Rotating convection: recent developments

摘要

Convection in a rotating layer was a subject close to Professor Chandrasekhar's interest, and graces the cover of the Dover edition of his famous book on Hydrodynamic and Hydromagnetic Stability. In this article I will briefly review the results described in this book, which concern primarily linear stability theory for an unbounded layer. I will then discuss in detail the differences between the theory for the unbounded layer and that for a finite cylindrical container heated from below and rotating about its axis. These differences can be traced to the differences in the symmetries of the two systems, and lead one to expect time-dependent convection at onset, regardless of the value of the Prandtl number. In addition the boundaries are responsible for the presence of a new nonaxisymmetric mode of instability, called a wall mode. The theory predicts that these wall modes will precess at onset and set in before the body modes characteristic of the unbounded layer. These predictions have been confirmed in elegant experiments by R. Ecke and colleagues. In the nonlinear regime perhaps the most interesting phenomenon is the Kuppers--Lortz instability. In this instability a pattern of parallel rolls becomes unstable to another set of rolls oriented at an angle with respect to the first, once the rotation rate exceeds a critical value. This new set is itself unstable in the same fashion etc., resulting in complex dynamics right at onset. I will describe the theory behind this instability and identify similar instabilities for oscillatory convection whether in the form of standing or traveling rolls. Such instabilities are triggered by the formation of heteroclinic orbits connecting two sets of standing or traveling rolls with different orientation. Structurally stable heteroclinic cycles involving four traveling roll states are possible, and result in quite unusual dynamical behavior.