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Magnetic Stability Analysis for the Geodynamo

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Magnetic Stability Analysis for the Geodynamo Magnetic Stability Analysis for the Geodynamo by Philip William Livermore Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds School of Earth Sciences December 2003 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. To my parents i Acknowledgements First and foremost I would like to thank my supervisor, Andy Jackson, for his help, support and attentiveness to my work over my last three years. I am also grateful for his thorough reading of this manuscript. Over the duration of my Ph.D. I have had enlightening discussions with Rich Kerswell, John Lister and Mike Proctor, which have greatly contributed to the path of my studies. In addition, I would like to acknowledge discussions on linear algebra with Brian Stewart, transient behaviour with David Gerard-Varet and Glenn Ierley and general geodynamo topics with Dave Gubbins, Ashley Willis, Chris Finlay and Nick Teanby. I thank Steve Gibbons for supplying LEOPACK, an abundance of FORTRAN codes on which some of my routines are based, in particular those producing graphical outputs. I also used his kinematic dynamo codes to benchmark my results. I also thank Alan Thomson at BGS for supplying the Ørsted data on which Appendix A is based. I would like to thank everyone in the School of Earth Sciences for making my time here so enjoyable, including James Wookey, Steve A, Trudi, Lindsey, Simon, Tanya, Dave, Ian, Kristof, Andy C, James While, Susan, Nicola, Bryony, Neil and Andy G. I would also like to acknowledge some very fine ales consumed at the Eldon on many a friday night post-seminar pub session. I made many friends in the Ballroom and Latin American Dancesport Society, in par- ticular I would like to thank the coaches John and Julie, and some of the dancers: Fiona, Tash, Lauren, Mat, Mary, Sophie and others for making my time so enjoyable. I have also been in- volved in various campanology activities, for which I thank the Burley and Leeds Parish Church ringers who are too numerous to list here. I would like to thank Rob for his company when climbing Munros and on other long distance walks, and for often supplying transport to get us out there. I thank my parents and my brother Stephen for their continual support, both financial where necessary and otherwise, over the whole duration of my education. Last but by no means least, I give a special thank you to my fiancee´ Fiona, whose love and support over the last three years has been unfailing, particularly in the ‘dark’ months of writing up. ii Abstract Kinematic dynamo modelling addresses the growth of magnetic fields under the ac- tion of fluid flow, effected in the geophysical case by the action of the convecting electrically conducting outer core and magnetic diffusion. In this study, we prescribe the flow to be station- ary and geophysically motivated by some large scale process, for example, differential rotation or convection. Historically, workers have applied eigenvalue stability analysis and although in some cases Earth-like solutions have been found, the results hinge critically on the precise choice of flow. We therefore cannot attribute physical mechanisms in such cases since it is rather case dependent. Following the success of more generalised stability techniques applied to the transition to turbulence in various non-magnetic fluid dynamical problems, we investigate both the onset of and subsequent maximised transient growth of magnetic energy. These approaches differ from the eigenvalue methodology due to the non-normality of the underlying operator, which means that superposition of non-orthogonal decaying eigenmodes can result in sub-critical growth. The onset or instantaneous instability problem can be formulated using variational techniques which result in an equation amenable to a numerical Galerkin method. For the suite of flows studied, we find robust results that can be physically explained by field line stretching. Convectively driven flows exhibit the greatest instability, the field structures giving this maximal instability being axisymmetric. All flows indicate an apparent asymptotic dependence on the magnetic ¡ Reynolds number Rm, which is reached when Rm O 1000 ¢ . For the flows studied, we find improved lower bounds on Rm for energetic instability of between 5 and 14 times, compared to that resulting from the analytic analysis of Proctor (1977a). In all the flows studied, without exception the geophysically dominant axisymmetric dipole field symmetry is preferentially transiently amplified. The associated physical mecha- nisms are either shearing of poloidal field into toroidal field by differential rotation, or advection into locations of radial upwelling followed by field line stretching in the convective case. Tran- ¡ sient energy growth of O 1000 ¢ , which can be obtained when Rm 1000 is robust and may explain the recovery of field intensity after a magnetic reversal. Assuming the flow to be a sta- tionary solution of the geostrophic balance equation where buoyancy, pressure and the Coriolis forces are in equilibrium, we computed the geophysically scaled ratio of the Lorentz to the Cori- ¡ olis forces and found it to be O 1 ¢ for flows with a large convective component. This indicates that transient growth, in particular of axisymmetric fields that are ostensibly precluded by the theorem of Cowling (1933), can explain the entry into the non-linear regime without the need for eigenmode analysis. iii Contents Preamble i Acknowledgements . i Abstract . ii Contents . iii List of figures . viii List of tables . x Abbreviations . xii 1 Introduction 1 1.1 A brief history . 1 1.2 Paleomagnetism . 3 1.3 Reversals . 4 1.4 The field at Epoch 2000 . 6 1.5 Thesis outline . 7 2 Kinematic dynamo modelling 8 2.1 Introduction . 8 2.2 Equations of Electromagnetism . 8 2.2.1 Maxwell’s equations . 8 2.2.2 The induction equation . 10 2.3 The general dynamo problem . 11 2.3.1 The Navier-Stokes equations . 11 2.3.2 The Lorentz force . 12 2.3.3 A few comments on energy . 12 2.3.4 Rationale of the kinematic assumption . 13 2.4 The kinematic dynamo problem . 14 2.4.1 Some notation . 15 2.5 Anti-dynamo theorems . 16 2.6 A review of kinematic modelling . 17 2.6.1 After Cowling’s theorem . 17 2.6.2 The Bullard and Gellman model . 19 2.6.3 Non geophysical dynamos . 21 2.6.4 Further work on spherical dynamos . 21 2.6.5 Work based on the flow of Kumar and Roberts . 23 2.6.6 Simple roll flows . 24 2.6.7 The influence of an insulating boundary . 25 2.6.8 Motivating the choice of flow . 26 2.6.9 Non-linear effects . 26 2.6.10 Summary . 27 2.7 Dynamo experiments . 29 iv 2.7.1 Introduction . 29 2.7.2 Two working dynamos . 30 2.7.3 Spherical dynamos . 30 3 Theoretical considerations 33 3.1 The defining equations and geometry . 33 3.1.1 The induction equation . 33 3.2 Vector harmonics . 34 3.2.1 Poloidal and toroidal form . 34 3.2.2 Orthogonality and normalisation . 36 3.3 Boundary conditions . 37 3.3.1 Continuity conditions . 37 3.3.2 The magnetic field in an electrical insulator . 38 3.3.3 Electrically insulating boundary conditions . 39 3.3.4 Regularity at the origin . 39 3.4 The diffusion operator . 41 3.4.1 Properties of the Laplacian . 41 3.4.2 The diffusion problem . 42 3.5 The magnetic energy equation . 44 3.5.1 Magnetic energy . 44 3.5.2 Derivation of the magnetic energy equation . 44 3.6 Symmetry . 46 3.6.1 Symmetry of vectors . 46 3.6.2 Symmetry of operators . 47 3.6.3 Symmetry of harmonics . 48 3.6.4 Separation of harmonics . 48 3.6.5 Selection rules . 49 3.7 Existing bounds . 51 3.7.1 Backus’s bound on Rm . 52 3.7.2 Proctor’s bound . 53 3.7.3 Other bounds on Rm . 53 3.7.4 A note on the trivial bound . 54 3.7.5 Bounds on motion . 54 3.7.6 Upper bounds on time-dependence . 55 3.8 Adjoints . 56 3.8.1 The magnetic energy adjoint . 57 3.8.2 The adjoint of the induction operator . 57 3.8.3 Self adjoint operators . 58 3.8.4 Non self-adjoint operators . 59 3.9 Magnetic energy instability . 59 3.9.1 Measures of magnetic energy growth . 59 3.9.2 The onset of magnetic energy instability . 61 λ 3.9.3 Using E to bound the eigenvalue growth rate . 62 3.9.4 An adjoint derivation of magnetic energy instability . 63 3.9.5 Finite time growth of magnetic energy . 64 3.9.6 A variational method applied to Proctor’s bound . 65 3.9.7 The singular electrically insulating limit . 67 3.9.8 The differential equations of instantaneous magnetic energy growth . 68 4 Numerical methods 69 4.1 A simple example . 69 4.1.1 A range of numerical approaches . 70 v 4.1.2 A Galerkin method . 71 4.2 Comparison of.
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