  This paper establishes the global regularity of classical solution to the 2D
MHD system with only horizontal dissipation and horizontal magnetic diffusion
in a strip domain $\mathbb{T}\times\mathbb{R}$ when the initial data is
suitable small. To prove this, we combine the Littlewood-Paley decomposition
with anisotropic inequalities to establish a crucial commutator estimate. We
also analysis the asymptotic behavior of the solution. In addition, the global
existence and uniqueness of classical solution is obtained for the 2D
simplified tropical climate model with only horizontal dissipations.
