Global controllability of Boussinesq channel flows only through the temperature

We show the global approximate controllability of the Boussinesq system with viscosity and diffusion in a planar periodic channel by using only a temperature control supported in a thin strip. At the walls, a slip boundary condition is chosen for the fluid and the normal derivative of the temperature is assumed to vanish. This contributes a first global controllability result of such type for the Boussinesq system in the presence of non-periodic boundary conditions. We resort to a small-time scaling argument to control the vorticity through a large initial temperature. Moreover, relying on the special choice of the domain, we employ J.-M. Coron's return method in order to steer the temperature without significantly impacting the vorticity.