  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.
