Jonathan Kingslake Department of Geography University of Sheffield Modelling Ice-Dammed Lake Drainage PhD Thesis Submitted in accordance with the requirements for the degree of Doctor of Philosophy 9th January 2013 Abstract The drainage of ice-dammed lakes produces floods that can pose hazards, waste water resources and modulate ice flow. In this thesis I investigate several aspects of ice-dammed lake drainage through the development and analysis of mathematical models. After an introduction in the first chapter and a description of the mathematical background to the thesis in the second, the third chapter investigates the mechanisms behind observed variability in the size and timing of subglacial floods from ice-dammed lakes. In particular, I examine how environmental controls like the weather and the shape of glaciers affect floods. In the next chapter, I quantify how well simple models can predict the dates of floods from an ice-marginal lake in Kyrgyzstan. I find that incorporating environmental controls into models improves their prediction ability. Next I investigate the coupling between subglacial drainage and glacier motion during ice-dammed lake drainage by developing and analysing a model which couples a marginal lake, glacier sliding, subglacial drainage through a channel and subglacial drainage through a distributed system of cavities. I show how changes in lake level cause the rate at which a glacier slides to increase during the first half of floods and decrease during the second half. The next two chapters are concerned with two lake-drainage scenarios that involve water flowing as an open stream: firstly, the subglacial open- channel flow that occurs after a marginal lake drains completely during a flood, and secondly, the drainage of supraglacial lakes across the surface of ice sheets. I end the thesis with a summary of my findings and some suggestions of theoretical and field-based investigations that are worthwhile pursing in the future. Acknowledgements Thank you first to my supervisor Felix Ng for being so incredibly generous with his time and ideas. This thesis would not have been possible without his guidance and training. I am also grateful to my second supervisor Grant Bigg for comments on the thesis, to Andrew Sole for bringing his real-world experience of supraglacial lake drainage to bear on Chapter 7 and to Darrel Swift for many heated conversations about subglacial hydrology. Thank you to all the PhD students, post-docs and staff at Sheffield who have helped make my time in Sheffield so enjoyable. I also acknowledge the financial support of a University of Sheffield Postgraduate Studentship. Thank you to my parents Peter and Clare Kingslake for their constant support and to my Grandma Una Kingslake for assisting me financially throughout my time as a student. Lastly, thank you to my fiancée Jen for her fantastic support. The thesis would certainly not have been possible without it. Modelling Ice-dammed Lake Drainage Contents Chapter 1 Introduction .......................................................................... 1 Chapter 2 Mathematical background .................................................. 13 2.1 Introduction .................................................................................................................................. 13 2.2 Channels ........................................................................................................................................ 13 2.2.1 Steady-state drainage through an ice-walled channel ........................................................... 14 2.2.2 Time-dependent drainage through an ice-walled channel: Nye’s model .............................. 17 Channel geometry evolution ............................................................................................ 17 Mass conservation ............................................................................................................ 18 Momentum balance ......................................................................................................... 18 Energy balance .................................................................................................................. 18 Heat transfer ..................................................................................................................... 19 2.2.3 Fowler’s modification of Nye’s model .................................................................................... 21 2.2.4 The Nye-Fowler jökulhlaup model ......................................................................................... 24 Coupling the channel to a lake.......................................................................................... 24 Non-dimensionalisation .................................................................................................... 25 Boundary conditions ......................................................................................................... 27 Summary ........................................................................................................................... 28 2.2.5 Numerical methods ................................................................................................................ 29 Relaxation method ............................................................................................................ 29 Newton’s method ............................................................................................................. 32 Boundary layer method .................................................................................................... 37 Summary ........................................................................................................................... 39 2.3 Cavities .......................................................................................................................................... 40 2.3.1 Enhanced creep and regelation .............................................................................................. 40 2.3.2 Sliding with cavitation ............................................................................................................ 41 2.3.3 Linked-cavity drainage ............................................................................................................ 43 2.4 Summary and outlook ................................................................................................................... 44 Chapter 3 Environmental controls on flood cycles .............................. 47 3.1 Introduction .................................................................................................................................. 47 3.2 Flood cycles ................................................................................................................................... 49 3.2.1 Model setup ........................................................................................................................... 49 3.2.2 Control simulation: unstable growth in flood-cycle size ........................................................ 51 3.2.3 Stable flood cycles .................................................................................................................. 53 Negative basic hydraulic gradient near the lake suppresses flood cycle growth ............. 53 Supplying the channel with water along its length suppresses flood cycle growth ......... 56 Summary ........................................................................................................................... 59 3.3 Controls on the characteristics of flood cycles ............................................................................. 59 3.3.1 Sensitivity analysis of the reduced Nye-Fowler Model .......................................................... 60 3.3.2 Physics of the divide’s migration ............................................................................................ 63 3.4 Mode-locking, resonance and chaotic dynamics of the Nye-Fowler model ................................. 65 3.4.1 Model setup: time-varying model forcings ............................................................................ 66 3.4.2 Results .................................................................................................................................... 67 3.5 Discussion ...................................................................................................................................... 75 3.6 Conclusions ................................................................................................................................... 79 Chapter 4 Quantifying the predictability of the timing of jökulhlaups from Merzbacher Lake, Kyrgyzstan .......................................................... 81 I Modelling Ice-dammed Lake Drainage 4.1 Introduction ................................................................................................................................... 81 4.2 Flood-date prediction with the threshold model .......................................................................... 82 4.2.1 ‘Real’ and ‘Simple’ Schemes of weather forcing ..................................................................... 86 4.3 Study site and data sources ........................................................................................................... 87 4.3.1 The Inylchek Glaciers and Merzbacher Lake ..........................................................................