Strong interactions, local instabilities and turbulence: a postscript

Short waves and long waves

Small-wavelength disturbances may ride on large-amplitude long gravity waves. The orbital velocities of fluid particles due to the long waves provide a variable surface current through which the short waves propagate. When this current is comparable with the propagation velocity of the short waves relative to the long ones, their interaction is no longer weak. Nevertheless, the characteristics of the short waves may still be described, at least in part, by Whitham's theory of slowly-varying wave-trains in an inhomogeneous medium (see §11.3). Phillips (1981b) deduced from wave action conservation that capillary waves are likely to be ‘blocked’ by steep gravity waves. In much the same way, Gargett & Hughes (1972) earlier showed that short gravity waves may be trapped by long internal waves, so leading to caustic formation and local wave breaking. Untrapped modes also undergo amplitude modulations by the straining of the dominant wave field.

Computations of Longuet-Higgins (1978a, b) and McLean et al. (1981), already described in §22.2, display generation of short waves by high-order instability of steep gravity waves. In addition, finite-amplitude wave-trains necessarily contain bound harmonics, which travel with the fundamental Fourier component. Weakly-nonlinear interaction of neighbouring frequency components may also give rise to phase-locking of modes.

For these reasons, and doubtless others besides, measurements of the phase speeds of Fourier components of wave fields often reveal significant departures from the linear dispersion relation, even after allowance is made for wave-induced mean currents.

Introduction to nonlinear theory

Introductory remarks

Nonlinear theories are of three more or less distinct kinds. In one, properties of arbitrarily-large disturbances are deduced directly from the full Navier–Stokes equations. Consideration of integral inequalities yields bounds on flow quantities, such as the energy of disturbances, which give stability criteria in the form of necessary or sufficient conditions for growth or decay with time. An admirable account of such theories is given by Joseph (1976). They have the advantage of supplying mathematically rigorous results while incorporating very few assumptions regarding the size or nature of the disturbances. Sometimes, these criteria correspond quite closely to observed stability boundaries. The bounds for onset of thermal (Bénard) instability and centrifugal (Rayleigh–Taylor) instability in concentric rotating cylinders are particularly notable successes. Often, however, the bounds are rather weak: this is especially so for shear-flow instabilities, where local details of the flow typically play an important rôle which cannot be (or, at least, has not been) incorporated into the global theory.

The second class of theories relies on the idea that linearized equations provide a satisfactory first approximation for those finite-amplitude disturbances which are, in some sense, sufficiently small. Successive approximations may then be developed by expansion in ascending powers of a characteristic dimensionless wave amplitude. These are known as weakly nonlinear theories, and they have proved successful in revealing many important physical processes.

Introduction

Waves occur throughout Nature in an astonishing diversity of physical, chemical and biological systems. During the late nineteenth and the early twentieth century, the linear theory of wave motion was developed to a high degree of sophistication, particularly in acoustics, elasticity and hydrodynamics. Much of this ‘classical’ theory is expounded in the famous treatises of Rayleigh (1896), Love (1927) and Lamb (1932).

The classical theory concerns situations which, under suitable simplifying assumptions, reduce to linear partial differential equations, usually the wave equation or Laplace's equation, together with linear boundary conditions. Then, the principle of superposition of solutions permits fruitful employment of Fourier-series and integral-transform techniques; also, for Laplace's equation, the added power of complex-variable methods is available.

Since the governing equations and boundary conditions of mechanical systems are rarely strictly linear and those of fluid mechanics and elasticity almost never so, the linearized approximation restricts attention to sufficiently small displacements from some known state of equilibrium or steady motion. Precisely how small these displacements must be depends on circumstances. Gravity waves in deep water need only have wave-slopes small compared with unity; but shallow-water waves and waves in shear flows must meet other, more stringent, requirements. Violation of these requirements forces abandonment of the powerful and attractive mathematical machinery of linear analysis, which has reaped such rich harvests. Yet, even during the nineteenth century, considerable progress was made in understanding aspects of weakly-nonlinear wave propagation, the most notable theoretical accomplishments being those of Rayleigh in acoustics and Stokes for water waves.

When, over four years ago, I began writing on nonlinear wave interactions and stability, I envisaged a work encompassing a wider variety of physical systems than those treated here. Many ideas and phenomena recur in such apparently diverse fields as rigid-body and fluid mechanics, plasma physics, optics and population dynamics. But it soon became plain that full justice could not be done to all these areas – certainly by me and perhaps by anyone.

Accordingly, I chose to restrict attention to incompressible fluid mechanics, the field that I know best; but I hope that this work will be of interest to those in other disciplines, where similar mathematical problems and analogous physical processes arise.

I owe thanks to many. Philip Drazin and Michael McIntyre showed me partial drafts of their own monographs prior to publication, so enabling me to avoid undue overlap with their work. My colleague Alan Cairns has instructed me in related matters in plasma physics, which have influenced my views. General advice and encouragement were gratefully received from Brooke Benjamin and the series Editor, George Batchelor.

Various people kindly supplied photographs and drawings and freely gave permission to use their work: all are acknowledged in the text. Other illustrations were prepared by Mr Peter Adamson and colleagues of St Andrews University Photographic Unit and by Mr Robin Gibb, University Cartographer. The bulk of the typing, from pencil manuscript of dubious legibility, was impeccably carried out by Miss Sheila Wilson, with assistance from Miss Pat Dunne.

The purpose of this book is to describe in detail the theory known as classical dynamics. This is a theory which is very well known and has a large number of important practical applications. It is regarded as one of the most basic scientific theories, with many other theories being direct developments or extensions of it. However in spite of its acknowledged importance, it is not always as well understood as it ought to be.

In the English speaking world at least, the emphasis in the teaching of classical dynamics is largely on its relevance to idealised applications. Thus most traditional mechanics textbooks are example orientated, and students are required to work through large numbers of artificial exercises. Other textbooks are available which emphasise the mathematical techniques that are appropriate in the applications of the subject. These approaches are of course of great importance. An understanding of this subject can only be achieved by working through numerous examples in which the theory is applied to practical problems. However, I feel that a discussion of the fundamental concepts of the theory and its basic structure has largely been neglected.

Classical dynamics is far more than an efficient tool for the solution of physical and engineering problems. It is a fascinating scientific theory in its own right. Its basic concepts of space, time and motion have fascinated some of the greatest intellects over many centuries.

Up until this point the theory of classical dynamics has been described more or less in the form in which it had been developed by the end of the eighteenth century. The analytic approach of Lagrangian dynamics had been developed, as well as the vectorial approach of Newtonian mechanics. In practice, either of these approaches may be applied to a large class of practical problems, but ultimately they both yield sets of second-order differential equations which it is necessary to solve. Sometimes it is possible to obtain some simple first integrals of these equations, but in many situations complete integrals in analytic form are difficult or impossible to obtain. Thus, although these approaches enable the motion of many systems to be described in terms of equations of motion, certain mathematical problems prevent their complete analysis.

Because of this situation the development of the theory of classical dynamics has always been associated with the development of appropriate mathematical techniques. In particular, some of the techniques developed have enabled the equations of motion to be formulated in new ways, and thus they have contributed to a deeper understanding of the theory itself. Such results must therefore be included in a discussion of the foundations of the subject.

It is the purpose of this chapter to describe some of the alternative approaches to the subject which were developed in the nineteenth century. Of particular interest are the Hamiltonian approach and the development of the Hamilton–Jacobi theory.

The subject of classical dynamics deals with the motion of bodies in space. Its aim is to provide models which are capable of accurately describing the way in which bodies change their position in space as time progresses. Thus the starting point for a study of dynamics must be a set of initial assumptions about the nature of space and time. These are primitive or foundational concepts which are necessary for the development of any theory of dynamics. Newton himself found it necessary to discuss these concepts in the first chapter of his Principia, although it is not necessary here to introduce his concepts of an absolute space and absolute time.

Writing at the end of the twentieth century, it is possible to assess the theory of classical dynamics in the light of the modern theories of relativity and quantum mechanics. The classical theory has ultimately been refuted, and the new theories indicate those aspects that require modification. According to the theories of relativity some of the weakest points of the classical theory are in fact its assumptions about space and time. It is therefore appropriate here to clarify the way in which these concepts are used in classical dynamics, and to contrast this with their use in the theories of relativity, before going on to describe the theory itself. We shall also take the opportunity in this chapter of introducing the vector notation which is so useful in classical dynamics as a consequence of its initial assumptions about space and time.