  In this paper, we consider the Cauchy problem to the TROPIC CLIMATE MODEL
derived by Frierson-Majda-Pauluis in [Comm. Math. Sci, Vol. 2 (2004)] which is
a coupled system of the barotropic and the first baroclinic modes of the
velocity and the typical midtropospheric temperature. The system considered in
this paper has viscosities in the momentum equations, but no diffusivity in the
temperature equation. We establish here the global well-posedness of strong
solutions to this model. In proving the global existence of strong solutions,
to overcome the difficulty caused by the absence of the diffusivity in the
temperature equation, we introduce a new velocity $w$ (called the pseudo
baroclinic velocity), which has more regularities than the original baroclinic
mode of the velocity. An auxiliary function $\phi$, which looks like the
effective viscous flux for the compressible Navier-Stokes equations, is also
introduced to obtain the $L^\infty$ bound of the temperature. Regarding the
uniqueness, we use the idea of performing suitable energy estimates at level
one order lower than the natural basic energy estimates for the system.
