Introduction

When two engineering surfaces are loaded together there will always be some distortion of each of them. These deformations may be purely elastic or may involve some additional plastic, and so permanent, changes in shape. Such deflections and modifications in the surface profiles of the components can be viewed at two different scales. For example, consider the contact between a heavily loaded roller and the inner and outer races in a rolling-element bearing. In examining the degree of flattening of the roller we could choose to express the deflections as a proportion of their radii, that is, to view the distortions on a relatively macroscopic scale. On the other hand, as we have seen in Chapter 2, at the microscale no real surface, such as those of the roller or the race, can be truly smooth, and so it follows that when these two solid bodies are pushed into contact they will touch initially at a discrete number of points or asperities. The sum of the areas of all these contact spots, the ‘true’ area of contact, will be a relatively small proportion of the ‘nominal’ or geometric contact area–perhaps as little as only a few per cent of it. Some deformation of the material occurs on a very small scale at, or very close to, these areas of true contact. It is within these regions that the stresses are generated whose total effect is just to balance the applied load.

Introduction

A hydrostatic bearing is one in which the loaded surfaces are separated by a fluid film which is forced between them by an externally generated pressure. Formation of the film, and so successful operation of the bearing, requires the supply pump to operate continuously, but it does not depend on the relative motion of the surfaces (hence the term ‘hydrostatic’). Such bearings have a great attraction to engineers; machine elements supported in this way move with incomparable smoothness and the only restriction to motion arises from the small viscous losses in the fluid. A mass supported on a hydrostatic bearing will glide silently down the slightest incline.

The essential features of a typical hydrostatic single-pad thrust bearing are shown in Fig. 6.1 (a). The bearing is supplied with fluid under pressure ps which, before entering the central pocket or recess, passes through some form of restrictor or compensator in which its pressure is dropped to some lower value pr. The fluid then passes out of the bearing through the narrow gap, shown of thickness h, between the bearing land and the opposing bearing surface or slider which is also often known as the bearing runner. The depth of the pocket is very much greater than the gap h. The restrictor is an essential feature of the bearing since it allows the pocket pressure pr to be different from the supply pressure; this difference, between pr and ps, depends on the load applied W.

In this chapter the theory of the generation of oceanwaves by wind will be developed resulting in an expression for the wind-input source function Sin of the action balance equation. As will be seen from the subsequent discussion, this problem has led to many debates and much controversy. There may be several reasons for this. On the one hand, from the theoretical point of view it should be realized that one is dealing with an extremely difficult problem because it involves the modelling of a turbulent airflow over a surface that varies in space and time. Although there has been much progress in understanding turbulence over a flat plate in steady-state conditions, modelling attempts of turbulent flow over (nonlinear) gravity waves are only beginning and, as will be seen, there is still a considerable uncertainty regarding the validity of these models.

On the other hand, from an experimental point of view it should be pointed out that it is not an easy task to measure growth rates of waves by wind. First of all, one cannot simply measure growth rates by studying time series of the surface elevation since the time evolution of ocean waves is determined by a number of processes such as wind input, nonlinear interactions and dissipation. In order to measure the growth of waves by wind one therefore has to make certain assumptions regarding the process that causes wave growth.

In this chapter we study the effects of nonlinearity on the evolution of deep-water gravity waves. Eventually this will result in an expression for the source function for nonlinear wave–wave interactions and dissipation (presumably by white capping), which completes the description of the energy balance equation.

We shall begin with a fairly extensive discussion of nonlinear wave–wave interactions, followed by a brief treatment of dissipation of wave energy by white capping. The latter treatment is only very schematic, however, because this process involves steep waves which only occur sporadically. At best the choice of the white-capping source function can be made plausible. It turns out that the overall dissipation rate is in agreement with observed dissipation rates. Much more is known regarding nonlinear wave–wave interactions. An important reason for this is that ocean waves may be regarded most of the time as weakly nonlinear, dispersive waves. Because of this there is a small parameter present which permits the study of the effect of nonlinearity on wave evolution by means of a perturbation expansion with starting point linear, freely propagating ocean waves. In addition, it should be pointed out that the subject of nonlinear ocean waves has conceptually much in common with nonlinear wave phenomena arising in diverse fields such as optics and plasma physics. In particular, since the beginning of the 1960s many people have contributed to a better understanding of the properties of nonlinear waves, and because of the common denominater we have seen relatively rapid progress in the field of nonlinear ocean waves.

The subject of ocean waves and their generation by wind has fascinated me greatly since I started towork in the Department of Oceanography at the Royal Netherlands Meteorological Institute (KNMI) at the end of 1979. The wind-induced growth of water waves on a pond or a canal is a daily experience for those who live in the lowlands, yet it appeared that this process was hardly understood. Gerbrand Komen, who arrived 2 years earlier at KNMI and who introduced me to this field, pointed out that the most prominent theory explaining wave growth by wind was the Miles (1957) theory which relied on a resonant interaction between wind and waves. Since I did my Ph.D. in plasma physics, I noticed immediately an analogy with the problem of the interaction of plasma waves and electrons; this problem has been studied extensively both experimentally and theoretically. The plasma waves problem has its own history. It was Landau (1946), who discovered that depending on the slope of the particle distribution function at the location where the phase velocity of the plasma wave equals the particle velocity, the plasma wave would either grow or damp. Because of momentum and energy conservation this would result in a modification of the particle-velocity distribution. For a spectrum of growing plasma waves with random phase, this problem was addressed in the beginning of the 1960s by Vedenov et al. (1961) and by Drummond and Pines (1962).

This is a book about ocean waves, their evolution and their interaction with the environment. It presents a summary and unification of my knowledge of wave growth, nonlinear interactions and dissipation of surface gravity waves, and this knowledge is applied to the problem of the two-way interaction of wind and waves, with consequences for atmosphere and ocean circulation.

The material of this book is, apart from my own contributions, based on a number of sources, ranging from the works of Whitham and Phillips to the most recent authoritative overview in the field of ocean waves, namely the work written by the WAM group, Dynamics and Modelling of Ocean Waves. Nevertheless, the present book is limited in its scope because it will hardly address interesting issues such as the assimilation of observations, the interpretation of satellite measurements from, for example, the radar altimeter, the scatterometer and the synthetic-aperture radar, nor will it address shallow-water effects. These are important issues but I felt that the reader would be served more adequately by concentrating on a limited number of subjects, emphasizing the role of ocean waves in practical applications such as wave forecasting and illuminating their role in the air–sea momentum exchange.

I started working on this book some 8 years ago. It would never have been finished had it not been for the continuous support of my wife Danielle Mérelle.

In this chapter we shall try to derive, from first principles, the basic evolution equation for ocean-wave modelling which has become known as the energy balance equation. The starting point is the Navier–Stokes equations for air and water. The problem of wind-generated ocean waves is, however, a formidable one, and several approximations and assumptions are required to arrive at the desired result. Fortunately, there are two small parameters in the problem, namely the steepness of the waves and the ratio of air density to water density. As a result of the relatively small air density, the momentum and energy transfer from air to water is relatively small so that, because of wind input, it will take many wave periods to have an appreciable change of wave energy. In addition, the steepness of the waves is expected to be relatively small. In fact, the assumption of small wave steepness may be justified a posteriori. Hence, because of these two small parameters one may distinguish two scales in the time–space domain, namely a short scale related to the period and wavelength of the ocean waves and a much longer time and length scale related to changes due to small effects of nonlinearity and the wind-induced growth of waves.

Using perturbation methods, an approximate evolution equation for the amplitude and the phase of the deep-water gravity waves may be obtained.

In this book we have given an overview of the role that ocean waves play in the problem of the interaction of atmosphere and ocean. However, in order to appreciate this role we had to elaborate on how ocean waves evolve in space and time. It was found that ocean waves evolve according to the well-known energy balance equation which states that the wave spectrum changes due to advection with the group velocity, and due to physical processes such as generation by wind, nonlinear transfer by four-wave interactions and dissipation by, for example, white capping. A detailed exposition of the derivation of the physical source functions was given, followed by a study of the impact of ocean waves on the atmospheric circulation and one aspect of the ocean circulation, namely storm surges. It was also pointed out that the study of the effects of ocean waves on ocean circulation is only beginning, but that promising improvements on its wind-driven component are expected in the near future.

This book was concluded by an extensive discussion of the verification of the ECMWF forecast of wave parameters, such as significant wave height. The impression from this verification study is that the quality of the ECMWF wave analysis and wave forecast is high, certainly if the results are put in a historical perspective.

This chapter is devoted to a synthesis of what we have previously learned about the physics of wind–wave interaction, culminating in a numerical wave prediction system. After a brief discussion of the numerics of such a model, we illustrate the combined effects of wind input, nonlinear interactions and dissipation on the simple case of the growth of surface gravity waves by wind for an infinite ocean. The resulting growth laws for parameters such as wave height, peak frequency and the Phillips parameter are compared with empirical growth laws. Also, the simulated wave-age dependence of ocean roughness is compared with a number of empirical fits. In addition, we discuss effects of gustiness on wave evolution and roughness of the sea surface which gives an indication of the well-known sensitive dependence of wave results on the forcing wind field.

The resulting wave prediction system turns out to be a promising tool for forecasting purposes and in the remainder of this chapter we discuss a number of applications of the wave forecasting system as it has been implemented at the European Centre for Medium-Range Weather Forecasts (ECMWF). Historically, ECMWF played an important role in the development of the third-generation wave prediction system, called WAveModel (WAM). They provided the necessary infrastructure (such as supercomputers and data-handling facilities), high-quality surface wind fields and support.

Introduction

In this chapter we wish to consider the stability of steady two-dimensional or axisymmetric flows with parallel streamlines. Flows of this type were first studied experimentally by Reynolds (1883), who observed that instability could occur in quite different ways depending on the form of the basic velocity distribution. Thus, when the velocity profile is of the form shown in Fig. 4.1(a) he observed that ‘eddies showed themselves reluctantly and irregularly’ whereas when the profile is as shown in Fig. 4.1(b) the ‘eddies appeared in the middle regularly and readily’. From these observations he was led to consider the role of viscosity in flows of this type. By comparing the flow of a viscous fluid with that of an inviscid fluid, both flows being assumed to have the same basic velocity distribution, he was led to formulate two fundamental hypotheses which can be stated as follows: