Foundations of Geomagnetism

Transcription

1 Foundations of Geomagnetism GEORGE BACKUS University of California, San Diego ROBERT PARKER University of California, San Diego CATHERINE CONSTABLE University of California, San Diego m.m CAMBRIDGE UNIVERSITY PRESS

2 CONTENTS Preface po.ge xi 1 The Main Field A Whirlwind Tour History Spatial Variations Time Variations 8 2 Classical Electrodynamics Helmholtz's Theorem and Maxwell's Equations A Simple Solution: The Static Case Maxwell's Equations in a Polarized Medium On Judicious Neglect of Terms in Equations Internal and External Fields Solving Maxwell's Equations as an Initial Value Problem 37 3 Spherical Harmonics Completeness on 5(1) Homogeneous and Harmonic Polynomials The Laplacian of a Certain Homogeneous Polynomial An Expansion in Harmonic Polynomials Orthogonality in Ve The Self-Reproducing Kernel on He Inner Product Spaces Inner Products and Linear Functionals A Special Linear Functional on He A Rotational Symmetry of q~e Properties of Q e An Orthonormal Basis for He Application of the Surface Curl, A 65

3 vi Contents Lifting and Lowering Operators Explicit Expressions of a Basis Normalizing the Natural Basis Axisymmetric Spherical Harmonics Behavior Near the z Axis Calculating the Kernel Function The Generating Function for Legendre Polynomials Green's Function for V The Character of the Natural Basis Nodal Lines of Re Y m e (r) on 5(1) General Appearance of Y m e for large 91 m Horizontal Wavelength of Y g Numerical Calculations and the Like Explicit Formulas for P^in) Three-Term Recurrence Relationships Numerical Calculations Gauss' Theory of the Main Field Finding All the Harmonics in a Shell Uniqueness of the Coefficients Observing the Sources in Principle Measuring the Gauss Coefficients Nonuniqueness of Fields Based on Total Field Observations The Spectrum Crustal Signals Inferences about the Field on the Core: Averaging Kernels The Mie Representation The Helmholtz Representation Theorem Solving the Surface Form of Poisson's Equation Integral Form of the Solution The Helmholtz Representation Theorem on S(r) and S(a, b) Divergence and Curl in the Helmholtz Representation The Mie Representation of Vector Fields Solenoidal Vector Fields Poloidal and Toroidal Fields 177

4 Contents vii Continuity of the Mie Scalars Summary Application to Sources Mie Sources of a Magnetic Field Internal and External Fields: A Complication Separation of Poloidal Fields The Generalization of Gauss' Resolution Induction in the Mantle and the Core Equations for the Mie Scalars Application of Boundary Conditions: Toroidal Field Application of Boundary Conditions: Magnetic Sounding Free Decay of Fields in the Core Ohmic Heating in the Core Hydromagnetics of the Core The Bullard Disk Dynamo Hydromagnetics in an Ohmic Conductor Ohm's Law for a Moving Conductor Equations Governing the Geodynamo The Kinematic Problem: Limiting Case with u Eulerian and Lagrangian Descriptions The Kinematic Problem: Limiting Case with77 = Frozen-Flux Condition Some Simple Dynamic Problems The Maxwell Stress Tensor Sunspots Alfven Waves Application of Perfect Conductor Theory to the Core The Hypothesis of Roberts and Scott Null-Flux Curves Kinematic Dynamos Cowling's Theorem Elsasser; Blackett and Runcorn; Bullard and Gellman Rigorous Dynamos Early Numerical Dynamos Mean Field Dynamos 276

5 viii Contents 6.6 The Dynamics of Dynamos The Taylor Theorem Bullard Dynamo, Poincare-Bendixson Theorem, and Chaos Data Possibly Relevant to the Dynamics Appendix: Mathematical Background Linear Algebra Arrays Index Conventions Properties of the Kronecker Delta and the Alternator Applications of Delta and the Alternator to Vector Algebra Vector Analysis: Differential Calculus Scalar and Vector Fields Scalar Linear Operators Sums and Products of Scalar Linear Operators Scalar Linear Operators Acting on Vector Fields Vector Linear Operators Linear Combinations of Vector Linear Operators Products of Vector Linear Operators Dot and Cross Products of Vector Linear Operators FODOs Arithmetic with FODOs Commutation An Important FODO and Its Commutation Properties Curvilinear Coordinates and V Spherical Polar Coordinates Vector Analysis: Integral Calculus The Theorems of Stokes and Gauss Jump Discontinuities Sources of a Vector Field Scalar and Vector Fields on Orientable Surfaces Projection of a Vector onto a Plane Vector Fields on an Oriented Surface Surface Gradient and Normal Derivative Surface Curl 339

6 Contents ix Applying the FODOs V s and A s to Vector Fields on Surface Forms of the Theorems of Gauss and Stokes Representation of Tangent Vector Fields 346 References 351 Index 361