  This paper is concerned with an initial-boundary value problem of the
two-dimensional inhomogeneous primitive equations with density-dependent
viscosity. The global well-posedness of strong solutions is established,
provided the initial horizontal velocity is suitably small, that is, $\|\nabla
u_{0}\|_{L^{2}}\leq \eta_{0}$ for suitably small $\eta_{0}>0$. The initial data
may contain vacuum. The proof is based on the local well-posedness and the
blow-up criterion proved in \cite{0}, which states that if $T^{*}$ is the
maximal existence time of the local strong solutions $(\rho,u,w,P)$ and
$T^{*}<\infty$, then \begin{equation*}
  \sup_{0\leq t<T^{*}}(\left\|\nabla
\rho(t)\right\|_{L^{\infty}}+\left\|\nabla^{2}\rho(t)\right\|_{L^{2}}+\left\|\nabla
u(t)\right\|_{L^{2}})=\infty. \end{equation*} To complete the proof, it is
required to make an estimate on a key term $\|\nabla
u_{t}\|_{L_{t}^{1}L_{\Omega}^{2}}$. We prove that it is bounded and could be as
small as desired under certain smallness conditions, by making use of the
regularity result of hydrostatic Stokes equations and some careful time
weighted estimates.
