In this chapter we introduce the equations of motion for both the atmosphere and ocean and develop some simplifications for later use. While the atmosphere and ocean are both fluids, and therefore, despite their difference in density, obey the same basic fluid equations, there are some essential differences that make their treatment and simplification very different. We will derive the equations of motion on a rotating sphere and show how the equations can be written on an f-plane tangent to the rotating sphere. The basic simplifications of hydrostatic and geostrophic balance will be motivated and introduced and the Boussinesq approximations, where differences of density are important only when coupled to gravity, are introduced for both the atmosphere and ocean. For the ocean, the existence of standing vertical modes leads to a profoundly useful simplification, the shallow-water equations (SWEs). The SWEs turn out to be an effective model for the atmosphere as well, though the interpretation is not straightforward and there are a number of different ideas about why it works as well as it does, as discussed in Chapter 5.

The material in this chapter, familiar to those with a background in atmospheric or ocean dynamics, is a necessary prerequisite for the mathematical treatments that follow. Aside from a few idiosyncrasies, we claim no great originality or excitement here and those who know this material are invited to skip it. The reader should recognize that necessary notation, concepts and derivations are collected here.