A Student's Guideto Geophysical Equations

WILLIAM LOWRIEInstitute of Geophysics

Swiss Federal Institute of TechnologyZurich, Switzerland

Preface page xiAcknowledgments xiii

Mathematical background 11.1 "Cartesian and spherical coordinates 11.2 Complex numbers 11.3 Vector relationships 41.4 Matrices and tensors . 81.5 Conservative force, field, and potential 171.6 The divergence theorem (Gauss's theorem) 181.7 The curl theorem (Stokes'theorem) 201.8 Poisson's equation 231.9 Laplace's equation 261.10 Power series 281.11 Leibniz's rule 321.12 Legendre polynomials 321.13 The Legendre differential equation 341.14 Rodrigues'formula 411.15 Associated Legendre polynomials . 431.16 Spherical harmonic functions 491.17 Fourier series, Fourier integrals, and Fourier transforms 52Further reading 58Gravitation 592.1 Gravitational acceleration and potential 592.2 Kepler's laws of planetary motion 602.3 Gravitational acceleration and the potential of a solid

sphere 662.4 Laplace's equation in spherical polar coordinates 692.5 MacCullagh's formula for the gravitational potential 74Further reading 85

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Contents

Gravity3.1 The ellipticity of the Earth's figure3.2 The geopotential3.3 The equipotential surface of gravity3.4 Gravity on the reference spheroid3.5 Geocentric and geographic latitude3.6 The geoidFurther readingThe tides4.1 Origin of the lunar tide-raising forces4.2 Tidal potential of the Moon4.3 Love's numbers and the tidal deformation4.4 Tidal friction and deceleration of terrestrial and lunar

"rotationsFurther readingEarth's rotation5.1 Motion in a rotating coordinate system5.2 The Coriolis and Eotvos effects5.3 Precession and forced nutation of Earth's rotation axis5.4 The free, Eulerian nutation of a rigid Earth5.5 The Chandler wobbleFurther readingEarth's heat6.1 Energy and entropy6.2 Thermodynamic potentials and Maxwell's relations6.3 The melting-temperature gradient in the core6.4 The adiabatic temperature gradient in the core6.5 The Griineisen parameter6.6 Heat flowFurther readingGeomagnetism7.1 The dipole magnetic field and potential7.2 Potential of the geomagnetic field7.3 The Earth's dipole magnetic field7.4 Secular variation7.5 Power spectrum of the internal field '7.6 The origin of the internal fieldFurther readingFoundations of seismology8.1 Elastic deformation

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Contents

8.2 Stress 2288.3 Strain 2338.4 Perfectly elastic stress-strain relationships 2398.5 The seismic wave equation 2448.6 Solutions of the wave equation 2528.7 Three-dimensional propagation of plane P- and S-waves 254Further reading 258

Appendix A Magnetic poles, the dipole field, and current loops 259Appendix B Maxwell's equations of electromagnetism 265References 276Index 278