Mode saturation in ideal plasma and inviscid fluid

In the 1800s, Rayleigh and Kelvin initiated the theoretical discussion of instabilities of shear flows in fluid dynamics. They posed the question of whether such flows could suffer linear instability and exposed some mathematical problems that arose if the fluid was inviscid. Specifically, although Rayleigh's work suggested that these flows could be linearly unstable, Kelvin was deeply worried by the fact that the flows could not support freely propagating waves. Half a century or so later, related problems arose in discussions of oscillations in ideal plasma. The mathematical difficulties were largely surmounted in the 1950s and 60s when it was appreciated that the linear theory of these systems demanded the inclusion of a continuous spectrum. Nevertheless, Rayleigh's instabilities appear by bifurcating from this continuous spectrum, and this feature of the linear problem resisted the development of a theory describing the nonlinear saturation of the unstable modes, a ``pattern theory", for several decades thereafter.

Pattern formation theories describe phenomena as diverse as morphology of living organisms, crystal structure and river meanders. These theories consider the appearance of distinct modes of instability (i.e. simple structures, such as hexagonal convection cells in a heated fluid), which are the eigenmodes of the linear stability problem and provide a natural set of tools to describe the forming patterns. However in systems for which instability appears when a discrete mode bifurcates from a continuous spectrum, standard pattern theories fail because the continuous spectrum prevents application of standard techniques such as centre-manifold reduction. Instead, one must follow a more convolution path using matched asymptotic expansions. The result is a unified description of the instability, and is referred to as the single-wave model in plasma physics.