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

Global in Time Weak Solutions to Singular 3D Quasi-Geostrophic Systems