Coherent structures on a boundary layer in Rayleigh-Bénard turbulence

Observations of Rayleigh-Bénard convection have shown the existence of various coherent structures in the boundary layer region of the convection cell in the regime of hard turbulence (Zocchi et al 1990). These structures include long-lived waves, plumes and propagating regions of rotational fluid, termed ‘swirls’. Besides providing visualizations of these flow structures, the experimenters have conjectured that the observed waves are normal modes associated with the buoyant shear layer at the wall, and have provided a measured dispersion relation. We study here a simple model of such an unstably stratified boundary layer. The model consists of a two-dimensional layer of constant vorticity against a wall; this models the effect of viscosity and the large-scale rolling motion in creating shear at the wall. The effect of buoyancy is included as a sharp density jump across the boundary of the shear layer. We study both the linear analysis of the model, and its nonlinear behaviour through numerical simulation. The model reproduces much of the behaviour observed in the experiment. We observe the formation of both plumes and swirls, similar in form and dynamics to those in the experiment. We find that plumes and swirls arise from the same instability mechanism, only differing in the amount of shear acting upon them. The linear and nonlinear behaviour of the model's neutrally stable normal modes are inconsistent with that of the observed waves on the boundary layer; neutrally stable waves of the model obey a different dispersion relation, and are nonlinearly unstable to superharmonic Rayleigh-Taylor instabilities. We suggest that the observed waves reflect some other collective action of the system.