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In this paper we establish global well-posedness and instantaneousregularization results for the primitive equations with transport noise ofH\"{o}lder regularity $ \gamma>\frac{1}{2}$. It is known that if $\gamma<1$,then the noise is too rough for a strong formulation of primitive equations inan $L^2$-based setting. To handle rough noise, we crucially use$L^q$-techniques with $q> 2$. Interestingly, we identify a family of criticalanisotropic Besov spaces for primitive equations, which is new even in thedeterministic case. The behavior of these spaces reflects the intrinsicanisotropy of the primitive equations and plays an essential role inestablishing global well-posedness and regularization. Our results coverKraichnan's type noise with correlation greater than one, and as a by-product,a 2D noise reproducing the Kolmogorov spectrum of turbulence. Moreover, theinstantaneous regularization is new also in the widely studied case of$H^1$-data and $\gamma>1 $.

The primitive equations with rough transport noise: Global well-posedness and regularity