  We prove the existence and polynomial stability of periodic mild solutions
for Boussinesq systems in critical weak-Morrey spaces for dimension
$n\geqslant3$. Those systems are derived via the Boussinesq approximation and
describe the movement of an incompressible viscous fluid under natural
convection filling the whole space $\mathbb{R}^{n}$. Using certain dispersive
and smoothing properties of heat semigroups on Morrey-Lorentz spaces as well as
Yamazaki-type estimate on block spaces, we prove the existence of bounded mild
solutions for the linear {systems} corresponding to the Boussinesq systems.
Then, we establish a Massera-type theorem to obtain the existence and
uniqueness of periodic solutions to corresponding linear {systems} on the half
line time-axis by using a mean-ergodic method. Next, using fixed point
arguments, we can pass from linear {systems} to prove the existence uniqueness
and polynomial stability of such solutions for Boussinesq systems. Finally, we
apply the results to Navier-Stokes equations.
