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
CONTENTS Preface po.ge xi 1 The Main Field 1 1.1 A Whirlwind Tour 1 1.2 History 2 1.3 Spatial Variations 5 1.4 Time Variations 8 2 Classical Electrodynamics 15 2.1 Helmholtz's Theorem and Maxwell's Equations 15 2.2 A Simple Solution: The Static Case 17 2.3 Maxwell's Equations in a Polarized Medium 23 2.4 On Judicious Neglect of Terms in Equations 29 2.5 Internal and External Fields 35 2.6 Solving Maxwell's Equations as an Initial Value Problem 37 3 Spherical Harmonics 41 3.1 Completeness on 5(1) 42 3.1.1 Homogeneous and Harmonic Polynomials 42 3.1.2 The Laplacian of a Certain Homogeneous Polynomial 44 3.1.3 An Expansion in Harmonic Polynomials 46 3.2 Orthogonality in Ve 48 3.3 The Self-Reproducing Kernel on He 54 3.3.1 Inner Product Spaces 54 3.3.2 Inner Products and Linear Functionals 56 3.3.3 A Special Linear Functional on He 57 3.3.4 A Rotational Symmetry of q~e 59 3.3.5 Properties of Q e 61 3.4 An Orthonormal Basis for He 64 3.4.1 Application of the Surface Curl, A 65
vi Contents 3.4.2 Lifting and Lowering Operators 70 3.4.3 Explicit Expressions of a Basis 75 3.4.4 Normalizing the Natural Basis 78 3.5 Axisymmetric Spherical Harmonics 81 3.5.1 Behavior Near the z Axis 82 3.5.2 Calculating the Kernel Function 83 3.5.3 The Generating Function for Legendre Polynomials 84 3.5.4 Green's Function for V 2 87 3.6 The Character of the Natural Basis 88 3.6.1 Nodal Lines of Re Y m e (r) on 5(1) 88 3.6.2 General Appearance of Y m e for large 91 m 3.6.3 Horizontal Wavelength of Y g 101 3.7 Numerical Calculations and the Like 104 3.7.1 Explicit Formulas for P^in) 104 3.7.2 Three-Term Recurrence Relationships 106 3.7.3 Numerical Calculations 112 4 Gauss' Theory of the Main Field 119 4.1 Finding All the Harmonics in a Shell 119 4.2 Uniqueness of the Coefficients 123 4.3 Observing the Sources in Principle 128 4.4 Measuring the Gauss Coefficients 136 4.4.1 Nonuniqueness of Fields Based on Total Field Observations 138 4.4.2 The Spectrum 138 4.4.3 Crustal Signals 150 4.4.4 Inferences about the Field on the Core: Averaging Kernels 153 5 The Mie Representation 161 5.1 The Helmholtz Representation Theorem 161 5.1.1 Solving the Surface Form of Poisson's Equation 161 5.1.2 Integral Form of the Solution 163 5.1.3 The Helmholtz Representation Theorem on S(r) and S(a, b) 167 5.1.4 Divergence and Curl in the Helmholtz Representation 169 5.2 The Mie Representation of Vector Fields 173 5.2.1 Solenoidal Vector Fields 173 5.2.2 Poloidal and Toroidal Fields 177
Contents vii 5.2.3 Continuity of the Mie Scalars 179 5.2.4 Summary 181 5.3 Application to Sources 182 5.3.1 Mie Sources of a Magnetic Field 183 5.3.2 Internal and External Fields: A Complication 185 5.3.3 Separation of Poloidal Fields 188 5.3.4 The Generalization of Gauss' Resolution 189 5.4 Induction in the Mantle and the Core 192 5.4.1 Equations for the Mie Scalars 192 5.4.2 Application of Boundary Conditions: Toroidal Field 195 5.4.3 Application of Boundary Conditions: Magnetic Sounding 196 5.4.4 Free Decay of Fields in the Core 200 5.5 Ohmic Heating in the Core 204 6 Hydromagnetics of the Core 211 6.1 The Bullard Disk Dynamo 213 6.2 Hydromagnetics in an Ohmic Conductor 217 6.2.1 Ohm's Law for a Moving Conductor 219 6.2.2 Equations Governing the Geodynamo 224 6.2.3 The Kinematic Problem: Limiting Case with u 0 226 6.2.4 Eulerian and Lagrangian Descriptions 228 6.2.5 The Kinematic Problem: Limiting Case with77 = 0 231 6.2.6 Frozen-Flux Condition 235 6.3 Some Simple Dynamic Problems 239 6.3.1 The Maxwell Stress Tensor 240 6.3.2 Sunspots 243 6.3.3 Alfven Waves 243 6.4 Application of Perfect Conductor Theory to the Core 247 6.4.1 The Hypothesis of Roberts and Scott 247 6.4.2 Null-Flux Curves 254 6.5 Kinematic Dynamos 260 6.5.1 Cowling's Theorem 262 6.5.2 Elsasser; Blackett and Runcorn; Bullard and Gellman 267 6.5.3 Rigorous Dynamos 272 6.5.4 Early Numerical Dynamos 276 6.5.5 Mean Field Dynamos 276
viii Contents 6.6 The Dynamics of Dynamos 285 6.6.1 The Taylor Theorem 285 6.6.2 Bullard Dynamo, Poincare-Bendixson Theorem, and Chaos 286 6.6.3 Data Possibly Relevant to the Dynamics 286 7 Appendix: Mathematical Background 291 7.1 Linear Algebra 291 7.1.1 Arrays 291 7.1.2 Index Conventions 292 7.1.3 Properties of the Kronecker Delta and the Alternator 293 7.1.4 Applications of Delta and the Alternator to Vector Algebra 296 7.2 Vector Analysis: Differential Calculus 298 7.2.1 Scalar and Vector Fields 298 7.2.2 Scalar Linear Operators 299 7.2.3 Sums and Products of Scalar Linear Operators 301 7.2.4 Scalar Linear Operators Acting on Vector Fields 303 7.2.5 Vector Linear Operators 304 7.2.6 Linear Combinations of Vector Linear Operators 306 7.2.7 Products of Vector Linear Operators 306 7.2.8 Dot and Cross Products of Vector Linear Operators 309 7.2.9 FODOs 312 7.2.10 Arithmetic with FODOs 314 7.2.11 Commutation 316 7.2.12 An Important FODO and Its Commutation Properties 317 7.2.13 Curvilinear Coordinates and V 320 7.2.14 Spherical Polar Coordinates 322 7.3 Vector Analysis: Integral Calculus 328 7.3.1 The Theorems of Stokes and Gauss 328 7.3.2 Jump Discontinuities 331 7.3.3 Sources of a Vector Field 332 7.4 Scalar and Vector Fields on Orientable Surfaces 334 7.4.1 Projection of a Vector onto a Plane 334 7.4.2 Vector Fields on an Oriented Surface 336 7.4.3 Surface Gradient and Normal Derivative 336 7.4.4 Surface Curl 339
Contents ix 7.4.5 Applying the FODOs V s and A s to Vector Fields on 5 340 7.4.6 Surface Forms of the Theorems of Gauss and Stokes 343 7.4.7 Representation of Tangent Vector Fields 346 References 351 Index 361