Controllability of Boussinesq flows driven by finite-dimensional and physically localized forces

We show approximate controllability of Boussinesq flows in $\mathbb{T}^2 = \mathbb{R}^2 / 2\pi\mathbb{Z}^2$ driven by finite-dimensional controls that are supported in any fixed region $\omega \subset \mathbb{T}^2$. This addresses a Boussinesq version of a question by Agrachev and provides the first known example of incompressible fluids with this property. In this context, we complement results obtained for the Navier--Stokes system by Agrachev--Sarychev (Comm. Math. Phys. 265, 2006), where the controls are finite-dimensional but not localized in physical space, and Nersesyan--Rissel (Comm. Pure Appl. Math. 78, 2025), where physically localized controls admit for special $\omega$ a degenerate but not finite-dimensional structure. For our proof, we study controllability properties of tailored convection equations governed by time-periodic degenerately forced Euler flows that provide a twofold geometric mechanism: transport of information through $\omega$ versus non-stationary mixing effects transferring energy from low-dimensional sources to higher frequencies. The temperature is then controlled by using Coron's return method, while the velocity is mainly driven by the buoyant force. When $\omega$ contains two cuts of $\mathbb{T}^2$, our approach allows to effectively construct low-dimensional control spaces of dimensions that are independent of the choice of $\omega$ within this class of control regions.