Global Warming is a topic that increasingly occupies the attention of the world. Is it really happening? If so, how much of it is due to human activities? How far will it be possible to adapt to changes of climate? What action to combat it can or should we take? How much will it cost? Or is it already too late for useful action? This book sets out to provide answers to all these questions by providing the best and latest information available.

I was privileged to chair or co-chair the Scientific Assessments for the Intergovernmental Panel on Climate Change (IPCC) from its inception in 1988 until 2002. During this period the IPCC published three major comprehensive reports – in 1990, 1995 and 2001 – that have influenced and informed those involved in climate change research and those concerned with the impacts of climate change. In 2007, a fourth assessment report was produced, and in 2014 the fifth assessment report was published. It is the extensive new material in this latest report that has provided the basis for the substantial revision necessary to update this fifth edition.

The IPCC reports have been widely recognised as the most authoritative and comprehensive assessments on a complex scientific subject ever produced by the world's scientific community. On the completion of the first assessment in 1990, I was asked to present it to Prime Minister Margaret Thatcher's cabinet – the first time an overhead projector had been used in the Cabinet Room in Number 10 Downing Street. In 2005, the work of the IPCC was cited in a joint statement urging action on climate change presented to the G8 meeting in that year by the Academies of Science of all G8 countries plus China, India and Brazil. The world's top scientists could not have provided stronger approval of the IPCC's work. An even wider endorsement came in 2007 when the IPCC was awarded a Nobel Peace Prize.

The significance of the vorticity field has been stressed at a number of points in the discussion thus far. For example, the Helmholtz decomposition showed that the vorticity can be used with the dilatation to re-create the velocity field. Furthermore, vorticity does often concentrate into small regions, as in a tornado, and so by following the vorticity dynamics, one has an economical and insightful means of capturing the essence of the flow field. Another significant example is in the case of turbulent flows, where in many ways the dynamics of the vorticity field offers the most direct means of understanding such key processes in the flow as momentum transport and the passage of energy between scales.

An equation that governs the physics of the vorticity field in a moving fluid can be obtained by taking a curl of the Navier-Stokes equation (16.2). Under many circumstances, as in the case of incompressible flow, the pressure is removed by the action of taking the curl, so that the resulting vorticity equation offers a route toward determining the flow without directly taking the pressure into account. This may represent an advantage in the analysis of some flows by providing a more straightforward approach toward understanding the physics of the flow than can occur with the coupled velocity and pressure. In particular, the pressure continuously adapts to the evolution of the velocity field so that it may be more difficult to establish the origin and cause of flow behaviors in this case than through examining the vorticity field and its evolution. In flows for which vortical structures dominate the physics, as in turbulent flow, there very well may be a definite advantage to analyzing the flow through numerical schemes that model the vorticity equation. In this case, the terms in the vorticity equation represent the physical processes that are most relevant for understanding the dynamics of the flow field.

What Is a Fluid?

What distinguishes fluids from solids? Because we all have an intuitive understanding of the difference, the question is really about precisely identifying the formal differences between them. In this regard, it is useful to consider the different ways that fluids and solids react to applied forces. For example, in the case of a solid, the material offers resistance if we press down on it. If the applied force is not so large as to shatter the solid, then it is clear that the solid is quite capable of resisting the force so as to reach a state of equilibrium. The solid arrives at a state where it ceases to move or deform.

Consider now specifically the case of a gas, as shown in Fig. 1.1. In the figure, a piston is pushing down on the gas, and although initially the gas might compress because of the applied force, it is also able to eventually reach a point for any given applied force where it does not compress further. In other words, the gas is capable of resisting the downward force in the same way as the solid. We may conclude that resistance to a normal force is not a good candidate for framing the distinguishing properties of fluids and solids.

The situation for fluids and solids is different if we consider an applied shear force, as in the experiment indicated in Fig. 1.2. Following the application of a shear force to the top surface of the solid, as shown in Fig. 1.2(a), an equilibrium is reached in which the body has deformed a fixed amount. Alternatively, if the container holds a fluid, as in Fig. 1.2(b), and a shear force is applied to the top lid, the fluid cannot prevent the lid from sliding to the side. This is true no matter how small the applied force may be. This is not to say that the fluid does not offer resistance – it does – but the resistance it offers cannot be enough to create a stationary equilibrium.

This book is inspired by a graduate-level course in fluid dynamics that I have taught at the University of Maryland for many years. The typical student taking this course, which is the starting point for graduate studies in fluid mechanics, has had one undergraduate course on fluids and a limited exposure to vector and tensor analysis. Consequently, the goal of this book is to provide a background in the physics and mathematics of fluid mechanics necessary for the pursuit of advanced studies and research at the graduate level. It is my experience that an effective route to these objectives is via a synthesis of the best features of two very excellent books, namely, An Introduction to Fluid Dynamics by George Batchelor, which presents the physics of fluid mechanics with exceptional clarity, and An Introduction to Continuum Mechanics by M. E. Gurtin (and now expanded and revised as The Mechanics and Thermodynamics of Continua by Gurtin, Fried, and Anand), which demonstrates the advantages of direct tensor notation in simplifying the expression of physical laws. Thus, to a large extent, this book combines the physics of Batchelor with the mathematics of Gurtin. The hope is that, in this way, an environment is created that helps make the subject of fluid dynamics clear, focused, and readily understandable. As a practical matter, this book should serve as an effective stepping-stone for new graduate students to enhance their accessibility to the books by Batchelor and Gurtin as well as those by many others.

Stylistically, this book follows an arc through the material that builds steadily toward the derivation and then application of the Navier-Stokes equations. The sequence of topics is also chosen so as to provide some significant exposure to examples of fluid flow and problem solving, before a relatively long and unavoidable set of chapters that deal in detail with the derivation of the flow equations.