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.

The concept of a rigid body

Most simple elements and compounds exist in three possible states, gaseous, liquid and solid. According to the methods described in the previous chapter, the motion of bodies composed of matter in any of these states can be described in terms of a large number of small particles. The motions of gases and liquids have been briefly discussed at the end of the last chapter, and we turn now to consider the motion of solid bodies.

In general, solid bodies have a very complicated structure, and in their motion some parts move relative to other parts. However, in many applications, particularly in mechanical problems, the component parts of a body move in an approximately rigid way. Thus it is convenient to introduce the concept of a rigid body as a theoretical idealisation. This may be defined as follows.

Definition 8.1.A body is said to be rigid if the distance between each of its constituent points remains constant, irrespective of the motion of the body as a whole or the forces that act upon it.

Of course no physical body is exactly rigid. Even objects made of solid steel bend or deform slightly when acted on by external forces. But when considering the motion of solid objects, it is often convenient to regard them as being perfectly rigid, at least as a first approximation. Thus, in all applications of the study of rigid body dynamics, there must always be an initial simplifying assumption that the body under consideration should be regarded as being perfectly rigid.

It has been emphasised in previous chapters that position, and hence also velocity and acceleration, are relative concepts. Thus when stating or defining certain positions some frame of reference is always, at least implicitly, assumed. For example, a child may be told to sit still in a car, even though, together with the car, he may be moving at a great speed. In this case it is obviously implied that the child is required not to move relative to the car. Alternatively, the positions of the pieces in a game of chess are usually stated relative to the chessboard, and are therefore unambiguously defined, even if the board were to be moved from one room to another. In the same way, the items of furniture in a room can be described as being located in fixed positions relative to the floor and walls, in spite of the fact that they have a large velocity due to the rotation of the earth and its orbital velocity about the sun.

Clearly then, positions and dynamical properties such as velocities and accelerations can only be stated or determined relative to an assumed frame of reference. When an individual is required to state a position, he is free to choose an arbitrary frame of reference for convenience. However, when he attempts to measure a dynamical property such as the velocity or acceleration of a particle, in practice he would usually make the measurements relative to himself or to the instruments he is using.

Systems of several particles as a model

Up to this point we have been considering mainly the motion of a single particle relative to a frame of reference that has been somehow determined. However, some concepts associated with the possible interaction of one particle with another have been introduced. We are now in a position to consider systems of several particles relative to some frame of reference. The particles may be considered to be interacting with each other and each particle may also be acted on by other external forces. Each particle may still be considered individually, but, in addition, certain general properties of the motion of the whole system may also be considered, without necessarily determining the motion of every individual particle.

Such an approach may be used, for example, to analyse the motion of stars in a galaxy. Each star may be represented as a particle since a galactic scale of distance is being used, and the motion of each star relative to an inertial frame is affected by the gravitational forces induced by all the other stars. In such a case the dominant forces are the mutual interactions between the particles.

Of course, it is possible also to consider just a cluster of stars. In this case the mutual interaction between the individual stars is important, but so also is the effect of the gravitational field caused by the other stars in the galaxy which are not part of the particular cluster under consideration.

The approach to classical dynamics proposed so far has been a more or less direct application of Newton's laws of motion. In such an approach the motion of a body or particle can be predicted on the assumption of a given set of external forces which act on it, simply by integrating the equations of motion. However, for complex systems of particles or rigid bodies, it is not always easy to determine appropriate equations for each component, let alone perform the required integration. In practice, using this approach, it is found that each individual type of problem requires its own particular insights and techniques.

In this chapter the Lagrangian approach to classical dynamics is developed. This approach is based upon two scalar properties of a system, its kinetic energy and work. It leads to a powerful and general method for the solution of dynamical problems which is found to be particularly useful in the analysis of mechanical systems which contain a number of rigid bodies that are connected in some way, but which may move relative to each other. In the traditional approach each component would have to be treated separately in terms of the forces acting on it. However, the Lagrangian approach enables such a system to be considered as a whole.

The aim here is to develop a general approach which may be applied to any dynamical system. It is found that the equations of motion can be presented in a standard and convenient form.

As with other scientific theories, theories of dynamics aim to contribute to an understanding and explanation of phenomena that occur in the real world. In the theories of dynamics, it is the general subject of motion that comes under investigation. The aim is to describe how objects move, and to suggest physical reasons as to why they move. In particular they should provide methods for analysing or predicting the motion of specific bodies, and also possibly suggest techniques for controlling the motion of some objects.

The theory known as classical or Newtonian dynamics, the subject of this book, is one such theory. Before describing it in detail, however, it is convenient to describe in general terms the way in which the theory is used. This is the purpose of this first chapter.

The technique of mathematical modelling

The basic method by which any theory of dynamics is applied can be described in terms of three distinct phases. The first phase consists essentially of constructing a simplified model. This is an idealised imaginary representation of some physical situation in the real world. In phase two this theoretical representation is analysed mathematically and its consequences are deduced on the basis of some assumed theory. Finally, in phase three, the theoretical results of phase two are interpreted and compared with observations of the real physical situation. This whole process can of course be repeated many times using different initial representations or different basic theories.

The classical theory of dynamics is that scientific theory which was developed through the seventeenth, eighteenth and nineteenth centuries to describe the motion of physical bodies. The theory was originally developed in two distinct parts, one dealing with terrestrial bodies such as a projectile or a pendulum, and the other dealing with celestial bodies, in particular the planets. These two apparently distinct subjects were first brought together by Isaac Newton whose book, known as the Principia, is rightly acknowledged as the first complete formulation of the theory that is now referred to as classical dynamics.

Since those early days the theory has been thoroughly developed and extended so that it can now be applied to a very wide range of physical situations. Some of the early concepts have been clarified, and others have been added. But, although the theory has now been formulated in many different ways, it is still essentially the same as that originally proposed by Newton. The most significant advances have in fact been those associated with the development of new mathematical techniques, which have subsequently enabled the theory to be applied to situations which previously had proved too difficult to analyse.

Now, as is well known, a revolution occurred in scientific thinking in the first part of the present century. Classical theories were disproved, and exciting new theories were put forward. The theory of relativity was suggested in order to explain the results obtained when the speed of light was measured.