The dynamics of the ocean and the atmosphere deals with the characteristics of, interaction between and generation mechanisms for a wide range of motions covering a very broad spectrum of spatial and temporal scales. The modes of motion include many distinct types of wave phenomena, quasi-permanent circulation, vertical convection phenomena, and turbulent vortex motion. These can be distinguished by their kinematic and/or dynamic properties (such as vorticity, divergence, restoring forces) or by their spatial and temporal scales. For the large scale horizontal circulation, the motion is nearly geostrophic and the vertical distribution of pressure nearly hydrostatic. This is also true of planetary Rossby waves. For long gravity waves like tides, the hydrostatic approximation is still justifiable but the motion is not geostrophic (although the Coriolis acceleration must still be included). At shorter horizontal scales vertical acceleration becomes important as in short surface gravity waves and internal waves (where density stratification and associated buoyancy force is vital). For acoustic waves bulk divergence as well as acceleration is vital. Finally, at very short scales surface capillary waves can exist in which surface tension is the dominant restoring force.
In the following section we summarize the basic hydrodynamic and thermodynamic equations governing in principle the whole spectrum of problems. These are given in fairly complete form, including viscous and diffusive terms. Following this we summarize certain important relations among the physical properties of the fluid (treated as a binary system). We consider not only sea water but also the atmosphere, since a familiarity with both is essential for those concerned with the dynamics of either medium. We devote a special section to the discussion of thermal stability and neutral surfaces, since it is essential to the later discussion of baroclinic modes in subsequent chapters. Boundary conditions for the case of an inviscid, non-conducting, non-diffusive fluid are considered in the next section. In this connection we discuss the concept of the well-posed problem.
One of the major problems in the dynamics of fluids is that of the non-linearity of the basic system of equations. In analytical studies one frequently resorts to a solution technique known as the method of perturbations in which one seeks solutions of a linearized form of the basic equations where the new variables are perturbations of some adopted basic state, which in many cases is a relatively simple steady solution of the original equations. We consider such a linearization of the basic equations and boundary conditions relative to a basic state which is exactly hydrostatic. The purpose is not to examine detailed solutions in this chapter, but to examine certain integral properties related to the energetics. We conclude the chapter with a discussion of an alternate linearization for a basic state having velocity shear, to illustrate the qualitative differences one can expect and to introduce the notion of possible dynamic instability and the concept of exchange of energy among modes by non-linearity.