  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
second is annular electroconvection in a thin semetic film, where experiments
reveal the birth of convection-like vortices in the plane as the electric field
intensity is increased. This is modeled by two dimensional Navier-Stokes
equations coupled with a simplified version of 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.
