Strong pathwise solution and large deviation principle for the stochastic Boussinesq equations with partial diffusion term

We establish the existence and uniqueness of local strong pathwise solutions to the stochastic Boussinesq equations with partial diffusion term forced by multiplicative noise on the torus in $\mathbb{R}^{d},d=2,3$. The solution is strong in both PDE and probabilistic sense.In the two dimensional case, we prove the global existence of strong solutions to the Boussinesq equations forced by additive noise using a suitable stochastic analogue of a logarithmic Gronwall's lemma. After the global existence and uniqueness of strong solutions are established, the large deviation principle (LDP) is proved by the weak convergence method. The weak convergence is shown by a tightness argument in the appropriate functional space.