The mechanisms responsible for mixing in the interior of a stratified fluid are even less well understood than those described in chapter 9, since the sources of energy are not so obvious, and several different processes must be taken into account simultaneously. The important ideas have already been introduced in earlier sections, but it seems appropriate in this final chapter to take a broader and less detailed view, and to consider together the whole array of mixing phenomena which can be relevant in large natural bodies of stratified fluid. Geophysical examples have figured prominently in this book, and indeed the range of subject matter has been chosen with this final synthesis in mind. The basic facts requiring explanation are somewhat scattered, however, and in order to collect them together and to define the problems to be treated here, a brief summary will now be given of the observed structure of the ocean and atmosphere.

This field is developing rapidly, and the interpretation given here must necessarily be a somewhat tentative and personal one. Nevertheless, it seems useful to sketch how our present knowledge of the separate components can be fitted into a self-consistent picture, at the same time extending some of the earlier arguments so that they can be applied in this wider context.

The observational data

Routine density profiles made using reversing bottles and thermometers show that the ocean is everywhere stably stratified, except for limited regions where bottom water forms intermittently as the water column becomes convectively unstable.

A comparatively recent development in the field of convection has been the study of fluids in which there are gradients of two (or more) properties with different molecular diffusivities. When the concentration gradients have opposing effects on the vertical density distribution, a number of surprising things can happen, and these are the subject of the present chapter. The phenomena were first studied with an application to the ocean in mind (see §8.2.4), and because heat and salt (or some other dissolved substance) are then the relevant properties, the process has been called ‘thermohaline’ or ‘thermosolutal’ convection. Related effects have now been observed in the laboratory using a pair of solutes, and in solidifying metal alloys, and the name ‘double-diffusive convection’ has been chosen to encompass this wider range of phenomena.

The stability problem will first be reviewed, somewhat more fully than was done in previous chapters because of the comparative novelty of the double-diffusive phenomena. It will then be shown that when two components contribute to the vertical density gradient, a series of steps tends to form, with well-mixed layers separated by sharper density interfaces. The detailed structure of these interfaces and measurements of the coupled fluxes across them will also be described.

The stability problem

The mechanism, of instability

In such a system with opposing gradients, the existence of a net density distribution which decreases upwards is not a guarantee of stability.

Buoyancy forces arise as a result of variations of density in a fluid subject to gravity, and produce a wide range of phenomena of importance in many branches of fluid mechanics. Progress in this field has been made largely through the desire to solve very practical problems, arising for instance in meteorology or in hydraulic engineering. This emphasis on particular applications has meant that parallel developments have often been made in different disciplines without much cross reference to related work, and some results, well understood in one context, are less familiar in another where they might be used to advantage. In this book I have attempted to write a coherent account of the various fluid motions which can be driven or influenced by the presence of small density differences. It is intended as a general introduction to the subject and its literature, in which the physical understanding of the phenomena is emphasized, rather than the applications on the one hand or detailed mathematical theory on the other.

The selection of subject matter must always be a personal one, however, and my own research interests have certainly influenced the topics chosen and the amount of space given to each of them. I have worked with laboratory models of small scale processes in the ocean and atmosphere, and so laboratory and geophysical examples come most readily to mind, but comparisons have also been made with results from various fields of engineering where possible.

In this chapter we consider various mechanisms whereby laminar flows of a stably stratified fluid can break down and become turbulent. The first task is to summarize some results of hydrodynamic stability theory as it applies in this context, that is, the investigation of the conditions under which small disturbances to the motion can grow. The logical development of the preceding chapters will be followed by restricting the discussion to dynamic instabilities due to shearing motions of a statically stable initial stratification. ‘Convective’ instability associated with an increase of density with height will be left until chapter 7, and two-component systems, in which different diffusivities play a vital role, will be treated separately in chapter 8.

The problem of instability of layers across which both density and velocity are rapidly varying functions of height is given special attention here, since such shear layers are very common in the atmosphere and ocean, and the vertical transports of properties such as heat and salt depend strongly on what happens near them. The topics covered now will, however, go beyond what is usually meant by the term ‘instability’ in the strict sense. It seems useful at the same time to outline other ways in which energy can be fed into limited regions of a more extensive flow, at a rate sufficient to cause a local breakdown. Some of the phenomena have already been mentioned in the context of steady flows, and the wider geophysical implications of the results will be discussed in chapter 10.

The topics to be discussed

It seems useful to begin by outlining the range of subjects covered in this book, to give a broad picture of the way in which the several parts of the field have developed, and at the same time some explanation of the theme which has been used to connect them. The phenomena studied all depend on gravity acting on small density differences in a non-rotating fluid. Often the undisturbed fluid has a density distribution which varies in the vertical but is constant in horizontal planes; this will be called a stratified system whether the density changes smoothly or discontinuously. Special attention will be given to the problems of buoyant convection (arising from an unstable density distribution) and to various mechanisms of mixing when the stratification is stable.

Chapters 2 and 3 summarize relevant results on internal waves, and these were also historically the first phenomena to be studied. The original applications of the methods of perfect fluid theory to motion under gravity were to the problems of small amplitude surface waves and tides (subjects which will not be discussed here). These were soon extended to the case of two layers of uniform density with a density discontinuity between them. Some of the basic results had already been obtained by 1850 (notably by Stokes 1847), and they were applied to phenomena such as the drag experienced by a ship when it creates a wave on an interface close to the surface (Ekman 1904), and to internal seiches in lakes.

Natural bodies of fluid such as the atmosphere, the oceans and lakes are characteristically stably stratified: that is, their mean (potential) density decreases as one goes upwards, in most regions and for most of the time. When they are disturbed in any way, internal waves are generated. These ubiquitous motions take many forms, and they must be invoked to explain phenomena ranging from the temperature fluctuations in the deep ocean to the formation of clouds in the lee of a mountain. In this chapter we summarize the results which can be obtained using linear theory (i.e. when the amplitudes are assumed to be small), and in §3.1 extend some of them to describe waves of large amplitude.

Many of the elementary properties of infinitesimal wave motions in stratified fluids can be introduced conveniently by considering waves at an interface between two superposed layers, and so this case is treated first in §2.1. These waves are analogous to waves on a free water surface, and therefore seem very familiar. It should be emphasized at the outset, however, that they are not the most general wave motions which can occur in a continuously stratified fluid. Energy can propagate through such a fluid at an angle to the horizontal, not just along surfaces of constant density, and our intuition based on surface waves is of little help here. The more general theory, and a comparison between the two descriptions, is given in §2.2.

The previous chapter was based on equations of motion made linear by assuming that the amplitude of wave-like disturbances of the fluid remained infinitesimal. We now consider various large amplitude phenomena which require the inclusion of the non-linear terms for their explanation. First, some of the inviscid wave problems already treated will be extended to finite amplitude, and the essentially non-linear phenomenon of internal solitary waves will be discussed. Then various quasi-steady flows which arise in nature and in civil engineering applications will be treated, using a generalization of free surface hydraulic theory (and thus relating the properties of such flows to the waves which can form on them). Internal hydraulic jumps, the flow of a thin layer down a slope, and the nose at the front of a gravity current come under this heading. Finally we introduce the effects that viscosity and diffusion can have on slow steady motions in a stratified fluid, describing upstream wakes and boundary layers and the process of selective withdrawal.

Internal waves of finite amplitude

Interfacial waves

We refer again to the statement made in §2.1.2, that (2.1.8) is valid for finite amplitude long waves. (Cf. Lamb 1932, p. 278.) This implies that the highest point of any disturbance will move fastest, and so the forward slope of a wave of finite amplitude will tend to steepen, an effect called ‘amplitude dispersion’. This is in contrast with the result of the frequency or wavenumber dispersion previously described by (2.1.7) which is valid for general depths.

The buoyancy effects discussed so far have for the most part been stabilizing, or have been assumed to produce a small modification of an existing turbulent flow. Now we turn to convective flows, in which buoyancy forces play the major role because they are the source of energy for the mean motion itself. The usual order of presentation will be reversed: it is convenient to set aside for the present the discussion of the mean properties of a convecting region of large horizontal extent, and flows near solid bodies, and in this chapter to treat various models of the individual convective elements which carry the buoyancy flux. (See Turner (1969a) for a review of this work and a more extensive bibliography.)

Such models can be broadly divided into two groups, those which assume the motion to be in the form of ‘plumes’ or of ‘thermals’. (See fig. 6.1.) In both of them motions are produced under gravity by a density contrast between the source fluid and its environment; the velocity and density variations are interdependent, and occupy a limited region above or below the source. Plumes, sometimes called buoyant jets, arise when buoyancy is supplied steadily and the buoyant region is continuous between the source and the level of interest. The term thermal is used in the sense which has become common in the meteorological literature to denote suddenly released buoyant elements.

The theme of this chapter will be a more detailed discussion of various kinds of turbulent shear flows, which are (at least at the beginning of the period of interest) well past the state of marginal stability. Some knowledge of the properties of a turbulent shear flow in a homogeneous fluid must be assumed (see, for example, Townsend 1956), and we consider here the additional effects introduced by the presence of density gradients. Turbulent flows in which gravity plays an essential role in driving the mean motion (e.g. turbulent gravity currents) will be treated separately in chapter 6.

These flows will be discussed against the background of the classification introduced in §4.3.1, emphasizing the turbulent features, but also referring to the wave aspects when necessary. We consider first a shear flow near a horizontal boundary, and the effect of a vertical density gradient on the velocity and density profiles, and on the rates of transport. Next we discuss the few theoretical results which are available to describe ‘boundary’ and ‘interior’ turbulence in stratified shear flows. Finally, we present and discuss laboratory and larger scale observations which can be used to test these ideas.

Velocity and density profiles near a horizontal boundary

The most important example of a shear flow near a boundary is the wind near the ground, in a shallow enough layer for the effects of the earth's rotation to be ignored.

The problems which arise when the buoyancy sources are distributed over a surface, rather than being localized, are rather different from those treated in the previous chapter. Often the temperature difference between the fluid and the surface or between two surfaces is now specified, and the heat flux is not given but is a derived quantity which it is desired to predict. Attention is shifted from the individual elements to the properties of the mean flow and buoyancy fields, since it is not clear at the outset what form the buoyant motions will take. As we shall see later in this chapter, however, the details of the observed flows can only be properly understood if we again consider the buoyant elements explicitly and try to understand how their form and scale is determined by the boundary conditions (which in many cases are nominally uniform over the solid surface).

There is also the new feature of stability to be discussed here. A point source of buoyancy in an otherwise unconstrained perfect fluid always gives rise to convective motions above or below it, with an associated production of vorticity, and the same is true of a horizontal interface with the lighter fluid below (see §1.2). In a real fluid, however, the criterion based on (1.2.1) is no longer sufficient to determine whether a horizontal layer of fluid is gravitationally unstable or not; even when the fluid is lighter below, molecular viscosity and diffusion can act to damp out small disturbances, so that no overturning occurs.

Scope and purpose of the book

This book grew out of an undergraduate course in the University of Manchester, in which the author attempted to expound the most useful facts about Fourier series and integrals. It seemed to him on planning the course that a satisfactory account must make use of functions like the delta function of Dirac which are outside the usual scope of function theory. Now, Laurent Schwartz in his Théorie des distributions* has evolved a rigorous theory of these, while Professor Temple has given a version of the theory (Generalised functions) which appears to be more readily intelligible to students. With some slight further simplifications the author found that the theory of generalised functions was accessible to undergraduates in their final year, and that it greatly curtails the labour of understanding Fourier transforms, as well as making available a technique for their asymptotic estimation which seems superior to previous techniques. This is an approach in which the theory of Fourier series appears as a special case, the Fourier transform of a periodic function being a ‘row of delta functions’.

The book which grew out of the course therefore covers not only the principal results concerning Fourier transforms and Fourier series, but also serves as an introduction to the theory of generalised functions, whose general properties as well as those useful in Fourier analysis are derived, simply but without any departure from rigorous standards of mathematical proof.