Pattern formation in annular convection: An equivariant bifurcation analysis
This study of spatio-temporal pattern formation in an annulus is motivated by two physical problems on vastly different scales. The first is atmospheric convection in the equatorial plane between the warm surface of the Earth and the cold tropopause, modeled by the two-dimensional Boussinesq equations. The Boussinesq approximation may also capture features of the more complex atmospheric dynamics in a gas-giant planet atmosphere. The second is annular electroconvection in a thin smectic film, where experiments reveal the birth of convection-like vortices in the plane as the electric field intensity is increased. Modeling this last phenomenon involves the two-dimensional Navier-Stokes equations coupled with Maxwell's equations. The two models share fundamental mathematical properties and satisfy the prerequisites for application of O(2)-equivariant bifurcation theory. We show this can give predictions of interesting dynamics, including stationary and spatio-temporal patterns. In particular, full numerical computations are developed for annular thermoconvection at various Prandtl number values (for Earth's atmosphere or oceans, or for a gas-giant planet's atmosphere) and the steady-state patterns as well as rotating waves are identified and classified.