The discussion so far has treated the ocean circulation as though it were steady-apart from concerns about aliasing from components regarded mainly as noise. But one does not really expect a fluid system as complex as the coupled ocean and atmosphere, driven externally by the sun, to remain steady. Indeed, if there were certain elements, describable as a finite band of frequencies and/or wavenumbers, which were truly steady, it would be both a remarkable phenomenon and a very powerful theoretical tool. For this would be a statement that a spectral gap existed, permitting the separation of the flow field into different dynamical regimes, as sometimes assumed in turbulence theories (e.g., Monin & Yaglom, 1975).

Treatment of the ocean circulation as though it were time invariant carries the weight of history and tradition, and there is no doubt that a great deal has been learned about the ocean this way. But as oceanographers become more quantitative in their description of the circulation, it must be recognized that the steadiness assumption has been a consequence of necessity dictated by the paucity of data and intellectual inertia rather than a defensible deduction from fluid dynamics. One must question whether a steady, laminar flow field (as in Figures 2–7, 2–29) could conceivably depict the physics of a fluid that is turbulent in the sense suggested by Figure 1–1.

Physical oceanography is a branch of fluid dynamics and is a part of classical physics. As such, the oceanographer's job is to produce quantitative descriptions and explanations of the behavior of the fluid ocean. The movement of oceanic water has consequences for a bewildering variety of applicationsclimate change; biological productivity; sealevel change; weather forecasting; fisheries prosperity; the chemical history of the earth; the dynamics of the earth-moon system; the movement of pollutants; and so forth. Understanding of the fluid circulation and the properties it carries is of great and growing importance.

Fluid flows are diverse and often very complicated. For this reason, most understanding of particular situations has resulted from an intimate partnership of theory with observation and with laboratory experimentation. But as compared to fluid dynamics as practiced in its innumerable applications-meteorology, aerodynamics, hydraulics, heat transfer, etc.-the problem of observing the ocean is particularly difficult, rivaled perhaps only by the observational problems of inferring the fluid properties of the earth's interior, or of other planets and of stellar interiors. The ocean is very large, turbulent, and inaccessible to electromagnetic radiation. Armed mainly with slow-moving, expensive ships, and instruments which have to work in a corrosive, high-pressure environment, oceanographers have over the years built up in somewhat painful fashion a picture of how the ocean operates. But the picture is known to be badly distorted by the very limited observational base, leading for example, to the need to assume that the large-scale fluid flow is steady with time, so that measurements obtained over many years could be combined in the inferential process. Much of the description available is only qualitative.

This chapter describes a potpourri of ideas, techniques, and applications not covered in Chapter 3. A number of issues raised in Chapter 4-for example, how to impose inequality constraints-are discussed. We take up the problem of understanding the errors in the coefficient matrices E, which have not hitherto been much addressed. Further application of the SVD to a variety of problems, including how to best make maps of the ocean properties, is described. Few of the results are in any sense complete (many books exist devoted to linear programming alone, for example), but they are intended to indicate some of the ideas and methods that can be employed for quantitative description of the ocean circulation, which remain for future work.

The problems discussed in the last chapter raise a number of issues about the models that are difficult to address with the mathematical machinery already available. Among those which seem most important to deal with are the nonlinearities, which are in turn of at least two types, and the grossly underdetermined nature of the conventional large-scale box inversions.

Inequality Constraints; Nonnegative Least Squares

There are a number of aspects of the estimation problem of the general circulation that suggest the usefulness of being able to impose inequality constraints upon the solutions. Problems involving tracer concentrations usually demand that they remain positive; eddy coefficients are sometimes regarded as acceptable only when positive; in some problems we may wish to impose directions, but not magnitudes, upon the flow fields.

Some of the figures in Chapter 2 (e.g., Figure 2–2) show huge tongues of properties extending for thousands of kilometers throughout the ocean.

The purpose of this chapter is to record a number of results that are essential tools for the discussion of the problems already described. Much of this material is elementary and is discussed here primarily to produce a consistent notation for later use. Reference will be made to some of the good available textbooks. But some of the material is given what may be an unfamiliar interpretation, and I urge everyone to at least skim the chapter.

Our basic tools are those of matrix and vector algebra as they relate to the solution of simultaneous equations, and some elementary statistical ideas mainly concerning covariance, correlation, and dispersion. Least squares is reviewed, with an emphasis placed upon the arbitrariness of the distinction between knowns, unknowns, and noise. The singular-value decomposition is a central building block, producing the clearest understanding of least squares and related formulations. I introduce the Gauss-Markov theorem and its use in making property maps, as an alternative method for obtaining solutions to simultaneous equations, and show its relation to and distinction from least squares. The chapter ends with a brief discussion of recursive least squares and estimation as essential background for the time-dependent methods of Chapter 6.

Matrix and Vector Algebra

This subject is very large and well developed, and it is not my intention to repeat material better found elsewhere (e.g., Noble & Daniel, 1977; Strang, 1988). Only a brief survey of central results is provided.

Basic Physical Elements

Equations of Motion

The ocean circulation is governed by Newton's laws of motion plus those of the thermodynamics of a heat and salt stratified fluid. That these physics govern the system is a concise statement that a great deal is known about it. Our problem is to exploit this knowledge to the fullest extent possible in the context of anything we can observe about the ocean. Elaborate theoretical studies of the ocean circulation exist; here we assume that the reader has a knowledge of this theory at a basic level, with a working knowledge of Ekman layers, geostrophy, the simplest theories of western boundary currents, and the existence of internal waves and similar phenomena. Extended treatments are provided by Fofonoff (1962), Phillips (1963), Veronis (1981), Gill (1982), Pedlosky (1987a), and others.

The full equations of motion describing the ocean are the Navier-Stokes equations for a thin shell of temperature and salinity-stratified fluid on a bumpy near-spheroidal body undergoing rapid rotation. Appropriate boundary conditions are those of no flow of fluid into the bottom and sides, statements about the stress exerted on these boundaries, and those appropriate to exchange of momentum, heat, and moisture with the atmosphere at the surface. It is not really possible to separate the study of the ocean and atmosphere; a rigorous treatment must describe the movement of both fluids together, but both meteorology and oceanography remain at a stage of understanding where a great deal is to still to be learned by discussing them separately.

In this chapter we will deal with the elements of matrix algebra which we will use throughout the rest of this book. The material presented here is selfcontained. However, the student who has little prior knowledge of most of the topics in this chapter may find it helpful to refer to some of the books indicated at the end in our “For Further Reading” list. In classical mechanics we will be mainly dealing with vectors (and matrices) whose elements will be real. Hence unless explicitly stated, we shall deal in this book with real vectors and real matrices, as opposed to vectors and matrices whose elements may be complex numbers.

The chapter is divided into two main sections. The first is entitled Preliminaries, the second is called Generalized Inverse of a Matrix. We recommend that the reader begin by going to the second section after skimming over the two subsections that precede it. The details will become clearer as greater familiarity with the topics is gained. It is then that (s)he may go back to the section entitled Preliminaries, so that a better groundwork for the chapters to follow will be laid.

Preliminaries

This section contains some of the preliminary material related to vector spaces. The reader is welcome to skip the material here if (s)he is familiar with it. For the beginner, it may be better to simply skim over this section lightly at the first reading.

In this chapter we will introduce the reader to the fundamentals of Lagrangian mechanics. The reader has by now had a fair exposure to the fundamental equation, and has seen how to describe constrained motion in the framework of Cartesian coordinates. Here we will delve deeper into the basics of mechanics and introduce some of the key concepts: the principle of virtual work, and the concept of generalized coordinates. We will be taking on a more rigorous approach and amplifying on several of the issues which we had to gloss over in earlier chapters so that the reader could grasp the big picture better. The first part of this chapter is in a sense going back to better appreciate the issues involved. The second part of the chapter deals with Lagrange's equations.

The chapter introduces, in a rigorous way, what we mean by an “unconstrained” system described in terms of the Lagrangian coordinates chosen to describe the system's configuration. We show that, for a given physical situation, the analyst may exercise considerable choice in what (s)he considers to be an unconstrained system. For any specific choice of an unconstrained system, we then show how to obtain the explicit equations of motion of the constrained mechanical system in terms of the chosen Lagrangian coordinates.

Virtual Displacements

The concept of virtual displacements plays a central role in mechanics. We have already seen that equality constraints of interest to analytical dynamics are of the holonomic type or the nonholonomic type.

This book primarily deals with developing the equations of motion for mechanical systems. It is a problem which was first posed at least as far back as Lagrange, over 200 years ago, and has been vigorously worked on since by many physicists and mathematicians. The list of scientists who have contributed and attempted this problem is truly staggering. A recent monograph on the subject by Neimark and Fufaev lists more than 500 recent references.

The first major step in the understanding of constrained motion was taken by Lagrange when he formulated and developed the technique of using, what are called today, the Lagrange multipliers. The next step took about a century in the making when Gibbs and Appell developed the Gibbs–Appell approach in the late eighteen hundreds. As mentioned in Pars's book (written in 1965) this approach is considered by most to provide the simplest and most comprehensive way of setting up the equations of motion for systems with nonintegrable constraints. In 1964, P. A. M. Dirac attempted to solve the problem anew and, for Hamiltonian systems with singular Lagrangians, developed a procedure for obtaining the equations of motion for constrained systems by ingeniously extending the concept of a Poisson bracket. Dirac considered constraints which were not explicitly dependent on time.

Dirac's approach has not been discussed in this book. It is well documented in advanced treatises on analytical mechanics and may be found in the list of references that we have provided; besides, it requires background material which would go well beyond an introductory text in mechanics.

There are many treatises in the field of analytical mechanics that have been written in this century. This book is different in that it presents a new and fresh approach to the central problem of the motion of discrete mechanical systems. A system of point masses differs from a set of point masses in that the masses of a system satisfy certain constraints. This book primarily deals with the statement and analytical resolution of the problem of constrained motion and we provide the explicit equations of motion that govern large classes of constrained mechanical systems. The simplicity of the results has encouraged us to write a text which we hope will be well within the grasp of the average college senior in science and engineering.

We assume that the student has had an elementary level course dealing with statics and dynamics, and some exposure to elementary linear algebra, though the latter is not essential, because most of what is needed is contained in Chapter 2 of this book. Being pitched at the junior/senior undergraduate level, we have tried to take pains in introducing concepts slowly, gradually building them up in depth through a continual process of revisitation. We have also restricted our “For Further Reading List” at the end of each chapter principally to two books (those by Pars and Rosenberg), though there are also many other excellent treatises on analytical dynamics.

The reader will recall that we began our study of constrained motion in Chapter 3 by invoking Gauss's principle. There, we simply stated the principle as a basic principle of mechanics. We obtained the fundamental equation which described the motion of systems constrained by holonomic and nonholonomic constraints using this principle. It behooves us to understand Gauss's principle in greater depth now, moving full circle, as it were, by coming around to where we started from. This will be our agenda for this chapter.

We will begin with a simple proof of Gauss's principle in Cartesian coordinates, based on equation (5.18), the basic equation of analytical mechanics which we introduced in Chapter 5. We will next interpret this equation physically to demonstrate its aesthetic beauty, and then move on to prove the principle in terms of generalized coordinates. Using this principle, we will then provide an alternative proof for the fundamental equation in generalized coordinates.

Statement of Gauss's Principle in Cartesian Coordinates

Consider a system of n particles. The inertial Cartesian coordinates of the n particles can be described by the 3n-vector x = [x1x2x3 … x3n−1x3n]T in which the first three elements of the vector x correspond to the X-, Y- and Z- components of the position of the first particle, the next three elements correspond to the X-, Y- and Z-components of the position of the second particle, etc.