Global small solutions to a tropical climate model without thermal diffusion

We obtain the global well-posedness of classical solutions to a tropical climate model derived by Feireisl-Majda-Pauluis in \cite{FMP} with only the dissipation of the first baroclinic model of the velocity ($-\eta \Delta v$) under small initial data. The main difficulty is the absence of thermal diffusion as the work by Li-Titi in \cite{LT}. To overcome it, we exploit the structure of the equations coming from the coupled terms, dissipation term and damp term. Then we find the hidden thermal diffusion. In addition, based on the Littlewood-Palay theory, we establish a generalized commutator estimate, which may be applied to other partial differential equations.