Introduction

The observation of the geophysical flows in the atmosphere and the ocean reveal the existence of large-scale coherent flow structures. Examples of these structures are the atmospheric cyclonic and anti-cyclonic flow patterns, mesoscale ocean eddies, currents, and jets. These structures develop under fairly broad conditions and are characterized by their essentially steady nature, as well as their robustness and persistence in time. Possibly the most dramatic example of such coherent flow structures is exemplified by the Great Red Spot of Jupiter, discovered by Robert Hooke in 1664, which has persisted for at least 300 years.

From a dynamical point of view, such robust and persistent steady states must be non-linearly stable; small but finite initial perturbations of the steady states must remain small in time for the coherent flow structures to be observable. It is therefore clear that a fundamental problem is the study of the non-linear dynamical stability of the steady geophysical flows under small initial perturbations of the flow. This chapter and the next are devoted to the study of non-linear stability or instability of several classes of steady flows introduced earlier in Chapter 1. This study considers geophysical flows with topography and beta-plane effects, but without external forcing and dissipation mechanisms. In particular, we are interested in gaining a better understanding of what is the role played in the stability of the steady states by the beta-plane effect, and by the non-linear interaction of the large-scale mean flow and the small-scale flow through topographic stress.