6.1 Introduction

In this chapter we will lay down the foundations for statistical theories of geophysical flows and predictability that will be developed in the following chapters. The mathematical underpinning for these applications is a systematic use of information theory following Shannon (1948) Shannon and Weaver (1949) and Jaynes (1957). We develop such ideas beginning with elementary examples in this chapter. The motivation for the statistical studies of barotropic quasi-geostrophic flows lies in the generic occurrence of coherent large-scale flow patterns in physical observations and numerical simulations of flows that are approximately two dimensional. Examples of these flows are large isolated eddies in the atmosphere (a well-known example is the great red spot in Jupiter s atmosphere, see Chapter 13), and the discovery of mesoscale eddies in the ocean. Large-scale organized flow patterns emerge under a wide variety of initial conditions of the flow and topography. The robustness of these patterns seems to indicate that these largescale coherent flows do not depend on the fine details in the dynamics of the flow or the topography. In addition, these flow patterns persist for a long time, and are essentially steady in nature. Therefore, it is plausible that an explanation for the observed two-dimensional coherent patterns can be found with considerations from equilibrium statistical mechanics, where we are interested in the large-scale features of the flow rather than all the fine details, and where only a few bulk properties of the flow, such as averaged conserved quantities like the energy and enstrophy enter...