The balance equations describe changes of extensive quantities, the amounts of water, salt, momentum, and internal energy within a (infinitesimal) volume. They represent basic physical principles. Since the internal energy is not a directly measured quantity and since sea water can for many purposes be regarded as a nearly incompressible fluid it is convenient to introduce pressure and temperature as prognostic variables, instead of the density and specific internal energy. The basic equations for oceanic motions then consist of prognostic equations for the:

• pressure;

• velocity vector;

• temperature; and

• salinity.

There are six equations. They contain:

• “external” fields and parameters like the gravitational potential and earth's rotation rate, which need to be specified (or calculated);

• molecular diffusion coefficients, which need to be specified;

• thermodynamic coefficients, which need to be specified or are given by the equilibrium thermodynamic relations of Chapter 2. Most importantly, the density or equation of state must be specified.

These equations have to be augmented by appropriate boundary conditions. The equations, together with the boundary conditions, then determine the time evolution of any initial state. Nothing else is needed. It can, of course, be elucidating to study the evolution of other quantities, such as the circulation and vorticity. The evolution of such quantities is governed by theorems that are consequences of the basic equations. One important theorem is Ertel's potential vorticity theorem.

In this chapter we discuss the gravitational potential whose gradient enters the momentum equation. The potential is the sum of two terms, the potential of the Earth and the tidal potential caused by the Moon and Sun. For many problems, the gradient of the Earth's potential can be assumed to be a constant gravitational acceleration g0. However, the actual equipotential surfaces (the geoid) have a fairly complicated shape. The gravitational potential is determined by the mass distribution, as the solution of a Poisson equation. For a prescribed mass distribution this solution can be expressed in terms of the Green's function of the Poisson equation. The determination of the gravitational potential then becomes a mere matter of integration, with well-known solutions for a sphere and other simple distributions. For a self-attracting rotating body, like the Earth, the mass distribution is not known a priori but needs to be determined simultaneously with the gravitational potential. For a fluid body of constant density, the solution to this implicit problem is MacLaurin's ellipsoid. For the Earth, the geoid needs to be measured.

The basic geometry of the geoid is an oblate ellipsoid. This suggests oblate spheroidal coordinates as the most convenient coordinate system. Since the eccentricity of the geoid is small one can approximate the metric coefficients of this coordinate system such that they look like the metric coefficients of spherical coordinates.

In this chapter we discuss free linear waves on a sphere, again for an ideal fluid. Waves are a basic mechanism by which a fluid adjusts to changes and propagates momentum and energy. Many of the approximations that are applied to the basic equations of oceanic motions and that form the remainder of this book can be characterized by the wave types that they eliminate. Free linear waves, as opposed to forced waves, are solutions of homogeneous linear equations. They can be superimposed. Mathematically, these homogeneous linear equations form an eigenvalue problem. The eigenfunctions determine the wave form, the eigenvalues the dispersion relation. Specifically, we consider linear waves on a stably stratified motionless background state. The linearized equations have separable eigensolutions. The vertical eigenvalue is characterized by the speed of sound, gravitation, and stratification. The horizontal eigenvalue problem is characterized by rotation and the spherical geometry of the earth. We then classify the various wave solutions into:

• sound (or acoustic) waves;

• surface and internal gravity waves; and

• barotropic and baroclinic Rossby waves;

and consider the limit of short-wave solutions. This chapter owes much to Kamenkovich (1977) where more details and proofs can be found.

Basic kinematic concepts of waves, such as propagating and standing waves, and the geometric optics approximation are covered in Appendix E.

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.