  This article studies the global regularity problem of the two-dimensional
zero thermal diffusion tropical climate model with fractional dissipation,
given by $(-\Delta)^{\alpha}u$ in the barotropic mode equation and by
$(-\Delta)^{\beta}v$ in the first baroclinic mode of the vector velocity
equation. More precisely, we show that the global regularity result holds true
as long as $\alpha+\beta\geq2$ with $1<\alpha<2$. In addition, with no
dissipation from both the temperature and the first baroclinic mode of the
vector velocity, we also establish the global regularity result with the
dissipation strength at the logarithmically supercritical level. Finally, our
arguments can be extended to obtain the corresponding global regularity results
of the higher dimensional cases.
