As we scrutinize the landforms that surround us, we develop a sense of appreciation for the multitude of processes that shaped our planet. At the outset, we may think it is improbable that, out of this chaotic combination of physical processes, there could emerge any sense of abiding order. Nevertheless, we have come to appreciate during the preceding decades that the collective interaction of many different ingredients, as demonstrated by the Earth, yields manifestations of a new class of behavior which we refer to as “nonlinearity.” It is important that we distinguish between the nonlinearity that we associate with the kinds of differential equations that we have discussed – i.e. the presence of terms that are of higher order than linear – from the collective behavior and self-organization that we sometimes observe. We reviewed this collective behavior in the context of solitary waves. Another aspect sometimes maintained by nonlinearity has roots in geometry and, as a consequence, influences in a profound way a variety of physical systems. Later, we shall survey features of percolation and fractal geometry as illustrations of this. We observed earlier how a “cascade” picture for energy transfer between different length scales at the same rate, as a form of self-organization, could help us understand turbulence and the emergence of power-law scalings. Perhaps this and other kinds of nonlinearity can help us understand the nature of earthquakes.

Most of this book, in its pursuit of studying continuum mechanics, focused on linearized problems and their decomposition into relatively simple superpositions of solutions. Real-world problems, in contrast, are much more complicated. The calculation of exact solutions to linearized problems in the face of complicated geometries very often can become computationally intensive. Further, realistic problems often introduce substantial nonlinearity – this is especially true in applications involving fluids – rendering such calculations inaccurate, if not incorrect. Moreover, the application of computational methods, however, is not trivial. Computer arithmetic, unlike computer algebra such as that performed by Mathematica and Maple, is executed with a finite number of digits of accuracy (Higham, 2002). Typically, there are 16 digits in double precision arithmetic in programming languages such as C++ and FORTRAN 95, but as great as 25 in MATLAB. Continuum mechanical problems involving matrices, particularly those of large rank, can lose many digits of accuracy due to ill-conditioned matrices. Adding to the arithmetic limitations of computers, the more common tasks of solving nonlinear ordinary and, especially, partial differential equations of continuum mechanics present a formidable issue. The operative differential equations were formulated in the mathematical limit of certain differential quantities going to zero, i.e. infinitesimal quantities that emerge in the evaluation of derivatives. Computers, on the other hand, only work in the realm of the finite quantities.

Body and surface forces

We begin by examining the nature of forces on continuous media. We will pursue this theme later by examining material response. This topic is a truly venerable one with significant references made by Newton (Chandrasekhar, 1995) and many others before and after. Early treatments of this topic employ modes of notation very similar to ours, but largely focus on two-dimensional problems since much of the algebra reduces to that associated with quadratic equations. We will show that in three dimensions, the algebraic problem corresponds to a cubic polynomial with real roots which can be easily determined by analytical means. A medium is homogeneous if its properties are the same everywhere.

Homogeneity, however, can be of two types: regular or random. A regular homogeneous medium has the same underlying character everywhere, e.g. a piece of metal whose atoms are organized in a lattice. A random homogeneous medium has the same underlying statistical distribution of properties, but may lack regularity. For example, a rock composed of many different grains cemented together can be said to be homogeneous if the statistical properties of the mix do not vary. A homogeneous material is also said to be isotropic, i.e. looks the same in all directions. The material looks the same because it is the same. However, an isotropic material is not necessarily homogeneous. For example, the Earth appears to be (crudely) isotropic as viewed from its center but the core, mantle, and lithosphere are very distinct from each other.

This book is the outcome of an introductory graduate-level course that I have given at the University of California, Los Angeles for a number of years as part of our program in Geophysics and Space Physics. Our program is physics-oriented and draws many of its students from the ranks of undergraduate physics and, sometimes, mathematics majors, in addition to geophysics and occasionally geology majors. Accordingly, this text approaches the subject by promoting a physics-based understanding of the basic principles with a relatively rigorous mathematical approach. Since the needs of this course were rather unique, blending concepts in physics and mathematics with the Earth sciences, I approached teaching the subject by drawing on many sources in developing the necessary material. (Throughout this volume, I refer to materials that provide more complete treatments of the topics which we only have time to overview.) In contrast to other sources, I wanted this course to treat not only classical methods but survey some of the ideas emerging in the geosciences that were drawn directly from current ideas in physics, especially nonlinear dynamics. Over time, the material developed more coherence and my lecture notes for this academic quarter-long course evolved into this text.

The subject of continuum mechanics is predicated on the notion that many natural phenomena have a fundamentally smooth, continuous nature. This constitutes the basis for solid and fluid mechanics, major components of this course.

Fluid mechanics occupies an important niche in the study of the Earth. Fluid motions describe the behavior of the interior of this and other planets, as well as the motion of our respective oceans and atmospheres. Remarkably, fluid behavior can manifest some truly amazing properties. Van Dyke (1982) provides a visual compendium of these behaviors, describing the richness of flow patterns that can emerge. The study of fluid mechanics remains a venerable topic and there are a number of excellent textbooks available. Batchelor (1967) provides an authoritative introduction to the subject from the perspective of an applied mathematician, while Landau and Lifshitz (1987) does so from the viewpoint of theoretical physicists. Faber (1995) provides a modern treatment which is encyclopedic in scope but remains a relatively easy read. Fowler (2011) has published an encyclopedic volume addressing many flow problems encountered in geophysics.

Owing to their intrinsic nature, fluids can respond rather dramatically to subtle, almost imperceptible changes in their environment. We now appreciate that sensitivity to an initial set of conditions is the hallmark of chaos. Indeed, thermal convection is often cited as a source of chaos (Drazin, 1992) and the Lorenz model, a skeletal description of a fluid heated from below, as is the case in earth's atmosphere and mantle, has become the paradigm for chaotic behavior. The Lorenz model consists of three coupled ordinary differential equations (Strogatz, 1994; Drazin, 1992).

In the previous chapter, we focused on the behavior of fluids in an inertially stationary environment. The Earth, as well as other planets and stars, on the other hand, rotate at a relatively constant rate thereby introducing a time scale and a length scale, namely the length of day and the Earth's radius, respectively, into the problem. Moreover, rotation introduces non-inertial forces, as we saw in section 1.6, that interact with other forces that are present, particularly gravity. This is further complicated by issues ranging from heat flow, variations in density, the role of viscosity, and the presence of topographically complex boundaries. Pedlosky (1979) provides an exhaustive survey of the field of geophysical fluid dynamics (GFD). Many arenas of investigation have emerged in response to these theoretical advances. Atmospheric and oceanographic flow, owing to their practicality, have preserved a prominent place in contemporary science. Houghton (2002) focuses on the physics of atmospheres while Holton (2004) presents a more meteorologically-based perspective. Marshall and Plumb (2008) explore the combined roles of atmosphere and ocean and their interaction, exploring also their long-term contributions to climate dynamics. Gill (1982) focuses on the atmosphere–ocean dynamics. Given that Earth's hydrosphere has almost three orders of magnitude more mass than its atmosphere, these interaction effects can be profound. Ghil and Childress (1987) investigate a set of topics emerging from atmospheric dynamics, dynamo theory, and climate dynamics.

Scalars, vectors, and Cartesian tensors

Geometry is a vital ingredient in the description of continuum problems. Our treatment will focus on the mathematically simplest representation for this subject. Although curvilinear coordinates can be more natural, they introduce complications that go beyond the scope of this book. The initial part of our treatment will parallel the Cartesian approach of Mase and Mase (1990) rather than the curvilinear approach of Narasimhan (1993) and Fung (1965). Hence, we will adhere to a Cartesian description of problems and be spared the need to distinguish between covariant and contravariant notation. Moreover, we will generally employ second-rank tensors which are matrices that possess some very special and important (coordinate) transformation properties.

We will distinguish between three classes of objects: namely, scalars, vectors, and tensors. In reality, all quantities may be regarded as tensors of a specific rank. Scalar (nonconstant) quantities, such as density and temperature, are zero rank or order tensors, while vector quantities (which have an associated direction, such as velocity) are first-rank tensors. Second-rank tensors, such as the stress tensor, are a special case of square matrices. We will usually denote vector quantities by bold-face lower-case letters, while second-rank tensors will be denoted by bold-face upper-case letters.