A Student's Guide to Geophysical Equations

Contents......Page 8
Preface......Page 12
Acknowledgments......Page 14
1.2 Complex numbers......Page 16
1.3.1 Scalar and vector products......Page 19
1.3.2 Vector differential operations......Page 20
1.4.1 The rotation matrix......Page 23
1.4.2 Eigenvalues and eigenvectors......Page 24
1.4.3 Tensor notation......Page 29
1.4.4 Rotation of coordinate axes......Page 30
1.4.5 Vector differential operations in tensor notation......Page 31
1.5 Conservative force, field, and potential......Page 32
1.6 The divergence theorem (Gauss’s theorem)......Page 33
1.7 The curl theorem (Stokes’ theorem)......Page 35
1.8 Poisson’s equation......Page 38
1.9 Laplace’s equation......Page 41
1.10.1 MacLaurin series......Page 43
1.10.2 Taylor series......Page 44
Infinite series......Page 45
1.10.4 Linear approximations......Page 46
1.12 Legendre polynomials......Page 47
1.13 The Legendre differential equation......Page 49
1.13.1 Orthogonality of the Legendre polynomials......Page 52
1.13.2 Normalization of the Legendre polynomials......Page 54
1.14 Rodrigues’ formula......Page 56
1.15 Associated Legendre polynomials......Page 58
1.15.1 Orthogonality of associated Legendre polynomials......Page 61
1.15.2 Normalization of associated Legendre polynomials......Page 62
1.16 Spherical harmonic functions......Page 64
1.16.1 Normalization of spherical harmonic functions......Page 65
1.16.2 Zonal, sectorial, and tesseral spherical harmonics......Page 66
1.17.1 Fourier series......Page 67
1.17.3 Fourier sine and cosine transforms......Page 69
Further reading......Page 73
2.1 Gravitational acceleration and potential......Page 74
2.2 Kepler’s laws of planetary motion......Page 75
2.2.2 Kepler’s First Law......Page 77
2.2.3 Kepler’s Third Law......Page 80
2.3.1 Outside a solid sphere, using Laplace’s equation......Page 81
2.3.2 Inside a solid sphere, using Poisson’s equation......Page 82
2.4 Laplace’s equation in spherical polar coordinates......Page 84
2.4.1 Azimuthal (longitudinal) solution......Page 85
2.4.2 Polar (latitudinal) solution for rotational symmetry......Page 86
2.4.3 Radial solution......Page 87
2.4.5 General solution of Laplace’s equation......Page 88
2.5 MacCullagh’s formula for the gravitational potential......Page 89
2.5.1 Gravitational potential of a spheroid......Page 96
2.5.2 MacCullagh’s formula and the figure of the Earth......Page 99
Further reading......Page 100
3.1 The ellipticity of the Earth’s figure......Page 101
3.2 The geopotential......Page 103
3.2.1 Gravitational potential......Page 105
3.3 The equipotential surface of gravity......Page 106
3.3.1 Relationship of J2, J4, f, and m......Page 109
3.3.2 Inferred increase of density with depth in the Earth......Page 110
3.4 Gravity on the reference spheroid......Page 111
3.4.1 Polar component of gravity......Page 112
3.4.2 Radial component of gravity......Page 113
3.4.3 Variation of reference gravity with geocentric latitude......Page 115
3.5 Geocentric and geographic latitude......Page 117
3.5.1 Normal gravity on the reference ellipsoid......Page 119
3.6 The geoid......Page 121
3.6.1 The potential of a geoid undulation......Page 122
3.6.2 Stokes’ formula for the height of the geoid......Page 123
3.6.3 Evaluation of the function F(θ)......Page 126
Further reading......Page 130
4.1 Origin of the lunar tide-raising forces......Page 131
4.2 Tidal potential of the Moon......Page 134
Potential W2......Page 135
Potential W3......Page 136
4.2.3 The solar tide-raising acceleration......Page 137
4.3.1 Tidal height......Page 139
4.3.2 Tidal gravity anomaly......Page 140
4.3.3 Tidal deflection of the vertical......Page 144
4.4 Tidal friction and deceleration of terrestrial and lunar rotations......Page 145
4.4.1 Angular momentum of the Earth–Moon system......Page 146
4.4.2 Slowing of terrestrial and lunar rotations......Page 147
4.4.3 Development of the Earth–Moon separation......Page 148
Further reading......Page 151
5 Earth’s rotation......Page 152
5.1.1 Velocity......Page 153
5.1.2 Acceleration......Page 154
5.2 The Coriolis and Eötvös effects......Page 155
5.2.2 Horizontal component: the Coriolis effect......Page 156
5.3.1 Effects of the torque due to the Sun’s attraction......Page 157
5.3.2 Comparison of vectors in the coordinate systems of Earth and Sun......Page 161
5.3.3 Computation of the Sun’s torque on the Earth......Page 162
5.3.4 Equations of solar-induced precession and nutation......Page 163
5.3.5 Simplification of the equations of motion......Page 165
5.3.6 Precession and nutation induced by the Sun......Page 167
5.3.7 Precession and nutation induced by the Moon......Page 168
5.3.8 Nutation due to precession of the Moon’s orbit......Page 169
5.4 The free, Eulerian nutation of a rigid Earth......Page 170
5.5 The Chandler wobble......Page 172
5.5.2 Computation of the products of inertia......Page 178
5.5.4 Period of the Chandler wobble......Page 181
5.5.5 Calculation of Love’s number k from the period of the Chandler wobble......Page 182
Further reading......Page 184
6 Earth’s heat......Page 185
6.1 Energy and entropy......Page 186
6.2.1 Thermodynamic potentials......Page 187
6.3 The melting-temperature gradient in the core......Page 191
6.4 The adiabatic temperature gradient in the core......Page 193
6.5 The Grüneisen parameter......Page 194
6.5.1 Temperature and density in the Earth......Page 196
6.6 Heat flow......Page 197
6.6.2 The thermal-conduction equation......Page 198
6.6.3 Penetration of solar heat in the Earth......Page 200
6.6.4 Cooling of a semi-infinite half-space......Page 205
6.6.5 Cooling of oceanic lithosphere......Page 210
Further reading......Page 212
7.1 The dipole magnetic field and potential......Page 213
7.2.1 The fields of internal and external origin......Page 215
7.2.2 Determination of the Gauss coefficients......Page 217
7.3.2 The geocentric inclined dipole......Page 220
7.3.3 Axial dipole with axial offset......Page 223
7.3.4 Axial dipole with equatorial offset......Page 225
7.3.5 Best-fitting eccentric inclined dipole......Page 226
7.4 Secular variation......Page 228
7.5 Power spectrum of the internal field......Page 229
7.5.1 Estimation of the source depth of the main field......Page 231
7.6 The origin of the internal field......Page 232
7.6.1 Electromagnetic model......Page 233
7.6.2 The magnetohydrodynamic model......Page 235
7.6.3 The frozen-flux theorem......Page 237
Further reading......Page 240
8.1 Elastic deformation......Page 242
8.2 Stress......Page 243
8.2.1 Symmetry of the stress tensor......Page 245
8.2.2 Equation of motion......Page 247
8.3 Strain......Page 248
8.3.1 Normal strain......Page 250
8.3.2 Shear strain......Page 252
Young’s modulus......Page 254
8.4.1 The Lamé constants......Page 255
8.5 The seismic wave equation......Page 259
8.5.1 Primary waves (P-waves)......Page 261
8.5.2 Secondary waves (S-waves)......Page 264
8.5.3 Displacement potentials......Page 265
S-waves......Page 266
8.6.1 One-dimensional solution for plane P-waves......Page 267
8.7 Three-dimensional propagation of plane P- and S-waves......Page 269
8.7.1 P-wave propagation......Page 271
8.7.2 S-wave propagation......Page 272
Further reading......Page 273
A1. The concept of magnetic poles and Gauss’s law......Page 274
A2. The magnetic dipole......Page 275
A3. The Lorentz force......Page 277
A4. Torque on a current loop in a magnetic field......Page 278
1. Coulomb’s law......Page 280
1.1. The effect of bound charges......Page 282
2. Ampère’s law......Page 283
3. Gauss’s law for magnetism......Page 285
3.1. The magnetic field inside a magnetizable material......Page 287
4. Faraday’s law......Page 289
References......Page 291
Index......Page 293