Stated most simply, fluids are ‘things that flow’. This definition distinguishes between liquids and gases (both fluids) and solids, where the atoms are held more or less rigidly in some form of lattice. Of course, it is always possible to think of substances whose status is ambiguous in this regard, such as those, normally regarded as solids, which exhibit ‘creep’ over sufficiently long timescales (glass would fall into this category). Such borderline cases do not undermine the fact that the vast majority of substances can be readily classified as fluid or not. If they are fluids, then it is important to understand the general problem of how they flow, and under what circumstances they attain equilibrium (i.e. do not flow). These issues, in an astronomical context, form the subject of this book.

There is also a more subtle point about the sorts of systems that can be described as fluids. Although fluids are always in practice composed of particles at a microscopic level, the equations of hydrodynamics treat the fluid as a continuous medium with well-defined macroscopic properties (e.g. pressure or density) at each point. Such a description therefore presupposes that we are dealing with such large numbers of particles locally that it is meaningful to average their properties rather than following individual particle trajectories.

The material in this book is based on lecture notes of a course on astrophysical fluid dynamics which has been given for several years to third-year students at the University of Cambridge. There are several excellent books which cover fluid dynamics from a terrestrial standpoint, but very few provide a full introduction to the concepts and methods used to deal with the highly compressible flows which arise in astrophysical contexts. Our aim with this book is to provide just such an introduction, and we hope that it will also serve as a reference volume for advanced undergraduate and graduate students.

Several people have provided input at various stages of the preparation of this book. In particular we thank Jim Pringle, Donald Lynden-Bell and Giuseppe Lodato for their help. We are also grateful to the students who have taken the course at Cambridge for correcting typographical errors in the lecture notes, drawing our attention to parts where the description was less clear than it should have been, and helping us to develop the exercises.

In the last chapter, sound waves were introduced as the way that low amplitude (i.e. linear) disturbances propagate in a fluid. But what if a piece of fluid is subject to a non-linear disturbance, e.g. compression by a large factor or acceleration to velocities that are large compared with the sound speed? The result of such a disturbance is the propagation of a shock. On Earth, shocks are produced, for example, by the rapid pushing of a piston into a cold gas, or by the passage of a supersonic aircraft.

Astrophysics abounds in shock phenomena because gravity is an effective way of accelerating gas to high velocities: for example, gas free-falling onto the surface of stars or gas orbiting in a spiral galaxy like the Milky Way travels at hundreds of kilometres per second; in clusters of galaxies, the free-fall speed of the gas may attain thousands of kilometres per second. Such speeds correspond to the speed of sound in gas that is respectively at ~106 K and ~108 K; for gas that is cooler than this, any relative motion between fluid elements at these sorts of speeds must result in a shock. Note that acceleration to supersonic velocities (in the frame of the galaxy, say) does not itself generate a shock, since in its own rest frame the gas is at rest!

Consider a fluid in a steady state, i.e. one which satisfies the hydrodynamic equations with ∂/∂t = 0 everywhere. If we find that small perturbations to this configuration grow with time, then our chances of finding the initial configuration in nature are very small, and the configuration is said to be unstable with respect to those perturbations. A stable configuration is one where either the perturbations diminish, or there is the possibility of oscillations or waves about the equilibrium configuration.

In this chapter we will be examining a variety of fluid instabilities. Such instabilities can often be invoked to explain the wealth of structure in astronomical images, as seen in e.g. Figures 9.7 and 9.8. More profoundly, the instabilities we discuss are responsible for such fundamental processes as convection in stars and the creation of the multi-phase state of the interstellar medium. Most importantly, the Jeans instability (Section 10.2) is responsible for the formation of the most important building blocks of the Universe – galaxies and stars.

Rayleigh–Taylor instability

In a Rayleigh–Taylor instability, an unstably stratified configuration of fluid layers overturns under the influence of gravity. This phenomenon is observed in many settings, including astrophysics, atmospheric and ocean science, and industrial processes.

Convective instability

The stability condition in the first case we consider is one which may be obtained by fairly simple arguments, and in particular without doing a perturbation analysis of the full hydrodynamic equations.

One of the most important applications of shock wave theory in astronomy is to the problem of a supernova exploding in an interstellar cloud. Supernovae dump around 1044 joules of thermal and kinetic energy of ejecta into a small region around the star on an astronomically minute timescale (a day or so). The shocked medium expands and sweeps up more gas, creating a large bubble in the interstellar medium (ISM). It is now known that the continuous injection of energy into the ISM in this way by successive supernovae is responsible for giving the ISM a ‘Swiss cheese’ structure, with bubbles of hot gas alternating with sheets and filaments of cooler material swept up in the cavity walls. Astronomical opinion is divided as to whether the net effect of supernova action is to suppress further star formation (through hot gas in the bubbles breaking out of the galactic potential) or to promote it (through enhanced fragmentation of gas swept up in the bubble walls). At any rate, the effect of supernovae is evidently fundamental to understanding the ISM in galaxies. Before we are in a position to estimate the global effects of successive supernovae, we must first construct an idealised model of a single supernova exploding in a uniform medium.

The title of the next section however indicates that the problem was first studied in another and more menacing context: the impetus for developing the elegant theory laid out below came from the need to model the effects of thermonuclear weapons exploding in the Earth's upper atmosphere.

By far the most important application of the Navier–Stokes equations in astrophysics is to the case of circular shear flows, also known (for reasons that will become obvious once we have derived their properties) as accretion discs.

Such flows are encountered in many astronomical environments where gas is in nearly circular orbit around a massive central object. The scales of these flows vary immensely according to the nature of the central objects involved, which range from planets to stars to supermassive black holes. Figure 12.1 shows an example of what is believed to be an accretion disc around a black hole at the centre of a galaxy. Although, as we shall see, there are many qualitative aspects of these flows that are similar in all cases, the manner in which such rotating shear flows are created differs according to the type of astronomical system involved. For example, in the case of the primordial solar nebula, from which the planets in our Solar System were formed, the flow inherited its angular momentum from the slowly rotating cloud core which collapsed to form the Solar System. In the case of discs around supermassive black holes in the cores of galaxies, the origin of the gas orbiting the hole is not entirely understood, but virtually any explanation one could think of – be it gas shed from stars in the inner galaxy, or debris from stars tidally shredded by the black hole, or the swallowing of a small satellite galaxy – would involve gas that possessed significant angular momentum with respect to the central black hole.