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4.3 Brine Rejection Velocity

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4.3 Brine Rejection Velocity Ice Growth and Platelet Crystals in Antarctica by Jonathan Crook A thesis submitted to the Victoria University of Wellington in fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics. Victoria University of Wellington 2010 Abstract First-year land-fast sea ice growth in both the Arctic and the Antarctic is characterised by the formation of an initial ice cover, followed by the di- rect freezing of seawater at the ice-water interface. Such growth usually results, through geometric selection, in congelation ice. This is, in general, the typical crystal structure observed in first-year ice growth in the Arctic. However, in certain regions of the Antarctic, platelet crystals are observed to contribute significantly to the ice growth, beyond a depth of 1 m. This thesis will investigate a number of ideas as to why the platelet crystals only appear in the ice after a significant amount of congelation growth has occurred. One of the key premises will be that platelet ice forms when smaller frazil crystals, beneath the ice, rise up and attach to the interface. They are then incorporated into the ice cover and become the platelets seen in ice cores. The Shields criterion is used to find the strength of turbulence, asso- ciated with tidal flow, required to keep a frazil crystal from adhering to the interface. It is shown that the sub-ice flow is sufficient to keep most crystals in motion. However, this turbulence may weaken or dissipate completely as the tide turns. The velocity associated with brine rejection is suggested as an alternative to keep the crystals in suspension during these periods of low shear turbulence. Brine rejection occurs as the sea ice grows, rejecting salt into the seawater below. By comparing this velocity with a model for the frazil rise velocity it is shown that brine rejection has sufficient strength to keep crystals in suspension. This effect weakens as the ice gets thicker, allowing larger frazil crystals to rise to the interface. The early work in this thesis shows that a flow can keep a single crystal from adhering to the interface. This can be regarded as the competence of a flow to keep a crystal in suspension. However, of equal importance is the capacity of a flow to keep a mass of crystals in suspension. It is shown that, given a sufficiently large mass of crystals beneath the ice, the same flow that can hold a single crystal in suspension will not be able to keep all the crystals in motion. The deposition of crystals is predicted to occur in a gradual manner if there is a steady build-up of crystals beneath the ice. The largest crystals, close to the interface, will settle against the ice as the flow is unable to support the entire mass of crystals Also considered is whether frazil crystals may be similar to cohesive sediments. If this is the case, a sudden influx of crystals from outside of the system may lead to the formation of a layer of unattached crystals beside the ice-water interface. This can cause a critical collapse of the turbulent field, resulting in the settling of a large quantity of frazil crystals. Though the emphasis of much of this thesis is on the effect of the flow on the crystals, it is also found that a mass of crystals can have a stabilising effect on the flow. The change in the density profile induced by an increase in the frazil concentration towards the ice-water interface (and hence a decrease in the density of the ice-water mixture) damps the turbulence produced by shear. The mass and size of crystals in suspension play major roles in the strength of stabilisation. Measurements of turbulence and the suspension of frazil crystals be- neath sea ice are difficult to make. This thesis aims to present and analyse a number of models which may explain the platelet puzzle - the delayed appearance of the platelet crystals in ice cores. These are compared with the observations which are available, and conclusions made on the valid- ity of the theories presented. Acknowledgments First and foremost I must thank Mark McGuinness, my supervisor, who has guided and mentored me through this process of producing my thesis. He has always retained a positive attitude to the work, never failing in making time for me whenever I needed it. Similarly, I know that there was always an open door and a friendly ear in the office of my co-supervisor, Matt Visser. Also of great importance are the significant financial contributions which various groups have made to support me in my work, and in al- lowing me to present it, both within New Zealand and overseas. Many thanks go to the New Zealand Institute of Mathematics and its Applica- tions (NZIMA) who have funded the final two years of my thesis. I also appreciate the important grants awarded to me by the Johnsonville Rotary Club and the New Zealand Mathematical Society. I must thank Victoria University here in Wellington, especially the Maths department: not just for the standard support that one often takes for granted, but also for financial support, especially in allowing me to gain experience through travel. I was also fortunate to be invited to spend time at the Mathematics Applications Consortium for Science and Indus- try (MACSI) at the University of Limerick in Ireland. They also helped me with my travel there and were quick to welcome and involve me in the work (and Guinness - which is part of the work!) there. Writing a thesis is often a lonely process and so family and friends are invaluable. Special thanks to mum, for giving me a house to live in and allowing me to focus solely on the work. And also for proofreading what I’m sure wasn’t the lightest reading before bed! Last, but not least, all those people at Uni who know exactly what you are going through; those that I have shared a room with, both physically (Sione and Nicole, cheers) and virtually (Ros, gidday); and those that I have just been able to share a laugh or a kick-around with when things seemed tough - the whole Physics group. I could probably write another thesis listing all the people who deserve thanks, so I will stop here, but to those who have helped me along my way, know that you have my deepest and sincerest gratitude. i Contents List of Figures vi List of Tables x 1 Introduction 1 2 Background 4 2.1 Initial Formation . .5 2.2 Sea Ice Growth . .9 2.2.1 Congelation Growth . 10 2.2.2 Analytic Modelling . 11 2.2.3 Numerical Modelling . 14 2.2.4 Mushy Zones . 16 2.2.5 Frazil and Platelet Ice . 19 2.3 Salinity . 22 2.4 Turbulence . 26 2.4.1 Turbulence and frazil suspension . 27 2.5 Frazil induced turbulence damping . 28 2.6 Summary . 29 3 Shear flow, frazil and platelet ice 31 3.1 Frazil ice or platelet ice? . 31 3.2 Mechanisms for growth . 33 3.3 Quantity of growth . 36 ii 3.4 Platelet inclusion in sea ice . 39 3.5 The Shields Criterion . 41 3.6 Shear results . 46 3.6.1 Tidal and friction velocity . 46 3.6.2 Frazil in suspense? . 49 3.7 Summary . 50 4 Stirring due to brine rejection 52 4.1 Brine Rejection . 53 4.2 Stokes’ flow . 55 4.2.1 Crystal buoyancy . 56 4.2.2 Rise velocities . 59 4.3 Brine rejection velocity . 60 4.3.1 Density of seawater . 62 4.4 Ice growth data . 64 4.5 Brine plumes and frazil . 66 4.5.1 Air temperature and seawater salinity . 70 4.5.2 Varying mixed layer depth . 72 4.6 Brine rejection in a turbulent environment . 74 4.7 Torque on a frazil crystal . 77 4.8 Summary . 81 5 A two-phase sea ice model 83 5.1 Stefan solution . 84 5.2 Model regions . 87 5.2.1 In the ice . 87 5.2.2 The intermediate zone . 91 5.2.3 Below the ice . 92 5.3 The two-phase model . 94 5.3.1 Ice growth in the presence of shear flow . 95 5.3.2 Results . 96 5.3.3 Role of salt . 98 iii 5.3.4 Model Sensitivity . 99 5.4 The effects of brine rejection on ice growth . 107 5.5 Heat flux . 109 5.5.1 Frazil production . 113 5.6 Summary . 115 6 Shear stabilisation of platelet ice 117 6.1 Bedload or suspended load . 117 6.1.1 Suspended load criteria . 118 6.1.2 Suspension criterion . 124 6.2 The effect of frazil on the flow . 130 6.2.1 The flux Richardson number . 132 6.2.2 Reduction in u∗ ...................... 138 6.3 Collapse of turbulence . 142 6.3.1 Cohesive or non-cohesive? . 143 6.3.2 Criterion for collapse . 144 6.4 Critical flux Richardson number . 149 6.5 Summary . 154 7 Conclusions 157 7.1 Shear flow . 157 7.2 Rise velocity . 158 7.3 Brine rejection . 159 7.4 Ice growth . 161 7.5 Shear stabilisation . 162 Glossary 164 A Conservation equations 168 A.1 Conservation of salt . 168 A.2 Conservation of energy . 170 iv B Matlab M-files 171 B.1 M-file ‘RungeKuttarealTa’ . 171 B.2 M-file ‘RealTaeqns’ . 175 C Stress Due to a Suspended Load 180 D Concentration profiles 181 D.1 Linear eddy viscosity . 181 D.2 Parabolic eddy viscosity . 183 D.3 Eddy viscosity varying with Rif ................
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