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We prove global well-posedness of the ocean primitive equations coupled toadvection-diffusion equations of the oceanic tracers temperature and salinitythat are supplemented by the eddy parametrization model due to Gent-McWilliamsand Redi. This parametrization forms a milestone in global ocean modelling andconstitutes a central part of any general ocean circulation model computation.The eddy parametrization adds a secondary transport velocity to the tracerequation and renders the original Laplacian operators in theadvection-diffusion equations nonlinear, with a diffusion matrix that dependsvia the equation of state in a nonlinear fashion on both tracerssimultaneously. The eddy parametrization of Gent-McWilliams-Redi augments thecomplexity of the mathematical analysis of the whole system which we presenthere. We show first that weak solutions exist globally in time, provided theparametrization uses a regularized density. Then we prove by a detailedanalysis of the eddy operators the global well-posedness. Our results applyalso to the ``small-slope approximation'' that is commonly used in global oceansimulations.

Global Well-Posedness of the Primitive Equations of Large-Scale Ocean Dynamics with the Gent-McWilliams-Redi Eddy Parametrization Model