Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity

We address long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity in the cases of the torus, ${\mathbb{R}}^2$, and on a bounded domain with Lions or Dirichlet boundary conditions. In all the cases, we obtain bounds on the long time behavior for the norms of the velocity and the vorticity. In particular, we obtain that the norm $\Vert (u,\rho)\Vert_{H^2\times H^{1}}$ is bounded by a single exponential, improving earlier bounds.