INTRODUCTION

This chapter is devoted to understanding the nature of the transitions that are possible in rotating systems. Rotation is implicated in most instabilities of astrophysical and geophysical interest. These include, for example, the baroclinic instability responsible for the formation of weather fronts in the earth's atmosphere, the instability that forms the spiral arms of galaxies, and of course the dynamo instability. The approach we take emphasizes generic, i.e., model-independent, behaviour. As a result the discussion that follows focuses on the symmetries of the system which are often responsible for much of the observed behaviour. As such we do not address specific physical mechanisms that give rise to the instabilities, or even specific model equations that might be used to describe them. Nonetheless we find that the approach used provides a number of new insights into the type of dynamics that are characteristic of rotating systems. In addition it points out the shortcomings of local studies of rotating systems that have been used to simplify the analysis. Moreover, since the results are model-independent, they apply to any system sharing the same symmetry properties. Thus our results shed light not only on the possible transitions in dynamo theory, but also on those occurring in baroclinic and other rotating flows.

We begin by pointing out that a rotating cylinder and a rotating sphere have the same symmetry: both are invariant under proper rotations about the rotation axis. In fact any figure of revolution rotating about its axis has this symmetry. For a solid body the meaning of this statement is quite intuitive.