Global strong solutions and time decay of 2D tropical climate model with zero thermal diffusion

The global small solutions of the tropical climate model are obtained with the fractional dissipative terms ?(alpha)u in the equation of the barotropic mode u and ?(alpha)v in the equation of the first baroclinic mode v. More precisely, we prove for 1<alpha <= 2 that the couple system has global unique strong solutions for small initial data with critical regularities. Moreover, the smallness assumption imposed on the initial barotropic mode of the velocity can be removed if alpha=2. We also study the large time behavior of the constructed solutions and obtain optimal time decay rates by a pure energy argument.