  We establish global-posedness in time for the viscous Boussinesq equations in
two dimensions of space with temperature-dependent diffusivity in the framework
of a smooth vortex patch. We also provide the inviscid limit for velocity,
temperature, and associated flow toward the system studied very recently in
\cite{Paicu-Zhu} as soon as the viscosity goes to zero, and quantify the rate
of convergence.
