1 Introduction

A phenomenon that may satisfy all conservation laws of nature exactly, may still be unobservable. For the phenomenon to occur in nature, it has to satisfy one more condition, namely, it must be stable to small disturbances. In other words, infinitesimal disturbances, which are invariably present in any real system, must not amplify spontaneously. A perfectly vertical rod satisfies all equations of motion, but it does not occur in nature. A smooth ball resting on the surface of a hemisphere is stable (and therefore observable) if the surface is concave upwards, but unstable to small displacements if the surface is convex upwards Figure 12.1. In fluid flows, smooth laminar flows are stable to small disturbances only when certain conditions are satisfied. For example, in flows of homogeneous viscous fluids in a channel, the Reynolds number must be less than some critical value, and in a stratified shear flow, the Richardson number must be larger than a critical value. When these conditions are not satisfied, infinitesimal disturbances grow spontaneously. Sometimes the disturbances can grow to a finite amplitude and reach equilibrium, resulting in a new steady state. The new state may then become unstable to other types of disturbances, and may grow to yet another steady state, and so on. Finally, the flow becomes a superposition of various large disturbances of random phases, and reaches a chaotic condition that is com-monly described as "turbulent." Finite amplitude effects, including the development of chaotic solutions, will be examined briefly later in...