The basic equations of oceanic motion assume local thermodynamic equilibrium. The ocean is viewed as consisting of many fluid parcels. Each of these fluid parcels is assumed to be in thermodynamic equilibrium though the ocean as a whole is far from thermodynamic equilibrium. Later we make the continuum hypothesis and assume that these parcels are sufficiently small from a macroscopic point of view to be treated as points but sufficiently large from a microscopic point of view to contain enough molecules for equilibrium thermodynamics to apply. This chapter considers the equilibrium thermodynamics that holds for each of these fluid parcels or points. The thermodynamic state is described by thermodynamic variables. Most of this chapter defines these thermodynamic variables and the relations that hold among them. An important point is that sea water is a two-component system, consisting of water and sea salt. Gibbs' phase rule then implies that the thermodynamic state of sea water is completely determined by the specification of three independent thermodynamic variables. Different choices can be made for these independent variables. Pressure, temperature, and salinity are one common choice. All other variables are functions of these independent variables. In principle, these functions can be derived from the microscopic properties of sea water, by means of statistical mechanics. This has not been accomplished yet. Rather, these functions must be determined empirically from measurements and are documented in figures, tables, and numerical formulae.We do not present these figures, tables, and formulae in any detail.

This book derives and classifies the most common dynamic equations used in physical oceanography, from the planetary geostrophic equations that describe the wind and thermohaline driven circulations to the equations of small-scale motions that describe three-dimensional turbulence and double diffusive phenomena. It does so in a systematic manner and within a common framework. It first establishes the basic dynamic equations that describe all oceanic motions and then derives reduced equations, emphasizing the assumptions made and physical processes eliminated.

The basic equations of oceanic motions consist of:

• the thermodynamic specification of sea water;

• the balance equations for mass, momentum, and energy;

• the molecular flux laws; and

• the gravitational field equation.

These equations are well established and experimentally proven. However, they are so general and so all-encompassing that they become useless for specific practical applications. One needs to consider approximations to these equations and derive equations that isolate specific types or scales of motion. The basic equations of oceanic motion form the solid starting point for such derivations.

In order to derive and present the various approximations in a systematic manner we use the following concepts and organizing principles:

• distinction between properties of fluids and flows;

• distinction between prognostic and diagnostic variables;

• adjustment by wave propagation;

• modes of motion;

• Reynolds decomposition and averaging;

• asymptotic expansion;

• geometric, thermodynamic, and dynamic approximations; and

• different but equivalent representations,

which are discussed in the remainder of this introduction.

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.