Geostrophic flows require that the Rossby and Ekman numbers are small, Ro, Eh, Ev ≪ 1, so that the momentum balance reduces to the geostrophic balance. Quasi-geostrophic motions (or small-scale geostrophic motions) are geostrophic motions for which additionally:

• the horizontal length scale is much smaller than the radius of the Earth, γ := L/r0 ≪ 1; and

• the vertical displacement Z of isopycnals is much smaller than the vertical scale D, σ := Z/D ≪ 1.

The second condition is equivalent to requiring the vertical strain to be smaller than one. The quasi-geostrophic theory also disregards temperature–salinity effects and assumes a one-component fluid. The smallness of γ is exploited by applying the midlatitude beta-plane approximation. We will carry out the perturbation expansion with respect to all of the small parameters explicitly since the zeroth order is degenerate and one has to go to the first order. The evolution of quasi-geostrophic flows is again governed by the potential vorticity equation. The quasi-geostrophic vorticity consists of the relative vorticity, planetary vorticity, and a vertical strain contribution. The boundary conditions also include zeroth and first order contributions. Because of the inherent approximations, the quasi-geostrophic equations cannot include all dissipation and forcing processes. Linearization of the quasigeostrophic potential vorticity equation about a motionless stratified background state yields the Rossby wave solutions discussed in Chapter 8.

In this chapter we establish the equations that govern the evolution of a fluid that is not in thermodynamic, but in local thermodynamic equilibrium. These equations are the balance or budget equations for water, salt, momentum, angular momentum, and energy within arbitrary volumes. We formulate these equations by making the continuum hypothesis. The balance equations then become partial differential equations in space and time. They all have the same general form. The rate of change of the amount within a volume is given by the fluxes through its enclosing surface and by the sources and sinks within the volume. This balance is self-evident. Specific physics enters when the fluxes and the sources and sinks are specified. If no sources and sinks are assumed then the balance equations become conservation equations. It will turn out that six balance equations, the ones for the mass of water, the mass of salt, the three components of the momentum vector, and energy completely determine the evolution of oceanic motions, given appropriate initial and boundary conditions. The vector and tensor calculus required for this chapter is briefly reviewed in Appendix B.

Continuum hypothesis

The ocean is assumed to be in local thermodynamic equilibrium. Each fluid parcel can be described by the usual thermodynamic variables such as temperature, pressure, etc., and the usual thermodynamic relations hold among these variables, but the values of these variables depend on the fluid particle. The ocean as a whole is not in thermodynamic equilibrium.

Here we present the Boussinesq approximation. It consists of two steps: the anelastic approximation and a set of additional approximations exploiting the characteristics of the oceanic density field.

In many oceanographic problems one is only interested in time scales slower than those associated with acoustic phenomena. The simplest and most straightforward way to eliminate sound waves is to regard sea water as incompressible, i.e., to consider the limit c2 → ∞ in the basic equations of oceanic motions. While this limit is adequate for many purposes compressibility (though not sound waves) plays a crucial role in the deep thermohaline circulation. For this reason one introduces the anelastic approximation, which assumes that the pressure field adjusts instantaneously. The anelastic approximation eliminates sound waves but does not remove compressibility effects. It is the lowest order in an expansion with respect to the small parameter ∈a := Ta/T where Ta is the fast time scale of acoustic waves and T the slow time scale of the motions under consideration. The main structural change in the equations brought about by the anelastic approximation is that the pressure and vertical velocity become diagnostic variables.

The density field of the ocean is characterized by the facts that:

• the density of the ocean at a point does not differ very much from a reference density; and

• the reference density does not change very much from surface to bottom.

The basic equations of oceanic motions describe all fluid motions on all space and time scales. All these motions and scales cannot be resolved simultaneously, neither observationally nor computationally. In the remainder of this book we therefore consider approximations to these equations. These approximations reduce the physics. They suppress certain processes, modes of motions, and scales. All these approximations can be obtained in a systematic manner by asymptotic expansions. These expansions clearly identify the circumstances under which a certain approximation holds and the errors involved in making this approximation. These approximations should thus be distinguished from idealizations. These merely assign specific values to parameters that describe the fluid and its environment. Typical idealizations are ideal fluids (λ = D = D′ = ν = ν′ = 0), non-rotating systems (Ω = 0), incompressible fluids (c = ∞), and two-dimensional flows (ω = ∂/∂z = 0). Such idealized systems might be much more amenable to analysis. However, it is unclear how they relate to the real ocean. Real fluids are not ideal. They diffuse salt, momentum, and heat. Ideal fluids do not. Real fluid flows are irreversible. Ideal fluid flows are time-reversible. Similar remarks apply to the other idealizations. There are, of course, oceanic motions that behave approximately as if the fluid were ideal or non-rotating or incompressible or two-dimensional. One would like to understand the circumstances under which this is the case and the errors involved in any such approximation. Asymptotic expansions address this question in a systematic manner.

Medium-scale motions are defined here as motions whose horizontal space scales L are much smaller than the Earth's radius r0 but still large enough that they are not directly affected by molecular diffusion. They are the middle component of our triple decomposition into large-, medium-, and small-scale motions that arises from two Reynolds decompositions. The first decomposition separates large- from medium- and small-scale motions. The second decomposition separates large- and medium-scale motions from small-scale motions. Large-scale motions must thus parametrize the eddy fluxes caused by medium- and small-scale motions. Mediumscale motions must prescribe the large-scale fields and parametrize the eddy fluxes caused by small-scale motions. Small-scale motions must prescribe the large- and medium-scale motions. Diffusion is molecular. Medium-scale motions thus face two closure problems, one with respect to larger scale motions and one with respect to smaller scale motions. The only simplification is one of geometry. The smallness of the parameter γ := L/r0 allows the spherical geometry to be approximated by a variety of “planar” geometries. These approximations are only valid locally and include:

• midlatitude beta-plane approximation;

• equatorial beta-plane approximation;

• f-plane approximation; and

• polar plane approximation.

These are geometric approximations that are similar to the spherical approximation that relied on the smallness of Earth's eccentricity d20 /r20 and of the parameter H/r0, and led to the introduction of pseudo-spherical coordinates.

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).