This book about the equations of oceanic motions grewout of the course “Advanced Geophysical Fluid Dynamics” that I have been teaching for many years to graduate students at the University of Hawaii. In their pursuit of rigorous understanding, students consistently asked for a solid basis and systematic derivation of the dynamic equations used to describe and analyze oceanographic phenomena. I, on the other hand, often felt bogged down by mere “technical” aspects when trying to get fundamental theoretical concepts across. This book is the answer to both. It establishes the basic equations of oceanic motions in a rigorous way, derives the most common approximations in a systematic manner and uniform framework and notation, and lists the basic concepts and formulae of equilibrium thermodynamics, vector and tensor analysis, curvilinear coordinate systems, and the kinematics of fluid lows and waves. All this is presented in a spirit somewhere between a textbook and a reference book. This book is thus not a substitute but a complement to the many excellent textbooks on geophysical fluid dynamics, thermodynamics, and vector and tensor calculus. It provides the basic theoretical background for graduate classes and research in physical oceanography in a comprehensive form.

The book is about equations and theorems, not about solutions. Free wave solutions on a sphere are only included since the emission of waves is a mechanism by which fluids adjust to disturbances, and the assumption of instantaneous adjustment and the elimination of certain wave types forms the basis of many approximations.

Introduction

The main goal of the mathematical sections of this book is to show that linked twist maps have the Bernoulli property on all of their domain (except for possibly a set of measure zero). Before discussing the theory that will be necessary to attack this problem, we start with an easier, preliminary result. Namely, we give Devaney's proof of a theorem that a linked twist map has a Smale horseshoe (Devaney (1978)). This is a somewhat ambiguous, albeit commonly used statement in the literature. The Smale horseshoe map is a homeomorphism (it need not be area preserving) having the property that it has an invariant set on which the map is topologically conjugate to the Bernoulli shift, i.e., it has the Bernoulli property on an invariant set. A slight confusion may arise since occasionally the invariant set itself is referred to as the horseshoe. Smale horseshoe (or just “horseshoe”) maps are ubiquitous in the sense that they can always be constructed near transverse homoclinic points. This is the content of the Smale–Birkhoff homoclinic theorem. All of this is described in detail, and from an elementary point of view, in Wiggins (2003).

Mixing is a common phenomenon in everyday life. A blob of white cream placed in a cup of black coffee and gently stirred with a spoon forms, if one looks carefully, intricately shaped striated structures, until the mixture of coffee and cream homogenizes into a fluid that is uniformly brown in colour. This common phenomenon serves to illustrate some of the key features of mixing; namely, the interplay between advection and diffusion. If the coffee is at rest when the cream is added (and assuming that the insertion of the cream into the coffee only causes negligible disturbance of the surrounding coffee) then, in the absence of stirring, the cream mixes with the coffee by the mechanism of molecular diffusion. Experience tells us that in this particular situation the mixing takes much longer than we would typically be willing to wait. Therefore we stir the admixture of coffee and cream with a spoon, and observe it to homogenize very quickly. This stirring illustrates the role that advection plays in homogenizing the cream and coffee. In fact, in this particular example (as well as many others) the role of molecular diffusion in achieving the desired final mixed state may very well be negligible.

Introduction

We will mostly be interested in the simplest mixing problem: the mixing of a fluid with itself. This serves as a foundation for all mixing problems. Practically, we can think of placing a region, or ‘blob’ of dye in the fluid, and asking how long it takes for the dye to become evenly, or uniformly, distributed throughout the entire domain of the flow. We will need the mathematical machinery to make this question precise and quantitative. We will first need a framework to mathematically describe, measure and move regions of fluid. To do so we will introduce simple ideas and definitions from topology, measure theory and dynamical systems. In particular, notions of set theory from topology correspond naturally to properties of an arbitrary region of fluid, such as its boundary, interior and connectedness. Measure theory provides the tools necessary to measure the size of a region in a generalized and consistent way. The basis of the field of dynamical systems is the study of the evolution of some system with time, and these ideas can be applied directly to the application of moving fluid. These are by now all well-established techniques in the study of fluid mechanics (see for example Ottino (1989a)).

To discuss mixing of fluid we will need the area of mathematics known as ergodic theory. This provides a framework in which many physically relevant pieces of the mixing problem can be fruitfully studied.

Where is mixing important?

Mixing processes occur in a variety of technological and natural applications, with length and time scales ranging from the very small (as in microfluidic applications), to the very large (mixing in the Earth's oceans and atmosphere). The spectrum is quite broad; the ratio of the contributions of inertial forces (dominant in the realm of the very large) to viscous forces (dominant on the side of the very small) spans more than twenty orders of magnitude.

Theoretical and experimental developments over the last two decades have provided a strong foundation for the subject, yet much remains to be done. Earlier work focused on mixing of liquids and considerable advances have been made. The basic theory can be extended in many directions and the picture has been augmented in various ways. One strand of the expansion has been an incursion into new applications such as oceanography, geophysics and applications to the design of new mixing devices, as in microfluidics. Asecond strand is incursion into new types of physical situations, such as mixing of dry granular systems and liquid granular systems (in which air is replaced by a liquid). These applications clearly put us on a different plane – new physics – since, in contrast to mixing of liquids, a complicating factor in the flow of granular material is the tendency for materials to segregate or demix as a result of differences in particle properties, such as density, size, or shape. Mixing competes with segregation: mixtures of particles with varying size (S-systems) or varying density (D-systems) often segregate leading to what, on first viewing, appear to be baffling results.

Introduction

In this final chapter we discuss some of the open questions connected with the linked twist map approach to mixing. It is apparent that in terms of designing, creating and optimizing mixers this approach is still very much in its infancy. Translating the ideas of previous chapters into ‘design’ principles is itself a sizeable task, and may result in the theory being extended in a variety of interesting directions. We mention only a few here.

In Section 9.2 we consider the question of how to optimize the size of domain on which the Bernoulli property is present. We have seen in previous chapters that, providing certain conditions on the properties of the twists are satisfied, the Bernoulli property is enjoyed by almost all points in the union of the annuli. However, we have also seen that the required conditions on the twists are more likely to be satisfied for smaller size domains. For example, recall that a counter-rotating toral linked twist map with linear single-twists is periodic with period 6 when the annuli are equal in size to the torus (see Example 6.4.2), while a linear counter-rotating TLTM may be Bernoulli for a choice of smaller annuli.

Introduction

In Chapter 2 we discussed the connection between linked twist maps and fluid flow, and observed that linked twist maps on the plane arise naturally in a number of existing experimental constructions, such as blinking flows and duct flows. However the extension of the results for toral linked twist maps to planar linked twist maps is not entirely straightforward. The situation for toral linked twist maps is relatively simple (at least in comparison to linked twist maps on other objects) because we can express twist maps in two independent directions in the same (Cartesian) coordinate system. The situation for planar annuli is more complicated.

As in Chapter 4, annuli in the plane and twist maps on such annuli are naturally described in polar coordinates. However, to create a linked twist map we require a pair of annuli with different centres. There is no simple coordinate system which then describes twist maps in both annuli. We therefore require additional transformations to move from one coordinate frame to another.

The following work in this chapter is mainly due to Wojtkowski (1980). This work predates Katok et al. (1986) by some six years, and so the author could not appeal to the Katok–Strelcyn version of Pesin theory for systems with singularities.

Introduction

As discussed in Chapter 3, the property of ergodicity is a long way down in the ergodic hierarchy. For the strongest mixing behaviour, we would like our linked twist maps to possess the Bernoulli property. Fortunately, the Katok–Strelcyn version of Pesin theory given in Chapter 5 gives conditions to show exactly that. Recall that if the Katok–Strelcyn conditions are satisfied, and Lyapunov exponents are non-zero for every tangent vector, and for almost every point, we have the existence of local stable and unstable manifolds for almost all points in our domain for a smooth dynamical system with singularities. Furthermore, if some forward iterate of the local unstable manifold of some point intersects some backward iterate of the local stable manifold of another point, for almost every pair of points (the Manifold Intersection Property), then the ergodic partition we showed to exist in the previous chapter has only one component, and so our linked twist map is ergodic. Moreover, if every (far enough) forward iterate of the local unstable manifold intersects some (far enough) backward iterate of the local stable manifold, again for almost every pair of points (the Repeated Manifold Intersection Property), then the linked twist map has the Bernoulli property.

In this chapter we prove that both these conditions hold for toral linked twist maps, following the work of Wojtkowski (1980) and, mainly, Przytycki (1983).

Introduction

In Chapter 4 we gave Devaney's construction of a horseshoe for a linked twist map on the plane. The existence of the horseshoe and the accompanying subshift of finite type implies that the linked twist map contains a certain amount of complexity. However, topological features such as horseshoes may not be of interest from a statistical, observable, or measure-theoretic point of view, as they occur on invariant sets of measure zero. The subshift of finite type occurs on just such an invariant set of measure zero and is therefore arguably not of significant statistical interest. Nevertheless it is possible that similar behaviour is shared by points in the vicinity of the horseshoe, meaning that complex behaviour is present in a significant (that is, positive measure) domain. Easton (1978) conjectures that this may indeed be the case, and that in fact linked twist maps may be ergodic.

Three papers provide the framework for applying the results of Pesin (1977) connecting hyperbolicity and ergodicity. In this and the following two chapters we draw heavily on each of Burton & Easton (1980), Wojtkowski (1980) and Przytycki (1983).

Introduction

The central theme of this book is that the mathematical notion of a linked twist map, and attendant dynamical consequences, is naturally present in a variety of different mixing situations. In this chapter we will define what we mean by a linked twist map, and then give a general idea of why they capture the essence of ‘good mixing’. To do this we will first describe the notion of a linked twist map as first studied in the mathematical literature. This setting may at first appear to have little to do with the types of situations arising in fluid mechanics, but we will argue the contrary later. However, this more mathematically ideal setting allows one to rigorously prove strong mixing properties in a rather direct fashion that would likely be impossible for the types of maps arising in typical fluid mechanical situations. We will then consider a variety of mixers and mixing situations and show how the linked twist map structure naturally arises.

Introduction

Hyperbolic dynamics, loosely speaking, concerns the study of systems which exhibit both expanding and contracting behaviour. Hyperbolicity is one of the most fundamental aspects of dynamical systems theory, both from the point of view of pure dynamical systems, in which it represents a widely studied and thoroughly understood class of system, and from the point of view of applied dynamical systems, in which it gives one of the simplest models of complex and chaotic dynamics. However the pay-off for this amount of knowledge and (apparent) simplicity is severe. While hyperbolic objects (for example certain fixed points and periodic orbits, and horseshoes, like that constructed in the previous chapter) are common enough occurrences, these are arguably of limited practical importance, as all these objects comprise sets of zero (Lebesgue) measure. There are only a handful of real systems for which the strongest form of hyperbolicity (uniform hyperbolicity) has been shownto exist on a set of positive measure. Typically, uniformly hyperbolic systems tend to be restricted to model systems, such as the Arnold Cat Map (Arnold & Avez (1968)), or idealized mechanical examples, such as the triple linkage of Hunt & Mackay (2003).

Weaker forms of hyperbolicity have been studied in great detail, and powerful results exist linking these to mixing properties, but still any sort of hyperbolicity is not a straightforward property to demonstrate.