WAVES ARE EVERYWHERE: on the sea-shore, on piano wires, in football stadiums, and filling the space between the distant stars and Earth. This chapter provides an introduction to their properties, paying particular attention to a wave that is especially important to the large scale flow in both ocean and atmosphere — the Rossby wave. We start with an elementary introduction to wave kinematics, discussing such basic concepts as phase speed and group velocity. Then, beginning with Section 6.4, we discuss the dynamics of Rossby waves, and this part may be considered to be the natural follow-on from the geostrophic theory of the previous chapter. Finally, in Section 6.7, we return to group velocity in a more general way and illustrate the results using Poincaré waves, with more applications to gravity and Rossby waves in later chapters.

The reason for such an ordering of topics is that wave kinematics without a dynamical example is jejune and dry, yet understanding wave dynamics of any sort is hardly possible without appreciating at least some of its formal structure, and readers should flip pages back and forth through the chapter as needed. Those readers who wish to cut to the chase may skip the first few sections and begin at Section 6.4, referring back as needed. (Many of the key elementary results are summarized in the shaded box on page 218.) Other readers may wish to skip the sections on Rossby waves altogether and, after absorbing the sections on the wave theory move on to Chapter 7 on gravity waves, returning to Rossby waves (or not) later on. The Rossby wave and gravity wave discussions are largely independent of each other, although they both require that the reader is familiar with such ideas as group velocity and phase speed. Rossby waves and gravity waves can co-exist and close to the equator the two kinds of waves become more intertwined; we deal with the ensuing waves in Chapter 8. We also extend our discussion of Rossby waves in a global atmospheric context in Chapter 16.

FUNDAMENTALS AND FORMALITIES

Definitions and Kinematics

A wave may be more easily recognized than defined. Loosely speaking, a wave is a propagating disturbance that has a characteristic relationship between its frequency and size, and a linear wave may be defined as a disturbance that satisfies a dispersion relation.