  Geophysicists have studied 3D Quasi-Geostrophic systems extensively. These
systems describe stratified flows in the atmosphere on a large time scale and
are widely used for forecasting atmospheric circulation. They couple an
inviscid transport equation in $\mathbb{R}_{+}\times\Omega$ with an equation on
the boundary satisfied by the trace, where $\Omega$ is either $2D$ torus or a
bounded convex domain in $\mathbb{R}^2$. In this paper, we show the existence
of global in time weak solutions to a family of singular 3D quasi-geostrophic
systems with Ekman pumping, where the background density profile degenerates at
the boundary. The proof is based on the construction of approximated models
which combine the Galerkin method at the boundary and regularization processes
in the bulk of the domain. The main difficulty is handling the degeneration of
the background density profile at the boundary.
