The linear wave theory described in Chapters 2 and 3 is useful for interpreting small-amplitude wave phenomena occurring near oceanic bottom features or near density fronts. The nonlinear approach considered in Chapter 4 shows how large the difference can be between the linear and nonlinear waves when the external generating forcing is sufficiently strong. Several examples of the generating processes were studied for both subcritical (Fr < 1) and supercritical (Fr > 1) cases for typical oceanic conditions. However, when considering the large variety of oceanic background stratifications, bottom profiles, or values of external forcings, it is often not possible to apply the results described in Chapter 4 directly to the specific realistic conditions encountered in the field.

In this chapter, we concentrate on phenomena that are new when compared with those described in Chapter 4. These phenomena involve the influence of the vertical and horizontal fluid stratification on the nonlinear mechanism of internal wave generation, as well as the effects related to the rotation of the Earth (especially at high-latitude seas). Nonlinear baroclinic tides over steep bottom topography are also considered. Note that all topics in Chapter 4 were motivated by observational results obtained during field measurements at different sites of the World Ocean. This will also be the approach taken here; thus, all theoretical reasonings will be accompanied by illustrations and interpretations of real in situ data.

One type of interesting phenomenon which makes the dynamics of the World Ocean rather complicated and surprisingly manifold is a “hydrological front.” The term front characterizes regions in which the hydrodynamic fields possess large horizontal gradients. These manifest themselves in the thermohaline characteristics. Fronts in the ocean or in the atmosphere can be defined as regions where background long-term averaged properties of the medium change substantially over a relatively short distance. Such sharp boundaries usually separate two adjacent water masses with different properties. In the World Ocean, however, hydrological fronts can also reach more than 100 km width [233].

We emphasize that a strict standardized definition of the term “oceanic front” does not exist. As a quantitative measure for the systematization of oceanic fronts, the change of any hydrological property across a localized area – say temperature, or salinity, or both – can be used. If such gradients are an order of magnitude larger than similar changes adjacent to this area, then the presence of a front can be identified.

From the viewpoint of water dynamics, the hydrological fronts in the World Ocean can be classified by their influence on the distribution of the density field.

The scientific literature on baroclinic tides in the ocean is as abundant as is the general mathematical theory of internal gravity waves. The majority of research papers on this topic written since the late 1970s deal mainly with the attempts to provide a theoretical description of the vast number of experimental data obtained on the dynamics of baroclinic tides. Various analytical and semianalytical models were developed for the case of infinitesimal waves when the external tidal forcing is negligible. However, observations show that strong nonlinear baroclinic tides are the more common phenomena; they demand not only theoretical but also numerical methods to describe the wave processes adequately.

The idea of the book is to present the theoretical basis of baroclinic tides at all stages of their genesis and evolution (generation, transformation, dissipation), and to give, by comparison with experiments, advice and help regarding the theoretical interpretation and prognosis of baroclinic tides. Such an exhaustive and complete theoretical review does not exist at present, nor is there a scientific book which describes this phenomenon multilaterally. So we are intending to fill this gap.

A second reason for writing a book on this topic is to gather in one place our experience, which extends over more than 20 years, in investigating internal waves in general and baroclinic tides in particular.

The basic relations of the linear theory presented in Chapters 2 and 3, and the results obtained with them, are valid within the framework of the linear approximation; i.e. the assumption that the amplitudes of the internal waves are infinitely small is assumed to be satisfied with sufficient accuracy. In such circumstances, the role of the nonlinear terms of the governing equations describing the dynamics of waves could be shown to be negligible, and the terms marked by boxes in system (1.39) could be omitted. This assumption is commonly used at low intensity of the external tidal forcing. However, such conditions are not valid for the whole area of the World Ocean. In many places they are not satisfied, because of either the strength of the forcing or the steepness of the topography, and numerous in situ data reveal a nonlinear wave response.

Obviously, a mathematical model which adequately describes nonlinear wave dynamics must be based on the complete system of nonlinear equations; it includes both the full advective nonlinearities and the nonhydrostatic law for the pressure. The model must also incorporate all those factors which control the structure of the wave fields: variable bottom topography, fluid stratification, external forcing, parameterization of background mixing, etc.

The World Ocean, considered as an active dynamical system, is in permanent motion. Most of its manifestations can be related to wave phenomena. Besides the well known surface waves, there are also waves of other nature; among these, internal gravity waves are particularly important. They exist due to the presence of vertical fluid stratification, they are permanently generated, and they evolve and are destroyed again in the deep ocean. The amplitudes of internal waves are usually much larger than those of surface waves, due to the weak returning force, and their amplitudes can sometimes reach values of 100 m and more (see refs. [5], [23], [120], [128], [180], [192], and [193]).

Numerous in situ measurements, carried out in all regions of the World Ocean (see, for instance, refs. [63], [119], [123], and [157]), have shown that internal gravity waves exist wherever a stable vertical stratification of a fluid is observed. They were discovered more than 100 years ago, and were understood by the scientists of the day to be a disappointing obstacle disturbing the “correct” structure and dynamics of the oceanic water masses. More than half a century passed before the importance of internal waves to the global dynamics of the ocean was realized.

This chapter is devoted to a study of the dynamics of infinitesimal waves in a continuously stratified ocean of variable depth. Small-amplitude baroclinic tides are usually generated when the intensity of the barotropic tidal forcing is small. As mentioned earlier, depending on the wave amplitude, the analysis can be carried out by means of either the linear or nonlinear theory. A quantitative estimation of the efficiency of the tidal generation of internal waves and the discussion of the necessity to use the full nonlinear system of equations to study baroclinic tides is given in refs. [73] and [75]. We return to this latter point in detail in Chapter 4. Here we simply assume that the amplitudes of the considered waves are so small that with sufficient accuracy the advective terms in the governing system (1.39) can be neglected.

Historically, the first linear models of baroclinic tides were developed for a two-layer ocean. In the models in refs. [199] and [265], the internal tide was generated as a result of the interaction of the barotropic tidal wave with a Heaviside-like bottom step, an obstacle, which approximates the transition zone between the deep and shallow parts of the ocean. In these works, reflection of the waves from the coastal line was used as a boundary condition.

In the previous chapters we have only considered the two-dimensional nature of baroclinic tides. This simplification of the three-dimensional behavior is a valid approximation in shelf zones, at least as a tendency, due to the oblongness of the continental margins. It is, however, often violated when one tries to study the wave-generation process near three-dimensional bottom features like oceanic banks or abrupt changes of the shelf break topography in the along shore direction. Such places constitute a remarkable sink of barotropic tidal energy into baroclinic wave components if only a substantial along slope tidal flux interacts with along shore bottom variations. Under such conditions, the correct prediction of the total scattering of the barotropic tidal energy into internal tides is impossible without using three-dimensional global tidal models, which can predict the dynamics in those shelf areas where such a remarkable along slope forcing of internal tides may exist. Figure 7.1 illustrates this point. The broad gray bands show regions where the tide, derived from the barotropic model of Schwiderski [214], propagates along the coast as a Kelvin-type wave, and where the amplitude of its elevation is larger than 0.4 m. If the depth of the ocean near shore is estimated typically as 2000 m, then the value of the maximum barotropic tidal flux along the slope is of the order of 60 m2 s-1.

In this chapter we consider the evolutionary stage of baroclinic tides, i.e. the behavior of the tidally generated internal waves beyond the source of generation. During propagation, the long baroclinic tidal waves which are radiated from this area of generation – usually bottom features – are subjected to nonlinear effects, as shown in Chapter 4. If the nonlinearity is sufficiently strong, these waves are usually transformed into a sequence of solitary internal waves or wave trains. So, we shall now concentrate on the dynamic structure of solitary internal waves. First, we will give a brief summary of a number of analytical theories and consider the stationary solutions of weakly nonlinear equations. Then, we will move on to consider strong solitary internal waves; even though any analytical theory fails to describe their structure and dynamics, nevertheless they are a common feature of the real ocean. Using several numerical procedures, the governing equations can be handled and a physical understanding can be extracted. Using this approach, we will study the spatial–temporal structure of strong waves and indicate their differences from strict analytical solutions of equations describing weaker nonlinear interactions. Finally, we will present the wave transformation appropriate over variable bottom topography that includes strong effects such as wave overturning and breaking.

Background

In 1999, a major programme on turbulence was held at the Isaac Newton Institute (INI) at Cambridge, England, which was aimed at taking an overview of the current situation on turbulent flows with particular reference to the prediction of such flows in engineering systems. Though the programme spanned the range from the very fundamental to the applied, a very important feature was the involvement and support (through the UK Royal Academy of Engineering) of key players from industry. This volume, which has evolved from the INI programme, aims to address the needs of people in industry and academia who carry out calculations on turbulent systems.

It should be recognised that the prediction of turbulent flows is now of paramount importance in the development of complex engineering systems involving flow, heat and mass transfer and chemical reactions (including combustion). Whereas, in the past, the developer had to rely on experimental studies, based usually on small scale model systems, more and more emphasis is being placed nowadays on the use of computation, often through the use of commercial computational fluid dynamics (CFD) codes. Superficially, the use of such computational methods seems ideal; they allow painless extension to large scale and can often give information on fine details of the flow that are not economically accessible to experimental measurement. Furthermore, the results can be presented in an easily accessible and attractive form using the sophisticated computer graphics now generally available.

Summary

This chapter begins with a review of the principles underlying general purpose turbulence models and the assumptions and procedures involved in applying them to calculate the kind of complex flows that are analysed in practical engineering and environmental problems. Secondly we develop, from considerations of basic mechanisms of turbulence and the different types of statistical turbulence model, a new guideline ‘map’ based on characteristic statistical parameters, which can be derived from standard models. This indicates in principle which types of turbulent flow can and cannot be approximately calculated with the current generation of ‘CFD’, one-point turbulence models, including those using k–ε and second order closure equations. No attempt is made to identify any one optimum model scheme. Thirdly, the proposed guidelines for the likely accuracy of turbulent modelling are tested by comparing them with the results of previous test-case studies for a range of complex turbulent flows, where standard models fail or need special adaptation. These include thermal convection, free stream turbulence, aeronautical flows and flows round bluff bodies. The relative merits of advanced models (e.g. involving two-point statistics) and numerical simulations are also discussed, but the CFD practitioner should note that the emphasis here is on why current models will not work in all circumstances. The technical level of this chapter is most suitable for readers with some formal training in fluid dynamics. These general guidelines are complementary to user guidelines for computational fluid dynamics codes.

Abstract

Recent research is making progress in framing more precisely the basic dynamical and statistical questions about turbulence and in answering them. It is helping both to define the likely limits to current methods for modelling industrial and environmental turbulent flows, and to suggest new approaches to overcome these limitations. This chapter had its basis in the new results that emerged from more than 300 presentations during the programme held in 1999 at the Isaac Newton Institute, Cambridge, UK, and on research reported elsewhere. The objective of including this material (which is a revised form of an article which appeared in the Journal of Fluid Mechanics – Hunt et al., 2001) in the present volume is to give a background to the current state of the art. The emphasis is on the physics of turbulence and on how this relates to modelling. A general conclusion is that, although turbulence is not a universal state of nature, there are certain statistical measures and kinematic features of the small-scale flow field that occur in most turbulent flows, while the large-scale eddy motions have qualitative similarities within particular types of turbulence defined by the mean flow, initial or boundary conditions, and in some cases, the range of Reynolds numbers involved. The forced transition to turbulence of laminar flows caused by strong external disturbances was shown to be highly dependent on their amplitude, location, and the type of flow.

Introduction

Over 80% of the world's energy is generated by the combustion of hydrocarbon fuels, and this is likely to remain the case for the foreseeable future. In addition to the release of heat, this combustion is accompanied by the emission, in the exhaust stream, of combustion generated pollutants such as carbon monoxide, unburnt hydrocarbons and oxides of nitrogen, NOx. The former two quantities arise as a result of incomplete combustion, whereas NOx is formed from the reaction of nitrogen present in the air or fuel with oxygen, usually at high temperatures. An unavoidable outcome of the burning of hydrocarbon fuels is the formation of carbon dioxide, CO2, which is a ‘greenhouse’ gas that may contribute to global warming. While the amount of carbon dioxide generated depends on the fuel burnt, any improvements which can be achieved to combustion efficiency will clearly contribute to an overall reduction in the emissions of CO2. Because of the growing need to reduce the emissions of combustion generated pollutants and improve combustion efficiencies, there is increased interest in accurate methods for predicting the properties of combustion systems. The combustion in the vast majority of practical systems is turbulent and this poses a number of difficulties for prediction. The development of accurate methods for predicting turbulent combusting flows remains a largely unresolved problem which continues to attract a large number of researchers.