A Student's Guide to Geophysical Equations

by William Lowrie

• ISBN13: 9780521183772
• ISBN10: 0521183774
• Format: Paperback
• Copyright: 2011-06-30
• Publisher: Cambridge University Press

Summary

The advent of accessible student computing packages has meant that geophysics students can now easily manipulate datasets and gain first-hand modeling experience - essential in developing an intuitive understanding of the physics of the Earth. Yet to gain a more in-depth understanding of physical theory, and to develop new models and solutions, it is necessary to be able to derive the relevant equations from first principles. This compact, handy book fills a gap left by most modern geophysics textbooks, which generally do not have space to derive all of the important formulae, showing the intermediate steps. This guide presents full derivations for the classical equations of gravitation, gravity, tides, earth rotation, heat, geomagnetism and foundational seismology, illustrated with simple schematic diagrams. It supports students through the successive steps and explains the logical sequence of a derivation - facilitating self-study and helping students to tackle homework exercises and prepare for exams.

Table of Contents

Preface	p. xi
Acknowledgments	p. xiii
Mathematical background	p. 1
Cartesian and spherical coordinates	p. 1
Complex numbers	p. 1
Vector relationships	p. 4
Matrices and tensors	p. 8
Conservative force, field, and potential	p. 17
The divergence theorem (Gauss's theorem)	p. 18
The curl theorem (Stokes' theorem)	p. 20
Poisson's equation	p. 23
Laplace's equation	p. 26
Power series	p. 28
Leibniz's rule	p. 32
Legendre polynomials	p. 32
The Legendre differential equation	p. 34
Rodrigues' formula	p. 41
Associated Legendre polynomials	p. 43
Spherical harmonic functions	p. 49
Fourier series, Fourier integrals, and Fourier transforms	p. 52
Further reading	p. 58
Gravitation	p. 59
Gravitational acceleration and potential	p. 59
Kepler's laws of planetary motion	p. 60
Gravitational acceleration and the potential of a solid sphere	p. 66
Laplace's equation in spherical polar coordinates	p. 69
MacCullagh's formula for the gravitational potential	p. 74
Furhter reading	p. 85
Gravity	p. 86
The ellipticity of the Earth's figure	p. 86
The geopotential	p. 88
The equipotential surface of gravity	p. 91
Gravity on the reference spheroid	p. 96
Geocentric and geographic latitude	p. 102
The geoid	p. 106
Further reading	p. 115
The tides	p. 116
Origin of the lunar tide-raising forces	p. 116
Tidal potential of the Moon	p. 119
Love's numbers and the tidal deformation	p. 124
Tidal friction and deceleration of terrestrial and lunar rotations	p. 130
Further reading	p. 136
Earth's rotation	p. 137
Motion in a rotating coordinate system	p. 138
The Coriolis and Eotvos effects	p. 140
Precession and forced nutation of Earth's rotation axis	p. 142
The free, Eulerian nutation of a rigid Earth	p. 155
The Chandler wobble	p. 157
Further reading	p. 169
Earth's heat	p. 170
Energy and entropy	p. 171
Thermodynamic potentials and Maxwell's relations	p. 172
The melting-temperature gradient in the core	p. 176
The adiabatic temperature gradient in the core	p. 178
The Gruneisen parameter	p. 179
Heat flow	p. 182
Further reading	p. 197
Geomagnetism	p. 198
The dipole magnetic field and potential	p. 198
Potential of the geomagnetic field	p. 200
The Earth's dipole magnetic field	p. 205
Secular variation	p. 213
Power spectrum of the internal field	p. 214
The origin of the internal field	p. 217
Further reading	p. 225
Foundations of seismology	p. 227
Elastic deformation	p. 227
Stress	p. 228
Strain	p. 233
Perfectly elastic stress-strain relationships	p. 239
The seismic wave equation	p. 244
Solutions of the wave equation	p. 252
Three-dimensional propagation of plane P- and S-waves	p. 254
Further reading	p. 258
Magneti poles, the dipole field, and current loops	p. 259
Maxwell's equations of elecromagnetism	p. 265
References	p. 276
Index	p. 278
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