Parametrization of the wind stress (drag) over the ocean is an essential issue in numerical analysis of the ocean–atmosphere interactions for climate modelling, satellite observations of air–sea fluxes, and other purposes. While wind stress over the ocean has been the subject of study for 50 years, the parametrization of the ocean surface momentum flux by a drag coefficient is still an uncertain process. There are many uncertainties not only in the way of proper parametrization, but in understanding the physical processes of the generation of stress by the wind system over the complicated nature of the ocean surface.

The drag coefficient has traditionally been treated as a function of the mean wind speed at a certain level, say at 10 m. Alternatively the coefficient can be represented by an aerodynamic roughness parameter. However, the spread of the observed values indicates that the question is not so simple. In 1955 Henry Charnock proposed a disarmingly simple expression for aerodynamic roughness, which was expressed in terms of the air friction velocity and the acceleration of gravity, independent of the state of ocean waves. While this expression has been widely used, there are important deviations and these were studied as a function of the wave age, a parameter representing the state of growth of wind waves relative to the local wind speed. Alarmingly, the trend of the observational values of aerodynamic roughness could be interpreted as opposite to the theoretical prediction according to the experiments considered.

Introduction

Before addressing the issues of the effects of surface tension on drag over the ocean, it is necessary to describe the physical properties and their controlling influences that make surface tension important to the air–sea boundary conditions. Surface tension is a special case of interfacial tension, that is, the pull between the molecules at the interface between two immiscible fluids. This pull exists because the molecules in the interfacial layer have fewer nearest neighbouring molecules of their own kind than do molecules in the bulk phase of either fluid. It is a physical quantity important to air–sea interaction because it affects many hydrodynamic phenomena, most notably, capillary and capillary–gravity waves. The single most important reason that these short waves (having wavelengths between about 0.1 to 30 cm) are influenced by surface tension is not really directly due to surface tension, but to surface elasticity, which is caused by the lowering of surface tension introduced by the addition of surfactants. Surfactant is a term coined by F. D. Snell that is short for surface-active agent. It describes molecular species that are more thermodynamically favoured to reside at the surface of a liquid. Typically, molecules that act as surfactants in aqueous solutions have two moieties, one hydrophilic (water-liking) and one hydrophobic (water-avoiding). In both terrestrial and aquatic environments that contain biological organisms, molecules composed of two such moieties are common.

This monograph is an attempt to address the theory of turbulence from the points of view of several disciplines. The authors are fully aware of the limited achievements here as compared with the task of understanding turbulence. Even though necessarily limited, the results in this book benefit from many years of work by the authors and from interdisciplinary exchanges among them and between them and others. We believe that it can be a useful guide on the long road toward understanding turbulence.

One of the objectives of this book is to let physicists and engineers know about the existing mathematical tools from which they might benefit. We would also like to help mathematicians learn what physical turbulence is about so that they can focus their research on problems of interest to physics and engineering as well as mathematics. We have tried to make the mathematical part accessible to the physicist and engineer, and the physical part accessible to the mathematician, without sacrificing rigor in either case. Although the rich intuition of physicists and engineers has served well to advance our still incomplete understanding of the mechanics of fluids, the rigorous mathematics introduced herein will serve to surmount the limitations of pure intuition. The work is predicated on the demonstrable fact that some of the abstract entities emerging from functional analysis of the Navier–Stokes equations represent real, physical observables: energy, enstrophy, and their decay with respect to time.

Introduction

As mentioned earlier in this text, we take for granted that the Navier–Stokes equations (NSE), together with the associated boundary and initial conditions, embody all the macroscopic physics of fluid flows. In particular, the evolution of any measured property of a turbulent flow must be relatable to the solutions of those equations. In turbulent flow regimes, the physical properties are universally recognized as randomly varying and characterized by some suitable probability distribution functions. In this and the following chapter, we discuss how those probability distribution functions (also called probability distributions or measures, or Borel measures, in the mathematical terminology; see Appendix A.1) are determined by the underlying Navier–Stokes equations. Although in many cases such distributions may not be known explicitly, their existence and many useful properties may be readily established. For many practical purposes, such partial knowledge may be all that is needed. Thus we note that the issue of an explicit form of the distribution function – in particular, whether this measure is unique or depends on the initial data – is still an incompletely solved mathematical problem. But there are enough firm results available assuring that many of the widely accepted experimental results are meaningful and in consonance with the theory of the Navier–Stokes equations.

For instance, measurements of various aspects of turbulent flows (e.g., the turbulent boundary layer) are actually measurements of time-averaged quantities.

Introduction

In principle, the idea that solutions of the Navier–Stokes equations (NSE) might be adequately represented in a finite-dimensional space arose as a result of the realization that the rapidly varying, high-wavenumber components of the turbulent flow decay so rapidly as to leave the energy-carrying (lower-wavenumber) modes unaffected. With the understanding gained from Kolmogorov's [1941a,b] phenomenological theory (see also Section 3), it appeared that, in 3-dimensional turbulent flows, only wavenumbers up to the cutoff value κd = (∈/ν3)1/4 need be considered. This is the boundary between the inertial range, which is dominated by the inertial term in the equation, and the dissipation range, which is dominated by the viscous term. As explained by Landau and Lifshitz [1971], the question is then reduced to finding the number of resolution elements needed to describe the velocity field in a volume – say, a cube of length ℓ0 on each side. Clearly, if the smallest resolved distance is to be ℓd = 1/κd, then the number of resolution elements is simply (ℓ0/ℓd)3. On adducing some phenomenological and intuitive arguments, it was argued that this ratio is Re9/4, where Re is the Reynolds number. An alternate way to count the number of active modes is as follows: since these modes are those in the inertial range, their frequency κ satisfies κ0 < κ < κd, with κ0 = 1/ℓ0; we conclude that, for κd/κ0 large, that number is of the order of (κd/κ0)3 = (ℓ0/ℓd)3.

Introduction

This long and technical chapter aims at providing some basic connections between the mathematical theory of the Navier–Stokes equations (NSE) and the conventional theory of turbulence. As stated earlier, the conventional theory of turbulence (including the famous Kolmogorov spectrum law) is based principally on physical and scaling arguments, with little reference to the NSE. We believe that it is instructive to connect turbulence more precisely with the Navier–Stokes equations.

It is commonly accepted that turbulent flows are necessarily statistical in nature. Indeed, if a flow is turbulent, then all physical quantities are rapidly varying in space and time and we cannot determine the actual instantaneous values of these quantities. Instead, one usually measures the moments, or some averaged values of physical quantities; that is, only a statistical description of the flow is available. The first task in this chapter is to establish, in a more precise way, the time evolution of the probability distribution functions associated with the fluid flow – that is, the statistical solutions of the Navier–Stokes equations. Although the discussion is relevant to deterministic data (initial values of the velocities and volume forces), we extend our discussion to the case of random data; however, we will not examine the more involved case of very irregular forcing (such as white or colored volume forces), since deterministic or moderately irregular stochastic data suffice, in practice, to generate complex turbulent flows.

Introduction

The purpose of this chapter is to recall some elements of the classical mathematical theory of the Navier–Stokes equations (NSE). We try also to explain the physical background of this theory for the physics-oriented reader.

As they stand, the Navier–Stokes equations are presumed to embody all of the physics inherent in the given incompressible, viscous fluid flow. Unfortunately, this does not automatically guarantee that the solutions to those equations satisfy the given physics. In fact, it is not even guaranteed a priori that a satisfactory solution exists. This chapter addresses the means for specifying function spaces – that is, the ensembles of functions consistent with the physics of the situation (such as incompressibility, boundedness of energy and enstrophy, as well as the prescribed boundary conditions) – that can serve as solutions to the Navier–Stokes equations. An important point is made that the kinematic pressure, p, is determined uniquely by the velocity field up to an additive constant. Hence, one cannot specify independently the initial boundary conditions for the pressure. This observation leads naturally to a representation of the NSE by an abstract differential equation in a corresponding function space for the velocity field.

Two types of boundary conditions are considered: no-slip, which are relevant to flows in domains bounded by solid impermeable walls; and space-periodic boundary conditions, which serve to study some idealized flows (including homogeneous flows) far away from real boundaries.

Introduction

In this chapter we first briefly recall, in Section 1, the derivation of the Navier–Stokes equations (NSE) starting from the basic conservation principles in mechanics: conservation of mass and momentum. Section 2 contains some general remarks on turbulence, and it alludes to some developments not presented in the book. For the benefit of the mathematically oriented reader (and perhaps others), Section 3 provides a fairly detailed account of the Kolmogorov theory of turbulence, which underlies many parts of Chapters III–V. For the physics-oriented reader, Section 4 gives an intuitive introduction to the mathematical perspective and the necessary tools. A more rigorous presentation appears in the first half of Chapter II and thereafter as needed. For each of the aspects that we develop, the present chapter should prove more useful for the nonspecialist than for the specialist.

Viscous Fluids. The Navier–Stokes Equations

Fluids obey the general laws of continuum mechanics: conservation of mass, energy, and linear momentum. They can be written as mathematical equations once a representation for the state of a fluid is chosen. In the context of mathematics, there are two classical representations. One is the so-called Lagrangian representation, where the state of a fluid “particle” at a given time is described with reference to its initial position.

In this chapter we discuss applications of the perturbation approach to stability analysis of elastic bodies subjected to large deformations. Various ideas commonly used in the perturbation approach are explained by using simple examples. Two types of bifurcations are distinguished: bifurcations at a non-zero critical mode number and bifurcations at a zero critical mode number. For each type we first explain with the aid of a model problem how stability analysis can be carried out and then explain how the analysis could be extended to problems in Finite Elasticity. Although the present analysis focuses on the perturbation approach, the dynamical systems approach is also discussed briefly and references are made to the literature where more details can be found. In the final section, we carry out a detailed analysis for the necking instability of an incompressible elastic plate under stretching.

Introduction

This chapter is concerned with nonlinear stability analysis of elastic bodies subjected to large elastic deformations. A typical problem we have in mind is the stability of a cylindrical rubber tube that is compressed either by an external pressure or by forces at the two flat ends. In general terms, we consider an elastic body which has an undeformed configuration Br in a three-dimensional Euclidean point space. This elastic body is then subjected to some external forces. It is now customary to refer to such an elastic body as pre-stressed in the Finite Elasticity literature (the pre-stress considered in the present context is not therefore that induced in a manufacturing process).

This chapter is an overview of a theory of a class of nonlinear elastic materials for which the deformation is subject to an internal material constraint described in experiments by James F. Bell on the finite plastic deformation of a variety of annealed metals. Research by Bell and his associates published since about 1979 is reviewed, and Bell's empirically deduced rules and laboratory data are compared with analytical results obtained within the context of nonlinear elasticity theory. First, Bell's empirical characterization of the constrained response of polycrystalline annealed metals in finite plastic strain is sketched. A few kinematical consequences of Bell's constraint, an outline of the constitutive theory developed to characterize the isoteopic, nonlinearly elastic response of Bell materials, and theoretical results that lead to Bell's empirical parabolic laws within the structure of isotropic, elastic and hyperelastic Bell constrained materials are presented. The study concludes with discussion of Bell's empirically based incremental theory of plasticity.

Introduction

It is common in technical writing to begin with a sketch of related research assembled to set the stage for the work ahead. But I'm not going to follow the usual path. There is more to this account than just its technical side - teachers and students, colleagues and associates, family and friends, places and events, life and death - the ingredients of the human side of the story. A reader who feels no interest in this sort of personal, anecdotal retrospection, however, will find immediate relief and surely suffer no loss in skipping ahead to Section 2.3 where Bell's important experiments and his internal material constraint are introduced. We'll return to this shortly.

A deformation or a motion is said to be a universal solution if it satisfies the balance equations with zero body force for all materials in a given class, and is supported in equilibrium by suitable surface tractions alone. On the other hand for a given deformation or motion, a local universal relation is an equation relating the stress components and the position vector which holds at any point of the body and which is the same for any material in a given class. Universal results of various kinds are fundamental aspects of the theory of finite elasticity and they are very useful in directing and warning experimentalists in their exploration of the constitutive properties of real materials. The aim of this chapter is to review universal solutions and local universal relations for isotropic nonlinearly elastic materials.

Introduction

One of the main problems encountered in the applications of the mechanics of continua is the complete and accurate determination of the constitutive relations necessary for the mathematical description of the behavior of real materials.

After the Second World War the enormous work on the foundations of the mechanics of continua, which began with the 1947 paper of Rivlin on the torsion of a rubber cylinder published in the Journal of Applied Physics, has allowed important insights on the above mentioned problem (Truesdell and Noll, 1965).

In this chapter we provide an introductory exposition of singularity theory and its application to nonlinear bifurcation analysis in elasticity. Basic concepts and methods are discussed with simple mathematics. Several examples of bifurcation analysis in nonlinear elasticity are presented in order to demonstrate the solution procedures.

Introduction

Singularity theory is a useful mathematical tool for studying bifurcation solutions. By reducing a singular function to a simple normal form, the properties of multiple solutions of a bifurcation equation can be determined from a finite number of derivatives of the singular function. Some basic ideas of singularity theory were first conjectured by R. Thorn, and were then formally developed and rigorously justified by J. Mather (1968, 1969a, b). The subject was extended further by V. I. Arnold (1976, 1981). In two volumes of monographs, M. Golubitsky and D. G. Schaeffer (1985), and M. Golubitsky, I. Stewart and D. G. Schaeffer (1988) systematized the development of singularity theory, and combined it with group theory in treating bifurcation problems with symmetry. Their work establishes singularity theory as a comprehensive mathematical theory for nonlinear bifurcation analysis.

The purpose of this chapter is to give a brief exposition of singularity theory for researchers in elasticity. The emphasis is on providing a working knowledge of the theory to the reader with minimal mathematical prerequisites. It can also serve as a handy reference source of basic techniques and useful formulae in bifurcation analysis.

In this chapter we provide a brief overview of the main ingredients of the nonlinear theory of elasticity in order to establish the basic background material as a reference source for the other, more specialized, chapters in this volume.

Introduction

In this introductory chapter we summarize the basic equations of nonlinear elasticity theory as a point of departure and as a reference source for the other articles in this volume which are concerned with more specific topics.

There are several texts and monographs which deal with the subject of nonlinear elasticity in some detail and from different standpoints. The most important of these are, in chronological order of the publication of the first edition, Green and Zerna (1954, 1968, 1992), Green and Adkins (1960, 1970), Truesdell and Noll (1965), Wang and Truesdell (1973), Chadwick (1976, 1999), Marsden and Hughes (1983, 1994), Ogden (1984a, 1997), Ciarlet (1988) and Antman (1995). See also the textbook by Holzapfel (2000), which deals with viscoelasticity and other aspects of nonlinear solid mechanics as well as containing an extensive treatment of nonlinear elasticity. These books may be referred to for more detailed study. Subsequently in this chapter we shall refer to the most recent editions of these works. The review articles by Spencer (1970) and Beatty (1987) are also valuable sources of reference.

Section 1.2 of this chapter is concerned with laying down the basic equations of elastostatics and it includes a summary of the relevant geometry of deformation and strain, an account of stress and stress tensors, the equilibrium equations and boundary conditions and an introduction to the formulation of constitutive laws for elastic materials, with discussion of the important notions of objectivity and material symmetry.