This book is to a large extent the second edition of The Ocean Circulation Inverse Problem, but it differs from the original version in a number of ways. While teaching the basic material at MIT and elsewhere over the past ten years, it became clear that it was of interest to many students outside of physical oceanography — the audience for whom the book had been written. The oceanographic material, instead of being a motivating factor, was in practice an obstacle to understanding for students with no oceanic background. In the revision, therefore, I have tried to make the examples more generic and understandable, I hope, to anyone with even rudimentary experience with simple fluid flows.

Also many of the oceanographic applications of the methods, which were still novel and controversial at the time of writing, have become familiar and almost commonplace. The oceanography, now confined to the two last chapters, is thus focussed less on explaining why and how the calculations were done, and more on summarizing what has been accomplished. Furthermore, the time-dependent problem (here called “state estimation” to distinguish it from meteorological practice) has evolved rapidly in the oceanographic community from a hypothetical methodology to one that is clearly practical and in ever-growing use.

Background

The purpose of this chapter is to record a number of results that are useful in finding and understanding the solutions to sets of usually noisy simultaneous linear equations and in which formally there may be too much or too little information. A lot of the material is elementary; good textbooks exist, to which the reader will be referred. Some of what follows is discussed primarily so as to produce a consistent notation for later use. But some topics are 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 linear 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. Minimum variance estimation is introduced through the Gauss—Markov theorem as an alternative method for obtaining solutions to simultaneous equations, and its relation to and distinction from least-squares is discussed. The chapter ends with a brief discussion of recursive least-squares and estimation; this part is essential background for the study of time-dependent problems in Chapter 4.

The origins of this book date to a conversation between the authors a short time before both were due to (formally) retire. Sadly little had been achieved before the more experienced author died. As a consequence any shortcomings of the book must be attributed to the surviving author.

Exact solutions of any system of partial differential equations attract attention. This must be particularly true of the Navier–Stokes equations which, for the best part of 200 years, have been the foundation for the significant and worldwide study of the behaviour of fluids in motion. The subject burgeoned in the twentieth century from stimuli as diverse as international conflict, and a desire to create a better understanding of the environment. In the nineteenth century theoretical advance was slow, and until the approximate or, as we would rather view them, asymptotic theories of Stokes and Prandtl for small and large values of the Reynolds number were devised, only exact solutions, and few at that, were available. In spite of the advances in asymptotic methods during the first half of the twentieth century, and the increasing use of computational methods in its later decades, exact solutions of the Navier–Stokes equations have been pursued. At best these provide an insight into the behaviour of fluids in motion; they may also provide a vehicle for novel mathematical methods or a useful check for a computer code. Some, it must be admitted, provide little of value in either of these senses.

This monograph elaborates a fundamental topic of the theory of fluid dynamics which is introduced in most textbooks on the theory of flow of a viscous fluid. A knowledge of this introductory background, for which reference may be made to Batchelor (1967), will be assumed here. However, it will be helpful to summarise a little of the background wherever we need it. In particular, we begin by introducing the scope of the book by loosely defining the terms of the title.

The Navier–Stokes equations are the system of non-linear partial differential equations governing the motion of a Newtonian fluid, which may be liquid or gas. In essence, they represent the balance between the rate of change of momentum of an element of fluid and the forces on it, as does Newton's second law of motion for a particle, where the stress is linearly related to the rate of strain of the fluid. Newton himself did not understand well the nature of the forces between elemental particles in a continuum, but he did (Newton 1687, Vol. II, Section IX, Hypothesis, Proposition LI) initiate the theory of the dynamics of a uniform viscous fluid in an intuitive and imaginative way. It was many years later that the Navier–Stokes equations, as we now know them, were deduced from various physical hypotheses, and in various forms, by Navier (1827), Poisson (1831), Saint-Venant (1843) and Stokes (1845).

Introduction

In the previous chapters we discussed various equilibrium statistical theories. These statistical theories are developed for the idealized inviscid unforced geophysical flows. However, as we have discussed in Section 10.4, virtually all practical geophysical flows are subject to both forcing and dissipation. For instance, the earth's atmosphere is subject to intense random small-scale forcing from convective storms, and the ocean is subject to forcing from unresolved baroclinic instability processes on a small length scale. Thus, a natural question to ask is whether the equilibrium statistical theories can be applied in a forced and damped environment. The purpose of this chapter is to address this question. More precisely, we want to provide answers to the applied issue (A-4) and theoretical issue (T-5) from Chapter 10.

As was discussed in Section 10.4, equilibrium statistical theories will not be able to approximate geophysical flows in a statistical sense for all forcing. This is not a surprise, since intuitively we could only expect equilibrium theories to succeed when the flow is near equilibrium. What we are interested in here are external forcing which is random and small scale in space and kicks in time. This kind of forcing mimics the small-scale random forcing in the atmosphere and ocean as discussed above and in Chapter 10. The “quasi-equilibrium” state of the geophysical flow is achieved if the inverse cascade of energy from the small scales, where the external forcing occurs, to the large scales (the scale characterized by the equilibrium statistical theory) is sufficiently weak.