The Earth's dynamical behaviour is a complex and fascinating subject with many practical ramifications. Its description requires the language of mathematics and computation. In this book, we attempt to make the theoretical foundations of the description of Earth's dynamics as complete as possible, and we accompany the theoretical descriptions with computer code and graphics for the implementation of the theory.

Scalar, vector and tensor analysis

We will make extensive use of scalars, vectors and tensors throughout the book. In this section, we will summarise the properties most often used. It is assumed that the reader is familiar with the elementary results of vector analysis summarised in Appendix A.

Scalars

Physical quantities determined by a single number such as mass, temperature and energy are scalars. Scalars are invariants under a change of co-ordinates, they remain the same in all co-ordinate systems. They are sometimes simply referred to as invariants. A scalar field is a function of space and time.

Vectors

Vectors require both magnitude and direction for their specification. They may be described by their components, their projections on the co-ordinate axes. An arbitrary vector then associates a scalar with each direction in space through an expression that is linear and homogeneous in the direction cosines.

In general, a vector can be defined in a space of arbitrary dimensions numbering two or greater. Our applications will be confined to a space of three dimensions and we will adopt this limitation.

The study of Earth's figure and gravitation goes back to the very roots of the physical sciences. We begin with a description of its very interesting historical roots.

Historical development

Although ancient Greek and Egyptian philosophers and astronomers believed the Earth to be spherical and had made rough measurements of its size, the modern theory of the Earth's figure originates with the work of Newton. Using his newly developed laws of dynamics and gravitation, and making the remarkable assumptions that the Earth's figure is nearly an oblate spheroid (the surface generated by an ellipse revolved about its minor axis) and that the Earth behaves as a fluid, he was able to show that the ellipticity of figure is directly given by the ratio of centrifugal force to gravitational force at the equator. As a result of the neglect of the concentration of mass towards the Earth's centre, Newton's estimate of the ellipticity was more than 30% too large. However, the oblate spheroid continues as the figure of reference for geodesy, and the assumption of hydrostatic equilibrium remains as the basis of much of the modern theory. Perhaps Newton's most important contribution, though, was his unequivocal demonstration that any theory of the Earth's figure must take account of both its gravitation and its rotation.

Newton published his calculation in the first edition of the Principia in 1687, but the meridian arc length per angular unit of geographical latitude, as measured on European baselines, appeared to increase with increasing latitude, implying that the Earth was prolate rather than oblate. It was not until expeditions had been dispatched by the French Academy to Peru and Lapland that the matter was settled in Newton’s favour in the 1740s.

New discoveries in Earth dynamics can only be made through the comparison of theory with observations. Often we are looking for signals close to or below the noise level; otherwise, they would have already been observed. Thus, the analysis of observations in both time and frequency domains is of crucial importance.

For several decades now, observations in the time domain have been represented by discrete samples. The samples may be equally spaced along the time axis or unequally spaced. Unequally spaced samples may result from inherent properties of the measurement technique, or from fundamental restrictions such as the visibility of sources at particular times. Unequally spaced samples may also be the result of digitiser failure or other instrument problems, leaving gaps in otherwise equally spaced time sequences. We include the analysis of unequally spaced time sequences and the application of singular value decomposition to their study. Most sequences of interest were originally continuous physical signals. Thus, we examine the effects of the sampling process itself on the results of the analysis.

Often observations are made at several locations and it is desired to bring out common features of the records from different observatories. For this purpose, we describe in detail, the product spectrum. This may be regarded as a kind of generalisation of the cross spectrum between two records. As is the case in any spectral analysis, the estimation of confidence intervals is of prime importance in establishing the significance of the results.

The inversion of seismological observations leads to the construction of models of the physical properties of Earth. The basic models take Earth to be spherically symmetric with properties, such as density, the Lamé coefficients of elasticity, and gravity, listed as functions of radius alone. These models form the numerical basis for the calculation of Earth's dynamics, including its short-period and long-period free oscillations, its response to tidal forcing and the dynamics of its rotation.

The Earth models

We list here the files cal8.dat, 1066a.dat, prem.dat and core11.dat for four wellknown Earth models, Cal8 (Bullen and Bolt, 1985, pp. 471–473), 1066A (Gilbert and Dziewonski, 1975), PREM (Dziewonski and Anderson, 1981) and Core11 (Widmer et al., 1988), respectively. The first line in each case gives the name of the Earth model in the format 10A8. The second line gives the number of Love numbers to be calculated in the format I10. The third line gives the number of model points and the initial number of integration steps for the inner core, outer core, mantle and crust in the format 8I10. Although a variable stepsize Runge– Kutta method is used in calculations, initial stepsizes are specified. The columns tabulate the radius, density, the Lamé coefficients of elasticity, and gravity in the format 1X,F10.1,F10.2,F10.1,F10.1,F10.1. Following tradition, radius is expressed in kilometres, density in grams per cubic centimetre, the Lamé coefficients in kilobars and gravity in centimetres per second per second. After they are read in, these Earth properties are scaled to SI units before proceeding to calculations.

The study of Earth's dynamics, from near surface earthquake displacement fields to the translational modes of the solid inner core, has long been a fascination for me. In the present work I have been influenced by Numerical Recipes, published by Cambridge University Press, to include computer code so often omitted in scientific publications. I have gone one step further to include, on the website www.cambridge.org/smylie, open source downloadable software through the Oracle Virtual Machine, allowing a full Fedora Linux operating system to be installed on users’ machines along with the TRIUMF graphics system, giving full access to Fortran, LaTeX and TeX as well as codes from the book itself and possible updates.

Throughout the writing of this book I have been very ably assisted by Dr Gary Henderson in every aspect. While I take full responsibility for any remaining errors, his high skills in English, theory and Fortran have been very much appreciated. I have been fortunate to have benefited from many teachers and professors in my studies. Reg Daniels kindled my interest in mathematics in high school, Fraser Grant introduced me to mathematical geophysics, while Tuzo Wilson hosted the geophysics laboratory at 49 St. George St. in the University of Toronto.

In the subseismic description of core dynamics, presented in the previous chapter, the changes in density caused by adiabatic compression or expansion due to transport through the hydrostatic pressure field are included, but those arising from flow pressure fluctuations are ignored in comparison. While this leads to the subseismic wave equation governing long-period core oscillations, the geophysicist's favourite analytical tool, the representation of solutions by expansion in spherical harmonics, is poorly convergent for such modes, due to tight Coriolis coupling between harmonics of like azimuthal number but differing zonal number (Johnson and Smylie, 1977). Instead, we use local polynomial basis functions to represent the generalised displacement potential. We show that solutions of the governing subseismic wave equation and boundary conditions have either purely even or purely odd symmetry across the equatorial plane. The basis functions then take the forms developed in Section 1.6.3. Solutions to the subseismic wave equation are found through the development of a variational principle, which includes the continuity of the normal component of displacement as a natural boundary condition of the problem (Smylie et al., 1992). The remaining elasto-gravitational boundary conditions are incorporated through the load Love numbers described in the previous chapter.

A subseismic variational principle

A variational principle for the subseismic wave equation, (7.24), will involve stationarity of the functional with respect to variations in the generalised displacement potential, χ, subject to boundary conditions. The vector displacement field is given by expression (7.16), entirely in terms of the gradient of the generalised displacement potential.

The study of Earth's rotation would not be of much interest if the Earth rotated uniformly about a fixed axis. Variations in the speed of rotation and changes in the orientation of the rotation axis, both within the body of the Earth and in space, make the subject deeply fascinating and rewarding. The subject has a long and interesting history that is well reviewed in the classic treatise The Rotation of the Earth (Munk and MacDonald, 1960), which did much to revitalise modern interest. More recently, a second authoritative treatise entitled The Earth's Variable Rotation, by Kurt Lambeck, has appeared (Lambeck, 1980), giving a modern overview of the subject.

Observations of the rotation are generally made by observatories attached to the Earth, measuring motions with reference to stars and other celestial objects. Thus, both a terrestrial reference frame and a celestial reference frame need to be defined to make such observations.

Reference frames

To the lowest order of approximation, observations are made in a rigid, uniformly rotating frame. Of course, the subject is of interest because the actual frame is neither perfectly rigid nor perfectly uniform in its rotation. The observer's frame is usually defined by a prescribed method of adjusting the frame to the mean motion of a set of observatories, in such a way as to approximately minimise the variance of the relative motions over all the observatories. For example, the Bureau International de l'Heure (BIH) defines the 1968 BIH (Guinot and Feissel, 1969) reference system in terms of the latitudes and longitudes assigned to 68 observatories, with each having an assigned weight in latitude and time.

In addition to generating seismic waves, tsunamis and free oscillations of the Earth, earthquakes generate static displacements, strains and tilts both locally and at teleseismic distances. The nearly global extent of these static deformations was brought out clearly by Press (1965) following the great Alaska earthquake of 27 March 1964. The basis for modelling the displacement fields is Volterra's theory of elastic dislocations, published in 1907, dealing with the elasticity theory of surfaces across which displacements are discontinuous. The theory was revived by Steketee (1958a), and applied to geophysical problems by him (Steketee, 1958b) and his student at the time (Rochester, 1956). The surprising extent of the displacement fields demonstrated by Press led to a re-evaluation of their effect on the polar motion (Mansinha and Smylie, 1967). In this chapter, we first present the elasticity theory of dislocations, as it is now known, in an infinite uniform elastic half-space, and then in realistic Earth models. We conclude with the calculation of the effects of earthquakes on the polar motion.

The elasticity theory of dislocations

The starting point for the elasticity theory of dislocations is the reciprocal theorem of Betti (1.266) (Sokolnikoff, 1956, pp. 390–391). This states that for two systems of surface tractions and body forces the work done by the first system, acting through the displacements caused by the second system, is equal to the work done by the second system, acting through the displacements caused by the first system.