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In this paper we introduce and study the primitive equations with$\textit{non}$-isothermal turbulent pressure and transport noise. They arederived from the Navier-Stokes equations by employing stochastic versions ofthe Boussinesq and the hydrostatic approximations. The temperature dependenceof the turbulent pressure can be seen as a consequence of an additive noiseacting on the small vertical dynamics. For such model we prove globalwell-posedness in $H^1$ where the noise is considered in both the It\^{o} andStratonovich formulations. Compared to previous variants of the primitiveequations, the one considered here present a more intricate coupling betweenthe velocity field and the temperature. The corresponding analysis is seriouslymore involved than in the deterministic setting. Finally, the continuousdependence on the initial data and the energy estimates proven here are new,even in the case of isothermal turbulent pressure.

The stochastic primitive equations with non-isothermal turbulent pressure