Transition to Turbulence in Thermal Convection with and without Rotation

Abstract

This chapter discusses the transition to turbulence in thermal convection with and without rotation. Thermal convection in a layer heated from below and Taylor vortices between differentially rotating coaxial cylinders are the classical examples of fluid systems that exhibit a slow onset of turbulence. A fluid layer heated from below is horizontally isotropic and homogenous at least in the limit when a layer of infinite extent is approached. This causes an infinite degeneracy of the solutions even at the onset of convection. The physical conditions on flow between rotating cylinders are anisotropic and the onset of Taylor vortices can be described by a simple bifurcation from the basic state. In the case of convection, this behavior can be modeled by using layers in which height and width are comparable. This permits a removal of the degeneracy to an experimentally relevant extent. Mathematical models for the description of the onset of turbulence have usually been developed on the basis of the theory of ordinary differential equations. The concept of a mathematical description of turbulence by a large manifold of solutions, each of which is unstable with respect to some other solution of the manifold, is not a new one. Convection in a rotating layer offers the simplest example of such a model of turbulence with the additional advantage of a weak nonlinearity of the basic equations.