is as a vector u (as for the velocity), and then the appropriate generalisation of u x is ∇.u. The generalisation of (8.7) is thus Ht = ∇. H n+2 |∇ H |n−1 ∇ H + M.

(8.8)

∂ ∂ . It is a Note that the gradient operator here is in the plane, i. e., ∇ = , ∂x ∂ y straightforward matter to work through the equations to show that this is so; for example, the shear stress approximation is (τ13 , τ23 ) ≈ −ρg(s − z)∇s, and so on. If we can solve (8.8) numerically, then we have a useable model for the Barnes ice cap in Canada, for example [1]. The Barnes ice cap is a particularly simple ice sheet on a rather flat bed.

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8.2.2

189

Analogy with the Heat Equation

A step toward understanding the SIA model is to recognise its analogy with the betterknown heat equation. All numerical methods for solving (8.8) can be understood as modifications of well-known heat equation methods. Consider the temperature T (x, t) of a conducting rod. It satisfies the onedimensional (1D) heat equation Tt = κ Tx x ,

(8.9)

at least when material properties are constant. The constant κ is the thermal diffusivity. This equation has a smoothing effect on the solution T as it evolves in time, because any local maximum in the temperature is flattened (i. e., Tx x < 0 implies T decreases), while any local minimum is also flattened (i. e., Tx x > 0 implies T increases). The 2D heat equation, which we will see is analogous to Eq. (8.8), describes the temperature T (x, y, t) at position x, y in a planar object at time t. Recall Fourier’s law is the formula q = −k∇T for heat flux (k is the thermal conductivity). If we allow an additional heat source f , conservation of internal energy says ρcTt = f − ∇.q, where ρ is density and c is specific heat capacity. Combining these we get the 2D heat equation: ρcTt = f + ∇.(k∇T ). Assuming ρc is constant, define the thermal diffusivity κ = F=

(8.10) k and source term ρc

f ; generally these may vary. The revised 2D heat equation is ρc Tt = F + ∇.(κ∇T ),

(8.11)

which extends (8.9). Figure 8.1 shows a numerical solution of the heat equation (detailed later), where the initial condition is a localized ‘hot spot’. Solutions of the heat equation involve the spreading, in all directions, of any local heat maxima or minima by the process of diffusion. The SIA equation (8.8) and the heat equation (8.11) are both diffusive, timeevolving partial differential equations (PDEs). The diffusivity in the SIA equation, which appears in the same position as κ in (8.11), is the coefficient D = H n+2 |∇ H |n−1 .

(8.12)

In fact a non-sliding shallow ice flow diffuses the ice sheet profile. When or H or |∇ H | are large, the diffusion acts more quickly because D is larger. This analogy explains generally why the surfaces of ice sheets are smooth, at least if we ignore non-fluid processes like crevassing and wind-driven snow features. However, the equation (8.8) is degenerate, because D may go to zero at margins, where H → 0, and at divides and domes, where |∇ H | → 0. In fact, because we expect the ice flux (which is −D∇ H ) to be zero at a land-based margin, it is generally the case that the

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Fig. 8.1 A solution of the heat equation (8.11) with κ = 1 and F = 0. Left: initial condition T (x, y, 0). Right: solution T (x, y, t) at t = 0.02

ice sheet slope at the margin, as described by (8.8), is infinite. This causes issues for numerical methods. Nonetheless we will provide a verified numerical scheme for (8.8) in Sect. 8.4.

8.3

Finite Difference Numerics

As already noted, numerical schemes for the heat equation are a good starting place for solving the SIA equation (8.8) numerically. Finite difference numerical schemes, in particular, replace derivatives in a differential equation by arithmetic. The basic fact is Taylor’s theorem, which says that for a smooth function f (x), f (x + ) = f (x) + f (x) +

1 1 f (x)2 + f (x)3 + · · · . 2 3!

(8.13)

You can replace by its multiples, for example 1 1 f (x)2 − f (x)3 + · · · 2 3! 4 f (x + 2) = f (x) + 2 f (x) + 2 f (x)2 + f (x)3 + · · · . (8.14) 3 f (x − ) = f (x) − f (x) +

The idea for constructing finite difference schemes is to combine such expressions to give approximations of derivatives. Consequently, function values on a grid combine to approximate the differential equation.

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191

Here we want partial derivative approximations, so we apply the Taylor’s expansions one variable at a time. For example, with a general function u = u(x, t), u(x, t + t) − u(x, t) + O(t), t u(x, t + t) − u(x, t − t) + O{(t)2 }, ut = 2t u(x + x, t) − u(x − x, t) ux = + O{(x)2 }, 2x u(x + x, t) − 2 u(x, t) + u(x − x, t) + O{(x)2 }. uxx = (x)2 ut =

(8.15)

The first two approximations (both for u t ) show that finite difference approximations are not unique and may have different accuracies (a term such as O(t) is called an order term, and represents a quantity of size ∝ t); (8.15)1 is termed a forward difference, while the other approximations are called central differences: there can also be backward differences.

8.3.1

Explicit Scheme for the Heat Equation

Now we can state the simplest explicit scheme which approximates the 1D heat equation (8.9) (in the following numerical exposition, we use D rather than κ to represent the diffusion coefficient):

T (x + x, t) − 2T (x, t) + T (x − x, t) T (x, t + t) − T (x, t) . ≈D t (x)2 (8.16) The method is called explicit because, as we shall see, it provides an explicit formula to update T . (Implicit methods require a further solution step.) The following finite difference scheme is not just an approximation of the PDE, but an actual method for computing numbers on a grid. Let (x j , tn ) denote the time-space grid points. Denoting our approximation1 of the solution value T (x j , tn ) by T jn , the finite difference scheme is n n T j+1 − 2 T jn + T j−1 T jn+1 − T jn . (8.17) =D t (x)2 To get a computable formula, let μ = Dt/(x)2 and rearrange the equation: n n T jn+1 = μT j+1 + (1 − 2μ)T jn + μT j−1 .

(8.18)

1 Evaluating the exact solution at a grid point would give a different value from the one we compute

by the numerical scheme! Of course we plan that these numbers will be close, but that needs checking or proof.

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Fig. 8.2 Left: space-time stencil for the explicit scheme (8.18) for the 1D heat equation. Right: two spatial dimension stencil for scheme (8.20)

Note that T jn+1 is given explicitly in terms of values at time tn . Figure 8.2 shows the ‘stencil’ for scheme (8.18): three values at the current time tn are combined to update the one value at the next time tn+1 . How accurate is the scheme (8.18)? Its construction tells us that the difference between the scheme and the PDE (8.9) (with κ = D) is O(t) + O{(x)2 }. This difference between scheme and PDE goes to zero as we refine the grid in space and time (this is called consistency). With care about the smoothness of boundary conditions, and using mathematical facts about the heat equation itself, one can show that the difference between the numerical estimate T jn and the exact solution T (x j , tn ) is also O(t) + O{(x)2 }, which is convergence; see Sect. 8.9. However, to get convergence, the PDE problem must be known to generate smooth solutions and the scheme (8.18) must be stable, which we address below. Then the main theorem for numerical PDE schemes is: consistency plus stability implies convergence; see Sect. 8.9. In these notes we do something rather practical, namely we find problems for which we already know exact values T (x, t), and we compute the differences |T jn − T (x j , tn )|. Thus we determine directly whether they go to zero. This is verification; more on this below.

8.3.2

A First Implemented Scheme

For the first Matlab implementation we consider the two spatial dimension equation (8.11) with κ = D constant and F = 0: Tt = D(Tx x + Tyy ).

(8.19)

n ≈ T (x , y , t ), the 2D explicit scheme is Writing T jk j k n n+1 n T jk − T jk

t

=D

n n + Tn T j+1,k − 2 T jk j−1,k

(x)2

+

n n + Tn − 2 T jk T j,k+1 j,k−1

(y)2

. (8.20)

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193

Fig. 8.3 Numerically-computed temperature at time t = 0.05 on 40 × 40 grids. Everything in the two runs is the same, except that the left figure has t = 0.0005 so Dt/(x)2 = 0.2, while the right has t = 0.001 so Dt/(x)2 = 0.4

The stencil for the right-hand side of (8.20) is in Fig. 8.2. The scheme in (8.20) is implemented in the Matlab m-file heat.m below. We use T = 0 values on the boundary of the square −1 < x < 1, −1 < y < 1. The initial condition is a Gaussian: T (x, y, 0) = exp(−30(x 2 + y 2 )). The code uses Matlab ‘colon’ notation to remove loops over spatial variables. Here is an example run: >>

heat(1.0,30,30,0.001,20);

This sets D = 1.0 and uses a 30 × 30 spatial grid. We take N = 20 time steps of t = 0.001. The result was shown in Fig. 8.1, right. This has the look of success. heat.m function T = heat(D,J,K,dt,N) dx = 2 / J; dy = 2 / K; [x,y] = meshgrid(-1:dx:1, -1:dy:1); T = exp(-30*(x.*x + y.*y)); mu_x = dt * D mu_y = dt * D for n=1:N T(2:J,2:K) mu_x * mu_y * end surf(x,y,T),

/ (dx*dx); / (dy*dy); = T(2:J,2:K) + ... ( T(3:J+1,2:K) - 2 * T(2:J,2:K) + T(1:J-1,2:K) ) + ... ( T(2:J,3:K+1) - 2 * T(2:J,2:K) + T(2:J,1:K-1) ); shading(’interp’),

xlabel x,

ylabel y

Very similar runs seem to fail, however. Compare the results shown in Fig. 8.3: >> heat(1.0,40,40,0.0005,100); >> heat(1.0,40,40,0.001,50);

% Figure 8.3, left % Figure 8.3, right

Both runs compute temperature T on the same spatial grid, at the same final time, but they have different time steps. The second run clearly shows instability.

194

8.3.3

E. Bueler

Stability Criteria and Adaptive Time Stepping

A process is stable if small perturbations do not affect it dramatically. We say solutions of differential equations are stable if small perturbations cause small changes: a hanging pendulum is stable; an upside-down pendulum is unstable. Numerical instability is similar: small changes to initial conditions, or merely rounding errors, cause large changes in the numerical solution. Typically such instability is manifested by oscillations at the scale of the grid: this is what is seen in Fig. 8.3 on the right. How do we avoid the instability? We need to understand the scheme better. It turns out we have not made an implementation error, but we must be more careful with the choice of space and time steps. n + (1 − 2μ)T n + μT n . Recall our 1-D explicit scheme (8.18): T jn+1 = μT j+1 j j−1 The new value T jn+1 is an average of the old values, in the sense that the coefficients add to one. Actually, this is an average only if the middle coefficient is positive. A linear combination with coefficients which add to one is not an average if any coefficients are negative! (For example, we would not accept 13 as an “average” of 5 and 7, but of course we can write 13 = −3 × 5 + 4 × 7, and −3 + 4 = 1.) Averaging is stabilising because averaged wiggles are always smaller than the original wiggles. We want this property. So, what would follow from requiring the middle coefficient in (8.18) to be positive?: 1 − 2μ ≥ 0

⇐⇒

1 Dt ≤ (x)2 2

⇐⇒

t ≤

(x)2 . 2D

(8.21)

This condition on the size of t is a sufficient stability criterion. (It is enough to guarantee stability, though something weaker might do.) For given x, shortening the time step so that t ≤ (x)2 /(2D) will make the scheme into an averaging process. Applying this same idea to the 2D heat equation (8.19) leads to the stability condition that 1 − 2μx − 2μ y ≥ 0 where μx = Dt/(x)2 and μ y = Dt/(y)2 . In the cases shown in Fig. 8.3 with x = y, this condition requires Dt/(x)2 ≤ 0.25, which precisely distinguishes between the two parts of the figure. In summary, instability in runs of heat.m happens if the time step t is too big relative to the spacing x. The stability criterion above is easily satisfied by making each time step shorter. Doing so in a program is an adaptive implementation, with guaranteed stability. To show how easy it is to implement, heatadapt.m (not shown) is the same as heat.m except that the time step is chosen from the stability criterion. It cannot generate the instability seen in Fig. 8.3. However, if the diffusivity D is large or the spatial steps x, y are small, then explicit, adaptive implementations like this one must take short time steps to assure stability. In general, this is the disadvantage of explicit schemes: the time steps need to be very short.

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8.3.4

195

Implicit Schemes

There is an alternative stability fix instead of adaptivity, namely implicitness. For example, the finite difference scheme T jn+1 − T jn t

=D

n+1 n+1 T j+1 − 2 T jn+1 + T j−1

(x)2

(8.22)

(cf. (8.17)) is an O(t) + O{(x)2 } accurate implicit scheme for (8.9). Such implicit schemes for the heat equation are stable for any positive time step t > 0 (they are unconditionally stable). Another well-known example of an implicit scheme is the Crank-Nicolson scheme, which is unconditionally stable for the heat equation and which has yet smaller error O{(t)2 } + O{(x)2 }. But implicit schemes are harder to implement. The unknown solution values at time step tn+1 are treated as a vector in a large system of equations which must be formed and solved at each time step. If the PDE is nonlinear—the SIA equation is highly nonlinear—then the system of equations may be hard to solve. In this chapter we stay with the above ideas of adaptive explicit schemes. Generally, there is a tradeoff between the implementability of adaptive explicit schemes and the stability of implicit schemes.

8.3.5

Numerical Solution of Diffusion Equations

We aim to model ice flows numerically, not just heat conduction. We have an analogy, however, which says that the SIA equation is like the heat equation: both are diffusive. In this section, because we wish to solve the SIA equation on a non-uniform bed z = b(x, y), for which the appropriate version of (8.8) is, bearing in mind (8.5), Ht = ∇. H n+2 |∇s|n−1 ∇s + M,

(8.23)

where s = H + b, we construct a numerical scheme for a generalized diffusion which additionally has a ‘shift’ inside the gradient, namely Tt = F + ∇. (D∇(T + b)) .

(8.24)

In equation (8.24), the source term F(x, y), the diffusivity D(x, y), and the “shift” b(x, y) may all vary in space. We now provide a code that solves (8.24) and is then used in SIA solutions.

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An adaptive explicit method for the diffusion equation in (8.24) is conditionally stable, with the same time step restriction as for the constant diffusivity case, as long as we evaluate D(x, y) at staggered grid points. That is, we use this expression for the second derivative: ∇. (D∇ X ) ≈ +

D j+1/2,k (X j+1,k − X j,k ) − D j−1/2,k (X j,k − X j−1,k ) (x)2 D j,k+1/2 (X j,k+1 − X j,k ) − D j,k−1/2 (X j,k − X j,k−1 ) , (y)2

(8.25)

where X = T + b. Figure 8.4 shows the stencil. The code diffusion.m (below) solves (8.24) numerically. The user supplies the diffusivity D(x, y) on the staggered grid. The initial temperature T (x, y, 0), source term F(x, y), and shift b(x, y) are supplied on the regular grid. When using this code for standard diffusions, or for the flat-bed SIA equation, we would take b = 0.

k+1

k+1

k

k

k-1

k-1 j-1

j

j+1

j-1

j

j+1

Fig. 8.4 Left: spatial stencil for staggered grid evaluation of diffusivity (at triangles) in the diffusion equation (8.24). Right: stencil showing how the staggered-grid diffusivity (triangle) can be evaluated in the SIA equation, from surface elevation (diamonds) and thicknesses (squares)

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197

diffusion.m function [T,dtav] = diffusion(Lx,Ly,J,K,Dup,Ddown,Dright,Dleft,T0,tf,F,b) dx = 2 * Lx / J; dy = 2 * Ly / K; [x,y] = ndgrid(-Lx:dx:Lx, -Ly:dy:Ly); T = T0; if nargin < 11, F = zeros(size(T0)); end if nargin < 12, b = zeros(size(T0)); end t = 0.0; count = 0; while t < tf maxD = [max(max(Dup)) max(max(Ddown)) max(max(Dleft)) max(max(Dright))]; maxD = max(maxD); if maxD 0. We calculate this exact solution by a method which generalizes to the SIA equation. The Green’s function of the heat equation is self-similar over time, in the sense that it changes shape only by shrinking the output (vertical) axis and lengthening the input (horizontal) axis, as shown in Fig. 8.5. These scalings are related to each other by the conservation of energy, which says that the total heat energy is independent of time. In particular, the Green’s function of the 1D heat equation is

x2 1 . (8.26) exp − T (x, t) = (4π Dt)1/2 4Dt Similarity variables for this solution, the above-mentioned scalings, are input scaling

s

=

x √ , t

output scaling

T (x, t)

=

φ(s) √ . t

(8.27)

The invariant shape function is φ(s) = (4π D)−1/2 e−s /4D . Note that all time dependence is in the input and output scalings. In a spatial domain [−L, L], where L is sufficiently large, the prescription of boundary conditions of T = 0 at x = ±L is in practice consistent with (8.26); in 2

delta function δ(x) is a generalised function having the property

∞ that it is zero everywhere, except at zero, where it is infinite in such a way that the integral −∞ δ(x) d x = 1. 2 The

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that case a numerical solver for the 1D heat equation should produce numbers which are close to this exact solution: see the exercises.

8.5.2

Halfar’s Exact Similarity Solution to the SIA

Now we jump forward from Green’s idea in about 1830 to the year 1983 in which P. Halfar published the similarity solution of the SIA equation in the case of cylindrical symmetry, a flat bed, and zero surface mass balance [35]. Halfar’s solution to the SIA model (8.8), using a Glen exponent n = 3, has scalings s=

r t 1/18

,

H (r , t) =

φ(s) , t 1/9

(8.28)

where r = (x 2 + y 2 )1/2 , and φ(s) is given below. The scalings in (8.28) imply that, quite differently from heat, the diffusion of ice slows down severely as the shape flattens out; the powers t −1/9 and t −1/18 imply very slow change for large times t. The Halfar solution can be written in terms of a single dimensional variable, which we may take to be the conserved volume of the ice sheet V0 . There is no intrinsic time scale, but it is useful to define a characteristic time scale t0 , and then to write the solution in essentially dimensionless form. While t0 and V0 are independent, it is yet more convenient to use instead two independent measures of depth and radius, H0 and R0 , these being the maximum depth and the ice sheet radius at the time t0 . Some laborious algebra leads to the definition (for arbitrary n) R0n+1 2n + 1 n , (8.29) t0 = n+1 (5n + 3) H02n+1 and then the Halfar solution is 2α α (n+1)/n n/(2n+1) t0 t0 r H = H0 , 1− t R0 t

(8.30)

where α=

1 . 5n + 3

(8.31)

Evidently the volume is V0 = O(R02 H0 ),3 and the margin position r0 is given by α t . (8.32) r 0 = R0 t0 2π n 2n 3n + 1 R02 H0 , where B(z, w) is the beta B , n+1 n + 1 2n + 1 function. For n = 3, this gives V0 = 1.974R02 H0 .

3 Specifically,

we can compute V0 =

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Numerical Modelling of Ice Sheets, Streams, and Shelves

7000

7000

7000

7000

201 7000

6000

6000

6000

6000

6000

5000

5000

5000

5000

5000

4000

4000

4000

4000

4000

3000

3000

3000

3000

3000

2000

2000

2000

2000

2000

1000

1000

1000

1000

1000

0

0

0

0

0

300

600

900

0

300

600

900

0

300

600

900

0

300

600

900

0

0

300

600

900

Fig. 8.6 Halfar’s solution H (r , t) (8.30) of the SIA equation (8.8) on a flat bed, with n = 3 and zero mass balance. The solution is shown on H (m) versus r (km) axes for times t = 1, 10, 100, 1000, 10000 years, in a case where t0 = 422 years

The ice thickness given by (8.30) is plotted in Fig. 8.6, using n = 3. Note that the slope is infinite at the margin. The decay of the ice sheet is very slow at large times, due to the small value of the exponent α.

8.5.3

Using Halfar’s Solution

Formula (8.30) is simple enough to use for verifying time-dependent SIA models. The code verifysia.m (not shown) takes as input the number of grid points in each (x, y) direction. It uses the Halfar solution at 200 years as the initial condition, does a numerical run to time 20,000 y, using siaflat.m above, and then compares the result to the Halfar formula for that time: >> verifysia(20) average thickness >> verifysia(40) average thickness >> verifysia(80) average thickness >> verifysia(160) average thickness

error

= 22.310

error

= 9.490

error

= 2.800

error

= 1.059

We see that the thickness error, namely the absolute value of the difference between the numerical and exact thickness solutions at t = 20, 000 y, decreases with increasing grid resolution. This is as expected for a correctly implemented code. What is less obvious, perhaps, is that almost any numerical implementation mistake—almost any bug—will break this property, and these errors will not shrink.

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One might ask, is the Halfar solution also useful for modelling real ice masses? Nye et al. [2] compared the long-time consequences of different flow laws for the south polar cap on Mars. In particular, they evaluated CO2 ice and H2 O ice softness parameters by comparing the long-time behaviour of the corresponding Halfar solutions. Their conclusions: …none of the three possible [CO2 ] flow laws will allow a 3000-m cap, the thickness suggested by stereogrammetry, to survive for 107 years, indicating that the south polar ice cap is probably not composed of pure CO2 ice [but rather] water ice, with an unknown admixture of dust.

This theoretical result has been confirmed by the observation and sampling of the polar geology of Mars. Are exact solutions always available when needed? No; but many ice flow models already have exact solutions which are relevant to verification. For example, we will use van der Veen’s solution for ice shelves in a later section. Sources for additional exact solutions appear in Sect. 8.9.

8.5.4

A Test of Robustness

Verification is a reasonable start to testing a code. Another kind of test is for robustness: does the model break when you ask it to do hard things? In contrast to verification, we might not have precise knowledge of what it should do, but we know the program should act in a ‘reasonable’ way. The robustness test in the program roughice.m (not shown) demonstrates that siaflat.m can handle an ice sheet with extraordinarily large driving stresses. Recall that the driving stress is τ b = −ρg H ∇s. This quantity appears in the SIA model as the value of the shear stress (τ13 , τ23 ) at the base of the ice. We give siaflat.m a randomly-generated initial ice sheet which is of the worst possible sort. It is both thick—mean thickness 3000 m—and it has large surface slopes. Such an initial sheet is shown in the left side of Fig. 8.7. During the run of 50 model years, the time step is determined adaptively, increasing from 0.0002 to 0.2 years. The maximum value of the driving stress decreases from 5.7 × 106 to 3.6 × 105 Pa.

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203

Fig. 8.7 siaflat.m evolves the worst possible initial ice sheet (left), an ice cap with huge driving stresses, to the ice cap at right after 50 model years.

At the end of the run the ice cap has the shape shown at right in Fig. 8.7. The shape is rather close to a Halfar solution; indeed Halfar proved that all solutions of the zero mass balance SIA on a flat bed asymptotically approach the Halfar solution.

8.6

Applying Our Numerical Ice Sheet Model

Finally we apply the model to the Antarctic ice sheet. First we modify siaflat.m to allow non-flat bedrock elevation b(x, y) and time-independent surface mass balance M(x, y). Also we calve floating ice, and we enforce non-negative thickness at each timestep. The result is siageneral.m (not shown). We use measured accumulation, bedrock elevation, and surface elevation from ALBMAPv1 data [3]. Melt is not modelled so the surface mass balance is equal to the accumulation rate. These input data are read from a NetCDF file by additional codes buildant.m and netcdf.m (not shown). The code ant.m (not shown) calls siageneral.m to do the simulation in blocks of 500 model years. The volume is computed at the end of each block. Figure 8.8 shows the initial and final surface elevations from a run of 40,000 model years on a x = y = 50 km grid. The runtime on a typical laptop is a few minutes. Note that areas of low-slope and (actually) fast-flowing ice experience thickening in the model, while near-divide ice in East Antarctica thins. Assuming the present-day Antarctic ice sheet is near steady state, these thickness differences reflect the lack of a sliding mechanism in the former case and the lack of thermomechanical coupling in the latter case. Figure 8.9 compares the ice volume time series for 50, 25, and 20 km grids. This result, namely grid dependence of the ice volume, is typical. One cause is that the steep gradients near the ice margin are poorly resolved with any of these coarse resolutions. Mainly this result is a warning about the interpretation of model runs on fixed grids. Even if the data are available only on a fixed grid, the model should be run at different resolutions to evaluate the robustness of the model results.

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Fig. 8.8 Left: initial surface elevation (m) of the Antarctic Ice Sheet. Right: final surface elevation at the end of a 40 ka model run on 50 km grid Fig. 8.9 Ice volume time series of the modelled Antarctic ice sheet, in units of 106 km3 , from runs on 50 km (red), 25 km (green), and 20 km (blue) grids

8.7

Shelves and Streams

The shallow ice approximation applies to shallow flows in which the principal resistive stress to the gravitational driving force is the shear stress. However, there are other situations in which the small ratio of depth to length is also used. Two such situations are those which occur in ice streams and ice shelves, in each of which the principal resistive stress is the longitudinal stress. The shallow shelf approximation (SSA) stress balance applies to ice shelves as its name suggests. It applies best to parts of ice shelves which are away from grounding lines and calving fronts. The SSA also applies reasonably well to ice streams, like those in Fig. 8.10 whose bed topography is gentle, and for which there is low basal resistance. But what is, and what is not, an ice stream? Ice streams slide at 50–1,000 m y−1 , they flow predominantly by sliding, and typically they flow into ice shelves. Pres-

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Fig. 8.10 Left: satellite-radar-derived surface speeds (in colour) of ice streams on the Siple Coast in Antarctica. The SSA model is applicable to such ice streams. Right: Cross sections, without vertical exaggeration, of the Jakobshavn Isbræ outlet glacier in Greenland (a) and the Whillans Ice Stream on the Siple Coast (b); this is Fig. 1 of Truffer and Echelmeyer [4], reprinted from the Annals of Glaciology with permission of the International Glaciological Society

surised liquid water at their beds plays a critical role enabling their fast flow. However, there are other fast-flowing grounded parts of ice sheets called outlet glaciers. They can have even faster surface speed (up to 10 km y−1 ), but it is often uncertain how much of this speed is from sliding at the base. In an outlet glacier there is substantial vertical shear within the ice column, sometimes caused by soft temperate ice in a significant fraction of the thickness. Finally, outlet glaciers are strongly controlled by fjord-like, high slope bedrock topography. Figure 8.10 (right) compares the shallowness and bedrock topography of an outlet glacier and an ice stream. Few simplifying assumptions are appropriate for outlet glaciers, and the SSA may not be an adequate model.

8.7.1

The Shallow Shelf Approximation (SSA)

The principal assumption behind the SSA, other than the shallowness of the flow, is that the basal shear stress is ‘small’ (in the case of an ice shelf it is zero), so that there is no shear in the flow. In a two-dimensional ‘flow line’ model, the horizontal velocity u is a function of x and t only. In this case the stress balance equation is 1

(2B H |u x | n −1 u x )x − C|u|m−1 u = ρg H sx .

(8.33)

The first term is the vertically-integrated longitudinal stress gradient, also called the ‘membrane’ stress when there are two horizontal variables. The second term is the basal resistance, which is zero in an ice shelf. The negative of the term on the right is the driving stress. Thus the SSA equation describes a balance wherein longitudinal strain rates are determined by the integrated ice stiffness (i. e., the coefficient B H ), the slipperiness of the bed (i. e., by the coefficient C and the power m) and the geometry of the ice sheet (i. e., the thickness H and the surface slope sx ).

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We assume that the ice stiffness B = A−1/n is independent of the vertical coordinate z. (Models which are not isothermal must compute an appropriate vertical average of the stiffness.) The basal resistance term τb = −C|u|m−1 u is the sliding law, here of Weertman power law type (see Sect. 3.2). 1 Denoting the coefficient ν¯ = B|u x | n −1 , called the effective viscosity, (8.33) can be written (2ν¯ H u x )x − C|u|m−1 u = ρg H sx ;

(8.34)

it is understood that the viscosity ν¯ depends on the velocity solution u. Equation (8.34) simplifies if the ice is floating, because we assume zero resistance is applied by the ocean (and by the atmosphere, for that matter). Also, the ice surface elevation is proportional to the thickness if the ice is floating. Where the bed lies below the sea-surface level z = 0, we have b < 0 and the inequality ρ H < −ρw b defines the flotation criterion. For grounded ice we know ρ H ≥ −ρw b and that the driving stress is −ρg H sx = −ρg H (Hx + bx ). On the floating side we know ρ H < −ρw b and that s = (1 − ρ/ρw )H so the driving stress is −ρg H sx = −ρ(1 − ρ/ρw )g H Hx . Thus the SSA becomes ρ H Hx (8.35) (2ν¯ H u x )x = ρg 1 − ρw for floating ice. A useful observation about this flow line equation is that both left and right hand expressions are derivatives; this fact can be used to build a 1D exact solution. Specifically, a first integral of (8.35) yields ux =

ρg 4B

1−

ρ ρw

n H n,

(8.36)

where the integration constant can be taken as zero on applying a force balance at the (calving) front. For a steady 1D ice shelf, in which Ht = 0, the mass continuity equation (8.8) reduces to M = (u H )x and thus x u H = Q(x) = M dx + QI , (8.37) xg

where x g is the grounding line position and Q I is the ice flux delivered from the ice sheet. Then H = Q/u, and (8.36) reduces to a simple first order differential equation for u, whose solution can be written down as an integral. This exact solution depends on the ice thickness Hg and velocity u g at the grounding line; note Q I = Hg u g . For the surface mass balance M we will choose a positive constant M0 . Supposing Hg = 500 m, u g = 50 m y−1 and M0 = 0.3 m y−1 , we find the results in Fig. 8.11, which are from the code exactshelf.m (not shown). We will use this exact solution to verify a numerical SSA code. Note that driving stresses are much higher near the grounding line than away from it, and thus the highest longitudinal stresses, strain rates, and thinning rates all occur near the grounding line.

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Fig. 8.11 The upper and lower surface elevation (m; left) of the exact ice shelf solution and its velocity (m y−1 ; right); x = 0 is the grounding line

8.7.2

Numerical Solution of the SSA

Suppose the ice thickness is a fixed function H (x). To find the velocity we must solve the nonlinear PDE (8.34) or (8.35) for the unknown u(x). When we do this numerically an iteration is needed because of the nonlinearity. The simplest iterative idea is to use an initial guess at the velocity, which allows us to compute an effective viscosity and then get a new velocity solution from a linear PDE problem. Then we recompute the effective viscosity, solve for a new velocity, and repeat until things stop changing. This is a Picard-type iteration. Specifically, denote the previous velocity iterate as u (k−1) and the current iterate (k−1) n1 −1 | and define W (k−1) = 2ν¯ (k−1) H . Solve the as u (k) . Compute ν¯ (k−1) = B|u x following linear elliptic PDE for the unknown u (k) at each iteration: W (k−1) u (k) − C|u (k−1) |m−1 u (k) = ρg H sx . (8.38) x x

If the difference between u (k−1) and u (k) were ever to be zero then we would have a solution of (8.34). In practice we stop the iteration when the difference is small enough. Equation (8.38) is a linear boundary value problem. We can write it in the general form (W (x)u x )x − α(x)u = β(x)

(8.39)

where the functions W (x), α(x), β(x) are known in this context. Equation (8.39) applies on an interval of the x-axis. For boundary conditions we will suppose that x = x g is a location where the velocity is known, u(x g ) = u g , as in Fig. 8.11. In the ice shelf case (where C = 0) we also have the calving front condition ρ gH2 2B H |u x |1/n−1 u x = 21 ρ 1 − (8.40) ρw at the end of the ice shelf x = xc ; this was also used in writing (8.36). The boundary condition (8.40) can be written in the form u x (xc ) = γ , where γ is given in terms of known quantities, including the known thickness at the calving front.

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How do we get an initial guess u (0) ? Generally this may require effort, but we can give brief answers for the 1D cases. For floating ice, an initial velocity comes from assuming a uniform strain rate provided by the calving front condition: u (0) (x) = γ (x − x g ) + u g . For grounded ice, we may assume ice is held by basal resistance 1/m only: u (0) (x) = −C −1 ρg H sx .

8.7.3

Numerics of the Linear Boundary Value Problem

Suppose Eq. (8.39) applies on [x g , xc ] = [0, L]. We discretize with equal spacing x and index j = 1, 2, . . . , J + 1, so that x1 = 0 and x J +1 = L are endpoints. The coefficient W (x) is needed on a staggered grid, for stability and accuracy reasons similar to those for the SIA diffusivity. Our finite difference approximation of (8.39) is W j+ 1 (u j+1 − u j ) − W j− 1 (u j − u j−1 ) 2 2 − αju j = βj (8.41) (x)2 For the left end boundary condition we have u 1 = u g given, which is easy to include in the linear system (below). For the right end boundary condition we have u x (L) = γ , which requires more thought. First introduce a notional point x J +2 . Now require (u J +2 − u J )/(2x) = γ . Using Eq. (8.41) for j = J + 1, we then eliminate the u J +2 variable by hand. This determines the form of the last equation in our linear system. Thus each iteration to solve the SSA stress balance has the form Av = b.

(8.42)

Indeed, at each location x1 , . . . , x J +1 we can write an equation, basically a row of the matrix A in (8.42), involving the unknown velocities. (Note that A here does not symbolise the ice flow rate constant.) In more detail it is the following linear system of J + 1 equations: ⎤⎡ ⎤ ⎡ ⎤ ⎡ 1 ug u1 ⎥ ⎢ u 2 ⎥ ⎢ β2 (x)2 ⎥ ⎢W3/2 A22 W5/2 ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ u 3 ⎥ ⎢ β3 (x)2 ⎥ ⎢ W5/2 A33 ⎥⎢ ⎥ ⎢ ⎥ ⎢ (8.43) ⎥ ⎢ .. ⎥ = ⎢ ⎥ ⎢ .. .. .. ⎥⎢ . ⎥ ⎢ ⎥ ⎢ . . . ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎣ WJ− 1 AJ J W J + 1 ⎦ ⎣ u J ⎦ ⎣β J (x)2 ⎦ 2 2 u J +1 b J +1 A J +1,J A J +1,J +1 The diagonal entries are A22 = −(W3/2 + W5/2 + α2 (x)2 ), . . .

A J J = −(W J − 1 + W J + 1 + α J (x)2 ). 2

2

There are special cases for the coefficients in the last equation, A J +1,J = 2W J + 1 , 2

A J +1,J +1 = −(2W J + 1 + α J +1 (x)2 ). 2

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For the right side of the last equation, b J +1 = −2γ x W J + 3 + β J +1 (x)2 . We 2 have assumed that W J + 3 = W J + 1 . 2 2 Equation (8.43) is a tridiagonal linear system, which is easy to solve numerically, but it is not really necessary to find out how to do this. It is fully appropriate to give the system (8.43) to Matlab’s linear solver, the ‘backslash’ operator v = A \ b, and not worry further about solving finite linear systems at this initial implementation stage. Thus we now have a code to solve (8.39) by finite differences and linear algebra, namely flowline.m below. flowline.m function u = flowline(L,J,gamma,W,alpha,beta,V0) dx = L / J; rhs = dxˆ2 * beta(:); rhs(1) = V0; rhs(J+1) = rhs(J+1) - 2 * gamma * dx * W(J+1); A = sparse(J+1,J+1); A(1,1) = 1.0; for j=2:J A(j,j-1:j+1) = [ W(j-1), -(W(j-1) + W(j) + alpha(j) * dxˆ2), W(j) ]; end A(J+1,J) = W(J) + W(J+1); A(J+1,J+1) = - (W(J) + W(J+1) + alpha(J+1) * dxˆ2); scale = full(max(abs(A),[],2)); for j=1:J+1, A(j,:) = A(j,:) ./ scale(j); rhs = rhs ./ scale;

end

u = A \ rhs;

By manufacturing exact solutions to (8.39) (see further notes in Sect. 8.9) we can test this code. In fact, results from testflowline.m (not shown) demonstrate that our implemented numerical scheme converges at the expected rate O{(x)2 }. We do this test before proceeding to solve the nonlinear SSA problem.

8.7.4

Solving the Stress Balance for an Ice Shelf

The code ssaflowline.m (below) numerically computes the velocity for an ice shelf. The thickness is assumed to be given, so we are not solving the full, coupled ice shelf problem. This code implements Picard iteration to solve the nonlinear equation (8.35). It calls ssainit.m (not shown) to get the initial iterate u (0) (x), as already described, and it calls flowline.m at each iteration. It also calls small helper functions stagav,regslope,stagslope (not shown) to compute certain gridded values. Now we can ask: does ssaflowline.m work correctly? The exact velocity solution shown in Fig. 8.11 allows us to compare the numerical and exact solutions by finding the maximum difference between them. This uses the exact thickness shown in Fig. 8.11, from exactshelf.m. A convergence comparison, shown in Fig. 8.12, is done by codes testshelf.m and shelfconv.m (not shown). At

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Fig. 8.12 The numerical SSA velocity solution from ssaflowline.m converges to the exact solution, at nearly the expected rate O((x)2 ), as the grid is refined from spacing x = 4 km to x = 62 m. Each circle gives the maximum velocity error for a solution on a given grid. The red line shows the linear best-fit through the circles. Even on the coarsest grid the error is less than 1 m y−1 while the maximum velocity is ∼ 300 m y−1

screen resolution, the numerical velocity solution from a coarse grid computation is identical to that shown in the right part of Fig. 8.11, so there is no need to show the numerical solution. ssaflowline.m function [u,u0] = ssaflowline(p,J,H,b,ug,initchoice) if nargin ˜= 6, error(’exactly 6 input arguments required’), end dx = p.L / J; x = (0:dx:p.L)’; xstag = (dx/2:dx:p.L+dx/2)’; alpha = p.C * ones(size(x)); h = H + b; hx = regslope(dx,h); beta = p.rho * p.g * H .* hx; gamma = ( 0.25 * p.Aˆ(1/p.n) * (1 - p.rho/p.rhow) *... p.rho * p.g * H(end) )ˆp.n; u0 = ssainit(p,x,beta,gamma,initchoice); u = u0; Hstag = stagav(H); tol = 1.0e-14; eps_reg = (1.0 / p.secpera) / p.L; maxdiff = Inf; W = zeros(J+1,1); iter = 0; while maxdiff > tol uxstag = stagslope(dx,u); sqr_ux_reg = uxstag.ˆ2 + eps_regˆ2; W(1:J) = 2 * p.Aˆ(-1/p.n) * Hstag .* sqr_ux_reg.ˆ(((1/p.n)-1)/2.0); W(J+1) = W(J); unew = flowline(p.L,J,gamma,W,alpha,beta,ug); maxdiff = max(abs(unew-u)); u = unew; iter = iter + 1; end

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Fig. 8.13 Results from tuning the single, constant ice stiffness parameter B. Left: observed radarderived (white) and modelled (black) ice velocities are nearly coincident across the whole Ross ice shelf. The grounding line, at which observed velocities were applied as boundary conditions, is the thin black curve. Right: observed versus modelled flow speeds (in metres per year) for the same locations

8.7.5

Realistic Ice Shelf Modelling

Real ice shelves have two horizontal dimensions. They are frequently confined in bays, and thus they experience side drag. Ice flow velocities vary spatially and temporally along their grounding lines (the curves where the flotation criterion first becomes an equality). Furthermore, real ice shelves have interesting boundary processes, including high basal melt near grounding lines, marine ice basal freeze-on (see Chap. 5) and fracturing which can cause full-depth penetration near the calving front. Nonetheless, diagnostic (i. e., fixed geometry) ice shelf modelling in two horizontal variables, done in a similar manner to the above example, where the velocity is the model unknown but the thickness is known, is quite successful with the isothermal SSA model. For example, Fig. 8.13 shows a Parallel Ice Sheet Model (PISM) result for the Ross ice shelf, compared to observed velocities. There is only one tuned parameter, the constant value of the ice stiffness B. Many ice shelf models yield comparable matches [5].

8.8

A Summary of Numerical Ice Flow Modelling

In this chapter we have illustrated some general principles about numerical modelling. First of all, it is worth spending time with the continuum model, to understand its behaviour. Second, it is helpful to modularize your codes, so the component parts can be tested: is a particular component robust? Does it show convergence to exact solutions? Regarding the specific problems of ice flow, we have seen that distributed stress balance equations like the SSA determine horizontal velocity from geometry and boundary conditions. These equations are nonlinear so iteration is necessary. At each iteration a sparse matrix ‘inner’ problem is solved; non-experts should give this

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matrix problem to a solver package. The SIA stress balance is simpler in that the velocity follows immediately from integration of the driving stress. Note that in ice sheet models, the mass continuity equation contains the time derivative. It describes how the ice geometry evolves. The numerical nature of this equation depends on which is the stress balance which supplies the velocity. It is a diffusion equation for frozen bed, large scale flows (SIA). It is more nearly hyperbolic (i. e., like the wave equation) for membrane stresses (e. g., SSA), especially when there is no basal resistance, as in ice shelves. It has some diffusiveness for ice streams (but how much?), and more for outlet glaciers. In any case, it is a kind of transport equation, but with diffusive character at larger spatial scales.

8.9

Notes

The SIA model, which was derived by several authors (Fowler and Larson [6], Morland and Johnson [7], Hutter [8]), follows by scaling the Stokes equations using the aspect ratio ε = [H ]/[L], where [H ] is a typical thickness of an ice sheet and [L] is a typical horizontal dimension. After scaling one drops the terms that are O(ε) or smaller (e. g., [8]); this is a ‘small-parameter argument’; for example, see the treatment in Sects. 1.9 and 1.11. In one such derivation there are no O(ε) terms in the scaled equations, so one only drops O(ε2 ) terms (e. g., [9]). The SIA model is re-formulated as a well-posed free boundary problem by Jouvet and Bueler [10], which provides the correct boundary condition at grounded margins. The Mahaffy scheme for diffusivity used here [1] is not the only possible scheme [11]. The SSA model [12] was derived by Morland [13] for ice shelves and by MacAyeal [14] for ice streams. A well-posed steady whole ice sheet model using only the SSA is given by Schoof [15]. As noted in Chap. 3, basal water is required for ice streaming. To model the supply to subglacial water drainage, one must at least compute the ice base temperature and the basal melt rate through the energy conservation equation (see Chap. 2). In fact, a key modelling issue omitted in the current chapter is thermomechanical coupling (see Chap. 2). This is important because the ice softness A(T ) varies by three orders of magnitude in the temperature range relevant to ice sheet modelling. Ice temperature therefore gives ice sheet dynamics a long memory of past climate (e. g., Johnsen et al. [16]). Both dissipation of gravitational potential energy and geothermal flux are major heat sources which produce basal melt. For example, each year the Jakobshavn drainage basin in Greenland dissipates enough gravitational potential energy to melt more than 1 km3 of ice. Using an enthalpy variable is a good way to track the energy content of polythermal ice masses [17], but there are other methods for this [18]. One of the most significant issues in modelling ice sheets using shallow models is to describe the switch, in space and time, between shear-dominated and membranestress-dominated flow. It is not a good idea to abruptly switch from the SIA model to the SSA model at the edge of an ice stream, by whatever criterion that switch might be applied, though this has been attempted [19,20]; these might be called ‘horizontal hybrid’ schemes. Instead, ‘vertical’ hybrid schemes exist which solve the SIA and

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SSA everywhere in the ice sheet, combining the results without such an either/or decision (e. g., [21–24]). The latter hybrids can be used at high spatial resolution and long time scales because the SSA model is an easier stress balance to solve than higher order alternatives. Note that higher order approximations of the Stokes equations, such as the BlatterPattyn model [25,26], are also shallow approximations. Indeed, as both the SIA and the SSA are derived by small-parameter arguments from the Stokes equations, one might ask whether there is a common shallow antecedent model. Schoof and Hindmarsh’s [27] answer to this question is that the Blatter-Pattyn model is just such a model. However, computational limitations generally restrict either the spatial extent, the spatial resolution, or the model run duration of the more complete higher order models. Questions remain regarding what are the most important deficiencies, relative to the Stokes model, when using either higher-order [28] or shallow hybrid models. Solving the Stokes model itself requires explicit accounting for the constraint of incompressibility through a pressure variable. Numerical approximations of this stress balance are indefinite, and thus harder to solve. Along a flow line, however, one can address the incompressibility constraint by using stream functions [29]. Which are the best numerical models for moving grounding lines? Even when the minimal SSA stress balance is used, this is still a significant open question [30,31]. The physics requires that at least the quantities H and u are continuous at x = x g . Where can we find exact solutions for ice flow models? Textbooks by Greve and Blatter [32] and van der Veen [33] have a few. Halfar’s similarity solution to the SIA equation [34,35] was generalised to non-zero mass balance by Bueler et al. [36]. There are flowline [37–39] and cross-flow [15] solutions to the SSA model. For the Stokes equations themselves there are flowline solutions for constant viscosity [29]. For numerical verification purposes, one may manufacture solutions by starting with a proposed solution and then deriving a source term (or right-hand side) so that the specified function is actually a solution [40]. There are such manufactured solutions to the thermomechanically-coupled SIA model [41], the flowline Blatter model [42], and even to the Glen’s flow law Stokes equations [43,44]. The material on finite differences in this chapter should probably be read along with a textbook like that by Morton and Mayers [45] or LeVeque [46]. The ‘main theorem’ for numerical PDE schemes cited at the end of Sect. 8.3.1 is the Lax equivalence theorem. Alternative numerical discretization techniques include the finite element [47], finite volume [48], and spectral [49] methods. Newton iteration for the nonlinear discrete equations is superior to Picard iteration used here, in terms of rapid convergence once the iterates are near the solution, but methods to encourage global convergence are needed [50]. Finally, this chapter is based on a set of notes by the author which accompany the numerical codes, which are available at https://github.com/bueler/karthaus.

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Exercises 8.1 Assume f has continuous derivatives of all orders. Show using Taylor’s theorem that f (x + ) − f (x − ) f (x) = + O(2 ) 2 and f (x + ) − 2 f (x) + f (x − ) + O(2 ). f (x) = 2 Sometimes we want finite difference approximations for derivatives between grid points. Show that f (x + 21 ) =

f (x + ) − f (x) + O(2 ).

8.2 Rewrite heat.m using for loops instead of colon notation. (The only purpose here is to help understand colon notation.) 8.3 The 1D explicit scheme (8.18) for the heat equation, namely n n T jn+1 = μT j+1 + (1 − 2μ)T jn + μT j−1 ,

is averaging if the stability criterion (8.21) holds. But of course we must be stepping forward in time. Show that the scheme is not averaging if t < 0. Try running heat.m backward in time to see what happens. In general there are no consistent stable numerical schemes for unstable PDE problems. 8.4 This multi-part exercise concerns the numerical treatment of ∇.(D∇u). (a) Show that if D = D(x, y) and u = u(x, y) then ∇.(D∇u) = D∇ 2 u + ∇ D.∇u. (b) Write down the centered O(t) + O{(x)2 } explicit finite difference method for the equation u t = D0 u x x + E 0 u x , assuming D0 > 0 and E 0 are constant. Solve the scheme for the unknown u n+1 j . (c) Stability for your method will occur if the right hand side from the scheme in part (b) has all positive coefficients. If |E 0 | D0 , what does this say about t? (d) Why do we use the staggered grid for ∇.(D∇u), instead of expanding by the product rule as in part (a)? 8.5 Derive the Green’s function of the 1D heat equation, namely

1 x2 , T (t, x) = √ exp − 4Dt 2 π Dt

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∞ which is a solution to Tt = DTx x , satisfying T → 0 as x → ±∞ and −∞ T d x = 1. Start by supposing there is a solution of the form T (t, x) = t −1/2 φ(s) where s = t −1/2 x is the similarity variable. Write down an ordinary differential equation for φ, and solve it. 8.6 In the text, and in the code verifysia.m, we used Halfar’s solution to verify our numerical scheme siaflat.m. Create the analogous code verifyheat.m to use the Green’s function of the 2D heat equation (8.19), namely

1 (x 2 + y 2 ) , T (x, y, t) = exp − 4π Dt 4Dt to verify heatadapt.m. You can use the high quality approximation e−A ≈ 0 for |A| > 10 to choose a rectangular domain in space, for which you may use the Dirichlet boundary condition T = 0. 2

8.7 Show that formula (8.30) solves (8.8) in the case of zero climatic mass balance (M = 0). You will want to express divergence and gradient in polar coordinates. 8.8 In the text it is claimed that any modification of siaflat.m will make the output of verifysia.m show non-convergence, e. g., the reported average thickness error will not go to zero as the grid is refined. By randomly altering lines of siaflat.m, or by other methods of your choice, evaluate this claim. 8.9 Some output from verifysia.m has been suppressed in the text, including a map-plane view of the numerical ice thickness error. Near the grounded margin of an ice sheet this error is much larger than elsewhere. Why? Would a Stokes model with moving margin on the same grid have significantly smaller thickness error, supposing we knew an exact solution, so that we could evaluate the error? n 8.10 Let Cs = A 41 ρg(1 − r ) and assume x g = 0 is the location of the grounding line. Derive the two parts of the van der Veen exact ice shelf solution, namely

1/(n+1) Cs n+1 n+1 u = u n+1 (M + x + u H ) − (u H ) , 0 g g g g g M0 H=

M0 x + u g Hg . u

Start from equation (8.35), and use the fact that (H 2 )x = 2H Hx to generate the first integral. Also use boundary condition (8.40). On the other hand, note that the mass continuity equation M0 = (u H )x can be integrated to give u H = M0 (x − x g ) + u g Hg for the flux. Find u(x) first, and then H (x) in terms of u(x) as above. These exact solutions from van der Veen [39] are used in the code exactshelf.m.

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References 1. Mahaffy MW (1976) A three-dimensional numerical model of ice sheets: tests on the Barnes Ice Cap, Northwest Territories. J Geophys Res 81:1059–1066 2. Nye JF, Durham WB, Schenk PM, Moore JM (2000) The instability of a South Polar Cap on Mars composed of carbon dioxide. Icarus 144:449–455 3. Le Brocq AM, Payne AJ, Vieli A (2010) An improved Antarctic dataset for high resolution numerical ice sheet models (ALBMAP v1). Earth Syst Sci Data 2:247–260 4. Truffer M, Echelmeyer KA (2003) Of isbræ and ice streams. Ann Glaciol 36:66–72 5. MacAyeal DR, Rommelaere V, Huybrechts P, Hulbe C, Determann J, Ritz C (1996) An iceshelf model test based on the Ross ice shelf. Ann Glaciol 23:46–51 6. Fowler AC, Larson DA (1978) On the flow of polythermal glaciers I. Model and preliminary analysis. Proc R Soc Lond A 363:217–242 7. Morland LW, Johnson IR (1980) Steady motion of ice sheets. J Glaciol 25:229–246 8. Hutter K (1983) Theoretical glaciology. Reidel, Dordrecht 9. Fowler A (2011) Mathematical geoscience. Springer-Verlag, London 10. Jouvet G, Bueler E (2012) Steady, shallow ice sheets as obstacle problems: well-posedness and finite element approximation. SIAM J Appl Math 72:1292–1314 11. Hindmarsh RCA, Payne AJ (1996) Time-step limits for stable solutions of the ice-sheet equation. Ann Glaciol 23:74–85 12. Weis M, Greve R, Hutter K (1999) Theory of shallow ice shelves. Continuum Mech Thermodyn 11:15–50 13. Morland LW (1987) Unconfined ice-shelf flow. In: van der Veen CJ, Oerlemans J (eds) Dynamics of the West Antarctic ice sheet. Reidel, Dordrecht, pp 99–116 14. MacAyeal DR (1989) Large-scale ice flow over a viscous basal sediment: theory and application to ice stream B, Antarctica. J Geophys Res 94:4071–4087 15. Schoof C (2006) A variational approach to ice stream flow. J Fluid Mech 556:227–251 16. Johnsen SJ, Dahl-Jensen D, Dansgaard W, Gundestrup N (1995) Greenland paleotemperatures derived from GRIP bore hole temperature and ice core isotope profiles. Tellus 47B:624–629 17. Aschwanden A, Bueler E, Khroulev C, Blatter H (2012) An enthalpy formulation for glaciers and ice sheets. J Glaciol 58:441–457 18. Greve R (1997) A continuum-mechanical formulation for shallow polythermal ice sheets. Phil Trans R Soc Lond A 355:921–974 19. Hulbe CL, MacAyeal DR (1999) A new numerical model of coupled inland ice sheet, ice stream, and ice shelf flow and its application to the West Antarctic Ice Sheet. J Geophys Res 104:25349–25366 20. Ritz C, Rommelaere V, Dumas C (2001) Modeling the evolution of Antarctic ice sheet over the last 420,000 years: implications for altitude changes in the Vostok region. J Geophys Res 106:31943–31964 21. Bueler E, Brown J (2009) Shallow shelf approximation as a "sliding law" in a thermodynamically coupled ice sheet model. J Geophys Res 114:F03008 22. Goldberg D (2011) A variationally derived, depth-integrated approximation to a higher-order glaciological flow model. J Glaciol 57:157–170 23. Pollard D, DeConto RM (2007) A coupled ice-sheet/ice-shelf/sediment model applied to a marine-margin flowline: forced and unforced variations. In: Hambrey MJ, Christoffersen P, Glasser NF, Hubbard B (eds) Glacial sedimentary processes and products, Special Publication number 39 of the International Association of Sedimentologists, Blackwell, Oxford, pp 37–52 24. Winkelmann R, Martin MA, Haseloff M, Albrecht T, Bueler E, Khroulev C, Levermann A (2011) The Potsdam Parallel Ice Sheet Model (PISM-PIK) part 1: model description. The Cryosphere 5:715–726 25. Blatter H (1995) Velocity and stress fields in grounded glaciers: a simple algorithm for including deviatoric stress gradients. J Glaciol 41:333–344

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26. Pattyn F (2003) A new three-dimensional higher-order thermomechanical ice sheet model: basic sensitivity, ice stream development, and ice flow across subglacial lakes. J Geophys Res 108(B8):2382 27. Schoof C, Hindmarsh RCA (2010) Thin-film flows with wall slip: an asymptotic analysis of higher order glacier flow models. Quart J Mech Appl Math 63:73–114 28. Pattyn F and 20 others (2008) Benchmark experiments for higher-order and full Stokes ice sheet models (ISMIP-HOM). The Cryosphere 2:95–108 29. Balise M, Raymond C (1985) Transfer of basal sliding variations to the surface of a linearlyviscous glacier. J Glaciol 31:308–318 30. Goldberg D, Holland DM, Schoof C (2009) Grounding line movement and ice shelf buttressing in marine ice sheets. J Geophys Res 114:F04026 31. Schoof C (2007) Marine ice-sheet dynamics. Part 1. The case of rapid sliding. J Fluid Mech 573:27–55 32. Greve R, Blatter H (2009) Dynamics of ice sheets and glaciers. Springer, Berlin 33. van der Veen CJ (2013) Fundamentals of glacier dynamics, 2nd edn. CRC Press, Rotterdam 34. Halfar P (1981) On the dynamics of the ice sheets. J Geophys Res 86:11065–11072 35. Halfar P (1983) On the dynamics of the ice sheets 2. J Geophys Res 88:6043–6051 36. Bueler E, Lingle CS, Kallen-Brown JA, Covey DN, Bowman LN (2005) Exact solutions and numerical verification for isothermal ice sheets. J Glaciol 51:291–306 37. Bodvardsson G (1955) On the flow of ice-sheets and glaciers. Jökull 5:1–8 38. Bueler E (2014) An exact solution for a steady, flowline marine ice sheet. J Glaciol 60:1117– 1125 39. van der Veen CJ (1983) A note on the equilibrium profile of a free floating ice shelf. IMAU Report V83-15. University of Utrecht, Utrecht 40. Roache P (1998) Verification and validation in computational science and engineering. Hermosa Publishers, Albuquerque 41. Bueler E, Brown J, Lingle C (2007) Exact solutions to the thermomechanically coupled shallow ice approximation: effective tools for verification. J Glaciol 53:499–516 42. Glowinski R, Rappaz J (2003) Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology. Math Model Numer Anal 37:175–186 43. Jouvet G, Rappaz J (2011) Analysis and finite element approximation of a nonlinear stationary Stokes problem arising in glaciology. Adv Numer Anal 2011:164581 44. Sargent A, Fastook JL (2010) Manufactured analytical solutions for isothermal full-Stokes ice sheet models. The Cryosphere 4:285–311 45. Morton KW, Mayers DF (2005) Numerical solutions of partial differential equations: an introduction, 2nd edn. C.U.P., Cambridge 46. LeVeque RJ (2007) Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. SIAM, Philadelphia 47. Braess D (2007) Finite elements: theory, fast solvers, and applications in solid mechanics, 3rd edn. C.U.P., Cambridge 48. LeVeque RJ (2002) Finite volume methods for hyperbolic problems. C.U.P., Cambridge 49. Trefethen LN (2000) Spectral methods in MATLAB. SIAM, Philadelphia 50. Kelley CT (1987) Solving nonlinear equations with Newton’s method. SIAM, Philadelphia

9

Least-Squares Data Inversion in Glaciology Doug MacAyeal

9.1

Preamble

Virtually all of glaciology rests on the interaction of models and observation. While everyone is confident that models can be tested or validated by this juxtaposition, advanced researchers also use the process to improve and develop models. This research activity often goes by the name of “data inversion”. Every chapter in this book involves some form of data inversion or another, and in this chapter, we provide a simple example of one of the most enduring methods in data inversion: least-squares estimation. The vehicle used here to convey the basic ideas of least-squares data inversion is concerned with geodesy and the use of satellite navigation signals (such as ‘GPS’ (global positioning system)) as a means of observing ice flow and other phenomena. The advantage of using this particular example of data inversion is that it will be useful for understanding the underpinning of all glaciological remote sensing (particularly in knowing the time and place of observation). The treatment presented here simplifies a learning process that might take weeks or months to master. All glaciologists should, however, have at least a basic awareness of the technology on which so much of glaciology rests.

9.2

Introduction

Inverse methods have become widely used in glaciology whenever observations are used to infer parameters or variables that cannot be observed directly, and where numerical models must be guided by the assimilation of data. The complexity of

D. MacAyeal (B) Department of Geophysical Sciences, University of Chicago, Chicago, USA e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_9

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the methods varies from simplistic to profound. In the chapters on geophysics and glacial isostasy, inverse methods are implicitly present in the discussion, because they are the data-processing techniques needed to, for example, process an active seismic data set for englacial and subglacial geometry, or to determine the history of past ice-sheet retreat from observations of coastal sea level at disparate locations around the world. The simplest kind of inverse method is one which merely requires the reversal of the order of algebraic operations in the definitions of an observed quantity in terms of a quantity to be inferred. An example would be the conversion of an observed strain rate tensor, ε˙ , into an inferred deviatoric stress tensor, τ , by means of the relation τ = 2η(˙ε)˙ε ,

(9.1)

where the viscosity η is a function of strain rate. A more serious example is that of determining bed undulations from observed surface undulations. This was discussed in Sect. 1.8.3 of Chap. 1, where a simple model yielded the relationship ˆ ˆ S(k) = Kˆ (k) B(k),

(9.2)

where the hats denote the Fourier transform, k is the transform variable (the wave number), and K is a transfer kernel. Determination of | Kˆ (k)| showed that it is a decreasing function of κ = kh, where h is ice depth, and Kˆ ∼ 4e−κ as κ → ∞. The forward problem is the computation of S given B. The inverse problem is that of computing B (which is not directly observable) given S. This is of course easily done by computing the transform of B: ˆ S(k) ˆ ; B(k) = Kˆ (k)

(9.3)

but now note that imprecise estimates of Sˆ at large k cause severe inaccuracy in ˆ since Kˆ is very small; this is a sign of ill-posedness. The inverse relation repB, resented by (9.3) is an exact representation of the desired quantity B that can be derived from observations S. The ill-posedness comes in to play when the typical S observed is contaminated with observational noise, or is simply discretised (spatially discontinuous) in such a way as to introduce artifacts that have no origin whatsoever from any possible B. If the S is observed by an aircraft overflight in which a laser altimeter pulses the surface at discrete time intervals, the resulting data will contain “data-point-to-data-point” oscillations caused by the aircraft’s bumping position and the characteristics of the snow surface. If these point-to-point oscillations have a short horizontal length scale, an unrealistically large B would be required to create them. This defect of the data thus motivates the fitting of an inferred B to the S by some sort of procedure that does not demand that the inferred B re-constitutes the observed S exactly. The data analyst may wish that the inferred B specifically does

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not reproduce the point-to-point noise. At this juncture, the data analysis has entered the realm of inverse methods. A yet more complex inverse problem may involve the inference of variables or functions that depend in a very complex way on the observed quantities. An example is the inference of the basal traction, τ b (x, y) of an ice sheet, where x and y are horizontal coordinates, from observations of ice thickness h(x, y) (which can be derived from measurements of ice surface elevation and the presumed known bed elevation) and surface velocity us (x, y). A well-posed mathematical problem is that of computing the velocity u in a fixed domain (since the depth is prescribed) as the solution of a nonlinear set of equations and boundary conditions, which we may write schematically in the form S (u; x, t; τ b ) = 0;

(9.4)

here u is the unknown, to be found as a function of position x = (x, y, z) and t. We may suppose in this forward problem that the basal stress τ b is prescribed, perhaps through a sliding law. In principle, this problem can be solved to find u = U(x, t; τ b ),

(9.5)

and then the surface velocity can be determined: us = Us (x, y, t; τ b ).

(9.6)

The inverse problem consists of using measurements of, say, us to determine τ b directly; mathematically, we invert (9.6) as τ b = Us−1 (x, y, t; us ),

(9.7)

and as for the example in Chap. 1 this is possible to do; by choosing to satisfy the surface velocity condition rather than the basal stress condition, the problem still has a unique forward solution and this can be inverted. The issue with this lies in the fact that, just as for the bed undulation example, the inverse problem is not well-posed, and high wavenumber components in the surface velocity observations are amplified. This is the essential challenge of solving inverse problems. As an example, Sergienko and Hindmarsh [1] solved this type of inverse problem to infer the patterns of subglacial traction for various Antarctic ice streams, such as that shown in Fig. 9.1. An important constituent of the problem they solved was the fact that the data and assumptions on which their inference was based were both imperfect, containing noise and model simplifications, respectively. In an idealised sense, the goal of this chapter is to demonstrate the key concepts of inverse methods, so as to provide a useful foundation for working glaciologists. The starting point is a statement of the forward problem relating the observable, d, to the desired quantity to be inferred, m: A(m) + U (m) = d − ε;

(9.8)

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Fig. 9.1 Inferred basal traction magnitude (the colour scale represents a range of 0–450 kPa from blue to red) in the grounding zone of Thwaites Glacier, Antarctica (Olga Sergienko, personal communication, 2018). Surface relief is shown in grayscale. The grounding line cuts across the image where the surface crevassing abruptly appears in the lower left

here ε is unknown observation error, A(·) is a function that describes the assumed physics of how perfect, error-free data d should be related to m, and U (·) represents the unknown effects of oversimplification in the assumed physics A(·). The inverse problem associated with this forward problem is symbolically written m = A−1 (d),

(9.9)

where A−1 (·) is used to denote whatever algebraic or other process is needed to convert d into m. Note that neither ε nor U (·) appear in the inverse problem, and this means that one or both of the following two qualifications can apply to the inferred m: (i) no single m can exactly satisfy all the d, i. e., the physics predicts data, d˜ = A(m), that do not necessarily equal some or all of the observed data d; (ii) the data may all be satisfied, i. e., d˜ = d, but for a multitude of m, i. e., m is not uniquely determined. To help glaciologists more immediately understand the concepts and practices of inverse methods on a simple level, this chapter is restricted to examining inverse problems that can be described with linear algebra, and which are subject only to the first of the above problems, i.e., that d˜ = d. For this purpose, I have chosen to develop the technique for solving the multi-lateration (position fixing) problem that arises in processing Global Positioning System (GPS) data using least squares. In this problem, m = m will be a column vector of four unknowns: the position of the GPS receiver antenna in geodetic coordinates and the error of the GPS receiver’s onboard clock. The data, d = d, will be a set of many (typically ten to thirteen) observed

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distances between orbiting GPS satellites and the GPS receiver. The physics, A(·) = A will be (after linearisation, to be developed below) a matrix operator that maps the four-component vector m into the N -component vector d with N > 4. An additional goal of this chapter is to provide a brief tutorial on the basics of GPS observation used in glaciology.

9.3

The Roots of GPS in Glaciology

Glaciology has its roots in the measurements made possible by the increasingly precise technology of measuring distance and location. Prior to the 1970s, velocity and strain rate in Antarctica could only be measured using optical instruments and surveyor’s tapes. By the 1980s, the implementation of a constellation of GPS satellite transmitters in orbit around Earth allowed precise location fixing, and the subsequent ability to use time series of these precise locations to determine ice flow. Today, longterm average ice flow is easily measured by remote sensing (which also depends on the GPS constellation), but short-term fluctuations in ice flow, such as the stick-slip sliding motion of the Whillans Ice Stream in Antarctica discussed below [3], still require GPS measurements conducted on the ground. The idea of GPS is to use the observations of the time-of-flight of an electromagnetic wave (in this case, a radio wave) from a known benchmark (in this case, a satellite in a precisely determined orbit) to determine distance. This is not a new idea, as it was first proposed by Michelson and Morley [2] (see Fig. 9.2), and put to work using visible light to produce the first observations on many important Earth science related phenomena, such as the rigidity of the Earth, the Earth’s rotation, and Earth movement during earthquakes. An example of GPS use in modern glaciological context involves the observation of ice-shelf flexure and rebound in response to moving meltwater loads [4].

Fig. 9.2 Albert Michelson (left) and his colleague Edward Morley (right) proposed [2] that the light wave could become the ultimate standard of length. Michelson went on to set the stage for the uses of electromagnetic radiation as a standard in measuring Earth motions by determining the rigidity of the Earth, the rotation of the Earth, and differential Earth movements resulting from earthquakes. While glaciologists no longer use light waves to observe ice deformation, they use radio waves from GPS satellites

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Introduction to GPS

The GPS system consists of a constellation of 32 satellites (at the time of writing) and a network of several high-precision ground tracking stations that precisely monitor the orbital geometry and clocks aboard the satellites. The GPS satellites transmit signals using two carrier phases (referred to as L1 and L2), on which various coded signals and orbit ephemeris data are modulated. Typical GPS receivers in the hands of glaciologists are equipped with oscillators that replicate the L1 and L2 carrier phase oscillations, and with electronics that can interpret incoming code signals. An example is shown in Fig. 9.3. Position and time fixing can be done in real time by typical GPS receivers to an accuracy that is suitable for navigation and for time keeping at a precision needed by instruments like seismometers. Interferometric analysis of the L1 and L2 carrier phases with the replica oscillations in the receiver’s electronics can be used for precise, millimetre accuracy position fixing. This interferometric analysis is best done in post-processing, after a several week latency period when precision information about satellite orbit geometry and clock errors are published by the various international geodetic services.

Fig. 9.3 A GPS antenna and solar panel deployed on the Ross Ice Shelf in 2005. The receiver itself is buried just beneath the snow surface to prevent it from being blown away by wind storms

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225

History

The development of the GPS system was motivated by the U. S. military’s desire to provide an accurate and global radio-wave based method for determining location and telling precise time. Initially, the GPS system was restricted to military use. This restriction was changed in 1983 after a Korean passenger jet liner, KAL flight 007, accidentally strayed into Soviet airspace and was shot down. This incident allowed civilian use of an intentionally degraded form of GPS known as “selective availability”. This intentional degradation was suspended in 2000, allowing civilians to fix position and time to within several metres and microseconds using hand-held receivers processing the GPS signals in real time (as they are received). The U. S. military retains a classified form of GPS technology that is accurate to within several tens of centimetres. In addition to real-time operation, GPS provides a means to fix location and time very accurately (within millimetres and nanoseconds) through post-processing of data with a several week latency period. This is accomplished through precise tracking of GPS carrier signals (on which the information allowing immediate, realtime location and time fixing is modulated) and the application of interferometry methodology. This highly-accurate technique is what is used for most geodeticquality GPS observations in glaciology. Currently, the U. S. GPS system is joined by similar systems operated by Russia, China and the European Union, going by the names of GLONASS, Bei Dou, and Galileo, respectively.

9.4.2

Coarse Acquisition (C/A) Code

As described above, GPS satellites broadcast specific signals (e. g., Fig. 9.4) allowing a variety of processing methods to determine results either in real time or after the fact, and these are either highly accurate or of limited accuracy. The least accurate, but real-time, functionality of the GPS system involves what is called the Coarse Acquisition code (C/A code) that is broadcast on the carrier phase L1. This is the signal that is used by navigation-quality GPS receivers, such as hand-held Garmin receivers and smart phones used to fix the location of glaciological field camps and to navigate from site to site in an aircraft. Each GPS satellite is assigned a single, publicly known pseudo-random number (PRN), also called a “Gold code” after Robert Gold, the mathematician who invented the technology that generates and uses them. (Another PRN is assigned for military purposes, and is called the P-code, but this is not available to glaciologists.) The PRN is a 1023 term sequence of ones and zeroes that is phase modulated onto the L1 carrier and is broadcast repeatedly every millisecond. Generated by the electronics of the GPS receiver is a replica PRN for each of the 32 GPS satellites. The software of the GPS receiver cross-correlates the replicas of the PRN codes with the PRN codes it receives to both differentiate between the satellites and determine the timeof-flight of the C/A signal as it travels the distance between the satellite’s broadcast

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Fig. 9.4 A random-looking but distinct (and pre-determined) code is transmitted by GPS satellites so that a their signals can be distinguished from each other and b the receiver can measure time differences. Here is a graphical representation of the code transmitted by GPS satellite number 24

antenna and the GPS’s receiving antenna. The key property of the PRNs designed by Robert Gold is that they do not cross-correlate, and only auto-correlate when they are lined up in time. The GPS receiver thus performs a time-lagged correlation, which means lagging or leading the digits of the replica code relative to the code that is received from the satellite; only one such correlation will yield a large magnitude. The correlation between the correct replica code matching the satellite’s code yields a time lag reflecting the time difference between the broadcast of the PRN by the satellite and its receipt by the GPS receiver. This time lag is typically several hundredths of a second.

9.5

The Equations of Pseudorange

Using the C/A code, a GPS receiver deduces the apparent time-of-flight for the C/A signal to traverse the distance from the satellite to the receiver. The procedure is illustrated in Fig. 9.5. This observed time-of-flight, t, is influenced by errors: t = tr − ts + τ − ε;

(9.10)

here, tr is the time the receiver receives the GPS signal and ts is the time the satellite sent the signal, but τ and ε are the clock errors on the receiver’s and satellite’s clocks, respectively, that enter the observation. The observed t can be converted to the distance between the satellite and receiver using the speed of radio waves in a vacuum, c: c = 299,729,458 ≈ 3 × 108 m s−1 .

(9.11)

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position clock error

227

Data:

travel time satellite coordinates

satellite orbit in ECEF coordinates

precisely known satellite position

Fig. 9.5 Geometry of one of 32 GPS satellites in orbit around Earth in Earth Centered, Earth Fixed (ECEF) coordinates. The purpose of GPS technology is to determine location (X , Y , Z ) and clock error τ for a GPS receiver located at an arbitrary position on or above the Earth’s surface. The radio signal transmitted by the satellite is used to determine travel time of the radio wave from the satellite to the receiver. This travel time, and also up to 12 other travel times for other satellites (not shown) visible to the receiver, comprise the data that is inverted to obtain (X , Y , Z , τ )

The apparent distance or range P (called the pseudorange in Global Navigation Satellite System (GNSS) parlance) from satellite to receiver is P = ct = where the quantity

(xr − ξ )2 + (yr − η)2 + (zr − ζ )2 + c (τ − ε) ,

(9.12)

(xr − ξ )2 + (yr − η)2 + (zr − ζ )2

(9.13)

is the true range between the satellite, at coordinates (ξ, η, ζ ), and the receiver, at coordinates (xr , yr , zr ). By convention, the Cartesian coordinates of the satellite and receiver are expressed in the Earth Centered, Earth Fixed (ECEF) reference frame. This reference frame has its origin at the center of mass of the Earth, z is aligned with the rotation axis, and the reference frame rotates at the same rate as the Earth. With N or more GPS satellites visible to the receiver (above the receiver’s horizon) at any one time, there will be N simultaneous observations of the pseudorange of the type seen above:

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P1 = ct1 = .. . PN = ct N =

(xr − ξ1 )2 + (yr − η1 )2 + (zr − ζ1 )2 + c (τ − ε1 ) , (xr − ξ N )2 + (yr − η N )2 + (zr − ζ N )2 + c (τ − ε N ) .

(9.14)

For each pseudorange observed, there is additional error associated with precision of satellite orbit geometry, index of refraction due to the differing signal paths through the atmosphere, ionospheric effects and precision within the electronics of both the receiver and the satellite. In general, these errors can be treated as random. For simplicity, we shall not concern ourselves with these errors. For now, we shall assume that ε1 , . . . , ε N are determined by the GPS satellite monitoring system (and hence are known). The unknowns are thus the GPS receiver’s location (xr , yr , zr ) and clock error τ . Under most circumstances, there will be more than N = 4 satellites in the GPS receiver’s view, and this will provide pseudorange data sufficient to determine the four unknowns, but the problem is then overdetermined, as there are more constraints than are needed to determine the unknowns. The non-linearity of the pseudorange expressions given above is overcome by iterative linearisation. Starting with an initial guess of the unknowns, denoted by x g , yg , z g and τg , we can write a Taylor series expression for Pn , the nth satellite’s observed pseudorange, in terms of the difference between the initial guess and the actual position and clock error of the GPS receiver: Pn = Pn +

∂ Pn ∂ Pn ∂ Pn ∂ Pn x + y + z + τ, ∂ xg ∂ yg ∂z g ∂τg

(9.15)

where Pn = r gn + cτg , 2 2 2 r gn = x g − ξn + yg − ηn + z g − ζn ,

(9.16)

and x = xr − x g , y = yr − yg , z = zr − z g , τ = τ − τg .

(9.17)

The partial derivatives in the above equation are evaluated using the chain rule: x g − ξn ∂ Pn = , ∂ xg r gn

yg − ηn ∂ Pn = , ∂ yg r gn

z g − ζn ∂ Pn = , ∂z g r gn

∂ Pn = c. ∂τg (9.18)

We note that the clock errors of the satellites, and any other factors which affect the observed Pn s, do not appear in the expression for Pn . Issues associated with these factors will be dealt with later.

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We now manipulate the Taylor series expression given in (9.15) to yield a familiar system of linear equations. Subtracting Pn from both sides, and writing ⎡

x

⎢ y ⎢ m=⎢ ⎣ z τ

⎡

⎤ ⎥ ⎥ ⎥, ⎦

(P1 − P1 )

⎤

⎢ ⎥ ⎢ (P2 − P2 ) ⎥ ⎢ ⎥ d=⎢ ⎥, .. ⎢ ⎥ . ⎣ ⎦ (PN − P N )

(9.19)

⎤ ∂ P1 ∂ P1 ∂z g ∂τg ⎥ ⎥ ⎥ ⎥ ∂ P2 ∂ P2 ⎥ ⎥ ∂z g ∂τg ⎥ ⎥ ⎥, ⎥ .. .. ⎥ . . ⎥ ⎥ ⎥ ⎥ ∂ PN ∂ PN ⎦ ∂z g ∂τg

(9.20)

and ⎡

∂ P1 ∂ P1 ⎢ ∂ x g ∂ yg ⎢ ⎢ ⎢ ⎢ ∂ P2 ∂ P2 ⎢ ⎢ ∂ x g ∂ yg ⎢ A=⎢ ⎢ ⎢ .. .. ⎢ . . ⎢ ⎢ ⎢ ⎣ ∂ PN ∂ PN ∂ x g ∂ yg

the system of equations may be written in vector form as Am = d.

(9.21)

This is a set of N equations for four unknowns. If N < 4, there is generally no solution. If N = 4, the desired set of corrections to the initial guess may be immediately determined by inversion: m = A−1 d,

(9.22)

where A−1 is the inverse of A. If N > 4, the system is over-determined, and an exact solution is again no longer possible; however, in this case it is possible to gain an approximate solution of the form (9.22), where A is a ‘generalised inverse’ which is discussed further below. Setting aside temporarily what the above expression means computationally, and putting aside the question of how to formulate A−1 , we correct the initial guess to (under all but the most pathological conditions) obtain a new estimate of the position and clock error of the GPS receiver as x g(2) = x g + m 1 ,

(2) yg(2) = yg + m 2 , z (2) g = z g + m 3 , τg = τg + m 4 , (9.23)

or, on defining the four-component vector ψ g = (x g , yg , z g , τg )T , ψ (2) g = ψ g + m,

(9.24)

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where we use the superscript to indicate an improved second guess. This can be used to repeat the process to obtain an even better estimate ψ (3) g by composing and solving a new set of linear equations A(2) m(2) = d(2) ,

(9.25)

where A(2) and d(2) are composed in the same way as A and d except that the new (3) estimate ψ (2) g is used. The solution of (9.25) gives ψ g : (2) (2) ψ (3) g = ψg + m .

(9.26)

The above sequence of solutions is computed until the solution converges to the best estimate of the GPS position and clock error (xr , yr , zr , τ ) ≈ ψ r = lim ψ [k] g . k→∞

(9.27)

Typically, the processing of coarse acquisition data from GNSS satellites yields convergence with just a few iterations. This will be demonstrated in exercise 9.2.

9.6

Least-Squares Solution of an Overdetermined System of Linear Equations

In a typical GPS observation, eight to thirteen satellites are visible and provide pseudorange observations to be used to constrain the four unknowns in the problem formulated above. In a perfect world, where no other sources of uncertainty affect an individual satellite’s pseudorange to the GPS receiver, only data from four satellites would be sufficient to determine the GPS receiver’s position and clock error exactly and unambiguously. In practice, it is beneficial to use all eight to thirteen observed pseudorange observations to constrain the four unknowns as best as possible. This means that there will be more equations than unknowns, and the matrix A has more rows than columns. This system of equations is said to be over-determined, and there is generally no solution. (For a simple illustration, think of solving two equations for one unknown: ax = b and cx = d.) One way to solve such a system for a solution that is ‘best’ in some sense is to use least squares, i. e., to minimise a scalar quantity J that measures the misfit between the solution chosen and the data in the common least-squares sense: J = (Am − d)T (Am − d) ,

(9.28)

where (Am − d)T is the transpose of (Am − d). (The expression for J is just the familiar dot product between two vectors, each of which is the difference between the linear operator A acting on the desired solution vector m and the vector of data from the satellite observation d.) The name ‘least squares’ comes from the fact that J is designed as the sum of squares of the individual elements of a vector (this vector could be called the ‘data misfit’ vector).

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231

Equation (9.28) can be written in component notation as ⎛ ⎞2 N 4 ⎝ J= An j m j − dn ⎠ . n=1

(9.29)

j=1

It is convenient in the following to use the Einstein summation convention, in which summation over once repeated suffixes is understood. Thus (9.29) can be written in the form J = (An j m j − dn )(Ank m k − dn ).

(9.30)

Multiplying this out, we obtain J = An j Ank m j m k − 2 Anp m p dn + dn dn ,

(9.31)

where, according to the Einstein notation, summation over indices n, j, k and p is assumed. One condition for a minimum of J is that its partial derivatives with respect to the individual components of the solution vanish. This allows us to write ∂J = 0 for q = 1, . . . , 4. ∂m q

(9.32)

Evaluating these partial derivatives (it is useful to note that ∂m j /∂m q = δ jq , the Kronecker delta symbol, defined by δi j = 1, i = j δi j = 0, i = j),

(9.33)

∂J = 2 Anq An j m j − dn ∂m q

(9.34)

we find

for each value of q. On setting the partial derivatives to zero, these four equations become the q-th components of a vector equation, which takes the form AT Am = AT d.

(9.35)

−1 AT d. m = AT A

(9.36)

The solution of this equation is

Note that if there are N satellites and 4 unknowns in the problem, then A is a N × 4 matrix and AT is a 4 × N matrix. This means that AT A is a 4 × 4 matrix,

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D. MacAyeal

which (provided the determinant is not zero)1 has a formal inverse. This means that T −1 A A actually exists. The 4 × N matrix

−1

AT A

AT

(9.37)

is called the ‘least-squares inverse’ of A.

9.7

Observational Techniques to Improve GPS Accuracy

In the previous section, the main emphasis was to solve the multi-lateration (pseudorange) problem to determine four essential unknowns, the position and clock error of the GPS receiver. The measurement error associated with the basic method discussed so far is typically too large for many glaciological applications. Position processed from the C/A code is commonly several metres in error, and this is too large for the study of, for example, ice-stream stick slip motions of Whillans Ice Stream shown in Fig. 9.6. In this example, GPS accuracy of a few centimetres or less is needed to resolve the daily motion of the ice stream. The principal method to gain accuracy with GPS observations is to use carrier phase tracking of the two carrier signals, denoted L1 and L2, to observe pseudorange to within a few percent of the two carrier signal wavelengths, 19.05 and 24.45 cm, respectively.2 A geodetic quality GPS receiver intended to perform phase tracking maintains two reference oscillations within its circuitry, with which it can compare the frequencies of L1 and L2 that it receives as the satellite passes above. In phase tracking, the phase of a carrier signal is the argument of the periodic sinusoidal function that represents the carrier wave. This phase, φ, is equal to an integer number times 2π plus some fraction of 2π . In tracking the L1 and L2 phases, often the fraction of 2π of phase is precisely measured, however the integer number of 2π phase of the signal being received is unknown. Because of Doppler shift, i. e., the satellite moving toward or away from the receiver, the phase differences between the received L1 and L2 signals and the reference oscillation will change over time. One 2π interval of phase change in L1, for example, denotes a 19.05 cm change in the range between the satellite and receiver. The phase change in L1 can be measured down to 1% or less of 2π , thus range changes can ideally be determined to about 0.1905 cm or 1.9 mm.

1 The determinant can only be zero if A is singular, in which case some of the N equations are dependent on the others. These can then be eliminated until the problem is reduced to solving M < N equations; in practice this will never occur, however. 2 Two carrier signals are used as a means of correcting for electromagnetic wave dispersion in the ionosphere; but this detail will not be explored further because our present focus is on data processing methods. Interested GPS users should consult a typical GPS text or website (such as the ESA supported web text: https://gssc.esa.int/navipedia/GNSS_Book/ESA_GNSS-Book_TM-23_ Vol_I.pdf) to learn more about the “ionosphere free” combination signal that involves combining L1 and L2.

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Fig. 9.6 Stick-slip motion of ice stream B (now called the Whillans Ice Stream), Antarctica. The dots represent individual epochs of a single GPS receiver’s location during several days. The hour of day is denoted by colour. The clusters of points represent periods when the ice stream was motionless. The forward motion, about 60 cm per day, of the ice stream occurs as a result of two short 30-minute periods when the dots jump about 30 cm to the next cluster. This stick-slip motion would not have been discovered without the use of on-the-ground GPS survey, because observation of the ice stream’s velocity from space, or by ground-based methods that cannot resolve position on a minute-by-minute basis, averages the velocity over multiple days. This motion is driven by effects of the ocean tide in the Ross Ice Shelf [3]

9.7.1

The Ionosphere-Free Combination

A crucial design element of the GPS system is that military (P-code) coarse acquisition and phase tracking can be simultaneously done on two carrier frequencies, L1 and L2. This was an intentional design, because the following combinations of the two carrier frequencies eliminate about 99% of noise due to ionospheric effects: φ (IF) =

2 φ (L1) − f 2 φ (L2) f L1 L2 2 − f2 f L1 L2

(9.38)

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for phase tracking, and P (IF) =

2 P (L1) − f 2 P (L2) f L1 L2 2 − f2 f L1 L2

(9.39)

for P-code pseudorange (military coarse acquisition code that is not available for civilian use). Here φ is the phase and P is the pseudorange, f L1 and f L2 are the frequencies of the L1 and L2 carrier phases, 1575.42 and 1227.60 MHz, respectively, and the superscript IF denotes the ionosphere-free combination. In what follows, we shall assume that the ionosphere combination is being referred to when we write φ. The wavelength λ IF = 21.7 cm of the IF combination carrier phase is the mean of L1’s and L2’s wavelengths.

9.7.2

Carrier-Phase Determined Range and Integer Wavelength Ambiguity

The main challenge with processing GPS phase tracking data is that there is an integer ambiguity in determining the pseudorange from the phase differences. At time ti , when the n-th satellite’s ionosphere-free combination signal is first acquired by the GPS receiver, the pseudorange is Pn (ti ) = λ IF φn (ti ) + In λ IF + c (τ − εn ) ,

(9.40)

where λ IF is the wavelength of the IF carrier-combination signal, φn (ti ) is the observed phase difference between the received signal and the receiver’s selfgenerated reference signal, In is an integer representing the phase ambiguity due to the fact that the number of integral wavelengths on the path between satellite and receiver cannot be directly observed, and τ and εn are the receiver and satellite clock errors, respectively. At the point when the satellite first comes into view of the receiver, the distance to the satellite will be In λ IF + φn (t = 0), where 0 ≤ φn (t = 0) ≤ 2π , and t = 0 denotes the initial time the satellite is tracked.

9.7.3

Resolving Range Ambiguity by Phase Tracking

To deal with ambiguity in the phase difference derived pseudorange, a least-squares inversion process is used to compute the unknown receiver position (xr , yr , zr ), clock error τ , and integer ambiguities I1 , . . . , I Ns from multiple observations of the satellite pseudorange over a range of times, t = t1 , . . . , t M . Here Ns is the number of satellites. It is assumed that the satellite clock errors, ε1 , . . . , ε Ns , are known. The number of unknowns is 4 + Ns , so relatively few time epochs need to be used in the process; however, for accurate results and well-conditioned matrix arithmetic in the inversion process, it is good for t M − t1 to be as large as practicable to allow satellite geometry to evolve. If the satellite geometry did not evolve, i. e., if ξn , ηn , ζn

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235 3

2 4

5

Ambiguous (unobservable) initial phase at t = t1

Movement of single satellite

Receiver

Fig. 9.7 The phase tracking method for resolving ambiguous phase. A fixed receiver tracks a single satellite across various epochs of the satellite’s movement. For the initial epoch, the total phase length (i. e., the number of integer plus fractional wavelengths) of the carrier frequency along the travel path from satellite to receiver is unknown. At that initial time, the receiver only knows the fraction of a single wavelength that must be added to the integer number of wavelengths to get the true range to the satellite. Following the initial acquisition of the satellite by the receiver, the subsequent change in phase is measured very precisely (down to a fraction of 2π ) and this allows the addition or subtraction of length to the initial ambiguous distance to be measured precisely. Least-squares analysis of multiple epochs allows the ambiguity of the initial phase to be resolved

remained the same, rows of the matrix A would be the same, and this would lead to singularity of the matrix AT A. The procedure is illustrated in Fig. 9.7. As an aside: for ease of understanding, the clock errors are not assumed to change with time. In a typical glaciological application, where the data is being processed three weeks or more after the data were collected, the clock errors on the satellites are known, and are provided along with the satellite geometry data needed to specify satellite position ξn , ηn , ζn . We also assume that the clock error on the GPS receiver changes sufficiently slowly that it may be taken as constant during each epoch of phase tracking. A final assumption is that the range ambiguities I1 , . . . , I Ns do not change during each epoch of phase tracking. This is actually not a good assumption, as cycle slips3 in detecting phase difference between the receiver’s replica carrier oscillator and the incoming carrier phase are a common problem. The detection and treatment of cycle slips is a very complex procedure and is beyond the scope of this chapter. Hence, in this section, we develop the solution procedure only for idealised, cycle-slip free data.

3 Cycle

slips occur when the receiver phase-change detection misses or miscounts changes of complete cycles (multiples of 2π ) between successive measurements of φ.

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To lay out the phase-tracking solution explicitly, we write ⎡

⎡

x

⎢ y ⎢ ⎢ ⎢ z ⎢ ⎢ τ m=⎢ ⎢ ⎢ I1 ⎢ ⎢ . ⎢ . ⎣ . I N s

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

(P1 − P1 )t=t1

⎢ (P − P ) 2 2 t=t1 ⎢ ⎢ .. ⎢ ⎢ . ⎢ ⎢ ⎢ PNs − P Ns t=t1 ⎢ ⎢ (P1 − P1 ) t=t2 ⎢ ⎢ ⎢ (P2 − P2 )t=t2 ⎢ ⎢ .. ⎢ . d=⎢ ⎢ ⎢ PNs − P Ns ⎢ t=t2 ⎢ .. ⎢ ⎢ . ⎢ ⎢ (P − P ) 1 1 t=t M ⎢ ⎢ ⎢ (P2 − P2 )t=t M ⎢ ⎢ .. ⎢ . ⎣ PNs − P Ns t=t

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(9.41)

M

and the matrix A will be composed (for ease of display) by M submatrices: ⎡

A1

⎤

⎢ ⎥ ⎢ A2 ⎥ ⎢ ⎥ A = ⎢ . ⎥, ⎢ .. ⎥ ⎣ ⎦ AM

(9.42)

where ⎤ ∂ P1 (ti ) ∂ P1 (ti ) ∂ P1 (ti ) c λI F 0 . . . 0 ⎥ ⎢ ∂ xg ∂ yg ∂z g ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂ P2 (ti ) ∂ P2 (ti ) ∂ P2 (ti ) ⎢ c 0 λI F . . . 0 ⎥ ⎥ ⎢ ∂ xg ∂ y ∂z g g ⎥ ⎢ Ai = ⎢ ⎥. ⎥ ⎢ ⎢ .. .. .. .. .. .. .. .. ⎥ ⎢ . . . . . . . . ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ ∂ P Ns (ti ) ∂ P Ns (ti ) ∂ P Ns (ti ) c 0 0 . . . λI F ∂ xg ∂ yg ∂z g ⎡

(9.43)

The solution for the unknown receiver position, clock errors and integer ambiguities can be determined iteratively as described in Sect. 9.5.

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9.7.4

237

Differential GPS

Range errors are introduced by ionospheric and tropospheric effects that are hard to characterise and correct for. The approach most commonly taken to mitigate their effects is called ‘differential GPS’, or dGPS. This involves setting up a GPS ‘base station’ on a known survey benchmark that remains fixed through the entire measurement campaign involving other, “roving” GPS receivers deployed at positions of interest. The purpose of the base station is to allow estimation of the satellite-toreceiver path errors introduced by the atmosphere. The accuracy of the correction depends on how close the base station is to the roving GPS, and hence how similar the paths are between the satellite and the two receivers. As a rule of thumb, separation by several kilometres will still yield an improvement in accuracy. For simplicity, let us assume that there is just 1 roving GPS receiver, and we denote its pseudorange data with the superscript r , likewise we denote the pseudorange data from the base station receiver with the superscript b. An advantage of dGPS is that it removes all errors that are common to both base and roving stations. This includes satellite clock errors εi . If the clock error on the base station is not known, only the difference in clock error between the roving and base GPS receivers can be resolved. The dGPS solution is set up in the same manner as in Sect. 9.5. Ignoring, for the time being, the complexity associated with phase tracking, considering only one roving GPS station, and considering the pseudorange at only one observation time epoch, the solution for the unknown roving GPS station positions and the clock error difference between the rover and base receivers (four unknowns) can be found using the iterative technique for the equation m = (AT A)−1 AT d, where

⎛

(b−r )

P1

(b−r )

− P1

⎜ (b−r ) ⎜ P2 − P2(b−r ) ⎜ d=⎜ .. ⎜ . ⎝

(9.44)

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

Pn(b−r ) = Pn(b) − Pn(r )

(9.45)

PN(b−r ) − P N(b−r ) is the observed pseudorange difference between the base and the roving station for satellite n, and

2 2 2 x (b) − ξn + y (b) − ηn + z (b) − ζn 2 2 2 ) x g(r ) − ξn + yg(r ) − ηn + z (r − + cτg(b−r ) , g − ζn

Pn(b−r ) =

(9.46)

where x (b) , y (b) , z (b) are the known location coordinates of the base station, and τg(b−r ) is the difference in clock errors (only the difference in errors between the two

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clocks matters in the pseudorange problem). The matrix A is constructed in the usual fashion from partial derivatives, which take the form

An1

∂(Pn(b−r ) ) = = ∂ xg

− x g − ξn(r ) (t) x g(r )

− ξn

2

+

yg(r )

− ηn

2

+

) z (r g

− ζn

2 , etc.,

An4 = c.

(9.47)

Exercises 9.1 Exercises in linear algebra In this exercise we use the summation convention, which implies summation over the range of indices if they are repeated in any product term. We also define the Kronecker delta δi j to be zero unless i = j, in which case it is equal to one. (Written as a matrix, (δi j ) is thus the unit matrix.) We also write the i-th component of a vector v as vi , its magnitude |v| as v, and the i j-th component of the matrix A as Ai j . Show that vi vi = |v|2 , ∂ xi /∂ x j = δi j , ∇x = x/x, ∇.u = ∂u i /∂ xi . If A is an n × m matrix and m is a vector in Rm , thus an m × 1 (column) vector, show that the i-th component of the vector Am is (Am)i = Ai j m j . It is required to minimise the quantity J = (Am − d)T (Am − d) ; show that J = An j Ank m j m k − 2 Anp m p dn + dn dn , and deduce that at a minimum (a stationary point where ∂ J /∂m q = 0), AT Am = AT d. Hint: note that (AT )i j = A ji . 9.2 Identical twin experiment: processing C/A pseudorange data for an idealised flat Earth Before tackling the full Earth/satellite geometry, it is instructive to examine an idealised satellite navigation system that operates in a universe where the Earth is nothing more than a straight line. In this idealisation, Earth is represented by a line segment of length 2π Re which we take to be 40,000 km (Re = 6380 km is the radius of the Earth). The geometry is shown in Fig. 9.8. In the space surrounding the Earth, there is a constellation of satellites, also arranged on a straight line segment, that is 40,000 km long and displaced from the Earth’s surface by an altitude of z = Z s = 20,200 km (this is about the altitude of the current GPS satellites above the Earth’s surface). Due to the way the

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239

Fig. 9.8 Geodetic coordinate system for a Flat Earth idealisation

satellites are placed in the sky above the idealised Earth, there are two gaps of satellite positioning about the two polar end-points, each gap being ≈ 6,125 km in length. These gaps represent an account of the 55◦ inclination of the real GPS satellite orbits. Given that the idealised universe has no gravity (for now), we shall assume that it is possible for the satellites to hold fixed positions above the planet. In Fig. 9.8, a constellation of 5 navigation satellites is shown. The x coordinate of the satellites is chosen randomly (but care is taken to exclude them from the polar gap regions), and the altitude is fixed. A single GPS receiver is located at position x = X r and at elevation z = Z r somewhere near the South Pole. This receiver records radio signals transmitted from the constellation of 5 satellites, and compares the time when these signals are received to an accurate clock, thereby determining the ‘time of flight’ for the radio transmissions. Assuming that the radio signals travel at the speed of light, c = 299,792,458 m s−1 , determine the receiver’s antenna location (X r , Z r ) from the pseudorange and satellite geometry data provided. The positions of the Ns = 5 satellites, X s and Z s are known and also provided. You may find it useful to express c in metres per millisecond and interpret clock error in milliseconds. This will make the matrix A better conditioned. Note that x = 0 is taken to be the equator, and z = 0 is taken to be the Earth’s surface. Also note that pseudorange is defined to be the ‘apparent distance’ between the receiver’s antenna and the satellite’s antenna. The term ‘pseudorange’ is used, because the ability to time the signals from each satellite contains random errors corresponding to errors in distance of up to 20 m, due to the resolution of the receiver’s hardware. An additional complication is that the receiver’s clock, albeit a very accurate one, does not necessarily tell the right time, because it may be offset by an amount τ from the standard of time used by the satellites. For now, assume that the clock errors on the satellite are all zero.

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For this problem, you are asked to perform an ‘identical twin’ data processing experiment. First, create ‘artificial data’ that represents what a GPS receiver would measure if it were located at a particular place (X r , Z r ). Do this by randomly choosing (or otherwise choosing) the ‘true’ location (X r , Z r ). To create the artificial data, you will need to create (randomly, or by other choice) the ‘true’ positions for the 5 satellites (X s , Z s ), as well as satellite clock errors τs . You will additionally need to account for error in the pseudorange observed by the GPS receiver and for the clock error τ on the GPS receiver. This can be done by adding appropriately scaled random numbers to the pseudorange artificial data. Once you have created the artificial data, proceed to invert it using the methods of this chapter to obtain an apparent (X r , Z r ) and τ . Now you can compare the two ‘identical twins’: determine how closely your apparent (X r , Z r ) and τ match the exact (X r , Z r ) and τ that you assumed to create the artificial data. Discuss your results and pay particular attention to how the differences between the identical twins depend on aspects of the error you assumed.

References 1. Sergienko OV, Hindmarsh RCA (2013) Regular patterns in frictional resistance of ice-stream beds seen by surface data inversion. Science 342(6162):1086–1089 2. Michelson AA, Morley EW (1889) On the feasibility of establishing a light-wave as the ultimate standard of length. Amer J Sci 38(3):181–185 3. Bindschadler RA, King MA, Alley RB, Anandakrishnan S, Padman L (2000) Tidally controlled stick-slip discharge of a West Antarctic Ice Stream. Science 301(5636):1087–1089 4. Banwell AF, Willis IC, Macdonald GJ, Goodsell B, MacAyeal DR (2019) Direct measurements of ice-shelf flexure caused by surface meltwater ponding and drainage. Nat Commun 10:730

Analytical Models of Ice Sheets and Ice Shelves

10

Hans Oerlemans

10.1

Introduction

The modelling of present-day or palaeo-ice sheets in a realistic way requires numerical methods with high spatial resolution and a comprehensive description of the relevant physical processes. Nevertheless, a basic understanding of the interaction between ice sheets and climate can be obtained with simple models. Various simple models have been developed and presented in the glaciological literature. Anyone undertaking an advanced numerical modelling project is advised to study them ﬁrst, because they shed important insight into the processes that control ice sheet behaviour. In this chapter, we examine a range of simple physical models which have been used to describe ice sheets and ice shelves.

10.2

Perfectly-Plastic Ice Sheet Model

The concept of perfect plasticity can be used to derive the proﬁle of an ice sheet or cap [1]. A plastic medium is one where there is no deformation if the stress is lower than a critical yield stress, but when the stress reaches this value, it cannot increase further, and indeﬁnite deformation takes place. We recall (from Sect. 1.9) that the stress balance at the base of an ice sheet is approximated as dh qi g H ¼ s0 : dx

ð10:1Þ

H. Oerlemans (&) IMAU, Utrecht University, Utrecht, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_10

241

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H. Oerlemans

Here h denotes the ice surface elevation, H is the ice thickness, and s0 is the (constant) yield stress. The shape of a simple symmetric ice cap resting on a flat bed can be derived easily from Eq. (10.1). In this case H ¼ h, and for the part of the ice cap where dh=dx [ 0 we have d(h2 Þ 2s0 ¼ : dx qi g

ð10:2Þ

It follows that the proﬁle is parabolic and is described by h2 ðxÞ h2 ðx0 Þ ¼

2s0 ðx x0 Þ: qi g

ð10:3Þ

In order to construct the solution for an ice cap, we need to prescribe its size (its horizontal span L) and the thickness at the boundaries (which we take to be zero). By deﬁning rp ¼ 2s0 =qi g (the plasticity parameter), we obtain hðxÞ ¼

pﬃﬃﬃﬃﬃﬃﬃﬃ rp x

for 0 x

L ; 2

hðxÞ ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ L rp ðL xÞ for x L: 2

ð10:4Þ

height (normalized)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ The peak elevation is thus hmax ¼ rp L=2. Yield stresses used in the literature are typically in the range 0.5 105 to 2 105 Pa (e.g., Li et al. [2]), where the smaller values apply to large ice sheets with low mass turnover, and the higher values to active valley glaciers (Ng et al. [3]). The corresponding range for rp is 10–40 m. Figure 10.1 shows the resultant proﬁle. Although it provides a reasonable ﬁrst approximation to the shape of an ice cap, there are notable deﬁciencies. For instance, the solution is not valid at the ice divide. In reality, here the surface slope is very small and longitudinal stresses are important.

1

0

distance

Fig. 10.1 The surface proﬁle of a perfectly plastic ice cap on a flat bed (solid line). Dashed lines show the surface and bed proﬁles if local isostasy is included

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243

The perfectly plastic ice sheet proﬁle can be modiﬁed to include the effect of local isostatic depression of the bed. If the bed elevation is denoted by b (< 0), isostasy implies that H ¼ ð1 þ fÞ h;

ð10:5Þ

where the constant f ¼ b=h ¼ qi =ðqm qi Þ is determined by the ice density qi and mantle density qm. Equation (10.5) can be substituted in (10.1), and the new solution of the ice-sheet proﬁle is as shown in Fig. 10.1: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rp x; hðxÞ ¼ ð1 þ fÞ HðxÞ ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rp ð1 þ fÞ x:

ð10:6Þ ð10:7Þ

An important conclusion can be drawn from this analysis: bedrock sinking reduces the mean surface elevation of the ice sheet, but increases the ice thickness. It should be stressed that the adjustment of the bed to an ice load depends very much on the flexural length scale dl of the lithosphere. There can be a local isostatic balance only when L dl . This point will be taken up again in Chap. 15 on ice sheets and geodynamics. There we will also discuss the time scale over which isostatic adjustment takes place. By integrating (10.6) with respect to x, it is found that the mean surface elevation is pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ h ¼ 2 rpi L 3

ð10:8Þ

where rpi = rp/(1 + f). The mean surface elevation of an ice sheet thus varies with the square root of its size. This is an important result, because h determines to a large extent the total surface mass budget of an ice sheet.

10.3

The Height–Mass Balance Feedback

Except for very cold climates (as currently found in the Antarctic), the speciﬁc balance normally increases with height. For a growing ice sheet the mean surface elevation increases; therefore a positive feedback mechanism is apparent. Without the height– mass balance feedback, the Northern Hemisphere ice sheets would never have become so large during the ice age. The height–mass balance feedback may lead to hysteresis, and this can be illustrated in a simple way with the perfectly plastic ice-sheet model.

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We ﬁrst consider the (very idealised) case of a bounded domain of size L with a horizontal bed at sea level. The size of the ice sheet, if it can exist, is thus equal to L. We assume that the balance rate B is a linear function of height z: BðzÞ ¼ b ðz EÞ:

ð10:9Þ

The mean speciﬁc balance for an ice sheet is then obtained by integrating B(z) over the ice-sheet surface, yielding for the net surface balance rate Bn ¼ b ðh EÞ:

ð10:10Þ

Here h is the mean surface elevation of the ice sheet. Since the shape of a perfectly plastic ice sheet is entirely determined by its size L, we have: ðiÞ the ice sheet vanishes if Bn \ 0;

i:e:; E [ h;

ð10:11Þ

ðiiÞ the ice sheet remains if Bn [ 0;

i:e:; E \ h:

ð10:12Þ

For E [ 0, L ¼ 0 always represents a stable steady state. This leads to the solution diagram in Fig. 10.2. Apparently, over the range of values of the equilibrium-line altitude 0\E\h, two stable steady states exist. The occurrence of hysteresis (arrows) is typical for this ice-sheet model. The importance of the height–mass balance feedback was earlier pointed out by Bodvardsson [4]. Weertman [1, 5] used the perfectly-plastic model to study the dynamics and stability of the Pleistocene northern hemisphere ice sheets. The theory summarised here is similar to his original analysis. Figure 10.3 shows the geometric set-up. The x-axis is north-south oriented; the domain is bounded in the north (Arctic Ocean) but not in the south. The equilibrium line slopes upward in the southerly direction. It is assumed that all ice accumulating on the northern half of the ice sheet flows into the Arctic Ocean or melts close to it. The ice sheet’s evolution is therefore governed by the mass balance condition on its southern half. Weertman

•

Fig. 10.2 Equilibrium states for a perfectly plastic ice sheet subject to a balance rate that varies linearly with height size of ice sheet

L

•

0

_ h

0 cold

E

warm

Analytical Models of Ice Sheets and Ice Shelves

245

z (m)

10

equilibrium line

0

Arctic ocean

P

0

x (south)

L

Fig. 10.3 Conﬁguration of a northern hemisphere ice sheet with a tilted equilibrium line

assumed constant accumulation and runoff rates above and below the equilibrium line. Here we take the mass balance to be a linear function of height again, and write B ¼ b vðP xÞ þ b z;

ð10:13Þ

in which b is the elevational balance gradient and b v is the horizontal balance gradient. The intersection of the equilibrium line with sea level, at x ¼ P, is called the climate point. The ice sheet will be in steady state when the net balance over its southern half (Bn) is zero, i.e., 2 Bn ¼ L

Z

2b B dx ¼ L L=2 L

Z

L

½v ðP xÞ þ z dx ¼ 0:

ð10:14Þ

1=2 hðxÞ ¼ rpi ðL xÞ note rpi ¼ rp =ð1 þ fÞ :

ð10:15Þ

L=2

For the ice sheet proﬁle we write

By substituting this for z in Eq. (10.14) we ﬁnd 2b Bn ¼ L

Z

L

h

1=2 i vðP xÞ þ rpi ðL xÞ dx ¼ F1 þ F2 L1=2 þ F3 L; ð10:16Þ

L=2

where F1 ¼ b vP;

ð10:17aÞ

pﬃﬃﬃ 2 pﬃﬃﬃﬃﬃﬃ b rpi ; F2 ¼ 3

ð10:17bÞ

246

H. Oerlemans

3 F3 ¼ b v: 4

ð10:17cÞ

Equation (10.16) shows that, if b and v are positive (balance increasing with latitude and height), lim Bn ðL ! 1Þ ¼ 1. This implies that the equilibrium ice-sheet size is ﬁnite. Figure 10.4 shows how Bn varies with L for different positions of the climate point for parameter values b = 0.0007 (m ice) yr−1m−1, v = 0.001 and rpi = 5 m. For a range of negative values of P (the climate point lies in the Arctic Ocean) there are two equilibrium states (red curve/dots in the ﬁgure). The smaller one is unstable, the larger one stable. For positive values of P (the climate point lies on land) there is only one equilibrium state, and it is stable. Adding to these considerations that L ¼ 0 is always an equilibrium state if P\ 0, we are now able to construct a solution diagram. The equilibrium states are obtained by setting the net balance to zero, i.e., Bn ¼ F1 þ F2 L1=2 þ F3 L ¼ 0:

ð10:18Þ

This is a quadratic equation that can be solved directly for L, yielding L1;2 ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ F22 2F1 F3 F2 F22 4F1 F3 : 2F32

ð10:19Þ

Fig. 10.4 The net balance as a function of ice sheet size L, for three values of P (labels on the curves). Equilibrium states are indicated by dots. Where a curve intersects Bn = 0 with negative slope, the equilibrium state is stable because the ice sheet would lose (gain) mass if its size is perturbed to increase (decrease). An analogous argument shows that equilibrium states at positive-sloped intersections are unstable

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247

3500 3000 2500

L (km)

2000 1500 1000 500

Pc

• •

0 -600

-400

-200

0

200

400

600

P (km)

Fig. 10.5 Equilibrium ice-sheet size L as a function of the position of the climate point P. Model parameters as in Fig. 10.4

The solution with the + sign refers to the stable equilibrium state, the solution with the − sign (only applicable for P < 0) to the unstable equilibrium state. The solution diagram is shown in Fig. 10.5. Bifurcation points are indicated by dots, and the arrows indicate the implied hysteresis. The unstable branch is dashed. The solution in (10.19) represents a simple fold with one bifurcation point at the nose of the curve. In dynamical systems, this bifurcation is referred to as a saddle-node bifurcation. However, because L ¼ 0 is always a stable solution for P\ 0, there is another stable branch, making the solution qualitatively similar to the cusp catastrophe with two bifurcation points [6]. Note that the solution diagram is qualitatively similar to that for a glacier on a sloping bed with a horizontal equilibrium line. This case will be considered later.

10.4

Ice-Sheet Profile for Plane Shear with Glen’s Law

Ice-sheet proﬁles can also be derived analytically for the case of simple shearing flow that obeys Glen’s law. We assume that the balance rate B is constant (and positive) and consider an axially symmetric geometry (Fig. 10.6) with a horizontal bed. The starting point is the expression for the vertically-averaged radial ice velocity for plane-shearing flow (see also Chap. 8, speciﬁcally Sects. 8.2.1 and 8.5.2):

248

H. Oerlemans

Fig. 10.6 Geometry of the axially symmetric ice sheet model

n dH Ur ¼ A0 H n þ 1 ; dr

ð10:20Þ

which can be derived by integrating the proﬁle of horizontal velocity over the glacier thickness, and dividing the result by the thickness. Phenomenological constants have been absorbed into the effective flow parameter A′ which is given by A′ = 2A(qig)n/(n + 2). The continuity equation reads (in the case of constant ice density) @H @ðr H Ur Þ ¼ þ B: @t r @r

ð10:21Þ

For a steady state we thus have dðr H Ur Þ ¼ Br dr

!

Ur ¼

B 1 H r: 2

ð10:22Þ

Combining Eqs. (10.20) and (10.22) yields H

1 þ 2=n

1=n B dH ¼ r 1=n dr: 2A0

ð10:23Þ

This expression can readily be integrated from the centre of the ice sheet (r ¼ 0, H ¼ Hc ) to distance r to give 1=n n 2 þ 2=n n B H Hc2 þ 2=n ¼ r 1 þ 1=n ; 2n þ 2 n þ 1 2A0

ð10:24Þ

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249

or H 2 þ 2=n Hc2 þ 2=n ¼ 2

B 2A0

1=n

r 1 þ 1=n :

ð10:25Þ

To ﬁnd Hc we have to formulate a boundary condition at the edge of the ice sheet (r ¼ R). The simplest condition is HðRÞ ¼ 0:

ð10:26Þ

It follows that the ice thickness at the centre is given by Hc2 þ 2=n ¼ 2

B 2A0

1=n

R1 þ 1=n ;

ð10:27Þ

and the full solution becomes HðrÞ ¼ 2

n=ð2n þ 2Þ

B 2A0

1=ð2n þ 2Þ n

R1 þ 1=n r 1 þ 1=n

on=ð2n þ 2Þ

:

ð10:28Þ

This expression looks more friendly if we substitute a value for n. With n ¼ 3, HðrÞ ¼ 21=4

1=8 n o3=8 B 4=3 4=3 R r ; A0

ð10:29Þ

and Hc ¼ 2

1=4

1=8 B R1=2 : A0

ð10:30Þ

The expression in (10.29) represents a surface shaped much like an ellipse. The solution is shown in Fig. 10.7, with the units normalised so that the maximum radius and depth are both equal to one. Derived by Vialov [7] originally, Eq. (10.28) is referred to as the Vialov solution. Figure 10.7 compares it with the perfectly-plastic solution, with the latter scaled in such a way that the mean ice thickness is the same as for the plane-shear solution. The plane-shear solution is better than the perfectly-plastic solution at the dome, because its surface slope vanishes there. Nearer the edge of the ice sheet, the perfectly-plastic proﬁle is lower and somewhat more realistic (when compared to actual proﬁles along flowlines for the Antarctic Ice Sheet). However, this is fortuitous. Large ice sheets tend to have shallower proﬁles close to the margin because they drain ice more efﬁciently due to the presence of outlet glaciers and/or ice streams, and because extensive sliding is more widespread. It is noteworthy that the

250

H. Oerlemans 1.2

H (normalised)

1 0.8 plane shear 0.6 perfectly plastic 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

r (normalised)

Fig. 10.7 A comparison of plane-shear and perfectly-plastic solutions for an ice sheet on a flat bed

ice thickness in the centre varies with R1/2 for the plane-shear model as well as for the perfectly-plastic model.

10.5

Ice Shelves

In cold regions where large amounts of ice flow into the sea, ice tongues and ice shelves may form. Ice shelves can be very thick, up to 800 m thick in some places in West Antarctica. The Antarctic Ice Sheet is buttressed by ice shelves almost everywhere, but it is only in large-scale embayments where very extensive shelves form (Fig. 10.8). The largest ice shelves are the Ross Ice Shelf and the Ronne-Filchner Ice Shelf. Ice shelves exist outside the Antarctic continent also. Some glaciers in northeast Greenland have large floating tongues. Because forces acting on the top and base of an ice shelf are small, it can be expected that the horizontal ice velocity does not vary much with depth. Accordingly, if the basal drag is set to zero and the ice shelf flows in the x-direction, the balance of forces can be approximated as @ @h 0 ð2H W s xx Þ þ ðsy2 þ sy1 ÞH ¼ qi g H W; @x @x

ð10:31Þ 0

where W(x) is the width of the ice shelf, h is its surface elevation, s xx is the depth-averaged deviatoric longitudinal stress (in the flow direction), and sy1 and sy2 are the depth-averaged shear stresses exerted by the lateral boundaries on the shelf flow. This equation is used to study idealised cases for which an analytical solution can be derived straightforwardly. First we consider an ice shelf with a constant width that does not experience side drag (see Fig. 10.9). In this case the force balance becomes

10

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251

Fig. 10.8 Topography of the Antarctic Ice Sheet (contours in m; courtesy of J. Bamber). Some of the major ice shelves are indicated

Fig. 10.9 Geometry of the one-dimensional ice-shelf model

@ @h 0 ð2H s xx Þ ¼ qi g H : @x @x

ð10:32Þ

As an ice shelf is floating, it is simple to relate the ice thickness to the surface elevation by writing

qi h¼ 1 H: qw

ð10:33Þ

252

H. Oerlemans

Integrating Eq. (10.32) with respect to x and using Eq. (10.33) yields

1 q 0 2 H s xx ¼ qi g 1 i H 2 ; 2 qw

ð10:34Þ

1 q 0 s xx ¼ qi g 1 i H: 4 qw

ð10:35Þ

or

Thus the longitudinal stress deviator varies linearly with the ice thickness. The next step is to use Glen’s law for simple stretching: 0 n @U ¼ A s xx : @x

ð10:36Þ

Combining this with Eq. (10.35) gives @U ¼ CH n @x

with

C¼A

qi g ðqw qi Þ n : 4qw

ð10:37Þ

The continuity equation for this case reads @H @ ¼ ðHUÞ þ B: @t @x

ð10:38Þ

The pair of equations (10.37) and (10.38) can be used to calculate the evolution of an ice shelf. For the case in which B is a function of x and/or t, this can only be done with numerical methods. However, for the special case of constant balance rate and steady-state conditions, an analytical solution can be obtained (due to Oerlemans and Van der Veen [8]). Setting the time derivative in Eq. (10.38) to zero and integrating from the grounding line we ﬁnd UH ¼ U0 H0 þ Bx;

ð10:39Þ

where U0 and H0 are the ice velocity and the ice thickness at the grounding line, respectively. With the chain rule we can write d dU dH ðHUÞ ¼ H þU : dx dx dx

ð10:40Þ

We multiply this equation by H and use Eq. (10.39) to obtain H2

dU dH þ ðU0 H0 þ BxÞ ¼ BH: dx dx

ð10:41Þ

10

Analytical Models of Ice Sheets and Ice Shelves

253

Now the velocity can be eliminated with the aid of Eq. (10.37), yielding CH n þ 2 þ ðU0 H0 þ BxÞ

dH ¼ BH: dx

ð10:42Þ

The simplest case to consider is the case with B = 0, i.e., changes in ice thickness are only due to longitudinal stretching. We then have H ðn þ 2Þ

dH C ¼ : dx U0 H0

ð10:43Þ

Integrating from the grounding line yields ðn þ 1Þ

H ðn þ 1Þ H0

¼

Cðn þ 1Þ x: U0 H0

ð10:44Þ

The ice-thickness proﬁle thus becomes HðxÞ ¼

Cðn þ 1Þ ðn þ 1Þ x þ H0 U0 H0

1=ðn þ 1Þ

:

ð10:45Þ

For n ¼ 3 this expression becomes HðxÞ ¼

4C x þ H04 U0 H0

1=4

:

ð10:46Þ

This solution shows that the ice thickness drops off very rapidly with distance close to the grounding line (Fig. 10.10). Without giving a derivation (this can be found in [9]), we mention that the solution for a positive balance rate B > 0 is given by

Fig. 10.10 Steady-state solution for a two-dimensional ice shelf with zero balance rate

254

H. Oerlemans

( HðxÞ ¼

)1=ðn þ 1Þ C U0n þ 1 CB H0n þ 1 1 ; B ðBx þ H0 U0 Þn þ 1

ð10:47Þ

and that for a negative balance B < 0 by ( HðxÞ ¼

)1=ðn þ 1Þ C U0n þ 1 CB H0n þ 1 þ 1 þ : B ðBx þ H0 U0 Þn þ 1

ð10:48Þ

Note that these solutions do not provide any information on the length of the ice shelf, which is essentially determined by the effectiveness of the calving process. Real ice shelves normally display less rapid ice thinning as one moves away from the grounding line. The detailed topography there depends on the distribution of melt rates and accretion rates at the base of the ice shelf. In reality, the grounding line is more like a grounding zone. Upstream of this zone the basal drag balances the driving stress, and downstream of this zone the longitudinal stress gradient balances the driving stress. Within the grounding zone, there is a gradual transition between these two regimes.

Exercise 10:1 For the model in Sect. 10.3, examine how the location of the critical point Pc (shown in Fig. 10.5) depends on the model parameters.

References 1. Weertman J (1961) Stability of ice-age ice sheets. J Geophys Res 66:3783–3792 2. Li H, Ng F, Li Z, Qin D, Cheng G (2012) An extended ‘perfect-plasticity’ method for estimating ice thickness along the flow line of mountain glaciers. J Geophys Res Earth Surf. 117:F01020 3. Ng F, Barr ID, Clark CD (2010) Using the surface proﬁles of modern ice masses to inform palaeo-glacier reconstruction. Quat Sci Rev 29(23–24):3240–3255 4. Bodvardsson G (1955) On the flow of ice-sheets and glaciers. Jökull 5:1–8 5. Weertman J (1976) Milankovitch solar radiation variations and ice age ice sheet sizes. Nature 261:17–20 6. Gilmore R (1981) Catastrophe theory for scientists and engineers. Wiley, New York 7. Vialov SS (1958) Regularities of glacial shields movement and the theory of plastic viscous flow. Int Ass Hydrol Sci Publ. 47:266–275 8. Oerlemans J, van der Veen CJ (1984) Ice sheets and climate. Reidel, Dordrecht 9. van der Veen CJ (1999) Fundamentals of glacier dynamics. Balkema, Rotterdam

11

Firn Christo Buizert and Michiel Helsen

11.1

Introduction

Firn is the transitional stage between snow and glacial ice. After some initial drifting, disintegration and packing of snow crystals, near-surface snow typically has a density of 280–420 kg m−3 , meaning that by volume it consists of roughly two thirds air, and one third ice. The overburden pressure from shallower firn strata gradually densifies the firn by squeezing out the interstitial air, until the firn density approaches that of ice (ρi = 917 kg m−3 ). Gravity is the driving force behind the densification process. Not all air is expelled during densification, and some air bubbles are permanently trapped between the ice grains. Measurements in mature ice show an air content of around 10% by volume, implying a mean bubble close-off density around ρco = 830 kg m−3 . Firn is commonly defined as having densities below ρco , although densification does continue below the corresponding depth. The firn zone is situated in the accumulation zone of a glacier or ice sheet, since accumulation must outweigh ablation for a firn layer to be formed. Apart from the densification process, which brings the centres of adjacent firn grains closer together, firn metamorphism also occurs—this refers to the alteration of the firn grains themselves through grain growth and recrystallisation. The mechanical properties of firn are determined by the arrangement and size distribution of its constituent grains, as well as by the viscosity of ice. A complete description of the densification process would therefore include grain metamorphism. In practice, most

C. Buizert (B) College of Earth, Ocean and Atmospheric Sciences, Oregon State University, Corvallis, USA e-mail: [email protected] M. Helsen School of Education, Rotterdam University of Applied Sciences, Rotterdam, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_11

255

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densification models use an empirical approach where the underlying microscopic firn structure is not explicitly described. Much of the firn literature focusses on the dry snow zone, where the surface temperature remains below the freezing point all year round. Surface melt has a large impact on firn density. Meltwater either refreezes at the surface, or percolates through the snowpack to form ice lenses at depth. Both situations give rise to positive density anomalies of essentially air-free, solid ice layers. A description of firn densification in the percolation zone is beyond the scope of this chapter; we refer the interested reader to the paper by Reeh [1]. A wide variety of firn models exist, ranging from empirical models to physically based models that try to describe microscopic processes in detail. In Sect. 11.2.1 we discuss the physics of firn densification, and in Sect. 11.2.2 we describe the way it is commonly represented in models. Modelling of firn densification has diverse applications in glaciology and paleoclimate studies. We describe two of the more important applications in this chapter. The first application is that of correcting satellite altimetry data for changes in firn column thickness. This step is necessary when translating surface elevation changes into an ice sheet mass balance estimate, and has implications for global sea level rise. The second application is the calculation of the bubble lock-in depth at times in the past. Since air bubbles are occluded at the bottom of the firn column, they are younger than the surrounding ice by an amount age. Knowledge of age is important for dating records of paleo-atmospheric composition obtained from air bubbles in ice cores, and for determining the relative timing of greenhouse gas variations and temperature changes. We discuss these applications in detail in Sect. 11.3. Note that firn densification models are also relevant when describing the flow of cold glaciers and ice shelves that are sufficiently thin for the firn layer to impact the flow dynamics significantly.

11.2

Firn Densification

Within the firn layer, between the surface and the close-off depth, compaction occurs and air is expelled. The rate of densification depends on the rheology of the material and the relative importance of the different processes at play. Large spatial differences are present in depth-density relationships (Fig. 11.1), primarily as a result of the contrasting climatological conditions at different sites.

11.2.1 Mechanisms of Firn Densification Distinct stages of firn compaction, in which different processes dominate, occur during the transformation from fresh snow to glacial ice (Fig. 11.2). In general, the transformation from snow to ice involves (1) the mutual displacement of crystals to

11

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Fig. 11.1 a Spatial pattern of Antarctic pore close-off depth (colour scale in metres) and b density profiles in the firn at South Pole and at Byrd Station, Antarctica, from Van den Broeke [2] Fig. 11.2 Different stages of firn densification (figure courtesy of Michiel van den Broeke)

optimise packing, (2) changes in the size and shape of the crystals and (3) internal deformation of crystals. Here we outline these processes as if they occur sequentially, which is a simplification; in reality, the depth intervals where these processes occur overlap.

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Fig. 11.3 a Relationship between snow crystal shapes and the temperature and water vapour pressure during formation; b examples of snow flakes; see Libbrecht [3,4]. Reproduced with the kind permission of Kenneth Libbrecht

Snow crystals can have a large variety of distinct shapes; the shape depends on the temperature and (water vapour) pressure at which they are formed (Fig. 11.3). These often complex shapes of the snow flakes can lead to very low density values for freshly accumulating snow (40–200 kg m−3 ), as the snow flakes are packed inefficiently. During and after a snowfall event, wind action induces breakup of the complex snow crystals. This causes disintegration of the snow flakes, turning them into more rounded snow grains, which can settle efficiently by sliding into a more organised, stacked pattern. In addition, recrystallisation by diffusion and sublimation occurs immediately after snowfall events. Molecular diffusion is the process whereby molecules move through the ice lattice or along the crystal surface, and this is facilitated when the snow is near its melting point. Moreover, sublimation occurs readily: water molecules can move by sublimating from one grain, and condensing onto another grain. The associated water vapour transport is driven by temperature gradients in the snow pack and the geometry (curvature) of the ice surfaces via the so-called Kelvin effect. The distinctly-shaped crystals turn into round grains because of the tendency for the free energy of the system to be minimised. A reduction in surface area reduces the free energy, so the formation of spherical grains is favoured. Large snow grains grow at the expense of smaller grains because this also reduces the free energy. Together, grain growth and the settling effects of wind reworking and the more effective stacking of grounded grains lead to a fast consolidation of snow after a snowfall event. Aged surface snow has typical densities of 200–400 kg m−3 . A phenomenon related to sublimation is the formation of depth hoar. This is a low-density layer (100–300 kg m−3 ) consisting of coarse, pyramidal or cup-shaped crystals. These layers are formed when cooling occurs at the surface, while the snow pack below remains warmer. The steep temperature gradient that results causes rapid vapour flux towards the surface, and recrystallisation occurring along this pathway leads to the formation of faceted, coarser grains.

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Settling and grain growth dominate the relatively fast firn compaction process in the first stage, until a density of ∼ 550 kg m−3 is reached. At this density the spherical grains are organised in their most effective configuration. Further compaction occurs by an increase of the contact area between the neighbouring ice crystals. This process is often referred to as sintering, and involves the transfer of material to the contact points between grains to form bonds. Initially, this process is also dominated by vapour transport, but with increasing density, the porosity decreases and hence the influence of vapour transport is greatly reduced. As the overburden pressure and the contact area between grains both increase, recrystallisation is favoured: molecular diffusion changes the size and shape of crystals, thereby reducing the stress at the contact points. This last process is controlled by temperature and the overburden pressure, and the corresponding compaction rates are much slower than during the initial settling phase. At a density of ∼ 730 kg m−3 , the compaction rate further slows down. The area of contact between grains has reached a maximum, and the remaining air occupies thin channels between the grain boundaries. The main process that can further increase the density of the medium is internal deformation (creep) of crystals. The final stage of densification of the ice matrix occurs below pore close-off: due to increasing overburden pressure, the air bubbles are reduced in volume. At the close-off density the hydrostatic pressure in the ice is around 4–7 times as large as the (near-ambient) air pressure in the bubbles. The bubbles will be compacted until their pressure matches that of the surrounding ice. At depths below ∼ 600 m this process eventually results in the formation of clathrates, when the nitrogen, oxygen and trace gas molecules that were previously present in gaseous state are pressed into cage-like crystalline inclusions in the glacial ice.

11.2.2 Firn Densification Models A range of firn densification models exist that serve different applications and purposes. In increasing order of complexity, we can sub-classify them as steady-state empirical models, time-dependent empirical models, and models that quantify the individual physical processes in a time-dependent way. Of the first type (steady-state empirical), we here discuss the densification model by Herron and Langway [5], which is the most widely known and used model in glaciological research.

11.2.2.1 The Herron and Langway (H-L) Model By comparing a suite of firn density profiles obtained from different cores from Greenland and Antarctica, Herron and Langway [5] concluded that the densification process can be divided into two distinct stages: a rapid phase during which grain settling is dominant until the density reaches a threshold value of ∼ 550 kg m−3 , and a subsequent, slower phase of densification due to grain growth and sintering until pore close-off. This crossover at the critical density is visible as a kink in the firn density profiles (Fig. 11.1b). They use the postulate by Robin [6] that the proportional

260

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Fig. versus 11.4 Depth ρ ln profiles in ρi − ρ Antarctic firn, from Herron and Langway [5]. Reprinted from the Journal of Glaciology with permission of the International Glaciological Society

change in air space is linearly related to the change in stress due to the weight of overlying snow, which can be expressed as dv p = −kρ dz, vp

(11.1)

where v p denotes pore space fraction, ρ denotes firn density, z is depth and k is a densification rate constant. The minus sign indicates that the pore space decreases with increasing stress. We can express v p in terms of the density difference between ice (ρi ) and firn: ρi − ρ vp = . (11.2) ρi Substitution of (11.2) into (11.1), rearranging, and integrating gives ln

ρ ρi − ρ

= kρi z,

(11.3)

which suggests that a linear relation is expected between the term on the left and depth. This is indeed seen at various sites (Fig. 11.4 shows sites from inland Antarctica), and these observations constrain the value of k in (11.1) for the two densification stages above and below the critical density of 550 kg m−3 . The differential equation model expressing the density change with depth can be derived from (11.1), and is dρ = kρ(ρi − ρ). dz

(11.4)

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In steady state, the vertical velocity of material in the firn (dz/dt) is equal to the accumulation rate A divided by ρ (Sorge’s Law, e. g., [7]); hence one obtains the equation for the densification rate: dz A = , dt ρ

dρ = k A(ρi − ρ). dt

(11.5)

It is assumed that the temperature and accumulation-rate dependences of the rate can be separated. The rate constant k is made a function of temperature T , following an Arrhenius-type law. Note that in their original paper, Herron and Langway [5] used units for A of metre water equivalent per year, and expressed the density in Mg m−3 . In the following we use kg m−2 year−1 for accumulation and kg m−3 for ρ, ⎧ α A 10,160 ⎪ ⎪ k0 , α = 1, ρ < 550, (ρi − ρ), k0 = 11 exp − ⎪ ⎪ ρw RT ⎨

dρ = β ⎪ dt ⎪ A 21,400 ⎪ ⎪ ⎩ k1 , β = 0.5, ρ > 550, (ρi − ρ), k1 = 575 exp − ρw RT (11.6) where R is the gas constant and ρw is the density of water. In this model, two different expressions of k are found above and below the critical density, and there are different values of the exponent of accumulation A for those regions. The fact that the exponent β is less than one means that in the second stage of densification, the accumulation A exerts a weaker influence on the rate than the effect considered in (11.5). Equation (11.6) is a generally applicable firn densification model, which fits observations quite well (Fig. 11.5). It allows for a fully analytical solution of ρ(z), which can be found in the original publication of the model.

11.2.2.2 Time-Dependent Firn Densification Models The Herron-Langway model assumes both A and T to be constant. In reality both vary on the sub-annual time scales of seasonality and weather, as well as on the decadal to millennial time scales of climate change. To incorporate this variability in our description of firn densification, time-dependent models are needed. We shall first consider seasonality. The Herron-Langway steady state model produces smooth (or bulk) depth/density curves. By contrast, high-resolution density measurements in the firn column show pronounced small-scale variability, or layering, suggesting a seasonality in densification rates. Zwally and Li [8] proposed that the variability is caused by seasonal temperature variations, and they introduced a temperature dependence into the densification rate factor k by relating it to grain growth rates: E , k = β K 0G exp − RT

K 0G = 8.36|Tc |−2.061 ,

E = 883.8|Tc |−0.885 ; (11.7)

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Fig. 11.5 Model depth/density curves compared to data collected in the firn at Greenland and Antarctic sites, from Herron and Langway [5]. ρ0 denotes the firn density at the surface. Reprinted from the Journal of Glaciology with permission of the International Glaciological Society

Tc is the time-variable temperature in Celsius, and β is an empirical constant (> 1) that scales the effect of grain growth to total densification. Helsen et al. [9] derived a parameterisation for β by fitting a suite of Antarctic density profiles and optimising for the pore close-off depth, and found β = max (1, 76.138 − 0.28965T ).

(11.8)

This model produces annual density peaks in good agreement with observed seasonal density variability. An example of its application to the summit area of Greenland is shown in Fig. 11.6. Using a slightly different approach, Arthern et al. [11] quantified the temperature dependence of the densification rate by making in situ measurements of Antarctic firn layer compaction rates using strain-meters. These measurements also show increased snow compaction with high temperatures, and the results support a model that combines the effect of grain growth and creep based on lattice diffusion (Nabarro-Herring creep). The optimised formulation of these authors is

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Fig. 11.6 Observed (black) and simulated (red) depth/density curves obtained using the Zwally and Li [8] model with a sinusoidal seasonal temperature forcing and a constant accumulation appropriate for Summit, Greenland conditions. The observations are taken from Bolzan and Strobel [10]

⎧ Eg Ec ⎪ ⎪ 0.07Ag exp − (ρi − ρ), ρ < 550, + ⎪ ⎪ ⎨ RT RT dρ = ⎪ dt ⎪ Eg Ec ⎪ ⎪ (ρi − ρ), ρ > 550, + ⎩ 0.03Ag exp − RT RT

(11.9)

where g is gravity, and the optimised activation energies for creep and grain growth are E c = 60 kJ mol−1 and E g = 42.4 kJ mol−1 , respectively. These formulations were subsequently tested against a large set of Antarctic firn density profiles by Ligtenberg et al. [12]. These authors found a clear relation between ratios (called MO-ratios) of modelled versus observed critical density and pore closeoff depths and accumulation, and thus they concluded that the effect of accumulation on densification was too large in the model of Arthern et al. [11]. They suggested making a correction by multiplying the densification rate expressions of Arthern et al. by the following MO-ratios: ¯ ρ < 550, MO550 = max(0.25, 1.435 − 0.151 ln A), ¯ MO830 = max(0.25, 2.366 − 0.293 ln A), ρ > 550,

(11.10)

where A¯ is the annual average accumulation rate. To describe the firn evolution on the longer time scales of climatic change, different time-dependent densification models are commonly used. As discussed in Sect. 11.3.2 below, the main application of such models is to reconstruct changes in firn thickness and age back in time. Two frequently used models are the BarnolaPimienta (B-P) model [13] and the Arnaud-Goujon (A-G) model [14,15]. Both models distinguish three stages of densification, which are separated at ρ = 550 and

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ρ = 830 kg m−3 . The B-P model is identical to the Herron-Langway model for the first stage; densification in stages 2 and 3 is described by dρ Ec , = A0 ρ f (ρ)(P)n exp − dt RT

(11.11)

where A0 is a constant and E c is the activation energy for mechanical creep. The factor f depends on ρ, and describes the geometry of the pores. P gives the overburden pressure minus the bubble pressure in the pores. The dependence on the overburden pressure allows the model to be run as a time-dependent model, because the densification rate at any given depth depends only on local parameters (ρ, T , P). In contrast to (semi-)empirical models such as the H-L and B-P models, the A-G model is a physically based model that explicitly describes the underlying processes. The first stage is based on the grain sliding model of Alley [16]; the second stage is based on a sintering model for spherical metallic powders due to Arzt [17]; the third stage describes the compression of bubbles in a way that is very similar to the B-P model.

11.2.3 Firn Layering and Microstructure Polar firn is a layered medium, exhibiting pronounced density fluctuations with depth. The layering is caused by a number of factors, including variability in accumulation, deposition density, surface temperature, impurity loading, wind scouring, insolation and hoar formation. Surface redeposition processes introduce lateral variability, for example due to the formation of sastrugi. The layering is clearly visible in snowpits dug into the firn (Fig. 11.7). Fig. 11.7 Firn layering observed in a (covered) backlit snowpit at the West Antarctic Ice Sheet (WAIS) Divide deep drilling site, Antarctica. Photograph courtesy of Kendrick Taylor

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Depth (m) Fig. 11.8 Density variability and two proposed mechanisms. a Density (grey curve) and density variability (orange curve); data from Miller and Schwager [32]. b Variability minimum explained by layer microstructure crossover. Green and blue dashed lines represent the coarse-grained (initially low ρ) and fine-grained (initially high ρ) layers, respectively. c Variability minimum explained by a combination of dust and deposition noise. Blue dashed line shows how deposition noise vanishes with depth due to densification; green dashed line shows the increase in variability with depth due to (seasonal) impurity forcing

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In Sect. 11.2.2 we outlined the time-dependent densification model of Zwally and Li [8], in which layering is introduced through changes in temperature forcing. Although this model, and others derived from it, are reasonably successful in modelling surface elevation changes, a growing body of evidence indicates that temperature and accumulation alone cannot explain all of the observed density variability. Recent observations show that firn microstructure and impurity content play important roles in controlling densification rates and producing firn layering. Here we review some of the evidence and conceptual models that have been proposed to explain the observations. Subtle yet compelling evidence for the influence of microstructure is provided by the change in density variability with depth. In Fig. 11.8a, we plot high-resolution density data from the B26 core in Greenland. After subtracting the bulk density curve, we take the standard deviation over a 2 m running window. The largest density variability is found near the surface. At around 25 m depth, there is a clear minimum in variability, after which the variability increases again to reach a second maximum between 40 and 60 m. This variability minimum is consistently observed at densities of 600–650 kg m−3 in high resolution density records taken over a wide range of climatic conditions. It is impossible to explain this minimum when assuming densification is driven by T , A and the surface (firn/snow) density ρ0 and their temporal variations. All densification models discussed in this chapter predict a monotonic decrease in density variability with depth. For example, in the Herron and Langway model the densification rate scales as dρ/dt ∝ (ρi − ρ); this causes low density layers to densify faster, reducing the density contrast between adjacent layers. Clearly, there must be another property intrinsic to the firn that is able to increase its density variability. A first conceptual model to explain the variability structure is a joint densitymicrostructure layering in which the near-surface high density layers ‘cross over’ to become the low density layers at depth (Fig. 11.8b). In this hypothesis, first formulated by Gerland et al. [18] and refined by Freitag et al. [19], the coarse-grained firn that makes up the low density layers near the surface densifies at a faster rate than fine-grained high density layers, thus ending up as (coarse-grained) high density layers at depth. The different densification trajectories for coarse- and fine-grained firn are shown as green and blue dashed lines, respectively, in Fig. 11.8b. Gerland et al. [18] based their hypothesis on the observation that the anticorrelation between density and electrical conductivity in shallow firn changes to a positive correlation below the density variability minimum. A similar crossover in correlation is observed for other structural properties, such as grain size, specific surface area, optical brightness and structural anisotropy [20,21]. A second proposed mechanism is based on the observation that impurities can influence firn densification rates [22,23], as shown in Fig. 11.9. High resolution density measurements are shown for three sections of a Greenland firn core, together with the logarithm of the Ca2+ ion concentration. Near the surface (Fig. 11.9a) the signals are uncorrelated, yet they increasingly co-vary in deeper firn (Fig. 11.9b–c). At a depth of ∼ 28 m w.e. (m water a clear seasonal signal has developed

equivalent), in ρ(z) that closely tracks log Ca2+ . Hörhold et al. [22] argued that the impurities

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Fig. 11.9 The influence of impurity content on firn densification. Firn density (black) and the logarithm of the Ca2+ concentration (red, ng g−1 ), for three sections of 2 m length from Greenland firn core B29. All depths (horizontal axes) are in metres water equivalent (m w.e.). Figure from Hörhold et al. [22]. Reprinted from Earth and Planetary Science Letters ©2012 with permission from Elsevier

soften the firn, and because this more ductile firn densifies at a higher rate, positive density anomalies develop that correlate with the impurity content. Note that the mechanistic link between impurities and densification is currently unknown, and it is unclear whether the softening is caused by the ionic content of firn itself or, for example, by particulate dust for which Ca2+ is a proxy. Impurities can lead to the observed density variability minimum in the following way (Fig. 11.8c). Near the surface there is high variability due to seasonality and deposition noise (i. e., variable A and ρ0 ); this density variability will decay with depth following the blue dashed line. Under the influence of impurities a new layering develops with depth, following the green dashed line. The combined effect

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of deposition noise and impurities will be a minimum in the density variability, as observed in real firn. From the above discussions, it is clear that the causes of firn layering are not fully understood, but probably involve a combination of seasonality, deposition noise, microstructure and impurities. Note that the processes discussed so far are not mutually exclusive, and might all contribute to the observed variability. An understanding of firn layering aids the study of gas records from firn and ice cores (see Chap. 12). The dense layers will reach the close-off density before the open layers do, thereby forming sealing layers that can inhibit vertical diffusion of gas molecules. Sampling of interstitial firn air indicates the existence of a lock-in depth below which gravitational enrichment ceases (indicative of a strongly reduced diffusive flux), yet sufficient connected porosity exists to extract large volumes of air. This lock-in depth is thought to be associated with the formation of such sealing layers. The volume fraction not occupied by ice is the porosity s, given by s = 1 − ρ/ρi . The porosity is divided into open and closed pores (s = sop + scl )—the former still being connected with each other and the overlying atmosphere, and the latter being closed off. Closed porosities can be measured by determining the amount of air a sample displaces in a calibrated volume; results from Greenland Summit are shown in Fig. 11.10a. The data show high scl (z) variability with depth. However, when the scl data are plotted versus the (local) sample density rather than depth, they fall on a single curve (Fig. 11.10b). Clearly it is the local density, and not the bulk density, that controls the bubble trapping process. This relationship allows scl to be parameterised as a function of local density. The density contrast between summer and winter layers leads to staggered trapping depths, and thereby a small age difference between the gases trapped in both layers. Etheridge et al. [25] noted that at the coastal Law Dome site in East Antarctica, the air trapped in the denser winter layers is on average about two years older

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than air trapped in the lower density summer layers. This difference is well explained by the notion that the denser winter layers close off earlier on, at shallower depths.

11.3

Applications of Firn Models

11.3.1 Ice Sheet Surface Mass Balance from Altimetry As introduced in Chap. 7, one way to monitor ice sheet mass changes is satellite altimetry (see Sects. 7.3.2 and 13.4). This method measures changes in ice volume, which contain a signal from firn layer thickness variability that can be of comparable magnitude to the satellite-derived elevation change observations. Therefore firn thickness changes can obscure the underlying ice dynamical effects. The effect of firn thickness changes on surface elevation is illustrated in Fig. 11.11 (cf. Fig. 7.4): the vertical movement of the snow surface (dh/dt) is influenced by accumulation and sublimation (both included in vacc ), snowmelt (vme ), firn compaction (v f c ), ice flow beneath the firn layer (vice ) and bedrock movement (vbed ) or buoyancy and ocean effects on floating ice shelves (vby ): dh = vacc + vme + v f c + vice + vbed + vby . dt

(11.12)

Ligtenberg et al. [26] have computed the seasonal cycle in Antarctic firn thickness changes, using the model described in Sect. 11.2.2. They show that each year, the volume of the Antarctic firn layer gradually increases in thickness during autumn, Fig. 11.11 Different components contributing to ice sheet elevation changes: after Zwally and Li [8]. Reprinted from the Journal of Glaciology with permission of the International Glaciological Society

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Fig. 11.12 Monthly surface elevation changes (vtot , black) and its components; accumulation (vacc , green), firn compaction (v f c , red), snowmelt (vme , blue), vertical downward movement of ice (vice , brown), and buoyancy effects (vby , orange), from Ligtenberg et al. [26]. Reprinted with permission from John Wiley and Sons, ©2012 American Geophysical Union. All rights reserved

Fig. 11.13 Periodic changes in accumulation rate (a) induce variability in firn layer thickness (b). Reproduced from Helsen et al. [9], reprinted with permission from AAAS

winter and spring, and quickly decreases in the austral summer (Fig. 11.12). The main driver of this asymmetric behaviour is seasonal variability in accumulation rate (green curve in Fig. 11.12), acting in concert with the effect of summer heat that penetrates the firn. From March until October the surface steadily rises due to enhanced precipitation, the absence of melt, and reduced firn compaction. On interannual time scales, variability of firn layer thickness is even more strongly controlled by accumulation changes. To illustrate the effects of accumulation rate anomalies at different time scales on the firn column thickness, Fig. 11.13b shows simulated firn thickness changes using a synthetic time series of accumulation rate (Fig. 11.13a). Clearly, firn thickness is not determined by the actual sign of the accumulation trend, but by the period and sign of the accumulation anomaly. In this

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Fig. 11.14 Antarctic elevation changes over the period 1995–2003: a observed, b firn thickness component and c residual. Reproduced from Helsen et al. [9], reprinted with permission from AAAS

particular case, a 100-year cycle in accumulation rate dominates the resulting dh/dt pattern. The effect of a 25-year accumulation rate fluctuation is smaller and superimposed on the 100-year signal (Fig. 11.13b). When only short-term observations are available, this might lead one to interpret surface elevation changes incorrectly. This is indicated by the blue and green sections of both the accumulation and the firn depth anomaly time series: these opposing surface elevation trends are entirely caused by positive and negative accumulation rate anomalies with respect to a long-term mean. However, if data were only available over a 25-year period, one might attribute the observed surface height trend to ice dynamical changes. When satellite altimetry observations are corrected for firn depth variations over the same period, ice dynamic effects and long-term accumulation change effects can be identified. As an example of such a correction, we show the results of Helsen et al. [9], who have done this for Antarctic altimetry observations over the period 1995–2003 (Fig. 11.14). The simulated pattern of thickening and thinning firn layer (Fig. 11.14b) is subtracted from the observed elevation changes (Fig. 11.14a). When doing so, these authors assumed that the mean accumulation over the period 1980– 2004 was in balance with the ice velocity. Wherever this is not the case, the firncorrected elevation changes (Fig. 11.14c) are non-zero. This residual signal is then in turn due either to recent change in glacial flow velocity, or to changes in accumulation rate on time scales longer than the 25-year period. These results show that a thickening firn layer partly obscured a dynamical thinning signal over the area bordering the Amundsen Sea Embayment. Besides that, an extensive area of slightly increasing elevation in East Antarctica suggests a positive accumulation rate anomaly over the period 1980–2004 with respect to the longerterm average (over centuries to millennia in dry areas like these). For this latter case, a conceptual analogy can be found in the blue lines of Fig. 11.13: as the accumulation is known only for a relatively short period, the variability within the 25-year period produces both increases and decreases in firn depth, while the dominant signal is a positive firn depth trend, given that the 25-year average accumulation exceeds the accumulation need to balance the ice velocity. Hence, positive dh/dt values in East Antarctica are not caused by a positive trend in accumulation over the period of observation, but instead by a positive accumulation anomaly over a longer (> 25 year) time scale.

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11.3.2 Delta Age Calculations in Deep Ice Cores Air bubbles trapped during the firn-ice transition preserve a sample of the atmosphere at the time of bubble closure. This archive is accessible through ice cores drilled in the Greenlandic and Antarctic ice sheets, allowing a reconstruction of past atmospheric composition. The pore network within the firn facilitates continued gas exchange with the overlying atmosphere through diffusion and convection, keeping the air in the firn pores younger than the surrounding ice grains. Because the bubbles are occluded at the base of the firn column, the air they contain is younger than the surrounding ice by an amount age. Depending on the climatic conditions of the site, age can range from several decades (e. g., ∼ 35 years at the high-accumulation Law Dome site in East Antarctica) to several thousand years (e. g., during glacial conditions on the Antarctic plateau). Knowledge of past age evolution is important for the dating of ice core gas records, investigating the relative timing of temperature and CO2 changes and for the synchronisation of ice core records between both hemispheres. Firn densification models are used to estimate past age values from temperature and accumulation reconstructions. We first review some basic concepts of firn air transport. A powerful tracer of firn air processes is the isotopic composition of molecular nitrogen (N2 ), which is constant in the atmosphere on the time scales considered here. The abundance of the heavy 15 N isotope is given in delta notation as δ 15 N =

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with R 15 = [15 N]/[14 N] being the ratio of the isotopes in either the sample (SA) or the standard (ST). For 15 N the modern atmosphere is used as the reference standard. As detailed in the next chapter, the delta notation measures or quantifies the deviation from the standard, where a positive value means that the sample is enriched in 15 N relative to the standard. Two processes alter the δ 15 N of firn air (Fig. 11.15). The first is gravitational separation, where the heavy 15 N isotopes are preferentially transported downwards, leading to an enrichment with depth following the isobaric equation Mgz Mgz − 1 × 103 ≈ δ 15 N = exp × 103 , RT RT

(11.14)

with M being the mass difference between the isotopes in kg mol−1 . The gravitational enrichment with depth is clearly visible in measurements of δ 15 N on air samples pumped from the firn layer (Fig. 11.15c, dashed black line). The second process affecting δ 15 N is thermal fractionation, which occurs only in the presence of temperature gradients in the firn. Thermal fractionation can be understood as a tendency of the heavy 15 N isotope to preferentially diffuse along the temperature gradient towards the colder regions. The seasonal temperature cycle in the upper firn (Fig. 11.15b) leads to an enrichment in δ 15 N at the coldest part of the firn (Fig. 11.15c). Under stable climatic conditions, thermal fractionation does not influence the deep firn, because the seasonal temperature variations only penetrate the top ∼15 m. The firn column is commonly divided into three zones based on the gravitational enrichment in δ 15 N. The top few metres of the firn are vigorously ventilated to the atmosphere by wind pumping and convection. We refer to this zone as the convective zone (CZ). Below the CZ we find the longest section of the firn column, where gas transport is dominated by molecular diffusion. This zone is called the diffusive zone (DZ), and extends from 4 to 49 m depth at Siple Dome (Fig. 11.15c). The effective diffusivity of the firn decreases with depth; once we reach the lock-in depth the diffusive flux becomes negligible. The zone below the lock-in depth is named the lock-in zone (LIZ), where advection with the ice matrix dominates the downward gas transport. Convection and advection are macroscopic transport phenomena, which do not discriminate between isotopes. Consequently the DZ is the only part of the firn column where gravitational enrichment occurs, as indicated by the dashed black line. Another example of a δ 15 N profile showing gravitational enrichment—from the NEEM ice core—is shown in Fig. 12.24. The lock-in depth (z lid ) is where gases are effectively isolated from the atmosphere, so to calculate age one has to calculate the age of both ice and gas at z lid . According to Buizert et al. [29], the mean gas age X of gas X at z lid is given empirically by the expression 1 X (z lid ) = γX

DCH2 + 4.05 , 0.934 · 0 DCO2

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in which γ X is the diffusion coefficient of gas X relative to CO2 , DCH is the diffusive 0 is the free air diffusivity of CO2 . The column height (the length of the DZ) and DCO 2 age of the ice at z lid needs to be calculated using a (time-dependent) firn densification model. Note that the gas age (5–30 years) is always small compared to the age of the ice. In Fig. 11.16 we have used the Herron-Langway model to calculate age and gravitational δ 15 N under a wide range of steady state climatic conditions. At higher temperature, densification is enhanced, resulting in a smaller age and δ 15 N. The influence of the accumulation rate is more complex; increasing A leads to both a thickening of the firn column (larger δ 15 N) and fewer annual layers per metre of firn. The net effect is a reduction of age when A is increased. We point out that δ 15 N by itself is not a good proxy for age; for example, an ice sample with δ 15 N = 0.4 can have a age ranging from tens of years to thousands of years. As we shall see, however, the thermal fractionation in δ 15 N provides very strong constraints on age in Greenland. In Fig. 11.15 we showed thermal fractionation in the upper firn due to seasonality. The seasonal signal does not penetrate down to the firn-ice transition where bubbles are trapped, and is therefore not recorded in the ice. Climate-induced changes in mean annual temperature, on the other hand, can induce temperature gradients in the firn that persist long enough to impart transient excursions on the δ 15 N record. An example of this from the central Greenland GISP2 ice core over the last deglaciation is shown in Fig. 11.17. The upper curve shows δ 18 O of precipitation as a proxy for site temperature. The deglacial sequence has two strong warmings at the onset of the Bølling-Allerød interstadial and of the Holocene, and a pronounced cooling at the onset of the Younger Dryas stadial (all indicated with grey shading). During a rapid δ 15N

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warming event the surface temperature is higher than the temperature at close-off, causing preferential diffusion of heavy δ 15 N molecules to the deep firn where they are trapped. This leads to positive δ 15 N excursions during warming episodes [30]. During an abrupt cooling the opposite happens, causing a negative anomaly in δ 15 N. Similar excursions in δ 15 N due to abrupt warming events during the last glacial have also been found (Fig. 12.25), as discussed in the next chapter (Sect. 12.5.1). For the GISP2 example considered in Fig. 11.17, the thermal signals in the gas phase are found ∼ 20 m deeper than the concurrent δ 18 O signals in the ice phase. This is due to the fact that the bubbles are trapped at ∼ 90 m below the surface where the δ 18 O signal is deposited; subsequent firn densification and ice flow reduce this number to the observed ∼ 20 m. This latter quantity, called depth, provides a very strong constraint on age, because we know the ice and gas phase signals are synchronous in time. Figure 11.17b shows a fit to the δ 15 N data using a timedependent firn densification model; the modelled evolution of age with time is shown in Fig. 11.17c. The fact that the model successfully reproduces the timing of the thermal δ 15 N signal means that the calculated age is correct at the transitions. The model shows that age decreases during the warm phases (Holocene, B-A), and increases during cold periods (YD). The thermal δ 15 N excursions are only observed in Greenland ice cores, because temperature change over Antarctica is much more gradual. −34 −36

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For Greenland ice cores, the firn densification models do a good job in reproducing both gravitational and thermal changes in δ 15 N in response to climatic change. For non-coastal Antarctic sites there is a pronounced mismatch between model and data under glacial conditions, with the models predicting a thickening of the firn column under colder conditions and the δ 15 N showing a thinning of the diffusive column instead. An increased convective zone thickness during glacial times is commonly proposed as an explanation for this discrepancy, and the anomalous low accumulation megadunes site could provide a modern analogue for Antarctic glacial conditions. Another explanation could be the softening effect of dust on firn (Sect. 11.2.3), which is currently not included in densification models. The high dust loading during glacial conditions would enhance firn densification, thereby reducing the firn column thickness, as indicated by δ 15 N data. Last, the model-data mismatch could simply reflect incorrect assumptions about the T and A model input scenarios. Finally, we briefly discuss the choice of densification model for age calculations. A first important requirement is that the model should be time-dependent (as opposed to steady-state), as we are interested in the evolution of age under changing climatic conditions. Secondly, the densification model should be coupled to a heat diffusion model to obtain a realistic response to surface temperature changes. Due to the large thermal mass of the underlying ice sheet, the firn column has a long response time to temperature changes (up to several times age). Thirdly, a 1-D ice flow model should be included to describe the vertical advection of heat and the influence of the geothermal heat flux at the bed. At high accumulation sites (such as Greenland or coastal Antarctica) this last component is not very important, because the strong downward advection of cold surface ice dominates the heat budget of the upper ice sheet. On the Antarctic plateau, however, the low A can cause the geothermal gradient to extend up into the firn column where it influences both densification rates and thermal fractionation of δ 15 N.

11.4

Summary and Conclusions

The firn layer is the top 40–120 m of unconsolidated snow on ice sheets and glaciers, which gradually densifies under its own weight in the transformation to glacial ice. The densification process consists of roughly three stages that somewhat overlap in depth: (1) grain growth and settling, (2) sintering, and (3) bubble compression. Firn densification is primarily controlled by temperature and overburden pressure, and to first order its steady state (bulk) behaviour can be adequately captured by simple empirical models, such as the Herron-Langway model. To incorporate variability due to seasonality or climatic change, time-dependent firn models are required, such as the Zwally and Li [8] model for sub-annual variability, or the Barnola et al. [13] model for long-term firn evolution. The importance of seasonality is clear from the strong density variability, or layering, that characterises polar firn. Apart from seasonality, microstructure and impurities are also thought to play an important rôle in establishing firn layering, as can be seen from the existence of a variability minimum

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at around ρ = 650 kg m−3 . Despite its importance, no complete understanding of firn layering exists yet. The first main application of firn modelling is in correcting satellite altimetry data for firn thickness changes that are masking underlying changes in ice dynamics and ice sheet mass balance. A second important application is in understanding the evolution of age back in time for the interpretation of ice core gas records. In conclusion, firn is both simple and complex: simple enough to allow for useful quantitative modelling in a range of Earth science applications, yet complex enough to leave many of the intricacies of its workings beyond satisfactory explanation.

References 1. Reeh N (2008) A nonsteady-state firn-densification model for the percolation zone of a glacier. J Geophys Res Earth Surf 113:F03023 2. van den Broeke MR (2008) Depth and density of the Antarctic firn layer. Arct Antarct Alp Res 40:432–438 3. Libbrecht KG (2005) The physics of snow crystals. Rep Prog Phys 68:855–895 4. Libbrecht KG (2007) The formation of snow crystals. Am Sci 95:52–59 5. Herron MM, Langway CC (1980) Firn densification: an empirical model. J Glaciol 25(93):373– 385 6. Robin GdeQ (1958) Glaciology. III. Seismic shooting and related investigations. NorwegianBritish-Swedish Antarctic Expedition, 1949–52. Scientific Results, vol. 5. Norsk-Polarinstitut 7. Bader H (1954) Sorge’s law of densification of snow on high polar glaciers. J Glaciol 2(15):319– 323 8. Zwally HJ, Li J (2002) Seasonal and interannual variations of firn densification and ice-sheet surface elevation at the Greenland summit. J Glaciol 48:199–207 9. Helsen MM, van den Broeke MR, van de Wal RSW, van de Berg WJ, van Meijgaard E, Davis CH, Li Y, Goodwin I (2008) Elevation changes in Antarctica mainly determined by accumulation variability. Science 320(5883):1626–1629 10. Bolzan JF, Strobel M (1994) Accumulation-rate variations around Summit, Greenland. J Glaciol 40:56–66 11. Arthern RJ, Vaughan DG, Rankin AM, Mulvaney R, Thomas ER (2010) In situ measurements of Antarctic snow compaction compared with predictions of models. J Geophys Res 115:F03011 12. Ligtenberg SRM, Helsen MM, van den Broeke MR (2011) An improved semi-empirical model for the densification of Antarctic firn. The Cryosphere 5(4):809–819 13. Barnola J-M, Pimienta P, Raynaud D, Korotkevich YS (1991) CO2 -climate relationship as deduced from the Vostok ice core—a re-examination based on new measurements and on a reevaluation of the air dating. Tellus 43B:83–90 14. Arnaud L, Barnola J-M, Duval P (2000) Physical modeling of the densification of snow/firn and ice in the upper part of polar ice sheets. In: Hondoh T (ed.) Physics of ice core records. Hokkaido University Press, Hokkaido, pp 285–305 15. Goujon C, Barnola J-M, Ritz C (2003) Modeling the densification of polar firn including heat diffusion: application to close-off characteristics and gas isotopic fractionation for Antarctica and Greenland sites. J Geophys Res Atmos 108(D24):4792 16. Alley RB (1987) Firn densification by firn boundary sliding—a first model. Journal de Physique 43:249–256 17. Arzt E (1982) The influence of an increasing particle coordination on the densification of spherical powders. Acta Metallurgica 3:1883–1890 18. Gerland S, Oerter H, Kipfstuhl J, Wilhelms F, Miller H, Miners W (1999) Density log of a 181 m long ice core from Berkner Island, Antarctica. Ann Glaciol 29:215–219

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19. Freitag J, Wilhelms F, Kipfstuhl J (2004) Microstructure-dependent densification of polar firn derived from X-ray microtomography. J Glaciol 50:243–250 20. Fujita S, Okuyama J, Hori A, Hondoh T (2009) Metamorphism of stratified firn at Dome Fuji, Antarctica: a mechanism for local insolation modulation of gas transport conditions during bubble close off. J Geophys Res Earth Surf 114:F03023 21. Hawley RL, Morris EM (2006) Borehole optical stratigraphy and neutron-scattering density measurements at Summit, Greenland. J Glaciol 52(179):491–496 22. Hörhold M, Laepple T, Freitag J, Bigler M, Fischer H, Kipfstuhl S (2012) On the impact of impurities on the densification of polar firn. Earth Planet Sci Lett 325:93–99 23. Freitag J, Kipfstuhl S, Laepple T, Wilhelms F (2013) Impurity-controlled densification: a new model for stratified polar firn. J Glaciol 59(218):1163–1169 24. Schwander J, Barnola J-M, Andrie C, Leuenberger M, Ludin A, Raynaud D, Stauffer B (1993) The age of the air in the firn and the ice at Summit, Greenland. J Geophys Res Atmos 98:2831– 2838 25. Etheridge D, Pearman G, Fraser P (1992) Changes in tropospheric methane between 1841 and 1978 from a high accumulation-rate Antarctic ice core. Tellus 44B:282–294 26. Ligtenberg SRM, Horwath M, van den Broeke MR, Legrésy B (2012) Quantifying the seasonal “breathing” of the Antarctic ice sheet. Geophys Res Lett 39:L23501 27. Severinghaus JP, Grachev A, Battle M (2001) Thermal fractionation of air in polar firn by seasonal temperature gradients. Geochem Geophys Geosyst 2(7):2000GC000146 28. Martinerie P, Lipenkov VY, Raynaud D, Chappellaz J, Barkov NI, Lorius C (1994) Air content paleo record in the Vostok ice core (Antarctica): a mixed record of climatic and glaciological parameters. J Geophys Res 99:10565–10576 29. Buizert C, Sowers T, Blunier T (2013) Assessment of diffusive isotopic fractionation in polar firn, and application to ice core trace gas records. Earth Planet Sci Lett 361:110–119 30. Severinghaus JP, Brook EJ (1999) Abrupt climate change at the end of the last glacial period inferred from trapped air in polar ice. Science 286:930–934 31. Buizert C and 13 others (2014) Greenland temperature response to climate forcing during the last deglaciation. Science 345:1177–1180 32. Miller H, Schwager M (2000) Density of ice core NGT37C95.2 from the North Greenland Traverse. PANGAEA. https://doi.org/10.1594/PANGAEA.57798

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12.1

Introduction

In the interior of polar ice sheets and on high-altitude glaciers, where dry snow prevails, a stratigraphically-ordered sequence of precipitation is archived. This archive can be accessed by ice-core drilling. In an ice core, proxy information about the past climate is preserved in the ice matrix itself, by particulate and dissolved tracers in the ice, and by gases in bubble enclosures. The annual snow accumulation on the ice surface integrates over many precipitation events, and a seasonal resolution is possible in those ice cores where the annual accumulation rate is relatively high (typically > 10 cm water equivalent (w.e.) per year). In contrast, the gas records retrieved from ice cores have a much lower resolution of decades to centuries, because the bubble enclosure process is slow. The maximum time sampled by an ice core depends on the ice thickness, the rates of local accumulation and basal melting, and whether the ice flow has been undisturbed at the drill site. To date, the oldest stratigraphically-ordered ice found is about 800,000 years old, derived from the EPICA (European Project for Ice Coring in Antarctica) ice core at Dome C (Fig. 12.1). The oldest stratigraphically-ordered ice in Greenland, from the NorthGRIP (North GReenland Ice core Project) ice core, is 124,000 years old. Older ice (128,000 years) can be reconstructed from the H. Fischer (&) Climate and Environmental Physics, Physics Institute, Oeschger Centre for Climate Change Research, University of Bern, Bern, Switzerland e-mail: hubertus.ﬁ[email protected] T. Blunier Centre for Ice and Climate, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark R. Mulvaney British Antarctic Survey, Cambridge, UK © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_12

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Fig. 12.1 Deep ice core locations in the polar regions

NEEM (North Eem) ice core from northwest Greenland, but interpreting it requires correction for stratigraphic disturbances. These deep ice cores have been described for example by EPICA community members [1], Jouzel et al. [2] and NEEM community members [3]. The diverse physical and chemical parameters measurable in ice cores provide vital data for understanding climatic variability and its forcing factors (e.g., greenhouse gases and solar activity). In this chapter, we describe how ice cores are dated (Sect. 12.2) and explain what information is provided by stable water isotopes (Sect. 12.3), aerosols (Sect. 12.4) and gases (Sect. 12.5) in the cores.

12.2

Dating Ice Cores

A prerequisite for the interpretation of palaeo-records from an ice core is accurate dating of the ice to establish its depth-age scale (e.g., Fig. 12.2), also called age scale or time scale. Annual layers become thinner with depth due to the ice-flow conﬁguration and ﬁrniﬁcation in the top *100 m (Chap. 11). This thinning limits the resolution of ice-core parameters in the deeper ice and makes its dating difﬁcult. In addition, different depth-age scales apply to the information archived in the ice (stable isotopes, aerosols) and to the gases enclosed in bubbles, as the bubbles are only closed off at a depth of 60–100 m below the snow surface. The major tools/techniques used to establish the age scale are as follows: 1. One-dimensional ice flow models: by making assumptions on how the vertical velocity w of the ice or its thinning rate –dw/dz varies with depth in a core, the age of the ice t can be calculated from

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Fig. 12.2 Depth-age scales calculated by the Nye, Dansgaard-Johnsen and easyFlow models (assuming constant interglacial accumulation rate A) for the last 60 kyr of the NGRIP ice core, compared against the independently-counted GICC05 age scale [4]. Beyond 60 kyr BP, the GICC05 scale is extended by a Dansgaard-Johnsen model with varying accumulation rate estimated from stable water isotopes. The GICC05 scale assigns older ages to ice below 1500 m above bedrock (depth of the glacial-interglacial transition) due to low accumulation rates during the glacial. The two right-hand panels show the corresponding vertical velocities and vertical strain rates

Z t¼

z H

1 dz ; w

ð12:1Þ

where z is distance above the bed and H is the thickness of the ice sheet (after correcting for density variations). As we will see shortly, the history of the snow accumulation rate A is needed as input. A one-dimensional (1-D) model is a good approximation for the flow at dome and divide locations. More complicated flow conﬁgurations, such as encountered in the outflow regions of the ice sheet, require accounting for horizontal motion, in some cases by using 3-D ﬁnite element models. The Nye model assumes a constant thinning rate and constant accumulation rate: wðzÞ ¼

A z: H

ð12:2Þ

Then integration of (12.1) gives Z tðzÞ ¼

z

H

H H z dz ¼ ln : Az A H

ð12:3Þ

An improved approach accounts for decline of the thinning rate as the bed is approached (assuming a cold base). In the Dansgaard-Johnsen model, the thinning rate is assumed to decrease linearly to zero below a height h, and A remains constant over time. The corresponding age model is

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2H h 2H h ln ; h z H; 2A 2z h 2H h 2H h 2H h h ln þ 1 ; tðzÞ ¼ 2A h A z

tðzÞ ¼

0 z h:

ð12:4Þ

Another simple model that has been applied successfully to ice cores at divide locations is what we call the ‘easyFlow’ model. It assumes wðzÞ ¼ A

z m þ 1 H

;

ð12:5Þ

where m is close to 0.5. As in the Dansgaard-Johnsen model, the thinning rate goes to zero at z = 0, as required for ice frozen to bedrock. Analytical integration of (12.5) gives Z tðzÞ ¼

z H

H m þ 1 m1 H z dz ¼ mA A

m H 1 : z

ð12:6Þ

Generally, the Dansgaard-Johnsen and easyFlow models assign older ages to ice at depth than the Nye model (Fig. 12.2). If melting occurs at the base, then w(z = 0) < 0 and is given by the annual melt rate M in metres of ice equivalent per year. Accounting for this in the easyFlow model leads to z m þ 1 wðzÞ ¼ ðA MÞ M: H

ð12:7Þ

The evaluation of (12.1) requires numerical integration in this case. Since A actually varied with time in the past, a precise determination of age necessitates numerical integration of (12.1) together with independent information about the past accumulation rate, derived for example by isotope-based temperature reconstructions. 2. Annual layer counting: by detecting seasonal cycles of d18O and chemical tracers in the ice, annual layers can be identiﬁed, and their age ‘counted’ directly, down a core. Typically, this method will work to some depth, until the layers become too thin. Deriving past accumulation rates from the layer thicknesses requires their progressive thinning with depth to be corrected. 3. Identiﬁcation of (independently-dated) time markers in the ice, such as those associated with volcanic eruptions (Fig. 12.3; Sect. 12.4.5), magnetic anomalies and nuclear weapon tests.

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Fig. 12.3 Volcanic eruptions imprinted in the sulphuric acid deposition in Greenland ice [5]

4. Matching of parameters measured down-core with other already-dated climate archives or orbital parameters. This approach is sound if the phase relationship of the matched parameters is known or the physical processes linking them to insolation can be ﬁrmly established. 5. Radioisotopic dating of materials in the ice: Numerous ‘clocks’ are being used but are not yet standard tools in ice-core studies. They include carbonaceous aerosols and 10Be in aerosol tracers, 36Cl/10Be and 26Al/10Be ratios, and the 40 Ar/38Ar ratio in air bubbles.

12.3

Stable Water Isotopes

12.3.1 Basics and Nomenclature Oxygen and hydrogen atoms have a natural abundance of heavier and lighter stable isotopes. Accordingly, the water molecule has several isotopologues1. In the 18 well-mixed ocean, the three major isotopes are H16 2 O, H2 O, and HDO (where D 2 stands for Deuterium, H), and these components have abundances of 99.76%, 0.201% and 0.031%, respectively. Standard measurements made on ice cores include the stable water isotope ratios R18 = 18O:16O and RD = D:H. During phase transitions (evaporation, condensation), these isotope ratios change slightly as a result of equilibrium fractionation (which depends on the water vapour pressures of the components) and kinetic fractionation (dependent on their diffusion constants). 1

An isotopologue is a molecule that differs in one or several isotopes of its atoms. For simplicity, in this chapter we speak of isotopes, when we mean isotopologues.

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Equilibrium fractionation is reversible: evaporation followed by condensation at the same temperature recovers the same isotope ratio. Kinetic fractionation is irreversible and causes the product of the phase transition to be depleted in the heavier isotopes. For technical reasons, isotope ratios are determined relative to those in a ‘standard water’ (rather than determined as absolute abundances) using mass spectrometry. The d-notation is used to express the difference: dD ¼

RD RD VSMOW

1

and d18 O ¼

R18 R18 VSMOW

1;

ð12:8Þ

where RVSMOW is the isotope ratio of Vienna Standard Mean Ocean Water (VSMOW). Since the deviations dD and d18O are very small, it is usual to report them in units of ‰ (parts per thousand). When ocean water evaporates, equilibrium fractionation depletes the vapour of the heavier isotopes. The equilibrium fractionation factor is ae ¼

R l pS ¼ ; Rv p0S

ð12:9Þ

where Rl and Rv are the isotope ratios in the liquid (l) and vapour (v), respectively, and pS and pʹS are the saturation water vapour pressures of the lighter and the heavier isotope, respectively. As shown in Table 12.1, ae is temperature dependent. Parameterisations of this dependence (with temperature T in Kelvin) for 18O and D are [6] 1137 0:4156 0:0020667; T2 T 24844 76:248 ln aD þ 0:052612: e ¼ T2 T

ln a18 e ¼

ð12:10Þ

In addition, kinetic depletion occurs due to undersaturation of the water vapour above the ocean surface. The kinetic fractionation factor is given approximately by ak ¼ h þ

D D0

n ð1 hÞ;

ð12:11Þ

where n 1, h is the relative humidity, and D and Dʹ are the diffusion constants of the normal and the heavy water isotopes, respectively (these depend on the molecular weight of the gases). In the evaporation scenario, the total fractionation factor is thus al!v ¼ ae ak ¼

RVSMOW Rv; e ; Rv; e Rv

ð12:12Þ

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Table 12.1 Equilibrium fractionation factors and respective d-values (from Roedel [7]) Temperature (°C)

ae (HDO)

deD (‰)

ae (H18 2 O)

d18 e O (‰)

deD/d18 e O

−20 −10 0 10 20 30 40 50

1.14920 1.12932 1.11231 1.09767 1.08502 1.07403 1.06446 1.05608

−129.83 −114.51 −100.97 −88.98 −78.36 −68.93 −60.55 −53.10

1.01413 1.01285 1.01172 1.01070 1.00979 1.00897 1.00823 1.00756

−13.93 −12.69 −11.58 −10.59 −9.70 −8.89 −8.17 −7.51

9.320 9.024 8.719 8.402 8.078 7.754 7.411 7.071

where Rv,e is the isotope ratio after equilibrium fractionation and Rv is the isotope ratio after both fractionation processes. During condensation in equilibrium, the opposite fractionation occurs and the condensate is enriched in the heavier isotope compared to the vapour. In clouds, where liquid and water vapour coexist in equilibrium, no kinetic fractionation occurs. But if the ice phase is present as well as liquid and vapour, the ice is undersaturated relative to the vapour so that kinetic fractionation occurs during snow formation, depleting the snow crystals of the heavier isotope. This process is important at low temperatures.

12.3.2 The Isotope Proxy Thermometer When water vapour is transported away from its evaporation source and cools, some of it is removed (from clouds) as rain or snow precipitation. The precipitate is enriched in the heavier isotopes compared to the vapour, leaving the vapour more depleted. The more water rains out of a cloud, the more depleted the vapour becomes, and so does the freshly-formed precipitate. Accordingly, the more a cloud is cooled, the more negative are the d-values of both vapour and precipitate. This process is called Rayleigh distillation (Fig. 12.4). The temperature control on the isotopic depletion motivates an isotopic ‘thermometer’, whose operation can be modelled as follows. Let us assume a constant fractionation factor and no kinetic fractionation during condensation (Rc = aeRv). The isotopic changes must satisfy the balances dðNv þ Nl þ Np Þ ¼ dNv þ dNl þ dNp ¼ 0; dðRv Nv þ Rl Nl þ Rp Np Þ ¼ Nv dRv þ Rv dNv þ Nl dRl þ Rl dNl þ Np dRp þ Rp dNp ¼ 0;

ð12:13Þ

where Nv, Nl and Np are the numbers of water molecules in the vapour, cloud water and precipitation, respectively. These balances lead to the differential equations

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Fig. 12.4 The Rayleigh distillation process for air parcels migrating towards the polar regions. d values are shown in units of ‰ (per mil). Owing to fractionation, vapour formed from the evaporation of ocean water is isotopically lighter than the ocean water. Precipitation from the vapour is isotopically heavier than the vapour, and so vapour reaching the pole is severely depleted in the heavier isotope

dRv ða 1ÞdNv Nl da ¼ Nv þ aNl Rv

dRp ða 1ÞdNv Nl da da : ¼ þ Nv þ aNl a Rp

and

ð12:14Þ

If we assume that all condensate is removed from the cloud as soon as it forms (Nl = 0), then (12.14) can be solved analytically to give Rv ¼ Rv;0

Nv Nv;0

a1

¼ f a1 and

Rp a ¼ f a1 ; Rp;0 a0

ð12:15Þ

where a is assumed to be constant during the distillation process, and the subscript 0 labels initial values. More generally, numerical integration of (12.14) is necessary, supplemented by knowledge of the temperature history and thus the change of a along the water vapour pathway. Imagine now an air parcel moving from the ocean to a polar ice sheet. With Rv,0 = RVSMOW/a0 and Rp,0 = RVSMOW, the results in (12.15) yield 1 a1 f a0

a a1 f ; a0

ð12:16Þ

Nv ps VT0 VT0 LDT ¼ exp ¼ : RTT0 Nv;0 ps;0 V0 T V0 T

ð12:17Þ

dv þ 1 ¼

and dp þ 1 ¼

in which (according to the gas law) f ¼

Here L is the latent heat of evaporation, R is the gas constant, T0 is the evaporation temperature over the (source) ocean, T is the condensation temperature over the ice

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sheet, and DT = T − T0. The last equality in (12.17) assumes changes in saturation water vapour pressure in equilibrium following the Clausius-Clapeyron equation. Thus we have a1 a VT0 LDT dp þ 1 ¼ exp : a0 V 0 T RTT0

ð12:18Þ

By expanding this result for small DT, it can be shown that dp / DT to the ﬁrst order. Typically, changes in T0 are small compared to those in the precipitation temperature T, so the latter dominates changes in DT. This causes the pattern of d-values across an ice sheet to be linearly related to its precipitation temperature pattern, as observed in both Antarctica and Greenland (Fig. 12.5). To derive the temperature dependence of dp, we need to account also for the cooling history of the air parcel. Notably, the loss of water vapour is more efﬁcient for isobaric cooling than for moist adiabatic cooling (where part of the heat is used for the expansion of the air parcel). Thus a stronger temperature gradient is expected for isobaric cooling: in this case, V/V0 = T/T0, and differentiating (12.18) with respect to T gives a1 a1 @dp 1 @a LDT a ða 1ÞL LDT exp ¼ þ exp : a0 @T RTT0 a0 RT 2 RTT0 @T

ð12:19Þ

If we consider 18O and take T0 = +20 °C with a0 = 1.00979 and T = −20 °C with a = 1.01413 (Table 12.1), then we obtain the temperature gradient 18 @d O ¼ 0:9&= C: @T isobar

ð12:20Þ

In the case of moist adiabatic cooling, V=V0 ¼ ðT0 =T ÞCv =R ¼ ðT0 =T Þ5=2 (here Cv is the heat capacity at constant volume), and we ﬁnd instead " #a1 T0 7=2 LDT exp RTT0 T " #a1 a ða 1Þð2L 7RTÞ T0 7=2 LDT þ exp ; a0 2RT 2 RTT0 T

@dp 1 @a ¼ a0 @T @T

ð12:21Þ

and the corresponding result is

@d18 O ¼ 0:6&= C: @T adiab

ð12:22Þ

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Fig. 12.5 Average d18O-value at shallow depths in ice cores plotted against mean annual temperature at the core sites, as determined by ﬁrn temperature below 15 m depth. The d18O-temperature gradient is approximately 0.67‰/°C for Greenland and 0.79‰/°C for Antarctica. Figure provided by V. Masson-Delmotte via personal communication

In practice, the temperature dependence of d18O lies between these two extremes. As shown in Fig. 12.5, the observed gradients are 0.67‰/°C for Greenland (close to the moist adiabatic value) and 0.79‰/°C in Antarctica (closer to isobaric). The temperature dependence of the d-values is reflected not only geographically, but also over time on both seasonal and glacial-interglacial time scales. Past temperature changes may thus be interpreted from the fluctuations in palaeo d archived in ice cores. However, the spatially-derived ‘d-temperature’ gradient (i.e., @d=@T) found over today’s ice sheets is not strictly applicable to the temporal variations. Modelling studies show that the spatial gradient represents a very good calibration for temporal changes for East Antarctica [2], but not so for glacial temperature changes in Greenland [8]. The likely reasons are a lack of winter precipitation during glacial conditions in Greenland, which biases the d-record to summer temperatures [9, 10], and signiﬁcant changes in the water vapour source temperature. For seasonal changes, complication also arises from diffusion of water molecules in the ice lattice and in the open pore space of the ﬁrn (Sect. 12.3.4). This causes a strong smoothing of the d-record, which is most pronounced at low accumulation sites. Recent studies also suggest that there is a signiﬁcant

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post-depositional net exchange of water vapour between the snow pack and the overlying atmosphere, which can alter the isotopic composition of the snow [11]. While dominated by the temperature pattern over polar ice, the d-values are not totally independent of the temperature variation of the ocean water vapour source. Information regarding this can be gained by interpreting d18O and dD jointly— speciﬁcally, via the deuterium excess d = dD − 8d18O. Due to the different temperature dependences of the equilibrium fractionation of D and 18O during evaporation (Table 12.1) and the different kinetic fractionation of these species, d reflects the evaporation conditions of the water vapour (i.e., evaporation temperature and relative humidity over the ocean surface). For example, for sea surface temperatures above 20 °C, equilibrium fractionation leads to a positive deuterium excess (Table 12.1). In Greenland, d changes with sea surface temperature by 0.35‰/°C [12]. By using the observed d-values and simple isotope models, the temperature changes at the ocean source and on the ice sheet can be disentangled [13]. Recent developments in isotopic analysis of H17 2 O also allows us to quantify the past relative humidity of the source region [14].

12.3.3 Examples of Isotope Records Long-term temperature variations are reflected in the ice-core stable water isotope records from Greenland and Antarctica. The EPICA Dome C (EDC) core covers eight glacial cycles (see the top panel in Fig. 12.27) and shows orbital climate variations with glacial/interglacial temperature changes of 8–10 °C (based on the spatial dD-temperature gradient) and a tendency for cooler interglacials prior to 450 kyr BP. Analysis of its deuterium excess indicates temperature changes of 2–3 °C in the water vapour source regions that were not in phase with those on the ice sheet. These changes may be caused by obliquity-driven variations in the latitude of the vapour source for East Antarctic precipitation. Ice-core records in Greenland (e.g., Figs. 12.6 and 12.20) show stable climate conditions through the Holocene and conspicuous, rapid climate changes (Dansgaard-Oeschger events) during the last ice age (see also Chap. 16). Shifts in precipitation seasonality during the glacial make it difﬁcult to convert the d18O changes directly to temperature change, but independent methods using borehole temperature and gas thermometry (see below) show that Dansgaard-Oeschger events involved temperature changes in Greenland of 10–15 °C in as little as a few decades, and that the glacial was 20–22 °C colder than the Holocene in Greenland. Quantitative temperature reconstructions using d18O from Greenland ice cores, accounting for source temperature effects and precipitation seasonality, yield results consistent with these ﬁndings. The longest continuous ice core from NGRIP reaches back to the end of the Eemian (the last interglacial) when temperatures in Greenland were 4–5 °C higher than today [15]. This warming includes a contribution from warming of the ice-core site due to reduced thickness of the ice sheet in

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Fig. 12.6 a Depth proﬁle of d18O in the GRIP ice core. b Depth proﬁle of d18O in the NGRIP ice core (red). The GRIP d18O record (blue) has been placed on the NGRIP depth scale using major transitions in d18O as tie points. c The difference between the two d18O records (black) on the GRIP time scale. For comparison, a sea-level record (green) and the smoothed NGRIP d18O timeseries (red) are shown. From North Greenland Ice Core Project members [15]. Reprinted from Nature with permission from Springer Nature ©2004

combination with the adiabatic lapse rate, as well as the warmer climate at that time. Results from the NEEM ice core reveal temperatures in Greenland that were on average 5 °C higher during most of the last interglacial, and potentially up to 8 °C higher at its onset, than today [3].

12.3.4 Isotope Diffusion in Firn and Ice The temperature changes during the course of a year lead to pronounced seasonal variations in the d-values, as exempliﬁed by the Camp Century ice core from Greenland (Fig. 12.7). A seasonal temperature amplitude of 30 °C, together with the spatial d18O-temperature gradient of 0.67‰/°C, would give a d18O amplitude of 20‰ —this corresponds to the observed amplitude in the top proﬁle in Fig. 12.7. However, we also see that the seasonal amplitude decreases with depth by a factor of three within the top 100 m, although the damping process deeper down is slower. The decay occurs because isotope concentration gradients between the summer and winter layers drive diffusion of isotopes in the pore space (top 100 m) and in the ice lattice (below 100 m).

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Fig. 12.7 High-resolution depth proﬁle of d18O in the Camp Century ice core, North Greenland, showing seasonal variations. From Johnsen et al. [16]. Reprinted from Nature with permission from Springer Nature ©1972

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For vertical diffusion in solid ice, we can write @d @2d ¼ Dice 2 ; @t @z

ð12:23Þ

where d is the d-value for H18 2 O or HDO, and Dice is the self diffusion constant of water molecules in ice. The general solution of (12.23) for an inﬁnite domain is 1 dðz; tÞ ¼ pﬃﬃﬃﬃﬃﬃ r 2p

# ðz nÞ2 dðn; 0Þ exp dn; 2r2 1

Z

"

1

ð12:24Þ

where r = (2Dicet)1/2 is the diffusion length (mean displacement of molecules after time t). Let us ignore densiﬁcation or strain of the layers and assume that surface temperature and thus the isotope signal vary sinusoidally. Moving with the ice, we suppose a sinusoidal depth proﬁle of d, dðz; tÞ ¼ d0 ekt sin

2pz : z0

ð12:25Þ

Substituting this into (12.23) and using z0 = 0.25 m ice equivalent for the annual layer thickness and Dice = 1.7 10−16 m2 s−1 (for H2O molecules in ice at −30 °C), we ﬁnd 2 2p k ¼ Dice ¼ 3:39 106 year1 ; z0

ð12:26Þ

which implies that the d18O amplitude decays by 50% in t1/2 = (ln2)/k 200,000 years. This decay is much slower than that inferred from Fig. 12.7 (*15 years). Including annual-layer thinning in the model in (12.23) reduces the time constant to 20,000 year, more in line with the observed data below 100 m. To explain the very fast damping in the top 100 m, diffusion in the pore space must be considered. Equation (12.23) is still applicable, but D is replaced by the effective diffusion constant of the vapour in ﬁrn Deff firn;i ¼ Dair;i

Nv;i sNs;i

ð12:27Þ

for each isotope i. The temperature-dependent diffusivity of each isotope in air— Dair,18 or Dair,D—is known experimentally. The ﬁrn tortuosity factor s (quantifying how much the diffusion path is enlarged due to meandering pore space) is a function of ﬁrn density; and Nv,i/Ns,i is the ratio of the number of water molecules in the vapour and the solid state, as governed by temperature-dependent fractionation processes and saturation vapour pressure in the ﬁrn. Detailed calculations show that for a constant ﬁrn density of 0.46 kg/l, the resulting constant k in (12.26) is

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1.06 10−9 s−1; the associated decay time scale is t1/2 = 20.7 years. Therefore most of the observed reduction in seasonal isotope amplitude in the ﬁrn column can be explained by diffusional smoothing.

12.3.5 Diffusion Thermometry While isotopic diffusion in the ﬁrn (and the ice matrix) implies an inherent loss of climatic information over time, the temperature dependence of the diffusional smoothing provides an opportunity for estimating the ﬁrn temperature from high-resolution isotope records. This is the idea behind isotope diffusion thermometry, which aims to reconstruct palaeotemperatures by estimating the effective diffusivities for H18 2 O and HDO from the records. For a non-deforming ﬁrn column with constant density, the squared diffusion lengths r218 and r2D obey dr2 ¼ 2Deff v;i ; dt

ð12:28Þ

but in the presence of vertical compaction where the column undergoes creep deformation at the effective strain rate e_ z , (12.28) is modiﬁed to dr2 2 ¼ 2Deff firn;i ðtÞ þ 2_ez ðtÞr : dt

ð12:29Þ

The strain rate changes the diffusion length because the thinning of layers sharpens their isotopic variations as these are degraded by diffusion. The diffusion equation in (12.23) is also extended to @d @2d @d ¼ Deff e_ z z : firn;i @t @z2 @z

ð12:30Þ

These differential equations are coupled to numerical models of ﬁrn densiﬁcation and ice flow. We briefly outline how palaeotemperatures can be reconstructed with this method, and refer the interested reader to the paper by Simonsen et al. [17] for the details. The coupled model is solved to calculate parameter ﬁelds quantifying how the squared diffusion lengths r218 and r2D depend on accumulation and temperature. Speciﬁcally, the parameter ﬁeld of the ‘differential squared diffusion length’ Dr2 ¼ r218 r2D is considered, because it is affected only by diffusion and not by vertical strain. For each interval in the ice-core record where palaeotemperature and accumulation can be assumed not to have varied signiﬁcantly over time, Dr2 can be found from the power spectra of d18O and dD in that interval. Then, given independent estimates for both the accumulation rate and thinning rate, unique palaeotemperatures can be inferred from the Dr2 parameter ﬁeld. By applying this

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method to the NGRIP ice core, Simonsen et al. [17] found that the Younger Dryas was about 11 °C colder than the preceding Bølling-Allerød warm period.

12.4

Aerosols in Ice

12.4.1 Introduction and Origin of Aerosols in Ice Besides affecting the Earth’s radiation balance, aerosol species in the atmosphere act as nucleation sites for snow precipitation. Falling ice crystals capture additional aerosols and gas phase chemicals before being deposited on the ice sheets and buried by snowfall. Ice cores thus contain an archive of aerosols in particulate and dissolved form. These tracers can be used to reconstruct the environmental or climate conditions in their source regions and large-scale atmospheric transport patterns. Some tracers record the history of emission of anthropogenic pollutants. Modern analytical techniques applied to ice cores measure the concentrations of a wide range of major ionic species (typically at *1–500 parts per billion: originating from sea water, the oxidation of marine biogenic gases, anthropogenic and volcanic emissions) and minor constituents (typically at *1–500 parts per trillion: e.g., photochemical oxidants, heavy metal pollutants, organic molecules, soot from industrial activity, cosmogenic nuclides and radioactive particles from nuclear weapon testing). Table 12.2 lists the typical concentrations of the key ionic species in ice cores, and Fig. 12.8 gives an overview of the sources of different species/constituents.

12.4.2 Aerosol Sources and Transport How do different types of aerosol particles form? Sea salt aerosol is produced by wind dispersion of seawater in liquid state (e.g., at wave crests producing large droplets) and small air bubbles bursting at the ocean surface, producing small aerosol particles. At high latitudes, sea salt aerosol is also derived from sea ice, e.g., from sea spray on snow-covered sea ice, and ‘frost flowers’ formed during the refreezing of open leads and polynyas. Mineral dust particles are produced by chemical and physical weathering of crustal material. They are entrained into the air by high surface winds, and by collision of larger particles with the surface (during saltation). The efﬁciency of this process depends on wind speed and soil moisture. Convection drives uplift of aerosol particles in the atmospheric boundary layer. Consequently, the atmospheric aerosol load depends on the climate conditions of the source region, and aerosol records reflect these conditions. For example, sea salt aerosol formation is closely linked to cyclonic activity, via surface wind strength. In the case of mineral dust, whose sources are often in arid and semi-arid regions far from the ice sheets, soil properties such as aridity and vegetation cover are

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Table 12.2 Typical concentrations of major ionic components in polar ice cores Concentrations in parts per million or mg kg−1 Na+ Cl−

Ca2+

SO2 4

NO 3

Mg2+

Dome C Holocene [18, BAS] 0.021 0.013 0.002 0.100 0.014 0.007 Dome C LGM [18, BAS] 0.100 0.140 0.044 0.220 0.056 0.038 GISP2 Holocene [19] 0.005 0.012 0.007 0.040 0.076 0.002 GISP2 LGM [19] 0.052 0.098 0.210 0.205 0.076 0.023 DML Holocene [20] 0.052 0.106 0.002 0.070 0.045 0.007 San Benedetto Aqua Minerale 6.9 1.9 48.2 3.8 8.2 29.4 Sea water [21] 10500 19000 400 2700 3 1350 Modern and Last Glacial Maximum values from Greenland and Antarctica are shown for comparison. LGM: last glacial maximum; GISP2: US Greenland Ice Sheet Program 2; Dome C, Antarctica; DML: Dronning Maud Land, Antarctica. BAS: British Antarctic Survey unpublished data

Fig. 12.8 Soluble aerosols and insoluble particles commonly measured in ice cores. Their primary origins and typical geographical source areas are indicated by the black and orange boxes, respectively. The box for cosmogenic nuclides refers to production in the stratosphere

additional factors. Dust storm events can lift dust to the high troposphere above the cloud level, increasing their atmospheric lifetime and enabling long-range transport. Sulphate aerosol is produced mainly by gas/particle conversion in the atmosphere during the oxidation of the precursor gas SO2 emitted by fossil fuel burning

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and volcanoes, or the oxidation of dimethyl sulphide and sulphur-bearing gases from biological activity in the ocean and on land. Whereas dispersion aerosol (e.g., sea salt, dust) is on the order of a few lm in size, particles formed from gaseous precursors (e.g., sulphate) are nanometre sized. The latter grow by coagulation. Typically, after travelling long distances to the polar ice sheets, the ‘aged’ aerosol particles and nucleation particles have grown to 1–2 lm in size; in contrast, most of the larger dispersion particles have been removed by dry deposition on the way. Long-range transported aerosol suffers deposition en route to the ice sheets (Fig. 12.9), where precipitation and aerosol deposition form a stratigraphicallyordered sequence. At any location, the total average deposition Jtot is the sum of dry and wet deposition; it can be approximated by Jtot ¼ cice A ¼ Jdry þ Jwet ¼ vdry cair þ Wcair A;

ð12:31Þ

and thus cice ¼ Wcair þ

vdry cair : A

ð12:32Þ

Here, cice is the (average) measured aerosol concentration in ice, A is the local precipitation rate, cair is the atmospheric aerosol concentration, vdry is dry deposition velocity, and W is an empirical scavenging efﬁciency for a given aerosol. Equation (12.32) shows that a climatological change in A can change the flux and modulate cice.

Reactions with other atmospheric constituents Long range transport, Cair, free troposphere

Deposition en route Uplift Cair, source

Low-/mid-latitude source

Deposition Cair, polar

Polar snow & ice Cice

Fig. 12.9 Schematic of evolution of aerosol concentration from source to deposition

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The atmospheric aerosol concentration decreases along the transport path due to the losses. If we ignore replenishment of the air parcel with freshly-generated aerosol, a simple model is that the concentration declines exponentially with the transport time t: cair ¼ cair;0 et=s ;

ð12:33Þ

where the atmospheric residence time is s

H : vdry þ W A

ð12:34Þ

H is the typical height of the air column (we assume cair to be independent of is the mean precipitation rate along the path, and W is deﬁned above. altitude), A In summary, both the transport path length and residence time of an aerosol in the atmosphere affect its concentration in ice cores for any given source region conditions. In Antarctica, for example, sea salt derived sodium (sourced from the surrounding oceans) is high in concentration at near-coastal, low-altitude sites and depleted in interior higher-altitude sites (Fig. 12.10). Accordingly, interpretation of the ice-core record of an aerosol for changes in its production history in source areas must account for along-transport modiﬁcation.

Fig. 12.10 Spatial concentrations of sea salt sodium across the Antarctic continent [22]. Reprinted from the Annals of Glaciology with permission of the International Glaciological Society

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12.4.3 Post-depositional Modification Post-depositional processes further confound the interpretation of ice-core aerosol records. Most non-volatile compounds bound to aerosol particles are deposited on the ice sheet irreversibly. But some compounds such as nitric acid (usually measured as − the NO 3 ion), hydrochloric acid (usually measured as the Cl ion), formic acid (HCHO) and hydrogen peroxide (H2O2) are observed to reduce in concentration on burial, implying post-depositional loss to the atmosphere or reaction within the ice matrix. This is particularly evident at low accumulation sites, where the opportunity for re-emission to the atmosphere is greater than at high accumulation sites, where burial rates are faster. Figure 12.11 shows an example from Antarctica, for nitrate. Its concentration decays with depth at both Dome C and South Pole―much more rapidly at the former site (accumulation rate 3 cm w.e. per year) than the latter site (accumulation rate 7 cm w.e. per year). As demonstrated by the South Pole proﬁle, seasonal cycles can also survive deeper at higher accumulation sites. Two processes that can cause post-depositional loss from a snowpack are (i) evaporation of volatile compounds (such as hydrochloric acid) and (ii) photolysis of photo-reactive chemicals (such as hydrogen peroxide and nitric acid). Post-depositional modiﬁcation of methanesulphonic acid or MSA (CH3SO3H: measured via its anion, methane sulphonate) was examined by Mulvaney et al. [23], who found that during burial, MSA appeared to relocate from the summer layers at shallow depths to winter layers deeper in the ice (Fig. 12.12). Tentatively, drainage of MSA in its liquid phase in the vein network may explain such relocation, but it remains unclear why non-sea salt sulphate (which may exist as liquid H2SO4) does not drain from one layer to another.

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Fig. 12.11 Nitrate losses in Antarctic snow. The initial high surface concentration of nitrate is depleted due to post-depositional loss of HNO3 to the atmosphere. A seasonal cycle is still evident in the South Pole core

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Fig. 12.12 Migration of MSA in ice. Near-surface layers show that MSA is co-deposited with summer season non-sea salt (‘nss’) sulphate and in phase with the summer stable water isotope signal. At depth, the non-sea salt sulphate remains in phase with the summer water isotope signal, but the MSA is located in the winter layers. Redrawn from Pasteur and Mulvaney [24]

12.4.4 Seasonal Cycles in Aerosol and Particle Constituents in Ice Many aerosols and particles deposited with snowfall or directly onto the snow/ice surface exhibit pronounced seasonal cycles. These cycles are often preserved in ice cores from coastal Antarctica, coastal and central Greenland and some non-polar regions, where the accumulation rate is sufﬁciently high ( ≳ 10 kg m−2 year−1 or 10 cm w.e. year−1) so that the signal is not destroyed by wind erosion of the surface. The cycles have diverse origins, such as changes in production rate, and changing weather patterns affecting wind speed/trajectory (and thus the source region), snow accumulation, or dry deposition. The main species known to show seasonal cycles in ice cores include (parentheses indicate their commonly measured ion/element/molecule): methane sulpho− nate (CH3SO 3 or MSA )—produced by oxidation of seasonal biogenic emissions and deposited with summer snow (Sect. 12.4.3); sulphate (SO2 4 )—from the oxidation of seasonal biogenic emissions (causing a prominent summer-time signal in coastal ice cores) and anthropogenic SO2 emissions; nitrate (NO 3 )—related to the atmospheric nitrogen cycle and typically peaking in summer but its level in ice is impacted by re-emission (Sect. 12.4.3); ammonium cation (NH4þ )—from soils and bacterial decomposition of excreta and from biomass burning, and typically appearing in summer layers; sodium (Na+)—from wind-blown sea salt aerosol, typically peaking in winter; chloride (Cl−)—the anion from sea salt (it exhibits a similar signal to Na+); calcium (Ca2+)—some originating from sea salt, but with a main seasonal signal from dust transported from the continents, normally in

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Fig. 12.13 High-resolution chemical aerosol records from the NGRIP ice core, showing pronounced seasonal cycles at a depth of 1465 m, from the early Holocene [27]. Reprinted with permission from John Wiley and Sons, ©2006 American Geophysical Union

winter/spring; insoluble dust particles—measured independently from Ca2+ but showing a similar signal as Ca2+; and hydrogen peroxide (H2O2)—product of photolytic reactions involving the OH• radical in the atmosphere, and deposited with clear seasonal cycles with a maximum in the summer and preserved (with smoothing) at high accumulation rate sites despite its moderate high chemical reactivity. See Fig. 12.8 for the geographical source areas of some of these species. The seasonal signal is exploited in annual layer counting for ice-core dating (Sect. 12.2). The longest records with successful annual layer counting are from the deep Greenland ice cores, where present-day accumulation rates are approximately 20 kg m−2 year−1. The GICC05 Greenland ice core chronology extends layer counting as far back as 60 kyr (Fig. 12.2; Svensson et al. [4]). Figure 12.13 illustrates a core section—covering the early Holocene—where seasonal cycles remain visible and can be counted. High resolution chemical measurements are needed in such exercises. The modern technique uses Continuous Flow Analysis (CFA), where a core section is melted continuously from one end, and the meltwater passed through different instruments for composition determination (e.g., [25]). While layer counting through ice from the full glacial period to the Eemian is not yet possible, seasonal cycles in chemistry have been observed in Eemian ice from Greenland [26]. Annual layer thickness determined from them has been valuable for estimating accumulation rate during the Eemian.

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12.4.5 The Volcanic Signal in Ice and Its Use for Chronological Control

Sulphate μEq L

-1

The sulphate ion (SO2 4 ) is arguably the best understood of the commonly measured chemical components in polar ice. It is incorporated into snow crystals in the atmosphere as sulphuric acid (H2SO4). Four sources of sulphuric acid are identiﬁable in ice-core records: (i) sea-salt sulphate from the incorporation of sea-water droplets (sea-water contains *2.7 g kg−1 of SO2 4 ); (ii) photolytic oxidation of dimethyl sulphide (DMS) emitted by the respiration of phytoplankton; (iii) oxidation of sulphur dioxide (SO2) emitted from industrial sources, particularly power generation; (iv) emissions from volcanoes. Most volcanic activity emits sulphur dioxide, but much of this is contained within the troposphere, oxidized to H2SO4, and deposited locally in precipitation. Large, explosive, volcanic eruptions can carry SO2 into the stratosphere, where it is dispersed globally, before descending into the troposphere and precipitating as H2SO4. The global dispersion of stratospheric SO2, normally over a period of a few months, means that volcanic H2SO4 may be deposited in polar snow. In the larger eruptions, deposition occurs both in Greenland and Antarctica. The volcanic signal, measured chemically (as the sulphate ion, and expressed as non-sea salt sulphate) or via the electrical properties of the ice (Sect. 12.4.9), is manifested by large sporadic peaks against the background seasonal sulphate level from sea-salt spray and marine biogenic emissions (Fig. 12.14). Sulphate from globally-dispersed volcanic emissions (often from Indonesia) make ideal chronological markers, either for age-control on ice cores where the

Tambora 1815 Unknown 1809

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Krakatoa 1883

Unknown 1688 Huaynaputina 1600

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Fig. 12.15 Synchronization of the chronologies of three deep ice cores from Dronning Maud Land (EDML), Dome C (EDC) and Talos Dome (TD), Antarctica, using unidentiﬁed volcanic peaks [28]. Each record spans the period 8.5–10.4 kyr BP

eruption date is known, or for tying chronologies between ice cores by matching the patterns of volcanic peaks (Fig. 12.15). Although volcanic matching between Greenland and Antarctic cores is possible, one of the largest recent events recorded in northern-hemisphere ice cores (Laki, erupted in 1783) is not recorded in Antarctica, as it was not sufﬁciently explosive to inject SO2 into the stratosphere.

12.4.6 Marine Biogenic MSA and Sea Salt as Sea-Ice Proxies Methanesulphonic acid CH3SO3H (commonly abbreviated as MSA) is a photolytic oxidation product of dimethyl sulphide DMS, which is produced by marine phytoplankton in the marginal sea ice zone in polar regions. It is transported with marine aerosols to the ice sheets, notably during the summer when there is high primary production and sunlight. While the other product of DMS oxidation—sulphuric acid —has other sources such as sea salt, volcanic and anthropogenic emissions, the source of MSA appears to be exclusively DMS. Hence its concentration in ice cores (normally measured as methane sulphonate, CH3 SO 3 , which is the conjugate base of MSA) may reflect marine biological productivity or sea ice extent. Curran et al. [29] demonstrated a correlation between annually-resolved MSA measurements

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Fig. 12.16 Methane sulphonate (MSA) and sea ice extent (SIE) in the Wilkes Land, Antarctica, from Curran et al. [29]. The annual record of MSA, its 3-year running mean and 20-year running mean are shown by the light grey, thick grey and black lines, respectively. The line with open circles plot the SIE from 1974 to 1995. The inset shows the correlation between SIE and MSA over that period. Figure reprinted with permission from AAAS

from the East Antarctic Law Dome ice core and winter sea ice extent observed by satellite in the nearby ocean (Fig. 12.16), suggesting that MSA might be used as a proxy for past sea ice extent. MSA also shows correlation with a multi-decadal temperature and sea-level pressure record from a meteorological station on the Antarctic Peninsula [30], again suggestive of its influence by sea ice extent, but more understanding of the production, transport and oxidation pathways for DMS and its products is necessary before MSA can be considered a reliable proxy. As introduced earlier, the concentration of sea salt aerosol in snowfall tends to decay with distance from the open ocean, owing to deposition during transport (e.g., Fig. 12.10). Its concentration in an ice core therefore reflects the distance to the ocean or to the sea ice region. Accordingly, its measured variations down-core might reflect expansion or contraction of the sea ice zone—with reduced concentration indicating expansion. While this idea is plausible, there are complicating factors. First, the strength of atmospheric circulation may vary between climatic states: higher wind speeds at the Last Glacial Maximum may compensate for the presumed greater distance to the ocean when the sea ice zone expanded. Second, some of the sea salt aerosol may derive from sources within the sea ice zone itself. A potential source is to be found in ‘frost flowers’ that form during the refreezing of open leads and polynyas, and which may wick sea water from the sea surface to form lightweight sea-salt rich structures that are easily lofted by wind. If such contributions dominate, then an expansion of the sea ice zone would increase the observed concentration in the core.

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12.4.7 The Record of Anthropogenic Pollution Although modern instruments can observe and track the concentration of diverse pollutants in the atmosphere to subtle levels, establishing the history of pollution from before the industrial era to the present relies on evidence archived in datable sediments, tree rings, and ice cores, etc. A classic achievement from ice cores was the record of lead accumulating in the atmosphere from the emissions of vehicles using fuels with lead additives (Fig. 12.17). Lead is toxic when absorbed by the human body. Evidence from Greenland snow that atmospheric lead concentration had increased signiﬁcantly since the late nineteenth century due to industrial activities (as opposed to natural sources such as volcanoes) was instrumental in persuading fuel companies to remove tetraethyl lead from vehicle fuel. The decision by the USA to phase out lead additives in fuels in 1970 prompted a fall in atmospheric lead concentration almost immediately afterwards; another factor behind the fall is increased use of ethanol in vehicle fuels in South America [31]. By 2000, lead additives had largely been phased out across the world. Ice cores have also shown that the modern industrial era was not the ﬁrst time when anthropogenic pollutants had global influence. For instance, heavy metals such as lead and copper derived from smelting by the early Greek and Roman civilizations have been detected in ice cores in Greenland. Measurements of the concentrations of the sulphate (SO2 4 ) and nitrate (NO3 ) ions in ice cores (Table 12.2) have aided research into the general increase in the acidity of precipitation—commonly referred to as acid rain—during the twentieth century. Acid rain harms land plants and freshwater ecosystems, and damages building facades and monuments. The key chemicals responsible for acid rain are sulphuric acid (H2SO4) and nitric acid (HNO3), which form from the oxidation of the precursors sulphur dioxide and nitrogen oxides, followed by dissolution in precipitation. Natural sources for these precursor species exist (e.g., from volcanic

Fig. 12.17 Lead concentration measured in a a Greenland ice core (redrawn after [32]) and b an Antarctic snow pit (redrawn after [31]), showing increasing atmospheric lead levels due to use of tetraethyl lead in vehicle fuels. Falls followed curbs in lead emissions. The much lower absolute concentrations in Antarctica indicate the predominance of emissions in the northern hemisphere and the effect of long-range transport to the remote Antarctic

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Fig. 12.18 Long-term trends in sulphate concentration (upper panel) and nitrate concentration (lower panel) in a Greenland ice core. Triangles show the industrial and vehicle emissions inventory from Eurasia and USA. Redrawn from Legrand and Mayewski [33] and Fischer et al. [5]

emissions and lightning, respectively), and their contributions are reflected in ice cores by the pre-industrial levels of sulphate and nitrate. Greenland ice cores document clear increases in both ions during the twentieth century that mimic emission inventories (Fig. 12.18). A major source of sulphate is electricity generation using coal; a major source of nitrate consists of vehicle emissions. Relatively recently, clean air laws designed to reduce emissions and the reduced use of coal for power production have led to a fall in emissions and deposition of sulphuric acid on the Greenland ice sheet. Due to the continued rise in vehicle use, however, nitrate has shown little evidence of a reduction. Antarctic ice cores show little or no industrial increase in sulphate or nitrate, due to lower emissions in the southern hemisphere and the long transport distance; the latter means that precipitation can cleanse the air of these highly soluble acidic aerosols before they reach the high southern latitudes. Black carbon (soot) from the burning of fossil fuels and biomass burning (predominantly in forest ﬁres) is also recorded in ice cores. Black carbon is believed to impact human health negatively and contribute to climate warming directly via its radiative effects. In terms of radiative forcing, its total warming effect since the beginning of the industrial era (1750–2005) has been about 1.1 W m−2, second only

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to carbon dioxide (*1.7 W m−2), and greater than that of methane [34]. Soot deposited onto glacier and sea ice surfaces also lowers their albedo, promoting melting. Ice-core records from central Greenland show that total black carbon emissions have varied widely [35]: industrial emissions increased seven-fold from 1850 to a maximum around the start of the twentieth century, but have reduced since 1950.

12.4.8 Long Aerosol Records from Greenland and Antarctica Both Antarctic and Greenland ice cores show that on glacial-interglacial time scales, some aerosol species show much higher concentrations during cold periods than warm periods (Figs. 12.19 and 12.20). These changes have been linked to changes in aerosol source strength, together with reduced washout during transport in colder climate conditions. In Antarctica (Fig. 12.19), non-sea salt calcium flux and iron flux are a factor of ten higher during glacials than interglacials [36]. Both species are derived from continental dust. For the Dome C core, South America (especially Patagonia) is considered to be the dominant dust source area. Model studies suggest that changing transport between Patagonia and Dome C can explain only part of the observed variations in deposition flux in the core, which should therefore mainly reflect changes in the source region [36]. Numerous local factors may explain increased dust production during glacials—such as aridity, temperature, vegetation cover, wind strength, glacial processes, and the area of exposed continental shelf (which depends on sea level). Also, the Dome C record implies a strongly elevated flux of iron to the southern oceans during glacials, which is expected to enhance iron-fertilization of the oceans and thus help draw down CO2. In contrast, the Dome C core documents no signiﬁcant change over time of non-sea salt sulphate from marine biogenic emissions (DMS-producing plankton species) (Fig. 12.19). This is surprising. Earlier proﬁles from the Vostok core had shown a high glacial flux of the marine biogenic ion methane sulphonate, suggesting that the sulphur cycle was sensitive to climate change; since the oxidation products of DMS (i.e., MSA and H2SO4) are cloud condensation nuclei, the sulphur cycle may have caused climate feedbacks. While the relative production of MSA and H2SO4 from DMS is sensitive to temperature and may vary with time, the dominant product measured in ice is likely to be sulphate, so the observation of stable non-sea salt sulphate in the Dome C core suggests that the export of marine biogenic sulphuric acid was approximately constant through eight glacial cycles. Recent studies on MSA concentrations in surface snow from Dronning Maud Land show that this component is subject to post-depositional loss (Sect. 12.4.3), so the higher MSA levels found for glacial periods at Vostok may be the result of better preservation. The Dome C record of sea salt sodium shows a two- to four-fold increase in glacial periods above the interglacial flux. Recognizing that the use of sea salt as a proxy for sea ice is still debated (Sect. 12.4.6), Wolff et al. [36] tentatively suggested that this record could be interpreted as a quantitative winter sea-ice record for the past 740 kyr.

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Fig. 12.19 Sea salt, mineral dust and biogenic sulphate fluxes in the EPICA Dome C ice core [36]. Reprinted from Nature with permission from Springer Nature ©2006

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Fig. 12.20 Sea salt and mineral dust concentrations in the GISP2 ice core [19]

As exempliﬁed in Fig. 12.20, Greenland ice cores show strong changes in sea salt and mineral dust fluxes in connection with Dansgaard-Oeschger events. Continental dust (represented by Ca concentration) at the Last Glacial Maximum was approximately 80 times its level during the Holocene, while dust concentration in the glacial was 15 times higher than observed in the Dansgaard-Oeschger warm periods. Correcting for the changes of aerosol deposition during transport, Schüpbach et al. [37] showed that both sea salt and in particular mineral dust changes were accompanied by changes in their source strengths.

12.4.9 Electrical Properties of Ice and Their Relationship to Chemistry The AC dielectric properties of ice are temperature- and density-dependent and modiﬁed by the dissolved chemical species (notably the acidic components) and the ice fabric. Trapped particles of dust are thought to neutralize acidity, reducing the electrical conductivity. Air bubbles in the ice, which are poor conductors, result in density variations in the more porous upper layers. Early radar surveys of ice sheets, aimed at ﬁnding basal returns to measure the ice thickness, imaged internal reflecting horizons that show natural variations in dielectric properties through the ice column [38]. The reflections arise from spatial changes in the ice permittivity (Sect. 14.3.1). Today we recognize that the permittivity above the ﬁrn-ice transition is largely controlled by density, whereas below the transition it is controlled by fabric and temperature, with chemical changes producing contrasts that probably cause the majority of the internal reflecting horizons seen by radar. As the strong reflectors are likely due to high levels of acidity from volcanic sulphuric acid, and as the deposition of aerosols

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Dronning Maud Land depth / m Fig. 12.21 Dielectric proﬁles of the EPICA Dome C core and a shallow ice core in Dronning Maud Land, Antarctica, showing volcanic spikes. The background signal reflects the high-frequency conductance of pure ice; its slow drift is caused by changes in ﬁrn density

across an ice sheet in each volcanic event occurred over a short time (*a year), the horizons are usually interpreted as isochronal surfaces: of equal time/age. As such, they have been exploited to map the age-depth relationship of ice sheets, decipher the history of ice flow, and link chronologies between ice cores. Dielectric Proﬁling (DEP) determines the dielectric properties of an ice core by measuring the capacitance and conductance at a range of frequencies on the full ice core held between two electrodes. The DEP responds weakly to sea salt ions (especially chloride, and ammonium ions dispersed in the ice fabric) but strongly to acidity (e.g., caused by volcanic input; see Fig. 12.21). Pure ice has a signiﬁcant high-frequency conductivity, which can be seen as a background response even when impurity concentrations are low (Fig. 12.21). Ice also conducts DC electric current. Two potential pathways for this are: (i) solid phase conduction, involving the displacement of ionic H3O+ defects in the lattice, and (ii) conduction through the network of liquid veins located at triple grain boundaries [39]. The discovery of acids with low eutectic temperatures at grain boundaries in ice suggests that the second mechanism should operate. Hammer [40] pioneered the use of the Electrical Conductivity Method (ECM), which continuously measures the resistance between a pair of brass electrodes (across which a high DC voltage is applied) as they are drawn across the surface of a core. This technique is routinely used in the early stages of ice-core analysis. As in the DEP, the measurement is rapid and non-destructive (no ice is lost), and clear seasonal signals are often observed in the record for cores with a relatively high accumulation rate. These signals result from the seasonality in acidic components such as sulphuric acid (H2SO4) and MSA (CH3SO3H). Since these acids are normally

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Fig. 12.22 EPICA Dome C ice core: AC dielectric conductance measured by DEP (bottom panel) and DC conductivity measured with the ECM (middle), compared to the climate signal dD (top)

deposited in the summer layers, peaks in the ECM signal may be used to count annual layers for developing the ice-core chronology. Strong peaks caused by volcanic input are used for chronological control and synchronizing the time scales of different cores (Sect. 12.4.5). Over glacial-interglacial timescales, the electrical properties of ice measured by DEP and ECM follow the general pattern of the bulk chemistry of the ice (compare Fig. 12.22 with Fig. 12.19). As mentioned earlier, radar reflecting horizons deep in the ice sheets originate from permittivity contrasts across layers of different acidity levels. Research has been undertaken to relate the pattern of isochrones seen in radargrams to the DEP and ECM records of ice cores obtained at the same location.

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311

Gases Enclosed in Ice

12.5.1 Firn Gas and Gas Occlusion During the ﬁrniﬁcation in the top 100 m of the ice sheet, the density increases with depth and the open pore space between snow grains is reduced; eventually, air is entrapped in bubbles in the ice (Fig. 12.23). This process was addressed in Chap. 11. Here we review its implications for the gas records in ice cores, extending the ideas in Sect. 11.3.2. At a density of approximately 810 kg m−3, the ﬁrst bubbles are sealed off from the pore space—this trapping process is called occlusion. Air in the pore space before occlusion communicates with the overlying atmosphere through molecular diffusion. The bubbles thus archive the atmospheric gas composition. The pore air at depth is somewhat older than the air in the atmosphere, despite the diffusive transport of air downward. Bubble enclosure is completed when the density reaches about 840 kg m−3. Due to increasing pressure, the air bubbles gradually shrink in size with depth. At a depth of around 700–1000 m, they disappear completely and the gases are included in the ice lattice as gas hydrates (also called clathrates). These considerations imply that the air enclosed in the bubbles is younger than the ice at the same depth. This ice age/gas age difference (Dage) has to be taken into account when comparing greenhouse gas concentration and temperature records

Fig. 12.23 Development of the gas archive in ice cores; adapted from Schwander [41]

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from the same ice core (Sect. 11.3.2). Also, the bubbles are not occluded at the same depth/time. Thus, even in a small sample, the age of the air in individual bubbles is different, and this causes a broadening of the atmospheric variation. Both molecular diffusion and the slow bubble close-off lead to low-pass ﬁltering of atmospheric gas signals in ice cores. As introduced in Sect. 11.3.2, the isotopic composition of a gas varies through the ﬁrn and two processes determine its vertical proﬁle: gravitational fractionation and thermal fractionation. In much of the ﬁrn thickness, a static air column is established. As derived in (11.13) and (11.14), the gravitational enrichment of a heavier isotope at depth z below the surface can be deduced from the barometric formula Dd þ 1 ¼

Rg ðzÞ DMgz DMgz ; ¼ exp 1þ RT RT Rg0

ð12:35Þ

where DM is the difference in molecular weight between the isotopes (kg mol−1), g is gravity, R is the gas constant, T is temperature, and Rg0 and Rg(z) are the isotope ratios at the surface and at depth z, respectively. The blue line in Fig. 12.23 shows the (idealized) isotopic proﬁle expected from gravitational fractionation. Consider N2, for example. Its amount and composition in the atmosphere can be assumed constant. The top zone of the ﬁrn (convective zone: CZ) is well ventilated and mixes rapidly with the atmosphere, so d15N is constant over this zone. Equation (12.35) implies that, in the static air column (diffusive zone: DZ), d15N increases with depth. In the lowest zone (non-diffusive, or lock-in zone: LIZ), diffusion is so slow that although the air is not yet enclosed, d15N no longer changes. Field measurement of the d15N proﬁle can be used to identify these zones. Figure 12.24 shows a proﬁle from the NEEM ice core site (77.45° N, 51.07° W) in Greenland that was studied by Buizert et al. [42]. As expected, a linear ﬁt to the d15N data does not yield a zero intercept; the convective zone inferred from these data is 4.5 m thick. The model curves in Fig. 12.24 are based on (12.35) (assuming constant d15N at the surface), accounting also for the effective diffusivity in the ﬁrn; the model uses the known history of atmospheric CO2 concentration to constrain the diffusivity proﬁle. Thermal fractionation (the second process) occurs when a temperature gradient sustained along the ﬁrn column causes the heavier isotopes to accumulate at its colder end—as a result of thermal diffusion. When the close-off region is colder than the surface, this process causes extra enrichment on top of the gravitational enrichment (Fig. 12.23, red line); in the reverse situation, it works against the gravitational enrichment. The thermal fractionation of two isotopes is described by Dd þ 1 ¼

Rb ¼ Rt

c Tt ; Tb

ð12:36Þ

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Fig. 12.24 Data and modelling of ﬁrn air from the NEEM project in northern Greenland [42]; compare with Fig. 11.15

where Rt and Rb are the isotope ratios at the top and the bottom of the ﬁrn column, and Tt and Tb denote the respective temperatures. The thermal diffusion factor c depends on the gas species (e.g., c 0.0033 for 14N2 and 14N15N molecules) and is determined experimentally. Enrichment due to thermal fractionation is observed in the ice-core records of d15N at the onset of Dansgaard-Oeschger events, when the surface temperature in Greenland rose by *10 °C in only a few years (Fig. 12.25). Heat conduction along the ﬁrn column is slow compared to gas transport through the open pore space, so the heavier isotopes are enriched at the still colder end of the gradient. This effect can be used to ﬁne-tune the reconstruction of temperature change during rapid climate variations from water isotopes (Fig. 12.25; [43]). Note that thermal fractionation only gives information near the time when rapid temperature changes occur.

12.5.2 Trace Gases On an absolute scale, water vapour is the most important greenhouse gas. Its amount in the atmosphere depends on temperature and cannot easily be influenced. In a climate change context, it is usually ‘disregarded’, and CO2 is frequently cited as the key greenhouse gas. This is appropriate when one considers variations against the natural background. CH4 and N2O are two other atmospheric trace gases (Table 12.3) that contribute signiﬁcantly to the greenhouse effect. Direct (instrumental) records of these greenhouse gases are restricted to the last few decades. Much of what we know about their changing concentrations over the

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Fig. 12.25 Reconstruction of rapid temperature changes [43]. Top: water isotopes (a proxy for the precipitation temperature). Middle: d15N excursions originating from thermal fractionation. Bottom: Methane concentrations together with estimates for the local temperature modelled from the d15N values. Figure reprinted from Earth and Planetary Science Letters ©2006 with permission from Elsevier

past centuries or longer time scales has been derived from ice cores. Here, we briefly review their histories and the causes of change. The last *1000 years Figure 12.26 (left) shows ice-core records of the history of CO2, CH4 and N2O concentrations over the last millennium. These concentrations had been relatively stable until about 1800 and they increased dramatically afterwards. The main reason for the 100 ppmv increase of CO2 over the last 200 years is emissions from fossil fuel burning. This rise is corroborated by the drop in atmospheric d13C from −6.4 to −7.8‰ over the same period (Fig. 12.26, right), because CO2 from fossil fuel is isotopically light (d13C −25‰), owing to its organic origin. Direct systematic measurements of atmospheric CO2 concentration began in 1958, and their good agreement with the ice-core measurements (where they overlap) attests to the reliability of the latter.

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Table 12.3 Composition of the dry atmosphere (2011) from the IPCC [44]

315

Gas

%

ppm

Nitrogen (N2) Oxygen (O2) Argon (Ar) Carbon dioxide (CO2) Neon (Ne) Helium (He) Methane (CH4) Krypton (Kr) Hydrogen (H2) Nitrous oxide (N2O) Carbon monoxide (CO) Xenon (Xe)

78.084 20.946 0.934 3.905 1.818 5.240 1.803 1.140 5.500 3.242 10−5 9.000

780,840.000 209,460.000 9340.000 390.500 18.180 5.240 1.803 1.140 0.550 0.324 0.100 0.090

10−2 10−3 10−4 10−4 10−4 10−5 10−5 10−6

In the case of methane, the increase is even more dramatic. Its concentration rose from about 700 ppbv during pre-industrial times to about 1800 ppbv today. A key cause is anthropogenic output related to food and energy production. About 60% of today’s emissions are human-related. Half of that comes from bacterial production in agricultural soil (e.g., rice production) and ruminants. The other half comes from energy production, emission from landﬁlls and biomass burning. Nitrous oxide is also emitted by human activities such as fertilizer use and fossil fuel burning. Natural processes in soils and the oceans also release N2O.

380

340

CO2

300

-6.2

1200

320

1000

CH4

310

800

300

600

290

7000

N2O

280

6000 5000

270

4000

260

3000 2000

World population 1000

1200

1400

1600

Age (yr AD)

1800

1000 2000

0

-6.4

380

CO2 concentration (ppm)

1400

330

CH4 (ppbv)

1600

280

N2O (ppbv)

400

1800

-6.6

Concentration and δ13C data from various drilling sites

360

Mauna Loa (atmosphere) GRIP EDML Law Dome South Pole Law Dome δ13C

340

-6.8 -7.0 -7.2

320

δ13C of CO2 (‰)

320

World population (mio)

CO2 (ppmv)

360

-7.4 300 -7.6 280

-7.8 1000

1200

1400

1600 years AD

1800

2000

Fig. 12.26 Left: CH4 record from Law Dome, Antarctica, including ﬁrn measurements and direct atmospheric measurements (open triangles) from Cape Grim, Tasmania [45]. N2O measurements from ice cores [46, 47] and ﬁrn air [48]. CO2 data from Antarctic ice cores compiled by Barnola [49]; the ice core measurements overlap with direct atmospheric measurements (solid line) [50]. World population from [51]. Right: CO2 and d13C variations over the last 1000 years [45, 52]

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The Holocene period Atmospheric CO2 concentration varied through the Holocene, but with much smaller amplitude and over a much longer time scale than in the recent anthropogenic increase. It increased by 20 ppmv over the last 7000 years, mainly as the result of slow re-equilibration of the carbonate budget in the deep ocean via carbonate sedimentation on the ocean floor and by coral reef growth [53]. This increase is a long-term response of the global carbon cycle to the preceding glacial-interglacial transition. CH4 concentration varied also, but again was relatively stable during the Holocene. In the early Holocene, it was near the pre-anthropogenic value of the last 1000 years; it dropped by 150 ppbv by the mid Holocene and slowly increased afterwards [54]. The hypothesis that this increase is due to early human influence [55] ﬁnds support in evidence that human activity has an imprint on the methane cycle through biomass burning and agricultural activities over at least the last two thousand years [56], but this contribution is too small to explain the observed increase since the mid-Holocene. Glacial-interglacial time scales On these time scales, atmospheric CO2 and CH4 concentrations covaried strongly with isotopic temperature in prominent *100 kyr cycles (Fig. 12.27). Concentrations were much lower during glacials—170–190 ppmv [57] and 350 ppb [58], respectively—than today. Key reasons for the reduced CO2 concentration include higher carbon storage in the deep ocean due to reduced Southern Ocean mixing, changes in the carbonate equilibrium in the ocean, and an increase in the marine biogenic ﬁxation of CO2 due to dust-induced iron fertilization in the Southern Ocean. In the case of CH4, the lower concentrations are mainly due to a decline in (boreal and tropical) wetland methane emissions. Methane isotope studies showed that not only the emission strengths of different sources of methane, but also the isotopic signature of the emissions, varied with climate [59]. The interglacials before 450 kyr BP had lower CO2 and (to some extent) CH4 concentrations than the recent ones. According to the dD record, they represent also cooler interglacials. This suggests that the temperature dependences of both greenhouse gases have remained approximately the same over the last 800 kyr. Spectrally, the gas records exhibit variations on the Earth’s orbital frequencies, like many other major climatic parameters. The 100-kyr eccentricity cycle modiﬁes the net incoming solar radiation (very slightly); the precession and obliquity cycles (20 and 40 kyr) affect the time distribution during the year and the geographical distribution of the incoming solar radiation, but not its total. Although the observed climatic variations are paced by Milankovitch forcing, feedback mechanisms are necessary to translate the small perturbations in insolation into prominent, asymmetric glacial-interglacial cycles. Chapter 16 provides further discussion of this topic.

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EDC3/GT4 age [yr BP] -340

0

200000

400000

600000

800000

δD [°/°°] -390 -440 CO2 300 [ppmv] 260 220 180 CH4 750 [ppbv] 600 450 300

Fig. 12.27 Carbon dioxide and methane concentrations over the last 800,000 years from the Vostok and EPICA Dome C ice cores

As described in Sect. 12.3.3 and as visible in the Greenland ice core records in Fig. 12.25, Dansgaard-Oeschger events also occurred during the last glacial in the northern hemisphere. Involving oceanic and atmospheric circulation changes, these variations are too rapid (and their periodicities are too short) to be of orbital origin. Figure 12.25 shows that CH4 concentration oscillated strongly with temperature over Greenland in these events. This correlation may stem from the effect of temperature and precipitation changes on methane emissions from the tropics to the high northern latitudes. Since the Greenland ice cores do not extend beyond 125 kyr BP, we currently have no direct records of the occurrence of Dansgaard-Oeschger type events before that time. However, methane oscillations observed in earlier records may be the signature of such events. For example, in the EPICA ice core, Dansgaard-Oeschger type methane fluctuations have been detected across the past eight glacial cycles ([58]; also see Fig. 12.27). This ﬁnding suggests that such oscillations are an intrinsic feature of the glacial climatic state.

12.6

Timing of Climate Events

For climatic changes straddling both hemispheres, an interesting question is whether their timing and characteristics are the same in both hemispheres. Antarctic and Greenland ice cores reveal that during Dansgaard-Oeschger type events, which have a signature across a large portion of the globe, temperature changes in

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Antarctica deviate somewhat—but not entirely—from those in Greenland (e.g., compare Fig. 12.27 with Figs. 12.25 and 12.20). Speciﬁcally, whereas these events are characterized by rapid temperature increase and gradual temperature decrease in Greenland, they are portrayed by gradual warming followed by gradual cooling in Antarctica. A similar pattern is seen during the last glacial termination, when Greenland warmed abruptly while Antarctic temperatures rose gradually. To study the precise timing relationships, synchronization of Greenland and Antarctic ice cores is necessary. The methane records in these cores allow unambiguous matching for this purpose [60, 61], because the interhemispheric mixing time of methane is short relative to its atmospheric lifetime, and because methane shows rapid variations in parallel to Dansgaard-Oeschger events (Fig. 12.28). A common ‘gas age scale’ can be established by accounting for the climate-dependent ice age/gas age difference (Dage) calculated by ﬁrniﬁcation models (Sects. 12.5.1 and 11.3.2). This has been achieved over the last 55,000 years for the EPICA Dronning Maud Land (EDML) and the NGRIP ice cores (Fig. 12.28). The results show that each Dansgaard-Oeschger event has a counterpart called the Antarctic Isotope Maximum (AIM) in the Southern Ocean— with Antarctica warming when Greenland is in a cold stadial state, and Antarctica cooling when Greenland is in a warm interstadial (Fig. 12.28). The current theory is that this bipolar seesaw is transmitted across hemispheres by changes in the Atlantic Meridional Overturning Circulation. During stadials, this circulation is reduced and exports less heat from the South to the North Atlantic. Accordingly, the North Atlantic is cold while heat accumulates in the large Southern Ocean heat reservoir, slowly warming this ocean basin. The amplitude of the Southern Ocean warming thus depends on the length of the stadial in Greenland [61, 62]. During interstadials, the opposite effect occurs. This seesaw behaviour between the hemispheres continued to operate through the deglacial transition into the Holocene. While CH4 concentrations closely followed the northern hemisphere signal of this bipolar seesaw, CO2 concentrations showed an accompanying change with the largest AIM. This points to a close link between Southern Ocean hydrography and biology and atmospheric CO2. The temperature increase during glacial-interglacial transitions cannot be explained without the increase in CO2 concentration. However, CO2 does not necessarily need to lead the temperature rise; it can be an ampliﬁer of the warming instead. Antarctic ice cores show that across the different glacial-interglacial transitions so far recorded, the CO2 increase was either synchronous with the temperature increase or lagged behind it by a few hundred years, although there are exceptions when CO2 led the transition [63, 64]. For the last transition (the last glacial termination), it is clear that the CO2 increase preceded temperature increase in Greenland. For this transition, Shakun et al. [64] suggested that the bipolar seesaw is what caused the rise in northern hemispheric temperature to lag behind both the southern hemispheric temperature rise and the CO2 increase.

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A2

A1

-410

Byrd -44

-46 AIM 1

AIM 12 AIM 8 AIM 4

-49

5 4.1 6

3

7 9

AIM 2

EDC

ACR

-52

-430

-48

EDML

10 11

Tsurf [°C]

-40

δ Dcorr [°/ ] °°

-43

-37

δ 18Ocorr [°/ ] °°

δ 18O [°/ ] °°

-34

-52

CH4 [ppbv]

H1

2

3 4 H2

5 6 7

H3 4.1

DO8

10 11

9 H4

DO12

NorthGRIP

-37

H5

-41

δ 18O [°/ ] °°

-33 DO1

-450

800 -45 650

Greenland CH4 composite

500

EDML 350 10000

20000

30000

40000

50000

60000

NGRIP age [yr BP]

Fig. 12.28 Comparison of Antarctic and Greenland ice core temperature records after CH4 synchronization [61]. Reprinted from Nature with permission from Springer Nature ©2006

Exercises 12:1 Depth-age model for an ice core (i) According to Nye’s formula in (12.2), what is the annual layer thickness at 1500, 500 and 0 m above bedrock of an ice sheet 2400 m thick (you may disregard the density change in the ﬁrn column) if the annual accumulation rate at the surface is 20 cm ice equivalent? (ii) What are the ages of the ice at these depths? Are these ages physically realistic? 12:2 Playing with isotope standards The d of a sample is measured against a working standard as –3‰. You know that the working standard has a value of +8‰ versus the reference standard. What is the d value of the sample versus the reference standard? 12:3 Firn gas fractionation The following graph presents a record of d15N2 during the rapid Younger Dryas/Preboreal and the Bølling-Allerød warming, with d15N2 = Rbubble/ Rfree air − 1 expressed in ‰. Note that d15N2 in free air may be taken to be constant on the time scales considered in the ﬁgure.

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(i) Using the graph, estimate the diffusive ﬁrn column depth when T = 230 K. (Use the following parameters: universal gas constant R = 8.314 J mol−1 K−1, and g = 9.81 m s−2.) (ii) How large was the temperature change at the top of the ﬁrn column during the rapid warmings if the thermal diffusive factor is a = 0.0033 for d15N2 and the temperature before the warming was 230 K? 12:4 Dating with an ECM record The following graph shows a conductivity curve of an Antarctic ﬁrn core drilled in the austral summer season 2006/07 (top: 11-point running mean; bottom: 51-point running mean). Try to date the core by using the graph (note the data gap at 18–19 m WE) and the volcanic eruptions listed in the

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table. Determine the average accumulation rate. Compare the age at 40 m depth found from your result against that found from a Nye model using this accumulation rate. 80 60

cond. [μS/m] 40 20 0 0

10

20

80

cond. [μS/m]

30

40

50

60

40

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60

depth WE [m]

60 40 20 0 0

10

20

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depth WE [m]

12:5 Isotope equilibrium Oxygen dissolved in ocean water is 0.7‰ heavier than atmospheric oxygen in equilibrium. The fractionation factor for oxygen leaving the water was measured as a = 0.9965 (personal communication: Boaz Luz). What is the resulting fractionation factor for the exchange between atmospheric oxygen and dissolved oxygen?

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12:6 Sea level at the Last Glacial Maximum (LGM) Mean ocean water had a d18O value of +1‰ during the LGM. Use this information and make some assumptions to calculate how much lower the sea level was compared to today. 12:7 Rayleigh distillation Ocean water is evaporated in equilibrium at 20 °C. The vapour is then cooled in equilibrium to 0 °C. (i) What is the fraction f of water vapour remaining in the cloud? (Assume the speciﬁc evaporation energy L = 45 kJ/mol and the universal gas constant 8.314 J mol−1 K−1.) (ii) What is the d18O and the deuterium excess relative to SMOW of the condensate, if the equilibrium fractionation factor during evaporation is a0 = 1.00979 for 18O (and 1.08502 for D) and the fractionation factor during equilibrium condensation is a = 1.01172 for 18O (and 1.11231 for D)? (iii) What is the d18O and the deuterium excess relative to SMOW of the initial water vapour at 20 °C if the relative humidity h is 80%? (iv) What was the glacial d18O relative to SMOW of the water vapour evaporated in equilibrium, if the d18OSW of the sea water was +1.5‰ (due to the expansion of continental ice sheets and the respective sea level lowering in the glacial)? (v) What was the glacial d18O of the condensate produced in equilibrium if the d18OSW of the sea water was +1.5‰ (as above)?

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11. Steen‐Larsen HC and 23 others (2011) Understanding the climatic signal in the water stable isotope records from the NEEM shallow ﬁrn/ice cores in northwest Greenland. J Geophys Res 116:D06108 12. Johnsen SJ, Dansgaard W, White JWC (1989) The origin of Arctic precipitation under present and glacial conditions. Tellus 41B:452–468 13. Stenni B and 13 others (2003) A late-glacial high-resolution site and source temperature record derived from the EPICA Dome C isotope records (East Antarctica). Earth Planet Sci Lett 217:183–195 14. Winkler R, Landais A, Sodemann H, Dümbgen L, Prie F, Masson-Delmotte V, Stenni B, Jouzel J (2012) Deglaciation records of 17O-excess in East Antarctica: reliable reconstruction of oceanic normalized relative humidity from coastal sites. Clim Past 8:1–16 15. North Greenland Ice Core Project Members (2004) High-resolution climate record of the northern hemisphere reaching into the last interglacial period. Nature 431:147–151 16. Johnsen SJ, Dansgaard W, Clausen HB, Langway CC Jr (1972) Oxygen isotope proﬁles through the Antarctic and Greenland Ice Sheets. Nature 435(5339):429–434 17. Simonsen SB, Johnsen SJ, Popp TJ, Vinther BM, Gkinis V, Steen-Larsen HC (2011) Past surface temperatures at the NorthGRIP drill site from the difference in ﬁrn diffusion of water isotopes. Clim Past 7:1327–1335 18. Röthlisberger R, Mulvaney R, Wolff EW, Hutterli MA, Bigler M, Sommer S, Jouzel J (2002) Dust and sea salt variability in central East Antarctica (Dome C) over the last 45 kyrs and its implications for southern high‐latitude climate. Geophys Res Lett 29(20): 24-1–24-4 19. Mayewski PA and 13 others (1994) Changes in atmospheric circulation and ocean ice cover over the North Atlantic during the last 41,000 years. Science 263:1747–1751 20. Stenberg M, Isaksson E, Hansson M, Karlén W, Mayewski PA, Twickler MS, Whitlow SI, Gundestrup N (1998) Spatial variability of snow chemistry in western Dronning Maud Land, Antarctica. Ann Glaciol 27:378–384 21. Finlayson-Pitts BJ and Pitts JN (2000) Chemistry of the upper and lower atmosphere. Academic Press, New York 22. Bertler N and 54 others (2005) Snow chemistry across Antarctica. Ann Glaciol 41:167–179 23. Mulvaney R, Wolff EW, Oates K (1988) Sulphuric acid at grain boundaries in Antarctic ice. Nature 331:247–249 24. Pasteur EC, Mulvaney R (2000) Migration of methane sulphonate in Antarctic ﬁrn and ice. J Geophys Res Atmos 105:11525–11534 25. Kaufmann PR, Federer U, Hutterli MA, Bigler M, Schüpbach S, Ruth U, Schmitt J, Stocker TF (2008) An improved continuous flow analysis system for high-resolution ﬁeld measurements on ice cores. Environ Sci Technol 42:8044–8050 26. Svensson A, Bigler M, Kettner E, Dahl-Jensen D, Johnsen S, Kipfstuhl S, Nielsen M, Steffensen JP (2011) Annual layering in the NGRIP ice core during the Eemian. Clim Past 7:1427–1437 27. Rasmussen SO and 15 others (2006) A new Greenland ice core chronology for the last glacial termination. J Geophys Res 111:D06102 28. Severi M, Udisti R, Becagli S, Stenni B, Traversi R (2012) Volcanic synchronisation of the EPICA-DC and TALDICE ice cores for the last 42 kyr BP. Clim Past 8:509–517 29. Curran MAJ, van Ommen TD, Morgan VI, Phillips KL, Palmer AS (2003) Ice core evidence for Antarctic sea ice decline since the 1950s. Science 302:1203–1206 30. Abram NJ, Thomas ER, McConnell JR, Mulvaney R, Bracegirdle TJ, Sime LC, Aristarain AJ (2010) Ice core evidence for a 20th century decline of sea ice in the Bellingshausen Sea, Antarctica. J Geophys Res Atmospheres 115:D23101 31. Wolff EW, Suttie ED (1994) Antarctic snow record of southern hemisphere lead pollution. Geophys Res Lett 21:781–784 32. McConnell JR, Edwards R (2008) Coal burning leaves toxic heavy metal legacy in the Arctic. Proc Natl Acad Sci 105:12140–12144

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Satellite Remote Sensing of Glaciers and Ice Sheets

13

Helmut Rott and Frank Paul

13.1

Introduction

Given the widespread distribution of snow and ice masses across the globe, satellite-based remote sensing techniques provide the only feasible means of observing them comprehensively. Over the past decades, data collected by remote sensing instruments on satellites and airborne platforms have greatly advanced our understanding of how the cryosphere interacts with the atmosphere and oceans [1, 2]. Whereas airborne sensors are used mainly for occasional/infrequent regional or national surveys, satellite-borne sensors can make regular repeat observations over much larger areas. This chapter focusses on the key satellite remote sensing techniques used to monitor the mass balance and dynamics of glaciers, ice caps, and ice sheets. Their principles and applications are considered. The sensors operate in the visible, infrared and microwave parts of the electromagnetic spectrum. A sensing technique is classiﬁed as passive when it exploits reflected solar or emitted thermal radiation, or active when it uses its own illumination source (such as in lidar, radar). Our treatment below is arranged in the order of passive optical sensors (Sect. 13.2),

H. Rott (&) Institute of Atmospheric and Cryospheric Sciences, University of Innsbruck, Innsbruck, Austria ENVEO IT GmbH, Innsbruck, Austria e-mail: [email protected] F. Paul Department of Geography, University of Zurich, Zurich, Switzerland e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_13

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active microwave sensors (imaging radars) (Sect. 13.3), and active altimetry sensors (Sect. 13.4). Passive microwave techniques (multi-channel microwave radiometers), used to monitor daily global snow and ice extent at coarse spatial resolution, are not covered.

13.2

Optical Sensors and Applications

13.2.1 Sensors and Satellites Space-borne optical sensors have been used to study glaciers since the 1970s. The ﬁrst sensor―the MultiSpectral Scanner (MSS)―was deployed on the Landsat 1 satellite (named ERTS1 at that time) and has been used to observe glacier dynamics and various ice and snow facies (e.g., [3]). Its spatial resolution of 80 m and four different spectral bands (located in the visible and near-infrared part of the spectrum) allowed different surface types (e.g., snow, ice, debris, ash, water) to be identiﬁed on account of their spectral characteristics. The potential to use MSS data to create glacier inventories was recognized early on, and MSS data contributed to the creation of the World Glacier Inventory (WGI) in some regions. These data had two main drawbacks, however. Their spatial resolution was too coarse to delineate small glaciers well, and they could not discriminate snow from clouds. These problems were largely resolved with the launch of the Landsat 4 satellite in 1982 and its new sensor Thematic Mapper (TM). This sensor had a ﬁner spatial resolution (30 m), one extra band in the blue part of the spectrum, two extra bands in the shortwave infrared (SWIR) and a thermal infrared (TIR) band with 120 m resolution (Table 13.1). With the SWIR bands, it was possible to discriminate snow from clouds and map glacier extents automatically (Sect. 13.2.2) because snow and ice have SWIR reflectances that are markedly different from those of clouds and other surface types. Field measurements and theoretical studies (e.g., [4]) revealed that the spectral reflectance of snow is very high in the visible part (VIS) of the spectrum, drops in the near-infrared (NIR), and is near-zero in the SWIR at 1.5 and 2 lm wavelength (Fig. 13.1). Reflectance also decreases with grain size—weakly in the VIS but strongly in the NIR and SWIR. Consequently, snow grain size can be determined from the measured spectral reflectances in these bands (found via conversion of the raw Digital Numbers (DN) from the sensor) after applying atmospheric and topographic correction. The spectral characteristics of bare/clean glacier ice―which forms from snow metamorphosis and compression (see Chap. 11)―resemble those of snow with large grain size; these characteristics are altered by the presence of debris or water on the glacier surface. Notably, contamination of snow or ice with darker materials (e.g., dust, soot) lowers the reflectance much more signiﬁcantly in the VIS than in the NIR or SWIR. In contrast, ice and water clouds have a comparably high reflectance in the SWIR [4] and can thus be discriminated from snow.

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Table 13.1 Spectral band ranges of the freely-available sensors that are used for glacier mapping Band Number Landsat Sentinel-2 Band TM ETM OLI MSI AST TM ETM+ OLI MSI Blue 1 1 2 2 0.45-0.52 0.45-0.52 0.45-0.51 0.46-0.52 Green 2 2 3 3 1 0.52-0.60 0.53-0.61 0.53-0.60 0.54-0.58 Red 3 3 4 4 2 0.63-0.69 0.63-0.69 0.63-0.68 0.65-0.68 NIR 4 4 5 8 3 0.76-0.90 0.76-0.90 0.85-0.89 0.78-0.90 SWIR 5 5 6 11 4 1.55-1.75 1.55-1.75 1.56-1.66 1.57-1.66 SWIR 7 7 7 12 5-9 2.08-2.35 2.09-2.35 2.10-2.30 2.10-2.28 Pan 8 8 0.52-0.90 0.50-0.68 -

Terra ASTER 0.52-0.60 0.63-0.69 0.76-0.86 1.60-1.70 a 2.15-2.43 -

The spatial resolution (in m) is colour-coded: 10, 15, 20, 30. NIR: near infrared; SWIR: shortwave infrared; Pan: panchromatic a The sum of ﬁve bands Sources http://geo.arc.nasa.gov/sge/health/sensor/cfsensor.html, https://www.usgs.gov/faqs/whatare-band-designations-landsat-satellites

Fig. 13.1 Modelled spectral reflectance curves of snow with three different grain sizes, and position of TM spectral bands. (Data from the ASTER spectral library at JPL.)

The separation of different surface types is improved by analyzing false-colour composites from the VIS, NIR and SWIR bands (shown in Fig. 13.2). As the grey values across these bands are largely uncorrelated, RGB composite images made by combining these bands are colourful (they show clouds in white, and ice and snow

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Fig. 13.2 Visualisation of reflectance values in a 9.5 km by 9 km sub-area of a Landsat TM scene covering Oberaargletscher (just below the centre of each image) in the Swiss Alps. Left: TM band 3. Centre: TM band 4. Right: TM band 5

in cyan). The TM band 5 image in Fig. 13.2 (right), whose contrast has been enhanced, shows how reflectance increases in this band as snow grain size decreases towards higher elevations (dark grey region to the middle left (of this panel)). The heavily debris-covered tongue of Unteraargletscher (near top of image) displays the same spectral characteristics as the surrounding terrain (bare rock), so it can be mapped correctly only when manual editing by an experienced analyst is applied (Sect. 13.2.2). The spectral bands described above for the Landsat TM sensor are available on several other satellites (e.g., Landsat 7 ETM+, Landsat 8 OLI, Terra ASTER, Sentinel-2 MSI) at spatial resolutions of 10–30 m (Table 13.1). Besides resolution, the choice of a sensor for a speciﬁc purpose has to consider spectral characteristics, data availability (which may be hampered by clouds), how the swath covers the area of interest, and pre-processing of the raw data [5]. For glaciological studies, data from the Landsat TM and ETM+ sensors have been the most intensively used. They are available free of charge, cover a wide swath (180 km), have the longest archive on record (since 1984 for TM), can be downloaded easily (e.g., from https:// earthexplorer.usgs.gov) in a common format (geotif), and have been orthorectiﬁed1 with a digital elevation model (DEM). This suits them especially well to large-scale studies, time-series analyses and change assessment, including those interpreting glacier variations as indicators of climate change. Since 2013 and 2015, Landsat 8 and Sentinel-2 data have also been freely available, offering enhanced spectral capabilities (more bands, 12-bit radiometric resolution) and, for Sentinel-2 data, also increased spatial resolution (10 m) and swath width (290 km).

1

Orthorectiﬁcation is the procedure of correcting image distortions (shift of pixels) resulting from sensor tilt and/or terrain elevation. It yields a planimetric image with a constant scale in the chosen map projection.

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13.2.2 Applications A key application of Landsat data is mapping glacier extents and how they change through time. The related automated methods were developed in the 1990s [6], and comparisons of different algorithms [7, 8] found that the simplest method―a TM4/TM5 band ratio using the raw DN and a threshold―yielded the most accurate results. Additionally, a TM3/TM5 ratio and the normalized difference snow index (NDSI) are widely used and as accurate [9]. With the ratio method, the strong contrast in spectral reflectance of ice and snow in the VIS or NIR compared to the SWIR (see Fig. 13.2) is utilized to identify glacier surfaces. Dividing the DNs in the VIS or NIR by the DNs in the SWIR yields high ratio values for ice and snow and very low ratios for all other materials (Fig. 13.3, left). By imposing a threshold, the ratio image is transformed to a binary glacier map (Fig. 13.3, centre), which is converted to glacier outlines with raster-vector conversion (Fig. 13.3, right). In this process, it is necessary to manually correct the debris or cloud-covered glacier parts and to exclude seasonal snow (e.g., [5]). The corrected outlines can be further divided into glacier entities using drainage divides derived by watershed analysis from a DEM, and topographic parameters can be obtained for each entity using statistics found from the DEM for those zones deﬁned by the outlines [8]. Thus, topographic information for detailed glacier inventories can be created automatically for inclusion in the Global Land Ice Measurements from Space (GLIMS) database (www.glims.org). In cases where the satellite images are accurately orthorectiﬁed and a DEM of sufﬁcient accuracy is available, the raw DNs of the individual bands can be converted to top of atmosphere reflectance (e.g., [10]), and further application of topographic and atmospheric corrections gives surface reflectance. This allows one to distinguish different glacier zones (or facies) spectrally, to discriminate ice from snow for mapping the snow-covered area at the end of the ablation season as a proxy for the equilibrium-line altitude, and to calculate glacier albedo using

Fig. 13.3 Key steps of glacier mapping. Left: TM4/TM5 band ratio. Centre: glacier map after a threshold of 1.9 is applied (grey); red pixels were removed and blue pixels added by the median ﬁlter. Right: glacier outlines derived from raster-vector conversion after manual correction of debris cover (added) and lakes (removed)

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Fig. 13.4 Elevation changes between 1985 and 1999 across glaciers in the Swiss Alps derived from subtracting the national DEM from the SRTM DEM

empirical conversion factors. The resulting albedo information can be used to improve distributed mass-balance modelling [11]. Imagery from optical stereo sensors such as ASTER are used to derive DEMs from ‘parallax matching’, i.e., the shifts in pixel position due to the terrain height are converted to elevation [12]. Such DEMs are widely used for calculating glacier elevation changes by subtracting DEMs from at least two points in time (e.g., [13]). On time scales of approximately a decade, this method yields overall glacier volume changes (e.g., Fig. 13.4) that complement ﬁeld-based measurements of glacier mass balance [14]. Important issues to consider are careful co-registration of the DEMs [15] and the proper handling of data voids and artefacts. The DEM from the Shuttle Radar Topography Mission (SRTM) typically serves as a base for glacier surface topography in the year 2000, and numerous studies have used it in combination with earlier (mostly national) or later DEMs to estimate glacier volume changes and their contribution to sea-level rise. New DEMs that were recently made available and cover large regions such as the ArcticDEM (north of 60° N) or the TanDEM-X DEM (globally) will be increasingly used in the future for glaciological applications and calculation of glacier volume changes. Image matching techniques can be applied to orthorectiﬁed satellite scenes to derive ice motion vectors or surface flow ﬁelds. When two images from different times have been co-registered accurately, their cross-correlation within a matching window of given size can be calculated and used to identify corresponding pixels in the two images and convert their displacement into a flow vector [16]. The method is preferably used with higher resolution sensors such as the 15 m ASTER and Landsat pan bands (e.g., [17–19]). Typically, images separated by about one year are used to give displacements for glacier surface velocities of *10–200 m/year. For faster flow and the now available 10 m resolution Sentinel-2 images, the temporal difference of the image pairs could be much shorter, down to a few days or

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weeks [12]. Whereas feature tracking with optical sensors is preferably used with summer images, microwave sensors are mainly used during winter (Sect. 13.3.3) and can track surface displacements also during the polar night [20].

13.3

SAR Methods and Applications

Imaging radar sensors (synthetic aperture radar or ‘SAR’) are widely used for mapping the surface topography and surface motion of glaciers and ice sheets. Some of their other applications include the monitoring of fronts of glaciers and ice shelves, and mapping of diagenetic glacier facies and melt area extent. In addition, low-resolution active microwave sensors (scatterometers) deliver maps of daily melt extent on ice sheets; they are complementary to satellite-borne microwave radiometers.

13.3.1 Radar Signal Interaction with Snow and Ice Radar waves incident on a snow or ice medium are subject to scattering at the surface and in the volume (at snow grains, ice lenses, air bubbles in ice, internal interfaces) and to absorption along the propagation path. The depth of the snow and ice layer contributing to the observed backscatter signal depends on the dielectric and structural properties of the medium and on the sensor wavelength. Using the radiative transfer approach, the one-way penetration depth dp (the depth where the intensity of the signal is attenuated to 37% of the incident signal) can be approximated by dp ¼

1 ; ks þ ka

ð13:1Þ

pﬃﬃﬃ where ks is the volume scattering coefﬁcient, ka ¼ 2pe00 =ðk0 e0 Þ is the volume absorption coefﬁcient, k0 is the wavelength in free space, and e = e′ − ie″ is the complex permittivity (see Chap. 14). In dry snow, ks is approximately proportional to the third power of the effective grain size and inverse to the third power of k. The imaginary part of the permittivity of snow, e″, shows a strong dependence on the liquid water content. Dielectric losses in dry snow are small, and dp is in the order of several hundred wavelengths―about 20 m in ﬁne-grained snow for C-band SAR (5 GHz) (Fig. 13.5, left). In wet snow, the absorption losses (ka) dominate, resulting in low backscatter coefﬁcients, and dp is of the order of one wavelength, due to the high dielectric loss of water. In C-band, for example, the depth dp for snow containing 5% by volume of liquid water is only 3 cm (Fig. 13.5, right). Consequently, for wet snow the microwave signal reflected or emitted from a melting snow pack originates from the snow surface and a (very)

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Fig. 13.5 Microwave one-way penetration depth in snow and ﬁrn (metres). Left: dp in dry snow and ﬁrn. Triangles—the Alps [22]; circles and rectangle—dry snow zone, Antarctica [23]. Right: dp in wet snow and its dependence on liquid water content (Vw, % by volume) for C-band (5.3 GHz) and X-band (9.6 GHz)

shallow subsurface layer. This signal is exploited for mapping melting snow areas on land surfaces and glaciers by means of C-band and X-band SAR [21]. In dry snow, the volume down to several metres depth contributes to the observed radar backscatter signal in C-band and X-band. On glacier ice, the backscatter signal of the rough ice surface dominates over the volume contribution. The surface topography retrieved by means of SAR interferometry (InSAR) refers to the position of the scattering phase centre in the snow or ﬁrn volume. This causes a difference, Dh, between the elevation measured by InSAR and the true surface elevation. Dh 0.5 dp if dp is small compared to the height of ambiguity, Ha (see deﬁnition in Sect. 13.3.3).

13.3.2 SAR Sensor and Image Characteristics Synthetic aperture radar (SAR) sensors are able to image the Earth’s surface at high spatial resolution. They generate their own illumination. Current satellite-borne SARs have phased-array antennas collecting image data at many different conﬁgurations in terms of illuminated swath (strip map mode, ScanSAR, spotlight SAR), polarization and incidence angle. TerraSAR-X, developed and operated in partnership by the German Aerospace Center (DLR) and industry, for example, offers about 100 different operation modes in order to match the optimum requirements for many applications [24]. Figure 13.6 shows the imaging geometry of a SAR sensor operating in strip map mode. The radar antenna illuminates a swath sideways along the satellite track. The swath width across track is constrained by the pulse repetition frequency and other system parameters. For TerraSAR-X strip map mode, the swath width is about

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Fig. 13.6 Left: SAR imaging geometry; rr—slant range resolution; rg—ground range resolution. Right: section of TerraSAR-X image of Hectoria-Green-Evans (HGE) glaciers, Larsen-B embayment, Antarctic Peninsula, 24 March, 2008. Arrow indicates radar illumination

30 km; for Sentinel-1 it is about 80 km. Larger swath width can be achieved by ScanSAR operation mode at the expense of spatial resolution. The SAR transmits short pulses of high energy at the radar frequency and records the signal reflected from the Earth’s surface as a time series; the travel time depends on the distance between the antenna and the illuminated object. The spatial resolution rr in slant range (line of sight of the radar beam, LOS) depends on the pulse duration Dt. The resolution across track on the surface rg (the ground range resolution) varies with the local incidence angle, hi. The synthetic array approach of SAR is based on the coherent combination of multiple received signals along the flight path, enabling the formation of a virtual aperture that is much longer than the physical antenna length. This approach provides high spatial resolution along the flight track, ra, that is independent of the distance to the observed object. The key equations are: rr ¼ cDt=2;

rg ¼ rr = sin hi ; ra ¼ L=2;

ð13:2Þ

where c is the speed of light and L is the antenna length. Technically, short pulse duration is achieved by modulation of the carrier frequency over a certain bandwidth. For example, TerraSAR-X in strip map mode has a nominal transmit bandwidth of 150 MHz, which results in rr = 1.0 m. Because a radar image line across track is formed according to the arrival time of the received signal (the distance between antenna and object), the image in radar geometry is distorted on slopes. This is evident in Fig. 13.6 on the nunataks

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between the glacier streams. Slopes facing towards the radar antenna (fore-slopes) appear shortened with bright return signal, whereas slopes facing away from the radar (back-slopes) appear elongated. Geometric rectiﬁcation, using precise orbit information and digital elevation data, is needed to transform the radar images to map projection. Another characteristic feature of SAR images is multiplicative noise (speckle). The SAR sensor records the complex return signal, characterized by amplitude and phase of the incoming wave. The complex signal is needed for aperture synthesis and for interferometric processing. For illuminated objects with many different scattering elements, such as snow and ice, the backscatter contributions of the individual scattering elements are added coherently. This causes the signal intensity to follow an exponential frequency distribution with the standard deviation corresponding to the mean value of intensity (speckle). For SAR applications based on intensity (e.g., mapping of snow melt area), low pass ﬁltering is required to reduce speckle. If the physical properties of the observed object do not change, the speckle pattern is preserved in repeat-pass SAR images (the signal in the image pair is coherent) so that interferograms can be produced. Table 13.2 lists several satellite-borne SAR systems whose data have been widely applied for radar interferometry and glaciological applications. The ERS satellites, in particular the tandem operation of ERS-1 and ERS-2, acquired an important InSAR data set for glacier research. From April 1995 to early 2000 both satellites orbited in 35-day repeat cycle, with an orbital time lag adjusted for imaging the same swath exactly with one day time difference. The Shuttle Radar Topography Mission (SRTM) delivered a complete topographic data set between 60° N and 56° S, applying single-pass SAR interferometry with active (transmit/receive) C-band and X-band SARs accommodated in the shuttle bay and passive antennas on a 60 m mast [25]. The SAR sensors on ERS, Radarsat, Envisat and Sentinel-1 operate at C-band frequencies, the PALSAR on ALOS at L-band, and COSMO-Skymed and TerraSAR-X at X-band. The Sentinel-1 mission, comprising two identical SAR satellites, is part of an ongoing European Sentinel satellite constellation series dedicated to operational observations of main Earth-system components. The interferometric wide swath mode, which is the default operation mode over land surfaces and ice sheets, offers excellent capabilities for comprehensive and long-term observations of the cryosphere [26].

13.3.3 InSAR Measurement Principles and Applications Repeat-pass SAR images, acquired over a certain time span depending on the orbit repeat cycle, are used to map glacier motion (by means of interferometry or offset tracking) and surface topography by means of interferometric processing. Signal coherence is a basic requirement for generating an interferogram. Temporal decorrelation (loss of coherence) is a major limiting factor for applying repeat-pass SAR interferometry over glaciers. It can be caused by changes in the phase of the

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Table 13.2 Speciﬁcations of selected satellite-borne SAR systems Satellite, sensor

Frequency (GHz)

Swath width (km)

Spatial resolution (m)

Repeat orbit (days)

Operation period

ERS-1/-2 SAR

5.3

100

25

35

ERS-1/-2 Tandem

5.3

100

25

1

SRTM, C-SAR

5.3

225

Single-pass

STRM, X-SAR Radarsat-1, SAR Radarsat-2, SAR Envisat, ASAR ALOS, PALSAR ALOS-2, PALSAR-2 COSMO-Skymed, SAR

9.6 5.3 5.4 5.3 1.27 1.26

50 50–500 10–500 70–400 40–350 25–500

30, 90 (DEM) 25 (DEM) 10–100 3–100 25–150 10–100 3–100

1991–2000/1995– 2011 April 1995–Jan 2000 11–22 Feb 2000

Single-pass 24 24 35 46 14

11–22 Feb 2000 1995–2013 2007– 2002–2012 2006–2011 2014–

9.6

10–200

1–30

16 (1 satellite)

TerraSAR-X TanDEM-X

9.6 9.6

10–100 10–100

1–16 1–16

Sentinel-1A/B

5.3

11 11, single-pass with TerraSAR-X 12/6

2007–; constellation of four satellites 2007– 2009–; in formation with TerraSAR-X 2014–/2016–

80, 250, 5, 20, 40 400 Different numbers for swath width and resolution refer to different SAR operation modes

reflected radar signal due to snowfall, wind drift, surface melt or surface deformation at sub-pixel scale. Single-pass InSAR conﬁgurations, such as the TerraSAR-X/TanDEM-X conﬁguration [24] and the Shuttle Radar Topography Mission (SRTM) [25], are therefore the preferred choice for producing DEMs. The TanDEM-X mission (TerraSAR-X add-on for Digital Elevation Measurements) of DLR, launched in June 2010, operates in close formation (helix) with the TerraSAR-X satellite, forming a single-pass interferometric SAR system. The main objective was the generation of a global DEM with 12 12 m2 horizontal sampling and 2 m vertical accuracy. An interferogram is generated from two complex, precisely co-registered SAR images that have been acquired from nearby antenna positions in space [27]. The complex backscatter signals of the two images V1, V2, are composed of an amplitude and phase component: V1 ¼ jV1 j expðiw1 Þ;

V2 ¼ jV2 j expðiw2 Þ:

ð13:3Þ

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Fig. 13.7 Left: geometry of across-track SAR interferometry. S1, S2: sensor positions; Bn: perpendicular baseline. Inset: simulated interferogram (topographic phase) of a mountain cone (500 m high, 5 km diameter) with C-band SAR parameters; Bn = 100 m. Each grey cycle corresponds to a phase shift of 2p (one fringe). Right: InSAR geometry for displacement measurements (assuming Bn = 0). DR: shift in LOS, corresponding to horizontal displacement across track Dy = DR/sinh, or to vertical displacement Dz = −DR/cosh

The multiplication yields the complex interferogram V ¼ V1 V2 ¼ jV1 jjV2 j exp½iðw1 w2 Þ;

ð13:4Þ

where * denotes the complex conjugate. If the scattering characteristics are the same for both image acquisitions, the phase difference between the two SAR images, also called the interferometric phase D/, is related to the difference in the propagation path of the two radar beams (Fig. 13.7): D/ ¼ w2 w1 ¼

4p ðR2 R1 Þ: k

ð13:5Þ

A repeat-pass interferogram is computed from two SAR images taken at different dates. The interferometric phase for repeat-pass interferometry includes the following contributions: D/ ¼ D/flat þ D/topo þ D/dis þ D/atm ;

ð13:6Þ

where D/flat and D/topo are the phase differences due to changes in the satellite-to-target distance for flat Earth and topography. D/flat can be computed from precise satellite orbit data and subtracted from the interferogram. D/dis is the

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phase difference due to displacement of the observed surface element in slant range (LOS). D/atm is the phase difference due to temporal changes in atmospheric propagation conditions, related mainly to the water vapour content of the atmosphere [27]. D/atm is usually neglected for InSAR analysis in cold regions because the water vapour content is small. This leaves D/topo and D/dis as unknowns in (13.6). This ambiguity can be resolved by differential processing of two interferograms with different baselines if the velocities over the time spans of the two interferograms are the same (DInSAR). Another option is to subtract a synthetic topographic interferogram, computed by means of a DEM and precise orbit data. In single-pass interferometry, which is the optimal conﬁguration for topographic mapping, the phase change across an image is only related to D/topo. Topographic mapping: for a dual-pass system, the phase difference due to an elevation change, Dz, is given by D/topo ¼ Dz

4p Bn : k R sin h

ð13:7Þ

Bn is the perpendicular baseline and h is the radar incidence angle. For a single-pass system with a single transmit antenna and two receive antennas, the term 4p is replaced by 2p. The height sensitivity of an interferogram can be speciﬁed by the height of ambiguity: Ha ¼

k R sin h : 2 Bn

ð13:8Þ

Ha is the height difference corresponding to a phase shift of 2p (one fringe). As Ha is inversely proportional to the perpendicular baseline, long baselines are more sensitive to topography. However, baseline length is subject to theoretical and practical limits, because long baselines make it difﬁcult to separate individual fringes and perform phase unwrapping [27]. Single-pass interferometry has great potential for determining volume changes and deriving glacier mass balance. This technique is not impaired by temporal decorrelation of the interferometric phase, variations in atmospheric propagation conditions, cloudiness, and variable illumination. Effects of SAR signal penetration (Sect. 13.3.1) during the determination of surface change can be largely eliminated if InSAR data are acquired at the same time of the year (to ensure similar penetration conditions). Figure 13.8 shows a DEM of the Drygalski Glacier region on Antarctic Peninsula, posted on a 6 m horizontal grid, derived from single-pass SAR images acquired by the TanDEM-X/TerraSAR-X formation. The difference of two DEMs, acquired over a time span of about two years, provides a detailed map of surface elevation change that can be converted into glacier mass balance. Two decades after collapse of Larsen-A Ice Shelf in January 1995, Drygalski Glacier is still experiencing signiﬁcant dynamic thinning.

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Fig. 13.8 Left: TanDEM-X DEM of the Drygalski Glacier region, Antarctic Peninsula, 21 April, 2013. The elevation scale refers to WGS-84. Right: map of surface elevation change dh/dt (m/year), derived from TanDEM-X data of 6 September, 2011 to 21 April, 2013. Background: Landsat image of 31 December, 2001. Figure redrawn from Rott et al. [28]

Retrieving surface motion: the displacement during the time interval of two repeat-pass SAR image acquisitions introduces a phase shift, D/dis, in LOS (Fig. 13.7) equal to D/dis ¼

4p DR: k

ð13:9Þ

DR is the component of the three-dimensional displacement vector projected onto the LOS of the radar beam. A phase shift of 2p corresponds to displacement of half a wavelength (D//2p DR/(k/2)). The phase measurement is relative. In order to obtain an absolute value for the motion, the measured phase has to be related to a spot of zero or known velocity in the image. Information about the orientation of the motion vector on the surface is needed to obtain the magnitude of velocity. Combining interferometric data from ascending and descending satellite orbits helps, but still does not provide the full 3D solution. On glaciers the surface parallel flow is often assumed to infer the 3D motion [29], but this only works well if subsidence and emergence are small compared to the surface-parallel flow components. Offset tracking with SAR images of crossing orbits delivers four different projections of the velocity vector, so that maps of full 3D ice motion can be derived [30]. Figure 13.9 shows a one-day repeat-pass interferogram of the ERS Tandem mission over the Skeiðarárjökull outlet glacier, Vatnajökull, Iceland [31]. One fringe (colour cycle) corresponds to a displacement of 2.8 cm (k/2) in LOS.

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Fig. 13.9 Left: ERS-1/ERS-2 tandem interferogram of Skeiðarárjökull outlet glacier of Vatnajökull, Iceland, 27–28 March 1996. The motion-related phase D/dis is shown. Right: derived map of surface velocity. The blue line indicates the flood path of the water outbreaks of the subglacial lake Grímsvötn [31]

This interferogram, acquired 9 days before the flood discharge peak of a jökulhlaup (Sect. 3.3.3), shows an increase of velocity up to three-fold relative to InSAR observations several weeks earlier. Offset tracking techniques deliver two components of the velocity vector (in LOS and along track) from repeat-pass data of single swath. They are less sensitive to displacement than InSAR and reveal less spatial detail because larger templates are needed to yield good correlation peaks. InSAR enables measuring displacements with the precision of millimetres, as fractions of a 2p phase cycle can be detected. As explained below, there are various options for the image correlation used in offset tracking, depending on whether the image pair is coherent or not (Table 13.3). Ice-sheet wide maps of surface velocity, such as the International Polar Year ice velocity map of Antarctica, are assembled from SAR data of different sensors by applying offset tracking and InSAR techniques [32]. Image cross-correlation methods can measure shifts at fractions of one pixel. The accuracy of velocity measurements can be improved by using SAR data over longer time spans if the features are stable. For applying cross-correlation of complex data (“speckle tracking” or “coherence tracking”) in areas without distinct amplitude features, the signal needs to be coherent. However, phase unwrapping is not needed, so decorrelation gaps can be bridged. Amplitude cross-correlation requires stable features and therefore often fails in accumulation zones. It can be applied also in the case of complete absence of coherence. For example, Fig. 13.10 shows an ice motion map of the terminus of Crane Glacier, Antarctic Peninsula, produced by matching TerraSAR-X amplitude images

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Table 13.3 Techniques for retrieving ice motion by using repeat-pass SAR data Method

Signal requirements

Sensitivity to displacement

Special characteristics

InSAR

Coherence

Cross-correlation of complex data or coherence images (speckle-tracking, coherence matching) Cross-correlation of amplitude or intensity images (feature tracking)

Coherence, but less sensitive than DInSAR

High (fractions of one wavelength, mm) Fractions of one pixel (decimetre to *1 m, depending on spatial resolution)

Motion in LOS; limitations due to decorrelation Two components of the velocity vector; does not require amplitude features

Fractions of one pixel (decimetre to *1 m, depending on spatial resolution)

Two components of the velocity vector; limited to areas of amplitude features

Conservative amplitude features

Fig. 13.10 Left: Ice velocity (magnitude) on Crane Glacier, Larsen-B embayment, from TerraSAR X images 2009-10-27 and 2009-11-07. Red line: flux gate. Right: surface velocity across the flux gate in 1995 and 1999 (ERS InSAR) and from 2007 to 2012 (TerraSAR-X offset tracking)

in strip map mode with pixel size 1.95 m 0.90 m (azimuth slant range). Image templates of 64 64 pixels size and a sampling step of 10 pixels were used. The accuracy of the derived displacement is of the order of 0.2 to 0.3 pixels, resulting in an uncertainty in surface speed of ±0.05 m/day for repeat-pass images spanning 11 days. Ice motion derived from SAR images and ice thickness data at the flux gates from radar soundings and bathymetric data were the basis for computing the temporal evolution of calving fluxes of glaciers after collapse of the Larsen-B Ice Shelf [33].

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343

Satellite Altimetry

Measuring volume changes with repeat observations by altimeters and converting them to mass changes by using the density of ice and estimates for the density change of the vertical snow/ﬁrn/ice column is a key method of deriving glacier and ice-sheet mass balance (see Sect. 7.3.2). Repeat mapping of glacier surfaces by airborne laser scanner has been used since the 1990s in many regions, including on the outlet glaciers of ice sheets. Whereas airborne measurements deliver surface topography and volume change at high spatial resolution over limited areas, satellite-borne altimeters provide regular coverage of the global ice surfaces, though with lower sampling density.

13.4.1 Altimetry Missions Satellite-borne altimeters emit short radar or laser pulses in nadir or near-nadir direction and detect the signal reflected from the surface. The surface height is calculated from the signal travel time, precise knowledge of the satellite position in space, and the signal propagation conditions. Most altimeter missions making ice-sheet observations have employed radar sensors (radar altimeter, RA) operating in the Ku-band (Table 13.4). With the exception of Cryosat, these sensors were mainly developed for oceanographic applications. S- and C-band channels help to

Table 13.4 Main altimetry missions for ice sheet observations Satellite Seasat Geosat ERS-1 ERS-2 Envisat Cryosat-2 Sentinel-3

Sensor

Ø (km)

Orbit inclination (degree)

Time period (MM/YYYY)

RA—Ku 1.7 108 07/1978–10/1978 RA—Ku 2.0 108 04/1985–11/1988 RA—Ku 1.7 98 07/1991–03/2000 RA—Ku 1.7 98 08/1995–06/2003 RA-2—Ku and S 1.7 98 04/2002–04/2012 92 10/2010– RA, SIRAL—Ku 0.3 1.5a RA, SRAL—Ku 0.8 98 02/2016– and C ICESat-1 LA, GLAS 0.07 94 02/2003–10/2009b c ICESat-2 LA, ATLAS 0.01 94 09/2018– RA: radar altimeter; Ku-band: 13.6 GHz; S-band: 3.2 GHz; C-band: 5.4 GHz. Ø: diameter of resolution cell. LA: laser altimeter. Time period refers to datasets over ice sheets a Includes SAR mode b 18 operation periods of 5 weeks average duration c Multiple beams

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improve ionospheric corrections. The RA antenna illuminates a surface footprint (the beam-limited footprint) several kilometres wide. Processing techniques involving frequency-modulated pulses are used to reduce the diameter of the resolution cell (the pulse-limited footprint) [34]. RA data sets of Seasat and Geosat provide information on elevation change of ice sheets up to 72° N and S latitudes in 1978 and 1985–1988, respectively. For 1991– 2012, a continuous and homogeneous time series of RA elevation measurements over ice sheets up to 82° latitudes is available from the ESA ERS-1, ERS-2, and Envisat missions [35]. Cryosat, launched in April 2010 and carrying the SAR/Interferometric Radar Altimeter (SIRAL), provides better coverage over ice sheets and glaciers than previous RA Missions because of (i) the satellite orbit reaching to 88° latitude and (ii) the SAR interferometer (SARIn) operation mode, which utilizes high pulse-repetition frequency and the Doppler effect in the forwardand aft-looking parts of the beam to reduce the cell size on sloping terrain [36]. Laser altimeters (LAs), operating at 532 nm and 1064 nm wavelengths, have smaller footprint size than RAs and better performance on sloping terrain. Furthermore, the laser signal is reflected from the skin surface layer of snow and ice, whereas the radar signal penetrates into the volume. The total operation period of the Geoscience Laser Altimeter System (GLAS) instrument aboard the NASA Ice, Cloud, and land Elevation Satellite (ICESat-1) has been rather limited, due to constraints in the laser life time.

13.4.2 Measuring Elevation Change The range, R, from the satellite to the reflecting surface is equal to half the time difference between reception (t = t2) and transmission (t1) of the radar pulse multiplied by the propagation speed c: Z

t2

R ¼ 0:5

c dt:

ð13:10Þ

t1

c varies along the path and depends on the atmospheric refractivity. The total atmospheric refractivity includes contributions from dry gases, water vapour, liquid water and ionospheric free electrons. Gridded meteorological data are used for the tropospheric corrections, which are required for both RA and LA sensors. For RA, ionospheric effects also need to be corrected, using Global Ionosphere Maps of total electron content derived from GPS data. Some RA sensors include additional channels (S-band or C-band) to improve the ionospheric corrections. The height with respect to the reference ellipsoid is obtained by subtracting the measured range from the satellite position [34]. For RA measurements, correcting for penetration of the radar signal into the snow pack is necessary. The range correction accounts for the relative contributions of the surface and volume [37]. The waveform shape (Fig. 13.11), representing the time sequence of the received signal, is used to derive the mean surface elevation of

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Fig. 13.11 Backscattered power received by a pulse-limited radar altimeter and its dependence on time (altimetric waveforms) for returns from glacier ice (black; dominated by surface echo) and from dry snow (blue; dominated by volume echo). The left inset shows the leading edge of the pulse arriving at the surface; cs is the pulse length. The retracking point refers to the retrieved average surface elevation of the footprint

an RA cell, through a procedure known as retracking. Another issue is the position of the retrieved surface point within the RA beam on sloping surfaces. Conventional RAs have an antenna beam width of 1° to 2°, corresponding to a footprint diameter on the Earth’s surface of *10–20 km (dependent upon the orbital altitude). The altimeter measures the range between the antenna and the nearest point on the surface that is shifted in the upslope direction when the surface is inclined. The correction procedure relocates the impact point in the upslope direction by means of sequential processing and use of a priori topography estimates. Nevertheless, good accuracy with conventional RA is achievable only on surfaces with slopes < ca. 1°. With the SARIn mode of Cryosat, the application can be extended to steeper slopes. For LAs, antenna pointing is another source of errors for sloping terrains, but the slope-induced error is less than for RAs [38]. Time series of altimeter measurements allow surface elevation change (dh/dt) to be derived for use in mass-balance estimations. In order to minimize the uncertainty, dh/dt is determined at crossing points of the satellite ground tracks. Correcting for glacial isostatic adjustment (GIA; Chap. 15) over the time period is necessary; the impact of GIA here is smaller than for gravimetric estimates of mass balance (Sect. 7.3.3). Firn densiﬁcation models (Chap. 11) driven by climate data are used to convert volume change to mass change. Figure 13.12 shows two maps of elevation change derived in this way. The map for coastal West Antarctica on the left is based on ﬁltered, spatially-averaged (10-km radius) dh/dt data from ICESat, with a Radarsat map as background image [39]; the estimated uncertainty of the spatially-averaged dh/dt values is ± 0.07 m/year at the

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Fig. 13.12 Maps of surface elevation change, dh/dt. Left: rate of elevation change of coastal West Antarctica over the period February 2003 to November 2007 from ICESat data [39]; ﬁgure reprinted from Nature with permission from Springer Nature ©2009. Inset: high-resolution data over Pine Island Glacier (PIG) with satellite tracks. Right: map of elevation change of Greenland between January 2011 and January 2014 [36]. The CryoSat LRM mode and SARIn mode are used inside and outside the black polygon, respectively

1r level. The map shows strong dynamic thinning for the glaciers draining into the Amundsen Sea Embayment (ASE) and the Getz Ice Shelf (GIS). The map of elevation change for Greenland on the right, derived from three full CryoSat-2 cycles between January 2011 and January 2014, reveals volume change that is a factor of 2.5 higher than during the ICESat-1 period [36]. Over the flat interior of the ice sheet, the low-resolution mode (LRM) is used—this corresponds to a conventional pulse-limited radar altimeter that integrates the backscattered energy over the full beam width. The pulse-limited footprint diameter is about 2.3 km. Near the ice-sheet margin, the altimeter operated in the interferometric SARIn mode, which decreases the along-track resolution to 305 m (at the expense of higher sensor noise). The estimated uncertainty in dh/dt is 0.1 m/year in the interior and higher for steeper slopes along the margin.

References 1. Mason R, Lubin D (2006) Polar remote sensing. Vol. II: Ice sheets. Springer, Berlin 2. Rees WG (2006) Remote sensing of snow and ice. Taylor and Francis, Boca Raton 3. Krimmel RM, Meier MF (1975) Glacier applications of ERTS-1 images. J Glaciol 15:391–402 4. Dozier J (1989) Spectral signature of alpine snow cover from Landsat 5 TM. Remote Sens Environ 28:9–22

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5. Paul F and 24 others (2015) The glaciers climate change initiative: methods for creating glacier area, elevation change and velocity products. Remote Sens Environ 162:408–426 6. Hall DK, Ormsby JP, Bindschadler RA, Siddalingaiah H (1987) Characterization of snow and ice zones on glaciers using Landsat Thematic Mapper data. Ann Glaciol 9:104–108 7. Albert T (2002) Evaluation of remote sensing techniques for ice-area classiﬁcation applied to the tropical Quelccaya Ice Cap, Peru. Polar Geogr 26(3):210–226 8. Paul F, Kääb A, Maisch M, Kellenberger TW, Haeberli W (2002) The new remote-sensingderived Swiss glacier inventory: I. Methods. Ann Glaciol 34:355–361 9. Racoviteanu AE, Paul F, Raup B, Khalsa SJS, Armstrong R (2009) Challenges in glacier mapping from space: recommendations from the Global Land Ice Measurements from Space (GLIMS) initiative. Ann Glaciol 50(53):53–69 10. Hall DK, Chang ATC, Siddalingaiah H (1988) Reflectances of glaciers as calculated using Landsat 5 Thematic Mapper data. Remote Sens Environ 25:311–321 11. Naegeli K, Huss M (2017) Mass balance sensitivity of mountain glaciers to changes in bare-ice albedo. Ann Glaciol 58(75):119–129 12. Kääb A, Winsvold SH, Altena B, Nuth C, Nagler T, Wuite J (2016) Glacier remote sensing using Sentinel-2. Part I: Radiometric and geometric performance, and application to ice velocity. Remote Sens 8(7):598 13. Brun F, Berthier E, Wagnon P, Kääb A, Treichler D (2017) A spatially resolved estimate of High Mountain Asia glacier mass balances from 2000 to 2016. Nat Geosci 10(9):668–673 14. Zemp M and 14 others (2019) Global glacier mass changes and their contributions to sea-level rise from 1961 to 2016. Nature 568:382–386 15. Nuth C, Kääb A (2011) Co-registration and bias corrections of satellite elevation data sets for quantifying glacier thickness change. The Cryosphere 5:271–290 16. Kääb A, Vollmer M (2000) Surface geometry, thickness changes and flow ﬁelds on creeping mountain permafrost: automatic extraction by digital image analysis. Permafr Periglac Process 11(4):315–326 17. Kääb A, Lefauconnier B, Melvold K (2005) Flow ﬁeld of Kronebreen, Svalbard, using repeated Landsat 7 and ASTER data. Ann Glaciol 42:7–13 18. Scherler D, Strecker MR (2012) Large surface velocity fluctuations of Biafo Glacier, central Karakoram, at high spatial and temporal resolution from optical satellite images. J Glaciol 58 (209):569–580 19. Dehecq A, Gourmelen N, Trouve E (2015) Deriving large-scale glacier velocities from a complete satellite archive: application to the Pamir–Karakoram–Himalaya. Remote Sens Environ 162:55–66 20. Strozzi T, Paul F, Wiesmann A, Schellenberger T, Kääb A (2017) Circum-Arctic changes in the flow of glaciers and ice caps from satellite SAR data between the 1990s and 2017. Remote Sens 9(6):947 21. Nagler T, Rott H (2000) Retrieval of wet snow by means of multitemporal SAR data. IEEE Trans Geosci Remote Sens 38(2):754–765 22. Mätzler C (1987) Applications of the interaction of microwaves with the natural snow cover. Remote Sens Rev 2:259–387 23. Rott H, Sturm K, Miller H (1993) Active and passive microwave signatures of Antarctic ﬁrn by means of ﬁeld measurements and satellite data. Ann Glaciol 17:337–343 24. Krieger G and 18 others (2013) TanDEM-X: a radar interferometer with two formation flying satellites. Acta Astronaut 89:83–98 25. Farr TG and 17 others (2007) The shuttle radar topography mission. Rev Geophys 45: RG2004 26. Nagler T, Rott H, Hetzenecker M, Wuite J, Potin P (2015) The Sentinel-1 mission: new opportunities for ice sheet observations. Remote Sens 7(7):9371–9389 27. Hanssen RF (2001) Radar interferometry: data interpretation and error analysis. Kluwer, Dordrecht

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28. Rott H, Floricioiu D, Wuite J, Scheiblauer S, Nagler T, Kern M (2014) Mass changes of outlet glaciers along the Nordensjköld Coast, northern Antarctic Peninsula, based on TanDEM-X satellite measurements. Geophys Res Lett 41:8123–8129 29. Joughin I, Kwok R, Fahnestock MA (1998) Interferometric estimation of three-dimensional ice-flow using ascending and descending passes. IEEE Trans Geosc Rem Sens 36:25–37 30. Nagler T, Rott H, Hetzenecker M, Scharrer K, Magnússon E, Floricioiu D, Notarnicola C (2012) Retrieval of 3D glacier movement by means of high resolution X-band SAR data. In: Proceedings of the 2012 IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Munich, Germany, 22–27 July 2012, pp 3233–3236. https://doi.org/10.1109/ IGARSS.2012.6350735 31. Magnússon E, Rott H, Björnsson H, Pálsson F (2007) The impact of jökulhlaups on basal sliding observed by SAR interferometry on Vatnajökull. J Glaciol 35(181):232–240 32. Rignot E, Mouginot J, Scheuchl B (2011) Ice flow of the Antarctic Ice Sheet. Science 333 (6048):1427–1430 33. Wuite J, Rott H, Hetzenecker M, Floricioiu D, De Rydt J, Gudmundsson GH, Nagler T, Kern M (2015) Evolution of surface velocities and ice discharge of Larsen B outlet glaciers from 1995 to 2013. The Cryosphere 9:957–969 34. Rosmorduc V and 13 others (2018) In: Benveniste J, Picot N (eds) Radar altimetry tutorial. Issue 3a. http://www.altimetry.info/ﬁlestorage/Radar_Altimetry_Tutorial.pdf 35. Schröder L, Horwath M, Dietrich R, Helm V, van den Broeke MR, Ligtenberg SRM (2019) Four decades of Antarctic surface elevation changes from multi-mission satellite altimetry. The Cryosphere 13:427–449 36. Helm V, Humbert A, Miller H (2014) Elevation and elevation change of Greenland and Antarctica derived from CryoSat-2. The Cryosphere 8:1539–1559 37. Davis CH (1996) Temporal change in the extinction coefﬁcient of snow on the Greenland Ice Sheet from an analysis of Seasat and Geosat altimeter data. IEEE Geosc Rem Sens 34 (5):1066–1073 38. Brenner AC, DiMarzio JP, Zwally HJ (2007) Precision and accuracy of satellite radar and laser altimeter data over the continental ice sheets. IEEE Trans Geosci Remote Sens 45 (2):321–331 39. Pritchard HD, Arthern RJ, Vaughan DG, Edwards LA (2009) Extensive dynamic thinning on the margins of the Greenland and Antarctic ice sheets. Nature 461(7266):971–975

Geophysics

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14.1

Geophysical Methods: Overview

Geophysical methods have become a widely used tool in glaciology and other fields of cryospheric research. Their strength lies in the ability to provide information about the sub-surface without the need to access it directly. In this sense, geophysical methods can be considered a subgroup of remote-sensing techniques. They are relatively fast applications to carry out in the field and they allow the possibility of obtaining information quickly about the spatial distribution of diverse physical properties. This chapter introduces the basic principles of the main geophysical methods, how they work and how they are applied in glaciology. Passive methods are those which measure naturally occurring fields: gravimetry measures variations in the Earth’s gravity field; magnetics measures the Earth’s magnetic field; and seismology detects seismic waves of natural origin. In contrast, active methods employ some sort of controlled source: these include radar, an acronym (radio detection and ranging) referring to the propagation and reflection of electromagnetic (radio) waves; seismics, which refers to active source seismology; geoelectrics, which determines resistivity distribution; and electromagnetic induction, which measures induced currents.

O. Eisen (B) Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research, Bremerhaven, Germany e-mail: [email protected] Department of Geosciences, University of Bremen, Bremen, Germany © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_14

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In this chapter we focus on active seismic and radar methods, which are the most widely used geophysical methods in glaciology. Active geoelectrics and eminduction are widely applied in periglacial and sea-ice research, respectively, but are not considered.

14.2

Passive Methods

14.2.1 Gravimetry Gravimetric methods determine the variation of gravitational acceleration as a function of space and time. From these variations, as measured with gravimeters (in most cases essentially sophisticated spring balances), one aims to identify variations in the density and shape of the underground material or of the Earth in general. Because gravimetric methods sense an existing field which is not artificially generated, they are classed as a passive approach. Moreover, as the relevant mathematical description involves the use of a gravitational potential, it belongs to a class of potential methods which also includes magnetics. In Sect. 14.7.2, we derive the fundamental equations. With the development of techniques for quasi-continuous gravimetric observations from satellites (e. g., CHAMP, GRACE, GOCE),1 gravimetric methods have become a basic tool for the determination of the mass-balance state of the world’s ice masses and their response to climate change (also see Sect. 7.3.3 and Fig. 7.16). On airborne platforms they are used to obtain information about subglacial geology and physical properties, especially spatial variation in geothermal heat flux. An example of a combined gravito-magnetic survey is shown in Figs. 14.1 and 14.2.

14.2.2 Magnetics Magnetics is the second passive potential field method in applied geophysics. It has much in common with gravimetry, so that several analogies between them can be exploited. In magnetics, the magnetic field and its spatio-temporal variations which are present in the environment are sensed by different kinds of magnetometers. Because of the non-magnetic nature of ice masses, this technique is only of indirect relevance to glaciological applications. We will therefore only treat this method briefly. Magnetics is of relevance for glaciology mostly in determining the geological properties of the subglacial substrate, e. g., the existence of elevated geothermal heat flux in volcanic areas. The big advantage of the two passive potential field methods, gravimetry and magnetics, is that they can be combined easily during surveys, as they do not influence each other.

1 The acronyms have the following meaning: CHAMP, Challenging Mini-satellite Payload; GRACE:

Gravity Recovery and Climate Experiment; GOCE: Gravity field and steady-state Ocean Circulation Explorer.

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Fig. 14.1 Results from a concurrent gravimetric and magnetic survey in Dronning Maud Land, Antarctica [1]. Gridded cell size is 15 km. Compilation of the free-air anomaly map, derived from gravimetry. Data were obtained from aero-geophysical surveys

Whereas in gravimetry, density is the relevant physical property of the subsurface deduced from acceleration measurements, the corresponding property in magnetics is the magnetisation deduced from the magnetic field. Relevant equations for the magnetostatic law for forces, as derived from the Maxwell equations and the Lorentz force, take a comparable form to those in gravimetry. One major difference is that density is a scalar property, whereas magnetisation is a vector quantity. That fact tremendously complicates the interpretation of magnetic field surveys, as not only do the magnitude of the magnetisation and the shape of the subsurface influence the measurements, but also the direction of magnetisation.

14.2.3 Seismology Seismology embodies the measurement of elastic (or acoustic) waves in continuous media. In the case of passive seismology, we only consider the application of seismic waves of natural origin. The physical principles of seismic wave propagation, which

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Fig. 14.2 As for Fig. 14.1: compilation of the magnetic anomaly map from the magnetic measurements

are identical for passive (natural) and active (man-made) seismic sources, will be treated in further detail in Sect. 14.3. Examples of the special use of seismology in glaciology include the detection of naturally occurring seismic waves from fractures (crevasse opening), motion (ice on ground, e. g., glacier basal stick-slip), calving events (ice on ice, ice on sea/lake bed), collapse (tunnels) and mass movements (ice avalanches). The measured distribution and characteristics of seismic wave sources are used to infer the local variation of friction at the ice/bed interface (sticky spots), assess the dynamic state of, for example, hanging glaciers, and quantify calving events at the terminus of ice shelves and tidewater glaciers. The first seismic events caused by glaciers were identified in the 1950s by Röthlisberger [2] and Crary [3]. The first dedicated microseismic survey on ice occurred on the Athabasca Glacier in Canada. The number of publications on cryoseismology (seismology related to ice in all forms) more than doubled from the mid 2000s to 2016 [4]. Nowadays, seismometers are routinely deployed on glaciers and ice streams for long periods of time, ranging from a single season to more than a year in some autonomous campaigns.

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Active Methods: Basics

The two main active methods used in glaciology, seismics and radar, both utilise waves as a tool. These are electromagnetic waves in the case of radar, and elastic waves in the case of seismics (depending on the context: the terms acoustic, elastic and seismic waves are commonly used synonymously). Since the general principles regarding wave properties, data processing and analysis are the same for radar and seismics, we cover them first before presenting the details of each method. Both methods employ a source at or above the glacier surface. The wave propagates downwards into the firn or ice and is reflected where physical properties and thus the impedance2 change (Fig. 14.3). The reflected wave travels back to the surface, where it is recorded with either a single receiving element (for radar) or multiple receiving elements (for seismics). From the analysis of the recorded wave’s travel time (also called ‘two-way travel time’), the characteristic properties of the medium can be inferred. A wave can be considered as a disturbance of the physical equilibrium of a medium. When disturbed, the medium produces a ‘restoring force’, and it is this which causes wave propagation. Waves are described by such properties as period, frequency, amplitude, phase, wavelength and pulse length. Physical properties of the medium, which can be deduced from observations, are wave speed, absorption and reflectivity, among others. The primary objective of field measurements is to determine the travel time of a certain waveform between the (active) source and a receiver, sometimes along different ray paths, such as a direct, reflected or refracted wave. Secondary objectives are to determine the wave properties and the way in which they change along the travel path, as evidenced for example by the shape of the waveform. Seismic waves can propagate within the medium (when they are called body waves) or along an interface (and are called surface waves). They come in two main types: P waves (pressure or primary), with particle motion parallel to the direction of propagation (so they are also called longitudinal waves), and S waves (shear or secondary), with particle motion perpendicular to the direction of propagation (so they are also called transverse waves). Electromagnetic waves are transverse waves, where instead of a particle the electric and magnetic field vectors E and H oscillate perpendicularly to the propagation direction. For the detailed mathematical description of seismic waves and electromagnetic waves, see Sects. 14.7.3 to 14.7.5. The propagation of waves can be described by Huyghens’ principle, which states that in the passage of a wave, each point on the wavefront acts as a source for further radial propagation; the resulting family of spheres (or circles in two dimensions) have an envelope which describes the further propagation of the wave front. For a simplistic

2 As

its name suggests, impedance is a measure of the resistance of the medium. In an electrical circuit, the resistance is the voltage divided by the current; in a uni-directional fluid flow such as in a river, we might define it as the hydraulic head divided by the velocity. Impedance generalises this idea to alternating currents where there is no net flow or current. Further detail is given in Sects. 14.3.1 and 14.7.

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measurement set-up

recorded signals

movement

TX air ice

RX

trace no.

surface transmitter pulse

crevasses

conduits

traveltime

internal layers

basal water layer bedrock

minimal offset

*

geophone spacing Δs spread s

seismic source

s/2

Fig. 14.3 Top: set-up of a common-offset radar survey. TX represents the transmitter, and RX the receiver. Traces on the right indicate the received signals schematically. Bottom: geometry and ray paths for a seismic set-up with one source (×) and 11 geophones ()

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understanding it is also possible to describe both seismic and electromagnetic waves using rays (see Yilmaz [5] for an in-depth treatment). Another useful approximation for deriving solutions for the wave equation is to use plane waves. Physically, these represent the propagation of waves from a point source at infinite distance.

14.3.1 Propagation Properties and Reflection Origin The propagation velocity or wave speed in a medium depends on the type of wave. For a seismic P wave and S wave, respectively, these speeds are vp =

k + 43 μ , ρ

vs =

μ , ρ

(14.1)

where k is the bulk modulus of elasticity, μ is the shear modulus, and ρ is the density. This explains why P waves are always faster than S waves, hence the term primary as opposed to secondary wave. For electromagnetic waves, firn and ice can be very well approximated by a socalled low-loss medium with wave speed c0 c= √ , ε

(14.2)

where ε is the real part of the ordinary relative permittivity of the medium and c0 is the velocity of light in a vacuum (see Sect. 14.7.5). Wave reflection, refraction and the ray paths for both wave types (seismics and radar) obey the same laws. A partial reflection of the wave’s energy occurs when the impedance Z of the medium changes across an interface. The reflection amplitude ratio and transmission amplitude ratio, a R and aT (usually called the reflection and transmission coefficients), are defined for a vertical angle of incidence by aR =

AR Z2 − Z1 AT 2Z 2 = , aT = = , AI Z2 + Z1 AI Z2 + Z1

(14.3)

where A I is the amplitude of the incident wave, A R,T are the amplitudes of the reflected (R) and transmitted (T ) waves, and the subscripts 1 and 2 indicate the medium in which the wave propagates first and second, respectively. The acoustic impedance √ Z s = ρv is appropriate for seismic waves and the dielectric impedance Z r = μ/ε for radar waves (here ε is the electric permittivity, and μ is the magnetic permeability). For arbitrary (non-vertical) angle of incidence and polarisations, the reflection coefficients are described by the Fresnel equations for radar waves [6] and the Zoeppritz equations for seismic waves [7]. Reflections of seismic and radar waves can have the same physical origin. Changes in seismic impedance can be caused by changes in density or in the elastic moduli

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of the medium. An important effect is the crystal orientation fabric (COF).3 For reflection of radar waves, changes in density ρ, conductivity σ and COF are the most important mechanisms. An increase in radar backscatter occurs even when only small amounts (a few percent) of liquid water are present in the ice, because the relative permittivity of water, ε ≈ 82, is so much larger than that of ice, ε ≈ 3.2. The reflection and refraction of waves at an interface obey Snell’s law: sin α = constant, v

(14.4)

where α is the angle of incidence of the incoming ray, that is, the angle between its direction of propagation and the normal to the interface. This also explains the conversion of seismic wave types from P to S waves during reflection and refraction: sin α1 sin α2 sin β1 sin β2 = = = , v p1 v p2 vs1 vs2

(14.5)

where α and β are the angles of incidence for P and S waves, respectively. As seismic wave speed generally increases with depth, then α increases, and the wave direction of propagation is refracted away from the (normal) vertical. In the firn column, which has a positive velocity gradient with depth because of the density increase, continuous refraction leads to what are known as diving waves. Furthermore, downward waves whose initial direction of propagation exceeds a certain critical angle from the vertical get refracted so much that they return to the surface as critically refracted waves. In contrast, radar wave speed in general decreases with depth because density, and thus permittivity, increases with depth. Radar waves are thus refracted towards the (normal) vertical at interfaces, and this leads to ray focussing. This effect increases the performance of radar systems and is called a geometric gain. When considering reflections (or backscatter) of waves, three approximations or assumptions are often made in applied geophysics: the wave propagates at normal incidence (i. e., the wave front is parallel and the ray path normal to the reflecting interface); the reflection originates at smooth interfaces as surface scatter; and no volume scatter occurs (see Chap. 13 on satellite remote sensing for further explanation).

14.3.2 Seismic System Set-Up Table 14.1 lists relevant physical properties and system parameters relating to seismic and radar measurements in glaciological studies. In terms of the system’s set-up,

3 Other terms are also used, such as LPO (lattice preferred orientation). An individual ice crystal has

considerable anisotropic properties [rheology, thermal conductance, seismic and electromagnetic wave speed (Table 14.1)]. The overall properties of the bulk medium depend on the orientation of the individual crystals within the bulk, e. g., whether they are randomly orientated (isotropic) or have a preferred (anisotropic) orientation.

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Table 14.1 Comparison of medium and system parameters of seismics and radar. Absorption and reflection coefficients are taken from Peters et al. [8] for seismics and Booth et al. [9] for radar Property

Seismics

Radar

Impedance properties Velocity in air Velocity in ice

v, ρ 340 m s−1 v p ≈ 3940 m s−1 vs ≈ 1850 m s−1 , Increasing ≥0

ε , σ 3 × 108 m s−1 1.68 × 108 m s−1

v for greater depth Mean gradient ∂v/∂z Anisotropy

Frequency f Sample interval dt Travel time (1000 m ice) Wavelength in ice Absorption (ice sheet) Reflection coefficients (internal layers) Reflection coefficient magnitude (base) Vertical resolution Horizontal resolution Sub-bed signals

Decreasing ≤0

≈ 0.01ε , vs − vs⊥ ≈ 120 m s−1 ≈ ε − ε⊥ 0.07vs (ε ≈ 3.1 − 3.2) −1 ≈ v p − v⊥ p ≈ 150 m s 0.03v p 10 Hz–10 kHz 1 MHz–10 GHz O(ms) O(ns) Sampling interval scaleable by factor of ∼106 O(0.5 s) O(10 µs) 1 m to 102 m 1 cm to 102 m O(10−4 m−1 ) O(10−3 m−1 ) −20 dB −40 to −80 dB

5 × 10−4 to 0.97 (−1 to −33 dB) Max(λ/4, bandwidth) First Fresnel zone Penetrates into bed No signals −0.7 to 0.6

until the end of the 20th century a key distinction of seismics from radar was that seismics operate with multiple receivers, called geophones, though there is only a single source. This means that multiple offsets between source and receiver are recorded simultaneously. The reason is that setting up a repeatable seismic source is comparatively difficult in contrast to radar, especially when explosives are used. As seismic waves are slow, with speeds of the order of km s−1 , simple electronics are sufficient to collect geophone data. Active seismic sources can be categorised as impulse or vibrating sources. Impulse sources create a short seismic impulse a few milliseconds long with a more or less undefined spectrum. In glaciology, explosives are mostly used; they are deployed in boreholes, at depths ranging from several metres to more than 50 m. The amount of explosives used varies from a few tens of grams to several tens of kilograms, depending on the distance between source and receivers and the target depth. A considerable disadvantage of explosives is that they involve large destructive forces, and the surroundings of the borehole are irreversibly altered with every charge, sometimes even destroyed. Explosive shots are therefore not repeatable. Other seismic impulse sources are hammers and drop weights. Although these allow experiments

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to be repeated since they are destruction-free, the characteristic of each hammer blow or weight drop is slightly different. Another kind of seismic source which was introduced to glaciology in the year 2010, but which has been used in industry for decades, is the vibroseis device. A baseplate is pushed onto the surface with a hold-down mass (100 kg–30 t) and set into vibrating motion. The vibrations follow a pre-defined frequency pattern, called a sweep, which typically lasts about ten seconds. The advantage of vibroseis over borehole explosives is that it can be operated from the surface, involves only weak forces, thus avoiding irreversible damage, and allows for repeated signals with known characteristics at the same location. The seismogram is created by correlating the source sweep with the recording of the geophones. By using either vertical or horizontal vibrations of the baseplate, either pressure or shear waves are excited, respectively. Typical sweep frequency ranges are between 10 and 350 Hz. For an example see Fig. 14.12 in Exercise 14.5.

14.3.3 Radar System Set-Up In contrast to seismic set-ups, traditional radar systems employ a single transmitterreceiver pair. The reasons for this are that it is easy to repeat recordings, as radar is a completely destruction-free method, and data quality is improved by stacking several recordings. Another reason is that—unlike in seismics—sophisticated and fast electronics are necessary to obtain data in sufficiently high temporal resolution. Only since about the year 2000 have systems been developed that use more than one receiver antenna to improve signal processing.

14.4

Data Acquisition and Processing

Seismic and radar signals are most commonly visualised by recording several traces and plotting these next to each other, with travel time or depth in the vertical and trace number (or horizontal position/offset) in the horizontal (Fig. 14.3). Different visual aspects of the signals are analysed—for example, the wiggles on them, their polarity and amplitude, and so on. In these displays, reflections can be identified as coherent signals in neighbouring traces, and these constitute events. The shapes or configurations of such events in the offset–travel time display are referred to as travel time curves. Each wave type (reflected wave, critically refracted wave, direct wave) has travel time curves of a distinct shape. The exact shape depends on the medium parameters, especially the wave velocity. For a single layer of thickness h and wave speed v, and with x denoting the distance between source and receiver at the surface, we can derive the relation between travel time t and offset x. Notably, for constant h and v, the two-way travel time curve for

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a reflection from the base of the layer has the form of a hyperbola in (x, t) space, given by √ 4h 2 + x 2 t= . (14.6) v An example of several reflection hyperbolae in real data is shown in Fig. 14.6. Direct waves (those that travelled along the surface from source to receiver) and refracted waves usually have simpler shapes than this, with almost linear dependences of time and offset (Fig. 14.4). In real data, deviations from linearity arise from inhomogeneities in the medium. The most characteristic parameters which can be derived from travel time curves are t(0), the time of reflection at zero offset; ti , the intercept time of a refracted wave (extrapolated from larger offsets); xcrit , the critical distance (offset at which critically refracted waves first appear); and xcr os , the cross-over distance (above which critically refracted waves arrive earlier than reflected waves). The difference between the time of reflection at zero offset t(0) and the total travel time along the travel paths (Fig. 14.5) yields the normal moveout correction (NMO) tNMO = t(x) − t(0) ∼

x2 2 2vNMO t(0)

,

(14.7)

t

reflected arrivals

critically refracted arrivals ( slope: 1/v2 ) t0 ti

direct arrivals ( slope: 1/v1 )

x crit

x cros

offset x

Fig. 14.4 Seismic travel time curves for a direct wave, reflection and critically refracted wave (adapted from van der Kruk [10])

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Offset x

x

A A

t

Δt NMO

t0

Fig. 14.5 Travel time curve before (left) and after (right) NMO correction (adapted from Yilmaz [5])

Fig. 14.6 Subglacial sediments imaged by seismics with a P waves and b Sv -waves [17]. The data are displayed as recorded. Events thus show up as reflection hyperbolae. Reprinted from Nature with permission from Springer Nature ©1986

which defines the normal moveout velocity vNMO . Fitting the hyperbolic approximation (14.7) to field data for a known offset x provides us with a means to estimate vNMO of the medium from measurements at the surface. If only one reflection hyperbola appears in the data, for example from the ice–bed interface, the NMO velocity is an estimate of the average velocity of the (single) depth interval between surface and bed. If more than one reflection hyperbola appears, one can derive the distribution of (multiple) interval velocities as a function of depth. Although raw seismic or radar data already contain many interesting visible features, signal enhancement and more sophisticated processing steps are needed to

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Table 14.2 Flow chart describing the sequential steps followed in seismic data processing. Shot gather refers to several recordings associated with a single shot. One obtains a CMP (common mid-point) gather by re-sorting the data belonging to a common reflection point at depth (using several shots and/or recordings)

Shot gather

CMP gather

geometry ↓ static correction/editing ↓ amplitude correction ↓ suppression of direct waves ↓ frequency filter ↓ deconvolution ↓ multiple suppression

↓ CMP sorting ↓ velocity analysis ↓ NMO correction ↓ stacking ↓ migration ↓ stratigraphic interpretation

prepare the data before their interpretation. As radar data are usually acquired in a common-offset geometry where the transmitter/receiver pair is towed along the surface at fixed separation (constant x), processing is comparatively simple. In contrast, for a seismic data set acquired with one source and multiple geophones (or a streamer4 ), more elaborate processing steps are involved; the flow chart in Table 14.2 indicates a standard processing sequence. The velocity analysis yields the distribution of wave velocity with depth and its variation along the profile. With high quality data, a number of significant parameters can be deduced from the velocity distribution, such as density distribution in the firn column. Note that in radar campaigns using a single transmitter and receiver channel, the velocity cannot be obtained from a common-offset survey. To retrieve the velocity distribution, dedicated common-midpoint (CMP) surveys need to be carried out at various locations along each radar profile. Once an estimate of the velocity distribution is known (either from geophysics or from firn cores), the data collected from common-offset profiling are converted from the travel time to the depth domain. This then enables the interpretation of stratigraphic features visible in the data, such as reflection horizons. A specialised technique, mainly established in seismics, is the analysis of amplitude variation with offset (AVO) or with angle of incidence (AVA). This can produce detailed information about the physical properties of the media above and below the reflecting interfaces. Other methods in seismics include the comparison of shear waves with pressure waves, from which elastic properties can be derived.

4A

streamer is a linear array of geophones towed along the ground or through the water.

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Seismic Applications in Ice

This section briefly presents the main established and emerging applications of seismic data analysis in glaciology. These concern the determinations of the following properties/unknowns: ice thickness (both in an ice sheet and an ice shelf); water thickness below an ice shelf; basal topography; subglacial bed structure; the internal structure of ice; rheological properties; temperature distribution; and anisotropy.

14.5.1 Ice Thickness and Basal Topography Determination of ice thickness is the most straightforward application of seismics as well as radar in glaciology. Both seismic and radar measurements have been applied for more than eighty years to make estimates of ice volume. In both geophysical methods, the essential approach is to examine the seismogram/radargram (the set of recorded traces along a profile) and identify or ‘pick’ the ice-base reflection, which often appears as a strong continuous reflector at depth. This is usually done after travel time has been converted to depth (see Sect. 14.4). In the case of airborne radar measurements, it is also necessary to identify the ice-surface reflection. Examples of seismic data for deducing ice thickness are provided in the exercises. Subtracting the ice thickness from known surface elevations measured by other methods yields the bed topography, and this is how key (observational) “bed map” datasets have been compiled for Greenland [11] and Antarctica [12]. More recently, inverse methods [13,14] involving mass conservation and the ice surface velocity have also been used to extend these ‘bed maps’ into areas where geophysical measurements are sparse.

14.5.2 Subglacial Structure and Properties Seismics constitute a useful tool to image materials beneath the lower/bottom surface of the ice (Fig. 14.6). In contrast to radar waves, the presence of water does not exclude further penetration of seismic waves. Although shear waves cannot penetrate liquids, the pressure waves continue to propagate. Additionally, whereas radar waves are damped significantly in rocks or sediments (but not ice, which is an extraordinarily transparent medium), the differences of attenuation between rock and ice for seismic waves are far less pronounced. Extensive knowledge about both the englacial and subglacial properties of ice streams, which are temperate at their base, has been obtained by seismics [15,16]. Of major importance for the dynamic behaviour of an ice mass with a wet base, especially for an ice stream, are the properties (e. g., the water content) of any underlying till. In addition to imaging stratigraphy, the application of seismic AVO enables one to elucidate the subglacial material properties. To this end, the angle of incidence of the seismic waves at the interface under consideration is varied by using a large range of offsets between shot and geophone. The incidence-angle dependence of the reflection coefficient is determined by the Zoeppritz equations, which describe the

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theoretical variation of the amplitude and phase of reflected/refracted P and S waves with offset. In an AVO analysis, a geological model of the subglacial interface, layer thickness and material properties is fitted to the data to obtain the best agreement with the observations [9,18,19].

14.5.3 Rheological and Other Englacial Properties In the firn column, where density increases with depth, seismic waves are refracted away from the normal, and some of these waves return to the surface. The larger the offset between shot point and geophone, the deeper these waves have penetrated into the firn. This provides an opportunity to deduce the rheological properties of the firn column from measurements at the surface. For example, King and Jarvis [20] developed an approach for determining the depth variation of Poisson’s ratio in the firn at one site. The new utilisation of vibroseismics means that imaging this material property continuously along a survey profile is also possible. The speed of seismic waves within an ice crystal is anisotropic, i. e., it depends on the direction of travel. Where the crystal orientation fabric of the bulk changes, the seismic wave velocity changes. Therefore, seismic waves are partly reflected at such interfaces. Although this behaviour has been known for decades, detailed or specific observation and scientific investigation of such internal seismic layers have not been pursued until recently [21]. The englacial temperature affects the attenuation of seismic waves. On this basis there have also been attempts to infer temperature from seismic data [22].

14.6

Radar Applications in Ice

As radar is extensively used by different communities in the Earth sciences, there exist a wide variety of terms for the application of radar, partly reflecting the underlying measurement principle, partly the area of application. Commonly used terms in the present context are: RES (radio-echo sounding), snow radar, ice radar, georadar, GPR (ground penetrating radar), IPR (ice penetrating radar), SPR (surface penetrating radar), FMCW (frequency modulated continuous wave), and SFR (stepped frequency radar). Since the first routine employment of radar in glaciology for ice-thickness measurements in the 1950s and 1960s, the variety of applications has continuously been increasing. Nowadays, radar is used for mapping ice thickness and subglacial topography (as described in Sect. 14.5.1), bed conditions (including the location of grounding lines), internal layer architecture, density distribution, polythermal boundaries (cold/temperate ice interfaces), liquid water content of snow and firn, and the location of englacial conduits, crevasses, and materials such as sediments, boulders, and even parts of airplanes. Before the availability of magnetic and digital storage devices, RES data were recorded on photographic films. Since a fair number of these films have meanwhile been digitised, they are again widely used for scientific

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studies because of their extensive coverage (e. g., [23]). Bogorodsky et al. [24] gave an overview of the early applications and techniques in radioglaciology.

14.6.1 Internal Layer Architecture and Ice Dynamics Imaging the internal structure in ice sheets over hundreds or even thousands of kilometres is a key area of application of radar. The results can be interpreted for different kinds of glaciological information. Most internal layers are isochrones, that is, each layer has a single age: an example would be the ash layer from a volcanic eruption. Shallow applications usually aim at determining the spatial pattern of accumulation from imaged layer geometry (for an overview see [25]), whereas deeper applications are often performed in conjunction with ice-core deep drilling to determine past flow conditions [23,26] and the stratigraphy in the vicinity of an ice-core borehole, and to link together the depth-age scales of different ice cores (Chap. 12). With increasing computer power, it has become feasible to use internal layer stratigraphy as a constraint for ice-dynamic modelling that utilises inverse approaches to find the history of forcings and boundary conditions (e. g., past accumulation rate) [27,28]. Ice dynamics has a particularly profound effect on the internal layer stratigraphy underneath divides and domes, where deviatoric stresses are different from the flank. The so-called Raymond effect leads to the formation of isochrone arches, an upwarping of layers underneath the divide, which is referred to as a ‘Raymond bump’. Progress over recent years and ongoing efforts make it possible to estimate the rheological properties, for example Glen’s flow index n of the ice at divides or domes, from radar measurements [28,29]. Whereas the Raymond bump itself is caused by the non-linear flow law, secondary features on the bump are likely caused by anisotropic crystal orientation fabric, as evident from numerical flow modelling [30].

14.6.2 Subglacial Conditions In contrast to seismic waves, radar waves do not penetrate the medium underlying the ice. However, the very high contrast in dielectric permittivity between ice or sediments and liquid water makes it feasible to determine whether the glacial bed is temperate, and thus liquid water is present, or whether it is frozen (e. g., [31]). Moreover, even without considering the reflection amplitude, the presence of subglacial lakes can be determined by simply investigating the geometry of the subglacial reflector, as the ice/water interface at such lakes is expected to be flat or almost flat. New radar systems with increased spatial resolution enable the imaging of details within structures in deep/basal ice which have so far been considered ‘low-backscatter zones’ or ‘echo-free zones’ (EFZ) [32]. In some regions of the Greenland and Antarctic Ice Sheets, complex englacial structures near the bed have been interpreted as resulting from freeze-on of subglacial water [33].

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Fig. 14.7 500 MHz radargram from Kongsvegen, Svalbard [36]; SI denotes superimposed ice. The vertical axis is the two-way travel time (TWT)

14.6.3 Englacial Conditions The ice itself can contain significant fractions of liquid water for temperatures at and even a few degrees below the pressure melting point. This makes it possible to use radar to delineate the cold-temperate transition surface (CTS), which separates temperate from cold parts in polythermal glaciers [34,35]. Especially in sub-polar temperate glaciers, where the full set of glacier facies is present, radar provides an impressive view of the internal structure of the ice (Fig. 14.7). Where considerable melting occurs at the surface, the water penetrates into the firn and collects at the firn-ice boundary. For shallow slopes, this can lead to the formation of a water table. Run-off from the surface of an ice mass often penetrates into the ice along pre-defined cracks or crevasses, and forms moulins by further melting. Crevasses and moulins can also be imaged by radar under favourable conditions, due to the dielectric contrast between air- or water-filled spaces and the surrounding ice.

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14.7

Notes and References

14.7.1 Further Reading There are a number of books which are useful on the subject of active seismics. For discussion of general exploration geophysics, see Robinson and Coruh [37]. On seismic data acquisition and processing, the book by Yilmaz [5] is a useful source. For applied geophysics for shallow applications, see Burger et al. [38]. Other useful information, especially terminology, is available from the Encyclopedic Dictionary of Applied Geophysics, hosted by the Society of Exploration Geophysicists, whose web page is at http://wiki.seg.org/ For further reading on radar, the following books are good sources. For general electromagnetics, see Hecht [39] and Jackson [40]. For general radar applications, the book by Ulaby et al. [6] covers almost everything. Ground-penetrating radar is discussed by Daniels [41], and radioglaciology by a number of books, amongst them those by Bogorodsky et al. [24] and Hubbard and Glasser [42]; see also the paper by Dowdeswell et al. [43], and the book chapter by Navarro and Eisen [44].

14.7.2 Gravimetry Measurements of variation of gravity allow one to derive information about the distribution of mass in the Earth. To see how this can be done, we consider the Newtonian law of gravitation. This states that the attraction between two point masses is proportional to the product of their masses, and inversely proportional to the square of the distance between them. Accordingly, for a distribution of matter having density ρ(s) at a point s in space, the acceleration (force per unit mass) at a point r due to a small volume5 ds at s is −Gρ(s) ds/|r − s|2 , where G = 6.67 × 10−11 N m2 kg−2 , and if we take the distribution of matter to be the Earth (denoted by the volume E), then the gravitational acceleration at a point is just Gρ(s)(r − s) ds . (14.8) a=− |r − s|3 E Note that this can be written as the gradient of a potential, Gρ(s) ds . a = ∇V , V (r) = E |r − s|

(14.9)

However, this is not in itself the acceleration experienced on the Earth, because the planetary rotation contributes a centrifugal acceleration. To understand the effect of this, we need to transform derivatives of vectors from a fixed frame to a rotating frame. If ei (i = 1, 2, 3) constitute a Cartesian set of coordinates in the rotating

5 We

use the notation ds = ds1 ds2 ds3 to denote volume elements.

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frame, then the angular velocity is defined to be such that e˙ i = × ei (the vector product); here the overdot denotes a time derivative. If w is a vector which varies in time, having coordinates wi in the rotating frame, thus w = wi ei (we use Einstein’s ˙ = summation convention which implies summation over repeated suffixes), then w w˙ i ei + wi × ei , and this can be written in the form ˙ R + × w, ˙ F = w| w|

(14.10)

where F refers to the fixed (absolute) frame, and R refers to the rotating frame. Putting w = r, we derive the relation between absolute velocity and velocity with respect to the rotating frame: v F = v R + × r,

(14.11)

and then repeating the exercise with w = v| F , we find that the absolute acceleration is related to the acceleration in the rotating frame by a F = a R + 2 × v R + × ( × r).

(14.12)

The second term on the right is the Coriolis force (per unit mass), and the third is the centrifugal acceleration, and can be written as −∇( 21 | × r|2 ). It follows from this that the acceleration experienced at a point r in the rotating frame due to gravity and the centrifugal acceleration can be written in terms of a potential: a R = ∇W , W = V + 21 | × r|2 = V + 21 2 R 2 ,

(14.13)

where R is the distance of the point from the axis of rotation (which gives the vector its direction). The rotation causes Earth’s oblateness, so that its average radius 6371 km is higher than its value 6357 km at the poles, and less than its value 6378 km at the equator. As the poles are closer to Earth’s centre than the equator, the gravimetric acceleration a in (14.9) is larger at the poles than at the equator. The centrifugal acceleration is zero at the poles and about 3.4 gal at the equator; the unit used here is the galileo.6 One gal is equal to 1 cm s−2 , so that the Earth’s gravitational acceleration is about 980 gal. Another unit is the gravity unit, 1 g.u. = 1 µm s−2 ≈ 0.1 mgal. The total difference gpole − gequator = 5.2 gal, or 0.5% of the average acceleration. If Earth was a liquid body of homogeneous density, its shape could be approximated by a rotational ellipsoid.7 The variation of the gravitational acceleration on Earth could then be derived from an approximation based on the solution of Poisson’s equation for the gravitational potential. This provides an empirical relation for the acceleration as a function of latitude. However, the presence of a solid crust and strong density variations within the Earth causes the Earth to have the approximate

6 In

reference to Galileo’s Pisa experiment on the fall of two spheres of unequal mass. is, an ellipse rotated about its minor axis.

7 That

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shape of the geoid, which is an equipotential surface for the effective gravitational potential W . In a simple conception the geoid can be considered as the sea surface without considering the influence of winds and tides. The geoid is for instance defined from measurements in the Geodetic Reference System from 1967, GRS67. To derive the spatial field of gravity anomalies from measured acceleration values, further corrections in addition to those of latitude include the effects of tides, topography (including known variations in density of the terrain), altitude of measurement point (e. g., in an airplane), isostatic corrections, accelerations due to a moving measurement platform (these are called Eötvös corrections, and are due to the Coriolis force, for example on a ship or airplane). These corrections then lead to the free-air anomaly or the Bouguer anomaly, which forms the basis for geoscientific interpretations. Long wave anomalies of Earth’s acceleration on the planetary scale of the order of several thousand kilometres originate in the mantle, whereas short wave anomalies originate in the crust.

14.7.3 General Wave Equation and Solution Both seismic and electromagnetic waves are ultimately described by the wave equation, which for a general quantity φ is given by φtt = c2 ∇ 2 φ;

(14.14)

∂2 ∂2 ∂2 + + is the ∂x2 ∂ y2 ∂z 2 Laplacian operator. A basic solution of (14.14) is the plane wave

here the subscripts t denote partial derivatives, and ∇ 2 =

φ = Aei(k.r−ωt) ;

(14.15)

this represents a travelling wave with (complex) amplitude A (of magnitude |A|), frequency ω and wavenumber k = |k|. The quantity χ = k.r − ωt is the phase of the wave, and in three dimensions, the surfaces with χ constant are planes orthogonal to the wave vector k, and these phase surfaces move at speed c = ω/k in the direction of the wave vector. Thus c in (14.14) is the wave speed. The quantity ω is sometimes called the circular frequency. It is related to the period P of the wave by P = 2π/ω; also a common definition of frequency (in Hz, or cycles per second) is f , and related to ω by ω = 2π f (thus P = 1/ f ). It should be added that while it is common to use the complex exponential in writing wave solutions as in (14.15), this is really a convenience. Because the wave equation is linear, solutions can be superposed; in particular the solution (14.15) shows that sin(k.r − ωt) and cos(k.r − ωt) are solutions (and really the same, other than a phase shift of 21 π ).

14.7.3.1 Transmission and Reflection Of particular interest is the process of transmission and reflection of rays at an interface. The geometry of the rays is shown in Fig. 14.8. An incoming ray at an

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y

Fig. 14.8 Transmission and reflection at an interface

θI

θR

x θT

angle of incidence θ I (to the normal) is reflected at an angle of reflection θ R , while a transmitted ray is transmitted at an angle of refraction θT . Note that for a ray travelling at (positive) angle clockwise from the upwards normal, k = (sin , cos ), and that for Fig. 14.8 we have I = π − θ I , R = θ R , T = π − θT . We can thus write, for a monochromatic wave of frequency ω, φ = A I ei[k I (x sin θ I −y cos θ I )−ωt] + A R ei[k I (x sin θ R +y cos θ R )−ωt] , φ = A T ei[kT (x sin θT −y cos θT )−ωt] ,

y < 0,

y > 0, (14.16)

subject to appropriate boundary conditions at the interface. Boundary conditions depend on the particular context, but it is simplest to illustrate them for ordinary sound waves, where the wave equation is a consequence of the linearised Euler equations in the form ρt + ρ0 ∇.u = 0, ρ0 ut = −c2 ∇ρ,

(14.17)

where ρ0 is the undisturbed density, ρ is the density perturbation, u is fluid velocity, and c2 = dp/dρ, and we derive the wave equation for ρ, for example. It is convenient to define φ = c2 ρ (φ is thus the pressure perturbation), and φ also satisfies the wave equation. Suitable boundary conditions at the interface are that the normal velocity 1 ∂φ and pressure are continuous, and these correspond to taking φ and continuous ρ0 ∂n at the interface. Continuity of φ at y = 0 leads to A I eik I x sin θ I + A R eik I x sin θ R = A T eikT x sin θT ,

(14.18)

and satisfaction of this for all x requires θ R = θ I (the law of reflection), and k I sin θ I = k T sin θT , or

sin θ I sin θT = cI cT

(14.19)

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(Snell’s law of refraction, with c I and cT the sound speeds on either side of the interface), and also A I + A R = AT .

(14.20)

Denoting the undisturbed densities on either side as ρ I and ρT , the condition of 1 ∂φ continuity of at the interface then leads to ρ0 ∂n − 1 + a R = −K aT ,

(14.21)

where a R = A R /A I , aT = A T /A I are the reflection and transmission coefficients, and Z I cos θT ; (14.22) K = Z T cos θ I here the quantities Z = ρ I c I , ρT cT on either side are the impedances.8 From this and (14.21), we have 1−K 2 , aT = . 1+K 1+K For normal incidence, we regain (14.3). aR =

(14.23)

14.7.3.2 Ray Theory It is beyond the scope of the present chapter to draw more than a sketch of the mathematical connection of ray theory with the wave equation, but it is useful to give the basic idea. Particularly, it illuminates the bending of rays in media where the wave speed is spatially variable, due for example to density increases with depth, and it is also useful to understand the process of diffraction, where rays bend to illuminate shadow regions. The idea is that in solving the wave equation in inhomogeneous media with variable wave speed, while ei(k.r−ωt) is no longer an exact solution, it will be approximately so providing the spatial variation of c is sufficiently slow (or equivalently, the wavenumber is sufficiently high). To be specific, we consider an incident wave train of frequency ω and define the local wavenumber to be k = ω/c, which is thus variable in space. We can write the solution as φ = (r)eiωt , and then satisfies the Helmholtz equation ∇ 2 + k 2 = 0.

8 Impedance

(14.24)

is a measure of resistance to motion. For sound waves it can be defined as the ratio of pressure perturbation to velocity perturbation. Since, from (14.17), with u and ρ ∝ ei(k.r−ωt) , ρ0 ut = −iωρ0 u = −c2 ∇ρ = −iρc2 k, and the pressure perturbation φ = ρc2 , and c = ω/k, it follows that φ/u = ρ0 c.

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When k is large, we put k = n(r)/ν and assume ν 1; then an asymptotic solution, motivated by the constant k solution, is to seek an approximation in the form9 ∼ A exp

iψ(r) , ν

(14.25)

where also A may be expanded in powers of ν. To leading order, the phase ψ satisfies the eikonal equation |∇ψ| = n,

(14.26)

and this can be solved using Charpit’s method. This equation is in fact hyperbolic, and its characteristics, along which information propagates, are precisely the rays of the wave field. If n varies in space, then the rays bend. Further, when a ray is tangent to an obstacle, the approximation breaks down and must be adjusted; this is when diffraction occurs, and the resultant theory is able to predict observed diffraction patterns.

14.7.4 Seismic Waves The equations of linear elasticity can be written in the form ρutt = ∇.σ ,

(14.27)

where σ is the stress tensor, related (essentially this is Hooke’s law) to the strain10 field u by

∂u j ∂u i + K − 23 μ ∇.u δi j , + (14.28) σi j = 2μ ∂x j ∂ xi where K is the bulk compressibility, μ is the shear modulus, and δi j is the Kronecker delta (= 1 if i = j, zero otherwise). Substituting this into (14.27), we obtain

ρutt = K + 43 μ ∇∇.u − μ∇ × ∇ × u.

(14.29)

= ∇.u, ω = ∇ × u,

(14.30)

Defining and using the identity ∇ ∇. a − ∇ × ∇ × a =

∇ 2 a,

we obtain

tt = v 2p ∇ 2 , ωtt = vs2 ∇ 2 ω,

(14.31)

9 The symbol ‘∼’ is used rather than ‘=’ because the approximation is asymptotic rather than convergent. 10 The strain at a point of an elastic material is its displacement from its position of equilibrium.

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where

vp =

K + 43 μ , vs = ρ

μ ; ρ

(14.32)

these are the P and S wave equations, respectively. For liquids μ = 0, and shear waves do not propagate. For a displacement field u = (u, v, w) propagating in the x direction, and where the displacement depends only on x and t, we have = (u x , 0, 0) and ω = (0, −wx , vx ) (the subscripts denote partial derivatives), which explains the physical interpretation of these waves as compressional and shear waves. Reflection and refraction at an interface can be calculated in a similar manner to that in Sect. 14.7.3.

14.7.4.1 Recording: Geophones Seismic waves are usually recorded with spiked geophones, either a single channel recording either the vertical or one horizontal component of the wave, or threecomponent geophones, which record the complete motion caused by the wave in three dimensions. Spiked geophones are pushed several tens of centimetres into the snow and covered with snow, to avoid wind-induced noise. The geophone cables are then clipped onto the so-called spread cable, which connects all geophones to the data acquisition unit. Spiked geophones yield good data quality because of good coupling, but are time-consuming to install, because every geophone has to be handled separately.

14.7.4.2 Recording: Snow Streamer A faster way to record seismic data on land, adopted from the offshore industry, is to use a snowstreamer, which consists of a number of gimballed geophones fixed to a single streamer cable. This allows quick movement of the whole streamer at once. For recording, the streamer stops to avoid induced noise. A disadvantage is that the coupling of the gimballed geophones can vary considerably, depending on the conditions of the ground surface. In addition to a single geophone per channel, it is also common to use several geophones per channel (a geophone chain), electrically connected in parallel in order to suppress the signal caused by surface waves using destructive interference.

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14.7.4.3 Stacking Chart The combination of shot and receiver positions for data analysis forms an essential part of seismic data analysis. So-called stacking charts indicate the geometric relation of different combinations. Four different combinations are possible, which are referred to as gathers: (i) common shot; (ii) common receiver; (iii) common midpoint; (iv) common offset (see Yilmaz [5] for further details).

14.7.5 Electromagnetic Waves Maxwell’s equations can be written in the form ∇. B = 0, ∇ × E = −Bt , ∇. D = ρ, ∇ × H − Dt = J.

(14.33)

In these equations, H is the magnetic field, E is the electric field, B is the magnetic flux density, and D is the electric displacement field. ρ is charge density, and J is the electric current. We suppose the medium through which the waves pass is linear, isotropic and non-dispersive, and thus write B = μH, D = εE,

(14.34)

where μ is the magnetic permeability and ε is the electric permittivity. Commonly one writes ε = ε0 ε , μ = μ0 μ , where the subscripts zero denote the values in vacuo and the primed quantities are the relative permeability and permittivity. When the material is electrically conductive, we write J = σ E,

(14.35)

where σ is the conductance (the inverse of the resistance). Charge conservation follows from Maxwell’s equations in the form ρt + ∇. J = 0, but additionally we find ρt = −σρ/ε, and ρ rapidly approaches zero. Assuming this, the equations can be manipulated to the form wtt + λwt = −c2 ∇ × ∇ × w, ∇. w = 0,

(14.36)

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where w can be either H or E, and 1 σ , λ= , εμ ε

(14.37)

wtt + λwt = c2 ∇ 2 w.

(14.38)

c2 = and thus

Thus both E and H satisfy a damped wave equation. As earlier, we can seek solutions proportional to ei(k.r−ωt) , and then we find k 2 c2 = ω2 + iλω.

(14.39)

The distinction now with the damping coefficient λ = 0 is that for real ω, k is complex, specifically (assuming λ is small) kc ≈ ±(ω + 21 iλ).

(14.40)

If we select a wave moving in the positive x direction, so that the positive sign is appropriate, then approximately iω λx exp w ∝ exp − (x − ct) , 2c c

(14.41)

and the wave is damped over a distance ∼ 2c/λ. Note that (14.39) can be written in the form k 2 = μεω2 + iωμσ,

(14.42)

and thus one can interpret the damping term in terms of a complex relative permittivity ε = ε0 (ε − iε ), ε =

c02 σ , ε = , 2 c ε0 ω

(14.43)

where, assuming μ ≈ μ0 for ice, c0 is the speed of light in vacuo. At an interface, transmission and reflection coefficients can be calculated as before. Assuming there are no surface currents or charge, we have that both H and E are continuous, but the coupling between them enables the coefficients to be determined.

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14.7.5.1 Mixing Formulae Radar processing involves converting the travel time to depth by using the wave speed. In snow, firn and ice, the wave speed is governed by the relative permittivity ε [see (14.2) or (14.43)], as the relative magnetic permeability of these materials is close to 1. The density ρ is the key control on ε . Accounting for the vertical density profile through the firn is thus critically necessary for an accurate time-depth conversion. Different relationships between ε and ρ are available for this purpose. They are often called mixing formulae because the medium is an air-ice mixture: as ρ increases from ≈ 0 (air) to ρi = 917 kg m−3 (solid ice), ε increases from 1 to its ice value, ≈ 3.2. A theoretical formula based on Looyenga’s [45] derivation assuming a two-component mixture with spherical particles is ε =

3 ρ 1/3 εi − 1 + 1 . ρi

(14.44)

In glaciological applications, empirical formulae determined by field experiments that measure the speed through firn of known density profiles are often used instead. For dry snow, firn, and ice, a widely-used formula was published by Robin et al. [46], and a later improved version of it by Kovacs et al. [47] is ε = (1 + 0.000845ρ)2 ,

(14.45)

where ρ is in kg m−3 . (The standard error in the regression fit of (14.45) to field data is ±0.031, or about 1% of ε .) This formula predicts ε = 1.6 for ρ = 300 kg m−3 (uncompacted dry snow). For ice at the pressure melting point or wet snow, alternative formulae accounting for the liquid water content must be used, because water has a high permittivity (ε = 82). Recent studies have explored more sophisticated formulae that include as many as four components: air, ice, water and rock.

14.7.5.2 Reflection Coefficients for Radar Waves in Ice In the firn (the upper ∼ 100 m of ice sheets), measured radar reflections are mainly caused by changes in ε originating from density contrasts. At greater depths, below the firn-ice transition, density contrasts vanish, and two other properties are responsible for changing ε to generate reflections. The first is layers of acidity in the ice inherited from the past deposition of volcanic aerosols (Sect. 12.4.9). The acidity increases the conductivity, and the imaginary part of the relative permittivity (ε ) depends on both conductivity and frequency; see Eq. (14.43). The second property is dielectric anisotropy of the ice. The effect of this can be significant and has been observed at the deeper levels (> 500–1000 m) of the ice sheets, where a spatiallyvarying anisotropic crystal fabric develops as a result of the ice-flow dynamics. Paren [48] derived approximate equations for the power reflection coefficient R p caused by small changes in the relative permittivity. For a small change in its real part, ε , Rp =

ε 4ε

2 ,

(14.46)

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whereas for a small change in its imaginary part, the power reflection coefficient is Rp =

1

2

4 (tan δ)

,

(14.47)

where tan δ = ε /ε is the electric loss tangent.

14.7.5.3 Radar Equation Both the design of radars and analysis of reflections portrayed by radargrams use the so-called radar equation or radar range equation to estimate the signal power detected by the receiver antenna. The equation accounts for the characteristics of the radar system as well as various physical factors affecting the wave propagation (e. g., [6]). Notably, geometric spreading causes the energy of the transmitted wave to decay with distance from the transmitter antenna; the reflected wave also experiences such spreading. Suppose the transmitter and receiver antennae are co-located on the surface, and a reflection occurs at a large depth z (the range) from an underlying smooth horizontal plane interface that has the power reflection coefficient R p — such as given by (14.46) or (14.47). If Pt is the transmitted power, then the received power Pr is approximately given by Pr =

Pt G 2 λ2 ε R p . (4π )3 z 4 L

(14.48)

Here, G is the gain of the receiving and transmitting antennae (assumed equal for simplicity), L is the propagation absorption loss (attenuation) due to impurities and inhomogeneity in the firn/ice, and λ is the wavelength. In the denominator, z 4 describes the geometric spreading. The factor ε (the ordinary relative dielectric permittivity) arises from refraction at the air-interface, and from the stratification and density distribution in firn, which cause bending of oblique rays and focus the wave towards the normal/vertical.

Exercises 14.1 Travel time curves: reflection Derive the travel time equation in (14.6). What is the asymptotic value of t for large x? What is the value of t for x = 0? Which quantity can be derived from it? What do lim x→∞ t(x) and t(0) represent in terms of the ray paths? 14.2 Travel time curves: refraction and critically refracted wave Derive the equation for the travel time curve of a refracted wave in a two-layer homogeneous medium. In Fig. 14.4, the travel time curve between x = 0 and x = xcrit is plotted as dashed. What is the reason? [Hints: what is the physical meaning of x = xcrit ? What are the conditions for refracted waves to occur?]

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14.3 Reflection of radar waves Derive the reflection coefficient for an ice–water boundary (e. g., over a lake): = 0; (i) for an electromagnetic wave with ε1 = 3.2, ε2 = 82, ε1,2 −1 (ii) for a seismic P wave with v1 = 3750 m s , v2 = 1500 m s−1 , ρ1 = 917 kg m−3 , ρ2 = 1030 kg m−3 . 14.4 Seismic wave types Figures 14.9 and 14.10 show raw data from a seismic line recorded with a snow streamer of 1.5 km length. The streamer has 60 geophones (channels). The geophones are separated by 25 m. How can the data from this streamer be re-sorted, to create a “virtual” streamer of 3.0 km length? How large is the offset for the first geophone? Shown in these figures are common-shot gathers of six virtual shots, no. 28, 29, 30, and no. 113, 114, 115. Shots 28–30 were recorded on the ice shelf, shots 113–115 on the grounded part of the ice sheet. The sequence number (seqno) indicates the number (1–120) of the virtual channel within the common-shot gather. Distance between shot and first (virtual) geophone is 150 m. Which type of events are visible? (An event is some form of coherent signal in neighbouring traces, related to certain wave types.) What differences can you spot between the adjacent shot gathers? Determine the velocities associated with the events, and estimate their depths. What problems do you expect when analysing this data set?

Fig. 14.9 Seismic profile, common-shot gathers 28–30 recorded on the ice shelf

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Fig. 14.10 Seismic profile, common-shot gathers 113–115 recorded on grounded ice

14.5 Seismic events In addition to the direct waves, single reflections and refractions, a large suite of other ray paths can be considered. These usually involve multiple reflections and are therefore termed ‘multiples’ (Fig. 14.11). The seismic line (processing of shot gathers and re-sorting to CMP gathers) consists of shot gathers (1–150), as shown in Exercise 14.4 and Fig. 14.12. Which events can you identify? From which physical cause are they likely to originate? Which parts of this seismic line show artefacts? Which processing step(s) could be responsible for these artefacts?

Double Multiple

Peg - Leg ( Type II )

Near - Surface Multiple

Long - Path Multiples

Peg - Leg ( Type I )

Near - Surface Multiple

Ghost

Simple Reflexion

Short - Path Multiples

FIRN

int. layer

ICE

WATER

BED

Fig. 14.11 A collection of possible multiples

SEAFLOOR

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Fig. 14.12 Processed data for a seismic line recorded with a 10–100 Hz vibroseismic sweep, Queen Maud Land, Antarctica. The data have been correlated, filtered, CDP-sorted, NMO-corrected, and finally stacked. (Ice flow is to the right and out of the page, the ice rise Halvfarryggen is to the left and the ice shelf Ekströmisen to the right.)

References 1. Riedel S, Jokat W (2007) A compilation of new airborne magnetic and gravity data across Dronning Maud Land, Antarctica. Tech. Rep., U. S. Geological Survey and The National Academies, USGS OF-2007-1047, extended abstract 149. At https://pubs.usgs.gov/of/2007/ 1047/ea/of2007-1047ea149.pdf 2. Röthlisberger H (1955) Studies on glacier physics on the Penny Ice Cap, Baffin Island, 1953: part III: seismic soundings. J Glaciol 2:539–552 3. Crary AP (1955) A brief study of ice tremors. Bull Seismol Soc Am 45(1):1–9 4. Podolskiy EA, Walter F (2016) Cryoseismology. Rev Geophys 54:708–758 5. Yilmaz Ö (2001) Seismic data analysis: processing, inversion, and interpretation of seismic data. Society of Exploration Geophysicists, Tulsa 6. Ulaby FT, Moore RK, Fung AK (1981/1982) Microwave remote sensing (2 vols.). AddisonWesley, Reading, Massachusetts 7. Aki K, Richards PG (1980) Quantitative seismology, theory and methods. W. H. Freeman, San Francisco 8. Peters ME, Blankenship DD, Morse DL (2005) Analysis techniques for coherent airborne radar sounding: application to West Antarctic ice streams. J Geophys Res 110:B06303 9. Booth AD, Clark RA, Kulessa B, Murray T, Carter J, Doyle S, Hubbard A (2012) Thin-layer effects in glaciological seismic amplitude-versus-angle (AVA) analysis: implications for characterising a subglacial till unit, Russell Glacier, West Greenland. The Cryosphere 6(4):909–922

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10. van der Kruk, J (2002) Reflection seismics. Lecture notes, eth-25462-01, ETH Zürich. https:// doi.org/10.3929/ethz-a-004363847 11. Bamber JL and 10 others (2013) A new bed elevation dataset for Greenland. The Cryosphere 7:499–510 12. Fretwell P and 59 others (2013) Bedmap2: improved ice bed, surface and thickness datasets for Antarctica. The Cryosphere 7:375–393 13. Morlighem M and 31 others (2017) BedMachine v3: complete bed topography and ocean bathymetry mapping of Greenland from multi-beam echo sounding combined with mass conservation. Geophys Res Lett 44:11051–11061 14. Morlighem M and 36 others (2020) Deep glacial troughs and stabilizing ridges unveiled beneath the margins of the Antarctic ice sheet. Nat Geosci 13:132–137 15. King EC, Woodward J, Smith AM (2007) Seismic and radar observations of subglacial bed forms beneath the onset zone of Rutford Ice Stream, Antarctica. J Glaciol 53:665–672 16. Peters LE, Anandakrishnan S, Alley RB, Winberry JP, Voigt DE, Smith AM, Morse DL (2006) Subglacial sediments as a control on the onset and location of two Siple Coast ice streams, West Antarctica. J Geophys Res 111:B01302 17. Blankenship DD, Bentley CR, Rooney ST, Alley RB (1986) Seismic measurements reveal a saturated porous layer beneath an active Antarctic ice stream. Nature 322(6074):54–57 18. Anandakrishnan S (2003) Dilatant till layer near the onset of streaming flow of Ice Stream C, West Antarctica, determined by AVO (amplitude vs offset) analysis. Ann Glaciol 36:283–286 19. Peters LE, Anandakrishnan S (2007) Subglacial conditions at a sticky spot along Kamb Ice Stream, West Antarctica. Tech. Rep., U. S. Geological Survey and The National Academies, USGS OF-2007-1047 20. King EC, Jarvis EP (2007) Use of shear waves to measure Poisson’s ratio in polar firn. J Environ Eng Geophys 12(1):15–21 21. Horgan HJ, Anandakrishnan S, Alley RB, Peters LE, Tsoflias GP, Voigt DE, Winberry JP (2008) Complex fabric development revealed by englacial seismic reflectivity: Jakobshavn Isbræ, Greenland. Geophys Res Lett 35(10):L10501 22. Peters LE, Anandakrishnan S, Alley RB, Voigt DE (2012) Seismic attenuation in glacial ice: a proxy for englacial temperature. J Geophys Res 117(F2):F02008 23. Siegert MJ, Welch B, Morse D, Vieli A, Blankenship DD, Joughin I, King EC, Leysinger-Vieli GJ-MC, Payne AJ, Jacobel R (2004) Ice flow direction change in the interior of West Antarctica. Science 305(5692):1948–1951 24. Bogorodsky VV, Bentley CR, Gudmandsen PE (1985) Radioglaciology. Reidel, Dordrecht 25. Eisen O and 15 others (2008) Ground-based measurements of spatial and temporal variability of snow accumulation in East Antarctica. Revs Geophys 46:RG2001 26. Waddington ED, Neumann TA, Koutnik MR, Marshall H-P, Morse DL (2007) Inference of accumulation-rate patterns from deep layers in glaciers and ice sheets. J Glaciol 53(183):694– 712 27. Leysinger-Vieli GJ-MC, Hindmarsh RCA, Siegert MJ (2007) Three-dimensional flow influences on radar layer stratigraphy. Ann Glaciol 46:22–28 28. Gillet-Chaulet F, Hindmarsh RCA (2011) Flow at ice-divide triple junctions: 1. Threedimensional full-Stokes modeling. J Geophys Res 116(F2):F02023 29. Drews R, Martín C, Steinhage D, Eisen O (2013) Characterization of glaciological conditions at Halvfarryggen ice dome, Dronning Maud Land, Antarctica. J Glaciol 59(213):9–20 30. Martín C, Gudmundsson GH, Pritchard HD, Gagliardini O (2009) On the effects of anisotropic rheology on ice flow, internal structure, and the age-depth relationship at ice divides. J Geophys Res 114:F04001 31. Carter SP, Blankenship DD, Young DA, Holt JW (2009) Using radar-sounding data to identify the distribution and sources of subglacial water: application to Dome C, East Antarctica. J Glaciol 55:1025–1040

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32. Drews R, Eisen O, Weikusat I, Kipfstuhl S, Lambrecht A, Steinhage D, Wilhelms F, Miller H (2009) Layer disturbances and the radio-echo free zone in ice sheets. The Cryosphere 3(2):195– 203 33. Bell RE and 11 others (2011) Widespread persistent thickening of the East Antarctic Ice Sheet by freezing from the base. Science 331(6024):1592–1595 34. Petterson R, Jansson P, Holmlund P (2003) Cold surface layer thinning on Storglaciären, Sweden, observed by repeated ground penetrating radar surveys. J Geophys Res 108(F1):6004 35. Eisen O, Bauder A, Riesen P, Funk M (2009) Deducing temperature distribution in the tongue of Gornergletscher, Switzerland, from radar surveys. Ann Glaciol 50:63–70 36. Brandt O (2007) Application of GPR as a tool for cryosphere characterizations. Ph. D. thesis, University of Oslo 37. Robinson ES, Coruh C (1988) Basic exploration geophysics. Wiley, Chichester 38. Burger HR, Sheehan AF, Jones CH (2006) Introduction to applied geophysics: exploring the shallow subsurface. W. W. Norton and Company, New York 39. Hecht E (2001) Optics, 4th edn. Addison-Wesley, Reading, Massachusetts 40. Jackson JD (1998) Classical electrodynamics. Wiley, Chichester 41. Daniels DJ (1996) Surface-penetrating radar, vol 6 of Radar, Sonar, Navigation and Avionics Series. IEE 42. Hubbard B, Glasser N (2005) Field techniques in glaciology and glacial geomorphology. Wiley, Chichester 43. Dowdeswell JA, Hodgkins R, Nuttall A-M, Hagen JO, Hamilton GS (1995) Mass balance changes as a control on the frequency and occurrence of glacier surges in Svalbard, Norwegian High Arctic. Geophys Res Lett 22(21):2909–2912 44. Navarro FJ, Eisen O (2010) Ground-penetrating radar. In: Pellikka P, Rees WG (eds.) Remote sensing of glaciers: techniques for topographic, spatial and thematic mapping. Taylor and Francis, London, pp 195–229 45. Looyenga H (1965) Dielectric constant of heterogeneous mixtures. Physica 31(3):401–406 46. Robin GdeQ, Evans S, Bailey JT (1969) Interpretation of radio echo sounding in polar ice sheets. Phil Trans R Soc Lond A 146:437–505 47. Kovacs A, Gow AJ, Morey RM (1995) The in-situ dielectric constant of polar firn revisited. Cold Reg Sci Technol 23:245–256 48. Paren JG (1981) PRC at a dielectric interface. J Glaciol 27(95):203–204

Glacial Isostatic Adjustment

15

Pippa Whitehouse, Glenn Milne, and Kurt Lambeck

15.1

Introduction

What happens to the Earth’s crust and mantle when ice sheets grow or decay? Can we observe the solid Earth response to ice loading with geological, geophysical or geodetic methods? How do we use such observations to constrain past ice-sheet evolution? In this chapter, we explore the relationship between ice sheets, the solid Earth and global sea level and consider how different kinds of data can be used to understand the feedbacks between them. These feedbacks are numerous. For example, growth of an ice sheet causes subsidence of the solid Earth, which in turn changes the ice sheet’s surface elevation and bed topography, modifying its flow. Similarly, change in the ice volume alters the sea level, but sea-level change influences the dynamics of marine-based ice sheets; thus, sea-level change is both a forcing of, and a response to, cryospheric dynamics. The coupled response of the Earth system to ice and ocean load changes is called Glacial Isostatic Adjustment (GIA). The word ‘isostatic’ reflects the assumption that the system tends to evolve towards a state of isostasy, or equilibrium. In Sects. 15.2 and 15.3 we focus on two core topics in the study of GIA: (i) how the Earth responds to surface loading, and (ii) the link between cryospheric changes and sea-level

P. Whitehouse (&) Department of Geography, Durham University, DH1 3LE Durham, UK e-mail: [email protected] G. Milne Department of Earth and Environmental Sciences, University of Ottawa, Ottawa, Ontario, Canada K. Lambeck Research School of Earth Sciences, The Australian National University, Canberra 0200, Australia © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_15

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changes. In Sect. 15.4, we outline a number of studies in which GIA concepts and ﬁeld observations have been brought together to infer past ice-sheet change.

15.2

Earth Response to Loading

Climate-induced changes in the distributions of ice and ocean water on the Earth’s surface constitute a loading that deforms the solid Earth and causes mass redistribution within it (Fig. 15.1). The forces associated with the loading are partly supported by the lithosphere (mechanically strong outer shell), and partly compensated by the deflection of layers having different densities within the solid Earth. The latter process causes lateral density gradients, which give rise to buoyancy forces that drive the restoring process of isostasy. The isostatic response depends on the density and rheology of the solid Earth, as well as the loading history. The deformation, which occurs at rates governed by the mantle viscosity, is very slow, so inertia is negligible.

15.2.1 Rheology of the Earth Deformation of the solid Earth is fundamentally governed by processes at the grain and sub-grain scales, which control its bulk rheology. Materials initially behave elastically under loading: the deformation is instantaneous and proportional to the stress applied. But once the applied stress exceeds an effective yield stress, the material fails. There are two modes of failure: brittle failure (which causes fracture) and ductile failure (which results in flow). The size of the yield stress and the failure mode depend on material properties (e.g., composition, structure) and environmental factors. Brittle failure is described by the Coulomb-Navier failure criterion: rs ¼ c þ arn :

ð15:1Þ

The material fails when the shear stress (rs) reaches the sum of the shear strength of the material (c) and the internal frictional resistance of the material (arn), where a is the coefﬁcient of internal friction and rn is the normal stress. The strain rate e_ s during ductile failure can be estimated using the Dorn equation: e_ s ¼ Ad

m

E þ pV n exp rs : RT

ð15:2Þ

A and m are material constants, d is the grain size, E* + pV* is the activation enthalpy, R is the universal gas constant, T is absolute temperature, n is the stress exponent, and rs is the shear stress required to maintain the strain rate.

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Fig. 15.1 Solid Earth response to ice loading and unloading. a Prior to glaciation. b Glaciation: water is transferred from the ocean to the land. Lithosphere flexure is accommodated by deformation in the mantle. Peripheral bulges (PB) form adjacent to the ice sheet. c Deglaciation: solid Earth relaxes towards equilibrium as water is transferred from the land back to the ocean. Note that in these schematic diagrams illustrating the GIA process, the geometry has been vertically exaggerated, and the amount and spatial extent of deformation and of land motion are not to scale

The failure mode of Earth materials depends on temperature and pressure. Both of these variables increase with depth (Fig. 15.2). The brittle strength increases with pressure and is virtually independent of temperature, whereas the ductile strength decreases with temperature, and is less dependent on pressure (unless V* is large). Whichever failure mode gives a lower yield stress dictates the style of deformation. At shallow depths, the ductile strength is very high due to relatively low temperatures (see (15.2)); the brittle strength is low and is hence the limiting factor, so failure occurs via fracture or fault reactivation, producing earthquakes. As the depth increases, increasing overburden pressure raises the brittle strength (via the normal stress in Eq. (15.1)) whereas rising temperature reduces the ductile strength. Consequently, for a material of uniform composition, there will be a transition from

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Fig. 15.2 Modelled temperature and pressure proﬁles within the Earth. In the left-hand panel, Ts and Tc are the temperatures at the surface and at the centre of the Earth, respectively. The thermal boundary layer at the core-mantle boundary at 2890 km depth is hypothetical. The horizontal axis is effectively reversed in the two panels. From Davies [48]. Reprinted with permission from Cambridge University Press

Fig. 15.3 Strength envelope and the brittle-ductile transition

brittle to ductile failure at some depth. Figure 15.3 shows the ‘strength envelope’ describing this transition.1 In reality, the Earth’s composition is not uniform with depth. Compositional differences between the crust and mantle, and water content, affect the strength envelope. The resulting vertical differentiation in rheology means that we can deﬁne different layers in the Earth. The lithosphere is the cold, outer shell that is mechanically strong (has a high yield stress), even over long time periods. A signiﬁcant proportion

1

Editors’ note: for a sceptical view of the use of this strength envelope, see Fowler [1], p. 517 ff.

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of this layer behaves elastically over geological time scales. Below the lithosphere, a steeply increasing downward temperature gradient causes the yield stress to drop. This region of sub-lithospheric mantle behaves as a viscous fluid on time scales of 104 yr. However, the duration of the applied stress strongly influences the behaviour of the mantle. For short-period loading ( 1 yr), the sub-lithospheric mantle behaves approximately elastically. For loading applied on intermediate time scales of hundreds to thousands of years, the deformational response includes an immediate elastic component and a viscous component, and a Maxwell model of a viscoelastic material is used to describe the rheology (next section). The time taken for the viscous component of strain to equal the magnitude of the elastic component of strain, under a constant applied stress, is known as the Maxwell time: g sM ¼ : l

ð15:3Þ

Here, sM is the Maxwell time, η is the mantle viscosity, and l is the elastic shear modulus of the mantle. Mantle viscosities are typically *1021 Pa s and the shear modulus of the mantle is *1011 Pa, so the Maxwell time is on the order of 1000 years.

15.2.2 Building an Earth Model When modelling the solid Earth response to loading during a glacial cycle (over a time period of *105 years), it is common to assume that the lithosphere behaves elastically and the mantle behaves viscoelastically (Fig. 15.4). In elastic deformation, the strain e occurs instantaneously and is proportional to the applied stress r: e¼

r : 2l

ð15:4Þ

l is the elastic shear modulus. The response has no time dependence. Elastic deformation can be visualised as the deformation of a spring. Viscous deformation is time dependent. Under an applied shear stress r, a viscous material with viscosity η deforms at the strain rate e_ ¼

r : 2g

ð15:5Þ

The strain thus increases linearly in time for a constant applied stress. Viscous deformation can be visualised as the response of a dashpot to loading. Equation (15.4) is applicable to the lithosphere, which is assumed to be elastic. However, as mentioned above, the mantle is considered to be a viscoelastic, or Maxwell, material. Its deformation can be visualised as the response of a spring and

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Fig. 15.4 Representation of the Earth rheology used in most modelling studies. A different elastic modulus (E) and viscosity (η) is assumed for each layer. Lithosphere thickness may vary

Fig. 15.5 Strain-time diagram for a viscoelastic (Maxwell) body—a system represented by a spring and a dashpot in series. A shear stress of magnitude r is applied between t0 and t1. The vertical jumps have the same height because the elastic strain r/2l is recovered at t1. Between t0 and t1, strain increases at the rate r/2η

a dashpot in series to loading (Fig. 15.5). The constitutive law for a Maxwell material can be written in the form e_ ¼

r r_ ; þ 2l 2g

ð15:6Þ

from this we see that the characteristics of a Maxwell material are that: (i) for a constant shear stress, strain increases with time linearly (i.e., creep), and (ii) if strain

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Fig. 15.6 Seismic velocities (P and S waves) and density as a function of depth. From Shearer [49]. Reprinted with permission from Cambridge University Press

is constant, stress decreases with time (i.e., relaxation). More complex nonlinear models for the mantle rheology have also been proposed. The elastic and viscous parameters in the chosen Earth model vary with depth and are typically constrained by geophysical and laboratory-based methods. The elastic properties (and density) of the mantle can be deduced from seismic travel times (Fig. 15.6; Exercise 15.2). The mantle viscosity is determined by two different approaches. In the ﬁrst approach, laboratory deformation experiments are carried out at high temperatures and pressures to measure the viscosity of rock samples that have been extruded from the mantle. The main limitations of this approach relate to the difﬁculty of simulating realistic (very low) strain rates and of obtaining suitable samples, especially from the deep mantle. The second approach infers mantle viscosities by inverting data related to GIA and mantle flow (Fig. 15.7; [2]) or GIA alone. This approach is limited by the resolving power of the data and by the simplifying assumptions used in the geophysical models to represent processes. Seismic velocity perturbations can also be used to constrain mantle viscosity perturbations (e.g., [3]).

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Fig. 15.7 An inferred envelope of mantle viscosities with depth, deduced by jointly inverting GIA and mantle convection data. From Mitrovica and Forte [2]. Reprinted from Earth and Planetary Science Letters ©2004 with permission from Elsevier

15.2.3 Earth Models Used in Glaciology and Glacial Isostatic Adjustment Having considered the Earth’s rheological structure, we now describe the typical models used to represent it in the disciplines of glaciology and glacial isostatic adjustment.

15.2.3.1 Glaciology In glaciological modelling studies, a “flat Earth” formulation―consisting of a lithosphere overlying a purely viscous half-space―is commonly used to determine the Earth’s response to loading by ice. It is assumed that the areal extent of most ice sheets is small enough for the Earth’s surface curvature to be neglected. Within this formulation, the lithosphere and sub-lithospheric mantle are commonly modelled in different ways. The lithosphere can be modelled by a “local lithosphere” (LL) approach or an “elastic lithosphere” (EL) approach [4]. In the LL approach, the flexural properties of the lithosphere are neglected, there is no lateral stress propagation, and deformation is assumed to be in response to loading at that point: w¼

qi hi : qm

ð15:7Þ

w is the downward deflection of the lithosphere, qi is the density of ice (*920 kg m−3), qm is the density of the mantle (*3300 kg m−3), and hi is the thickness of the local ice load. Equation (15.7) is equivalent to (10.5). In the EL

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approach, which is more realistic, the flexural rigidity D of the lithosphere is used to calculate the response to loading by solving the elastic plate problem [5]: Dr4 w ¼ q qm gw;

ð15:8Þ

where q = qighi is the point load, g is gravity, and qmgw is the buoyancy force exerted upward by the mantle below the lithosphere. Note that Eq. (15.8) reduces to Eq. (15.7) when D is small. A point load deflects the elastic plate downward within a characteristic distance of the load. Beyond this distance, a small upwards deflection, called the peripheral bulge, is produced. For the sub-lithospheric mantle, the time-dependent deformation rate can be deﬁned either via a diffusion equation (‘diffusive asthenosphere’; DA)2 or by an exponentially-decaying response function (‘relaxing asthenosphere’; RA). In both cases, a single parameter representing the diffusive or relaxation time controls the rate of viscous flow, and there is no scope for the deformation rate to be deﬁned as a function of the load size. Although different treatments of the lithosphere and sub-lithospheric mantle are often combined, the EL-RA model is the one most commonly adopted in large-scale ice-sheet modelling. Beyond the ‘flat Earth’ formulation, spherical self-gravitating viscoelastic Earth models (see next section) are also occasionally used in the ﬁeld of glaciology [6, 7]. They provide the most realistic 1-D representation of the Earth, but are computationally expensive and are only used when it is absolutely necessary to model the whole Earth, e.g., when modelling the solid Earth response to both ice and ocean loading during a glacial cycle.

15.2.3.2 Glacial Isostatic Adjustment (GIA) For most GIA calculations, a spherical self-gravitating viscoelastic Earth (SGVE) model is used. A spherical formulation must be used because GIA is calculated on a global domain: meltwater from ice sheets is redistributed gravitationally consistently throughout the global ocean (Sect. 15.3). Furthermore, the ‘self-gravitation’ component of the model accounts for the gravitational attraction that takes place within and between different components of the Earth system (ice, ocean, and solid Earth). Finally, in order to develop the most realistic representation of the Earth, the SGVE model adopts a viscoelastic rheology (Sect. 15.2.2), typically with the lithosphere assigned a very high viscosity, so that it essentially behaves elastically. In an SGVE model, three differential equations are solved simultaneously―the momentum equation, the continuity equation and Poisson’s equation―for the solid Earth response to an impulse force [8, 9]. Green’s functions are used to represent this response. The response to a general load is then determined by convolving the load history with the appropriate Green’s function (e.g., for radial motion of the Earth’s

2

The asthenosphere is the shallow part of the upper mantle (Fig. 15.4) where the effective viscosity is relatively low.

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surface). This approach makes use of viscoelastic Love numbers, which are a function of the elastic, viscous and density structure of the assumed Earth model [10]. The Love numbers encapsulate the response of the Earth to a point load and deﬁne the spectrum of decay times, which are dependent on the wavelength of loading. Due to the computational expense of SGVE models, they are not widely used in glaciological modelling, but hybrid models have been developed that encompass some of their features (e.g., [11]). Such models combine an elastic layered sphere with a viscous half space overlain by an elastic plate, to simulate the elastic response of a 1-D spherical Earth, while accounting for wavelength-dependent relaxation times. They are as computationally efﬁcient as the EL-RA model.

15.2.3.3 Comparison of Earth Models Different Earth model formulations yield different outputs for the same loading history. As an example, Le Meur and Huybrechts [4] compared ﬁve of the models discussed here (LL-DA, LL-RA, EL-DA, EL-RA, SGVE) in an experiment reconstructing Antarctic ice volume through the last glacial cycle. They found that the pattern of ice-sheet change, and hence the predicted pattern of postglacial rebound, varied signiﬁcantly between experiments due to the differing solid Earth response to ice loading. Differences in ice-sheet evolution, despite the use of identical climate forcing, are related to the strong feedbacks between ice surface elevation (partly determined by solid Earth deformation) and surface mass balance (Sect. 10.3). The differences in simulated total ice volume and bedrock response reach 2 million km3 and 40 m, respectively, with the SGVE model being most closely reproduced by the EL-RA model [4]. Van den Berg et al. [7] compared the EL-RA and SGVE model results in a study aimed at reproducing the Fennoscandian Ice Sheet at the Last Glacial Maximum (LGM), ﬁnding that these models cannot agree in both ice extent and ice volume

Fig. 15.8 Model reconstruction of the Fennoscandian ice sheet at the LGM using a the SGVE model, b an EL-RA model that best-ﬁts the ice extent (ice volume too small), and c the EL-RA model that best-ﬁts the ice volume (ice extent too large). Colours represent ice thickness. From van den Berg et al. [7]. Reprinted with permission from John Wiley and Sons, ©2008 American Geophysical Union

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simultaneously (Fig. 15.8). When their LGM ice extent is forced to be identical, the ice volume in the EL-RA experiment is *30% less than in the SGVE experiment. When the LGM ice volume is forced to be identical, the ice sheet in the EL-RA experiment spans a much larger area than that in the SGVE experiment. These differences, which greatly exceed those in the Antarctic study by Le Meur and Huybrechts, are probably related to the stronger sensitivity of the Fennoscandian Ice Sheet to surface mass balance. In addition to uncertainties about the style of Earth model to adopt, there are uncertainties in model parameters―notably lithospheric thickness and mantle viscosity―which can lead to differences in model output that are just as substantial. How such properties influence the response depends on multiple factors, including the temporal and spatial scales under consideration and the type of ice sheet being modelled. In particular, since geophysical observations show that such properties vary across different geographical regions, the use of a 1-D Earth model is fundamentally flawed. Accordingly, GIA models capturing a 3-D viscoelastic structure have been developed (e.g., [12, 13, 14]). These models are computationally expensive, so it is necessary to ascertain the sensitivity of GIA processes to lateral structure in order to determine when such models are required; lateral structure may be important in some regions, but not others. Generally, when deciding between 1-D and 3-D Earth models, there is a trade-off between model accuracy and computation time.

15.3

The Cryosphere and Sea Level

As an ice sheet grows and decays, the distribution of ice and water on the Earth’s surface changes. During the build up to full glaciation, water from the ocean becomes progressively locked up in the ice sheets; during deglaciation, there is enhanced flux of glacial meltwater to the ocean. In each scenario, the redistribution of surface mass deforms the Earth, altering the shape of the solid Earth and the shape of Earth’s gravity ﬁeld, both of which influence how water in the ocean is distributed. In this section, we discuss the intrinsic link between ice-sheet and sea-level changes and outline the physical processes that govern spatially variable sea-level change due to mass fluctuations of ice sheets.

15.3.1 Factors Affecting Sea-Level Change The growth and melting of ice sheets affects global mean sea level by altering both the mass and the volume of water in the ocean. Today, such changes may be measured using gravity or altimetry satellites, respectively (Sect. 15.4.5). When relating observations of global mean sea-level change to change in ice volume, it

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Fig. 15.9 Factors affecting sea-level change. From Milne et al. [16]. Reprinted from Nature Geoscience with permission from Nature Publishing Group ©2009

should be remembered that changes in terrestrial water storage also enter the mass conservation equation, and that the volume of water in the ocean depends on its density structure, which changes according to its distributions of temperature and salinity (Fig. 15.9; [15, 16]). When relating local observations of sea-level change―determined by tide gauges or geological and archaeological indicators―to ice-sheet changes, additional local factors must be considered. Sea level is deﬁned here to be the distance between the ocean surface and the solid Earth; if the vertical position of either of these changes, sea level will change. The ocean surface is perturbed by waves, tides, and changes in atmospheric pressure, ocean circulation, or the shape of the geoid. The last factor is modiﬁed by any redistribution of surface mass, including a change in the conﬁguration of the ice sheets. The height of the solid Earth will change due to the isostatic response to loading by ice, ocean and sediment, or as a result of tectonics. Some of these processes are unrelated to the cryosphere; others are intimately linked to changes in the conﬁguration of the ice sheets and hence must be considered when considering sea-level change within a GIA model.

15.3.2 Eustatic Sea-Level Change The global mean change in sea level due to a change in the volume of ice is often referred to as the glacio-eustatic change in sea level (henceforth shortened to eustatic). Eustatic sea level is calculated using mass conservation:

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DS ¼

qi DVice : qw Aocean

ð15:9Þ

Here, DS is the change in eustatic sea level, DVice is the change in ice volume, Aocean is the surface area of the ocean, and qi and qw are the densities of ice and water, respectively. Equation (15.9) assumes Aocean to be constant and that there is no solid Earth response to load changes. In practice, ice volume changes during a glacial cycle are large enough to impact the shape of the global coastline signiﬁcantly, so we must consider temporal changes in Aocean when calculating eustatic sea level in this case: q DSðtÞ ¼ i qw

Z

t t0

1

dVice dt: Aocean ðtÞ dt

ð15:10Þ

Eustatic sea level is a useful measure of the change in global ice volume; however, in reality sea-level changes due to a change in ice volume are highly non-uniform.

15.3.3 Departures from Eustasy and the Sea-Level Equation As mentioned above, sea level in the context of GIA is deﬁned to be the difference between the height of the ocean surface (deﬁned by the shape of the geoid) and the height of the ocean floor. Departures from eustasy stem from spatial variations in, and perturbations to, either of these surfaces. The geoid shape is deﬁned by the gravity ﬁeld, which may be perturbed directly by surface mass redistribution or a change in the rotational potential, or indirectly by the Earth’s deformation in response to these forcings. Meanwhile, the height of the ocean floor may change in response to surface mass redistribution or a change in the Earth’s rotation. The changes in both surfaces are non-uniform, and the processes governing these changes can be quantiﬁed into a single equation―called the sea-level equation―whose solution determines sea-level change across the oceans in response to ice-sheet fluctuations. The sea-level equation and the theory of sea-level change resulting from ice-ocean mass exchange are set out in Farrell and Clark [17]. The original formulation of the equation by these authors considers the effect of ice and ocean loading upon a viscoelastic Earth. More recent formulations incorporate further feedbacks between ice-loading, sea-level change and Earth rotation, as well as temporal variations of the coastline position. We expand on some of the detailed interactions featuring in the sea-level equation in the next few sections. Reviews of different approaches of solving the equation and the different loading implications of floating and marine-grounded ice have been given by Mitrovica and Milne [18], and Kendall et al. [19]. The structure of the sea-level equation forms the basis of a GIA model (Fig. 15.10). Two components must be deﬁned in this model: (i) the forcing applied to the Earth (primarily governed by changes in ice loading) and (ii) how the Earth responds to this forcing (the Earth model in Sect. 15.2.3). The geometry of the

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Fig. 15.10 Key components of a GIA model, including factors affecting the forces that are applied to the solid Earth, and factors affecting how the Earth responds to these forces

model domain and present-day topography must also be speciﬁed. After these are deﬁned, the sea-level equation is solved to determine the spatio-temporal evolution of the solid Earth and the geoid during the period of interest. Solutions are usually obtained iteratively because although the surface load governs the deformation, at the same time the deformation alters the geoid shape, and thus the load distribution. Model inputs―e.g., ice-loading history and Earth rheology―are constrained by matching the model outputs to observational data.

15.3.4 Rotational Feedback Besides loading by ice and water, changes in the Earth’s rotational state alter the shape of the geoid and the solid Earth (Fig. 15.10). Secular changes in the location of the rotation pole occur in response to surface mass redistribution, and thus are partially due to GIA. Consequently, feedbacks between surface mass redistribution and the Earth’s rotational potential must be considered within the sea-level equation. The mass redistribution during a glacial cycle causes the rotation pole to relocate by *10 km. Any change in rotation instantaneously modiﬁes the geoid (and hence sea level), but there is also a slower viscous response of the solid Earth. Together these changes result in a relatively simple pattern of change: on the spinning Earth, centripetal forces cause polar flattening and a bulge around the equator. As the rotation axis migrates, movement of the bulge causes vertical deflections of the geoid and solid Earth surface outwards and inwards in opposite quadrants (Fig. 15.11).

15.3.5 Spatial Pattern of Sea-Level Change As an ice sheet loses mass, three factors conspire to determine the global pattern in sea-level change (Fig. 15.12). Firstly, the loss of ice mass decreases the gravitational attraction between the ice sheet and the ocean; this lowers the height of the

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Fig. 15.11 Spatial pattern of the perturbation to the shape of the geoid and solid Earth surface due to a change in the orientation of Earth’s rotation vector between time t and t+. Vertical perturbations are positive (up) in the grey quadrants and negative (down) in the white quadrants. From Mitrovica et al. [50]. Reprinted with permission from Oxford University Press

Fig. 15.12 Schematic diagram illustrating the origin of the spatially-variable sea-level changes shown in Fig. 15.13. From Tamisiea et al. [51]. Reprinted from Space Science Reviews with permission from Springer Nature

sea surface within *3000 km of the ice sheet. A second consequence of the loss is isostatic rebound of the solid Earth, which enhances the drop in relative sea level close to the ice sheet. Thirdly, the overall mass gain by the ocean leads to far-ﬁeld sea-level rise; this is slightly ampliﬁed by solid Earth subsidence caused by the increase in ocean loading. Together, these factors govern the global pattern of sea-level change that would result from a loss of ice from Greenland (Fig. 15.13a) or West Antarctica (Fig. 15.13b). The sea-level changes depicted in Fig. 15.13 are normalised, with the eustatic change ﬁxed at 1 mm/yr. White, blue and orange areas respectively highlight the anticipated (i) near-ﬁeld sea-level fall, (ii) less than eustatic sea-level rise within *3000 km of the source of ice loss, and (iii) more than eustatic sea-level rise in the far ﬁeld. Such distinct patterns present us with a

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Fig. 15.13 Relative sea-level change ‘ﬁngerprints’ that would arise due to ice loss from a Greenland and b West Antarctica. From Milne et al. [16]. Reprinted from Nature Geoscience with permission from Nature Publishing Group ©2009

Fig. 15.14 Sea-level change due to ocean syphoning. a Peripheral bulges (PB) surround former glaciated regions immediately following deglaciation. b Peripheral bulges collapse over several thousand years, creating greater accommodation space within the ocean basins, and causing sea level to fall in the far ﬁeld. From Mitrovica and Milne [21]. Reprinted with permission from Elsevier

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potential methodology for using tide gauge records to determine the relative contributions to sea-level change from the Antarctic and Greenland ice sheets during the twentieth century (Sect. 15.4.4).

15.3.6 Viscous Effects: Ocean Syphoning The sea-level changes depicted in Fig. 15.13 only account for the instantaneous elastic response of the solid Earth to surface load changes. However, even in the absence of a change in the mass of the ocean, there will be continual viscous effects following ice unloading that causes sea-level change. For example, following ice unloading, the peripheral bulges surrounding the major ice sheets will gradually collapse, increasing the capacity of the ocean basins and causing a globally uniform sea-level fall that can dominate in the far ﬁeld (Fig. 15.14). This process, in addition to deformation around continents due to ocean loading, leads to an increase in the volume of the ocean basins and a lowering of the ocean surface. This change, termed ‘ocean syphoning’ [20], began during the late-Wisconsinan deglaciation (the period from *20 to *10 ka BP) and is ongoing today. It explains the widespread occurrence of mid-Holocene oceanic highstands throughout the equatorial oceans [21].

Fig. 15.15 Global records of relative sea-level change. a Barbados, b Cleveland Bay, Australia, c Pounawea, New Zealand, d Angerman River, Sweden, e Arisaig, Scotland. From Milne et al. [16]. Reprinted from Nature Geoscience with permission from Nature Publishing Group ©2009

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In fact, due to the spatial variability in the processes associated with GIA, and the time dependence of many of them, we expect widespread and signiﬁcant departures from eustasy throughout deglaciation, with the pattern of such departures varying over time [22]. Such departures are evident in records of relative sea-level change from around the world during the post-LGM period (Fig. 15.15). Importantly, they must be considered when projecting future sea-level change, as well as when interpreting past sea-level histories.

15.3.7 Sea-Level Change as a Controlling Factor on Ice-Sheet Evolution Water depth has a ﬁrst-order influence on the grounding line position, and indeed the dynamics, of a marine-grounded ice sheet [23]. In palaeo-ice-sheet modelling, sea-level change is commonly assumed to be eustatic, with the time series of the eustatic signal deduced from scaling the marine isotope record (Sect. 16.3; Fig. 16.8) or reconstructed from far-ﬁeld sea-level records (e.g., from Barbados). However, as explained in the last two sections, sea-level change during a glacial cycle is far from globally uniform. In particular, perturbations to the ocean floor and the sea surface in the near-ﬁeld of the ice sheets cause strong departures from eustasy adjacent to the marine grounded sectors of these ice sheets. Given that the ice-sheet models developed over the last decade include a proper treatment of grounding line evolution, it is logical to prescribe accurate (spatially-variable) sea-level forcing as their boundary condition. Gomez et al. [24] demonstrated that GIA processes associated with ice-mass loss―ocean floor rebound and sea surface fall―reduce the water depth adjacent to the retreating grounding line of a marine-grounded ice sheet. This acts to stabilise grounding line retreat, even potentially preventing runaway retreat on beds that slope gently downwards upstream (Fig. 15.16; [25]). This intercoupling between GIA and ice mass evolution is beginning to be included in the next generation of numerical ice-sheet models.

15.4

Constraining Cryospheric Changes with Observations

Observations of sea-level change, both in the near ﬁeld and far ﬁeld of former ice sheets, can provide powerful constraints on past changes in ice volume and extent, as well as the rheology of the solid Earth. Such observations may be combined with near-ﬁeld geomorphological records of former ice extent to understand past ice-sheet geometry and dynamics. In the following sections we explore the techniques that are used to reconstruct ice-sheet changes on a range of time scales.

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Fig. 15.16 Grounding-line retreat on a reverse bed slope. a No GIA-feedback. b With GIA feedbacks. Ice sheet proﬁles change from dark to light blue as time increases. Units on the horizontal axes are km. From Gomez et al. [25]. Reprinted with permission from John Wiley and Sons, ©2012 American Geophysical Union

15.4.1 Global Ice Volumes Global ice volumes can be directly equated to eustatic sea level (Sect. 15.3.2). However, sea level rarely tracks eustasy (Fig. 15.15), so we must turn to equatorial regions―in the far ﬁeld of the major ice sheets―to ﬁnd records for deducing global ice volumes. Such records still need to be corrected for isostatic (and tectonic) effects, even though their departure from eustasy is relatively small. In order to demonstrate this, Milne and Mitrovica [22] modelled the departure from eustasy across the ocean during the last deglaciation (Fig. 15.17). The locations of the key far-ﬁeld records (marked in Fig. 15.17a) in the predicted patterns of departure show that while they approximately record eustatic sea level, a correction for isostasy is still necessary and it varies with location and time. Piecing together

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Fig. 15.17 Model predictions of the departure from eustasy at a 21 ka BP and b 6 ka BP. Yellow-to-red/blue colours indicate areas where relative sea level was higher/lower than the contemporary eustatic sea level. From Milne and Mitrovica [22]. Reprinted with permission from Elsevier

Fig. 15.18 Standard deviation of the predicted relative sea level (RSL) at the LGM for an ensemble of 162 different Earth models. a Over the whole ocean. b Only in regions where the mean RSL prediction for the ensemble lies within 3 m of eustatic sea level at the LGM. From Milne and Mitrovica [22]. Reprinted with permission from Elsevier

the spatial pattern that gives rise to these differences thus enables tighter constraints to be placed on the relative contributions from the major ice sheets. Milne and Mitrovica [22] also studied the sensitivity of their results to the choice of Earth model, by examining the standard deviation of the LGM relative sea level predictions for 162 Earth models (Fig. 15.18a). Figure 15.18b plots the standard deviation where sea level is predicted to depart from eustasy by < 3 m. Regions showing a weak sensitivity to the Earth model, such as the Indian Ocean, are the best locations for seeking unambiguous constraints on global ice volume at the LGM.

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Accounting for GIA effects is important also when reconstructing eustatic sea levels at times before the LGM. For example, Raymo et al. [26] predicted the present-day elevation of shorelines from the Pliocene (*3 Ma BP), assuming that the Greenland and West Antarctic Ice Sheets were absent during this warm period. Palaeo-shorelines dating to this period are located at 10–40 m above present sea level, leading to diverse and incompatible estimates of global ice volume. Raymo et al. [26] found that these sea-level records can be largely reconciled by considering GIA effects, and concluded that they must be corrected for such effects when estimating the minimum in global ice volume during the Pliocene. Over such long periods, it may also be necessary to correct for the effects of tectonics, dynamic topography (due to mantle convection) and the isostatic response to long-term sediment loading, all of which add to the difﬁculty of piecing together the global implications of Pliocene sea-level records. Nearer the present, sea-level records from the Last Interglacial (LIG) (i.e., the Eemian; Sect. 16.5.1) are highly sought after due to their potential to constrain global ice volumes from *120,000 years ago. Global mean annual surface temperatures were 1–2 °C higher than present during the LIG, so the LIG may illustrate the potential state of the cryosphere under future global warming conditions. Kopp et al. [27] used probabilistic methods to reconcile a compilation of sea-level records from 132 to 116 ka BP with estimates of eustatic changes during this period. Their methodology accounts for the expected variations in relative sea level over space and time, and they concluded that eustatic sea level peaked at > 6.6 m above present with 95% likelihood (> 8.0 m with 67% likelihood and > 9.4 m with 33% likelihood) (Fig. 15.19). This result implies that one or more of today’s ice sheets was considerably smaller than present during the LIG, even though mean annual temperatures were only 1–2 °C above present. Since estimates for the eustatic contribution from the Greenland Ice Sheet during the LIG lie in the range 1.4 to 4.3 m, a sizeable contribution from Antarctica is probable (see Sect. 5.6.2 of IPCC [28], and Sect. 16.5.1 herein).

Fig. 15.19 Probability density plot of global sea level during the Last Interglacial. Darker colours indicate higher probabilities. Solid line indicates the median reconstruction; dashed and dotted lines indicate the 16th/84th and 2.5th/97.5th percentiles, respectively. From Kopp et al. [27]. Reprinted from Nature with permission from Springer Nature ©2009

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We have the most complete set of palaeo-sea-level records for the post-LGM period [29]. From these it has been deduced that the excess LGM ice volume was equivalent to 125 ± 10 m of eustatic sea-level change. Deglaciation commenced around 19 ka BP and major melting ceased around 7–6 ka BP. There was ongoing melt of 3 ± 1 m between *7 and *2 ka BP; since then, global ice volumes have been relatively stable (to within a few decimetres of present sea level) until the industrial revolution [30].

15.4.2 Meltwater Pulses During the last deglaciation, ice-sheet melting did not take place at a uniform rate; sea-level records indicate that the deglaciation was punctuated by several periods of accelerated sea-level rise at rates over 4 cm/yr (4 m per century) [31]. Constraining the source and input rate of the meltwater is important for understanding the climate forcing and dynamic processes that led to rapid ice-sheet collapse, and for forecasting the upper limit on potential future sea-level rise. The most rapid period of sea-level rise took place *14.5 ka BP and was the likely cause of 14–18 m of eustatic sea-level change over < 500 years [31]. This event, called Meltwater Pulse 1A (MWP-1A), is apparent in sea-level records at low-latitude sites including Barbados, Tahiti, and the Sunda Shelf. The source of meltwater can be deduced using the distinctive patterns of sea-level change―or ‘ﬁngerprints’ [32]―that are produced in response to geographically-localised ice melt (Sect. 15.3.5 and Fig. 15.13). The records from Barbados and the Sunda Shelf indicate a similar magnitude of rapid sea-level rise. An early explanation for this was that the meltwater source was predominantly from West Antarctica [32], but recent studies of the magnitude of melt available from West Antarctica conclude that this ice sheet contained < 10 m additional ice at the LGM [33, 34, 35] and so could not have been the sole source of MWP-1A. Evaluating the climatic impacts of rapid ice melt in different regions is another way of determining the source of MWP-1A. Weaver et al. [36] used a climate model to show that an Antarctic meltwater source can help to explain the onset of the Bølling-Allerød warm interval (14.7–12.7 ka BP), due to its influence on North Atlantic Deep Water formation. However, improvements in the age resolution of proxy climate records are needed to determine unequivocally the relative timing of the various climate and sea-level changes. Of further interest in understanding the origin of meltwater pulses are the dynamical processes that enable rapid ice-mass loss. It was originally thought that non-marine-based ice sheets could not undergo catastrophic collapse. However, ice-sheet modelling by Gregoire et al. [37] has shown that feedbacks between ice surface elevation change and surface mass balance can remove 9 m (eustatic equivalent) of ice from the Laurentide Ice Sheet in *500 years, thus generating meltwater pulses. Such modelling can be used to test the forcing and response mechanisms of MWP-1A after its meltwater partitioning has been better constrained.

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MWP-1A remains a subject of debate, as additional near-ﬁeld and far-ﬁeld data are required to better constrain the source partitioning of this major melt event [38].

15.4.3 Regional Ice-Sheet Histories GIA modelling can be used together with ﬁeld observations to constrain regional ice-sheet histories (and not just global ice volume; Sect. 15.4.1). The work of various groups of researchers (e.g., [35, 39, 40, 41, 42]) has led to the following approach for tackling this problem: (i) compile a regional data base of sea-level change and ice-extent change data; (ii) reconstruct the ice history using the ice-extent constraints; (iii) use a GIA model to calculate predictions of relative sea-level change; (iv) compare predictions and observations and tune the Earth model to minimise the misﬁt. With many parameters to tune that represent the ice-sheet dynamics and Earth rheology, non-uniqueness is an issue in such an exercise. One way to address this is to adopt independent sets of constraints (e.g., sea-level data and ice-extent data) and use these to constrain different components of the problem. Also, geodetic data quantifying present-day solid Earth deformation (measured by GPS receivers) and changes to the Earth’s gravity ﬁeld (measured by absolute gravimeters and GRACE) can be incorporated into the approach to provide additional constraints on the ice-loading history and Earth rheology. In the following, we briefly describe a case study of this kind from Antarctica. Figure 15.20 locates (across Antarctica) the constraints on ice extent and thickness change since the LGM, relative sea-level change during the Holocene, and present-day uplift rates as measured by GPS. Figure 15.21a shows the modelled change in ice thickness since the LGM, as reconstructed using the Glimmer ice-sheet model [35, 43] and Fig. 15.21b shows the misﬁt between the observed and reconstructed ice-sheet geometry at the LGM. Ice thickness near the coast is predicted to

Fig. 15.20 Map of constraints related to former ice thickness and extent, relative sea level, and present-day uplift across Antarctica. From Whitehouse et al. [35]. Reprinted with permission from Elsevier

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Fig. 15.21 a Modelled Antarctic ice thickness change between the LGM and present; positive values indicate thickening since the LGM and vice versa. b Modelled Antarctic LGM ice surface elevation and misﬁt to geological and glaciological observations of former ice extent. Adapted from Whitehouse et al. [35] and used with permission from Elsevier

have mostly decreased since the LGM due to a retreat in the grounding-line position, while ice in the interior of East Antarctica is predicted to have thickened since the LGM in response to an increase in accumulation rates (as implied by ice-core studies). Model output for the full deglacial reconstruction is then combined with a range of Earth models and used to drive a GIA model. Predictions of relative sea-level change around Antarctica are calculated for different ice model–Earth model combinations, and the misﬁt between these predictions and ﬁeld observations is used to determine the Earth-model parameters most suitable for Antarctica. The accuracy of the GIA model is quantiﬁed by comparing the present-day uplift rates predicted by the model against those measured by GPS [42]. Recently, some regional GIA studies have focussed on areas where the tectonic setting means that ice-covered regions are underlain by very weak Earth rheology (e.g., Iceland, Alaska, Patagonia, the Antarctica Peninsula). In such regions, the shorter relaxation time associated with low mantle viscosity means that the solid Earth response to the main post-LGM deglaciation has largely decayed, and present-day deformation is dominated by the response to ice-sheet changes during the last few thousand years. In these regions, and indeed throughout Greenland and Antarctica, the GIA signal is difﬁcult to interpret because it contains contributions from both past and ongoing ice-sheet changes. Separating these signals is an area of active research, and some groups have moved away from forward-modelling the GIA process and instead tried to infer the background GIA signal by combining satellite observations and model output that relate to changes in the mass, volume and surface mass balance of the ice sheet [44, 45].

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15.4.4 Twentieth Century Ice-Sheet and Sea-Level Changes It has been hypothesised that twentieth century ice-sheet changes could be inferred from observations of spatially variable sea-level change during this period, as recorded by tide gauges. Tide gauge data are available from over 2000 locations worldwide (see www.psmsl.org). They were ﬁrst collected in Amsterdam around 1700 AD, but records spanning > 100 years are only available from *75 sites. Most sites have data covering only the last 40 years (Fig. 15.22a, b), with considerable data gaps in the time series. The data must be corrected for background GIA effects―e.g., relative sea-level fall is recorded at Stockholm due to ongoing isostatic rebound (Fig. 15.22c)―and screened for other effects such as land subsidence due to groundwater pumping (e.g., Bangkok; Fig. 15.22c), sedimentation (e.g., Manila; Fig. 15.22c) and offsets due to a change in instrumentation or tectonic activity (e.g., Nezugaseki; Fig. 15.22c). Even sites which record an apparently steady rate of sea-level change may be complicated by the presence of long-term geological effects (e.g., Honolulu; Fig. 15.22c). Once the longest records have been screened and corrected, it is found that the mean rate of twentieth century sea-level rise lies in the range 1–2 mm/yr [46]. This indicates an increase in the meltwater input to the ocean since the late Holocene (when sea-level rise rates were 0.1–0.2 mm/yr), but the timing of the acceleration and the meltwater source is difﬁcult to pin down. One way of identifying the source is to use the scatter that remains in the GIA-corrected tide gauge rates, which may be attributed to the fact that meltwater from a speciﬁc region (e.g., Antarctica or Greenland) will not be distributed spatially uniformly throughout the ocean: as a

Fig. 15.22 a Global distribution of tide gauge sites. b Distribution of tide gauge sites with at least 40 years of data. c Examples of tide gauge records. All images are taken from the Permanent Service for Mean Sea Level website (www.psmsl.org)

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Fig. 15.23 Map of estimated linear trends in sea-level change due to thermal expansion for the period 1955–2003. Figure taken from Fig. 5.16b of the IPCC Fourth Assessment Report [52]

result of the change in the distribution of surface mass following ice melt, sea level will fall in the near ﬁeld of the ice loss and rise by greater than the eustatic-equivalent amount in the far ﬁeld, thus forming a unique ‘ﬁngerprint’ of sea-level change (Fig. 15.13). By considering how melt may be partitioned between different meltwater sources and minimising the misﬁt to the tide gauge rates, an estimate of twentieth century ice mass loss may be obtained. However, ice volume change is not the only climate-related contribution to sea-level change; steric and dynamic effects must also be considered when interpreting tide gauge data. Figure 15.23 shows the mean rate of sea-level change due to ocean temperature change for the period 1955–2003 as derived from ocean temperature observations. It can be seen that signiﬁcant trends are likely to be present in many tide gauge records due to thermosteric effects (note that these effects are not necessarily steady in space or time due to variations in atmospheric and ocean dynamics), and therefore attempts to use these data to constrain ice volume changes are likely to be biased. We next discuss whether the situation improves during the satellite era.

15.4.5 Satellite Era Ice-Sheet and Sea-Level Changes Satellite altimetry data indicate that the mean rate of sea-level rise seems to have increased since 1993, with current rates estimated to be 3.2 ± 0.4 mm/yr (for 1993–2010; [28]). It is thought that *40% of this rise is due to ocean warming

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Fig. 15.24 Map of observed trends in sea-level change derived from satellite altimetry data for the period 1993-2003. Figure taken from Fig. 5.15a of the IPCC Fourth Assessment Report [52], which was adapted from Cazenave and Nerem [47]

[28], and indeed, Fig. 15.24 shows that the pattern of sea-level trends derived from altimetry (e.g., [47]) resembles the predicted pattern due to thermosteric effects (Fig. 15.23) in some regions of the global ocean. The component of the observed trends due to ice-mass loss remains an open question, because the meltwater ‘ﬁngerprints’ considered in the last section are modulated by the dynamic effect of introducing meltwater into the ocean. Twentieth century ice-sheet changes are therefore best constrained by direct observations of the cryosphere rather than inferences from sea-level records, which are likely to be contaminated by other processes. Satellite observations of changes in ice elevation or ice velocity can be used to infer ice-mass change after accounting for the density of ﬁrn or mass gain via accumulation, respectively. Gravity changes measured by the GRACE satellites can also be used to directly measure the change in the mass of the ice sheet, although GRACE data must ﬁrst be corrected for hydrological and GIA-related processes. For more information about these methods, see Sects. 7.3.2, 7.3.3 and 13.4.

Exercises 15:1

A Maxwell rheology comprises an elastic element in series with a Newtonian viscous element. A mechanical analogue is a spring in line with a dashpot (Fig. 15.5).

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(a) Draw a graph showing the shear strain response of a homogeneous and isotropic Maxwell body to a constant deviatoric stress applied between the times t1 and t2. (b) What is the magnitude of the strain (in terms of the applied stress and the relevant material parameters) immediately after t1 and immediately before and after t2? (c) Write an equation that gives the shear strain response of a Maxwell body to: (i) a constant applied deviatoric stress, (ii) a general, time-varying deviatoric stress. 15:2

Estimation of elastic and viscous parameters for a Maxwell model Earth. (a) The shear modulus of rocks in the upper mantle is estimated from inferences of seismic velocity. Use the seismic velocity model in Fig. 15.6 to estimate a mean value of shear modulus l for upper mantle pﬃﬃﬃﬃﬃﬃﬃﬃ rock. Use the relationship ts ¼ l=q, where ts is the seismic shear wave velocity, l is the elastic shear modulus (for a homogeneous, isotropic material) and q is the density. (b) The viscosity of mantle rock can be inferred from modelling sea-level observations. Sea-level measurements from Scotland indicate a crustal uplift following the deglaciation of the British Ice Sheet of 120 m from 15 ka BP to present. Assuming a mean isostatic driving stress of 0.5 MPa during this period, and a linear viscous flow law, estimate the viscosity of the upper mantle. (Assume that the vertical component of the deformation extends to depths of 500 km in the Earth.)

15:3

Calculating the solid Earth response to ice loading with a Maxwell model. Consider a simple mantle model comprising a single layer with no lithosphere. (a) Use the parameter values you obtained in Exercise 15.2 to deﬁne a Maxwell model for the Earth’s upper mantle. (b) Assume a surface load (e.g., an ice sheet or a glacier) that has characteristic loading periods described by the time-varying deviatoric stress ﬁeld rD ðtÞ ¼ r0 cosð2pt=sÞ. What are the time dependences of the elastic and viscous components of the deformation? (c) Calculate the ratio of the amplitude (ignore phase differences) of elastic to viscous deformation for changes in the ice load with characteristic periods of 1 year (annual changes), 100 years (abrupt climate change), 10,000 years (deglacial scale changes) and 100,000 years (glacial cycle). For which time scales can the elastic component of the response be neglected (i.e., a viscous model would sufﬁce)?

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26. Raymo ME, Mitrovica JX, O’Leary MJ, DeConto RM, Hearty PL (2011) Departures from eustasy in Pliocene sea-level records. Nat Geosci 4:328–332 27. Kopp RE, Simons FJ, Mitrovica JX, Maloof AC, Oppenheimer M (2009) Probabilistic assessment of sea level during the last interglacial stage. Nature 462:863–867 28. IPCC (2013) Climate change 2013: the physical science basis. Contribution of working group I to the ﬁfth assessment report of the Intergovernmental Panel on Climate Change. In: Stocker TF, Qin D, Plattner G-K, Tignor M, Allen SK, Boschung J, Nauels A, Xia Y, Bex V, Midgley PM (eds). C.U.P., Cambridge 29. Lambeck K, Rouby H, Purcell A, Sun YY, Sambridge M (2014) Sea level and global ice volumes from the Last Glacial Maximum to the Holocene. Proc Nat Acad Sci 111:15,296– 15,303 30. Kopp RE, Kemp AC, Bittermann K, Horton BP, Donnelly JP, Gehrels WR, Hay CC, Mitrovica JX, Morrow ED, Rahmstorf S (2016) Temperature-driven global sea-level variability in the Common Era. Proc Nat Acad Sci 113:E1,434–E1,441 31. Deschamps P, Durand N, Bard E, Hamelin B, Camoin G, Thomas AL, Henderson GM, Okuno J, Yokoyama Y (2012) Ice-sheet collapse and sea-level rise at the Bølling warming 14,600 years ago. Nature 483:559–564 32. Clark PU, Mitrovica JX, Milne GA, Tamisiea ME (2002) Sea-level ﬁngerprinting as a direct test for the source of global meltwater pulse IA. Science 295:2438–2441 33. Golledge NR, Fogwill CJ, Mackintosh AN, Buckley KM (2012) Dynamics of the last glacial maximum Antarctic ice-sheet and its response to ocean forcing. Proc Nat Acad Sci 109 (40):16,052–16,056 34. Gomez N, Pollard D, Mitrovica JX (2013) A 3-D coupled ice sheet-sea level model applied to Antarctica through the last 40 ky. Earth Planet Sci Lett 384:88–99 35. Whitehouse PL, Bentley MJ, Le Brocq AM (2012) A deglacial model for Antarctica: geological constraints and glaciological modelling as a basis for a new model of Antarctic glacial isostatic adjustment. Quat Sci Rev 32:1–24 36. Weaver AJ, Saenko OA, Clark PU, Mitrovica JX (2003) Meltwater pulse 1A from Antarctica as a trigger of the Bølling-Allerød warm interval. Science 299:1709–1713 37. Gregoire LJ, Payne AJ, Valdes PJ (2012) Deglacial rapid sea level rises caused by ice-sheet saddle collapses. Nature 487:219–222 38. Liu J, Milne GA, Kopp RE, Clark PU, Shennan I (2016) Sea-level constraints on the amplitude and source distribution of Meltwater Pulse 1A. Nat Geosci 9:130–134 39. Lambeck K, Smither C, Johnston P (1998) Sea-level change, glacial rebound and mantle viscosity for northern Europe. Geophys J Int 134:102–144 40. Peltier WR (2004) Global glacial isostasy and the surface of the ice-age Earth: the ICE-5G (VM2) model and GRACE. Ann Rev Earth Planet Sci 32:111–149 41. Tarasov L, Dyke AS, Neal RM, Peltier WR (2012) A data-calibrated distribution of deglacial chronologies for the North American ice complex from glaciological modeling. Earth Planet Sci Lett 315:30–40 42. Whitehouse PL, Bentley MJ, Milne GA, King MA, Thomas ID (2012) A new glacial isostatic adjustment model for Antarctica: calibrating the deglacial model using observations of relative sea-level and present-day uplift rates. Geophys J Int 190:1464–1482 43. Rutt IC, Hagdorn M, Hulton NRJ, Payne AJ (2009) The Glimmer community ice sheet model. J Geophys Res 114:F02004 44. Gunter BC, Didova O, Riva REM, Ligtenberg SRM, Lenaerts JTM, King MA, van den Broeke MR, Urban T (2014) Empirical estimation of present-day Antarctic glacial isostatic adjustment and ice mass change. The Cryosphere 8:743–760 45. Martín-Español A and 11 others (2016) Spatial and temporal Antarctic Ice Sheet mass trends, glacio-isostatic adjustment, and surface processes from a joint inversion of satellite altimeter, gravity, and GPS data. J Geophys Res Earth Surf 121:182–200 46. Douglas BC (1997) Global sea rise: a redetermination. Surv Geophys 18:279–292

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47. Cazenave A, Nerem RS (2004) Present-day sea level change: observations and causes. Rev Geophys 42: RG3001 48. Davies GF (1999) Dynamic Earth: plates, plumes and mantle convection. C.U.P., Cambridge 49. Shearer PM (1999) Introduction to seismology. C.U.P., Cambridge 50. Mitrovica JX, Milne GA, Davis JL (2001) Glacial isostatic adjustment on a rotating Earth. Geophys J Int 147:562–578 51. Tamisiea ME, Mitrovica JX, Davis JL, Milne GA (2003) Long wavelength sea level and solid surface perturbations driven by polar ice mass variations: ﬁngerprinting Greenland and Antarctic ice sheet flux. Space Sci Rev 108:81–93 52. IPCC (2007) Climate change 2007: the physical science basis. Contribution of working group I to the fourth assessment report of the Intergovernmental Panel on Climate Change. In: Solomon S, Qin D, Manning M, Chen Z, Marquis M, Averyt KB, Tignor M, Miller HL (eds). C.U.P., Cambridge

Ice Sheets in the Cenozoic

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16.1

Introduction

In this chapter we give a historical overview of ice sheets in the Earth’s geological past. The main focus is on the Cenozoic era, which covers the past 65 million years (Myr) of Earth’s history. During the Eocene epoch, starting 55 Myr ago, the climate cooled significantly, and this epoch ended with the inception of the Antarctic Ice Sheet around 34 Myr ago. The next two epochs, the Oligocene and Miocene, witnessed a significant increase of the total global volume of ice, most of which resided in Antarctica. The continent has been covered by an ice sheet similar to its present-day form since the Late Miocene, ca. 10 Myr ago. Glaciation in the Northern Hemisphere was initiated around 2.8 Myr ago, when the global average temperature dropped significantly below the present day (PD) temperature. During the Pleistocene, the past 2.6 Myr, the global climate has oscillated strongly between glacial and interglacial stages. The former are characterised by greatly expanded ice sheets in both hemispheres, whereas during the latter, ice sheets were present only in Antarctica and (partially) in Greenland, and this is similar to the current interglacial: the Holocene. Information concerning past climate on these long time scales is derived predominantly from deep-sea sediment cores. Changes in temperature and sea level can be deduced from oxygen isotope ratios which are measured from fossilised calcite shells embedded in the sediment. Numerous core sites across different oceans have yielded an enormous amount of data, covering time periods ranging from hundreds

B. de Boer (B) Faculty of Science, Earth and Climate Cluster, VU Amsterdam, Amsterdam, The Netherlands e-mail: [email protected] R. van de Wal Institute for Marine and Atmospheric Research (IMAU), Utrecht University, Utrecht, The Netherlands © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_16

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to millions of years. Ice-core records from Greenland and Antarctica (Chap. 12) also provide a suite of information on past climate variability; for example, they link changes in greenhouse gases to changes in ice volume and temperature. However, existing ice-core records go back only to 800,000 years before present, whereas the deep-ocean sediment data provide information over the entire Cenozoic era.

16.2

Forcing Mechanisms

The occurrence of long-term climate variability involves the interaction between ice sheets and many different components of the climate system, such as the carbon and hydrological cycles, the ocean (thermohaline) circulation and the orbital variability in solar forcing. On time scales greater than a million years, topographic changes caused by mountain uplift, and the opening and closing of ocean gateways, also play a significant role. Accordingly, we develop our understanding of the Cenozoic evolution of ice sheets by first considering the main forcing mechanisms of the climate system and their interactions with ice sheets on geological time scales.

16.2.1 Changes in the Carbon cycle A primary mechanism for the control of temperature on the Earth’s surface is the greenhouse effect. Paleoclimate records, notably measurements of CO2 and CH4 in ice cores, show a strong link between temperature and greenhouse gas concentrations in the atmosphere (e. g., [1,2]), as is illustrated in Fig. 16.1. Changes in the carbon cycle are thus implicated in the observed long-term climatic variations. However, the interactions between greenhouse gases, temperature and ice volume are far from straightforward. An increase in atmospheric CO2 concentration, for example, leads to a stronger greenhouse effect. This enhances the absorption of longwave radiation from the Earth and warms the lower atmosphere and the surface. Naturally, changes in atmospheric CO2 concentrations also induce changes in the global carbon cycle, by altering the sources and sinks within the biosphere, soil and the ocean. In turn, positive and negative feedbacks in the transport between the different reservoirs influence CO2 in the atmosphere, which also depends on temperature. Climate change depends on numerous (nonlinear) feedbacks between the individual constituents of the climate system: ice sheets, oceans, atmosphere, and oceanic and terrestrial biomass. The key atmospheric factors behind the mass balance of ice sheets are temperature and precipitation, which themselves depend on the ice sheet elevation: this is the mass-balance height feedback considered in Chap. 10. Ice sheets also respond to ocean warming via enhanced melting of ice shelves. Atmospheric temperature responds to CO2 due to the greenhouse effect, and an additional feedback occurs due to the increased solubility of CO2 in the ocean at lower temperatures. Growth of ice sheets causes ocean lowering, which exposes coral reefs on continental shelves, thus promoting increased carbonate weathering, with a consequent enhanced input of carbon to the ocean. Despite these complexities, the general

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Fig. 16.1 From top to bottom: June 65◦ N insolation (black: [3]), temperature (green: [4]), and atmospheric CO2 concentration from the EPICA Dome C ice core (red: [1]). The dashed lines indicate the present day (PD) values; in the case of CO2 , this is the pre-industrial value of 278 ppmv

consensus is that atmospheric CO2 concentration has gradually decreased over the Cenozoic, and that this was the main forcing mechanism which caused the transition from the early Cenozoic (and Cretaceous: before 65 Myr ago) “greenhouse world” into the late Cenozoic “icehouse world”. One view of the cause of this cooling is that the tectonic collision of India with Asia, which caused the formation of the Himalayas and the Tibetan Plateau, led to increased weathering and thus drawdown of CO2 from the atmosphere.

16.2.2 Orbital Cycles and Climate Variability Apart from internal regulation of the Earth’s climate through the feedbacks described above, the external forcing due to incoming radiation from the sun also causes variability. The received solar radiation varies over very long time scales, but also (at any particular location) on shorter time scales (tens of thousands of years) due to variations in the parameters which describe the Earth’s orbit round the sun. Three key parameters are the eccentricity of the orbit, and the obliquity and precession of the axis of Earth’s rotation (see Fig. 16.2). Their variations are caused by the small perturbative effects of the other planets of the solar system on what would otherwise be perfect Keplerian elliptical orbits of the two-body sun–Earth system.

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Fig. 16.2 The three orbital parameters affecting long-term insolation receipt, as described by Milankovi´c [5]. From left to right: eccentricity refers to the ellipticity of the Earth’s orbit around the sun; precession refers to a secondary rotation of the Earth’s rotation axis; obliquity refers to variations in the tilt of this axis

The Milankovi´c theory [5] associates these orbitally induced changes with climatic variations, and specifically with the cause, or at least timing, of the Pleistocene ice ages. Particularly, Milankovi´c, and others since, consider the driving signal for ice sheet growth to be the received summer insolation at 65◦ N, with the idea that it is the summer melt rate in the Northern Hemisphere which controls whether ice sheets can grow there. The theory received a boost when it was discovered that the spectral peaks in the climate record derived from marine sediment records coincide with those which occur in the received insolation [6].

16.2.2.1 Eccentricity The orbit of the Earth around the sun is an ellipse, and eccentricity is a measure of the elliptic shape of the orbit. Eccentricity is defined as e = f /a; f is the distance from a focus of the ellipse (there are two) to the centre of the ellipse, while a is the length of the semi-major axis; see Fig. 16.2. The sun is located at one of the two focal points of the ellipse. Over time, the Earth’s orbital eccentricity varies around a mean value of 0.028, with a maximum value of 0.058 and a minimum, nearly a circle (for which e = 0), of 0.005. At present, the orbit has an eccentricity of 0.017. The major long-term period of the variations in eccentricity is 413 kyr, while the most well known frequency is that associated with a period of 100 kyr, which arises through a combination of 95 and 125 kyr cycles. Eccentricity is the only orbital parameter which influences the global mean annual insolation, although the impact is very small. Additionally, eccentricity modulates precession.

16.2.2.2 Precession The direction of the Earth’s axis of rotation wobbles around a fixed line, which is perpendicular to the orbital plane. This motion is caused by tidal forces of the Sun and Moon exerted on the solid Earth, modulated by the shift of the entire elliptically shaped orbit of the Earth. The combined effect is called climatic precession, or just

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precession. Naturally, if the orbital shape were circular, i. e., e = 0, this effect would disappear. The two major periods of precession are 19 and 23 kyr, and these regulate the position of the seasons on the orbit (e. g., [7]). Precessional effects are opposite for the two hemispheres.

16.2.2.3 Obliquity The angle of the axis of rotation (tilt) varies with respect to a fixed line perpendicular to the orbital plane of the Earth between a value of 22.1◦ and 24.5◦ . Currently, the axis is tilted at 23.44◦ and is slowly decreasing. The main period, the time it takes for the tilt to shift between the maximum and minimum values and back, is 41 kyr. The change in tilt mainly influences solar insolation over high latitudes. The higher the angle the more radiation is received at high latitudes. Although the main driver of the Earth’s climate is the sun, internal mechanisms related to the carbon cycle seem to be the main driver of long-term climate change. In this view, simulations over multiple glacial cycles with climate models still require an input of insolation and CO2 (e. g., [8]). More recent research is directed towards Earth system models which simulate ice age variability and which incorporate Milankovi´c variability [9].

16.3

From Benthic δ 18 O to Global Ice Volume

As has been shown in Chap. 12, stable oxygen isotopes are prime indicators of past climate variability. Rayleigh distillation (Fig. 12.4) provides the basis of an equation for converting the oxygen isotope ratios (specifically δ 18 O) measured in ice cores to paleotemperature. An equivalent approach can be used for the interpretation of oxygen isotopes extracted from fossilised benthic foraminiferic shells. The δ 18 O is measured from the calcite shell, with respect to the Vienna Pee Dee Belemnite (VPDB) standard. The VPDB reference value for calcite δ 18 O ratios is related to VSMOW (the standard for ocean water) by the equation VPDB = VSMOW − 0.28. The calcite δ 18 O ratio is influenced by several properties of the ocean environment. The two main components affecting the uptake of 18 O in the calcite shell are local temperature and the δ 18 O of the water [10]. Evaporation of sea water preferentially removes the lighter oxygen isotope from the water, so that moisture reaching the polar regions to form accumulation on ice masses has very negative δ 18 O (Fig. 12.5). Since net evaporation of the sea water enriches the water in the heavier isotope, the build-up of global (land) ice volume causes the δ 18 O of the ocean (henceforth denoted by δw ) to rise, and consequently the value of benthic δ 18 O increases. This implies that the responses of the oxygen isotope ratios in the ocean and in the ice to climate variations have opposite signs. A climatic cooling leads to an increase in δ 18 O of the ocean and of the benthic foraminifera, whereas the δ 18 O of the ice decreases. To calculate δw , one can use the

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mass conservation equation δw Vo + δ 18 Oi Vi = K ,

(16.1)

where K is a constant, Vo and Vi are respectively the ocean volume and global ice volume (in km3 water equivalent), δw is the ocean-water δ 18 O, and δ 18 Oi (which is negative) is the mean ice-sheet δ 18 O (both in VSMOW). The constant K is defined here as the PD contribution of δ 18 O stored in the ice, scaled with ice and ocean volume. Rewriting (16.1) as a function of δw relative to PD yields δ 18 Oi Vi δ 18 Oi Vi δw = − + Vo Vo

.

(16.2)

PD

The first term on the right-hand side contains the time-dependent variables influencing δw . Calculations show that δ 18 Oi varies only a little over a glacial cycle, so that the change in δw mainly depends on changes in global ice volume Vi . The second term is a constant reflecting the PD reference value. As mentioned earlier, the benthic δ 18 O depends on δw (and thus on ice volume change) but is also affected by the water temperature. In colder water the shells have a preferred uptake of the heavier isotope of oxygen, and this causes an additional increase of the δ 18 O value. This second important contribution, the influence of deep-water temperature, is parameterised as a linear relation between the change in deep-water temperature, Tdw , and the change in δ 18 O from temperature, δT , both relative to PD: δT = γ Tdw .

(16.3)

The value of γ is taken from the work of Duplessy et al. [14], who derived a linear relation between benthic δ 18 O and deep-water temperatures with a slope of −0.28 ◦ C−1 . The changes in benthic δ 18 O can be computed by combining the two contributions in (16.2) and (16.3): δ 18 Oi Vi δ 18 Oi Vi δ Ob = − + Vo Vo

+ γ Tdw ,

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(16.4)

PD

with denoting values relative to PD. The results of this modelling exercise will be described in Sect. 16.4.3.

16.4

Cenozoic Evolution of Ice Volume

The start of the Cenozoic is associated with the K-T (Cretaceous-Tertiary) Boundary. At this time a well-known mass extinction event occurred, in which all forms of dinosaurs other than birds died out: this occurred around 65 Myr ago. Although temperatures remained relatively warm thereafter, long-term cooling was initiated

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Fig. 16.3 a Stacked benthic δ 18 O from [11] indicating fluctuations in both ice volume and temperature. Labels along the top give the geological time scale: Plt: Pleistocene, Plioc: Pliocene. Horizontal bars indicate the presence of the Antarctic Ice Sheet (orange) and Northern Hemisphere glaciation (blue). MMCO: Mid-Miocene Climate Optimum; Mi-1: Miocene isotope excursion 1; Oi-1: Oligocene isotope excursion. b Proxy derived CO2 estimates from a range of different proxies indicated by the various colours [12]. The smoothed blue line with error margin shows the evolution of CO2 estimated from the compilation. Red colour band shows uncertainty in the CO2 threshold for growth of a continental scale Antarctic ice sheet [13]. Horizontal dashed lines in a and b indicate the pre-industrial values of 3.23 and 280 ppmv for δ 18 O and CO2 , respectively

around 49 Myr ago in the early Eocene. Through the course of the Cenozoic the climate cooled from a previously warm state dominated by high concentrations of greenhouse gases [12] to one controlled by ice sheets since the Eocene-Oligocene transition about 40 Myr ago. The long-term cooling trend, assumed to be associated with a decrease in atmospheric pCO2 (here p stands for partial pressure), led to the initiation of Antarctic ice during the late Eocene. The top panel of Fig. 16.3 shows benthic δ 18 O measurements from marine sediment cores, which indicate variations in temperature and sea level [11], while the lower panel shows a collection of pCO2 estimates derived from proxy data found in deep-sea sediments.

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16.4.1 Inception of Antarctica The most distinct climate transition during the Cenozoic is the Eocene-Oligocene transition, ∼ 34 Myr ago, also known as Oi-1. It is marked by a strong increase in benthic δ 18 O (Fig. 16.3a). This transition is well known as the first period with major glaciation on Antarctica. Several causes have been proposed for it, such as a threshold response to a decrease in atmospheric CO2 concentrations, or the opening of ocean gateways. As is indicated by the benthic δ 18 O records, the temperature dropped significantly both in the atmosphere and in the deep ocean. Although the exact values of pCO2 in the atmosphere cannot be deduced, proxy-based estimates indicate a significant decrease in pCO2 that coincided with lowering temperatures and an increase in ice volume. The estimated total increase in ice volume of the Antarctic Ice Sheet is a significant fraction of its present-day volume, but the point of inception is rather uncertain (Fig. 16.3b; [13]).

16.4.2 Oligocene and Miocene Variability Following the inception of the Antarctic Ice Sheet, the Oligocene and Miocene are characterised by large fluctuations in temperature, ice volume and analogous changes in atmospheric pCO2 . The Miocene is characterised by several oxygen isotope excursions in the benthic δ 18 O records which are related to fluctuations in Antarctic ice volume (Fig. 16.4). One of the strongest excursions is the Mi-1 event occurring at the Oligocene-Miocene boundary. The Mi-1 event marks a strong increase in ice volume and lower temperatures at ∼ 23 Myr ago. It can be seen in Fig. 16.4 that carbon and oxygen isotope data vary fairly coherently during the Miocene. Another distinct period is the Mid-Miocene Climate Optimum (MMCO) ∼ 15 Myr ago, a warm period lasting for about 2 Myr with reduced glaciation; temperatures then were higher than at the present day (PD). Following the MMCO, ice volume increased rapidly, leading to a permanent ice sheet on Antarctica, which is likely to have been of a similar size to the PD ice sheet. As for the Oi-1 event, several causes have been hypothesised for this rapid increase in ice volume, for example the effect of decreasing pCO2 in causing a hysteretic growth of the ice sheet due to the elevation–accumulation feedback (see Chaps. 10 and 18). Although little is known about the precise behaviour of sea ice during this era, a few studies have evaluated the existence of sea ice during the Cenozoic. The first sea ice is thought to have existed around Antarctica, in response to the growth of the Antarctic Ice Sheet. The strong increase in Antarctic ice volume after the MMCO seems to be a global signal of climate change, since there is apparent evidence of Arctic sea ice at the same time (e. g., [19]). After sea-ice cover was established in the Arctic, Antarctic ice volume remained rather constant whereas temperatures continued to decrease.

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Fig. 16.4 a Benthic δ 13 C from Site 1264, indicating variability in the carbon cycle. b Benthic δ 18 O indicating changes in ice volume and temperature. c A reconstruction of eustatic sea level relative to PD, and d a reconstruction of the global surface-air temperature anomaly relative to PD. Sea level and temperature are deduced from the benthic δ 18 O as described in [15] and [16]. The black lines indicate the 400 kyr running mean

16.4.3 Pleistocene Ice Ages Over the past 5 Myr, during the Pliocene and Pleistocene, the Earth’s climate fluctuated strongly, as indicated by the benthic δ 18 O data in Fig. 16.3. The benthic δ 18 O data mostly show values higher than PD values, indicating temperatures below those of the PD, and the presence of large ice sheets in the Northern Hemisphere (NH). Glaciation in the NH started during the Pliocene-Pleistocene transition, around 2.8 Myr ago. During the Pleistocene ice ages, vast ice sheets covered large parts of the NH continents and the Antarctic Ice Sheet expanded to the continental edge, as illustrated in Fig. 16.5. Due to the large increase of ice volume on land, regional sea level was as low as ∼130 m below PD sea level [20]. The occurrence of the Pleistocene ice ages is commonly attributed to Milankovi´c variations in solar forcing, but there is a striking dissimilarity between the deepsea records and insolation variations during this period. The main periodicity of

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Fig. 16.5 Ice coverage during the Pleistocene glacial cycles [17]. Maximum glaciation is shown in transparent grey; the PD ice coverage is shown in black. Topography is from the SRTM30_PLUS dataset [18] Fig. 16.6 Spectral analysis of insolation [3] and ice volume [21]. Summer insolation change at 65◦ N (top panel) has a strong precession and a weaker tilt/obliquity signature. In contrast, ice volume from 3.0 to 0.9 Myr ago shows a strong 41,000 year cycle in response to obliquity variations (bottom panel). After 0.9 Myr ago, ice volume shows a strong 100,000 year cycle (bottom panel), which is only very weakly present in the insolation variations

the climate signal over the last million years is that of the 100 kyr ice age cycle, whereas the insolation data show very little power at the corresponding frequency. Furthermore, the climate signal underwent a change in frequency at the so-called ‘Mid-Pleistocene Transition’ (MPT), as indicated in Figs. 16.6 and 16.7. The MPT indicates the transition from a dominant 41 kyr (obliquity) cycle to a 100 kyr glacial cycle between 0.8 and 1.2 Myr ago. Several explanations have been proposed to explain this transition, mainly associated with nonlinear responses within the climate system.

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Fig. 16.7 a The benthic δ 18 O LR04 stack (black: [22]). b Surface-air temperature anomaly over the NH continents, with the right-hand scale showing the deep-water temperature anomaly (green: [16]). c Sea level change associated with all ice-volume contributions (black, total ice volume in metres sea-level equivalent), and the component contributions from Eurasia (red), North America (blue), Greenland (green) and Antarctica (orange). Labels on top indicate the transition from the 41 kyr to 100 kyr glacial cycles across the Mid-Pleistocene Transition (MPT). NHG: Northern Hemisphere Glaciation

16.5

The Last Glacial Cycle

During the Pleistocene ice ages, and especially during the past million years, glaciations were largely dominated by the major expansion of ice sheets across Eurasia and North America, with an estimated combined ice volume corresponding to more than 100 m of eustatic sea level lowering. Furthermore, the Antarctic Ice Sheet extended towards the continental shelf, and its increased volume lowered sea level by an additional ∼ 10 m [19]. The last glacial period has been studied more extensively and in much greater detail than the rest of the Cenozoic era; the variations of observed greenhouse gases, temperature, sea level and ice volume for this period are shown in Fig. 16.8. Two prominent features of the climate over the last glacial cycle are the sawtoothshaped fluctuations superimposed on the long-term cooling from the previous inter-

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pCO2 (ppmv)

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Time (kyr ago) Fig. 16.8 Data and reconstructions spanning 130 kyr ago to present day. a Atmospheric pCO2 [1]. b Temperature over Greenland from the NEEM ice core (red: NEEM community members [23]) and the GRIP ice core (green: North Greenland Ice Core Project members [24], EPICA Community Members [25]), and temperature in Antarctica from the Dronning Maud Land ice core (orange: EPICA Community Members [25]). c Change in relative sea level from observations. A compilation of coral-reef data (black: Thompson and Goldstein [26]), Red Sea record derived from oxygen isotopes (red: Grant et al. [27]) and a reconstruction of the Mediterranean (green: Rohling et al. [28]). d Reconstructed ice volume in metres sea level equivalent (m s.e.) (black) with the constituent contributions (red, blue, green, orange) following the same colour scheme as in Fig. 16.7c (de Boer et al. [16])

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glacial, the Eemian (∼125 kyr ago), towards the Last Glacial Maximum (LGM; ∼18 kyr ago), and a fast termination of the glacial period within 10 kyr. Moreover, the high-resolution ice core records from Greenland show sharp jumps occurring at intervals of hundreds to thousands of years (Fig. 16.8b); these are known as Dansgaard-Oeschger events, and the sharp rises in temperature (of the order of 10 ◦ C) had a decadal time scale. These large peaks in temperature are also related to changes in Antarctic warm peaks [25]; see also Chap. 12. Additional (multi-)millennial scale climate variability during the glaciation is associated with the occurrence of Heinrich events that involved major ice discharge into the North Atlantic ocean (e. g., [29]).

16.5.1 The Previous Interglacial—The Eemian Over the past 800 kyr, there have been several interglacial periods that show high CO2 concentrations and high temperatures (Fig. 16.1). The previous interglacial, the Eemian (∼ 128–115 kyr ago), is a widely studied interval because it can be compared to the current interglacial. Temperature estimates for the Eemian indicate that there existed a warmer than PD climate, with sea level estimated to be 4–9 m higher than PD sea level. The extent of the Greenland Ice Sheet during the Eemian and its consequent contribution to the Eemian sea-level highstand are quite uncertain. Examples of model reconstructions of the configuration and volume of the Greenland Ice Sheet during the Eemian are shown in Fig. 16.9. The three simulations, all based on the use of numerical ice sheet models, provide an estimated contribution to sea-level

Fig. 16.9 The results of three modelling studies showing the extent of the Greenland Ice Sheet during the Eemian interglacial. a At 122.4 kyr ago, from Robinson et al. [30]. b At 121 kyr ago, from Helsen et al. [31]. c At 123.5 kyr ago, from Stone et al. [32]

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rise during the Eemian of between 0.3 and 4.4 m. These estimates depend on the geodynamical and glacio-dynamical models used, as well as the assumptions for the surface mass balance of the ice sheet. The ice-core records also provide important constraints on the extent of the Greenland Ice Sheet during the Eemian (and since then). Records from Dye-3 indicate the presence of Eemian ice at that location, corroborating the results in Fig. 16.9a,c. Furthermore, biological material from the basal ice at the Dye-3 site has been dated back to 400 kyr ago, and this suggests that the ice sheet has not retreated from this location since then. Lastly, results from the NEEM ice core indicate only minor changes in ice thickness in central/northern Greenland. Although uncertainties are quite large, an additional ice volume contribution from the West Antarctic Ice Sheet is needed to explain the total estimated rise in sea level.

16.5.2 The Last Glacial Maximum Another key period which has been studied intensely is the Last Glacial Maximum (LGM). As shown in Fig. 16.5, at that time the NH was covered by large ice sheets on both continents, which increased both the elevation and the albedo of the surface. Hence paleoclimate reconstructions show significant decrease in atmospheric temperature during this time interval (Fig. 16.10), which has also been identified in ice cores.

Fig. 16.10 Reconstruction of Last Glacial Maximum surface air temperature anomaly (◦ C) based on multi-modal regression. Source: Annan and Hargreaves [33]

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The timing of the LGM is generally given as ∼ 21 kyr before present. However, this largely depends on the location of the data. As can be seen from Fig. 16.8a, the minimum value of CO2 is around 20 kyr ago, whereas minimum temperatures on Greenland are reached well before; on the other hand they occur in Antarctica somewhat later (Fig. 16.8b). Moreover, the minimum sea level is estimated from proxy data to have occurred around 22 kyr ago (Fig. 16.8c), whereas reconstructions with ice-sheet models, based on benthic δ 18 O data, indicate a minimum at 18 kyr ago (Fig. 16.8d).

Exercise 16.1 It is commonly stated that the Milankovi´c theory provides a suitable explanation for the occurrence of the Pleistocene ice ages. Discuss the merits or demerits of this suggestion, with reference to Figs. 16.1, 16.6 and 16.7. In particular, can you use the last of these figures in conjunction with the ice sheet dynamical model of Fig. 10.5 to suggest a mechanism for the transition from the 41 kyr world to the 100 kyr world?

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15. Liebrand D, Lourens LJ, Hodell DA, de Boer B, van de Wal RSW, Pälike H (2011) Antarctic ice sheet and oceanographic response to eccentricity forcing during the early Miocene. Clim Past 7(3):869–880 16. de Boer B, van de Wal RSW, Lourens LJ, Bintanja R (2013) A continuous simulation of global ice volume over the past 1 million years with 3-D ice-sheet models. Clim Dyn 41:1365–1384 17. Ehlers J, Gibbard PL (2007) The extent and chronology of Cenozoic global glaciation. Quat Int 164–165:6–20 18. Becker JJ and 17 others (2009) Global bathymetry and elevation data at 30 arc seconds resolution: SRTM30_PLUS. Marine Geodesy 32(4):355–371 19. Krylov AA, Andreeva IA, Vogt C, Backman J, Krupskaya VV, Grikurov GE, Moran K, Shoji H (2008) A shift in heavy and clay mineral provenance indicates a middle Miocene onset of a perennial sea ice cover in the Arctic Ocean. Paleoceanography 23(1):PA1S06 20. Simms AR, Lisiecki L, Gebbie G, Whitehouse PL, Clark JF (2019) Balancing the last glacial maximum (LGM) sea-level budget. Quat Sci Rev 205:143-153 21. de Boer BL, Lourens J, van de Wal RSW (2014) Persistent 400,000-year variability of Antarctic ice volume and the carbon cycle is revealed throughout the Plio-Pleistocene. Nat Commun 5:2999 22. Lisiecki L, Raymo M (2005) A Pliocene-Pleistocene stack of 57 globally distributed benthic δ 18 O records. Paleoceanography 20:PA1003 23. NEEM Community Members (2013) Eemian interglacial reconstructed from a Greenland folded ice core. Nature 493(7433):489–494 24. North Greenland Ice Core Project Members (2004) High-resolution climate record of the Northern Hemisphere reaching into the last interglacial period. Nature 431:147–151 25. EPICA Community Members (2006) One-to-one coupling of glacial climate variability in Greenland and Antarctica. Nature 444:195–198 26. Thompson WG, Goldstein SL (2006) A radiometric calibration of the SPECMAP timescale. Quat Sci Rev 25(23–24):3207–3215 27. Grant KM and 10 others (2014) Sea-level variability over five glacial cycles. Nat Commun 5:5076 28. Rohling EJ, Foster GL, Grant KM, Marino G, Roberts AP, Tamisiea ME, Williams F (2014) Sea-level and deep-sea-temperature variability over the past 5.3 million years. Nature 508(7497):477–482 29. MacAyeal D (1993) Binge/purge oscillations of the Laurentide Ice Sheet as a cause of the North Atlantic’s Heinrich events. Paleoceanography 8:775–784 30. Robinson A, Calov R, Ganopolski A (2011) Greenland ice sheet model parameters constrained using simulations of the Eemian interglacial. Clim Past 7(2):381–396 31. Helsen MM, van de Berg WJ, van de Wal RSW, van den Broeke MR, Oerlemans J (2013) Coupled regional climate-ice sheet simulation shows limited Greenland ice loss during the Eemian. Clim Past 9:1773–1788 32. Stone EJ, Lunt DJ, Annan JD, Hargreaves JC (2013) Quantification of the Greenland ice sheet contribution to last interglacial sea level rise. Clim Past 9(2):621–639 33. Annan JD, Hargreaves JC (2013) A new global reconstruction of temperature changes at the Last Glacial Maximum. Clim Past 9:367–376

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Arjen Stroeven, Clas Hättestrand, Krister Jansson, and Johan Kleman

17.1

Introduction

Paleoglaciology, taken to mean the science of reconstructing the outline, volume and dynamics of former glaciers and ice sheets from geomorphology, has been revitalized during the past thirty years. This chapter aims to review basic knowledge of the landforms used in paleoglaciological reconstructions, as well as our limitations in understanding their formational processes (the forward problem); the possibilities and pitfalls of data acquisition, data reduction, and cartographic practices; and the methodology developed by Kleman and Borgström [1] for the reconstruction of ice sheets (the inversion problem). The integration of stratigraphical knowledge and absolute chronology is an essential ingredient of modern paleoglaciology; we will highlight the importance of these topics, with some focus on cosmogenic nuclide dating. We end the chapter with glaciological insights gained from paleoglaciology. Robust paleoglaciological interpretations and reconstructions require proﬁciency in dealing with the three fundamental domains of geomorphology: (genetic) processes, space and time. They further require an in-depth knowledge and efﬁcient handling of large volumes of data. The presence of landforms and landform assemblages, or spatial gradients in these, can provide information about glacial processes and help deﬁne glaciological parameters at the time of landform formation. The spatial range over which these parameters remain valid is established through the classiﬁcation, spatial delineation, and tracing of internally coherent landform systems. The temporal range (the age and age span) over which inferred spatial patterns and processes remain valid relies on relative-age relationships (from stratigraphy or cross-cutting of landforms), as well as the careful use of absolute age A. Stroeven (&) C. Hättestrand K. Jansson J. Kleman Department of Physical Geography, Stockholm University, Stockholm, Sweden Bolin Centre for Climate Research, Stockholm University, Stockholm, Sweden e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. Fowler and F. Ng (eds.), Glaciers and Ice Sheets in the Climate System, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-42584-5_17

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constraints. Each dating technique may necessitate the use of auxiliary models to clarify the relation between an absolute date and the geomorphological context for which an age assignment is sought.

17.2

Glacial Landforms

We begin with a brief tour of glacial landforms and their formative conditions and processes, considering matters relevant to the inversion context. Subglacial landforms are the principal source of information for reconstructing the glaciological parameters of former ice sheets. This is because the nature of the basal interface (between ice and substrate) depends on those parameters, and in particular the interface is insulated from short-term changes in climate. In contrast, ice-marginal landforms carry information about climatic, proglacial hydrological or oceanographic processes as well as the glaciological conditions.

17.2.1 Formation Time and Landform Size-Scale Subglacial landforms occur at a variety of length scales, and broadly speaking the smaller scale features have a shorter time of formation. It is useful to examine the different formation times of the landforms (Fig. 17.1) alongside their size-scale progression.

Fig. 17.1 Formation time of landforms used in paleoglaciological reconstructions

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Small-scale erosional landforms Figure 17.2 shows several small-scale erosional forms: friction cracks, crescentic gouges and striae. Friction cracks are small crescentic features, centimetres to metres in length, found on deglaciated bedrock/pavements (Fig. 17.2b), often on the stoss (upstream) side of roches moutonnées. Their long axes are transverse to the direction of ice flow. They are generally thought to be associated with a build-up of differential friction between the bed and the overlying (debris-ﬁlled) ice, which results in down-dipping fractures in the bedrock. When the fractures result in gouging of the rock surface, the features are known as crescentic gouges (Fig. 17.2a). Glacial striae (or striations) are linear marks scored into bedrock, and are due to the scraping of the bed by rocks embedded in the ice (Fig. 17.2c). They are thus aligned with the direction of ice flow. Glacial grooves have similar origin but have larger wavelength. Imprints such as these that form almost instantaneously (from minutes to weeks; Fig. 17.1) convey only a snapshot picture of the (basal) ice flow. Due to their abundance and the extensive ﬁeldwork needed to record them, they may be difﬁcult to assemble comprehensively and objectively over large spatial scales. It is moreover difﬁcult to ascertain that they formed synchronously and therefore show a truthful impression of the subglacial system at one time. Also, these small-scale imprints are often strongly dependent on subglacial topography; for example, their direction may locally deviate substantially from the mean ice flow direction, making them difﬁcult to analyze on an ice-sheet scale. However, where the topography is negligible compared to the ice thickness, these imprints usually portray a consistent direction, and different sets of ice flow directions may be preserved on different bedrock facets. Such locations are ideally suited to determine the relative age of ice flow events, because younger flow generations will have entirely or partly erased imprints of older flow generations, and under favourable conditions their erosional relationship may be determined—for example, younger striae cutting through older ones (Fig. 17.2c), or older sets of striae being preserved on lee-facets of younger flow directions [2]. Where such inferred flow directions can be linked to the flow directions inferred from larger landforms (such as lineations, see below), their stacked record can help us interpret the glaciological history over broader spatial scales. In addition, small-scale erosional landforms are sometimes the only ice flow directional indicator available, such as in landscapes of areal scouring. Large-scale erosional landforms Figures 17.2d–e show examples of larger-scale features which are formed through the erosion of bedrock. Roches moutonnées are formed through the abrasion and plucking (by fracture) of bedrock knobs, which yields a typical smooth stoss face and a more jagged lee face (left- and right-hand sides, respectively, in Fig. 17.2d). A rock drumlin is an example of a similarly-formed but smoother feature at a larger scale (hundreds rather than tens of metres). Unlike rivers, glaciers

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Fig. 17.2 Examples of glacial landforms mentioned in the text. Panels a to e: small-scale and large-scale glacial erosional landforms. a Crescentic gouge, Svartisen, northern Norway. b Friction cracks, Svartisen, northern Norway. c Striae/striations, Baillie Hamilton Island, Arctic Canada. d Roche moutonnée, Lake Sommen, southern Sweden. e Fjord and hanging valleys, Bafﬁn Island, Arctic Canada. Panels f to h: intermediate-sized subglacial landforms used in paleoglaciological reconstructions, presented in map form (Digital Elevation Model [DEM], left panels) and photographs (right panels). f Lineations, DEM and photo: Nilivaara, north-eastern Sweden. g Ribbed moraine, DEM and photo: Lake Rogen, west-central Sweden. h Eskers, DEM: Gävleborg, east-central Sweden; photo: Storuman, east-central Sweden. Arrows indicate ice flow direction during landform formation. Panels i to k (opposite): intermediate-sized ice-marginal landforms used in paleoglaciological reconstructions, presented in map form (DEM, left panels) and photographs (right panels). i End moraines, DEM: Skövde, south-western Sweden; photo: Keiva moraine, south-eastern Kola Peninsula, Russia. j Marginal meltwater channels, DEM and photo: Härjedalen, west-central Sweden. k Glacial lake features, DEM: Oviksfjällen, west-central Sweden (glacial lake shorelines, outlet channel and perched delta); photo: Sarek, north-western Sweden (outlet channel, perched delta). Arrows indicate ice flow direction during landform formation

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Fig. 17.2 (continued)

erode their valleys into a characteristic U-shape; when inundated by the sea, these form fjords (Fig. 17.2e). Other typical remnants of glacial erosion are cirques (bowl-shaped depressions) and hanging valleys (Fig. 17.2e). Features such as these typically require formation times in excess of 104 years (Fig. 17.1). They are difﬁcult to employ in the paleoglaciological reconstruction of a speciﬁc time slice (e.g., a time slice port