  In this paper we consider the following 2D Boussinesq-Navier-Stokes systems
  \partial_{t}u+u\cdot\nabla u+\nabla p+ |D|^{\alpha}u &= \theta e_{2}
\partial_{t}\theta+u\cdot\nabla \theta+ |D|^{\beta}\theta &=0 \quad with
$\textrm{div} u=0$ and $0<\beta<\alpha<1$. When $\frac{6-\sqrt{6}}{4}<\alpha<
1$, $1-\alpha<\beta\leq f(\alpha) $, where $f(\alpha)$ is an explicit function
as a technical bound, we prove global well-posedness results for rough initial
data.
