Eulerian mean flow from an instability of convective plumes

The dynamical origin of large-scale flows in systems driven by concentrated Archimedean forces is considered. A two-dimensional model of plumes, such as those observed in thermal convection at large Rayleigh and Prandtl numbers, is introduced. From this model, we deduce the onset of mean flow as an instability of a convective state consisting of parallel vertical flow supported by buoyancy forces. The form of the linear equation governing the instability is derived and two modes of instability are discussed, one of which leads to the onset of steady Eulerian mean flow in the system. We are thus able to link the origin of mean flow precisely to the profiles of the unperturbed plumes. The form of the nonlinear partial differential equation governing the Eulerian mean flow, including nonlinear effects, is derived in one special case. The extension to three dimensions is outlined. (c) 2000 American Institute of Physics.