The primitive equations are one of the fundamental models for geophysical flows used to describe oceanic and atmospheric dynamics. In this talk I will discuss some recent results on the primitive equations with noise of transport type. In addition to transport noise, we also consider non-isothermal turbulent pressure. The dependence of the turbulent pressure on the temperature is a consequence of a stochastic Boussinesq approximation. For the primitive equations with transport noise and non-isothermal turbulent pressure, we provide a physical derivation and we discuss the global well-posedness for data in the critical spaces $H^1$. The latter result gives a non-trivial extension of the celebrated work by C. Cao and E.S. Titi on the deterministic model. Our approach is based on recent developments of maximal regularity techniques in the context of stochastic parabolic PDEs. If time allows, then I will also discuss further results in presence of rough noise.
Based on joint works with M. Hieber (TU Darmstadt), A. Hussein (TU Kaiserslautern) and M. Saal (TU Darmstadt).