  We prove that an implicit time Euler scheme for the 2D-Boussinesq model on
the torus $D$ converges. Various moment of the $W^{1,2}$-norms of the velocity
and temperature, as well as their discretizations, are computed. We obtain the
optimal speed of convergence in probability, and a logarithmic speed of
convergence in $L^2(\Omega)$. These results are deduced from a time regularity
of the solution both in $L^2(D)$ and $W^{1,2}(D)$, and from an $L^2(\Omega)$
convergence restricted to a subset where the $W^{1,2}$-noms of the solutions
are bounded.
