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In this paper we consider the stochastic primitive equation for geophysicalflows subject to transport noise and turbulent pressure. Admitting very roughnoise terms, the global existence and uniqueness of solutions to thisstochastic partial differential equation are proven using stochastic maximal$L^2$-regularity, the theory of critical spaces for stochastic evolutionequations, and global a priori bounds. Compared to other results in thisdirection, we do not need any smallness assumption on the transport noise whichacts directly on the velocity field and we also allow rougher noise terms. Theadaptation to Stratonovich type noise and, more generally, to variableviscosity and/or conductivity are discussed as well.

The stochastic primitive equations with transport noise and turbulent pressure