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Waves in Shallow Water Magnetohydrodynamics Waves in Shallow Water Magnetohydrodynamics Samuel Hunter Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds Department of Applied Mathematics August 2015 ii The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. c 2015 The University of Leeds and Samuel Hunter The right of Samuel Hunter to be identified as Author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. iii Abstract The dynamics of planetary and stellar objects are dominated by the fluid motions of electrically conducting media. Often, such fluid is confined to a shallow layer, perhaps in an atmosphere or bounded by stratification. One such layer is the solar tachocline (Spiegel and Zahn 1992): a thin layer of high velocity shear in the Sun, which is permeated by strong magnetic fields. The discovery of the solar tachocline by Schou et al. (1997) has inspired the derivation of the equations of shallow water magnetohydrodynamics (SWMHD) by Gilman (2000), in which the small aspect ratio of vertical to horizontal length scales is used to simplify the governing equations. This thesis takes these equations as a base, and aims to build on knowledge of the wave-like dynamics supported in the shallow water system. It will be shown that the analogy between the shallow water and 2D compressible hydrodynamic systems is broken with the introduction of magnetic field, and the differences between the two systems discussed. An energy conservation law will be derived and be used to infer stability properties of the SWMHD system. We will then construct a multi-layer system, and consider linear wave-like perturbations to a motionless basic state with a uniform magnetic field. Particular focus will be on the 2- and 3-layer models, and the effect of magnetic field strength on wave properties. A weakly nonlinear analysis reveals that the single layer and 2-layer rigid lid weakly non- hydrostatic models support solitary and cnoidal waves. The effect of magnetic field in the single layer case translates to a long-time rescaling, but has much more of an effect on the supported modes in the 2-layer model. The phenomenon of three-wave resonance is also supported in the single layer system, and magnetic influence is discussed. In rotating SWMHD, we find that resonant triad interactions are supported only in the presence of magnetic field. Weakly nonlinear predictions are found to be accurate at low disturbance amplitudes, when compared to the results from a fully nonlinear numerical scheme. Exact nonlinear solutions are derived and categorised, and their stability addressed using numerics. iv v To Jill Hunter who taught me that if something’s worth doing, it’s worth doing right. vi vii Acknowledgements First and foremost I would like to thank my supervisors, David Hughes and Stephen Griffiths, for their support and direction in the production of this work. Their knowledge and expertise has been an invaluable asset, opening my eyes to such a rich and interesting subject. I would also like to thank Steve Tobias and Chris Jones for their questions and discussions, and their clear explanations. I am grateful to Julian Mak, who was like a mentor to me when I started my research; our frequent discussions helped me find my feet quickly. Thanks also to Laura Currie, for guidance when I started working with MATLAB, and my friends and colleagues who have created a pleasant working environment at Leeds. Thanks to my family for their continued encouragement and interest during my studies, and finally to my wife, Kim. Your support has helped me more than you know. viii ix Contents Abstract . iii Dedication . v Acknowledgements . vii Contents . ix List of figures . xiv List of tables . xix Notation . 1 1 Introduction 1 1.1 Waves........................................ 1 1.1.1 Magnetically influenced waves . 3 1.2 Astrophysical objects and stellar dynamics . 4 1.3 The shallow water approximation . 7 1.4 Outline of the thesis . 12 2 The single layer model 15 2.1 Introduction . 15 2.2 Derivation of the equations of shallow-water MHD . 16 2.2.1 The momentum equation . 17 2.2.2 The equation for conservation of mass . 19 CONTENTS x 2.2.3 The induction equation . 20 2.2.4 The solenoidal constraint . 21 2.2.5 Summary . 22 2.3 Energy conservation . 22 2.4 The analogy with compressible flow . 25 2.4.1 Linear waves . 27 2.4.2 Analysis of the dispersion relation (2.41), with f = 0 ........... 29 2.4.3 Analysis of the dispersion relation (2.41), with nonzero f ......... 32 2.5 Group Velocity . 34 2.6 Incompressible and irrotational waves . 40 2.6.1 Disturbance energy conservation . 41 2.7 Two-dimensional linear waves . 42 2.8 Summary and discussion . 44 3 The n-layer model 49 3.1 Introduction . 49 3.2 The n-layer model and governing equations . 50 3.2.1 Derivation of the governing equations . 53 3.3 Energy conservation . 56 3.3.1 Disturbance energy conservation . 60 3.4 Linear waves . 61 3.4.1 Nondimensionalisation . 64 3.5 Summary and discussion . 66 CONTENTS xi 4 The 2 and 3-layer models 69 4.1 Introduction . 69 4.2 The 2-layer model . 70 4.3 The 2-layer free-surface model . 73 4.3.1 The hydrodynamic regime . 77 4.3.2 Example 1: Equal Alfven´ speeds . 80 4.3.3 Example 2: Equal magnetic field strengths . 82 4.3.4 Example 3: Magnetic field in one layer only . 87 4.3.5 Group velocities . 91 4.4 The 2-layer rigid lid model . 97 4.4.1 Magnetic dominance . 98 4.4.2 The link to the free surface case . 99 4.5 Rotation in the 2-layer model . 102 4.6 The 3-layer model . 105 4.7 The 3-layer free-surface model . 106 4.7.1 Example 1: Equal Alfven´ speeds and the hydrodynamic reduction . 108 4.7.2 Example 2: Equal magnetic field strengths . 113 4.7.3 Example 3: Field in middle layer only . 118 4.7.4 Non-linear stratification . 122 4.7.5 Group velocities . 125 4.8 The 3-layer rigid lid model . 127 4.8.1 Parallels with the 2-layer models . 128 4.8.2 Field dependence . 128 4.8.3 Group Velocities . 130 4.9 Summary and discussion . 131 CONTENTS xii 5 Weakly nonlinear and dispersive waves 135 5.1 Introduction . 135 5.2 The KdV equation for single layer SWMHD . 137 5.2.1 Asymptotic analysis . 140 5.3 KdV for 2-layer SWMHD . 147 5.3.1 Leading order analysis . 149 5.3.2 Order analysis . 151 5.3.3 Analysis of the 2-layer rigid lid KdV equation . 153 5.4 Three-wave resonance . 158 5.4.1 Derivation of the equations for resonant triad interactions . 159 5.4.2 Solving the three-wave resonance equations . 163 5.4.3 Numerical solution of the three-wave resonance equations . 166 5.5 Summary and discussion . 170 6 Nonlinear waves with rotation 173 6.1 Introduction . 173 6.2 Three-wave resonance in rotating SWMHD . 175 6.2.1 Weakly nonlinear analysis . 178 6.2.2 Results . 181 6.2.3 Nonlinear solution with three-wave initialisation . 185 6.2.4 Numerical results . 186 6.2.5 Further results . 188 6.2.6 Other measures for the discrepancy . 189 6.3 Finite-amplitude travelling wave solutions . 193 6.3.1 Numerical generation of nonlinear wave solutions . 195 CONTENTS xiii 6.3.2 Numerical wave solutions – the shooting algorithm . 200 6.3.3 Initialising the numerical solver with an exact nonlinear solution . 203 6.4 Fully two-dimensional simulations . 206 6.4.1 Code verification . 207 6.4.2 Perturbations to an exact nonlinear solution . 208 6.5 Summary and Discussion . 212 7 Conclusion 215 7.1 Overview . 215 7.2 Further work . 219 Appendices 221 A The magnetic constraint and spurious modes . 221 B The n-layer inductively defined dispersion relation . ..
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