This chapter and the following one provide a review of some aspects of the qualitative theory of dynamical systems that we will need in our analyses of low-dimensional models derived from the Navier–Stokes equations. Dynamical systems theory is a broad and rapidly growing field which, in its more megalomaniacal forms, might be claimed to encompass all of differential equations (ordinary, partial, and functional), iterations of mappings (real and complex), devices such as cellular automata and neural networks, as well as large parts of analysis and differential topology. Here our aim will be merely the modest one of introducing, with simple examples, some tools for the analysis of non-linear ordinary differential equations that may not be as familiar as, say, perturbation and asymptotic methods.

The viewpoint of dynamical systems theory is geometric, and invariant manifolds play a central rôle, but we shall not assume or require familiarity with differential topology. In the same way, symmetries are crucial in determining the behavior, and permitting the analysis, of the low-dimensional models of interest, but we shall avoid appeals to the subtleties of group theory in our introduction to symmetric bifurcations. Thus, it should be clear that these two chapters cannot substitute for a serious course (or, more likely, courses) in dynamical systems theory. The makings of such a course can be found in the books of Arnold, Guckenheimer and Holmes, Arrowsmith and Place, or Glendinning, and in other references cited below.

Turbulence

Turbulence is the last great unsolved problem of classical physics. Although temporarily abandoned by much of the community in favor of particle physics, the current popularity of chaos and dynamical systems theory (as well as funding problems in particle physics) is now drawing the physicists back. During the interim and up to the present, turbulence has been avidly pursued by engineers.

Turbulence has enormous intellectual fascination for physicists, engineers, and mathematicians alike. This scientific appeal stems in part from its inherent difficulty – most of the approaches that can be used on other problems in fluid mechanics are useless in turbulence. Turbulence is usually approached as a stochastic problem, yet the simplifications that can be used in statistical mechanics are not applicable – turbulence is characterised by strong dependency in space and in time, so that not much can be modelled usefully as a simple Markov process, for example. The non-linearity of turbulence is essential – linearisation destroys the problem. Many problems in fluid mechanics can be approached by supposing that the flow is irrotational – that is, that the vorticity is zero everywhere. In turbulence, the presence of vorticity is essential to the dynamics. In fact, the non-linearity, rotationality and the dimensionality interact dynamically to feed the turbulence – hence, to suppose that a realisation of the flow is two-dimensional also destroys the problem. There is more, but this is enough to make it clear that one faces the turbulence problem stripped of the usual arsenal of techniques, reduced to hand-to-hand combat.

The methods developed in this book may be applied rather generally to model the dynamics of coherent structures in spatially extended systems. They are gaining acceptance in many areas in addition to fluid mechanics, including mechanical vibrations, laser dynamics, non-linear optics, and chemical processes. They are even being applied to studies of neural activity in the human brain. Numerous studies of closed flow systems have been done using empirical eigenfunctions, some of which were discussed in Section 3.7. A considerable amount of work has also been done on model PDEs for weakly non-linear waves, such as the Ginzburg–Landau and Kuramoto–Sivashinsky equations, which falls largely outside the scope of this book. We do not have the abilities (or space) to provide a survey of these multifarious applications, but we do wish to draw the reader's attention to some of the other recent work on open fluid flows.

We restrict ourselves to studies in which empirical eigenfunctions are used to construct low-dimensional models and some attempt is made to analyse their dynamical behavior. There is an enormous amount of work in which the POD is applied and its results assessed in a “static,” averaged fashion. Some of this we have reviewed in Section 3.7. Yet even thus restricted, our survey cannot pretend to be complete: new applications to fluid flows are appearing at an increasing rate. We have selected five problems on which a reasonable amount of work has been done, one of which (the jet) is a “strongly” turbulent flow.

In numerical simulations of turbulence, one can only integrate a finite set of differential equations or, equivalently, seek solutions on a finite spatial grid. One method that converts an infinite-dimensional evolution equation or partial differential equation into a finite set of ordinary differential equations is that of Galerkin projection. In this procedure the functions defining the original equation are projected onto a finite-dimensional subspace of the full phase space. In deriving low-dimensional models we shall ultimately wish to use subspaces spanned by (small) sets of empirical eigenfunctions, as described in the previous chapter. However, Galerkin projection can be used in conjunction with any suitable set of basis functions, and so we discuss it first in a general context.

After a brief description of the method in Section 4.1, we apply it in Section 4.2 to a simple problem: the linear, constant-coefficient heat equation in both one- and two-space-dimensions. We recover the classical solutions, which are often obtained by separation of variables and Fourier series methods in introductory applied mathematics courses. We then consider an equation with a quadratic non-linearity, Burgers' equation, which was originally introduced as a model to illustrate some of the features of turbulence. The remainder of the chapter is devoted to the Navier–Stokes equations. In Section 4.3 we describe Fourier mode projections for fluid flows in simple domains with periodic boundary conditions, paying particular attention to the way in which the incompressibility condition is addressed.

In this chapter we shall describe the qualitative structure, in phase space, of some of the low-dimensional models derived in the preceding chapter. We will also discuss the physical implications of our findings. Drawing on the material introduced in Chapters 5–8, we shall solve for some of the simpler fixed points (steady, time-independent flows and travelling waves) and discuss their stability and bifurcations under variation of the loss parameters αj introduced in Section 9.1. We focus on the five mode model (N = 1, K1 = 0, K3 = 5) introduced in the original paper of Aubry et al., and referred to there as the “six mode model,” the k3 = 0 mode being implicitly included in the model of the slowly varying mean flow. The full range of dynamical behavior of even such a draconian truncation as this is bewilderingly complex and still incompletely understood, but we are able to give a fairly complete account of a particular family of solutions – attracting heteroclinic cycles – which appear especially relevant to understanding the burst/sweep cycle which was described in Section 2.5.

In Sections 10.1 and 10.2 we use the nesting properties of invariant subspaces, noted in Section 9.5, to solve a reduced system, containing only two (even) complex modes, for fixed points. We exhibit the bifurcation diagram and discuss the stability of a particular branch of fixed points corresponding to streamwise vortices of the appropriate spanwise wavenumber.

On physical grounds there is no doubt that the Navier–Stokes equations provide an excellent model for fluid flow as long as shock waves are relatively thick (in terms of mean free paths), and in such conditions of temperature and pressure that we can regard the fluid as a continuum. The incompressible version is restricted, of course, to lower speeds and more moderate temperatures and pressures. There are some mathematical difficulties – indeed, we still lack a satisfactory existence-uniqueness theory in three dimensions – but these do not appear to compromise the equations' validity. Why then is the “problem of turbulence” so difficult? We can, of course, solve these nonlinear partial differential equations numerically for given boundary and initial conditions, to generate apparently unique turbulent solutions, but this is the only useful sense in which they are soluble, save for certain non-turbulent flows having strong symmetries and other simplifications. Unfortunately, numerical solutions do not bring much understanding.

However, three fairly recent developments offer some hope for improved understanding. (1) The discovery, by experimental fluid mechanicians, of coherent structures in certain fully developed turbulent flows, (2) the suggestion that strange attractors and other ideas from finite-dimensional dynamical systems theory might play a rôle in the analysis of the governing equations, and (3) the introduction of the statistical technique of Karhunen–Loéve or proper orthogonal decomposition. This book introduces these developments and describes how the three threads can be drawn together to weave low-dimensional models that address the rôle of coherent structures in turbulence generation.

As we near the end of our story, the reader will now appreciate that there are many steps in the process of reducing the Navier–Stokes equations to a low-dimensional model for the dynamics of coherent structures. Some of these involve purely mathematical issues, but most require an interplay among physical considerations, judgement, and mathematical tractibility. While our development of a general strategy for constructing low-dimensional models has been based on theoretical developments such as the POD and dynamical systems methods, the general theory is still sketchy and, in specific applications, many details remain unresolved.

The mathematical techniques we have drawn on lie primarily in probability and dynamical systems theory. In this closing chapter we review some aspects of the reduction process and attempt to put them into context. Some prospects for rigor in the reduction process are also mentioned. This is by no means a comprehensive review or discussion of future work; instead, we have chosen to highlight a few recent applications of dynamical and probabilistic ideas to illustrate lines along which a general theory might be further developed.

We start by discussing some desirable properties for low-dimensional models, and criteria by which they might be judged. We then outline in Section 12.2 an a priori short-term tracking estimate which describes, in a probabilistic context, how rapidly typical solutions of the model equations are expected to diverge from those of the full Navier–Stokes equations restricted to the model domain.

In the preceding eight chapters we have developed our basic tools and techniques. In this chapter and the next we shall illustrate their use in the derivation and analysis of low-dimensional models of the wall region of a turbulent boundary layer. First, the Navier–Stokes equations are rewritten in a form that highlights the dynamics of the coherent structures (CS) and their interaction with the mean flow. To do this, both the neglected (high) wavenumber modes and the mean flow must be modelled, unlike a large eddy simulation (LES), in which only the neglected high modes are modelled. Second, using physical considerations, we select a family of empirical subspaces upon which to project the equations. Galerkin projection is then carried out. In doing this, we restrict ourselves to a small physical flow domain, and so the response of the (quasi)local mean flow to the coherent structures must also be modelled. This chapter describes each step of the process in some detail, drawing on material presented in Chapters 2, 3, and 4. After deriving the family of low-dimensional models, in the last three sections we discuss in more depth the validity of assumptions used in their derivation. In Chapter 10 we shall describe the use of the dynamical systems ideas, presented in Chapters 5 through 8, in the analysis of these models, and interpret their solutions in terms of the dynamical behavior of the fluid flow.

Background

Ocean modelers, in the formal sense of the word, attempting to describe the ocean circulation have paid comparatively little attention in the past to the problems of working with real data. Thus, for example, one drives models, theoretical or numerical, with analytically prescribed wind or buoyancy forcing without worrying overly much about how realistic such assumed forms might be. The reasons for approaching the problem this way are good ones-there has been much to learn about how the models themselves behave, without troubling initially about the question of whether they describe the real ocean. Furthermore, there has been extremely little in the way of data available, even had one wished to use, say, realistic wind and buoyancy flux fields.

This situation is changing rapidly; the advent of wind measurements from satellite-borne instruments and other improvements in the ability of meteorologists to estimate the windfields over the open ocean, and the development of novel technologies for observing the ocean circulation, have made it possible to seriously consider estimating the global circulation in ways that were visionary only a decade ago. Technologies of neutrally buoyant floats, long-lived current meters, chemical tracer observations, satellite altimeters, acoustical methods, etc., are all either here or imminent.

The models themselves have also become so complex (e.g., Figure 1–1a) that special tools are required to understand them, to determine whether they are actually more complex than required to describe what we see (Figure 1–1b), or if less so, to what externally prescribed parameters or missing internal physics they are likely to be most sensitive.

The ocean is so difficult to observe that theoreticians intent upon explaining known phenomena have made plausible assumptions about the behavior

The purpose of this chapter is to bring to bear the mathematical machinery of Chapter 3 onto the problem of determining the oceanic general circulation. The focus is on the thermal wind equations, as outlined in Chapter 2, but the methods apply to a broad variety of problems throughout oceanography and science generally. The approach initially will be to work with a dataset that has toylike qualities-far smaller in size than one would use in practice and hence inaccurate as a representation of the ocean. Nonetheless, the data are taken from real observations, have the structure of a real situation, and raise many of the problems of practice. This approach permits the working through of a series of estimation problems in a form where complete results can be displayed readily. Realistic problems present the same issues, but if the uncertainty matrices are 1000 × 1000, there are problems of display and discussion. Following these simple examples, the results of a number of published computations addressing the ocean circulation will be described.

The central idea is very simple: The equations of motion (2.1.1)–(2.1.5) are to be combined with whatever observations are available so as to estimate the ocean circulation, along with an estimate of the errors. Recall from Chapter 2, Equation (2.5.1), that knowledge of the density field in a steady flow satisfying the thermocline equations (2.1.11)–(2.1.15) is formally adequate to determine completely the three components of the absolute velocity. But because the observed density field is corrupted owing to sampling errors, it cannot be assumed to be consistent with the equations of motion.