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Mathematical Modelling the Formation and Evolution of Melt Ponds on Sea Ice

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Mathematical Modelling the Formation and Evolution of Melt Ponds on Sea Ice Mathematical modelling the formation and evolution of melt ponds on sea ice A dissertation submitted for the degree of Doctor of Philosophy UCIL Paul Duncan Taylor Centre for Polar Observation and Modelling Department of Space & Climate Physics University College London October 2003 ProQuest Number: U643083 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest. ProQuest U643083 Published by ProQuest LLC(2015). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. Microform Edition © ProQuest LLC. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 ABSTRACT Abstract Sea ice is formed in the polar oceans by the freezing of seawater. It is a sensitive component of the global climate, and its influence on climate is significant through feedbacks, such as the albedo-feedback mechanism. Melt ponds are pools of melt water formed annually on the Arctic sea ice surface. Melt ponds form because of preferential absorption of solar radiation compared to bare sea ice and snow. This thesis investigates the evolution of melt ponds on the surface of sea ice, pri­ marily through numerical modelling, and also through simple studies of individual processes that are deemed to be important in the evolution of melt ponds. The theoretical model of melt ponds upon sea ice developed in this thesis is based upon the mushy layer equations and a relatively simple two-stream radiation model. A sea ice mushy layer consists of a solid ice matrix surrounded by brine. The two- stream radiation model has been used in previous sea ice studies, and incorporates a summertime parameterisation of optical properties based upon SHEBA field data. Stationary solutions of the model, without melt ponds or snow, are analysed, show­ ing potentially two stationary ice thicknesses for a given set of forcing data. However, a linear stability analysis reveals that only the larger of the two solutions is stable. Fundamental summertime processes are investigated and discussed. These include melt pond thermal stability, melt pond drainage, and the evolution of sea-ice lenses (ice formed at the interface of fresh drained water and the salty ocean). The full melt-pond-sea-ice thermo-radiative model is forced using primarily SHEBA data. Model sensitivity to processes and important parameters is investigated. The model simulates the evolution of ponds well and demonstrates the importance of radiative effects on summertime evolution. The potential application of the model to GCMs is discussed. ACKNOWLEDGEMENTS Acknowledgements I would like to thank my supervisor Daniel Feltham for the opportunity that he gave me to study sea ice at CPOM and his continued support throughout the course of my thesis. He has always given me the encouragement necessary to keep motivated. I would also like to thank my family for always providing me with books when I wanted them (and when I didn’t) and for encouraging me to go to university. Without the love, support, and generosity of Gerry, Jo, and Anna I would not have been able to complete my thesis. I am eternally grateful. Finally, I dedicate this thesis to Alice who inspires me to believe in myself. Her love, kindness, and support are inspirational and always motivate me to achieve my fullest potential. Contents A bstract 2 Acknowledgements 3 Contents 4 List of figures 11 List of tables 18 1 Background 20 1.1 Introduction to sea ic e ....................................................................................20 1.2 Importance of sea ice .......................................................................................21 1.3 Melt ponds ....................................................................................................... 22 1.4 Summertime evolution of Arctic melt p o n d s ............................................. 24 1.5 SHEBA field experim ent .................................................................................27 1.6 Thermodynamic sea ice m odels ....................................................................27 1.7 Aims and preview of model .......................................................................... 30 1.8 Structure of thesis .......................................................................................... 32 CONTENTS 2 Existing Theory 34 2.1 Introduction ....................................................................................................... 34 2.2 Mushy layers and sea i c e .................................................................................34 2.2.1 Conservation equations .........................................................................35 2.2.2 Boundary co n d itio n s ............................................................................39 2.2.3 Application of mushy layers to sea ice ................................................ 40 2.3 Optical properties and radiative models of sea ic e ..................................... 43 2.3.1 Optical properties .................................................................................. 44 2.3.2 Radiative m o d e ls .................................................................................. 45 2.3.3 Suitability of radiative m o d e l ........................................................ 52 3 Application of two-stream radiation model to melt-pond—sea-ice m odel 55 3.1 Introduction ....................................................................................................... 55 3.2 Simplification of the radiation m odel .......................................................... 56 3.3 Boundary conditions appropriate to sea ic e ..................................................57 3.4 Three-layer two-stream model overview .......................................................58 3.5 Albedo and e n e r g y .......................................................................................... 60 3.5.1 A lbedo ......................................................................................................61 3.5.2 E nergy ......................................................................................................61 3.6 Spectral considerations ....................................................................................62 3.7 Parameterisation of sea ice optical properties during ponding using SHEBA field d a ta ...............................................................................................63 CONTENTS 3.7.1 SHEBA melt pond d a t a ..................................................................... 63 3.7.2 Melt pond optical property parameterisation ..................................65 3.8 D iscussion ............................................................................................................ 67 4 Formulation of melt-pond-sea-ice model 69 4.1 Introduction .........................................................................................................69 4.2 Heat transport within sea ice .........................................................................72 4.3 Heat transport within melt pond and internal melt region ........................ 74 4.4 Heat transport within s n o w ............................................................................ 78 4.5 Boundary c o n d itio n s .........................................................................................79 4.5.1 Sea ice o n l y ............................................................................................ 79 4.5.2 Melt p o n d ............................................................................................... 84 4.5.3 Internal melt region ............................................................................... 86 4.5.4 Snow lay er ............................................................................................... 87 4.5.5 Open o c e a n ............................................................................................ 91 4.6 Drainage during p o n ding .................................................................................. 92 4.7 Method of solution ............................................................................................ 93 4.8 Forcing d a t a .........................................................................................................94 4.9 Some model lim itations ......................................................................................99 4.10 Summary .......................................................................................................... 101 5 Application of sea ice model formulation without melt ponds 103 5.1 Introduction ....................................................................................................... 103 5.2 The governing equation of temperature .......................................................106 CONTENTS 5.3 The Thickness E q u a tio n............................................................................... 108 5.4 Summary of the stationary problem .........................................................109 5.5 No short-wave r a d ia tio n ............................................................................... 109 5.5.1 Analysis of 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