Introduction

The properties of dispersants

The objective of this chapter is to describe some of the ideas and observations that have been devised to assess and quantify rates of turbulence dispersion in the ocean.

There are several reasons why dispersion, introduced in Section 1.5.1, is of importance in the ocean. It is dispersion that determines the distribution of the naturally occurring ‘tracers’, such as salinity, for example the area within the North Atlantic that is affected by the high salinity emanating from the Mediterranean through the Straits of Gibraltar (Fig. 5.1). The volume of the water column in the Pacific affected by the plume containing helium 3 (3He) coming from hydrothermal vents in the East Pacific Rise (Fig. 5.2) is a consequence of several processes of dispersion, notably in the initial buoyant ascent of the plume, including the entrainment of surrounding water, and the subsequent advection and spread in the stratified ocean of the water ‘labelled’ by the 3He. In many cases, especially those relating to the accidental discharge of toxic chemicals or oil into the sea or the development and spread of harmful algal blooms (HABs; Fig. 5.3), the dispersion of solutes or particles by turbulent motion may have dire consequences as the pollutants spread to sensitive regions, especially those near shore where there can be a detrimental effect on mariculture, human health and recreation. Prediction is therefore of great practical importance.

Introduction: processes, and types of boundary layers

This chapter is about turbulence in the two, very extensive, boundary layers of the ocean, the upper ocean boundary layer or region near the sea surface that is directly affected by the presence of the overlying atmosphere, and the benthic or bottom boundary layer (bbl) that lies above the underlying solid, but possibly rough and (in strong flows) mobile, seabed. Two fluxes imposed at the bounding surfaces have direct effects, those of buoyancy and momentum. The former is often dominated by a flux of heat, related by (2.8) to a flux of buoyancy. This is sometimes supplemented at the sea surface by the entry of buoyant freshwater in the form of rain or snow. The flux of momentum can be equated to the stress, as explained in Section 2.2.1. This stress, or horizontal force per unit area, may be exerted by the wind on the sea surface or, for example, by the frictional forces of an immobile sedimentary layer composed of sand or gravel on a current passing over the ocean floor.

It is useful here, as in preceding chapters, to refer to processes. By a ‘process’ is meant a physical mechanism, one that can be described in terms of its effects and its associated spatial and temporal structure, that generally involves the transfer of energy from one scale to another or from one part of the ocean to another.

Introduction

The study of ocean turbulence may be viewed as a key component in the investigation of the ocean's processes and their energetics: how the energy supplied from external sources is distributed and eventually dissipated by the external and internal processes of mixing referred to in Section 3.1. The ocean is driven mainly by forcing from the atmosphere at the sea surface and by the tidal body forces imposed by the gravitational attraction of the Moon and the Sun. Relatively insignificant are the localized, but spectacular, inputs of energy from hydrothermal vents in the deep ocean ridges, the fortunately infrequent seismic movements of the seabed that may generate devastating tsunamis, the flux of geothermal heat through the floor of the abyssal plains and the energy inputs from rivers and the break-up or melting of ice sheets. The tidal forces and atmospheric inputs are the dominant sources of energy responsible for the overall circulation of the ocean (the kinetic energy of the mean flow) and its density structure (containing potential energy), and are the principal cause of the waves and the turbulence within the ocean.

The discussion in this chapter focuses on how turbulent mixing in the deep ocean is maintained. Much of the energy provided by the atmosphere is used in driving surface waves and the processes that sustain the structure of the upper ocean boundary layer.

Measurement is at the heart of science. The measurement of turbulence in the ocean has proved difficult, and not all the technical and operational problems have been overcome. In this chapter we review the characteristics that are used to describe turbulent motion and its effects, and describe some of the methods of measuring and quantifying turbulence.

Characteristics of turbulence

Some of the characteristics of turbulence are described in general terms in Chapter 1. These can provide ways of quantifying turbulent motion as explained in this, and later, sections.

Structure

Figure 2.1 is a shadowgraph image of the development of a turbulent shear flow in a laboratory experiment. It shows large billows formed downstream of a ‘splitter plate’ dividing two streams of gases with different speeds and densities. As in the photograph of the surf zone (Fig. 1.4), Fig. 2.1 shows that the flow contains patterns or structures – the billows – that recur. Each billow extends over a finite region: the motions within are spatially coherent. Although the billows are transient, they also persist for times long enough to allow them to be identified: they are coherent for short periods of time. The structure within billows varies in detail from one to another, and consists of small-scale turbulent motions that lead to the fine ‘texture’ visible in the image.

Such patterns of relatively large-scale coherent eddies containing small-scale motions are commonly found in turbulent flows.

Introduction

Processes of turbulence generation

This chapter is about turbulence within the stratified body of the ocean beyond the direct effects, described in Chapter 3, of its boundaries. The ultimate sources of energy leading to mixing in the ocean are external. The processes causing mixing in the stratified regions of the ocean derive their energy internally, as illustrated in Fig. 3.2, from sources (e.g., radiating internal waves) that may themselves be directly or indirectly driven by external forcing at the boundaries.

Two very different processes usually dominate in the generation of turbulence and diapycnal mixing in the stably stratified ocean. The first is instability resulting from the shear or differential motion of water, i.e., the vertical gradient of the horizontal current, dU/dz, which is often caused by internal waves. This is described in Section 4.2 and some aspects and evidence of the related turbulent motion are presented in Sections 4.3–4.7. The second process is more subtle, a form of convection that results from the different molecular diffusion coefficients of heat and salinity. How these lead to instability is explained in Section 4.8.

The first observations of turbulence in the thermocline

The first published measurements of turbulence within the stratified waters of the thermocline were reported in 1968 by Grant, Moilliet and Vogel. They were made off the west coast of Vancouver Island using hot-film anemometers mounted on a submarine. Grant and his colleagues compared their measurements of turbulence with those made in the mixed layer near the sea surface.

These are possible answers to the problems set at the end of each chapter in An Introduction to Ocean Turbulence. Alternative or simpler answers may be possible in some cases. Please advise the author if any errors are found.

Chapter 1

P1.1. Integrating over the area of the tube, the net flow is ∫0au ⋅ 2π r. dr = 4π U ∫ (r − r3/a2)dr = 4π U[r2/2 − r4/(4a2)]0a=π Ua2. By definition, this must be equal to the mean flow times the cross-sectional area, π a2, so U is equal to the mean flow. By conservation of the volume flux, the mean flow downstream of the transition from laminar to turbulent flow must also be equal to U.

The flux of kinetic energy upstream of the transition from laminar to turbulent flow is u(ρ u2/2) integrated over the cross-section of the tube, i.e., ∫0a (ρ u3/2)⋅ 2π r dr = 8π ρ U3∫0ar(1 − r2/a2)3 dr = − π ρ U3a2[(1 − r2/a2)4]0a = π ρ U3a2.

In the same way, the flux of the kinetic energy of the mean flow is (ρ U3/2)π a2 within the turbulent flow downstream of the transition, so the reduction in the flux is the difference between the flux upstream of the transition and that downstream, or (ρ U3/2) π a2. This represents a flux of kinetic energy to the turbulent motion, ignoring any work done by pressure forces.

Introduction

Turbulence is the dominant physical process in the transfer of momentum and heat, and in dispersing solutes and small organic or inorganic particles, in the lakes, reservoirs, seas, oceans and fluid mantles of this and other planets. Oceanic turbulence has properties that are shared by turbulence in other naturally occurring fluids and in flows generated in civil, hydraulic and chemical engineering installations and in buildings. The study of turbulence consequently has applications well beyond the particular examples in the ocean that are selected for description below.

Figure 1.1 shows the sea surface in a wind of about 26 m s− 1. It is covered by waves, many of them breaking and injecting their momentum and bubbles of air from the overlying atmosphere into the underlying seawater. Immediately below the surface, and even at great depths, the water is generally in the state of irregular and variable motion that is referred to as ‘turbulence’, although there is no simple and unambiguous definition of the term. Turbulence has, however, characteristics that, as will be explained, can be quantified and which make it of vital importance. Many of the figures in this book illustrate the nature of turbulent motion, the processes that drive turbulence, or the measurements that can be made to determine its effects.

• Turbulence is generally accepted to be an energetic, rotational and eddying state of motion that results in the dispersion of material and the transfer of momentum, heat and solutes at rates far higher than those of molecular processes alone.

OUTLINE

In this chapter we consider the force equilibrium in a continuous body under the assumption that the underlying deformation is adequately described by the small strain hypothesis. The principle of virtual power occupies a central place in this treatment, since it offers a rational basis for developing equations that apply to a continuum in a state of equilibrium. Furthermore, the concept of stress arises naturally from this analysis as dual to small strain for a solid continuum. Stress equilibrium and traction boundary conditions also appear in the most convenient invariant form. For illustration, virtual power expressions are given for systems obeying different kinematics, such as inviscid fluid flow or beams in simple bending, and the resulting stresslike quantities and their equilibrium equations are readily derived. Cauchy's stress principle and the Cauchy–Poisson theorem are also given.

Once it is established that equlibrium stress states in continuum solids in the absence of body forces are given by divergence-free tensors, the representation of such tensor fields is addressed. Beltrami potential representation of divergence-free tensors is considered, and Donati's theorem is introduced to illustrate a certain duality that exists between stress equilibrium and strain compatibility conditions.

FORCES AND MOMENTA

A body may be subject to a system of exterior forces of the following types:

• Body force f is described by a vector field distributed over the entire volume of body Ω. Denoting body force by the vector f (x, t) represents the fact that the force f (x, t)dv acts on an infinitesimal volume dv at point x at time t. An example of the body force is the force of gravity.

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OUTLINE

In the preceding chapters we have discussed, on the one hand, the kinematics of deformation of continuous media, where the principal unknowns are the displacement vector field u and the strain tensor field ∊. On the other hand, we have introduced the dynamics of deformation, representing the balance of forces in terms of the stress tensor field σ as the principal unknown.

Until now we have made no attempt to relate the strain and stress fields to each other. Before we begin the discussion of the detailed nature of this relationship, we can make the following general remarks:

• Description so far is clearly incomplete, because we have at our disposal only 6 kinematic relations and 3 force balance equations for the determination of the 3 + 6 + 6 unknown functions, that is, the components of displacement u, strain ∊, and stress σ.

• We are so far unable to distinguish between different materials which might assume different deformed configurations under the same external loading. Clearly structures produced out of wood, steel, or ceramic may deform in different ways, so that the complete solutions are different.

The purpose of this chapter is to establish a class of relationships between strains and stresses known as the linear elastic constitutive law and to discuss a series of basic properties of these relationships.