  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.
