Relating Subglacial Water Flow to Surface Velocity Variations of Breiðamerkurjökull, Iceland. Tayo van Boeckel

Transcription

1 Relating Subglacial Water Flow to Surface Velocity Variations of Breiðamerkurjökull, Iceland Tayo van Boeckel Faculty of Earth Sciences University of Iceland 2015

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3 Relating Subglacial Water Flow to Surface Velocity Variations of Breiðamerkurjökull, Iceland Tayo van Boeckel 60 ECTS thesis submitted in partial fulfillment of a Magister Scientiarum degree in Geophysics Advisors Alexander H. Jarosch Guðfinna Aðalgeirsdóttir Examiner Tómas Jóhannesson Faculty of Earth Sciences School of Engineering and Natural Sciences University of Iceland Reykjavík, 20 June 2015

4 Relating Subglacial Water Flow to Surface Velocity Variations of Breiðamerkurjökull, Iceland Subglacial Hydrology & Glacier Velocity Variations 60 ECTS thesis submitted in partial fulfillment of a Magister Scientiarum degree in Geophysics Copyright 2015 Tayo van Boeckel All rights reserved Faculty of Earth Sciences School of Engineering and Natural Sciences University of Iceland Sturlugötu 7 101, Reykjavík Iceland Telephone: Bibliographic information: Tayo van Boeckel, 2015, Relating Subglacial Water Flow to Surface Velocity Variations of Breiðamerkurjökull, Iceland, Master s thesis, Faculty of Earth Sciences, University of Iceland, pp. 71. Printing: Háskólaprent Reykjavík, Iceland, 20 June 2015

5 Abstract The ways in which subglacial hydrology relates to velocity variations of glaciers has been a topic of discussion for several decades. Studies have revealed that changes in sliding, water pressure and water storage do not correlate in phase. In particular, observations have indicated that the overwhelming of the subglacial drainage systems results in increased sliding. Studies have therefore hypothesized that a dynamically evolving subglacial drainage system controls the sliding velocity by adjusting its capacity to variable surface water input. In addition, studies have hypothesized that velocities at the surface are also affected by longitudinal stress gradients caused by spatial variabilities in the drainage system capacities. For this study, GPS data from two locations on Breiðamerkurjökull, Iceland, during 2010, 2012 and 2013, are processed and analyzed. The combination of both hypotheses is tested against the surface velocity measurements. Raininduced speed-up events indicate that the overwhelming of the drainage system indeed causes rapid acceleration. The drainage system is capable of increasing its capacity within days because a series of rainstorms have decreasing response on glacier acceleration. Sliding velocities are derived after subtracting the ice deformation velocities. These creep velocities are estimated during typical winter weather conditions, when sliding is assumed to be negligible. Velocities calculated with a Full Stokes flowline model support the estimated winter velocities. Basal motion generally peaks in spring and decreases during summer, indicating that the drainage system evolves to greater efficiency during the melt season. Finally, a new conceptual model of the drainage system is presented in an attempt to address the problem of temporal variations of basal motion, water pressure and water storage. The model combines channels, cavities and sheets that dynamically interact on each other.

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7 Dedicated to Wilderness

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9 Table of Contents TABLE OF CONTENTS... IX LIST OF FIGURES... XI VARIABLE NAMES... XIII ACKNOWLEDGEMENTS... XV CHAPTER 1 INTRODUCTION General Introduction Study Area: Breiðamerkurjökull Bedrock Topography Surface Topography Historical Retreat and Thinning Theory of Subglacial Hydrology and Sliding The Challenge of Subglacial Hydrology Two Types of Drainage Systems Hypothesis: Dynamic Evolution of the Subglacial Drainage System Surface Velocity and Basal Conditions Physics of Glacier Flow The Global Position System (GPS) What is GPS? Relative Kinematic Positioning Long Baseline Processing Software: RTKLIB CHAPTER 2 METHODS Deriving Velocities from GPS Displacements Tuning the Parameters for GPS Processing Deriving Velocities from Displacements GPS Error Analysis Weather Data Temperature Ablation Precipitation Flow Model Modelling Ice Flow Boundary Conditions ix

10 2.3.3 Flowline Surface Topography Retreat of Breiðamerkurjökull Creating Grid CHAPTER 3 RESULTS Time Series of Velocities Daily Melt-Induced Fluctuations Speed-up Events Pre-Melt season: Winter Early Melt season: Spring Late Melt season: Fall Interpretation According to Hypotheses Ice Deformation Estimating Winter Velocities Modelling Ice Deformation Seasonal Evolution of the Drainage System CHAPTER 4 DISCUSSION Anomalous Correlations Comparing Ablation with Precipitation Multiple Linear Regression Theory Results of MLR Model Evaluation Lateral Drag (ew, drag) Surface Velocity at Calving Front (ew, velocity) Surface Topography (ew, profile) Calving Front Location (ew, margin) Summing of Model Errors Glen s Flow Parameters Discussion of Subglacial Hydrology Alternative Subglacial Hydrology Models Uplift of Breiðamerkurjökull Proposed Conceptual Model of the Drainage System CHAPTER 5 CONCLUSIONS REFERENCES x

11 List of figures Figure 1: Breiðamerkurjökull as seen from Jökulsárlón, picture courtesy of Tayo van Boeckel, Figure 2: Vatnajökull ice cap with important outlet glaciers on September 6, 2014 (false color Landsat 8 image, courtesy of NASA). The Breiðamerkurjökull watershed is outlined after Björnsson & Pálsson (2008). Note the Holuhraun eruption in the north Figure 3: Bedrock trough in Breiðamerkurjökull courtesy of Björnsson et al. (2001). Blue colors mark elevations below sea level Figure 4: The retreat and thinning of Breiðamerkurjökull over the past century. Zero distance marks the coast line. Graph courtesy of Björnsson et al. (2001) Figure 5: The retreat of Breiðamerkurjökull, drawn approximately every second year from 1983 to 2014, is overlain on a Landsat 8 image of September 6, 2014 (Courtesy of NASA). The retreat is measured from greatest to smallest extent along the pink line Figure 6: Schematic interpretation of the hysteresis in the temporal relationships between horizontal velocity and water pressure (Sugiyama & Gudmundsson 2004) and bed separation (Howat et al. 2008) during a glacier speed-up event Figure 7: Examples of the two types of drainage systems modified after Flowers (2015) Figure 8: Accuracy and precision. Graph courtesy of Langley (2010) Figure 9: Distribution of positions of a fixed single frequency receiver in Reykjavík derived from kinematic processing of 20 days of data. (a) Density function of positions and a Gaussian distribution. (b) Box plot including 90 % of the data within the whiskers. The black dot marks the mean Figure 10: Weather data from Breiðamerkurjökull during the year (a) Temperature. (b) Ablation. (c) Precipitation. Data from the HARMONIE model and Kvísker weather stations are from the IMO, provided by N. Nawri (Personal communication, October, 2014) and H. Ágústsson (Personal communication, October, 2014), respectively. Data from the AWS on Breiðamerkurjökull, including the ablation measurements are from the Institute of Earth Sciences, HÍ, provided by F. Pálsson, November, 2014) Figure 11: Map of the Breiðamerkurjökull area, including three estimated flowlines and locations providing weather data. Landsat 8 image of September 6, 2014 (Courtesy of NASA) Figure 12: Glacier bedrock and the 2010 and 2014 surface profiles. Solid surface profiles are measured while dashed profiles are estimated. Elevations are in WGS84 geoid reference frame where the lagoon level of 65.5 m is equal to sea level. The 2014 profile is extrapolated above the measurements by fitting to the ablation. Bedrock data are from Björnsson et al. (2001) Figure 13: (a) Outline of flowline cross section vertically exaggerated by a factor 2. (b) Mesh of flowline cross section near the terminus as prepared by Gmsh Figure 14: Year Upper tile: horizontal velocities of ICE-A and ICE-D smoothed with a 24 h sliding window. Interpolation of the horizontal velocities is drawn in gray and light blue color for both rovers. Middle tile: detrended uplift of ICE-A and ICE-D. Note that data gaps are not highlighted. Lower tile: Weather data including temperature, ablation, rain and snow with sources described in section Figure 15: Year See plotting scheme description in Figure Figure 16: Year See plotting scheme description in Figure 14. Note that at day 198, 2013, ICE- D was moved back upstream behind ICE-A xi

12 Figure 17: Daily melt-induced fluctuations during early August Alternating gray shading marks days. Upper tile: horizontal velocities of ICE-A and ICE-D smoothed with a 6 h sliding window. Interpolation of the horizontal velocities is drawn in gray and light blue color for both rovers. Middle tile: detrended uplift of ICE-A and ICE-D. Note that data gaps are not highlighted. Lower tile: Weather data including temperature, ablation, rain and snow with sources described in section 2.2. Water input is given in bins, where the area of the bin equals the amount of surface water input. However, note that the ablation measurements are not representable for the entire glacier and cannot be compared to the rain input (see section 2.2.3) Figure 18: Correlation between velocity and temperature between days 212 and 232, (a) The correlation with a linear regression fit. (b) The time lag between velocity and temperature variations calculated in steps of 1 h Figure 19: (a) Pre-melt season speed-up during The red dashed line marks the bend point. (b) Early melt season speed-up during See plotting scheme description in Figure Figure 20: (a) Late melt season speed-up event in (b) Late melt season speed-up event in See plotting scheme description in Figure 17. The red dashed line marks the bend point Figure 21: The late February 2010 interval shows velocities during typical winter conditions. The horizontal blue line marks the average velocity of 0.73 m/day. See plotting scheme description in Figure Figure 22: (a) Horizontal and (b) vertical velocities (black line and gray scatter) are plotted against the tidal rate (red line). The tide elevations are from the IMO, provided by S. Zóphóníasson (personal communication, November, 2014). ICE-A data plotted in the black solid line are the result of hourly spline-smoothed solutions that are derived from 6 h linear fit sliding windows. The gray scatter shows 10 min solutions from 2 h linear fit sliding windows. Alternating gray shading marks days Figure 23: Tides plotted against the correlation between vertical velocity and tidal rate. Blue shadings mark periods when the correlation is negative and red when it is positive Figure 24: Estimated winter surface profiles for 2010, 2012 and 2013, fitted to rover measurements. The rover tracks are color coded to mark the date of the year. The winter locations of the rovers are marked with a triangle that is filled with the same color as the profile of the corresponding year Figure 25: TerraSAR-X velocity field between August 11 and 22, 2010 (Nagler et al. 2012) plotted on top of the October 2010 lidar hillshade DEM (Jóhannesson et al. 2013) Figure 26: Comparison between measured velocities from TerraSAR-X (Nagler et al. 2012) and the Stokes model, including the locations of the ICE-A and ICE-D rovers Figure 27: Example of cross section showing the modelled velocities for the winter Note the vertical exaggeration of 15. The color of the filled triangle shows the measured velocity Figure 28: 10-day averaged sliding velocities during 2010, 2012 and The thick Gaussian smoothed line represents the seasonal cycle which dampens some of the peaks related to rainfall Figure 29: Bedrock hill diverting ice flow direction during a speed-up event (red arrow) Figure 30: Changes in velocity, water pressure and water storage during a typical speed-up event according to the conceptual model. Note the temporal offset in the peak of surface velocity, water pressure and water storage Figure 31: (a) to (d) Conceptual model of drainage system evolution. Plane view of one bedrock bump (brown) and its cavity (blue) that is linked to the subglacial drainage system. The water sheet surrounds the cavity in (b) and (c) (turquoise hashed). Glacier flow is directed upwards and the relative basal motion ub is shown on the right xii

13 Variable Names τ b σ ij N p i p w ρ i ρ w u u b u s g ε ij ε E τ ij τ E η n A Q α, β, γ, a, b, c h H z z bed z surf P a(t) r e E e N e U v w e v e w Basal drag Stress tensor Effective pressure Ice overburden pressure Water pressure Ice density Water density General glacier velocity used in equations Basal motion (sliding) Surface velocity gravitational acceleration Deviatoric strain rate tensor Effective strain rate Deviatoric stress tensor Effective stress Viscosity Creep exponent Creep parameter Water input rate Unknowns in Multiple Linear Regression Lagoon depth Ice thickness Height in glacier from bed Bed elevation Surface elevation Glacier surface profile Scaling factor Pearson correlation coefficient Error in easting Error in northing Error in vertical Measured velocity Modelled velocity Error in measured velocity Error in modelled velocity xiii

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15 Acknowledgements First and foremost I express my most sincere gratitude to Alexander H. Jarosch who supervised the scientific part of my thesis. I have been very much inspired by his manner of teaching and conducting research which has been both thorough and creative. It has been my honor that he invited me to work with him. He smoothly introduced me into the world of GPS and modelling. His clear explanations stem from the fact that they are given step by step, starting from the beginning. The grants that he helped me receive enabled me to attend the Karthaus course. I appreciated that during my research he has given me enormous amounts of freedom, while always being willing to answer my questions. I am grateful to Guðfinna Aðalgeirsdóttir who supervised me on the curriculum related to my Master s degree. She has been the most important contact person at the university with whom I could share my thoughts. She has fully devoted her energy to any challenges I faced from the moment I arrived in Iceland. I have had the privilege to assist her in teaching the glaciology course which was a fruitful and rewarding collaboration. My thesis would not have been possible without the Institute of Earth Sciences having collected invaluable data. Special thanks go to Eyjólfur Magnússon who has been a pivot in setting up and maintaining the GPS receivers on the glacier. Finnur Pálsson has contributed to my work in many ways, for example by providing data and being open for interesting discussions. With him and Helgi Björnsson I had many pleasant conversations. Many thanks to Dave Ostman, Einar Hjörleifsson, Vaiva Čypaitė and other fellow students and co-workers in Askja, for contributing to a perfect study environment. I am indebted to Nicolai Nawri, Hálfdán Ágústsson and Snorri Zóphóníasson from the Iceland Meteorological Office because no interpretation on the velocity variations would have been possible without the weather and tidal data. It has been my friends that have made Iceland feel like a home. Together with them I stayed hungry and foolish following the pursuit of life: adventure. Chris Novitsky, Niccolò Segreto, Daniel Ben-Yehoshua, David Harning, Benjamin Bickel, Victor Pajuelo Madrigal, Joaquín Muñoz-Cobo and Alexander Baker have especially been important to me. My flatmates have made Njarðargata 37 into one bustling and unforgettable house. My family has been of great support. With Mikis van Boeckel I have lived my outdoor dreams. Luca van Boeckel and Fokke van Breukelen have boosted me with an infinite source of good feelings. Jan van Boeckel and Ceciel Verhey gave inspiration on living a free life and their comments greatly improved the manuscript. Lastly, my time in Iceland has become ever more spectacular and enlightening thanks to my girlfriend Gabrielle Fraisse. Together we have explored all kinds of corners, peaks and undergrounds. Last year her support on distance has often been the most valuable one. xv

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17 Chapter 1 Introduction 1.1 General Introduction It has long been hypothesized that ice sheets might have the potential to lose mass through instabilities and that these would have dramatic consequences for sea level rise (Mercer 1968). However, it was not until the past two decades that observations have provided evidence that ice sheet instabilities may already be occurring (e.g. Zwally et al. 2002; Rott et al. 2002). The observations demonstrate that within a short time scale, ice sheets are inherently unstable and that high mass loss rates can be caused by positive feedbacks that can pass tipping points. This potential instability is very different from the processes governing mass balance of mountain glaciers: it is mostly attributed to changes in the amount of melt or precipitation. Increasing global mean temperatures, due to global warming, have led to mountain glaciers melting rapidly, resulting in alarming sea level rise projections for the next century (IPCC 2013). With the processes governing mass balance of mountain glaciers it would take ice sheets many centuries to cause equivalent sea level rise (Pollard & DeConto 2009). The melting of mountain glaciers currently contributes most to sea level rise (IPCC 2013), but has only a limited potential of rising the sea level of half a meter at most (Pfeffer et al. 2014). Ice sheet instabilities, on the other hand, have the possibility to rapidly increase the sea level by several meters (Fretwell et al. 2013). To estimate how fast the ice sheet instabilities may contribute to sea level rise it is important to understand the nature of the instabilities. Examples of described instabilities are: dynamic thinning (Zwally et al. 2002), marine terminus destabilization (Jenkins et al. 2010; Shepherd et al. 2002; Rignot 2008), tidewater-glacier instability (Joughin et al. 2014), ice-shelf buttressing (Rott et al. 2002) and cryo-hydrologic warming (Dunse et al. 2015). The dynamic thinning feedback is somewhat of an exception in the sense that the process is increasingly being debated in the past years, rather than proven. That this instability is questioned awkwardly highlights how one of glaciology s most fundamental processes glacier sliding is not well understood. The feedback relies on theories and observations that were made on smaller alpine glaciers already in the eighties (Iken et al. 1983; Iken & Bindschadler 1986). These studies show that surface melt reaching the bottom of the glacier reduces the friction at the bed and hence accelerates the flow, a process referred to as hydraulic lubrication. The observations imply a simple relationship between changes in glacier uplift (water storage), water pressure and basal motion. 1

18 Based on hydraulic lubrication, increased melt was thought to cause higher summer velocities in Greenland s land terminating glaciers (Zwally et al. 2002). This led to the hypothesis that the hydraulic lubrication would result in a positive feedback when more melt happens: it would cause ice to move faster down to lower elevations where it is subject the yet higher melt rates (Parizek & Alley 2004). Recent studies have questioned the cycle of positive feedback on annual scales because no correlation was found between warmer years in Greenland and the annual velocity of its land terminating glaciers (van de Wal et al. 2008; Sole et al. 2013; Tedstone et al. 2013). More importantly, the actual processes driving the hydraulic lubrication were not fully understood because the expected relationships between changes in glacier uplift, water pressure and basal motion were not observed in alpine glaciers (Iken & Bindschadler 1986; Kamb et al. 1994; Sugiyama & Gudmundsson 2004; Bartholomaus et al. 2008; Howat et al. 2008; Fudge et al. 2009). The presumed simple causal relationship of water pressure leading to increased basal motion therefore required adjustments. Newer studies tried to solve the problem by formulating qualitative models that suggested that the subglacial drainage system is dynamically evolving and never in a steady state (Kamb et al. 1994; Anderson et al. 2004; Bartholomaus et al. 2008). Moreover, it was suggested separately that glacier uplift was not only a result of water storage, but also of a longitudinal stress gradient (Anderson et al. 2004; Sugiyama & Gudmundsson 2004; Howat et al. 2008). To determine whether or not the qualitative models are able to simulate observed velocity variations, recent efforts are therefore being focused on numerical modelling (Schoof 2010; Pimentel et al. 2010; Werder et al. 2013; De Fleurian et al. 2014; Rosier et al. 2015). The models successfully explain part of the velocity variations. For example, they capture the evolution of the drainage system through the melt season which rejects the dynamic thinning feedback (Schoof 2010). However, there is still no consensus among modelers about the causal relationships between the evolution of the subglacial drainage system, the amount of water storage and the water pressure: for example, Werder et al. (2013) model water draining through cavities and channels on a hard bed, whereas De Fleurian et al. (2014) model water flowing through porous layers. It is also unclear how the presence of water relates to basal motion through the friction law (Hewitt 2013). After a wide range of models have been presented, it can be worth attempting to assess whether or not the basics, a dynamically evolving subglacial drainage system in combination with longitudinal stress gradients, are sufficient to explain the observed velocity variations. This study aims to make an attempt to determine the influence of an evolving subglacial drainage system in combination with longitudinal stress gradients on the variations in the surface velocities observed on the temperate tidewater glacier Breiðamerkurjökull in Iceland. 2

19 Because these relationships can change at different time scales, this study specifically targets the combination of inter-annual, seasonal as well as single event flow speed variations. The thesis therefore has two main objectives: 1. To study velocity variations at time scales of single events, often lasting up to a week, and to compare them to surface water input in an effort to improve the understanding of the relationship between water input, sliding and glacier uplift. 2. To study inter-annual and seasonal velocity variations and to compare them with temperature records with the aim to improve the understanding of how weather can affect long term average velocities for temperate tidewater glaciers specifically. In order to meet the first objective, surface velocities were measured with two high resolution Global Positioning System (GPS) receivers that were positioned near the terminus of Breiðamerkurjökull, Iceland, during 2010, 2012 and The choice of studying Breiðamerkurjökull and its characteristics is accounted for in section 1.2. The theory of GPS processing is discussed in section 1.5. The derivation of velocities is discussed in section 2.1. Finally weather data are discussed in section 2.2. The second objective requires more information because on longer time scales variations in surface velocity are the sum of variations in sliding and ice deformation that are both changing temporally and spatially. Spatial changes in ice deformation are related to the increasing creep flow towards the calving front, because buttressing stresses at the calving front (from the lagoon) are small. These effectively result in long term temporal changes because the GPS receivers move downstream or because they are manually repositioned once every other year. Only sliding variations are related to the seasonal and inter-annual evolution of the subglacial drainage system. Therefore ice deformation has to be subtracted from the surface velocities. The ice deformation is estimated in two ways. First, the creep flow can be represented by winter velocities that are steady during periods when air temperatures are below freezing point such that surface water input from melt and rain are likely negligible. This cannot be assumed however, because there is evidence that for temperate glaciers, even during winter, variations in the velocity and water pressure occur (Burgess et al. 2013; Schoof et al. 2014). Therefore the ice deformation is estimated with a second independent method: modelling the creep flow with a Full Stokes (section 1.4) flowline model (section 2.3) of which the results will be compared to the measured winter velocities. Note that for clarity, not quantified velocities for equations are denoted by u, measured velocities are denoted by v and modelled velocities are denoted by w. 3

20 1.2 Study Area: Breiðamerkurjökull Breiðamerkurjökull is Iceland s most dynamic and only calving glacier, see Figure 1. The combination of its size and relatively easy access makes it an excellent natural laboratory for which there is a treasure of data collected already such as the bedrock topography (Björnsson 1996; Björnsson et al. 2001), a lidar DEM (Jóhannesson et al. 2013), velocity fields (Nagler et al. 2012; Voytenko et al. 2015) and time series of summer season GPS measurements (Howat et al. 2008). Breiðamerkurjökull is an outlet glacier of Vatnajökull ice cap, Europe s largest contiguous mass of ice, that is situated in the south east of Iceland (see Figure 2). The outlet glacier is made up by three tributary glaciers: Norðlingalægðarjökull, Esjufjallajökull and Mávabyggðajökull. The first, Norðlingalægðarjökull, is by far the most dynamic glacier and calves in the lagoon Jökulsárlón. This study focusses only on Norðlingalægðarjökull, but it has been common terminology for this outlet to use Breiðamerkurjökull, which will be adopted in this study (e.g. Howat et al. 2008). The glacier extends from sea level to ~1700 m elevation. Just like all other Icelandic glaciers Breiðamerkurjökull is temperate. Breiðamerkurjökull has surged on average every 20 years since the beginning of recordings in 1794 with the latest surging event occurring 1978 (Björnsson et al. 2003). Mass balance has been negative since the end of the Little Ice Age, but there have been periods of thickening. Since 1995 Breiðamerkurjökull mass balance has been continuously negative, averaging 1 m w. eq. (Björnsson & Pálsson 2008). The glacier has been assumed to be soft-bedded with a layer of till between the ice and the hard bedrock. Bedrock and surface topography and the historical margin retreat are further discussed in sections 1.2.1, and 1.2.3, respectively. Figure 1: Breiðamerkurjökull as seen from Jökulsárlón, picture courtesy of Tayo van Boeckel,

21 Vatnajökull Figure 2: Vatnajökull ice cap with important outlet glaciers on September 6, 2014 (false color Landsat 8 image, courtesy of NASA). The Breiðamerkurjökull watershed is outlined after Björnsson & Pálsson (2008). Note the Holuhraun eruption in the north. 5

22 1.2.1 Bedrock Topography The bedrock topography has been measured by ground penetrating radar and is shown in Figure 3 (Björnsson 1996; Björnsson et al. 2001). Under Breiðamerkurjökull is a 25 km long trough with a maximum depth of ~300 m below sea level. The bedrock data have a spatial resolution of 200 m. Figure 3: Bedrock trough in Breiðamerkurjökull courtesy of Björnsson et al. (2001). Blue colors mark elevations below sea level Surface Topography Two Digital Elevation Models have been used for this study. A lidar DEM of Breiðamerkurjökull was measured on October 2010 (Jóhannesson et al. 2013). A second DEM has been created by flying a drone, equipped with a GPS and camera, over the calving front of Breiðamerkurjökull on 10 October 2014 (A. H. Jarosch, personal communication, November, 2014). 6

23 1.2.3 Historical Retreat and Thinning The retreat and thinning of Breiðamerkurjökull since the end of the Little Ice Age has been studied by Björnsson et al. (2001) and is shown in Figure 4. During the Little Ice Age maximum extent (~1870 AD) the glacier almost reached the coast line. Since then the glacier has continuously been thinning and retreating, albeit at different rates. Between 1903 and 2014, Breiðamerkurjökull has retreated ~7.5 km. A more recent summary of the retreat of Breiðamerkurjökull was established by the use of NASA Landsat images that show the retreat since Since then the total retreat has been 5.2 km, corresponding to an average of ~160 m/year (Figure 5). In the period from 1983 to 1992, the retreat was only 300 m or on average ~30 m/year. From then on, the glacier retreat rate increased, to about ~150 m/day on average between 1992 and 2002, but highest rates were reached between 2002 and 2010, when the glacier exposed ground and lagoon at ~325 m/year. During this period the location of the calving front was above the relatively deep bedrock discussed in section The high retreat rate can therefore be attributed to tidewater-glacier instability (Joughin et al. 2014). Since 2010, the position of the calving front reached shallower bedrock and retreat rate decreased to ~200 m/year. Figure 4: The retreat and thinning of Breiðamerkurjökull over the past century. Zero distance marks the coast line. Graph courtesy of Björnsson et al. (2001). 7

24 Figure 5: The retreat of Breiðamerkurjökull, drawn approximately every second year from 1983 to 2014, is overlain on a Landsat 8 image of September 6, 2014 (Courtesy of NASA). The retreat is measured from greatest to smallest extent along the pink line. 8

25 1.3 Theory of Subglacial Hydrology and Sliding The Challenge of Subglacial Hydrology Historical Introduction Glaciers slide because of lubricating water at the bedrock ice interface. A simple causal relationship between water and sliding (basal motion) would imply that more subglacial water reduces the basal drag and causes increased sliding. There should therefore be a direct relationship between measured values of sliding (from surface velocity variations), water pressure (in bore holes) and water storage (indirectly from glacier uplift and/or drainage output minus input). However, early observations have not been able to confirm that the variations are occurring in phase. Iken et al. (1983) observed that the peak in sliding coincided with maximum upward velocity, rather than with the peak uplift. They therefore concluded that sliding is related to water pressure rather than subglacial water storage. Iken & Bindschadler (1986) confirmed that there was an overall relationship between water pressure and velocity, but detailed studies suggested more complexity (Blake et al. 1994; Kamb et al. 1994). This was already predicted by modelling work of Iken (1981): her model revealed that the sliding velocity is larger when cavities are growing than when they had reached steady-state size for a given water pressure. The early observations therefore suggested a more complicated relationship between the basal motion, water pressure and water storage. New observations More recent observations have changed the perspective on the understanding of subglacial hydrology. Two important conclusions from these observations can be described as follows: 1. Increased horizontal surface velocities correlate with increasing rates of change in water pressure (Sugiyama & Gudmundsson 2004; Anderson et al. 2004), water storage (Bartholomaus et al. 2008) and bed separation (Anderson et al. 2004; Howat et al. 2008). The typical evolution of the horizontal surface velocity relative to the water pressure and water storage during a speed-up event is given in Figure 6. Increasing horizontal surface velocities correlate with increasing water pressures, but are relatively high compared to identical, but decreasing water pressures. Moreover velocities peak when the amount of bed separation is increasing most rapidly. Bed separation peaks later, and decreases less rapidly, than the horizontal surface velocities. 2. On seasonal time scales, sliding is greatest during spring and decreases as the melt season progresses despite surface water input remaining at the same level (Bartholomew et al. 2010). The drainage system is therefore not always as responsive to the amount of surface water input. 9

26 The observations lead to hypothesis that amount of sliding depends on an interplay between different kinds of drainage systems which will be discussed in section Subglacial drainage systems determine how water flows, where it is located and how fast it drains. Before discussing the theories on how drainage systems relate to sliding, two types of drainage systems are described in section Figure 6: Schematic interpretation of the hysteresis in the temporal relationships between horizontal velocity and water pressure (Sugiyama & Gudmundsson 2004) and bed separation (Howat et al. 2008) during a glacier speed-up event Two Types of Drainage Systems Distributed or Inefficient A distributed drainage system inefficiently drains water through cavities (gaps at the glacier ice and bed interface), channels cut into the bed, and possibly through a layer of permeable till (see Figure 7). Water filled cavities can exist because ice flows over bedrock obstacles in such a way that at the lee side ice pressures would be less and the cavities could be kept open by the balance between water pressure and ice overburden pressure (Lliboutry 1968; 1979). Cavity size therefore depends on the sliding rate and water pressure, because the ice overburden pressure is fairly constant. As discharge to the cavity increases, water pressure also builds up and cavities expand. In case of two adjacent cavities water pressures will equilibrate between the two such that both cavities remain intact. Lliboutry (1968) imagined that such cavities could be linked into a so-called distributed drainage system, which was later confirmed by observations (e.g. Iken & Bindschadler 1986). Distributed drainage systems reduce the contact between ice and bedrock and decrease basal drag and subsequently increase the sliding. Weertman (1972) argued that glaciers and ice sheets could slide on thin films of water, but Walder (1982) showed such water sheets would be unstable because water would prefer to flow through channels. Recent studies suggest that water can flow in sheets through the pore spaces in the subglacial till (e.g. Flowers & Clarke 2002). In section it will be discussed that such sheets are increasingly being favored in models. 10

27 Channelized or Efficient Röthlisberger (1972) and Nye (1976) described a subglacial drainage system consisting of channels that cut into the ice as tunnels at the glacier bed (see Figure 7). During steady state these tunnels can exist through a balance between the closing ice creep due to the ice overburden pressure and melting due to frictional heating that depends on the water discharge. The channels have the important property that as discharge increases, wall melt increases even more rapidly and the tunnels become larger leading to decreasing water pressure. This has important implications for the system in a steady state: when considering two adjacent tunnels, the larger one has a relatively low pressure which causes water from the smaller channel to flow towards it. As a consequence a channelized drainage system will evolve into a stable configuration consisting of a few conduits with great efficiency. Therefore, a channelized drainage system causes the ice to have a large area of contact with the bedrock which increases basal drag and reduces the sliding. Figure 7: Examples of the two types of drainage systems modified after Flowers (2015) Hypothesis: Dynamic Evolution of the Subglacial Drainage System Any hypothesis on the relationship between the state of the subglacial hydrology and sliding has to be in accordance with both observations described in section To account for the temporal offset in basal motion, water pressure and water storage, studies increasingly support the co-existence or switching between different types of drainage systems (Kamb & Engelhardt 1987; Nienow et al. 1996; Fountain & Walder 1998; Mair et al. 2001). Moreover, to account for the decreasing responsiveness of the drainage system during melt season, the drainage system should change over time (Bartholomew et al. 2010; Sole et al. 2013). The observations in section therefore support a hypothesis including a dynamically evolving subglacial drainage system. Bartholomaus et al. (2008) provided such a hypothesis when they observed that glacier speed-up is caused by the overwhelming of the subglacial drainage system with surface 11

28 water input. They came up with a conceptual model describing that when the subglacial drainage system becomes overwhelmed existing channels can no longer drain the water and the system becomes pressurized to such a degree that excessive water is driven back into the linked cavity system, causing increased speed-up. Glacier speed-up increases the cavity size which reduces the water pressure and hence serves as a negative feedback on the basal motion. Eventually water drains from the cavities through the channels and the velocity decreases back to original levels. Moreover, because the drainage system slowly develops to greater efficiency, sustained high rates of discharge do not result in persistent sliding. Rather than having a steady state drainage system, the drainage systems develops dynamically and is therefore affected by increasing rates rather than peaks in water input. The hypothesis of Bartholomaus et al. (2008) has been incorporated in models described in literature (Schoof 2010; Pimentel et al. 2010; Hewitt 2013; Werder et al. 2013). The frequent use of the hypothesis in models underlines the necessity to test the hypothesis with observations. For this thesis the hypothesis of the dynamic evolution of the subglacial drainage system is formulated as follows: Because the subglacial drainage system evolves dynamically, the overwhelming of the drainage system causes the glacier to speed-up temporarily while sustained high rates of water input cause levelling or slowdown of glacier velocities. Recently, other models included different drainage systems when interpreting the hypothesis (De Fleurian et al. 2014; Rosier et al. 2015). These models are discussed in section Surface Velocity and Basal Conditions The hypothesis of the dynamic evolution of the subglacial drainage system predicts how changes in surface water input affect velocity variations at the bed of the glacier (e.g. sliding). Because glacier velocities for this study were measured at the surface it needs to be considered how changes at the base relate to changes at the surface. First, sliding variations at the bed cause horizontal surface velocity variations. However, time is needed to transmit the stresses from the bed to the surface. The measured speed of stress propagation through ice streams due to tidal forcing is at least 1 m/s (Gudmundsson 2007). With these values the time needed to transmit stresses vertically through Breiðamerkurjökull is negligible: 500 m 1 m/s = 8 minutes. Second, the efficiency of drainage systems can vary spatially under a glacier (Fountain & Walder 1998) resulting in some parts of the glacier moving faster compared to others. As a result longitudinal stress gradients builds up. For example, during melt season the drainage system is often more distributed upstream compared to downstream (Fountain & Walder 1998). In case the subglacial drainage system becomes overwhelmed the velocity will increase more upstream, pushing against slower moving ice downstream, resulting in compressive flow. Longitudinal gradients therefore cause uplift/subsidence 12

29 and horizontal acceleration/deceleration of the glacier (Anderson et al. 2004; Sugiyama & Gudmundsson 2004; Howat et al. 2008). Third, water storage at the bed lifts the glacier by hydraulic jacking, resulting in bed separation. As a result: horizontal surface velocities variations are caused by changes in ice creep, sliding and compressional/extensional flow. vertical surface velocity variations (glacier uplift and subsidence) are caused by changes in water storage and longitudinal stress gradients. 13

30 1.4 Physics of Glacier Flow Ice flow can be described with the Stokes Equation because ice has a relatively high viscosity and creeps relatively slowly, implying that Reynolds numbers are low. The field equations describe conservation of mass, linear momentum and angular momentum and are given by: u i,i = 0, (1) σ ij,j + ρg i = 0, (2) σ ij σ ji = 0, (3) where u i is the velocity vector, σ ij is the stress tensor, ρ the ice density and g i the gravitational acceleration vector. Early lab experiments of Glen (1955) described a power law relation between strain rate ε ij and the deviatoric stress τ ij, referred to as Glen s flow law: ε ij = Aτ E n 1 τ ij, (4) where n is the creep exponent, A is the creep parameter, τ ij = σ ij 1 3 δ ijσ ij and τ E is the second invariant of the stress tensor, also referred to as the effective stress, described by τ E = τ ij τ ji /2. Inverting Glen s flow law yields the deviatoric stresses in terms of strain rates, leading to: τ ij = A 1/n ε E (1 n)/n ε ij, (5) Where ε E is the effective strain described by ε E = 1 ε ijε ji. Because the viscosity η = 2 τ ij / 2ε ij, inserting equation (5) in eq. (2), yields the Stokes Equation, in vector notation: [η( u + u T )] + p = ρg (6) where the viscosity η is given by: η = 1 2 A 1/n ε E (1 n)/n. (7) The coefficients of Glen s flow law depend on temperature, fabric, grain size and impurity content of ice. Experiments have been made to constrain them. They have revealed that values of n range from 1.5 to 4.2 although most temperate glacier observations seem to be consistent with n = 3 (Cuffey & Paterson 2010). Because Iceland s glaciers are always at pressure melting point the temperature dependence vanishes and the creep parameter is better confined. For such temperate glaciers Cuffey & Paterson (2010) have suggested a single value of A = s 1 Pa 3 (Cuffey & Paterson 2010) derived from an average of five calibrated models. 14

31 1.5 The Global Position System (GPS) What is GPS? GPS is a powerful geodetic tool allowing for full 3D positioning of a geographic point in an absolute reference frame. Once the receiver is measuring at continuous intervals it can track the surface displacement. Recent technological and processing improvements have resulted in positioning accuracies reaching the order of millimeters. GPS is not the only satellite constellation capable of positioning. All constellations are identified as Global Navigation Satellite Systems (GNSS). Because the GPS constellation is most widely used for geodetic purposes and is the only one used in this study, the thesis will therefore refer to GPS only. Before GPS existed as a geophysical tool tracking the glacier motion was a tedious challenge involving placement of stakes and measuring their location with for example theodolites. Moreover measuring stake positions was only possible if the topography was available (i.e. not on ice sheets). GPS has not only made positioning easier, faster and more accurate, it also allowed for continuous measurements. Instead of only being able to measure the position when a scientist is in the field, a GPS receiver can record positions at a custom time interval often on the order of seconds. A receiver placed on a glacier will move along with the glacier s flow (along a flowline) recording its surface motion. These profound improvements have among others made GPS a well-established tool in the field of glaciology Relative Kinematic Positioning The challenge is then to measure the number of full carrier-signal cycles, also called the integer ambiguities, between the satellite and the receiver. When the integer ambiguities are solved they are called fixed solutions while integer ambiguities estimated otherwise are called float solutions. Solving the position is affected by various sources of errors such as: delay of the signal through the troposphere and ionosphere, multipath, clock errors, relativistic effects, receiver noise and satellite orbits (ephemeris). There is a powerful method to reduce the amount of error sources and to simultaneously solve the integer ambiguities by introducing relative positioning. When two receivers are placed closed to each other, relative displacements between the two can be measured. Because the signal propagates through a similar atmosphere and because the reference station is assumed not to move, many errors can be reduced. However, reducing the errors this way only provides reliable positions when distances (or baselines) between the two receivers are short (less than a few tens of kilometers); then the paths through the atmospheres do not differ too much. In addition to reducing errors, relative positioning increases the amount of equations and can therefore facilitate overcoming the underdetermined problem of solving the integer ambiguities. On a glacier one of the receivers moves while the other is assumed to be fixed on nearby ice-free terrain. The moving GPS receiver is called a rover and the other serves as a base 15

32 station. Because one rover moves, the strategy of solving the integer ambiguities mathematically with sufficient epochs does not work. The kinematic positioning method takes this into account. It makes use of other sources of information to estimate the ambiguities, as explained below Long Baseline Processing Software: RTKLIB Takasu & Yasuda (2010) have made an important contribution to long baseline (more than 100 km) positioning. They claim that their approach of using the Kalman-Filter-Based Integer Ambiguity Resolution Strategy rather than conventional Ambiguity Resolution strategies provides better results for long baseline Real Time Kinematic (RTK) processing. Explaining the strategy in detail goes beyond the scope of this thesis but in short the strategy enables sources of errors (resulting from long baselines), such as tropospheric delay, to be modelled or included in the problem as additional unknown parameters. Their approach uses another feature which is relevant for longer baselines: in the longer baseline environment (important for large glaciers) they found that satellites which newly rise above the horizon need longer convergence time for the carrier-phase ambiguities. However, by default the strategy requires all ambiguities to be fixed at the same time. Takasu & Yasuda s (2010) method increases the fixed to float solution ratio for long baselines through only fixing and then holding ambiguities above certain elevation horizons. Having fewer satellites available to be used in the solving process there is a trade-off problem between accuracy and the fixing ratio. Takasu & Yasuda (2010) show how their method, when applied to long baselines, increases the fixing ration while only slightly reducing the accuracy. Although other competitive strategies are available, the software RTKLIB provided by Takasu 1 is open-source and user-friendly. All in all it is well-suitable for long baseline post processing of GPS receiver data placed on large glaciers. 1 Available on 16

33 Chapter 2 Methods 2.1 Deriving Velocities from GPS Displacements Tuning the Parameters for GPS Processing The GPS data from the rovers on Breiðamerkujökull were kinematically processed relative to the basestation HOFN which is an International GNSS Service GPS station located in Höfn í Hornafirði. The baseline is ~55 km. Finding the right parameters for the GPS processing is a process of trial and error. This trial and error approach revealed that solutions mostly depended on the Elevation Mask, Ionosphere correction, Troposphere correction and elevations to fix and hold the ambiguities. Especially too low (less than 7 ) or too high (more than 15 ) Elevation Masks have a negative impact on the results for the following reasons. Too low Elevations Masks include too many satellites that have low signal to noise ratios and therefore need more time to converge to a reliable amount of ambiguities. Too high elevations exclude too many satellites such that it is more difficult to overcome the underdetermined ambiguity problem. The example values of Takasu & Yasuda (2010) often yield good results (fixing ratios > 70 %). Optimal parameters yielding fixing ratios > 90 % are given in Table 1. Table 1: Optimal parameters used for processing the GPS data from the rovers on Breiðamerkurjökull relative to the HOFN base station Parameter name Setting 1 Setting 2 Positioning mode: kinematic Frequencies: L1+2 Combined Elevation mask: 10 REC dynamics: OFF Earth tides correction: OFF Ionosphere correction: Estimate STEC Troposphere correction: Broadcast Satellite ephemeris: Precise Integer Ambiguity Res (GPS/GLO): Fix and Hold ON Min Ratio to Fix Ambiguity: 3 Min Lock / Elevation ( ) to Fix Amb: 0 25 Min Fix / Elevation ( ) to Hold Amb: Solution Format X/Y/Z-ECEF Base station RINEX Header Position Satellite/Receiver Antenna PCV File ANTEX..\igs05.atx 17

34 2.1.2 Deriving Velocities from Displacements The processed data are selected by only taking positions that are derived from fixed solutions and that have errors that fall within the threshold of 0.1 m. The Earth centered Cartesian coordinates (ECEF) provided by RTK are projected in a local East, North, Up reference frame, pinned at the international HOFN reference (coordinates are on : longitude = , latitude = ). Data are resampled to 2.5 min intervals to reduce calculation time while ensuring high enough resolution to observe velocity changes happening in the order of hours. The steps are described in a python script 2. Once a year one or both of the rovers are positioned back upstream to prevent them from falling into crevasses and to have them further away from the calving front. This causes a jump in the rover s position which should not be interpreted as a large velocity taking place. To prevent this, the location of the jump is identified and in further calculations data too close to the jump are excluded 3. On July 3, 2010, both rovers ICE-A and ICE-D were moved upstream by ~500 m such that ICE-A was placed on the original location of ICE-D. Because the newly installed ICE-A rover follows the same track as ICE-D would have followed, after the jump, the rover s names were switched such that the name ICE- D represents one continuous track. The total horizontal displacement is calculated from the easting and northing with the Pythagoras equation. A python script 4 calculates a smooth velocity with an adjustable time window (of default 6 hours) around data points that only includes data points that are within plus/minus half the window time. A linear line was fitted through the displacement data in the window by the least squares method. The slope of the fit defines the velocity at the data points. The window slides through the whole data set at a custom time interval (default every 10 min). Rejection of data points occurs when a window includes a jump or when the sliding window contains too little data (the threshold is set to one third of the possible amount). Finally, data gaps are linearly interpolated

35 2.1.3 GPS Error Analysis Precision and Accuracy Errors in Positioning There are so many sources of errors in GPS positioning that it is a study in itself to go through every individual one. The bottom line is that the error is described by errors in precision and accuracy. The difference between the two is as follows when considering the positioning of a non-moving (i.e. fixed) GPS receiver (see Figure 8): Figure 8: Accuracy and precision. Graph courtesy of Langley (2010). Precision is the width of the spread of repeated measurements. If the data are normally distributed, the precision is defined by the standard deviation. Accuracy is the difference between the true value and the best estimate of it. The processing software RTKLIB provides precision errors with every measurement. During all of 2010 the median of the standard deviations from the ICE-D receiver were e E = 1.2 cm, e N = 2.1 cm, e U = 3.2 cm. The distance root mean squared (drms), of the standard deviations is drms = e 2 E + e 2 N = 2.5 cm. Estimating accuracy errors is not possible because the rovers move and the true position is therefore unknown. To demonstrate how (internal) accuracy errors could be estimated an analogous scenario is discussed. When a fixed receiver is processed kinematically the average is assumed to be the true position. By definition all deviations from it are accuracy errors. Because the only two available dual frequency receivers were on Breiðamerkurjökull during the entire year, it was not possible to estimate accuracies in this manner. A single frequency receiver was therefore used instead. Data from 20 days was processed kinematically relative to the IGS REYK basestation in Reykjavík, similar to the dual frequency processing described in section Because the dual frequency receiver is assumed to perform better than a single frequency receiver, this case could serve as an upper bound estimate. However, note that the baseline for processing the single frequency receiver was much shorter (~2 km) compared to the baseline on Breiðamerkurjökull ( ~ 55 km). A longer baseline increases accuracy errors and the upper bound estimate is therefore not very reliable. 19

36 With the processing settings described in section 2.1.1, adjusted for a single frequency receiver, fixing ratios of 75% were obtained. This mediocre result implied that some of the larger positioning deviations from the float positions were included. These outliers were removed by only taking positions that are within a certain threshold from the average: ±3 cm in easting, ±4 cm in northing and ±15 cm in the vertical. This excludes 3.3, 3.9 and 4.4 % of the total data. The distribution of the positions is shown in Figure 9. The data are normally distributed. Standard deviations are e E = 0.8 cm, e N = 1.1 cm, e U = 3.5 cm and the drms = 1.4 cm. Accuracy errors of the single frequency receiver are therefore similar to precision errors of the dual frequency receiver. This suggests that the errors for the receivers on Breiðamerkurjökull may actually be in the order of centimeters. Preferably one of dual frequency receiver on the glacier should have been placed off ice for a month such that more realistic accuracy errors could be estimated. Bartholomew et al. (2012) estimated comparable errors of ±1 cm in the horizontal and ±2 cm in the vertical when using similar receivers and long baselines (more than 100 km). Velocity Errors Velocity error estimates were hard to make considering the way velocities were derived from displacements (see section 2.1.2). Conservative velocity error estimates are therefore taken from Bartholomew et al. (2012): they estimate the horizontal velocity errors to be at most e v = 0.05 m/day. (a) (b) Figure 9: Distribution of positions of a fixed single frequency receiver in Reykjavík derived from kinematic processing of 20 days of data. (a) Density function of positions and a Gaussian distribution. (b) Box plot including 90 % of the data within the whiskers. The black dot marks the mean. 20

37 2.2 Weather Data Because glacier flow is enhanced by water lubricated sliding at the bed, knowing how much water is added to the bed is an important objective of this study. For Icelandic temperate glaciers, the source is primarily melt water and rain draining to the bed through crevasses and moulins. Therefore proxies to estimate the energy available to melt ice and snow, such as temperature or ablation measurements, and records of precipitation are needed to estimate the water input to the bed. However, because weather patterns vary strongly in southeast Iceland, collecting reliable weather data is challenging. Many sources of data were compared. Figure 10 shows the available weather data in the area. The map in Figure 11 shows the appropriate locations. The weather data sources are: Three weather stations in Kvísker (5316, and 740), situated 20 km south west of the glacier terminus. The data are from the Icelandic Met Office (IMO) and were provided by H. Ágústsson (personal communication, October, 2014). One Automatic Weather Station (AWS) located at the glacier terminus. The data are from the University of Iceland (HÍ) and were provided by F. Pálsson (personal communication, November, 2014). The HARMONIE weather forecast model (described in Seity et al. 2011; Brousseau et al. 2011). The data are from the IMO and were provided by N. Nawri (Personal communication, October, 2014). (a) (b) (c) Figure 10: Weather data from Breiðamerkurjökull during the year (a) Temperature. (b) Ablation. (c) Precipitation. Data from the HARMONIE model and Kvísker weather stations are from the IMO, provided by N. Nawri (Personal communication, October, 2014) and H. Ágústsson (Personal communication, October, 2014), respectively. Data from the AWS on Breiðamerkurjökull, including the ablation measurements are from the Institute of Earth Sciences, HÍ, provided by F. Pálsson, November, 2014). 21

38 2.2.1 Temperature Temperature can be used as a proxy for the energy available to melt ice at the surface of a glacier. Because temperatures are measured at different altitudes, they are corrected with a linear lapse rate of T/ h = 0.6 C/100 m to the rover altitudes. The HARMONIE model calculates hourly weather forecast data on km 2 grids. The model output is taken from one grid cell at the middle of Breiðamerkurjökull, representing an average location. The temperature at the Kvísker stations 5316 and and from the HARMONIE grid cell are very similar except during the summer. Temperatures in the model are higher than measured ones which could be the result of a poorly chosen lapse rate or because temperatures close to the coast are modulated by the ocean temperature. Most likely however, model temperatures are too high because the model does not take into account the zero degree ice surface that in reality modulates the temperatures. Moreover, temperatures on the glacier are affected by cooling katabatic winds. This dampening of the temperature fluctuations can be observed at the AWS. Fluctuations are smaller and a threshold maximum temperature is reached in the summer. This shows that temperatures measured on the glacier surface are a bad representation of the incoming energy that is available to melt ice. Because the AWS behaves poorly on ice and because the model does not take the glacier surface into account, the Kvísker is thought to give the best representation of the incoming energy and is therefore used by default throughout the study Ablation Rather than observed temperatures, ablation measurement from the AWS could provide a better representation of incoming energy available for melting ice. The cumulative ablation is measured at one hour time intervals with a sonic ranger on the AWS. However, hourly data are not representative of incoming energy because the glacier does not melt linearly in time. For example, as ice melts, it leaves air filled pores that form weak structures that can suddenly collapse. The sonic ranger interprets this as initially no melt occurring followed by a sudden large melt event. To account for this inconsistency F. Pálsson (personal communication, November, 2014) recommended extracting daily values, by subtracting cumulative ablation at the end of the day with levels at the beginning of the day, and subsequently taking a three day average. However, after the smoothing, the high temporal resolution of the ablation measurements is removed. The sonic ranger was not always operating during the study period Precipitation During the studied period the Kvísker 5316 precipitation gauge was not always working properly (H. Águstsson, personal communication, October, 2014). To assess the quality of the measurements of the station there is a manual precipitation gauge placed in the proximity consisting of a water container that is measured and emptied daily by the inhabitants of Kvísker. However, there could be certain days where measurements have 22

39 not been taken. The HARMONIE weather forecast model is capable of estimating very local precipitation data. A comparison in the third tile in Figure 10 shows a great discrepancy between the amounts of precipitation. This is to be expected because there is much topography around Breiðamerkurjökull with Öræfajökull in the west, including Iceland s highest mountain Hvannadalshnjúkur (2109 m), and Þverártindsegg (1554 m) in the east. Because only precipitation falling on the glacier is of interest, the local precipitation data from the HARMONIE model are likely to give the most reliable results (H. Hannesdóttir, personal communication, October, 2014) and are therefore used in this study. For Breiðamerkurjökull, modelled precipitation from the one grid cell is expected to fall in similar amounts on most of the glacier whereas ablation rates measured near the terminus can be assumed to decrease with altitude. Melt water input to the bed is therefore likely overestimated for higher elevations compared with the rain rate. Velocity variations at the location of the GPS rovers are caused by variations in surface water input rates higher upstream. Because ablation is much smaller for these higher areas upstream, the given values of ablation are not representable for the entire glacier. Both variables cannot be added up and it therefore is only possible to do a qualitative interpretation of variations in water input. When they are plotted in the same graph, the total amount is referred to as relative water input. 23

40 2.3 Flow Model Modelling Ice Flow Ice creep is described with the Stokes equation (eq. 6) and includes a stress dependent viscosity (eq. 7). Calculating the viscosity requires knowing stresses, but the stresses of a glacier are only known once the Stokes equation is solved. In other words, calculating the solution depends on having the solution. This numerical challenge can be solved iteratively with a finite element model. Such a model can converge towards the rheology of the glacier ice through iterative steps. Jarosch (2008) has developed a model called Icetools that solves the Stokes equation in such an iterative way. Here follows a summary of the model. In the first iterative step an initial viscosity of η = Pa s for a Newtonian fluid is prescribed. The glacier geometry is captured by a set of points on the boundary such that the open source software Gmsh 5 creates a mesh with a velocity vector field and a pressure scalar field. Surface and bedrock elevations are required as input. Setting up Gmsh with an appropriate boundary is described below. The generated velocity field can be used to calculate the strain rates on each node of the model with the equation: ε ij = 1 2 (u i,j + u j,i ). (8) The new viscosity can then be calculated with eq. (7). However, as deviatoric stresses can go to zero, for example at the glacier surface, the viscosity can become singular. A linear term with an upper limit for the viscosity is therefore introduced in the non-linear flow law. The upper limit is not well constrained, but there seems to be agreement on η max = Pa s Jarosch (2008). The derived viscosity is calculated on every node on the grid as a viscosity scalar field. The new viscosity scalar field is the input viscosity in a second iterative step. Repeating the steps of solving for new viscosity fields and implementing them back into the problem results in an iterative process that converges to a stress dependent viscosity according to Glen s constitutive equation. Each solution is compared to the previous solution and once the difference becomes smaller than a predefined value (by default m/year), the problem is said to have converged because velocities of the order mm/year cannot be observed by current GPS receivers. When the grid is too coarse for the specific problem it can occur that the problem will not converge and the iteration sequence is aborted after a custom set value of maximum iterations

41 2.3.2 Boundary Conditions The model needs boundary conditions to make the problem solvable. Defining Dirichlet boundary conditions at the glacier bed satisfies the requirement. It is convenient to define velocities at the bed that correspond to physical sliding velocities. Depending on the model, boundary conditions at the terminus and the upstream boundary need to be defined as well. Vertical boundary conditions can be defined as an isostatic pressure from the lagoon given by: p = ρ w gh. (9) The vertical boundary conditions can also be given by prescribing velocities. An estimate of the velocity at the vertical boundaries can be made by assuming laminar flow: u(z) = u b + 2A n + 1 τ b n H (1 ( z surf z )n+1 (10) ) H where u(z) is the velocity at height z {z bed, z surf }, u b is the sliding velocity, A is the creep factor, n is Glen s rheology exponent, τ b is the basal drag and H is the ice thickness z surf z bed (Cuffey & Paterson 2010, p.310). The basal drag τ b is calculated, given the sliding velocity and surface velocity, u s, by: Therefore eq. (10) reduces to: τ b n = (u s u b ) n A H. (11) u(z) = u b + (u s u b ) (1 ( z surf z n+1 H ) ) (12) The importance of the correct boundary conditions will be evaluated in section Flowline To avoid unnecessary complex models, this study solves the Stokes equation for a simple 2D glacier flowline. However, it is important to bear in mind that for complex glacier geometries, for example where a glacier is fed by a considerable amount of tributary glaciers, a full 3D solution would yield more realistic results. The flowline should pass the GPS rovers. The up-glacier flow path can be estimated in several ways: i. Surface parallel flow: The glacier flows perpendicular to surface elevation contours. ii. Bedrock trench: If there is a trench in the bedrock topography, the glacier flows parallel to the deepest depth of the trench. iii. Tephra layers: If there are volcanic ash layers, the glacier flows perpendicular to the tephra layers. 25

42 Figure 11: Map of the Breiðamerkurjökull area, including three estimated flowlines and locations providing weather data. Landsat 8 image of September 6, 2014 (Courtesy of NASA). 26

43 Flowlines according to the three flow assumptions are shown in Figure 11. The bedrock trench (ii) flowline changes direction at relatively sharp angles which are not expected in glacier flow. The flowline perpendicular to surface elevation contours and volcanic ash layers (i and iii, respectively) are similar and smooth. The main difference between the flowlines in the proximity of the rovers is that the tephra layer (iii) flowline is closer to the tributary glacier Esjufjallajökull to the west. The tributary glacier Esjufjallajökull reduces the lateral friction, which increases the flow speed at the lateral boundary. As a result, the glacier is expected to flow closer to the medial moraine than the surface parallel flowline would suggest. The tephra layer flowline therefore automatically includes the tributary glacier flow and is chosen as the most representable flowline. The flowline should start and end far enough from the rovers to not have the boundaries affect the outcome, unless boundary conditions are properly defined. Gudmundsson (2003) shows that transmission of basal variability only affects surface velocity several ice thicknesses away. Considering the ice thickness in Breiðamerkurjökull is approximately 500 m, to be on the safe side boundaries should therefore be set about 5 km upstream or downstream from the rovers. Upstream this condition can be met by extending the flowline 25 km towards the equilibrium line. Downstream, however, the glacier calves into the lagoon at only 3 5 km distance from the rovers and the velocities at the calving front will therefore affect rover velocities. The boundary condition described in equation (12), assuming laminar flow, can be a first estimate to take this into account Surface Topography For a good comparison between modelled and measured velocities during 2010 to 2014 the glacier surface needs to be adjusted to thinning and retreat. Annual surface profiles are created by inter- and extrapolation between two surface profiles. The first profile, P 2010, is from a high resolution DEM created by lidar in 2010 (Jóhannesson et al. 2013). The second partial profile, dp 2014, is from a DEM created by photogrammetry (A. H. Jarosch, personal communication, November, 2014). This partial profile extends 670 m upstream from the calving front. The full 2014 profile was calculated by: P 2014 = P 2010, (13) where is the thinning that occurred between 2010 and However, only a part of the thinning could be estimated with the 2014 DEM because it was only measured near the calving front, hence 1 = P 2010 dp Other thinning data, 2, were derived from four mass balance measurements taken along a transect on Esjufjallajökull, 7 km towards the west of the flowline (F. Pálsson, personal communication, October, 2015). The locations of the measurements were projected on the flowline by minimizing the distance. The final thinning = 1 + 2, connects all thinning data linearly. Because the locations of the crevasses are unknown the final surface profiles were smoothed with a Gaussian filter. The P 2010 and P 2014 profiles are shown in Figure

44 With two full profiles the surface profiles at time t, P(t), can be estimated by inter- and extrapolation of the difference between the 2010 and 2014 profiles by: P(t) = P a(t) (P 2010 P 2014 ), (14) where a(t) is a parameter to fit the profile rover measurements and P 2014 is the estimated full 2014 profile. The advantage of this method is that all profiles will have the same realistic shape as the P 2010 lidar profile. A disadvantage is that the P 2014 profile strongly depends on mass balance measurements from Esjufjalljökull. Considering the circumstance that the measurements were taken on a distance of 7 km and that Esjufjallajökull is a land terminating outlet glacier, the measurements likely do not take into account thinning due to the ice dynamics of Norðlingalægðarjökull. However, the impact of the P 2014 profile on the final P(t) profiles is relatively little because the final profiles were fitted to the rover measurements with parameter a(t) in eq. (14). As a result, the elevation of the surface profiles near the rovers is constrained, reducing the effect of possible erroneous mass balance values further upstream. Figure 12: Glacier bedrock and the 2010 and 2014 surface profiles. Solid surface profiles are measured while dashed profiles are estimated. Elevations are in WGS84 geoid reference frame where the lagoon level of 65.5 m is equal to sea level. The 2014 profile is extrapolated above the measurements by fitting to the ablation. Bedrock data are from Björnsson et al. (2001). 28

45 2.3.5 Retreat of Breiðamerkurjökull Calving front locations at the time of the intermediate profiles P(t) were estimated with NASA Landsat 7/8 satellite images. Retreat is always measured relative to the P 2010 front and was 860 m for the P 2014 profile. Because Landsat imagery was not always available, for example due to cloud cover, accurate calving front positions can be hard to determine. Because the calving front retreats about ~100 m/year during (Voytenko et al. 2015), Landsat derived calving front positions that are one month off in time are expected to deviate less than ~10 m from the required front. However, the center of the glacier calves more during summer which results in a narrow, localized embayment during melt season and partially closes during winter (Voytenko et al. 2015). The embayment was especially large during Then the retreat was 360 m further in the central area of the calving front compared to the lateral areas Creating Grid The cross section along the flowline is made by surface and bedrock elevation data (Figure 13a). Taking into account the bedrock topography resolution and the assumption that small bedrock undulations have a minor effect on surface motion of a m thick glacier, the glacier outline is assembled at 200 m spaced intervals along the flowline. Because the vertical velocity gradient is largest at the bottom of the glacier, as shown with the exponent in eq. (12), the mesh is adjusted accordingly. Intervals of 40 m are chosen for the bed and 80 m at the surface. The final mesh used in the model computations is presented in Figure 13b. (a) (b) Figure 13: (a) Outline of flowline cross section vertically exaggerated by a factor 2. (b) Mesh of flowline cross section near the terminus as prepared by Gmsh. 29

46

47 Chapter 3 Results 3.1 Time Series of Velocities Two high resolution GPS rovers named ICE-A and ICE-D have operated on Breiðamerkurjökull from 2008 onwards. Figures 13, 14 and 14 show the time series of horizontal velocities and detrended vertical displacements compared to weather data during the most recent years 2010, 2012 and The melt season is assumed to start at days 118, 119 and 125 (first days of May) for years 2010, 2012 and 2013 respectively. For 2010 and 2013 there are no recorded data that mark a clear end of the melt season, but data from 2012 extend further in time and ablation rates did approach zero from day 290 and onwards. Melt-induced velocity variations were observed during melt season and are discussed in section 3.2. The time series include several speed-up events that in this study are limited to periods when horizontal velocities increased 1.5 times the original velocities and that last less than about a week. Such speed-up events mostly correlate with high rates of rainfall as observed in previous studies (Howat et al. 2008; Fudge et al. 2009). The speed-up events occurred mostly before or at the end of the melt season because there was no considerable measured rainfall (exceeding 50 mm/day) during the melt season. Speed-up events are described in section 3.3. During the melt season, velocities increase as ablation rates increase, but this correlation becomes less strong as the season progressed. Melt season velocities peaked in late May/early June for 2010 and 2013, although ablation rates remained high for another two months. This relative slowdown has been observed in other glaciers before, concluding that average annual temperature has little effect on average annual velocities (Sundal et al. 2011; Sole et al. 2011; Andersen et al. 2011; Tedstone et al. 2013). The seasonal evolution of the sliding velocity should be discussed after ice deformation are subtracted, as discussed in section 1.1. The ice deformation velocities are estimated in section 3.4 and are used to analyze the seasonal evolution of the drainage system in section

48

49 Figure 14: Year Upper tile: horizontal velocities of ICE-A and ICE-D smoothed with a 24 h sliding window. Interpolation of the horizontal velocities is drawn in gray and light blue color for both rovers. Middle tile: detrended uplift of ICE-A and ICE-D. Note that data gaps are not highlighted. Lower tile: Weather data including temperature, ablation, rain and snow with sources described in section

50 Figure 15: Year See plotting scheme description in Figure

51 Figure 16: Year See plotting scheme description in Figure 14. Note that at day 198, 2013, ICE-D was moved back upstream behind ICE-A. 35

52 3.2 Daily Melt-Induced Fluctuations During the melt season, velocities fluctuated with diurnal frequencies that followed temperature oscillations closely, as is demonstrated for August 2013 in Figure 17. The warmest time of the day occurred between three and four o clock, while the peak in velocity followed one or two hours later. Such melt-induced variations in surface velocity have been observed before (Iken & Bindschadler 1986; Howat et al. 2008). The melt-induced horizontal velocity variations correlate well with temporally shifted temperature measurements from the Kvísker weather station (Pearson correlation coefficient r = 0.73), but correlate worse with modelled temperatures (r = 0.63). This is to be expected because the model does not take the zero degree glacier surface into account (see section 2.2.1). The optimal correlations are found when the velocity lags the temperature by 2 3 h (Figure 18). A similar time lag of 2.2 h was found for the Kennicott glacier in Alaska and was deemed reasonable considering a mean supraglacial channel flow speed of 0.5 m/s and the ~ 2.5 km mean travel length to the first moulin (Bartholomaus et al. 2008). Because these velocity variations are caused by daily melt fluctuations, they were only observed during the melt season. 36

53 Figure 17: Daily melt-induced fluctuations during early August Alternating gray shading marks days. Upper tile: horizontal velocities of ICE-A and ICE-D smoothed with a 6 h sliding window. Interpolation of the horizontal velocities is drawn in gray and light blue color for both rovers. Middle tile: detrended uplift of ICE-A and ICE-D. Note that data gaps are not highlighted. Lower tile: Weather data including temperature, ablation, rain and snow with sources described in section 2.2. Water input is given in bins, where the area of the bin equals the amount of surface water input. However, note that the ablation measurements are not representable for the entire glacier and cannot be compared to the rain input (see section 2.2.3). (a) (b) Figure 18: Correlation between velocity and temperature between days 212 and 232, (a) The correlation with a linear regression fit. (b) The time lag between velocity and temperature variations calculated in steps of 1 h. 37

54 3.3 Speed-up Events Because glacier speed-up depends on the state of the drainage system, speed-up events before, at the beginning and at the end of the melt season are expected to differ. Therefore, pre-melt season, early melt season and late melt season speed-up events are described separately. The hysteresis in the relationships between horizontal velocity and water pressure and bed separation (see section 1.3.1) suggests that the acceleration, deceleration and uplift do not occur simultaneously. The events are therefore discussed in three stages: acceleration, deceleration and uplift. They are compared in section Pre-Melt season: Winter Acceleration. During February 9 12, 2013, about 130 mm rain precipitated on the glacier which resulted in speed-up by a factor of 2.7 as shown in Figure 19a. The speed-up occurred during the middle of winter where the subglacial drainage system is expected to be least efficient. The horizontal velocity acceleration relates to the rate of water input. This indicates that rainfall could have overwhelmed the drainage system which resulted in the speed-up. Deceleration. Velocities decreased in two phases. As precipitation ceased, initially horizontal velocities decreased rapidly (phase 1). At a linear rate it would have taken 24 h to decrease to 1/e of the maximum speed, but after February 11 the velocity decreased much slower (phase 2). The change in deceleration rate is marked by a point in time which will be referred to as the bend point. Uplift. The onset of uplift coincided with the onset of horizontal acceleration and reached a peak of 25 cm when horizontal velocities reached the bend point, half a day after rainfall had stopped. Part of the uplift could be related to water storage in cavities. The timing of uplift relates strongly to the horizontal velocities indicating that compressive flow due to stress gradients could also partially have caused the uplift. During the second phase extensional flow could have contributed to slow subsidence. The ice compression between the rovers cannot be determined because only one rover has been operating during the event Early Melt season: Spring Acceleration. During May 22 30, 2012, about 100 mm rain precipitated on Breiðamerkurjökull which was the main cause for the glacier to accelerate by a factor 1.8 as shown in Figure 19b. The rainfall seems to have overwhelmed the drainage system which resulted in speed-up. Deceleration. Because the melt season had just begun, the drainage system is expected to be inefficient with many linked cavities and few channels. This is confirmed by small horizontal velocity variations on top of the decreasing trend; they indicate that despite the rainstorm increasing the capacity, the subglacial drainage system was still prone to becoming overwhelmed with daily melt variations. It took a week for the glacier to slow 38

55 down to its original speed (May 30) and 4.3 days to decrease its speed by 1/e, which further indicates that the drainage system was not very efficient. Uplift. There was no significant uplift of the glacier. One possible explanation is that the drainage system is equally efficient throughout the glacier such that there is no compressive flow. (a) (b) Figure 19: (a) Pre-melt season speed-up during The red dashed line marks the bend point. (b) Early melt season speed-up during See plotting scheme description in Figure

56 3.3.3 Late Melt season: Fall The largest speed-up events occurred at the end of the melt season. Two particularly large speedup events were measured during September 25-29, 2010, and September 23-25, 2012, as discussed separately below. Another large speed-up during August 23-24, 2013 was also observed by Voytenko et al. (2015) who used terrestrial radar interferometry and TerraSAR-X. The latter speed-up event is not discussed in detail in this study. Acceleration. The late summer 2010 speed-up event was the most pronounced and wellrecorded event of all data and is shown in Figure 20a. Both rovers measured very similar (but not identical) velocities, reducing the probability that the processed GPS data can be interpreted ambiguously. Seven periods of rainfall were predicted by the weather forecast model, with total amounts of respectively 80, 35, 70, 100, 25, 20 and 40 mm. The initial rainfall relates to increasing velocities by a factor of 3. A second, smaller, rainfall resulted in an additional speed-up event, reaching 6.5 original velocities. In order for the glacier to speed-up so much considering the small rainfall rates, the subglacial water stored during the second rainfall seems to have added up to water stored during the first rainfall. Subsequent rainfall was heavier, but had less impact on the velocity. The three last rainfalls had no impact on speed-up at all. The drainage system could therefore have increased its capacity. This is in agreement with the hypothesis that increased velocities relate are caused only by the overwhelming of the drainage system. The late summer 2012 speed-up is similar to the 2010 speed-up event described above and is shown in Figure 20b. Rainfalls of 70, 15, 60 and 10 mm triggered the glacier to accelerate. The first rainfall increased the velocity, but the second, smaller, rainfall seems to have added up to water storage caused by the first one because it caused the main speed-up. The third and fourth rainfalls were of almost equal amount as the first two. Nevertheless the speed-up was less pronounced, indicating again that the drainage system increases in capacity and that only an overwhelming of the drainage system results in speed-up. Deceleration. Velocities decreased rapidly after each speed-up event, while the final slowdown occurred in two phases. For example, velocities decreased to 1/e of the last peak in 5 and 8 hours for the 2010 and 2012 speed-up events respectively (phase 1). This rapid deceleration lasted until a bend point from where velocities decreased at a slower rate (phase 2). It took ~6 days and ~2 days for both events respectively to reach original velocities after the last peak in horizontal velocity. Uplift. The onset of uplift coincided with the onset of horizontal acceleration. Peaks of 50 cm and 25 cm were reached for the 2010 and 2012 events respectively, when horizontal velocities reached the bend point, half a day after rainfall had stopped. Part of the uplift could be caused by hydraulic jacking of accumulated water from the large amounts of precipitation. The second rainfall during the 2010 event, triggered the largest speed-up and caused a rapid rise of ~10 cm. The close correlation between horizontal and vertical velocities suggests that the uplift could partially be caused by compressional flow because 40

57 of a longitudinal stress gradient. Indeed, the velocity of the upstream rover peaks earlier than the downstream rover. Along this thread, the second phase of decreasing horizontal velocities could partially be caused by extensional flow. The compression between the rover can be calculated quantitatively, because both rovers were operational, but is not done for this study (see section for explanation). Unlike early melt-season speed-up events, there is no sign of melt-induced contributions during the speed-up event. Diurnal melt variations may be expected to have been small because temperatures were relatively constant: 7 8 C. (a) (b) Figure 20: (a) Late melt season speed-up event in (b) Late melt season speed-up event in See plotting scheme description in Figure 17. The red dashed line marks the bend point. 41

58 3.3.4 Interpretation According to Hypotheses After each speed-up event, surface velocities decrease at different rates depending on the season. The winter 2010 speed-up event would have taken 24 h to decrease to 1/e of the peak velocity, and the spring 2010 speed-up event took more than 4 days to reach 1/e of the peak velocity. These slow deceleration rates indicate that the subglacial drainage system is relatively inefficient. Moreover, daily melt-induced contributions affect the horizontal velocities during the spring speed-up events significantly, indicating how easily the drainage system is repeatedly overwhelmed. During the fall speed-up events, it takes only 5 to 8 h for the velocity to decrease to 1/e of the peak velocity indicating that the drainage system is efficient. When comparing the estimated efficiencies during winter, spring and fall it is concluded that the drainage system evolves to greater efficiency through the year. Moreover, repeated fall rainstorms increase the capacity of the drainage system because the impact of the rain on horizontal velocities decreases for every rainfall. Horizontal velocity speed-up events can all be explained by rainfall overwhelming the drainage system. This causes channels to expand, which increases the capacity of the system. If excess water has been able to drain before the next rainfall event, the rain only has significant impact if the water input exceeds the increased drainage system capacity. If excess water could not drain, the newly incoming water adds to the previous amount of rainfall already stored in the glacier. Then the water input could cause further speed-up. Because the drainage system becomes overwhelmed, the subglacial water storage increases. The accumulation of water causes glacier uplift through hydraulic jacking. Part of the uplift can also be explained with longitudinal stress gradients. Faster ice movement upstream relative to downstream causes compression downstream, leading to uplift at the location of the rovers. The compression turns to extension which leads to slow subsidence and slowly decreasing horizontal velocities. The quantitative contribution of both factors is not determined in this study because the compression and extension can only be derived between the two GPS rovers that are located close to each other. For example, compression and extension on larger on spatial scales, beyond the two rovers, cannot be estimated: an entire block of ice containing both GPS rovers could be lifted or subsided. The measurements of speed-up events show that the described hypotheses on subglacial hydrology (see section 1.3) are capable of describing parts of the rover displacements on short time scales. However, if hydraulic jacking causes most of the glacier uplift, there is a large discrepancy between the amount of subglacial water storage and the horizontal surface velocities. In other words, this scenario would imply that large amounts of water can be stored under the glacier without causing increased sliding. This would oppose the described hypotheses on subglacial hydrology (see section 1.3) because they relate the amount of water storage to increased rates of sliding. The topic is discussed in section

59 3.4 Ice Deformation It is one of the objectives of this study to estimate ice deformation in order to derive sliding velocities. Two methods are used: using rover winter velocities and Full Stokes flow modelling Estimating Winter Velocities During winter conditions, the measured velocities are relatively constant. This is demonstrated for late February 2010 in Figure 21, where between days 48 and 60 of 2010 velocities vary around a stable 0.73 m/day for ICE-D, with only small variations of 0.05 m/day. This supports the hypothesis that winter flow is mostly the result of ice creep. For every year periods with winter weather conditions were used to derive winter velocities for each rover which are shown in Table 2. For 2010 and 2013 winter velocities are observed during the end of winter season. Because 2012 data do not include winter velocities from before the melt season, velocities from after the melt season are used. Even though there is no expected water input after the melt season, summer melt may have implications for (early) winter velocities due to the evolution of the drainage system (Burgess et al. 2013; Schoof et al. 2014). Evaluation of these winter velocities through modelling is therefore important. Because the rovers were not always operational due to the lack of battery power, for some rovers no winter velocities could be derived. To fill the gap an attempt to estimate those values was made. During each year there was some time interval where both rovers were measuring, obtaining v rover,1 and v rover,2. From this interval a difference between both rovers was established dv 1 2 (interval) = v rover,1 v rover,2. When assuming that this difference remains the same throughout the year, dv 1 2 (interval) = dv 1 2 (year), measured velocities can be transferred to obtain an estimate for velocity of the other nonoperational rover by v not measuring rover = v measuring rover dv 1 2. The velocities presented in Table 2 show significant annual variation, but it is assumed that this is mostly due to varying distances of the rovers to the calving front and this therefore does not necessarily represent annual acceleration or deceleration. The small variations in velocity that were measured during winter could be the result of (1) GPS errors, (2) natural variability, or (3) the tides in Jökulsárlón. Tide data was provided by S. Zóphóníasson from the IMO (personal communication, November, 2014). 1. Because horizontal velocity errors were estimated to be ~0.05 m/day (see section 2.1.3), they could fully explain the observed variability. 2. Natural variations can be caused by a remnant drainage system that is kept active by water input from various sources such as ground water, englacial water or melted ice by friction or geothermal heat (Schoof et al. 2014). Moreover the connection between a distributed subglacial water storage and a more efficient drainage system can cause instabilities in the water pressure (Schoof et al. 2014). 43

60 3. Vertical velocities anti-correlate with the tidal rate during days 70 and 75 (r = 0.55), days 80 and 91 (r = 0.54) and days 98 and 104 (r = 0.73), see Figure 22. However, vertical velocities correlate slightly positive during the days in between, see Figure 22. The correlation between vertical velocities and the tidal rate is positive in the transition from neap tide to spring tide, but negative in the transition the other way around. No correlations were measured between the horizontal velocities and the tides, even when introducing a time lag to account for a possible phase shift: for example, during days 98 and 104, r = 0.1. Voytenko et al. (2015) did not find any relation to the horizontal velocity of the tides either and argue that the small tidal signal has a negligible influence on the effective pressure of the grounded glacier on the bed. Table 2: Winter and average melt season velocities. Values with an asterisk are estimates of the velocity because no measurements were made. Year Interval v ICE A v ICE D DOY m/day m/day * * * * * Figure 21: The late February 2010 interval shows velocities during typical winter conditions. The horizontal blue line marks the average velocity of 0.73 m/day. See plotting scheme description in Figure

61 (a) (b) Figure 22: (a) Horizontal and (b) vertical velocities (black line and gray scatter) are plotted against the tidal rate (red line). The tide elevations are from the IMO, provided by S. Zóphóníasson (personal communication, November, 2014). ICE-A data plotted in the black solid line are the result of hourly spline-smoothed solutions that are derived from 6 h linear fit sliding windows. The gray scatter shows 10 min solutions from 2 h linear fit sliding windows. Alternating gray shading marks days. Figure 23: Tides plotted against the correlation between vertical velocity and tidal rate. Blue shadings mark periods when the correlation is negative and red when it is positive. 45

62 3.4.2 Modelling Ice Deformation Glacier flow can be modelled with the Stokes equations applied to ice, as summarized in section 2.3. Winter profiles for the model were estimated according to equation (14) for days that are within the winter intervals discussed in section 3.4.1, 20 February 2010, 17 October 2012 and 18 March The used profiles are plotted in Figure 24. The model results are validated along the flowline using a velocity field of Breiðamerkurjökull from Nagler et al. (2012), which was created from an InSAR image pair taken with the TerraSAR-X satellite at August 11 and 22, The model is then used to derive ice deformation velocities at the times given above. These modelled velocities justify the use of rover winter velocities as ice deformation velocities. For (Icelandic) temperate glaciers n = 3 and A = s 1 Pa 3 are recommended by Cuffey & Paterson (2010) and are therefore adopted in this study. An attempt to calibrate the model to the Glen s flow law coefficients is discussed in section Figure 24: Estimated winter surface profiles for 2010, 2012 and 2013, fitted to rover measurements. The rover tracks are color coded to mark the date of the year. The winter locations of the rovers are marked with a triangle that is filled with the same color as the profile of the corresponding year. 46

63 Spatial Evaluation: Model vs TerraSAR-X Velocity Field The TerraSAR-X velocity field from Nagler et al. (2012) was taken only one month after the lidar 2010 measurements such that the glacier surface profile is a reliable input in the Stokes model. A complication is that the measurements were done during melt season when the glacier is sliding. The magnitude of the sliding is therefore an unknown input variable in the model. From August 11 to 22 the average velocities of the ICE-A and ICE-D rovers were 0.92 m/day and 1.00 m/day. Because ICE-D moved in one continuous track, comparison to winter velocities is possible. The measured velocity is 0.21 m/day higher than the estimated winter velocity for early Between summer and winter the calving front has approached the rovers, which partially explains the increase in velocity. The glacier retreated about 200 meters between winter and summer, but in the same period ICE-D also moved 170 m downstream. As a result the front was relatively 370 m closer to the rover which caused velocities to increase by ~0.1 m/day. The glacier sliding at the rovers is therefore estimated to be 0.11 m/day. Velocities reached 2.08 m/day at the front and 0.6 m/day at the start of the flowline and are used for setting the boundary conditions. The boundaries at the front and the back velocity are assumed to decrease towards the sliding velocity at the bed according to eq. (12). The model correlates well (r 2 = 0.93) with measured velocities along the flowline and it reproduces even local variations, see Figure 26. This is an indication that the measured winter velocities do represent ice deformation and that sliding is estimated correctly. Although there is considerable noise in the TerraSAR-X data, the modelled velocities correlate with the average values. Note that the model performs poorly in two areas: near the calving front and at higher elevations. Near the calving front, the surface velocity field shows a more complex 2D structure with increasing velocities towards the eastern part of the calving front (see Figure 25). Velocities are expected to be higher there due to the ice that goes through the trough discussed in section This could drag the western part of the front along. Another explanation could be higher sliding velocities towards the calving front. Velocities are modelled too low at higher elevations. This is to be expected because sliding increases by altitude as the drainage system becomes less efficient higher up (Fountain & Walder 1998). Another explanation is the deeper trough to the east pulls the glacier along. Both could be corrected for through adding sliding at the base, as is shown in the advanced model in Figure 26. The correlation increases to r 2 = Despite the improved match it is not possible to conclude whether the added sliding is real or whether the lateral drag increases the ice deformation. 47

64 Figure 25: TerraSAR-X velocity field between August 11 and 22, 2010 (Nagler et al. 2012) plotted on top of the October 2010 lidar hillshade DEM (Jóhannesson et al. 2013). 48

65 Figure 26: Comparison between measured velocities from TerraSAR-X (Nagler et al. 2012) and the Stokes model, including the locations of the ICE-A and ICE-D rovers. 49

66 Modelling Winter Ice Deformation The Stokes flow model discussed in section 2.3 is used to validate the use of winter velocities as creep flow velocities. The velocity boundary condition at the bed is set to zero to model ice deformation only and shear stress at the upper surface is assumed to be zero. At the calving front and far upstream the glacier velocity is assumed to decrease towards the bed according to eq. (12). This required specifying velocities from the surface to the bed. The surface velocities at these two locations are taken from the TerraSAR-X velocity field minus the estimated sliding during that period and are 2 m/day at the terminus and 0.5 m/day at the upstream boundary. A comparison of the modelled velocities, w, at the location of the ICE-D rover, with the measured winter velocities of ICE-D itself, v, is shown in Table 3. The modelled velocities are similar to the measured winter velocities, although only a thorough error analysis of the model can justify the use of winter velocities to represent creep flow, see discussion in section 4.3. Table 3: Comparison between measured winter velocities, v, and modelled velocities, w, in m/day. Year DOY v w Figure 27: Example of cross section showing the modelled velocities for the winter Note the vertical exaggeration of 15. The color of the filled triangle shows the measured velocity. 50

67 3.5 Seasonal Evolution of the Drainage System One objective of this study is to relate water input forcing to long term changes in velocities. However, only glacier sliding is related to water input. The 10-day smoothed basal sliding is obtained by subtracting the modelled ice deformation (derived in section 3.4) from the surface velocities and are shown in Figure 28. The seasonal cycle is visualized by the thick line (Gaussian smoothed sliding with a width of 20), which removes some of the peaks related to rainfall. For 2010 and 2013 the cycle is as follows: velocities from February to April are relatively low with velocities being pure creep flow during winter weather conditions. Maximum velocities were reached during the spring speed-up around May/early June. Then sliding decreases as the melt season progresses. As discussed in section 1.3.3, this decrease in velocity is explained by the evolution of the subglacial drainage system. Sustained melt increases the size of subglacial channels such that the drainage system capacity increases. As the season progresses, melt water is therefore routed through channels with increasing efficiency such that the distributed drainage system decreases in size. This increases ice sole contact to the bed and hence decreases sliding. However, data from 2012, show a different seasonal cycle in velocity because velocities peak late, at the end of August. Average temperatures from the HARMONIE model (see section 2.2.1) in the early melt season (days ) were 7.8 C, 5.4 C and 5.9 C for the years in chronological order. Indeed, 2012 had the coldest spring, but the temperature is not significantly colder than during the spring of Consequently, the spring temperature alone cannot explain the slow onset of sliding in The average late winter (days ) temperatures were 0.4 C, 0.3 C and 3.5 C for the years in chronological order. This shows that although 2012 had a relatively cold start of the melt season, the temperatures were relatively higher in the time before the melt season, frequently above the freezing point. The combination of a relatively warm late winter and a cold melt season can explain the slow increase of sliding as follows: because relatively much melt water reached the glacier bed during late winter, the drainage system evolved to greater efficiency. Because the spring was relatively cold, surface water input was insufficient to overwhelm the relatively large capacity of the drainage system. Therefore, it was not until the relatively warm summer that enough ablation occurred to accelerate the glacier. To determine whether increased melt affects annual velocities requires having velocities for the entire year. The time series of the GPS rovers on Breiðamerkurjökull cover only February to October at the most, and are therefore insufficient to draw definite conclusions regarding this question. Moreover, 2012 and 2013 were on average 1 C colder than 2010, but there does not seem to be any consistent correlation between temperatures and sliding rates during summer. 51

68 Figure 28: 10-day averaged sliding velocities during 2010, 2012 and The thick Gaussian smoothed line represents the seasonal cycle which dampens some of the peaks related to rainfall. 52

69 Chapter 4 Discussion 4.1 Anomalous Correlations Some measured ablation and simulated rain does not correlate with increased surface velocities. A possible explanation for this is that the estimated surface water input is not a direct measurement of the rate of water input to the subglacial storage, both spatially and temporally. The shortcomings of the estimated precipitation and ablation are as follows. First, the rain input is based on the HARMONIE weather forecast model. Section discussed that modelled rain on the glacier is offset compared to measured rain at the Kvísker weather station, indicating high spatial variability. Because the rain is only estimated for one km 2 grid cell it may not be representative for other areas on the glacier. However, full incorporation of spatial weather forecast data is outside the scope of this study. Second, ablation input is based on vertical elevation changes of the glacier measured with a sonic ranger. These elevation changes do not accurately represent ice melt as discussed in section Moreover, elevation changes during winter may be the result of the redistribution of the snow pack due to snow drift or settling and densification of snow due to wind. A snow pack on the glacier inhibits percolation of rain to the bed because dry snow sucks up water which causes snow to become denser. Therefore neither modelled rain nor measured ablation represents water that actually reaches the bed of the glacier. Spatial variation in velocity may also relate to the spatial distribution of moulins (Mair et al. 2002). This variation is likely negligible on the spatial resolution covered with both GPS rovers because they are only separated by 500 m. Some anomalous correlations can be attributed to the sonic ranger (see section 2.2.2) failing. For example, measured ablation on March 16 (day 75), 2010, did neither coincide with positive temperatures, nor with glacier speed-up. The measured ablation could there be caused by instrument malfunction. Other explanations could be snow drift and snow compaction. 53

70 4.2 Comparing Ablation with Precipitation Multiple Linear Regression Theory As discussed in section it is difficult to quantitatively compare the effect of ablation and precipitation on ice velocity because there are spatial variations in both precipitation and ablation. Most important, melt water input to the bed is overestimated for higher elevations because it was only measured near the terminus. Precipitation is only estimated at one grid cell (see section 2.2.3), but it is possible that similar amounts fall on the entire glacier. Because in general only the total rate of water into the subglacial drainage system causes the velocity variations, ablation and precipitation are two variables that explain a single response. Assuming the causal relation is linear, Multiple Linear Regression (MLR) can be used to estimate how much of the measured ablation and estimated precipitation contribute to velocity variations (Moore et al. 2009). Note that no studies support a perfect linear relationship between the velocity of the glacier and the water input from the surface. Actually the hypothesis of dynamically evolving subglacial drainage system (discussed in section 1.3.3) implies that the drainage system evolves to greater efficiency directly after speed-up events. To reduce the impact of inconsistencies within a day, correlations are done with average daily values. The method is experimental and should be used with care, although high correlations could suggest there is some truth in the method. At a given time interval and a given location the melt water rate Q melt and rain rate Q rain, can be added linearly. The total water input is given by: Q total (γ) = γ Q melt + β Q rain. (15) When ablation is assumed to be overestimated and precipitation is assumed to fall equally over the glacier, γ is unknown and β = 1. Velocity variations relate to changes in Q total. Assuming a simple linear relationship between glacier surface velocity, u s, and the total rate of surface water to the bed, Q total, the glacier velocity could be described by: Substituting equation (15) into equation (16) yields: or alternatively: u s = α Q total + u creep. (16) u s = α (γ Q melt + Q rain ) + u creep (17) where c = u creep, a = α γ and b = α, implying that γ = a b. u s = a Q melt + b Q rain + c, (18) MLR of the explanatory variables melt and rain with the response variable glacier velocity solves for the correlation coefficients of Q melt and Q rain, a and b, respectively, such that the factor of over estimation can be derived when considering a linear model (Moore et al. 2009). 54

71 4.2.2 Results of MLR The interval of days of 2013 includes both strong precipitation events and daily melt-induced variability. The MLR yields very high correlation (r 2 = 0.85) which suggests a linear relation on daily timescales to some extent. Ablation is overrepresented by a factor of γ = 2 compared to precipitation. Creep velocities for rovers ICE-A and ICE-D are then estimated to be 0.84 and 0.67 m/day respectively, which is 0.10 and 0.19 m/day lower than estimated winter creep velocities. Other factors of overestimation are possible, such as for the days of 2012 where rain is contributing 5 times more to glacier variability (r 2 = 0.82). Creep velocities for rovers ICE-A and ICE-D are estimated to be 0.82 and 1.00 m/day respectively, which is 0.08 and m/day compared to estimated winter creep velocities. Because the factor of overestimation of ablation, γ, depends strongly on the spatial extent of each rainfall, the factor may expected to be highly variable. MLR will then not perform reliably over long time intervals. This can partially be attributed to the incorrect assumption of a linear relation between the rate of water to the bed and velocity variations. Iken & Bindschadler (1986) already showed a nonlinear relationship between water pressure and velocity. Moreover the relationship also depends on the evolution of the sub-glacial drainage system. MLR is therefore not suitable to accurately determine how much more precipitation compared to ablation enters the drainage system. 55

72 4.3 Model Evaluation The velocities from the Stokes flow model, w, are compared to measured winter velocities at ICE-D, v, in Table 3. At a first glance the values coincide well which suggests that the physics of glaciers is well understood. However, the model relies on estimated parameters that have their associated uncertainties. Five important parameters are considered in this section. 1. There is considerable spatial variation in the velocity field near the calving front due to the lateral drag from the fast flowing ice towards the east of the flowline, especially during 2010 (see section 3.4.2). 2. The glacier margin could be partially floating which causes perturbations in the velocity field. 3. The surface topography could have been poorly constrained and winter advances have been excluded (see discussion in section 2.3.4). 4. The calving front location was not straight which causes perturbations in the surrounding stress field that cannot be captured in a 2-D model. There was especially strong embayment during 2012 (see section 2.3.5). 5. The recommended creep parameter, A, for temperate glaciers could be inappropriate for Breiðamerkurjökull (see section 4.3.6). The uncertainty in each parameter yields an error in the modelled velocities, e w,i. The error in the velocity is calculated through a sensitivity analysis for the uncertainty in each parameter. Because the parameters are independent and the errors due to the first five parameters are assumed to be normally distributed the total error in the velocity, e model, may be calculated as: e w = e w,i 2 = e w,drag 2 + e w,velocity 2 + e w,profile 2 + e w,margin 2. (19) Lateral Drag (e w,drag ) The TerraSAR-X velocity field in section shows that ice flowed faster towards the north east of the flowline. As a result, the faster flowing ice exerts a lateral drag on the ice at the flowline. However, due to the inherent 2-D nature of flowline models, spatial variation cannot be accounted for. As the lateral drag cannot be included in the model, its impact can only be estimated. In comparison to the TerraSAR-X velocity field discussed in the previous section, the model calculates velocities 0.2 m/day lower than those measured with the rovers. During the summer of 2010, the measured values are therefore biased by 10 e w,drag 56 = 0.2 m/day. The discrepancy is explained by inhomogeneities of the bedrock geometry at the calving front. The flowline was created by assuming that the glacier flows perpendicular to volcanic ash layers. It therefore does not have to follow the observed bedrock trough (discussed in section 1.2.1). For the most part the flowline follows the trough except for the last 1.3 km from the 2010 calving front. Here the trough bends 500 m towards the east

73 with respect to the flowline. The flowline is located above bedrock elevations m higher than elevations within the trough. Because the ice to the north east flowed through an overdeepening, the ice was thicker and therefore yielded higher velocities exerting the lateral drag. During 2012 and 2013, however, the effect should have decreased because the calving front retreated. Therefore the trough excursion to the north east was no longer covered by ice. In other words, the intersection of the flowline and the calving front was located exactly at the deepest part of the trough. As a result, the opposite effect could instead have been taking place: the ice at the flowline is withheld by a negative lateral drag from thinner, slower moving surrounding ice. The bias in the drag was therefore substantially smaller than the bias during 2010, but cannot be determined quantitatively, hence: 12, m/day. The uncertainty would be reduced if the flowline following the e w,drag bedrock trough is chosen (assumption ii. in section 2.3.3) Surface Velocity at Calving Front (e w,velocity ) There was no floating ice in the summer of 2010 as shown in Figure 24. However, the DEM from October 2014 shows that about 200 m of the glacier front was floating and another 200 m could be partially floating, because elevations in crevasses are below the flotation criterion. Therefore, there could have been little basal drag in the first 400 m from the front of the glacier, or possibly it moved as a floating ice shelf. Therefore, it cannot be excluded that at least a part of the calving front was floating during winters of 2012 and Indeed, Voytenko et al. (2015) measured that about ~200 m of the terminus was floating during the summer of If the glacier is slightly afloat there is a small ice shelf in front of the glacier that has reduced or no friction at the bed such that it moves as one solid block. The small ice shelf changes two boundary conditions simultaneously: first, the velocity at the calving front which increases due to reduced friction and second, the calving front wall moved with the same velocity from surface to bottom. The second change implies that the surface velocities only decrease upstream of the grounding line such that the velocity field effectively shifts towards the rovers. As discussed in section 3.2, the winter 2010 surface velocities are well determined and are estimated to be 2 m/day. In addition, no floatation is observed during There is therefore no uncertainty to take into account such that e w,velocity = 0.00 m/day. For 2012 and 2013, however, neither summer nor winter velocities at the calving front have been measured. During this period the glacier retreated towards deeper waters and for this reason the glacier also became more prone to floatation. That the glacier picked up speed as it retreated was observed from the feature tracking of the photogrammetry used to also create the 2014 DEM. Velocities of 3 m/day were measured during October 2014 (A. H. Jarosch, personal communication, November, 2014). Assuming a linear increase between 2010 and 2014 velocities were approximately 2.5±

74 m/day during 2012 and The increase in velocity can be explained by floatation. When taking into account ~ 100 m floatation and increased frontal velocities, the 12, 13 associated error was estimated to be e w,velocity = 0.06 m/day. More velocity data on the calving front are needed to carefully estimated the boundary conditions Surface Topography (e w,profile ) The estimated winter surface profiles rely partially on thinning rates from distant mass balance measurements although their importance was reduced by fitting the profiles to rover measurements. Because the actual thinning was not measured, there are no unique solutions to the surfaces and as a result modelled velocities at the rovers locations are erroneous by e profile. The shape of the 2010 profile is not expected to differ very much 10 from the lidar profile because they only span an interval of 150 days, therefore e w,profile 0.00 m/day. For the 2012 and 2013 there is a range of likely surfaces near the calving front as they should resemble the shape of the lidar surface profile. Applying the most extreme surfaces for 2012 and 2013, velocities can differ by e 12, 13 w,profile = 0.02 m/day. This result agrees well with errors estimated by Voytenko et al. (2015). When assuming horizontal surfaces (worst case scenario) they derive errors of 3% (~0.03 m/day) for the first 500 m of the terminus and errors of 1 % (~0.01 m/day) further up glacier Calving Front Location (e w,margin ) Calving front locations have been derived from Landsat satellite imagery that usually is not available on exactly the right date where winter motions were defined in Measuring from the 2010 lidar calving front, the front advanced 70 m and retreated m and 490 m, for winters periods during 2010, 2012 and 2013, respectively (as defined in section 2.2). For 2010 and 2013, there is therefore an uncertainty of about ± 50 m. For 2012, the embayment centered at the flowline resulted in an uncertainty of the calving front of 360 m. The location of the calving front does not change the absolute magnitude of the velocity, but shift the entire velocity field spatially. For example, a retreat implies higher velocities at the location of the rovers, even though the glacier dynamics and sliding are the same. The associated error e w,margin = 0.01, 0.05 and 0.02 m/day for 2010, 2012 and 2013 respectively. 58

75 4.3.5 Summing of Model Errors The errors in velocity were added according to equation (19) and are shown in Table 5. The measured ICE-D winter velocities and modelled ice deformation velocities are compared in Table 5. Measured and modelled ice deformation velocities agree within the error margins, although measured values are consistently higher. The bias could indicate that some winter sliding occurs, but the errors are too large to determine this. Table 4: Summing of model errors in m/day. Year DOY e w,drag e w,velocity e w,profile e w,margin e w Table 5: Comparison between measured winter velocities, v, and modelled ice deformation velocities, w, in m/day, including error estimates. Year DOY v w ± ± ± ± ± ± Glen s Flow Parameters Results from a sensitivity analysis for the creep parameter are presented in Table 6. Velocities were calculated with the 2010 winter surface at the location of the ICE-D rover. The recommended creep flow parameter for temperate glaciers is A 0 = s 1 Pa 3 (Cuffey & Paterson 2010) and yields w 0 = 0.67 m/day. Velocities were calculated with creep parameters in the range of A = s 1 Pa 3. The sensitivity analysis shows that the changes of A relative to A 0 are the same as changes of w relative to w 0. This result is expected considering the linear relationship between the strain rates and the creep parameter described in Glen s flow law, equation (4). Winter velocities for ICE-D were in ±0.05 m/day. Adding to that the estimated error in the model during 2010 of ±0.10 m/day (Table 6), modelled velocities of 0.73 ± 0.15 m/day agree with the measurements. Therefore, the range of accepted creep parameters is A = s 1 Pa 3. 59

76 Table 6: Sensitivity analysis of the creep parameter. Values of A {2.1, 3.2} s 1 Pa 3 are consistent with observations and marked blue. Inconsistent values are marked red. A w (A A 0 )/A 0 (w w 0 )/w ( s 1 Pa 3 ) (m/day) (%) (%) A 0 =

77 4.4 Discussion of Subglacial Hydrology The results obtained from the GPS rovers on Breiðamerkurjökull indicate that surface velocity variations correlate with increasing amounts of water input. This important finding confirms that the hypothesis of the dynamic evolution of the subglacial drainage as described in section and thereby agrees with other studies (e.g. Bartholomaus et al. 2008; Fudge et al. 2009). Surface uplift can be explained by water storage and longitudinal stress gradients described in section also agreeing with other studies (Sugiyama & Gudmundsson 2004; Howat et al. 2008). However, the hypotheses described in sections and do not explain the discussed temporal offset in the basal motion, water pressure and water storage. Descriptive models fall short and recent efforts have therefore mostly been focused on numerical modelling to explain observed variations Alternative Subglacial Hydrology Models Two branches in the modelling efforts can be distinguished. First, is the coupling of distributed and channelized drainage systems where there effectively is no distinction discernable between the two drainage systems (Schoof 2010; Werder et al. 2013). The second direction of modelling drops the concept of cavities and channels altogether and captures the entire drainage system in one homogenous porous medium. This medium is modelled in two layers where one is an aquifer representing the inefficient drainage system and the second efficient layer becomes activated when the drainage system becomes overwhelmed (Flowers & Clarke 2002; De Fleurian et al. 2014; Rosier et al. 2015). The flow through the porous media is described by Darcy s law. The challenge remains to establish in what ways the drainage systems relates to the basal motion through the friction law. Fowler (1987) directly related effective pressure to basal drag because increasing water pressures lead to cavity expansion. Newer friction laws are based on the same principle and also directly relate basal drag to effective pressure (Schoof 2010; Werder et al. 2013; Hewitt 2013; De Fleurian et al. 2014; Rosier et al. 2015). However, measurements indicate that the water pressure does not change in phase with surface velocity variations (see section 1.3.1). Despite a widely expressed demand for a better description of the friction law (e.g. Hewitt 2013) most modelling work has focused on studying the evolution of subglacial drainage system and the effective pressure, while leaving the friction law aside. To conclude, despite recent modelling efforts, the relationships between changes in the water pressure, water storage and basal motion remain a fundamental challenge of subglacial hydrology. The data used in this study are insufficient to shed more light on the problem because no bore hole water pressures have been measured. 61

78 4.4.2 Uplift of Breiðamerkurjökull Because recent models still need improvement to explain the relationships between changes in the water pressure, water storage and basal motion, subtle details in observations may bring elucidation. As showed in section 3.3, during speed-up events of Breiðamerkurjökull, the glacier tends to lift up and remain at higher elevations much longer than the duration of the speed-up event itself. As discussed in section 1.3, longitudinal stress gradients and bed separation can cause such uplift. It could also be that another factor plays a role at the Breiðamerkurjökull. During the years when this study was carried out the glacier calved above a bedrock hill of ~60 m height, see Figure 29. Enhanced sliding could have caused the glacier to be pressed up against the hill causing glacier uplift (E. Magnússon, personal communication, May, 2015). For this thesis, the precise contributions of all factors were not derived, primarily because each factor causes a spatially inhomogeneous signal that cannot be resolved with only two rovers that are located nearby each other. Nevertheless, studies have indicated that bed separation is the main source of glacier uplift and that it lags behind the peak in surface velocity (cf. Iken et al. 1983; Anderson et al. 2004; Harper et al. 2007; Howat et al. 2008). Moreover, the amount of water that is stored under a glacier was also found to lag behind the peak in horizontal surface velocity (Bartholomaus et al. 2008). The lag and the hysteresis have important implications for theories on subglacial hydrology. For if the interpretation of the observations holds, water can apparently be stored under the glacier without causing significant increased sliding. Moreover, water pressure was observed to occur in hysteresis with the surface velocity, as discussed section The fundamental problem is accounted for in section Figure 29: Bedrock hill diverting ice flow direction during a speed-up event (red arrow). 62

79 4.4.3 Proposed Conceptual Model of the Drainage System A simple conceptual model that explicitly accounts for the temporal offset in the peak of basal sliding, water pressure and bed separation, based on a model suggested by Bartholomaus et al. (2008) and described in section 1.3.3, goes as follows: As water input exceeds the drainage system s capacity water fills and pressurizes the drainage system. When assuming a steady state distributed drainage system, increased water pressure leads to expansion of cavities. I suggest that water is pressed under the surrounding ice through pore spaces in subglacial till and/or by hydraulic jacking of the surrounding ice, resulting in a sheet of water (see Figure 31b). The sheet of water reduces basal drag, causing most of the increased sliding. The cavities are relatively small and therefore have only limited contribution to the reduction of the basal drag. Although sheet expansion is initiated through high water pressures, the flow of water away from the cavity has a negative feedback on the water pressure. The sheet can therefore expand in area while the water pressure remains relatively constant. The increased basal motion leads to increased cavity size causing water to flow back from the sheets into the cavities. Water pressures cannot decrease however, because changes in pressure are immediately accounted for by the inflow of the water sheet surrounding the cavity (see Figure 31c). The decrease in area of the sheet causes the basal motion to decrease until the sheet has completely disappeared and the sliding is only sustained by cavities. If, by then, the drainage system no longer is overwhelmed, water can flow out of the cavity, which can in turn lead to water pressures decreasing (see Figure 31d). In the meantime, channels have been increasing in size such that drainage becomes more efficient. After a few days a new stable state is reached because the ice overburden pressure compresses the cavities and channels to their original sizes. The peak horizontal surface velocities coincide with the maximum aerial extent of the water sheet, whereas the observed bend point, which discussed in section 3.3, marks the time at which the sheet is drained and sliding is caused by cavities only. Because it takes more time to decrease the size of the cavities compared to draining the sheet of water, horizontal velocities decrease with a slower rate after the bend point compared to before, see Figure 30. This study has mostly focused on temporal variations in surface velocities. With two GPS rovers it was not possible to study the spatial variations in surface velocities. However, the importance of spatial variation has been elucidated in other studies (Mair et al. 2001; Ryser et al. 2014). The conceptual model discussed above is consistent with spatial variations, because sliding only occurs at water filled areas of the cavities and sheets. Sticky and slippery spots used to explain spatial velocity variations in Ryser et al. (2014) are therefore predicted. Other models that use a homogenous porous medium layer for the drainage system do not explain such spatial variations. 63

80 Figure 30: Changes in velocity, water pressure and water storage during a typical speed-up event according to the conceptual model. Note the temporal offset in the peak of surface velocity, water pressure and water storage. Figure 31: (a) to (d) Conceptual model of drainage system evolution. Plane view of one bedrock bump (brown) and its cavity (blue) that is linked to the subglacial drainage system. The water sheet surrounds the cavity in (b) and (c) (turquoise hashed). Glacier flow is directed upwards and the relative basal motion u b is shown on the right. 64