Colloquium
4:00 p.m., Monday (Feb. 24th)
Math 203
Neil Balmforth
Department of Applied Mathematics
UC Santa Cruz
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 centremanifold 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 singlewave model in plasma physics.
