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.